The CFD Module User’s Guide - lost-contact.mit.edu home … · Visit the Contact COMSOL page at...

VERSION 4.4

User s Guide

CFD Module

C o n t a c t I n f o r m a t i o n

Visit the Contact COMSOL page at www.comsol.com/contact to submit general inquiries, contact Technical Support, or search for an address and phone number. You can also visit the Worldwide Sales Offices page at www.comsol.com/contact/offices for address and contact information.

If you need to contact Support, an online request form is located at the COMSOL Access page at www.comsol.com/support/case.

Other useful links include:

• Support Center: www.comsol.com/support

• Product Download: www.comsol.com/support/download

• Product Updates: www.comsol.com/support/updates

• COMSOL Community: www.comsol.com/community

• Events: www.comsol.com/events

• COMSOL Video Center: www.comsol.com/video

• Support Knowledge Base: www.comsol.com/support/knowledgebase

Part number: CM021301

C F D M o d u l e U s e r ’ s G u i d e © 1998–2013 COMSOL

Protected by U.S. Patents 7,519,518; 7,596,474; 7,623,991; and 8,457,932. Patents pending.

This Documentation and the Programs described herein are furnished under the COMSOL Software License Agreement (www.comsol.com/sla) and may be used or copied only under the terms of the license agreement.

COMSOL, COMSOL Multiphysics, Capture the Concept, COMSOL Desktop, and LiveLink are either registered trademarks or trademarks of COMSOL AB. All other trademarks are the property of their respective owners, and COMSOL AB and its subsidiaries and products are not affiliated with, endorsed by, sponsored by, or supported by those trademark owners. For a list of such trademark owners, see www.comsol.com/tm.

Version: November 2013 COMSOL 4.4

http://www.comsol.com/sla/

http://www.comsol.com/tm/

http://www.comsol.com/contact/

http://www.comsol.com/contact/offices/

http://www.comsol.com/support/case/

http://www.comsol.com/support/

http://www.comsol.com/support/download/

http://www.comsol.com/support/updates/

http://www.comsol.com/community/

http://www.comsol.com/events/

http://www.comsol.com/video/

http://www.comsol.com/support/knowledgebase/

C o n t e n t s

C h a p t e r 1 : I n t r o d u c t i o n

About the CFD Module 20

Why CFD is Important for Modeling . . . . . . . . . . . . . . . 20

How the CFD Module Helps Improve Your Modeling . . . . . . . . . 21

Where Do I Access the Documentation and Model Libraries? . . . . . . 22

Overview of the User’s Guide 26

C h a p t e r 2 : Q u i c k S t a r t G u i d e

Modeling and Simulations of Fluid Flow 30

Modeling Strategy . . . . . . . . . . . . . . . . . . . . . . 30

Geometric Complexities . . . . . . . . . . . . . . . . . . . . 31

Material Properties . . . . . . . . . . . . . . . . . . . . . . 31

Defining the Physics . . . . . . . . . . . . . . . . . . . . . . 32

Meshing . . . . . . . . . . . . . . . . . . . . . . . . . . 32

The Choice of Solver and Solver Settings. . . . . . . . . . . . . . 34

The CFD Module Physics Interface Guide . . . . . . . . . . . . . 35

C h a p t e r 3 : S i n g l e - P h a s e F l o w

Modeling Single-Phase Flow 46

Selecting the Right Physics Interface. . . . . . . . . . . . . . . . 46

The Single-Phase Flow Interface Options . . . . . . . . . . . . . . 47

Coupling to Other Physics Interfaces . . . . . . . . . . . . . . . 50

The Laminar Flow, Creeping Flow, and Turbulent Flow

Interfaces 52

The Laminar Flow Interface . . . . . . . . . . . . . . . . . . . 52

The Creeping Flow Interface . . . . . . . . . . . . . . . . . . 57

C O N T E N T S | 3

4 | C O N T E N T S

The Turbulent Flow, k- Interface . . . . . . . . . . . . . . . . 58

The Turbulent Flow, Low Re k- Interface . . . . . . . . . . . . . 60

The Turbulent Flow, k- Interface . . . . . . . . . . . . . . . . 62

The Turbulent Flow, SST Interface . . . . . . . . . . . . . . . . 63

The Turbulent Flow, Spalart-Allmaras Interface . . . . . . . . . . . 64

Domain, Boundary, Pair, and Point Nodes for Single-Phase Flow . . . . . 66

Fluid Properties . . . . . . . . . . . . . . . . . . . . . . . 67

Volume Force . . . . . . . . . . . . . . . . . . . . . . . . 71

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . . 71

Wall. . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Inlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Outlet . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 83

Open Boundary . . . . . . . . . . . . . . . . . . . . . . . 84

Boundary Stress . . . . . . . . . . . . . . . . . . . . . . . 85

Screen . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Vacuum Pump . . . . . . . . . . . . . . . . . . . . . . . . 88

Periodic Flow Condition . . . . . . . . . . . . . . . . . . . . 89

Fan . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Interior Fan . . . . . . . . . . . . . . . . . . . . . . . . . 92

Interior Wall . . . . . . . . . . . . . . . . . . . . . . . . 93

Grille . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Flow Continuity . . . . . . . . . . . . . . . . . . . . . . . 96

Point Mass Source . . . . . . . . . . . . . . . . . . . . . . 97

Line Mass Source . . . . . . . . . . . . . . . . . . . . . . . 98

Pressure Point Constraint . . . . . . . . . . . . . . . . . . . 99

More Boundary Condition Settings for the Turbulent Flow Interfaces . . . 99

The Rotating Machinery, Laminar and Turbulent Flow

Interfaces 103

The Rotating Machinery, Laminar Flow Interface . . . . . . . . . . 103

The Rotating Machinery, Turbulent Flow, k- Interface . . . . . . . . 105

Domain, Boundary, Point, and Pair Nodes for the Rotating

Machinery, Laminar and Turbulent Flow Interfaces . . . . . . . . 106

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 107

Rotating Domain . . . . . . . . . . . . . . . . . . . . . . 108

Rotating Wall . . . . . . . . . . . . . . . . . . . . . . . 109

Rotating Interior Wall . . . . . . . . . . . . . . . . . . . . 110

Theory for the Single-Phase Flow Interfaces 111

General Single-Phase Flow Theory . . . . . . . . . . . . . . . 112

Compressible Flow . . . . . . . . . . . . . . . . . . . . . 114

The Mach Number Limit . . . . . . . . . . . . . . . . . . . 114

Incompressible Flow . . . . . . . . . . . . . . . . . . . . 115

The Reynolds Number. . . . . . . . . . . . . . . . . . . . 116

Non-Newtonian Flow: The Power Law and the Carreau Model . . . . 116

The Boussinesq Approximation . . . . . . . . . . . . . . . . 118

Theory for the Wall Boundary Condition . . . . . . . . . . . . 119

Prescribing Inlet and Outlet Conditions . . . . . . . . . . . . . 122

Laminar Inflow. . . . . . . . . . . . . . . . . . . . . . . 123

Laminar Outflow . . . . . . . . . . . . . . . . . . . . . . 124

Mass Flow . . . . . . . . . . . . . . . . . . . . . . . . 125

No Viscous Stress . . . . . . . . . . . . . . . . . . . . . 126

Pressure, No Viscous Stress Boundary Condition . . . . . . . . . 126

Normal Stress Boundary Condition . . . . . . . . . . . . . . . 127

Pressure Boundary Condition . . . . . . . . . . . . . . . . . 128

Vacuum Pump Boundary Condition . . . . . . . . . . . . . . . 128

Fan Defined on an Interior Boundary . . . . . . . . . . . . . . 130

Theory for the Fan and Grille Boundary Conditions . . . . . . . . 131

Screen Boundary Condition. . . . . . . . . . . . . . . . . . 134

Mass Sources for Fluid Flow. . . . . . . . . . . . . . . . . . 136

Numerical Stability—Stabilization Techniques for Fluid Flow . . . . . 138

Solvers for Laminar Flow . . . . . . . . . . . . . . . . . . . 140

Pseudo Time Stepping for Laminar Flow Models . . . . . . . . . . 142

The Projection Method for the Navier-Stokes Equations . . . . . . . 144

Discontinuous Galerkin Formulation . . . . . . . . . . . . . . 145

Particle Tracing in Fluid Flow . . . . . . . . . . . . . . . . . 146

References for the Single-Phase Flow, Laminar Flow Interfaces. . . . . 147

Theory for the Turbulent Flow Interfaces 149

Turbulence Modeling . . . . . . . . . . . . . . . . . . . . 149

The k-Turbulence Model . . . . . . . . . . . . . . . . . . 153

The k- Turbulence Model . . . . . . . . . . . . . . . . . . 160

The SST Turbulence Model . . . . . . . . . . . . . . . . . . 162

The Low Reynolds Number k- Turbulence Model . . . . . . . . . 166

The Spalart-Allmaras Turbulence Model . . . . . . . . . . . . . 169

Inlet Values for the Turbulence Length Scale and Turbulent Intensity . . 171

C O N T E N T S | 5

6 | C O N T E N T S

Theory for the Pressure, No Viscous Stress Boundary Condition . . . 172

Solvers for Turbulent Flow . . . . . . . . . . . . . . . . . . 172

Pseudo Time Stepping for Turbulent Flow Models . . . . . . . . . 173

References for the Single-Phase Flow, Turbulent Flow Interfaces . . . . 173

Theory for the Rotating Machinery Interfaces 175

Frozen Rotor . . . . . . . . . . . . . . . . . . . . . . . 176

Setting Up a Rotating Machinery Model . . . . . . . . . . . . . 178

References . . . . . . . . . . . . . . . . . . . . . . . . 179

C h a p t e r 4 : H e a t T r a n s f e r a n d N o n - I s o t h e r m a l F l o w

Modeling Heat Transfer in the CFD Module 182

Selecting the Right Physics Interface. . . . . . . . . . . . . . . 182

Coupling to Other Physics Interfaces . . . . . . . . . . . . . . 185

The Heat Transfer Interface 186

Domain, Boundary, Edge, Point, and Pair Nodes for the Heat

Transfer Interfaces . . . . . . . . . . . . . . . . . . . . 189

Heat Transfer in Solids . . . . . . . . . . . . . . . . . . . . 190

Translational Motion . . . . . . . . . . . . . . . . . . . . 193

Heat Transfer in Fluids . . . . . . . . . . . . . . . . . . . . 194

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 197

Heat Source. . . . . . . . . . . . . . . . . . . . . . . . 198

Thermal Insulation . . . . . . . . . . . . . . . . . . . . . 200

Temperature . . . . . . . . . . . . . . . . . . . . . . . 201

Outflow . . . . . . . . . . . . . . . . . . . . . . . . . 202

Symmetry . . . . . . . . . . . . . . . . . . . . . . . . 202

Heat Flux. . . . . . . . . . . . . . . . . . . . . . . . . 202

Surface-to-Ambient Radiation . . . . . . . . . . . . . . . . . 204

Periodic Heat Condition . . . . . . . . . . . . . . . . . . . 205

Boundary Heat Source. . . . . . . . . . . . . . . . . . . . 205

Continuity . . . . . . . . . . . . . . . . . . . . . . . . 207

Thin Thermally Resistive Layer. . . . . . . . . . . . . . . . . 207

Line Heat Source . . . . . . . . . . . . . . . . . . . . . . 209

Point Heat Source . . . . . . . . . . . . . . . . . . . . . 210

Line Heat Source on Axis . . . . . . . . . . . . . . . . . . 210

Point Heat Source on Axis . . . . . . . . . . . . . . . . . . 211

Pressure Work . . . . . . . . . . . . . . . . . . . . . . 211

Viscous Heating . . . . . . . . . . . . . . . . . . . . . . 212

Inflow Heat Flux . . . . . . . . . . . . . . . . . . . . . . 212

Open Boundary . . . . . . . . . . . . . . . . . . . . . . 214

Convective Heat Flux . . . . . . . . . . . . . . . . . . . . 214

Out-of-Plane Heat Transfer Nodes 217

Out-of-Plane Convective Heat Flux . . . . . . . . . . . . . . . 217

Out-of-Plane Radiation . . . . . . . . . . . . . . . . . . . 218

Out-of-Plane Heat Flux . . . . . . . . . . . . . . . . . . . 219

Change Thickness . . . . . . . . . . . . . . . . . . . . . 220

The Heat Transfer in Porous Media Interface 221

Domain, Boundary, Edge, Point, and Pair Nodes for the Heat

Transfer in Porous Media Interface . . . . . . . . . . . . . . 222

Heat Transfer in Porous Media. . . . . . . . . . . . . . . . . 222

Thermal Dispersion . . . . . . . . . . . . . . . . . . . . . 226

Theory for the Heat Transfer Interfaces 227

What is Heat Transfer? . . . . . . . . . . . . . . . . . . . 227

The Heat Equation . . . . . . . . . . . . . . . . . . . . . 228

A Note on Heat Flux and Balance . . . . . . . . . . . . . . . 230

Heat Transfer Variables . . . . . . . . . . . . . . . . . . . 235

About the Boundary Conditions for the Heat Transfer Interfaces . . . 245

Radiative Heat Transfer in Transparent Media . . . . . . . . . . . 248

Consistent and Inconsistent Stabilization Methods for the Heat


References for the Heat Transfer Interfaces . . . . . . . . . . . . 252

Theory of Out-of-Plane Heat Transfer 253

Equation Formulation . . . . . . . . . . . . . . . . . . . . 253

Activating Out-of-Plane Heat Transfer and Thickness . . . . . . . . 254

Theory for the Heat Transfer in Porous Media Interface 255

Reference for the Heat Transfer in Porous Media Interface . . . . . . 256

C O N T E N T S | 7

8 | C O N T E N T S

About the Heat Transfer Coefficients 257

Heat Transfer Coefficient Theory . . . . . . . . . . . . . . . 258

Nature of the Flow—the Grashof Number . . . . . . . . . . . . 259

Heat Transfer Coefficients — External Natural Convection . . . . . . 260

Heat Transfer Coefficients — Internal Natural Convection . . . . . . 262

Heat Transfer Coefficients — External Forced Convection . . . . . . 263

Heat Transfer Coefficients — Internal Forced Convection . . . . . . 263

References for the Heat Transfer Coefficients . . . . . . . . . . . 264

The Non-Isothermal Flow and Conjugate Heat Transfer,

Laminar Flow and Turbulent Flow Interfaces 265

The Non-Isothermal Flow, Laminar Flow Interface . . . . . . . . . 265

The Conjugate Heat Transfer, Laminar Flow Interface . . . . . . . . 269

The Turbulent Flow, k- and Turbulent Flow Low Re k- Interfaces . . . 270

The Turbulent Flow, Spalart-Allmaras Interface . . . . . . . . . . 273

The Turbulent Flow, SST Interface . . . . . . . . . . . . . . . 274

The Turbulent Flow, k- Interface . . . . . . . . . . . . . . . 276

Domain, Boundary, Edge, Point, and Pair Nodes Settings for the

NITF Interfaces . . . . . . . . . . . . . . . . . . . . . 277

Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . 279

Wall. . . . . . . . . . . . . . . . . . . . . . . . . . . 285

Interior Wall . . . . . . . . . . . . . . . . . . . . . . . 287

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 288

Open Boundary . . . . . . . . . . . . . . . . . . . . . . 288

Pressure Work . . . . . . . . . . . . . . . . . . . . . . 289

Viscous Heating . . . . . . . . . . . . . . . . . . . . . . 290

Symmetry, Heat and Symmetry, Flow . . . . . . . . . . . . . . 290

Theory for the Non-Isothermal Flow and Conjugate Heat

Transfer Interfaces 293

Turbulent Non-Isothermal Flow Theory . . . . . . . . . . . . . 295

Theory for the Non-Isothermal Screen Boundary Condition . . . . . 299

References for the Non-Isothermal Flow and Conjugate Heat


C h a p t e r 5 : H i g h M a c h N u m b e r F l o w

The High Mach Number Flow Interfaces 302

The High Mach Number Flow, Laminar Flow Interface. . . . . . . . 303

The High Mach Number Flow, Turbulent Flow, k- Interface . . . . . 306

The High Mach Number Flow, Turbulent Flow, Spalart-Allmaras

Interface. . . . . . . . . . . . . . . . . . . . . . . . 309

Domain, Boundary, Edge, Point, and Pair Nodes for the High Mach

Number Flow Laminar and Turbulent Interfaces . . . . . . . . . 311

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 313

Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . 313

Inlet . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

Outlet . . . . . . . . . . . . . . . . . . . . . . . . . . 319

Theory for the High Mach Number Flow Interfaces 321

Compressible Flow for All Mach Numbers . . . . . . . . . . . . 321

Sutherland’s Law . . . . . . . . . . . . . . . . . . . . . . 323

Consistent Inlet and Outlet Conditions . . . . . . . . . . . . . 324

Pseudo Time Stepping for High Mach Number Flow Models . . . . . 328

References for the High Mach Number Flow Interfaces . . . . . . . 329

C h a p t e r 6 : M u l t i p h a s e F l o w

Modeling Multiphase Flow 332


The Multiphase Flow Interface Options . . . . . . . . . . . . . 333

The Relationship Between the Physics Interfaces . . . . . . . . . . 333


The Laminar Flow, Two-Phase, Level Set and Phase Field

Interfaces 338

The Laminar Two-Phase Flow, Level Set Interface . . . . . . . . . 338

The Laminar Two-Phase Flow, Phase Field Interface . . . . . . . . . 341

Domain, Boundary, Point, and Pair Nodes for the Laminar and

Turbulent Flow, Two-Phase, Level Set and Phase Field Interfaces. . . 343

C O N T E N T S | 9

10 | C O N T E N T S

Wall. . . . . . . . . . . . . . . . . . . . . . . . . . . 345

Fluid Properties . . . . . . . . . . . . . . . . . . . . . . 347

Gravity . . . . . . . . . . . . . . . . . . . . . . . . . 349

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 350

Initial Interface . . . . . . . . . . . . . . . . . . . . . . . 351

The Turbulent Flow, Two-Phase, Level Set and Phase Field

Interfaces 352

The Turbulent Flow, Two-Phase Flow, Level Set Interface . . . . . . . 352

The Turbulent Two-Phase Flow, Phase Field Interface . . . . . . . . 355

Mixing Length Limit . . . . . . . . . . . . . . . . . . . . . 356

The Bubbly Flow Interfaces 357

The Laminar Bubbly Flow Interface . . . . . . . . . . . . . . . 357

The Turbulent Bubbly Flow Interface . . . . . . . . . . . . . . 360

Domain and Boundary Nodes for the Laminar and Turbulent Bubbly

Flow Interfaces . . . . . . . . . . . . . . . . . . . . . 363


Gravity . . . . . . . . . . . . . . . . . . . . . . . . . 367

Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . 368

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 369

Wall. . . . . . . . . . . . . . . . . . . . . . . . . . . 369

Inlet . . . . . . . . . . . . . . . . . . . . . . . . . . . 371

Outlet . . . . . . . . . . . . . . . . . . . . . . . . . . 373

Symmetry . . . . . . . . . . . . . . . . . . . . . . . . 375

Gas Boundary Condition Equations . . . . . . . . . . . . . . . 375

The Mixture Model Interfaces 377

The Mixture Model, Laminar Flow Interface. . . . . . . . . . . . 377

The Mixture Model, Turbulent Flow Interface . . . . . . . . . . . 381

Domain and Boundary Nodes for the Mixture Model Laminar and

Turbulent Flow Interfaces . . . . . . . . . . . . . . . . . 383

Mixture Properties . . . . . . . . . . . . . . . . . . . . . 385

Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . 387

Gravity . . . . . . . . . . . . . . . . . . . . . . . . . 388

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 389

Wall. . . . . . . . . . . . . . . . . . . . . . . . . . . 390

Inlet . . . . . . . . . . . . . . . . . . . . . . . . . . . 392

Outlet . . . . . . . . . . . . . . . . . . . . . . . . . . 393

Symmetry . . . . . . . . . . . . . . . . . . . . . . . . 394

The Euler-Euler Model, Laminar Flow Interface 395

Domain, Boundary, Point, and Pair Nodes for the Euler-Euler Model,

Laminar Flow Interface . . . . . . . . . . . . . . . . . . 398

Phase Properties . . . . . . . . . . . . . . . . . . . . . . 399

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 401

Wall. . . . . . . . . . . . . . . . . . . . . . . . . . . 402

Inlet . . . . . . . . . . . . . . . . . . . . . . . . . . . 402

Outlet . . . . . . . . . . . . . . . . . . . . . . . . . . 404

Theory for the Two-Phase Flow Interfaces 405

Level Set and Phase Field Equations . . . . . . . . . . . . . . . 405

Conservative and Non-Conservative Formulations . . . . . . . . . 408

Phase Initialization . . . . . . . . . . . . . . . . . . . . . 408

Numerical Stabilization . . . . . . . . . . . . . . . . . . . 409

References for the Level Set and Phase Field Interfaces . . . . . . . 410

Theory for the Bubbly Flow Interfaces 411

The Bubbly Flow Equations . . . . . . . . . . . . . . . . . . 411

Turbulence Modeling in Bubbly Flow Applications . . . . . . . . . 415

References for the Bubbly Flow Interfaces . . . . . . . . . . . . 416

Theory for the Mixture Model Interfaces 418

The Mixture Model Equations . . . . . . . . . . . . . . . . . 418

Dispersed Phase Boundary Conditions Equations . . . . . . . . . 421

Turbulence Modeling in Mixture Models . . . . . . . . . . . . . 422

Slip Velocity Models . . . . . . . . . . . . . . . . . . . . . 423

References for the Mixture Model Interfaces . . . . . . . . . . . 425

Theory for the Euler-Euler Model, Laminar Flow Interface 426

The Euler-Euler Model Equations . . . . . . . . . . . . . . . . 426

References for the Euler-Euler Model, Laminar Flow Interface . . . . . 432

C O N T E N T S | 11


C h a p t e r 7 : P o r o u s M e d i a a n d S u b s u r f a c e F l o w

Modeling Porous Media and Subsurface Flow 436


The Porous Media Flow Interface Options . . . . . . . . . . . . 437


The Darcy’s Law Interface 441

Domain, Boundary, Edge, Point, and Pair Nodes for the Darcy’s

Law Interface . . . . . . . . . . . . . . . . . . . . . . 442

Fluid and Matrix Properties . . . . . . . . . . . . . . . . . . 443

Mass Source . . . . . . . . . . . . . . . . . . . . . . . 445

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 445

Pressure . . . . . . . . . . . . . . . . . . . . . . . . . 446

Mass Flux. . . . . . . . . . . . . . . . . . . . . . . . . 446

Inlet . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

Symmetry . . . . . . . . . . . . . . . . . . . . . . . . 447

No Flow . . . . . . . . . . . . . . . . . . . . . . . . . 448

Flux Discontinuity . . . . . . . . . . . . . . . . . . . . . 448

Outlet . . . . . . . . . . . . . . . . . . . . . . . . . . 449

The Brinkman Equations Interface 450

Domain, Boundary, Point, and Pair Nodes for the Brinkman

Equations Interface . . . . . . . . . . . . . . . . . . . . 452


Forchheimer Drag . . . . . . . . . . . . . . . . . . . . . 455

Mass Source . . . . . . . . . . . . . . . . . . . . . . . 455

Volume Force . . . . . . . . . . . . . . . . . . . . . . . 455

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 456

The Free and Porous Media Flow Interface 457

Domain, Boundary, Point, and Pair Nodes for the Free and Porous

Media Flow Interface . . . . . . . . . . . . . . . . . . . 459


Porous Matrix Properties. . . . . . . . . . . . . . . . . . . 460

Volume Force . . . . . . . . . . . . . . . . . . . . . . . 461

Forchheimer Drag . . . . . . . . . . . . . . . . . . . . . 462

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 462

Microfluidic Wall Conditions . . . . . . . . . . . . . . . . . 462

The Two-Phase Darcy’s Law Interface 464

Domain, Boundary, and Pair Nodes for the Two-Phase Darcy’s Law

Interface. . . . . . . . . . . . . . . . . . . . . . . . 465


Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 468

No Flux . . . . . . . . . . . . . . . . . . . . . . . . . 468

Pressure and Saturation . . . . . . . . . . . . . . . . . . . 469

Mass Flux. . . . . . . . . . . . . . . . . . . . . . . . . 469

Inlet . . . . . . . . . . . . . . . . . . . . . . . . . . . 470

Outlet . . . . . . . . . . . . . . . . . . . . . . . . . . 470

Theory for the Darcy’s Law Interface 471

Darcy’s Law—Equation Formulation . . . . . . . . . . . . . . 471

Theory for the Brinkman Equations Interface 473

About the Brinkman Equations . . . . . . . . . . . . . . . . 473

Brinkman Equations Theory. . . . . . . . . . . . . . . . . . 474

References for the Brinkman Equations Interface . . . . . . . . . . 475

Theory for the Free and Porous Media Flow Interface 476

Reference for the Free and Porous Media Flow Interface . . . . . . . 476

Theory for the Two-Phase Darcy’s Law Interface 477

Darcy’s Law—Equation Formulation . . . . . . . . . . . . . . 477

C h a p t e r 8 : C h e m i c a l S p e c i e s T r a n s p o r t

Modeling Chemical Species Transport 480



Adding a Chemical Species Transport Interface . . . . . . . . . . 483



The Transport of Diluted Species Interface 485

Domain, Boundary, and Pair Nodes for the Transport of Diluted

Species Interface. . . . . . . . . . . . . . . . . . . . . 488

Dynamic Transport Feature Node . . . . . . . . . . . . . . . 489

Turbulent Mixing . . . . . . . . . . . . . . . . . . . . . . 491

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 491

Line Mass Source . . . . . . . . . . . . . . . . . . . . . . 492

Reactions. . . . . . . . . . . . . . . . . . . . . . . . . 493

No Flux . . . . . . . . . . . . . . . . . . . . . . . . . 493

Concentration . . . . . . . . . . . . . . . . . . . . . . . 494

Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . 495

Inflow . . . . . . . . . . . . . . . . . . . . . . . . . . 495

Outflow . . . . . . . . . . . . . . . . . . . . . . . . . 496

Symmetry . . . . . . . . . . . . . . . . . . . . . . . . 496


Periodic Condition . . . . . . . . . . . . . . . . . . . . . 497

Point Mass Source . . . . . . . . . . . . . . . . . . . . . 498

Open Boundary . . . . . . . . . . . . . . . . . . . . . . 499

Thin Diffusion Barrier . . . . . . . . . . . . . . . . . . . . 499

Thin Impermeable Barrier . . . . . . . . . . . . . . . . . . 500

The Transport of Concentrated Species Interface 501

Domain, Boundary, and Pair Nodes for the Transport of

Concentrated Species Interface . . . . . . . . . . . . . . . 505

Transport Feature . . . . . . . . . . . . . . . . . . . . . 506


Reactions. . . . . . . . . . . . . . . . . . . . . . . . . 510

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 511

Mass Fraction . . . . . . . . . . . . . . . . . . . . . . . 511

Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . 512

Inflow . . . . . . . . . . . . . . . . . . . . . . . . . . 513

No Flux . . . . . . . . . . . . . . . . . . . . . . . . . 514

Outflow . . . . . . . . . . . . . . . . . . . . . . . . . 515

Symmetry . . . . . . . . . . . . . . . . . . . . . . . . 515


Open Boundary . . . . . . . . . . . . . . . . . . . . . . 516

The Reacting Flow Interfaces 518

The Reacting Flow, Laminar Flow Interface . . . . . . . . . . . . 518

The Reacting Flow, Turbulent Flow, k- Interface . . . . . . . . . . 522

The Reacting Flow, Turbulent Flow, k- Interface . . . . . . . . . 524

The Reacting Flow, Turbulent Flow, Low Re k-Interface. . . . . . . 525

Domain, Boundary, Point, and Pair Nodes for the Reacting Flow,

Laminar Flow and Turbulent Flow Interfaces . . . . . . . . . . 526

Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . 528

Wall. . . . . . . . . . . . . . . . . . . . . . . . . . . 531

Reactions. . . . . . . . . . . . . . . . . . . . . . . . . 531

Inflow . . . . . . . . . . . . . . . . . . . . . . . . . . 532

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 533

Reacting Boundary . . . . . . . . . . . . . . . . . . . . . 534

Open Boundary . . . . . . . . . . . . . . . . . . . . . . 535

The Reacting Flow in Porous Media Interfaces 537

The Reacting Flow in Porous Media (rfcs) Interface . . . . . . . . . 537

Domain, Boundary, Point, and Pair Nodes for the Reacting Flow in

Porous Media (rfcs) Interface . . . . . . . . . . . . . . . . 540

Transport Properties . . . . . . . . . . . . . . . . . . . . 541

Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . 542

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 542

Reacting Boundary . . . . . . . . . . . . . . . . . . . . . 543

The Reacting Flow in Porous Media (rfds) Interface . . . . . . . . . 544

Domain, Boundary, Point, and Pair Nodes for the Reacting Flow in

Porous Media (rfds) Interface . . . . . . . . . . . . . . . . 547

Transport Properties . . . . . . . . . . . . . . . . . . . . 548

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 549

Theory for the Transport of Diluted Species Interface 551

Mass Balance Equation . . . . . . . . . . . . . . . . . . . . 551

Convective Term Formulation . . . . . . . . . . . . . . . . . 552

Solving a Diffusion Equation Only . . . . . . . . . . . . . . . 553

Mass Sources for Species Transport . . . . . . . . . . . . . . . 553

Crosswind Diffusion . . . . . . . . . . . . . . . . . . . . 555

Reference . . . . . . . . . . . . . . . . . . . . . . . . 556

About Turbulent Mixing . . . . . . . . . . . . . . . . . . . 556



Theory for the Transport of Concentrated Species Interface 558

Multicomponent Mass Transport . . . . . . . . . . . . . . . . 558

Multicomponent Diffusion: Mixture-Average Approximation . . . . . 559

Multispecies Diffusion: Fick’s Law Approximation. . . . . . . . . . 560

Multicomponent Thermal Diffusion . . . . . . . . . . . . . . . 561


References for the Transport of Concentrated Species Interface . . . . 562

Theory for the Reacting Flow Interfaces 563

Pseudo Time Stepping for Mass Transport . . . . . . . . . . . . 563

The Stefan Velocity . . . . . . . . . . . . . . . . . . . . . 564

The Chemical Reaction Rate . . . . . . . . . . . . . . . . . 565

Turbulent Mass Transport Models . . . . . . . . . . . . . . . 566

Mass Transport Wall Functions . . . . . . . . . . . . . . . . 569

Turbulent Reactions . . . . . . . . . . . . . . . . . . . . . 570

The Reaction Feature . . . . . . . . . . . . . . . . . . . . 572

References for the Reacting Flow Interfaces. . . . . . . . . . . . 572

Theory for the Reacting Flow in Porous Media Interfaces 574

Theory for the Reacting Flow in Porous Media (rfcs) Interface. . . . . 574

Theory for the Reacting Flow in Porous Media (rfds) Interface . . . . 575

C h a p t e r 9 : T h i n - F i l m F l o w

Modeling Thin-Film Flow 578


The Thin-Film Flow Interfaces 579

The Thin-Film Flow, Shell Interface . . . . . . . . . . . . . . . 579

The Thin-Film Flow, Domain Interface . . . . . . . . . . . . . . 581

The Thin-Film Flow, Edge Interface . . . . . . . . . . . . . . . 582

Domain, Boundary, Edge, Point, and Pair Nodes for the Thin-Film

Flow Branch Interfaces . . . . . . . . . . . . . . . . . . 583

Fluid-Film Properties . . . . . . . . . . . . . . . . . . . . 583

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 589

Inlet . . . . . . . . . . . . . . . . . . . . . . . . . . . 589

Outlet . . . . . . . . . . . . . . . . . . . . . . . . . . 590

Border. . . . . . . . . . . . . . . . . . . . . . . . . . 590

Wall. . . . . . . . . . . . . . . . . . . . . . . . . . . 591

Symmetry . . . . . . . . . . . . . . . . . . . . . . . . 592

Theory for the Thin-Film Flow Interfaces 593

Thin-Film Flow. . . . . . . . . . . . . . . . . . . . . . . 593

The Reynolds Equation . . . . . . . . . . . . . . . . . . . 594

Flow Models . . . . . . . . . . . . . . . . . . . . . . . 598

The Modified Reynolds Equation—Gas Flows . . . . . . . . . . . 605

Flow Models for Rarefied Gases . . . . . . . . . . . . . . . . 606

Frequency Domain Formulation . . . . . . . . . . . . . . . . 609

Boundary Conditions . . . . . . . . . . . . . . . . . . . . 612

References for the Thin-Film Flow Interfaces . . . . . . . . . . . 612

C h a p t e r 1 0 : T h e M a t h e m a t i c s , M o v i n g I n t e r f a c e

B r a n c h

The Level Set Interface 614

Domain, Boundary, and Pair Nodes for the Level Set Interface . . . . 615

Level Set Model . . . . . . . . . . . . . . . . . . . . . . 616

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 617

Inlet . . . . . . . . . . . . . . . . . . . . . . . . . . . 617


No Flow . . . . . . . . . . . . . . . . . . . . . . . . . 618

The Phase Field Interface 619

Domain, Boundary, and Pair Nodes for the Phase Field Interface . . . . 620

Phase Field Model . . . . . . . . . . . . . . . . . . . . . 621

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 622

Inlet . . . . . . . . . . . . . . . . . . . . . . . . . . . 623


Wetted Wall . . . . . . . . . . . . . . . . . . . . . . . 624



Theory for the Level Set Interface 625

The Level Set Method . . . . . . . . . . . . . . . . . . . . 625

Conservative and Non-Conservative Form . . . . . . . . . . . . 627

Initializing the Level Set Function . . . . . . . . . . . . . . . . 627

Variables For Geometric Properties of the Interface . . . . . . . . 628

Reference for the Level Set Interface . . . . . . . . . . . . . . 629

Theory for the Phase Field Interface 630

About the Phase Field Method. . . . . . . . . . . . . . . . . 630

The Equations for the Phase Field Method . . . . . . . . . . . . 630

Conservative and Non-Conservative Forms . . . . . . . . . . . 632

Additional Sources of Free Energy . . . . . . . . . . . . . . . 632

Initializing the Phase Field Function . . . . . . . . . . . . . . . 633

Variables and Expressions . . . . . . . . . . . . . . . . . . 633

Reference for the Phase Field Interface . . . . . . . . . . . . . 634

C h a p t e r 1 1 : G l o s s a r y

Glossary of Terms 636

1

I n t r o d u c t i o n

This guide describes the CFD Module, an optional add-on package for COMSOL Multiphysics® that provides you with tools for computational fluid dynamics, CFD. The modeling of fluid flow is an increasingly important part in development of new equipment and processes.

This chapter introduces you to the capabilities of the module. A summary of the physics interfaces and where you can find documentation and model examples is also included. The last section is a brief overview with links to each chapter in this guide.

• About the CFD Module

• Overview of the User’s Guide

19

20 | C H A P T E R

Abou t t h e C FD Modu l e

In this section:

• Why CFD is Important for Modeling

• How the CFD Module Helps Improve Your Modeling

• Where Do I Access the Documentation and Model Libraries?

Why CFD is Important for Modeling

Computational fluid dynamics, CFD, is an integral part in a constantly growing number of development processes, and is a well established field within many different engineering disciplines; mechanical, chemical, civil, aeronautical, and also in more specialized areas such as biomedical engineering.

Often the flow itself is not the main focus in a simulation. Instead it is how the flow affects other process and application parameters that is important. The transport of species through the different parts of a chemical reactor, the effective cooling of a computer’s hard drive and electronics, the dispersion of energy within the damping film of an accelerometer, the extent of nuclear waste spreading from a subterranean repository—all of these are applications for which the flow must be fully understood and is an integral part of the process’s description and simulation.

In many situations, while the flow can add necessary operational parameters to a process or application, it is also affected by them. For example, a chemical reactor creates a pressure which disturbs the flow, the electronic heat affects the fluid’s density and flow properties, the accelerometer elasticity imposes an oscillation on the flow, while the subterranean environment’s poroelasticity changes the course of the flow.

A description combining several physics fields is often required to produce accurate models of real world applications involving fluid flow. Being able to effectively simulate such models increases the understanding of the studied processes and applications, which in turn leads to optimization of the flow and other parameters.

Historically, a sophisticated modeling tool was a privilege that only large companies could afford, for which the savings made in bulk production justified the computer software costs and need for CFD specialists. Today’s engineers are educated in the use of software modeling tools, and are often expected to create realistic models of

1 : I N T R O D U C T I O N

advanced systems on their personal computers. This is where COMSOL Multiphysics can improve your modeling capabilities.

How the CFD Module Helps Improve Your Modeling

The CFD Module is an optional package that extends the COMSOL Multiphysics modeling environment with customized physics interfaces and functionality optimized for the analysis of all types of fluid flow. It is developed for a wide variety of users including researchers, developers, teachers, and students. It is not just a tool for CFD experts; it can be used by all engineers and scientists who work with systems in which momentum transport through fluid flow is an important part of a process or application.

The module uses the latest research possible to simulate fluid flow and it provides the most user-friendly simulation environment for CFD applications. The solvers and meshes are optimized for fluid-flow applications and have built-in robust stabilization methods.

The readily available coupling of heat and mass transport to fluid flow enables modeling of a wide range of industrial applications such as heat exchangers, turbines, separation units, and ventilation systems.

Ready-to-use interfaces enable you to model laminar and turbulent single- or multiphase flow. Functionality to treat coupled free and porous media flow, stirred vessels, and fluid structure interaction is also included.

Together with COMSOL Multiphysics and its other optional packages, the CFD Module takes flow simulations to a new level, allowing for arbitrary coupling to physics interfaces describing other physical phenomena, such as structural mechanics, electromagnetics, or even user-defined transport equations. This allows for unparalleled modeling capabilities for multiphysics applications involving fluid flow.

Like all COMSOL modules, the interfaces described in this guide include all the steps available for the modeling process, which are described in detail in the COMSOL Multiphysics Reference Manual:

• Definitions of parameters and component variables

• Creating, importing and manipulating the geometry

• Specifying the materials to include in the component

• Defining the physics of the fluid flow in domains and on boundaries, and coupling it to other physics

A B O U T T H E C F D M O D U L E | 21

22 | C H A P T E R

• Set up an appropriate mesh for the modeling domain with consideration given to the fluid-flow system’s behavior

• Solving the equations that describe a system for stationary or dynamic behavior, or as a parametric or optimization study

• Collecting and analyzing results to present for further use in other analyses.

Once a model is defined, you can go back and make changes in all of the branches listed, while maintaining consistency in the other definitions throughout. You can restart the solver, for example, using the existing solution as an initial guess or even alter the geometry, while the equations and boundary conditions are kept consistent through the associative geometry feature.

Where Do I Access the Documentation and Model Libraries?

A number of Internet resources provide more information about COMSOL, including licensing and technical information. The electronic documentation, topic-based (or context-based) help, and the Model Libraries are all accessed through the COMSOL Desktop.

T H E D O C U M E N T A T I O N A N D O N L I N E H E L P

The COMSOL Multiphysics Reference Manual describes all core physics interfaces and functionality included with the COMSOL Multiphysics license. This book also has instructions about how to use COMSOL and how to access the electronic Documentation and Help content.

Opening Topic-Based HelpThe Help window is useful as it is connected to many of the features on the GUI. To learn more about a node in the Model Builder, or a window on the Desktop, click to highlight a node or window, then press F1 to open the Help window, which then

If you are reading the documentation as a PDF file on your computer, the blue links do not work to open a model or content referenced in a different guide. However, if you are using the Help system in COMSOL Multiphysics, these links work to other modules (as long as you have a license), model examples, and documentation sets.


displays information about that feature (or click a node in the Model Builder followed by the Help button ( ). This is called topic-based (or context) help.

Opening the Documentation Window

T H E M O D E L L I B R A R I E S W I N D O W

Each model includes documentation that has the theoretical background and step-by-step instructions to create the model. The models are available in COMSOL as MPH-files that you can open for further investigation. You can use the step-by-step

To open the Help window:

• In the Model Builder, click a node or window and then press F1.

• On any toolbar (for example, Home or Geometry), hover the mouse over a button (for example, Browse Materials or Build All) and then press F1.

• From the File menu, click Help ( ).

• In the upper-right part of the COMSOL Desktop, click the ( ) button.

To open the Help window:

• In the Model Builder, click a node or window and then press F1.

• On the main toolbar, click the Help ( ) button.

• From the main menu, select Help>Help.

To open the Documentation window:

• Press Ctrl+F1.

• From the File menu select Help>Documentation ( ).

To open the Documentation window:

• Press Ctrl+F1.

• On the main toolbar, click the Documentation ( ) button.

• From the main menu, select Help>Documentation.


24 | C H A P T E R

instructions and the actual models as a template for your own modeling and applications. In most models, SI units are used to describe the relevant properties, parameters, and dimensions in most examples, but other unit systems are available.

Once the Model Libraries window is opened, you can search by model name or browse under a module folder name. Click to highlight any model of interest and a summary of the model and its properties is displayed, including options to open the model or a PDF document.

Opening the Model Libraries WindowTo open the Model Libraries window ( ):

C O N T A C T I N G C O M S O L B Y E M A I L

For general product information, contact COMSOL at [email protected].

To receive technical support from COMSOL for the COMSOL products, please contact your local COMSOL representative or send your questions to [email protected]. An automatic notification and case number is sent to you by email.

The Model Libraries Window in the COMSOL Multiphysics Reference Manual.

• From the Home ribbon, click ( ) Model Libraries.

• From the File menu select Model Libraries.

To include the latest versions of model examples, from the File>Help menu, select ( ) Update COMSOL Model Library.

• On the main toolbar, click the Model Libraries button.

• From the main menu, select Windows>Model Libraries.

To include the latest versions of model examples, from the Help menu select ( ) Update COMSOL Model Library.


C O M S O L WE B S I T E S

COMSOL website www.comsol.com

Contact COMSOL www.comsol.com/contact

Support Center www.comsol.com/support

Product Download www.comsol.com/support/download

Product Updates www.comsol.com/support/updates

COMSOL Community www.comsol.com/community

Events www.comsol.com/events

COMSOL Video Gallery www.comsol.com/video

Support Knowledge Base www.comsol.com/support/knowledgebase


http://www.comsol.com

http://www.comsol.com/contact/

http://www.comsol.com/support/

http://www.comsol.com/support/download/

http://www.comsol.com/support/updates

http://www.comsol.com/community

http://www.comsol.com/events/

http://www.comsol.com/video/

http://www.comsol.com/support/knowledgebase/

26 | C H A P T E R

Ove r v i ew o f t h e U s e r ’ s Gu i d e

The CFD Module User’s Guide gets you started with modeling CFD systems using COMSOL Multiphysics. The information in this guide is specific to this module. Instructions on how to use COMSOL in general are included with the COMSOL Multiphysics Reference Manual. For theory relating to the physics interfaces, see the end of each chapter.

TA B L E O F C O N T E N T S , G L O S S A R Y , A N D I N D E X

To help you navigate through this guide, see the Contents, Glossary, and Index.

Q U I C K S T A R T G U I D E

The Quick Start Guide includes some basic modeling strategies to get you started modeling fluid flow in your particular application area. For example, it gives some tips about how to control your material properties and how to set the optimal mesh to make solving the model easier and quicker. It also includes a summary of all the physics interfaces included with the CFD Module.

T H E P H Y S I C S I N T E R F A C E S

The CFD Module both extends physics interfaces available with COMSOL Multiphysics and provides additional physics interfaces. As a result, the module contains a wide range of physics interfaces for modeling various types of momentum transport. You can simulate laminar and turbulent flow, Newtonian and non-Newtonian flow, isothermal and non-isothermal flow, multiphase flow, and flow in porous media. The CFD Module also provides interfaces for modeling flows that occur in thin-films or in bounded regions, and in stationary and rotating domains.

On top of this, the CFD Module includes physics interfaces for modeling heat transfer, and transport and reactions of chemical species. These are typical phenomena that occur in fluid flow and are strongly coupled to the flow field.

As detailed in the section Where Do I Access the Documentation and Model Libraries? this information can also be searched from the COMSOL Multiphysics software Help menu.


Single-Phase FlowThe Single-Phase Flow chapter describes the many physics interfaces available for laminar and turbulent flow. Modeling Single-Phase Flow helps you choose the best fluid flow interface for your particular application.

Heat Transfer and Non-Isothermal FlowThe CFD Module includes physics interfaces for the simulation of heat transfer in fluid flow. As with all other physics models simulated in COMSOL Multiphysics, any description involving heat transfer can be directly coupled to any other physical process. This is particularly relevant for systems with fluid flow and mass transfer. The interfaces also allow you to account for heat sources and sinks, such as energy evolving from chemical reactions.

The Heat Transfer and Non-Isothermal Flow chapter describes these physics interfaces in greater detail. To help you select which physics interface to use see Modeling Heat Transfer in the CFD Module.

High Mach FlowThe High Mach Number Flow chapter describes three versions of the same predefined multiphysics interface used to model laminar and turbulent compressible flow at high Mach numbers.

Multiphase FlowThe Multiphase Flow chapter describes physics interfaces to model flows with more than one phase, for example flows with two fluids or flows with dispersed droplets or particles. To help you select which physics interface to use see Modeling Multiphase Flow.

Porous Media and Subsurface FlowThe Porous Media and Subsurface Flow chapter describes the Darcy’s Law, Brinkman

Equations, and Free and Porous Media Flow interfaces. To help you select which physics interface to use see Modeling Porous Media and Subsurface Flow.

Thin-Film FlowThe Thin-Film Flow chapter describes physics interfaces that model flow in thin regions such as lubrication shells and fluid bearings. Modeling Thin-Film Flow helps you select the correct physics interface to use.

Chemical Species TransportThe Chemical Species Transport chapter describes physics interfaces that are used for the simulation of chemical reactions, and mass or material transport through diffusion

O V E R V I E W O F T H E U S E R ’ S G U I D E | 27

28 | C H A P T E R

and convection. Modeling Chemical Species Transport helps you select the best physics interface to use.

The Mathematics InterfacesThe Mathematics, Moving Interface Branch describes the Phase Field and Level Set interfaces found under the Mathematics>Moving Interface branch. In the CFD Module these physics features are integrated into the relevant physics interfaces.


2

Q u i c k S t a r t G u i d e

This chapter has some basic modeling strategies to get you started modeling fluid flow in your application area.

In this chapter:

• Modeling and Simulations of Fluid Flow

| 29

30 | C H A P T E R

Mode l i n g and S imu l a t i o n s o f F l u i d F l ow

In this section:

• Modeling Strategy

• Geometric Complexities

• Material Properties

• Defining the Physics

• Meshing

• The Choice of Solver and Solver Settings

• The CFD Module Physics Interface Guide

Modeling Strategy

Modeling and simulating fluid flow is a cost-effective way for engineers and scientists to understand, develop, optimize, and control designs and processes.

One of the most important things to consider before setting up a model is the accuracy that is required in the simulation results. This determines the level of complexity in the model.

Since fluid-flow simulations are often computationally demanding, a multi-stage modeling strategy is usually required. This implies using a simplified model as a starting point in the project. Complexities can then be introduced gradually so that the effect of each refinement of the model description is well understood before introducing new complexities.

Complexities in the modeling process can be introduced at different stages in order to achieve the desired accuracy. They can be introduced in the description of the geometry, the physical properties, and in the governing equations. The Model Builder,

The Physics Interfaces and Building a COMSOL Model in the COMSOL Multiphysics Reference Manual

2 : Q U I C K S T A R T G U I D E

which shows the model setup as a sequence of operations in the model tree, is designed with this modeling strategy in mind.

In addition to fluid flow, COMSOL Multiphysics and the CFD Module have predefined couplings for fluid flow and other phenomena. Examples of these couplings are heat transfer for free convection and transport of chemical species in simulations of reacting flows. Set up your own couplings by defining mathematical expressions of the dependent variables (velocity, pressure, temperature, and so forth) in the physics interfaces for arbitrary multiphysics combinations.

Geometric Complexities

A complicated 3D CAD drawing is usually not the best starting point for the modeling process. A 2D representation of a cross-section of the geometry can give valuable initial estimates of the flow field that can be used when setting up the full 3D model. For example, you might be able to determine the pressure variations and the nature of the flow, or whether or not a turbulence model is needed. This provides information about where in the final geometry the most amount of ‘change’ occurs, if a more advanced fluid-flow model and/or better resolution is required, and what parts of the modeling process are more sensitive than others.

Simplifying the geometry reduces the simulation time. Making use of symmetry planes can cut down the geometry to one half or even less of the original size. Rounding-off corners is another way to reduce mesh resolution. Resolving small geometric parts requires a fine mesh, but the parts themselves can have a negligible effects on the fluid field as a whole.

Material Properties

Depending on the accuracy required in a simulation, the effort put in acquiring data for the fluid properties can also vary. In many cases, the dependencies of the fluid properties on pressure and temperature have to be taken into account.

For a pressure-driven flow, it is usually a good approach to first set up a model using constant density and viscosity, to get a first estimate of the flow and pressure fields.

Geometry Modeling and CAD Tools in the COMSOL Multiphysics Reference Manual

M O D E L I N G A N D S I M U L A T I O N S O F F L U I D F L O W | 31

32 | C H A P T E R

Once the model works with constant properties it can be extended by adding the accurate expressions for density and viscosity.

For free convection the density variations drive the flow. The fluid properties’ dependencies on the modeled variables, for example temperature, then have to be accounted for from the beginning. In difficult cases, with large temperature variations, it can be beneficial to run a time-dependent simulation even if the purpose of the simulation is to get the results at steady-state.

Defining the Physics

The CFD Module has various physics interfaces for laminar and turbulent single-phase, multiphase, nonisothermal and high Mach number flow, and for thin-film flow and porous media flow.

The definition of the physics depends on the accuracy required in a simulation. A fluid which is weakly compressible could be approximated as incompressible if the required accuracy allows for it. A complex turbulence model can be replaced by a much simpler one, again if the resulting accuracy is sufficient. A first step to set up the physics is to make the initial model as simple as possible. The results from such a simulation can reveal useful pieces of information that help later when more complex steps are added to the physics.

The fluid-flow interfaces can also be coupled to any other physics interface in a multiphysics model. When setting up such a complex multiphysics component involving fluid flow and other coupled physics, it is a good strategy to first define and solve one physics at a time. This allows for verification of the model setup, for example to check if the intended domain and boundary settings are reflected in the solution of each decoupled physics. The alternative of debugging the model setup with several coupled physics interfaces could be very time-consuming.

In steady-state multiphysics simulations, it can also be a good strategy to solve the model for each physics in a decoupled setup as a first step. The solutions from the decoupled models can then be used as initial guesses for the fully coupled model. This is especially recommended for highly nonlinear models. The Study node in the Model

Builder is designed for this modeling strategy.

Meshing

The mesh used in a fluid-flow simulation depends on the fluid-flow model and on the accuracy required in the simulation. A fluid-flow model can inherently require a fine


resolution in order to converge, even though the results might not require a correspondingly high accuracy. In such cases, it can be a good idea to change the fluid-flow model. An example is the low-Reynolds number k- model which gives a very accurate description of the flow near solid walls, but requires a very fine mesh there. In many cases, the standard k- model with wall functions can deliver an accurate enough result at a much lower computational cost. In other cases, the requirement of accuracy in the results can limit the maximum element size.

There are a number of different mesh types and meshing strategies for fluid-flow modeling in COMSOL Multiphysics and these are briefly described next.

U N S T R U C T U R E D M E S H E S

Free-meshing techniques generate unstructured meshes that can be used for most types of geometries. The mesh-generating algorithms are highly automated, often creating a good quality mesh from minimal user input. This mesh type is therefore a good choice when the geometry of the domain is evident but the behavior of the mathematical model in it is unknown. Yet, unstructured meshes tend to be isotropic or homogenous in nature, so that they fail to take advantage of the different resolution requirements in the stream-wise and cross-stream directions.

S T R U C T U R E D M E S H E S

In many ways, the properties of structured meshes complement those of the unstructured type. Structured meshes provide high-quality meshes with few elements for sufficiently simple geometries. The properties of a structured mesh can furthermore be used to create very efficient numerical methods. Finally, it is often easier to control the mesh when high anisotropy or large variations in mesh size and distribution is required, as the size of a structured mesh can be easily increased linearly or geometrically with the dimensions of the computational domain.

S W E P T M E S H E S

Swept meshes are a particular form of structured meshes, sometimes denoted semi-structured. These are generated in 3D by creating a mesh at a source face and then sweeping it along the domain to a destination face, such as from a cut in the cylindrical part of a polymerization reactor to its outlet face. A swept mesh is structured in the sweep direction, while the mesh at the source and destination faces can be either structured or unstructured. As is the case for structured meshes, the model geometry determines if a swept mesh is applicable. Swept meshes are typically ideal when the cross section in the sweep direction is constant, which is the case for channels and


34 | C H A P T E R

pipes, for instance. Revolving a mesh around a symmetry axis is another useful sweep operation.

B O U N D A R Y L A Y E R M E S H E S

A boundary layer mesh is a mesh with an element distribution that is stacked or dense in the direction normal to a boundary. It is created by inserting structured layers of elements along specific boundaries and merging the outer layer with the surrounding structured or unstructured mesh. This type of mesh is very useful for many fluid-flow applications especially when coupled to mass- and energy transfer, where thin boundary layers need to be resolved. This is also the default physics-induced mesh for fluid flow.

M E S H C O N V E R G E N C E

Ideally, a mesh convergence analysis should be performed in order to estimate the accuracy of a simulation. This means that the mesh should be made twice as fine in each spatial direction and the simulation carried out once again on the refined mesh. If the change in critical solution parameters for the original mesh and the finer mesh is within the required tolerance, the solution can be regarded as being mesh-converged. For practical reasons, it is seldom possible to make the mesh twice as fine in each direction. Instead, some critical regions can be identified and the mesh is refined only there.

The Choice of Solver and Solver Settings

The default solver for the fluid-flow interfaces are optimized for a large variety of fluid-flow conditions and applications. Which solver that is suggested depends both on the physics interface and on the study type.

The default solver settings are trade-offs between performance and robustness. The more advanced a model is, the more tuning of the solver might be necessary to obtain a solution. This is another reason for why it is wise to start with a reduced model description rather than the complete description.

2D models and small 3D models get so called direct-solver suggestions. Direct solvers are very robust but their memory requirement scales as somewhere between N1.5and N2 where N is the number of degrees of freedom in the model. This means that a

Meshing in the COMSOL Multiphysics Reference Manual


direct solver becomes prohibitively expensive for large problems. Large 3D models therefore get iterative-solver suggestions per default. The memory requirement for an iterative solver optimally scales as N. The drawback with iterative solvers is that they are less robust than direct solvers. A model can converge with a direct solver but fail with an iterative solver. Large models therefore require more care when being set up than small models do.

For well-posed models, there are possibilities to tune the default solvers to gain performance. This is especially true for time-dependent models with a wide variety of solver settings providing many options to reduce the computational time.

The CFD Module Physics Interface Guide

The CFD Module extends the functionality of the physics interfaces of the base package for COMSOL Multiphysics. The details of the physics interfaces and study types for the CFD Module are listed in the table. The functionality of the COMSOL Multiphysics base package is listed in the COMSOL Multiphysics Reference Manual.

Studies and Solvers in the COMSOL Multiphysics Reference Manual

In the COMSOL Multiphysics Reference Manual:

• Studies and Solvers

• The Physics Interfaces

• For a list of all the core physics interfaces included with a COMSOL Multiphysics license, see Physics Guide.

INTERFACE ICON TAG SPACE DIMENSION

AVAILABLE PRESET STUDY TYPE

Chemical Species Transport

Transport of Diluted Species1 chds all dimensions stationary; time dependent

Transport of Concentrated Species

chcs all dimensions stationary; time dependent


36 | C H A P T E R

Reacting Flow

Laminar Flow rspf 3D, 2D, 2D axisymmetric

stationary; time dependent

Turbulent Flow

Turbulent Flow, k- rspf 3D, 2D, 2D axisymmetric


Turbulent Flow, k- rspf 3D, 2D, 2D axisymmetric


Turbulent Flow, Low Re k-

rspf 3D, 2D, 2D axisymmetric

stationary with initialization; transient with initialization

Reacting Flow in Porous Media

Transport of Diluted Species

rfds 3D, 2D, 2D axisymmetric


Transport of Concentrated Species

rfcs 3D, 2D, 2D axisymmetric


Fluid Flow

Single-Phase Flow

Laminar Flow1 spf 3D, 2D, 2D axisymmetric


Turbulent Flow

Turbulent Flow, k- spf 3D, 2D, 2D axisymmetric


Turbulent Flow, k- spf 3D, 2D, 2D axisymmetric


Turbulent Flow, SST spf 3D, 2D, 2D axisymmetric






spf 3D, 2D, 2D axisymmetric


Turbulent Flow, Spalart-Allmaras

spf 3D, 2D, 2D axisymmetric


Creeping Flow spf 3D, 2D, 2D axisymmetric


Rotating Machinery, Fluid Flow

Rotating Machinery, Laminar Flow

rmspf 3D, 2D frozen rotor; time dependent

Rotating Machinery, Turbulent Flow, k-

rmspf 3D, 2D frozen rotor; time dependent

Thin-Film Flow

Thin-Film Flow, Shell tffs 3D stationary; time dependent; frequency domain; eigenfrequency

Thin-Film Flow, Domain tff 2D stationary; time dependent; frequency domain; eigenfrequency

Thin-Film Flow, Edge tffs 2D and 2D axisymmetric

stationary; time dependent; frequency domain; eigenfrequency

Multiphase Flow

Bubbly Flow

Laminar Bubbly Flow bf 3D, 2D, 2D axisymmetric


Turbulent Bubbly Flow bf 3D, 2D, 2D axisymmetric





38 | C H A P T E R

Mixture Model

Mixture Model, Laminar Flow

mm 3D, 2D, 2D axisymmetric


Mixture Model, Turbulent Flow

mm 3D, 2D, 2D axisymmetric


Euler-Euler Model

Euler-Euler Model, Laminar Flow

ee 3D, 2D, 2D axisymmetric


Two-Phase Flow, Level Set

Laminar Two-Phase Flow, Level Set

tpf 3D, 2D, 2D axisymmetric

transient with initialization

Turbulent Two-Phase Flow, Level Set



Two-Phase Flow, Phase Field

Laminar Two-Phase Flow, Phase Field



Turbulent Two-Phase Flow, Phase Field



Porous Media and Subsurface Flow

Brinkman Equations br 3D, 2D, 2D axisymmetric


Darcy’s Law dl all dimensions stationary; time dependent

Two-Phase Darcy’s Law tpdl 3D, 2D, 2D axisymmetric


Free and Porous Media Flow

fp 3D, 2D, 2D axisymmetric





Non-Isothermal Flow

Laminar Flow nitf 3D, 2D, 2D axisymmetric


Turbulent Flow

Turbulent Flow, k- nitf 3D, 2D, 2D axisymmetric




Turbulent Flow, SST nitf 3D, 2D, 2D axisymmetric



nitf 3D, 2D, 2D axisymmetric





High Mach Number Flow

Laminar Flow hmnf 3D, 2D, 2D axisymmetric


Turbulent Flow, k- hmnf 3D, 2D, 2D axisymmetric



hmnf 3D, 2D, 2D axisymmetric


Heat Transfer

Heat Transfer in Fluids1 ht all dimensions stationary; time dependent

Heat Transfer in Porous Media ht all dimensions stationary; time dependent




40 | C H A P T E R

Conjugate Heat Transfer

Laminar Flow nitf 3D, 2D, 2D axisymmetric


Turbulent Flow








Turbulent Flow, SST nitf 3D, 2D, 2D axisymmetric





Mathematics

Moving Interface

Level Set ls all dimensions transient with initialization

Phase Field pf all dimensions time dependent; transient with initialization

1 This physics interface is included with the core COMSOL package but has added functionality for this module.




S H O W M O R E P H Y S I C S O P T I O N S

There are several general options available for the physics interfaces and for individual nodes. This section is a short overview of these options, and includes links to additional information.

To display additional options for the physics interfaces and other parts of the model tree, click the Show button ( ) on the Model Builder and then select the applicable option.

After clicking the Show button ( ), additional sections are displayed on the settings window when a node is clicked and additional nodes are made available.

Physics nodes are available from the Physics ribbon toolbar (Windows users), Physics context menu (Mac or Linux users), or right-click to access the context menu (all users).

The additional sections that can be displayed include Equation, Advanced Settings, Discretization, Consistent Stabilization, and Inconsistent Stabilization.

You can also click the Expand Sections button ( ) in the Model Builder to always show some sections or click the Show button ( ) and select Reset to Default to reset to display only the Equation and Override and Contribution sections.

For most nodes, both the Equation and Override and Contribution sections are always available. Click the Show button ( ) and then select Equation View to display the Equation View node under all nodes in the Model Builder.

Availability of each node, and whether it is described for a particular node, is based on the individual selected. For example, the Discretization, Advanced Settings, Consistent

Stabilization, and Inconsistent Stabilization sections are often described individually throughout the documentation as there are unique settings.

The links to the features described in the COMSOL Multiphysics Reference Manual (or any external guide) do not work in the PDF, only from the online help in COMSOL Multiphysics.

In general, to add a node, go to the Physics toolbar, no matter what operating system you are using.


42 | C H A P T E R

O T H E R C O M M O N S E T T I N G S

At the main level, some of the common settings found (in addition to the Show options) are the Interface Identifier, Domain Selection, Boundary Selection, Edge Selection, Point Selection, and Dependent Variables.

At the node level, some of the common settings found (in addition to the Show options) are Domain Selection, Boundary Selection, Edge Selection, Point Selection, Material Type, Coordinate System Selection, and Model Inputs. Other sections are common based on application area and are not included here.

SECTION CROSS REFERENCE

Show More Options and Expand Sections

Advanced Physics Sections

The Model Builder

Discretization Show Discretization

Discretization (Node)

Discretization—Splitting of complex variables

Compile Equations

Consistent and Inconsistent Stabilization Stabilization

Numerical Stabilization

Constraint Settings Weak Constraints and Constraint Settings

Override and Contribution Physics Exclusive and Contributing Node Types


Coordinate System Selection Coordinate Systems

Domain, Boundary, Edge, and Point Selection (geometric entity selection)

About Geometric Entities

About Selecting Geometric Entities

The Geometry Entity Selection Sections

Equation Physics Nodes—Equation Section

Interface Identifier Predefined and Built-In Variables

Variable Naming Convention and Namespace

Viewing Node Names, Identifiers, Types, and Tags

Material Type Materials


T H E L I Q U I D S A N D G A S E S M A T E R I A L S D A T A B A S E

The CFD Module includes an additional Liquids and Gases material database with temperature-dependent fluid properties.

Model Inputs About Materials and Material Properties

Selecting Physics

Model Inputs and Multiphysics Couplings

Pair Selection Identity and Contact Pairs

Continuity on Interior Boundaries


For detailed information about materials and the Liquids and Gases Material Database, see Materials in the COMSOL Multiphysics Reference Manual.


44 | C H A P T E R

3

S i n g l e - P h a s e F l o w

There are several fluid-flow interfaces available as listed in The CFD Module Physics Interface Guide. This chapter describes the fluid-flow groups under the Fluid

Flow>Single-Phase Flow branch ( ) when adding a physics interface. The section Modeling Single-Phase Flow helps to choose the best one to start with.

In this chapter:

• The Laminar Flow, Creeping Flow, and Turbulent Flow Interfaces

• The Rotating Machinery, Laminar and Turbulent Flow Interfaces

• Theory for the Single-Phase Flow Interfaces

• Theory for the Turbulent Flow Interfaces

• Theory for the Rotating Machinery Interfaces

• The Wall Distance Interface is also available and described in the COMSOL Multiphysics Reference Manual including the theory and how it relates to fluid flow.

45

46 | C H A P T E R

Mode l i n g S i n g l e - Pha s e F l ow

The descriptions in this section are structured based on the order displayed in the Fluid

Flow branch. All the physics interfaces described in this section are found under the Fluid

Flow>Single-Phase Flow branch ( ) when adding a physics interface. Because most of the physics interfaces are integrated with each other, many physics features described cross reference to other physics interfaces. For example, nodes are usually available in both the laminar flow (Laminar Flow and Creeping Flow) and turbulent flow (k-, k-, SST, low Reynolds number k-, and Spalart-Allmaras turbulence models) interfaces.

In this section:

• Selecting the Right Physics Interface

• The Single-Phase Flow Interface Options

• Coupling to Other Physics Interfaces

Selecting the Right Physics Interface

The Single-Phase Flow branch included with the CFD Module has a number of subbranches with physics interfaces that describe different types of single-phase fluid flow. One or more of these physics interfaces can be added, either singularly or in combination with other physics interfaces for mass transport and heat transfer, for example.

Different types of flow require different equations to describe them. If the type of flow to model is already known, then select it directly. However, when you are uncertain of the flow type, or because it is difficult to reach a solution easily, you can start instead with a simplified model and add complexity as the model is built. Then test your way forward and compare models and results. For single-phase flow, the Laminar Flow interface is a good place to start if this is the case.

In other cases, you may know exactly how a fluid behaves and which equations, models, or physics interfaces best describe it, but because the model is so complex it is difficult to reach an immediate solution. Simpler assumptions may need to be made to solve the problem, and other physics interfaces might be better to fine-tune the solution process for

The Wall Distance Interface in the COMSOL Multiphysics Reference Manual

3 : S I N G L E - P H A S E F L O W

the more complex problem. The next section gives you an overview of each of the Single-Phase Flow interfaces to help you choose.

The Single-Phase Flow Interface Options

Several of the interfaces vary only by one or two default settings (see Table 3-1) in the Physical Model section, which are selected either from a check box or drop-down list. For the Single-Phase Flow branch, all except the Rotating Machinery interfaces have the same Interface Identifier (spf). The differences are based on the default settings required to model that type of flow as described in Table 3-1. Figure 3-1 shows the Laminar Flow interface settings window where you choose the type of compressibility (incompressible or compressible at Mach numbers below 0.3) and the turbulence model (or none for laminar flow), and a check box to model Stokes flow by neglecting the inertial term.

TABLE 3-1: THE SINGLE-PHASE FLOW PHYSICAL MODEL DEFAULT SETTINGS

INTERFACE ID COMPRESSIBILITY TURBULENT MODEL TYPE

TURBULENCE MODEL

NEGLECT INERTIAL TERM (STOKES FLOW)

Laminar Flow spf Compressible flow (Ma<0.3)

None n/a None

Turbulent Flow, k- spf Compressible flow (Ma<0.3)

RANS k- None

Turbulent Flow, k- spf Compressible flow (Ma<0.3)

RANS k- None

Turbulent Flow, SST spf Compressible flow (Ma<0.3)

RANS SST None


spf Compressible flow (Ma<0.3)

RANS Low Reynolds number k-

None


spf Compressible flow (Ma<0.3)

RANS Spalart-Allmaras

None

Creeping Flow spf Compressible flow (Ma<0.3)

None n/a Stokes Flow

Rotating Machinery, Laminar Flow

rmspf Compressible flow (Ma<0.3)

None None None

Rotating Machinery, Turbulent Flow, k-

rmspf Compressible flow (Ma<0.3)

None RANS, k- None

M O D E L I N G S I N G L E - P H A S E F L O W | 47

48 | C H A P T E R

L A M I N A R F L O W

The Laminar Flow Interface ( ) is used primarily to model flow at small to intermediate Reynolds numbers. The interface solves the Navier-Stokes equations, and by default assumes the flow to be compressible; that is, the density is not assumed to be constant.

Compressible flow is possible to model in this physics interface provided that the Mach number is less than 0.3, but you have to maintain control of the density and any of the mass balances that are deployed to accomplish this. You can also choose to model incompressible flow and simplify the equations to be solved.

This physics interface also allows you to simulate a certain class of non-Newtonian fluid flows by modifying the dynamic viscosity in the Navier-Stokes equations. You can model the fluid using the power law and Carreau models or enter another expression that describes the dynamic viscosity appropriately.

You can also describe other material properties such as density by entering equations for its dependence on fluid composition, pressure, or temperature. Many materials in the material libraries use temperature- and pressure-dependent property values. If the density is affected by temperature, the Non-Isothermal Flow interface may be applicable (see The Non-Isothermal Flow and Conjugate Heat Transfer, Laminar Flow and Turbulent Flow Interfaces).


Figure 3-1: The settings window for the Laminar Flow interface. Model compressible or non-compressible flow, laminar or turbulent flow, and Stokes flow. Combinations are also possible.

TU R B U L E N T F L O W

The various forms of the Single-Phase Flow, Turbulent Flow interfaces ( ) model flow of large Reynolds numbers. The interfaces solve the Reynolds-averaged Navier-Stokes (RANS) equations and for the filtered velocity field and filtered pressure as well as models for the turbulent viscosity. See The Laminar Flow, Creeping Flow, and Turbulent Flow Interfaces for links to the physics interface information.

The following turbulence models are available—a basic k- model, a k- model, an SST model, a Low Reynolds number k- model, and the Spalart-Allmaras model. Each model has its merits and weaknesses. See the Theory for the Turbulent Flow Interfaces for more details.


50 | C H A P T E R

Similarly to the Laminar Flow interface, compressibility (Mach<0.3) is selected by default. If required this can be deactivated to simplify the model. Non-Newtonian fluid models are not available for the Turbulent Flow interfaces.

C R E E P I N G F L O W

The Creeping Flow Interface ( ) models the Navier-Stokes equations without the contribution of the inertia term. This is often referred to as Stokes flow and is appropriate for flow at small Reynolds numbers, such as in very small channels or microfluidic applications.

The Creeping Flow interface can also be activated by selecting a check box in the Laminar

Flow interface. This physics interface can also model non-Newtonian fluids, including the Power Law and Carreau models, but not turbulence.

R O T A T I N G M A C H I N E R Y

The Rotating Machinery, Laminar and Turbulent Flow Interfaces ( ) model fluid flow in geometries with rotating parts. For example, stirred tanks, mixers, propellers and pumps.

Unlike the other physics interfaces under the Single-Phase Flow branch, the Rotating

Machinery, Fluid Flow interface can not be activated directly from the Laminar Flow interface or any of the other physics interfaces. It supports compressible and incompressible flow, the flow of non-Newtonian fluids using the Power Law and Carreau models, as well as turbulence. This physics interface also supports creeping flow.

Coupling to Other Physics Interfaces

Often, you may want to simulate applications that couple fluid flow to another type of phenomenon described in another physics interface. Although this is not often another type of flow, it can still involve physics interfaces supported in the CFD Module and in the COMSOL Multiphysics base package. This is often the case for applications that include chemical reactions and mass transport (see Chemical Species Transport), or energy transport, found in the Heat Transfer and Non-Isothermal Flow chapter.

More extensive descriptions of heat transfer, including radiation, can be found in the Heat Transfer Module, while a wider variety of tools for modeling chemical reactions and mass transport are found in the Chemical Reaction Engineering Module. Fluid flow is an important component for cooling electromagnetic phenomena, such as heat created through induction and microwave heating, which are simulated using the AC/DC Module and RF Module, respectively. Many applications involve the effect of


fluid-imposed loads on structural applications, for example, fluid-structure interaction (FSI). The Structural Mechanics Module and MEMS Module have physics interfaces specifically for these multiphysics applications.


52 | C H A P T E R

Th e L am i n a r F l ow , C r e e p i n g F l ow , and Tu r bu l e n t F l ow I n t e r f a c e s

In this section:

• The Laminar Flow Interface

• The Creeping Flow Interface

• The Turbulent Flow, k- Interface

• The Turbulent Flow, Low Re k- Interface


• The Turbulent Flow, SST Interface

• The Turbulent Flow, Spalart-Allmaras Interface

The Laminar Flow Interface

The Laminar Flow (spf) interface ( ), found under the Single-Phase Flow branch ( ) when adding a physics interface, is used to compute the velocity and pressure fields for the flow of a single-phase fluid in the laminar flow regime. A flow will remain laminar as long as the Reynolds number is below a certain critical value. At higher Reynolds numbers, disturbances have a tendency to grow and cause transition to turbulence. This critical Reynolds number depends on the model, but a classical example is pipe flow where the critical Reynolds number is known to be approximately 2000.

The physics interface supports incompressible flows and compressible flows at low Mach numbers (typically less than 0.3). It also supports non-Newtonian fluids.

For 2D axisymmetric models, COMSOL Multiphysics takes the axial symmetry boundaries (at r0) into account and automatically adds an Axial Symmetry node that is valid on the axial symmetry boundaries only.

• Domain, Boundary, Pair, and Point Nodes for Single-Phase Flow



The equations solved by the Laminar Flow interface are the Navier-Stokes equations for conservation of momentum and the continuity equation for conservation of mass.

The Laminar Flow interface can be used for stationary and time-dependent analyses. Note that for higher Reynolds numbers, a flow becomes inherently time dependent and three-dimensional, and time-dependent studies have to be used.

When this physics interface is added, the following default nodes are also added in the Model Builder—Fluid Properties, Wall (the default boundary condition is No slip), and Initial

Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and volume forces. You can also right-click Laminar Flow to select physics from the context menu.

I N T E R F A C E I D E N T I F I E R

The interface identifier is used primarily as a scope prefix for variables defined by the physics interface. Refer to these variables in expressions using the pattern <identifier>.<variable_name>. In order to distinguish between variables belonging to different physics interfaces, the identifier string must be unique. Only letters, numbers and underscores (_) are permitted in the Identifier field. The first character must be a letter.

The default identifier (for the first physics interface in the component) is spf.

P H Y S I C A L M O D E L

By default the physics interface uses the Compressible flow (Ma<0.3) formulation of the Navier-Stokes equations. Select Incompressible flow to use the incompressible (constant density) formulation.

Turbulence Model TypeBy default, None is selected as the Turbulence model type. The flow state in a fluid-flow model, however, is not always known beforehand.

Selecting an option in this section switches between available Single-Phase Flow (spf) interfaces. For example, this physics interface changes to The Turbulent Flow, k- Interface when the Turbulence model type selected is RANS (Reynolds-averaged Navier–Stokes).

T H E L A M I N A R F L O W , C R E E P I N G F L O W , A N D TU R B U L E N T F L O W I N T E R F A C E S | 53

54 | C H A P T E R

Swirl Flow

Neglect Inertial Term (Stokes Flow)By default, the Neglect inertial term (Stokes flow) check box is not selected. If the check box is selected, the physics interface changes to The Creeping Flow Interface.

Use Shallow Channel Approximation

Such channels often have an almost rectangular cross section where the thickness is much smaller than the channel width, and simple 2D models often fail to give correct results because they exclude the boundaries that have the greatest effect on the flow. The shallow channel approximation takes the effect of these boundaries into account by adding a drag term as a volume force to the fluid-flow equation. The form of this term is

(3-1)

where is the fluid’s dynamic viscosity (SI unit: kg/(m·s)), u is the velocity field (SI unit: m/s), and dz is the channel thickness. This term represents the resistance that the parallel boundaries impose on the flow; however, it does not account for any changes in velocity due to variations in the cross-sectional area of the channel.

D E P E N D E N T VA R I A B L E S

The following dependent variables (fields) are defined for this physics interface—the Velocity field u (SI unit: m/s) and its components, and the Pressure p (SI unit: Pa).

For 2D axisymmetric models, select the Swirl flow check box to include the swirl velocity component, which is the velocity component in the azimuthal direction. While can be nonzero, there can be no gradients in the direction.

General Single-Phase Flow Theory

Select the Use shallow channel approximation check box for 2D models to model shallow channels in microfluidics applications. Enter a Channel

thickness dz (SI unit: m). The default is 1 m.

uu

F 12u

dz2

--------–=


The Projection Method for the Navier-Stokes Equations requires additional dependent variables. These are the Corrected velocity field uc (SI unit: m/s) and the Corrected pressure pc (SI unit: Pa).

If required, edit the field, component, and dependent variable names. Editing the name of a scalar dependent variable changes both its field name and the dependent variable name. If a new field name coincides with the name of another field of the same type, the fields share degrees of freedom and dependent variable names. A new field name must not coincide with the name of a field of another type, or with a component name belonging to some other field. Component names must be unique within a model except when two fields share a common field name.

C O N S I S T E N T S T A B I L I Z A T I O N

To display this section, click the Show button ( ) and select Stabilization.

The consistent stabilization methods applicable to the Navier-Stokes equations are Streamline diffusion and Crosswind diffusion. These check boxes are selected by default. If required, click to clear one or both of the Streamline diffusion and Crosswind diffusion check boxes. Observe that using P1+P1 elements requires Streamline diffusion to be active. If you deactivate Streamline diffusion, make sure that your model uses P2+P1 elements or higher.

I N C O N S I S T E N T S T A B I L I Z A T I O N

To display this section, click the Show button ( ) and select Stabilization. By default, the Isotropic diffusion check box is not selected for the Navier-Stokes equations. Click to select as required.

A D V A N C E D S E T T I N G S

To display this section, click the Show button ( ) and select Advanced Physics Options. Normally these settings do not need to be changed.

Select the Use pseudo time stepping for stationary equation form check box to add pseudo time derivatives to the equation when the Stationary equation form is used. When selected, also choose a CFL number expression—Automatic (the default) or Manual. Automatic sets the local CFL number (from the Courant–Friedrichs–Lewy condition) to the built-in variable

The Projection Method for the Navier-Stokes Equations


56 | C H A P T E R

CFLCMP which in turn triggers a PID regulator for the CFL number. If Manual is selected, enter a Local CFL number CFLloc (dimensionless).

D I S C R E T I Z A T I O N

To display this section, click the Show button ( ) and select Discretization. It controls the discretization (the element types used in the finite element formulation). From the Discretization of fluids list select the element order for the velocity components and the pressure: P1+P1 (the default), P2+P1, or P3+P2.

• P1+P1 (the default) means linear elements for both the velocity components and the pressure field. This is the default element order for the Laminar Flow and Turbulent Flow interfaces. Linear elements are computationally cheaper than higher-order elements and are also less prone to introducing spurious oscillations, thereby improving the numerical robustness. P1+P1 elements require streamline diffusion to be a numerically valid discretization. Make sure that Streamline Diffusion in the Consistent Stabilization section is selected when using P1+P1 elements.

• P2+P1 means second-order elements for the velocity components and linear elements for the pressure field. This is the default for the Creeping Flow interface because second-order elements work well for low flow velocities.

• P3+P2 means third-order elements for the velocity components and second-order elements for the pressure field. This can add additional accuracy but it also adds additional degrees of freedom compared to P2+P1 elements.

Specify the Value type when using splitting of complex variables—Real (the default) or Complex for each of the variables in the table.

• Pseudo Time Stepping for Laminar Flow Models

• Pseudo Time Stepping in the COMSOL Multiphysics Reference Manual

• Show More Physics Options

• Domain, Boundary, Pair, and Point Nodes for Single-Phase Flow



The Creeping Flow Interface

The Creeping Flow (spf) interface ( ), found under the Single-Phase Flow branch ( ) when adding a physics interface, is used for simulating fluid flows at very low Reynolds numbers for which the inertial term in the Navier-Stokes equations can be neglected. This single-phase flow type is also referred to as Stokes flow and occurs in systems with high viscosity or small geometrical length scales (for example, in microfluidics and MEMS devices). The fluid can be compressible or incompressible, and Newtonian or non-Newtonian.

The equations solved by the Creeping Flow interface are the Stokes equations for conservation of momentum and the continuity equation for conservation of mass.

The Creeping Flow interface can be used for stationary and time-dependent analyses.

The main feature is the Fluid Properties node, which adds the Stokes equations and provides an interface for defining the fluid material and its properties.


By default, the Neglect inertial term (Stokes flow) check box is selected whereby the physics interface models flow at small Reynolds numbers for which the inertial term in the Navier-Stokes equations can be neglected.


To display this section, click the Show button ( ) and select Discretization. Select a Discretization of fluids—P1+P1, P2+P1 (the default), or P3+P2.

Flow Past a Cylinder: model library path COMSOL_Multiphysics/Fluid_Dynamics/cylinder_flow

Except where noted below, see The Laminar Flow Interface for all other settings.

Contrary to general laminar and turbulent single-phase flow, P2+P1 elements are well suited for low flow velocities, so the default uses P2+P1 elements rather than purely linear P1+P1 elements.


58 | C H A P T E R

The first number is the element order for the fluid velocity, and the second number is the element order for the pressure. Specify the Value type when using splitting of complex

variables—Real (the default) or Complex.


To display this section, click the Show button ( ) and select Stabilization. Only the Streamline diffusion consistent stabilization method is available for the Navier-Stokes equations. This check box is selected by default and should remain selected for optimal performance. This consistent stabilization method does not perturb the original transport equation. If required, click to clear the Streamline diffusion check box.

Observe that using P1+P1 elements requires Streamline diffusion to be active. If you deactivate Streamline diffusion, make sure that your model uses P2+P1 elements or higher.

The Turbulent Flow, k- Interface

The Turbulent Flow, k-(spf)interface ( ), found under the Single-Phase Flow>Turbulent

Flow branch ( ) when adding a physics interface, is used for simulating single-phase flows at high Reynolds numbers. The physics interface is suitable for incompressible flows, and compressible flows at low Mach numbers (typically less than 0.3).

The equations solved by the Turbulent Flow, k-interface are the Navier-Stokes equations for conservation of momentum and the continuity equation for conservation of mass. Turbulence effects are modeled using the standard two-equation k-model with realizability constraints. Flow close to walls is modeled using wall functions.

The Turbulent Flow, k- interface can be used for stationary and time-dependent analyses.

The main feature is Fluid Properties, which adds the RANS equations and the transport equations for k and , and provides an interface for defining the fluid material and its properties. When this physics interface is added, the following default nodes are also added





in the Model Builder—Fluid Properties, Wall (the default boundary condition is Wall

functions), and Initial Values.


For this physics interface, the Turbulence model type defaults to RANS and the Turbulence

model defaults to k-. This enables the Turbulence Model Parameters section.

TU R B U L E N C E M O D E L P A R A M E T E R S

Turbulence model parameters are optimized to fit as many flow types as possible, but for some special cases, better performance can be obtained by tuning the model parameters.

For this physics interface the parameters are Ce1, Ce2, C, k, e, v, and B.

D E P E N D E N T V A R I A B L E S

The following dependent variables (fields) are defined for this physics interface:

• the Velocity field u (SI unit: m/s) and its components

• the Pressure p (SI unit: Pa)

• the Turbulent kinetic energy k (SI unit: m2/s2)

• the Turbulent dissipation rate ep (SI unit: m2/s3)



To display this section, click the Show button ( ) and select Advanced Physics Options. The Turbulence variables scale parameters subsection is available when the Turbulence model

type is set to RANS.

Except where included below, see The Laminar Flow Interface for all the other settings.

This property also disables the Neglect inertial term (Stokes flow) check box for the Turbulent Flow interfaces.

For 2D models, it also disables the Shallow Channel Approximation check box.


60 | C H A P T E R

In addition to the settings described for the Laminar Flow interface, enter a value for Uscale (SI unit: m/s) (the default is 1 m/s) and Lfact (dimensionless) (the default is 0.035) under the Turbulence variables scale parameters subsection.

The Uscale and Lfact parameters are used to calculate absolute tolerances for the turbulence variables. The scaling parameters must only contain numerical values, units or parameters defined under Global Definitions. The scaling parameters can not contain variables. The parameters are used when a new default solver for a transient study step is generated. If you change the parameters, the new values take effect the next time you generate a new default solver

The Turbulent Flow, Low Re k- Interface

The Turbulent Flow, Low Re k-(spf) interface ( ), found under the Single-Phase

Flow>Turbulent Flow branch, is used for simulating single-phase flows at high Reynolds numbers. The physics interface is suitable for incompressible flows and compressible flows at low Mach numbers (typically less than 0.3).

The equations solved by the Turbulent Flow, Low Re k- interface are the Navier-Stokes equations for conservation of momentum and the continuity equation for the conservation of mass. Turbulence effects are modeled using the AKN two-equation k- model with realizability constraints. The AKN model is a so-called low-Reynolds number model, which means that it resolves the flow all the way down to the wall. The AKN model depends on the distance to the closest wall. The physics interface therefore includes a wall distance equation.





• Airflow Over an Ahmed Body: model library path CFD_Module/Single-Phase_Benchmarks/ahmed_body

• Solar Panel in Periodic Flow: model library path CFD_Module/Single-Phase_Tutorials/solar_panel


The Turbulent Flow, Low Re k- interface can be used for stationary and time-dependent analyses.



model defaults to Low Reynolds number k-. This enables the Turbulence Model Parameters

section.



For this physics interface the parameters are Ce1, Ce2, C, k, e, and v.






• the Turbulent dissipation rate ep (SI unit: m2/s3)

• the Reciprocal wall distance G (SI unit: 1/m)

The Low Reynolds number k- interface requires a Wall Distance

Initialization study step in the study previous to the Stationary or Time Dependent study step.

For study information, see Stationary with Initialization, Transient with Initialization, and Wall Distance Initialization in the COMSOL Multiphysics Reference Manual.

See The Laminar Flow Interface and The Turbulent Flow, k- Interface for all the other settings.


62 | C H A P T E R


The Turbulent Flow, k-(spf) interface ( ), found under the Single-Phase Flow>Turbulent

Flow branch ( ) when adding a physics interface, is used for simulating single-phase flows at high Reynolds numbers. The physics interface is suitable for incompressible flows, and compressible flows at low Mach numbers (typically less than 0.3).

The equations solved by the Turbulent Flow, k- interface are the Navier-Stokes equations for conservation of momentum and the continuity equation for conservation of mass. Turbulence effects are modeled using the Wilcox revised two-equation k- model with realizability constraints. Flow close to walls is modeled using wall functions.

The Turbulent Flow, k- interface can be used for stationary and time-dependent analyses.

The main feature is Fluid Properties, which adds the RANS equations and the transport equations for the turbulent kinetic energy k and the specific dissipation , and provides an interface for defining the fluid material and its properties. When this physics interface is added, the following default nodes are also added in the Model Builder—Fluid Properties, Wall (the default boundary condition is Wall functions), and Initial Values.



model defaults to k-. This enables the Turbulence Model Parameters section.



For this physics interface the parameters are , k,w, 0, 0

,v and B.











• the Specific dissipation rate om (SI unit: 1/s)


The Turbulent Flow, SST Interface

The Turbulent Flow, SST (spf) interface ( ), found under the Single-Phase Flow>Turbulent

Flow branch ( ) when adding a physics interface, is used for simulating single-phase flows at high Reynolds numbers. The physics interface is suitable for incompressible flows and compressible flows at low Mach numbers (typically less than 0.3).

The equations solved by the Turbulent Flow, SST interface are the Navier-Stokes equation for conservation of momentum and the continuity equation for conservation of mass. Turbulence effects are modeled using the Menter shear-stress transport (SST) two-equation model from 2003 with realizability constraints. The SST model is a so-called low-Reynolds number model, which means that it resolves the flow all the way down to the wall. The SST model depends on the distance to the closest wall. The physics interface therefore includes a wall distance equation.

The Turbulent Flow, SST interface can be used for stationary and time-dependent analyses.


Flow Through a Pipe Elbow: model library path CFD_Module/Single-Phase_Benchmarks/pipe_elbow


64 | C H A P T E R

When this physics interface is added, the following default nodes are also added in the Model Builder—Fluid Properties, Wall (the default boundary condition is No slip), and Initial

Values.



model defaults to SST. This enables the Turbulence Model Parameters section.



For this physics interface the parameters are a1, 0,1, 2, 1, 2k1, k2, w1,and w2.






• the Specific dissipation rate om (SI unit: 1/s


The Turbulent Flow, Spalart-Allmaras Interface

The Turbulent Flow, Spalart-Allmaras (spf) interface ( ), found under the Single-Phase

Flow>Turbulent Flow branch ( ) when adding a physics interface, is used for simulating

The SST interface requires a Wall Distance Initialization study step in the study previous to the Stationary or Time Dependent study step.




single-phase flows at high Reynolds numbers. The physics interface is suitable for incompressible flows, and compressible flows at low Mach numbers (typically less than 0.3).

The equations solved by the Turbulent Flow, Spalart-Allmaras interface are the Navier-Stokes equation for conservation of momentum and the continuity equation for conservation of mass. Turbulence effects are modeled using the Spalart-Allmaras one-equation model. The Spalart-Allmaras model is a so-called low-Reynolds number model, which means that it resolves the flow all the way down to the wall. The Spalart-Allmaras model depends on the distance to the closest wall. The physics interface therefore includes a wall distance equation.

The Turbulent Flow, Spalart-Allmaras interface can be used for stationary and time-dependent analyses.

The main feature is Fluid Properties, which adds the RANS equations and the transport equations for the undamped turbulent kinematic velocity , and provides an interface for defining the fluid material and its properties. When this physics interface is added, the following default nodes are also added in the Model Builder—Fluid Properties, Wall (the default boundary condition is No slip), and Initial Values.



model defaults to Spalart-Allmaras. This enables the Turbulence Model Parameters section.



For this physics interface the parameters are Cb1, Cb2, Cv1, v, Cw2, Cw3, v, and Crot.

The Spalart-Allmaras interface requires a Wall Distance Initialization study step in the study previous to the Stationary or Time Dependent study step.



66 | C H A P T E R


To display this section, click the Show button ( ) and select Advanced Physics Options. Under the Turbulence variables scale parameters subsection, the default scale is 5 x 10-6 m2/s. This number can be compared to 1.5 x 10-6 m2/s which is the kinematic viscosity for air at 300 K. Enter another value or expression as required.






• the Undamped turbulent kinematic viscosity (SI unit: m2/s)


Domain, Boundary, Pair, and Point Nodes for Single-Phase Flow

The following nodes are for all physics interfaces found under the Fluid Flow>Single-Phase

Flow branch ( ) when adding a physics interface. Other physics interfaces also share many of these domain, boundary, pair, and point nodes.

These nodes, listed in alphabetical order, are available from the Physics ribbon toolbar (Windows users), Physics context menu (Mac or Linux users), or right-click to access the context menu (all users).




Fluid Properties

The Fluid Properties node adds the momentum equations solved by the physics interface, except for volume forces which are added by the Volume Force feature. The node also provides an interface for defining the material properties of the fluid.

M O D E L I N P U T S

Edit input variables to the fluid-flow equations if required. For fluid flow, these are typically introduced when a material requiring inputs has been selected.

Absolute PressureThis input appears when a material requires the absolute pressure as model input. The absolute pressure input controls the pressure used to evaluate material properties, but it also relates to the value of the pressure field. There are usually two ways to calculate the pressure when describing fluid flow. Either solve for the absolute pressure or for a pressure

• No Viscous Stress

• Fan

• Flow Continuity

• Fluid Properties

• Grille

• Initial Values

• Inlet

• Interior Fan

• Interior Wall

• Line Mass Source*

• Open Boundary

• Outlet

• Periodic Flow Condition

• Point Mass Source*

• Pressure Point Constraint

• Screen

• Symmetry

• Vacuum Pump

• Volume Force

• Wall

* A feature that might require an additional license

For 2D axisymmetric models, COMSOL Multiphysics takes the axial symmetry boundaries (at r0) into account and adds an Axial Symmetry node that is valid on the axial symmetry boundaries only.

For the Turbulent Flow interfaces, the Fluid Properties node also adds the equations for the turbulence transport equations.


68 | C H A P T E R

(often denoted gauge pressure) that relates to the absolute pressure through a reference pressure.

Which option to choose usually depends on the system and the equations being solved. For example, in a unidirectional incompressible flow problem, the pressure drop over the modeled domain is probably many orders of magnitude smaller than the atmospheric pressure, which, when included, reduces the stability and convergence properties of the solver. In other cases, you can solve for the absolute pressure, such as when the pressure is part of an expression for the gas volume or the diffusion coefficients.

The default Absolute pressure pA (SI unit: Pa) is ppref where p defaults to the pressure variable from the Navier-Stokes or RANS equations and pref to 1[atm] (1 atmosphere 101,325 Pa). The default setting is to solve for a gauge pressure.

If the pressure field instead is an absolute pressure field, clear the Reference pressure check box.

To model an incompressible fluid, set Absolute pressure pA to User defined and enter the desired pressure level in the edit field. The default value is 1[atm].

F L U I D P R O P E R T I E S

DensityThe default Density (SI unit: kg/m3) uses the value From material. Select User defined to enter a different value or expression.

This makes it possible to use a system-based (gauge) pressure as the pressure variable while automatically including the reference pressure in places where it is required, such as for gas flow governed by the gas law. While this check box maintains control over the pressure variable and instances when absolute pressure is required within this specific physics interface, it can not do so within other physics interfaces that it is coupled to. In such models, check the coupling between any physics interfaces using the same variable.

To define the Absolute Pressure, see the settings for the Heat Transfer in Fluids node.


Dynamic ViscosityThe default Dynamic viscosity (SI unit: Pa·s) uses the value From material and describes the relationship between the shear rate and the shear stresses in a fluid. Intuitively, water and air have low viscosities, and substances often described as thick (such as oil) have higher viscosities. Select User defined to define a different value or expression.

For laminar flow, select Non-Newtonian power law to use a power law expression to model the viscosity of a non-Newtonian fluid. For the Non-Newtonian power law enter the following model parameters

• the Fluid consistency coefficient m (SI unit: kg/(m·s)). The default is 1 kg/(m·s)

• the Flow behavior index n (dimensionless). The default is 1.

• the Lower shear rate limit (SI unit: 1/s). The default is 0.01 1/s.

For laminar flow, select Non-Newtonian Carreau model to use the Carreau model for the viscosity of a non-Newtonian fluid. Enter the Carreau model parameters:

• the Zero shear rate viscosity 0 (SI unit: Pa·s). The default is 1 x 10-3 Pa·s.

• the Infinite shear rate viscosity inf (SI unit: Pa·s). the default is 0 Pa·s.

• the Model parameters (SI unit: s), the default is 0, and n (dimensionless), the default is 0.

Using the built-in variable for the shear rate magnitude, spf.sr, makes it possible to define arbitrary expressions of the dynamic viscosity as a function of the shear rate.

The non-Newtonian fluids models have a shear-rate dependent viscosity. Examples of non-Newtonian fluids include yogurt, paper pulp, and polymer suspensions. See Non-Newtonian Flow: The Power Law and the Carreau Model.

·min


70 | C H A P T E R

M I X I N G L E N G T H L I M I T

Select how the Mixing length limit lmix,lim (SI unit: m) is defined—Automatic (default) or Manual:

• If Automatic is selected, the mixing length limit is automatically evaluated as the shortest side of the geometry bounding box. If the geometry is, for example, a complicated system of slim entities, this measure can be too high. In such cases, it is recommended that the mixing length limit is defined manually.

• Select Manual to define a different value or expression. The default is 1 (that is, one unit length of the model unit system).

D I S T A N C E E Q U A T I O N

Select how the Reference length scale lref (SI unit: m) is defined—Automatic (default) or Manual:

• If Automatic is selected, the wall distance is automatically evaluated to one tenth of the shortest side of the geometry bounding box. This is usually a good choice but it can sometimes give too great a value if the geometry consists of several slim entities. In such cases, it is recommended that the reference length scale is defined manually.

• Select Manual to define a different value or expression for the wall distance. The default is 1 m.

lref controls the solution to the distance equation. Objects that are much smaller than lref are effectively diminished while the distance to objects much larger than lref are accurately represented.

This section applies to the Turbulent Flow, k-, Turbulent Flow, k-, and Rotating Machinery, Turbulent Flow k-interfaces, for which an upper limit on the mixing length is required.

This section applies to a Turbulent Flow, Low Reynolds number k-e, a Turbulent flow, SST or a Turbulent Flow, Spalart-Allmaras interface since a Wall Distance interface is included.


Volume Force

The Volume Force node specifies the volume force F on the right-hand side of the momentum equation. Use it, for example, to incorporate the effects of gravity in a model.

If several volume force nodes are added to the same domain, then the sum of all contributions are added to the momentum equations.

VO L U M E F O R C E

Enter the components of the Volume force F (SI unit: N/m3). The defaults for all components are 0 N/m3.

Initial Values

The Initial Values node adds initial values for the velocity field and the pressure that can serve as initial conditions for a transient simulation or as an initial guess for a nonlinear solver.

C O O R D I N A T E S Y S T E M S E L E C T I O N

The Global coordinate system is selected by default. The Coordinate system list contains any additional coordinate systems that the component includes.

I N I T I A L V A L U E S

Enter values or expressions for the initial value of the Velocity field u (SI unit: m/s) and the Pressure p (SI unit: Pa). The default values are 0 m/s and 0 Pa, respectively.

The Boussinesq Approximation

In the Turbulent Flow interfaces, initial values for the turbulence variables are also specified. By default these are specified using the predefined variables defined by the expressions in Initial Values.

t

u u u+ pI– u u T+ 23--- u I–+ F+=


72 | C H A P T E R

For the k- and Low Reynolds number k- turbulence models, also define the Turbulent

kinetic energy k (SI unit: m2/s2) and the Turbulent dissipation rate ep (SI unit: m2/s3). The default values are spf.kinit and spf.epinit.

For the SST, Low Reynolds number k-and Spalart-Allmaras turbulence models, define the Reciprocal wall distance G (SI unit: 1/m). The default is spf.G0.

For the k- and SST turbulence models, define the Turbulent kinetic energy k (SI unit: m2/s2) and the Specific dissipation rate om (SI unit: 1/s). The default values are spf.kinit and spf.omInit.

For the Spalart-Allmaras turbulence model, define the Undamped turbulent kinematic

viscosity 0 (SI unit: m2/s). The default is spf.nutildeinit.

Wall

The Wall node includes a set of boundary conditions describing the fluid-flow condition at a wall.

• No Slip (the default for laminar flow the Low Reynolds number k-, the SST and Spalart-Allmaras turbulence models)

• Slip

• Sliding Wall

• Moving Wall

• Leaking Wall

• Slip Velocity

In addition to the Slip condition, the following are also available for k- and k- turbulence models:

• Wall Functions (the default for turbulent flow with a k- turbulence model)

• Sliding Wall (Wall Functions)

• Moving Wall (Wall Functions)

B O U N D A R Y S E L E C T I O N

For a default node, the setting inherits the selection from the parent node, and can not be edited; that is, the selection is automatically made and is the same as for the physics interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific boundaries or select All boundaries as required.


B O U N D A R Y C O N D I T I O N

Select a Boundary condition for the wall.

No SlipNo slip is the default boundary condition for a stationary solid wall for laminar flow (and for the SST, Low Reynolds number k- and Spalart-Allmaras turbulence models). The condition prescribes u = 0, that is, the fluid at the wall is not moving.

SlipThe Slip option prescribes a no-penetration condition, u·n. It hence implicitly assumes that there are no viscous effects at the slip wall and hence, no boundary layer develops. From a modeling point of view, this can be a reasonable approximation if the main effect of the wall is to prevent fluid from leaving the domain.

Sliding WallThe Sliding wall boundary condition is appropriate if the wall behaves like a conveyor belt; that is, the surface is sliding in its tangential direction. The wall does not have to actually move in the coordinate system.

The Moving Wetted Wall boundary condition is also available for the Wall node for The Laminar Two-Phase Flow, Phase Field Interface.

The Wetted wall boundary condition is also available for the Wall node for The Laminar Two-Phase Flow, Level Set Interface and The Laminar Two-Phase Flow, Phase Field Interface.

For 3D models, enter the components of the Velocity of the sliding wall uw (SI unit: m/s). If the velocity vector entered is not in the plane of the wall, COMSOL Multiphysics projects it onto the tangential direction. Its magnitude is adjusted to be the same as the magnitude of the vector entered.


74 | C H A P T E R

Moving WallIf the wall moves, so must the adjacent fluid. Hence, this boundary condition prescribes u = uw. Enter the components of the Velocity of moving wall uw (SI unit: m/s).

Leaking WallUse this boundary condition to simulate a wall where fluid is leaking into or leaving through a perforated wall u = ul. Enter the components of the Fluid velocity ul (SI unit: m/s).

Slip VelocityIn the microscale range, the flow at a boundary is seldom strictly no slip or slip. Instead, the boundary condition is something in between, and there is a Slip velocity at the boundary. Two phenomena account for this velocity: the noncontinuum effects and flow induced by a thermal gradient along the boundary. Enter the Velocity of moving wall: uw (SI unit: m/s). The defaults are 0 m/s.

If the Use viscous slip check box is selected, the default Slip length Ls (SI unit: m) is User

defined. Enter another value or expression if the default of 1·107 m is not applicable. If Maxwell’s Model is selected, enter values or expressions for the Tangential momentum

accommodation coefficient av and the Mean free path (SI unit: m).

For 2D models, the tangential direction is unambiguously defined by the direction of the boundary, but the situation becomes more complicated in 3D. For this reason, the sliding wall boundary condition has slightly different definitions in the different space dimensions. Enter the components of the Velocity of the tangentially moving wall Uw (SI unit: m/s). In axial symmetry, if Swirl flow is selected in the interface properties, also specify a velocity, vw, in the direction.

Specifying this boundary condition does not automatically cause the associated wall to move. An additional Moving Mesh interface needs to be added to physically track the wall movement in the spatial reference frame.

Tangential accommodation coefficients are typically in the range of 0.85 to 1.0 and can be found in G. Kariadakis, A. Beskok, and N. Aluru, Microflows and Nanoflows, Springer Science and Business Media, 2005.


If the Use thermal creep check box is selected, it activates the thermal creep component of the boundary condition. By default, the fluid Temperature T (SI unit: K) is 293.15 K and the default Thermal slip coefficient T (dimensionless) is 0.75.

Wall FunctionsThe Wall functions boundary condition applies wall functions to solid walls in a turbulent flow. Wall functions are used to model the thin region with high gradients in flow variables near the wall.

Click to select the Apply wall roughness check box to apply the roughness correction. When the check box is selected, the default Roughness model is Sand roughness, which is derived from the experiments by Nikuradse. Select Generic roughness to specify more general roughness types.

• If Sand roughness is selected, enter an Equivalent sand roughness height kseq (SI unit: m). The default is 3.2 micrometers.

• If Generic roughness is selected, enter a Roughness height ks (SI unit: m). The default is 3.2 micrometers. Then enter a Roughness parameter Cs (dimensionless). The default is 0.26.

Sliding Wall (Wall Functions)The Sliding wall (wall functions) boundary condition applies wall functions to a wall in a turbulent flow where the velocity magnitude in the tangential direction of the wall is prescribed. The tangential direction is determined in the same manner as in the Sliding Wall feature. Enter the component values or expressions for the Velocity of sliding wall uw (SI unit: m/s). The defaults are 0 m/s.

Thermal slip coefficients are typically between 0.3 and 1.0 and can be found in G. Kariadakis, A. Beskok, and N. Aluru, Microflows and Nanoflows, Springer Science and Business Media, 2005.

This boundary condition is not compatible with the projection method.

References for the Single-Phase Flow, Turbulent Flow Interfaces


76 | C H A P T E R

Click to select the Apply wall roughness check box to apply the roughness correction. See Wall Functions for details.

Moving Wall (Wall Functions)

The Moving wall (wall functions) boundary condition applies wall functions to a wall with prescribed velocity uw in a turbulent flow. Enter the component values or expressions for the Velocity of moving wall uw (SI unit: m/s). The defaults are 0 m/s.

Click to select the Apply wall roughness check box to apply the roughness correction. See Wall Functions for details.

C O N S T R A I N T S E T T I N G S

To display this section, click the Show button ( ) and select Advanced Physics Options.

For the No Slip, Moving Wall, and Leaking Wall boundary conditions, select an option from the Apply reaction terms on: list—All physics (symmetric) or Individual dependent

variables. The other types of wall boundary conditions with constraints use Individual

dependent variables constraints only.

Select the Use weak constraints check box (not available for the Sliding Wall condition) to use weak constraints and create dependent variables for the corresponding Lagrange multipliers.

Specifying this boundary condition does not automatically cause the associated wall to move.

• Theory for the Wall Boundary Condition

• Moving Mesh Interface in the COMSOL Multiphysics Reference Manual


Inlet

The Inlet node includes a set of boundary conditions describing the fluid-flow conditions at an inlet. The Velocity boundary condition is the default.


Select a Boundary condition for the inlet—Velocity (the default), Pressure, No Viscous Stress, Laminar Inflow, Mass Flow, or Normal Stress.

In many cases the Inlet boundary conditions are available, some of them slightly modified, for the Outlet type as well. For certain conditions there is nothing in the mathematical formulations to prevent the fluid from leaving the domain through boundaries where the Inlet type is specified.

• Prescribing Inlet and Outlet Conditions

• Pressure, No Viscous Stress Boundary Condition

• Normal Stress Boundary Condition

After selecting a Boundary Condition from the list, a section with the same name displays underneath. For example, if Velocity is selected, a Velocity section displays where further settings are defined for the velocity.

For the Velocity, Pressure, no viscous stress, Mass flow, and Normal stress sections, also enter the turbulent flow settings as described in More Boundary Condition Settings for the Turbulent Flow Interfaces.


78 | C H A P T E R

VE L O C I T Y

The Velocity boundary condition is available for the Inlet and Outlet boundary nodes.

• Select Normal inflow velocity (the default) to specify a normal inflow velocity magnitude u = nU0 where n is the boundary normal pointing out of the domain. Enter the velocity magnitude U0 (SI unit: m/s). The default is 0 m/s.

• If Velocity field is selected, it sets the velocity equal to a given velocity vector u0 when u = u0. Enter the components of u0 (SI unit: m/s). The defaults are 0 m/s.

• Also enter the turbulent flow settings as described in More Boundary Condition Settings for the Turbulent Flow Interfaces.

P R E S S U R E , N O V I S C O U S S T R E S S

The Pressure, no viscous stress boundary condition is available for the Inlet boundary node. It specifies vanishing viscous stress along with a Dirichlet condition on the pressure. Enter the Pressure p0 (SI unit: Pa) at the boundary. The default is 0 Pa.

Also enter the turbulent flow settings as described in More Boundary Condition Settings for the Turbulent Flow Interfaces.

N O R M A L S T R E S S

The Normal stress boundary condition is available for the Inlet, Outlet (via the Pressure condition), Open Boundary, and No Viscous Stress nodes. Enter the magnitude of Normal

stress f0 (SI unit: N/m2). This implicitly imposes . The default is 0 N/m2.

Also enter the turbulent flow settings as described in More Boundary Condition Settings for the Turbulent Flow Interfaces.

L A M I N A R I N F L O W

The Laminar inflow boundary condition is available for the Inlet node.

Depending on the pressure field in the rest of the domain, an inlet boundary with this condition can become an outlet boundary.

• Gravity and Boundary Conditions (outlet): model library path CFD_Module/Single-Phase_Tutorials/gravity_tutorial

• Non-Newtonian Flow (inlet): model library path CFD_Module/Single-Phase_Tutorials/non_newtonian_flow

p f0


Select a flow quantity for the inlet—Average velocity (the default), Flow rate, or Entrance

pressure.

• When Average velocity is selected, enter an Average velocity Uav (SI unit: m/s). The default is 0 m/s.

• If Flow rate is selected, enter the Flow rate V0 (SI unit: m3/s). The default is 0 m3/s.

• If Entrance pressure is selected, enter the Entrance pressure pentr (SI unit: Pa) at the entrance of the fictitious channel outside of the model. The default is 0 Pa.

Then for any selection, specify the entrance length and constraints:

• Enter the Entrance length Lentr (SI unit: m) to define the length of the inlet channel outside the model domain.

• Select the Constrain outer edges to zero (for 3D models) or Constrain endpoints to zero (for 2D and 2D axisymmetric models) check box to force the laminar profile to go to zero at the bounding points or edges of the inlet channel. Otherwise the velocity is defined by the boundary condition of the adjacent boundary in the model. For example, if one end of a boundary with a laminar inflow condition connects to a slip boundary condition, then the laminar profile has a maximum at that end.

M A S S F L O W

The Mass flow boundary condition is available for the Inlet node. However, depending on the sign of the specified mass flow or flux, an inlet boundary with this condition can become an outlet boundary.

Select a Mass flow type—Mass flow rate (the default), Pointwise mass flux, Standard flow rate, or Standard flow rate (SCCM).

Mass Flow RateThe Mass flow rate sets the integrated mass flow across the entire boundary to a specific value. The mass flow is assumed to be parallel to the boundary normal, so the tangential velocity is set to zero.

The Entrance length value must be large enough so that the flow can reach a laminar profile. For a laminar flow, Lentr should be significantly greater than 0.06ReD, where Re is the Reynolds number and D is the inlet length scale (this formula is exact if D is the diameter of a cylindrical pipe and approximate for other geometries). For turbulent flow the equivalent expression is 4.4Re1/6D. The default is 1 m.


80 | C H A P T E R

Enter the Normal mass flow rate m (SI unit: kg/s). The default value is 105 kg/s.

Pointwise Mass FluxThe Pointwise mass flux sets the mass flux parallel to the boundary normal. The flux perpendicular to the normal is set to zero. The mass flux is a model input, which means that COMSOL Multiphysics can take its value from another physics interface if applicable. If User defined is selected from the Mass flux list, enter a value for Mf (SI unit: kg/(m2·s)). The default value is 0 kg/(m2·s).

Standard Flow RateThe Standard flow rate sets a standard volumetric flow rate, according to the SEMI standard E12-0303. The flow rate is specified as the volumetric flow rate that would occur for the same number of moles per second, if the gas density were a standard value (the molar mass over a fixed standard volume). The flow occurs across the whole boundary in the direction of the boundary normal and is computed by a surface (3D) or line (2D) integral. The tangential flow velocity is set to zero.

The standard density can be defined directly, or by specifying a standard pressure and temperature, in which case the ideal gas law is assumed. Select an option from the Standard

flow rate defined by list—Standard density (the default) or Standard pressure and

temperature.

For either option enter the following values:

• Standard flow rate Qsv (SI unit: m3/s) to specify the standard volumetric flow rate through the boundary. The default value is 106 m3/s.

• Mean molar mass Mn (SI unit: kg/mol). This can be selected as a model input from another physics interface or, if User defined is selected, you can either keep the default (0.032 kg/mol) or enter another value or expression.

For 2D models, enter a Channel thickness dbc (SI unit: m). The default value is 1 m.

The Channel thickness is used only in defining the area across which the mass flow occurs—it is not a setting applied to the whole model. Line or surface integrals of the mass flux over the boundary evaluated during post-processing or using when integration coupling operators do not include this scaling automatically. Such results should be appropriately scaled when comparing them with the specified mass flow.


If Standard density is selected also enter the Standard molar volume Vm (SI unit: m3/mol). The default is 0.0224136 m3/mol.

If Standard pressure and temperature is selected, also enter the Standard pressure Pst (SI unit: Pa) (the default is 1 atm (101,325 Pa)) and the Standard temperature Tst (SI unit: K). The default is 273.15 K (0 degrees Celsius).


To display this section, click the Show button ( ) and select Advanced Physics Options. Select the Use weak constraints check box to use weak constraints and create dependent variables for the corresponding Lagrange multipliers.

When Velocity or Pressure, No Viscous Stress are selected as the Boundary condition, and to Apply reaction terms on all dependent variables, select All physics (symmetric). Or select Individual dependent variables to restrict the reaction terms as required.

Outlet

The Outlet node includes a set of boundary conditions describing fluid-flow conditions at an outlet. Pressure is the default. Other options are based on individual licenses. Selecting appropriate outlet conditions for the Navier-Stokes equations is not a trivial task. Generally, if there is something interesting happening at an outflow boundary, extend the computational domain to include this phenomenon.

For 2D models, enter a Channel thickness dbc (SI unit: m). The default value is 1 m.

The Channel thickness is used only in defining the area across which the mass flow occurs—it is not a setting applied to the whole model. Line integrals of the mass flux on the boundary evaluated during post-processing and the use of coupling variables still produce results per unit thickness and need to be scaled appropriately for comparison with the specified mass flow.

Some of the formulations for the Outlet type are also available, possibly slightly modified, in other boundary types. For certain conditions there is nothing in the mathematical formulations to prevent the fluid from entering a domain through boundaries where the Outlet boundary type is specified.


82 | C H A P T E R


Select a Boundary condition for the outlet—Pressure (the default), Laminar Outflow, or Velocity.

PressureThe Pressure condition specifies the normal stress which in most cases is approximately equal to the pressure. The tangential stress component is set to 0 N/m2.

• Enter the Pressure p0 (SI unit: Pa) at the boundary. The default is 0 Pa.

• Select the Normal flow check box to change the no tangential stress condition to a no tangential velocity condition. This forces the flow to exit (or enter) the domain perpendicularly to the outlet boundary.

• The Suppress backflow check box is selected by default. This option adjusts the outlet pressure in order to prevent fluid from entering the domain through the boundary.

Laminar OutflowThis section displays when Laminar outflow is selected as the Boundary condition. Select a flow quantity to specify for the inlet:

• If Average velocity is selected, enter an Average velocity Uav (SI unit: m/s). The default is 0 m/s.

• If Flow rate is selected, enter the Flow rate V0 (SI unit: m3/s). The default is 0 m3/s.

• If Exit pressure is selected, enter the Exit pressure pexit (SI unit: Pa) at the end of the fictitious channel following the outlet. The default is 0 Pa.

Then specify the Exit length and Constrain endpoints to zero parameters:

Prescribing Inlet and Outlet Conditions

The Velocity boundary condition is described for the Inlet node.


Enter the Exit length Lexit (SI unit: m) to define the length of the fictitious channel outside the model domain.

Select the Constrain outer edges to zero (3D models) or Constrain endpoints to zero (2D models) check box to force the laminar profile to go to zero at the bounding points or edges of the inlet channel. Otherwise the velocity is defined by the boundary condition of the adjacent boundary in the model. For example, if one end of a boundary with a Laminar

outflow condition connects to a Slip boundary condition, then the laminar profile has a maximum at that end.



When Velocity or Pressure is selected as the Boundary condition, and to Apply reaction terms

on all dependent variables, select All physics (symmetric). Or select Individual dependent

variables to restrict the reaction terms as required.

Symmetry

The Symmetry node adds a boundary condition that describes symmetry boundaries in a fluid-flow simulation. The boundary condition for symmetry boundaries prescribes no penetration and vanishing shear stresses. The boundary condition is a combination of a Dirichlet condition and a Neumann condition:

The Exit length value must be large enough so that the flow can reach a laminar profile. For a laminar flow, Lexit should be significantly greater than 0.06ReD, where Re is the Reynolds number and D is the outlet length scale (this formula is exact if D is the diameter of a cylindrical pipe and approximate for other geometries). For turbulent flow the equivalent expression is 4.4Re1/6D. The default is 1 m.

u n 0,= pI– u u T+ 23--- u I–

+ n 0=

u n 0,= pI– u u T+ + n 0=


84 | C H A P T E R

for the compressible and incompressible formulations. The Dirichlet condition takes precedence over the Neumann condition, and the above equations are equivalent to the following equation for both the compressible and incompressible formulations:


From the Selection list, choose the boundaries on which to apply the condition.



Open Boundary

The Open Boundary node adds boundary conditions describing boundaries in contact with large volumes of fluid. Fluid can both enter and leave the domain on boundaries with this type of condition.

B O U N D A R Y C O N D I T I O N S

Select a Boundary condition for the open boundaries—Normal Stress (the default) or Laminar Outflow. The Normal stress condition is described for the Inlet node.

For 2D axial symmetry, a boundary condition does not need to be defined. For the symmetry axis at r0, the software automatically provides a condition that prescribes ur0 and vanishing stresses in the z direction and adds an Axial Symmetry node that implements this condition on the axial symmetry boundaries only.

Also enter the additional settings described in More Boundary Condition Settings for the Turbulent Flow Interfaces.

u n 0,= K K n n– 0=

K u u T+ n=


No Viscous StressThe No Viscous Stress condition specifies vanishing viscous stress on the outlet. This condition does not provide sufficient information to fully specify the flow at the outlet and must at least be combined with pressure constraints on adjacent points.

If No viscous stress is selected, it prescribes vanishing viscous stress:

using the compressible and the incompressible formulations.

Boundary Stress

The Boundary Stress node adds a boundary condition that represents a very general class of conditions also known as traction boundary conditions.


Select a Boundary condition for the boundary stress—General stress (the default), Normal Stress (described for the Inlet node), or Normal stress, normal flow.

This condition can be useful in some situations because it does not impose any constraint on the pressure. A typical example is a model with volume forces that give rise to pressure gradients that are hard to prescribe in advance. To make the model numerically stable, combine this boundary condition with a point constraint on the pressure.

This boundary condition is not compatible with the projection method equation form.

Also enter the settings described in More Boundary Condition Settings for the Turbulent Flow Interfaces.

u u T+ 23--- u I–

n 0=

u u T+ n 0=


86 | C H A P T E R

General StressWhen General stress is selected, enter the components for the Stress F (SI unit: N/m2).The total stress on the boundary is set equal to a given stress F:


This boundary condition implicitly sets a constraint on the pressure that for 2D flows is

(3-2)

If unn is small, Equation 3-2 states that pn·F.

Normal Stress, Normal FlowIf Normal stress, normal flow is selected, enter the magnitude of the Normal stress f0 (SI unit: N/m2).

In addition to the stress condition set in the Normal Stress condition, the Normal stress,

normal flow condition also prescribes that there must be no tangential velocities on the boundary:


This boundary condition also implicitly sets a constraint on the pressure that for 2D flows is

(3-3)

pI– u u T+ 23--- u I–

+ n F=

pI– u u T+ + n F=

p 2unn---------- n F–=

pI– u u T+ 23--- u I–

+ n f0n,–= t u 0=

pI– u u T+ + n f0n,–= t u 0=

p 2unn---------- f0+=


If unn is small, Equation 3-3 states that pf0.



If Normal Stress, Normal Flow is selected as the Boundary condition, then to Apply reaction

terms on all dependent variables, select All physics (symmetric). Or select Individual

dependent variables to restrict the reaction terms as required.

Screen

Use the Screen node to model interior wire-gauzes, grilles, or perforated plates as thin permeable barriers. Common correlations are included for resistance and refraction coefficients.


From the Selection list, choose the boundaries to define. The Screen boundary condition can only be applied on interior boundaries.

S C R E E N TY P E

Select a Screen type—Wire gauze (the default), Square mesh, Perforated plate, or User

defined. Observe the equations shown based on the selection (excluding User defined). The selection also adjusts what is available in the Parameters section.

P A R A M E T E R S

If Wire gauze, Square mesh, or Perforated plate is selected as the Screen type, enter values or expressions for the Solidity s (dimensionless). The default is 0.5.

• If Wire gauze is selected as the Screen type, also enter a value or expression for the Wire

diameter d (SI unit: m). The default is 0.001 m (1 mm).

• If User defined is selected as the Screen type, enter a Resistance coefficient K (dimensionless). The default is 1.56.

This boundary condition is not compatible with the projection method equation form.


88 | C H A P T E R

For all screen types, the Refraction defaults to Wire gauze. Or select User defined to enter a Refraction coefficient (dimensionless). The default is 0.683.

Vacuum Pump

The Vacuum Pump node models the effective outlet pressure created by a vacuum pump device that is attached to the outlet.

P A R A M E T E R S

Select a Static pressure curve to define a lumped curve—Linear (the default), Static pressure

curve data, or User defined.

LinearIf Linear is selected, enter values or expressions for the Static pressure at no flow pnf (SI unit: Pa) and the Free delivery flow rate V0,fd (SI unit: m3/s). The defaults are 100 Pa and 0.01 m3/s, respectively.

The static pressure curve is equal to the static pressure at no flow rate when V00 and equal to 0 when the flow rate is larger than the free delivery flow rate.

Static Pressure Curve DataSelect Static pressure curve data to enter or load data under the Static Pressure Curve Data section that displays. The interpolation between points given in the Static Pressure Curve

Data table is defined using the Interpolation function type list in the Static Pressure Curve

Interpolation section.

User DefinedSelect User defined to enter different values or expressions. The flow rate across the selection where this boundary condition is applied is defined by phys_id.V0 where phys_id is the physics interface identifier (for example, phys_id is spf by default for this physics interface). In order to avoid unexpected behavior, the function used for the fan curve is the maximum of the user-defined function and 0.

S T A T I C P R E S S U R E C U R V E D A T A

This section is available when Static pressure curve data is selected as the Static pressure

curve. In the table, enter values or expressions for the Flow rate and Static pressure curve

• Screen Boundary Condition

• Theory for the Non-Isothermal Screen Boundary Condition


(or click the Load from file button ( ) under the table to import a text file). Select the Flow rate (the default SI unit is m3/s) and Static pressure curve (the default SI unit is Pa).

S T A T I C P R E S S U R E C U R V E I N T E R P O L A T I O N


curve. Select the Interpolation function type—Linear (the default), Piecewise cubic, or Cubic

spline.

Periodic Flow Condition

The Periodic Flow Condition splits its selection into a source group and a destination group. Fluid that leaves the domain through one of the destination boundaries enters the domain through the corresponding source boundary. This corresponds to a situation where the geometry is a periodic part of a larger geometry. If the boundaries are not parallel to each other, the velocity vector is automatically transformed.

The extrapolation method always returns a constant value. In order to avoid problems with an undefined function, the function used for the boundary condition is the maximum of the interpolated function and 0.

Vacuum Pump Boundary Condition

If the boundaries are curved, it is recommended to only include two boundaries.

No input is required when Compressible flow (Ma<0.3) is selected as the Compressibility option under the Physical Model section for the physics interface. Typically when a periodic boundary condition is used with a compressible flow the pressure is the same at both boundaries and the flow is driven by a volume force.


90 | C H A P T E R

P R E S S U R E D I F F E R E N C E

This section is available when Incompressible flow is selected as the Compressibility option under the Physical Model section for the physics interface.

Enter a value or expression for the pressure difference, psrcpdst (SI unit: Pa). This pressure difference can, for example, drive the fully developed flow in a channel. The default is 0 Pa.



Fan

Use the Fan node to define the flow direction (inlet or outlet) and the fan parameters on exterior boundaries. Use the Interior Fan node for interior boundaries.


To set up a periodic boundary condition select both boundaries in the Periodic Flow Condition node. COMSOL Multiphysics automatically assigns one boundary as the source and the other as the destination. To manually set the destination selection, add a Destination Selection node to the Periodic Flow Condition node. All destination sides must be connected.

Periodic Boundary Conditions in the COMSOL Multiphysics Reference Manual

This node is not available for the turbulent flow interfaces.


F L O W D I R E C T I O N

Select a Flow direction—Inlet or Outlet.

P A R A M E T E R S

When Inlet is selected as the Flow direction, enter the Input pressure pinput (SI unit: Pa) to define the pressure at the fan input. The default is 0 Pa.

When Outlet is selected as the Flow direction, enter the Exit pressure pexit (SI unit: Pa) to define the pressure at the fan outlet. The default is 0 Pa.

For either flow direction, select a Static pressure curve to specify a fan curve—Linear (the default), Static pressure curve data, or User defined.

LinearFor both Inlet and Outlet flow directions, if Linear is selected, enter values or expressions for the Static pressure at no flow pnf (SI unit: Pa), the default is 100 Pa, and the Free delivery flow rate V0,fd (SI unit: m3/s), the default is 0.01 m3/s.

User DefinedSelect User defined to enter a different value or expression for the Static pressure curve. The flow rate across the selection where this boundary condition is applied is defined by phys_id.V0 where phys_id is the physics interface identifier (for example, phys_id is spf by default for laminar single-phase flow). In order to avoid unexpected behavior, the function used for the fan curve is the maximum between the user-defined function and 0.

Static Pressure Curve DataSelect Static pressure curve data to enter or load data under the Static Pressure Curve Data section that displays. The interpolation between points given in the table is defined using the Interpolation function type list in the Static Pressure Curve Interpolation section.

After a boundary is selected, an arrow displays in the Graphics window to indicate the selected flow direction. To update the arrow if the selection changes, click any node in the Model Builder and then click the Fan node again to update the Graphics window.

The static pressure curve is equal to the static pressure at no flow rate when V00 and equal to 0 when the flow rate is larger than the free delivery flow rate.


92 | C H A P T E R

S T A T I C P R E S S U R E C U R V E D A T A


curve. In the table, enter values or expressions the Flow rate and Static pressure curve (or click the Load from file button ( ) under the table to import a text file). Select the unit for the Flow rate (the default SI unit is m3/s) and for the Static pressure curve (the default SI unit is Pa).

S T A T I C P R E S S U R E C U R V E I N T E R P O L A T I O N


curve. Select the Interpolation function type—Linear (the default), Piecewise cubic, or Cubic

spline.

The extrapolation method is always a constant value. In order to avoid problems with an undefined function, the function used for the boundary condition is the maximum of the interpolated function and 0.

Interior Fan

The Interior Fan node represents interior boundaries where a fan condition is set using the fan pressure curve to avoid an explicit representation of the fan. The Interior Fan defines a boundary condition on the slit. That means that the pressure and the velocity can be discontinuous across this boundary.

One side represents a flow inlet; the other side represents the fan outlet. The fan boundary condition ensures that the mass flow rate is conserved between its inlet and outlet:

This boundary condition acts like a Pressure, No Viscous Stress boundary condition on each side of the fan. The pressure at the fan outlet is fixed so that the mass flow rate is conserved. On the fan inlet the pressure is set to the pressure at the fan outlet minus the pressure drop due to the fan. The pressure drop due to the fan is defined by the static

Theory for the Fan and Grille Boundary Conditions

u ninlet u n

outlet+ 0=


pressure curve, which is usually a function of the flow rate. To define a fan boundary condition on an exterior boundary, use the Fan node instead.


From the Selection list, choose the boundaries to define.

I N T E R I O R F A N

Define the Flow direction by selecting Along normal vector (the default) or Opposite to

normal vector. This defines which side of the boundary is considered the fan’s inlet and outlet.

Interior Wall

The Interior Wall boundary condition includes a set of boundary conditions describing the fluid-flow conditions at an interior wall.

It is similar to the Wall boundary condition available on exterior boundaries except that it applies on both sides of an internal boundary. It allows discontinuities (velocity, pressure, or turbulence) across the boundary. Use the Interior Wall boundary condition to avoid

This node is not available for the turbulent flow interfaces.

After a boundary is selected, an arrow displays in the Graphics window to indicate the selected flow direction. To update the arrow if the selection changes, click any node in the Model Builder and then click the Interior fan node again to update the Graphics window.

The rest of the settings for this section are the same as for the Fan node. See Linear and Static Pressure Curve Data for details.

Fan Defined on an Interior Boundary


94 | C H A P T E R

meshing thin structures by applying no-slip conditions on interior curves and surfaces instead. You can also prescribe slip conditions and conditions for a moving wall.


From the Selection list, choose the boundaries on which to apply the condition. The Interior Wall condition can only be applied on interior boundaries.


Select a Boundary condition—No slip (available for laminar flow and is the default), Wall

functions (available for turbulent flow, and the default), Slip, or Moving wall.

No SlipNo slip is the default boundary condition for a stationary solid wall. The condition prescribes u = 0 on both sides of the boundary; that is, the fluid at the wall is not moving.

Wall FunctionsThe Wall functions boundary condition applies wall functions to solid walls in a turbulent flow. Wall functions are used to model the thin region with high gradients in flow variables near the wall.



• If Generic roughness is selected, enter a Roughness height ks (SI unit: m). The default is 3.2 micrometers. Then enter a Roughness parameter Cs (dimensionless). The default is 0.26.

The Interior Wall boundary condition is only available for single-phase flow. It is compatible with laminar and turbulent flows.



SlipThe Slip condition prescribes a no-penetration condition, u·n0. It hence implicitly assumes that there are no viscous effects on either side of the slip wall and hence, no boundary layer develops. From a modeling point of view, this can be a reasonable approximation if the important effect is to prevent the exchange of fluid between the regions separated by the interior wall.

Moving WallIf the wall moves, so must the fluid on both sides of the wall. Hence, this boundary condition prescribes u = uw. Enter the components of the Velocity of moving wall uw (SI unit: m/s).


To display this section, click the Show button ( ) and select Advanced Physics Options. For the No slip and Moving wall boundary conditions, and to Apply reaction terms on all dependent variables, select All physics (symmetric). Otherwise, select Current physics

(internally symmetric) or Individual dependent variables to restrict the reaction terms as required.

Grille

The Grille node models the pressure drop caused by having a grille that covers the inlet or outlet.

Specifying this boundary condition does not automatically cause the associated wall to move. An additional Moving Mesh interface must be added to physically track the wall movement in the spatial reference frame.

• Wall

• Moving Mesh Interface in the COMSOL Multiphysics Reference Manual

See Fan for all of the settings for the Laminar Flow interface, except for Quadratic loss, which is described here. This node is not available for the turbulent flow interfaces.


96 | C H A P T E R



P A R A M E T E R S

If Quadratic loss is selected as the Static pressure curve, enter the Quadratic loss coefficient to define qlc (SI unit: kg/m7). The default value is 0 kg/m7. qlc defines the static pressure curve that is a piecewise quadratic function equal to 0 when flow rate is < 0, equal to V0

2qlc when flow rate is > 0.

Flow Continuity

The Flow Continuity node is suitable for pairs where the boundaries match; it prescribes that the flow field is continuous across the pair.

A Wall subnode is added by default and it applies to the parts of the pair boundaries where a source boundary lacks a corresponding destination boundary and vice versa. The Wall feature can be overridden by any other boundary condition that applies to exterior boundaries. Right-click the Flow Continuity node to add additional subnodes.




• Continuity on Interior Boundaries

• Identity and Contact Pairs


Point Mass Source

The Point Mass Source feature models mass flow originating from an infinitely small domain centered around a point.

PO I N T S E L E C T I O N

The Point Mass Source feature is available in 3D where it can be added to any point and in 2D axisymmetry where it can be added to points on the symmetry axis.

S O U R C E S T R E N G T H

Enter the Mass flux, , for the source (SI unit: kgs). A positive value results in mass being ejected from the point into the computational domain. A negative value results in mass being removed from the computational domain.

Point sources located on a boundary or on an edge affects the adjacent computational domains. This has the effect, for example, that the physical strength of a point source located in a symmetry plane is twice the given strength.

For the Reacting Flow in Porous Media, Diluted Species interface, which is available with the CFD Module, Chemical Reaction Engineering Module, or Batteries & Fuel Cells Module, the Point Mass Source node is available as two nodes, one for the fluid flow (Fluid Point Source) and one for the species (Species Point Source).

This feature requires at least one of the following licenses: Batteries & Fuel Cells Module, CFD Module, Chemical Reaction Engineering Module, Corrosion Module, Electrochemistry Module, Electrodeposition Module, Microfluidics Module, Pipe Flow Module, or Subsurface Flow Module.

Mass Sources for Fluid Flow in the COMSOL Multiphysics Reference Manual

q·p


98 | C H A P T E R

Line Mass Source

The Line Mass Source feature models mass flow originating from a tube region with infinitely small radius.

S E L E C T I O N

The Line Mass Source feature is available for all dimensions, but the applicable selection differs between the dimensions.

S O U R C E S T R E N G T H

Enter the Mass flux, , for the source (SI unit: kgs·m)). A positive value results in mass being ejected from the line into the computational domain and a negative value means that mass is removed from the computational domain.

For the Reacting Flow in Porous Media, Diluted Species interface, which is available with the CFD Module, Chemical Reaction Engineering Module, or Batteries & Fuel Cells Module, the Line Mass Source node is available as two nodes, one for the fluid flow (Fluid Line Source) and one for the species (Species Line Source).


MODEL DIMENSION APPLICABLE GEOMETRICAL ENTITY

2D Points

2D Axisymmetry Points not on the symmetry axis and the symmetry axis

3D Edges

q· l


Line sources located on a boundary affect the adjacent computational domains. This, for example, has the effect that the physical strength of a line source located in a symmetry plane is twice the given strength.

Pressure Point Constraint

The Pressure Point Constraint node adds a pressure constraint at a point. If it is not possible to specify the pressure level using a boundary condition, the pressure level must be set in some other way, for example, by specifying a fixed pressure at a point.

P R E S S U R E C O N S T R A I N T

Enter a point constraint for the Pressure p0 (SI unit: Pa). The default is 0 Pa.


To display this section, click the Show button ( ) and select Advanced Physics Options. To Apply reaction terms on all dependent variables, select All physics (symmetric). Or select Individual dependent variables to restrict the reaction terms as required. Select the Use weak

constraints check box to replace the standard constraints with a weak implementation.

More Boundary Condition Settings for the Turbulent Flow Interfaces

For the Inlet, Outlet, and Boundary Stress features, the following settings are also required for the turbulent flow interfaces. The first sections (Turbulent Intensity and Turbulence Length Scale Parameters and Boundary Stress Turbulent Boundary Type) provide further information about the boundary conditions, and the additional settings information is described under Boundary Condition.

For The Turbulent Flow, Spalart-Allmaras Interface, the only additional parameter required is for the Undamped turbulent kinematic viscosity 0 (SI unit: m2/s). The default is 3*spf.nu.

Turbulent Intensity and Turbulence Length Scale ParametersThe Turbulent intensity IT and Turbulence length scale LT values are related to the turbulence variables via the following equations, Equation 3-4 for the Inlet and Equation 3-5 for the Open Boundary:

Mass Sources for Fluid Flow in the COMSOL Multiphysics Reference Manual


100 | C H A P T E

Inlet (3-4)

Open Boundary (3-5)

For the Open Boundary and Boundary Stress options, and with any turbulent flow interface, inlet conditions for the turbulence variables also need to be specified. These conditions are used on the parts of the boundary where u·n0, that is, where flow enters the computational domain.

For the k- and SST turbulence models the Turbulent intensity IT and Turbulence length

scale LT values are related to the turbulence variables via the following equations, Equation 3-6 for the Inlet and Equation 3-7 for the Open Boundary:

Inlet (3-6)

Open Boundary (3-7)

Boundary Stress Turbulent Boundary TypeFor Boundary Stress, first select a Turbulent boundary type to apply to the turbulence variables—Open boundary (the default), Inlet, or Outlet.

• If Open boundary is selected, then expect parts of the boundary to be an outlet and parts of the boundary to be an inlet.

• Select Inlet when the whole boundary is expected to be an inlet. Under Exterior

turbulence, the same options to specify turbulence variables as for the Open boundary option are available. The difference is that, for the Inlet option, the conditions are applied on the whole boundary.

• Select Outlet when the whole boundary is expected to be an outflow boundary. Homogeneous Neumann conditions are applied for the turbulence variables, that is, for k and

for k and

k 32--- U IT 2,= C

3 4 k3 2/

LT-----------=

k 32--- ITUref 2,=

C3 4

LT------------

3 ITUref 2

2----------------------------

32---

=

k 32--- U IT 2,= k

0* 1 4/ LT

--------------------------=

k 32--- ITUref 2,= 1

0* 1 4/ LT

--------------------------3 ITUref 2

2----------------------------

=

k n 0= n 0=

R 3 : S I N G L E - P H A S E F L O W

or for the Spalart-Allmaras model:


Click the Specify turbulence length scale and intensity button (the default) to enter values or expressions for the:

• Turbulent intensity IT (dimensionless)

• Turbulence length scale LT (SI unit: m)

• Reference velocity scale Uref (SI unit: m/s). This is available for most options (excluding Velocity for the Inlet node).

If Specify turbulence variables is selected, enter values or expressions for the:

• Turbulent kinetic energy k0 (SI unit: m2/s2)

• Turbulent dissipation rate, 0 (SI unit: m2/s3) for the k- interfaces, or Turbulent specific

dissipation rate 0 (SI unit: 1/s) for the k- and SST interfaces

For Open Boundary and Boundary Stress>Open boundary, the following settings are found under the Exterior turbulence subsection.

For Boundary Stress, first select a Turbulent boundary type to apply to the turbulence variables—Open boundary (the default), Inlet, or Outlet. Then for Open boundary and Inlet continue by entering the following parameters.

For the Turbulent Flow, Spalart-Allmaras interface, select a Turbulent

boundary type and enter a value or expression for the Undamped turbulent

kinematic viscosity 0 (SI unit: m2/s). The default is 3*spf.nu.

The default values are different for Inlet, Open Boundary, and Boundary Stress. See Table 3-2 (excluding the Spalart-Allmaras model).

Also, for recommendations of physically sound values see Inlet Values for the Turbulence Length Scale and Turbulent Intensity.

k n 0= n 0=

n 0=


102 | C H A P T E

TABLE 3-2: DEFAULT VALUES FOR THE TURBULENT INTERFACES

NAME AND UNIT VARIABLE INLET OPEN BOUNDARY

BOUNDARY STRESS

Turbulent intensity (dimensionless)

IT 0.05 0.005 0.01

Turbulence length scale (m)

LT 0.01 0.1 0.1

Reference velocity scale (m/s)

Uref 1 1 1

Turbulent kinetic energy (m2/s2)

k0 0.005 2.5 x 10-3 1 x 10-2

Turbulent dissipation rate (m2/s3)

0 0.005 1.1 x 10-4 1 x 10-3

Turbulent specific dissipation rate (1/s)

0 20 0.5 0.5


Th e Ro t a t i n g Ma ch i n e r y , L am i n a r and Tu r bu l e n t F l ow I n t e r f a c e s

The Rotating Machinery, Laminar Flow (rmspf) and Rotating Machinery, Turbulent Flow

(rmspf) interfaces, found under the Single-Phase Flow>Rotating Machinery branch ( ) when adding a physics interface, are used for modeling flow where one or more of the boundaries rotate in a periodic fashion. This is used for mixers and propellers.

The physics interfaces support compressible and incompressible flow, the flow of non-Newtonian fluids described by the Power Law and Carreau models, and also turbulent flow. The physics interfaces also support creeping flow, although the shallow channel approximation is redundant.

In this section:

• The Rotating Machinery, Laminar Flow Interface

• The Rotating Machinery, Turbulent Flow, k- Interface

• Domain, Boundary, Point, and Pair Nodes for the Rotating Machinery, Laminar and Turbulent Flow Interfaces

The Rotating Machinery, Laminar Flow Interface

The Rotating Machinery, Laminar Flow (rmspf) interface ( ), found under the Single-Phase

Flow>Rotating Machinery branch ( ) when adding a physics interface, is used to simulate flow at low to moderate Reynolds numbers in geometries with one or more rotating parts. The physics interface supports incompressible and compressible flows at low Mach numbers (typically less than 0.3). It also supports modeling of non-Newtonian fluids. The physics interface is available for 3D and 2D models.

There are two study types available for this physics interface. Using the Time Dependent study type, rotation is achieved through moving mesh functionality, also known as sliding mesh. Using the Frozen Rotor study type, the rotating parts are kept frozen in position, and rotation is accounted for by the inclusion of centrifugal and Coriolis forces. In both types, the momentum balance is governed by the Navier-Stokes equations, and the mass conservation is governed by the continuity equation. See Theory for the Rotating Machinery Interfaces .

T H E R O T A T I N G M A C H I N E R Y , L A M I N A R A N D TU R B U L E N T F L O W I N T E R F A C E S | 103

104 | C H A P T E

When this physics interface is added, the following default physics nodes are also added in the Model Builder—Fluid Properties, Wall, Rotating Wall, and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and volume forces. You can also right-click Rotating Machinery, Laminar Flow to select physics from the context menu. See Rotating Domain, Initial Values, and Rotating Wall.



The default identifier (for the first physics interface in the component) is rmspf.



Pseudo time steppingSelect the Use pseudo time stepping for stationary equation form check box to add pseudo time derivatives to the equation when the Frozen Rotor equation form is used. (Frozen rotor is a pseudo stationary formulation.) When selected, also choose a CFL number

expression—Automatic (the default) or Manual. Automatic sets the local CFL number (from the Courant–Friedrichs–Lewy condition) to the built-in variable CFLCMP which in turn triggers a PID regulator for the CFL number. If Manual is selected, enter a Local CFL number CFLloc (dimensionless).

In addition to the settings described below, see The Laminar Flow, Creeping Flow, and Turbulent Flow Interfaces for all the other settings available.

Also go to Domain, Boundary, Point, and Pair Nodes for the Rotating Machinery, Laminar and Turbulent Flow Interfaces for links to all the physics nodes.


The Rotating Machinery, Turbulent Flow, k- Interface

The Rotating Machinery, Turbulent Flow, k- (rmspf) interface ( ), found under the Single-Phase Flow>Rotating Machinery branch ( ) when adding a physics interface, is used to simulate flow at high Reynolds numbers in geometries with one or more rotating parts. The physics interface is suitable for incompressible and compressible flows at low Mach numbers (typically less than 0.3).

The momentum balance is governed by the Navier-Stokes equations, and the mass conservation is governed by the continuity equation. Turbulence effects are modeled using the standard two-equation k- model with realizability constraints. Flow close to walls is modeled using wall functions.

There are two study types available for this physics interface. Using the Time Dependent study type, the rotation is achieved through moving mesh functionality, also known as sliding mesh. Using the Frozen Rotor study type, the rotating parts are kept frozen in position, and the rotation is accounted for by the inclusion of centrifugal and Coriolis forces. See Theory for the Rotating Machinery Interfaces .

When this physics interface is added, the following physics nodes are also added in the Model Builder—Fluid Properties, Wall, Rotating Wall, and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and volume forces. You can also right-click Rotating Machinery, Turbulent Flow, k- to select physics from the context menu. See Rotating Domain, Initial Values, and Rotating Wall in



This physics interface changes to the Rotating Machinery, Turbulent Flow, k-interface when the Turbulence model type selected is Rans, k-.

Laminar Flow in a Baffled Stirred Mixer: model library path CFD_Module/Single-Phase_Tutorials/baffled_mixer


106 | C H A P T E

this section for details.

Domain, Boundary, Point, and Pair Nodes for the Rotating Machinery, Laminar and Turbulent Flow Interfaces

The Rotating Machinery, Laminar Flow Interface and The Rotating Machinery, Turbulent Flow, k- Interface have the following domain, boundary, point, and pair physics nodes, which are available from the Physics ribbon toolbar (Windows users), Physics context menu (Mac or Linux users), or right-click to access the context menu (all users).

See The Laminar Flow, Creeping Flow, and Turbulent Flow Interfaces for all the settings. Also go to Domain, Boundary, Point, and Pair Nodes for the Rotating Machinery, Laminar and Turbulent Flow Interfaces for links to all the physics nodes.

This physics interface changes to Rotating Machinery, Laminar Flow when the Turbulence model type selected is None.


• Initial Values

• Rotating Domain

• Rotating Wall

• Rotating Interior Wall


The following nodes (listed in alphabetical order) are described for the Laminar Flow interface:

Initial Values

The Initial Values node adds initial values for the velocity field and pressure that can serve as initial conditions for a transient simulation or as an initial guess for a nonlinear solver.

D O M A I N S E L E C T I O N

For a default node, the setting inherits the selection from the parent node, and can not be edited; that is, the selection is automatically made and is the same as for the physics interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific domains or select All domains as required.


Enter values or expressions for the initial value of the Velocity field u (SI unit: ms) (the default is 0 ms) and for the Pressure p (SI unit: Pa) (the default value is 0 Pa).

For The Rotating Machinery, Turbulent Flow, k- Interface also enter values for the Turbulent kinetic energy (SI unit: m2s2) and Turbulent dissipation rate (SI unit: m2s3).


• Flow Continuity


• Inlet

• Interior Wall

• Line Mass Source

• Open Boundary

• Outlet


• Point Mass Source


• Screen

• Symmetry

• Volume Force

• Wall


108 | C H A P T E

Rotating Domain

Use the Rotating Domain node to specify the rotational frequency and direction in the Rotating Machinery, Fluid Flow interfaces.

The angular displacement, (SI unit: rad), of the rotating domain is computed from a specified angular velocity w (SI unit: rads) by solving the ODE

(3-8)

Since the angular displacement is solved for, the domain rotation can be specified using any type of angular velocity (constant, analytic, interpolation function, and so forth).

If there is more than one rotating domain, these must not intersect.

ddt-------- w t =


R O T A T I N G D O M A I N

Rotating Wall

The Rotating Wall node is a boundary condition for The Rotating Machinery, Laminar Flow Interface and The Rotating Machinery, Turbulent Flow, k- Interface. The feature applies to external boundaries of a Rotating Domain. It applies flow conditions corresponding to no slip (laminar flow) or wall functions (for turbulent flow) as defined by the Wall node , but accounts for the velocity of the Rotating Domain.

For 3D models, select the Axis of rotation, the z-axis is the default. If the x-axis is selected, it corresponds to a rotational axis (1, 0, 0) with the origin as the base point. This is the same for the y-axis and z-axis. If User

defined is selected, enter values for the Rotation axis base point rbp (SI unit: m) and Rotation axis direction rax.

Select a Rotational frequency—Revolutions per time (the default) or Angular

velocity.

• If Revolutions per time is selected, enter a value or expression in the field (SI unit: 1s) and select a Rotational direction—Positive angular velocity

(the default) or Negative angular velocity. The angular velocity in this case is defined as the input multiplied by 2·.

• If Angular velocity is selected, enter an Angular velocity w (SI unit: rads). The default is 0 rads.

For 2D models, enter coordinates for the Rotation axis base point rbp (SI unit: m). The default is the origin (0, 0).

Select a Rotational frequency—Revolutions per time (the default) or Angular

velocity.

• If Revolutions per time is selected, enter a value or expression in the field (SI unit: 1s) and select a Rotational direction—Clockwise (the default) or Counterclockwise. The angular velocity in this case is defined as the input multiplied by 2·.

• If Angular velocity is selected, enter an Angular velocity w (SI unit: rads). The default is 0 rads.


110 | C H A P T E





Rotating Interior Wall

The Rotating Interior Wall node is a boundary condition for The Rotating Machinery, Laminar Flow Interface and The Rotating Machinery, Turbulent Flow, k- Interface. The feature applies Rotating Wall conditions to internal boundaries of a Rotating Domain.

The feature is similar to the Rotating Wall boundary condition, available on exterior boundaries, except that it applies on both sides of an internal boundary. It allows discontinuities (velocity, pressure, turbulence) across the boundary. You can use the Rotating Interior Wall boundary condition to avoid meshing thin structures by instead applying conditions on interior curves and surfaces.


For a default node, the setting inherits the selection from the parent node, and can not be edited; that is, the selection is automatically made and is the same as for the physics interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific fluid boundaries subjected to a rotating interior wall, or select All boundaries as required.




Th eo r y f o r t h e S i n g l e - Pha s e F l ow I n t e r f a c e s

The Single-Phase Flow, Laminar Flow theory is described in this section:

• General Single-Phase Flow Theory

• Compressible Flow

• The Mach Number Limit

• Incompressible Flow

• The Reynolds Number

• Non-Newtonian Flow: The Power Law and the Carreau Model

• The Boussinesq Approximation

• Theory for the Wall Boundary Condition

• Prescribing Inlet and Outlet Conditions

• Laminar Inflow

• Laminar Outflow

• Mass Flow




• Vacuum Pump Boundary Condition

• Fan Defined on an Interior Boundary

• Theory for the Fan and Grille Boundary Conditions

• Screen Boundary Condition

• Mass Sources for Fluid Flow

• Numerical Stability—Stabilization Techniques for Fluid Flow

• Solvers for Laminar Flow

Also see Theory for the Turbulent Flow Interfaces and Theory for the Rotating Machinery Interfaces.

T H E O R Y F O R T H E S I N G L E - P H A S E F L O W I N T E R F A C E S | 111

112 | C H A P T E


• The Projection Method for the Navier-Stokes Equations

• Discontinuous Galerkin Formulation

• Particle Tracing in Fluid Flow

• References for the Single-Phase Flow, Laminar Flow Interfaces

General Single-Phase Flow Theory

The single-phase fluid-flow interfaces are based on the Navier-Stokes equations, which in their most general form read

(3-9)

(3-10)

(3-11)

where

• is the density (SI unit: kg/m3)

• u is the velocity vector (SI unit: m/s)

• p is pressure (SI unit: Pa)

• is the viscous stress tensor (SI unit: Pa)

• F is the volume force vector (SI unit: N/m3)

• Cp is the specific heat capacity at constant pressure (SI unit: J/(kg·K))

• T is the absolute temperature (SI unit: K)

• q is the heat flux vector (SI unit: W/m2)

• Q contains the heat sources (SI unit: W/m3)

• S is the strain-rate tensor:

The theory about most boundary conditions is found in Ref. 2.

t------ u + 0=

ut------- u u+ pI– + F+=

CpTt------- u T+ q – :S T

---- T-------

p

pt------ u p+ – Q+ +=


The operation “:” denotes a contraction between tensors defined by

(3-12)

This is sometimes referred to as the double dot product.

Equation 3-9 is the continuity equation and represents conservation of mass. Equation 3-10 is a vector equation which represents conservation of momentum. Equation 3-11 describes the conservation of energy, formulated in terms of temperature. This is an intuitive formulation that facilitates boundary condition specifications.

To close the equation system, Equation 3-9 through Equation 3-11, constitutive relations are needed. For a Newtonian fluid, which has a linear relationship between stress and strain, Stokes (Ref. 1) deduced the following expression:

(3-13)

The dynamic viscosity, (SI unit: Pa·s), for a Newtonian fluid is allowed to depend on the thermodynamic state but not on the velocity field. All gases and many liquids can be considered Newtonian. Examples of non-Newtonian fluids are honey, mud, blood, liquid metals, and most polymer solutions. With the CFD Module, you can model flows of non-Newtonian fluids using the predefined power law and Carreau models, which describe the dynamic viscosity for non-Newtonian fluids.

Other commonly used constitutive relations are Fourier’s law of heat conduction and the ideal gas law.

In theory, the same equations describe both laminar and turbulent flows. In practice, however, the mesh resolution required to simulate turbulence with the Laminar Flow interface makes such an approach impractical.

There are several books where derivations of the Navier-Stokes equations and detailed explanations of concepts such as Newtonian fluids can be found. See, for example, the classical text by Batchelor (Ref. 3) and the more recent work by Panton (Ref. 4).

S 12--- u u T+ =

a:b anmbnm

m

n=

2S 23--- u I–=


114 | C H A P T E

Many applications describe isothermal flows for which Equation 3-11 is decoupled from Equation 3-9 and Equation 3-10. Non-isothermal flow and the temperature equation are described in the Heat Transfer and Non-Isothermal Flow chapter.

2 D A X I S Y M M E T R I C F O R M U L A T I O N S

A 2D axisymmetric formulation of Equation 3-9 and Equation 3-10 requires to be zero. That is, there must be no gradients in the azimuthal direction. A common additional assumption is however that u0. In such cases, the -equation can be removed from Equation 3-10. The resulting system of equations is both easier to converge and computationally less expensive compared to retaining the -equation. The default 2D axisymmetric formulation of Equation 3-9 and Equation 3-10 therefore assumes that

You can activate the Swirl Flow property which reduces the above assumptions to and reintroduces the -equation into Equation 3-10.

Compressible Flow

The Navier-Stokes equations solved by default in all Single-Phase Flow interfaces apply the compressible formulation of the continuity equation:

(3-14)

and the momentum equation:

(3-15)

These equations are applicable for incompressible as well as compressible flow with density variations.

The Mach Number Limit

An important dimensionless number in fluid dynamics is the Mach number, Ma, defined by

0=

u 0=

0=

t------ u + 0=

ut------- u u+ p– u u T+ 2

3--- u I–

F++=

Ma ua

-------=


where a is the speed of sound. A flow is formally incompressible when Ma0. This is theoretically achieved by letting the speed of sound tend to infinity. The Navier-Stokes equations then have the mathematical property that pressure disturbances are instantaneously propagated throughout the entire domain. This results in a parabolic equation system.

The momentum equation, Equation 3-15, is parabolic for unsteady flow and elliptic for steady flow, whereas the continuity equation, Equation 3-14, is hyperbolic for both steady and unsteady flow. The combined system of equations is thus hybrid parabolic-hyperbolic for unsteady flow and hybrid elliptic-hyperbolic for steady flow. An exception occurs when the viscous term in Equation 3-15 becomes vanishingly small, such as at an outflow boundary, in which case the momentum equation becomes locally hyperbolic. The number of boundary conditions to apply on the boundary then depends on the number of characteristics propagating into the computational domain. For the purely hyperbolic system, the number of characteristics propagating from the boundary into the domain changes as the Mach number passes through unity. Hence, the number of boundary conditions required to obtain a numerically well posed system must also change. The compressible formulation of the laminar and turbulent interfaces uses the same boundary conditions as the incompressible formulation, which implies that the compressible interfaces are not suitable for flows with Mach number larger than or equal to one.

The practical Mach number limit is lower than one, however. The main reason is that numerical scheme (stabilization and boundary conditions) of the Laminar Flow interface does not recognize the direction and speed of pressure waves. The fully compressible Navier-Stokes equations do, for example, start to display very sharp gradients already at moderate Mach numbers. But the stabilization in the Single-Phase Flow interface does not necessarily capture these gradients. It is impossible to give an exact limit where the low Mach number regime ends and the moderate Mach number regime begins, but a rule of thumb is that the Mach number effects start to appear at Ma 0.3. For this reason the compressible formulation is referred to as compressible flow (Ma<0.3) in COMSOL Multiphysics. To model high Mach number flows, use on of The High Mach Number Flow Interfaces.

Incompressible Flow

When the temperature variations in a flow are small, a single-phase fluid can often be assumed incompressible; that is, is constant or nearly constant. This is the case for all liquids under normal conditions and also for gases at low velocities. For constant , Equation 3-14 reduces to


116 | C H A P T E

(3-16)

and Equation 3-15 becomes

(3-17)

The Reynolds Number

A fundamental characteristic in analyses of fluid flow is the Reynolds number:

where U denotes a velocity scale, and L denotes a representative length. The Reynolds number represents the ratio between inertial and viscous forces. At low Reynolds numbers, viscous forces dominate and tend to damp out all disturbances, which leads to laminar flow. At high Reynolds numbers, the damping in the system is very low giving small disturbances the possibility to grow by nonlinear interactions. If the Reynolds number is high enough, the flow field eventually ends up in a chaotic state called turbulence.

Observe that the Reynolds number can have different meanings depending on the length scale and velocity scale. To be able to compare two Reynolds numbers, they must be based on equivalent length and velocity scales.

The Fluid Flow interfaces automatically calculate the local cell Reynolds number Recuh2 using the element length h for L and the magnitude of the velocity vector u for the velocity scale U. This Reynolds number is not related to the character of the flow field, but to the stability of the numerical discretization. The risk for numerical oscillations in the solution increases as Rec grows. The cell Reynolds number is a predefined quantity available for visualization and evaluation (typically it is available as: spf.cellRe).

Non-Newtonian Flow: The Power Law and the Carreau Model

The viscous stress tensor is directly dependent on the shear rate tensor and can be written as:

u 0=

t

u u u+ pI– u u T+ + F+=

Re UL

------------=

· 23--- u I–=

·=


using the compressible and incompressible formulations. Here denotes the strain-rate tensor defined by:

Its magnitude, the shear rate, is:

where the contraction operator “:” is defined by

For a non-Newtonian fluid, the dynamic viscosity is assumed to be a function of the shear rate:

The Laminar Flow interfaces have the following predefined models to prescribe a non-Newtonian viscosity—the power law and the Carreau model.

POW E R L A W

The power law model is an example of a generalized Newtonian model. It prescribes

(3-18)

where m and n are scalars that can be set to arbitrary values. For n1, the power law describes a shear thickening (dilatant) fluid. For n1, it describes a shear thinning (pseudoplastic) fluid. A value of n equal to one gives the expression for a Newtonian fluid.

Equation 3-18 predicts an infinite viscosity at zero shear rate for n1. This is however never the case physically. Instead, most fluids have a constant viscosity for shear rates smaller than 10-2 s-1 (Ref. 19). Since infinite viscosity also makes models using Equation 3-18 difficult to solve, COMSOL Multiphysics implements the power law model as

(3-19)

where is a lower limit for the evaluation of the shear rate magnitude. The default value for is 10-2 s-1, but can be given an arbitrary value or expression using the corresponding edit field.

·

· u u T+ =

· · 12---· :·= =

a:b anmbnm

m

n=

· =

m· n 1–=

mmax · ·min n 1–=

·min

·min


118 | C H A P T E

C A R R E A U M O D E L

The Carreau model defines the viscosity in terms of the following four-parameter expression

(3-20)

where is a parameter with the unit of time, 0 is the zero shear rate viscosity, inf is the infinite shear-rate viscosity, and n is a dimensionless parameter. This expression is able to describe the viscosity for most stationary polymer flows.

The Boussinesq Approximation

The Boussinesq approximation is a way to treat certain simple cases of buoyant flows without having to use a compressible formulation of the Navier-Stokes equations.

The Boussinesq approximation assumes that variations in density have no effect on the flow field except that they give rise to buoyancy forces. The density is assigned a reference value, 0, everywhere except in the body force term, which is set to

(3-21)

where g is the gravity vector. A further simplifications is often possible. Since g can be written in terms of a potential, , it is possible to write Equation 3-21 as:

The first part can be canceled out by splitting the true pressure, p, into a hydrodynamic component, P, and a hydrostatic component, 0. Equation 3-16 and Equation 3-17 are expressed in terms of the hydrodynamic pressure Pp0:

(3-22)

(3-23)

To obtain the Boussinesq approximation in this form, enter the expression for g for the Volume Force feature.

In practice, the shift from p to P can be ignored except where the pressure appears in boundary conditions. The pressure that is specified at boundaries is the hydrodynamic pressure in this case. For example, on a vertical outflow or inflow boundary, the

0 inf– 1 · 2+ n 1–

2-----------------

+=

F 0 + g=

F 0 – g+=

u 0=

0ut------- 0u u+ P– u u T+ g ++=


hydrodynamic pressure is typically a constant, while the true pressure is a function of the vertical coordinate.

The system that Equation 3-22 and Equation 3-23 form has its limitations. The main assumption is that the density fluctuations must be small; that is, 01. There are also some more subtle constraints that, for example, make the Boussinesq approximation unsuitable for systems of very large dimensions. An excellent discussion of the Boussinesq approximation and its limitations appears in Chapter 14 of Ref. 10.

Theory for the Wall Boundary Condition

S L I P

The Slip condition assumes that there are no viscous effects at the slip wall and hence, no boundary layer develops. From a modeling point of view, this is a reasonable approximation if the important effect of the wall is to prevent fluid from leaving the domain. Mathematically, the constraint can be formulated as:

The no penetration term takes precedence over the Neumann part of the condition and the above expression is therefore equivalent to

expressing that there is no flow across the boundary and no viscous stress in the tangential direction.

For turbulent flow, any turbulence variable is subject to a homogeneous Neumann condition. For example

See Volume Force for the node settings.

See Wall for the node settings.

u n 0,= pI– u u T+ + n 0=

u n 0,= K K n n– 0=

K u u T+ n=

k n 0= n 0=


120 | C H A P T E

for a k- model.

S L I D I N G WA L L

The Sliding Wall boundary condition is appropriate if the wall behaves like a conveyor belt; that is, the surface is sliding in its tangential direction. The wall does not have to actually move in the coordinate system.

S L I P VE L O C I T Y

In the microscale range, the flow at a boundary is seldom strictly no slip or slip. Instead, the boundary condition is something in between, and there is a slip velocity at the boundary. Two phenomena account for this velocity: violation of the continuum hypothesis for the viscosity and flow induced by a thermal gradient along the boundary.

The following equation relates the viscosity-induced jump in tangential velocity to the tangential shear stress along the boundary:

In 2D, the tangential direction is unambiguously defined by the direction of the boundary, but the situation becomes more complicated in 3D. For this reason, this boundary condition has slightly different definitions in the different space dimensions.

For 2D and 2D axisymmetric models, the velocity is given as a scalar Uw and the condition prescribes

where t(ny , nx) for 2D and t(nznr) for axial symmetry.

For 3D models, the velocity is set equal to a given vector uw projected onto the boundary plane:

The normalization makes u have the same magnitude as uw even if uw is not exactly parallel to the wall.

u n 0,= u t Uw=

uuw n uw n–

uw n uw n–-------------------------------------------- uw=

u 1---n t=


For gaseous fluids, the coefficient is given by

where is the fluid’s dynamic viscosity (SI unit: Pa·s), v represents the tangential momentum accommodation coefficient (TMAC) (dimensionless), and is the molecules’ mean free path (SI unit: m). The tangential accommodation coefficients are typically in the range of 0.85 to 1.0 and can be found in Ref. 15.

A simpler expression for is

where Ls, the slip length (SI unit: m), is a straight channel measure of the distance from the boundary to the virtual point outside the flow domain where the flow profile extrapolates to zero. This equation holds for both liquids and gases.

Thermal creep results from a temperature gradient along the boundary. The following equation relates the thermally-induced jump in tangential velocity to the tangential gradient of the natural logarithm of the temperature along the boundary:

where T is the thermal slip coefficient (dimensionless) and is the density of the fluid. The thermal slip coefficients range between 0.3 and 1.0 and can be found in Ref. 15.

Combining the previous relationships results in the following equation:

Relate the tangential shear stress to the viscous boundary force by

where the components of K are the Lagrange multipliers that are used to implement the boundary condition. Similarly, the tangential temperature gradient results from the difference of the gradient and its normal projection:

2 v–

v---------------

-------------------------=

Ls------=

u T---t Tlog=

u uw t–Ls

------n t T

T-------tT+=

n t K n K n–=

tT T n T n–=


122 | C H A P T E

Use Viscous SlipWhen viscous slip is used, select Maxwell’s model to calculate Ls using:

Prescribing Inlet and Outlet Conditions

The Navier-Stokes equations can show large variations in mathematical behavior, ranging from almost completely elliptic to almost completely hyperbolic. This has implications when it comes to prescribing admissible boundary conditions. There is also a discrepancy between mathematically valid boundary conditions and practically useful boundary conditions.

I N L E T C O N D I T I O N S

An inlet requires specification of the velocity components. The most robust way to do this is to prescribe a velocity field using a Velocity condition.

A common alternative to prescribing the complete velocity field is to prescribe a pressure, in which case the normal velocity component is specified indirectly via the continuity equation. The pressure can be specified pointwise, which is a mathematically over-constraining but numerically robust formulation. This is achieved using the Pressure, No Viscous Stress condition. Alternatively, the pressure can be specified via a stress condition:

(3-24)

where unn is the normal derivative of the normal velocity component. Equation 3-24 is prescribed by the Normal stress condition. Equation 3-24 is mathematically more

Also see Wall for the node settings.

• Also see Inlet and Outlet for the node settings.



Ls2 v–

v--------------- =

p– 2unn

---------+ Fn=


stringent compared to specifying the pressure pointwise, can at the same time not guarantee that p obtains the desired value. In practice, p will however be close to Fn, except for low Reynolds number flows where viscous effects are the only effects that balance the pressure.

O U T L E T C O N D I T I O N S

The most common approach is to prescribe a pressure via a normal stress condition on the outlet. This is often accompanied by a vanishing tangential stress condition:

where utn is the normal derivative of the tangential velocity field. It is also possible to prescribe ut to be zero. The latter option should be used with care since it can have a significant effect on the upstream solution.

The elliptic character of the Navier-Stokes equations mathematically permit specifying a complete velocity field at an outlet. This can however be difficult to apply in practice. The reason being that it is hard to prescribe the outlet velocity so that it at each point is consistent with the interior solution. The adjustment to the specified velocity then occurs across an outlet boundary layer. The thickness of this boundary layer depends on the Reynolds number; the higher the Reynolds number, the thinner the boundary layer.

A L T E R N A T I V E F O R M U L A T I O N S

COMSOL provides several specialized boundary conditions that either provide detailed control over the flow at the boundary or that simulate specific devices. In practice they often prescribe a velocity or a pressure, but calculate the prescribed values using for example ODEs.

Laminar Inflow

In order to prescribe a fully developed inlet velocity profile, this boundary condition adds a weak form contribution and constraints corresponding to unidirectional flow perpendicular to the boundary. The applied condition corresponds to the situation shown in Figure 3-2: a fictitious domain of length Lentr is assumed to be attached to the inlet of the computational domain. The domain is an extrusion of the inlet boundary, which means that laminar inflow requires the inlet to be flat. The boundary condition uses the assumption that the flow in this fictitious domain is fully developed laminar flow. The “wall” boundary conditions for the fictitious domain is inherited from the real domain, ,

utn

-------- 0=


124 | C H A P T E

unless the option to constrain outer edges or endpoints to zero is selected in which case the fictitious “walls” are no-slip walls.

Figure 3-2: An example of the physical situation simulated when using the Laminar inflow boundary condition. is the actual computational domain while the dashed domain is a fictitious domain.

If an average inlet velocity or inlet volume flow is specified instead of the pressure, COMSOL Multiphysics adds an ODE that calculates a pressure, pentr, such that the desired inlet velocity or volume flow is obtained.

Laminar Outflow

In order to prescribe an outlet velocity profile, this boundary condition adds a weak form contribution and constraints corresponding to unidirectional flow perpendicular to the boundary. The applied condition corresponds to the situation shown in Figure 3-3: assume that a fictitious domain of length Lexit is attached to the outlet of the computational domain. The domain is an extrusion of the outlet boundary, which means that laminar outflow requires the outlet to be flat. The boundary condition uses the assumption that the flow in this fictitious domain is fully developed laminar flow. The “wall” boundary conditions for the fictitious domain is inherited from the real domain, , unless the option to constrain outer edges or endpoints to zero is selected in which case the fictitious “walls” are no-slip walls.

Figure 3-3: An example of the physical situation simulated when using the Laminar outflow boundary condition. is the actual computational domain while the dashed domain is a fictitious domain.

If the average outlet velocity or outlet volume flow is specified instead of the pressure, the software adds an ODE that calculates pexit such that the desired outlet velocity or volume flow is obtained.

Lentr

pentr

pexit

Lexit


Mass Flow

The Mass Flow boundary condition constrains the mass flowing into the domain across an inlet boundary. The mass flow can be specified in a number of ways.

PO I N T W I S E M A S S F L U X

The pointwise mass flux sets the velocity at the boundary to:

where mf is the normal mass flux and is the density.

M A S S F L O W R A T E

The mass flow rate boundary condition sets the total mass flow through the boundary according to:

where dbc (only present in the 2D Cartesian axis system) is the boundary thickness normal to the fluid-flow domain and m is the total mass flow rate.

In addition to the constraint on the total flow across the boundary, the tangential velocity components are set to zero on the boundary

(3-25)

S T A N D A R D F L O W R A T E

The standard flow rate boundary condition specifies the mass flow as a standard volumetric flow rate. The mass flow through the boundary is set by the equation:

where dbc (only present in the 2D model Cartesian axis system) is the boundary thickness normal to the fluid-flow domain, st is the standard density, and Qsv is the standard flow rate. The standard density is defined by one of the following equations:

umf

-------n–=

dbc u n Sd– m=

u n 0=

dbcst------- u n Sd

– Qsv=

stMn

Vn--------=


126 | C H A P T E

where Mn is the mean molar mass of the fluid, Vn is the standard molar volume, pst is the standard pressure, R is the universal molar gas constant, and Tst is the standard temperature.

Equation 3-25 or Equation 3-26 is also enforced for compressible and incompressible flow, respectively, ensuring that the normal component of the viscous stress and the tangential component of the velocity are zero at the boundary.

No Viscous Stress

For this module, and in addition to the Pressure, No Viscous Stress Boundary Condition, the viscous stress condition sets the viscous stress to zero:

using the compressible and the incompressible formulation, respectively.

The condition is not a sufficient outlet condition since it lacks information about the outlet pressure. It must hence be combined with pressure point constraints on one or several points or lines surrounding the outlet.

This boundary condition is numerically the least stable outlet condition, but can still be beneficial if the outlet pressure is nonconstant due to, for example, a nonlinear volume force.

Pressure, No Viscous Stress Boundary Condition

The Pressure, No Viscous Stress boundary condition specifies vanishing viscous stress along with a Dirichlet condition on the pressure:


stpstMnRTst

-----------------=

u u T+ 23--- u I–

n 0=

u u T+ n 0=

u u T+ 23--- u I–

n 0,= p p0=

u u T+ n 0,= p p0=


This boundary condition physically corresponds to flow entering from a large container. It is numerically stable and admits total control of the pressure level along the entire boundary; however, it can give artifacts on the boundary if the viscous stresses just downstream of the inlet are non-zero. In such situations there are two choices. Either move the boundary farther away to a location where the artifacts do not matter or use another stress type boundary condition present in the Boundary Stress feature.

While the Pressure, no viscous stress boundary condition is numerically more robust than the Normal stress condition (which also specifies the pressure), it is also theoretically over-constraining of the flow field (Ref. 4). This theoretical “flaw” is often ignored since it in most cases has no practical implication.

Normal Stress Boundary Condition

The total stress on the boundary is set equal to a stress vector of magnitude f0, oriented in the negative normal direction:


This implies that the total stress in the tangential direction is zero. This boundary condition implicitly sets a constraint on the pressure which for 2D flows is

(3-26)

If unn is small, Equation 3-26 states that pf0.

The Normal Stress condition is the mathematically correct version of the Pressure, No Viscous Stress condition (Ref. 4), but it is numerically less stable.

Inlet and Outlet for the node settings. Note that some modules have additional theory sections describing options available with that module.

pI– u u T+ 23--- u I–

+ n f0n–=

pI– u u T+ + n f0n–=

p 2unn---------- f0+=


128 | C H A P T E

Pressure Boundary Condition

For single-phase flow, a mathematically correct natural boundary condition for outlets is

(3-27)

This is a normal stress condition together with a no tangential stress condition. When Equation 3-27 can be supplemented with a tangential velocity condition

ut (3-28)

If so, the no tangential stress condition is overridden. An issue with Equation 3-27 is that it does not strongly enforce outflow on the boundary. If the prescribed pressure is too high, parts of the outlet can actually have inflow. This is not as much of an issue for the Navier-Stokes equations as it is an issue for scalar transport equations solved along with Navier-Stokes equations. Hence, when applying the Pressure boundary condition at an outlet you can further constrain the outflow. With the Suppress backflow option

(3-29)

the normal stress is adjusted to keep

(3-30)

Equation 3-29 effectively means that the prescribed pressure is p0 if un , but smaller at locations where un . This means that Equation 3-29 does not completely prevent backflow, but the backflow is substantially reduced

Vacuum Pump Boundary Condition

Vacuum pumps (devices) can be represented using lumped curves implemented as boundary conditions. These simplifications also imply some assumptions. In particular, it is assumed that a given boundary can only be either an inlet or an outlet. Such a boundary should not be a mix of inlets/outlets nor switch between them during a simulation.

Inlet, Outlet, Open Boundary, and No Viscous Stress for the individual node settings. Note that some modules have additional theory sections describing options available with that module.

pI– u u T+ + n p0n–=

pI– u u T+ + n p0n–=

p0 p0

u n 0


Manufacturers usually provide curves that describe the static pressure as a function of flow rate for a vacuum pump.

D E F I N I N G A D E V I C E A T A N O U T L E T

In this case (see Figure 3-4), the device’s inlet is the interior face situated between the blue (cube) and green (circle) domains while its outlet is an external boundary, here the circular boundary of the green domain. The lumped curve gives the flow rate as a function of the pressure difference between the interior face and the external boundary. This boundary condition implementation follows the Pressure Boundary Condition for outlets with the Suppress backflow option:

(3-31)

Here, V0 is the flow rate across the boundary and pvacuum pumpV0 is the static pressure function of flow rate for the vacuum pump. Equation 3-31 corresponds to the compressible formulation. For incompressible flows, the term 23u vanishes.

Also see Vacuum Pump for the node settings.

In 2D the thickness in the third direction, Dz, is used to define the flow rate. Vacuum pumps are modeled as rectangles in this case.

pI– u u T+ 23--- u I–+

n p0n–=

p0 pvacuum pump V0


130 | C H A P T E

Figure 3-4: A vacuum pump at the outlet. The arrow represents the flow direction, the green circle represents the vacuum pump (that should not be part of the model), and the blue cube represents the modeled domain with an outlet boundary condition described by a lumped curve for the attached vacuum pump.

Fan Defined on an Interior Boundary

In this case, the inlet and outlet of the device are both interior boundaries (see Figure 3-5). The boundaries are called dev_in and dev_out. The boundary conditions are described as follows:

• The inlet of the device is an outlet boundary condition for the modeled domain. For this outlet boundary condition, on dev_in, the value of the pressure variable is set to the sum of the mean value of the pressure on dev_out and the pressure drop across the device. The pressure drop is calculated from a lumped curve using the flow rate evaluated on dev_in.

• For the inlet boundary condition, on dev_out, the pressure value is set so that the flow rate is equal on dev_in and dev_out. An ODE is added to compute the pressure value.

In both cases, the boundary condition implementation follows the Pressure Boundary Condition for outlet or inlet with the Suppress backflow option.


Figure 3-5: A device between two boundaries. The red arrows represent the flow direction, the cylindrical part represents the device (that should be not be part of the model), and the two cubes are the domain that are modeled with a particular inlet boundary condition to account for the device.


Fans, pumps, or grilles (devices) can be represented using lumped curves implemented as boundary conditions. These simplifications also imply some assumptions. In particular, it is assumed that a given boundary can only be either an inlet or an outlet. Such a boundary should not be a mix of inlets/outlets, nor switch between them during a simulation.

Manufacturers usually provide curves that describe the static pressure as a function of flow rate for a fan.

D E F I N I N G A D E V I C E A T A N I N L E T

In this case, the device’s inlet is an external boundary, represented by the external circular boundary of the green domain on Figure 3-6. The device’s outlet is an interior face

See Interior Fan for node settings.

See Fan and Grille for node settings.


132 | C H A P T E

situated between the green and blue domains in Figure 3-6. The lumped curve gives the flow rate as a function of the pressure difference between the external boundary and the interior face. This boundary condition implementation follows the Pressure Boundary Condition for inlets with the Suppress backflow option:

(3-32)

The Grille boundary condition sets the following conditions:

(3-33)

Here, V0 is the flow rate across the boundary, pinput is the pressure at the device’s inlet, and pfanV0 and pgrilleV0 are the static pressure functions of flow rate for the fan and the grille. Equation 3-32 and Equation 3-33 correspond to the compressible formulation. For incompressible flows, the term 23u vanishes.

Figure 3-6: A device at the inlet. The arrow represents the flow direction, the green circle represents the device (that should not be part of the model), and the blue cube represents the

In 2D the thickness in the third direction, Dz, is used to define the flow rate. Fans are modeled as rectangles in this case.

pI– u u T+ 23--- u I–+

n p0n–=

p0 pinput pfan V0 +

pI– u u T+ 23--- u I–+

n p0n–=

p0 pinput pgrille V0 +


modeled domain with an inlet boundary condition described by a lumped curve for the attached device.

D E F I N I N G A D E V I C E A T A N O U T L E T

In this case (see Figure 3-7), the fan’s inlet is the interior face situated between the blue (cube) and green (circle) domain while its outlet is an external boundary, here the circular boundary of the green domain. The lumped curve gives the flow rate as a function of the pressure difference between the interior face and the external boundary. This boundary condition implementation follows the Pressure Boundary Condition for outlets with the Suppress backflow option:

(3-34)

The Grille boundary condition sets the following conditions:

(3-35)

Here, V0 is the flow rate across the boundary, pexit is the pressure at the device outlet, and pfanV0 and pgrilleV0 are the static pressure function of flow rate for the fan and the grille. Equation 3-34 and Equation 3-35 correspond to the compressible formulation. For incompressible flows, the term 23u vanishes.

In 2D the thickness in the third direction, Dz, is used to define the flow rate. Fans are modeled as rectangles in this case.

pI– u u T+ 23--- u I–+

n p0n–=

p0 pexit pfan V0 –

pI– u u T+ 23--- u I–+

n p0n–=

p0 pexit pgrille V0 –


134 | C H A P T E

Figure 3-7: A fan at the outlet. The arrow represents the flow direction, the green circle represents the fan (that should not be part of the model), and the blue cube represents the modeled domain with an outlet boundary condition described by a lumped curve for the attached fan.

Screen Boundary Condition

The word “screen” refers to a barrier with distributed perforations such as a wire gauze, grille, or perforated plate. The screen is assumed to have a width, which is small compared to the resolved length-scales of the flow field and can thus be modeled as an edge (in 2D) or surface (in 3D). This idea permits an economic implementation of the screen, where the details of the barrier need not be resolved. The general influence of a screen on the flow field is a loss in the normal momentum component, a change in direction (related to a suppression of the tangential velocity component), attenuation of the turbulence kinetic energy and preservation of the turbulence length scale (Ref. 16). The conditions across the screen are expressed as,

(3-36)

(3-37)

(3-38)

(3-39)

u n -+ 0=

u n 2 p nT T+ u u T 23--- u I–+

23---kI–

n–+-

+

K2----- u- n 2–=

n u + n u - =

k+ 2k-=


and, depending on the turbulence model in use, either,

(3-40)

or,

(3-41)

and refer to the upstream and downstream side of the screen, whereas K and are the screen resistance and refraction coefficients. The attenuation of the turbulence kinetic energy (Equation 3-39) is based on the suppression of the tangential velocity (Equation 3-38) and the changes in and are determined by the assumption of preservation of the turbulence length-scale across the screen.

The Screen feature provides three commonly used correlations for K (Ref. 17). The following correlation is valid for wire gauzes

(3-42)

Here sis the solidity (ratio of blocked area to total area of the screen) and d is the diameter of the wires. For a square mesh, the following correlation is applied,

(3-43)

and for a perforated plate,

(3-44)

The following correlation for wire gauzes (Ref. 18) gives reasonable values for for a wide range of applications and has been included in the implementation,

(3-45)

See Screen for the node settings. Also see Theory for the Non-Isothermal Screen Boundary Condition for the non-isothermal version of these physics interfaces.

+ 3-=

+ -=

K 0.52 0.66/Red4 3/+ 1 s– 2– 1– Red u d = =

K 0.98 1 s– 2– 1– 1.09=

K 0.94 1 s– 2– 1– 1.28=

K2

16------- 1+

K4----–=


136 | C H A P T E

Mass Sources for Fluid Flow

There are two types of mass sources in the Single-Phase Flow interface: point sources and line sources.

PO I N T S O U R C E

A point source is theoretically formed by taking a mass injection/ejection, (SI unit: kg/(m3·s)), in a small volume V and then letting the size of the volume tend to zero while keeping the total mass flux constant. Given a point source strength, (SI unit: kg/s), this can be expressed as

(3-46)

An alternatively way to form a point source is to assume that mass is injected/extracted through the surface of a small object. Letting the object surface area tend to zero while keeping the mass flux constant, results in the same point source. For this alternative approach, effects resulting from the physical object volume, such as drag and fluid displacement, need to be neglected.

The weak contribution

is added to a point in the geometry. As can be seen from Equation 3-46, must tend to plus or minus infinity as V tends to zero. This means that in theory the pressure also tends to plus or minus infinity.

Observe that “point” refers to the physical representation of the source. A point source can therefore only be added to points in 3D models and to points on the symmetry axis in 2D axisymmetry models. Other geometrical points in 2D models represent physical lines.

The finite element representation of Equation 3-46 corresponds to a finite pressure in a point with the effect of the point source spread out over a region around the point. The size of the region depends on the mesh and on the strength of the source. A finer mesh

These features require at least one of the following licenses: Batteries and Fuel Cells Module, CFD Module, Chemical Reaction Engineering Module, Corrosion Module, Electrochemistry Module, Electrodeposition Module, Microfluidics Module, Pipe Flow Module, or Subsurface Flow Module.

Q·

q·p

V 0lim Q

·

V q·p=

q·ptest p

Q·


gives a smaller affected region, but also a more extreme pressure value. It is important not to mesh too finely around a point source since the resulting pressure can result in unphysical values for the density, for example. It can also have a negative effect on the condition number for the equation system.

L I N E S O U R C E

A line source can theoretically be formed by assuming a source of strength (SI unit: kg/(m3·s)), located within a tube with cross-section area S and then letting S tend to zero while keeping the total mass flux per unit length constant. Given a line source strength, (SI unit: kg/(m·s)), this can be expressed as

(3-47)

As in the point source case, an alternative approach is to assume that mass is injected/extracted through the surface of a small object. This results in the same mass source, but requires that effects on the fluid resulting from the physical object volume are neglected.


is added to lines in 3D or to points in 2D (which represent cut-through views of lines). Line sources can also be added to the axisymmetry line in 2D axisymmetry models. It can not, however, be added to geometrical lines in 2D since those represent physical planes.

As with a point source, it is important not to mesh too finely around the line source.

For feature node information, see Line Mass Source and Point Mass Source in the COMSOL Multiphysics Reference Manual.

Q·

q· l

S 0lim Q

·

S q· l=

q· ltest p


138 | C H A P T E

Numerical Stability—Stabilization Techniques for Fluid Flow

The momentum equation (Equation 3-15 or Equation 3-17) is a (nonlinear) convection-diffusion equation. Such equations can easily become unstable if discretized using the Galerkin finite element method. Stabilized finite element methods are usually necessary in order to obtain physical solutions. The stabilization settings are found in the main fluid-flow features. To display this section, click the Show button ( ) and select Stabilization.

There are three types of stabilization methods available for Navier-Stokes—streamline diffusion, crosswind diffusion, and isotropic diffusion. Streamline diffusion and crosswind diffusion are consistent stabilization methods, whereas isotropic diffusion is an inconsistent stabilization method.

For optimal functionality, the exact weak formulations of and constants in the streamline diffusion and crosswind diffusion methods depend on the order of the shape functions (basis functions) for the elements. The values of constants in the streamline diffusion and crosswind diffusion methods follow Ref. 5 and Ref. 6.

S T R E A M L I N E D I F F U S I O N

For strongly coupled systems of equations, the streamline diffusion method must be applied to the system as a whole rather than to each equation separately. These ideas were first explored by Hughes and Mallet (Ref. 7) and were later extended to Galerkin least-squares (GLS) applied to the Navier-Stokes equations (Ref. 8). This is the streamline

For the Reacting Flow in Porous Media, Diluted Species interface, which is available with the CFD Module, Chemical Reaction Engineering Module, or Batteries & Fuel Cells Module, these shared physics nodes are renamed as follows:

• The Line Mass Source node is available as two nodes, one for the fluid flow (Fluid Line Source) and one for the species (Species Line Source).

• The Point Mass Source node is available as two nodes, one for the fluid flow (Fluid Point Source) and one for the species (Species Point Source).


• Numerical Stabilization

• Iterative


diffusion formulation that COMSOL Multiphysics supports. The time-scale tensor is the diagonal tensor presented in Ref. 9.

Streamline diffusion is active by default because it is necessary when convection is dominating the flow.

The governing equations for incompressible flow are subject to the Babuska-Brezzi condition, which states that the shape functions (basis functions) for pressure must be of lower order than the shape functions for velocity. If the incompressible Navier-Stokes equations are stabilized by streamline diffusion, it is possible to use equal-order interpolation. Hence, streamline diffusion is necessary when using first-order elements for both velocity and pressure. This applies also if the model is solved using geometric multigrid (either as a solver or as a preconditioner) and at least one multigrid hierarchy level uses linear Lagrange elements.

C R O S S W I N D D I F F U S I O N

Crosswind diffusion can also be formulated for systems of equations, and when applied to the Navier-Stokes equations it becomes a shock-capturing operator. COMSOL Multiphysics supports the formulation in Ref. 8 with a shock capturing viscosity of the Hughes-Mallet type Ref. 7.

Incompressible flows do not contain shock waves, but crosswind diffusion is still useful for introducing extra diffusion in sharp boundary layers and shear layers that otherwise would require a very fine mesh to resolve.

Crosswind diffusion is active by default as it makes it easier to obtain a solution even if the problem is fully resolved by the mesh.

I S O T R O P I C D I F F U S I O N

Isotropic diffusion adds diffusion to the Navier-Stokes equations. Isotropic diffusion significantly reduces the accuracy of the solution but does a very good job at reducing oscillations. The stability of the continuity equation is not improved.

Crosswind diffusion also enables the iterative solvers to use inexpensive presmoothers. If crosswind diffusion is deactivated, more expensive preconditioners must be used instead.


140 | C H A P T E

Solvers for Laminar Flow

The Navier-Stokes equations constitute a nonlinear equation system. A nonlinear solver must hence be applied to solve the problem. The nonlinear solver iterates to reach the final solution. In each iteration, a linearized version of the nonlinear system is solved using a linear solver. In the time-dependent case, a time marching method must also be applied. The default suggestions for each of these solver elements are discussed below.

N O N L I N E A R S O L V E R

The nonlinear solver depends if the model solves a stationary or a time-dependent problem.

Stationary SolverIn the stationary case, a fully coupled, damped Newton method is applied. The initial damping factor is low since a full Newton step can be harmful unless the initial values are close to the final solution. The nonlinear solver algorithm automatically regulates the damping factor in order to reach a converged solution.

For advanced models, the automatically damped Newton method might not be robust enough. A pseudo time-stepping algorithm can then be invoked. See Pseudo Time Stepping for Laminar Flow Models.

Time-Dependent SolverIn the time-dependent case, the initial guess for each time step is (loosely speaking) the previous time step, which is a very good initial value for the nonlinear solver. The automatic damping algorithm is then not necessary. The damping factor in the Newton method is instead set to a constant value slightly smaller than one. Also for the same reason, it suffices to update the Jacobian once per time-step.

It is seldom worth the extra computational cost to update the Jacobian more than once per time step. For most models it is more efficient to restrict the maximum time step or possibly lower the damping factor in the Newton method.

L I N E A R S O L V E R

The linearized Navier-Stokes equation system has saddle point character, unless the density depends on the pressure. This means that the Jacobian matrix has zeros on the diagonal.

Stationary Solver in the COMSOL Multiphysics Reference Manual


Even when the density depends on the pressure the equation system effectively shares many numerical properties with a saddle point system.

For small 2D models and 3D models, the default solver suggestion is a direct solver. Direct solvers can handle most non-singular systems and are very robust and also very fast for small models. Unfortunately, they become slow for large models and their memory requirement scales as somewhere between N1.5and N2 where N is the number of degrees of freedom in the model. The default suggestion for large 3D models is therefore the iterative GMRES solver. The memory requirement for an iterative solver optimally scales as N.

Geometric Multigrid (GMG) is used to accelerate GMRES. GMG needs smoothers but the saddle point character of the linear system restricts the number of applicable smothers. The choices are further restricted by the anisotropic meshes frequently encountered in fluid-flow problems. Pointwise smoothers, such as SOR, are not very efficient on anisotropic meshes.

The efficiency of the smoothers is highly dependent on the numerical stabilization. Iterative solvers perform at their best when both Streamline Diffusion and Crosswind Diffusion are active.

The default smoother for P1+P1 elements is SCGS. This is an efficient and robust smoother specially designed to solve saddle point systems on meshes that contain anisotropic elements. The SCGS smoother works well even without crosswind diffusion. SCGS can sometimes work for higher order elements, especially if Method in the SCGS settings is set to Mesh element lines. But there is no guarantee for this, so the default smoother for P2+P1 elements and P3+P2 elements is an SOR Line smoother. SOR Line handles mesh anisotropy, but does not formally address the saddle point character. It does however function in practice provided that streamline diffusion and crosswind diffusion are both active.

A different kind of saddle point character can arise if the equation system contains ODE variables. Some advanced boundary conditions, for example Laminar Inflow, can add equations with such variables. These variables must be treated with the Vanka algorithm. SCGS includes an option to invoke Vanka. Models with higher order elements must either


142 | C H A P T E

apply SCGS, or use the Vanka smoother. The latter is the default suggestion for higher order elements, but it does not work optimally for anisotropic meshes.

T I M E - D E P E N D E N T S O L V E R

The default time-dependent solver for Navier-Stokes is the BDF method with maximum order set to two. Higher BDF orders are not stable for transport problems in general nor for Navier-Stokes in particular.

BDF methods have been used for a long time and are known for their stability. However, they can have severe damping effects, especially the lower-order methods. Hence, if robustness is not an issue, a model can benefit from using the generalized- method instead. Generalized- is a solver which has properties similar to those of the second-order BDF solver but it is much less diffusive.

Both BDF and generalized- are per default set to automatically adjust the time step. While this works well for many models, extra efficiency and accuracy can often be gained by specifying a maximum time step. It is also often beneficial to specify an initial time step to make the solver progress smoothly in the beginning of the time series.

Pseudo Time Stepping for Laminar Flow Models

A stationary formulation has per definition no time derivatives and Equation 3-17 reduces to:

(3-48)


• Multigrid

• Direct

• Iterative

• SCGS

• SOR Line

• Vanka

Time-Dependent Solver in the COMSOL Multiphysics Reference Manual

u u pI– u u T+ + F+=


Solving Equation 3-48 requires a starting guess that is close enough to the final solution. If no such guess is at hand, the fully transient problem can be solved instead. This is, however, a rather costly approach in terms of computational time. An intermediate approach is to add a fictitious time derivative to Equation 3-48:

where is a pseudo time step. Since unojac(u) is always zero, this term does not affect the final solution. It does, however, affect the discrete equation system and effectively transforms a nonlinear iteration into a step of size of a time-dependent solver.

Pseudo time stepping is not active per default. The pseudo time step can be chosen individually for each element based on the local CFL number:

where h is the mesh cell size. A small CFL number means a small time step. It is practical to start with a small CFL number and gradually increase it as the solution approaches steady state.

If the automatic expression for CFLloc is set to the built-in variable CFLCMP. The automatic setting then suggests a PID regulator for the pseudo time step in the default solver. The PID regulator starts with a small CFL number and increases CFLloc as the solution comes closer to convergence.

The default manual expression is

(3-49)

The variable niterCMP is the nonlinear iteration number. It is equal to one for the first nonlinear iteration. CFLloc starts at 1.3 and increases by 30% each iteration until it reaches

. It remains there until iteration number 20 at which it starts to increase until it reaches approximately 106. A final increase after iteration number 40 then takes it to

For details about the CFL regulator, see Pseudo Time Stepping in the COMSOL Multiphysics Reference Manual.

u nojac u –

t--------------------------------- + u u pI– u u T+ + F+=

t

t

t

t CFLlochu-------=

1.3min niterCMP 9 +

if niterCMP 20 9 1.3min niterCMP 20– 9 0 +

if niterCMP 40 90 1.3min niterCMP 40– 9 0

1.39 10.6


144 | C H A P T E

1060. Equation 3-49 can for some advanced flows increase CFLloc too slowly or too quickly. CFLloc can then be tuned for the specific application.

The Projection Method for the Navier-Stokes Equations

A well-known approach to solve the Navier-Stokes equations is the pressure-correction method. This type of method is a so-called segregated method, and it generally requires far less memory than the COMSOL Multiphysics default formulation. Several versions of the original method have been developed (see Ref. 12, for example). COMSOL uses incremental pressure-correction schemes.

This method reformulates the Navier-Stokes equations so that it is possible to solve for one variable at a time in sequence. Let u and p be the velocity and pressure variables and uc and pc the corrected velocity and pressure variables, respectively. The pressure-correction algorithm solves the Navier-Stokes equations using the following steps:

1 Solve in sequence for all u components following equation:

where the superscript denotes the time-step index, and is discretized using a BDF method up to second order where the u values from previous time steps are replaced by uc values. To first order it is discretized as:

2 Solve Poisson’s equation to adjust the pressure:

(3-50)

3 Update the corrected velocity:

This formulation is only available for time-dependent problems and requires the time discrete solver. It is available for the Laminar Flow and Turbulent Flow, k-e interfaces.

ut------- uc

n un 1++ pn– un 1+ un 1+ T+ 2

3--- un 1+ I–

F++=

ut-------

un 1+ ucn

–

timestep---------------------------

timestep pn 1+ pn–

t------ un 1+–=


Due to the specific time discretization scheme, this algorithm is only available with the time discrete solver.

Because the velocity components and the pressure are solved in a segregated way, some boundary conditions have a different implementation or might not be available with the projection method. In such cases, this is mentioned in the documentation for each boundary condition.

When the projection method is used for turbulent flows or with multiphysics couplings, the same algorithm is used for the velocity and pressure variables. Extra steps are needed to solve the other variables. By default the equation form used for these variables is the time-dependent form, and the time derivative is automatically discretized using a second-order BDF method.

Discontinuous Galerkin Formulation

Some boundary conditions are implemented using a discontinuous Galerkin formulation. These boundary conditions include

• Wall – Slip


• Flow Continuity

The formulation used in the fluid-flow interfaces in COMSOL Multiphysics is the Symmetric Interior Penalty Galerkin method (SIPG). The SIPG method can be regarded to satisfy the boundary conditions in an integral sense rather than pointwise. More information on SIPG can be found in Ref. 13.

In particular, the SIPG formulation includes a penalty parameter that must be large enough for the formulation to be coercive. The higher the value, the better the boundary

For incompressible flows, the

term in Equation 3-49 and the term in Equation 3-50 are excluded.

ucn 1+ un 1+ timestep

--------------------- pn 1+ pn– –=

23--- un 1+ I

t------


146 | C H A P T E

condition is fulfilled, but a too high value results in an ill-conditioned equation system. The penalty parameter in COMSOL is implemented according to Ref. 14.

Particle Tracing in Fluid Flow

It is possible to model particle tracing with COMSOL Multiphysics provided that the impact of the particles on the flow field is negligible. First compute the flow field, and then, as an analysis step, calculate the motion of the particles. The motion of a particle is defined by Newton’s second law

where x is the position of the particle, m the particle mass, and F is the sum of all forces acting on the particle. Examples of forces acting on a particle in a fluid are the drag force, the buoyancy force, and the gravity force. The drag force represents the force that a fluid exerts on a particle due to a difference in velocity between the fluid and the particle. It includes the viscous drag, the added mass, and the Basset history term. Several empirical expression have been suggested for the drag force. One of those is the one proposed by Khan and Richardson (Ref. 11). That expression is valid for spherical particles for a wide range of particle Reynolds numbers. The particle Reynolds number is defined as

where u is the velocity of the fluid, up the particle velocity, r the particle radius, the fluid density, and the dynamic viscosity of the fluid. The empirical expression for the drag force according to Khan and Richardson is

The Particle Tracing Module is available to assist with these types of modeling problems.

The model Flow Past a Cylinder (model library path COMSOL_Multiphysics/Fluid_Dynamics/cylinder_flow) demonstrates how to add and set up particle tracing in a plot group using the Particle Tracing

with Mass node. It uses the predefined Khan-Richardson model for the drag force and neglects gravity and buoyancy forces.

mt2

2

dd x F t x

tddx

=

Repu up– 2r

------------------------------=


References for the Single-Phase Flow, Laminar Flow Interfaces

1. G.G. Stokes, Trans. Camb. Phil. Soc., 8, 287-305, 1845

2. P.M. Gresho and R.L. Sani, Incompressible Flow and the Finite Element Method, Volume 2: Isothermal Laminar Flow, John Wiley & Sons, 2000.

3. G.K. Batchelor, An Introduction To Fluid Dynamics, Cambridge University Press, 1967.

4. R.L. Panton, Incompressible Flow, 2nd ed., John Wiley & Sons, 1996.

5. I. Harari and T.J.R. Hughes, “What are C and h? Inequalities for the Analysis and Design of Finite Element Methods,” Comp. Meth. Appl. Mech. Engrg, vol. 97, pp. 157–192, 1992.

6. Y. Bazilevs, V.M. Calo, T.E. Tezduyar, and T.J.R. Hughes, “YZ Discontinuity Capturing for Advection-dominated Processes with Application to Arterial Drug Delivery,” Int.J.Num. Meth. Fluids, vol. 54, pp. 593–608, 2007.

7. T.J.R. Hughes and M. Mallet, “A New Finite Element Formulation for Computational Fluid Dynamics: III. The Generalized Streamline Operator for Multidimensional Advective-Diffusive System,” Comp. Meth. Appl. Mech. Engrg, vol. 58, pp. 305–328, 1986.

8. G. Hauke and T.J.R. Hughes, “A Unified Approach to Compressible and Incompressible Flows,” Comp. Meth. Appl. Mech. Engrg, vol. 113, pp. 389–395, 1994.

9. G. Hauke, “Simple Stabilizing Matrices for the Computation of Compressible Flows in Primitive Variables,” Comp. Meth. Appl. Mech. Engrg, vol. 190, pp. 6881–6893, 2001.

10. D.J. Tritton, Physical Fluid Dynamics, 2nd ed., Oxford University Press, 1988.

11. J.M. Coulson and J.F. Richardson, “Particle Technology and Separation Processes,” Chemical Engineering, Volume 2, Butterworth-Heinemann, 2002.

12. J.L. Guermond, P. Minev, and J. Shen, “An overview of projection methods for incompressible flows,” Comp. Meth. Appl. Mech. Engrg, vol. 195, pp. 6011–6045, 2006.

13. B. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations, SIAM, 2008.

F r2 u up– u up– 1.84Rep-0.31 0.293Rep

0.06+ 3.45

=


148 | C H A P T E

14. Y. Epshteyn and B. Rivière, “Estimation of penalty parameters for symmetric interior penalty Galerkin methods,” J. Computational and Applied Mathematics, vol. 206, pp. 843–872, 2007.

15. G. Kariadakis, A. Beskok, and N. Aluru, Microflows and Nanoflows, Springer Science and Business Media, 2005.

16. G.B. Schubauer, W.G. Spangenberg, and P.S. Klebanoff, “Aerodynamic Characteristics of Damping Screens,” NACA Technical note 2001, Jan. 1950.

17. P.E. Roach, “The Generation of Nearly Isotropic Turbulence by Means of Grids,” Int. J. Heat and Fluid Flow, vol. 8, pp. 82–92, 1986.

18. J.C. Gibbings, “The Pyramid Gauze Diffuser,” Ing. Arch., vol. 42, pp. 225–233, 1973.

19. R.P. Chhabra and J.F. Richardson, Non-Newtonian Flow and Applied Rheology, 2nd ed., Elsivier, 2008.


Th eo r y f o r t h e Tu r bu l e n t F l ow I n t e r f a c e s

The theory for the Single-Phase Flow, Turbulent Flow interfaces is described in this section:

• Turbulence Modeling

• The k-Turbulence Model

• The k- Turbulence Model

• The SST Turbulence Model

• The Low Reynolds Number k- Turbulence Model

• The Spalart-Allmaras Turbulence Model

• Inlet Values for the Turbulence Length Scale and Turbulent Intensity

• Theory for the Pressure, No Viscous Stress Boundary Condition

• Solvers for Turbulent Flow

• Pseudo Time Stepping for Turbulent Flow Models

• References for the Single-Phase Flow, Turbulent Flow Interfaces

• References for the Single-Phase Flow, Turbulent Flow Interfaces

Turbulence Modeling

Turbulence is a property of the flow field and it is characterized by a wide range of flow scales: the largest occurring scales, which depend on the geometry, the smallest, quickly fluctuating scales, and all the scales in between. The propensity for an isothermal flow to become turbulent is measured by the Reynolds number

(3-51)

Theory for the Single-Phase Flow Interfaces

Re UL

------------=

T H E O R Y F O R T H E TU R B U L E N T F L O W I N T E R F A C E S | 149

150 | C H A P T E

where is the dynamic viscosity, the density, and U and L are velocity and length scales of the flow, respectively. Flows with high Reynolds numbers tend to become turbulent. Most engineering applications belong to this category of flows.

The Navier-Stokes equations can be used for turbulent flow simulations, although this would require a large number of elements in order to capture the wide range of scales in the flow. An alternative approach is to divide the flow into large, resolved scales and small, unresolved scales. The small scales are then modeled using a turbulence model with the goal that this model is numerically less expensive than resolving all present scales. Different turbulence models invoke different assumptions on the unresolved scales resulting in different degrees of accuracy for different flow cases.

This module includes Reynolds-averaged Navier-Stokes (RANS) models which is the model type most commonly used in industrial flow applications.

R E Y N O L D S - A V E R A G E D N A V I E R - S T O K E S ( R A N S ) E Q U A T I O N S

The following assumes that the flowing fluid is incompressible and Newtonian in which case the Navier-Stokes equations take the form:

(3-52)

Once the flow has become turbulent, all quantities fluctuate in time and space. It is seldom worth the extreme computational cost to obtain detailed information about the fluctuations. An averaged representation often provides sufficient information about the flow.

The Reynolds-averaged representation of turbulent flows divides the flow quantities into an averaged value and a fluctuating part,

where can represent any scalar quantity of the flow. In general, the mean value can vary in space and time. This is exemplified in Figure 3-8, which shows time averaging of one component of the velocity vector for nonstationary turbulence. The unfiltered flow has a time scale t1. After a time filter with width t2 t1 has been applied, there is a fluctuating part, ui, and an average part, Ui. Because the flow field also varies on a time

t

u u u+ pI– u u T+ + F+=

u 0=

+=


scale longer than t2, Ui is still time-dependent but is much smoother than the unfiltered velocity ui.

Figure 3-8: The unfiltered velocity component ui, with a time scale t1, and the averaged velocity component, Ui, with time scale t2.

Decomposition of the flow field into an averaged part and a fluctuating part, followed by insertion into the Navier-Stokes equation, and averaging, gives the Reynolds-averaged Navier-Stokes (RANS) equations:

(3-53)

where U is the averaged velocity field and is the outer vector product. A comparison with Equation 3-52 indicates that the only difference is the appearance of the last term on the left-hand side of Equation 3-53. This term represents the interaction between the fluctuating parts of the velocity field and is called the Reynolds stress tensor. This means that to obtain the mean flow characteristics, information about the small-scale structure of the flow is needed. In this case, that information is the correlation between fluctuations in all three directions.

E D D Y V I S C O S I T Y

The most common way to model turbulence is to assume that the turbulence is of a purely diffusive nature. The deviating part of the Reynolds stress is then expressed as

t

U U U u' u' + + P U U T+ F+ +–=

U 0=

u' u' 3---trace u' u' I– T– U U T+ =


152 | C H A P T E

where T is the eddy viscosity, also known as the turbulent viscosity. The spherical part can be written

where k is the turbulent kinetic energy. In simulations of incompressible flows, this term is included in the pressure, but when the absolute pressure level is of importance (in compressible flows, for example) this term must be explicitly included.

TU R B U L E N T C O M P R E S S I B L E F L OW

If the Reynolds average is applied to the compressible form of the Navier-Stokes equations, terms of the form

appear and need to be modeled. To avoid this, a density-based average, known as the Favre average, is introduced:

(3-54)

It follows from Equation 3-54 that

(3-55)

and a variable, ui, is decomposed into a mass-averaged component, , and a fluctuating component, ui, according to

(3-56)

Using Equation 3-55 and Equation 3-56 along with some modeling assumptions for compressible flows (Ref. 7), Equation 3-15 and Equation 3-16 can be written on the form

(3-57)

The Favre-averaged Reynolds stress tensor is modeled using the same argument as for incompressible flows:

3---trace u' u' I 2

3---k=

u

ui˜ 1

--- 1

T----

T lim x ( , )ui x ( , ) d

t

t T+

=

ui ui=

ui

ui ui ui+=

t------

xi------- ui + 0=

uit

-------- ujuixj

--------+pxi

-------–xj

------- uixj

--------ujxi

--------+ 2

3---

ukxk

---------ij– ujui–

Fi++=


where k is the turbulent kinetic energy. Comparing Equation 3-57 to its incompressible counterpart (Equation 3-53), it can be seen that except for the term

the compressible and incompressible formulations are exactly the same, except that the free variables are instead of

More information about modeling turbulent compressible flows can be found in Ref. 1 and Ref. 7.

The turbulent transport equations are used in their fully compressible formulations (Ref. 8).

The k-Turbulence Model

The k- model is one of the most used turbulence models for industrial applications. This module includes the standard k- model (Ref. 1). The model introduces two additional transport equations and two dependent variables: the turbulent kinetic energy, k, and the turbulent dissipation rate, . The turbulent viscosity is modeled as

(3-58)

where C is a model constant.

The transport equation for k reads:

(3-59)

where the production term is

(3-60)

The transport equation for reads:

ujui– Tuixj

--------ujxi

--------+ 2

3--- T

ukxk

--------- k+

ij–=

2 3 kij–

ui

Ui ui=

T Ck2

------=

kt

------ + u k Tk------+

k Pk –+=

Pk T u: u u T+ 23--- u 2–

23---k u–=


154 | C H A P T E

(3-61)

The model constants in Equation 3-58, Equation 3-59, and Equation 3-61 are determined from experimental data (Ref. 1) and the values are listed in Table 3-3.


Equation 3-59 and Equation 3-61 cannot be implemented directly as written. There is, for example, nothing that prevents division by zero. The equations are instead implemented as suggested in Ref. 9. The implementation includes an upper limit on the mixing length,

:

(3-62)

The mixing length is used to calculated the turbulent viscosity. should not be active in a converged solution but is merely a tool to obtain convergence.

R E A L I Z A B I L I T Y C O N S T R A I N T S

The eddy-viscosity model of the Reynolds stress tensor can be written

where ij is the Kronecker delta and Sij is the strain-rate tensor. The diagonal elements of the Reynolds stress tensor must be nonnegative, but calculating T from Equation 3-58 does not guarantee this. To assert that

the turbulent viscosity is subjected to a realizability constraint. The constraint for 2D and 2D axisymmetry without swirl is:

TABLE 3-3: MODEL CONSTANTS

CONSTANT VALUE

C 0.09

C1 1.44

C2 1.92

k 1.0

1.3

t

----- + u T------+

C1

k---P

kC2

2

k-----–+=

lmixlim

lmix max Ck3 2/

----------- lmix

lim =

lmixlim

uiuj 2TSij–23---kij+=

uiui 0 i


(3-63)

and for 3D and 2D axisymmetry with swirl flow it reads:

(3-64)

Combining equation Equation 3-63 with Equation 3-58 and the definition of the mixing length gives a limit on the mixing length scale:

(3-65)

Equivalently, combining Equation 3-64 with Equation 3-58 and Equation 3-62 gives:

(3-66)

This means there are two limitations on lmix: the realizability constraint and the imposed limit via Equation 3-62.

The effect of not applying a realizability constraint is typically excessive turbulence production. The effect is most clearly visible at stagnation points. To avoid such artifacts, the realizability constraint is always applied for the RANS models. More details can be found in Ref. 4, Ref. 5, and Ref. 6.

M O D E L L I M I T A T I O N S

The k- turbulence model relies on several assumptions, the most important of which is that the Reynolds number is high enough. It is also important that the turbulence is in equilibrium in boundary layers, which means that production equals dissipation. These assumptions limit the accuracy of the model because they are not always true. It does not, for example, respond correctly to flows with adverse pressure gradients and can result in under-prediction of the spatial extent of recirculation zones (Ref. 1). Furthermore, in simulations of rotating flows, the model often shows poor agreement with experimental data (Ref. 2). In most cases, the limited accuracy is a fair trade-off for the amount of computational resources saved compared to using more complicated turbulence models.

WA L L F U N C T I O N S

The flow close to a solid wall is for a turbulent flow very different from the free stream. This means that the assumptions used to derive the k- model are not valid close to walls.

Tk 2

3 SijSij

-----------------------

Tk

6 SijSij

---------------------------

lmix23---

kSijSij

-------------------

lmix16

------- kSijSij

-------------------


156 | C H A P T E

While it is possible to modify the k- model so that it describes the flow in wall regions (see The Low Reynolds Number k- Turbulence Model), this is not always desirable because of the very high resolution requirements that follow. Instead, analytical expressions are used to describe the flow near the walls. These expressions are known as wall functions.

The wall functions in COMSOL Multiphysics are such that the computational domain is assumed to be located a distance w from the wall (see Figure 3-9).

Figure 3-9: The computational domain is located a distance w from the wall for wall functions.

The distance w is automatically computed so that

where uC1/4k is the friction velocity, is 11.06. This corresponds to the distance from

the wall where the logarithmic layer meets the viscous sublayer (or to some extent would meet it if there were no buffer layer in between). w is limited from below so that it never becomes smaller than half the height of the boundary mesh cell. This means that can become larger than 11.06 if the mesh is relatively coarse.

Always investigate the solution to check that w is small compared to the dimensions of the geometry. Also check that is 11.06 on most of the walls.

If is much larger than 11.06 over a significant part of the walls, the accuracy might become compromised. Both the wall lift-off, w, and the wall lift-off in viscous units, , are available as results and analysis variables.

Solid wall

w

Mesh cells

w+ uw =

w+

w+

w+

w+


The boundary conditions for the velocity is a no-penetration condition u n = 0 and a shear stress condition

where

is the viscous stress tensor and

where in turn, v, is the von Kárman constant (default value 0.41) and B is a constant that by default is set to 5.2.

The turbulent kinetic energy is subject to a homogeneous Neumann condition n k = 0 and the boundary condition for reads:

See Ref. 9 and Ref. 10 for further details.

WA L L F U N C T I O N S F O R R O U G H WA L L S

The physics interfaces: Wall, Interior Wall, Rotating Wall, and Rotating Interior Wall have an option to apply wall roughness by modifying the wall functions. Cebeci (Ref. 18) suggested a model which adjusts the friction velocity for surface roughness,

(3-67)

where

n n n n– uuu-------max C

1 4/ k u –=

u u T+ =

uu

1v----- w

+ln B+-----------------------------=

C

3 4/ k3 2/

vw-----------------------=

uu

1v----- w

+ln B B–+-------------------------------------------=


158 | C H A P T E

is the roughness height in viscous units,

The roughness height, ks, is the peak-to-peak value of the surface variations and the wall is relocated to their mean level.

Figure 3-10: Definitions of the roughness height and the modified wall location.

Hence, when Equation 3-67 is used the lift-off is modified according to,

where h+ is the height of the boundary mesh cell in viscous units. Cs is a parameter that depends on the shape and distribution of the roughness elements. When the turbulence parameters and B have the values 0.41 and 5.2, respectively, and Cs0.26, ks corresponds to the equivalent sand roughness height, kseq, as introduced by Nikuradse (Ref. 19). A few characteristic values of the equivalent sand roughness height are given in Table 3-4 below,

TABLE 3-4: EQUIVALENT SAND ROUGHNESS HEIGHTS kseq

SURFACE TYPE EQUIVALENT SAND ROUGHNESS HEIGHT

Steel, new 50 m

Galvanized steel 0.13 mm

Riveted steel 0.9-9 mm

Cast iron, new 0.25 mm

Galvanized iron 0.15-0.30 mm

B

0 ks+ 2.25

1v-----

ks+ 2.25–

87.75------------------------ Csks

++ 0.4258 ks+ 0.811–ln sinln 2.25 ks

+ 90

1v----- 1 Csk+ s

+ ln ks+ 90

=

ks+

ks+

C1 4/ k

-----------------------ks=

w+ max 11.06, ks

+ 2 h+ 2 =


Use other values of the roughness parameter Cs and roughness height ks to specify generic surface roughnesses.


The default initial values for a stationary simulation are (Ref. 9),

where is the mixing length limit. For time-dependent simulations, the initial value for k is instead

S C A L I N G F O R T I M E - D E P E N D E N T S I M U L A T I O N S

The k- equations are derived under the assumption that the flow has a high enough Reynolds number. If this assumption is not fulfilled, both k and have very small magnitudes and behave chaotically in the manner that the relative values of k and can change by large amounts due to small changes in the flow field.

A time-dependent simulation of a turbulent flow can include a period when the flow is not fully turbulent. A typical example is the startup phase when for example an inlet velocity or a pressure difference is gradually increased. To sort out numerical fluctuations in k and

Cast iron, rusted 1.0-1.5 mm

Bituminized steel or iron, new 30-50 m

Glass 0.3 m

Drawn tubing 1.5 m

Wood, new 0.5 mm

Concrete, new 0.3-3 mm

TABLE 3-4: EQUIVALENT SAND ROUGHNESS HEIGHTS kseq

SURFACE TYPE EQUIVALENT SAND ROUGHNESS HEIGHT

u 0=

p 0=

k 10 0.1 lmix

lim ------------------------------- 2

=

Ckinit

3 2/

0.1 lmixlim

-----------------------=

lmixlim

k 0.1 lmix

lim ------------------------------- 2

=


160 | C H A P T E

during such periods, the default time-dependent solver for the k- model employs unscaled absolute tolerances for k and . The tolerances are set to

(3-68)

where Uscale and Lfact are input parameters available in the Advanced Settings section of the physics interface node. Their default values are 1 ms and 0.035 respectively. lbb,min is the shortest side of the geometry bounding box. Equation 3-68 is closely related to the expressions for k and on inlet boundaries (see Equation 3-91).

The practical implication of Equation 3-68 is that variations in k and smaller than kscale and scale respectively, are regarded as numerical noise.

The k- Turbulence Model

The k- model solves for the turbulent kinetic energy, k, and for the dissipation per unit turbulent kinetic energy, . is also commonly know as the specific dissipation rate. The CFD Module has the Wilcox revised k- model (Ref. 1)

(3-69)

where

(3-70)

kscale 0.01Uscale 2=

scale 0.09ksclae3 2/ Lfact l bb min =

kt

------ u k+ Pk *k *T+ k +–=

t

------- u + k----Pk 2 T+ +–=

T k----=

1325------= 0f= * 0

*f= 12---= * 1

2---=

013125----------= f

1 70+

1 80+-----------------------=

ijjkSki

0* 3

--------------------------=


(3-71)

where in turn ij is the mean rotation-rate tensor

and Sij is the mean strain-rate tensor

Pk is given by Equation 3-60. The following auxiliary relations for the dissipation, , and the turbulent mixing length, l, are also used:

(3-72)

M I X I N G L E N G T H L I M I T A N D R E A L I Z A B I L I T Y C O N S T R A I N T S

The implementation of the k- model relies on the same concepts as the k- model (Ref. 9 This means that the following approximations have been used:

where lr is the limit given by the realizability constraints (Equation 3-65 and Equation 3-66).

0* 9

100----------= f*

1 k 0

1 680k2+

1 400k2+

-------------------------- k 0

= k13------- k =

ij12---

uixj

--------ujxi

--------–

=

Sij12---

uixj

--------ujxi

--------+

=

*k= lmixk

-------=

k----

max T eps ---------------------------------

1---- k

k------------

lmix

k--------- lmix

lmixT

------------- lmix

2 max T eps ---------------------------------= = =

lmix min k

------- lmixlim lr

=


162 | C H A P T E

WA L L F U N C T I O N S

Wall boundaries are treated with the same type of boundary conditions as for the k- model (see Wall Functions with C replaced by and the boundary condition for given by

(3-73)


The default initial values are the same as for the k- model (see Initial Values) but with the initial value of given by


The k- model applies absolute scales of the same type as the k- model (see Scaling for Time-Dependent Simulations) except that the scale for is given by

M O D E L P R O P E R T I E S

The k- model can in many cases give results that are superior to those obtained with the k- model (Ref. 1). It behaves, for example, much better for flat plate flows with adverse or favorable pressure gradients. However, there are two main drawbacks. The first is that the k- model can display a relatively strong sensitivity to free stream inlet values of . The other is that the k- model is numerically less robust than the k- model.

The SST Turbulence Model

To combine the superior behavior of the k- model in the near-wall region with the robustness of the k- model, Menter (Ref. 15) introduced the SST (Shear Stress Transport) model which interpolates between the two. The version of the SST model in the CFD Module includes a few well-tested (Ref. 14, Ref. 16) modifications, such as production limiters for both k and , the use of S instead of in the limiter for T and a sharper cut-off for the cross-diffusion term.

0*

wk

w+

--------------=

initkinit

0.1 lmixlim

-----------------------=

scalekscale

Lfact l bb min---------------------------------=


It is also a low Reynolds number model, that is, it does not apply wall functions. “Low Reynolds number” refers to the region close to the wall where viscous effects dominate. The model equations are formulated in terms k and

(3-74)

where,

(3-75)

and Pk is given in Equation 3-60. The turbulent viscosity is given by,

(3-76)

where S is the characteristic magnitude of the mean velocity gradients,

(3-77)

The model constants are defined through interpolation of appropriate inner and outer values,

(3-78)

The interpolation functions F1 and F2 are defined as,

(3-79)

and,

kt

------ u k+ P 0*k kT+ k +–=

t

------- u +T------P 2 T+ 2 1 F1–

2

------------- k++–=

P min Pk 100*k =

Ta1k

max a1 SF2 ----------------------------------------=

S 2SijSij=

F11 1 F1– 2+= for k =

F1 14 tanh=

1 min max k0

*lw

---------------- 500lw

2--------------

42k

CDklw2

---------------------=

CDk max22

----------------- k 10-10 =


164 | C H A P T E

(3-80)

where lw is the distance to the closest wall.

Realizability Constraints are applied to the SST model.

WA L L D I S T A N C E

The wall distance variable, lw, is provided by a mathematical Wall Distance interface that is included when using the SST model. The solution to the wall distance equation is controlled using the parameter lref. The distance to objects larger than lref is represented accurately, while objects smaller than lref are effectively diminished by appearing to be farther away than they actually are. This is a desirable feature in turbulence modeling since small objects would get too large an impact on the solution if the wall distance were measured exactly.

The most convenient way to handle the wall distance variable is to solve for it in a separate study step. A Wall Distance Initialization study type is provided for this purpose and should be added before the actual Stationary or Transient study step.

The default model constants are given by,

(3-81)

WA L L B O U N D A R Y C O N D I T I O N S

In a low Reynolds number model the equations are integrated all the way through the boundary layer to the wall, which allows for a no slip condition to be applied to the velocity, that is u0.


• The Wall Distance Interface

• Stationary with Initialization, Transient with Initialization, and Wall Distance Initialization

F2 22 tanh=

2 max 2 k0

*lw

---------------- 500lw

2--------------

=

1 0.075,= 1 5 9,= k1 0.85,= 1 0.5=

2 0.0828,= 2 0.44,= k2 1.0,= 2 0.856=

0* 0.09,= a1 0.31=


Since all velocities must disappear on the wall, so must k. Hence, k0 on the wall.

The corresponding boundary condition for is

(3-82)

To avoid the singularity at the wall, is not solved for in the cells adjacent to a solid wall. Instead, its value in those cells is prescribed by Equation 3-82. Accurate solutions in the near-wall region require that,

(3-83)

where u is the friction velocity which is calculated from the wall shear-stress w,

(3-84)

The boundary variable Dimensionless distance to cell center, , is available to ensure that the mesh is fine enough. Observe that very small values of can reduce the convergence rate.

I N L E T V A L U E S F O R T H E TU R B U L E N C E L E N G T H S C A L E A N D I N T E N S I T Y

The guidelines given in Inlet Values for the Turbulence Length Scale and Turbulent Intensity for selecting the turbulence length scale, LT, and the turbulence intensity, IT, apply also to the SST model.

F A R - F I E L D B O U N D A R Y C O N D I T I O N S

The SST model was originally developed for exterior aerodynamic simulations. The recommended far-field boundary conditions (Ref. 15) can be expressed as,

(3-85)

where L is the approximate length of the computational domain.


The SST model has the same default initial guess as the standard k- model (see Initial Values) but with replaced by lref.

61lw

2----------------=

lw 0lim

lw+ ulw 1=

u w =

lc+

lc+

10-5U2

ReL-------------------- k

0.1U2

ReL-----------------

UL

-------- 10UL

--------

lmixlim


166 | C H A P T E

The default initial value for the wall distance equation (which solves for the reciprocal wall distance) is 2lref.


The SST model applies absolute scales of the same type as the k- model (see Scaling for Time-Dependent Simulations).

The Low Reynolds Number k- Turbulence Model

When the accuracy provided by wall functions in the k- model is not enough, a so called low Reynolds number model can be used. “Low Reynolds number” refers to the region close to the wall where viscous effects dominate.

Most low Reynolds number k- models adapt the turbulence transport equations by introducing damping functions. This module includes the AKN model (after the inventors Abe, Kondoh, and Nagano; Ref. 11). The AKN k- model for compressible flows reads (Ref. 8 and Ref. 11):

(3-86)

where

(3-87)

and

(3-88)

kt

------ + u k Tk------+

k Pk –+=

t

----- + u T------+

C1

k---P

kfC2

2

k-----–+=

Pk T u: u u T+ 23--- u 2–

23---k u–=

T fCk2

------=

f 1 e l* 14–– 2 1 5Rt

3 4/------------e Rt 200 2–+

=

f 1 e l* 3.1–– 2 1 0.3e Rt 6.5 2–– =

l* ulw = Rt k2 = u 1 4/=

C1 1.5= C2 1.9= C 0.09= k 1.4= 1.4=


lw is the distance to the closest wall.

Realizability Constraints are applied to the low Reynolds number k- model.

WA L L D I S T A N C E

The wall distance variable, lw, is provided by a mathematical Wall Distance interface that is included when using the low Reynolds number k- model. The solution to the wall distance equation is controlled using the parameter lref. The distance to objects larger than lref is represented accurately, while objects smaller than lref are effectively diminished by appearing to be farther away than they actually are. This is a desirable feature in turbulence modeling since small objects would get too large an impact on the solution if the wall distance were measured exactly.

The most convenient way to handle the wall distance variable is to solve for it in a separate study step. A Wall Distance Initialization study type is provided for this purpose and should be added before the actual Stationary or Transient study step.


The damping terms in the equations for k and allow for a no slip condition to be applied to the velocity, that is u0.

Since all velocities must disappear on the wall, so must k. Hence, k0 on the wall.

The correct wall boundary condition for is

where n is the wall normal direction. This condition is however numerically very unstable. Therefore, is not solved for in the cells adjacent to a solid wall and instead the analytical relation

(3-89)

is prescribed in those cells. Equation 3-89 can be derived as the first term in a series expansion of


• The Wall Distance Interface

• Stationary with Initialization, Transient with Initialization, and Wall Distance Initialization

2 k n 2

2--- k

lw2

-----=


168 | C H A P T E

For the expansion to be valid, it is required that

is the distance, measured in viscous units, from the wall to the center of the wall adjacent cell. The boundary variable Dimensionless distance to cell center is available to ensure that the mesh is fine enough. Observe that it is unlikely that a solution is obtained at all if

I N L E T V A L U E S F O R T H E TU R B U L E N C E L E N G T H S C A L E A N D I N T E N S I T Y

The guidelines given in Inlet Values for the Turbulence Length Scale and Turbulent Intensity for selecting the turbulence length scale, LT, and the turbulence intensity, IT, apply also to the low-Reynolds number k- model.


The low-Reynolds number k- model has the same default initial guess as the standard k- model (see Initial Values) but with replaced by lref.

The default initial value for the wall distance equation (which solves for the reciprocal wall distance) is 2lref.

In some cases, especially for stationary solutions, a fast way to convergence is to first solve the model using the ordinary k- model and then to use that solution as an initial guess for the low-Reynolds number k- model. The procedure is then as follows:

1 Solve the model using the k- model.

2 Switch to the low-Reynolds number k- model.

3 Add a new Stationary with Initialization study.

4 In the Wall Distance Initialization study step, set Values of variables not solved for to Solution from the first study. This is to propagate the solution from the first study down to the second step in the new study.

5 Solve the new study.


The low-Reynolds number k- model applies absolute scales of the same type as the k- model (see Scaling for Time-Dependent Simulations).

2 k n 2

lc* 0.5

lc*

lc* 0.5»

lmixlim


The Spalart-Allmaras Turbulence Model

The Spalart-Allmaras turbulence model is a one-equation turbulence model designed mainly for aerodynamic applications. It is a low Reynolds number model, that is, it does not utilize wall functions. “Low Reynolds number” refers to the region close to the wall where viscous effects dominate.

The model gives satisfactory results for many engineering applications, in particular for airfoil and turbine blade applications for which it is calibrated. It is however not appropriate for applications involving jet-like free shear regions. It also has some nonphysical properties. For example, it predicts zero decay rate for the eddy viscosity in a uniform free-stream (Ref. 1).

Compared to the low Reynolds number k- model, the Spalart-Allmaras model is generally considered more robust and is often used as a way to obtain an initial solution for more advanced models. It can give reasonable results on relatively coarse meshes for which the low Reynolds number k- model does not converge or even diverges.

This module includes the standard version of the Spalart-Allmaras model without the trip term (see Ref. 1 and Ref. 12). The model solves for the undamped turbulent kinematic viscosity, :

(3-90)

The model includes the following auxiliary variables

where

t

------ u + cb1S cw1fwlw-----

2 1--- +

cb2

-------- + +–=

cw1cb1

v2

--------1 cb2+

-----------------+=

---= fv1

3

3 cv13+

--------------------=

fv2 1 1 fv1+--------------------–= fw g

1 cw36+

g6 cw36+

--------------------- 1 6/

= g r cw2 r6 r– +=

r min Sv

2lw2

----------------- 10 = S max CRotmin 0 S –

v2lw

2------------fv2+ + 0.3

=

S 2SijSij= 2ijij=


170 | C H A P T E

are the mean strain rate and mean rotation rate tensors, lw, is the distance to the closest wall and is the kinematic viscosity. The turbulent viscosity is calculated by

The default values for the modeling parameters are:

The implementation of the production term includes the rotation correction suggested in Ref. 12. See also Ref. 13. The terms r and are furthermore regularized according to Ref. 12.

Pseudo Time Stepping for Turbulent Flow Models is by default applied to the stationary form of the Spalart-Allmaras model.


The Spalart-Allmaras model is consistent with a no slip boundary condition, that is u0. Since, there can be no fluctuations on the wall, the boundary condition for is .

The Spalart-Allmaras model can be considered to be well resolved at a wall if is of order unity. is the distance, measured in viscous units, from the wall to the center of the wall adjacent cell and can be evaluated as the boundary variable: Dimensionless distance to cell center. See also Wall for boundary condition details.


The default initial values for the Spalart-Allmaras interface are:


The Spalart-Allmaras model applies absolute scales of the same type as the k- model (see Scaling for Time-Dependent Simulations) except that the scale for is given directly by

Sij 0.5 u uT+ = ij 0.5 u uT– =

T fv1=

cb1 0.1355= cb2 0.622= cv2 7.1= 2 3=

cw2 0.3= cw3 2= v 0.41= CRot 2.0=

SS

0=

lc*

lc*

u 0=

p 0=

---=


the scale parameter available in the Advanced section of the physics interface node. The default value for scale is 5·106 m2s.

Inlet Values for the Turbulence Length Scale and Turbulent Intensity

If inlet data for the turbulence variables are not available, crude approximations for k, and can be obtained from the following formulas:

(3-91)

where IT is the turbulent intensity and LT is the turbulence length scale.

A value of 0.1% is a low turbulent intensity IT. Good wind tunnels can produce values of as low as 0.05%. Fully turbulent flows usually have intensities between five and ten percent.

The turbulence length scale LT is a measure of the size of the eddies that are not resolved. For free-stream flows these are typically very small (on the order of centimeters). The length scale cannot be zero, however, because that would imply infinite dissipation. Use Table 3-5 as a guideline when specifying LT (Ref. 3) where lw is the wall distance, and

TABLE 3-5: TURBULENCE LENGTH SCALES FOR TWO-DIMENSIONAL FLOWS

FLOW CASE LT L

Mixing layer 0.07L Layer width

Plane jet 0.09L Jet half width

Wake 0.08L Wake width

Axisymmetric jet 0.075L Jet half width

Boundary layer (px0)

– Viscous sublayer and log-layer

– Outer layer 0.09L

Boundary layer thickness

Pipes and channels

(fully developed flows)

0.07L Pipe radius or channel half width

k 32--- U IT 2=

C3 4 k3 2/

LT-----------=

k0

* 1 4/ LT

--------------------------=

lw+ lw l*=

lw 1 lw+ 26– exp–


172 | C H A P T E

Theory for the Pressure, No Viscous Stress Boundary Condition

For this module, in addition to the information in the Pressure, No Viscous Stress Boundary Condition (described in the COMSOL Multiphysics Reference Manual), the turbulent intensity IT, turbulence length scale LT, and reference velocity scale Uref values are related to the turbulence variables via

Solvers for Turbulent Flow

The nonlinear system that the Navier-Stokes (RANS) and turbulence transport equations constitute can become ill-conditioned if solved using a fully coupled solver. Turbulent flows are therefore solved using a segregated approach (Ref. 17): Navier-Stokes in one group and the turbulence transport equations in another.

For each iteration in the Navier-Stokes group, two or three iterations are performed for the turbulence transport equations. This is necessary to make sure that the very nonlinear source terms in the turbulence transport equations are in balance before performing another iteration for the Navier-Stokes group.

The default iterative solver for the turbulence transport equations is GMRES accelerated by Geometric Multigrid. The default smoother is SOR Line.

For recommendations of physically sound values see Inlet Values for the Turbulence Length Scale and Turbulent Intensity.

Also see Inlet and Outlet for the node settings.

• Pseudo Time Stepping for Turbulent Flow Models


• Multigrid

• Stationary Solver

• Iterative

• SOR Line

k 32--- ITUref 2,=

C3 4

LT------------

3 ITUref 2

2----------------------------

32---

,= 32---

ITUref

0* 1 4/ LT

--------------------------=


Pseudo Time Stepping for Turbulent Flow Models

The default stationary solver applies pseudo time stepping for both 2D and 3D models. This improves the robustness of the nonlinear iterations as well as the condition number for the linear equation system. The latter is especially important for large 3D models where iterative solvers must be applied. The turbulence equations use the same as the momentum equations.

The default manual expression for CFLloc is, for 2D models:

and for 3D models:


1. D.C. Wilcox, Turbulence Modeling for CFD, 2nd ed., DCW Industries, 1998.

2. D.M. Driver and H.L. Seegmiller, “Features of a Reattaching Turbulent Shear Layer in Diverging Channel Flow,” AIAA Journal, vol. 23, pp. 163–171, 1985.

3. H.K. Versteeg and W. Malalasekera, An Introduction to Computational Fluid Dynamics, Prentice Hall, 1995.

4. A. Durbin, “On the k- Stagnation Point Anomality,” Int. J. Heat and Fluid Flow, vol. 17, pp. 89–90, 1986.

5. A, Svenningsson, Turbulence Transport Modeling in Gas Turbine Related Applications,” doctoral dissertation, Department of Applied Mechanics, Chalmers University of Technology, 2006.

6. C.H. Park and S.O. Park, “On the Limiters of Two-equation Turbulence Models,” Int. J. Computational Fluid Dynamics, vol. 19, no. 1, pp. 79–86, 2005.

7. J. Larsson, Numerical Simulation of Turbulent Flows for Turbine Blade Heat Transfer, doctoral dissertation, Chalmers University of Technology, Sweden, 1998.

t

1.3min niterCMP-1 9 +



1.3min niterCMP-1 9 +




174 | C H A P T E

8. L. Ignat, D. Pelletier, and F. Ilinca, “A Universal Formulation of Two-equation Models for Adaptive Computation of Turbulent Flows,” Computer Methods in Applied Mechanics and Engineering, vol. 189, pp. 1119–1139, 2000.

9. D. Kuzmin, O. Mierka, and S. Turek, “On the Implementation of the k- Turbulence Model in Incompressible Flow Solvers Based on a Finite Element Discretization,” Int.J. Computing Science and Mathematics, vol. 1, no. 2–4, pp. 193–206, 2007.

10. H. Grotjans and F.R. Menter, “Wall Functions for General Application CFD Codes,” ECCOMAS 98, Proceedings of the Fourth European Computational Fluid Dynamics Conference, John Wiley & Sons, pp. 1112–1117, 1998.

11. K. Abe, T. Kondoh, and Y. Nagano, “A New Turbulence Model for Predicting Fluid Flow and Heat Transfer in Separating and Reattaching Flows—I. Flow Field Calculations,” Int. J. Heat and Mass Transfer, vol. 37, no. 1, pp. 139–151, 1994.

12. “The Spalart-Allmaras Turbulence Model,” http://turbmodels.larc.nasa.gov/spalart.html.

13. J. Dacles-Mariani, G.G. Zilliac and J.S. Chow, “Numerical/Experimental Study of a Wingtip Vortex in the Near Field”, AIAA Journal, vol. 33, no. 9, 1995.

14. “The Menter Shear Stress Transport Turbulence Model,” http://turbmodels.larc.nasa.gov/sst.html.

15. F.R. Menter, “Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications,” AIAA Journal, vol. 32, no. 8, 1994.

16. F.R. Menter, M. Kuntz, and R. Langtry, “Ten Years of Industrial Experience with the SST Turbulence Model,” Turbulence Heat and Mass Transfer, vol. 4, 2003.

17. M. Vázquez, M. Ravachol, F. Chalot, and M. Mallet, “The Robustness Issue on Multigrid Schemes Applied to the Navier-Stokes Equations for Laminar and Turbulent, Incompressible and Compressible Flows,” Int. J.for Numerical Methods in Fluids, vol. 45, pp. 555–579, 2004.

18. T. Cebeci, Analysis of Turbulent Flows, 2nd ed., Elsevier, Amsterdam, 2004.

19. J. Nikuradse, “Strömungsgesetze in rauhen Rohren”, Forschg. Arb. Ing.-Wes., no. 361, 1933.


http://turbmodels.larc.nasa.gov/spalart.html

http://turbmodels.larc.nasa.gov/sst.html

Th eo r y f o r t h e Ro t a t i n g Ma ch i n e r y I n t e r f a c e s

Both the Rotating Machinery, Laminar Flow and Rotating Machinery, Turbulent Flow interfaces model flow in geometries with rotating parts. For example, stirred tanks, mixers, and pumps.

The Navier-Stokes equations formulated in a rotating coordinate system read (Ref. 1 and Ref. 2)

(3-92)

(3-93)

where v is the velocity vector in the rotating coordinate system, r is the position vector and is the angular velocity vector. The relation between v and the velocity vector in the stationary coordinate system is

The Rotating Machinery, Fluid Flow interfaces solve Equation 3-92 and Equation 3-93, but reformulated in terms of a non-rotating coordinate system, that is they solve for u. This is achieved by invoking the Arbitrary Lagrangian-Eulerian Formulation (ALE) machinery. In rotating domains, xxt as prescribed in the Rotating Domain features. The Navier-Stokes equations on rotating domains then read

(3-94)

(3-95)

t------ v + 0=

vt------ v v 2 v+ + =

pI– + F t

------- r r + –+

u v rt

------+=

T------- x

T------- – u + 0=

uT------- x

T------- u–

u u+ pI– + F+=

T H E O R Y F O R T H E R O T A T I N G M A C H I N E R Y I N T E R F A C E S | 175

176 | C H A P T E

The derivative operator is the mesh time derivative of the density and appears in the equation view as d(rmspf.rho,TIME). Analogously, is the mesh time derivative of the velocity. The variable TIME replaces t as the variable for time.

The user input for a rotating domain prescribes the angular frequency, w. To calculate , the physics interfaces set up an ODE variable for the angular displacement . The equation for is

(3-96)

, is defined as times the normalized axis of rotation. In 2D, the axis of rotation is the z direction while it in 3D is specified in the Rotating Domain features. If the model contains several rotating domains, each domain has its own angular displacement ODE variable.

In non-rotating domains, the ordinary Navier-Stokes equations are solved. The rotating and fixed parts need to be coupled together by an identity pair, where a continuity boundary condition is applied.

Boundary conditions in the rotating parts must be specified correctly. For example walls that rotate must be prescribed as rotating walls. Walls in rotating domains can also be prescribed as non-rotating, but in that case, , where n is the wall normal, must be tangential to the wall.

Frozen Rotor

Equation 3-94 through Equation 3-96 must be solved using a Time Dependent study. This can be computationally expensive. The Rotating Machinery, Fluid Flow interfaces therefore support the so-called frozen rotor approach. The frozen rotor approach assumes that the flow in the rotating domain, expressed in the rotating coordinate system, is fully developed. Equation 3-92 then reduces to

(3-97)

and Equation 3-93 to

Arbitrary Lagrangian-Eulerian Formulation (ALE) in the COMSOL Multiphysics Reference Manual

Tu T

ddt-------- w=

n

v 0=


(3-98)

Frozen rotor is both a study type and an equation form. When solving a rotating machinery model using a Frozen Rotor study step, the Rotating Machinery, Fluid Flow interfaces effectively solve Equation 3-98 and Equation 3-98 in a “rotating” domain, but “rotating” domains do not rotate at all. Boundary conditions remain transformed as if the domains were rotating, but the domains remain fixed, or frozen, in position. As in the time-dependent case, the Rotating Machinery, Fluid Flow interfaces solve for the velocity vector in the stationary coordinate system, u, rather than for v.

To make Equation 3-94 and Equation 3-95 equivalent to Equation 3-97 and Equation 3-98, the Frozen Rotor study step defines a parameter TIME, which by default is set to zero (TIME appears in the Parameters node under Global Definitions). Equation 3-96 is replaced by

Since TIME is a parameter and x is a function of TIME, evaluates to its correct value. Finally, and the mesh time derivative of the velocity is replaced by

In non-rotating domains, the ordinary, stationary Navier-Stokes equations are solved. The Frozen Rotor study step invokes a stationary solver to solve the resulting equation system.

The frozen rotor approach can in special cases give the same solution as solving Equation 3-94 through Equation 3-96 to steady state. This is the case if, for example, the whole geometry is rotating, or if the model is invariant with respect to the position of the rotating domain relative to the non-rotating domain. The latter is the case for a fan placed in the middle of a straight, cylindrical duct.

In most cases however, there is no steady state solution to the rotating machinery problem. Only a pseudo-steady state where the solution varies periodically around some average solution. In these cases, the frozen rotor approach gives an approximate solution to the pseudo-steady state. The approximation depends on the position in which the rotor is frozen and the method cannot capture transient effects (see Ref. 3 and Ref. 4). An estimate of the effect of the rotor position can be obtained by making a parametric sweep over TIME.

v v 2 v+ =

pI– + F r –+

wTIME=

x T T 0=

uT------- u=


178 | C H A P T E

The frozen rotor approach is very useful for attaining initial values for time-dependent simulations. Starting from a frozen rotor solution, the pseudo-steady state can be reached within a few revolutions, while staring from u0 can require tens of revolutions. See for example Ref. 5.

Only interfaces that explicitly support frozen rotors are included in a Frozen Rotor study step.

Setting Up a Rotating Machinery Model

The Rotating Machinery, Fluid Flow interfaces primarily handle two types of geometries with rotating parts.

The first type is where the whole geometry rotates. Typical examples are individual parts in turbo machinery and lab-on-a-chip devices. For such cases, apply a Rotating Domain feature to all domains in the geometry.

The other type is geometries where it is possible to divide the modeled device into rotationally invariant geometries. The operation can be, for example, to rotate an impeller in a baffled tank, as in Figure 3-11 where the impeller rotates from position 1 to 2.

The first step to set up these type of models is to divide the geometry into two parts, as shown in Step 1a. Draw the geometry using separate domains for fixed and rotating parts. If you intend to do a time-dependent simulation, activate the assembly (using an assembly instead of a union, see Geometry Modeling and CAD Tools in the COMSOL Multiphysics Reference Manual) and create identity pairs, which makes it possible to treat the domains as separate parts in an assembly.

The second step is to specify the parts to model using Rotating Domain features and the ones to model using a fixed frame (Step 1b).

Once this is done, proceed to the usual steps of setting the fluid properties, boundary conditions. Apply a Flow Continuity to assembly pairs (Step 2a). Then mesh and solve the problem.



Figure 3-11: The modeling procedure in the Rotating Machinery, Fluid Flow interface.

References

1. H.P. Greenspan, The Theory of Rotating Fluids, Breukelen Press, 1990.

2. G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, 2000.

3. CFD Online, Best practice guidelines for turbomachinery CFD, http://www.cfd-online.com/Wiki/Best_practice_guidelines_for_turbomachinery_CFD.

4. A. Brucato, M. Ciofalo, F. Grisafi, and G. Micale, “Numerical Prediction of Flow Fields in Baffled Stirred Vessels: A Comparison of Alternative Modelling Approaches,” Chemical Engineering Science, vol. 53, no. 21, pp. 3653–3684, 1998.

5. J.-P. Torré, D.F. Fletcher, T. Lasuye, and C. Xuereb, “Single and Multiphase CFD Approaches for Modelling Partially Baffled Stirred Vessels: Comparison of Experimental Data with Numerical Predictions,” Chemical Engineering Science, vol. 62, no. 22, pp. 6246–6262, 2007.

1 2

1a 1b 2a


http://www.cfd-online.com/Wiki/Best_practice_guidelines_for_turbomachinery_CFD

180 | C H A P T E

4

H e a t T r a n s f e r a n d N o n - I s o t h e r m a l F l o w

There are several physics interfaces included in the CFD Module to model heat transfer and non-isothermal flow. This chapter describes the Heat Transfer branch ( ) when adding a physics interface. It also describes the Non-Isothermal

Flow branch ( ) that can be found under the Fluid Flow main branch. The Modeling Heat Transfer in the CFD Module section helps you choose the best physics to start with.

In this chapter:

• The Heat Transfer Interface

• Out-of-Plane Heat Transfer Nodes

• The Heat Transfer in Porous Media Interface

• Theory for the Heat Transfer Interfaces

• Theory of Out-of-Plane Heat Transfer

• Theory for the Heat Transfer in Porous Media Interface

• About the Heat Transfer Coefficients

• The Non-Isothermal Flow and Conjugate Heat Transfer, Laminar Flow and Turbulent Flow Interfaces

• Theory for the Non-Isothermal Flow and Conjugate Heat Transfer Interfaces

181

182 | C H A P T E

Mode l i n g Hea t T r a n s f e r i n t h e C FD Modu l e

Heat transfer is an important phenomenon in many industrial processes. Often, a fluid plays a major role in transporting the heat and a detailed description of the flow field is necessary to accurately describe such processes. Typical examples are heating and cooling operations. Chemical reactions and phase changes are other commonly occurring phenomena. The temperature, in turn, affects the fluid properties and can alter the flow field. Natural convection is an example of this.

In this section:




The Heat Transfer branch ( ) included with this module has a number of physics interfaces that can be used to model energy transport. One or more of these can be added; either by themselves or together with other physics interfaces, typically flow physics interfaces.

While, the standard COMSOL Multiphysics package includes physics interfaces for simulating heat transfer through conduction and convection, this module provides extra functionality within the standard Heat Transfer interfaces. The CFD Module also includes additional Heat Transfer interfaces such as The Heat Transfer in Porous Media interface.

For heat transfer in single-phase flows, the CFD Module provides the Non-Isothermal

Flow ( ) (located in the Fluid Flow branch) and Conjugate Heat Transfer ( ) interfaces. There are several versions of these interfaces (all with the interface identifier nitf), that combine heat transfer with either laminar or turbulent flow. The multiphysics interfaces automatically couple the flow and heat equations and they also provide functionality, such as support for turbulent heat transfer, that is not readily available when adding the interfaces separately.

R 4 : H E A T TR A N S F E R A N D N O N - I S O T H E R M A L F L O W

H E A T TR A N S F E R I N S O L I D S O R F L U I D S , A N D J O U L E H E A T I N G

The Heat Transfer in Solids ( ), Heat Transfer in Fluids ( ) (general convection and conduction), and Electromagnetic Heating>Joule Heating interfaces ( ), all belong to the COMSOL Multiphysics base package.

H E A T TR A N S F E R I N PO R O U S M E D I A

The Heat Transfer in Porous Media interface ( ) is an extension of a the generic Heat

Transfer interface that includes modeling heat transfer through convection, conduction and radiation, conjugate heat transfer, and non-isothermal flow. The ability to define material properties, boundary conditions, and more for porous media heat transfer is also activated by selecting the Heat transfer in porous media check box in the Heat

Transfer interface.

N O N - I S O T H E R M A L F L O W A N D C O N J U G A T E H E A T TR A N S F E R

The Non-Isothermal Flow ( ) and Conjugate Heat Transfer ( ) interfaces solve the Navier-Stokes equations together with an energy balance. They can also solve for heat transfer in solids.

Non-Isothermal Flow and Conjugate Heat Transfer differ by their default features. The default domain feature for Non-Isothermal Flow is a Fluid node while Conjugate Heat

Transfer has a Heat Transfer in Solids node as its default domain feature.

There are several entries for these physics interfaces in the Model Wizard. The entries differ by one or two default settings (Table 4-1). These settings can be changed from a check box or list under the Physical Model section in the physics interface. The Laminar

Flow versions of the physics interfaces are used primarily to model slowly flowing fluids in environments where energy transport is also an important part of the system and application, and must be coupled or connected to the fluid flow in some way. Stokes’ law (creeping flow) can be activated from the Non-Isothermal Flow, Laminar Flow and Conjugate Heat Transfer, Laminar Flow interfaces if wanted. If you expect the flow to become turbulent, select one of the Turbulent Flow interfaces. Each interface includes a RANS turbulence model to calculate the turbulence and algebraic models for the turbulent Prandtl number.

M O D E L I N G H E A T TR A N S F E R I N T H E C F D M O D U L E | 183

184 | C H A P T E

The Non-Isothermal Flow and Conjugate Heat Transfer interfaces can solve the fully compressible form of the Navier-Stokes equations. But boundary conditions and stabilizations are provided for Mach numbers less than 0.3. If you expect the Mach number in your model to become higher than 0.3, use The High Mach Number Flow Interfaces.

TABLE 4-1: THE NON-ISOTHERMAL FLOW AND CONJUGATE HEAT TRANSFER PHYSICAL MODEL DEFAULT SETTINGS*

INTERFACE (NITF) TURBULENCE MODEL TYPE

TURBULENCE MODEL

ICON

Non-Isothermal Flow, Laminar Flow

None N/A

Non-Isothermal Flow, Turbulent Flow, k-

RANS k-

Non-Isothermal Flow, Turbulent Flow, k-

RANS k-

Non-Isothermal Flow, Turbulent Flow, SST

RANS SST

Non-Isothermal Flow, Turbulent Flow, Low Re k-


Non-Isothermal Flow, Turbulent Flow, Spalart Allmaras


Conjugate Heat Transfer, Laminar Flow

None N/A

Conjugate Heat Transfer, Turbulent Flow, k-

RANS k-

Conjugate Heat Transfer, Turbulent Flow, k-

RANS k-

Conjugate Heat Transfer, Turbulent Flow, SST

RANS SST

Conjugate Heat Transfer, Turbulent Flow, Low Re k-


Conjugate Heat Transfer, Turbulent Flow, Spalart Allmaras


*For all the physics interfaces, the Neglect initial term (Stokes flow) check box is not selected by default.



Often, you are simulating applications that couple heat transfer in turbulent flow to another type of phenomenon described by another physics interface. This can, for example, include chemical reactions and mass transport, as covered by the physics interfaces in the Chemical Species Transport branch.

Furthermore, the Chemical Reaction Engineering Module includes, not only support for setting up and simulating chemical reactions, but also for simulating reaction kinetics through the temperature-dependent Arrhenius Expression and Mass Action Law. This interface also includes support for including and calculating thermodynamic data as temperature-dependent expressions, for both reaction kinetics and fluid-flow.

In addition, if you also have the Heat Transfer Module, it includes more detailed descriptions and tools for simulating energy transport, such as surface-to-surface and participating media radiation.


• The Joule Heating Interface in the COMSOL Multiphysics Reference Manual

Fluid Damper: model library path CFD_Module/Non-Isothermal_Flow/fluid_damper

M O D E L I N G H E A T TR A N S F E R I N T H E C F D M O D U L E | 185

186 | C H A P T E

Th e Hea t T r a n s f e r I n t e r f a c e

After selecting a version of the physics interface , default nodes are added under the main node, which then defines which version of the Heat Transfer interface is added.

Heat Transfer in SolidsThe Heat Transfer in Solids ( ) interface is used to model heat transfer by conduction, convection, and radiation. A Heat Transfer in Solids model is active by default on all domains. All functionality for including other domain types, such as a fluid domain, is also available.

The temperature equation defined in solid domains corresponds to the differential form of the Fourier's law that may contain additional contributions like heat sources.

When this version of the physics interface is added, these default nodes are added to the Model Builder—Heat Transfer in Solids, Thermal Insulation (the default boundary condition), and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and sources. You can also right-click Heat Transfer is Solids to select physics from the context menu.

Heat Transfer in FluidsThe Heat Transfer in Fluids ( ) is used to model heat transfer by conduction, convection, and radiation. A Heat Transfer in Fluids model is active by default on all domains. All functionality for including other domain types, such as a solid domain, is also available.

The temperature equation defined in fluid domains corresponds to the convection-diffusion equation that may contain additional contributions like heat sources.

When this version of the physics interface is added, these default nodes are added to the Model Builder—Heat Transfer in Fluids, Thermal Insulation (the default boundary condition), and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and sources. You can also right-click Heat Transfer in Fluids to select physics from the context menu.

Depending on the version of the physics interface selected, the default nodes vary.


Benefits of the Different Heat Transfer InterfacesThe benefit of the different versions of the Heat Transfer interfaces, with ht as the common default identifier, is that it is easy to add the default settings when selecting the interface. At any time, add a Heat Transfer in Fluids or Heat Transfer in Solids node from the Physics toolbar—the functionality is always available.



The default identifier (for the first physics interface in the model) is ht.


The default setting is to include All domains in the model to define heat transfer and a temperature field. To choose specific domains, select Manual from the Selection list.


Select the Heat transfer in porous media check box to add the Porous matrix model. See Modeling Heat Transfer in the CFD Module and The Heat Transfer in Porous Media Interface for details.


To display this section, click the Show button ( ) and select Stabilization. The Streamline diffusion check box is selected by default and should remain selected for optimal performance for heat transfer in fluids or other applications that include a

If required for 2D or 1D components, select the Out-of-plane heat transfer check box and then enter the Thickness of the plane (dz). The default is 1 m and applies to the entire geometry. If another thickness is specified for some of the domains, use the Change Thickness node.

About the Heat Transfer Coefficients

T H E H E A T TR A N S F E R I N T E R F A C E | 187

188 | C H A P T E

convective or translational term. Crosswind diffusion provides extra diffusion in regions with sharp gradients. The added diffusion is orthogonal to the streamlines, so streamline diffusion and crosswind diffusion can be used simultaneously. The Crosswind diffusion check box is selected by default.


To display this section, click the Show button ( ) and select Stabilization. The Isotropic diffusion check box is not selected by default.


To display this section, click the Show button ( ) and select Discretization.

• Select an element order (shape function order) for the Temperature—Quadratic (the default), Linear, Cubic, Quartic, or Quintic.

• The Compute boundary fluxes check box is selected by default so that COMSOL Multiphysics computes accurate boundary flux variables.

• The Apply smoothing to boundary fluxes check box is selected by default. The smoothing can provide a more well-behaved flux value close to singularities.

• In the table, specify the Value type when using splitting of complex variables—Real (the default) or Complex.


The Heat Transfer interfaces have the dependent variable Temperature T.The dependent variable names can be changed. Editing the name of a scalar dependent variable changes both its field name and the dependent variable name. If a new field name coincides with the name of another field of the same type, the fields share degrees of freedom and dependent variable names. A new field name must not coincide with the name of a field of another type, or with a component name belonging to some other field.


Add both a Heat Transfer (ht) and a Moving Mesh (ale) interface (found under the Mathematics>Deformed Mesh branch when adding a physics interface) then click the Show button ( ) and select Advanced Physics Options to display this section.

When the component contains a moving mesh, the Enable conversions between material

and spatial frame check box is selected by default.

This option has no effect when the component does not contain a moving frame since the material and spatial frames are identical in such cases. With a moving mesh, and


when this option is active, the heat transfer features automatically account for deformation effects on heat transfer properties. In particular the effects of volume changes on the density are considered. Rotation effects on the thermal conductivity of an anisotropic material and, more generally, deformation effects on an arbitrary thermal conductivity, are also covered. When the Enable conversions between material

and spatial frame check box is not selected, the feature inputs (for example, Heat Source, Heat Flux, Boundary Heat Source, and Line Heat Source) are not converted and are instead defined on the Spatial frame.

Domain, Boundary, Edge, Point, and Pair Nodes for the Heat Transfer Interfaces

The Heat Transfer Interface has these domain, boundary, edge, point, and pair nodes and subnodes (including out-of-plane features) available. These nodes, listed in alphabetical order, are available from the Physics ribbon toolbar (Windows users), Physics context menu (Mac or Linux users), or right-click to access the context menu (all users).

• About Frames in Heat Transfer in the COMSOL Multiphysics Reference Manual


• Domain, Boundary, Edge, Point, and Pair Nodes for the Heat Transfer Interfaces

• Consistent and Inconsistent Stabilization Methods for the Heat Transfer Interfaces


• Stabilization in the COMSOL Multiphysics Reference Manual

In general, to add a node, go to the Physics toolbar, no matter what operating system you are using. However, to add subnodes, right-click the parent node.


190 | C H A P T E

Heat Transfer in Solids

The Heat Transfer in Solids node uses the heat equation, Equation 4-1, to model heat transfer in solids:

(4-1)

For a steady-state problem the temperature does not change with time and the first term disappears. The equation includes the material properties: density , heat capacity Cp, and thermal conductivity k (a scalar or a tensor when the thermal conductivity is

• Boundary Heat Source

• Convective Heat Flux

• Continuity

• Heat Flux

• Heat Source

• Heat Transfer in Fluids

• Heat Transfer in Solids

• Initial Values

• Inflow Heat Flux

• Line Heat Source

• Line Heat Source on Axis

• Open Boundary

• Outflow

• Periodic Heat Condition

• Point Heat Source

• Point Heat Source on Axis

• Pressure Work

• Surface-to-Ambient Radiation

• Symmetry, Heat and Symmetry, Flow

• Temperature

• Thermal Insulation (the default boundary condition)

• Thin Thermally Resistive Layer

• Translational Motion

• Viscous Heating

If you also have the Heat Transfer Module, there are several other feature nodes available and described in the Heat Transfer Module User’s Guide.

For axisymmetric components, COMSOL Multiphysics takes the axial symmetry boundaries into account and automatically adds an Axial

Symmetry node that is valid on the axial symmetry boundaries only.

CpTt------- – kT Q=


anisotropic), and a heat source (or sink) Q—one or more heat sources can be added separately.

When parts of the model are moving in the material frame, right-click the Heat Transfer

in Solids node to add a Translational Motion subnode to take this into account.




This section contains fields and values that are inputs to expressions defining material properties. If such user-defined materials are added, the model inputs appear here. Initially, this section is empty.


The Global coordinate system is selected by default. The Coordinate system list contains any additional coordinate systems that the component includes (except for boundary coordinate systems). The coordinate system is used to define directions for orthotropic and anisotropic thermal conductivities.

H E A T C O N D U C T I O N , S O L I D

The default setting is to use the Thermal conductivity k (SI unit: W/(m·K)) From

material. If User defined is selected, choose Isotropic, Diagonal, Symmetric, or Anisotropic

With the CFD Module, right-click to add a Pressure Work subnode.

With the Heat Transfer Module, right-click to add a Pressure Work or an Opaque subnode. The Opaque subnode is automatically added to the entire selection when Surface-to-surface radiation is activated. The selection can be edited.


192 | C H A P T E

based on the characteristics of the thermal conductivity, and enter another value or expression.

T H E R M O D Y N A M I C S , S O L I D

The default Density (SI unit: kg/m3) and Heat capacity at constant pressure Cp (SI unit: J/(kg·K)) use values From material. Select User defined to enter other values or expressions. The heat capacity at constant pressure describes the amount of heat energy required to produce a unit temperature change in a unit mass.

Thermal DiffusivityIn addition, the thermal diffusivity , defined as k/(Cp) (SI unit: m2/s), is also a predefined quantity. The thermal diffusivity can be interpreted as a measure of thermal inertia (heat propagates slowly where the thermal diffusivity is low, for example). The components of the thermal diffusivity , when given on tensor form (xx, yy, and so on, representing an anisotropic thermal diffusivity) are available as ht.alphaTdxx, ht.alphaTdyy, and so on (using the default interface identifier ht). The single scalar mean thermal diffusivity ht.alphaTdMean is the mean value of the diagonal elements

The thermal conductivity describes the relationship between the heat flux vector q and the temperature gradient T as in q = kT, which is Fourier’s law of heat conduction. Enter this quantity as power per length and temperature.

The components of the thermal conductivity k when given on tensor form (kxx, kyy, and so on, representing an anisotropic thermal conductivity) are available as ht.kxx, ht.kyy, and so on (using the default interface identifier ht). The single scalar mean effective thermal conductivity ht.kmean is the mean value of the diagonal elements kxx, kyy, and kzz.

Fourier’s law assumes that the thermal conductivity tensor is symmetric. A non symmetric tensor can lead to unphysical results.


xx, yy, and zz. The denominator Cp is the effective volumetric heat capacity which is also available as a predefined quantity, ht.C_eff.

Translational Motion

Right-click the Heat Transfer in Solids node to add the Translational Motion subnode, which provides movement by translation to the model for heat transfer in solids. It adds the following contribution to the right-hand side of Equation 4-1, defined in the parent node:

The contribution describes the effect of a moving coordinate system which is required to model, for example, a moving heat source.


From the Selection list, choose the domains on which to apply the translational motion.

TR A N S L A T I O N A L M O T I O N

Enter values for x, y, and z (in 3D) components of the Velocity field utrans (SI unit: m/s).

• Axisymmetric Transient Heat Transfer: model library path COMSOL_Multiphysics/Heat_Transfer/heat_transient_axi

• 2D Heat Transfer Benchmark with Convective Cooling: model library path COMSOL_Multiphysics/Heat_Transfer/heat_convection_2d

– Cpu T

Special care must be taken on boundaries where n·u0. The Heat Flux boundary condition does not, for example, work on boundaries where n·u0.

By default, the selection is the same as for the Heat Transfer in Solids node that it is attached to, but it is possible to use more than one Heat Translation subnode, each covering a subset of the Heat Transfer in Solids node’s selection.


194 | C H A P T E

Heat Transfer in Fluids

The Heat Transfer in Fluids interface uses the following version of the heat equation to model heat transfer in fluids:

(4-2)

For a steady-state problem the temperature does not change with time and the first term disappears. This equation includes the following material properties, fields, and sources:

• Density (SI unit: kg/m3)

• Heat capacity at constant pressure Cp (SI unit: J/(kg·K))—describes the amount of heat energy required to produce a unit temperature change in a unit mass.

• Thermal conductivity k (SI unit: W/(m·K))—a scalar or a tensor if the thermal conductivity is anisotropic.

• Velocity field u (SI unit: m/s)—either an analytic expression or a velocity field from a fluid-flow interface.

• The heat source (or sink) Q—one or more heat sources can be added separately.

• The Ratio of specific heats (dimensionless)— the ratio of the heat capacity at constant pressure, Cp, to heat the capacity at constant volume, Cv.

CpTt------- Cpu T+ kT Q+=

When using the ideal gas law to describe a fluid, specifying is sufficient to evaluate Cp. For common diatomic gases such as air, 1.4 is the standard value. Most liquids have 1.1 while water has 1.0. is used in the streamline stabilization and in the variables for heat fluxes and total energy fluxes. It is also used if the ideal gas law is applied. See Thermodynamics, Solid.

Right-click to add Viscous Heating (for heat generated by viscous friction) or Pressure Work subnodes to the Heat Transfer in Fluids feature.

Heat Transfer by Free Convection: model library path COMSOL_Multiphysics/Multiphysics/free_convection





This section has fields and values that are inputs to expressions that define material properties. If such user-defined property groups are added, the model inputs appear here.

There are also two standard model inputs—Absolute pressure and Concentration. The absolute pressure is used in some predefined quantities that include the enthalpy (the energy flux, for example).

Absolute PressureThis section controls the variable itself as well as any property value (reference pressures) used when solving for the pressure. There are usually two ways to calculate the pressure when describing fluid flow with mass- and heat transfer. Solve for the absolute pressure or a pressure (often denoted gauge pressure) that relates to the absolute pressure through a reference pressure.

The default Absolute pressure pA (SI unit: Pa) is User defined and is 1 atm (101,325 Pa). When additional physics interfaces are added to the model, the pressure variables solved for can also be selected from the list. For example, if a fluid-flow interface is added you can select Pressure (spf/fp) from the list.

Absolute pressure is also used if the ideal gas law is applied. See Thermodynamics, Solid.

Which option to choose usually depends on the system and the equations being solved. For example, in a unidirectional incompressible flow problem, the pressure drop over the modeled domain is probably many orders of magnitude smaller than the atmospheric pressure, which, when included, reduces the stability and convergence properties of the solver. In other cases, such as when pressure is part of an expression for the gas volume or the diffusion coefficients, you might need to solve for the absolute pressure.


196 | C H A P T E

When a Pressure variable is selected, the Reference pressure check box is selected by default and the default value of pref is 1[atm] (1 atmosphere).

Velocity FieldThe default Velocity field u (SI unit: m/s) is User defined. When User defined is selected, enter values or expressions for the components based on space dimensions. The defaults are 0 m/s. Or select an existing velocity field in the component (for example, Velocity field (spf/fp1) from a Laminar Flow interface).


The Global coordinate system is selected by default. The Coordinate system list contains any additional coordinate systems that the component includes (except for boundary coordinate systems). The coordinate system is used to define directions for orthotropic and anisotropic thermal conductivities.

H E A T C O N D U C T I O N , F L U I D

The default Thermal conductivity k (SI unit: W/(m·K)) is taken From material. If User

defined is selected, choose Isotropic, Diagonal, Symmetric, or Anisotropic based on the characteristics of the thermal conductivity, and enter another value or expression.

T H E R M O D Y N A M I C S , F L U I D

The default Density (SI unit: kg/m3), Heat capacity at constant pressure Cp (SI unit: J/(kg·K)), and Ratio of specific heats (dimensionless) for a general gas or liquid use values From material. Select User defined to enter other values or expressions.

This makes it possible to use a system-based (gauge) pressure as the pressure variable while automatically including the reference pressure in places where it is required, such as for gas flow governed by the gas law. While this check box maintains control over the pressure variable and instances when the absolute pressure is required within this specific physics interface, it might not do so with other physics interfaces that it is coupled to. In such models, check the coupling between any physics interfaces using the same variable.

The thermal conductivity describes the relationship between the heat flux vector q and the temperature gradient T as in q = kT which is Fourier’s law of heat conduction. Enter this quantity as power per length and temperature.


Select a Fluid type—Gas/Liquid or Ideal gas.

Gas/LiquidSelect Gas/Liquid to specify the Density, the Heat capacity at constant pressure, and the Ratio of specific heats for a general gas or liquid. The default settings are to use data From material. Select User defined to enter another value for the density, the heat capacity, or the ratio of specific heats.

Ideal GasSelect Ideal gas to use the ideal gas law to describe the fluid. Then:

• Select a Gas constant type—Specific gas constant Rs (SI unit: J/(kg·K)) or Mean

molar mass Mn (SI unit: kg/mol). For both properties, the default setting is to use the property value from the material. Select User defined to enter another value for either of these material properties.

• From the list under Specify Cp or , select Heat capacity at constant pressure Cp (SI unit: J/(kg·K)) or Ratio of specific heats (dimensionless). For both properties, the default setting is to use the property value From material. Select User defined to define another value for either of these material properties.

Initial Values

The Initial Values node adds an initial value for the temperature that can serve as an initial condition for a transient simulation or as an initial guess for a nonlinear solver. Add additional Initial Values nodes from the Physics toolbar.



If Mean molar mass is selected, the software uses the universal gas constant R 8.314 J/(mol·K), which is a built-in physical constant.

For an ideal gas, specify either Cp or the ratio of specific heats, , but not both since, in this case, they are dependent.


198 | C H A P T E


Enter a value or expression for the initial value of the Temperature T (SI unit: K). The default value is approximately room temperature, 293.15 K (20 ºC).

Heat Source

The Heat Source describes heat generation within the domain. You express heating and cooling with positive and negative values, respectively. Add one or more nodes as required—all heat sources within a domain contribute to the total heat source. Specify the heat source as the heat per unit volume, as a linear heat source, or as a total heat source (power).


From the Selection list, choose the domains to add the heat source to.

H E A T S O U R C E

Click the General source (the default), Linear source, or Total power button.

• If General source is selected, enter a value for the distributed heat source Q (SI unit: W/m3) when the default option, User defined, is selected. The default is 0 W/m3(that is, no heat source). See also Additional General Source Options.

• If Linear source (Qqs·T) is selected, enter the Production/absorption coefficient qs (SI unit: W/(m3·K)). The default is 0 W/(m3·K).

• If Total power is selected, enter the total heat source, Ptot, (SI unit: W). The default is 0 W. In this case Q = Ptot/V, where V is the total volume of the selected domains.

In 3D and 2D axial symmetry, .

In 2D and 1D axial symmetry:

where dz is the out-of-plane thickness. If the out-of-plane property is not active, a text field is available for defining dz.

V 1=

V dz 1=


Additional General Source OptionsFor the general heat source Q, there are predefined heat sources available (in addition to a User defined heat source) when simulating heat transfer together with electrical or electromagnetic physics interfaces. Such sources represent, for example, ohmic heating and induction heating.

• With the addition of an Electric Currents interface, the Total power dissipation

density (ec/cucn1) heat source is available from the General source list.

• With the addition of any version of the Electromagnetic Waves interface (which requires the RF Module), the Total power dissipation density (emw/wee1) and Electromagnetic power loss density (emw/wee1) heat sources are available from the General source list.

• With the addition of a Magnetic Fields interface (a 3D model requires the AC/DC Module), the Electromagnetic heating (mf/al1) heat source is available from the General source list.

• With the addition of a Magnetic and Electric Fields interface (which requires the AC/DC Module), the Electromagnetic heating (mef/alc1) heat source is available from the General source list.

• For the Heat Transfer in Porous Media interface, with the addition of interfaces from the Batteries & Fuel Cells Module, Corrosion Module, or Electrodeposition

In 1D:

where Ac is the cross-sectional area. If the out-of-plane property is not active, a text field is available for defining Ac.

The advantage of writing the source on the second form is that it can be accounted for in the streamline diffusion stabilization. The stabilization applies when qs is independent of the temperature, but some stability can be gained as long as qs is only weakly dependent on the temperature.

V Ac 1=

The following options are also available from the General source list above but require additional physics interfaces and/or licenses as indicated.


200 | C H A P T E

Module, heat sources from the electrochemical current distribution interfaces are available.

F R A M E S E L E C T I O N

To display this section, add both a Heat Transfer (ht) and a Moving Mesh (ale) interface (found under the Mathematics>Deformed Mesh branch when adding a physics interface). Then click the Show button ( ) and select Advanced Physics Options.

When the model contains a moving mesh, the Enable conversions between material and

spatial frame check box is selected by default in the Heat Transfer interface, which in turn enables further options. Use Frame Selection to select the frame where the input variables are defined. If Spatial is selected, the variables take their values from the edit fields. If Material (the default) is selected, a conversion from the material to the spatial frame is applied to the edit field values.

Thermal Insulation

The Thermal Insulation node is the default boundary condition for all Heat Transfer interfaces. This boundary condition means that there is no heat flux across the boundary:

and hence specifies where the domain is well insulated. Intuitively, this equation says that the temperature gradient across the boundary is zero. For this to be true, the temperature on one side of the boundary must equal the temperature on the other side. Because there is no temperature difference across the boundary, heat cannot transfer across it.


For a default node, the setting inherits the selection from the parent node, and can not be edited; that is, the selection is automatically made and is the same as for the physics



• Stabilization Techniques in the COMSOL Multiphysics Reference Manual

n kT 0=


interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific boundaries or select All boundaries as required.

Temperature

Use the Temperature node to specify the temperature somewhere in the geometry, for example, on boundaries.


From the Selection list, choose the boundaries on which to define the temperature.

TE M P E R A T U R E

The equation for this condition is T = T0 where T0 is the prescribed temperature on the boundary. Enter the value or expression for the Temperature T0 (SI unit: K). The default is 293.15 K.


To display this section, click the Show button ( ) and select Advanced Physics Options.

• By default Classic constraints is selected. To Apply reaction terms on all dependent variables, select All physics (symmetric). Otherwise, select Current physics (internally

symmetric) or Individual dependent variables to restrict the reaction terms as required.

• Select the Use weak constraints check box to replace the standard constraints with a weak implementation.

• Select the Discontinuous Galerkin constraints button when Classic constraints do not work satisfactorily.

The Discontinuous Galerkin constraints option is especially useful to prevent oscillations on inlet boundaries where convection dominates. Unlike the Classic constraints, these constraints do not enforce the temperature on the boundary extremities. This is relevant on fluid inlets where the temperature condition should not be enforced on the walls at the inlet extremities.

Show More Physics Options


202 | C H A P T E

Outflow

The Outflow node provides a suitable boundary condition for convection-dominated heat transfer at outlet boundaries. In a model with convective heat transfer, this condition states that the only heat transfer occurring across the boundary is by convection. The temperature gradient in the normal direction is zero, and there is no radiation. This is usually a good approximation of the conditions at an outlet boundary in a heat transfer model with fluid flow.


Symmetry

The Symmetry node provides a boundary condition for symmetry boundaries. This boundary condition is similar to a Thermal Insulation condition, and it means that there is no heat flux across the boundary.


Heat Flux

Use the Heat Flux node to add heat flux across boundaries. A positive heat flux adds heat to the domain. This feature is not applicable to inlet boundaries.



H E A T F L U X

Click to select the General inward heat flux (the default), Inward heat flux, or Total heat

flux button.

In most cases, the Outflow node does not require any user input. If required, select the boundaries that are convection-dominated outlet boundaries.

In most cases, the node does not require any user input. If required, define the symmetry boundaries.


General Inward Heat FluxIf General inward heat flux q0 (SI unit: W/m2) is selected, it adds to the total flux across the selected boundaries. Enter a value for q0 to represent a heat flux that enters the domain. For example, any electric heater is well represented by this condition, and its geometry can be omitted. The default is 0 W/m2.

Inward Heat FluxIf Inward heat flux is selected, enter the Heat transfer coefficient h (SI unit: W/(m2·K)). The default is 0 W/(m2·K). Also enter an External temperature Text (SI unit: K). The default is 293.15 K. The value depends on the geometry and the ambient flow conditions. Inward heat flux is defined by q0 hText T.

Total Heat FluxIf Total heat flux is selected, enter the total heat flux qtot (SI unit: W) across the boundaries where the Heat Flux node is active. The default is 0 W. In this case q0 = qtot/A, where A is the total area of the selected boundaries.

For a thorough introduction about how to calculate heat transfer coefficients, see Incropera and DeWitt in Ref. 1.



where dz is the out-of-plane thickness. If the out-of-plane property is not active, a text field is available to define dz.

A 1=

A dz 1=


204 | C H A P T E


The settings are the same as for the Heat Source node and are described under the corresponding Frame Selection section.

Surface-to-Ambient Radiation

Use the Surface-to-Ambient Radiation condition to add surface-to-ambient radiation to boundaries. The net inward heat flux stemming from surface-to-ambient radiation is

where is the surface emissivity, is the Stefan-Boltzmann constant (a predefined physical constant), and Tamb is the ambient temperature.





S U R F A C E - T O - A M B I E N T R A D I A T I O N

The default Surface emissivity (a dimensionless number between 0 and 1) is taken From material. An emissivity of 0 means that the surface emits no radiation at all and an emissivity of 1 means that it is a perfect blackbody.

In 1D:

where Ac is the cross-sectional area. If the out-of-plane property is not active, a text field is available to define Ac.

A Ac 1=



q Tamb4 T4

– =


Enter an Ambient temperature Tamb (SI unit: K). The default is 293.15 K.

Periodic Heat Condition

Use the Periodic Heat Condition to add periodic heat conditions to boundary pairs. Right-click to add a Destination Selection subnode.



Boundary Heat Source

The Boundary Heat Source models a heat source (or heat sink) that is embedded in the boundary. When selected as a Pair Boundary Heat Source, it also prescribes that the temperature field is continuous across the pair.



P A I R S E L E C T I O N

When Pair Boundary Heat Source is selected from the Pairs menu, choose the pair to define. An identity pair has to be created first. Ctrl-click to deselect.

B O U N D A R Y H E A T S O U R C E

Click the General source (the default) or Total boundary power button.

• If General source is selected, enter a value for the boundary heat source Qb (SI unit: W/m2) when the default option, User defined, is selected. A positive Qb corresponds to heating and a negative Qb corresponds to cooling. The default is 0 W/m2. For the general boundary heat source Qb, there are predefined heat sources available when simulating heat transfer together with electrical or electromagnetic


• Periodic Condition and Destination Selection

• Periodic Boundary Conditions


206 | C H A P T E

interfaces. Such sources represent, for example, ohmic heating and induction heating.

• If Total boundary power is selected, enter the total power (total heat source) Pb, tot (SI unit: W). The default is 0 W. In this case Qb = Pb, tot/A, where A is the total area of the selected boundaries.





where dz is the out-of-plane thickness. If the out-of-plane property is not active, a text field is available to define dz.

In 1D:

where Ac is the cross-sectional area. If the out-of-plane property is not active, a text field is available to define Ac.

A 1=

A dz 1=

A Ac 1=




Continuity

The Continuity node can be added to pairs. It prescribes that the temperature field is continuous across the pair. Continuity is only suitable for pairs where the boundaries match.


The selection list in this section shows the boundaries for the selected pairs.


When this node is selected from the Pairs menu, choose the pair on which to apply the condition. An identity pair has to be created first. Ctrl-click to deselect.

Thin Thermally Resistive Layer

Use the Thin Thermally Resistive Layer node to define the thickness and thermal conductivity of a resistive material located on boundaries. It can be added to pairs by selecting Pair Thin Thermally Resistive Layer from the Pairs menu. The resistive material can also be defined through the Thermal Resistance:

The heat flux across the Thin Thermally Resistive Layer is defined by




Rsds

ks-----=

nd kdTd– – ksTu Td–

ds--------------------–=

nu kuTu– – ksTd Tu–

ds--------------------–=


208 | C H A P T E

where the u and d subscripts refer to the up- and downside of the layer, respectively.


From the Selection list, choose the boundaries on which to define the layer.




If this node is selected from the Pairs menu, choose the pair on which to define the layer. An identity pair has to be created first. Ctrl-click to deselect.

T H I N T H E R M A L L Y R E S I S T I V E L A Y E R

Select Layer properties (the default) or Thermal resistance from the Specify list.

• If Layer properties is selected, enter a value or expression for the Layer thickness ds (SI unit: m). The default is 0.005 m. The default Thermal conductivity ks (SI unit: W/(m·K)) is taken From material. Select User defined to enter another value or expression. The default is 0.01 W/(m·K).

• If Thermal resistance is selected, enter a value or expression for the Thermal resistance Rs (SI unit: s3·K/kg). The default is s3·K/kg.

When using the Pair Thin Thermally Resistive Layer node, then the u and d subscripts refer to the upside and the downside of the pair, respectively, instead of the layer.

Like any pair feature, the Pair Thin Thermally Resistive Layer condition contributes to any other pair feature. However, do not use two conditions of the same type on the same pair. In order to model a thin resistive layer made of several materials, use the Multiple layers option, which is available with the Heat Transfer Module.


Line Heat Source

The Line Heat Source node models a heat source (or sink) that is so thin that it has no thickness in the model geometry. Select this node from the Edges or Points submenu.

In theory, the temperature in a line source in 3D is plus or minus infinity (to compensate for the fact that the heat source does not have any volume). The finite element discretization used in COMSOL Multiphysics returns a finite temperature distribution along the line, but that distribution must be interpreted in a weak sense.

In 2D and 2D axisymmetric geometries the Line Heat Source is available on point level.

E D G E / P O I N T S E L E C T I O N

From the Selection list, choose the edges or points to define.

L I N E H E A T S O U R C E

Click the General source (the default) or Total line power button.

• When General source is selected, enter a value for the distributed heat source, Ql (SI unit: W/m) in unit power per unit length. A positive Ql corresponds to heating while a negative Ql corresponds to cooling. The default is 0 W/m.

• If Total line power is selected, enter the total power (total heat source) Pl,tot (SI unit: W). The default is 0 W.



The Line Heat Source node is available in 3D on edges. In 2D and 2D axisymmetric it is available on points.




210 | C H A P T E

Point Heat Source

The Point Heat Source node models a heat source (or sink) that is so small that it can be considered to have no spatial extension. Select this node from the Points menu.

In theory, the temperature in a point source in 3D is plus or minus infinity (to compensate for the fact that the heat source does not have a spatial extension). The finite element discretization used in COMSOL Multiphysics returns a finite value, but that value must be interpreted in a weak sense.


From the Selection list, choose the points on which to define the condition.

PO I N T H E A T S O U R C E

Enter the Point heat source Qp (SI unit: W) in unit power. A positive Qp corresponds to heating while a negative Qp corresponds to cooling. The default is 0 W.

Line Heat Source on Axis

The Line Heat Source on Axis node models a heat source (or sink) that is so thin that it has no thickness in the model geometry. Select this node from the Edges submenu.

E D G E S E L E C T I O N

From the Selection list, choose the edges on which to define the source. Note that the edges on the symmetry axis are the only applicable entities.

L I N E H E A T S O U R C E O N A X I S

The settings are the same as for the Line Heat Source node.

The Point Heat Source is available only in 3D

The Line Heat Source on Axis is available only in 2D axisymmetric models.


Point Heat Source on Axis

The Point Heat Source on Axis node models a heat source (or sink) that is so small that it can be considered to have no spatial extension. Select this node from the Points menu.


From the Selection list, choose the points on which to apply the condition. Note that the points on the symmetry axis are the only applicable entities.

PO I N T H E A T S O U R C E O N A X I S

The settings are the same for the Point Heat Source node.

Pressure Work

Right-click the Heat Transfer in Solids or Heat Transfer in Fluids node to add the Pressure Work subnode.

When added under the Heat Transfer in Solids node, the Pressure Work node adds the following term to the right-hand side of the Heat Transfer in Solids equation:

(4-3)

where Sel is the elastic contribution to entropy.

When added under theHeat Transfer in Fluids node, the Pressure Work feature adds the following contribution to the right-hand side of the Heat Transfer in Fluids equation:

(4-4)

The software computes the pressure work using the absolute pressure.

The Point Heat Source on Axis is available only in 2D axisymmetric models.

T t-----Sel–

T---- T-------

p

pt------ u p+ –


212 | C H A P T E


From the Selection list, choose the domains on which to include the term. By default, the selection is the same as for the parent node (Heat Transfer in Solids or Heat Transfer in Fluids) it is attached to.

P R E S S U R E WO R K

For the Heat Transfer in Solids model, enter a value or expression for the Elastic

contribution to entropy Ent (SI unit: Jm3·K)). The default is 0 Jm3·K).

For the Heat Transfer in Fluids model, select a Pressure work formulation—Full

formulation (the default) or Low Mach number formulation. The low Mach number formulation excludes the term u · p from Equation 4-46, which is small for most flows with a low Mach number.

Viscous Heating

Right-click the Heat Transfer in Fluids node to add the Viscous Heating subnode, which adds the following term to the right-hand side of the heat transfer in fluids equation:

(4-5)

where is the viscous stress tensor and S is the strain-rate tensor. Equation 4-5 represents the heating caused by viscous friction within the fluid.


From the Selection list, choose the domains on which to include the term. By default, the selection is the same as for the Heat Transfer in Fluids feature that it is attached to.

D Y N A M I C V I S C O S I T Y

The Dynamic viscosity (SI unit: Pa·s) uses the value of the viscosity From material. Select User defined to enter another value or expression. The default is 0 Pa·s. COMSOL Multiphysics uses the dynamic viscosity and the velocity field to compute the viscous stress tensor, .

Inflow Heat Flux

Use the Inflow Heat Flux node to model inflow of heat through a virtual domain with a heat source. The temperature at the outer boundary of the virtual domain is known. This boundary condition estimates the heat flux through the system boundary

:S


(4-6)

where

(4-7)

A positive heat flux adds heat to the domain. This feature is applicable to inlet boundaries. The second integral in Equation 4-45 is neglected if the feature is applied to the boundary of a solid domain.



I N F L O W H E A T F L U X

Select the Inward heat flux (the default) or Total heat flux buttons.

• When Inward heat flux is selected, define q0 (SI unit: W/m2) to add to the total flux across the selected boundaries. The default value is 0 W/m2.

• When Total heat flux is selected, define qtot. In this case q0qtot/A, where A is the total area of the selected boundaries. The default is 0 W.

For either selection, enter a value or expression for the External temperature Text (SI unit: K) (the default is 273.15 K) and the External absolute pressure pext (SI unit: Pa) (the default is 1 atm).

n kT– – q0 u n– 1

u n------------------- hin hext– u n+=

hin hext– Cp TdText

Tin

1--- 1 T

----

T-------

p

+

pdpext

pA

+=

In 3D and 2D axial symmetry, .A 1=


214 | C H A P T E

Open Boundary

The Open Boundary node adds a boundary condition for modeling heat flux across an open boundary; the heat can flow out of the domain or into the domain with a specified exterior temperature. Use this node to limit a modeling domain that extends in an open fashion.



O P E N B O U N D A R Y

Enter the exterior Temperature T0 (SI unit: K) outside the open boundary.

Convective Heat Flux

The Convective Heat Flux node adds the following heat flux contribution to its boundaries:

In 2D and 1D axial symmetry

where dz is the out-of-plane thickness. If the out-of-plane property is not active, a text field is available to define dz or Ac.

In 1D

where Ac is the cross-sectional area. If the out-of-plane property is not active, a text field is available to define dz or Ac.

A dz 1=

A Ac 1=

This feature was previously called Convective Cooling.

h Text T–


where the heat transfer coefficient, h, can be user-defined or by using a library of predefined coefficients described in About the Heat Transfer Coefficients.



H E A T F L U X

Select a Heat transfer coefficient h (SI unit: W/(m2·K)) to control the type of convective heat flux to model—User defined (the default), External natural convection, Internal natural convection, External forced convection, or Internal forced convection.

• For all options, enter an External temperature, Text (SI unit: K). The default is 293.15 K.

• For all options (except User defined), follow the individual instructions below and select an External fluid—Air (the default), Transformer oil, or water. If Air is selected, also enter an Absolute pressure, pA (SI unit: Pa). The default is 1 atm.

External Natural ConvectionIf External natural convection is selected, choose Vertical wall, Inclined wall, Horizontal

plate, upside, or Horizontal plate, downside from the list under Heat transfer coefficient.

• If Vertical wall is selected, enter a Wall height L (SI unit: m). The default is 0 m.

• If Inclined wall is selected, enter a Wall height L (SI unit: m) and the Tilt angle (SI unit: rad). The tilt angle is the angle between the wall and the vertical direction, for vertical walls. The default is 0 rad.

• If Horizontal plate, upside or Horizontal plate, downside is selected, define the Plate

diameter (area/perimeter) L (SI unit: m). L is approximated by the ratio between the surface area and its perimeter. The default is 0 m.

Internal Natural ConvectionIf Internal natural convection is selected, choose Narrow chimney, parallel plates or Narrow chimney, circular tube from the list under Heat transfer coefficient.

• If Narrow chimney, parallel plates is selected, enter a Plate distance L (SI unit: m) and a Chimney height H (SI unit: m). The defaults are each 0 m.

• If Narrow chimney, circular tube is selected, enter a Tube diameter D (SI unit: m) and a Chimney height H (SI unit: m). The defaults are 0 m.


216 | C H A P T E

External Forced ConvectionIf External forced convection is selected, choose Plate, averaged transfer coefficient or Plate, local transfer coefficient from the list under Heat transfer coefficient.

• If Plate, averaged transfer coefficient is selected, enter a Plate length L (SI unit: m) and a Velocity, external fluid Uext (SI unit: m/s). The defaults are 0 m and 0 m/s.

• If Plate, local transfer coefficient is selected, enter a Position along the plate xpl (SI unit: m) and a Velocity, external fluid Uext (SI unit: m/s). The defaults are 0 m and 0 m/s.

Internal Forced ConvectionIf Internal forced convection is selected, the only option is Isothermal tube. Enter a Tube

diameter D (SI unit: m); the default is 0 m, and a Velocity, external fluid Uext (SI unit: m/s); the default is 0 m/s.


Ou t - o f - P l a n e Hea t T r a n s f e r Nod e s

The following nodes are available for 1D and 2D Heat Transfer models.

In this section:

• Out-of-Plane Convective Heat Flux

• Out-of-Plane Radiation

• Out-of-Plane Heat Flux

• Change Thickness

Out-of-Plane Convective Heat Flux

Use the Out-of-Plane Convective Heat Flux node to model upside and downside cooling or heating caused by the presence of an ambient fluid.

This feature adds the following contribution

to the right-hand side of Equation 4-8 or Equation 4-9

(4-8)

Theory of Out-of-Plane Heat Transfer

Select the Out-of-plane heat transfer check box on a Heat Transfer interface to add these nodes to 2D and 1D models.

hu Text,u T– h+ d Text,d T–

dzCpTt------- – dzkT dzQ=

O U T - O F - P L A N E H E A T TR A N S F E R N O D E S | 217

218 | C H A P T E

(4-9)


Select the domains where you want to add an out-of-plane convective heat flux contribution.

U P S I D E H E A T F L U X

Select a Heat transfer coefficient hu (SI unit: W/(m2·K)) to control the type of convective heat flux to model—User defined (the default), External natural convection, Internal natural convection, External forced convection, or Internal forced convection. If convective flux is only required on the downside, use the default, which sets hu0.

• For all of the options, enter an External temperature, Text,u (SI unit: K).

• For all of the options (except User defined), follow the individual instructions in the Heat Flux section described for the Convective Heat Flux feature, then select an External fluid—Air, Transformer oil, or water. If Air is selected, also enter an Absolute

pressure, pA (SI unit: Pa). The default is 1 atm.

D O W N S I D E H E A T F L U X

The controls in the Downside Heat Flux section are the same as those in the Upside Heat

Flux section with subscript d referring to the downside, for example, the Heat transfer

coefficient is hd.

Out-of-Plane Radiation

The Out-of-Plane Radiation node models surface-to-ambient radiation on the upside and downside. The feature adds the following contribution to the right-hand side of Equation 4-8 or Equation 4-9:

CpdzTt------- u T+ dzkT dzQ+=


About the Heat Transfer Coefficients



Select the domains where you want to add an out-of-plane surface-to-ambient heat transfer contribution.

U P S I D E P A R A M E T E R S

The default Surface emissivity u (a dimensionless number between 0 and 1) is taken From material. Select User defined to enter another value. An emissivity of 0 means that the surface emits no radiation at all while an emissivity of 1 means that it is a perfect blackbody. The default is 0.

Enter an Ambient temperature Tamb,u (SI unit: K). The default is 293.15 K.

D O W N S I D E P A R A M E T E R S

Follow the instructions for the Upside Parameters section to define the downside parameters d and Tamb,d.

Out-of-Plane Heat Flux

The Out-of-Plane Heat Flux node adds a heat flux q0,u for the upside heat flux and a heat flux q0,d for the downward heat flux to the right-hand side of Equation 4-8 or Equation 4-9:


Select the domains where you want to add an out-of-plane heat flux.

u Tamb u4 T4– d Tamb d

4 T4– +


dsq0 u dsq0 d+


O U T - O F - P L A N E H E A T TR A N S F E R N O D E S | 219

220 | C H A P T E

U P S I D E I N W A R D H E A T F L U X

Select between specifying the upside inward heat flux directly or as a convective term using a heat transfer coefficient. The General inward heat flux button is selected by default. Enter a value or expression for the inward (or outward, if the quantity is negative) heat flux through the upside (SI unit: W/m2) in the q0,u field. The default is 0 W/m2.

Click the Inward heat flux button to specify an inward (or outward, if the quantity is negative) heat flux through the upside (SI unit: W/m2) as hu·(Text,uT). Enter a value or expression for the heat transfer coefficient in the hu field (SI unit: W/(m2·K) and a value or expression for the external temperature in the Text,u field (SI unit: K). The default value for the external temperature is 293.15 K.

D O W N S I D E I N W A R D H E A T F L U X

The controls in the Downside Inward Heat Flux section are identical to those in the Upside Inward Heat Flux section except that they apply to the downside instead of the upside.

Change Thickness

Use the Change Thickness node to model domains with another thickness than the overall thickness that is specified in the Heat Transfer interface Physical Model section.


Select the domains where you want to use a different thickness.

C H A N G E T H I C K N E S S

Specify a value for the Thickness dz (SI unit: m). The default value is 1 m. This value replaces the overall thickness in the domains that are selected in the Domain Selection

section.


Th e Hea t T r a n s f e r i n Po r ou s Med i a I n t e r f a c e

The Heat Transfer in Porous Media interface ( ), found under the Heat Transfer

branch ( ) when adding a physics interface, is used to model heat transfer by conduction, convection, and radiation in porous media. A Heat Transfer in Porous Media model is active by default on all domains. All functionality for including other domain types, such as a solid domain, is also available.

The temperature equation defined in porous media domains corresponds to the convection-diffusion equation with thermodynamic properties averaging models to account for both solid matrix and fluid properties.

The physics interface is an extension of the generic Heat Transfer interface. When this interface is added, the following default nodes are added in the Model Builder—Heat

Transfer in Porous Media, Thermal Insulation (the default boundary condition), and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions. You can also right-click Heat Transfer in Porous Media

to select physics from the context menu.

The capability to define material properties, boundary conditions, and more for porous media heat transfer is activated by selecting the Heat transfer in porous media check box. (Figure 4-1).

Figure 4-1: The capability to model porous media heat transfer is activated by selecting the Heat transfer in porous media check box in any Heat Transfer (ht) settings window under Physical Model.

The rest of the settings for this physics interface are the same as for the Heat Transfer interface.

T H E H E A T TR A N S F E R I N P O R O U S M E D I A I N T E R F A C E | 221

222 | C H A P T E

Domain, Boundary, Edge, Point, and Pair Nodes for the Heat Transfer in Porous Media Interface

The Heat Transfer in Porous Media Interface has the following nodes described in this section:

• Heat Transfer in Porous Media

• Thermal Dispersion

The rest of the domain, boundary, edge, point, and pair nodes are described for The Heat Transfer Interface and listed in the section Domain, Boundary, Edge, Point, and Pair Nodes for the Heat Transfer Interfaces.

Heat Transfer in Porous Media

The Heat Transfer in Porous Media node is used to specify the thermal properties of a porous matrix. Right-click to add a Thermal Dispersion subnode. The Heat Transfer in Porous Media model uses the following version of the heat equation to model heat transfer in fluids:

(4-10)

For a steady-state problem the temperature does not change with time and the first term disappears. It has the following material properties:

• Density (SI unit: kg/m3)

• Heat capacity at constant pressure Cp (SI unit: J/(kg·K)): This describes the amount of heat energy required to produce a unit temperature change in a unit mass.

• Thermal conductivity k (SI unit: W/(m·K)): A scalar or a tensor if the thermal conductivity is anisotropic.

• Velocity field u (SI unit: m/s): Either an analytic expression or a velocity field from a fluid-flow interface.


• Domain, Boundary, Edge, Point, and Pair Nodes for the Heat Transfer in Porous Media Interface

• Theory for the Heat Transfer in Porous Media Interface




• Heat source (or sink) Q: One or more heat sources can be added separately.

• Ratio of specific heats (dimensionless): The ratio of the heat capacity at constant pressure, Cp, to the heat capacity at constant volume, Cv.




This section is the same as for Heat Transfer in Fluids.


The Global coordinate system is selected by default. The Coordinate system list contains any additional coordinate systems that the model includes (except boundary coordinate systems). The coordinate system is used for interpreting directions of orthotropic and anisotropic thermal conductivity.

F L U I D M A T E R I A L

Select any component material from the list to define Fluid material. The default uses the Domain material.

When using the ideal gas law to describe a fluid, specifying is sufficient to evaluate Cp. For common diatomic gases such as air, 1.4 is the standard value. Most liquids have 1.1 while water has 1.0. is used in the streamline stabilization and in the variables for heat fluxes and total energy fluxes. It is also used if the ideal gas law is applied. See Thermodynamics, Porous Matrix.

With the Heat Transfer Module, the Opaque subnode is automatically added to the entire selection when Surface-to-surface radiation is activated. The selection can be edited.


224 | C H A P T E

H E A T C O N D U C T I O N , F L U I D

The default Thermal conductivity k (SI unit: W/(m·K)) is taken From material. If User

defined is selected, choose Isotropic, Diagonal, Symmetric, or Anisotropic based on the characteristics of the thermal conductivity, and enter another value or expression.

T H E R M O D Y N A M I C S , F L U I D

The default Density (SI unit: kg/m3), Heat capacity at constant pressure Cp (SI unit: J/(kg·K)), and Ratio of specific heats (dimensionless) for a general gas or liquid use values From material. Select User defined to enter other values or expressions.

Select a Fluid type—Gas/Liquid or Ideal gas.

Gas/LiquidSelect Gas/Liquid to specify the Density, the Heat capacity at constant pressure, and the Ratio of specific heats for a general gas or liquid. The default settings are to use data From material. Select User defined to enter another value for the density, the heat capacity, or the ratio of specific heats.

Ideal GasSelect Ideal gas to use the ideal gas law to describe the fluid. Then:

• Select a Gas constant type—Specific gas constant Rs (SI unit: J/(kg·K)) or Mean

molar mass Mn (SI unit: kg/mol). For both properties, the default setting is to use the property value from the material. Select User defined to enter another value for either of these material properties.

• From the list under Specify Cp or , select Heat capacity at constant pressure Cp (SI unit: J/(kg·K)) or Ratio of specific heats (dimensionless). For both properties,


If Mean molar mass is selected, the software uses the universal gas constant R 8.314 J/(mol·K), which is a built-in physical constant.


the default setting is to use the property value From material. Select User defined to define another value for either of these material properties.

I M M O B I L E S O L I D S

This section contains fields and values that are inputs to expressions defining material properties. The Solid material list can point to any material in the component. Enter a Volume fraction p (dimensionless) for the solid material.

H E A T C O N D U C T I O N , PO R O U S M A T R I X

The default Thermal conductivity kp (SI unit: W/(m·K)) uses values From material. If User defined is selected, choose Isotropic, Diagonal, Symmetric, or Anisotropic based on the characteristics of the thermal conductivity and enter another value or expression in the field or matrix.

T H E R M O D Y N A M I C S , PO R O U S M A T R I X

The default Density p (SI unit: kg/m3) uses values From material. If User defined is selected, enter another value or expression.

The default Specific heat capacity Cp,p (SI unit: J/(kg·K)) uses values From material. If User defined is selected, enter another value or expression.

The equivalent volumetric heat capacity of the solid-liquid system is calculated from

For an ideal gas, specify either Cp or the ratio of specific heats, , but not both since these, in that case, are dependent.

The thermal conductivity of the material describes the relationship between the heat flux vector q and the temperature gradient T as q = kpT, which is Fourier’s law of heat conduction.

The specific heat capacity describes the amount of heat energy required to produce a unit temperature change in a unit mass of the solid material.

Cp eq ppCp p LCp+=


226 | C H A P T E

Thermal Dispersion

Right-click the Heat Transfer in Porous Media node to add the Thermal Dispersion subnode. This adds an extra term ·kdT to the right-hand side of the heat equation

and specifies values for the longitudinal and transverse dispersivities.


From the Selection list, choose the domains to define.

D I S P E R S I V I T I E S

Define the Longitudinal dispersivity lo (SI unit: m) and Transverse dispersivity tr (SI unit: m).

For the Transverse vertical dispersivity the Thermal Dispersion node defines the tensor of dispersive thermal conductivity

where Dij is the dispersion tensor

and ijkl is the fourth order dispersivity tensor

Cp eqTt------- Cpu T+ keqT Q+=

kijd LCp L Dij=

Dij ijklukul

u------------=

ijkl trijkllo tr–

2-------------------- ikjl iljk+ +=


Th eo r y f o r t h e Hea t T r a n s f e r I n t e r f a c e s

The Heat Transfer Interface theory is described in this section. This section reviews the theory about the heat transfer equations in COMSOL Multiphysics and heat transfer in general. For more detailed discussions of the fundamentals of heat transfer, see Ref. 1 and Ref. 3.

In this section:

• What is Heat Transfer?

• The Heat Equation

• A Note on Heat Flux and Balance

• Heat Transfer Variables

• About the Boundary Conditions for the Heat Transfer Interfaces

• Radiative Heat Transfer in Transparent Media

• Consistent and Inconsistent Stabilization Methods for the Heat Transfer Interfaces

• References for the Heat Transfer Interfaces

What is Heat Transfer?

Heat transfer is defined as the movement of energy due to a difference in temperature. It is characterized by the following mechanisms:

• Conduction—Heat conduction occurs as a consequence of different mechanisms in different media. Theoretically it takes place in a gas through collisions of molecules; in a fluid through oscillations of each molecule in a “cage” formed by its nearest neighbors; in metals mainly by electrons carrying heat and in other solids by molecular motion which in crystals take the form of lattice vibrations known as phonons. Typical for heat conduction is that the heat flux is proportional to the temperature gradient.

• Convection—Heat convection (sometimes called heat advection) takes place through the net displacement of a fluid which transports the heat content with its velocity. The term convection (especially convective cooling and convective heating)

T H E O R Y F O R T H E H E A T TR A N S F E R I N T E R F A C E S | 227

228 | C H A P T E

also refers to the heat dissipation from a solid surface to a fluid, typically described by a heat transfer coefficient.

• Radiation—Heat transfer by radiation takes place through the transport of photons. Participating (or semitransparent) media absorb, emit and scatter photons. Opaque surfaces absorb or reflect them.

The Heat Equation

The fundamental law governing all heat transfer is the first law of thermodynamics, commonly referred to as the principle of conservation of energy. However, internal energy, U, is a rather inconvenient quantity to measure and use in simulations. Therefore, the basic law is usually rewritten in terms of the temperature, T. For a fluid, the resulting heat equation is:

(4-11)

where





• q is the heat flux by conduction (SI unit: W/m2)

• p is the pressure (SI unit: Pa)


• S is the strain-rate tensor (SI unit: 1/s):

• Q contains heat sources other than viscous heating (SI unit: W/m3)

CpTt------- u T+ q – :S T

---- T-------

p

pt------ u p+ – Q+ +=

S 12--- u u T+ =


For a detailed discussion of the fundamentals of heat transfer, see Ref. 1.

In deriving Equation 4-11, a number of thermodynamic relations have been used. The equation also assumes that mass is always conserved, which means that the density and the velocity must be related through:

The Heat Transfer interfaces use Fourier’s law of heat conduction, which states that the conductive heat flux, q, is proportional to the temperature gradient:

(4-12)

where k is the thermal conductivity (SI unit: W/(m·K)). In a solid, the thermal conductivity can be anisotropic (that is, it has different values in different directions). Then k becomes a tensor

and the conductive heat flux is given by

Specific heat capacity at constant pressure is the amount of energy required to raise one unit of mass of a substance by one degree while maintained at constant pressure. This quantity is also commonly referred to as specific heat or specific heat capacity.

t v + 0=

qi kTxi--------–=

k

kxx kxy kxz

kyx kyy kyz

kzx kzy kzz

=

qi kijTxj--------

j–=

Fourier’s law applies for symmetric thermal conductivity tensors. Non symmetric tensors lead to unphysical results.


230 | C H A P T E

The second term on the right-hand side of Equation 4-11 represents viscous heating in the fluid. An analogous term arises from the internal viscous damping of a solid. The operation “:” is a contraction and can in this case be written on the following form:

The third term represents pressure work and is the result of heating under adiabatic compression as well as some thermoacoustic effects. It is generally small for low Mach number flows. A similar term can be included to account for thermoelastic effects in solids.

Inserting Equation 4-12 into Equation 4-11, reordering the terms and ignoring viscous heating and pressure work put the heat equation into a more familiar form:

The Heat Transfer in Fluids interface solves this equation for the temperature, T. If the velocity is set to zero, the equation governing purely conductive heat transfer is obtained:

A Note on Heat Flux and Balance

The concept of heat flux is not as simple as it might first appear. The reason is that heat is not a conserved quantity. The conserved quantity is instead the total energy. Hence, there is both a heat flux and an energy flux that are similar but not identical.

This section briefly describes the theory for the variables for Total Energy Flux and Total Heat Flux, used when computing heat balance. The definitions of these postprocessing variables do not affect the computational results, only variables available for results analysis and visualization.

TO T A L E N E R G Y F L U X

The total energy flux for a fluid is equal to (Ref. 4, chapter 3.5)

(4-13)

Above, H0 is the total enthalpy

a:b anmbnm

m

n=


CpTt------- k– T + Q=

u H0 + k T – u qr+–


where in turn H is the enthalpy. In Equation 4-13 is the viscous stress tensor and qr is the radiative heat flux. in Equation 4-13 is the force potential. It has a simple form in some special cases, for example, for gravitational effects (Chapter 1.4 in Ref. 4), but it is in general rather difficult to derive. Potential energy is therefore often excluded and the total energy flux is approximated by

(4-14)

For a simple compressible fluid, the enthalpy, H, has the form (Ref. 5)

(4-15)

where p is the absolute pressure. The reference enthalpy, Href, is the enthalpy at reference temperature, Tref, and reference pressure, pref. Tref is 298.15 K and pref is one atmosphere. In theory, any value can be assigned to Href (Ref. 7), and COMSOL Multiphysics sets it to 0 J/kg by default.

The two integrals in Equation 4-15 are sometimes referred to as the sensible enthalpy (Ref. 7). These are evaluated by numerical integration. The second integral is only included for gas/liquid since it is commonly much smaller than the first integral for solids and it is identically zero for ideal gases.

TO T A L H E A T F L U X

The total heat flux vector is defined as (Ref. 6):

(4-16)

where E is the internal energy. It is related to the enthalpy via

H0 H 12--- u u +=

u H 12--- u u +

k T – u qr+–

H Href Cp Td

Tref

T

1--- 1 T

----

T-------

p

+

pd

pref

p

+ +=

For the evaluation of H to work, it is important that the dependencies of Cp, , and on the temperature are prescribed either via model inputs or as functions of the temperature variable. If Cp, , or depends on the pressure, that dependency must be prescribed either via a model input or by using the variable pA, which is the variable for the absolute pressure in COMSOL Multiphysics.

uE k T qr+–


232 | C H A P T E

(4-17)

Compared to the total energy flux, the total heat flux do not have viscous- and pressure-related terms.

C O N S E R V A T I O N E Q U A T I O N S I N D I F F E R E N T I A L F O R M S

According to the First Law of Thermodynamics, internal energy is the conserved quantity at the microscopic scale:

(4-18)

Here, Q and Wnc are the path-dependent heat and work contributions to the variation of internal energy dE. Some mass and momentum conservation equations are often solved together with the heat equation. They may take the form of the Navier-Stokes equations in Fluid Dynamics, Newton’s Laws of Motion in Solid Mechanics or Maxwell's equations in Electromagnetism for instance, and can be expressed by the following differential form of energy equation:

(4-19)

where dEk and dEp are the kinetic and potential energy variations, respectively. The variation of mechanical work from nonconservative forces, Wnc, is transmitted to the internal energy by the corresponding work, Wnc, as the last term of Equation 4-18. The sources Wnc include, amongst others, viscous heating and pressure work in Fluid Dynamics, Joule heating in Electromagnetism, friction and plastic deformation in Solid Mechanics. By combining Equation 4-18 and Equation 4-19, the complete First Law reads:

(4-20)

where W, equal to dEp, is the mechanical work from conservative forces.

H E A T B A L A N C E

This paragraph assumes a heat transfer model that only solves for the temperature T. The velocity field u and pressure field p are user-defined or computed from another physics interface. In this case, the heat balance in a domain follows the identity below (chapter 11.2 in Ref. 8), derived from Equation 4-18. It expresses the idea that internal energy variations in time and net heat flux are balanced by external heat and work sources.

H E p---+=

Ed Q Wnc–=

dEk dEp+ Wnc=

d E Ek+ Q W+=


(4-21)

The different variables in this formula are defined in Total Heat Flux and Total Energy Flux. For this equality to be true, the provided velocity field u and pressure field p must satisfy a mass and a momentum conservation equation such as the Navier-Stokes Equations or governing equations of continuum mechanics. The nonconservative work from the Navier-Stokes equations, WnsInt (definition in Table 4-2), contains both pressure work and viscous heating. The heat sources QInt include domain sources, interior boundary, edge and point sources, and radiative source at interior boundaries.

Four kinds of compensating heat power contributions are thus distinguished and available as COMSOL Multiphysics predefined variables:

• The total accumulated heat power (SI unit: W), dEiInt,

• the total net heat power (SI unit: W), ntfluxInt, integral on exterior boundaries of the total heat flux,

• the total heat source (SI unit: W), QInt,

• the total fluid losses (SI unit: W), WnsInt.

Table 4-2 summarizes the mathematical definitions of these variables.

tdd E d

uE kT– qr+ n d

ext+ QInt Wns Int–=

In 2D and 3D models, if isolated point or edge source is not adjacent to a boundary, these are not included in QInt. In this case, these need to be computed separately.

TABLE 4-2: GLOBAL POST-PROCESSING VARIABLE FOR TOTAL HEAT BALANCE

VARIABLE NAME MATHEMATICAL DEFINITION

dEiInt

ntfluxInt

QInt

WnsInt

tdd E d

uE kT– qr+ n dext

Q d Qb d

int Qr d

""+ +

pA u d – : u d

+


234 | C H A P T E

Here, ext and int denote the exterior and interior boundaries, respectively.

According to Equation 4-21, the following equality between COMSOL Multiphysics variables holds:

dEiInt + ntfluxInt = QInt - WnsInt

This is the most general form that can be used for time-dependent models. At steady-state, the formula is simplified. The accumulated heat power equals zero so the total net heat power, sum of incoming and outgoing powers, should correspond to the heat and work sources:

ntfluxInt = QInt - WnsInt

The sign convention used in COMSOL Multiphysics for QInt is the following: positive when energy is produced (as for a heater) and negative when energy is consumed (as for a cooler). For WnsInt, the losses that heat up the system are positive and the gains that cool down the system are negative.

For stationary models with convection by an incompressible flow, the heat balance becomes:

ntfluxInt = QInt

which corresponds to the conservation of convective and conductive flux as in:

E N E R G Y B A L A N C E

When the temperature T is solved together with additional mass and momentum equations from Fluid Dynamics for u and p, the total energy flux also becomes a conserved quantity and the following equation holds (chapter 11.1 in Ref. 8):

(4-22)

The different variables in this formula are defined in Total Heat Flux and Total Energy Flux. The work sources WInt are the contributions from custom volume forces. Three new variables are then useful to describe the energy powers involved in the system energy balance:

• The total accumulated energy power (SI unit: W), dEi0Int,

uE n dext kT n d

ext– QInt=

tdd E0 d

uH0 kT– u– qr+ n d

ext+ QInt WInt+=


• The total net energy power (SI unit: W), ntefluxInt, that in this case includes nonconservative work previously in WnsInt,

• The total work source (SI unit: W), WInt.

Their definitions are given in Table 4-3.

According to Equation 4-22, the following equality between COMSOL Multiphysics predefined variables holds:

dEi0Int + ntefluxInt = QInt + WInt

In stationary models, dEi0Int is zero so the energy balance simplifies into:

ntefluxInt = QInt + WInt

At steady-state, and without any additional heat source or volume force (QInt and WInt equal to zero), the integral of the net energy flux on all boundaries of the flow domain, ntefluxInt, vanishes. The corresponding integral of the net heat flux, on the other hand, does not, in general, vanish. It corresponds instead to the losses from mass and momentum equations, such as WnsInt for pressure work and viscous heating in fluids. Hence, energy is the conserved quantity, not heat.

Heat Transfer Variables

This section lists some predefined variables that are available for evaluating heat fluxes, sources and integral quantities used in energy balance. All the variable names start with the physics interface prefix. By default the Heat Transfer interface prefix is ht. As an

TABLE 4-3: GLOBAL POST-PROCESSING VARIABLE FOR TOTAL ENERGY BALANCE

VARIABLE NAME MATHEMATICAL DEFINITION

dEi0Int

ntefluxInt

QInt

WInt

tdd E0 d

uH0 kT– u– qr+ n dext

Q d Qb d

int Qr d

""+ +

W d


236 | C H A P T E

example, the variable named tflux can be analyzed using ht.tflux (as long as the physics interface prefix is ht).

TABLE 4-4: HEAT FLUX VARIABLES

VARIABLE NAME GEOMETRIC ENTITY LEVEL

dEiInt Total Accumulated Heat Power Global

ntfluxInt Total Net Heat Power Global

QInt Total Heat Source Global

WnsInt Total Fluid Losses Global

dEi0Int Total Accumulated Energy Power Global

ntefluxInt Total Net Energy Power Global

WInt Total Work Source Global

tflux Total Heat Flux Domains, boundaries

dflux Conductive Heat Flux Domains, boundaries

turbflux Turbulent Heat Flux Domains, boundaries

cflux Convective Heat Flux Domain, boundaries

trlflux Translational Heat Flux Domains, boundaries

teflux Total Energy Flux Domains, boundaries

chflux_u

chflux_d

chflux_z

Convective Out-of-Plane Heat Flux Out-of-plane domains (1D and 2D)

rflux_u

rflux_d

rflux_z

Radiative Out-of-Plane Heat Flux Out-of-plane domains (1D and 2D), boundaries

q0_u

q0_d

q0_z

Out-of-Plane Inward Heat Flux Out-of-plane domains (1D and 2D)

ntflux Normal Total Heat Flux Boundaries

ndflux Normal Conductive Heat Flux Boundaries

ncflux Normal Convective Heat Flux Boundaries

ntrlflux Normal Translational Heat Flux Boundaries

nteflux Normal Total Energy Flux Boundaries

ndflux_u Internal Normal Conductive Heat Flux, Upside

Interior boundaries

ndflux_d Internal Normal Conductive Heat Flux, Downside

Interior boundaries


G L O B A L V A R I A B L E S

In this paragraph, the variables presented are defined by integrals. A concise notation denotes the different domains of integration: is the geometry domain, ext stands for the exterior boundaries and int for the interior boundaries.

Total Accumulated Heat PowerThe total accumulated heat power variable, dEiInt, is the variation of internal energy per unit time in the domain:

ncflux_u Internal Normal Convective Heat Flux, Upside

Interior boundaries

ncflux_d Internal Normal Convective Heat Flux, Downside

Interior boundaries

ntrlflux_u Internal Normal Translational Heat Flux, Upside

Interior boundaries

ntrlflux_d Internal Normal Translational Heat Flux, Downside

Interior boundaries

ntflux_u Internal Normal Conductive Heat Flux, Upside

Interior boundaries

ntflux_d Internal Normal Conductive Heat Flux, Downside

Interior boundaries

nteflux_u Internal Normal Total Energy Flux, Upside

Interior boundaries

nteflux_d Internal Normal Total Energy Flux, Downside

Interior boundaries

rflux Radiative Heat Flux Boundaries

chflux Boundary Convective Heat Flux Boundaries

Qtot Domain Heat Sources Domains

Qbtot Boundary Heat Sources Boundaries

Qltot Line heat source (Line and Point Heat Sources)

Edges, Points (2D, 2Daxi)

Qptot Point heat source (Line and Point Heat Sources)

Points

TABLE 4-4: HEAT FLUX VARIABLES

VARIABLE NAME GEOMETRIC ENTITY LEVEL

dEiInttd

d E d=


238 | C H A P T E

Total Net Heat PowerThe total net heat power, ntfluxInt, is the integral of Total Heat Flux over all external boundaries:

It thus indicates the sum of incoming and outgoing total heat flux through the system.

Total Heat SourceThe total heat source, QInt, accounts for all domain sources, interior boundary, edge and point sources, and radiative sources at interior boundaries:

Total Fluid LossesThe total fluid losses, WnsInt, correspond to the work lost by a fluid by degradation of energy. These works are transmitted to the system through pressure work and viscous heating:

Total Accumulated Energy PowerThe total accumulated energy power, dEi0Int, is the variation of total internal energy per unit time in the domain:

where the total internal energy, E0, is defined as

Total Net Energy PowerThe total net heat power, ntefluxInt, is the integral of Total Energy Flux over all external boundaries:

ntfluxInt uE kT– qr+ n dext=

QInt Q d Qb d

int Qr d

""+ +=

WnsInt pA u d – : u d

+=

dEi0Inttd

d E0 d=

E0 E u u2

------------+=

ntefluxInt uH0 kT– u– qr+ n dext=


It thus indicates the sum of incoming and outgoing total energy flux through the system.

Total Work SourceThe total work source, WInt, sums all work contributions from custom forces:

D O M A I N H E A T F L U X E S

On domains the heat fluxes are vector quantities. Their definition can vary depending on the active physics nodes and selected properties.

Total Heat FluxOn domains the total heat flux, tflux, corresponds to the conductive and convective heat flux. For accuracy reasons the radiative heat flux is not included.

For solid domains, for example heat transfer in solids and biological tissue domains, the total heat flux is defined as:

For fluid domains (for example, Heat Transfer in Fluids), the total heat flux is defined as:

Conductive Heat FluxThe conductive heat flux variable, dflux, is evaluated using the temperature gradient and the effective thermal conductivity:

When the out-of-plane property is activated (1D and 2D only) the conductive heat flux is defined as follows:

• In 2D (dz is the domain thickness):

WInt W d=

See Radiative Heat Flux to evaluate the radiative heat flux.

tflux trlflux dflux+=

tflux cflux dflux+=

dflux keff T–=


240 | C H A P T E

• In 1D (Ac is the cross-section area):

In the general case keff is the thermal conductivity, k.

For heat transfer in fluids with turbulent flow, keff = k + kT, where kT is the turbulent thermal conductivity.

For heat transfer in porous media, keff = keq, where keq is the equivalent conductivity defined in the Heat Transfer in Porous Media feature.

Turbulent Heat FluxThe turbulent heat flux variable, turbflux, enables access to the part of the conductive heat flux that is due to turbulence.

Convective Heat FluxThe convective heat flux variable, cflux, is defined using the internal energy, E:

When the out-of-plane property is activated (1D and 2D only) the convective heat flux is defined as follows:


• In 1D (Ac is the domain thickness):

The internal energy, E, is defined as:

• ECpT for solid domains

• ECpT for ideal gas fluid domains

• EHp for other fluid domains

where H is the enthalpy defined in Equation 4-15.

dflux dzkeff T–=

dflux Ackeff T–=

turbflux kT T–=

cflux uE=

cflux dzuE=

cflux AcuE=


Translational Heat FluxSimilar to convective heat flux but defined for solid domains with translation. The variable name is trlflux.

Total Energy FluxThe total energy flux, teflux, is defined when viscous heating is enabled:

where the total enthalpy, H0, is defined as

O U T - O F - P L A N E D O M A I N F L U X E S

When the out-of-plane property is activated (1D and 2D only), out-of-plane domain fluxes are defined. If there are no out-of-plane physics features, they are evaluated to zero.

Convective Out-of-Plane Heat FluxThe convective out-of-plane heat flux, chflux, is generated by the Out-of-Plane Convective Heat Flux feature.

• In 2D:

upside:

downside:

• In 1D:

Radiative Out-of-Plane Heat FluxThe radiative out-of-plane heat flux, rflux, is generated by the Out-of-Plane Radiation feature.

• In 2D:

upside:

teflux uH0 dflux u+ +=

H0 H u u2

------------+=

chflux_u hu Text uT– =

chflux_d hd Text d T– =

chflux_z hz Text z T– =

rflux_u u Tamb u4 T4

– =


242 | C H A P T E

downside:

• In 1D:

Out-of-Plane Inward Heat FluxThe convective out-of-plane heat flux, q0, is generated by the Out-of-Plane Heat Flux feature.

• In 2D:

upside:

downside:

• In 1D:

B O U N D A R Y H E A T F L U X E S

All the domain heat fluxes (vector quantity) are also available as boundary heat fluxes. The boundary heat fluxes are then equal to the mean value of the heat fluxes on adjacent domains. In addition, normal boundary heat fluxes (scalar quantity) are available on boundaries.

Normal Total Heat FluxThe variable ntflux is defined as:

Normal Conductive Heat FluxThe variable ndflux is defined by on exterior boundaries as:

• ndfluxdflux_spatialT if the adjacent domain is on the downside,

• ndfluxuflux_spatialT if the adjacent domain is on the upside,

and, on interior boundaries, as: ndfluxuflux_spatialTdflux_spatialT2

Normal Convective Heat FluxThe variable ncflux is defined as:

rflux_d d Tamb d4 T4

– =

rflux_z z Tamb z4 T4

– =

q0_u hu Text u T– =

q0_d hd Text d T– =

q0_z hz Text z T– =

ntflux ndflux ncflux ntrlflux+ +=


Normal Translational Heat FluxThe variable ntrlflux is defined as

Normal Total Energy FluxThe variable nteflux is defined as:

Radiative Heat FluxOn boundaries the radiative heat flux, rflux, is a scalar quantity defined as:

where the terms respectively account for surface-to-ambient radiative flux.

Boundary Convective Heat FluxBoundary convective heat flux, chflux, is defined as the contribution from the Convective Heat Flux boundary condition:

When the out-of-plane property is activated (1D and 2D only) the boundary convective heat flux is defined as follows:


• In 1D (Ac is the cross section area):

I N T E R N A L B O U N D A R Y H E A T F L U X E S

The internal normal boundary heat fluxes (scalar quantity) are available on interior boundaries. They are calculated using the upside and the downside value of heat fluxes from the adjacent domains.

Internal Normal Conductive Heat Flux, UpsideThe variable ndflux_u is defined as:

ncflux mean cflux n=

ntrlflux mean trlflux n=

nteflux mean teflux n mean dflux n ndflux+–=

rflux Tamb4 T4

– G T4– qr net+ +=

chflux h Text T– =

chflux dzh Text T– =

chflux Ach Text T– =


244 | C H A P T E

Internal Normal Conductive Heat Flux, DownsideThe variable ndflux_d is defined as:

Internal Normal Convective Heat Flux, UpsideThe variable ncflux_u is defined as:

Internal Normal Convective Heat Flux, DownsideThe variable ncflux_d is defined as:

Internal Normal Translational Heat Flux, UpsideThe variable ntrlflux_u is defined as:

Internal Normal Translational Heat Flux, DownsideThe variable ntrlflux_d is defined as:

Internal Normal Total Heat Flux, UpsideThe variable ntflux_u is defined as:

Internal Normal Total Heat Flux, DownsideThe variable ntflux_d is defined as:

Internal Normal Total Energy Flux, UpsideThe variable nteflux_u is defined as:

Internal Normal Total Energy Flux, DownsideThe variable nteflux_d is defined as:

ndflux_u uflux_spatial T =

ndflux_d dflux_spatial T =

ncflux_u up cflux un=

ncflux_d down cflux dn=

ntrlflux_u up trlflux un=

ntrlflux_d down trlflux dn=

ntflux_u ndflux_u ncflux_u ntrlflux_u+ +=

ntflux_d ndflux_d ncflux_d ntrlflux_d+ +=

nteflux_u up teflux un up dflux un ndflux_u+–=


D O M A I N H E A T S O U R C E S

The sum of the domain heat sources added by different physics features is available in the variable, Qtot (SI unit: W/m3). This variable Qtot is the sum of:

• Q’s which are the heat sources added by the Heat Source (described for the Heat

Transfer interface) and Electromagnetic Heat Source (described for the Joule Heating interface in the COMSOL Multiphysics Reference Manual) features.

B O U N D A R Y H E A T S O U R C E S

The sum of the boundary heat sources added by different boundary conditions is available in the variable, Qb,tot (SI unit: W/m2). This variable Qbtot is the sum of:

• Qb which is the boundary heat source added by the Boundary Heat Source boundary condition.

• Qsh which is the boundary heat source added by the Boundary Electromagnetic Heat Source boundary condition (described for the Joule Heating interface in the COMSOL Multiphysics Reference Manual).

L I N E A N D PO I N T H E A T S O U R C E S

The sum of the line heat sources is available in a variable called Qltot (SI unit: W/m).

The sum of the point heat sources is available in a variable called Qptot (SI unit: W).

About the Boundary Conditions for the Heat Transfer Interfaces

TE M P E R A T U R E A N D H E A T F L U X B O U N D A R Y C O N D I T I O N S

The heat equation accepts two basic types of boundary conditions: specified temperature and specified heat flux. The specified condition is of constraint type and prescribes the temperature on a boundary:

while the latter specifies the inward heat flux

where

• q is the conductive heat flux vector (SI unit: W/m2), q = kT.

nteflux_d down teflux dn down dflux dn ndflux_d+–=

T T0= on

n– q q0= on


246 | C H A P T E

• n is the normal vector on the boundary.

• q0 is the inward heat flux (SI unit: W/m2), normal to the boundary.

The inward heat flux, q0, is often a sum of contributions from different heat transfer processes (for example, radiation and convection). The special case q0 0 is called thermal insulation.

A common type of heat flux boundary conditions is one for which q0h·TinfT, where Tinf is the temperature far away from the modeled domain and the heat transfer coefficient, h, represents all the physics occurring between the boundary and “far away.” It can include almost anything, but the most common situation is that h represents the effect of an exterior fluid cooling or heating the surface of a solid, a phenomenon often referred to as convective cooling or heating.

O V E R R I D I N G M E C H A N I S M F O R H E A T TR A N S F E R B O U N D A R Y C O N D I T I O N S

Many boundary conditions are available in heat transfer. Some of them can coexist (for example, Heat Flux and Highly Conductive Layer). Others cannot coexist (for example, Heat Flux and Thermal Insulation).

Several categories of boundary condition exist in heat transfer. Table 4-5 gives the overriding rules for these groups.

1 Temperature, Convective Outflow, Open Boundary, Inflow Heat Flux

2 Thermal Insulation, Symmetry, Periodic Heat Condition

3 Highly Conductive Layer

4 Heat Flux, Convective Heat Flux

5 Boundary Heat Source, Radiation Group

6 Surface-to-Surface Radiation, Diffuse Mirror, Prescribed Radiosity, Surface-to-Ambient Radiation

This module contains a set of correlations for convective heat flux and heating. See About the Heat Transfer CoefficientsAbout the Heat Transfer Coefficients.

This section includes information for features that might require additional modules.


7 Opaque Surface, Incident Intensity, Continuity on Interior Boundaries

8 Thin Thermally Resistive Layers, Thermal Contact

When there is a boundary condition A above a boundary condition B in the model tree and both conditions apply to the same boundary, use Table 4-5 to determine if A is overridden by B or not:

• Locate the line that corresponds to the A group (see above the definition of the groups). In the table above only the first member of the group is displayed.

• Locate the column that corresponds to the group of B.

• If the corresponding cell is empty A and B contribute. If it contains an X, B overrides A.

Example 1

TABLE 4-5: OVERRIDING RULES FOR HEAT TRANSFER BOUNDARY CONDITIONS

A\B 1 2 3 4 5 6 7 8

1-Temperature X X X X

2-Thermal Insulation X X X

3-Highly Conductive Layer

X X

4-Heat Flux X X

5-Boundary heat source

6-Surface-to-surface radiation

X X

7-Opaque Surface X

8-Thin Thermally Resistive Layer

X X

Group 4 and group 5 boundary conditions are always contributing. That means that they never override any other boundary condition. But they might be overridden.

Surface-to-Surface radiation requires the Heat Transfer Module.


248 | C H A P T E

Consider a boundary where Temperature is applied. Then a Surface-to-Surface Radiation boundary condition is applied on the same boundary afterward.

• Temperature belongs to group 1.

• Surface-to-surface radiation belongs to group 6.

• The cell on the line of group 1 and the column of group 6 is empty so Temperature and Surface-to-Surface radiation contribute.

Example 2Consider a boundary where Convective Heat Flux is applied. Then a Symmetry boundary condition is applied on the same boundary afterward.

• Convective Heat Flux belongs to group 4.

• Symmetry belongs to group 2.

• The cell on the line of group 4 and the column of group 2 contains an X so Convective Heat Flux is overridden by Symmetry.

Radiative Heat Transfer in Transparent Media

This discussion so far has considered heat transfer by means of conduction and convection. A third mechanism for heat transfer is radiation. Consider an environment with fully transparent or fully opaque objects. Thermal radiation denotes the stream of electromagnetic waves emitted from a body at a certain temperature.

In Example 2 above, if Symmetry followed by Convective Heat Flux is added, the boundary conditions contribute.


D E R I V I N G T H E R A D I A T I V E H E A T F L U X

Figure 4-2: Arriving irradiation (left), leaving radiosity (right).

Consider Figure 4-2. A point is located on a surface that has an emissivity , reflectivity , absorptivity , and temperature T. Assume that the body is opaque, which means that no radiation is transmitted through the body. This is true for most solid bodies.

The total arriving radiative flux at is named the irradiation, G. The total outgoing radiative flux is named the radiosity, J. The radiosity is the sum of the reflected radiation and the emitted radiation:

(4-23)

The net inward radiative heat flux, q, is then given by the difference between the irradiation and the radiosity:

(4-24)

Using Equation 4-23 and Equation 4-24 J can be eliminated and a general expression is obtained for the net inward heat flux into the opaque body based on G and T.

(4-25)

Most opaque bodies also behave as ideal gray bodies, meaning that the absorptivity and emissivity are equal, and the reflectivity is therefore obtained from the following relation:

(4-26)

Thus, for ideal gray bodies, q is given by:

G

,T ,T

J =G + T4

x x

x

xx

J G T4+=

q G J–=

q 1 – G T4–=

1 –= =


250 | C H A P T E

(4-27)

This is the expression used for the radiation boundary condition.

R A D I A T I O N TY P E S

It is common to differentiate between two types of radiative heat transfer: surface-to-ambient radiation and surface-to-surface radiation. Equation 4-27 holds for both radiation types, but the irradiation term, G, is different for each of them. The Heat Transfer interface supports radiation.

S U R F A C E - T O - A M B I E N T R A D I A T I O N

Surface-to-ambient radiation assumes the following:

• The ambient surroundings in view of the surface have a constant temperature, Tamb.

• The ambient surroundings behave as a blackbody. This means that the emissivity and absorptivity are equal to 1, and the reflectivity is 0.

These assumptions allow the irradiation to be explicitly expressed as

(4-28)

Inserting Equation 4-28 into Equation 4-27 results in the net inward heat flux for surface-to-ambient radiation

(4-29)

For boundaries where a surface-to-ambient radiation is specified, COMSOL Multiphysics adds this term to the right-hand side of Equation 4-29.

Consistent and Inconsistent Stabilization Methods for the Heat Transfer Interfaces

The different versions of the Heat Transfer interface have the advanced option to set the stabilization method parameters. This section provides information pertaining to

q G T4– =

Surface-to-surface radiation requires the Heat Transfer Module.

G Tamb4

=

q Tamb4 T4

– =


these options. To display the stabilization sections, click the Show button ( ) and select Stabilization.


This section contains two consistent stabilization methods: streamline diffusion and crosswind diffusion. These are consistent stabilization methods, which means that they do not perturb the original transport equation.

The consistent stabilization methods are activate by default. A stabilization method is active when the corresponding check box is selected.

Streamline DiffusionStreamline diffusion is active by default and should remain active for optimal performance for heat transfer in fluids or other applications that include a convective or translational term.

Crosswind DiffusionStreamline diffusion introduces artificial diffusion in the streamline direction. This is often enough to obtain a smooth numerical solution provided that the exact solution of the heat equation does not contain any discontinuities. At sharp gradients, however, undershoots and overshoots can occur in the numerical solution. Crosswind diffusion addresses these spurious oscillations by adding diffusion orthogonal to the streamline direction—that is, in the crosswind direction.


This section contains a single stabilization method: isotropic diffusion. Adding isotropic diffusion is equivalent to adding a term to the physical diffusion coefficient. This means that the original problem is not solved, which is why isotropic diffusion is an inconsistent stabilization method. Although, the added diffusion definitely attenuates spurious oscillations, try to minimize the use of isotropic diffusion.

By default there is no isotropic diffusion. To add isotropic diffusion, select the Isotropic

diffusion check box. The field for the tuning parameter id then becomes available. The default value is 0.25; increase or decrease the value of id to increase or decrease the amount of isotropic diffusion.


• Stabilization Techniques

• Stabilization


252 | C H A P T E

References for the Heat Transfer Interfaces

1. F.P. Incropera, D.P. DeWitt, T.L. Bergman and A.S. Lavine, Fundamentals of Heat and Mass Transfer, John Wiley & Sons, 6th ed., 2006.

2. R. Codina, “Comparison of Some Finite Element Methods for Solving the Diffusion-Convection-Reaction Equation,” Comp. Meth.Appl. Mech. Engrg, vol. 156, pp. 185–210, 1998.

3. A. Bejan, Heat Transfer, John Wiley & Sons, 1993.

4. G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, 2000.


6. M. Kaviany, Principles of Convective Heat Transfer, 2nd ed., Springer, 2001.

7. T. Poinsot and D. Veynante, Theoretical and Numerical Combustion, 2nd ed., Edwards, 2005.

8. R.B. Bird, W.E. Stewart and E.N. Lightfoot, Transport Phenomena, 2nd ed., John Wiley & Sons, 2007.

9. W. Wagner, and H-J Kretzschmar, International Steam Tables, 2nd ed., Springer, 2008.


Th eo r y o f Ou t - o f - P l a n e Hea t T r a n s f e r

When the object to model in COMSOL Multiphysics is thin or slender enough along one of its geometry dimensions, there is usually only a small variation in temperature along the object’s thickness or cross section. For such objects, it is computationally more efficient to reduce the model geometry to 2D or even 1D and to use the out-of-plane heat transfer mechanism. Figure 4-3 shows examples of possible situations in which this type of geometry reduction can be applied.

Figure 4-3: Geometry reduction from 3D to 1D (top) and from 3D to 2D (bottom).

The reduced geometry does not include all the boundaries of the original 3D geometry. For example, the reduced geometry does not represent the upside and downside surfaces of the plate in Figure 4-3 as boundaries. Instead, heat transfer through these boundaries appears as sources or sinks in the thickness-integrated version of the heat equation used when out-of-plane heat transfer is active.

Equation Formulation

When out-of-plane heat transfer is enabled, the equation for heat transfer in solids, Equation 4-1 is replaced by

(4-30)

qdown

qup

q

Out-of-Plane Heat Transfer Nodes

dzCpTt------- – dzkT dzQ=

T H E O R Y O F O U T - O F - P L A N E H E A T TR A N S F E R | 253

254 | C H A P T E

where dz is the thickness of the domain in the out-of-plane direction. The equation for heat transfer in fluids, Equation 4-2, is replaced by

(4-31)

The Pressure Work attribute for Solids and Fluids and the Viscous Heating attribute for Fluids are not available when out-of-plane heat transfer is activated.

Activating Out-of-Plane Heat Transfer and Thickness

Using a 1D or 2D model, activate the physics features for out-of-plane heat transfer and the thickness property by clicking the main Heat Transfer node and selecting the Out-of-plane heat transfer check box under Physical Model.

CpdzTt------- u T+ dzkT dzQ+=

Heat Source nodes that are added to a model with out-of-plane heat transfer enabled are multiplied by the thickness, dz. Boundary conditions are also adjusted.


Th eo r y f o r t h e Hea t T r a n s f e r i n Po r ou s Med i a I n t e r f a c e

The Heat Transfer in Porous Media Interface uses the following version of the heat equation (Ref. 1):

(4-32)

with the following material properties:

• is the fluid density.

• Cp is the fluid heat capacity at constant pressure.

• (Cp)eq is the equivalent volumetric heat capacity at constant pressure.

• keq is the equivalent thermal conductivity (a scalar or a tensor if the thermal conductivity is anisotropic).

• u is the fluid velocity field, either an analytic expression or the velocity field from a fluid-flow interface. u should be interpreted as the Darcy velocity, that is, the volume flow rate per unit cross-sectional area. The average linear velocity (the velocity within the pores) can be calculated as uLuL, where L is the fluid’s volume fraction, or equivalently the porosity.

• Q is the heat source (or sink). Add one or several heat sources as separate physics features.

The equivalent thermal conductivity of the solid-fluid system, keq, is related to the conductivity of the solid kp and to the conductive of the fluid, k by

The equivalent volumetric heat capacity of the solid-fluid system is given by

Here p denotes the solid material’s volume fraction, which is related to the volume fraction of the liquid L (or porosity) by

Cp eqTt------- Cpu T+ keqT Q+=

keq pkp Lk+=

Cp eq ppCp p LCp+=

L p+ 1=

T H E O R Y F O R T H E H E A T TR A N S F E R I N P O R O U S M E D I A I N T E R F A C E | 255

256 | C H A P T E

For a steady-state problem the temperature does not change with time, and the first term on the left-hand side of Equation 4-32 disappears.

Reference for the Heat Transfer in Porous Media Interface

1. J. Bear and Y. Bachmat, Introduction to Modeling of Transport Phenomena in Porous Media, Kluwer Academic Publisher, 1990.


Abou t t h e Hea t T r a n s f e r C o e f f i c i e n t s

One of the most common boundary conditions when modeling heat transfer is convective cooling or heating whereby a fluid cools or heats a surface by natural or forced convection. In principle, it is possible to model this process in two ways:

• Use a heat transfer coefficient on the surfaces

• Extend the model to describe the flow and heat transfer in the surrounding fluid

The second approach is the correct approach if the geometry or the external flow is complicated. The Heat Transfer Module includes the Conjugate Heat Transfer interface for this purpose. However, such a simulations can become costly, both in terms of computational time and memory requirement.

The first method is simple, yet powerful and efficient. The convective heat flux on the boundaries in contact with the fluid is then modeled as being proportional to the temperature difference across a fictitious thermal boundary layer. Mathematically, the heat flux is described by the equation

where h is a heat transfer coefficient and Tinf the temperature of the external fluid far from the boundary.

The main difficulty in using heat transfer coefficients is in calculating or specifying the appropriate value of the h coefficient. That coefficient depends on the fluid’s material properties, and the surface temperature—and, for forced-convection, also on the fluid’s flow rate. In addition, the geometrical configuration affects the coefficient. The Heat Transfer interface provides built-in functions for the heat transfer coefficients. For most engineering purposes, the use of such coefficients is an accurate and numerically efficient modeling approach.

In this section:

• Heat Transfer Coefficient Theory

• Nature of the Flow—the Grashof Number

• Heat Transfer Coefficients — External Natural Convection

• Heat Transfer Coefficients — Internal Natural Convection

• Heat Transfer Coefficients — External Forced Convection

n– k– T h Tinf T– =

A B O U T T H E H E A T TR A N S F E R C O E F F I C I E N T S | 257

258 | C H A P T E

• Heat Transfer Coefficients — Internal Forced Convection

• References for the Heat Transfer Coefficients

Heat Transfer Coefficient Theory

It is possible to divide the convective heat flux into four main categories depending on the type of convection condition (natural or forced) and on the type of geometry (internal or external flow). In addition, these four cases can all experience either laminar or turbulent flow conditions, resulting in a total of eight types of convection, as in Figure 4-4.

Figure 4-4: The eight possible categories of convective heat flux.

The difference between natural and forced convection is that in the latter case an external force such as a fan creates the flow. In natural convection, buoyancy forces induced by temperature differences together with the thermal expansion of the fluid drives the flow.



Natural Forced

External

Internal

Laminar Flow

Turbulent Flow


Heat transfer handbooks generally contain a large set of empirical and theoretical correlations for h coefficients. This module includes a subset of them. The expressions are based on the following set of dimensionless numbers:

• The Nusselt number, NuL(Re, Pr, Ra)hL/k

• The Reynolds number, ReLU L/

• The Prandtl number, PrCp/k

• The Rayleigh number, RaGr Pr 2 gCp T L3/(k)

where

• h is the heat transfer coefficient (SI unit: W/(m2·K)).

• L is the characteristic length (SI unit: m).

• T is the temperature difference between the surface and the external fluid bulk (SI unit: K).

• g is the acceleration of gravity (SI unit: m/s2).

• k is the thermal conductivity of the fluid (SI unit: W/(m·K)).

• is the fluid density (SI unit: kg/m3).

• U is the bulk velocity (SI unit: m/s).

• is the dynamic viscosity (SI unit: Pa·s).

• Cp is the heat capacity of the fluid (SI unit: J/(kg·K)).

• is the thermal expansivity (SI unit: 1/K)

Further, Gr refers to the Grashof number, which is the squared ratio of the viscous time scale to the buoyancy time scale multiplied by the Reynolds number.

Nature of the Flow—the Grashof Number

In cases of externally driven flow, such as forced convection, the nature of the flow is characterized by the Reynolds number, Re, which describes the ratio of the inertial forces to the viscous forces. However, the velocity scale is initially unknown for internally driven flows such as natural convection. In such cases the Grashof number, Gr, characterizes the flow. It describes the ratio of the time scales for viscous diffusion in the fluid and the internal driving force (the buoyancy force). Like the Reynolds number it requires the definition of a length scale, the fluid’s physical properties, and the temperature scale (temperature difference). The Grashof number is defined as:


260 | C H A P T E

where g is the acceleration of gravity, is the fluid’s coefficient of volumetric thermal expansion, Ts denotes the temperature of the hot surface, T0 equals the temperature of the surrounding air, L is the length scale, represents the fluid’s dynamic viscosity, and its density.

In general, the coefficient of volumetric thermal expansion is given by

which for an ideal gas reduces to

The transition from laminar to turbulent flow occurs at a Gr value of 109; the flow is turbulent for larger values.

Heat Transfer Coefficients — External Natural Convection

VE R T I C A L WA L L

The following correlations correspond to equations 9.26 and 9.27 in Ref. 1:

(4-33)

where L, the height of the wall, is a correlation input and

(4-34)

GrLg Ts T0– L3

2--------------------------------------=

1---

T-------

p

–=

1 T=

h

kL---- 0.68

0.67RaL1 4/

1 0.492kCp

------------------- 9 16/

+ 4 9/----------------------------------------------------------+

RaL 109

kL---- 0.825

0.387RaL1 6/

1 0.492kCp

------------------- 9 16/

+ 8 27/-------------------------------------------------------------+

2

RaL 109

=

RaL

g T p Cp T Text– L3

k---------------------------------------------------------------------------=


where in turn g is the acceleration of gravity equal to 9.81 m/s2. All material properties are evaluated at TText2.

I N C L I N E D WA L L

The following correlations correspond to equations 9.26 and 9.27 in Ref. 1 (the same as for a vertical wall):

(4-35)

where L, the length of the wall, is a correlation input and is the tilt angle (the angle between the wall and the vertical direction, for vertical walls). These correlations are valid for 60° 60°.

The definition of Raleigh number, RaL, is analogous to the one for vertical walls and is given by the following:

(4-36)

where in turn g denotes the gravitational acceleration, equal to 9.81 m/s2.

For turbulent flow, 1 is used instead of in the expression for h, because this gives better accuracy (see Ref. 2).

The laminar-turbulent transition depends on (see Ref. 2). Unfortunately, little data is available about transition. There is some data available in Ref. 2 but this data is only approximative, according to the authors. In addition, data is only provided for water

h

kL---- 0.68

0.67 Racos L 1 4/

1 0.492kCp

------------------- 9 16/

+ 4 9/----------------------------------------------------------+

RaL 109

kL---- 0.825

0.387RaL1 6/

1 0.492kCp

------------------- 9 16/

+ 8 27/-------------------------------------------------------------+

2

RaL 109

=

0=

RaL


k---------------------------------------------------------------------------=

cos

According to Ref. 1., correlations for inclined walls are only satisfactory for the top side of a cold plate or the down face of a hot plate. Hence, these correlations are not recommended for the bottom side of a cold face and for the top side of a hot plate.


262 | C H A P T E

(Pr around 6). For this reasons, we define a flow as turbulent, independently of the value, when

All material properties are evaluated at TText2.

H O R I Z O N T A L P L A T E , U P S I D E

The following correlations correspond to equations 9.30–9.32 in Ref. 1 but can also be found as equations 7.77 and 7.78 in Ref. 2.

If TText, then

(4-37)

while if T Text, then

(4-38)

RaL is given by Equation 4-34, and L, the plate diameter (defined as area/perimeter, see Ref. 2) is a correlation input. The material data are evaluated at TText2.

H O R I Z O N T A L P L A T E , D O W N S I D E

Equation 4-37 is used when T Text and Equation 4-38 is used when T Text. Otherwise it is the same implementation as for Horizontal plate, upside.

Heat Transfer Coefficients — Internal Natural Convection

N A R R O W C H I M N E Y, P A R A L L E L P L A T E S

If RaL HL and T Text, then

(4-39)

where L, the plate distance, and H, the chimney height, are correlation inputs (equation 7.96 in Ref. 2). RaL is given by Equation 4-34. The material data are evaluated at TText2.


k--------------------------------------------------------------------------- 109

h

kL----0.54RaL

1 4/ RaL 107

kL----0.15RaL

1 3/ RaL 107

=

h kL----0.27RaL

1 4/=

h kH----- 1

24------RaL=


N A R R O W C H I M N E Y, C I R C U L A R TU B E

If RaDHD, then

where D, the tube diameter, and H, the chimney height, are correlation inputs (table 7.2 in Ref. 2 with DhD). RaD is given by Equation 4-34 with L replaced by D. The material data are evaluated at TText2.

Heat Transfer Coefficients — External Forced Convection

P L A T E , A V E R A G E D TR A N S F E R C O E F F I C I E N T

This correlation is a combination of equations 7.34 and 7.41 in Ref. 1:

(4-40)

where Prcpk and ReLUextL. L, the plate length and Uext, the exterior velocity are correlation inputs. The material data are evaluated at TText2.

P L A T E , L O C A L TR A N S F E R C O E F F I C I E N T

This correlation corresponds to equations 5.79b and 5.131 in Ref. 2:

(4-41)

where Prcpk and RexUextx. x, the position along the plate, and Uext, the exterior velocity are correlation inputs. The material data are evaluated at TText2.

Heat Transfer Coefficients — Internal Forced Convection

I S O T H E R M A L TU B E

This correlation corresponds to equations 8.55 and 8.61 in Ref. 1:

h kH----- 1

128----------RaD=

h2k

L----

0.3387Pr1 3/ ReL1 2/

1 0.0468 Pr 2 3/+ 1 4/--------------------------------------------------------------- ReL 5 10 5

2kL----Pr1 3/ 0.037ReL

4 5/ 871– ReL 5 10 5

=

h

kmax x eps ----------------------------------0.332Pr1 3/ Rex

1 2/ Rex 5 10 5

kmax x eps ----------------------------------0.0296Pr1 3/ Rex

4 5/ Rex 5 10 5

=


264 | C H A P T E

(4-42)

where Prcpk, ReDUextD and n0.3 if TText and n0.4 if T Text. D, the tube diameter and Uext, the exterior velocity, are correlation inputs. All material data are evaluated at Text except T which is evaluated at the wall temperature, T.

References for the Heat Transfer Coefficients

1. F.P. Incropera and D.P. DeWitt, Fundamentals of Heat and Mass Transfer, 5th ed. John Wiley & Sons, 2002.

2. A. Bejan, Heat Transfer, John Wiley & Sons, 1993.

h

kD----3.66 ReD 2500

kD----0.027ReD

4 5/ Prn T ------------ 0.14

ReD 2500

=


T H E N O N - I S O T

Th e Non - I s o t h e rma l F l ow and Con j u g a t e Hea t T r a n s f e r , L am i n a r F l ow and Tu r bu l e n t F l ow I n t e r f a c e s

The following sections list all the physics interfaces and the associated physics nodes under the Non-Isothermal Flow branch. The descriptions follow a structured order as defined by the order in the branch. Because many of the physics interfaces are integrated with each other, some features described also cross reference to other physics interfaces. At the end of this section is a summary of the theory that goes towards deriving the physics interfaces under the Non-Isothermal Flow branch.

In this section:

• The Non-Isothermal Flow, Laminar Flow Interface

• The Conjugate Heat Transfer, Laminar Flow Interface

• The Turbulent Flow, k- and Turbulent Flow Low Re k- Interfaces

• The Turbulent Flow, Spalart-Allmaras Interface

• The Turbulent Flow, SST Interface


The Non-Isothermal Flow, Laminar Flow Interface

The Non-Isothermal Flow (nitf) version of the Laminar Flow interface ( ), found under the Fluid Flow>Non-Isothermal Flow branch ( ) when adding a physics interface, combines the heat equation with the equations for laminar flow.

• Modeling Heat Transfer in the CFD Module


• Domain, Boundary, Edge, Point, and Pair Nodes Settings for the NITF Interfaces


H E R M A L F L O W A N D C O N J U G A T E H E A T TR A N S F E R , L A M I N A R F L O W A N D TU R B U L E N T F L O W I N T E R F A C E S |

266 | C H A P T E

The physics interface can be used to simulate fluid flows where the fluid properties depend on temperature. Models can also include heat transfer in solids as well as surface-to-surface radiation and radiation in participating media, with the Heat Transfer Module. The physics interface supports low Mach number (typically less than 0.3) flows, as well as non-Newtonian fluids.

The Non-Isothermal Laminar Flow interface solves for conservation of energy, mass and momentum in fluids and for conservation of energy in solids. The physics interface can be used for stationary and time-dependent analyses.

When this physics interface is added, the following default nodes are also added in the Model Builder—Non-Isothermal Flow, Fluid, Wall, Thermal Insulation, and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions, volume forces, or heat sources. You can also right-click the node to select physics from the context menu.


The interface identifier is used primarily as a scope prefix for variables defined by the physics interface. Refer to such interface variables in expressions using the pattern <identifier>.<variable_name>. In order to distinguish between variables belonging to different physics interfaces, the identifier string must be unique. Only letters, numbers and underscores (_) are permitted in the Identifier field. The first character must be a letter.

The default identifier (for the first physics interface in the model) is nitf.


The default setting is to have All domains in the component define the dependent variables and the equations. To choose specific domains, select Manual from the Selection list.


Define physics interface properties to control the overall type of model.

Neglect Inertial Term (Stokes Flow)—All InterfacesSelect the Neglect inertial term (Stokes flow) check box to model flow at very low Reynolds numbers for which the inertial term in the Navier-Stokes equations can be neglected. The physics interface then solves the linear Stokes equations. This flow type is referred to as creeping flow or Stokes flow and can occur in microfluidics (and MEMS devices), where the flow length scales are very small.



Turbulence Model Type

Shallow Channel Approximation


The dependent variables (field variables) are the Velocity field u (SI unit: m/s), the Pressure p (SI unit: Pa), and the Temperature T (SI unit: K). The default values are 0 m/s for the velocity, 0 Pa for the pressure, and 293.15 K for the temperature. The names can be changed but the names of fields and dependent variables must be unique within a component.

For turbulence modeling and heat radiation, there are additional dependent variables for the turbulent dissipation rate, turbulent kinetic energy, reciprocal wall distance, and surface radiosity.



Select a Discretization of fluids—P1+P1 (the default), P2+P1, or P3+P2. The first term describes the element order for the velocity components, and the second term is the order for the pressure. The element order for the temperature is set to follow the velocity order, so the temperature order is 1 for P1+P1, 2 for P2+P1, and 3 for P3+P2.

Select a Surface radiosity—Linear (the default), Quadratic, Cubic, Quartic, or Quintic (2D axisymmetric and 2D models only).

By definition, no turbulence model is needed when studying laminar flows. The default Turbulence model type is None. However, if the default Turbulence model type selected is RANS, the additional turbulence model settings are made available. The node is still called Non-Isothermal Flow or Conjugate Heat Transfer with a number added at the end of the name to indicate the change.

For 2D models, the Shallow channel approximation and out-of-plane heat

transfer check box simultaneously activates the shallow channel approximation for the Navier-Stokes equations and the out-of-plane features for heat transfer. When selected, enter the domain Thickness dz (SI unit: m).


268 | C H A P T E

Specify the Value type when using splitting of complex variables—Real (the default) or Complex.

C O N S I S T E N T A N D I N C O N S I S T E N T S T A B I L I Z A T I O N

To display this section, click the Show button ( ) and select Stabilization. Any settings unique to this physics interface are listed below.

• The consistent stabilization methods are applicable to the Heat and flow equations.

• The Isotropic diffusion inconsistent stabilization method can be activated for both the Heat equation and the Navier-Stokes equations.



The Use pseudo time stepping for stationary equation form check box is active per default for the Non-Isothermal Flow interface. It adds pseudo time derivatives to the momentum and heat equations when the Stationary equation form is used. When selected, also choose a CFL number expression—Automatic (the default) or Manual. Automatic sets the local CFL number (from the Courant–Friedrichs–Lewy condition) to the built-in variable CFLCMP which in turn triggers a PID regulator for the CFL number. If Manual is selected, enter a Local CFL number CFLloc.

By default the Enable conversions between material and spatial frames check box is selected.

• About Frames in Heat Transfer and Pseudo Time Stepping in the COMSOL Multiphysics Reference Manual



• For settings window details for the Heat Transfer in Solids feature, see The Heat Transfer Interface

• The Conjugate Heat Transfer, Laminar Flow Interface



The Conjugate Heat Transfer, Laminar Flow Interface

The Conjugate Heat Transfer version of the Laminar Flow (nitf) interface ( ), found under the Heat Transfer>Conjugate Heat Transfer branch ( ), combines the heat equation with the equations for laminar flow.

The physics interface can be used to simulate fluid flows where the fluid properties depend on temperature. Models can also include heat transfer in solids as well as surface-to-surface radiation and radiation in participating media, with the Heat Transfer Module. The physics interface supports low Mach number (typically less than 0.3) flows, as well as non-Newtonian fluids.

A Heat Transfer in Solids model is active by default on all domains. A Fluid model is also added by default with an empty domain selection. All functionality for including other models, like turbulent flow or surface-to-surface radiation, is also available.

The Conjugate Heat Transfer, Laminar Flow interface solves for conservation of energy, mass and momentum in fluids and for conservation of energy in solids. The physics interface can be used for stationary and time-dependent analyses.

When this interface is added, the following default nodes are also added in the Model

Builder—Conjugate Heat Transfer, Heat Transfer in Solids (with domain selection set to All domains), Thermal Insulation, Wall, Fluid (with empty initial domain selection), and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions, volume forces, or heat sources. You can also right-click a node to select physics from the context menu.


• The Non-Isothermal Flow, Laminar Flow Interface


• The Heat Transfer Interface for settings window details for the Heat

Transfer in Solids feature.

Fluid Damper: model library path CFD_Module/Non-Isothermal_Flow/fluid_damper


270 | C H A P T E

The Turbulent Flow, k- and Turbulent Flow Low Re k- Interfaces

These predefined multiphysics couplings consist of a Turbulent Flow interface, using a compressible formulation, in combination with a Heat Transfer interface.

Most of the settings options are the same as for The Non-Isothermal Flow, Laminar Flow Interface, except where noted below. From the Physics toolbar, add other nodes that implement, for example, boundary conditions, volume forces, or heat sources. You can also right-click the node to select physics from the context menu.

The Conjugate Heat Transfer and Non-Isothermal Flow, Turbulent Flow, k- Interfaces The Non-Isothermal Turbulent Flow, k- and the Conjugate Heat Transfer, Turbulent Flow,

k- interface, combine the heat equation with the equations for turbulent flow.

The physics interfaces can be used to model energy transport in single-phase flows at high Reynolds numbers. Models can also include heat transfer in solids as well as surface-to-surface radiation and radiation in participating media, with the Heat Transfer Module. The physics interface supports low Mach number (typically less than 0.3) flows.

For the Conjugate Heat Transfer, Turbulent Flow, k- interface, a Heat Transfer in Solids model is active by default on all domains. A Fluid model is also added by default with an empty domain selection. All functionality for including other models, like different turbulent flow models or surface-to-surface radiation, is also available.

The Non-Isothermal Turbulent Flow, k- and Conjugate Heat Transfer, Turbulent Flow, k- interfaces solve for conservation of energy, mass and momentum in fluids and for conservation of energy in solids. Turbulence effects are modeled using the standard two-equation k- model with realizability constraints. Flow and heat transfer close to walls are modeled using wall functions.

The physics interfaces can be used for stationary and time-dependent analyses.

The Conjugate Heat Transfer and Non-Isothermal Flow, Turbulent Flow, Low Re-k- Interfaces The Non-Isothermal Turbulent Flow, Low Re k- and the Conjugate Heat Transfer,

Turbulent Flow, Low Re k- interfaces combine the heat equation with the equations for turbulent flow.

The physics interfaces can be used to model energy transport in single-phase flows at high Reynolds numbers. Models can also include heat transfer in solids as well as surface-to-surface radiation and radiation in participating media, with the Heat



Transfer Module. The physics interface supports low Mach number (typically less than 0.3) flows.

For the Conjugate Heat Transfer, Turbulent Flow, Low Re k- interface, a Heat Transfer in

Solids model is active by default on all domains. A Fluid model is also added by default with an empty domain selection. All functionality for including other models, like different turbulent flow models or surface-to-surface radiation, is also available.

The Non-Isothermal Turbulent Flow, Low Re k- and the Conjugate Heat Transfer,

Turbulent Flow, Low Re k- interfaces solve for conservation of energy, mass, and momentum in fluids and for conservation of energy in solids. Turbulence effects are modeled using the AKN two-equation k- model with realizability constraints. The AKN model is a so-called low-Reynolds number model, which means that it resolves the flow all the way down to the wall. The AKN model depends on the distance to the closest wall. The physics interfaces therefore include a wall distance equation.



The default Turbulence model for the Turbulent flow, k- interface is k-. For the Turbulent flow, Low Re k- interface it is Low Reynolds number k-.

For all the turbulent interfaces, the default Turbulence model type is RANS and the default Heat transport turbulence model is Kays-Crawford. Other Heat transport

turbulence model options are Extended Kays-Crawford or User-defined turbulent Prandtl

number.

The Extended Kays-Crawford model requires a Reynolds number at infinity. That input is given in the Model Inputs section of the Fluid feature node.

It is always possible to specify a user-defined model for the turbulent Prandtl number. Enter the user-defined value or expression for the turbulent Prandtl number in the Model Inputs section of the Fluid feature node.


Turbulence model parameters are optimized to fit as many flow types as possible, but for some special cases, better performance can be obtained by tuning the model parameters. For a description of the turbulence model and the included model parameters see Theory for the Turbulent Flow Interfaces.


272 | C H A P T E


The dependent variables (field variables) are the Velocity field u (SI unit: m/s), the Pressure p (SI unit: Pa), and the Temperature T (SI unit: K). The default values are 0 m/s for the velocity, 0 Pa for the pressure, and 293.15 K for the temperature. The names can be changed but the names of fields and dependent variables must be unique within a component.

For turbulence modeling and heat radiation, there are additional dependent variables for the transported turbulence properties and also a dependent variable for the Reciprocal wall distance if the Low-Reynolds number k- model or the Spalart-Allmaras model is employed.



• The consistent stabilization methods are applicable to the Heat and flow equations and the Turbulence equations.

• When the Crosswind diffusion check box is selected, enter a Tuning parameter Ck for one or both of the Heat and flow equations and Turbulence equations. The default for the Heat and flow equations is 0.5, and 1 for the Turbulence equations.

• The Isotropic diffusion inconsistent stabilization method can be activated for the Heat

equation, Navier-Stokes equations, and the Turbulence equations.

• By default there is no isotropic diffusion selected. If required, select the Isotropic

diffusion check box and enter a Tuning parameter id for one or all of Heat equation, Navier-Stokes equations, and Turbulence equations. The defaults are 0.25.



The Use pseudo time stepping for stationary equation form check box adds pseudo time derivatives not only to the momentum and heat equations but also to the turbulence equations when the Stationary equation form is used.

The Turbulence variables scale parameters subsection contains the parameters Uscale and Lfact that are used to calculate absolute tolerances for the turbulence variables. The section is only visible when Turbulence model type in the Physical Model section is set to RANS.



The scaling parameters must only contain numerical values, units or parameters defined under Global Definitions. The scaling parameters can not contain variables. The parameters are used when a new default solver for a transient study step is generated. If you change the parameters, the new values take effect the next time a new default solver is generated.


The Turbulent Flow, Spalart-Allmaras Interface

The Non-Isothermal Turbulent Flow, Spalart-Allmaras and Conjugate Heat Transfer,

Turbulent Flow, Spalart-Allmaras interfaces combine the heat equation with the equations for turbulent flow.

The physics interfaces can be used to model energy transport in single-phase flows at high Reynolds numbers. Models can also include heat transfer in solids as well as surface-to-surface radiation and radiation in participating media, with the Heat Transfer Module. The physics interfaces support low Mach number (typically less than 0.3) flows.

For the Conjugate Heat Transfer, Turbulent Flow, Spalart-Allmaras interface, a Heat

Transfer in Solids model is active by default on all domains. A Fluid model is also added by default with an empty domain selection. All functionality for including other models, like different turbulent flow models or surface-to-surface radiation, is also available.




• Turbulent Non-Isothermal Flow Theory

When this option is selected when adding a physics interface, the node name that displays in the Model Builder is either Non-Isothermal Flow or Conjugate Heat Transfer. The default Turbulence model is Spalart-Allmaras in both cases.


274 | C H A P T E

The Non-Isothermal Turbulent Flow, Spalart-Allmaras and the Conjugate Heat Transfer,

Turbulent Flow, Spalart-Allmaras interfaces solve for conservation of energy, mass, and momentum in fluids and for conservation of energy in solids. Turbulence effects are modeled using the Spalart-Allmaras one-equation model. The Spalart-Allmaras model is a so-called low-Reynolds number model, which means that it resolves the flow all the way down to the wall. The Spalart-Allmaras model depends on the distance to the closest wall. The physics interfaces therefore include a wall distance equation.


The Difference Between the InterfacesThe turbulent versions of the Non-Isothermal Flow and Conjugate Heat Transfer interfaces differ by where they are selected when adding a physics interface and the default domain feature selected—Fluid or Heat transfer in solids.

Most of the settings options are the same as for The Non-Isothermal Flow and Conjugate Heat Transfer, Laminar Flow and Turbulent Flow Interfaces and The Turbulent Flow, k- and Turbulent Flow Low Re k- Interfaces sections except the Turbulence model that defaults to Spalart-Allmaras and the Scaling parameters for

turbulence variables section in the Advanced Settings section that features a scale instead of Uscale and Lfact.

From the Physics toolbar, add other nodes that implement, for example, boundary conditions, volume forces, or heat sources. You can also right-click any node to select physics from the context menu.

The Turbulent Flow, SST Interface

The Non-Isothermal Turbulent Flow, SST interface and Conjugate Heat Transfer, Turbulent

Flow, SST interfaces combine the heat equation with the equations for turbulent flow.



When this option is selected when adding a physics interface, the node name that displays in the Model Builder is either Non-Isothermal Flow or Conjugate Heat Transfer. The default Turbulence model is SST in both cases.



The physics interfaces can be used to model energy transport in single-phase flows at high Reynolds numbers. Models can also include heat transfer in solids as well as either surface-to-surface radiation or radiation in participating media. The physics interfaces support low Mach number (typically less than 0.3) flows.

For the Conjugate Heat Transfer, Turbulent Flow, SST interface, a Heat Transfer in Solids model is active by default on all domains. A Fluid model is also added by default with an empty domain selection. All functionality for including other models, like different turbulent flow models or surface-to-surface radiation, is also available.

The Non-Isothermal Turbulent Flow, SST and the Conjugate Heat Transfer, Turbulent Flow,

SST interfaces solve for conservation of energy, mass, and momentum in fluids and for conservation of energy in solids. Turbulence effects are modeled using the Menter shear-stress transport (SST) two-equation model from 2003 with realizability constraints. The SST model is a so-called low-Reynolds number model, which means that it resolves the flow all the way down to the wall. The SST model depends on the distance to the closest wall. The physics interfaces therefore include a wall distance equation.


The Difference Between the InterfacesThe turbulent versions of the Non-Isothermal Flow and Conjugate Heat Transfer interfaces differ based on from which branch they are added and by the default domain feature selected—Fluid or Heat transfer in solids.

Most of the settings options are the same as for The Non-Isothermal Flow and Conjugate Heat Transfer, Laminar Flow and Turbulent Flow Interfaces and The Turbulent Flow, k- and Turbulent Flow Low Re k- Interfaces sections.





276 | C H A P T E


The Non-Isothermal Turbulent Flow, k- and Conjugate Heat Transfer, Turbulent Flow k-interfaces combine the heat equation with the equations for turbulent flow at low Mach numbers.

The physics interfaces can be used to model energy transport in single-phase flows at high Reynolds numbers. Models can also include heat transfer in solids as well as surface-to-surface radiation and radiation in participating media, with the Heat Transfer Module. The physics interfaces support low Mach number (typically less than 0.3) flows.

For the Conjugate Heat Transfer, Turbulent Flow, k- interface, a Heat Transfer in Solids model is active by default on all domains. A Fluid model is also added by default with an empty domain selection. All functionality for including other models, like different turbulent flow models or surface-to-surface radiation, is also available.

The Non-Isothermal Turbulent Flow, k- and Conjugate Heat Transfer, Turbulent Flow,

k- interfaces solve for conservation of energy, mass and momentum in fluids and for conservation of energy in solids. Turbulence effects are modeled using the Wilcox revised two-equation k- model with realizability constraints. Flow and heat transfer close to walls are modeled using wall functions.


The Difference Between the InterfacesThe turbulent versions of the Non-Isothermal Flow and Conjugate Heat Transfer interfaces differ by where they are selected and the default domain feature selected—Fluid or Heat transfer in solids.

Most of the settings options are the same as for The Non-Isothermal Flow and Conjugate Heat Transfer, Laminar Flow and Turbulent Flow Interfaces and The Turbulent Flow, k- and Turbulent Flow Low Re k- Interfaces sections.

When this option is selected when adding a physics interface, the node name that displays in the Model Builder is either Non-Isothermal Flow or Conjugate Heat Transfer. The default Turbulence model is k- (nitf) in both cases.




Domain, Boundary, Edge, Point, and Pair Nodes Settings for the NITF Interfaces

All the versions of the Non-Isothermal Flow and Conjugate Heat Transfer interfaces have shared domain, boundary, edge, point, and pair features based on the selections made for the model. The domain, boundary, edge, point, and pair features nodes, listed in alphabetical order, are available from the Physics ribbon toolbar (Windows users), Physics context menu (Mac or Linux users), or right-click to access the context menu (all users).

The following features are described in this section:



In general, to add a node, go to the Physics toolbar, no matter what operating system you are using. However, to add subnodes, right-click the parent node.

• Fluid

• Initial Values

• Interior Wall

• Open Boundary

• Pressure Work

• Symmetry, Flow


• Viscous Heating

• Wall


278 | C H A P T E

The following features (listed in alphabetical order) are described for the Laminar Flow interface:

The following features (listed in alphabetical order) are described for the Heat Transfer interface:


• Interior Fan

• Flow Continuity

• Inlet

• Interior Wall

• Outlet



• Screen

• Volume Force



• Continuity

• Heat Flux

• Heat Source


• Inflow Heat Flux


• Outflow




• Temperature

• Thermal Insulation


If you also have the Heat Transfer Module license, the Thermal Contact feature is also available and described in the Heat Transfer Module User’s Guide.


• The Non-Isothermal Flow and Conjugate Heat Transfer, Laminar Flow and Turbulent Flow Interfaces




Fluid

The Fluid node adds both the momentum equations and the temperature equation but without volume forces, heat sources, pressure work, or viscous heating. You can add volume forces and heat sources as separate features, whereas Viscous Heating and Pressure Work can be added as subnodes.

When the turbulence model type is set to RANS, the Fluid node also adds the equations for k and , or k and or t.




This section controls the variable as well as any property value (reference pressures) used when solving for the pressure. There are usually two ways to calculate the pressure when describing fluid flow with mass and heat transfer. Solve for the absolute pressure or for a pressure (often denoted gauge pressure) that relates to the absolute pressure through a reference pressure.

The best option to choose usually depends on the system and the equations being solved. For example, in a unidirectional incompressible flow problem, the pressure drop over the modeled domain is probably many orders of magnitude smaller than the atmospheric pressure, which, if included, reduces stability and convergence properties of the solver. In other cases, such as when pressure is part of an expression for the gas volume or the diffusion coefficients, you might be required to solve for the absolute pressure.


280 | C H A P T E

The default Absolute pressure p (SI unit: Pa) is Pressure (nitf/fluid). The Reference

pressure check box is selected by default and the default value of pref is 1[atm] (101,325 Pa).

H E A T C O N D U C T I O N

The default Thermal conductivity k (SI unit: W/(m·K)) uses values From material. If User defined is selected, choose Isotropic, Diagonal, Symmetric, or Anisotropic based on the characteristics of the thermal conductivity and enter other values or expressions in the field or matrix.


When the turbulence model type is set to RANS, the conductive heat flux includes the turbulent contribution: q = k+TI)T where k is the thermal conductivity tensor, I the identity matrix and T the thermal turbulent conductivity defined as

T H E R M O D Y N A M I C S

Select a Fluid type—Gas/Liquid (the default), Ideal Gas, or Moist Air.

This makes it possible to use a system-based (gauge) pressure as the pressure variable while automatically including the reference pressure in places where it is required, such as for gas flow governed by the gas law. While this check box maintains control over the pressure variable and instances when the absolute pressure is required within this specific physics interface, it can not do so with physics interfaces that it is coupled to. In such models, check the coupling between any physics interfaces using the same variable.

TCpT

PrT--------------=

The Heat capacity at constant pressure Cp describes the amount of heat energy required to produce a unit temperature change in a unit mass.

The Ratio of specific heats is the ratio of the heat capacity at constant pressure, Cp, to the heat capacity at constant volume, Cv.



Gas/LiquidIf Gas/Liquid is selected as the Fluid type, properties of a non-ideal gas or liquid can be used.

By default the Density (SI unit: kg/m3), Heat capacity at constant pressure Cp (SI unit: J/(kg·K)), and Ratio of specific heats (dimensionless) use values From

material. Select User defined to enter other values or expressions.

Ideal GasIf Ideal gas is selected as the Fluid type, the ideal gas law is used to describe the fluid. For an ideal gas the density is defined by the following equation where pA is the absolute pressure, and T the temperature:

Select a Gas constant type—Specific gas constant Rs (SI unit: J/(kg·K)) or Mean molar

mass Mn (SI unit: kg/mol). In both cases, the default uses data From material. Select User defined to enter other values or expressions. If Mean molar mass is selected, the universal gas constant R 8.314 J/(mol·K), which is a built-in physical constant, is also used.

Select an option from the Specify Cp or list—Heat capacity at constant pressure Cp (SI unit: J/(kg·K)) or Ratio of specific heats (dimensionless). For both options, the default uses the value From material. Select User defined to enter another value or expression.

MnpA

RT----------------

pA

RsT-----------= =

For an ideal gas, choose to specify either Cp or , but not both since they, in this case, are dependent.

When using the ideal gas law to describe a fluid, specifying is enough to evaluate Cp. For common diatomic gases such as air, 1.4 is the standard value. Most liquids have 1.1 while water has 1.0. is used in the streamline stabilization and in the results and analysis variables for heat fluxes and total energy fluxes. It is also used in the ideal gas law.


282 | C H A P T E

Moist AirIf Moist air is selected as the Fluid type, the thermodynamics properties are defined as a function of the quantity of vapor in the moist air. Select an Input quantity:

• Vapor mass fraction (the default) (SI unit: dimensionless, (kg of water vapor)/(total mass in kg = kg of dry air + kg of water vapor))

• Concentration (SI unit: mol/m3)

• Moisture content xvap (SI unit: dimensionless, kg of water vapor/ kg of dry air).

• Select Relative humidity to define the quantity of vapor from the following reference values. These are used to estimate the mass fraction of vapor that is used to define the thermodynamic properties of the moist air:

- Reference relative humidity ref (dimensionless). The default is 0.

- Reference temperature Tref (SI unit: K). The default is 293.15 K.

- Reference pressure pref (SI unit: Pa). The default is 1 atm.


The dynamic viscosity describes the relationship between the shear rate and the shear stresses in a fluid. Intuitively, water and air have low viscosities, and substances often described as thick, such as oil, have higher viscosities. Non-Newtonian fluids have a shear-rate dependent viscosity. Examples of non-Newtonian fluids include yoghurt, paper pulp and polymer suspensions.

When Concentration is selected, a Concentration model input is automatically added in the Models Inputs section.

This section is not available if Moist air is selected as the Fluid type.



The default Dynamic viscosity (SI unit: Pa·s) uses values From material. Select User

defined to use a built-in variable for the shear rate magnitude, spf.sr, which makes it possible to define arbitrary expressions for the dynamic viscosity.

Non-Newtonian Power Law

If Non-Newtonian power law is selected, enter the Power law model parameter m and Model parameter n (both dimensionless). This selection uses a power law model for the dynamic viscosity of a non-Newtonian fluid:

where is the shear rate and is a lower limit for the shear rate evaluation. is per default 1·10-2 1s.

Non-Newtonian Carreau Model

If you also have the following modules, additional options are available—Non-Newtonian power law and Non-Newtonian Carreau model.

• CFD Module

• Heat Transfer Module plus the Microfluidics Module, or

• Microfluidics Module plus either the CFD Module or the Heat Transfer Module

Non-Newtonian Flow: The Power Law and the Carreau Model

This option is available with the CFD Module or the Heat Transfer Module plus the Microfluidics Module.

m max · ·min n 1–=

· ·min ·min

This option is available with the CFD Module or the Heat Transfer Module plus the Microfluidics Module.


284 | C H A P T E

If Non-Newtonian Carreau model is selected, enter the following Carreau model parameters:

• Zero shear rate viscosity 0 (SI unit: Pa·s)

• Infinite shear rate viscosity inf (SI unit: Pa·s)

• Model parameters (SI unit: s) and n (dimensionless)

This selection uses the Carreau model for the dynamic viscosity of a non-Newtonian fluid:

M I X I N G L E N G T H L I M I T ( TU R B U L E N C E M O D E L S O N L Y )

Select a Mixing length limit—Automatic (the default) or Manual.

• If Automatic is selected, the mixing length limit is automatically evaluated as:

(4-43)

where lbb is the shortest side of the geometry bounding box. If the geometry is, for example, a complicated system of very slender entities, Equation 4-43 tends to give a result that is too large. In such cases define manually.

• If Manual is selected, enter a value or expression for the Mixing length limit (SI unit: m).

D I S T A N C E E Q U A T I O N ( TU R B U L E N C E M O D E L S O N L Y )

Select a Reference length scale—Automatic (the default) or Manual.

• If Automatic is selected, the reference length scale is automatically evaluated as:

0 inf– 1 · 2+ n 1–

2-----------------

+=

This section is available for the k- and k- models, which need an upper limit on the mixing length.

lmixlim 0.5lbb=

lmixlim

lmixlim

This section is available for the low-Reynolds number k- model and the Spalart-Allmaras model, which need the distance to the closest wall.



(4-44)

where lbb is the shortest side of the geometry bounding box. If the geometry is, for example, a complicated system of very slender entities, Equation 4-43 tends to give a result that is too large. In such cases define manually.

• If Manual is selected, enter a value or expression for the Reference length scale (SI unit: m).

Wall

For laminar flow, the low Reynolds number k- turbulence model, the SST turbulence model and the Spalart-Allmaras turbulence model, the settings for the Wall node are identical to those for the corresponding single-phase flow node (the Boundary

condition defaults to No slip). In these cases, continuity of the temperature is enforced on internal walls separating a fluid and a solid domain.

About the Thermal Wall FunctionWhenever wall functions are used, there is a theoretical gap between the solid wall and the computational domain for the fluid. This gap is often ignored when the computational geometry is drawn, but it must nevertheless be considered in the equations for the temperature field.

Figure 0-1 shows the difference between internal and external walls. The approach is slightly different depending on what type of wall the condition applies to. Any wall feature that utilizes wall functions automatically detects internal and external walls.

On internal walls, there are two temperatures, one for the solid, Ts, and one for the fluid, Tf. If a temperature is prescribed on an internal wall, the constraint is applied to the temperature in the solid, that is, to Ts.

On external walls, the temperature T is the temperature of the fluid while the wall temperature is represented by an added dependent variable Tw. Tw is obtained from the flux condition:

where qtot is the total heat flux prescribed on the boundary.

lref 0.1lbb=

lref

lref

The settings below are for the k- or the k- turbulence model.

qwf qtot=


286 | C H A P T E

Figure 4-5: A simple example that includes both an external wall and an internal wall.

If a temperature is prescribed on an external wall, a constraint is applied to the wall temperature Tw, and instead the fluid temperature is obtained from the flux condition.


When using the k- turbulence model or the k- turbulence model, the Boundary

condition defaults to Wall functions. The other options available are Slip, Sliding wall

(wall functions), and Moving wall (wall functions).

If any one of the following options is selected—Wall functions, Sliding wall (wall

functions), or Moving wall (wall functions)—the wall function for the temperature field is also prescribed. This is called a thermal wall functions.

• If Sliding wall (wall functions) is selected, enter the components for the Velocity of

sliding wall uw (SI unit: m/s).

• If Moving wall (wall functions) is selected, enter the components for the Velocity of

moving wall uw (SI unit: m/s).

External wall

OutflowInflow

Solid

Fluid Internal wall

Any other heat boundary condition applied to an external wall is wrong in the sense that it acts on the fluid temperature, Ts, instead of the wall temperature, Tw.




Interior Wall

For laminar flow, the low Reynolds number k- turbulence model, the SST turbulence model and the Spalart-Allmaras turbulence model, the settings for the Interior Wall node are identical to those for the corresponding single-phase flow node (the Boundary

condition defaults to No slip). In these cases, continuity of the temperature is enforced across internal walls separating two fluid domains.

About the Thermal Wall FunctionWhenever wall functions are used, there is a theoretical gap between the internal wall and the computational domain for the fluid. This gap is often ignored when the computational geometry is drawn, but it must nevertheless be considered in the equations for the temperature field.

On internal walls, there are three temperatures, one for the wall, Tw, and one each for the fluid on the up and down sides of the wall, Tf, u and Tf, d. If a temperature is prescribed on an internal wall, the constraint is applied to the temperature of the wall, that is, to Tw.


When using the k- turbulence model or the k- turbulence model, the Boundary

condition defaults to Wall functions. The other options available are Slip, and Moving wall

(wall functions).

If any one of the following options is selected—Wall functions or Moving wall (wall

functions)—the wall functions for the temperature field is also prescribed. This is called a thermal wall functions.

About the Thermal Wall Function

The settings below are for the k- or the k- turbulence model.


288 | C H A P T E

If Moving wall (wall functions) is selected, enter the components for the Velocity of

moving wall uw (SI unit: m/s).

Initial Values

The Initial Values node adds initial values for the velocity field, the pressure and the temperature that can serve as an initial condition for a transient simulation or as an initial guess for a nonlinear solver. For turbulent flow there are also initial values for the turbulence model variables.




Enter values or expressions for the initial value of the Velocity field u (SI unit: m/s), the Pressure p (SI unit: Pa), and the Temperature T (SI unit: K). The default values are 0 m/s for the velocity, 0 Pa for the pressure, and 293.15 K for the temperature.

For turbulence models, default initial values for the turbulence variables, defined for each turbulence model in Theory for the Turbulent Flow Interfaces, are applied.

Open Boundary

Use the Open Boundary node to set up heat and momentum transport across boundaries where both convective inflow and outflow can occur. The node specifies a


About the Thermal Wall Function



fluid flow condition, together with an exterior temperature to be applied on the parts of the boundary where fluid flows into the domain.




From the Boundary Condition list, choose a fluid flow condition to apply on the open boundaries. Select

• Select Normal stress (the default) and enter the normal stress f0 (SI unit: N/m2). For large Reynolds numbers this implicitly specifies that

• Select No viscous stress to prescribe a vanishing normal viscous stress on the boundary.

E X T E R I O R TE M P E R A T U R E

Enter a value or expression for the external temperature (SI unit: K).

Pressure Work

Right-click the Heat Transfer in Solids or Fluid node to add this subnode.

When added under a Heat Transfer in Solids node, the Pressure Work node adds the following term to the right-hand side of the Heat Transfer in Solids equation:

(4-45)

where Sel is the elastic contribution to entropy.

When added under a Fluid node, the Pressure Work feature adds the following contribution to the right-hand side of the Heat Transfer in Fluids equation:

(4-46)

The feature computes the pressure work using the absolute pressure.

The direction of the flow across the boundary is typically calculated by a Fluid Flow branch interface and is entered as Model Inputs.

p f0

T t-----Sel–

T---- T-------

p

pt------ u p+ –


290 | C H A P T E


From the Selection list, choose the domains to define. By default, the selection is the same as for the parent node to which it is attached (Heat Transfer in Solids or Fluid node).

P R E S S U R E WO R K

For the Heat Transfer in Solids model, enter a value or expression for the Elastic

contribution to entropy Ent (SI unit: Jm3·K)). The default is 0 Jm3·K).

For the Fluid model, select a Pressure work formulation—Full formulation (the default), or Low Mach number formulation. The latter excludes the term u · p from Equation 4-46.

Viscous Heating

The Viscous Heating subnode adds the following term to the right-hand side of the heat transfer in fluids equation:

(4-47)

Here is the viscous stress tensor and S is the strain rate tensor. Equation 4-47 represents the heating caused by viscous friction within the fluid.


From the Selection list, choose the domains on which to add viscous heating. By default, the selection is the same as for the Fluid node that it is attached to.

Symmetry, Heat and Symmetry, Flow

S Y M M E T R Y, H E A T

The Symmetry, Heat node provides a boundary condition for symmetry boundaries. This boundary condition is similar to a Thermal Insulation condition, and it enforces that there is no heat flux across the boundary.

If you apply a Symmetry, Heat feature to a boundary adjacent to a fluid domain, you should also consider adding a Symmetry, Flow node to that boundary since physically, you cannot have symmetry in the temperature without having symmetry in the velocity and pressure as well.

:S



Boundary Selection

S Y M M E T R Y , F L O W

The Symmetry, Flow node adds a boundary condition that describes symmetry boundaries in a fluid flow simulation. The boundary condition for symmetry boundaries prescribes no penetration and vanishing shear stresses. The boundary condition is a combination of a Dirichlet condition and a Neumann condition:

for the compressible and the incompressible formulation respectively. The Dirichlet condition takes precedence over the Neumann condition, and the above equations are equivalent to the following equations for both the compressible and incompressible formulations:

If you apply a Symmetry, Flow feature to a boundary, the boundary should also be supplemented with a Symmetry, Heat and Symmetry, Flow node because it is not possible to have symmetry in the velocity and pressure without having symmetry in the temperature as well.

Boundary SelectionFrom the Selection list, choose the boundaries on which to apply the condition.

In most cases, the node does not require any user input. If required, define the symmetry boundaries.

u n 0,= pI– u u T+ 23--- u I–

+ n 0=

u n 0,= pI– u u T+ + n 0=

u n 0,= K K n n– 0=

K u u T+ n=

For 2D axial symmetry, a boundary condition does not need to be defined. For the symmetry axis at r0, the software automatically provides a condition that prescribes ur0 and vanishing stresses in the z direction and adds an Axial Symmetry node that implements this condition on the axial symmetry boundaries only.


292 | C H A P T E

Constraint SettingsTo display this section, click the Show button ( ) and select Advanced Physics Options. Select the Use weak constraints check box to use weak constraints and create dependent variables for the corresponding Lagrange multipliers.


Th eo r y f o r t h e Non - I s o t h e rma l F l ow and Con j u g a t e Hea t T r a n s f e r I n t e r f a c e s

In industrial applications it is common that the density of a process fluid varies. These variations can have a number of different sources but the most common one is the presence of an inhomogeneous temperature field. This module includes the Non-Isothermal Flow predefined multiphysics coupling to simulate systems in which the density varies with temperature.

Other situations where the density might vary includes chemical reactions, for instance where reactants associate or dissociate.

The Non-Isothermal Flow and Conjugate Heat Transfer interfaces contain the fully compressible formulation of the continuity and momentum equations:

(4-48)

where




• is the dynamic viscosity (SI unit: Pa·s)

• F is the body force vector (SI unit: N/m3)

It also solves the heat equation, which for a fluid is given by

where in addition to the quantities above

• Cp is the specific heat capacity at constant pressure (SI unit: J/(kgK))

t------ u + 0=

ut------- u u+ p– u u T+ 2

3--- u I–

F++=

CpTt------- u T+ q – :S T

---- T-------

p

pt------ u p+ – Q+ +=

T H E O R Y F O R T H E N O N - I S O T H E R M A L F L O W A N D C O N J U G A T E H E A T TR A N S F E R I N T E R F A C E S | 293

294 | C H A P T E


• q is the heat flux by conduction (SI unit: W/m2)


• S is the strain-rate tensor (SI unit: 1/s)

• Q contains heat sources other than viscous heating (SI unit: W/m3)

The pressure work term

and the viscous heating term

are not included by default because they are usually negligible. These terms can, however, be added as subnodes to the Fluid node. For a detailed discussion of the fundamentals of heat transfer in fluids, see Ref. 3.

The physics interface also supports heat transfer in solids:

where E is the elastic contribution to entropy (SI unit: J/(m3·K)).

As in the case of fluids, the pressure work term

is not included by default but must be added as a subfeature.

S 12--- u u T+ =

T---- T-------

p

pt------ u p+

:S

CpTt------- q – T E

t-------– Q+=

T Et

-------

• The Heat Equation


• References for the Non-Isothermal Flow and Conjugate Heat Transfer Interfaces


Turbulent Non-Isothermal Flow Theory

Turbulent energy transport is conceptually more complicated than energy transport in laminar flows since the turbulence is also a form of energy.

Equations for compressible turbulence are derived using the Favre average. The Favre average of a variable T is denoted and is defined by

where the bar denotes the usual Reynolds average. The full field is then decomposed as

With this notation the equation for total internal energy, e, becomes

(4-49)

where h is the enthalpy. The vector

(4-50)

is the laminar conductive heat flux and

is the laminar, viscous stress tensor. Notice that the thermal conductivity is denoted .

The modeling assumptions are in large part analogous to those for incompressible turbulence modeling. The stress tensor

is modeled using the Boussinesq approximation:

T˜

T˜ T

-------=

T T˜

T''+=

t

----- euiui

2-----------+

ui''ui''

2-------------------+

xj

------- uj huiui

2-----------+

ujui''ui''

2-------------------+

=+

xj

------- qj– uj''h''– ijui''uj''ui''ui''

2---------------------------–+

xj

------- ui ij ui''uj''– +

qj Txj

-------–=

ij 2Sij23---

ukxk

---------ij–=

ui''u''j–


296 | C H A P T E

(4-51)

where k is the turbulent kinetic energy, which in turn is defined by

(4-52)

The correlation between and in Equation 4-49 is the turbulent transport of heat. It is modeled analogously to the laminar conductive heat flux

(4-53)

The molecular diffusion term,

and turbulent transport term,

are modeled by a generalization of the molecular diffusion and turbulent transport terms found in the incompressible k equation

(4-54)

Inserting Equation 4-50, Equation 4-51, Equation 4-52, Equation 4-53 and Equation 4-54 into Equation 4-49 gives

(4-55)

The Favre average can also be applied to the momentum equation, which, using Equation 4-51, can be written

(4-56)

ui''u''j– Tij 2T S

˜ij

13---

ukxk

---------ij– 2

3---kij–= =

k 12---ui''ui''=

uj'' h''

uj''h'' qTj T

T˜

xj

-------–TCpPrT

-------------- T˜

xj

-------–= = =

ijui''

uj''ui''ui'' 2

ijui''uj''ui''ui''

2---------------------------–

T

k------+

kxj

-------=

t

----- euiui

2----------- k+ +

xj------- uj h

uiui2

----------- k+ +

=+

xj

------- qj– qTj–

Tk------+

kxj

-------+

xj------- ui ij T

ij+ +

t

----- ui xj

------- ujui +pxj

-------–xj

------- ij Tij+ +=


Taking the inner product between and Equation 4-56 results in an equation for the resolved kinetic energy, which can be subtracted from Equation 4-55 with the following result:

(4-57)

where the relation

has been used.

According to Wilcox (Ref. 1), it is usually a good approximation to neglect the contributions of k for flows with Mach numbers up to the supersonic range. This gives the following approximation of Equation 4-57 is

(4-58)

Larsson (Ref. 2) suggests to make the split

Since

for all applications of engineering interest, it follows that

and consequently

(4-59)

where

ui

t

----- e k+ xj

------- uj e k+ pujx

--------j

– +=+

xj

------- qj– qTj–

Tk------+

kxj

-------+

xj------- ui ij T

ij+ +

h e p +=

t

----- e xj

------- uje pujx

--------j

–xj

------- qj– qTj–

xj------- ui ij T

ij+ + +=+

ij ij ij''+=

ij ij''»

ij ij

t

----- e xj

------- uje pujx

--------j

–xj

------- T+ T˜

xj

-------

xj------- uiij

Tot + +=+


298 | C H A P T E

Equation 4-59 is completely analogous to the laminar energy equation and can be expanded using the same theory (see for example Ref. 3):

which is the temperature equation solved in the turbulent Non-Isothermal Flow and Conjugate Heat Transfer interfaces.

TU R B U L E N T C O N D U C T I V I T Y

Kays-CrawfordThis is a relatively exact model for PrT, while still quite simple. In Ref. 4, it is compared to other models for PrT and found to be a good approximation for most kinds of turbulent wall bounded flows except for turbulent flow of liquid metals. The model is given by

(4-60)

where the Prandtl number at infinity is PrT0.85 and is the conductivity.

Extended Kays-CrawfordWeigand and others (Ref. 5) suggested an extension of Equation 4-60 to liquid metals by introducing

where Re, the Reynolds number at infinity must be provided either as a constant or as a function of the flow field. This is entered in the Model Inputs section of the Fluid feature.

TE M P E R A T U R E WA L L F U N C T I O N S

Analogous to the single-phase flow wall functions (see Wall Functions described for the Wall boundary condition), there is a theoretical gap between the solid wall and the

ijTot T+ 2S

˜ij

23---

ukxk

---------ij–

=

CpT˜

t

------- uj˜ T

˜xj

-------+

xj------- T+ T

˜xj

-------

ijS˜

ijT˜

----

T˜

-------

p

pt

------ uj˜ p

xj-------+

–+=

PrT1

2PrT----------------- 0.3

PrT

------------------CpT

-------------- 0.3CpT

--------------

21 e 0.3CpT PrT –– –+

1–

=

PrT 0.85 100CpRe

0.888--------------------------------+=


computational domain for the fluid and temperature fields. This gap is often ignored when the computational geometry is drawn.

The heat flux between the fluid with temperature Tf and a wall with temperature Tw, is:

where is the fluid density, Cp is the fluid heat capacity, C is a turbulence modeling constant, and k is the turbulent kinetic energy. T is the dimensionless temperature and is given by (Ref. 6):

where in turn

is the thermal conductivity, and is the von Karman constant equal to 0.41.

The computational results should be checked so that the distance between the computational fluid domain and the wall, w, is everywhere small compared to any geometrical quantity of interest. The distance w is available for evaluation on boundaries.

Theory for the Non-Isothermal Screen Boundary Condition

For non-isothermal flow, the conditions that apply across a screen in isothermal flow are complemented by:

qwfCpC

1 4/ k1 2/ Tw Tf–

T+-----------------------------------------------------------=

T+

Prw+ for w

+ w1+

15Pr2 3/ 500w

+2----------–

for w1+ w

+ w2+

PrT

--------- lnw+ + for w2

+ w+

=

w+

w C1 2/ k

------------------------------= w1

+ 10Pr1 3/-------------=

w2+ 10 10

PrT---------= Pr

Cp

----------=

15Pr2 3/PrT2--------- 1 ln 1000

PrT---------

+ –=


300 | C H A P T E

(4-61)

where H0 is the total enthalpy.

References for the Non-Isothermal Flow and Conjugate Heat Transfer Interfaces

1. D.C. Wilcox, Turbulence Modeling for CFD, 2nd ed., DCW Industries, 1998.

2. J. Larsson, Numerical Simulation of Turbulent Flows for Turbine Blade Heat Transfer, Doctoral Thesis for the Degree of Doctor of Philosophy, Chalmers University of Technology, Sweden, 1998.


4. W.M. Kays, “Turbulent Prandtl Number — Where Are We?”, ASME Journal of Heat Transfer, vol. 116, pp. 284–295, 1994.

5. B. Weigand, J.R. Ferguson, and M.E. Crawford, “An Extended Kays and Crawford Turbulent Prandtl Number Model,” Int. J. Heat and Mass Transfer, vol. 40, no. 17, pp. 4191–4196, 1997.

6. D. Lacasse, È. Turgeon, and D. Pelletier, “On the Judicious Use of the k— Model, Wall Functions and Adaptivity,” Int. J. Thermal Sciences, vol. 43, pp. 925–938, 2004.

H0 -+ 0=

See Screen for the feature node details.

Also see Screen Boundary Condition described for the single-phase flow interfaces.


5

H i g h M a c h N u m b e r F l o w

There are several fluid-flow interfaces available with the CFD Module. The fluid-flow interfaces are grouped by type under the Fluid Flow main branch. This chapter discusses applications involving the High Mach Number Flow branch ( ) when adding a physics interface.

In this chapter:

• The High Mach Number Flow Interfaces

• Theory for the High Mach Number Flow Interfaces

301

302 | C H A P T E

Th e H i g h Ma ch Numbe r F l ow I n t e r f a c e s

There are three versions of the same predefined multiphysics interface (all with the interface identifier hmnf) that combine the heat equation with either the laminar or turbulent flow equations. The advantage of using the multiphysics interfaces—compared to adding the individual physics interfaces separately—is that a set of two-way couplings has been predefined. In particular, the physics interfaces use the same definition of the density, which can therefore be a function of both pressure and temperature. Solving this coupled system of equations usually requires numerical stabilization, which the predefined multiphysics interface also sets up.

These physics interfaces vary only by one or two default settings (see Table 5-1) or selections from check boxes or lists under the Physical Model section for the physics interface.

TABLE 5-1: THE HIGH MACH NUMBER FLOW PHYSICAL MODEL DEFAULT SETTINGS

INTERFACE ID TURBULENCE MODEL TYPE

TURBULENCE MODEL

HEAT TRANSPORT TURBULENCE MODEL

Laminar Flow hmnf None N/A N/A

Turbulent Flow, k- hmnf RANS k- Kays-Crawford

Turbulent Flow, Spalart- Allmaras

hmnf RANS Spalart-Allmaras Kays-Crawford

Most of the other physics nodes share the same setting options as described in this section and in Domain, Boundary, Edge, Point, and Pair Nodes for the High Mach Number Flow Laminar and Turbulent Interfaces. See also The Heat Transfer Interface for details about the Heat

Transfer in Solids physics node.

R 5 : H I G H M A C H N U M B E R F L O W

The High Mach Number Flow, Laminar Flow Interface

The Laminar Flow (hmnf) interface ( ), found under the Fluid Flow>High Mach Number

Flow branch ( ) when adding a physics interface, is used to model gas flows at low and moderate Reynolds number where the velocity magnitude is comparable to the speed of sound, that is, laminar flows in the transonic and supersonic range. This state is often connected to very low pressures.

The physics interface solves for conservation of energy, mass and momentum. The interface also supports heat transfer in solids as well as surface-to-surface radiation.

This physics interface is a predefined multiphysics coupling consisting of a Laminar Flow interface, applied to compressible flow, in combination with a Heat Transfer interface.

When this physics interface is added, the following default nodes are also added in the Model Builder—Fluid, Wall, Thermal Insulation, and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions, volume forces, and heat sources. You can also right-click the node to select physics from the context menu.



The default identifier (for the first physics interface in the component) is hmnf.

• The High Mach Number Flow, Laminar Flow Interface

• The High Mach Number Flow, Turbulent Flow, k- Interface

• The High Mach Number Flow, Turbulent Flow, Spalart-Allmaras Interface

• Theory for the High Mach Number Flow Interfaces

T H E H I G H M A C H N U M B E R F L O W I N T E R F A C E S | 303

304 | C H A P T E


The default setting is to have All domains in the model define the dependent variables and governing equations. To choose specific domains, select Manual from the Selection list.


Define physics interface properties to control the overall type of model.

Turbulence Model TypeBy definition, no turbulence model is needed when studying laminar flows. The default Turbulence model type is None.

The flow state in a fluid-flow model is not, however, always known beforehand. Select RANS as the Turbulence model type and select k- as Turbulence model in order to account for turbulence. This changes the physics interface into the turbulent version.

S U R F A C E - T O - S U R F A C E R A D I A T I O N

R A D I A T I O N S E T T I N G S

If the default Turbulence model type selected is RANS, the additional turbulence model settings are made available. However, the node is still called High Mach Number Flow (hmnf) with a number added at the end of the name to indicate the change.

This section requires an additional Heat Transfer Module license and displays when the Surface-to-surface radiation check box is selected. See the Heat Transfer Module User’s Guide for details.

Select the Surface-to-surface radiation check box to enable the Radiation

Settings section.

This section requires an additional Heat Transfer Module license and displays when the Surface-to-surface radiation check box is selected. See the Heat Transfer Module User’s Guide for details.



The dependent variables (field variables) are the Velocity field u (SI unit: m/s), the Pressure p (SI unit: Pa), and the Temperature T (SI unit: K). The names can be changed but the names of fields and dependent variables must be unique within a component.


To display this section, click the Show button ( ) and select Discretization. Select an option for Discretization of fluids—P1+P1 (the default), P2+P1, or P3+P2.

Select a Surface radiosity—Linear (the default), Quadratic, Cubic, Quartic, or Quintic (2D axisymmetric and 2D models only).




• The consistent stabilization methods are applicable to the Heat and flow equations.

• The Isotropic diffusion inconsistent stabilization method can be activated for both the Heat equation and the Navier-Stokes equations.



Select the Use pseudo time stepping for stationary equation form check box to add pseudo time derivatives to the equation when the Stationary equation form is used. When selected, also choose a CFL number expression—Automatic (the default) or Manual. Automatic calculates the local CFL number (from the Courant–Friedrichs–Lewy condition) from a built-in expression. If Manual is selected, enter a Local CFL

number CFLloc.


306 | C H A P T E


The High Mach Number Flow, Turbulent Flow, k- Interface

The Turbulent Flow, k- (hmnf) interface ( ), found under the High Mach Number

Flow>Turbulent Flow branch ( ) when adding a physics interface, is used to model gas flows at high Reynolds number where the velocity magnitude is comparable to the speed of sound, that is, turbulent flows in the transonic and supersonic range.

The physics interface solves for conservation of energy, mass, and momentum. Turbulence effects are modeled using the standard two-equation k- model with realizability constraints. Flow and heat transfer close to walls are modeled using wall functions. The physics interface also supports heat transfer in solids as well as surface-to-surface radiation.

This is a predefined multiphysics coupling consisting of a Turbulent Flow k- interface, applied to compressible flow, in combination with a Heat Transfer interface. As shown in Table 5-1, the turbulent versions of the interfaces differ by where they are selected when adding a physics interface and the default Turbulence model selected—k- for this physics interface.

When this physics interface is added, the following default nodes are also added in the Model Builder—Fluid, Wall, Thermal Insulation, and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions, volume forces, and heat sources. You can also right-click the node to select physics from the context menu.



• Domain, Boundary, Edge, Point, and Pair Nodes for the High Mach Number Flow Laminar and Turbulent Interfaces







The default identifier (for the first physics interface in the component) is hmnf.


The default setting is to have All domains in the model define the dependent variables and governing equations. To choose specific domains, select Manual from the Selection list.


The default Turbulence model for the Turbulent Flow, k- interface is k-.

The default Turbulence model type is RANS and the default Heat transport turbulence

model is Kays-Crawford. Alternatively, select User-defined turbulent Prandtl number. The turbulent Prandtl number model describes the influence of the turbulent fluctuations on the temperature field. It is always possible to have a user-defined model for the turbulence Prandtl number. Enter the user-defined value or expression for the turbulence Prandtl number in the Model Inputs section of the Fluid feature node.


Edit the model parameters of the k- model as required. Turbulence model parameters are optimized to fit as many flow types as possible, but for some special cases, better performance can be obtained by tuning the model parameters. For a description of the turbulence model and the included model parameters see Theory for the Turbulent Flow Interfaces.


The dependent variables (field variables) are the Velocity field u (SI unit: m/s), the Pressure p (SI unit: Pa), and the Temperature T (SI unit: K). For turbulence modeling and heat radiation, the Turbulent kinetic energy k (SI unit: m2/s2) and Turbulent

dissipation rate ep (SI unit: m2/s3) variables are also available.

The names can be changed but the names of fields and dependent variables must be unique within a model.


308 | C H A P T E



• The consistent stabilization methods are applicable to the Heat and flow equations and the Turbulence Equations.

• The Isotropic diffusion inconsistent stabilization method can be activated for the Heat

equation, Navier-Stokes equations, and the Turbulence equations.

• By default there is no isotropic diffusion selected. If required, select the Isotropic

diffusion check box and enter a Tuning parameter id for one or all of Heat equation, Navier-Stokes equations, and Turbulence equations. The defaults are 0.25.



In addition to the settings for the laminar flow version of the physics interface, there is a Scaling parameters for turbulence variables subsection that contains the parameters Uscale and Lfact that are used to calculate absolute tolerances for the turbulence variables. The section is only visible when Turbulence model type in the Physical Model section is set to RANS.

The scaling parameters must only contain numerical values, units or parameters defined under Global Definitions. The scaling parameters can not contain variables. The parameters are used when a new default solver for a transient study step is generated. If you change the parameters, the new values take effect the next time you generate a new default solver.



The High Mach Number Flow, Turbulent Flow, Spalart-Allmaras Interface

The Turbulent Flow, Spalart-Allmaras (hmnf) interface ( ), found under the High Mach

Number Flow>Turbulent Flow branch ( ) when adding a physics interface, is used to model gas flows at high Reynolds number where the velocity magnitude is comparable to the speed of sound, that is, turbulent flows in the transonic and supersonic range.

The physics interface solves for conservation of energy, mass, and momentum. Turbulence effects are modeled using the one-equation Spalart-Allmaras turbulence model. The Spalart-Allmaras model is a so-called low-Reynolds number model, which means that it resolves the velocity, pressure, and temperature fields all the way down to the wall. The Spalart-Allmaras model depends on the distance to the closest wall. The physics interface therefore includes a wall distance equation. It also supports heat transfer in solids as well as surface-to-surface radiation.

This is a predefined multiphysics coupling consisting of a Turbulent Flow,

Spalart-Allmaras interface, applied to compressible flow, in combination with a Heat

Transfer interface. As shown in Table 5-1, the turbulent versions of the physics interfaces differ by where they are selected when adding a physics interface and the default Turbulence model selected—Spalart-Allmaras for this interface.

When this physics interface is added, the following default nodes are also added in the Model Builder—Fluid, Wall, Thermal Insulation, and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions, volume








310 | C H A P T E

forces, and heat sources. You can also right-click the node to select physics from the context menu.


The dependent variables (field variables) are the Velocity field u (SI unit: m/s), the Pressure p (SI unit: Pa), and the Temperature T (SI unit: K). For turbulence modeling and heat radiation, the Reciprocal wall distance G (SI unit: 1/m) and Undamped

turbulent kinematic viscosity nutilde (SI unit: m2/s) variables are also available.



In addition to the settings for the laminar flow version of the physics interface, there is a Scaling parameters for turbulence variables subsection that contains the parameter scale which is used to calculate absolute tolerances for the turbulence variable. The section is only visible when Turbulence model type in the Physical Model section is set to RANS.

The scaling parameter must only contain numerical values, units or parameters defined under Global Definitions. The scaling parameter can not contain variables. The parameter is used when a new default solver for a transient study step is generated. If you change the parameter, the new value takes effect the next time you generate a new default solver.

Except for the Dependent Variables and the Advanced Settings section, the rest of the settings are the same as for The High Mach Number Flow, Turbulent Flow, k- Interface.



Domain, Boundary, Edge, Point, and Pair Nodes for the High Mach Number Flow Laminar and Turbulent Interfaces

The High Mach Number Flow Interfaces has these domain, boundary, edge, point, and pair nodes available from the Physics ribbon toolbar (Windows users), Physics context menu (Mac or Linux users), or right-click to access the context menu (all users).

The following nodes are slightly different from those in the other physics interfaces and are described in this section:

• Initial Values

• Inlet

• Fluid

• Outlet







Transonic Flow in a Sajben Diffuser: model library path CFD_Module/High_Mach_Number_Flow/sajben_diffuser



312 | C H A P T E

Also see the Non-Isothermal Flow interfaces for:

• Symmetry, Flow



• Flow Continuity


• Volume Force

• Wall

The following physics nodes and subnodes (listed in alphabetical order) are described for the Heat Transfer interface:

For The High Mach Number Flow, Turbulent Flow, k- Interface, the same nodes are selected from the Turbulent Flow, k- submenus.

For The High Mach Number Flow, Turbulent Flow, Spalart-Allmaras Interface, the same nodes are selected from the Turbulent Flow,

Spalart-Allmaras submenus.



• Continuity

• Heat Flux

• Heat Source






• Temperature

• Thermal Insulation


If you also have the Heat Transfer Module license, the Thermal Contact feature is also available and described in the Heat Transfer Module User’s Guide.


Initial Values

The Initial Values node adds initial values for the velocity field, the pressure and the temperature that can serve as initial conditions for a transient simulation or as an initial guess for a nonlinear solver. For turbulent flow there are also initial values for the turbulence model variables. The surface radiosity is only applicable for surface-to-surface radiation.




Enter values or expressions for the initial value of the Velocity field u (SI unit: m/s), the Pressure p (SI unit: Pa), and the Temperature T (SI unit: K). The default values are 0 m/s for the velocity, 0 Pa for the pressure, and 293.15 K for the temperature.

Fluid

The Fluid node adds the continuity, momentum and temperature equations for an ideal gas but omits volume forces and heat sources. Volume forces and heat sources can be added as separate physics features. Viscous heating and pressure work terms are added by default to the temperature equation.

When the turbulence model type is set to RANS, the Fluid node also adds the equations for k and , or the undamped turbulent kinematic viscosity, depending on the turbulence model used.

The thermal conductivity describes the relationship between the heat flux vector q and the temperature gradient T as in q = kT, which is Fourier’s law of heat conduction. Enter this quantity as power per length and temperature.

In a turbulent flow interface, initial values for the turbulence variables are also specified. By default these are specified using the predefined variables defined by the expressions described in Theory for the High Mach Number Flow Interfaces under Initial Values.


314 | C H A P T E




Define the model inputs. If no model inputs are required, this section is empty.

H E A T C O N D U C T I O N

Select a Thermal conductivity k (SI unit: W/(m·K)) from the list—From material, Sutherlands Law (the default), or User defined. If User defined is selected, choose Isotropic, Diagonal, Symmetric, or Anisotropic based on the characteristics of the thermal conductivity and enter another value or expression in the field or matrix.

Sutherland’s Law If Sutherlands Law is selected, enter the following model parameters:

• Conductivity at reference temperature kref (SI unit: W/(m·K))

• Reference temperature Tk,ref (SI unit: K)

• Sutherland constant Sk (SI unit: K)

Sutherland’s law describes the relationship between the thermal conductivity and the total temperature of an ideal fluid according to

T H E R M O D Y N A M I C S

The High Mach Number Flow interface is applicable for ideal gases. Specify the thermodynamics properties by selecting a gas constant type and selecting between entering the heat capacity at constant pressure or the ratio of specific heats. For an ideal gas the density is defined as


k krefT

Tk ref-------------- 3 2/ Tk ref Sk+

T Sk+---------------------------=

MnpA

RT----------------

pARsT-----------= =


where pA is the absolute pressure, and T is the temperature.

• Select a Gas constant type—Specific gas constant Rs (SI unit: J/(kg·K)) or Mean molar

mass Mn (SI unit: kg/mol). The default setting is to use the property value From

material. Select User defined to enter another value or expression for either material property. In both cases, the default uses data From material. Select User defined to enter other values or expressions. If Mean molar mass is selected, the universal gas constant R 8.314 J/(mol·K), which is a built-in physical constant, is also used.

• From the Specify Cp or list, select Heat capacity at constant pressure Cp (SI unit: J/(kg·K)) or Ratio of specific heats (dimensionless). The default setting is to use the property value From material. Select User defined to enter another value or expression for either material property.


The dynamic viscosity describes the relationship between the shear rate and the shear stresses in a fluid.

Select a Dynamic viscosity (SI unit: Pa·s) from the list—From material, Sutherlands Law (the default), or User defined.

Sutherland’s LawIf Sutherlands Law is selected, enter the following model parameters:

• Dynamic viscosity at reference temperature ref (SI unit: Pa·s)

• Reference temperature T,ref(SI unit: K)

• Sutherland constant S (SI unit: K)

Sutherland’s law describes the relationship between the dynamic viscosity and the total temperature of an ideal fluid according to


refT

T ref-------------- 3 2/ T ref S+

T S+---------------------------=

This section is available for the Turbulent Flow, k- model only.


316 | C H A P T E

The k- turbulence model needs an upper limit on the mixing length to be numerically robust. Select a Mixing length limit—Automatic (the default) or Manual.

• If Automatic is selected, this limit is automatically evaluated as:

(5-1)

where lbb is the shortest side of the geometry bounding box. If the geometry is a complicated system of very slender entities, for example, Equation 5-1 tends to give a result that is too large. In such cases, define manually.


D I S T A N C E E Q U A T I O N

Select how the Reference length scale lref (SI unit: m) is defined—Automatic (default) or Manual:

• If Automatic is selected, the wall distance is automatically evaluated to one tenth of the shortest side of the geometry bounding box. This is usually quite accurate but it can sometimes give a too high value if the geometry consists of several slim entities. In such cases, define the reference length scale manually.

• Select Manual to define a different value or expression for the length scale. The default is 1 m.

lref controls the result of the distance equation. Objects that are much smaller than lref are effectively be diminished while the distance to objects much larger than lref are accurately represented.

lmixlim 0.5lbb=

lmixlim

lmixlim

This section is available for Turbulent Flow, Spalart-Allmaras since a Wall

Distance interface is then included.


Inlet

The Inlet node includes a set of boundary conditions describing the fluid flow and temperature conditions at an inlet. The applied conditions are controlled by the Flow

Condition.



F L O W C O N D I T I O N

Select a Flow condition—Characteristics based (the default) or Supersonic.

If a Characteristics based condition is selected, the current flow situation is analyzed using the inviscid flow characteristics at the inlet. This can be used to specify either a subsonic (Ma < 1) inlet or a supersonic inlet (Ma > 1). This is the default condition.

If Supersonic is selected, the inlet flow is assumed to be supersonic.

F L O W P R O P E R T I E S

Specify the flow properties at the inlet in terms of the static or total pressure, static or total temperature, Mach number, and flow direction. By default Static input variables are used.

Select an Input state—Static (the default) or Total. For either selection, also enter values or expressions for the Mach number Ma0 (dimensionless) at the inlet. The default is 1.5

• If Static is selected, enter values or expressions for the Static pressure p0,stat (SI unit: Pa) and Static temperature T0,stat (SI unit: K).

• If Total is selected, enter values or expressions for the Total pressure p0,tot (SI unit: Pa) and Total temperature T0,tot (SI unit: K).

The relationships between the static and total states are:

For background on the derivation and implementation of the conditions, see Theory for the High Mach Number Flow Interfaces


318 | C H A P T E

(5-2)

Select a Flow direction—Normal inflow (the default) or User defined. To specify an arbitrary flow direction, select User defined and enter the components of the direction normal nM (dimensionless).

TU R B U L E N C E P R O P E R T I E S

Using a turbulence model, specify the turbulence properties at an inlet. For the Turbulent Flow, k- model, specify turbulence quantities according to one of the following options:

• Select Specify turbulence length scale and intensity to enter values or expressions for the Turbulent intensity IT (dimensionless) and Turbulence length scale LT (SI unit: m). IT and LT values are related to the turbulence variables via

• If Specify turbulence variables is selected, enter values or expressions for the Turbulent

kinetic energy k0 (SI unit: m2/s2) and Turbulent dissipation rate, 0 (SI unit: m2/s3).

ptotpstat---------- 1 1–

2-----------Ma2

+

1–-----------

=

TtotTstat----------- 1 1–

2-----------Ma2

+ =

This section displays when RANS is selected as Turbulence Model Type.

k 32--- U IT 2,= C

3 4 k3 2/

LT-----------=

For The High Mach Number Flow, Turbulent Flow, Spalart-Allmaras Interface, also enter a value or expression for the Undamped turbulent

kinematic viscosity 0 (SI unit: m2/s).

For recommendations of physically sound values see Inlet Values for the Turbulence Length Scale and Turbulent Intensity.


Outlet

The Outlet node includes a set of boundary conditions describing fluid flow and temperature conditions at an outlet. The applied conditions are controlled by the Flow

Condition.

F L O W C O N D I T I O N

Select a Flow condition—Hybrid (the default), Supersonic, or Subsonic.

• Using a Hybrid condition, both subsonic (Ma < 1) and supersonic flow (Ma > 1) conditions at the outlet are supported.

• Select Supersonic when the flow at the outlet is known to be supersonic.

• Select Subsonic when the flow at the outlet is known to be subsonic.

F L O W P R O P E R T I E S

HybridIf Hybrid is selected, an outlet pressure is specified. This pressure is enforced at the outlet when the flow is subsonic. Select an Input state—Static (the default) or Total.

• If Static is selected, enter a value or expression for the Static pressure p0,stat (SI unit: Pa).

• If Total is selected, enter a value or expression for the Total pressure p0,tot (SI unit: Pa).

For a background on the derivation and implementation of the conditions, see Theory for the High Mach Number Flow Interfaces.

This section displays when the Hybrid or Subsonic flow condition is selected.

The relation between the static and total pressure is defined in Equation 5-2.


320 | C H A P T E

SubsonicIf Subsonic is selected, choose the Boundary condition—Normal Stress or Pressure. Then enter a value or expression for the Normal stress f0 (SI unit: N/m2) or Pressure p0 (SI unit: Pa).

These conditions correspond to the ones for the Single-Phase Flow, Laminar Flow interface. See Inlet and Outlet.

Selecting appropriate outlet conditions for the Navier-Stokes equations is not a trivial task. Generally, if there is something interesting happening at an outflow boundary, extend the computational domain to include this phenomenon.


Th eo r y f o r t h e H i g h Ma ch Numbe r F l ow I n t e r f a c e s

In some industrial applications involving fluid flow, the flow velocity is large enough to introduce significant changes in the density and temperature of the fluid. This occurs because the thermodynamic properties of the fluid are coupled. Appreciable changes in the fluid properties are encountered as the flow velocity approaches, or exceeds, the speed of sound. As a rule of thumb, velocities greater than 0.3 times the speed of sound are considered to be high Mach number flows.

The High Mach Number Flow interface theory is described in this section:

• Compressible Flow for All Mach Numbers

• Sutherland’s Law

• Consistent Inlet and Outlet Conditions

• Pseudo Time Stepping for High Mach Number Flow Models

• References for the High Mach Number Flow Interfaces

Compressible Flow for All Mach Numbers

The High Mach Number Flow interfaces solve the following equations

(5-3)

(5-4)

(5-5)

where





t------ u + 0=

ut------- u u+ pI– + F+=

CpTt------- u T+ q – :S T

---- T-------

p

pt------ u p+ – Q+ +=

T H E O R Y F O R T H E H I G H M A C H N U M B E R F L O W I N T E R F A C E S | 321

322 | C H A P T E

• F is the volume force vector (SI unit: N/m3)



• q is the heat flux vector (SI unit: W/m2)

• Q contains the heat sources (SI unit: W/m3)

• S is the strain-rate tensor:

These are the fully compressible Navier-Stokes equations for a simple compressible fluid. As can be seen, the same set of equations can be assembled using, for example, the Non-Isothermal Flow interface or by manually coupling a Single-Phase Flow interface with a Heat Transfer interface. The difference is that the High Mach Number Flow interface can handle flow of any Mach numbers, while the other physics interfaces are subject to The Mach Number Limit. The Mach number is defined as

where a is the speed of sound. Equation 5-3 is hyperbolic whereas Equation 5-4 and Equation 5-5 are parabolic for time-dependent flow and elliptic for stationary flow. If diffusive effects can be neglected, as is usually the case for high-speed flows, the entire system of equations becomes hyperbolic. When the Mach number passes through unity, the direction of the characteristics associated with the hyperbolic system changes. This means that new phenomena not observed for incompressible flows, such as shock waves and expansion fans, can occur (Ref. 2). The stabilization and boundary conditions must be adapted to the change in direction of the characteristics.

Note that the diffusive effects do not disappear entirely unless these terms are explicitly excluded from the equations. Instead, they are confined to either boundary layers or to “shock-waves” which are really thin regions with steep gradients. In the High Mach

Number Flow interfaces these thin regions are assumed to be under resolved, which results in the hyperbolic nature of the system. If the details of these regions are of physical interest they must be adequately resolved. Use a Non Isothermal Flow interface to solve the physics in such cases.

The physics interface assumes that the fluid is an ideal gas. This is necessary for the formulation of the Consistent Inlet and Outlet Conditions. The ideal gas law relates density and specific heats to the pressure and temperature. The viscosity and thermal

S 12--- u u T+ =

Ma ua

-------=


conductivity of an ideal gas can be accurately approximated using Sutherland’s Law, which is included as an option in the High Mach Number Flow interface.

Sutherland’s Law

Sutherland’s law, or Sutherland’s formula, is an approximation for how the viscosity of gases depends on the temperature. This law is based on an idealized intermolecular-force potential and reads (Ref. 5)

(5-6)

where S is an effective temperature called the Sutherland constant. Each gas has its own Sutherland constant. Equation 5-6 is strictly valid only for single-component gases at low pressure. It does, however, work well for air because air is mainly composed of nitrogen and oxygen, which have very similar properties. Parameter values for some common gases are given in Table 5-2 (Ref. 5).

The SI unit for 0 in Table 5-2 is N·sm2. The SI unit for T0 and S is Kelvin (K).

Sutherland’s law can also be formulated for thermal conductivity (Ref. 5):

(5-7)

The Mach Number Limit

TABLE 5-2: SUTHERLAND’S LAW PARAMETERS FOR DYNAMIC VISCOSITY

GAS 0 T0 S

Air 1.716·10-5 273 111

Argon 2.125·10-5 273 114

C02 1.370·10-5 273 222

CO 1.657·10-5 273 136

N2 1.663·10-5 273 107

O2 1.919·10-5 273 139

H2 8.411·10-5 273 97

Steam 1.12·10-5 350 1064

0------ T

T0------ 3 2/ T0 S+

T S+--------------------=

kk0------ T

T0------ 3 2/ T0 Sk+

T Sk+-------------------=


324 | C H A P T E

Values for k0, T0 and Sk for some common gases are given in Table 5-3(Ref. 5).

The unit for k0 in Table 5-3 is W(m·K). The unit for T0 and Sk is Kelvin (K).

Consistent Inlet and Outlet Conditions

In order to provide consistent inlet and outlet conditions for high Mach number flow, the flow situation needs to be monitored at the boundary. Because all flow properties are coupled, the number and combinations of boundary conditions needed for well posedness depend on the flow state—that is, with which speeds the different flow quantities are propagated at the boundary. For a detailed specification on the number of physical boundary conditions needed for well posedness, see Ref. 1.

P L A N E WA V E A N A L Y S I S O F I N V I S C I D F L O W

On inlets a plane wave analysis of the inviscid part of the flow is used in order to apply a consistent number of boundary conditions. The method used here is described in Ref. 3.

Inviscid flow is governed by Euler’s equations, which, provided that the solution is smooth and neglecting the gravity terms, can be written as

Considering a small region close to a boundary, the Jacobian matrices can be regarded as constant, which leads to a system of linear equations

TABLE 5-3: SUTHERLAND’S LAW PARAMETERS FOR THERMAL CONDUCTIVITY

GAS K0 T0 S,k

Air 0.0241 273 K 194

Argon 0.0163 273 170

C02 0.0146 273 1800

CO 0.0232 273 180

N2 0.0242 273 150

O2 0.0244 273 240

H2 0.168 273 120

Steam 0.0181 300 2200

tQ Fj

Q---------Q

xj--------+ 0=

tQ Fj

Q---------

0

Qxj--------+ 0=


where the subscript 0 denotes a reference state at the boundary. Assuming that the state at the boundary, described by a surface normal vector i (pointing out from the domain), is perturbed by a plane wave, the linear system of equations can be transformed to

where

and corresponds to the direction normal to the boundary. In the unsteady case, Euler’s equations are known to be hyperbolic in all flow regimes: subsonic, sonic, and supersonic flow (Ref. 4). This implies that A0 has real-valued eigenvalues and corresponding eigenvectors, and it can therefore be diagonalized according to

The matrix T contains the (left) eigenvectors, and the matrix is a diagonal matrix containing the eigenvalues. The eigenvalues are given exactly by

where cs is the speed of sound. Using the primitive variables

The characteristic variables on the boundary are

tQ A0

Q-------+ 0=

A0 jFj

Q---------

0=

TA0T 1– = ii 1 2 3 4 5 =

1 iui=

2 1=

3 1=

4 1 cs+=

5 1 cs–=

Q

uvwp

=


326 | C H A P T E

(5-8)

Each characteristic can be interpreted to describe a wave transporting some quantity. The first one is an entropy wave while the next two correspond to vorticity waves. The fourth and fifth, in turn, are sound waves.

Evaluating the primitive variables in Equation 5-8, the values are taken from the outside (specified values) or from the inside (domain values) depending on the sign of the eigenvalue corresponding to that characteristic variable. At inlets, a negative eigenvalue implies that the characteristic is pointing into the domain and hence outside values are used. Correspondingly, for a positive eigenvalue the inside values are used.

Variables in Equation 5-8 with a superscript A are computed as averages of the inside and outside values.

The characteristic variables are then transformed to consistent face values of the primitive variables on the boundary in the manner of

w1 p

csA

2----------------–=

w21v 2u–

11 22+-------------------------------------=

w31ii

--------------- w 11 22+ 31u 2v+

11 22+-------------------------------------

–=

w412--- A

csA

--------iui

ii

--------------- p

csA

2----------------+

=

w512--- A

csA

--------–iui

ii

--------------- p

csA

2----------------+

=


(5-9)

Characteristics Based InletsApplying this condition implies using the plane wave analysis described in Consistent Inlet and Outlet Conditions. With this condition, a varying flow situation at the inlet can be handled. This means that changes due to prescribed variations at the boundary, due to upstream propagating sound waves or spurious conditions encountered during the nonlinear solution procedure, can be handled in a consistent manner. The full flow condition at the inlet is specified by the following properties

(5-10)

from which the density is computed using the ideal gas law. The dependent variables defined in Equation 5-10 are applied as the outside values used in Equation 5-8, and the boundary values of the dependent variables are obtained from Equation 5-9.

Supersonic InletsApplying a supersonic inlet, the full flow at the inlet is specified using the inlet properties in Equation 5-10. Because the flow is supersonic, all characteristic at the inlet are known to be directed into the domain, and the boundary values of the dependent variables are computed directly from the inlet properties.

Hybrid OutletWhen building a model, it is recommended that it is constructed so that as little as possible happens at the outlet. In the high Mach number flow case this implies keeping the conditions either subsonic or supersonic at the outlet. This is, however, usually not possible. For example, often one boundary adjacent to the outlet consists of a no slip wall, in which case a boundary layer containing a subsonic region is present. The hybrid outlet feature adds the following weak expression:

b w1 w2 w3+ +=

u1 b2w2

11 22+-------------------------------------–

1

ii

---------------3w3

11 22+------------------------------------- aA

pA------ w4 w5+ –

–=

u2 b2w2

11 22+-------------------------------------

1

ii

---------------3w3

11 22+------------------------------------- aA

pA------ w4 w5+ –

–=

u3 b1ii

---------------w3

11 22+------------------------------------- 3

aA

pA------ w4 w5+ +

=

pb aA 2

w4 w5+ =

Min pin Tin


328 | C H A P T E

where û is the test function for the velocity vector. This corresponds to a pressure, no viscous stress condition in regions with subsonic flow and a no viscous stress condition in regions with supersonic flow. When the static pressure at the outlet is not known beforehand, it is recommended that it is set to the inlet pressure. When a converged solution has been reached, the solution can be analyzed to find the pressure level just outside the sonic point (Ma = 1) along the boundary. You can then apply this pressure level instead.

Supersonic OutletWhen the outlet condition is known to be fully supersonic, the viscous stress is specified in accordance with the equations and hence no physical condition is applied. This is done by prescribing the boundary stress using the full stress vector:

It is often possible to use the supersonic condition at outlets that are not strictly supersonic but mainly supersonic (the main part of the outlet boundary contains supersonic flow).

Pseudo Time Stepping for High Mach Number Flow Models

Pseudo time stepping is per default applied to all governing equations for stationary problems, for 2D models as well as 3D models. The momentum, continuity, energy and turbulence equations (when present) use the same expression for the pseudo time step symbol .

For laminar models the automatic expression for CFLloc is

while for models with turbulent flow it is

WNS0.5 p– pout– n u Ma 1

p– n u else

=

WNS pI– u u T+ 23--- u I–+ n u =

t

1 +

if niterCMP 10 1.2min niterCMP 10– 12 0 +




References for the High Mach Number Flow Interfaces

1. T. Poinsot and D. Veynante, Theoretical and Numerical Combustion, 2nd ed., Edwards, 2005.

2. J.D. Anderson, Modern Compressible Flow, 3rd ed., McGraw-Hill, 2003.

3. J. Larsson, Numerical Simulation of Turbulent Flows for Turbine Blade Heat Transfer Applications, Ph.D thesis, Chalmers University of Technology, 1998.

4. J.D. Tannehill, D.A. Anderson, and R.H. Pletcher, Computational Fluid Mechanics and Heat Transfer, 2nd ed., Taylor & Francis, 1997.

5. F.M. White, Viscous Fluid Flow, 3rd ed., McGraw-Hill, 2006.

1 +



if niterCMP 220 1.3min niterCMP 220– 9 0

• The Projection Method for the Navier-Stokes Equations



330 | C H A P T E

6

M u l t i p h a s e F l o w

The fluid-flow interfaces are grouped by type under the Fluid Flow main branch when adding a physics interface. This chapter discusses applications involving the Multiphase Flow branch ( ) when adding a physics interface. The section Modeling Multiphase Flow helps you choose the best one to start with.

In this chapter:

• The Laminar Flow, Two-Phase, Level Set and Phase Field Interfaces

• The Turbulent Flow, Two-Phase, Level Set and Phase Field Interfaces

• The Bubbly Flow Interfaces

• The Mixture Model Interfaces

• The Euler-Euler Model, Laminar Flow Interface

• Theory for the Two-Phase Flow Interfaces

• Theory for the Bubbly Flow Interfaces

• Theory for the Mixture Model Interfaces

• Theory for the Euler-Euler Model, Laminar Flow Interface

331

332 | C H A P T E

Mode l i n g Mu l t i p h a s e F l ow

In this section:


• The Multiphase Flow Interface Options

• The Relationship Between the Physics Interfaces



The Multiphase Flow branch ( ) included with this module has a number of subbranches to describe momentum transport for multiphase flow. One or more physics interfaces can be added; either singularly or in combination with other physics interfaces for applications such as mass transfer and energy (heat) transfer.

Different types of flow require different equations to describe them. If you know the type of flow to model, then select it directly. However, when you are not certain of the flow type, or when it is difficult to obtain a solution, you can instead start with a simplified model and add complexity as you build the model. Then you can successively advance forward, comparing models and results.

The Bubbly Flow and the Mixture Model interfaces are appropriate when you want to simulate a flow with many particles, droplets, or bubbles immersed in a liquid. With these physics interfaces, you do not track each particle in detail. Instead you solve for the averaged volume fraction. If you are interested in the exact motion of individual bubbles,

Certain types of multiphase flow can be described using the Phase Field and Level Set interfaces found under the Mathematics>Moving Interface

branch ( ). In this module these physics features are already integrated into the relevant fluid-flow interfaces.

• The Physics Interfaces in the COMSOL Multiphysics Reference Manual

• The Mathematics, Moving Interface Branch

R 6 : M U L T I P H A S E F L O W

including how the fluid interface deforms due to, for instance, surface tension, use any of the Two-Phase Flow interfaces.

To model the detailed dynamics of fluid interfaces, either use the level set method or the phase field method. In general, it is not obvious which one of these to use, but it is easy to switch between the two models within the Two-Phase Flow interfaces. You can also easily switch turbulence modeling on and off or modify compressibility assumptions.

The Multiphase Flow Interface Options

For any of the multiphase flow interfaces, you can assume either laminar or turbulent flow as the starting point. This enables you to make the appropriate mathematical model assumptions required to solve the flow. Turbulence is modeled using the standard k- model

The Relationship Between the Physics Interfaces

Several of the interfaces vary only by one or two default settings (see Table 6-1, Table 6-2, and Table 6-3) in the Physical Model section, which are selected either from a check box or a list. For the Multiphase Flow branch, the Bubbly Flow (bf) and Mixture Model (mm) subbranches have two physics interfaces each and both have the same Interface Identifier. All the Two-Phase Flow interfaces also have the same Interface Identifier (tpf). The differences are based on the default settings required to model that specific type of flow as described in this section. Figure 6-1 shows the Turbulent Two-Phase Flow, Phase Field settings window, which has options to select whether to use the Two-Phase Flow, Phase Field

M O D E L I N G M U L T I P H A S E F L O W | 333

334 | C H A P T E

or Two-Phase Flow, Level Set method, whether the flow is compressible or incompressible and whether it is best described as Stokes flow, laminar or turbulent flow.

Figure 6-1: The settings window for the Turbulent Two-Phase Flow, Phase Field interface. Select whether to model compressible or incompressible flow, and choose between Stokes, laminar and turbulent flow. Combinations are also possible using either the phase field or level set methods.

B U B B L Y F L O W

The Bubbly Flow ( ) branch interfaces are used primarily to model two-phase flow where the fluids are gas-liquid mixtures, and gas content is less than 10%. The Laminar Bubbly Flow Interface ( ) and The Turbulent Bubbly Flow Interface ( ) solve the flow

TABLE 6-1: BUBBLY FLOW PHYSICAL MODEL DEFAULT SETTINGS

INTERFACE ID LOW GAS CONCENTRATION

TURBULENT MODEL TYPE

SOLVE FOR INTERFACIAL AREA

Laminar Bubbly Flow bf Yes None No

Turbulent Bubbly Flow bf Yes RANS, k- No


equations, whether described by the Navier-Stokes equations or the RANS equations with the k- turbulence model, and where the momentum equation is corrected by a term induced by the slip velocity. The slip velocity can be described by the Hadamard-Rybczynski drag law for small spherical bubbles, a nonlinear drag law taking surface tension into account for larger bubbles, or by defining it on your own.

By default, the physics interfaces assume that the volume fraction of the gas is less than 0.1. It is then valid to approximate the liquid velocity as incompressible. This is significantly easier to solve numerically. It is possible, though, to use the complete continuity equation.

The physics interfaces also allow you to define your own relations for the density of both phases and for the dynamic viscosity of the gas phase. Definitions of non-Newtonian fluid flow through the power law and Carreau models are however not possible. You can also model mass transfer between the two phases, using the two-film theory or your own expression for interfacial mass transfer.

M I X T U R E M O D E L F L O W

The Mixture Model ( ) branch interfaces are similar to the Bubbly Flow interfaces except that both phases are assumed to be incompressible. Examples include solid particles dispersed in a liquid, and liquid droplets dispersed in another liquid when the two liquids are immiscible.

Like the Bubbly Flow interfaces, The Mixture Model, Laminar Flow Interface ( ) and The Mixture Model, Turbulent Flow Interface ( ) solve the flow equations, whether described by the Navier-Stokes equations or the RANS equations with the k- turbulence model, and where the momentum equation is corrected by a term induced by the slip velocity. The slip velocity can be described by the Hadamard-Rybczynski, Schiller-Naumann or Haider-Levenspiel method, or by defining it on your own.

These physics interfaces also allow you to define your own relations for the dynamic viscosity and density of both phases. Definitions of non-Newtonian fluid flow through the power law and Carreau models are however not possible. The dynamic viscosity of the

TABLE 6-2: MIXTURE MODEL PHYSICAL MODEL DEFAULT SETTINGS

INTERFACE ID DISPERSED PHASE

SLIP MODEL TURBULENCE MODEL TYPE

SOLVE FOR INTERFACIAL AREA

Mixture Model, Laminar Flow

mm Solid particles

Homogeneous flow

None No

Mixture Model, Turbulent Flow

mm Solid particles

Homogeneous flow

RANS, k- No


336 | C H A P T E

mixture can either be of Krieger type (which uses a maximum packing concentration), volume-averaged (for gas-liquid, liquid-liquid systems), or a user-defined expression.

You can also describe other material properties such as density by entering equations that describe this term as a function of other parameters like material concentration, pressure, or temperature. The physics interfaces also enable you to model mass transfer between the two phases, using the two-film theory or your own expression for interfacial mass transfer.

TW O - P H A S E F L O W I N T E R F A C E S

Two-Phase Flow, Level SetThe Laminar Two-Phase Flow, Level Set Interface ( ) and The Turbulent Flow, Two-Phase Flow, Level Set Interface ( ), found under the Two-Phase Flow, Level Set branch ( ), are used primarily to model two fluids separated by a fluid interface. The moving interface is tracked in detail using the level set method. Surface tension acting on the fluid interface is automatically included in the fluid-flow equations.

Like for other fluid-flow interfaces, compressible flow is possible to model for speeds less than Mach 0.3 in the Two-Phase Flow, Level Set interface. You can also choose to model incompressible flow, and simplify the equations to be solved. Stokes’ law is an option.

Specify the density and viscosity for each of the two fluids. For any of the two fluids, you can easily use non-Newtonian models based on the power law or Carreau model, or by using an arbitrary user-defined expression.

TABLE 6-3: TWO-PHASE FLOW PHYSICAL MODEL DEFAULT SETTINGS

INTERFACE ID MULTIPHASE FLOW MODEL

COMPRESSIBILITY TURBULENCE MODEL TYPE

NEGLECT INERTIAL TERM (STOKES FLOW)

Laminar, Two-Phase Flow, Level Set

tpf Two-phase flow, level set

Incompressible flow

None False

Turbulent, Two-Phase Flow, Level Set

tpf Two-phase flow, level set

Incompressible flow

RANS, k- False

Laminar, Two-Phase Flow, Phase Field

tpf Two-phase flow, phase field

Incompressible flow

None False

Turbulent, Two-Phase Flow, Phase Field

tpf Two-phase flow, phase field

Incompressible flow

RANS, k- False


Two-Phase Flow, Phase Field The Laminar Two-Phase Flow, Phase Field Interface ( ) and The Turbulent Two-Phase Flow, Phase Field Interface ( ) found under the Two-Phase Flow, Phase Field branch ( ), also model two fluids separated by a fluid interface. You can easily switch between the physics interfaces, which can be useful if you are not sure which physics interface provides the best description. A library surface tension coefficients between a number of common substances is also available.

Like for other fluid-flow interfaces, compressible flow is possible to model for speeds less than Mach 0.3 in the Two-Phase Flow, Phase Field interfaces. You can also model incompressible flow and simplify the equations to be solved. Stokes flow is also an option.

Specify the density and viscosity for each of the two fluids. For any of the two fluids, you can easily use non-Newtonian models based on the power law or Carreau model, or by using an arbitrary user-defined expression.

E U L E R - E U L E R M O D E L , L A M I N A R F L O W

The Euler-Euler Model, Laminar Flow Interface is used to model the flow of two continuous and fully interpenetrating phases. For both phases the conservation equations are averaged over volumes, which are small compared to the computational domain, but large compared to the dispersed phase particles, droplets or bubbles.


Often, you are simulating applications that couple fluid flow to another type of phenomenon described in another physics interface. Although this is not often another type of flow, it can still involve physics interfaces supported in the CFD Module or in the COMSOL Multiphysics base package. This typically occurs in cases where applications include chemical reactions and mass transport, as included in Chemical Species Transport, or energy transport, found in the Heat Transfer and Non-Isothermal Flow chapter.

More extensive descriptions of heat transfer, such as those involving radiation, can be found in the Heat Transfer Module, while tools for modeling chemical reactions and mass transport are available in the Chemical Reaction Engineering Module.


338 | C H A P T E

Th e L am i n a r F l ow , Two -Pha s e , L e v e l S e t and Pha s e F i e l d I n t e r f a c e s

The following sections list all the physics interfaces and the physics features associated with them under the Multiphase Flow branch ( ). The descriptions follow a structured order as defined by the order in the branch. Because many of the physics interfaces are integrated with each other, some nodes described also cross reference to other physics interfaces.

This section includes the following topics:

• The Laminar Two-Phase Flow, Level Set Interface

• The Laminar Two-Phase Flow, Phase Field Interface

• Domain, Boundary, Point, and Pair Nodes for the Laminar and Turbulent Flow, Two-Phase, Level Set and Phase Field Interfaces

The Laminar Two-Phase Flow, Level Set Interface

The Laminar Two-Phase Flow, Level Set (tpf) interface ( ), found under the Multiphase

Flow>Two-Phase Flow, Level Set branch ( ) when adding a physics interface, is used to track the interface between two immiscible fluids. The flow is assumed to be laminar, that is, to be of low to moderate Reynolds number. The fluids can be incompressible or compressible.

The physics interface solves Navier-Stokes equations for the conservation of momentum and a continuity equation for the conservation of mass. The interface position is tracked by solving a transport equation for the level-set function.


• The Mathematics, Moving Interface Branch

• The Turbulent Flow, Two-Phase, Level Set and Phase Field Interfaces


Simulations using the Laminar Two-Phase Flow, Level Set interface are always time-dependent since the position of an interface is almost always dependent of its history.

The main node is the Fluid Properties feature, which adds the Navier-Stokes equations and the level set equation and provides an interface for defining the properties of the fluids and the surface tension.

When this physics interface is added, the following default nodes are also added in the Model Builder—Fluid Properties, Wall (with a default No slip boundary condition), Initial

Interface, and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and volume forces. You can also right-click Laminar Two-Phase Flow, Level Set to select physics from the context menu.



The default identifier (for the first physics interface in the component) is tpf.


The default setting is to include All domains in the model. To choose specific domains, select Manual from the Selection list.


The Laminar Two-Phase Flow, Level-Set interface uses a level set method to track the fluid-fluid interface.

Except where included below, the Laminar Two-Phase Flow, Level Set has the same sections and settings as the Laminar Flow interface.

This physics interface changes to a Laminar Two-Phase Flow, Phase Field

interface if the Multiphase flow model selected is Two-phase flow, phase

field. See The Laminar Two-Phase Flow, Phase Field Interface for details.

T H E L A M I N A R F L O W , TW O - P H A S E , L E V E L S E T A N D P H A S E F I E L D I N T E R F A C E S | 339

340 | C H A P T E

The Compressibility defaults to Incompressible flow (constant density flow). Select Compressible flow (Ma<0.3) to use the compressible formulation of the Navier-Stokes equations.

Select the Neglect inertial term (Stokes flow) check box to model flow at very low Reynolds numbers for which the inertial term in the Navier-Stokes equations can be neglected. The physics interface then solves the linear Stokes equations instead. The Stokes equations are valid for creeping flow, which can occur in microfluidics and MEMS devices, where the flow speed or length scales are very small.


The dependent variables (field variables) are:

• the Velocity field (SI unit: m/s)

• the Pressure (SI unit: Pa)

• the Level set variable

• the Reciprocal initial interface distance (SI unit: 1/m)

The names can be changed but the names of fields and dependent variables must be unique within a component.


To display this section, click the Show button ( ) and select Advanced Physics Options. Normally these settings do not need to be changed. Select an option for the Convective

term—Non-conservative form (the default) or Conservative form.

By default the Turbulence model type selected is None. If RANS, k- is selected, the interface changes to The Turbulent Flow, Two-Phase Flow, Level Set Interface.

Conservative and Non-Conservative Formulations





• The consistent stabilization methods are applicable to the Navier-Stokes equations and

Level set equation—Streamline diffusion and Crosswind diffusion. These check boxes are selected by default. If required, click to clear one or more of the Streamline diffusion and Crosswind diffusion check boxes.

• The Isotropic diffusion inconsistent stabilization method can be activated for the Navier-Stokes equations.

• The Anisotropic diffusion inconsistent stabilization method can be activated for the Navier-Stokes equations. If this check box is selected, also enter a value for the Tuning

parameter sd. The default is 0.25.

The Laminar Two-Phase Flow, Phase Field Interface

The Laminar Two-Phase Flow, Phase Field (tpf) interface ( ), found under the Multiphase

Flow>Two-Phase Flow, Phase Field branch ( ) when adding a physics interface, is used to track the interface between two immiscible fluids. The flow is assumed to be laminar, that is, to be of low to moderate Reynolds number. The fluids can be incompressible or compressible.

See The Brinkman Equations Interface for these settings.




Filling of a Capillary Channel—Phase Field: model library path CFD_Module/Multiphase_Tutorials/capillary_filling_ls


342 | C H A P T E

The physics interface solves Navier-Stokes equations for the conservation of momentum and a continuity equation for the conservation of mass. The interface position is tracked by solving two additional transport equations, one for the phase field variable and one for the mixing energy density. The movement of the surface is determined by minimization of free energy.

Simulations using the The Laminar Two-Phase Flow, Phase Field interface are always time-dependent since the position of an interface is almost always dependent of its history.

The main node is the Fluid Properties feature, which adds the Navier-Stokes equations and the phase field equations and provides an interface for defining the properties of the fluids and the surface tension.

When this physics interface is added, the following default nodes are also added in the Model Builder—Fluid Properties, Wall (with a default No slip boundary condition), Initial

Interface, and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and volume forces. You can also right-click Laminar Two-Phase Flow, Phase Field to select physics from the context menu


The Laminar Two-Phase Flow, Phase Field interface uses a phase field method to track the fluid-fluid interface.

Except for the Physical Model section, the settings are the same as for The Laminar Two-Phase Flow, Level Set Interface.

This physics interface changes to a Laminar Two-Phase Flow, Level Set

interface if the Multiphase flow model selected is Two-phase flow, level set. See The Laminar Two-Phase Flow, Level Set Interface for details.

By default the selected Turbulence model type is None. If RANS, k- is selected, the physics interface changes to The Turbulent Two-Phase Flow, Phase Field Interface.


Domain, Boundary, Point, and Pair Nodes for the Laminar and Turbulent Flow, Two-Phase, Level Set and Phase Field Interfaces

The Laminar Flow, Two-Phase, Level Set and Phase Field Interfaces has these domain, boundary, point, and pair nodes, which are available from the Physics ribbon toolbar (Windows users), Physics context menu (Mac or Linux users), or right-click to access the context menu (all users).

These nodes, listed in alphabetical order, are described in this section:


• Gravity

• Initial Interface




Filling of a Capillary Channel—Phase Field: model library path CFD_Module/Multiphase_Tutorials/capillary_filling_pf



344 | C H A P T E

• Initial Values

• Wall


The Wall node represents the walls in a two-phase flow simulation. The Wall feature for this physics interface includes slip, no slip, and slip velocity boundaries as well as sliding, leaking, moving, and wetted walls. The No

slip boundary condition is the default boundary condition. These are all described for the Laminar Flow interface.

The Wetted wall and Moving, wetted wall options are described in this section. See Wall.

• Flow Continuity

• Inlet


• Outlet




• Symmetry

• Volume Force

• Wall

In addition to the surface level boundary conditions listed below, the pressure point constraint boundary condition is also available and should be used in addition to the boundary conditions listed when the pressure level is not specified by any other boundary condition.


Wall

Wetted Wall The Wetted wall boundary condition is suitable for walls in contact with the fluid-fluid interface. If this boundary condition is used, the fluid-fluid interface can move along the wall. For applications where the interface is fixed on the wall, the no slip condition is suitable.

The implementation of the wetted wall boundary condition differs depending on the method used to track the fluid-fluid interface (the level set method or the phase field method).

Level Set For the level set method, this boundary condition enforces the no-penetration condition u nwalland adds a frictional force of the form

where is the slip length. For numerical calculations it is suitable to set h, where h is the mesh element size. The boundary condition does not set the tangential velocity component to zero; however, the extrapolated tangential velocity component is 0 at a distance outside the wall (see Figure 6-2).

Finally, the boundary condition adds the following weak boundary term:

The boundary term results from the partial integration of the surface tension force in the momentum equation. Define the contact angle w (that is, the angle between the fluid

The Wall node represents the walls in a two-phase flow simulation. The Wall feature for this physics interface includes slip, no slip, and slip velocity boundaries as well as sliding, leaking, moving, and wetted walls. The No

slip boundary condition is the default boundary condition. These boundary conditions are described for the Laminar Flow interface.

The Wetted wall and Moving, wetted wall boundary conditions are described in this section. See the Laminar Flow interface for the other settings (Wall).

Ffr---– u=

test u nwall n wcos – Sd


346 | C H A P T E

interface and the wall). Figure 6-2 illustrates the definition of the contact angle.

Figure 6-2: Definition of the contact angle at interface/wall contact points (left) and an illustration of the slip length (right).

With the level set method, define the following two properties for the wetted wall:

• Enter a value or expression for the Contact angle w. The default is pi/2 (/2) rad.

• Enter a value or expression for the Slip length (SI unit: m). The default is h, which is the variable for the local mesh element size h.

Phase Field The motion of the interface on the boundary due to advection is zero and so the no slip boundary condition, u = 0, is used in the momentum equation. The following boundary condition defines the contact angle between Fluid 2 and the wall:

(6-1)

where w is the user-defined contact angle and n is the unit vector normal to the wall. The phase field help variable is assigned the boundary condition:

(6-2)

With the phase field method, enter a value or expression for the Contact angle w. The default value is pi/2 (/2) rad.

Moving Wetted Wall When the wetted wall is moving, Equation 6-1 and Equation 6-2 are still valid but the boundary condition for the momentum equation becomes u = uw. Enter a value or expression for:

• The components of the Velocity of the moving wall uw (SI unit: m/s).

• The Contact angle w. The default is pi/2 (2) rad.

Fluid 1

Fluid 2

Wall

u Wall

n 2 2 w cos =

n

2------ 0=


Fluid Properties

The Fluid Properties node adds the flow equations and the equations for the level set variable or the phase field variable (where applicable). For the level set and phase field interfaces, specify the fluid that each domain is initially filled with using the Initial Values node.




Specify the temperature to be used in the model:

• Select User defined to enter a value or expression for the temperature (SI unit: K).

• Alternatively, select a temperature defined by a heat transfer interface present in the component (if any). For example, select Temperature (ht/fluid1) to use the temperature defined by the Fluid1 node in the Heat Transfer interface ht.

F L U I D 1 A N D F L U I D 2 P R O P E R T I E S

Specify the Density and the Dynamic viscosity of Fluid 1 and Fluid 2 in the same way as in a single-phase flow interface (see Fluid Properties).

The material properties are obtained from the domain irrespective of the location of the interface. If two different materials are selected in domains 1 and 2, with the phase boundary initially coincident with the domain boundary, the model has convergence issues once the phase boundary moves away from the domain boundary. This is because a density discontinuity and a viscosity discontinuity occurs at the boundary separating the two


Care should be taken when using the Domain Material setting for the material properties for Fluid 1 and Fluid 2.


348 | C H A P T E

fluids. For this reason selecting the material directly is recommended when setting the material properties for Fluid 1 and Fluid 2.

The fluid defined as Fluid 1 affects the wetting characteristics on wetted walls. See the Wall node for details.

S U R F A C E TE N S I O N

Select the Neglect surface tension in momentum equation check box to neglect surface tension.

Select a Surface tension coefficient (SI unit: N/m):

• To use a predefined expression, select Library coefficient, liquid/gas interface or Library

coefficient, liquid/liquid interface. Then select an option from the list that displays below (for example, Water/Air, Glycerol/Air and so forth).

• If User defined is selected, enter a value or expression for the surface tension coefficient (SI unit: N/m).

L E V E L S E T P A R A M E T E R S

This section is available with The Laminar Two-Phase Flow, Level Set Interface and when turbulence is active.

Specify the parameters for the level set method:

• Enter a value or expression for the Reintialization parameter (SI unit: m/s). The default value is 1 m/s. Use an approximate value for the maximum speed occurring in the flow. Because the flow speed is not always known in advance, an initial computation might be needed to find a proper value for .

• Enter a value or expression for the Parameter controlling interface thickness ls (SI unit: m). The default expression is tpf.hmax/2, which corresponds to half of the maximum mesh element size in the model. In general, the results are optimal if the default value is used for this parameter.

P H A S E F I E L D P A R A M E T E R S

This section is available with The Laminar Two-Phase Flow, Phase Field Interface and when turbulence is active.

This section is not available for the turbulent versions of these physics interfaces.


Define the parameters for the phase field method.

Enter a value or expression for the Parameter controlling interface thickness pf (SI unit: m). The default expression is tpf.hmax/2, which corresponds to half of the maximum mesh element size in the model. In general, simply use the default value for optimal accuracy.

Enter a value or expression for the Mobility tuning parameter (SI unit: m·s/kg). The default is 1 m·s/kg, which is a good starting point for most models. This parameter determines the time scale of the Cahn-Hilliard diffusion and thereby also governs the diffusion-related time scale of the interface.

E X T E R N A L F R E E E N E R G Y

This section is available with The Laminar Two-Phase Flow, Phase Field Interface and when turbulence is active.

Add a source of external free energy as an external force term in the flow equation (the Fext term in Equation 3-2). The external free energy fext is a user-defined free energy. In most cases, the external free energy is zero.


This section is available for The Turbulent Flow, Two-Phase, Level Set and Phase Field Interfaces. By default, the Mixing length limit lmix, lim (SI unit: m) is Automatic. Select Manual to enter a different value or expression.

Gravity

The Gravity node adds the force Fg in Equation 3-2, which is equal to g, where g is the gravity vector.

Keep the parameter value high enough to maintain a constant interface thickness yet low enough not to damp the convective motion. A too high mobility value can also lead to excessive diffusion of droplets.

The expression for the external free energy must be manually differentiated with respect to and then entered into the —derivative

of external free energy field for (SI unit: J/m3). f


350 | C H A P T E


From the Selection list, choose the domains where gravity acts on the fluid.

G R A V I T Y

Enter the coordinates for the Gravity vector g (SI unit: m/s2).

Initial Values

The Initial Values node adds initial values for the flow variables that serve as an initial condition for a transient simulation. Also specify which of the fluids initially occupies the domain selection.



For 3D and 2D axisymmetric models, the default value is -g_const in the z component. g_const is the acceleration of gravity (a predefined physical constant).

For 2D models, the default value is -g_const in the y component. g_const is the acceleration of gravity (a predefined physical constant).

If the Transient with Initialization ( ) study is being used, for the initialization to work it is crucial that there are two Initial Value nodes and one Initial Interface node. One of the Initial Values nodes should use Fluid

initially in domain: Fluid 1and the other Fluid initially in domain: Fluid 2. The Initial Interface node should have all interior boundaries where the interface is initially present as selection. If the selection of the Initial Interface node is empty, the initialization fails.

See Phase Initialization.



Enter values or expressions for the initial value of the Velocity field u and for the Pressure p. The default values are all 0.

In The Turbulent Flow, Two-Phase, Level Set and Phase Field Interfaces, initial values for the turbulence variables are also specified. By default these are specified using the predefined variables defined by the expressions in Initial Values.

For the Level Set and Phase Field interfaces, click the Fluid 1 button or the Fluid 2 button under Fluid initially in domain to prescribe the fluid initially in the current domain selection. Add another Initial Values node to specify the fluid initially in another domain.

Initial Interface

Use the Initial Interface node to define the initial position as a boundary condition on interior boundaries. During the initialization step, this boundary condition sets the level set function to 0.5 or the phase field function to 0. Transient simulations of the fluid flow treats the boundary as an interior boundary.



If the Transient with Initialization ( ) study is being used, it is crucial that there are two Initial Value nodes and one Initial Interface node for the initialization to work. One of the Initial Values nodes should use Fluid

initially in domain: Fluid 1and the other Fluid initially in domain: Fluid 2. The Initial Interface node should have all interior boundaries where the interface is initially present as selection. If the selection of the Initial Interface node is empty, the initialization fails.

See Phase Initialization.


352 | C H A P T E

Th e Tu r bu l e n t F l ow , Two -Pha s e , L e v e l S e t and Pha s e F i e l d I n t e r f a c e s

In this section:

• The Turbulent Flow, Two-Phase Flow, Level Set Interface

• The Turbulent Two-Phase Flow, Phase Field Interface

• Mixing Length Limit

The Turbulent Flow, Two-Phase Flow, Level Set Interface

The Turbulent Two-Phase Flow, Level Set (tpf) interface ( ), found under the Multiphase

Flow>Two-Phase Flow, Level Set branch ( ), is used to track the interface between two immiscible fluids. The flow is assumed to be turbulent; that is, its Reynolds number is high. The fluids can be incompressible or compressible.

The physics interface solves Navier-Stokes equations for the conservation of momentum and a continuity equation for the conservation of mass. The interface position is tracked by solving a transport equation for the level-set function. Turbulence effects are modeled using the standard two-equation k- model with realizability constraints. Flow close to walls is modeled using wall functions.

Simulations using the Turbulent Two-Phase Flow, Level Set interface are always time-dependent since the position of an interface is almost always dependent of its history.

The main physics node is the Fluid Properties feature, which adds the RANS (Reynolds Averaged Navier-Stokes) equations, turbulence transport equations and the level set equation, and provides an interface for defining the properties of the fluids and the surface tension.

When this physics interface is added, the following default nodes are also added in the Model Builder—Fluid Properties, Wall (with a default Wall functions boundary condition), Initial Interface, and Initial Values. Then, from the Physics toolbar, add other nodes that


implement, for example, boundary conditions and volume forces. You can also right-click Turbulent Two-Phase Flow, Level Set to select physics from the context menu.


The Turbulent Two-Phase Flow, Level Set interface uses a level set method to track the fluid-fluid interface.


Enter values for the model parameters Ce1, Ce2, C, k, e, and v. Turbulence model parameters are optimized to fit as many flow types as possible, but for some special cases, better performance can be obtained by tuning the model parameters. For a description of the turbulence model and the included model parameters see Theory for the Turbulent Flow Interfaces.



• the Velocity field (SI unit: m/s)

• the Pressure (SI unit: Pa)

• the Level set variable (dimensionless)

• the Reciprocal initial interface distance (SI unit: 1/m)

• the Turbulent kinetic energy (SI unit: m2/s2)

• the Turbulent dissipation rate (SI unit: m2/s3)

Except for the sections included below, the rest of the settings for this physics interface are the same as for The Laminar Two-Phase Flow, Level Set Interface.

This physics interface changes to The Turbulent Two-Phase Flow, Phase Field Interface if the Multiphase flow model selected is Two-phase flow,

phase field.

By default the Turbulence model type selected is RANS, k-. If None is selected, the physics interface changes to The Laminar Two-Phase Flow, Level Set Interface.

T H E TU R B U L E N T F L O W , TW O - P H A S E , L E V E L S E T A N D P H A S E F I E L D I N T E R F A C E S | 353

354 | C H A P T E

The names can be changed but the names of fields and dependent variables must be unique within a model.



• The consistent stabilization methods are applicable to the Navier-Stokes equations, the Turbulence Equations and the Level set equation.

• The Isotropic diffusion inconsistent stabilization method is available for the Navier-Stokes

equations and the Turbulence equations.

• The Anisotropic diffusion inconsistent stabilization method can be activated for the Navier-Stokes equations. If this check box is selected, also enter a value for the Tuning

parameter sd. The default is 0.25.


To display this section, click the Show button ( ) and select Advanced Physics Options. Normally these settings do not need to be changed. Select an option for the Convective

term—Non-conservative form (the default) or Conservative form.

The Scaling parameters for turbulence variables subsection contains the parameters Uscale and Lfact that are used to calculate absolute tolerances for the turbulence variables. The section is only visible when Turbulence model type in the Physical Model section is set to RANS, k-.







The Turbulent Two-Phase Flow, Phase Field Interface

The Turbulent Two-Phase Flow, Phase Field (tpf) interface ( ), found under the Multiphase

Flow>Two-Phase Flow, Phase Field branch ( ), is used to track the interface between two immiscible fluids. The flow is assumed to be turbulent; that is, its Reynolds number is high. The fluids can be incompressible or compressible.

The physics interface solves Navier-Stokes equations for the conservation of momentum and a continuity equation for the conservation of mass. The interface position is tracked by solving two additional transport equations, one for the phase field variable and one for the free energy. The movement of the surface is determined by minimization of free energy. Turbulence effects are modeled using the standard two-equation k- model with realizability constraints. Flow close to walls is modeled using wall functions.

Simulations using the The Turbulent Two-Phase Flow, Phase Field interface are always time-dependent since the position of an interface is almost always dependent of its history.

The main node is the Fluid Properties feature, which adds the RANS (Reynolds Averaged Navier-Stokes) equations, turbulence transport equations and the phase field equations, and provides an interface for defining the properties of the fluids and the surface tension.

When this physics interface is added, the following default nodes are also added in the Model Builder— Fluid Properties, Wall (with a Wall functions boundary condition), Initial

Interface, and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and volume forces. You can also right-click Turbulent Two-Phase Flow, Phase Field to select physics from the context menu.


The Turbulent Two-Phase Flow, Phase Field interface uses a phase field method to track the fluid-fluid interface.

This physics interface changes to The Turbulent Flow, Two-Phase Flow, Level Set Interface if the Multiphase flow model selected is Two-phase flow,

level set.

T H E TU R B U L E N T F L O W , TW O - P H A S E , L E V E L S E T A N D P H A S E F I E L D I N T E R F A C E S | 355

356 | C H A P T E

Mixing Length Limit

This section appears in the Fluid Properties node settings when using a k- turbulence model.

The k- turbulence model needs an upper limit on the mixing length to be numerically robust. Select a Mixing length limit—Automatic (the default) or Manual.

• If Automatic is selected, the mixing length limit is automatically evaluated as:

(6-3)

where lbb is the shortest side of the geometry bounding box. If the geometry is a complicated system of very slender entities, for example, Equation 6-3 tends to give a result that is too large. In such cases, define manually.


By default the Turbulence model type selected is RANS, k-. If None is selected, the physics interface changes to The Laminar Two-Phase Flow, Phase Field Interface.

Except for the Physical Model section, the rest of the settings for this physics interface are the same as for The Turbulent Flow, Two-Phase Flow, Level Set Interface and The Laminar Two-Phase Flow, Level Set Interface.




lmixlim 0.5lbb=

lmixlim

lmixlim


Th e Bubb l y F l ow I n t e r f a c e s

In this section:

• The Laminar Bubbly Flow Interface

• The Turbulent Bubbly Flow Interface

• Domain and Boundary Nodes for the Laminar and Turbulent Bubbly Flow Interfaces

The Laminar Bubbly Flow Interface

The Laminar Bubbly Flow (bf) interface ( ), found under the Multiphase Flow>Bubbly Flow branch ( ) when adding a physics interface, is used to model the flow of liquids with dispersed bubbles at low and moderate Reynolds numbers.

It is assumed that the bubbles only occupy a small volume fraction and that they always travel with their terminal velocity. It is thereby possible to solve only one set of Navier-Stokes equations for the liquid phase and to let the velocity of the bubbles be guided by a slip model. The pressure distribution is computed from a mixture-averaged continuity equation. The volume fraction of bubbles is tracked by solving a transport equation for the effective gas density.

The physics interface can also model the distribution of the number density, that is, the number of bubbles per unit volume which in turn can be used to calculate the interfacial area, useful when simulating chemical reactions in the mixture.

The main physics node is the Fluid Properties feature, which adds the equations for laminar bubbly flow and provides an interface for defining the fluid materials for the liquid and the gas and the slip velocity model to use.

When this physics interface is added, the following default physics nodes are also added in the Model Builder—Laminar Bubbly Flow, Fluid Properties, Wall (the default boundary types are No slip for the liquid and No gas flux for the gas), and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and volume forces. You can also right-click Laminar Bubbly Flow to select physics from the context menu.


The interface identifier is used primarily as a scope prefix for variables defined by the physics interface. Refer to these variables in expressions using the pattern

T H E B U B B L Y F L O W I N T E R F A C E S | 357

358 | C H A P T E

<identifier>.<variable_name>. In order to distinguish between variables belonging to different physics interfaces, the identifier string must be unique. Only letters, numbers and underscores (_) are permitted in the Identifier field. The first character must be a letter.

The default identifier (for the first physics interface in the component) is bf.


The default setting is to have All domains in the component define the bubbly flow liquid and gas velocities, pressure, and the averaged volume fraction along with the equations that describe those fields. To choose specific domains, select Manual from the Selection list.


Specify if the gas concentration is low and whether or not to solve for the interfacial area.

Low Gas ConcentrationThe Low gas concentration check box is selected by default. This approximation is valid if the gas volume fraction is low and its density does not have any significant effects on the continuity equation in The Bubbly Flow Equations (Equation 6-10 becomes Equation 6-11).


Solve For Interfacial AreaTo add a transport equation for the bubble density in order to determine the interfacial area, select the Solve for interfacial area check box.

By default no turbulence model is used. However, the physics interface changes to The Turbulent Bubbly Flow Interface when the Turbulence

model type selected is RANS (k-).


Swirl Flow

R E F E R E N C E P R E S S U R E

Enter the Reference pressure pref (SI unit: Pa). The default is 1 atm (1 atmosphere or 101,325 Pa).


The dependent variables (field variables) are the Velocity field, liquid phase u (SI unit: m/s), the Pressure p (SI unit: Pa), the Effective gas density rhogeff (SI unit: kg/m3), and the Number density, gas phase nd (SI unit: 1/m3). The names can be changed but the names of fields and dependent variables must be unique within a component.


To display this section, click the Show button ( ) and select Stabilization. This section contains the settings for stabilization of the momentum transport (the fluid flow) in the Momentum transport area and stabilization of the equation for the dispersed phase in the Gas phase transport area.

Isotropic diffusion (shock capturing, O(h2)) requires a scale for the effective gas density. An appropriate scale is the maximum value of the gas volume fraction.

For 2D axisymmetric models, select the Swirl flow check box to include the swirl velocity component—that is, the velocity component in the azimuthal direction. While can be nonzero, there can be no gradients in the direction.

General Single-Phase Flow Theory (2D Axisymmetric Formulations)

uu


360 | C H A P T E


To display this section, click the Show button ( ) and select Discretization. Select a Discretization of fluids—P1+P1 (the default), P2+P1 or P3+P2. Specify the Value type when

using splitting of complex variables—Real (the default) or Complex.

The Turbulent Bubbly Flow Interface

The Turbulent Bubbly Flow (bf) interface ( ), found under the Multiphase Flow>Bubbly

Flow branch ( ) when adding a physics interface, is used to model the flow of liquids with dispersed bubbles at high Reynolds numbers.

It is assumed that the bubbles only occupy a small volume fraction and that they always travel with their terminal velocity. It is thereby possible to solve only one set of Navier-Stokes equations for the liquid phase and to let the velocity of the bubbles be guided by a slip model. The pressure distribution is calculated from a mixture-averaged continuity equation. The volume fraction of bubbles is tracked by solving a transport equation for the effective gas density. Turbulence effects are modeled using the standard two-equation k- model with realizability constraints and bubble-induced turbulence production. Flow close to walls is modeled using wall functions.

The physics interface can also model the distribution of the number density (that is, the number of bubbles per unit volume), which in turn can be used to calculate the interfacial area, useful when simulating chemical reactions in the mixture.

The main physics node is the Fluid Properties feature, which adds the equations for turbulent bubbly flow and provides an interface for defining the fluid materials for the liquid and the gas and the slip velocity model to use.

When this physics interface is added, the following default physics nodes are also added in the Model Builder—Turbulent Bubbly Flow, Fluid Properties, Wall (the default boundary types are Wall functions for the liquid and No gas flux for the gas), and Initial Values.




• The Turbulent Bubbly Flow Interface


Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and volume forces. You can also right-click Turbulent Bubbly Flow to select physics from the context menu.


Specify if the gas concentration is low and whether or not to solve for the interfacial area.

Low Gas ConcentrationSelect the Low gas concentration check box if the gas volume fraction is low ( less than a few percent). It is then generally valid to replace the continuity equation in The Bubbly Flow Equations see Equation 6-11 with Equation 6-13). This option is selected by default.



The dependent variables (field variables) are the Velocity field, liquid phase u (SI unit: m/s), the Pressure p (SI unit: Pa), the Effective gas density rhogeff (SI unit: kg/m3), the Turbulent dissipation rate ep (SI unit: m2/s3), the Turbulent kinetic energy k (SI unit: m2/s2), and the Number density, gas phase nd (SI unit: 1/m3).

The names of variables can be changed but the names of fields and dependent variables must be unique within a component.


To display this section, click the Show button ( ) and select Stabilization. This section contains the settings for stabilization of the momentum transport (the fluid flow) in the Momentum transport area, stabilization of the equation for the dispersed phase in the Gas

phase transport area, and stabilization for the turbulence variables in the Turbulence

equations area.

Except where indicated below, the nodes settings for this physics interface are the same as for The Laminar Bubbly Flow Interface.

By default a k- turbulence model is used. However, this physics interface changes to The Laminar Bubbly Flow Interface when the Turbulence

model type selected is None.

g


362 | C H A P T E

Isotropic diffusion (shock capturing, O(h2)) requires a scale for the effective gas density. An appropriate scale is the maximum value of the gas volume fraction.








When using a turbulence model, streamline and crosswind diffusion are by default applied to the turbulence equations.

P1+P1 is not permitted unless streamline diffusion is active for the momentum transport.




• The Laminar Bubbly Flow Interface


Domain and Boundary Nodes for the Laminar and Turbulent Bubbly Flow Interfaces

For both The Laminar Bubbly Flow Interface and The Turbulent Bubbly Flow Interface these domain and boundary nodes are available from the Physics ribbon toolbar (Windows users), Physics context menu (Mac or Linux users), or right-click to access the context menu (all users).

D O M A I N


• Gravity

• Initial Values

• Mass Transfer

• Volume Force

B O U N D A R Y

The boundary types for the liquid flow variables, described in this section, are:

• Inlet

Flow in an Airlift Loop Reactor: model library path CFD_Module/Multiphase_Benchmarks/airlift_loop_reactor


The Volume Force node is described for The Laminar Flow Interface.

The sections describe the available boundary conditions for the liquid and the gas. In all equations, n denotes the outward pointing unit vector normal to the boundary.


364 | C H A P T E

• Outlet

• Symmetry

• Wall (the default boundary condition)

In addition to the boundary conditions for the liquid, the following boundary conditions for the gas are available for all boundary condition types except symmetry:

• Gas Concentration (the default condition for inlets)

• Gas Flux

• Gas Outlet (the default condition for outlets)

• No Gas Flux (the default condition for walls)

• Symmetry

Fluid Properties

The Fluid Properties node contains the material properties for the liquid and the gas. It also contains settings for the slip model.



Gas Boundary Condition Equations

For 2D axisymmetric models, COMSOL Multiphysics takes the axial symmetry boundaries (at r = 0) into account and automatically adds an Axial Symmetry node that is valid on the axial symmetry boundaries only.



Specify the temperature to be used in the physics interface:

• Select User defined to enter a value or expression for the temperature (SI unit: K) in the corresponding field. This input is always available.

• Alternatively, select a temperature defined by a Heat Transfer interface present in the component (if any). For example, select Temperature (ht/fluid1) to use the temperature defined by the node Fluid1 in the Heat Transfer interface ht.

M A T E R I A L S

Select the materials to use for the material properties of the liquid and the gas (when they are set to take their value from the material). The default is to use the Domain material for both the Liquid and the Gas. Select another material to use that material’s properties for the liquid or gas as required.

L I Q U I D P R O P E R T I E S

The default Density, liquid phase l (SI unit: kg/m3) uses values From material. If User

defined is selected, enter another value or expression.

The default Dynamic viscosity, liquid phase l (SI unit: Pa·s) uses values From material; the value is then defined for the material selected in the Materials section for the continuous phase. If User defined is selected, enter another value or expression.

The dynamic viscosity describes the relationship between the shear stresses and the shear rate in a fluid. Intuitively, water and air have a low viscosity, and substances often described as thick, such as oil, have a higher viscosity.

G A S P R O P E R T I E S

The default Density, gas phase g (SI unit: kg/m3) uses values From material. If User defined is selected, enter another value or expression.

Enter the Bubble diameter db (SI unit: m). The default value is 103 m (1 mm).

S L I P M O D E L

Select a Slip model—Homogeneous flow (the default), Pressure-drag balance, or User defined.



366 | C H A P T E

Homogeneous flow assumes that the velocity of the two phases are equal; that is, uslip0. Select User defined to enter different values or expressions for the components of the Slip

velocity uslip (SI unit: m/s).

If Pressure-drag balance is selected, it uses a model based on the assumption that the pressure forces on a bubble are balanced by the drag force:

Here db (SI unit: m) is the bubble diameter, and Cd (dimensionless) is the drag coefficient.

Select a Drag coefficient model:

• Select Small spherical bubbles (Hadamard-Rybczynski) for bubbles with a diameter smaller than 2 mm.

• Select Large bubbles for gas bubbles with a diameter larger than 2 mm. Then enter the Surface tension coefficient (SI unit N/m). The default is 0.07 N/m.

• Select Air bubbles in tap water (Schwarz-Turner) for air bubbles of 1–10 mm mean diameter in water.

• Select User defined to enter a different value or expression for the Drag coefficient Cd (dimensionless). The default value is 1.

See the Slip Model theory section.

34---

Cddb-------l uslip uslip p–=



Gravity

The Gravity node adds the force g to the right-hand side of the momentum transport equation.


From the Selection list, choose the domains on which to add gravity.

G R A V I T Y

Enter the components of the Gravity vector, g (SI unit: m/s2). The default value is 0-g_const where g_const is a physical constant equal to 9.8066 m/s2.

This section is available for The Turbulent Bubbly Flow Interface.

By default, the Mixing length limit lmix, lim (SI unit: m) is Automatic. Select Manual to enter a different value or expression.

This physics node is also described here for the Euler-Euler Model, Laminar

Flow interface, for which it adds gravity contributions to both momentum equations. Gravity affects both phases. Gravity physics nodes are mutually exclusive, that is, there can only be one active gravity physics node per domain.

For 2D models, gravity acts in the negative y direction by default. For example, in a 2D model, the y component is -g_const and the other component is 0. In this setting, g_const is a predefined physical constant for the standard gravity (acceleration due to gravity at sea level).

For 3D and 2D axisymmetric models, gravity acts in the negative z direction by default. For example, in a 3D model, the z component is -g_const and the other components are 0. In this setting, g_const is a predefined physical constant for the standard gravity (acceleration due to gravity at sea level).


368 | C H A P T E

Mass Transfer

Add a Mass Transfer node to include mass transfer from the gas to the liquid.


From the Selection list, choose the domains on which to add mass transfer.

M A S S TR A N S F E R

Select a Mass transfer model—Two-film theory model or User defined. If User defined is selected, enter a value or expression for the Mass transfer from gas to liquid mgl (SI unit: kg/(m3·s)). The default is 0 kg/(m3·s).

If Two-film theory is selected, enter values for the Mass transfer coefficient k (SI unit: m/s), Henry’s constant H (SI unit: Pa·m3/mol), the Dissolved gas concentration c (SI unit: mol/m3), and the Molecular weight of species M (SI unit: kg/mol). Refer to the theory below for more information.

Henry’s law gives the equilibrium concentration c* of gas dissolved in liquid:

where H is Henry’s constant. The molar flux per interfacial area, N (SI unit: mol/(s·m2)), is determined by

where k is the Mass transfer coefficient and c is the Dissolved gas concentration in liquid.

The mass transfer from gas to liquid, mgl, is given by

where M is the Molecular weight of species and a is the interfacial area per volume (SI unit: m2/m3).

For Two-film theory, you also need to solve for the concentration of the dissolved gas;

Also see Theory for the Bubbly Flow Interfaces for details about how a is computed.

cp pref+

H-------------------=

N k c c– =

mgl NMa=


which can be done by adding a Transport of Diluted Species interface.

Initial Values

The Initial Values node adds initial values for the flow variables and the effective gas density that can serve as initial conditions for a transient simulation or as an initial guess for a nonlinear solver.




Enter initial values or expressions for the Velocity field. liquid phase u (SI unit: m/s) and for the Pressure p (SI unit: Pa). The default values are 0. Also enter a value or expression for the Effective gas density rhogeff (SI unit: kg/m3). The default is 0 kg/m3.

If the Solve for interfacial area check box is selected in The Laminar Bubbly Flow Interface or The Turbulent Bubbly Flow Interface enter an initial value for the Number density, gas

phase nd (SI unit: 1/m3). The default is 0 1/m3.

For The Turbulent Bubbly Flow Interface also enter initial values for the Turbulent

dissipation rate ep (SI unit: m2/s3), and Turbulent kinetic energy k (SI unit: m2/s2).

Wall

The Wall node adds a selection of boundary conditions that describe the existence of a solid wall.The Wall node by default specifies no gas flux for the gas phase.


tc cul + Dc

mglM

---------+=


370 | C H A P T E



L I Q U I D B O U N D A R Y C O N D I T I O N

The Liquid boundary condition is specified as follows.

• For The Laminar Bubbly Flow Interface, select No slip (the default) or Slip.

• For The Turbulent Bubbly Flow Interface, select Wall functions (the default) or Slip.

No SlipSets the liquid velocity to zero at the wall:

This is the default boundary condition for the liquid.

SlipSets the velocity component normal to the wall to zero:

Wall FunctionsThis boundary condition models a no slip condition for solid walls in a turbulent flow. Wall functions are used to model the thin region with large gradients in the flow variables near the wall.



ul 0=

ul n 0=


• If Generic roughness is selected, enter a Roughness height ks (SI unit: m). The default is 3.2 micrometers. Then enter a Roughness parameters Cs (dimensionless). The default is 0.26.

G A S B O U N D A R Y C O N D I T I O N

From the list, select a Gas boundary condition for the gas phase on the wall—No gas flux (the default), Gas concentration, Gas outlet, Gas flux, or Symmetry.

Gas ConcentrationIf Gas concentration is selected, enter the Effective gas density (SI unit: kg/m3). The default is 0 kg/m3.If Solve for interfacial area is selected, also select either the Bubble

number density (the default) or the Bubble diameter and gas density button.

• If Bubble number density is selected, enter the Bubble number density n0 (SI unit: 1/m3). The default is 1000 1/m3.

• If Bubble diameter and gas density is selected, enter the Bubble diameter db (SI unit: m) (the default is 1 mm) and Density, gas phase g (SI unit: kg/m3) (the default is 1 kg/m3).

Gas FluxIf Gas flux is selected, enter the Gas mass flux (SI unit: kg/(m2·s)) (the default is 0 kg/(m2·s)) and if Solve for interfacial area is selected, the Number density flux Nn (SI unit: 1/(m2·s)). The default is 0.1/(m2·s)).

Inlet

The Inlet node adds a selection of boundary conditions that describe inlets in fluid-flow simulations.


The formulations in this boundary type also appear, some of them slightly modified, in the Outlet type. This means that there is nothing in the mathematical formulations that prevents a fluid from leaving the domain through boundaries where the Inlet type is specified.

gg0

Ngg


372 | C H A P T E




Select a Liquid boundary condition—Velocity (the default), Pressure, no viscous stress. For The Laminar Bubbly Flow Interface, Laminar inflow is also an option.

VelocityIf Velocity is selected, select an option from the Velocity field componentwise list—Normal

inflow velocity (the default) or Velocity field.

• If Normal inflow velocity is selected, enter a value or expression for the Normal inflow

velocity U0 (SI unit: m/s). The default is 0 m/s.

• If Velocity field is selected, specify that the velocity at the boundary is equal to a given Velocity field u0 (SI unit: m/s) and enter the components in the matrix:

Pressure, No Viscous StressIf Pressure, no viscous stress is selected, enter a value or expression for the Pressure p0 (SI unit: Pa). The default is 0 Pa. To make the viscous stress zero at the boundary:

,

L A M I N A R I N F L O W

This section is available for The Laminar Bubbly Flow Interface and when Laminar inflow is selected as the Liquid boundary condition.

Select a Laminar inflow option—Average velocity (the default), Flow rate, or Entrance

pressure.

For any selection, also choose the Constrain outer edges to zero check box to force the laminar profile to go to zero at the bounding points or edges of the inlet channel. Otherwise the velocity is defined by the boundary condition of the adjacent boundary in the model. For example, if one end of a boundary with a laminar inflow condition connects to a slip boundary condition, the laminar profile has a maximum at that end.


ul u0=

p p0= l l T+ ul ulT 2

3--- ul I–+

n 0=


Then:

• If Average velocity is selected, enter an Average velocity Uav (SI unit: m/s). The default is 0 m/s. Enter an Entrance length Lentr (SI unit: m). The default is 1 m.

• If Flow rate is selected, enter an Entrance length Lentr (SI unit: m). The default is 1 m. Enter a Flow rate V0 (SI unit: m3/s). The default is 0 m3/s.

• If Entrance pressure is selected, enter an Entrance length Lentr (SI unit: m). The default is 1 m. Enter an Entrance pressure pentr (SI unit: Pa). The default is 0 Pa.


TU R B U L E N C E P R O P E R T I E S

The turbulent length scale LT and turbulence intensity IT are related to the turbulence variables via

(6-4)

Outlet

The Outlet node adds a set of boundary conditions that describe outlets in fluid-flow simulations; that is, the conditions at boundaries where the fluid exits the domain.

These settings are the same as for Wall. See Gas Boundary Condition. The only difference is that Gas concentration is the default.

This section is available for The Turbulent Bubbly Flow Interface so that the inlet conditions for the turbulence variables (k and ) can be specified. Except for Equation 6-4, of the settings information is the same as that described for single-phase flow. See More Boundary Condition Settings for the Turbulent Flow Interfaces.

k 32--- U IT 2,= C

3 4 k3 2/

LT-----------=


374 | C H A P T E

The Pressure, no viscous stress boundary condition is the default liquid boundary condition for outlet boundaries.




Select a Liquid boundary condition—Pressure, no viscous stress (the default) or Velocity. For The Laminar Bubbly Flow Interface, Laminar outflow is also an option.

Except for the default, the settings are the same as for the Inlet node (see Liquid Boundary Condition).


L A M I N A R O U T F L O W

This section is available for The Laminar Bubbly Flow Interface when Laminar outflow is selected as the Liquid boundary condition.

Select a Laminar outflow option—Average velocity (the default), Flow rate, or Exit pressure. For any selection, also choose the Constrain outer edges to zero check box to force the laminar profile to go to zero at the bounding points or edges of the inlet channel. Otherwise the velocity is defined by the boundary condition of the adjacent boundary in the model.

For example, if one end of a boundary with a laminar inflow condition connects to a slip boundary condition, then the laminar profile has a maximum at that end.

• If Average velocity is selected, enter an Average velocity Uav (SI unit: m/s) the default is 0. Enter an Exit length Lexit (SI unit: m). The default is 1 m.

The formulations for the Outlet type are also available, possibly slightly modified, in other boundary types. There is nothing in the mathematical formulation that prevents a fluid from entering the domain through boundaries where the Laminar outflow boundary type is specified.

These settings are the same as for Wall. See Gas Boundary Condition.


• If Flow rate is selected, enter an Exit length Lexit (SI unit: m). The default is 1 m. Enter a Flow rate V0 (SI unit: m3/s).

• If Entrance pressure is selected, enter an Exit length Lexit (SI unit: m). The default is 1 m. Enter an Exit pressure pexit (SI unit: Pa).

Symmetry

The Symmetry node adds boundary conditions that describe symmetry boundaries in fluid-flow simulations. The boundary condition for symmetry boundaries prescribes no penetration and vanishing shear stresses:




In addition to the boundary conditions for the liquid, specify boundary conditions for the gas on Wall, Inlet, and Outlet nodes. Select a Gas Boundary Condition:

Gas ConcentrationUsing this boundary condition, specify the effective gas density.

Gas OutletThis boundary condition is appropriate for boundaries where the gas phase flows outward with the gas velocity, ug, at the boundary.

Gas FluxUsing this boundary condition, specify the gas mass flux through the boundary:

A boundary condition for 2D axial symmetry is not required.

For the symmetry axis at r0, the software automatically provides a condition that prescribes ur0 and vanishing stresses in the z direction and adds an Axial Symmetry node that is valid on the axial symmetry boundaries only.

ul n 0,= tT l l T+ ul ulT 2

3--- ul I–+

n 0=

g g0

=


376 | C H A P T E

SymmetryThis boundary condition, which is useful on boundaries that represent a symmetry line for the gas flow, sets the gas flux through the boundary to zero:

No Gas FluxThis boundary condition represents boundaries where the gas flux through the boundary is zero:

n gug – Ngg=

n gug 0=

n dud 0=


Th e M i x t u r e Mode l I n t e r f a c e s

In this section:

• The Mixture Model, Laminar Flow Interface

• The Mixture Model, Turbulent Flow Interface

• Domain and Boundary Nodes for the Mixture Model Laminar and Turbulent Flow Interfaces

The Mixture Model, Laminar Flow Interface

The Mixture Model, Laminar Flow (mm) interface ( ), found under the Multiphase

Flow>Mixture Model branch ( ) when adding a physics interface, is used to model the flow at low and moderate Reynolds numbers of liquids containing a dispersed phase. The dispersed phase can be bubbles, liquid droplets, or solid particles, which are assumed to always travel with their terminal velocity.

The Mixture Model, Laminar Flow interface solves one set of Navier-Stokes equations for the momentum of the mixture. The pressure distribution is calculated from a mixture-averaged continuity equation and the velocity of the dispersed phase is described by a slip model. The volume fraction of the dispersed phase is tracked by solving a transport equation for the volume fraction.

The physics interface can also model the distribution of the number density, which in turn can be used to calculate the interfacial area, which is useful when simulating chemical reactions in the mixture.

The main physics node is the Mixture Properties feature, which adds the equations for the mixture and provides an interface for defining the fluid materials for the continuous and dispersed phases as well as which slip model and mixture viscosity model to use.

When this physics interface is added, the following default physics nodes are also added in the Model Builder—Mixture Model, Laminar Flow, Mixture Properties, Wall (the default boundary conditions for laminar flow are No slip and No dispersed phase flux) and Initial

Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and volume forces. You can also right-click Mixture Model, Laminar

Flow to select physics from the context menu.

T H E M I X T U R E M O D E L I N T E R F A C E S | 377

378 | C H A P T E



The default identifier (for the first physics interface in the component) is mm.


The default setting is to have All domains in the component define the mixture’s velocity, pressure and dispersed phase volume fraction, along with the equations that describe these fields. To choose specific domains, select Manual from the Selection list.


Specify the characteristics of the dispersed phase, the model for the slip velocity, and whether or not to solve for the interfacial area.

Dispersed PhaseTo characterize the Dispersed phase, select Solid particles (the default) or Liquid

droplets/bubbles.

Slip ModelTo compute the slip velocity uslip (SI unit: m/s), select a Slip model—Homogeneous flow (the default), Hadamard-Rybczynski, Schiller-Naumann, Haider-Levenspiel, or User defined.

• The Homogeneous flow model assumes that the velocities of the two phases are equal, that is, uslip = 0.

• In most cases there is a significant difference in the velocity fields due to the buoyancy of the dispersed phase. Use one of the predefined slip models for such cases.

• Select User defined to specify an arbitrary expression for the relative velocity. For example, give a constant velocity based on experimental data.

The selection from the Dispersed Phase list changes the physics available for the Mixture Properties node, under the section Mixture Model.



Mass Transfer and Interfacial AreaBy default, the Solve for interfacial area check box is not selected. For the Mass Transfer rate, use a two-film theory model, which includes the interfacial area per unit volume between the two phases. It is possible to compute the interfacial area per unit volume if the number density n (that is, the number of dispersed particles per volume) is known. Select the Solve for interfacial area check box to add the following equation for the number density n:

This equation states that a dispersed phase particle cannot disappear, appear, or merge with other particles, although it can expand or shrink.

The Mixture Model, Laminar Flow Interface calculates the interfacial area a (SI unit: m2/m3) from

The default selection is None. However, this physics interface changes to The Mixture Model, Turbulent Flow Interface when the Turbulence model

type selected is RANS (k-).

tn nud + 0=

a 4n 1 3 3d 2 3=


380 | C H A P T E

Swirl Flow


Enter values for the dependent variables (field variables):

• the Velocity field, Mixture u (SI unit: m/s)


• the Volume fraction, dispersed phase phid (dimensionless)

• the Squared slip velocity slipvel

• the Number density, dispersed phase nd (SI unit: 1/m3).


C O N S I S T E N T S T A B I L I Z A T I O N A N D I N C O N S I S T E N T S T A B I L I Z A T I O N

To display this section, click the Show button ( ) and select Stabilization. By default, consistent streamline and crosswind diffusion are applied to the Navier-Stokes and gas

transport equations. The Inconsistent Stabilizations section contains the settings for stabilization of the momentum transport (the fluid flow) under Momentum transport and for stabilization of the equation for the dispersed phase under Dispersed phase transport.

Isotropic shock capturing, O(h2) for the dispersed phase needs a scale for the volume fraction of the dispersed phase. A suitable scale is the maximum value of the dispersed phase volume fraction.

For 2D axisymmetric models, select the Swirl flow check box to include the swirl velocity component—that is, the velocity component in the azimuthal direction. While can be nonzero, there can be no gradients in the direction.

General Single-Phase Flow Theory (2D Axisymmetric Formulations)

uu





The Mixture Model, Turbulent Flow Interface

The Mixture Model, Turbulent Flow (mm) interface ( ), found under the Multiphase

Flow>Mixture Model branch ( ) when adding a physics interface, is used to model the flow at high Reynolds numbers of liquids containing a dispersed phase. The dispersed phase can be bubbles, liquid droplets, or solid particles, which are assumed to always travel with their terminal velocity.

The Mixture Model, Turbulent Flow interface solves one set of Navier-Stokes equations for the momentum of the mixture. The pressure distribution is calculated from a mixture averaged continuity equation and the velocity of the dispersed phase is described by a slip model. The volume fraction of the dispersed phase is tracked by solving a transport equation for the volume fraction. Turbulence effects are modeled using the standard two-equation k- model with realizability constraints. Flow close to walls is modeled using wall functions.

P1+P1 can only be used if consistent streamline diffusion is active for the momentum equation.



• The Mixture Model, Turbulent Flow Interface

• Slip Velocity Models and Theory for the Mixture Model Interfaces

Two-Phase Flow Modeling of a Dense Suspension: model library path CFD_Module/Multiphase_Benchmarks/dense_suspension


382 | C H A P T E

The physics interface can also model the distribution of the number density, which in turn can be used to calculate the interfacial area, which is useful when simulating chemical reactions in the mixture.





Enter values for the dependent variables (field variables):

• the Velocity field, Mixture u (SI unit: m/s)


• the Volume fraction, dispersed phase phid (dimensionless)

• the Squared slip velocity slipvel

• the Number density, dispersed phase nd (SI unit: 1/m3)


• the Turbulent dissipation rate (SI unit: m2/s3)


C O N S I S T E N T S T A B I L I Z A T I O N A N D I N C O N S I S T E N T S T A B I L I Z A T I O N

To display these sections, click the Show button ( ) and select Stabilization. See The Mixture Model, Laminar Flow Interface.

Except where indicated below, the settings for this physics interface are the same as for The Mixture Model, Laminar Flow Interface.

The default Turbulence model type is RANS, k-. However, this physics interface changes to The Mixture Model, Laminar Flow Interface when the Turbulence model type selected is None.



To display this section, click the Show button ( ) and select Advanced Physics Options. Normally these settings do not need to be changed. In addition, the consistent stabilizations are also applied to the Turbulence.

Penalty Diffusion can be used to suppress negative values of the dispersed volume fraction. Including this term has been observed to slow down convergence and it is therefore disabled by default.



Domain and Boundary Nodes for the Mixture Model Laminar and Turbulent Flow Interfaces

For both The Mixture Model, Laminar Flow Interface and The Mixture Model, Turbulent Flow Interface the following domain and boundary nodes are available from the Physics ribbon toolbar (Windows users), Physics context menu (Mac or Linux users), or right-click to access the context menu (all users).



• Slip Velocity Models

• Theory for the Mixture Model Interfaces



384 | C H A P T E

D O M A I N

• Gravity

• Initial Values

• Mass Transfer

• Mixture Properties

• Volume Force

B O U N D A R Y

Boundary conditions for the mixture velocity, pressure and dispersed phase volume fraction need to be specified. The following boundary condition types are available for The Mixture Model, Laminar Flow Interface and The Mixture Model, Turbulent Flow Interface:

• Inlet

• Outlet

• Symmetry

• Wall (the default boundary condition feature).

For the flow variables, the velocity and the pressure, the boundary conditions correspond to those defined for single phase flow (see Domain, Boundary, Pair, and Point Nodes for Single-Phase Flow). There is also a point constraint for the pressure, which is useful if no other boundary condition in the model includes a pressure level constraint.

The following boundary conditions for the dispersed phase are available for all boundary condition types except symmetry. These are described in more detail in the Theory for the Mixture Model Interfaces.

• Dispersed phase concentration (the default condition for inlets)

• Dispersed phase flux

• Dispersed phase outlet (the default condition for outlets)

The Volume Force node is described for The Laminar Flow Interface.


• No dispersed phase flux (the default condition for walls)

• Symmetry

Mixture Properties

The Mixture Properties node contains the material properties for the continuous phase and the dispersed phase. It also contains settings for the viscosity model.



M A T E R I A L S

Select the fluid materials to use for the material properties. The default material used for both Continuous phase and Dispersed phase is the Domain material. Select another material (when available).

C O N T I N U O U S P H A S E P R O P E R T I E S

The default Density, continuous phase c (SI unit: kg/m3) uses values From material (as selected in the Materials section). If User defined is selected, enter another value or expression. In this case the default is 0 kg/m3.

The default Dynamic viscosity, continuous phase c (SI unit: Pa·s), uses values From material. It describes the relationship between the shear stresses and the shear rate in a fluid. Intuitively, water and air have a low viscosity, and substances often described as thick, such as oil, have a higher viscosity. If User defined is selected, enter another value or expression. In this case, the default is 0 Pa·s.

For 2D axisymmetric models, COMSOL Multiphysics takes the axial symmetry boundaries (at r = 0) into account and automatically adds an Axial Symmetry node to the model that is valid on the axial symmetry boundaries only.


386 | C H A P T E

D I S P E R S E D P H A S E P R O P E R T I E S

The default Density, dispersed phase d (SI unit: kg/m3) uses values From material (as selected in the Materials section). If User defined is selected, enter another value or expression. In this case, the default is 0 kg/m3.

Enter the Diameter of particles/droplets dd (SI unit: m). The default is 103 m (1 mm). If Haider-Levenspiel is selected for the Slip model under Physical Model, enter a value between 0 and 1 for the Sphericity (dimensionless). The default is 1.

M I X T U R E M O D E L

Select the Mixture viscosity model. The options available are based on the selection made for either The Mixture Model, Laminar Flow Interface or The Mixture Model, Turbulent Flow Interface from the Dispersed phase list.

• When Solid particles is selected, choose either Krieger type (the default) or User defined.

• When Liquid droplets/bubbles is selected, choose Krieger type (the default), User defined, or Volume averaged.

User DefinedIf User defined is selected, enter a value or expression for the Dynamic viscosity (SI unit: Pas). The default is 0 Pas. When using this option, make sure to limit the viscosity to bounded positive values.

Krieger TypeWhen Krieger type is selected, enter a value or expression for the Maximum packing

concentration max (dimensionless). The default is 0.62.

Select Krieger type to model the most generally valid expression for the mixture viscosity:

where max is the maximum packing concentration, which for solid particles is approximately 0.62. The dimensionless parameter *1 for solid particles and

If Liquid droplets/bubbles is selected from the Dispersed phase list for either The Mixture Model, Laminar Flow Interface or The Mixture Model, Turbulent Flow Interface, then Dynamic viscosity, dispersed phase d (SI unit: Pa·s) is also available. The default uses values From material (as selected in the Materials section) or select User defined to enter another value or expression. In this case, the default is 0 Pa·s.

c 1dmax------------–

2,5max–

=


for droplets and bubbles. When applying the Krieger type viscosity model, d is replaced by min(d, 0.999max) for better robustness.

Volume AveragedSelect Volume averaged to model the mixture viscosity of liquid-liquid mixtures, which uses the following equation for the viscosity:

The Mixture Model interfaces always employ the mixture viscosity in the particle Reynolds number expression used to calculate the slip velocity, thereby accounting for the increase in viscous drag due to particle-particle interactions.


This section is available for The Mixture Model, Turbulent Flow Interface.

Select how the Mixing length limit lmix,lim (SI unit: m) is defined—Automatic (default) or Manual:

• If Automatic is selected, the mixing length limit is automatically evaluated as half the shortest side of the geometry bounding box. If the geometry is, for example, a complicated system of slim entities, this measure can give a too high value. In such cases, it is recommended that it is defined manually.

• Select Manual to define a different value or expression. The default is 1 (that is, one unit length of the model unit system).

Mass Transfer

Use the Mass Transfer node to include mass transfer from the dispersed phase to the continuous phase.



M A S S TR A N S F E R

Select a Mass transfer model— Two-film theory or User defined (the default). If User defined

is selected, enter a value or expression for the Mass transfer from dispersed to continuous

phase mdc (SI unit: kg/(m3·s)). The default is 0 kg/(m3·s).

d 0,4c+

d c+---------------------------=

dd cc+=


388 | C H A P T E

Two-film TheoryIf Two-film theory is selected enter values or expressions for each of the following:

• Mass transfer coefficient k (SI unit: m/s). The default is 0 m/s.

• Species concentration in dispersed phase cd (SI unit: mol/m3). The default is 0 mol/m3.

• Species concentration in continuous phase cc (SI unit: mol/m3). The default is 0 mol/m3.

• Molecular weight of species M (SI unit: kg/mol). The default is 0 kg/mol.

The mass transfer is modeled as

where k denotes the mass transfer rate, and cd and cc are the species concentrations in the dispersed and the continuous phase, respectively. M is the species’ molecular weight, and a is the interfacial area per unit volume between the two phases.

Interfacial AreaWhen the Solve for interfacial area check box is selected for either The Mixture Model, Laminar Flow Interface or The Turbulent Bubbly Flow Interface (under the Physical Model section), it is also possible to compute the interfacial area per unit volume. Let n denote the number density, that is, the number of dispersed particles per unit volume. This feature adds the following equation for the number density n:

This equation states that a dispersed phase particle cannot disappear, appear, or merge with other particles, although it can expand or shrink.

The Mixture Model interfaces calculate the interfacial area a (SI unit: m2/m3) from the following equation:

Gravity

The Gravity node adds the force g to the right-hand side of the momentum transport equation.

mdc k cd cc– Ma=

tn nud + 0=

a 4n 1 3 3d 2 3=




G R A V I T Y

Enter the components of the Gravity vector, g (SI unit: m/s2). The default value is (0,-g_const) where g_const is a physical constant equal to 9.8066 m/s2.

Initial Values

The Initial Values node adds initial values for the mixture velocity, pressure, and volume fraction of the dispersed phase, that can serve as an initial condition for a transient simulation or as an initial guess for a nonlinear solver.




Enter values or expressions for the following dependent variables:

• Velocity field, mixture, the components of u (SI unit: m/s). The defaults are 0 m/s.

• Pressure p (SI unit: Pa). The default is 0 Pa.

• Volume fraction, dispersed phase (dimensionless). The default is 0.

For 2D models, gravity acts in the negative y direction by default. For example, in a 2D model, the y component is -g_const and the other component is 0. In this setting, g_const is a predefined physical constant for the standard gravity (acceleration due to gravity at sea level).

For 3D and 2D axisymmetric models, gravity acts in the negative z direction by default. For example, in a 3D model, the z component is -g_const and the other components are 0. In this setting, g_const is a predefined physical constant for the standard gravity (acceleration due to gravity at sea level).


390 | C H A P T E

• If the Solve for interfacial area check box is selected for the physics interface, enter an initial value for the Number density, dispersed phase nd (SI unit: 1/m3). The default is 0 1/m3.

• If a Schiller-Naumann or a Haider-Levenspiel slip model is used in the physics interface, enter an initial value for the Squared slip velocity slipvel (SI unit: m2/s2). The default is 0.081 m2/s2.

For The Mixture Model, Turbulent Flow Interface also enter values or expressions for:

• the Turbulent dissipation rate ep (SI unit: m2/s3). The default is mm.epinit.

• the Turbulent kinetic energy k (SI unit: m2/s2). The default is mm.kinit.

Wall

The Wall node has boundary conditions available that describe the existence of a solid wall.



M I X T U R E B O U N D A R Y C O N D I T I O N

Select a Mixture boundary condition—No slip, Slip, or Wall functions (for turbulent flow).

SlipFor either physics interface, select Slip to set the velocity component normal to the wall to zero .

No SlipNo slip is the default boundary condition for The Mixture Model, Laminar Flow Interface and prescribes u = 0, that is, the fluid at the wall is not moving.

The following sections describe the available boundary conditions for the mixture and the dispersed phase volume fraction. In all equations, n denotes the outward pointing unit vector normal to the boundary.

u n 0=


Wall FunctionsWall functions is the default boundary condition for The Mixture Model, Turbulent Flow Interface. It models a no slip condition for a solid wall in turbulent flow. Wall functions are used to model the thin region with high gradients in the flow variables near the wall.

This boundary condition models a no slip condition for solid walls in a turbulent flow. Wall functions are used to model the thin region with large gradients in the flow variables near the wall.



• If Generic roughness is selected, enter a Roughness height ks (SI unit: m). The default is 3.2 micrometers. Then enter a Roughness parameters Cs (dimensionless). The default is 0.26.

D I S P E R S E D P H A S E B O U N D A R Y C O N D I T I O N

Select an option from the Dispersed phase boundary condition list—No dispersed phase flux (the default for the Wall node), Dispersed phase concentration (the default for the Inlet node), Dispersed phase outlet (the default for the Outlet node), Dispersed phase flux, or Symmetry.

• If Dispersed phase concentration is selected, enter a Dispersed phase volume fraction d0 (dimensionless). The default is 0. When Solve for interfacial area is selected under the Physical Model section, is select either the Dispersed phase number density n0 (SI unit: 1/m3) or Diameter of particles/droplets dd (SI unit: m) button and enter a value or expression for n0 (the default is 5 x 106 1/m3) or dd (the default is 1 mm).

• If Dispersed phase flux is selected, enter values or expression for the Dispersed phase flux

d (SI unit: m/s) and, if Solve for interfacial area is selected under the Physical Model



392 | C H A P T E

section, the Number density flux n (SI unit: 1/(m2s)). The defaults are 0 m/s and 0 1/(m2s), respectively.

Inlet

The Inlet node adds a set of boundary conditions that describe inlets in fluid-flow simulations. Define the Velocity boundary condition (the default mixture boundary condition for inlet boundaries) or the Pressure, no viscous stress condition.




Select a Mixture boundary condition for the inlet—Velocity (the default) or Pressure, no

viscous stress.

• If Velocity is selected, enter the components of u0 (SI unit: m/s) to specify the velocity at the boundary u = u0. The defaults are 0 m/s.

• If Pressure, no viscous stress is selected, enter the pressure p0 (SI unit: Pa) at the boundary to set the pressure equal to p0 and also to make the viscous stress vanish at the boundary:

,

Mixture Boundary Condition (Turbulent Flow)For The Mixture Model, Turbulent Flow Interface the following options are also available.

Dispersed Phase Boundary Conditions Equations

The formulations in this boundary type all appear, some of them slightly modified, in the Outlet type as well. This means that there is nothing in the mathematical formulation that prevents the fluid from leaving the domain through boundaries where the Inlet type is specified.

p p0= T+ u u T+ n 0=


Select the Specify turbulent length scale and intensity or Specify turbulence variables button.

• If Specify turbulent length scale and intensity is selected, enter a value for the Turbulent

intensity IT (dimensionless) and Turbulence length scale LT (SI unit: m). When Pressure,

no viscous stress is selected, also enter a Reference velocity scale Uref (SI unit: m/s).

• If Specify turbulence variables is selected, enter values or expressions for the Turbulent

kinetic energy k0 (SI unit: m2/s2) and Turbulent dissipation rate 0 (SI unit: m2/s3).


Outlet

The Outlet node adds a selection of boundary conditions that describe outlets in fluid-flow simulations; that is, the conditions at boundaries where the fluid exits the domain. Select Velocity, Pressure, no viscous stress (the default), or No viscous stress for outlet boundaries.




Select a Mixture boundary condition for the outlet—Pressure, no viscous stress (the default), Velocity, or No viscous stress.

• If Pressure, no viscous stress is selected, enter the pressure p0 (SI unit: Pa) at the boundary to set the pressure equal to p0 and also to make the viscous stress vanish at the boundary:

The default is Dispersed phase concentration. See Wall for the settings and Dispersed Phase Boundary Conditions Equations for more information about the options.

All of the formulations for the Outlet type are also available, possibly slightly modified, in other boundary types. There is nothing in the mathematical formulation that prevents the fluid from entering the domain through boundaries where the Outflow boundary type is specified.

p p0 T+ u uT+ n 0= =


394 | C H A P T E

• If Velocity (SI unit: m/s) is selected, enter the components of u0 to specify the velocity at the boundary u = u0.

• If No viscous stress is selected, the viscous stress is set equal to zero at the boundary:


Symmetry

The Symmetry node adds boundary conditions that describe symmetry boundaries in fluid-flow simulations. The boundary condition for symmetry boundaries prescribes no penetration and vanishing tangential stress:


From the Selection list, choose the symmetry boundaries.

The default is Dispersed phase outlet. See Wall for the settings and Dispersed Phase Boundary Conditions Equations for more information about the options.

A boundary condition for 2D axial symmetry is not required.

For the symmetry axis at r0, the program automatically provides a condition that prescribes ur0 and vanishing stresses in the z direction and adds an Axial Symmetry node that is valid on the axial symmetry boundaries only.

T+ ul ulT

+ n 0=

u n 0,= tT cd 1 cd– uslipuslip Gm+ n 0=


Th e Eu l e r - E u l e r Mode l , L am i n a r F l ow I n t e r f a c e

The Euler-Euler Model, Laminar Flow (ee) interface ( ), found under the Multiphase

Flow>Euler-Euler Model branch ( ) when adding a physics interface, can be used to simulate the flow of two continuous and fully interpenetrating incompressible phases (see Ref. 1 under the Theory for the Euler-Euler Model, Laminar Flow Interface). The physics interface can model flow at low and moderate Reynolds numbers. Typical applications are fluidized beds (solid particles in gas), sedimentation (solid particles in liquid), or transport of liquid droplets or bubbles in a liquid.

The physics interface solves two sets of Navier-Stokes equations, one for each phase, in order to calculate the velocity field for each phase. The phases interchange momentum as described by a drag model. The pressure is calculated from a mixture-averaged continuity equation and the volume fraction of the dispersed phase is tracked with a transport equation.

When this physics interface is added, the following default physics nodes are also added in the Model Builder— Phase Properties, Wall, and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and volume forces. You can also right-click Euler-Euler Model, Laminar Flow to select physics from the context menu.



The default identifier (for the first physics interface in the component) is ee.


The default setting is to have All domains in the component define the velocities, pressure, and volume fractions along with the equations that describe those fields. To choose specific domains, select Manual from the Selection list.

T H E E U L E R - E U L E R M O D E L , L A M I N A R F L O W I N T E R F A C E | 395

396 | C H A P T E

D I S P E R S E D P H A S E

Select a Dispersed phase—Solid particles or Liquid droplets/bubbles.



• the Velocity field, continuous phase uc

• the Velocity field, dispersed phase ud

• the Pressure p

• the Volume fraction, dispersed phase phid




Consistent Streamline diffusion and Crosswind diffusion are activated by default for Navier-Stokes and gas transport equations.



Inconsistent stabilization can be activated independently for the momentum equation for the continuous phase, the momentum equation for the dispersed phase, and for the dispersed phase transport equation respectively by selecting the corresponding check box. Each inconsistent stabilization contribution has a tuning parameter.



When Solid particles is selected, the Solid Viscosity Model is available in the Phase Properties node.

Also see Dispersed Phase in the theory section.


The Euler-Euler Model, Laminar Flow interface supports three options for the basis functions: P1+P1 (the default option), P2+P1, and P3+P2. They all represent Lagrangian basis functions of different orders:

• P1+P1 – Linear basis functions for all degrees of freedom. Linear basis functions are computationally less expensive than the higher-order options and are also more robust. This option requires that Streamline diffusion is activated for both of the momentum equations.

• P2+P1 – Quadratic basis functions for all degrees of freedom except the pressure which is described by linear basis functions. Higher order elements, as compared to linear elements, are a computationally effective way to obtain high accuracy but only if the flow is well resolved. This requirement is most likely fulfilled for flows with very low velocities and/or small length scales.

• P3+P2 – Cubic basis functions for all degrees of freedom except the pressure which is described by quadratic basis functions. This option is computationally very expensive and the least robust one but it is also the option with the highest formal accuracy.



• Iterative and Isotropic Diffusion in the COMSOL Multiphysics Reference Manual

• Domain, Boundary, Point, and Pair Nodes for the Euler-Euler Model, Laminar Flow Interface

• Theory for the Euler-Euler Model, Laminar Flow Interface

Circulating Fluidized Bed: model library path CFD_Module/Multiphase_Tutorials/fluidized_bed


398 | C H A P T E

Domain, Boundary, Point, and Pair Nodes for the Euler-Euler Model, Laminar Flow Interface

The Euler-Euler Model, Laminar Flow Interface has the following domain, boundary, point, and pair nodes available from the Physics ribbon toolbar (Windows users), Physics context menu (Mac or Linux users), or right-click to access the context menu (all users).

The following nodes are described for the Laminar Flow interface:


• Symmetry

• Volume Force


• Gravity

• Initial Values

• Inlet

• Outlet

• Phase Properties

• Wall

The Gravity domain physics node is described for the Bubbly Flow interface and it adds gravity contributions to the momentum equations. Gravity affects both phases. Gravity physics nodes are mutually exclusive, that is, there can only be one active gravity node per domain.

The Volume Force node is described for the Laminar Flow interface. It adds a user-defined volume force to the momentum equation for each phase. If several volume force nodes are added to the same domain, the sums of all contributions are added to the momentum equations.





Phase Properties

The Phase Properties node has the material properties settings for the continuous and dispersed phases. It also has settings for the solid viscosity model and the drag model. When the dispersed phase consists of solid particles, it also has settings for the solid pressure model.



M A T E R I A L S

Select the materials to use for the material properties of the continuous phase and the dispersed phase respectively (when they have the selection to take their value From

material). By default, both the Continuous phase and Dispersed phase use the Domain

material, which uses the material defined for the domain.

If required, select another material from the Continuous phase or Dispersed phase lists to use that material’s properties for the liquid or gas, respectively.

C O N T I N U O U S P H A S E P R O P E R T I E S

Density, Continuous PhaseThe default Density, continuous phase c (SI unit: kg/m3) uses values From material. If User

defined is selected, enter another value or expression. The default is 0 kg/m3.

Dynamic Viscosity, Continuous PhaseThe dynamic viscosity describes the relationship between the shear stresses and the shear rate in a fluid. Intuitively, water and air have a low viscosity, and substances often described as thick, such as oil, have a higher viscosity.

The default Dynamic viscosity, continuous phase c (SI unit: Pa·s) uses values From material. The value of the viscosity is then the value defined for the material selected in the Materials section for the continuous phase. Select User defined to define a different value or expression. The default is 0 Pa·s.

D I S P E R S E D P H A S E P R O P E R T I E S

The default Density, dispersed phase d (SI unit: kg/m3) uses values From material. If User

defined is selected, enter another value or expression. The default is 0 kg/m3.


400 | C H A P T E

Enter a value or expression for the Diameter of particles/droplets dd (SI unit: m). The default is 103 m (1 mm).

S O L I D V I S C O S I T Y M O D E L

This section is available when Solid particles is selected as the Dispersed phase in The Euler-Euler Model, Laminar Flow Interface.

Select a Solid viscosity model—Calculate from mixture velocity (the default) or User defined.

• If Calculate from mixture viscosity is selected, also choose a Mixture viscosity model—Krieger type (the default) or User defined. For Krieger type, enter an expression or value for the Maximum packing concentration max (dimensionless). The default is 0.62. For User defined, enter a value or expression for the Dynamic viscosity (SI unit: Pa·s)

• If User defined is selected, enter a value or expression for the Dynamic viscosity, dispersed

phase d (SI unit: Pa·s). The default expression is max(phid*0.5, 5e-4)[Pa*s].

D R A G M O D E L

Select a Drag model—Gidaspow, Schiller-Naumann, Haider-Levenspiel, Hadamard-Rybczynzki, or User defined. If User defined is selected, enter a value or expression for the Drag force

coefficient SI unit: kg·s2/m5). If Haider-Levenspiel is selected, enter a value between 0 and 1 for the Sphericity Sp (dimensionless).

The Dynamic viscosity, dispersed phase d (SI unit: Pa·s) field is available when Liquid droplets/bubbles is selected as the Dispersed phase in The Euler-Euler Model, Laminar Flow Interface.

The default uses values From material. The value of the viscosity is then the value defined for the material selected in the Materials section for the dispersed phase. Select User defined to define a different value or expression.

The models for the interphase drag force Fdrag are available and described in Drag Model. When the dispersed phase corresponds to Solid

particles, the Gidaspow, Schiller-Naumann, Haider-Levenspiel, and Hadamard-Rybczynzki models are available. For Liquid droplets/bubbles, the Schiller-Naumann, Haider-Levenspiel and Hadamard-Rybczynzki models are available.


S O L I D P R E S S U R E M O D E L

This section is available when Solid particles is selected as the Dispersed phase in The Euler-Euler Model, Laminar Flow Interface. Select a Solid pressure model ps (SI unit: N/m3)—No solid pressure (the default), Gidaspow-Ettehadieh, Gidaspow, Ettehadieh, or User defined. If User defined is selected, enter values or expressions in the table for each component.

Initial Values

The Initial Values node adds initial values for the velocity fields for the continuous and dispersed phases, the pressure, and the volume fraction of the dispersed phase that can serve as initial conditions for a transient simulation or as an initial guess for a nonlinear solver.




Enter initial values or expressions in the tables for each component of:

• Velocity field, continuous phase uc (SI unit: m/s). The defaults are 0 m/s.

• Velocity field, dispersed phase ud (SI unit: m/s). The defaults are 0 m/s.

Enter initial values or expressions for:

• Pressure p (SI unit: Pa). The default is 0 Pa.

• Volume fraction, dispersed phase phid (dimensionless). The default is 0.

The solid pressure models the particle dispersion due to collisions and friction between the solid particles. Details of the implemented models are described in Solid Pressure.


402 | C H A P T E

Wall

The Wall node includes a set of boundary conditions to describe the flow conditions at a wall. The conditions for each phase are selected separately.




If this node is selected from the Pairs menu, choose the pair on which to apply the condition. An identity pair has to be created first. Ctrl-click to deselect.

C O N T I N U O U S P H A S E B O U N D A R Y C O N D I T I O N

Select a Continuous velocity boundary condition—No slip (uc0) (the default), or Slip (uc·n0 and no viscous stresses in the tangential directions).


Select a Dispersed velocity boundary condition—No slip (ud0) (the default), or Slip (ud·n0 and no viscous stresses in the tangential directions).

Select a Dispersed phase boundary condition for the dispersed phase volume fraction— No

dispersed phase flux (a zero flux condition) (the default) or Dispersed phase concentration. If Dispersed phase concentration is selected, enter a Dispersed phase volume fraction d0 (dimensionless). The default is 0.


To display this section, click the Show button ( ) and select Advanced Physics Options. To Apply reaction terms on all dependent variables, select All physics (symmetric). Otherwise, select Individual dependent variables to restrict the reaction terms. Select the Use weak


Inlet

The Inlet node includes a set of conditions to prescribe the flow at a boundary where one or both phases enter a domain.




TW O - P H A S E I N L E T TY P E

Select a Two-phase inlet type for the inlet boundary—Mixture (for both continuous and dispersed phases), Continuous phase, or Dispersed phase.

C O N T I N U O U S P H A S E

The options available in this section are based on the Two-phase inlet type selected.

Mixture or Continuous PhaseWhen Mixture or Continuous phase is selected as the Two-phase inlet type, enter values or expressions for the components of Velocity field, continuous phase uc0 (SI unit: m/s).

Dispersed PhaseWhen Dispersed phase is selected as the Two-phase inlet type, also select a Continuous phase

boundary condition—No slip (uc0) (the default) or Slip (uc·n0 and no viscous stresses in the tangential directions).

D I S P E R S E D P H A S E

The options available in this section are based on the Two-phase inlet type selected.

Mixture or Dispersed PhaseWhen Mixture or Dispersed phase is selected as the Two-phase inlet type, enter values or expressions for the components of Velocity field, dispersed phase ud0 (SI unit: m/s).

Also enter a value or expression for the Dispersed phase volume fraction d0 (dimensionless). The default is 0.

Continuous PhaseWhen Continuous phase is selected as the Two-phase inlet type, also select a Dispersed phase

boundary condition—No slip (ud0) (the default) or Slip (ud·n0 and no viscous stresses in the tangential directions). The condition for is a no-flux condition for this two-phase inlet type.




d


404 | C H A P T E

Outlet

The Outlet condition is always a condition for the mixture. You can select between Pressure, no viscous stress (the default), Pressure, normal flow, and Velocity.




Select a Mixture boundary condition—Pressure, no viscous stress (the default), Pressure, normal flow, or Velocity.

When Pressure, no viscous stress or Pressure, normal flow is selected as the Mixture boundary

condition, enter a value or expression for the Pressure p0 (SI unit: Pa). Any prescribed pressure condition must be consistent with the volume forces including gravity effects.

When Velocity is selected as the Mixture boundary condition, enter values or expressions in the tables for Velocity field, continuous phase uc0 (SI unit: m/s) and Velocity field, dispersed

phase ud (SI unit: m/s).




Pressure, no viscous stress is the least constraining option and allows the flow to exit the domain in non-normal directions.

If you want the flow to be completely normal to the outlet boundary, select Pressure, normal flow. This option can influence the flow quite a bit upstream of the inlet.


Th eo r y f o r t h e Two -Pha s e F l ow I n t e r f a c e s

The Level Set and Phase Field interfaces can be used to model the flow of two different, immiscible fluids, where the exact position of the interface between the two fluids is of interest. The fluid-fluid interface is tracked using an auxiliary function on a fixed mesh. These methods account for differences in the two fluids’ densities and viscosities and include the effects of surface tension and gravity.

The Laminar Flow, Two-Phase, Level Set and Phase Field Interfaces theory is described in this section:

• Level Set and Phase Field Equations

• Conservative and Non-Conservative Formulations

• Phase Initialization

• Numerical Stabilization

• References for the Level Set and Phase Field Interfaces

Level Set and Phase Field Equations

by default, the Level Set and Phase Field interfaces use the incompressible formulation of the Navier-Stokes equations:

(6-5)

(6-6)

U S I N G T H E L E V E L S E T M E T H O D

If the level set method is used to track the interface, it adds the following equation:

(6-7)

where is the reinitialization parameter (set to 1 by default) and is the interface thickness controlling parameter (set to hmax/2 where hmax is the maximum element size in the component). The density is a function of the level set function defined as

t

u u u+ p– I u uT+ + Fg Fst Fext F++ + +=

u 0=

t

------ u + 1 – ----------–

=

T H E O R Y F O R T H E TW O - P H A S E F L O W I N T E R F A C E S | 405

406 | C H A P T E

and the dynamic viscosity is given by

where 1 and 2 are the constant densities of Fluid 1 and Fluid 2, respectively, and 1 and 2 are the dynamic viscosities of Fluid 1 and Fluid 2, respectively. Here, Fluid 1 corresponds to the domain where , and Fluid 2 corresponds to the domain where

.

Further details of the theory for the level set method are in Ref. 1.

U S I N G T H E P H A S E F I E L D M E T H O D

If the phase field method is used to track the interface, it adds the following equations:

(6-8)

(6-9)

where the quantity (SI unit: N) is the mixing energy density and (SI unit: m) is a capillary width that scales with the thickness of the interface. These two parameters are related to the surface tension coefficient, (SI unit: N/m), through the equation

and is related to through 2 where is the mobility tuning parameter (set to 1 by default). The volume fraction of Fluid 2 is computed as

where the min and max operators are used so that the volume fraction has a lower limit of 0 and an upper limit of 1. The density is then defined as

and the dynamic viscosity according to

1 2 1– +=

1 2 1– +=

0.5 0.5

t u +

2------=

2 – 2 1– 2

-----

fext+ +=

2 23

--------------=

Vf min max 1 + 2 0 1 =

1 2 1– Vf+=

1 2 1– Vf+=


where 1 and 2 are the densities and 1 and 2 are the dynamic viscosities of Fluid 1 and Fluid 2, respectively.

The mean curvature (SI unit: 1/m) can be computed by entering the following expression:

where G is the chemical potential defined as:

Details of the theory for the phase field method are in Ref. 2.

F O R C E TE R M S

The four forces on the right-hand side of Equation 6-5 are due to gravity, surface tension, a force due to an external contribution to the free energy (using the phase field method only), and a user-defined volume force.

The Surface Tension Force for the Level Set MethodFor the level set method, the surface tension force acting on the interface between the two fluids is Fst = n where is the surface tensions coefficient (SI unit: N/m), is the curvature, and n is the unit normal to the interface (SI unit: 1/m) is a Dirac delta function located at the interface. depends on second derivatives of the level set function

. This can lead to poor accuracy of the surface tension force. Therefore, the following alternative formulation is used:

For a derivation of this formulation, see Appendix A in Ref. 3. In the weak formulation of the momentum equation, it is possible to move the divergence operator, using integration by parts, to the test functions for the velocity components.

The -function is approximated by a smooth function according to

The Surface Tension Force for the Phase Field MethodThe surface tension force for the phase field method is implemented as a body force

2 1 + 1 – G----=

G 2– 1–

2--------------------+

f

------+=

Fst I nnT – =

6 1 – =


408 | C H A P T E

where G is the chemical potential (SI unit: J/m3) defined in The Equations for the Phase Field Method and is a user-defined source of free energy.

The Gravity ForceThe gravity force is Fg = g where g is the gravity vector. Add this as a Gravity feature to the fluid domain.

The User Defined Volume ForceWhen using a phase field interface, a force arising due to a user-defined source of free energy is computed according to:

This force is added when a -derivative of the external free energy has been defined in the External Free Energy section of the Fluid Properties feature.


When the velocity field is divergence free, you can use either the conservative or the non-conservative formulation of the level set or phase field equation. The conservative form perfectly conserves the mass of each fluid, but the computational time is generally longer.

Phase Initialization

If the study type Transient with Initialization is used in the model, the level set or phase field variable is automatically initialized. For this study, two study steps are created, Phase Initialization and Time Dependent. The Phase Initialization step solves for the distance to the initial interface, Dwi. The Time Dependent step then uses the initial condition for the level set function according to the following expression:

in domains initially filled with Fluid 1 and

Fst G f

------– =

f

Fext f =

01

1 eDwi

+------------------------=


in domains initially filled with Fluid 2.

Correspondingly, for the phase field method the following expressions are used:

in Fluid 1 and

in Fluid 2. The initial condition for the help variable is 0 = 0. These expressions are based on the analytical solution of the steady state solution of Equation 6-7, Equation 6-8, and Equation 6-9 for a straight, non-moving interface.

Numerical Stabilization

Four types of stabilization methods are available for the flow (Navier-Stokes), turbulence, and interface (level set or phase field) equations. Two are consistent stabilization methods—Streamline diffusion and Crosswind diffusion—and two are inconsistent—Isotropic diffusion and Anisotropic diffusion.

For the initialization to work it is crucial that there are two Initial Values nodes and one Initial Interface node. One of the Initial Values nodes should use Fluid initially in domain: Fluid 1and the other Fluid initially in

domain: Fluid 2. The Initial Interface node should have all interior boundaries where the interface is initially present as selection. If the selection of the Initial Interface node is empty, the initialization fails.


• Studies and Solvers and Transient with Initialization in the COMSOL Multiphysics Reference Manual

01

1 eDwi–

+--------------------------=

0Dwi

2---------- tanh–=

0Dwi

2---------- tanh=


410 | C H A P T E


References for the Level Set and Phase Field Interfaces

1. E. Olsson and G. Kreiss, “A Conservative Level Set Method for Two Phase Flow,” J. Comput. Phys., vol. 210, pp. 225–246, 2005.

2. P. Yue, J.J. Feng, C. Liu, and J. Shen, “A Diffuse-interface Method for Simulating Two-phase Flows of Complex Fluids,” J. Fluid Mech., vol. 515, pp. 293–317, 2004.

3. B. Lafaurie, C. Nardone, R. Scardovelli, S. Zaleski, and G. Zanetti “Modelling Merging and Fragmentation in Multiphase Flows with SURFER.” J. Comput. Phys., vol. 113, no. 1, pp. 134–147, 1994.


• Stabilization in the COMSOL Multiphysics Reference Manual


Th eo r y f o r t h e Bubb l y F l ow I n t e r f a c e s

The Bubbly Flow interfaces are suitable for macroscopic modeling of mixtures of liquids and gas bubbles. These physics interfaces solve for the averaged volume fraction occupied by each of the two phases, rather than tracking each bubble in detail. Each phase has its own velocity field.

In this section:

• The Bubbly Flow Equations

• Turbulence Modeling in Bubbly Flow Applications

• References for the Bubbly Flow Interfaces

The Bubbly Flow Equations

The two-fluid Euler-Euler Model is a general, macroscopic model for two-phase fluid flow. It treats the two phases as interpenetrating media, tracking the averaged concentration of the phases. One velocity field is associated with each phase. A momentum balance equation and a continuity equation describe the dynamics of each of the phases. The bubbly flow model is a simplification of the two-fluid model, relying on the following assumptions:

• The gas density is negligible compared to the liquid density.

• The motion of the gas bubbles relative to the liquid is determined by a balance between viscous drag and pressure forces.

• The two phases share the same pressure field.

Based on these assumptions, the momentum and continuity equations for the two phases can be combined and a gas phase transport equation is kept in order to track the volume fraction of the bubbles. The momentum equation is

(6-10)ll t

ul llul ul+ =

p– l l T+ ul ulT 2

3--- ul I–+

llg F+ ++=

T H E O R Y F O R T H E B U B B L Y F L O W I N T E R F A C E S | 411

412 | C H A P T E

In Equation 6-10, the variables are as follows:

• ul is the velocity vector (SI unit: m/s)


• is thephase volume fraction (SI unit: m3/m3)


• g is the gravity vector (SI unit: m/s2)

• F is any additional volume force (SI unit: N/m3)

• l is the dynamic viscosity of the liquid (SI unit: Pa·s), and

• T is the turbulent viscosity (SI unit: Pa·s)

The subscripts “l” and “g” denote quantities related to the liquid phase and the gas phase, respectively.

The continuity equation is

(6-11)

and the gas phase transport equation is

(6-12)

where mgl is the mass transfer rate from the gas to the liquid (SI unit: kg/(m3·s)).

For low gas volume fractions ( ) you can replace the continuity equation, Equation 6-11, by

(6-13)

By default, the Laminar Bubbly Flow interface uses Equation 6-13. To switch to Equation 6-11, click to clear the Low gas concentration check box under the Physical Model section.

The interface solves for ul, p, and

the effective gas density. The gas velocity ug is the sum of the following velocities:

(6-14)

t ll gg+ llul ggug+ + 0=

ggt

--------------- ggug + mgl–=

g 0.01

l ul 0=

g gg=

ug ul uslip udrift+ +=


where uslip is the relative velocity between the phases and udrift is a drift velocity (see Turbulence Modeling in Bubbly Flow Applications). The interface calculates the gas density from the ideal gas law:

where M is the molecular weight of the gas (SI unit: kg/mol), R is the ideal gas constant (8.314472 J/(mol·K)), pref a reference pressure (SI unit: Pa), and T is temperature (SI unit: K). pref is a scalar variable, which by default is 1 atm (1 atmosphere or 101,325 Pa). The liquid volume fraction is calculated from

When there is a drift velocity, it has the form

(6-15)

Where is an effective viscosity causing the drift. Inserting Equation 6-15 and Equation 6-14 into Equation 6-12 gives

That is, the drift velocity introduces a diffusive term in the gas transport equation. This is how the equation for the transport of the volume fraction of gas is actually implemented in the physics interface.

The bubbly-flow equation formulation is relatively simple, but it can display some nonphysical behavior. One is artificial accumulation of bubbles, for example, beneath walls where the pressure gradient forces the bubbles upward, but the bubbles have no place to go and there is no term in the model to prevent the volume fraction of gas from growing. To prevent this from happening, is set to l in the laminar case. The only apparent effect of this in most cases where the bubbly-flow equations are applicable is that nonphysical accumulation of bubbles is reduced. The small effective viscosity in the transport equation for has beneficial effects on the numerical properties of the equation system.

M A S S TR A N S F E R A N D I N T E R F A C I A L A R E A

It is possible to account for mass transfer between the two phases by specifying an expression for the mass transfer rate from the gas to the liquid mgl (SI unit: kg/(m3·s)).

gp pref+ M

RT------------------------------=

l 1 – g=

udriftl----gg

----------–=

ggt

--------------- gg ul uslip+ + gl

---------g mgl–=

g


414 | C H A P T E

The mass transfer rate typically depends on the interfacial area between the two phases. An example is when gas dissolves into a liquid. In order to determine the interfacial area, it is necessary to solve for the bubble number density (that is, the number of bubbles per volume) in addition to the phase volume fraction. The Bubbly Flow interface assumes that the gas bubbles can expand or shrink but not completely vanish, merge, or split. The conservation of the number density n (SI unit: 1/m3) then gives

The number density and the volume fraction of gas give the interfacial area per unit volume (SI unit: m2/m3):

S L I P M O D E L

The simplest possible approximation for the slip velocity uslip is to assume that the bubbles always follow the liquid phase; that is, uslip0. This is known as homogenous flow.

A better model can be obtained from the momentum equation for the gas phase. Comparing size of different terms, it can be argued that the equation can be reduced to a balance between the viscous drag force, fD and the pressure gradient (Ref. 3), a so called pressure-drag balance:

(6-16)

Here fD can be written as

(6-17)

where in turn db (SI unit: m) is the bubble diameter, and Cd (dimensionless) is the viscous drag coefficient. Given Cd and db, Equation 6-16 can be used to calculate the slip velocity.

Schwarz and Turner (Ref. 4) proposed a linearized version of Equation 6-17 appropriate for air bubbles of 1–10 mm mean diameter in water:

(6-18)

The Hadamard-Rybczynski model is appropriate for small spherical bubbles with diameter less than 2 mm. The model uses the following expression for the drag coefficient (Ref. 5):

tn nug + 0=

a 4n 1 3 3g 2 3=

g p fD=

fD dCd34---ldb------ uslip uslip–=

fD dCWuslip–= CW 5 104 kgm3s----------=


For bubbles with diameter larger than 2 mm, the model suggested by Sokolichin, Eigenberger, and Lapin (Ref. 1) is a more appropriate choice:

where Eö is the Eötvös number

Here, g is the magnitude of the gravity vector and the surface tension coefficient.

Turbulence Modeling in Bubbly Flow Applications

For most bubbly flow applications the flow field is turbulent. In that case, use a turbulence model and solve for the averaged velocity field. For the Turbulent Bubbly Flow interface, the k- turbulence model is used. In addition to the options of the single-phase flow model, it is also possible to account for bubble-induced turbulence—that is, extra production of turbulence due to relative motion between the gas bubbles and the liquid.

The k- model solves two extra transport equations for two additional variables: the turbulent kinetic energy, k (SI unit: m2/s2) and the dissipation rate of turbulent energy, (SI unit: m/s3). The turbulent viscosity is then


The transport equation for the turbulent kinetic energy, k, is

(6-19)


Cd16

Reb----------,= Reb

dbl uslipl

---------------------------=

Cd0,622

1Eö------- 0,235+-----------------------------=

Eögldb

2

---------------=

T lCk2

------=

l tk lul k+

Tk------+

k Pk+ l Sk+–=


416 | C H A P T E

and the evolution of the turbulent energy’s dissipation rate is given by:

(6-20)

In all the previous equations, the velocity, ul, is the liquid phase averaged velocity field.

The standard k- turbulence model uses the following constants:

The included source term Sk accounts for bubble-induced turbulence and is given by

Suitable values for the model parameters Ck and C are not as well established as the parameters for single-phase flow. In the literature, values within the ranges 0.01 < Ck < 1 and 1C1.92 have been suggested (Ref. 1). The turbulent viscosity appears in the momentum equation and when adding a drift term to the gas velocity:

References for the Bubbly Flow Interfaces

1. A. Sokolichin, G. Eigenberger, and A. Lapin, “Simulations of Buoyancy Driven Bubbly Flow: Established Simplifications and Open Questions,” AIChE Journal, vol. 50, no. 1, pp. 24–49, 2004.

2. D. Kuzmin and S. Turek, Efficient Numerical Techniques for Flow Simulation in Bubble Column Reactors, Institute of Applied Mathematics, University of Dortmund, 2000.

CONSTANT VALUE

C 0.09

C1 1.44

C2 1.92

k 1.0

1.3

Pk T ul: ul ul T+ 23--- ul 2–

23---k ul–=

l t lul +

T------+

C1k---C

1Pk lC2

2

k-----–+= CSk

k---+

Sk Ckg p– uslip=

udriftTl------

gg

----------–=


3. D. Kuzmin, S. Turek, and H. Haario, Finite Element Simulation of Turbulent Bubbly Flows in Gas-liquid Reactors, Ergebnisberichte Angew, Math, 298, University of Dortmund, 2005.

4. M.P. Schwarz and W.J. Turner, “Applicability of the Standard k- Turbulence Model to Gas-stirred Baths,” Applied Mathematical Modelling, vol. 12, pp. 273–279, 1988.

5. C. Crowe, M. Sommerfeld, and Y. Tsuji, Multiphase Flows with Droplets and Particles, CRC Press, 1998.


418 | C H A P T E

Th eo r y f o r t h e M i x t u r e Mode l I n t e r f a c e s

The mixture model is a macroscopic two-phase flow model, in many ways similar to the bubbly flow model. It tracks the averaged phase concentration, or volume fraction, and solves a single momentum equation for the mixture velocity. It is suitable for mixtures consisting of solid particles or liquid droplets immersed in a liquid.

The Mixture Model interface theory is described in this section:

• The Mixture Model Equations

• Dispersed Phase Boundary Conditions Equations

• Turbulence Modeling in Mixture Models

• Slip Velocity Models

• References for the Mixture Model Interfaces

The Mixture Model Equations

Just as for the Bubbly Flow interfaces, the Mixture Model interfaces are based on the two fluid Euler-Euler Model. The two phases consist of one dispersed phase and one continuous phase. The mixture model is valid if the continuous phase is a liquid, and the dispersed phase consists of solid particles, liquid droplets, or gas bubbles. For gas bubbles in a liquid, however, the bubbly flow model is preferable. The mixture model relies on the following assumptions.

• The density of each phase is approximately constant.

• Both phases share the same pressure field.

• The relative velocity between the two phases is essentially determined assuming a balance between the pressure gradient and viscous drag.

The momentum equation for the mixture is

ut u u+ – p Gm g F+ +–=

cd 1 cd– uslipDmd

1 cd– d-------------------------- d–

uslipDmd

1 cd– d-------------------------- d–

–


where:




• cd is the mass fraction of the dispersed phase (SI unit: kg/kg)

• uslip is the relative velocity vector between the two phases (SI unit: m/s)

• Gm is the sum of the viscous and turbulent stresses (SI unit: kg/(m·s2))

• g is the gravity vector (SI unit: m/s2), and

• F is any additional volume force (SI unit: N/m3)

The velocity u used here is the mass-averaged mixture velocity (SI unit: m/s), defined as

where

• c and d denote the volume fractions of the continuous phase and the dispersed phase (SI unit: m3/m3), respectively

• uc is the continuous phase velocity vector (SI unit: m/s)

• ud is the dispersed phase velocity vector (SI unit: m/s)

• c is the continuous phase density (SI unit: kg/m3)

• d is the dispersed phase density (SI unit: kg/m3), and

• is the mixture density (SI unit: kg/m3)

The relationship between the velocities of the two phases is defined by

(6-21)

Here, uslip (SI unit: m/s) denotes the relative velocity between the two phases. For different available models for the slip velocity, see Slip Velocity Models. Dmd is a turbulent dispersion coefficient (SI unit: m2/s) (see Turbulence Modeling in Mixture Models), accounting for extra diffusion due to turbulent eddies. When a turbulence model is not used, Dmd is zero.

The mixture density is given by

uccuc ddud+

--------------------------------------------=

ud uc– ucd uslipDmd

1 cd– d--------------------------d–= =

cc dd+=

T H E O R Y F O R T H E M I X T U R E M O D E L I N T E R F A C E S | 419

420 | C H A P T E

where c and d (SI units: kg/m3) are the densities of the two phases. The mass fraction of the dispersed phase cd is given by

The sum of the viscous and turbulent stresses is

where (SI unit: Pa·s) is the mixture viscosity and T (SI unit: Pa·s) the turbulent viscosity. If no turbulence model is used, T equals zero.

The transport equation for d, the dispersed phase volume fraction, is

(6-22)

where mdc (SI unit: kg/(m3·s)) is the mass transfer rate from the dispersed to the continuous phase and ud (SI unit: m/s) is the dispersed phase velocity according to Equation 6-21. Assuming constant density for the dispersed phase, Equation 6-22 can be rewritten as

(6-23)

The continuous phase volume fraction c is

The continuity equation for the mixture is

(6-24)

In the Mixture Model interfaces it is assumed that the densities of both phases, c and d, are constant, and therefore the following alternative form of the continuity equation for the mixture is used

(6-25)

Equation 6-25 is derived from Equation 6-22 and Equation 6-24.

cddd

------------=

Gm T+ u uT+ =

t dd ddud + mdc–=

t d dud +

mdcd

----------–=

c 1 d–=

t u + 0=

c d– d 1 cd– uslip Dmdd– mdcd

----------+ c u + 0=


Dispersed Phase Boundary Conditions Equations

In addition to the boundary conditions for the mixture, specify boundary conditions for the dispersed phase on Wall, Inlet, and Outlet nodes. For these boundary types, the boundary condition’s settings window contains a Dispersed Phase Boundary Condition section.

D I S P E R S E D P H A S E C O N C E N T R A T I O N

This is the default for the Inlet node. Specify the dispersed phase volume fraction:

Enter the dispersed phase volume fraction (dimensionless) in the d0 field.

D I S P E R S E D P H A S E O U T L E T

This is the default for the Outlet node. This boundary condition is appropriate for boundaries where the dispersed phase leaves the domain with the dispersed phase velocity ud. No condition is imposed on the volume fractions at the boundary.

D I S P E R S E D P H A S E F L U X

Using this boundary condition, specify the dispersed phase flux through the boundary:

Enter the dispersed phase flux (SI unit: m/s) in the field.

S Y M M E T R Y

This boundary condition, which is useful on boundaries that represent a symmetry line for the dispersed phase, sets the dispersed phase flux through the boundary to zero:

N O D I S P E R S E D F L U X

This is the default for the Wall node. This boundary condition represents boundaries where the dispersed phase flux through the boundary is zero:

d d0=

n dud – Nd=

Nd

n dud 0=

n dud 0=


422 | C H A P T E

Turbulence Modeling in Mixture Models

For turbulence modeling, use the k- turbulence model. Turbulence modeling is particularly relevant for dilute flows, that is, for flows with a low dispersed phase volume fraction. For dense flows, the mixture viscosity usually becomes high. In such cases, the flow is laminar and no turbulence modeling is necessary.

The k- turbulence model solves two extra transport equations for two additional variables: the turbulent kinetic energy, k (SI unit: m2/s2), and the dissipation rate of turbulent kinetic energy, (SI unit: m2/s3). The turbulent viscosity is given by


The transport equation for the turbulent kinetic energy k is


The evolution of the dissipation rate of the turbulent kinetic energy, , is determined by

By default, the following variables are used for the dimensionless parameters:

The turbulence must also be accounted for in the calculation of the dispersed phase velocity. This is accomplished by introducing a turbulent dispersion coefficient Dmd

CONSTANT VALUE

C 0.09

C1 1.44

C2 1.92

k 1.0

1.3

T Ck2

------=

kt

------ + u k Tk------+

k Pk –+=

Pk T u: u u T+ 23--- u 2–

23---k u–=

t

----- + u T------+

C1

k---P

kC2

2

k-----–+=


(SI unit: m2/s) in Equation 6-21 as

where T is the turbulent particle Schmidt number (dimensionless). The particle Schmidt number is usually suggested a value ranging from 0.35 to 0.7. In the physics interface the default value is 0.35. The so-called drift velocity is included in Equation 6-23 as a diffusion term

where in turn

Slip Velocity Models

The Mixture Model interfaces contain three predefined models for the relative velocity between the two phases uslip (SI unit: m/s):

• The Schiller-Naumann model

• The Haider-Levenspiel model

• The Hadamard-Rybczynski model

All three models use the following relation for the slip velocity:

(6-26)

where Cd (dimensionless) is the particle drag coefficient. Essentially, interpret the relation as a balance between viscous drag and buoyancy forces acting on the dispersed phase.

The Schiller-Naumann model models the drag coefficient according to

where Rep is the particle Reynolds number

DmdTT----------=

t d dd + Dmd d

mdcd

----------–=

d u uslip 1 cd– +=

34---

Cddd-------c uslip uslip

d–

--------------------p–=

Cd

24Rep---------- 1 0,15Rep

0,687+ Rep 1000

0,44 Rep 1000

=


424 | C H A P T E

Because the particle Reynolds number depends on the slip velocity, an implicit equation must be solved to obtain the slip velocity. Therefore, the Mixture Model interfaces add an additional equation for

when the Schiller-Naumann slip model is used. The Schiller-Naumann model is particularly well-suited for solid particles in a liquid.

The Haider-Levenspiel model is applicable to non-spherical particles. It models the drag coefficient according to

where A, B, C and D are empirical correlations of the particle sphericity. The sphericity is defined as the ratio of the surface area of a volume equivalent sphere to the surface area of the considered non-spherical particle

The correlation coefficients are given by

The diameter used in the particle Reynolds number is that of the volume equivalent sphere. The equation for the squared slip velocity is also added when the Haider-Levenspiel slip model is used.

The Hadamard-Rybczynski drag law is valid for particle Reynolds numbers less than 1, for particles, bubbles, and droplets. The drag coefficient for liquid droplets or bubbles is

Repddc uslip

---------------------------=

uslip2

Cd24

Rep---------- 1 A Sp Rep

B Sp +

C Sp 1 D Sp Rep+-----------------------------------------+=

0 SpAsphereAparticle------------------ 1=

A Sp e2.3288-6.4581Sp 2.4486Sp

2+=

B Sp 0.0964+0.5565Sp=

C Sp e4.905-13.8944Sp 18.4222Sp

2 10.2599Sp3–+

=

D Sp e1.4681 12.2584S+ p 20.7322Sp

2– 15.8855Sp3+

=


which yields the following explicit expression for the slip velocity

For solid particles, the slip velocity is given by

when Rep1. For very small gas bubbles, the drag coefficient is observed to be closer to the solid-particle value. This is believed to be caused by surface-active impurities collecting on the bubble surface.

References for the Mixture Model Interfaces

1. M. Manninen, V. Taivassalo, and S. Kallio, On the Mixture Model for Multiphase Flow, VTT Publications, 288, VTT Energy, Nuclear Energy, Technical Research Center of Finland (VTT), 1996.

2. C. Crowe, M. Sommerfeld, and Y. Tsuji, Multiphase Flows with Droplets and Particles, CRC Press, 1998.

Cd24

Rep----------

1 23---cd------+

1cd------+

--------------------

=

uslip d– dd

2

18c---------------------------

1cd------+

1 23---cd------+

--------------------

p–=

uslip d– dd

2

18c---------------------------– p=


426 | C H A P T E

Th eo r y f o r t h e Eu l e r - E u l e r Mode l , L am i n a r F l ow I n t e r f a c e

The Euler-Euler Model, Laminar Flow interface is based on averaging the Navier-Stokes equations for each phase over a volume that is small compared to the computational domain but large compared to the dispersed phase particles, droplets, or bubbles. The present phases, the continuous and the dispersed phase, are assumed to behave as two continuous and interpenetrating fluids. The physics interface solves one momentum equation for each phase.

In this section:

• The Euler-Euler Model Equations

• References for the Euler-Euler Model, Laminar Flow Interface

The Euler-Euler Model Equations

M A S S B A L A N C E

Assuming that the mass transfer between the two phases is zero, the following continuity relations hold for the continuous and dispersed phases (Ref. 3):

(6-27)

(6-28)

Here (dimensionless) denotes the phase volume fraction, (SI unit: kg/m3) is the density, and u (SI unit: m/s) the velocity of each phase. The subscripts c and d denote quantities relating to the continuous and the dispersed and phase, respectively. The following relation between the volume fractions must hold

Both phases are considered to be incompressible, in which case Equation 6-27 and Equation 6-28 can be simplified as:

t cc ccuc + 0=

t dd ddud + 0=

c 1 d–=


(6-29)

(6-30)

If Equation 6-29 and Equation 6-30 are added together, a continuity equation for the mixture is obtained:

(6-31)

In order to control the mass balance of the two phases, the Euler-Euler Model, Laminar Flow interface solves Equation 6-29 together with Equation 6-31. Equation 6-29 is used to compute the volume fraction of the dispersed phase, and Equation 6-31 is used for the mixture pressure.

M O M E N T U M B A L A N C E

The momentum equations for the continuous and dispersed phases, using the non-conservative forms of Ishii (Ref. 4), are:

(6-32)

(6-33)

Here p (SI unit: Pa) is the mixture pressure, which is assumed to be equal for the two phases. In the momentum equations the viscous stress tensor for each phase is denoted by (SI unit: Pa), g (SI unit: m/s2) is the vector of gravitational acceleration, Fm (SI unit: N/m3) is the interphase momentum transfer term (that is, the volume force exerted on the phase by the other phase), and F (SI unit: N/m3) is any other volume force term.

In these equations, the influence of surface tension in the case of liquid phases has been neglected, and they do not consider the potential size distribution of the dispersed particles or droplets.

For fluid-solid mixtures, Equation 6-33 is modified in the manner of Enwald (Ref. 5)

(6-34)

where ps (SI unit: Pa) is the solid pressure.

td dud + 0=

tc cuc + 0=

dud uc 1 d– + 0=

cc t uc uc+ uc cp– cc ccg Fm,c cFc+ + + +=

dd t ud ud+ ud dp– dd ddg Fm,d dFd+ + + +=

dd t ud ud+ ud sp– d ps– ddg Fm,d dFd+ + + +=

T H E O R Y F O R T H E E U L E R - E U L E R M O D E L , L A M I N A R F L O W I N T E R F A C E | 427

428 | C H A P T E

The fluid phases in the above equations are assumed to be Newtonian and the viscous stress tensors are defined as

where (SI unit: Pa·s) is the dynamic viscosity of the respective phase.

In order to avoid singular solutions when the volume fractions tend to zero, the governing equations above are divided by the corresponding volume fraction. The implemented momentum equation for the continuous phase is:

The implemented momentum equations for the dispersed phase in the case of dispersed liquid droplets or bubbles is:

(6-35)

and in the case of dispersed solid particles:

(6-36)

D I S P E R S E D P H A S E V I S C O S I T Y

The Newtonian viscosities of interpenetrating media are not readily available. Instead empirical and analytical models for the dynamic viscosity of the two-phase mixture have been developed by various researchers, usually as a function of the dispersed volume fraction. Using an expression for the mixture viscosity, the Euler-Euler Model, Laminar Flow interface retrieves the dispersed phase viscosity using the following linear relationship proposed by Enwald and others (Ref. 5):

(6-37)

c c u c u c T 23--- uc I–+

=

d d u d u d T 23--- ud I–+

=

c t uc( ) cuc uc + p– c

ccc

---------------+ cgFm,cc

----------- Fc+ + + +=

d t ud( ) dud ud + p– d

ddd

----------------+ dgFm,dd

------------ Fd+ + + +=

d t ud( ) dud ud + p– d

ddd

----------------+ ps–+= +

dgFm,dd

------------ Fd+ +

mix cc dd+=


A simple mixture viscosity covering the entire range of particle concentrations is the Krieger type model (Ref. 5):

(6-38)

Here d,max is the maximum packing limit, by default 0.62. Equation 6-38 can be applied when .

I N T E R P H A S E M O M E N T U M TR A N S F E R

In all the equations, Fm denotes the interphase momentum transfer, that is the force imposed on one phase by the other phase. Considering a particle, droplet, or bubble in a fluid flow, it is affected by a number of forces, for example, the drag force, the added mass force, the Basset force, and the lift force. The most important force is usually the drag force, especially in fluids with a high concentration of dispersed solids, and hence this is the predefined force included in the Euler-Euler model. The drag force added to the momentum equation is defined as:

(6-39)

where is a drag force coefficient and the slip velocity is defined as

Dense FlowsFor dense flows with a high concentration of the dispersed phase—for example, in fluidized bed models—the Gidaspow model (Ref. 6) for the drag coefficient can be used. It combines the Wen and Yu (Ref. 7) fluidized state expression:

For c > 0.8

(6-40)

with the Ergun (Ref. 8) packed bed expression:

For c < 0.8

The drag force on the dispersed phase is equal to the one on the continuous phase but is working in the opposite direction.

mix c 1d

d,max---------------–

2.5d,max–

=

c d«

Fdrag,c F·

– drag,d uslip= =

uslip ud uc–=

3cdcCd

4dd---------------------------- uslip c

2.65–=


430 | C H A P T E

(6-41)

In the above equations, dd (SI unit: m) is the dispersed particle diameter, and Cd is the drag coefficient for a single dispersed particle. The drag coefficient is in general a function of the particle Reynolds number

No universally valid expression for the drag coefficient exists. Using the implemented Gidaspow model, Cd is computed using the Schiller-Naumann relation

(6-42)

Dilute FlowsFor dilute flows the drag force coefficient can be modeled as:

(6-43)

In this case the drag coefficient can be computed from the Schiller-Naumann model in Equation 6-42, the Haider-Levenspiel model, or by using the Hadamard-Rybczynski drag law. The Haider-Levenspiel model is applicable to nonspherical particles. It models the drag coefficient according to

where A, B, C, and D are empirical correlations of the particle sphericity (see Slip Velocity Models for further details). The Hadamard-Rybczynski drag law is valid for particle Reynolds number less than one for particles, bubbles, and droplets and is defined as:

150cd

2

cdd2

------------ 1.75dcdd

----------- uslip+=

Rep

cddc uslip

c--------------------------------=

Cd

24Rep---------- 1 0.15Rep

0.687+ Rep 1000

0.44 Rep 1000

=

3dcCd

4dd----------------------- uslip=

Cd24

Rep---------- 1 A Sp Rep

B Sp +

C Sp 1 D Sp Rep+-----------------------------------------+=

Cd24

Rep----------

1 23---c

d------+

1cd------+

--------------------

=


for bubbles and droplets and as

for solid particles (the Stokes limit). For very small gas bubbles, the drag coefficient is observed to be closer to the solid-particle value. This is believed to be caused by surface-active impurities collecting on the bubble surface.

S O L I D P R E S S U R E

For fluid-solid mixtures, a model for the solid pressure, ps in Equation 6-36, is needed. The solid pressure models the particle interaction due to collisions and friction between the particles. The solid pressure model implemented uses a gradient diffusion based assumption:

where the empirical function G is given by

(6-44)

described in Ref. 5. The available predefined models (all defined in Ref. 5) are those of Gidaspow and Ettehadieh,

(6-45)

Ettehadieh,

(6-46)

and Gidaspow,

(6-47)

N O T E S O N T H E I M P L E M E N T A T I O N

There are several equations with potentially singular terms and expressions such as the term

Cd24

Rep----------=

ps G c c–=

G c 10B1c B2+

=

G c 10-8.76c 5.43+=

G c 10-10.46c 6.577+=

G c 10-10.5c 9.0+=

Fm,dd

------------


432 | C H A P T E

in Equation 6-35. Also, is a degree of freedom and can therefore, during computations, obtain nonphysical values in small areas. This can, in turn, lead to nonphysical values of material properties such as viscosity and density. To avoid these problems, the implementation uses the following regularizations:

• 1 k, kc,d is replaced by1 k,pos where

• is evaluated as .

• The variable is only used directly in Equation 6-29. It is everywhere else replaced by

Note that this includes the way is calculated; that is,

• An extra diffusion term is added to the right-hand side of Equation 6-29 to minimize the occurrence of negative values of . The “barrier” viscosity is calculated as

where the index k indicates that the viscosity and density are taken from the continuous phase if the dispersed phase is a solid and from the dispersed phase otherwise. Note that b is only nonzero if is less than zero.

References for the Euler-Euler Model, Laminar Flow Interface

1. M.J.V. Goldschmidt, B.P.B. Hoomans, and J.A.M. Kuipers, “Recent Progress Towards Hydrodynamic Modelling of Dense Gas-Particle Flows,” Recent Research Developments in Chemical Engineering, Transworld Research Network, India, pp. 273–292, 2000.

2. A. Soulaimani and M. Fortin, “Finite Element Solution of Compressible Viscous Flows Using Conservative Variables,” Computer Methods in Applied Mechanics and Engineering, vol. 118, pp. 319–350, 1994.

3. C. Crowe, M. Sommerfeld, and Y Tsuji, Multiphase Flows with Droplets and Particles, CRC Press, Boca Raton, 1998.

4. B.G.M. van Wachem, J.C. Schouten, C.M. van den Bleek, R. Krishna, and J.L. Sinclair, “Comparative Analysis of CFD Models of Dense Gas-Solid Systems,” AIChE Journal, vol. 47, no. 5, pp. 1035–1051, 2001.

d

k pos 0.001 0.999 flc1hs k 5e 4–– 5e 4– +=

k k log k pos

d

d reg min max d 0 1 =

c

c 1 d reg–=

d

b emax d 0.0025– 0 1– kk------=

d


5. H. Enwald, E. Peirano, and A.-E. Almstedt, “Eulerian Two-Phase Flow Theory Applied to Fluidization,” Int. J. Multiphase Flow, vol. 22, pp. 21–66, 1996.

6. D. Gidaspow, Multiphase Flow and Fluidization, Academic Press, San Diego, 1994.

7. C.Y. Wen and Y.H. Yu, “Mechanics of Fluidization,” Chemical Engineering Progress Symposium Series, vol. 62, pp. 100–110, 1966.

8. S. Ergun, “Fluid Flow Through Packed Columns,” Chemical Engineering Progress, vol. 48, pp. 89–94, 1952.


434 | C H A P T E

7

P o r o u s M e d i a a n d S u b s u r f a c e F l o w

The fluid-flow interfaces are grouped by type under the Fluid Flow main branch. This chapter discusses applications involving the Porous Media and Subsurface Flow branch ( ) when adding a physics interface. The section Modeling Porous Media and Subsurface Flow helps you choose the best one to start with.

In this chapter:

• The Darcy’s Law Interface

• The Brinkman Equations Interface

• The Free and Porous Media Flow Interface

• The Two-Phase Darcy’s Law Interface

• Theory for the Darcy’s Law Interface

• Theory for the Brinkman Equations Interface

• Theory for the Free and Porous Media Flow Interface

• Theory for the Two-Phase Darcy’s Law Interface

435

436 | C H A P T E

Mode l i n g Po r ou s Med i a and S ub s u r f a c e F l ow

In this section:


• The Porous Media Flow Interface Options



The Porous Media and Subsurface Flow branch ( ) included with the CFD Module has a number of subbranches to describe momentum transport. These can be added either singularly or in combination with other physics interfaces modeling mass and energy transfer, and even chemical reactions.

Different types of flow require different equations to describe them. If the flow type to model is known, then select it directly. However, when you are not certain of the flow type, or when it is difficult to obtain a solution, you can instead start with a simplified model and add complexity as you build the model. Then you can successively advance forward, comparing models and results. For porous media flow, the Darcy’s Law interface is a good place to start if this is the case.

In other cases, you might know exactly how a fluid behaves and which equations, models, or physics interfaces best describe it, but because the model is so complex it is difficult to reach convergence. Simplifying assumptions can be made to solve the problem, and other physics interfaces can be better at fine-tuning the solution process for the more complex problem. The next section gives you an overview of each of the physics interfaces to help you choose.

TABLE 7-1: THE POROUS MEDIA FLOW DEFAULT SETTINGS

INTERFACE ID COMPRESSIBILITY NEGLECT INERTIAL TERM

PORE SIZE

Darcy's Law dl n/a n/a Low porosity and low permeability, slow flow

Two-Phase Darcy’s law

tpdl n/a n/a Low porosity and low permeability, slow flow

R 7 : P O R O U S M E D I A A N D S U B S U R F A C E F L O W

Figure 7-1 is an example of the Brinkman Equations settings window where you can select either Compressible or Incompressible flow, and either normal or Stokes Brinkman flow.

The Porous Media Flow Interface Options

D A R C Y ’ S L A W

The Darcy’s Law Interface ( ) is used for modeling fluid movement through interstices in a porous medium by homogenizing the porous and fluid media into a single medium. Together with the continuity equation and equation of state for the pore fluid (or gas) this interface can be used to model low velocity flows, for which the pressure gradient is the major driving force. The penetration of reacting gases into a tight catalytic layer, such as a washcoat or membrane, is a classic example for the use of Darcy’s Law.

Darcy’s law can be used in porous media where the fluid is mostly influenced by the frictional resistance within the pores. It applies to very slow flows, or media where the pore size is very small.

B R I N K M A N E Q U A T I O N S

Where the size of the interstices are larger, and the fluid is also influenced by internal shear or shear stresses on boundaries, the viscous shear within the fluid must be considered. This is done in the Brinkman Equations interface. Fluid penetration of filters and packed beds are applications for this mode. The Brinkman Equations Interface ( ) is used to model compressible flow at speeds less than Mach 0.3, but you have to maintain control over the density and any of the mass balances that are deployed to accomplish this. You can also choose to model incompressible flow, and simplify the equations to be solved. Furthermore, you can select the Stokes-Brinkman flow feature to reduce the influence of inertial effects (see Figure 7-1).

Brinkman Equations

br Incompressible flow

Yes - Stokes-Brinkman

High permeability and porosity, faster flow

Free and Porous Media Flow

fp Incompressible flow

Not selected High permeability and porosity, fast flow

TABLE 7-1: THE POROUS MEDIA FLOW DEFAULT SETTINGS

INTERFACE ID COMPRESSIBILITY NEGLECT INERTIAL TERM

PORE SIZE

M O D E L I N G P O R O U S M E D I A A N D S U B S U R F A C E F L O W | 437

438 | C H A P T E

Figure 7-1: The settings window for the Brinkman Equations interface. You can model compressible or incompressible flow as well as Stokes-Brinkman flow. Combinations are also possible.

The Brinkman equations extend Darcy’s law to describe the dissipation of momentum by viscous shear, similar to the Navier-Stokes equation. Consequently, they are well suited to model transitions between slow flow in porous media, governed by Darcy’s law, and fast flow in channels described by the Navier-Stokes equations.

The Brinkman Equations interface also includes the possibility to add a Forchheimer drag term, which is a viscous drag on the porous matrix proportional to the square of the flow velocity.

F R E E A N D PO R O U S M E D I A F L O W

The Free and Porous Media Flow Interface ( ) is useful for modeling equipment that contain domains with both free flow and porous media flow, such as packed-bed reactors and catalytic converters. It should be noted that if the porous medium is large in comparison with the free channel, and you are not primarily interested in results in the vicinity of the interface, you can always couple a fluid-flow interface to a Darcy’s

Law interface, to make your overall model computationally cheaper.


The Free and Porous Media Flow interface is used on at least two different domains; a free channel and a porous medium. The interface adds functionality that allows the equations to be optimized according to the definitions of the material properties of the relevant domain. For example, you can select the Stokes-Brinkman flow feature to reduce the dependence on inertial effects in the porous domain, or just the Stoke’s flow feature to reduce the dependence on inertial effects in the free channel.

Compressible flow is also possible to model in this physics interface at speeds less than Mach 0.3, but you have to maintain control over the density and any of the mass balances that are deployed to accomplish this. You can also choose to model incompressible flow, and simplify the equations to be solved.

As always, the physics interface gives you provides you with options to define, either by constants or expressions, the material properties that describe the porous media flow. This includes the density, dynamic viscosity, permeability, porosity, and matrix properties.

TW O - P H A S E D A R C Y ’ S L A W

The Two-Phase Darcy’s Law Interface ( ) has the equations and boundary conditions for modeling two-phase fluid movement through interstices in a porous medium using Darcy’s law. The two fluids are considered immiscible, and in general, have different densities and viscosities.

As for the single phase Darcy's Law, the total velocity field is determined by the total pressure gradient and the structure of the porous medium, but the average viscosity and average density are calculated from the saturation of each immiscible phase and their fluid properties. An extra equation is computed—the fluid content of one phase—in order to calculate the saturation transport.


Often, you are simulating applications that couple fluid flow in porous or subsurface media to another type of phenomenon described in another physics interface. This can include chemical reactions and mass transport, as described in Chemical Species Transport, or energy transport in porous media described in the Heat Transfer and Non-Isothermal Flow chapter.

More extensive descriptions of modeling chemical reactions and mass transport are found in the Chemical Reaction Engineering Module. Furthermore, some applications that involve electrochemical reactions and porous electrodes, particularly in

M O D E L I N G P O R O U S M E D I A A N D S U B S U R F A C E F L O W | 439

440 | C H A P T E

electrochemical power source applications, are supported in the Batteries & Fuel Cells Module.

Fluid flow is an important phenomenon for cooling in electromagnetic applications, such as heat created through induction and microwave heating, which are simulated in the AC/DC Module and RF Module, respectively. Other applications can involve the effect of fluid-imposed momentum on structural applications; poroelasticity. The Structural Mechanics Module and Subsurface Flow Module have interfaces specifically for these multiphysics applications.

The following sections list all the physics interfaces and the features associated with them under the Porous Media Subsurface Flow branch. The descriptions follow a structured order as defined by the order in the branch. Because many of the physics interfaces are integrated with each other, some features described also cross reference to other physics interfaces. At the end of this section is a summary of the theory for the physics interfaces under the Porous Media Subsurface Flow branch.


Th e Da r c y ’ s L aw I n t e r f a c e

The Darcy’s Law (dl) interface ( ), found under the Porous Media and Subsurface Flow branch ( ) when adding a physics interface, is used to simulate fluid flow through interstices in a porous medium. It can be used to model low-velocity flows or media where the permeability and porosity are very small, and for which the pressure gradient is the major driving force and the flow is mostly influenced by the frictional resistance within the pores. Set up multiple Darcy's Law interfaces to model multiphase flows involving more than one mobile phase.

The Darcy's Law interface can be used for stationary and time-dependent analyses.

The main feature is the Fluid and Matrix Properties node, which provides an interface for defining the fluid material along with the porous medium properties.

When this physics interface is added, the following default nodes are also added in the Model Builder— Fluid and Matrix Properties, No Flow (the default boundary condition), and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and mass sources. You can also right-click Darcy's Law

to select physics from the context menu.



The default identifier (for the first physics interface in the component) is dl.


The default setting is to have All domains in the component define the pressure and the Darcy’s law and continuity equations. To choose specific domains, select Manual from the Selection list.


The dependent variable (field variable) is the Pressure. The name can be changed but the names of fields and dependent variables must be unique within a model.

T H E D A R C Y ’ S L A W I N T E R F A C E | 441

442 | C H A P T E


To display this section, click the Show button ( ) and select Discretization. Select a Pressure—Quadratic (the default), Linear, Cubic, or Quartic. Specify the Value type when


Domain, Boundary, Edge, Point, and Pair Nodes for the Darcy’s Law Interface

The Darcy’s Law Interface has the following domain, boundary, edge, point, and pair nodes, These nodes available from the Physics ribbon toolbar (Windows users), Physics context menu (Mac or Linux users), or right-click to access the context menu (all users).

D O M A I N

• Fluid and Matrix Properties

• Mass Source

• Initial Values

B O U N D A R Y , E D G E , A N D PO I N T

The following nodes are available on exterior boundaries:

• Flux Discontinuity


• Domain, Boundary, Edge, Point, and Pair Nodes for the Darcy’s Law Interface

• Theory for the Darcy’s Law Interface


For axisymmetric models, COMSOL Multiphysics takes the axial symmetry boundaries (at r = 0) into account and automatically adds an Axial Symmetry node that is valid on the axial symmetry boundaries only.


• Inlet

• Mass Flux

• Pressure

• No Flow (the default boundary condition)

• Symmetry

The relevant physics interface condition at interior boundaries is continuity:

The continuity boundary condition ensures that the pressure and mass flux are continuous. In addition, the Pressure boundary condition is available on interior boundaries.

Fluid and Matrix Properties

The Fluid and Matrix Properties node adds the equations for Darcy’s law, Equation 7-1 and Equation 7-2 (excluding any mass sources), and contains settings for the fluid properties and the porous matrix properties such as the effective porosity.



n 1u1 2u2– 0=




The links to the nodes described in the COMSOL Multiphysics Reference Manual do not work in the PDF, only from the on line help in COMSOL Multiphysics.


444 | C H A P T E


Define model inputs, for example, the temperature field if the material model uses a temperature-dependent material property. If no model inputs are required, this section is empty.


The Global coordinate system is selected by default. The Coordinate system list contains any additional coordinate systems that the model includes (except boundary coordinate systems). The coordinate system is used to define directions for an anisotropic permeability.


Select the material to use for the fluid properties. Select Domain material from the Fluid

material list (the default value) to use the material defined for the domain. Select another material from the Fluid material list to use that material’s properties for the fluid.

DensityThe default Density (SI unit: kg/m3) uses values From material based on the Fluid

material selection. If User defined is selected, enter another value or expression. The default is 0 kg/m3.

Ideal GasSelect Ideal gas to use the ideal gas law to describe the fluid. In this case, specify the thermodynamics properties by selecting a gas constant type and selecting between entering the heat capacity at constant pressure or the ratio of specific heats.

Select a Gas constant type—Specific gas constant Rs or Mean molar mass Mn (SI unit: J/(mol·K)). If Mean molar mass is selected, the universal gas constant R 8.314 J/(mol·K) is used as the built-in physical constant. For both properties, the defaults use values From material. If User defined is selected, enter another value or expression.

Dynamic ViscositySelect a Dynamic viscosity (SI unit: Pa·s). The default uses values From material as defined by the Fluid material selected. If User defined is selected, enter another value or expression. The default is 0 Pa·s.

To define the absolute pressure, see the settings for the Heat Transfer in Fluids node.


M A T R I X P R O P E R T I E S

Select the material to use as porous matrix. Select Domain material from the Porous

material list (the default) to use the material defined for the porous domain. Select another material from the Porous material list to use that material’s properties.

PorosityThe default Porosity p (SI unit: a dimensionless number between 0 and 1) uses the value From material, defined by the Porous material selected. If User defined is selected, enter another value or expression The default is 0.

PermeabilityThe default Permeability (SI unit: m2) uses the value From material, as defined by the Porous material selected. If User defined is selected, choose Isotropic to define a scalar value or Diagonal, Symmetric, or Anisotropic to define a tensor value and enter another value or expression in the field or matrix.

Mass Source

The Mass Source node adds a mass source Qm, which appears on the right-hand side of the Darcy’s Law equation (Equation 7-1, the equation for porosity).

(7-1)



M A S S S O U R C E

Enter a value or expression for the Source term Qm (SI unit: kg/(m3·s)).

Initial Values

The Initial Values node adds an initial value for the pressure that can serve as an initial condition for a transient simulation or as an initial guess for a nonlinear solver.


From the Selection list, choose the domains on which to apply the condition.

t u + Qm=


446 | C H A P T E


Enter a value or expression for the initial value of the Pressure p (SI unit: Pa). The default value is 0 Pa.

Pressure

Use the Pressure node to specify the pressure on a boundary. In many cases the distribution of pressure is known, giving a Dirichlet condition p = p0 where p0 is a known pressure given as a number, a distribution, or an expression involving time, t, for example.





P R E S S U R E

Enter a value or expression for the Pressure p (SI unit: Pa).


To display this section, click the Show button ( ) and select Advanced Physics Options. To Apply reaction terms on all dependent variables, select All physics (symmetric). Otherwise, select Current physics (internally symmetric) or Individual dependent

variables to restrict the reaction terms as required. Select the Use weak constraints check box to replace the standard constraints with a weak implementation.

Mass Flux

Use the Mass Flux node to specify the mass flux into or out of the model domain through some of its boundaries. It is often possible to determine the mass flux from the pumping rate or from measurements. With this boundary condition, positive values correspond to flow into the model domain:

where N0 is a value or expression for the specified inward (or outward) Darcy flux.

n --- p N0=



From the Selection list, choose the boundaries on which to specify the mass flux.

M A S S F L U X

Enter a value or expression for the Inward mass flux N0 (SI unit: kg/(m2·s)). A positive value of N0 represents an inward mass flux.

The Mass Flux node is also available as a condition on Edges (3D models) and Points (2D and 3D models). Enter a value or expression for the Inward mass flux N0 (SI unit for edges: kg/(m·s); SI unit for points: kg/s)). A positive value of N0 represents an inward mass flux whereas a negative value represents an outward mass flux.

Inlet

The Inlet node adds a boundary condition for the inflow (or outflow) perpendicular (normal) to the boundary:

where U0 is a value or expression for the specified inward (or outward) Darcy velocity. A positive value of the velocity U0 corresponds to flow into the model domain whereas a negative value represents an outflow.


From the Selection list, choose the boundaries on which to define the inlet.

I N L E T

Enter a value or expression for the Normal inflow velocity U0 (SI unit: m/s). A positive value of U0 represents an inflow velocity. A negative value represents an outflow velocity.

Symmetry

The Symmetry node describes a symmetry boundary. The following condition implements the symmetry condition on an axis or a flow divide:

n ---p U0=

n --- p 0=


448 | C H A P T E


From the Selection list, choose the boundaries on which to apply the symmetry condition.

No Flow

The No Flow node is the default boundary condition stating that there is no flow across impervious boundaries. The mathematical formulation is:

where n is the vector normal to the boundary.


From the Selection list, choose the boundaries on which to apply the no flow condition.

Flux Discontinuity

Use the Flux Discontinuity node to specify a mass flux discontinuity through an interior boundary. The condition is represented by the following equation:

In this equation, n is the vector normal (perpendicular) to the interior boundary, is the fluid density, u1 and u2 are the Darcy’s velocities in the adjacent domains (as defined in Equation 7-2) and N0 is a specified value or expression for the flux discontinuity.

(7-2)

For this boundary condition, a positive value of N0 corresponds to a flow discontinuity in the opposite direction to the normal vector of the interior boundary.


n --- p 0=

n– u1 – u2 N0=

u ---p–=






M A S S F L U X

Enter a value or expression for the Inward mass flux N0 (SI unit: kg/(m2·s)). A positive value of N0 represents a mass flux discontinuity in the opposite direction to the normal vector of the interior boundary.

Outlet

The Outlet node adds a boundary condition for the outflow (or inflow) perpendicular (normal) to the boundary:

where U0 is a specified value or expression for the outward (or inward) Darcy velocity. A positive value of the velocity U0 corresponds to flow out of the model domain whereas a negative value represents an inflow.


From the Selection list, choose the boundaries on which to define the outlet.

O U T L E T

Enter a value or expression for the Normal outflow velocity U0 (SI unit: m/s). A positive value of U0 represents an outflow velocity whereas a negative value represents an inflow velocity.

n– ---p U0=


450 | C H A P T E

Th e B r i n kman Equa t i o n s I n t e r f a c e

The Brinkman Equations (br) interface ( ), found under the Porous Media and

Subsurface Flow branch ( ) when adding a physics interface, is used to compute fluid velocity and pressure fields of single-phase flow in porous media in the laminar flow regime. The physics interface extends Darcy's law to describe the dissipation of the kinetic energy by viscous shear, similar to the Navier-Stokes equations. Fluids with varying density can be included at Mach numbers below 0.3. Also the viscosity of a fluid can vary, for example, to describe non-Newtonian fluids. To simplify the equations, select the Stokes-Brinkman flow feature to reduce the dependence on inertial effects when the Reynolds number is significantly less than 1.

The physics interface can be used for stationary and time-dependent analyses.

The main node is the Fluid and Matrix Properties feature, which adds the Brinkman equations and provides an interface for defining the fluid material and the porous matrix.

When this physics interface is added, the following default nodes are also added in the Model Builder—Fluid and Matrix Properties, Wall (the default boundary type, using No

slip as the default boundary condition), and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and volume forces. You can also right-click Brinkman Equations to select physics from the context menu.



The default identifier (for the first physics interface in the component) is br.


The default setting is to have All domains in the component define the fluid pressure and velocity and the Brinkman equations that describe those fields. To choose specific domains, select Manual from the Selection list.



This node specifies the properties of the Brinkman Equations interface, which describe the overall type of fluid flow model.

CompressibilityBy default the physics interface uses the Incompressible flow formulation of the Brinkman equations to model constant density flow. Alternatively, select Compressible

flow (Ma<0.3) from the Compressibility list if there are small variations in the density, typically dependent on the temperature (non-isothermal flow). For compressible flow modeled with the Brinkman Equations interface, the Mach number must be below 0.3.

Neglect Inertial Term (Stokes-Brinkman Flow)The Neglect inertial term (Stokes-Brinkman) check box is selected by default to model flow at very low Reynolds numbers for which the inertial term can be neglected in the Brinkman equations. This results in the linear Stokes-Brinkman equations.

Swirl Flow


To display this section, click the Show button ( ) and select Discretization. Select a Discretization of fluids—P2+P1 (the default), P1+P1, or P3+P2.

• P2+P1 uses second-order elements for the velocity components and linear elements for the pressure field. Second-order elements work well for low flow velocities. This is the default element order.

• P1+P1 uses linear elements for both the velocity components and the pressure field. Linear elements are computationally cheaper than higher-order elements and are also less prone to introducing spurious oscillations, thereby improving the numerical robustness.

• P3+P2 uses third-order elements for the velocity components and second-order elements for the pressure field. This can add additional accuracy but it also adds additional degrees of freedom compared to P2+P1 elements.

For 2D axisymmetric models, select the Swirl flow check box to include the swirl velocity component, that is the velocity component in the azimuthal direction. While can be nonzero, there can be no gradients in the direction.

uu

T H E B R I N K M A N E Q U A T I O N S I N T E R F A C E | 451

452 | C H A P T E


Domain, Boundary, Point, and Pair Nodes for the Brinkman Equations Interface

The Brinkman Equations Interface has the following domain, boundary, point, and pair nodes, listed in alphabetical order, available from the Physics ribbon toolbar (Windows users), Physics context menu (Mac or Linux users), or right-click to access the context menu (all users).

These nodes are described in this section:


• Forchheimer Drag

• Initial Values

The other sections (Dependent Variables, Consistent Stabilization, Inconsistent Stabilization, and Advanced Settings) are the same as for The Laminar Flow Interface.


• Domain, Boundary, Point, and Pair Nodes for the Brinkman Equations Interface

• Pseudo Time Stepping for Laminar Flow Models and Pseudo Time Stepping in the COMSOL Multiphysics Reference Manual

• Theory for the Brinkman Equations Interface



• Mass Source

• Volume Force

The following nodes (listed in alphabetical order) are described for the Laminar Flow interface :


The Fluid and Matrix Properties node adds the Brinkman equations: Equation 7-3 and Equation 7-4 (excluding any mass sources), and provides an interface for defining the properties of the fluid material and the porous matrix Also right-click to add a Forchheimer Drag subnode.



The boundary conditions are essentially the same as for the Laminar Flow interface. Differences exist for the following boundary types: Outlet, Symmetry, Open Boundary, and Boundary Stress where the viscous part of the stress is divided by the porosity to appear as

1p----- u u T+ 2

3--- u I–


• Flow Continuity

• Inlet


• Outlet

• Open Boundary




• Symmetry

• Wall


454 | C H A P T E




The default Fluid material uses the Domain material (the material defined for the domain). Select another material as required.

Both the default Density (SI unit: kg/m3) and Dynamic viscosity (SI unit: Pa·s) use values From material based on the Fluid material selection. If User defined is selected, enter another value or expression. In this case, the default is 0 kg/m3 for the density and 0 Pa·s for the dynamic viscosity.

PO R O U S M A T R I X P R O P E R T I E S

The default Porous material uses the Domain material (the material defined for the domain) for the porous matrix. Select another material as required.

Both the default Porosity p (a dimensionless number between 0 and 1) and Permeability (SI unit: m2) use values From material as defined by the Porous material selection. If User defined is selected, choose Isotropic, Diagonal, Symmetric, or Anisotropic based on the characteristics of the thermal conductivity, and enter another value or expression. The components of a permeability in the case that it is a tensor (xx, yy, and so on, representing an anisotropic permeability) are available as br.kappaxx, br.kappayy, and so on (using the default interface identifier br).

To define the Absolute Pressure, see the settings for the Heat Transfer in Fluids node described in the COMSOL Multiphysics Reference Manual.

The dynamic viscosity describes the relationship between the shear stresses and the shear rate in a fluid. Intuitively, water and air have a low viscosity, and substances often described as thick, such as oil, have a higher viscosity. Non-Newtonian fluids have a viscosity that is shear-rate dependent. Examples of non-Newtonian fluids include yoghurt, paper pulp, and polymer suspensions.


Forchheimer Drag

Right-click the Fluid and Matrix Properties node to add a Forchheimer Drag subnode. While the drag of the fluid on the porous matrix in the basic Brinkman equations is proportional to the flow velocity, in the same way as for Darcy’s law, the Forchheimer drag is proportional to the square of the fluid velocity.


From the Selection list, choose the domains on which to apply the drag. The default is to add All domains.

F O R C H H E I M E R D R A G

Enter a value for the Forchheimer coefficient F (SI unit: kg/m4). The default is 0 kg/m4.

Mass Source

The Mass Source node adds a mass source (or mass sink) Qbr to the right-hand side of the continuity equation: Equation 7-3. This term accounts for mass deposit and/or mass creation in porous domains. The physics interface assumes that the mass exchange occurs at zero velocity.

(7-3)


From the Selection list, choose the domains on which to add the mass source. Only Porous Matrix domains are available.

M A S S S O U R C E

Enter a value or expression for the Source term Qbr (SI unit: kg/(m3·s)). The default is 0 kg/(m3·s).

Volume Force

Use the Volume Force node to specify the force F on the right-hand side of Equation 7-4. It then acts on each fluid element in the specified domains. A common application is to include gravity effects.

t p u + Qbr=


456 | C H A P T E

(7-4)


From the Selection list, choose the domains where the volume force acts on the fluid.


Enter the components of Volume force F (SI unit: N/m3).

Initial Values

The Initial Values node adds initial values for the velocity field and the pressure that can serve as an initial condition for a transient simulation or as an initial guess for a nonlinear solver.




Enter initial values or expressions for the Velocity field u (SI unit: m/s) and the Pressure p (SI unit: Pa). The default values are 0 m/s and 0 Pa, respectively.

p-----

tu u u

p-----+

=

– p +1p----- u u T+ 2

3--- u I–

1– Qbr

p2

---------+

u F+–


Th e F r e e and Po r ou s Med i a F l ow I n t e r f a c e

The Free and Porous Media Flow (fp) interface ( ), found under the Porous Media and

Subsurface Flow branch ( ) when adding a physics interface, is used to compute fluid velocity and pressure fields of single-phase flow where free flow is connected to porous media. The Free and Porous Media Flow interface is used over at least two different domains: a free channel and a porous medium. The physics interface is well suited for transitions between slow flow in porous media, governed by the Brinkman equations, and fast flow in channels described by the Navier-Stokes equations. Fluids with varying density can be included at Mach numbers below 0.3. Also the viscosity of a fluid can vary, for example, to describe non-Newtonian fluids. The physics interface can be used for stationary and time-dependent analyses.

When this physics interface is added, the following default nodes are also added in the Model Builder— Fluid Properties, Wall, and Initial Values. Then, from the Physics toolbar, add a Porous Matrix Properties node to be used on the domain selection corresponding to the porous media, or add other nodes that implement, for example, boundary conditions and volume forces. You can also right-click Free and Porous Media Flow to select physics from the context menu.



The default identifier (for the first physics interface in the component) is fp.


The default setting is to have All domains in the component define the fluid pressure and velocity. To choose specific domains, select Manual from the Selection list.

T H E F R E E A N D P O R O U S M E D I A F L O W I N T E R F A C E | 457

458 | C H A P T E


CompressibilityBy default the physics interface uses the Incompressible flow formulation of the Navier-Stokes and Brinkman equations to model constant density flow. If required, select Compressible flow (Ma<0.3) from the Compressibility list, to account for small variations in the density, typically dependent on the temperature (non-isothermal flow). For compressible flow modeled with this physics interface, the Mach number must be below 0.3.

Neglect Inertial TermSelect the Neglect inertial term in free flow (Stokes flow) check box if the inertial forces are small compared to the viscous forces. This is typical for creeping flow, where

.

Select the Neglect inertial term in porous media flow (Stokes-Brinkman) check box to model flow at very low Reynolds numbers in the porous media, for which the inertial term in the Brinkman equations can be neglected. The physics interface then solves the linear Stokes-Brinkman equations.

Swirl Flow

Re 1«

For 2D axisymmetric models, select the Swirl flow check box to include the swirl velocity component, that is the velocity component in the azimuthal direction. While can be nonzero, there can be no gradients in the direction.

The other sections (Dependent Variables, Discretization, Consistent

Stabilization, Inconsistent Stabilization, and Advanced Settings) are the same as for The Laminar Flow Interface.


• Domain, Boundary, Point, and Pair Nodes for the Free and Porous Media Flow Interface

• Theory for the Free and Porous Media Flow Interface

uu


Domain, Boundary, Point, and Pair Nodes for the Free and Porous Media Flow Interface

The Free and Porous Media Flow Interface has the following domain, boundary, point, and pair nodes, listed in alphabetical order, available from the Physics ribbon toolbar (Windows users), Physics context menu (Mac or Linux users), or right-click to access the context menu (all users).


Fluid Properties

Use the Fluid Properties node to define the fluid material, density, and dynamic viscosity.




• Initial Values

• Mass Source

• Microfluidic Wall Conditions

• Porous Matrix Properties

• Volume Force


• Flow Continuity

• Inlet


• Outlet

• Open Boundary




• Symmetry

• Wall



460 | C H A P T E






The default Fluid material uses the Domain material (the material defined for the domain). Select another material as required.

The default Density (SI unit: kg/m3) uses values From material based on the Fluid

material selection. If User defined is selected, enter another value or expression. The default is 0 kg/m3.

The Dynamic viscosity (SI unit: Pa·s) uses values From material based on the Fluid

material selection. If User defined is selected, enter another value or expression. The default is 0 Pa·s.

Porous Matrix Properties

Use the Porous Matrix Properties node to define which domains contain porous material and to define the porous matrix properties, such as the porosity and permeability in these domains. Right-click to add a Forchheimer Drag subnode.


From the Selection list, choose the domains to solve for porous media flow governed by the Brinkman equations. In the domains not selected, the Free and Porous Media Flow interface solves for laminar flow governed by the Navier-Stokes (or Stokes) equations.

To define the Absolute Pressure, see the settings for the Heat Transfer in Fluids node as described in the COMSOL Multiphysics Reference Manual.


PO R O U S M A T R I X P R O P E R T I E S

The default Porous material uses the Domain material (the material defined for the domain) for the porous matrix. Select another material as required.

PorosityThe default Porosity p (a dimensionless number between 0 and 1) uses values From

material as defined by the Porous material selection. If User defined is selected, enter another value or expression. The default is 0.

PermeabilityThe default Permeability br (SI unit: m2) uses values From material as defined by the Porous material selection. If User defined is selected, choose Isotropic, Diagonal, Symmetric, or Anisotropic from the list and then enter other values or expressions. The components of a permeability in the case that it is a tensor (xx, yy, and so on, representing an anisotropic permeability) are available as fp.kappaxx, fp.kappayy, and so on (using the default interface identifier fp). The defaults is 0 m2.

Source TermEnter a value or expression for an optional mass source (or sink) Source term Qbr (SI unit: kg/(m3·s)). This term accounts for mass deposit and mass creation within domains. The physics interface assumes that the mass exchange occurs at zero velocity.

Effective Mass Transport ParametersA correction factor (Bruggeman, No Correction or User defined) to the mass transport parameters (defined in the Transport Properties node) can be applied for the porous domain. Species diffusivities and mobilities are automatically adjusted by the porous media corrections. If User defined is selected, enter a value or expression for the Conversion factor feff. The default is 1. Species diffusivities and mobilities are automatically adjusted by the porous media corrections.

Volume Force

The Volume Force node specifies the force F on the right-hand side of the Navier-Stokes or Brinkman equations, depending on whether the Porous Matrix Properties node is active for the domain. Use it, for example, to incorporate the effects of gravity in a model.


From the Selection list, choose the domains where the volume force acts on the fluid.


462 | C H A P T E


Enter the components of the Volume force F (SI unit: N/m3).

Forchheimer Drag

Right-click the Porous Matrix Properties node to add a Forchheimer Drag subnode to be used on the domain selection that corresponds to the porous medium. For the Brinkman equations the drag of the fluid on the porous matrix is proportional to the flow velocity, in the same way as for Darcy’s law. Add a Forchheimer drag, proportional to the square of the fluid velocity, as required.


From the Selection list, choose the domains on which to apply the drag. The default is to add All domains.

F O R C H H E I M E R D R A G

Enter a value for the Forchheimer coefficient F (SI unit: kg/m4).

Initial Values

The Initial Values node adds initial values for the velocity field and the pressure that can serve as an initial condition for a transient simulation or as an initial guess for a nonlinear solver.




Enter initial values or expressions for the Velocity field u (SI unit: m/s) and for the Pressure p (SI unit: Pa). The default values are 0 m/s and 0 Pa, respectively.

Microfluidic Wall Conditions

Use the Microfluidic Wall Conditions node to add boundary conditions to the moving wall and specify whether to use viscous slip or thermal creep.



From the Selection list, choose the boundaries on which to apply the conditions.


If this node is selected from the Pairs menu, choose the pair on which to apply the conditions. An identity pair has to be created first. Ctrl-click to deselect.


The default Boundary condition for the wall is Slip velocity. Enter values or expressions for the components of the Velocity of moving wall uw (SI unit: m/s).

Use Viscous SlipSelect the Use viscous slip check box to define the slip length:

• The default is User defined. Enter a value for Ls (SI unit: m). The default is 1e-7 m.

• Select Maxwell’s model to calculate it from:

• and then enter values or expressions for the Tangential momentum accommodation

coefficient (TMAC) av (dimensionless) (the default is 0.9) and the Mean free path (SI unit: m) (the default is 1e-6 m).

Use Thermal CreepSelect the Use thermal creep check box to activate the thermal creep component of the boundary condition. Enter the fluid’s Temperature T (SI unit: K) and the Thermal slip

coefficient T. The default temperature is 293.15 K and the default thermal slip coefficient is 0.75.

Ls2 v–

v--------------- =

If you also have a license for the MEMS Module, an additional Boundary

condition option Electroosmotic velocity is available. This is described in the MEMS Module User’s Guide.


464 | C H A P T E

Th e Two -Pha s e Da r c y ’ s L aw I n t e r f a c e

The Two-Phase Darcy’s Law (tpdl) interface ( ), found under the Porous Media and

Subsurface Flow branch ( ) when adding a physics interface, is used to simulate fluid flow through interstices in a porous medium. It solves Darcy's law for the total pressure and the transport of the fluid content for one fluid phase. The physics interface can be used to model low velocity flows or media where the permeability and porosity are very small, for which the pressure gradient is the major driving force and the flow is mostly influenced by the frictional resistance within the pores. The physics interface can be used for stationary and time-dependent analyses.

The main feature is the Fluids and Matrix Properties node, which provides an interface for defining the two immiscible fluids properties along with the porous medium properties. The interface is available in 2D, 2D axisymmetric, and 3D.

When this physics interface is added, the following default nodes are also added in the Model Builder— Fluids and Matrix Properties, No Flow (the default boundary condition), and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and mass sources. You can also right-click Two-Phase

Darcy’s Law to select physics from the context menu.



The default identifier (for the first physics interface in the component) is tpdl.


The default setting is to have All domains in the component define the pressure and the Darcy’s law and continuity equations. To choose specific domains, select Manual from the Selection list.



The dependent variables (field variables) are the Pressure and Fluid content 1. The name can be changed but the names of fields and dependent variables must be unique within a component.


The Discretization settings are the same as for The Brinkman Equations Interface.

Domain, Boundary, and Pair Nodes for the Two-Phase Darcy’s Law Interface

The Two-Phase Darcy’s Law Interface has these domain, boundary, and pair nodes, listed in alphabetical order, available from the Physics ribbon toolbar (Windows users), Physics context menu (Mac or Linux users), or right-click to access the context menu (all users).


• Domain, Boundary, and Pair Nodes for the Two-Phase Darcy’s Law Interface

• Theory for the Two-Phase Darcy’s Law Interface



• Inlet

• Initial Values

• Mass Flux

• No Flux (the default boundary condition)

• Outlet

• Pressure and Saturation


T H E TW O - P H A S E D A R C Y ’ S L A W I N T E R F A C E | 465

466 | C H A P T E


The Fluids and Matrix Properties node adds Equation 7-5 and Equation 7-6 and defines properties including density, dynamic viscosity, relative permeability, and porosity.

(7-5)

(7-6)



C A P I L L A R Y M O D E L

From the list, choose Isotropic to define a scalar or Diagonal, Symmetric, or Anisotropic to enter a tensor, and enter values or expressions in the Capillary diffusion Dc (SI unit: m2/s) field(s).

F L U I D 1 P R O P E R T I E S

Select the material to use for fluid 1. Select Domain material (the default value) from the Fluid 1 list to use the material defined for the domain. Select another material from the Fluid 1 list to use that material’s properties.

Density Fluid 1The default Density 1 (SI unit: kg/m3) uses values From material based on the Fluid 1 selection. If User defined is selected, enter another value or expression. The default is 0 kg/m3.




t p u + 0=

t c1p c1u + Dcc1=


Dynamic Viscosity Fluid 1Select a Dynamic viscosity 1(SI unit: Pa·s). The default uses values From material as defined by the Fluid 1 selection. If User defined is selected, enter another value or expression. The default is 0 Pa·s.

Relative Permeability Fluid 1Select the Relative permeability r1(a dimensionless number between 0 and 1) for fluid 1. The default is 1.

F L U I D 2 P R O P E R T I E S

Select the material to use for the fluid 2. Select Domain material (the default value) from the Fluid 2 list to use the material defined for the domain. Select another material from the Fluid 2 list to use that material’s properties.

Density Fluid 2The default Density 2 (SI unit: kg/m3) uses values From material based on the Fluid 2 selection. If User defined is selected, enter another value or expression. The default is 0 kg/m3.

Dynamic Viscosity Fluid 2Select a Dynamic viscosity 2(SI unit: Pa·s). The default uses values From material as defined by the Fluid 2 selection. If User defined is selected, enter another value or expression. The default is 0 Pa·s.

Relative Permeability Fluid 2Select the Relative permeability r2(a dimensionless number between 0 and 1) for fluid 2. The default is 1.

M A T R I X P R O P E R T I E S

Select the material to use as porous matrix. Select Domain material (the default) from the Porous material list to use the material defined for the porous domain. Select another material from the Porous material list to use that material’s properties.

PorosityThe default Porosity p (a dimensionless number between 0 and 1) uses the value From

material, defined by the Porous material selection. If User defined is selected, enter another value or expression The default is 0.

PermeabilityThe default Permeability (SI unit: m2) uses the value From material, as defined by the Porous material selection. If User defined is selected, choose Isotropic to define a scalar


468 | C H A P T E

or Diagonal, Symmetric or Anisotropic to enter a tensor and enter other values or expressions in the field or matrix.

Initial Values

The Initial Values node adds initial values for the pressure and the saturation of fluid 1 (that is, the fraction of that fluid inside the pore space) that can serve as initial conditions for a transient simulation or as an initial guess for a nonlinear solver.




Enter a value or expression for the initial value of the Pressure p (SI unit: Pa). The default value is 0 Pa. Enter a value or expression for the initial value of the Saturation

fluid 1 s1 (a dimensionless number between 0 and 1). The default value is 0. The saturation of Fluid 2 is then calculated as s2 = 1 s1.

No Flux

The No Flux node is the default boundary condition stating that there is no flow across impervious boundaries. The mathematical formulation is:

where n is the vector normal to the boundary.

Also, the No Flux boundary enforces a zero gradient condition for the fluid content across the selected boundary



n u 0=

n c1 0=


Pressure and Saturation

Use the Pressure and Saturation node to specify the pressure and fluid volume fractions on a boundary. In many cases the distribution of pressure and saturation are known as numbers, distributions, or expressions involving time, t, for example.





P R E S S U R E A N D S A T U R A T I O N

Enter a value or expression for the Pressure p (SI unit: Pa), and for the Saturation

fluid 1 s1 (a dimensionless number between 0 and 1). The default values are 0. The saturation of Fluid 2 is then calculated as s2 = 1 s1.

Mass Flux

Use the Mass Flux node to specify the mass flux into or out of the model domain through some of its boundaries. It is often possible to determine the mass flux from the pumping rate or from measurements. With this boundary condition, positive values correspond to flow into the model domain

where N0 is a value or expression for the inward (or outward) Darcy’s flux that is specified.


From the Selection list, choose the boundaries on which to apply the flux.

M A S S F L U X A N D S A T U R A T I O N

Enter a value or expression for the Inward mass flux N0 (SI unit: kg/(m2·s)). A positive value of N0 represents an inward mass flux, whereas a negative value represents an outward mass flux.

Enter a value or expression for the Saturation fluid 1 s1 (a dimensionless number between 0 and 1) in the mass flux. The default value is 0.

n– u N0=


470 | C H A P T E

Inlet

The Inlet node adds a boundary condition for the inflow (or outflow) perpendicular (normal) to the boundary:

where U0 is a specified value or expression for the inward (or outward) Darcy’s velocity. A positive value of the velocity U0 corresponds to flow into the model domain whereas a negative value represents an outflow.


From the Selection list, choose the boundaries on which to define the Inlet.

I N L E T

Enter a value or expression for the Normal inflow velocity U0 (SI unit: m/s). A positive value of U0 represents an inflow velocity, whereas a negative value represents an outward velocity.

Enter a value or expression for the Saturation fluid 1 s1 (a dimensionless number between 0 and 1). The default value is 0.

Outlet

The Outlet node adds a boundary condition for the outflow perpendicular (normal) to the boundary:

where Dc is the capillary diffusion (SI unit: m2/s) and c1 = s11 is the fluid 1 content (SI unit: kg/m3). This means that the normal gradient of fluid saturation does not change through this boundary.


From the Selection list, choose the boundaries on which to define the Outlet.

P R E S S U R E

Enter a value or expression for the Pressure p. The default value is 0.

n– u s11 s21+ U0=

n– Dcc1 0=


Th eo r y f o r t h e Da r c y ’ s L aw I n t e r f a c e

The Darcy’s Law Interface theory is described in this section.

In a porous medium, the global transport of momentum by shear stresses in the fluid is often negligible, because the pore walls impede momentum transport between fluid occupying different pores. In most applications, a detailed description, down to the resolution of every pore, is not practical. A homogenization of the porous and fluid media into a single medium is a common alternative approach. Darcy’s law, together with the continuity equation and equation of state for the pore fluid (see Darcy’s Law—Equation Formulation), provides a complete mathematical model suitable for a wide variety of applications involving porous media flows when the pressure gradient is the major driving force.

Darcy’s Law—Equation Formulation

Darcy’s law states that the velocity field is determined by the pressure gradient, the fluid viscosity, and the structure of the porous medium:

(7-7)

In this equation, (SI unit: m2) denotes the permeability of the porous medium, (SI unit: kg/(m·s)) the dynamic viscosity of the fluid, p (SI unit: Pa) the pressure, and u (SI unit: m/s) the Darcy velocity.

The Darcy’s Law interface combines Darcy’s law with the continuity equation:

(7-8)

In the above equation, (SI unit: kg/m3) is the density of the fluid, (dimensionless) is the porosity, and Qm (SI unit: kg/(m3·s)) is a mass source term. Porosity is defined as the fraction of the control volume that is occupied by pores. Thus, the porosity can vary from zero for pure solid regions to unity for domains of free flow.

If the Darcy’s Law interface is coupled to an energy balance, then the fluid density can be a function of the temperature, pressure, and composition (for mixture flows). For gas flows in porous media, the relation is given by the ideal gas law:

u ---p–=

t u + Qm=

T H E O R Y F O R T H E D A R C Y ’ S L A W I N T E R F A C E | 471

472 | C H A P T E

(7-9)

where R= 8.314 J/(mol·K) is the universal gas constant, M (SI unit: kg/mol) is the molecular weight of the gas, and T (SI unit: K) is the temperature.

pMRT---------=


Th eo r y f o r t h e B r i n kman Equa t i o n s I n t e r f a c e

The Brinkman Equations Interface theory is described in this section:

• About the Brinkman Equations

• Brinkman Equations Theory

• References for the Brinkman Equations Interface

About the Brinkman Equations

The Brinkman equations describe fluids in porous media for which the momentum transport within the fluid due to shear stresses is of importance. This mathematical model extends Darcy’s law to include a term that accounts for the viscous transport in the momentum balance, and it treats both the pressure and the flow velocity vector as independent variables. Use the Free and Porous Media Flow interface to model combinations of porous media and free flow domains. These types of problems are often encountered in applications such as monolithic reactors and fuel cells.

In porous domains, the flow variables and fluid properties are defined at any point inside the medium by means of averaging of the actual variables and properties over a certain volume surrounding the point. This control volume must be small compared to the typical macroscopic dimensions of the problem, but it must be large enough to contain many pores and solid matrix elements.

Porosity is defined as the fraction of the control volume that is occupied by pores. Thus, the porosity can vary from zero for pure solid regions to unity for domains of free flow.

The physical properties of the fluid, such as density and viscosity, are defined as intrinsic volume averages that correspond to a unit volume of the pores. Defined this way, they present the relevant physical parameters that can be measured experimentally,

The Free and Porous Media Flow Interface

T H E O R Y F O R T H E B R I N K M A N E Q U A T I O N S I N T E R F A C E | 473

474 | C H A P T E

and they are assumed to be continuous with the corresponding parameters in the adjacent free flow.

The flow velocity is defined as a superficial volume average, and it corresponds to a unit volume of the medium including both the pores and the matrix. It is sometimes called the Darcy velocity, defined as the volume flow rate per unit cross section of the medium. Such a definition makes the velocity field continuous across the boundaries between porous regions and regions of free flow.

Brinkman Equations Theory

The dependent variables in the Brinkman equations are the Darcy velocity and the pressure. The flow in porous media is governed by a combination of the continuity equation and the momentum equation, which together form the Brinkman equations:

(7-10)

(7-11)

In these equations:

• (SI unit: kg/(m·s)) is the dynamic viscosity of the fluid

• u (SI unit: m/s) is the velocity vector

• (SI unit: kg/m3) is the density of the fluid

• p (SI unit: Pa) is the pressure

• p is the porosity

• (SI unit: m2) is the permeability tensor of the porous medium, and

• Qbr (SI unit: kg/(m3·s)) is a mass source or mass sink

Influence of gravity and other volume forces can be accounted for via the force term F (SI unit: kg/(m2·s2)).

When the Neglect inertial term (Stokes-Brinkman) check box is selected, the term (u ·up) on the left-hand side of Equation 7-4 is disabled.

t p u + Qbr=

p-----

tu u u

p-----+

=

– p +1p----- u u T+ 2

3--- u I–

1– Qbr

p2

---------+

u F+–


The mass source, Qbr, accounts for mass deposit and mass creation within the domains. The mass exchange is assumed to occur at zero velocity.

The Forchheimer drag option, F (SI unit: kg/m4), adds a viscous force proportional to the square of the fluid velocity, FFFuu, to the right-hand side of Equation 7-4.

In case of a flow with variable density, Equation 7-3 and Equation 7-4 must be solved together with the equation of state that relates the density to the temperature and pressure (for instance the ideal gas law).

For incompressible flow, the density stays constant in any fluid particle, which can be expressed as

and the continuity equation (Equation 7-3) reduces to

References for the Brinkman Equations Interface

1. D. Nield and A. Bejan, Convection in Porous Media, 3rd ed., Springer, 2006.

2. M. Le Bars and M.G. Worster, “Interfacial Conditions Between a Pure Fluid and a Porous Medium: Implications for Binary Alloy Solidification,” J. of Fluid Mechanics, vol. 550, pp. 149–173, 2006.

t p u + 0=

u Qbr=

T H E O R Y F O R T H E B R I N K M A N E Q U A T I O N S I N T E R F A C E | 475

476 | C H A P T E

Th eo r y f o r t h e F r e e and Po r ou s Med i a F l ow I n t e r f a c e

The Free and Porous Media Flow Interface uses the Navier-Stokes equations to describe the flow in open regions, and the Brinkman equations to describe the flow in porous regions.

The same fields, u and p, are solved for in both the free flow domains and in the porous domains. This means that the pressure in the free fluid and the pressure in the pores is continuous over the interface between a free flow domain and a porous domain. It also means that continuity is enforced between the fluid velocity in the free flow and the Darcy velocity in the porous domain. This treatment is one of several possible models for the physics at the interface. Examples of other models can be found in Ref. 1.

The continuity in u and p implies a stress discontinuity at the interface between a free-flow domain and a porous domain. The difference corresponds to the stress absorbed by the rigid porous matrix, which is a consequence implicit in the formulations of the Navier-Stokes and Brinkman equations.

Reference for the Free and Porous Media Flow Interface

1. M.L. Bars and M.G. Worster, “Interfacial Conditions Between a Pure Fluid and a Porous Medium: Implications for Binary Alloy Solidification,” J. Fluid Mech., vol. 550, pp. 149–173, 2006.


Th eo r y f o r t h e Two -Pha s e Da r c y ’ s L aw I n t e r f a c e

The Two-Phase Darcy’s Law interface theory about the Darcy’s Law—Equation Formulation is described in this section.

Darcy’s Law—Equation Formulation

Darcy’s law states that the velocity field is determined by the pressure gradient, the fluid viscosity, and the structure of the porous medium. According to Darcy’s law, the velocity field is given by

(7-12)

In this equation:

• u (SI unit: m/s) is the Darcy velocity vector

• (SI unit: m2) is the permeability of the porous medium

• (SI unit: Pa·s) is the fluid’s dynamic viscosity

• p (SI unit: Pa) is the fluid’s pressure, and

• (SI unit: kg/m3) is its density.

Here the permeability, , represents the resistance to flow over a representative volume consisting of many solid grains and pores.

The average density and average viscosity are calculated from the fluids properties and the saturation of each fluid

(7-13)

(7-14)

(7-15)

The Two-Phase Darcy’s Law interface combines Darcy’s law with the continuity equation

u --- p–=

1 s1 s2+=

s11 s22+=

1--- s1

r11-------- s2

r22--------+=

T H E O R Y F O R T H E TW O - P H A S E D A R C Y ’ S L A W I N T E R F A C E | 477

478 | C H A P T E

(7-16)

and the transport equation for the fluid content c1 = s11

(7-17)

here, p is the porosity, defined as the fraction of the control volume that is occupied by pores, and Dc (SI unit: m2/s) is the capillary diffusion coefficient.

Inserting Darcy’s law (Equation 7-12) into the continuity equation (Equation 7-5) produces the generalized governing equation

(7-18)

If either of the fluids is compressible, its density must be related to the pressure (for instance using the ideal gas law).

t p u + 0=

t c1p c1u + Dcc1=

t p +

--- p– 0=


8

C h e m i c a l S p e c i e s T r a n s p o r t

The interfaces in the Chemical Species Transport branch ( ) when adding a physics interface accommodate all types of material transport that can occur through diffusion and convection. The section Modeling Chemical Species Transport helps you choose the best one to start with.

In this chapter:

• The Transport of Diluted Species Interface

• The Transport of Concentrated Species Interface

• The Reacting Flow Interfaces

• The Reacting Flow in Porous Media Interfaces

• Theory for the Transport of Diluted Species Interface

• Theory for the Transport of Concentrated Species Interface

• Theory for the Reacting Flow Interfaces

• Theory for the Reacting Flow in Porous Media Interfaces

479

480 | C H A P T E

Mode l i n g Ch em i c a l S p e c i e s T r a n s po r t

In this section:



• Adding a Chemical Species Transport Interface


The behavior of chemical reactions in real environments is often not adequately described by the assumptions of perfectly mixed or controlled environments. This means that the transport of material through both time and space need to be considered. Physics interfaces in the Chemical Species Transport branch accommodate all types of material transport that can occur through diffusion and convection—either alone or in combination with one another. The branch includes physics interfaces solving equations for diluted as well as concentrated mixtures, where the species propagation can occur in solids, free flowing fluids, or through porous media.

The Transport of Diluted Species Interface ( ) is applicable for solutions (either fluid or solid) where the transported species have concentrations at least one order of magnitude less than the solvent. The settings for this interface can be chosen to simulate chemical species transport through diffusion (Fick’s law) and convection (when coupled to fluid flow).

The Transport of Concentrated Species Interface ( ) is used for modeling transport within mixtures where no single component is clearly dominant. Often the concentrations of the participating species are of the same order of magnitude, and the molecular effects of the respective species on each other need to be considered. This interface supports transport through Fickian diffusion, a mixture average diffusion model, and as described by the Maxwell-Stefan equations.

R E A C T I N G F L O W

The Reacting Flow, Laminar Flow Interface ( ) combines the functionality of the Laminar Flow and Transport of Concentrated Species interfaces. Using this physics interface the mass and momentum transport in a reacting fluid can be modeled from a single physics interface, with the couplings between the velocity field and the mixture

R 8 : C H E M I C A L S P E C I E S TR A N S P O R T

density set up automatically. This physics interface is applicable for fluid flow in the laminar regime.

The Reacting Flow, Turbulent Flow, k- Interface ( ) combines the functionality of the Turbulent Flow, k- and Transport of Concentrated Species interfaces. Using this physics interface, the mass and momentum transport in reacting turbulent fluid flow can be modeled from a single physics interface, with the couplings between the velocity field and the mixture density set up automatically. The physics interface solves for the mean velocity and pressure fields, together with an arbitrary number of mass fractions. The fluid-flow turbulence is modeled using the standard k- model, solving for the turbulent kinetic energy k and the rate of dissipation of turbulent kinetic energy .

The Reacting Flow, Turbulent Flow, k- Interface ( ) combines the functionality of the Turbulent Flow, k- and Transport of Concentrated Species interfaces. Using this physics interface, the mass and momentum transport in reacting turbulent fluid flow can be modeled from a single physics interface, with the couplings between the velocity field and the mixture density set up automatically. The physics interface solves for the mean velocity and pressure fields, together with an arbitrary number of mass fractions. The fluid-flow turbulence is modeled using the Wilcox revised k- model, solving for the turbulent kinetic energy k and the rate of specific dissipation of turbulent kinetic energy .

The Reacting Flow, Turbulent Flow, Low Re k-Interface ( ) combines the functionality of the Turbulent Flow, Low Re k- and Transport of Concentrated Species interfaces. Using this physics interface, the mass and momentum transport in reacting turbulent fluid flow can be modeled from a single physics interface, with the couplings between the velocity field and the mixture density set up automatically. The physics interface solves for the mean velocity and pressure fields, together with an arbitrary number of mass fractions. The fluid-flow turbulence is modeled using the AKN low-Reynolds number k- model, solving for the turbulent kinetic energy k and the rate of dissipation of turbulent kinetic energy . The physics interface also includes a wall distance equation that solves for the reciprocal wall distance.

R E A C T I N G F L O W I N PO R O U S M E D I A

The Reacting Flow in Porous Media (rfds) Interface ( ), merges the functionality of the Transport of Diluted Species and the Free and Porous Media Flow interfaces into a multiphysics interface. This way, coupled mass and momentum transport in free and porous media can be modeled from a single physics interface, with the component coupling for the velocity field set up automatically. In addition, the effective transport

M O D E L I N G C H E M I C A L S P E C I E S TR A N S P O R T | 481

482 | C H A P T E

coefficients in a porous matrix domain can be derived based on the corresponding values in for a non-porous domain.

The Reacting Flow in Porous Media (rfcs) Interface ( ), combines the Transport of

Concentrated Species and the Free and Porous Media Flow interfaces. This means that mass and momentum transport can be modeled from a single physics interface, with the couplings between the velocity field and the mixture density set up automatically. Also, the effective transport coefficients in a porous matrix domain are derived based on the corresponding values for a non-porous domain. This physics interface is applicable for fluid flow in the laminar regime.


When you are simulating applications that can be described by the material transport interfaces in the Chemical Species Transport branch, there is often a need to couple the material transport to other physics. Convection is often the cause of the material transport, so couplings to fluid-flow interfaces are required. The CFD Module includes interfaces for laminar flow and porous media flow as well as more advanced descriptions of fluid flow, such as turbulent and multiphase flow.

Moreover, most chemical reactions or other types of material processing, such as casting, either require or produce heat, which in turn affects both the reaction and other physical processes connected to the system. This module includes physics interfaces for heat transfer through conduction and convection as well as through porous media. More extensive description of heat transfer, such as surface-to-surface radiation, can be found in the Heat Transfer Module.

Finally, COMSOL Multiphysics supports simulations of electrostatics or DC-based physical phenomena, even when the conductivity is nonlinear. If the electric field is AC/DC in nature, or if your system is affected by electromagnetic waves, then the AC/DC Module and RF Module include appropriate physics interfaces for such phenomena. Furthermore, some applications of electrochemical reactions, particularly in electrochemical power source applications, are better handled by the Batteries & Fuel Cells Module.


Adding a Chemical Species Transport Interface

A chemical species transport physics interface can be added when first creating a new model, or at any time during the modeling process.

1 To add a physics interface:

- Select New to open the Model Wizard, after selecting a space dimension, go to the Select Physics page.

- From the Home ribbon (Windows users) or the Model Toolbar (Mac and Linux users), click Add Physics ( ). Or right-click the Component node in the Model

Builder and select Add Physics. Go to the Add Physics window.

2 Under Chemical Species Transport, navigate to the interface to add and double-click it.

There are other ways to add an interface depending on whether you are in the Model Builder or Add Physics window:

- In the Model Wizard, click Add or right-click and select Add Physics ( ). The physics interface displays under Added physics.

- In the Add Physics window, click Add to Component ( ) or right-click and select Add to Component.

3 Specify the number of species (concentrations or mass fractions) and the names:

- In the Model Wizard, on the Review Physics page under Dependent Variables.

- In the Add Physics window, click to expand the Dependent Variables section.

- After adding the physics interface, you can also edit this information—click the node in the Model Builder, then, on the settings window under Dependent

Variables, specify the information.

4 Continue by adding more interfaces and specifying the number of species (concentrations or mass fractions) that are to be simulated in a mass transport physics interface when adding that interface.

5 In the Dependent Variables section, enter the Number of species. To add a single species, click the Add Concentration button ( ) underneath the table or enter a

Creating a New Model in the COMSOL Multiphysics Reference Manual

M O D E L I N G C H E M I C A L S P E C I E S TR A N S P O R T | 483

484 | C H A P T E

value into the Number of species field. Click the Remove Concentration button ( ) underneath the table if required.

The Transport of Concentrated Species interface needs to contain at least two species (the default). Also edit the strings or names directly in the table. The names must be unique for all species (and all other dependent variables) in the model.


Th e T r a n s po r t o f D i l u t e d S p e c i e s I n t e r f a c e

The Transport of Diluted Species (chds) interface ( ), found under the Chemical

Species Transport branch ( ) when adding a physics interface, is used to compute the concentration field of a dilute solute in a solvent. Transport and reactions of the species dissolved in a gas, liquid or solid can be computed. The driving forces for transport can be diffusion by Fick's law, convection, when coupled to fluid flow, and migration, when coupled to an electric field.

The physics interface supports simulation of transport by convection and diffusion in 1D, 2D, and 3D as well as for axisymmetric models in 1D and 2D. The dependent variable is the molar concentration, c. Modeling multiple species transport is possible, whereby the physics interface solves for the molar concentration, ci, of each species i.

When this physics interface is added, the following default nodes are also added in the Model Builder—Convection and Diffusion, No Flux, and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and rate expression terms. You can also right-click Transport of Diluted Species to select physics from the context menu.



The default identifier (for the first physics interface in the component) is chds.

T H E TR A N S P O R T O F D I L U T E D S P E C I E S I N T E R F A C E | 485

486 | C H A P T E



TR A N S P O R T M E C H A N I S M S

Diffusion is always included. By default, the Convection check box is selected under Additional transport mechanisms. Click to clear the check box to only model diffusion. The Dynamic Transport Feature Node name changes according to the selection made.


Add or remove species in the model and also change the names of the dependent variables that represent the species concentrations.

Enter the Number of species. Use the Add concentration ( ) and Remove

concentration ( ) buttons as required.


To display these sections, click the Show button ( ) and select Stabilization. Any settings unique to this physics interface are listed below.

• When the Crosswind diffusion check box is selected, a weak term that reduces spurious oscillations is added to the transport equation. The resulting system is nonlinear. There are two options for the Crosswind diffusion type:

- Do Carmo and Galeão—the default option. This type of crosswind diffusion reduces undershoots and overshoots to a minimum but can in rare cases give equation systems that are difficult to fully converge.

- Codina. This options is less diffusive compared to the Do Carmo and Galeão option but can result in more undershoots and overshoots. It is also less effective

There might be a domain in the model that is not described by mass transfer, such as a reactor’s solid wall. In this case, remove that domain selection from here.

The dependent variable is Concentration c. The species are dependent variables, and their names must be unique with respect to all other dependent variables in the component.


for anisotropic meshes. The Codina option activates a text field for the Lower

gradient limit glim (SI unit: mol/m4). It defaults to 0.1[mol/m^3)/chds.helem, where chds.helem is the local element size.

• For both consistent stabilization methods, select an Equation residual. Approximate

residual is the default setting and it means that derivatives of the diffusion tensor components are neglected. This setting is usually accurate enough and is computationally faster. If required, select Full residual instead.


To display this section, click the Show button ( ) and select Advanced Physics Options. Normally these settings do not need to be changed. Select a Convective term—Non-conservative form (the default) or Conservative form. The conservative formulation should be used for compressible flow.



• Select an element order (shape function order) for the Concentration—Linear (the default), Quadratic, Cubic, or Quartic.

• The Compute boundary fluxes check box is selected by default so that COMSOL computes predefined accurate boundary flux variables. The computation of the following boundary flux variables changes so that:

- ndflux_c (where c is the dependent variable for the concentration) is the normal diffusive flux and corresponds to the accurate boundary flux when diffusion is the only contribution to the flux term. When transport in an electric field is included or when the conservative form of the equation is used, ndflux_c is instead computed directly from the dependent variable.

- ntflux_c (where c is the dependent variable for the concentration) is the normal total flux and corresponds to the accurate boundary flux plus additional transport terms, for example, the convective flux when you use the non-conservative form.

If you clear the Compute boundary fluxes check box, COMSOL instead compute the flux variables from the dependent variables using extrapolation.


488 | C H A P T E


• Specify the Value type when using splitting of complex variables—Real (the default) or Complex.

Domain, Boundary, and Pair Nodes for the Transport of Diluted Species Interface

The Transport of Diluted Species Interface has the following domain, boundary, and pair nodes, listed in alphabetical order, available from the Physics ribbon toolbar (Windows users), Physics context menu (Mac or Linux users), or right-click to access the context menu (all users).

• Convective Term Formulation


• Domain, Boundary, and Pair Nodes for the Transport of Diluted Species Interface

• Theory for the Transport of Diluted Species Interface

• Effective Diffusivity in Porous Materials: model library path COMSOL_Multiphysics/Diffusion/effective_diffusivity

• Thin-Layer Diffusion: model library path COMSOL_Multiphysics/

Diffusion/thin_layer_diffusion



Dynamic Transport Feature Node

This node is dynamically dependent on the Transport Mechanisms chosen in the Transport of Diluted Species interface (convection and diffusion), and includes only the input fields required by the activated transport mechanisms. It has all the equations defining transport of diluted species as well as inputs for the material properties. The name of the node also changes the activated transport mechanisms, and can be one of the following:

• Diffusion

• Concentration

• Dynamic Transport Feature Node

• Flux


• Inflow

• Initial Values



• Open Boundary

• Outflow

• Periodic Condition


• Reactions

• Symmetry

• Thin Diffusion Barrier

• Thin Impermeable Barrier

• Turbulent Mixing

For axisymmetric models, COMSOL Multiphysics takes the axial symmetry boundaries (at r0) into account and automatically adds an Axial Symmetry node that is valid on boundaries representing the symmetry axis.




A boundary pair occurs when the solutions on two separate surfaces within a simulation are related (such as when two components in an assembly are in contact).


490 | C H A P T E

• Convection and Diffusion


For a default node, the setting inherits the selection from the parent node, and cannot be edited; that is, the selection is automatically made and is the same as for the physics interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific domains to define material properties and other parameters that govern the transport equations, or select All domains as required.


If transport by convection is active the velocity field of the solvent needs to be specified as a model input.

Select the source of the Velocity field u:

• Select User defined to enter values or expressions for the velocity components (SI unit: m/s) in the fields or table that appears below the drop-down menu. This input option is always available.

• Select the velocity field solved by a fluid flow interface that has also been added to the model. These physics interfaces have their own tags or Interface Identifier, and they are available to choose in the Velocity field drop-down menu, if they are active in the domains. This lists the variable names related to the fluid flow interface in the table underneath the drop-down menu.


The Global coordinate system is selected by default. The Coordinate system list contains any additional coordinate systems that the model includes (except boundary coordinate systems). The coordinate system is used to define directions for an anisotropic diffusion coefficient.

Also right-click the Convection and Diffusion node to add the Turbulent Mixing subnode.

If there is more than one type of domain, with different material properties, it might be necessary to deselect some of the domains. These are then defined in an additional Convection and Diffusion node.


D I F F U S I O N

Select an option from the Bulk material list. The default is None.

Under Diffusion coefficient select the appropriate scalar or tensor type to describe the diffusion transport—Isotropic, Diagonal, Symmetric, or Anisotropic—then enter the values for Dc (SI unit: m2/s) in the corresponding field.

Turbulent Mixing

Right-click a Dynamic Transport Feature Node to add a Turbulent Mixing subnode. This option is available if Convection is selected as a transport mechanism. Use this node to account for the turbulent mixing caused by the eddy diffusivity. An example is when the specified velocity field corresponds to a RANS solution.


From the Selection list, choose the domains in which to add the turbulent mixing. The default is to add it in All domains.

TU R B U L E N T M I X I N G

Some physics interfaces provide the turbulent kinematic viscosity, and these appear as options in the Turbulent kinematic viscosity T (SI unit: m2/s) list. The list always contains the User defined option where any value or expression can be entered.

The default Turbulent Schmidt number ScT is 0.71 (dimensionless). Enter another value or expression as required.

Initial Values

The Initial Values node specifies the initial values for the concentration of each species. These serve as an initial guess for a stationary solver or as initial conditions for a transient simulation.


For a default node, the setting inherits the selection from the parent node, and cannot be edited; that is, the selection is automatically made and is the same as for the physics

About Turbulent Mixing


492 | C H A P T E

interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific domains or select All domains as required.


Enter a value or expression for the initial value of the Concentration or concentrations ci. The default value is 0 mol/m3.

Line Mass Source

The Line Mass Source feature models mass flow originating from a tube region with infinitely small radius.

S E L E C T I O N

The Line Mass Source feature is available for all dimensions, but the applicable selection differs between the dimensions.

If there are several types of domains with different initial values defined, it might be necessary to remove some domains from the selection. These are then defined in an additional Initial Values node.


For the Reacting Flow in Porous Media, Diluted Species interface, which is available with the CFD Module, Chemical Reaction Engineering Module, or Batteries & Fuel Cells Module, the Line Mass Source node is available in two versions, one for the fluid flow (Fluid Line Source) and one for the species (Species Line Source).

MODEL DIMENSION APPLICABLE GEOMETRICAL ENTITY

2D Points

2D Axisymmetry Points not on the symmetry axis and the symmetry axis

3D Edges


S P E C I E S S O U R C E

Enter the source strength, , for each species (SI unit: molm·s)). A positive value results in species injection from the line into the computational domain, and a negative value means that the species is removed from the computational domain.

Line sources located on a boundary affect the adjacent computational domains. This effect makes the physical strength of a line source located in a symmetry plane twice the given strength.

Reactions

Use the Reactions node to account for the consumption or production of species. Define the rate expression as required, which display on the right-hand side of the species transport equations in the Convection and Diffusion node. In other words, these two nodes are integrated.


From the Selection list, choose the domains on which to define rate expression that govern the source term in the transport equations.

R E A C T I O N S

Add a rate expression, Rc (SI unit: mol/(m3·s)), for the species to be solved for. Enter a value or expression in the field .

No Flux

The No Flux node, which is the default boundary condition on exterior boundaries, represents boundaries where no mass flows in or out of the boundaries. Hence, the total flux is zero:

q· l,c

Mass Sources for Species Transport in the COMSOL Multiphysics Reference Manual

If there are several types of domains, with subsequent and different reactions occurring within them, it might be necessary to remove some domains from the selection. These are then defined in an additional Reactions node.


494 | C H A P T E


For a default node, the setting inherits the selection from the parent node, and cannot be edited; that is, the selection is automatically made and is the same as for the physics interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific boundaries or select All boundaries as required.

N O F L U X

Select Apply for all species (the default) to specify that the boundary is completely impervious for all species. If Apply for... is selected, click to select the check box for the species to specify the condition.

Concentration

The Concentration node adds a boundary condition for the species concentration. For example, a c = c0 condition specifies the concentration of species c.





C O N C E N T R A T I O N

Specify the concentration for each species individually. Select the check box for the Species to specify the concentration, and then enter a value or expression in the corresponding field. To use another boundary condition for a specific species, click to clear the check box for the concentration of that species.




n– cu Dc– N0=


Flux

The Flux node can be used to specify the total species flux across a boundary. The total flux of species c is defined accordingly:

where N0 is an arbitrary user-specified flux expression (SI unit: mol/(m2·s)). For example, N0 can represent a flux from or into a much larger surrounding environment, a phase change, or a flux due to chemical reactions.

When diffusion is the only transport mechanism present, the flux condition is extended to include a mass transfer term to describe flux into a surrounding environment:

where kc is a mass transfer coefficient (SI unit: m/s), and cb is the concentration (SI unit: mol/m3) in the surroundings of the modeled system (the bulk concentration). The mass transfer coefficient (to be specified) is often given by boundary-layer theory.



I N W A R D F L U X

Specify the flux of each species individually. Select the check box for the Species to specify the Inward flux N0,c (SI unit: mol/(m2·s)), and enter a value or expression in the corresponding field. To use another boundary condition for a specific species, click to clear the check box for the mass fraction of that species.

Inflow

The Inflow node adds a boundary condition for an inflow boundary, where the concentration of all species c = c0 is specified. This condition is similar to the Concentration node, except the concentrations of all species must be specified.

n– cu Dc– N0=

n– Dc– N0 kc cb c– +=

Use a minus sign appropriately when specifying a flux leaving the system.


496 | C H A P T E



C O N C E N T R A T I O N

For the concentration of each species c0,c (SI unit: mol/m3) enter a value or expression.


These settings are the same as for the Concentration node.

Outflow

Set the Outflow condition at outlets where species are transported out of the model domain by a fluid flow. It is assumed that convection is the dominating transport mechanism across outflow boundaries, and therefore that diffusive transport can be ignored, that is:



Symmetry

The Symmetry node can be used to represent boundaries where the species concentration is symmetric, that is, where there is no mass flux in the normal direction across the boundary.

This boundary condition is identical to that of the No Flux node, but applies to all species and cannot be applied to individual species.


From the Selection list, choose the boundaries on which to apply the symmetry condition.

n D c– 0=

In models where only diffusion occurs, this boundary condition is not available.


Flux Discontinuity

The Flux Discontinuity node represents a discontinuity in the mass flux across an interior boundary:

where the value N0 specifies the jump in flux evaluated at the boundary.


From the Selection list, choose the boundaries on which to apply the flux discontinuity.


If this node is selected from the Pairs menu, choose the pair on which to apply the flux discontinuity. An identity pair has to be created first. Ctrl-click to deselect.

F L U X D I S C O N T I N U I T Y

Specify the jump in species mass flux.

Select the check boxes for the species to specify a Flux discontinuity N0,c, (SI unit: mol/(m2·s)) and enter a value or expression for the mass flux jump in the corresponding field. To use a different boundary condition for a specific species, click to clear the check box for the flux discontinuity of that species.

Periodic Condition

The Periodic Condition node can be used to define periodicity or antiperiodicity between two boundaries. The node can be activated on more than two boundaries, in which case the feature tries to identify two separate surfaces that can each consist of several connected boundaries. For more complex geometries it might be necessary to right-click and add the Destination Selection subnode. With this subnode the boundaries that constitute the source and destination surfaces can be manually specified.

n– Nd Nu– N0= N cu Dc– =

Use a positive value for increasing flux when going from the downside to the upside of the boundary. The boundary normal points in the direction from the downside to the upside of an interior boundary and can be plotted for visualization.


498 | C H A P T E



Point Mass Source

The Point Mass Source feature models mass flow originating from an infinitely small domain around a point.


The Point Mass Source feature is available in 3D where it can be added to any point and in 2D axisymmetry where it can be added to points on the symmetry axis.

S P E C I E S S O U R C E

Enter the source strength, , for each species (SI unit: molm·s)). A positive value results in species injection from the point into the computational domain, and a negative value means that the species is removed from the computational domain.




This feature requires at least one of the following licenses: Batteries and Fuel Cells Module, CFD Module, Chemical Reaction Engineering Module, Corrosion Module, Electrochemistry Module, Electrodeposition Module, Microfluidics Module, Pipe Flow Module, or Subsurface Flow Module.

For the Reacting Flow in Porous Media, Diluted Species interface, which is available with the CFD Module, Chemical Reaction Engineering Module, or Batteries & Fuel Cells Module, the Point Mass Source node is available in two versions, one for the fluid flow (Fluid Point Source) and one for the species (Species Point Source).

q· l,c


Point sources located on a boundary or on an edge affect the adjacent computational domains. This has the effect, for example, that the physical strength of a point source located in a symmetry plane is twice the given strength.

Open Boundary

Use the Open Boundary node to set up mass transport across boundaries where both convective inflow and outflow can occur. Use this boundary condition to specify an exterior species concentration on parts of the boundary where fluid flows into the domain. A condition equivalent to the Outflow node applies to the parts of the boundary where fluid flows out of the domain.



E X T E R I O R C O N C E N T R A T I O N

Enter a value or expression for the Exterior concentration c0,c (SI unit: mol/m3). The default is 0 mol/m3.

Thin Diffusion Barrier

Use the Thin Diffusion Barrier boundary condition to model a thin layer through which mass is transported by diffusion only. To set up the node, specify the layer thickness and a diffusion coefficient for each transported species.


From the Selection list, choose the boundaries on which to define the barrier.

Mass Sources for Species Transport in the COMSOL Multiphysics Reference Manual

The direction of the flow across the boundary is typically calculated by a Fluid Flow branch interface and is provided as a model input to the Transport of Diluted Species interface.


500 | C H A P T E


If this node is selected from the Pairs menu, choose the pair to define. An identity pair has to be created first. Ctrl-click to deselect. A No Flux node is added by default to the Thin Diffusion Barrier pair.

T H I N D I F F U S I O N B A R R I E R

Enter a Layer thickness ds (SI unit: m). The default is 0.005 m (5 mm).

Enter a Diffusion coefficient Ds,c (SI unit: m2/s). The default is 0 m2/s.

Thin Impermeable Barrier

Use the Thin Impermeable Barrier feature node to model a thin mass transfer barrier. The feature is available on interior boundaries and introduces a discontinuity in the concentration across the boundary. On each side of the boundary a no flux condition is prescribed for the mass transport. The Thin Impermeable Barrier boundary feature can be used to avoid meshing thin structures.

Solving a model involving coupled fluid flow and mass transfer, the Thin Impermeable

Barrier feature can be combined with an Interior Wall or Rotating Interior Wall feature in order to model a thin solid wall.


From the Selection list, choose the boundaries on which to define the barrier.


Th e T r a n s po r t o f C on c en t r a t e d S p e c i e s I n t e r f a c e

The Transport of Concentrated Species (chcs) interface ( ), found under the Chemical

Species Transport branch ( ) when adding a physics interface, is used to study gaseous and liquid mixtures where the species concentrations are of the same order of magnitude and none of the species can be identified as a solvent. In this case, properties of the mixture depend on the composition, and the molecular and ionic interactions between all species need to be considered. The physics interface includes models for multicomponent diffusion, where the diffusive driving force of each species depends on the mixture composition, temperature, and pressure.

The physics interface solves for the mass fractions of all participating species. Transport through convection, diffusion, and migration in an electric field can be included.

It supports simulations of transport by convection and diffusion in 1D, 2D, and 3D as well as for axisymmetric models in 1D and 2D. The physics interface defines the equations for the species mass fractions, including a diffusion model (Mixture-averaged or Fick’s law).

Some examples of what can be studied with this physics interface include:

• The evolution of chemical species transported by convection and diffusion.

• Concentrated solutions or gas mixtures, where the concentration of all participating species are of the same order of magnitude, and their molecular and ionic interaction with each other therefore must be considered. This implies that the diffusive transport of a single species is dependent on the mixture composition, and possibly on the temperature, the electric potential, the pressure, or any combination.

The default transport mechanism is the Convection and Diffusion node, which is dynamic and derived from the transport mechanism activated in this physics interface.

When this physics interface is added, the following default nodes are also added in the Model Builder— Convection and Diffusion (which applies a Mixture-average diffusion model), No Flux, and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and reactions. You can also right-click Transport of Concentrated Species to select physics from the context menu.

T H E TR A N S P O R T O F C O N C E N T R A T E D S P E C I E S I N T E R F A C E | 501

502 | C H A P T E



The default identifier (for the first physics interface in the component) is chcs.


The default setting is to have All domains in the component define the species equations. To choose specific domains, select Manual from the Selection list.

E Q U A T I O N

The basic equation for an individual species i is:

(8-1)

The displayed formulation changes depending on the active transport mechanisms and the selected diffusion model.

TR A N S P O R T M E C H A N I S M S

Select a Diffusion model—Mixture-averaged (the default) or Fick’s law.

• The Mixture-averaged option employs a more detailed diffusion model which requires that the multicomponent Maxwell-Stefan diffusivities for all component pairs are known. It is more computationally expensive than the Fick’s law option.

• The Fick’s law model is a general model that should be used when the diffusion is assumed Fickian, or when no multicomponent diffusivities are available. Also, when molecular diffusion is not the dominating transport mechanism and a robust but low order model is wanted, the Fick’s law options should be used.

Under Additional transport mechanisms, click to select or clear any combination of check boxes as required. Diffusion is always active and the Transport Feature node name changes according to the selection. The Convection check box is selected by default. The second term on the left-hand side of Equation 8-1 represents mass transport by convection.

t i iu + ji R+ i–=


Mixture-Averaged Diffusion ModelWhen using the Mixture-averaged diffusion model the relative mass flux vector is

The mixture-averaged diffusion coefficient is computed as

where Dik (SI unit: m2/s) is the multicomponent Maxwell-Stefan diffusivities, which are supplied as inputs to the model.

Fick’s Law Diffusion ModelWhen using the Fick’s law diffusion model the relative mass flux vector is

where (SI unit: m2/s) is a user-defined diffusion coefficient (isotropic, diagonal, or symmetric).

S P E C I E S

Select the species that this physics interface solves for using the mass constraint in Equation 8-2 (that is, its value comes from the fact that the sum of all mass fractions must equal 1). In the From mass constraint list, select the preferred species. To minimize the impact of any numerical errors, use the species with the highest concentration. By default, the first species is used.

(8-2)


Add or remove species in the model and also change the names of the dependent variables that represent the species concentrations.

Specify the Number of species. There must be at least two species. To add a single species, click the Add concentration button ( ) under the table. To remove a species,

ji Dim i iDi

m MM

--------- DiT T

T--------+ +

–=

Dim

Dim 1 i–

xk

Dik---------

k i

N--------------------------=

ji DiF i iDi

F MM

--------- DiT T

T--------+ +

–=

DiF

1 1 i

i 2=

Q

–=


504 | C H A P T E

select it in the list and click the Remove concentration button ( ) under the table. Edit the names of the species directly in the table.



• There are two consistent stabilization methods that are available when using the Mixture-Averaged Diffusion Model or Fick’s Law Diffusion Model—Streamline

diffusion and Crosswind diffusion. Both are active by default.

The Residual setting applies to both the consistent stabilization methods. Approximate residual is the default setting and it means that derivatives of the diffusion tensor components are neglected. This setting is usually accurate enough and computationally faster. If required, select Full residual instead.

• There is one inconsistent stabilization method, Isotropic diffusion, which is available when using the Mixture-Averaged Diffusion Model or Fick’s Law Diffusion Model.



From the Regularization list, select On (the default) or Off. When turned On, regularized mass fractions are calculated such that

Regularized mass fractions are used for the calculation of composition-dependent material properties, such as the density.

The species are dependent variables, and their names must be unique with respect to all other dependent variables in the component.

Adding a Chemical Species Transport Interface

0 wi reg 1




• Select an element order (shape function order) for the Mass fraction—Linear (the default), Quadratic, Cubic, or Quartic.

• The Compute boundary fluxes check box is selected by default so that COMSOL Multiphysics computes predefined accurate boundary flux variables. The computations of the following boundary flux variables change so that:



If you clear the Compute boundary fluxes check box, COMSOL Multiphysics instead computes the flux variables from the dependent variables using extrapolation.


• Specify the Value type when using splitting of complex variables—Real (the default) or Complex.

Domain, Boundary, and Pair Nodes for the Transport of Concentrated Species Interface

The Transport of Concentrated Species Interface has these domain, boundary, and pair nodes, listed in alphabetical order, available from the Physics ribbon toolbar (Windows


• Domain, Boundary, and Pair Nodes for the Transport of Concentrated Species Interface

• Theory for the Transport of Concentrated Species Interface


506 | C H A P T E

users), Physics context menu (Mac or Linux users), or right-click to access the context menu (all users).

These are described in this section:

The Thin Impermeable Barrier node is described for the Transport of Diluted Species interface.

Transport Feature

The dynamic Transport node adds the equations for transport of concentrated species and provides inputs for the material properties. The feature name changes as the options are selected, and the node includes the input fields required by the active


Periodic Condition in the COMSOL Multiphysics Reference Manual

For axisymmetric models, COMSOL Multiphysics takes the axial symmetry boundaries (at r = 0) into account and automatically adds an Axial Symmetry node to the model that is valid on the axial symmetry boundaries only.

• Flux


• Inflow

• Initial Values

• Mass Fraction


• Open Boundary

• Outflow

• Reactions

• Symmetry


• Transport Feature


transport mechanisms and diffusion model. The name of the transport feature is composed of the included transport mechanisms, and can be one of the following:

• Diffusion

• Convection and Diffusion




Specify the velocity field, the pressure, and the temperature to be used in the physics interface. The velocity becomes the model input for the convective part of the transport. The temperature model input is used when calculating the density from the ideal gas law, but also when thermal diffusion is accounted for by supplying thermal diffusion coefficients.

Velocity FieldSelect the source of the Velocity field u:

• Select User defined to enter values or expressions for the velocity components (SI unit: m/s). This input is always available.

• In addition, select velocity fields defined by a fluid flow interface present in the model (if any). For example, select Velocity field (spf/fp1) to use the velocity field defined by the Fluid Properties node fp1 in a Single-Phase Flow, Laminar Flow interface with spf as the identifier.

Also right-click the Convection and Diffusion node to add the Turbulent Mixing subnode.


508 | C H A P T E

TemperatureSelect the source of the Temperature field T:

• Select User defined to enter a value or an expression for the temperature (SI unit: K). This input is always available.

• If required, select a temperature defined by a Heat Transfer interface present in the model (if any). For example, select Temperature (ht/fluid1) to use the temperature defined by the Heat Transfer in Fluids interface with the ht interface identifier.

Absolute PressureSelect the source of the Absolute pressure p:

• Select User defined to enter a value or an expression for the absolute pressure (SI unit: Pa). This input is always available.

• In addition, select a pressure defined by a fluid flow interface present in the model (if any). For example, select Pressure (spf/fp1) to use the pressure defined by the Fluid

Properties node fp1 in a Single-Phase Flow, Laminar Flow interface with spf as the identifier. Selecting a pressure variable also activates a check box to define the reference pressure, where 1 [atm] is the default. This allows for the use of a system-based (gauge) pressure, while automatically including the reference pressure in the absolute pressure.

D E N S I T Y

Define the density of the mixture and the molar masses of the participating species.

Mixture DensitySelect a way to define the density from the Mixture density list—Ideal gas (the default) or User defined:

• When the default Ideal gas is selected, it uses the ideal gas law

to compute the mixture density using the absolute pressure and temperature defined in the Model Inputs section.

• Select User defined to enter a value or an expression for the Mixture density (SI unit: kg/m2).

Molar MassEnter a value or expression for the Molar mass Mw (SI unit: kg/mol) for each species. The default value for each species is 0.032 kg/mol, which is the molar mass of O2 gas.

pMRgT-----------=


D I F F U S I O N

Specify the molecular and thermal diffusivities of the present species based on the Diffusion model selected.

When using a Mixture-Averaged Diffusion Model, specify the Maxwell-Stefan diffusivity

matrix Dik and the Thermal diffusion coefficients .

When using a Fick’s Law Diffusion Model, specify the Diffusion coefficient and the Thermal diffusion coefficients for each of the species.

Maxwell-Stefan Diffusivity MatrixUsing a Mixture-Averaged Diffusion Model, the Maxwell-Stefan diffusivity matrix Dik (SI unit: m2/s) can be specified. For a simulation involving Q species the Maxwell-Stefan diffusivity matrix is a Q-by-Q symmetric matrix, where the diagonal components are 1. Enter values for the upper triangular components, Dij, which describe the inter-diffusion between species i and j. The numbering of the species corresponds to the order, from top to bottom, used for all the input fields for species properties (see for example the molar mass fields in the Density section).

Diffusion CoefficientUsing a Fick’s Law Diffusion Model, the diffusion is by default assumed to be isotropic and governed by one Diffusion coefficient (SI unit: m2/s) for each species. To allow for a general representation, it is also possible to use diffusion matrices (diagonal, symmetric, or anisotropic).

Thermal Diffusion CoefficientTo model thermal diffusion, prescribe the Thermal diffusion coefficients (SI unit: m2/s), by entering one thermal diffusion coefficient for each species in the corresponding field. In a multicomponent mixture, the sum of the thermal diffusion coefficients is zero. The default value for all thermal diffusion coefficients is 0.

Turbulent Mixing

Right-click a Transport Feature node to add the Turbulent Mixing subnode and define the turbulent kinematic viscosity and turbulent Schmidt number. This feature is available if Convection is selected as a transport mechanism and if the Diffusion model is Mixture-averaged or Fick’s law. Use this node to account for the turbulent mixing caused by the eddy diffusivity. An example is when the specified velocity field corresponds to a RANS solution.

DwT

DwF

DwT

DwF

DiT


510 | C H A P T E


From the Selection list, choose the domains in which to add the turbulent mixing. The default is to add All domains.

TU R B U L E N T M I X I N G

Some physics interfaces provide the turbulent kinematic viscosity, and these appear as options in the Turbulent kinematic viscosity T (SI unit: m2/s) list. The list always contains the User defined option that makes it possible to enter any value or expression.

The default Turbulent Schmidt number ScT is 0.71 (dimensionless). Enter another value or expression as required.

Reactions

In order to account for consumption or production of species due to reactions, the Reactions node adds rate expressions to the right-hand side of the species transport equations.


From the Selection list, choose the domains in which to use the rate expressions defined in the node.

R E A C T I O N S

Add a rate expression, Ri (SI unit: kg/(m3·s)), for all individual species present except for the one computed from the mass constraint (see Species). Enter a value or expression in the field for the corresponding species.

Enable the Mass transport to other phases check box if mass is leaving or entering the fluid as a result of the reactions, for instance due to condensation or vaporization in a porous matrix. This allows you to also set a reaction rate for the species calculated from the mass constraint.



Initial Values

The Initial Values node adds initial values for the mass fractions that can serve as an initial condition for a transient simulation, or as an initial guess for a nonlinear solver. If required, add additional Initial Values nodes from the Physics toolbar.

The initial mass fractions can be specified from the following quantities:

• The mass fraction: 0

• The mole fraction: xx0

• The molar concentration: cc0

• The number density, which describes the number of particles per volume: nn0

• The density: 0




Select the type of input from the Mixture specification list. Select:

• Mass fractions (the default) to enter mass fractions ( for example)

• Mole fractions to enter mole fractions ( for example)

• Molar concentrations (SI unit: mol/m3) to enter molar concentrations ( for example)

• Number densities (SI unit: 1/m3) to enter number densities ( for example)

• Densities (SI unit: kg/m3) to enter densities ( for example)

Enter a value or expression in the field for each species except for the one computed from the mass constraint.

Mass Fraction

The Mass Fraction node adds boundary conditions for the species mass fractions. For example, the following condition specifies the mass fraction of species i: i = i,0.

0 1

x0 1

c0 1

n0 1

0 1


512 | C H A P T E

Set the mass fractions of all species except the one computed from the mass constraint. This ensures that the sum of the mass fractions is equal to one (see Species). This node is available for exterior and interior boundaries.





M A S S F R A C T I O N

Specify the mass fraction for each species individually. Select the check box for the species to specify the mass fraction, and enter a value or expression in the corresponding field. To use another boundary condition for a specific species, click to clear the check box for the mass fraction of that species.




Flux

The Flux node can be used to specify the total mass flux across a boundary. The total inward flux is defined in the manner of:

(8-3)

In Equation 8-3, N0,i (SI unit: Pa·s/m) is an arbitrary flux expression for species i and can be a function of i, temperature, pressure or even electric potential. Set the mass flux of all species except the one computed from the mass constraint. This ensures that

n– iu ji+ N0 i=


the sum of the mass fractions is equal to one (see Species). This node is available for exterior boundaries.



I N W A R D F L U X

Specify the Inward flux for each species individually. Select the check box for the species to prescribe a flux and enter a value or expression for the flux in the corresponding field. To use another boundary condition for a specific species, click to clear the check box for the flux of that species.

Inflow

The Inflow node adds a boundary condition for an inflow boundary where one condition for each species is specified. The condition can be specified using the following quantities:





• The density: 0

A concentration quantity other than the mass fractions can only be used when all species are defined, as in this boundary condition. The other quantities are composition dependent and therefore unambiguous only when all species are defined. For this reason the Mass Fraction node, which allows some species to use a different boundary condition, only includes inputs for the mass fractions. The Inflow node is available for exterior boundaries.

The Flux node can, for example, be used to describe a heterogeneous reaction or a separation process occurring at the boundary.

Use a positive value for an inward flux.


514 | C H A P T E





I N F L O W











No Flux

The No Flux node, which is the default boundary condition available for exterior boundaries, represents boundaries where no mass flows in or out; that is, the total flux is zero:



0 1

x0 1

c0 1

n0 1

0 1

n iu ji+ 0=


interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific boundaries or select All boundaries as required.

N O F L U X

Select Apply for all species to specify no flux for all species. Select Apply for to specify no flux for each species individually. To use another boundary condition for a specific species, click to clear the check box for that species.

Outflow

The Outflow node is the preferred boundary condition at outlets where the species are to be transported out of the model domain. It is useful, for example, in mass transport models where it is assumed that convection is the dominating effect driving the mass flow through the outflow boundary. This node is available for exterior boundaries. The boundary condition is applied to all species and corresponds to one of the following equations depending on the selected diffusion model:

• For the Mixture-Averaged Diffusion Model:

• For the Fick’s Law Diffusion Model:





Symmetry

The Symmetry node can be used to represent boundaries where the species concentration is symmetric; that is, there is no mass flux in the normal direction across the boundary:

n– ji n iDim M

M--------- Di

T TT

-------- izium i + + =

n– ji n iDif M

M--------- Di

T TT

-------- izium i + + =

n iu ji+ 0=


516 | C H A P T E

This boundary condition is identical to the No Flux node, but applies to all species and cannot be applied to individual species. The Symmetry node is available for exterior boundaries.



Flux Discontinuity

The Flux Discontinuity node represents a discontinuity in the mass flux across an interior boundary:

where the value of N0 specifies the size of the flux jump evaluated from the down to the upside of the boundary. This boundary condition is only available on interior boundaries.





F L U X D I S C O N T I N U I T Y

Specify the jump in species mass flux. Use a positive value for increasing flux when going from the downside to the upside of the boundary. The boundary normal points in the direction from the downside to the upside of an interior boundary and can be plotted for visualization.

Select the Species check boxes to specify a flux discontinuity, and enter a value or expression for the Flux discontinuity N0 (SI unit: kg/(m2·s)) in the corresponding field, N0, w1 for example. To use a different boundary condition for a specific species, click to clear the check box for the flux discontinuity of that species.

Open Boundary

Use the Open Boundary node to set up mass transport across boundaries where both convective inflow and outflow can occur. Use the node to specify an exterior species

n– Nd Nu– N0= N iu ji+ =


composition on parts of the boundary where fluid flows into the domain. A condition equivalent to the Outflow node applies to the parts of the boundary where fluid flows out of the domain.



E X T E R I O R C O M P O S I T I O N

Enter a value or expression for the species composition. Select:

• Mass fractions (the default) to enter mass fractions (0, 1, for example)

• Mole fractions to enter mole fractions (x0, 1, for example)

• Molar concentrations (SI unit: mol/m3) to enter molar concentrations (c0, 1, for example)

• Number densities (SI unit: 1/m3) to enter number densities (n0, 1, for example) and to describe the number of particles per volume n n0

• Densities (SI unit: kg/m3) to enter densities (0, 1, for example)


A concentration quantity other than the mass fractions can only be used when all species are defined.


518 | C H A P T E

Th e R e a c t i n g F l ow I n t e r f a c e s

The following interfaces and features are described in this section:

• The Reacting Flow, Laminar Flow Interface

• The Reacting Flow, Turbulent Flow, k- Interface


• The Reacting Flow, Turbulent Flow, Low Re k-Interface

The Reacting Flow, Laminar Flow Interface

The Laminar Flow version of the Reacting Flow (rspf) interface ( ) is used to compute the fluid flow and chemical composition of a gas or liquid in the laminar flow regime. The chemical species can either be low-concentration solutes, present in a solvent of significantly higher concentration, or a mixture of species where the concentrations are of comparable order of magnitude.

This physics interface solves for the instantaneous velocity and pressure fields, together with an arbitrary number of mass fractions. The governing fluid flow equations are the Navier-Stokes equations for conservation of momentum and the continuity equation for total conservation of mass. For the chemical species, the governing transport equations include convection, diffusion and, optionally, migration in an electric field.

The physics interface combines the functionality of the Laminar Flow and Transport of

Concentrated Species interfaces.

When this physics interface is added, the following default nodes are also added in the Model Builder—Fluid, No Flux, Wall, and Initial Values. Then, from the Physics toolbar,

The Reacting Flow (rspf) interface is added to the Model Builder after selecting any option under the Chemical Species Transport>Reacting Flow>

branch when adding a physics interface.


add other nodes that implement, for example, boundary conditions. You can also right-click the node to select physics from the context menu.



The default identifier (for the first physics interface in the component) is rspf.



S P E C I E S

Select the type of fluid mixture to study.

• Select Concentrated species (the default) when studying a mixture of species with comparable concentrations. Then, from the From mass constraint list, select the species not solved for. Use the species with the highest concentration in order to minimize the impact of any numerical errors.

• Select Dilute solution when studying low concentration solutes present in a solvent with significantly higher concentration. Then, from the Solvent list, choose the species not solved for.

• The Laminar Flow, Creeping Flow, and Turbulent Flow Interfaces

• The Transport of Concentrated Species Interface

The mass fraction for one species can be calculated from the constraint that the sum of all mass fractions must be equal to one. There is then no need to solve a transport equation for this species.

T H E R E A C T I N G F L O W I N T E R F A C E S | 519

520 | C H A P T E


The default Turbulence model type is None.

To enable a turbulence model, select RANS as the Turbulence model type. The Reacting

Flow interface supports the k-, Low-Re k-, and the k- turbulence models.

When a turbulence model is used, a model for the turbulent mass transport is also needed. Select Mass transport turbulence model—Kays-Crawford (the default), High

Schmidt number model (available for the Low-Re k- turbulence model), or User defined

Schmidt number.

Select the Neglect inertial term (Stokes flow) check box to model flow at very low Reynolds numbers where the inertial term in the Navier-Stokes equations can be neglected. Select the Use shallow channel approximation check box to model shallow channels in microfluidics applications. For more information see the settings for The Laminar Flow Interface.

Select a Diffusion model—Mixture-averaged(the default), or Fick’s law. See the settings for The Transport of Concentrated Species Interface for more information.


To display these sections, click the Show button ( ) and select Stabilization.

There are two consistent stabilization methods available—Streamline diffusion and Crosswind diffusion. Both are active by default.

The Residual setting applies to both the consistent stabilization methods for the mass fractions. Approximate residual (the default) means that derivatives of the diffusion tensor components are neglected. This setting is usually accurate enough and computationally faster. If required, select Full residual.

There is one inconsistent stabilization method, Isotropic diffusion.

In the case of an ideal gas, the fluid mixture type decides how the density is evaluated. It, however, does not effect the mass transport equations used. These solve for the species mass fractions for both mixture types.




• From the Regularization list, select On (the default) or Off. See the settings for the Transport of Concentrated Species interface for more information.

• The Use pseudo time stepping for stationary equation form check box is selected by default. See Pseudo Time Stepping for Mass Transport and the settings for The Laminar Flow Interface for more information.


To display this section, click the Show button ( ) and select Discretization. It controls the discretization (the element types used in the finite element formulation) and some related properties:

• Select an element order (shape function order) for the Mass fraction—Linear (the default), Quadratic, Cubic, or Quartic. See the settings for Transport of Concentrated

Species for more information.






• From the Discretization of fluids list select the element order for the velocity components and the pressure: P1+P1 (the default), P2+P1, or P3+P2 (where the first


522 | C H A P T E

term is the order of the velocity field and the second term is the order of the pressure). See the settings for Laminar Flow for more information.

• Specify the Value type when using splitting of complex variables for each dependent variable in the table—Real (the default) or Complex.


Specify the Number of species. There must be at least two species. To add a single species, click the Add mass fraction button ( ) under the table. To remove a species, select it in the list and click the Remove mass fraction button ( ) under the table. Edit the names of the species mass fraction directly in the table.

In addition to the chemical species mass fractions (dimensionless), this physics interface defines the following dependent variables (fields):

• Velocity field u and its components (SI unit: m/s)

• Pressure p (SI unit: Pa)

The Reacting Flow, Turbulent Flow, k- Interface

The Turbulent Flow, k-version of the Reacting Flow (rspf) interface ( ), found under the Reacting Flow>Turbulent Flow branch ( ) when adding a physics interface, is used to study the flow and chemical composition of a gas or liquid in the turbulent flow regime. The chemical species can either be low-concentration solutes, present in a solvent of significantly higher concentration, or a mixture of species where the concentrations are of comparable order of magnitude.

This physics interface solves for the averaged velocity and pressure fields, together with an arbitrary number of averaged mass fractions. The equations governing the fluid flow are the averaged Navier-Stokes equations for conservation of momentum and the continuity equation for total conservation of mass. The fluid flow turbulence is modeled using the standard two-equation k- model with realizability constraints. Flow and mass transport close to solid walls are modeled using wall functions. For the


• Domain, Boundary, Point, and Pair Nodes for the Reacting Flow in Porous Media (rfds) Interface

• Adding a Chemical Species Transport Interface

• Theory for the Reacting Flow Interfaces


chemical species, the governing transport equations include convection, diffusion and, optionally, migration in an electric field. Turbulent reactions are modeled using the eddy dissipation model.

The physics interface combines the functionality of the Turbulent Flow k- and Transport of Concentrated Species interfaces.

When this physics interface is added, the following default nodes are also added in the Model Builder—Fluid, Wall (the default boundary condition is Wall functions), and Initial

Values.




The default Turbulence model is k- for this physics interface. For all other settings see The Reacting Flow, Laminar Flow Interface.






• Turbulent kinetic energy k (SI unit: m2/s2)

• Turbulent dissipation rate ep (SI unit: m2/s3)

Except for the settings described below, see The Reacting Flow, Laminar Flow Interface. Also see Domain, Boundary, Point, and Pair Nodes for the Reacting Flow in Porous Media (rfds) Interface.


524 | C H A P T E

The Reacting Flow, Turbulent Flow, k- Interface

The Turbulent Flow k-version of the Reacting Flow (rspf) interface ( ), found under the Reacting Flow>Turbulent Flow branch ( ) when adding a physics interface, is used to study the flow and chemical composition of a gas or liquid in the turbulent flow regime. The chemical species can either be low-concentration solutes, present in a solvent of significantly higher concentration, or a mixture of species where the concentrations are of comparable order of magnitude.

This physics interface solves for the averaged velocity and pressure fields together with an arbitrary number of averaged mass fractions. The equations governing the fluid flow are the averaged Navier-Stokes equations for conservation of momentum and the continuity equation for total conservation of mass. The fluid flow turbulence is modeled using the Wilcox revised k- model with realizability constraints. Flow and mass transport close to solid walls are modeled using wall functions. For the chemical species, the governing transport equations include convection, diffusion and, optionally, migration in an electric field. Turbulent reactions are modeled using the eddy dissipation model.

The physics interface combines the functionality of the Turbulent Flow k- and Transport of Concentrated Species interfaces.


Values.


The default Turbulence model is k- for this interface. For all other settings see The Reacting Flow, Laminar Flow Interface.











• Specific turbulent dissipation rate om (SI unit: 1/s)

The Reacting Flow, Turbulent Flow, Low Re k-Interface

The Turbulent Flow, Low Re k-version of the Reacting Flow (rspf) interface ( ), found under the Reacting Flow>Turbulent Flow branch ( ) when adding a physics interface, is used to study the flow and chemical composition of a gas or liquid in the turbulent flow regime. The chemical species can either be low-concentration solutes, present in a solvent of significantly higher concentration, or a mixture of species where the concentrations are of comparable order of magnitude.

This physics interface solves for the averaged velocity and pressure fields, together with an arbitrary number of averaged mass fractions. The equations governing the fluid flow are the averaged Navier-Stokes equations for conservation of momentum and the continuity equation for total conservation of mass. The fluid flow turbulence is modeled using the AKN low-Reynolds number k- model. The low-Reynolds number model resolves the velocity, pressure, and mass fractions all the way down to the wall. For that reason this physics interface is suited for studying mass transfer at high Schmidt numbers. The AKN model depends on the distance to the closest wall, and the interface therefore includes a wall distance equation. For the chemical species, the governing transport equations include convection, diffusion and, optionally, migration in an electric field. Turbulent reactions are modeled using the eddy dissipation model.

This physics interface combines the functionality of the Turbulent Flow, Low Re k- and Transport of Concentrated Species interfaces.


526 | C H A P T E


Values.


The default Turbulence model is Low Reynolds number k- for this physics interface. For all other settings see The Reacting Flow, Laminar Flow Interface.









• Turbulent dissipation rate ep (SI unit: m2/s3)

• Reciprocal wall distance G (SI unit: 1/m)

Domain, Boundary, Point, and Pair Nodes for the Reacting Flow, Laminar Flow and Turbulent Flow Interfaces

The Reacting Flow Interfaces has these domain, boundary, point, and pair nodes, listed in alphabetical order, available from the Physics ribbon toolbar (Windows users),



Physics context menu (Mac or Linux users), or right-click to access the context menu (all users).

The following nodes are described in this section:

The following nodes (listed in alphabetical order) are described for the Transport of

Concentrated Species or the Transport of Diluted Species interface:


• Fluid

• Wall

• Reactions

• Inflow

• Initial Values

• Reacting Boundary

• Open Boundary

• Flux

• Mass Fraction

• No Flux

• Outflow


• Symmetry

• Thin Impermeable Barrier


528 | C H A P T E

The following nodes(listed in alphabetical order) are described for the Laminar Flow interface :

Fluid

Use the Fluid node to define the fluid properties; the density, the viscosity, the molar masses of the individual species, the Maxwell-Stefan diffusivity matrix or the diffusion coefficients, dependent on the diffusion model, and the thermal diffusion coefficient.




• Flow Continuity

• Inlet

• Interior Wall

• Outlet



• Symmetry

• Vacuum Pump

• Volume Force



• Theory of Laminar Flow







Select the source of the Temperature field T. Select User defined to enter a value or an expression for the temperature (SI unit: K). The default is 293.15 K.

When Ideal gas is selected for the Density, or when a material requires an absolute pressure input, select the source of the Absolute pressure p (SI unit: Pa). To apply a custom value or expression, select User defined.

Selecting a pressure solved for, Pressure rspf/rfluid for example, which corresponds to the pressure from the Reacting Flow interface, a reference pressure can be added. By default a Reference pressure (pref) of 1 atm (101,325 Pa) is applied.


DensityThe default Density (SI unit: kg/m3) uses the value From material. Select Ideal gas or User defined to enter different values or expressions. If Ideal gas is selected, additional options are available in the Model Inputs section in order to define the absolute pressure. The ideal gas density depends on the mixture type specified (see Species). When a Dilute solution mixture is selected, the density is computed using the molar mass of the Solvent. If a Concentrated species mixture is selected the density is computed using the mean molar mass of the entire mixture.

Dynamic ViscositySelect a Dynamic viscosity (SI unit: Pa·s)—From material (the default), User defined, Non-Newtonian power law, or Non-Newtonian Carreau model.

• From material is the default and describes the relationship between the shear rate and the shear stresses in a fluid. Intuitively, water and air have low viscosities, and substances often described as thick (such as oil) have higher viscosities.

• Select User defined to define a different value or expression. Using a built-in variable for the shear rate magnitude, spf.sr, makes it possible to define arbitrary expressions for the dynamics viscosity as a function of the shear rate.

When User defined turbulent Schmidt number is selected under Physical Model for any of the turbulent flow interfaces, also enter a Turbulent

Schmidt number ScT (dimensionless). The default is 0.71.


530 | C H A P T E

• If Non-Newtonian power law is selected, enter values or expressions for the Power law

model parameter m (SI unit: kg/(m·s)) and Model parameter n (dimensionless).

• If Non-Newtonian Carreau model is selected, enter values or expressions for the Zero

shear rate viscosity 0 (SI unit: Pa·s), Infinite shear rate viscosity inf (SI unit: Pa·s), Model parameter (SI unit: s), and Model parameter n (dimensionless).

Molar MassEnter a value or expression for the Molar mass Mw (SI unit: kg/mol) for each species. The default value for each species is 0.032 kg/mol, which is the molar mass of O2 gas.

Note that the individual molar masses for all species should be specified, independent of the mixture type (Dilute solution or Concentrated species, see Species). The reason for this is that the interface solves the same equations, governing the species mass fractions, for both mixture types.

D I F F U S I O N

The following settings are described in detail for The Transport of Concentrated Species Interface.

Maxwell-Stefan Diffusivity MatrixUsing a Mixture-Averaged Diffusion Model, the Maxwell-Stefan diffusivity matrix Dik (SI unit: m2/s) can be specified. For a simulation involving Q species, the Maxwell-Stefan diffusivity matrix is a Q-by-Q symmetric matrix, where the diagonal components are 1. Enter values for the upper triangular components, Dij, which describe the inter-diffusion between species i and j. The numbering of the species corresponds to the order, from top to bottom, used for all the input fields for species properties (see for example the molar mass fields in the Density section).

Diffusion CoefficientUsing a Fick’s Law Diffusion Model, the diffusion is by default assumed to be isotropic and governed by one Diffusion coefficient (SI unit: m2/s) for each species. To allow for a general representation, it is also possible to use diffusion matrices (diagonal, symmetric, or anisotropic).

Thermal Diffusion CoefficientTo model thermal diffusion, prescribe the Thermal diffusion coefficients (SI unit: m2/s), by entering one thermal diffusion coefficient for each species in the corresponding field. In a multicomponent mixture, the sum of the thermal diffusion coefficients is zero. The default value for all thermal diffusion coefficients is 0.

DwF

DiT


Wall

Using the Reacting Flow interfaces, The Stefan Velocity is automatically defined and applied on outer boundaries. Any Mass Fraction or Flux feature applied on a selection shared by a Wall feature adds its corresponding mass flux contribution. The resulting Stefan velocity is applied in the wall normal direction, in addition to the wall condition specified by the selected boundary condition.

Except for the application of the Stefan velocity, the Wall node is described for the Laminar Flow interface

Reactions

Use the Reactions node to define the production or destruction of species resulting from a chemical reaction. For turbulent flow the Reactions node supports non-premixed reactions. Implying that the reactants are not mixed when entering the model.


From the Selection list, choose the domains in which to use the rate expressions defined in the node.

R E A C T I O N R A T E

Select a Reaction rate—Automatic (the default), or User defined. Selecting Automatic the reaction rate is computed using the mass action law. When using a turbulence model the reaction rate is limited by the turbulent mixing as described in The Reaction Feature section.

Select User defined to input a custom expression or constants for the Reaction rate r (SI unit: mol/(m3s)).

Specify the reaction stoichiometry by entering values for the stoichiometric coefficients (dimensionless) of each species. Enter negative values for reactants and positive values for products.

The Chemical Reaction Rate


532 | C H A P T E

R A T E C O N S T A N T S

Enter values for the Turbulent reaction model parameters and (dimensionless) used in the Eddy Dissipation Concept model.

The Use Arrhenius expressions check box is selected by default. Enter values for both the Forward and Reverse variables as follows:

• Forward frequency factor Af and Reverse frequency factor Ar (dimensionless)

• Forward activation energy Ef and Reverse activation energy Er (SI unit: J/mol)

• Forward temperature exponent nf and Reverse temperature exponent nr

(dimensionless)

Click to clear the Use Arrhenius expressions check box to input custom expressions or constants for the Forward rate constant kf and Reverse rate constant kr (SI unit: mol/(m3s)).

Inflow

The Inflow node adds a boundary condition for an inflow boundary where one condition for each species is specified. The condition can be specified using the following quantities:





• The density: 0

A concentration quantity other than the mass fractions can only be used when all species are defined, as in this boundary condition. The other quantities are composition dependent and therefore unambiguous only when all species are defined. For this reason the Mass Fraction node, which allows some species to use a different boundary condition, only includes inputs for the mass fractions. The Inflow node is available for exterior boundaries.






I N F L O W




• Molar concentrations (SI unit: mol/m3) to enter molar concentrations ( for example).







Initial Values

The Initial Values node prescribes initial values for the mass fractions, velocity field, and pressure that can serve as initial conditions for a transient simulation, or as an initial guess for a nonlinear solver. If required, add additional Initial Values nodes from the Physics toolbar.






• The density: 0

0 1

x0 1

c0 1

n0 1

0 1


534 | C H A P T E







• Molar concentrations (SI unit: mol/m3) to enter molar concentrations ( for example).



Enter a value or expression in the field for each species, except for the one computed from the mass constraint.

Enter components for the initial value of Velocity field u (SI unit: m/s) and Pressure p (SI unit: Pa)

Reacting Boundary

Use the Reacting Boundary node to define the inward flux.



I N W A R D F L U X

Specify the Inward flux for each species individually. Select the check box for the species to prescribe a flux and enter a value or expression for the flux in the corresponding

0 1

x0 1

c0 1

n0 1

0 1

Additional initial values are available for the turbulent flow versions of this physics interface. See the individual interfaces for additional details.



• The Reacting Flow, Turbulent Flow, Low Re k-Interface


field. To use another boundary condition for a specific species, click to clear the check box for the flux of that species.

Open Boundary

Use the Open Boundary node to set up mass and momentum transport across boundaries where both convective inflow and outflow can occur. The node specifies a fluid flow condition, together with an exterior species composition to be applied on the parts of the boundary where fluid flows into the domain.




From the Boundary Condition list, choose a fluid flow condition to apply on the open boundary. Select

• Select Normal stress (the default) and enter the normal stress f0 (SI unit: N/m2). For high Reynolds numbers this implicitly specifies that

• Select No viscous stress to prescribe a vanishing normal viscous stress on the boundary

E X T E R I O R C O M P O S I T I O N

Enter a value or expression for the species composition. Select:

• Mass fractions (the default) to enter mass fractions (0, 1, for example)

• Mole fractions to enter mole fractions (x0, 1, for example)

• Molar concentrations (SI unit: mol/m3) to enter molar concentrations (c0, 1, for example)

Use a positive value for an inward flux.


p f0


536 | C H A P T E

• Number densities (SI unit: 1/m3) to enter number densities (n0, 1, for example) and to describe the number of particles per volume n n0

• Densities (SI unit: kg/m3) to enter densities (0, 1, for example)

A concentration quantity other than the mass fractions can only be used when all species are defined.


Th e R e a c t i n g F l ow i n Po r ou s Med i a I n t e r f a c e s

In this section:

• The Reacting Flow in Porous Media (rfcs) Interface

• Domain, Boundary, Point, and Pair Nodes for the Reacting Flow in Porous Media (rfcs) Interface

• The Reacting Flow in Porous Media (rfds) Interface

• Domain, Boundary, Point, and Pair Nodes for the Reacting Flow in Porous Media (rfds) Interface

The Reacting Flow in Porous Media (rfcs) Interface

The Reacting Flow in Porous Media (rfcs) interface ( ) is used to study the flow and chemical composition of a gas or liquid moving through the interstices of a porous medium. The fluid can consist of a mixture of species where the individual concentrations are of comparable order of magnitude. Apart from porous media regions, the flow system may also include regions with free flow.

This physics interface solves for the velocity and pressure fields, together with an arbitrary number of mass fractions. In porous media regions, the Brinkman equations governing the fluid momentum are solved. In free-flow regions these are replaced by the Navier-Stokes equations. The continuity equation governing the total conservation of mass is solved in all regions. For the chemical species, the governing transport equations include convection, diffusion and, optionally, migration in an electric field.

When this physics interface is added, the following default nodes are also added in the Model Builder—Transport Properties, No Flux, Wall, and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions.

The Reacting Flow in Porous Media (rfcs) interface is added to the Model

Builder after selecting Transport of Concentrated Species under the Chemical

Species Transport>Reacting Flow in Porous Media> branch when adding a physics interface.

T H E R E A C T I N G F L O W I N P O R O U S M E D I A I N T E R F A C E S | 537

538 | C H A P T E

You can also right-click Reacting Flow in Porous Media to select physics from the context menu.



The default identifier (for the first physics interface in the component) is rfcs.



S P E C I E S

This section is described for The Transport of Concentrated Species Interface.


Diffusion model is described under Transport Mechanisms for The Transport of Concentrated Species Interface. Compressibility is described under Physical Model for The Free and Porous Media Flow Interface.



• Select an element order (shape function order) for the Mass fraction—Linear (the default), Quadratic, Cubic, or Quartic. See the settings for for more information.


- ndflux_c (where c is the dependent variable for the concentration) is the normal diffusive flux and corresponds to the accurate boundary flux when diffusion is the only contribution to the flux term. When transport in an electric field is included


or when the conservative form of the equation is used, ndflux_c is instead computed directly from the dependent variable.




• From the Discretization of fluids list select the element order for the velocity components and the pressure: P1+P1 (the default), P2+P1, or P3+P2 (where the first term is the order of the velocity field and the second term is the order of the pressure). See the settings for The Laminar Flow Interface for more information.



To display this section, click the Show button ( ) and select Advanced Physics Options. Normally these settings do not need to be changed. From the Regularization list, select On (the default) or Off.

When turned On, regularized mass fractions are calculated such that

Regularized mass fractions are used for the calculation of composition-dependent material properties, such as the density.


The dependent variables (field variables) are the Velocity field u (SI unit: m/s), the Pressure p (SI unit: Pa), and the Mass fractions w (dimensionless). Use the Add

concentration ( ) and Remove concentration ( ) buttons as required. Add or remove species in the model and also change the names of the dependent variables that represent the species concentrations. Enter the Number of species.

0 wi reg 1


540 | C H A P T E


Domain, Boundary, Point, and Pair Nodes for the Reacting Flow in Porous Media (rfcs) Interface

The Reacting Flow in Porous Media (rfcs) Interface has these domain, boundary, point, and pair nodes, listed in alphabetical order, available from the Physics ribbon toolbar (Windows users), Physics context menu (Mac or Linux users), or right-click to access the context menu (all users).

The following nodes are described in this section:

• Diffusion

• Initial Values

• Reacting Boundary

• Transport Properties

The following nodes are described for the Free and Porous Media Flow interface:



More information on the applicability of the physics interface can be found in the individual documentation for the Transport of Concentrated

Species and Free and Porous Media Flow interfaces. See Domain, Boundary, Point, and Pair Nodes for the Reacting Flow in Porous Media (rfcs) Interface for links to the feature nodes, most of which are shared with other physics interfaces.






• Volume Force

The following nodes are described for the Transport of Concentrated Species interface:

The following nodes are described for the Laminar Flow interface:

Transport Properties

Use the Transport Properties node to define the fluid properties, the thermal diffusion coefficient, and the Maxwell-Stefan diffusivity matrix.




This section is described for the Free and Porous Media Flow interface, Fluid Properties node.

• Flux

• Inflow

• Mass Fraction

• No Flux

• Open Boundary

• Outflow

• Reactions

• Symmetry


• Flow Continuity

• Inlet

• Outlet



• Symmetry

• Symmetry



542 | C H A P T E


Fluid material, Density, and Dynamic viscosity are described for the Free and Porous Media

Flow interface, Fluid Properties node.

Molar mass is described in the Density section for the Transport of Concentrated Species interface, Transport Feature node.

D I F F U S I O N

Maxwell-Stefan diffusivity matrix and Thermal diffusion coefficient are described for the Transport of Concentrated Species interface, Transport Feature node.

Diffusion

Use the Diffusion node to define the mixture density, molar mass, Maxwell-Stefan diffusivity matrix, and thermal diffusion coefficient.


This section is described for the Free and Porous Media Flow interface, Fluid Properties node.

D E N S I T Y

Mixture density and Molar mass are described for the Transport of Concentrated Species interface, Transport Feature node.

D I F F U S I O N

Maxwell-Stefan diffusivity matrix and Thermal diffusion coefficient are described for the Transport of Concentrated Species interface, Transport Feature node.

Initial Values

The Initial Values node prescribes initial values for the mass fractions, velocity field, and pressure that can serve as initial conditions for a transient simulation, or as an initial guess for a nonlinear solver.







• The density: 0










Enter a value or expression in the field for each species, except for the one computed from the mass constraint.

Enter components for the initial value of Velocity field u (SI unit: m/s) and Pressure p (SI unit: Pa).

Reacting Boundary

Use the Reacting Boundary node to define the boundary conditions for a reactive surface over which there is a net mass flux. The flow velocity across the boundary is set up automatically based on the sum of the species fluxes.



If there are several types of domains with different initial values. These can be defined adding multiple Initial Values nodes.

0 1

x0 1

n0 1

n0 1

0 1


544 | C H A P T E

I N W A R D F L U X

Specify the Inward flux N0 (SI unit: kg/(m2·s)) for each species individually by selecting the check box for the species to prescribe a flux. Then enter a value or expression for the flux in the corresponding field. Click to clear the check box for the flux of a species to set the flux to zero. Use a positive value for an inward flux.




The Reacting Flow in Porous Media (rfds) Interface

The Reacting Flow in Porous Media (rfds) interface ( ) is used to study the flow and chemical composition of a gas or liquid moving through the interstices of a porous medium. The chemical species are assumed to be solutes, dissolved in a solvent of significantly higher concentration. Apart from porous media regions, the flow system may also include regions with free flow.

This physics interface solves for the velocity and pressure fields, together with an arbitrary number of molar concentrations. In porous media regions, the Brinkman equations governing the fluid momentum are solved. In free-flow regions these are replaced by the Navier-Stokes equations. The continuity equation governing the total conservation of mass is solved in all regions. For the chemical species, the governing transport equations include convection, diffusion and, optionally, migration in an electric field.

The physics interface combines the functionality of the Transport of Diluted Species and the Free and Porous Media Flow interfaces.

When this physics interface is added, the following default nodes are also added in the Model Builder—Transport Properties, No Flux, Wall, and Initial Values. Then, from the

The Reacting Flow in Porous Media (rfds) interface is added to the Model

Builder after selecting Transport of Diluted Species under the Chemical

Species Transport>Reacting Flow in Porous Media> branch when adding a physics interface.


Physics toolbar, add other nodes that implement, for example, boundary conditions. You can also right-click Reacting Flow in Porous Media to select physics from the context menu.



The default identifier (for the first physics interface in the component) is rfds.




Compressibility, Neglect inertial term in free flow (Stokes flow), Swirl flow (2D axisymmetric models), and Neglect inertial term in porous media flow (Stokes-Brinkman) are described for The Free and Porous Media Flow Interface.



• Select an element order (shape function order) for the Concentration—Linear (the default), Quadratic, Cubic, or Quartic. See the settings for for more information.


- ndflux_c (where c is the dependent variable for the concentration) is the normal diffusive flux and corresponds to the accurate boundary flux when diffusion is the only contribution to the flux term. When transport in an electric field is included


546 | C H A P T E

or when the conservative form of the equation is used, ndflux_c is instead computed directly from the dependent variable.




• From the Discretization of fluids list select the element order for the velocity components and the pressure: P1+P1 (the default), P2+P1, or P3+P2 (where the first term is the order of the velocity field and the second term is the order of the pressure). See the settings for The Laminar Flow Interface for more information.



The dependent variables (field variables) are the Velocity field u (SI unit: m/s) and the Pressure p (SI unit: Pa). The name can be changed but the names of fields and dependent variables must be unique within a component.

Add or remove species in the model and also change the names of the dependent variables that represent the species concentrations. Enter the Number of species. Use the Add concentration ( ) and Remove concentration ( ) buttons as required.

More information on the applicability of the physics interface can be found in the individual documentation for the Transport of Diluted Species and the Free and Porous Media Flow interfaces. See Domain, Boundary, Point, and Pair Nodes for the Reacting Flow in Porous Media (rfds) Interface for links to the feature nodes, most of which are shared with other interfaces.




Domain, Boundary, Point, and Pair Nodes for the Reacting Flow in Porous Media (rfds) Interface

The Reacting Flow in Porous Media (rfds) Interface has these domain, boundary, point, and pair nodes, listed in alphabetical order, available from the Physics ribbon toolbar (Windows users), Physics context menu (Mac or Linux users), or right-click to access the context menu (all users).

The Initial Values and Transport Properties nodes are described in this section. The following are described for the Free and Porous Media Flow or Brinkman Equations

interface:

The following nodes (listed in alphabetical order) are described for the Transport of

Diluted Species interface:




• Mass Source



• Volume Force

For the Reacting Flow in Porous Media (rfds) interface, the Point Mass Source node is available in two versions: one is used for the fluid flow (Fluid Point

Source) and one for the species (Species Point Source).

The same applies to the Line Mass Source node, one is used for the fluid flow (Fluid Line Source) and one for the species (Species Line Source).


548 | C H A P T E

The following nodes (listed in alphabetical order) are described for the Laminar Flow interface :

Transport Properties

Use the Transport Properties node to define the fluid properties, including density and dynamic viscosity. Also define the diffusion coefficient and, if applicable, the mobility and the charge number.



• Concentration

• Flux

• Inflow


• No Flux

• Open Boundary

• Outflow



• Reactions

• Symmetry

• Thin Diffusion Barrier


• Flow Continuity

• Inlet

• Outlet



• Symmetry

• Wall








Define model inputs, for example, when the temperature field of the material model uses a temperature-dependent material property. If no model inputs are required, this section is empty.


Fluid material, Density, and Dynamic viscosity are described for the Free and Porous Media

Flow interface, Fluid Properties node.

D I F F U S I O N

If materials have been added to the Materials node, select an option from the Bulk

material list. Otherwise the list defaults to None and values or expressions need to be entered in the Diffusion coefficient Dc (SI unit: m2/s) field.

Initial Values

Enter values or expressions for the Initial Values of variables and concentrations solved for in the physics interface. These can serve as initial conditions for a time dependent simulation or as an initial guess for a nonlinear solver.



The Diffusion coefficient is described for the Transport of Diluted Species interface, Dynamic Transport Feature Node.

If there are several types of domains, with subsequent and different initial values occurring within them, it might be necessary to remove some of the domains from this selection. These are then defined in an additional Initial Values node.


550 | C H A P T E


Enter a value or expression for the Concentration c (SI unit: mol/m3). The default is 0 mol/m3. Enter the components for the initial value of Velocity field u (SI unit: m/s) and for the Pressure p (SI unit: Pa).


Th eo r y f o r t h e T r a n s p o r t o f D i l u t e d S p e c i e s I n t e r f a c e

The Transport of Diluted Species Interface provides a predefined modeling environment for studying the evolution of chemical species transported by diffusion and convection. The physics interface assumes that all species present are dilute; that is, that their concentration is small compared to a solvent fluid or solid. As a rule of thumb, a mixture containing several species can be considered dilute when the concentration of the solvent is more than 90 mol%. Due to the dilution, mixture properties such as density and viscosity can be assumed to correspond to those of the solvent.

Fick’s law governs the diffusion of the solutes, dilute mixtures or solutions. The Transport of Diluted Species interface supports the simulations of chemical species transport by convection and diffusion in 1D, 2D, and 3D as well as for axisymmetric models in 1D and 2D.

In this section:

• Mass Balance Equation

• Convective Term Formulation

• Solving a Diffusion Equation Only

• Mass Sources for Species Transport

• Crosswind Diffusion

• Reference

Mass Balance Equation

The default node attributed to the Transport of Diluted Species interface models chemical species transport through diffusion and convection and implements the mass balance equation:

(8-4)

Equation 8-4 includes the following quantities:

• c is the concentration of the species (SI unit: mol/m3)

ct

----- u c+ D c R+=

T H E O R Y F O R T H E TR A N S P O R T O F D I L U T E D S P E C I E S I N T E R F A C E | 551

552 | C H A P T E

• D denotes the diffusion coefficient (SI unit: m2/s)

• R is a reaction rate expression for the species (SI unit: mol/(m3·s))


The first term on the left-hand side of Equation 8-4 corresponds to the accumulation (or indeed consumption) of the species.

The second term accounts for the convective transport due to a velocity field u. This field can be expressed analytically or be obtained from coupling this physics to one that describes fluid flow (momentum balance). To include convection in the mass balance equation, an expression that includes the space and time variables, or the velocity vector component variable names from a fluid-flow interface, can be entered into the appropriate field. The velocity fields from existing fluid-flow interfaces are available directly as predefined fields (model inputs) for multiphysics couplings.

On the right-hand side of the mass balance equation (Equation 8-4), the first term describes the diffusive transport, accounting for the interaction between the dilute species and the solvent. A field for the diffusion coefficient is available, and any expression containing other variables such as pressure and temperature can be entered here. The node has a matrix that you can use to describe anisotropic diffusion coefficients.

Finally, the second term on the right-hand side of Equation 8-4 represents a source or sink term, typically due to a chemical reaction. In order for the chemical reaction to be specified, another node must be added to the Transport of Diluted Species interface—the Reaction node, which has a field for specifying a reaction equation using the variable names of all participating species.

Convective Term Formulation

The default node attributed to The Transport of Diluted Species Interface assumes chemical species transport through diffusion and convection and implements the mass balance equation in Equation 8-4.

There are two ways to present a mass balance where chemical species transport occurs through diffusion and convection. These are the non-conservative and conservative formulations of the convective term:

non-conservative: (8-5)ct

----- u c+ D c R+=


conservative: (8-6)

and each is treated slightly differently by the solver algorithms. In these equations D (SI unit: m2/s) is the diffusion coefficient, R (SI unit: mol/(m3·s)) is a production or consumption rate expression, and u (SI unit: m/s) is the solvent velocity field. The diffusion process can be anisotropic, in which case D is a tensor.

If the conservative formulation is expanded using the chain rule, then one of the terms from the convection part, c·u, would equal zero for an incompressible fluid and would result in the non-conservative formulation above. This is in fact the default formulation in this physics interface and ensures that nonphysical source terms do not emerge from a solution for the flow field.

Solving a Diffusion Equation Only

Remove the convection term from Equation 8-5 and Equation 8-6 by clearing the Convection check box in the Transport Mechanisms section for The Transport of Diluted Species Interface. The equation then becomes

Mass Sources for Species Transport

There are two types of mass sources in the Transport of Diluted Species interface: point sources and line sources.

ct

----- cu + D c R+=

To switch between the two formulations, click the Show button ( ) and select Advanced Physics Options.

ct

----- D c R+=

These features require at least one of the following licenses: Batteries & Fuel Cells Module, CFD Module, Chemical Reaction Engineering Module, Corrosion Module, Electrochemistry Module, Electrodeposition Module, Microfluidics Module, Pipe Flow Module, or Subsurface Flow Module.


554 | C H A P T E

PO I N T S O U R C E

A point source is theoretically formed by assuming a mass injection/ejection, (SI unit: mol/(m3·s)), in a small volume V and then letting the size of the volume tend to zero while keeping the total mass flux constant. Given a point source strength, (SI unit: mol/s), this can be expressed as

(8-7)

An alternative way to form a point source is to assume that mass is injected/extracted through the surface of a small object. Letting the object surface area tend to zero while keeping the mass flux constant, results in the same point source. For this alternative approach, effects resulting from the physical object’s volume need to be neglected.


is added at a point in the geometry. As can be seen from Equation 8-7, must tend to plus or minus infinity as V tends to zero. This means that in theory the concentration also tends to plus or minus infinity.

Observe that “point” refers to the physical representation of the source. A point source can therefore only be added to points in 3D models and to points on the symmetry axis in 2D axisymmetry models. Other geometrical points in 2D models represent physical lines.

The finite element representation of Equation 8-7 corresponds to a finite concentration at a point with the effect of the point source spread out over a region around the point. The size of the region depends on the mesh and on the strength of the source. A finer mesh gives a smaller affected region, but also a more extreme concentration value. It is important not to mesh too finely around a point source since this can result in unphysical concentration values. It can also have a negative effect on the condition number for the equation system.

L I N E S O U R C E

A line source can theoretically be formed by assuming a source of strength (SI unit: mol/(m3·s)), located within a tube with cross-section S and then letting S tend to zero while keeping the total mass flux per unit length constant. Given a line source strength, (SI unit: mol/(m·s)), this can be expressed as

Q·

c

q·p,c

V 0lim Q

·c

V q·p,c=

q·p,ctest c

Q·

c

Q·

l,c

q· l ,c


(8-8)

As in the point source case, an alternative approach is to assume that mass is injected/extracted through the surface of a small object. This results in the same mass source, but requires that effects resulting from the physical object’s volume are neglected.


is added on lines in 3D or at points in 2D (which represent cut-through views of lines). Line sources can also be added on the axisymmetry line in 2D axisymmetry models. It cannot, however, be added on geometrical lines in 2D since those represent physical planes.

As with a point source, it is important not to mesh too finely around the line source.

Crosswind Diffusion

Transport of diluted species applications can often result in models with very high cell Péclèt number—that is, systems where convection or migration dominates over diffusion. Streamline diffusion and crosswind diffusion are of paramount importance to obtain physically reasonable results. The Transport of Diluted Species interface provides two crosswind diffusion options using different formulations. Observe that

S 0lim Q· l,c

S q· l,c=

q· l,ctest c

For feature node information, see Line Mass Source and Point Mass Source in the COMSOL Multiphysics Reference Manual.

For the Reacting Flow in Porous Media, Diluted Species interface, which is available with the CFD Module, Chemical Reaction Engineering Module, or Batteries & Fuel Cells Module, these shared physics nodes are renamed as follows:

• The Line Mass Source node is available as two nodes, one for the fluid flow (Fluid Line Source) and one for the species (Species Line Source).

• The Point Mass Source node is available as two nodes, one for the fluid flow (Fluid Point Source) and one for the species (Species Point Source).


556 | C H A P T E

crosswind diffusion makes the equation system nonlinear even if the transport equation is linear.

D O C A R M O A N D G A L E Ã O

This is the formulation described in Numerical Stabilization. The method reduces over- and undershoots to a minimum, even for anisotropic meshes.

In some cases, the resulting nonlinear equation system can be difficult to converge. This can happen when the cell Péclèt number is very high and the model contains many thin layers such as contact discontinuities. You then have three options:

• Refine the mesh, especially in regions with thin layers.

• Use a nonlinear solver with a constant damping factor less than one.

• Switch to the Codina crosswind formulation.

C O D I N A

The Codina formulation is described in Ref. 1. It adds diffusion strictly in the direction orthogonal to the streamline direction. Compared to the do Carmo and Galeão formulation, the Codina formulation adds less diffusion but is not as efficient at reducing over- and undershoots. It also does not work as well for anisotropic meshes. The advantage is that the resulting nonlinear system is easier to converge and that under-resolved gradients are less smeared out.

Reference

1. R. Codina, “A discontinuity-capturing crosswind-dissipation for the finite element solution of the convection-diffusion equation,” Computer Methods in Applied Mechanics and Engineering, vol. 110, pp. 325–342, 1993.


A flow field obtained using a turbulence model does not explicitly contain the small eddies. These unresolved eddies still have a profound effect on the species transport, an effect known as turbulent mixing.

The Transport of Diluted Species interface supports the inclusion of turbulent mixing via the gradient-diffusion hypothesis which adds the following contribution to the diffusion coefficient tensor:


(8-9)

where T is the turbulent kinematic viscosity, ScT is the turbulent Schmidt number and I is the unit matrix. While having the form of a diffusive contribution, Equation 8-9 really models a convective phenomenon and it can therefore only be applied along with convection.

The Schmidt number is typically given a value between 0.7 and 0.72, but it can range between 0.3 and 1.3 (at least) depending on the application.

TScT----------I


558 | C H A P T E

Th eo r y f o r t h e T r a n s po r t o f C on c en t r a t e d S p e c i e s I n t e r f a c e

The Transport of Diluted Species Interface theory is described in this section:

• Multicomponent Mass Transport

• Multicomponent Diffusion: Mixture-Average Approximation

• Multispecies Diffusion: Fick’s Law Approximation

• Multicomponent Thermal Diffusion


• References for the Transport of Concentrated Species Interface

Multicomponent Mass Transport

Suppose a reacting flow consists of a mixture with i1, …, Q species and j = 1, …, N reactions. Equation 8-1 then describes the mass transport for an individual species:

(8-10)

where, (SI unit: kg/m3) denotes the mixture density and u (SI unit: m/s) the mass averaged velocity of the mixture. The remaining variables are specific for each of the species, i, being described by the mass transfer equation:

• i is the mass fraction (1)

• ji (SI unit: kg/(m2·s)) is the mass flux relative to the mass averaged velocity, and

• Ri (SI unit: kg/ (m3·s)) is the rate expression describing its production or consumption.

The relative mass flux vector ji can include contributions due to molecular diffusion and thermal diffusion.

Summation of the transport equations over all present species gives Equation 8-11 for the conservation of mass

(8-11)

t i iu + ji R+ i–=

t u + 0=


assuming that

, ,

Using the mass conservation equation, the species transport for an individual species, i, is given by:

(8-12)

Q 1 of the species equations are independent and possible to solve for using Equation 8-12. To compute the mass fraction of the remaining species, COMSOL Multiphysics uses the fact that the sum of the mass fractions is equal to 1:

(8-13)

Multicomponent Diffusion: Mixture-Average Approximation

The mixture-averaged diffusion model assumes that the relative mass flux due to molecular diffusion is governed by a Fick’s law type expression

(8-14)

where i is the density and xi the mole fraction of species i. The diffusion hence depends on a single concentration gradient and is proportional to a diffusion coefficient . The diffusion coefficient describes the diffusion of species i relative to the remaining mixture and is referred to as the mixture-averaged diffusion coefficient. Equation 8-14 can be expressed in terms of mass fractions as

using the definition of the species density and mole fraction

,

i

i 1=

Q

1= ji

i 1=

Q

0= Ri

i 1=

Q

0=

t i u i+ – ji Ri+=

1 1 i

i 2=

Q

–=

jmd i iDim xi

xi---------–=

Dim

jmd i Dim i

i

M-----Di

m M+ –=

i i= xii

Mi-------M=

T H E O R Y F O R T H E TR A N S P O R T O F C O N C E N T R A T E D S P E C I E S I N T E R F A C E | 559

560 | C H A P T E

Assuming isobaric and isothermal conditions, the following expression for the mixture-averaged diffusion coefficient can be derived from the Maxwell-Stefan equations (Ref. 3):

The mixture-averaged diffusivities are explicitly given by be the multicomponent Maxwell-Stefan diffusivities Dik. For low-density gas mixtures, the Dik components can be replaced by the binary diffusivities for the species pairs present.

When using the mixture-averaged diffusion model, the species mass transport equations are

Apart from molecular diffusion, transport due to thermal diffusion is accounted for through the third term within the parenthesis on the right-hand side. Here (SI unit: kg/(m·s)) is the thermal diffusion coefficient.

Multispecies Diffusion: Fick’s Law Approximation

Using a Fick’s law approximation, the relative mass flux due to molecular diffusion is assumed to be governed by

(8-15)

where represents a general diffusion matrix (SI unit: m2/s) describing the diffusion of species i into the mixture. This form makes it possible to use any diffusion coefficient, matrix, or empirical model based on Fick’s law. For example, in situations when the mass transport is not dominated by diffusion, an alternative is to use the diffusion coefficients at infinite dilution,

Dim 1 i–

xkDik---------

k i

N--------------------------=

t i u i+ =

Dim i iDi

m MM

--------- DiT T

T--------+ +

Ri+

DiT

jmd i iDi klF xi

xi---------–=

Di klF

Di kkF Di

0=


These coefficients are typically more readily available compared to the binary diffusion diffusivities, especially for liquid mixtures.

When using multicomponent diffusivities based on Fick’s law, as described above, or when using mixture-averaged coefficients, the sum of the relative mass fluxes is not strictly constrained to zero. To reduce the relative error it is preferable to choose the species with the highest mass fraction as the species that is not solved for, and which is instead computed from the mass conservation constraint in Equation 8-2. It is not always necessary to know in advance which species has the highest mass fraction—it is possible to change the species solved for by the mass conservation constraint in Equation 8-2.

When using the Fick’s law approximation for the diffusion model, the species mass transport equations are:

Apart from molecular diffusion, transport due to thermal diffusion is accounted for through the third term within the parenthesis on the right-hand side. Here is the thermal diffusion coefficient (SI unit: kg/(m·s)).

Multicomponent Thermal Diffusion

Mass diffusion in multicomponent mixtures due to temperature gradients is referred to as the Soret effect. This occurs in mixtures with high temperature gradients and large variations in molecular weight (or size) of the species. Typically species with high molecular weight accumulate in lower temperature regions while the diffusion due to the Soret effect transports species with low molecular weight to higher temperature regions. In COMSOL Multiphysics, thermal diffusion is included by prescribing the thermal diffusion coefficients . In a multicomponent mixture, the sum of the thermal diffusion coefficients is zero:

t i u i+ =

DiF i iDi

F MM

--------- DiT T

T--------+ +

Ri+

DiT

DiT

DiT

i 1=

Q

0=

T H E O R Y F O R T H E TR A N S P O R T O F C O N C E N T R A T E D S P E C I E S I N T E R F A C E | 561

562 | C H A P T E

Turbulent Mixing

A flow field obtained using a turbulence model does not explicitly contain the small eddies. These unresolved eddies still have a profound effect on the species transport, an effect known as turbulent mixing.

The Transport of Concentrated Species Interface supports the inclusion of turbulent mixing via the gradient-diffusion hypothesis. In the case of mixture-averaged diffusion, it adds the following contribution to the diffusion coefficient for each species:

(8-16)

while for Fick’s law diffusion, it adds the following contribution to the diffusion coefficient tensor:

(8-17)

Here, T is the turbulent kinematic viscosity, ScT is the turbulent Schmidt number and I is the unit matrix. While having the form of diffusive contributions, Equation 8-16 and Equation 8-17 really model a convective phenomenon and they can therefore only be applied along with convection.

The Schmidt number is typically given a value between 0.7 and 0.72, but it can range between 0.3 and 1.3 (at least) depending on the application.

References for the Transport of Concentrated Species Interface

1. C.F. Curtiss and R.B. Bird, “Multicomponent Diffusion,” Ind. Chem. Res., vol. 38 pp. 2515–2522, 1999.

2. R.B. Bird, W.E. Stewart, and E.N. Lightfoot, Transport Phenomena, 2nd ed., John Wiley & Sons, 2005.

3. R.J. Kee, M.E. Coltrin, and P. Glarborg, Chemically Reacting Flow, John Wiley & Sons, 2003.

4. A. Soulaïmani and M. Fortin, “Finite Element Solution of Compressible Viscous Flows Using Conservative Variables,” Computer Methods in Applied Mechanics and Engineering, vol. 118, pp. 319–350, 1994.

TScT----------

TScT----------I


Th eo r y f o r t h e R e a c t i n g F l ow I n t e r f a c e s

The Reacting Flow Interfaces equations are basically the same as for the Transport of Concentrated Species and the Laminar Flow interfaces.

The following sections describe theory specific for the Reacting Flow interfaces:

• Pseudo Time Stepping for Mass Transport

• The Stefan Velocity

• The Chemical Reaction Rate

• Turbulent Mass Transport Models

• Mass Transport Wall Functions

• Turbulent Reactions

• The Reaction Feature

• References for the Reacting Flow Interfaces

Pseudo Time Stepping for Mass Transport

In order to improve the solution robustness, pseudo time stepping can be used (and is enabled by default) when solving a stationary model. Using pseudo time stepping, a fictitious time derivative term:


• The Single-Phase Flow Interface Options


i nojac i –

t------------------------------------

T H E O R Y F O R T H E R E A C T I N G F L O W I N T E R F A C E S | 563

564 | C H A P T E

is added to the left-hand side of the species transport equations. Here is the fluid density (SI unit: kg/m3), i is the mass fraction (dimensionless) of species i, and is the pseudo time step. Since inojac(i) is always zero, this term does not affect the final solution. It does, however, affect the discrete equation system and effectively transforms a nonlinear iteration into a time step of size .

The Stefan Velocity

Heterogeneous reactions on fluid-solid surfaces can affect the mass, momentum, and energy balances at the interface separating the fluid and the solid. On the reacting surface, the production or destruction rate, rs,i (SI unit: mol/(m2·s)), of a fluid phase species is balanced by the total mass flux of the species. The mass balance for species i on a boundary representing a fluid-solid interface is given by:

(8-18)

Here, n is the unit normal pointing out of the fluid domain, u is the mass averaged velocity of the fluid mixture (SI unit: m/s), ud,i denotes the diffusion velocity (SI unit: m/s), the velocity of species i relative to the mixture, and Mi is the species molar mass (SI unit: kg/mol). Summing the mass balances at the surface, over all species, results in an effective mixture velocity:

(8-19)

referred to as the Stefan velocity, here denoted us. To reach Equation 8-19 the fact that the sum of all mass fractions is one, and that the sum of all relative diffusive fluxes is zero, was used.

Equation 8-19 implies that surface reactions result in a net flux between the surface and the domain. A net flux in turn corresponds to an effective convective velocity at the domain boundary; the Stefan velocity. It should be noted here that when solving for mass transport inside a fluid domain, an outer boundary of the domain corresponds

t

t

For a description of the pseudo time step term for the Navier-Stokes equations and the pseudo time step see Pseudo Time Stepping for Laminar Flow Models and Pseudo Time Stepping in the COMSOL Multiphysics Reference Manual.

n i u ud i+ rs,iMi=

n us rs,iMi

i 1=

N

=


to a position just outside of the a actual physical wall (on the fluid side). The domain boundary does not coincide with the physical wall.

In the Reacting Flow interfaces, the species mass fractions in the fluid domain are solved for. Since the surface concentrations (mol per area) on exterior walls is not included, the surface reaction rates are often not explicitly known. In this case, surface reactions can be modeled either by applying a mass flux or prescribing the mass fraction or a combination of both, on fluid boundaries adjacent to the reacting surface. The Stefan velocity on a fluid domain boundary is then defined as the net mass flux resulting from the boundary conditions applied:

(8-20)

Here, the first sum contains flux contributions resulting from boundary conditions prescribing the mass fractions, and the second sum contains contributions from flux conditions. The resulting Stefan velocity based on mass transport boundary conditions is computed as:

(8-21)

Using the Reacting Flow interfaces, the Stefan velocity, defined in the manner of Equation 8-21, is automatically computed and applied on boundaries corresponding to walls. A Mass Fraction or Flux feature applied on a selection shared by a Wall feature adds a corresponding contribution to Equation 8-21. The resulting Stefan velocity is prescribed, in the wall normal direction, by the Wall feature.

The Chemical Reaction Rate

For laminar flow, the default (automatic) reaction rate used by the Reactions feature is based on the mass action law. Consider a general reaction belonging to a set of j reactions and involving i species:

n us n ius Ni d+ n N j tot

j 1=

nN

+

i 1=

nD

=

n u s

n N i d n N i tot

i 1=

nN

+

i 1=

nD

1 i

i

nD

–

----------------------------------------------------------------------------=


566 | C H A P T E

(8-22)

For such a reaction set, the reaction rates rj (SI unit: mol/(m3·s)), can be described by the mass action law:

(8-23)

Here, and denote the forward and reverse rate constants, respectively. The concentration of species i is denoted ci (SI unit: mol/m3). The stoichiometric coefficients are denoted ij, and are defined to be negative for reactants and positive for products. In practice, a reaction seldom involves more than two species colliding in a reacting step, which means that a kinetic expression is usually of order 2 or less (with respect to the involved concentrations).

In addition to the concentration dependence, the temperature dependence of reaction rates can be included by using the predefined Arrhenius expressions for the rate constants:

Here, A denotes the frequency factor, n the temperature exponent, E the activation energy (SI unit: J/mol) and Rg the gas constant, 8.314 J/(mol·K). The pre-exponential factor, including the frequency factor A and the temperature factor Tn, is given the units (m3/mol)1/s, where is the order of the reaction (with respect to the concentrations).

Turbulent Mass Transport Models

The turbulence models included in the Reacting Flow interfaces are based on averaging of the fluid flow equations. Applying a corresponding decomposition of the mass fraction into mean and fluctuating parts, and averaging the mass transport equations, additional unclosed terms are introduced in the equations. These terms need to be modeled in order to close the set of equations. The most important terms, containing the correlations of the velocity and mass fraction fluctuations, referred to as the turbulent mass transport fluxes, are given by:

+kj

bBaA

f

kjr

... + yYxX + ...+

rj kjf ci

– ij k– jr ci

ij

i prod

i react=

kjf kj

r

k ATn ERgT-----------–

exp=


Here the double primes denote Favre (density-based) fluctuations. In the case of varying density flow, Favre averaging is favored over Reynolds averaging since it reduces the number of unclosed terms and renders the equation on the same form as the incompressible RANS equations. For more background on averaging, see Turbulence Modeling.

The most common way to model this term is to use a gradient based assumption, where the additional turbulent transport is related to the turbulent viscosity through a turbulent Schmidt number ScT:

(8-24)

Here denotes the Favre averaged mass fraction which is the quantity solved for.

Using a RANS turbulence model, the turbulent mass flux is defined from Equation 8-24, and the equation solved for each species is:

where the molecular diffusion coefficient, Di, is given by the diffusion model (Mixture-Average or Fick’s Law).

K A Y S - C R A W F O R D

Assuming that the turbulent transport mechanisms of heat and mass processes are analogous, the turbulent Schmidt number is defined by (Ref. 1):

where the Schmidt number at infinity is ScT0.85, and the turbulent Peclet number is defined as the ratio of the turbulent to molecular viscosity times the Schmidt number:

ui xi

---------------------------

uiTScT---------

xi--------– DT

xi--------–= =

t i u i+ =

DiTScT---------+

i iDiM

M--------- Di

T TT

-------- izium i F V+ + + Ri+

ScT1

2ScT----------------

0.3PeT

ScT

------------------- 0.3PeT 2 1 e 1 0.3PeT ScT –– –+ 1–

=


568 | C H A P T E

H I G H S C H M I D T N U M B E R M O D E L

In the case of high Schmidt numbers, which is typical for mass transport in liquids, the mass transfer near walls can be significantly different than that for Schmidt numbers of order unity. A diffusion layer near a solid wall, due to, for example, a reaction on the wall, does not have the same properties as the (momentum) boundary layer. Most importantly the diffusion layer thickness is significantly smaller than the boundary layer thickness for high Schmidt numbers. In order to correctly capture the mass flux at the wall, the wall resolution required is dictated by the diffusion layer rather than by the boundary layer.

The High Schmidt number model is based on the model by Kubacki and Dick (Ref. 2) and is available when using the Low-Reynolds k- turbulence model (see The Reacting Flow, Turbulent Flow, Low Re k-Interface). In this case the fluid flow is resolved all the way to the physical wall and consequently the species boundary equations are applied directly on the wall.

In the near wall region, where the species transport is limited by diffusion, the mass diffusivity is modeled using an analytical function of the non-dimensional wall distance due to Na and Hanratty (Ref. 3):

The non-dimensional wall distance applied, , corresponds to that defined by the Low-Reynolds k- model. The value of the constants b and m are given in Table 8-1.

Further out from the wall, where the mass transport is governed by the turbulent transport, the transport is modeled using a turbulent Schmidt number of the form (Ref. 2):

TABLE 8-1: NA AND HANRATTY PARAMETERS

SCHMIDT NUMBER b m

0.1 < Sc < 5 7.3·10-4 3

5 < Sc < 50 5.3·10-4 3

Sc > 50 4.63·10-4 3.38

PeTT

------Sc=

DT,w

------------ b l* m

=

l*

ScT,c 0.5882 0.228T

------ 0.0441

T

------

31 e 5.165 T –– –+

1–

=


In order to combine the two descriptions, the blending function by Kader (Ref. 4) is used:

where

Mass Transport Wall Functions

Analogous to the single-phase flow wall functions (see Wall Functions applied for the Wall boundary condition), there is a theoretical gap between the solid wall and the computational domain of the fluid. This gap is often ignored when the computational geometry is drawn.

Assuming that the turbulent heat and mass transfer in the near-wall region are analogous, the same type of wall functions used for the temperature (Ref. 8) is also applicable for the mass transport. The mass transfer wall function is formulated as a function of the molecular and turbulent Schmidt numbers of each species, instead of the corresponding Prandtl numbers.

Assume that there is a mass fraction i,w just outside the wall and that it is in equilibrium with the surface chemistry. The mass flux, for species i, between the walland a fluid with a mass fraction of i,f at the lift-off position is:

where is the fluid density, C is a turbulence modeling constant, and k is the turbulent kinetic energy. is the dimensionless mass fraction given by (Ref. 8):

where in turn

DTT

ScT,c------------e – DT,we 1 –

+=

0.01 l* 4

1 5l*+ ------------------------=

mwfC

1 4/ k1 2/ i w i f–

i+

------------------------------------------------------------=

i+

i+

Scw+ for w

+ w1+

15Sc2 3/ 500w

+2----------–

for w1+ w

+ w2+

ScT

--------- lnw+ + for w2

+ w+

=


570 | C H A P T E

v is the von Karman constant, and Di,m the mean species diffusion coefficient. The latter corresponds directly to the mixture-averaged diffusion coefficient when using the Mixture-average diffusion model. Using the Fick’s law diffusion model, the mean diffusion coefficient is computed as 1N·tr(Di), where Di is the diffusion tensor and N is the dimension of the model.

The computational result should be checked so that the distance between the computational fluid domain and the wall, w, is almost everywhere small compared to any geometrical quantity of interest. The distance w is available as a post processing variable (delta_w) on boundaries.

Turbulent Reactions

In turbulent flow the reaction rate is significantly affected by the turbulence. Turbulent fluctuations can increase the reaction rate due to the increased mixing, but it can also quench it through removal of species or the heat required for ignition. The interaction between the flow and the reactions can be quantified through the Damköhler number Da:

which compares the time scale of the largest turbulent flow structures (the integral time scale), to the chemical time scale. In the limit of large Damköhler numbers Da >> 1, the reactive time scale is significantly smaller than that of the turbulence. In this regime the inner structure of a reaction zone is thin enough not to be affected by turbulence. The reacting regions are, however, convected and wrinkled by the turbulence. In this case the global reaction rate is proportional to the chemical reaction rate times the surface of the reacting regions. In the limit of small Damköhler numbers Da << 1, the chemical time scale is much larger than that of the turbulent flow. In this case the global reaction rate is controlled by the chemistry, while the turbulence acts

w+

w C1 2/ k

------------------------------= w1

+ 10Sc1 3/-------------=

w2+ 10 10

ScT---------= Sc

Di m----------------=

15Sc2 3/ScT2v--------- 1 ln 1000

ScT---------

+ –=

Da =Tc------


by continuously mixing the species. This regime is referred to as a “perfectly stirred reactor”.

M E A N V A L U E C L O S U R E

A mean turbulent production rate can be obtained by directly applying the mean concentrations and temperature in the kinetic reaction rate expressions:

where i denotes the species stoichiometric coefficient, Mi the molar mass (SI unit: kg/mol), and rEDC,i is the reaction rate from Equation 8-23 using mean quantities.

This is referred to as mean value closure of the turbulent reaction rate, and corresponds to keeping the first term in a Taylor series expansion of the turbulent reaction rate following a Reynolds decomposition of the fluctuating variables (Ref. 5). The mean value closure is directly applicable in the perfectly-stirred-reactor limit (Da << 1). In this case the turbulence is able to mix the species in the sense of changing the mean concentration, but the turbulent fluctuations do not affect the chemical reaction rate. The mean value closure is also applicable in the limit of low turbulence levels since the turbulent fluctuations tend to zero as the laminar flow regime is approached.

E D D Y D I S S I P A T I O N C O N C E P T

The eddy dissipation concept (EDC) is a model for the mean reaction rate originally developed by Magnusson and Hjertager (Ref. 6) for non-premixed combustion. Using this reaction model, the mean production rate of species i is defined as:

The model assumes that both the Reynolds and Damköhler numbers are sufficiently high for the reaction rate to be limited by the turbulent mixing time-scale T. The reaction can hence at most progress at the rate at which fresh reactants are mixed, at the molecular level, by the turbulence present. Furthermore the reaction rate is limited by the deficient reactant; the reactant with the lowest local concentration. When the model parameter is finite, the existence of product species is also required for reaction, modeling the activation energy required for reaction (ignition). For gaseous non-premixed combustion the model parameters have been found to correspond to (Ref. 6):

Ri iMirMVC,i=

Ri iMiT------ min min

rrMr-------------

ppMp--------------

p iMirEDC,i= =


572 | C H A P T E

,

For liquid reactions it is recommended that the model parameters are calibrated against experimental or simulation data. Using a mix of the mean value closure and a modified version of the EDC model Bakker and Fasano (Ref. 7) found the following parameter values

,

to give good results for a competitive reaction pair when compared with experimental results.

The Reaction Feature

For turbulent flow, the default (automatic) species production rate is computed as the minimum of the mean value closure and the eddy dissipation concept model:

This approach combines the regimes of the separate models. For fast reactions the reaction rate is limited by the turbulent mixing. At the same time, in regions with low turbulence levels, or low kinetic reactions rates, the latter limits the production rate.

References for the Reacting Flow Interfaces

1. W.M. Kays, “Turbulent Prandtl Number—Where Are We?” ASME Journal of Heat Transfer, vol. 116, pp. 284–295, 1994.

2. S. Kubacki and E. Dick, “Simulation of Impinging Mass Transfer at High Schmidt Number with Algebraic Models,” Progress in Computational Fluid Dynamics, vol. 11, no. 1, pp. 30–41, 2011.

3. Y. Na and T.J. Hanratty, “Limiting Behavior of Turbulent Scalar Transport Close to the Wall”, Int. J. Heat Mass Transfer, vol. 43, pp. 1749–1758, 2000.

4. B.A. Kader, “Temperature and Concentration Profiles in Fully Turbulent Boundary Layers”, Int. J. of Heat and Mass Transfer, vol. 24, pp. 1541–1544, 1981.

5. T. Poinsot and D. Veynante, Theoretical and Numerical Combustion, 2nd. ed., R.T. Edwards, Philadelphia, 2005.

4= 0.5=

2 0.08= 1 1 2 = = =

Ri iMi min rMVC,i rEDC,i=


6. B.F. Magnussen and B.H. Hjertager, “On Mathematical Modeling of Turbulent Combustion with Special Emphasis on Soot Formation and Combustion,” 16th Symp. (Int.) on Combustion. Comb. Inst., Pittsburg, Pennsylvania, pp.719–729, 1976.

7. A. Bakker and J.B. Fasano, “Time Dependent, Turbulent Mixing and Chemical Reactions in Stirred Tank,” AIChE Symposium Series 299, vol. 90, pp. 71–78, 1993.

8. D. Lacasse, È. Turgeon, and D. Pelletier, “On the Judicious Use of the k— Model, Wall Functions and Adaptivity,” Int.J. of Thermal Sciences, vol. 43, pp. 925–938, 2004.


574 | C H A P T E

Th eo r y f o r t h e R e a c t i n g F l ow i n Po r ou s Med i a I n t e r f a c e s

In this section:

• Theory for the Reacting Flow in Porous Media (rfcs) Interface

• Theory for the Reacting Flow in Porous Media (rfds) Interface

Theory for the Reacting Flow in Porous Media (rfcs) Interface

The Reacting Flow in Porous Media (rfcs) Interface solves for the mass fractions (wi) of an arbitrary number of species, as well as for the flow field (u) and pressure (p).

D O M A I N E Q U A T I O N S

The governing equations are basically the same as for the Transport of Concentrated

Species and the Free and Porous Media Flow interfaces, whereas this physics interface adds the ability to apply correction factors to calculate effective mass transport parameters in a porous domain. The effective mass transport parameters are treated the same way as for the Reacting Flow in Porous Media, Diluted Species interface.

C O M B I N E D B O U N D A R Y C O N D I T I O N S

The combined boundary conditions for velocity and flux, for reactions occurring on a boundary surface, are as follows.

Reactions occurring on a surface in contact with the fluid give rise to mass fluxes, and if there is a net mass transport in or out of the fluid domain due to the reactions (for example if a fluid condenses or evaporates) they impose a flow velocity at the surface.

Assigning the outward boundary normal as n and the inward mass flux for each species over the boundary as ni (SI unit: kg/(m2s)), the flow velocity, u (SI unit: m/s) at the boundary is coupled to the total mass flux of the species at the reacting surface in the following way:

(8-25)

where (SI unit: kg/m3) is the density of the fluid.

u n 1--- ni n

i=


Applying no slip conditions for the surface results in

(8-26)

This combination of boundary conditions can be applied using a Reacting Boundary node.

Theory for the Reacting Flow in Porous Media (rfds) Interface

The Reacting Flow in Porous Media (rfds) Interface equations are basically the same as for the Transport of Diluted Species and the Free and Porous Media Flow interfaces, the addition being the possibility to apply correction factors to calculate effective mass transport parameters in a porous domain. The Effective Mass Transport Parameters in Porous Media are discussed in this section.

E F F E C T I V E M A S S TR A N S P O R T P A R A M E T E R S I N PO R O U S M E D I A

The effective mass transport in a porous matrix is affected by the porosity of the porous media as the matrix of solid material lowers the volume available for transport. The tortuosity of the porous structure increases the transport length, and the species might interact with the pore walls. Effective transport parameters for the diffusivities (Deff) and mobilities (um,eff) are usually hard to measure. As an approximation, the parameter values for a non-porous domain (D, um) can be used to calculate the corresponding effective values by multiplying with a correction factor feff:

(8-27)

A common way to calculate feff is to relate it to the porosity, with the following Bruggeman relation:

(8-28)

The settings for applying effective species transport parameter corrections to a domain are found under the Porous Matrix Properties node.

u 1--- ni

i

– n=

Deff feffD=

um eff feffum=

feff 3 2=

T H E O R Y F O R T H E R E A C T I N G F L O W I N P O R O U S M E D I A I N T E R F A C E S | 575

576 | C H A P T E

9

T h i n - F i l m F l o w

The fluid-flow interfaces are grouped by type under the Fluid Flow main branch. The physics interfaces described in this section are found under the Thin-Film Flow

branch ( ) when adding a physics interface.

In this chapter

• Modeling Thin-Film Flow

• The Thin-Film Flow Interfaces

• Theory for the Thin-Film Flow Interfaces

577

578 | C H A P T E

Mode l i n g Th i n - F i lm F l ow


The physics interfaces found under the Thin-Film Flow branch ( ) when adding a physics interface, describe momentum transport. These physics interfaces can be added either singularly or in combination with other physics interfaces describing mass and energy transfer. The thin-film flow interfaces are shown in Table 9-1.

Each of the physics interfaces can be configured to solve either the standard Reynolds

Equation (with a material or user-defined density specified at the ambient pressure), or to solve the Modified Reynolds Equation, which should be used for gas flows.The Modified Reynolds Equation assumes the ideal gas law and applies when gas pressure changes in the flow itself result in significant density changes.

TABLE 9-1: THE THIN-FILM FLOW INTERFACES DEFAULT SETTINGS

PHYSICS INTERFACE ID SPACE DIMENSION GEOMETRIC ENTITY LEVEL

Thin-Film Flow, Shell tffs 3D Boundaries

Thin-Film Flow, Edge tffs 2D, 2D axisymmetric Boundaries

Thin-Film Flow, Domain tff 2D Domains

The Thin-Film Flow, Shell Interface was previously called the Lubrication Shell interface (for both the MEMS Module and the CFD Module, solving the Reynolds Equation) and the Film Damping Shell interface (MEMS Module, solving the Modified Reynolds Equation).

In 2D, The Thin-Film Flow, Edge Interface was previously called the Lubrication Shell interface (for both the MEMS Module and the CFD Module, solving the Reynolds Equation) and the Film Damping Shell interface (for the MEMS Module, solving the Modified Reynolds

Equation). For 2D axisymmetric The Thin-Film Flow, Domain Interface was previously called the Thin-Film Flow interface (for both the MEMS Module and the CFD Module, solving the Reynolds Equation) and the Thin-Film Gas Flow interface (MEMS Module, solving the Modified

Reynolds Equation).

R 9 : T H I N - F I L M F L O W

Th e Th i n - F i lm F l ow I n t e r f a c e s

In this section:

• The Thin-Film Flow, Shell Interface

• The Thin-Film Flow, Domain Interface

• The Thin-Film Flow, Edge Interface

• Domain, Boundary, Edge, Point, and Pair Nodes for the Thin-Film Flow Branch Interfaces

The Thin-Film Flow, Shell Interface

The Thin-Film Flow, Shell (tffs) interface ( ), found under the Thin-Film Flow

branch ( ) when adding a physics interface, is used to solve the Reynolds equation or the modified Reynolds equation in a narrow channel that is represented by a surface within the geometry. It is used for lubrication, elastohydrodynamics, or gas damping simulations when the fluid channel is thin enough for the Reynolds equation or the Modified Reynolds equation to apply. The physics interface is available for 3D geometries.

This physics interface is a boundary mode, which means that the boundary level is the highest level for which this physics interface has equations and features; it does not have any features or equations on the domain level. The boundary level represents a reference surface, on which the flow is solved. On one side of the boundary level there is a wall and on the other a base surface, neither of which is represented in the geometry. The wall and base surfaces are orientated with respect to the reference surface normal as shown in Figure 9-1. Fluid flows in the gap between the wall and the base.



T H E T H I N - F I L M F L O W I N T E R F A C E S | 579

580 | C H A P T E

Figure 9-1: Diagram illustrating the orientation of the wall and the base surfaces with respect to the reference surface in the Thin-Film Flow interfaces. A vector from the reference surface to the corresponding point on the wall always points in the nref direction, where nref is the reference surface normal. Similarly a vector from the reference surface to the corresponding point on the base points in the nref direction. The height of the wall above the reference surface (hw) and the height of the base below the reference surface (hb) are also shown in the figure.

Using equations on the reference surface, the physics interface computes the pressure in a narrow gap between the wall and the base. When modeling the flow, it is assumed that the total gap height, h=hw+hb, is much smaller than the typical lateral dimension L of the reference surface. The physics interface is used to model laminar flow in thin gaps or channels. A lubricating oil between two rotating cylinders is an example of this.

When this physics interface is added, the following default nodes are also added in the Model Builder—Fluid-Film Properties, Border, and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions. You can also right-click Thin-Film Flow, Shell to select physics from the context menu.



The default identifier for the first physics interface in the component is tffs.


The default setting is to include All boundaries in the model. To choose specific boundaries, select Manual from the Selection list.



Select an Equation type—Reynolds equation (the default) or Modified Reynolds equation.


The dependent variable (field variable) is the Pressure pf. The name can be changed but the names of fields and dependent variables must be unique within a component.


To display this section, click the Show button ( ) and select Discretization. Select a Pressure—Quadratic (the default), Linear, Cubic, or Quartic. Specify the Value type when

using splitting of complex variables—Real or Complex (the default).

The Thin-Film Flow, Domain Interface

The Thin-Film Flow, Domain (tffs) interface ( ), found under the Thin-Film Flow

branch ( ) when adding a physics interface, is used to solve the Reynolds equation or the modified Reynolds equation in a narrow channel that is represented by a domain within the geometry. It is used for lubrication or gas damping simulations when the fluid channel is thin enough for the Reynolds equation or the Modified Reynolds equation to apply. The physics interface is available for 2D geometries.

When this physics interface is added, the following default nodes are also added in the Model Builder—Fluid-Film Properties, Border (the default boundary condition), and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions. You can also right-click Thin-Film Flow, Domain to select physics from the context menu.




• Journal Bearing: model library path CFD_Module/Thin-Film_Flow/

journal_bearing

• Tilted Pad Thrust Bearing: model library path CFD_Module/

Thin-Film_Flow/tilted_pad_bearing


582 | C H A P T E



The default identifier (for the first physics interface in the component) is tff.


The default setting is to include All domains in the model. To choose specific domains, select Manual from the Selection list.

The Thin-Film Flow, Edge Interface

The Thin-Film Flow, Edge (tff) interface ( ) is used to solve the Reynolds equation or the modified Reynolds equation in a narrow channel that is represented by an edge within the geometry. It is used for lubrication, elastohydrodynamics, or gas damping simulations when the fluid channel is thin enough for the Reynolds equation or the Modified Reynolds equation to apply. The physics interface is for 2D and 2D axisymmetric components.

The rest of the settings are the same as for The Thin-Film Flow, Shell Interface.




The rest of the settings are the same as for The Thin-Film Flow, Shell Interface.


Domain, Boundary, Edge, Point, and Pair Nodes for the Thin-Film Flow Branch Interfaces

For the physics interfaces under the Thin-Film Flow branch, the following domain, boundary, edge, point, and pair nodes (listed in alphabetical order) are described in this section:

Fluid-Film Properties

Use the Fluid-Film Properties node to set the wall properties, base properties, fluid properties, and the film flow model.

D O M A I N O R B O U N D A R Y S E L E C T I O N

For a default node, the setting inherits the selection from the parent node, and can not be edited; that is, the selection is automatically made and is the same as for the physics interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific domains or boundaries, or select All domains or All

boundaries as required.




• Border

• Fluid-Film Properties

• Initial Values

• Inlet

• Outlet

• Symmetry

• Wall






584 | C H A P T E


This section contains field variables that appear as model inputs, if the current settings include such model inputs. By default, this section includes the ambient pressure and the temperature.

R E F E R E N C E S U R F A C E P R O P E R T I E S

The Reference normal orientation setting allows you to reverse the direction of the reference surface normal used by the physics interface. By default this is set to Same

direction as geometry normal. To reverse the direction, select Opposite direction to

geometry normal.

WA L L P R O P E R T I E S

Enter a value or expression for the Height of wall above reference plane hw1 (SI unit: m). The default is 10 m.

By default the Additional wall displacement is User defined. Enter values or expressions for uw (SI unit: m). The defaults are 0 m. Alternatively select a feature input (defined by a separate physics interface) or None from the list.

To determine the reference normal orientation before fully solving the model, click Show Default Solver ( ) on the Study toolbar (or right-click the Study 1 node and

To define the absolute pressure, see the settings for the Heat Transfer in Fluids node.

When using a feature input to define an additional displacement based on the movement of a structure, refer to Figure 9-1 to decide whether the additional displacement should be added to the wall or the base.

• If the reference normal points out of the structure for which the displacement is provided, the structure should be considered to be the wall, and its displacement should be added to the additional wall displacement.

• If the reference normal points into the structure for which the displacement is provided, the structure is the base and its displacement should be added to the additional base displacement.


select Show Default Solver). Then expand the Study 1>Solver Configurations>Solver 1 node, right-click the Dependent variables sub-node and choose Compute to Selected.

You can then add a 2D or 3D plot group with an Arrow Surface plot (in 3D) ( ) or Arrow Line plot (in 2D) ( ), and use the Replace Expression ( ) button to plot the Reference surface normal.

Note that the orientation of the reference surface normal can be changed using the Reference normal orientation setting in the Reference Surface Properties section.

Select a Wall velocity vw (SI unit: m/s).—None (the default), Calculate from wall

displacement, or User defined. If User defined is selected, enter values or expressions for the components of vw (SI unit: m/s). The defaults are 0 m/s.

B A S E P R O P E R T I E S

Enter a value or expression for the Height of base below reference plane hb1 (SI unit: m). The default is 0 m.

By default Additional base displacement is User defined. Enter values or expressions for ub (SI unit: m). The defaults are 0 m. Or select None from the list.

Select a Base velocity vb (SI unit: m/s)—None (the default), Calculate from base

displacement, or User defined. If User defined is selected, enter values or expressions for the components of vb (SI unit: m/s). The defaults are 0 m/s.

Arrow Line and Arrow Surface in the COMSOL Multiphysics Reference Manual

See Wall Properties above for details to help you decide whether the additional displacement should be added to the wall or the base.


586 | C H A P T E


The default Dynamic viscosity (SI unit: Pa·s) is taken From material. Select User defined to enter a different value or expression. The default is 0 Pa·s.

The default Density (SI unit: kg/m3) is taken From material. Select User defined to enter a different value or expression. The default is 0 kg/m3. The density is not required if the Modified Reynolds Equation is being solved.

F I L M F L O W M O D E L

Select a Film flow model—Non slip walls (the default), Slip at walls, User defined-relative

flow function, or User defined-general. The film flow model is used to compute the mean fluid velocity as a function of the pressure gradient, the wall velocity, and the base velocity. Within the gap the fluid velocity profile is a linear combination of the Poiseuille and Couette velocity profiles.

Non Slip WallsThis flow model assumes no slip at both the wall and base surfaces. Thus the average fluid velocity is computed by assuming that the fluid velocity at the wall and base is equal to the wall and base velocity, respectively.

Slip at WallsUse Slip at walls when slip occurs at the wall and/or the base. In this case the difference between the wall or base velocity and the fluid velocity is proportional to the tangential part of the of the normal stress tensor component. The slip length divided by the fluid viscosity is the constant of proportionality in this relationship. The mean fluid velocity is computed using this assumption, given the pressure gradient and the wall and base velocities.

Enter a Slip length, wall Lsw (SI unit: m). The default is 0.1 m. Select the Use different

slip length for base check box to enter a Slip length, base Lsb (SI unit: m). The default is 0.1 m.

For the Modified Reynolds Equation it is possible to use the gas mean free path to specify the slip length. Change the Type of Slip setting (which defaults to Slip Length with the


settings described above) to Mean free path and same accommodation coefficients or to Mean free path and different accommodation coefficients.

• If Mean free path and same accommodation coefficients is chosen, enter a value for the Wall and base accommodation coefficient (dimensionless). The default is 0.9.

• If Mean free path and different accommodation coefficients is chosen, enter values for the Wall accommodation coefficient w (dimensionless, default 0.9) and the Base

accommodation coefficient b (dimensionless, default 0.9).

Select an option to define the Mean free path—Compute from material properties (the default), User defined expression, or User defined with reference pressure.

• If User defined expression is selected, enter an expression for the Mean free path (SI unit: m). The default expression is ((70[nm])´(1[atm]))/(tffs.ptot).

• If User defined with reference pressure is selected, enter values for the Mean free path

at reference pressure 0 (SI unit: m, the default is 70 nm), and for the Mean free path

reference pressure p0 (SI unit: Pa, the default is 1 atm).

Rarefied-Total Accommodation (Modified Reynolds Equation Only)The Rarefied-total accommodation option provides a rarefied gas model that assumes total accommodation at the wall and the base. This model is accurate to within 5% over the range 0Kn880 (here Kn is the Knudsen number, which is the ratio of the film thickness to the mean free path). An empirical function, fitted to stationary solutions of the Boltzmann equation, is used to define the Poiseuille component of the flow.


• If User defined expression is selected, enter an expression for the Mean free path (SI unit: m). The default expression is ((70[nm])´(1[atm]))/(tffs.ptot).


at reference pressure 0 (SI unit: m, the default is 70 nm) and for the Mean free path


Select a Force model. Select:

• Normal-pressure forces only (the default) to include only the normal pressure forces in the model.


588 | C H A P T E

• Couette (slide film) forces only to include only the shear forces generated from an empirical model of the rarefied flow developed for pure Couette flows.

• Shear and normal forces to include both the shear and pressure forces, combining the other two force models.

Rarefied-General Accommodation (Modified Reynolds Equation Only)The Rarefied-general accommodation option provides a rarefied gas model that assumes the same accommodation coefficient, , at the wall and the base. This model is accurate to within 1% over the ranges 0.71 and 0.01Kn100 (here Kn is the Knudsen number, which is the ratio of the film thickness to the mean free path). An empirical function, fitted to stationary solutions of the Boltzmann equation, is used to define the Poiseuille component of the flow.


• If User defined expression is selected, enter an expression for the Mean free path (SI unit: m). The default expression is 70[nm]*1[atm]/tffs.ptot.


at reference pressure 0 (SI unit: m, the default is 70 nm) and for the Mean free path


Select a Force model. Select:

• Normal-pressure forces only (the default) to include only the normal pressure forces in the model.

• Couette (slide film) forces only to include only the shear forces generated from an empirical model of the rarefied flow developed for pure Couette flows.

• Shear and normal forces to include both the shear and pressure forces, combining the other two force models.

User Defined-relative Flow FunctionThe User defined-relative flow function option enables user-defined models in which an effective fluid viscosity is employed. The fluid viscosity is divided by an additional factor Qch, which can be defined as an arbitrary expression in the GUI. It is also possible to define the expressions for the fluid forces on the wall and on the base (these are included as feature inputs in other physics interfaces).

Enter values or expressions for:

• Relative flow rate function Qch (dimensionless). The default is 0.


• Fluid load on wall fw (SI unit: N/m2). The defaults are 0 N/m2.

• Fluid load on base fb (SI unit: N/m2). The defaults are 0 N/m2.

User Defined - GeneralThe User defined-general option allows for arbitrary flow models to be defined. Both the Poiseuille and Couette terms in the mean velocity can be defined arbitrarily. It is also possible to define the expressions for the fluid forces on the wall and on the base, (these are included as feature inputs in other physics interfaces).

Enter values or expressions for:

• Poiseuille mean fluid velocity coefficient vave,P (SI unit: m3s/kg). The default is 0 m3s/kg.

• Couette mean fluid velocity component vave,C (SI unit: m/s). The defaults are 0 m/s.

• Fluid load on wall fw (SI unit: N/m2). The defaults are 0 N/m2.

• Fluid load on base fb (SI unit: N/m2). The defaults are 0 N/m2.

Initial Values

The Initial Values node adds an initial value for the pressure that can serve as an initial condition for a transient simulation or as an initial guess for a nonlinear solver. If more than one set of initial values is required, add additional Initial Values nodes from the Physics toolbar.

D O M A I N O R B O U N D A R Y S E L E C T I O N

For a default node, the setting inherits the selection from the parent node, and can not be edited; that is, the selection is automatically made and is the same as for the physics interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific domains or boundaries, or select All domains or All

boundaries as required.


Enter a value or an expression for the initial value of the Pressure pf (SI unit: Pa).

Inlet

Use the Inlet node to define an edge or point where fluid enters the gap.


590 | C H A P T E

B O U N D A R Y , E D G E , O R PO I N T S E L E C T I O N

From the Selection list, choose the boundaries, edges, or points at which to define the inlet.

I N L E T S E T T I N G S

Select an Inlet condition—Zero pressure (the default), Pressure, or Normal inflow velocity.

• If Zero pressure is selected, pf0 applies on the boundary, edge, or point.

• If Pressure is selected, enter a Pressure pf0 (SI unit: Pa) to impose pfpf0 on the boundary, edge, or point. The default is 0 Pa.

• If Normal inflow velocity is selected, enter a Normal inflow velocity U0 (SI unit: m/s). The default is 0 m/s.

Outlet

The Outlet node defines an edge or point where fluid leaves the gap.


From the Selection list, choose the boundaries, edges, or points at which to apply the condition.

O U T L E T S E T T I N G S

Select an Outlet condition— Zero pressure (the default), Pressure, or Normal outflow

velocity.

• If Zero pressure is selected, pf0 applies on the boundary, edge, or point.

• If Pressure is selected, enter a Pressure pf0 (SI unit: Pa) to impose pfpf0 on the boundary, edge, or point. The default is 0 Pa.

• If Normal outflow velocity is selected, enter a Normal outflow velocity U0 (SI unit: m/s). The default is 0 m/s.

Border

Use the Border node to set the pressure condition at the border, and the border flow type.




interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific geometric entity (boundaries, edges, or points), or select All boundaries, All edges, or All points as required.

B O R D E R S E T T I N G S

Select a Border condition—Zero pressure (the default), Pressure, or Border flow.

If Pressure is selected, enter a Pressure pf0 (SI unit: Pa) to define pfpf0 on the boundary, edge, or point. The default is 0 Pa.

If Border flow is selected, also choose a Border flow type—Acoustic boundary

condition-Absolute elongation, Acoustic boundary condition-Relative elongation, or User

defined.

• If Acoustic Boundary condition-Absolute elongation is selected, enter a Border

elongation L (SI unit: m). The default is 0.1 m.

• If Acoustic boundary condition-Relative elongation is selected, enter a Relative border

elongation Lr (dimensionless). The default is 0.7.

• If User defined is selected, enter a Normal pressure gradient (SI unit: N/m3). The default is pf/(0.1 m) N/m3.

Wall

The Wall node prevents fluid flow in the direction perpendicular to the boundary.

B O U N D A R Y, E D G E O R PO I N T S E L E C T I O N


The Border flow condition is used to account for the pressure drop caused by the flow converging into the gap, outside the thin layer. It does this by adding additional thickness to the thin layer beyond the edge of the geometry (assuming a pressure gradient in this layer equal to that at the boundary). This is known as an acoustic boundary condition. Alternatively the pressure gradient at the boundary can be specified using an arbitrary expression, which allows for more sophisticated, user-defined models.


592 | C H A P T E

Symmetry

The Symmetry node sets the perpendicular component of the average velocity in the gap to zero.

B O U N D A R Y , E D G E O R PO I N T S E L E C T I O N



Th eo r y f o r t h e Th i n - F i lm F l ow I n t e r f a c e s

The theory for the different versions of the Thin-Film Flow interface is described in this section:

• Thin-Film Flow

• The Reynolds Equation

• Flow Models

• The Modified Reynolds Equation—Gas Flows

• Flow Models for Rarefied Gases

• Frequency Domain Formulation

• Boundary Conditions

• References for the Thin-Film Flow Interfaces

Thin-Film Flow

Figure 9-2 shows a typical configuration for the flow of fluid in a thin layer. The upper boundary is referred to as the wall, and the lower boundary is referred to as the base. Damping or lubrication forces operate on both surfaces.

In many cases the gap is sufficiently small for the flow in the thin film to be isothermal. Usually the gap thickness, h, is much smaller than the lateral dimensions of the geometry, L. If this is the case it is possible to neglect inertial effects in the fluid in comparison to viscous effects (for MEMS devices this assumption is reasonable below MHz frequencies). Additionally, the curvature of the reference surface can be ignored when h/L«1. Under these assumptions the Reynolds equation applies. For gas flows under the same conditions it is possible to derive a modified form of the Reynolds Equation, which uses the ideal gas law to eliminate the density from the equation system. Such a modified Reynolds equation can even be adapted to model the flow of rarefied gases.

Different terminology is used for thin-film flow in different fields of physics. In tribology the term lubrication is frequently used, especially when the fluid is a liquid. In resonant MEMS devices, flow in the thin layer of gas separating a device from the substrate on which it is fabricated often provides significant damping. In this case

T H E O R Y F O R T H E T H I N - F I L M F L O W I N T E R F A C E S | 593

594 | C H A P T E

there is usually a distinction between squeezed-film damping, when the direction of motion of the structure is predominantly perpendicular to the reference plane, and slide-film damping for motion predominantly parallel to the reference plane.

Figure 9-2: An example illustrating a typical configuration for thin-film flow. A reference surface with normal nref sits in a narrow gap between a wall and base. In COMSOL Multiphysics the vector nref points into the base and out of the wall. The wall moves with a displacement uwall and velocity vwall from its initial position. Similarly the base moves from its initial position with displacement ubase and velocity vbase. The compression of the fluid results in an excess pressure, pf, above the ambient pressure, pA, and a fluid velocity in the gap. At a point on the reference surface the average value of the fluid velocity along a line perpendicular to the surface is given by the in plane vector vave. The motion of the fluid results in forces on the wall (Fwall) and on the base (Fbase). The distance to the wall above the reference surface is hw while the base resides a distance hb below the reference surface. The total size of the gap is h=hw+hb. At a given time hwhw1nref uwall and hbhb1nref uwall where hw1 and hb1 are the initial distances to the wall and base respectively.

The Reynolds Equation

The equations of fluid flow in thin films are usually formulated on a reference surface in the Eulerian frame. Consider a cylinder, fixed with respect to a stationary reference surface, as shown in Figure 9-3.


Figure 9-3: Reference cylinder, fixed with respect to the stationary reference surface, in a small gap between two surfaces (the wall and the base). The cylinder has outward normal, nc.

The cylinder is fixed with respect to the reference surface, but its height can change due to changes in the position of the base and the wall. The rate of change of the total mass enclosed by the cylinder is determined by the density of the fluid, the net flow of fluid into or out of the cylinder through the curved surface, and the rate of change of volume of the cylinder.

(9-1)

where is the density of the fluid, dAc is an area element on the cylinder, and n is the outward unit normal on the cylinder. On the curved surface of the cylinder v is the bulk velocity of the fluid (which is a liquid, gas, or even a rarefied gas) measured in the stationary frame of the control volume.

Equation 9-1 is expressed so that only quantities relating to the reference surface are involved. First, it is assumed that the density does not vary in the direction of the normal to the reference surface It is either constant or can be assigned a continuous, differentiable average value on the reference surface. Therefore:

t Vd

V v n Acd

Ac

+ 0=


596 | C H A P T E

(9-2)

where h=hb+hw and A is the projection of the cylinder onto the reference plane.

The area integral in Equation 9-1 can be split into three parts, integrals over the upper and lower surfaces of the cylinder, and an integral over the curved surface, or mantle, of the cylinder:

where Aw is the area of the cylinder adjacent to the wall, Ab is the area adjacent to the base, Am is the mantle of the cylinder, vw is the velocity of the wall, vb is the velocity of the base, and nc is the outward pointing normal on the cylinder.

Consider the flow through the cylinder mantle. It is assumed that the flow can be represented by a continuous, differentiable, tangential velocity (vav) defined on the reference surface. vav represents the mean velocity of the flow in the reference plane. The flow across the mantle can then be written as:

(9-3)

where the divergence theorem was applied in the second step.

Figure 9-4 shows an element of area of the projection of the cylinder onto the reference plane. Coordinates x’, y’, and z’ are introduced with the x’-y’ plane being in the reference plane (see Figure 9-3). In the x’-z’ plane the wall directed normal is given by:

Applying the same argument in the y-z plane and then generalizing, gives the directed element of area on the wall as:

(9-4)

t Vd

V hb hw+

t Ad

A h

t Ad

A= =

v n AcdAc

vw nc AwdAw

vb nc AbdAb

v nc AmdAm

+ +=

v nc AmdAm

h vav nc lmdlm

t hvav AdA= =

ncdAc ncdx'

cos------------- nwx' nwz', dx'

cos------------- sin– cos, dx'

cos-------------= = =

ncdAc tan– 1, dx'dhw

dx'-----------– 1,

dx'= =

nc Acdx'd

dhc–y'd

dhc– 1,, dx'dy' nr– thw– dA= =


Similarly for the base of the cylinder the following relationship is obtained:

(9-5)

Figure 9-4: An example illustrating the angles between the normal on the reference surface and that of the wall in the x and y directions on the reference surface.

The integral over the intersection of the wall and the cylinder is therefore given by:

(9-6)

where vw is the velocity of the wall. For the base the following integral is obtained:

(9-7)

nb Abd nr thw– dA=

vw nc AcdAw

vw n– r thw– dAA vw thw–

thw+

AdA= =

vb nc AcdAb

vb nr thb– dAA vb thb–

thb+

dAA= =


598 | C H A P T E

Substituting Equations 9-2 to 9-7 into Equation 9-1 gives:

Note that:

so:

Since the cylinder was arbitrary, the equation must hold point-wise, and the following final form of the Reynolds equation is derived:

(9-8)

The definition of the average flow velocity, vav, together with the forces that act on the structure, remain to be specified.

Flow Models

The Navier stokes equations can be non-dimensionalized for a domain whose width (h0) is much smaller than its lateral dimension(s) (l0) (see Ref. 1 for a detailed discussion). When Re(h0/l0)2<<1, and terms of order (h0/l0)2 are neglected, the Navier Stokes equations reduce to a modified form of the Stokes equation, which must be considered in conjunction with the continuity relation. Once again, the equations are most conveniently expressed by considering a local coordinate system in which x’ and y’ are tangent to the plane of the reference surface, and z’ is perpendicular to the surface. Using this coordinate system:

ht

t hu + v– w thw vb thb–t

hw

thb+ +

+ Ad

A 0=

t hb hw+ hb hw+

t

thw

thb+

+=

t h t hvav vw thw vb thb+ –+

AdA 0=

t h t hvav vw thw vb thb+ –+ 0=


Here pf is the pressure resulting from the fluid flow, is the fluid viscosity, and (vx’,vy’) is the fluid velocity in the reference plane (which varies in the z’ direction). These equations can be integrated directly, yielding the in-plane velocity distributions, by making the assumption that the viscosity represents the mean viscosity through the film thickness. The following equations are derived:

(9-9)

(9-10)

The constants C1x’, C2x’, C1y’, and C2y’ are determined by the boundary conditions. Equation 9-9 shows that the flow is a linear combination of laminar Poiseuille and Couette flows. The velocity profile is quadratic in form, as shown in Equation 9-10.

The average flow rate in the reference plane, vav , is given by:

The forces acting on the walls are determined by the normal component of the viscous stress tensor, , at the walls (n - where n is the normal that points out of the fluid domain). The viscous stress tensor takes the form:

x'pf

z'

z'vx'

=

y'pf

z'

z'vy'

=

z'p 0=

z'vx' z'

---

x'pf C1x'

----------+=

z'vy' z'

---

y'pf C1y'

----------+=

vx'z2

2-------

x'pf C1x'z'

-------------- C2x'+ += vy'

z2

2-------

y'pf C1y'z'

-------------- C2y'+ +=

vav

1h--- vx' z'd

hb–

hw

1h--- vy' z'd

hb–

hw

1h---

hw2 hwhb– hb

2+

6---------------------------------------------

x'pf hw hb–

2--------------------C

1x'C2x'+ +

hw2 hwhb– hb

2+

6---------------------------------------------

y'pf hw hb–

2--------------------C

1y'C2y'+ +

= =


600 | C H A P T E

Neglecting the gradient terms, which are of order h0/l0, results in the following form for the stress tensor:

(9-11)

The components of the stress tensor can be expressed in terms of the velocity and pressure gradients using Equation 9-9. Note that the normals to both the wall and the base are parallel to the z’ direction, to zeroth order in h0/l0. The forces acting on the base and the wall are therefore given by:

G E N E R A L S L I P B O U N D A R Y C O N D I T I O N

Assuming a slip length of Lsw at the wall and a slip length of Lsb at the base, the general slip boundary conditions are given by:

v v T + pI–

pf– 2x'

vx'+ x'

vy'

y'vx'+

x'

vz'

y'vx'+

y'

vx'

x'vy'+

pf– 2y'

vy'+ y'

vz

z'vy'+

z'

vx'

x'vz'+

z'

vy'

y'vz+

pf– 2z'

vz'+

= =

pf– 0 z'

vx'

0 pf– z'

vy'

z'

vx' z'

vy' pf–

fw

hw

-------x'

pf C1x'

----------+

hw

-------

y'pf C1y'

----------+

pf

= fb

hb------

x'pf C1x'

----------+

hb

------

y'pf C1y'

----------+

pf–

=


For non-identical slip lengths the constants C1x’, C2x’, C1y’, and C2y’ take the following values:

The average flow rate becomes:

which can be expressed in vector notation as:

vx' z' hb–= vx'b

–Lsb

---------z'x' z' hb–= Lsb z'vx' z' hb–= = =

vy' z' hb–= vy'b

–Lsb

---------z'y' z' hb–= Lsb z'vy' z' hb–= = =

vx' z' hw= vx'w

–Lsw

----------– z'x' z' hw= Lsw z'

vx' z' hw= –= =

vy' z' hw= vy'w

–Lsw

----------– z'y'

· z' hw= Lsw z'vy' z' hw= –= =

C1x'

vx'w vx'

b–

h Lsw Lsb+ +------------------------------------

hb2 hw

2– 2Lswhw– 2Lsbhb+

2 h Lsb Lsw+ + --------------------------------------------------------------------------

x'pf –=

C2x'

vx'w hb Lsb+ vx'

b hw Lsw+ +

h Lsw Lsb+ +----------------------------------------------------------------------------=

hw

2 hb Lsb+ hb2 hw Lsw+ 2hwhb Lsw Lsb+ 2hLswLsb+ + +

2 h Lsw Lsb+ + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------

x'pf–

C1y'

vy'w vy'

b–

h Lsw Lsb+ +------------------------------------

hb2 hw

2– 2Lswhw– 2Lsbhb+

2 h Lsw Lsb+ + --------------------------------------------------------------------------

y'pf –=

C2y'

vy'w hb Lsb+ vy'

b hw Lsw+ +

h Lsw Lsb+ +----------------------------------------------------------------------------=

hw

2 hb Lsb+ hb2 hw Lsw+ 2hwhb Lsw Lsb+ 2hLswLsb+ + +

2 h Lsw Lsb+ + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------

y'pf–

vav

h vx'w vx'

b+ 2 Lswvx'

w Lsbvx'b

+ +

2 h Lsw Lsb+ + -----------------------------------------------------------------------------------

h h2 4h Lsw Lsb+ 12LsbLsw+ + 12 h Lsw Lsb+ +

---------------------------------------------------------------------------------------------x'

pf–

h vy'w vy'

b+ 2 Lswvy'

w Lsbvy'b

+ +

2 h Lsw Lsb+ + -----------------------------------------------------------------------------------


---------------------------------------------------------------------------------------------y'

pf–

=


602 | C H A P T E

The above equation can be written on the form:

(9-12)

where vav,c is a term associated with Couette flow and vav,p is a coefficient associated with Poiseuille flow (see Table 9-2 below).

The forces acting on the two boundaries are given by:

(9-13)

Note that the z’ direction corresponds to the -nr direction. The x’ and y’ directions correspond to the two tangent vectors in the plane. Using vector notation the forces become:

In Equation 9-13 it is assumed that nw=-nr and nb=nr. In COMSOL Multiphysics the accuracy of the force terms is improved slightly over the usual approximation (which

vav1

2 h Lsw Lsb+ + -------------------------------------------- I nrnr

T– h 2Lsw+ vw h 2Lsb+ vb+ =


---------------------------------------------------------------------------------------------tpf–

vav vav c vav p tpf–=

fw

h h 2Lsb+ 2 h Lsw Lsb+ + --------------------------------------------–

x'pf vx'

b vx'w

– h Lsw Lsb+ +

----------------------------------------+

h h 2Lsb+ 2 h Lsw Lsb+ + --------------------------------------------–

y'pf vy'

b vy'w

– h Lsw Lsb+ +

----------------------------------------+

pf

=

fb

h h 2Lsw+ 2 h Lsw Lsb+ + --------------------------------------------–

x'pf vx'

w vx'b

– h Lsw Lsb+ +

----------------------------------------+

h h 2Lsw+ 2 h Lsw Lsb+ + --------------------------------------------–

y'pf vy'

w vy'b

– h Lsw Lsb+ +

----------------------------------------+

pf–

=

fwh h 2Lsb+

2 h Lsw Lsb+ + --------------------------------------------– tpf

h Lsw Lsb+ +------------------------------------ I nrnr

T– vb vw

– pfnw+ +=

fbh h 2Lsw+

2 h Lsw Lsb+ + --------------------------------------------– tpf

h Lsw Lsb+ +------------------------------------ I nrnr

T– vw vb

– + pfnb+=


neglects the slope of the wall and base as it is of order h0/l0) by using the following equations for nw and nb:

These definitions are derived from Equation 9-4 and Equation 9-5 and include the additional area that the pressure acts on as a result of the wall slope.

Once again, the force terms can be written on the form:

(9-14)

where fw,p is the Poiseuille coefficient for the force on the wall and fw,c incorporates the Couette and normal forces (due to the pressure) on the wall. Similarly, fb,p is the Poiseuille coefficient for the force on the base and fb,c incorporates the Couette and normal forces (due to the pressure) on the base.

The cases of identical slip length and non-slip are limiting cases of the formulae derived above. The main results are summarized in Table 9-2, where the constants defined in Equation 9-12 and Equation 9-14 are used.

nw nr– thw– =

nb nr thw– =

fw fw p– tpf fw c+=

fb f– b p tpf fb c+=


604 | C H A P T E

TABLE 9-2: EQUATION VARIABLES FOR VARIOUS FLOW MODELS

VARIABLE DEFINITION

General Slip Flow Model

Equal Slip Lengths Flow Model Lsw=Lsb=L

vav c 12 h Lsw Lsb+ + -------------------------------------------- I nrnr

T– h 2Lsb+ vw h 2Lsw+ vb+

vav p h h2 4h Lsw Lsb+ 12LsbLsw+ + 12 h Lsw Lsb+ +

---------------------------------------------------------------------------------------------

fw c

h Lsw Lsb+ +------------------------------------ I nrnr

T– vb vw

– pfnw+

fw p h h 2Lsb+ 2 h Lsw Lsb+ + --------------------------------------------

fb c h Lsw Lsb+ +------------------------------------ I nrnr

T– vw vb

– pfnb+

fb p h h 2Lsw+ 2 h Lsw Lsb+ + --------------------------------------------

vav c 12--- I nrnr

T– vw vb+

vav p h2 6Lsh+ 12

fw c h 2Ls+-------------------- I nrnr

T– vb vw

– pfnw+

fw p h 2

fb c h 2Ls+-------------------- I nrnr

T– vw vb

– pfnb+

fb p h 2


The Modified Reynolds Equation—Gas Flows

Thin-film gas flows are often isothermal, and in many cases the ideal gas law can be assumed. Under these circumstances the ideal gas law can be written on the form:

where T0 is the (constant) temperature of the gas, Mn is the molar mass of the gas, and R is the universal gas constant. Here the total gas pressure, ptot=pA+pf, where pA is the ambient pressure and pf is the pressure developed as a result of the flow. Substituting this relation into Equation 9-8 and dividing through by the constant Mn/RT0 results in a modified form of the Reynolds equation:

(9-15)

This equation can be used to model isothermal flows of ideal gases. The average flow rate and the forces acting on the bearings are computed in the same manner as for the standard Reynolds equation.

S L I P B O U N D A R Y C O N D I T I O N S F O R G A S E S

For a gas, the slip length is often expressed using the mean free path, , and a tangential momentum accommodation coefficient, . For compatibility with the existing

Non-Slip Flow Model Lsw=Lsb=0

TABLE 9-2: EQUATION VARIABLES FOR VARIOUS FLOW MODELS

VARIABLE DEFINITION

vav c 12--- I nrnr

T– vw vb+

vav p h2 12

fw c h--- I nrnr

T– vb vw

– pfnw+

fw p h 2

fb c h--- I nrnr

T– vw vb

– pfnb+

fb p h 2

ptotRT0Mn-----------=

t ptoth t hptotvav ptot vw thw vb thb+ –+ 0=


606 | C H A P T E

literature on thin-film gas flow the following definition of the mean free path is used by COMSOL Multiphysics in the Thin-Film Flow interfaces:

The slip length is then defined as:

Values for the tangential-momentum-accommodation coefficients for various gas surface combinations are given in Ref. 2

Flow Models for Rarefied Gases

For gases at low pressure, the ratio of the gas mean free path, , to the gap size (known as the Knudsen number: Kn=/h) grows. For Knudsen numbers greater than 0.1 the gas can not be treated using the continuum Navier Stokes equations and the Boltzmann equation must be solved instead.

At steady state, the solutions to the linearized Boltzmann equation for isothermal flow in a narrow gap between parallel plates can be expressed as a combination of Poiseuille and Couette flows. This is analogous to the solutions of the Navier-Stokes equations in the limit of small h0/l0. Provided that the surfaces of the wall and base are identical (which is normally the case in many practical applications) the Couette contribution to the bulk fluid velocity is unchanged (it remains the mean of the wall and base velocities for identical surfaces). The Poiseuille contribution to the flow is more complicated for a rarefied gas. A practical approach, pioneered by Fukui and Kaneko (Ref. 3) is to solve the linearized Boltzmann BGK equation over a range of Knudsen numbers and to provide an empirical fit to the flow. This results in the following form for the average velocity of the flow:

(9-16)

where Q(Kn, w , b) is a non-dimensional function of the Knudsen number (Kn) and the tangential momentum accommodation coefficient at the wall (w) and base (b).

p--- 2RT

Mn------------

12---

=

Ls

2------- 2 –

------------- =

vav12--- I nrnr

T– vw vb+ h2

12eff----------------tp–=

eff

Q Kn w b,, ----------------------------------------=


Q(Kn, w , b) is obtained by solving the linearized Boltzmann BGK equation for steady Poiseuille flow with a range of Knudsen numbers and slip coefficients. This approach assumes that stationary solutions of the Boltzmann equation apply inside the gap, that is, that the flow can be treated as quasi-static.

Fukui and Kaneko provided data on Q(Kn, w , b) for the case where w=b (Ref. 4), which was subsequently fitted to different empirical formulae by Veijola et al. (Ref. 5). Note also that additional, more accurate, data is available in Ref. 6. Veijola provided two empirical formulae, which apply under different circumstances with various degrees of accuracy:

1 w=b=1 (available as the option Rarefied-Total Accommodation in COMSOL Multiphysics):

(accurate to within 5% in the range 0<Kn<880)

2 w=b=(available as the option Rarefied-General Accommodation in COMSOL):

(accurate to within 1% in the ranges D>0.01, 0.7<<1 and 0.01<Kn<100).

Both of these empirical models are available as flow models with the options listed. Additionally a user-defined relative flow rate function can be defined (which could, for example, be based on an interpolation function derived from the original data in Ref. 6). Data on the tangential momentum accommodation coefficients for various gas-surface combinations is available in Ref. 2.

In this definition the mean free path and Knudsen number are defined as:

Q Kn 1 1,, 1 9,638Kn1,159+=

Q Kn ,, 6Q˜

D D

-------------------------= D 2Kn------------=

Q˜

D D6---- 1

1,34 -------------------- 1

D---- 4,1+

6,4D------------- 1,3 1 –

D 0,08D1,83+

------------------------------------- 0,64D0,17

1 1,12D0,72+

-----------------------------------+ + +ln+=

Various definitions of the mean free path are used in the literature, frequently without explanation. For compatibility with the existing literature on thin-film flow COMSOL Multiphysics employs the same mean free path definition as Veijola et al..


608 | C H A P T E

(9-17)

where is the gas viscosity, pis the gas pressure, R is the molar gas constant, T is the temperature, and Mn is the molar mass of the gas. Ref. 2 also employs this definition of the mean free path.

In many applications the forces acting on the wall and base are important. The pressure in the gas can be computed correctly by solving Equations 9-15 and 9-16. However, this approach provides only the normal component of the traction acting on the wall and base. To obtain the shear forces, the approach adopted by Torczynski and Gallis (Ref. 7) is used. They produced an empirical expression for the shear force that has the correct behavior in the free molecular flow and continuum limits as well as in the limits for the accommodation coefficient. Torczynski and Gallis solve the problem of pure Couette flow and derive an empirical function for the slip length that predicts the correct forces for the flow in the gap in several limiting cases. Their empirical expression for the slip length is given by:

(9-18)

where d10.15 and d20.59. In principle d1 and d2 are variables themselves, but were found to be constant to within the accuracy of the DSMC experiments used to derive their values.

Equation 9-12 gives the following expressions for the shear forces on the wall and base for pure Couette flow:

p--- 2RT

Mn------------

12---

= Kn ph------- 2RT

Mn------------

12---

=

Ls2 –

------------- 2

----------- 1d1

1 2d2Kn +---------------------------------------------+

=

The slip length in Equation 9-18 (which is used in COMSOL Multiphysics) differs slightly from the equivalent expression in Ref. 7 as a result of a different definition of the mean free path (Torczynski and Gallis’ mean free path is different from Equation 9-17 by a factor of /2).

fwshear

h 2Ls+-------------------- I nrnr

T– vb vw

– =

fbshear

h 2Ls+-------------------- I nrnr

T– vw vb

– =


where Ls is taken from Equation 9-18.

A general flow incorporates both Poiseuille and Couette terms. Assuming that the Poiseuille and Couette flows can be superposed, the forces become:

(9-19)

here Ls is derived from Equation 9-18 and p is obtained by solving Equation 9-15 with Equation 9-16. Strictly speaking, Torczynski and Gallis’ result applies for Couette flow only, and was derived for a more general variable-soft-sphere gas rather than for the linearized BGK equations, using numerical simulations. From a practical perspective, it seems likely that solutions of the linearized BGK equations would also be fitted by these expressions, and in that case it should be possible to combine the forces using superposition. In the absence of a detailed proof Equation 9-19 is not the default option for the force model, but is available as an additional option.

Frequency Domain Formulation

In the frequency domain it is necessary to make additional simplifications to the equation system to produce a fully linearized equation set. In the general case the physical quantities in the Reynolds and flow equations take the form:

Here the tilde denotes a harmonically varying component. The components marked with the subscript 1 are static offsets to the harmonic terms. In order to linearize the equation system it is necessary to assume that the offsets are much larger than the

fwh2---tp–

h 2Ls+-------------------- I nrnr

T– vb vw

– + pnr–=

fbh2---– tp

h 2Ls+-------------------- I nrnr

T– vw vb

– + pnr+=

Both Veijola et. al. (Ref. 5) and Cercignani et al. (Ref. 5) provided data for the relative flow rate Q(Kn, sw, sb) in specific cases where the wall and base have different accommodation coefficients. Since no details on how to compute the forces acting on the walls for highly rarefied gases were published, these models are not currently supported in COMSOL Multiphysics.

hw hw1 hw+= hb hb1 hb+= h hw1 hb1 hw hb+ + + h1 h+= =

pf pf˜= ptot pA pf

˜+=

vw vw= vb vb=


610 | C H A P T E

harmonic components. Practically speaking this means that the harmonically varying pressures induced by the flow should be significantly smaller than the ambient pressure, and the harmonic changes in the gap size due to the wall and base displacements should be significantly less than the gap height itself. Given these assumptions, the average fluid velocity can also be written on the form:

since in general both the Couette and Poiseuille terms vary harmonically to first order accuracy.

Substituting these terms into the Reynolds equation gives:

In COMSOL Multiphysics is usually defined as a function of pA only, and so contains no small harmonic component. The products of small harmonic terms result in second order effects (at double the frequency of interest) and can be neglected provided the harmonic terms are much smaller than the static terms. The Reynolds equation therefore reduces to:

Note that:

(9-20)

So the following result holds:

(9-21)

Dropping the tildes leads to the form of the equation shown in the equation display:

The modified Reynolds Equation takes the following form:

vav vav=

t h1 h+ t h1 h+ vav vw t hw1 hw+ vb t hb1 hb+ + –+

0=

t h( ) t h1vav vw thw1 vb thb1+ –+ 0=

t h vb nref vw nref–=

vb nref vw nref– t h1vav vw thw1 vb thb1+ –+ 0=

vb nref vw nref– t h1vav vw thw1 vb thb1+ –+ 0=


(9-22)

Linearizing Equation 9-22 gives:

Equation 9-20 can be used to substitute for the time derivative of the harmonic component of h in the above equation, yielding:

(9-23)

Dropping the tildes and using complex notation to express the time derivative of pf gives:

Equation 9-21 and Equation 9-23 are used as the basis for the frequency domain formulation in the Thin-Film Flow interfaces. It is important to note that these equations are independent of the harmonic components of the displacement, since those terms lead to second order contributions to the frequency domain response (that is they produce a response proportional to the square of the harmonic term, which results in a signal at twice the driving frequency).

t pA pf

˜+ h1 h+ t pA pf˜+ h1 h+ vav +

pA pf˜+ vw t hw1 hw+ vb t hb1 hb+ + – 0=

pA t h( ) h1 t

pf˜( ) t pAh1vav pA vw t hw1 vb t hb1 + –+ +

0=

pA vb nref vw nref– h1 t pf

˜( ) t pAh1vav + +

pA vw t hw1 vb t hb1 + – 0=

pA vb nref vw nref– ih1pf t pAh1vav + +

pA vw t hw1 vb t hb1 + – 0=

Since the frequency domain results do not depend on the variation in the displacement, the additional displacement setting has no effect on the solution in a frequency domain problem (unless the velocity is computed from it by selecting Calculate from wall displacement or Calculate from base

displacement in the wall and base velocity settings respectively). This is reflected in the equations given above (and in the equation display in the physics interface) but can be non-intuitive, particularly when setting up a model that is coupled with a structural analysis. If the velocity of both the wall and the base is set to zero, then in the frequency domain there is no response.


612 | C H A P T E

Boundary Conditions

Most of the boundary conditions either constrain the flow into the system by prescribing a fluid velocity or constrain the pressure at the boundary. The border flow condition is slightly more complex, requiring the gradient of the pressure to be set according to the equation:

A user-defined boundary condition is also available, allowing for arbitrary normal pressure gradients.

References for the Thin-Film Flow Interfaces

1. B.J. Hamrock, S.R. Schmid and B.O. Jacobson, Fundamentals of Fluid Film Lubrication, Marcel Dekker, 2004.

2. F. Sharipov, “Data on the Velocity Slip and Temperature Jump on a Gas-Solid Interface,” J. Phys. Chem. Ref. Data, vol. 40, no. 2, 023101, 2011.

3. S. Fukui and R. Kaneko, “Analysis of Ultra-Thin Gas Film Lubrication Based on Linearized Boltzmann Equation: First Report—Derivation of a Generalized Lubrication Equation Including Thermal Creep Flow”, J. Tribology, vol. 110, no. 2, pp. 253–261, 1988.

4. S. Fukui and R. Kaneko, “A Database for Interpolation of Poiseuille Flow Rates for High Knudsen Number Lubrication,” Transactions of the ASME, vol. 112, pp. 78–83, 1990.

5. T. Veijola, H. Kuisma, and J. Lahdenperä, “The Influence of Gas-surface Interaction on Gas Film Damping in a Silicon Accelerometer,” Sensors and Actuators, vol. A 66, pp. 83–92, 1998.

6. C. Cercignani, Maria Lampis, and Silvia Lorenzani “Variational Approach to Gas Flows in Microchannels”, Physics of Fluids, vol. 16, no. 9, 2004.

7. J.R. Torczynski and M.A. Gallis, “DSMC-Based Shear-Stress/Velocity-Slip Boundary Condition for Navier-Stokes Couette-Flow Simulations”, 27th International Symposium on Rarefied Gas Dynamics (2010), AIP Conf. Proc. vol. 1333, pp. 802–807, 2011.

n p–pFL

-------=


10

T h e M a t h e m a t i c s , M o v i n g I n t e r f a c e B r a n c h

The fluid-flow interfaces are grouped by type under the Fluid Flow main branch when adding a physics interface. In addition, the Level Set and Phase Field Moving interfaces are available under the Mathematics>Moving Interface branch ( ). Also see Modeling Multiphase Flow to help you choose the best one to start with.

In this chapter:

• The Level Set Interface

• The Phase Field Interface

• Theory for the Level Set Interface

• Theory for the Phase Field Interface

613

614 | C H A P T E

Th e L e v e l S e t I n t e r f a c e

The Level Set (ls) interface ( ), found under the Mathematics>Moving Interface branch ( ) when adding an interface, is used to track moving interfaces in fluid-flow models by solving a transport equation for the level set function. Simulations using the Level Set interface are always time dependent since the position of an interface almost always depends on its history.

The main node is the Level Set Model feature, which adds the level set equation and provides an interface for defining the level set properties and the velocity field.

When this physics interface is added, the following default nodes are also added in the Model Builder—Level Set Model, No Flow (the default boundary condition) and Initial

Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions. You can also right-click Level Set to select physics from the context menu.



The default identifier (for the first physics interface in the component) is ls.


The default setting is to have All domains in the component define the level set function and the level set equation that describes it. To choose specific domains, select Manual from the Selection list.


The dependent variable (field variable) is the Volume fraction of fluid 2 phi. The name can be changed but the names of fields and dependent variables must be unique within a model.

R 1 0 : T H E M A T H E M A T I C S , M O V I N G I N T E R F A C E B R A N C H

S T A B I L I Z A T I O N

To display this section, click the Show button ( ) and select Stabilization. There are two types of stabilization methods and both are active by default and should remain active for optimal performance. If required, click to clear one or both of the Streamline

diffusion and Crosswind diffusion check boxes.


To display this section, click the Show button ( ) and select Advanced Physics Options. Normally these settings do not need to be changed. Select a Convective term—Non-conservative form (the default) or Conservative form.


To display this section, click the Show button ( ) and select Discretization. Select a Volume fraction of fluid—Linear, Quadratic (the default), Cubic, Quartic, or Quintic (order 1–5). Specify the Value type when using splitting of complex variables—Real (the default) or Complex.

Domain, Boundary, and Pair Nodes for the Level Set Interface

The Level Set Interface has the following domain, boundary and pair nodes described.


• Initial Values

• Inlet

• Level Set Model

• No Flow (the default boundary condition)

The Outlet and Symmetry nodes are described for the Laminar Flow interface.


• Conservative and Non-Conservative Form

• Domain, Boundary, and Pair Nodes for the Level Set Interface

• Theory for the Level Set Interface


T H E L E V E L S E T I N T E R F A C E | 615

616 | C H A P T E

Level Set Model

The Level Set Model node adds the following transport equation governing a level set function

and provides the options to define the associated level set parameters and the velocity field.



L E V E L S E T P A R A M E T E R S

Enter a value or expression for the Reinitialization parameter (SI unit: m/s). The default is 1 m/s.

Enter a value or expression for the Parameter controlling interface thickness els (SI unit: m). The default expression is ls.hmax/2, which means that the value is half of the maximum mesh element size in the region through which the interface passes.

C O N V E C T I O N

Enter values or expressions for the components (u, v, and w in 3D, for example) of the Velocity field u (SI unit: m/s). The applied velocity field transports the level set function through convection.

Boundary conditions for axial symmetry boundaries are not required. For the symmetry axis at r0, the software automatically provides a suitable boundary condition and adds an Axial Symmetry node that is valid on the axial symmetry boundaries only.

t u + 1 –

----------–

=


Initial Values

Use the Initial Values node to set up the initial conditions.




In order to be able to use the automatic initialization, select an option from the Domain

initially: list—Inside interface (the default), Outside interface, or Specify level set function

explicitly if automatic initialization is not required.

Inlet

The Inlet node adds a boundary condition for inlets (inflow boundaries). At inlets a value of the level set function must be specified. Typically set to either 0 or 1.


From the Selection list, choose the boundaries on which to define the Inlet.

S E T T I N G S

Enter a value for the Level set function value . The value must be in the range from 0 to 1, and the default is 0.

If the Transient with Initialization ( ) study is being used, for the initialization to work it is crucial that there are two Initial Values nodes and one Initial Interface node. One of the Initial Values nodes should use Domain initially: Inside interface and the other Domain initially: Outside

interface. The Initial Interface node should have all interior boundaries where the interface is initially present as selection. If the selection of the Initial interface node is empty, the initialization fails.

See Initializing the Level Set Function.

T H E L E V E L S E T I N T E R F A C E | 617

618 | C H A P T E

Initial Interface

The Initial Interface node defines the boundary as the initial position of the interface = 0.



No Flow

The No Flow node adds a boundary condition that represents boundaries where there is no flow across the boundary. This is the default boundary condition.





If the Transient with Initialization ( ) study is being used, for the initialization to work it is crucial that there are two Initial Values nodes and one Initial Interface node. One of the Initial Values nodes should use Domain initially: Inside interface and the other Domain initially: Outside

interface. The Initial Interface node should have all interior boundaries where the interface is initially present as selection. If the selection of the Initial interface node is empty, the initialization fails.

See Initializing the Level Set Function.


Th e Pha s e F i e l d I n t e r f a c e

The Phase Field (pf) interface ( ), found under the Mathematics>Moving Interface branch ( ) when adding a physics interface, is used to track moving interfaces by solving two transport equations, one for the phase field variable, and one for the mixing energy density, . The position of the interface is determined by minimizing the free energy.

The main node is the Phase Field Model feature, which adds the phase field equations and provides an interface for defining the phase field model properties.

When this physics interface is added, the following default nodes are also added in the Model Builder—Phase Field Model, Wetted Wall (the default boundary condition) and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions. You can also right-click Phase Field to select physics from the context menu.



The default identifier (for the first physics interface in the component) is pf.


The default setting is to have All domains in the component define the phase field functions and the equations. To choose specific domains, select Manual from the Selection list.


This interface defines the dependent variables (fields) Phase field variable and Phase

field help variable . If required, edit the name, but dependent variables must be unique within a model.

T H E P H A S E F I E L D I N T E R F A C E | 619

620 | C H A P T E


To display this section, click the Show button ( ) and select Advanced Physics Options. Normally these settings do not need to be changed. Select a Convective term—Non-conservative form (the default) or Conservative form.


To display this section, click the Show button ( ) and select Discretization. Select a Phase field variable—Linear, Quadratic (the default), Cubic, Quartic, or Quintic (order 1 – 5). Specify the Value type when using splitting of complex variables—Real (the default) or Complex.

Domain, Boundary, and Pair Nodes for the Phase Field Interface

The Phase Field Interface includes the following domain, boundary, and pair nodes, listed in alphabetical order, available from the Physics ribbon toolbar (Windows users), Physics context menu (Mac or Linux users), or right-click to access the context menu (all users).

• Conservative and Non-Conservative Forms


• Domain, Boundary, and Pair Nodes for the Phase Field Interface

• Theory for the Phase Field Interface

Phase Separation: model library path CFD_Module/Multiphase_Tutorials/

phase_separation

Compared to the single-phase flow interfaces, the two-phase flow interfaces include two additional boundary conditions—the Wall boundary conditions Wetted wall and Moving wetted wall.




• Initial Values

• Inlet

• Phase Field Model

• Wetted Wall (the default boundary condition)

The Outlet and Symmetry nodes are described for the Laminar Flow interface.

Phase Field Model

The Phase Field Model node adds the equations described in The Equations for the Phase Field Method. The node defines the associated phase field parameters including surface tension and interface thickness.



P H A S E F I E L D P A R A M E T E R S

Define the following phase field parameters. Enter a value or expression for the:

• Surface tension coefficient (SI unit: N/m).

• Parameter controlling interface thickness epf (SI unit: m). The default expression is pf.hmax/2, which means that the value is half of the maximum mesh element size in the region through which the interface passes.

• Mobility tuning parameter (SI unit: m·s/kg). The default is 1 m·s/kg, which is a good starting point for most models. This parameter determines the time scale of the Cahn-Hilliard diffusion and it thereby also governs the diffusion-related time scale for the interface.

Keep the parameter value large enough to maintain a constant interface thickness but still low enough to not damp the convective motion. A too high mobility can also lead to excessive diffusion of droplets.


622 | C H A P T E

E X T E R N A L F R E E E N E R G Y

Add a source of external free energy to the phase field equations. This modifies the last term on the right-hand side of the equation:

The external free energy f (SI unit: J/m3) is a user-defined free energy. In most cases, the external free energy can be set to zero. Manually differentiate the expression for the external free energy with respect to and then enter it into the -derivative of

external free energy field .

C O N V E C T I O N

Enter values or expressions for the components (u, v, and w in 3D, for example) of the Velocity field u (SI unit: m/s). The applied velocity field transports the phase field variables through convection.

Initial Values

The Initial Values node adds initial values for the phase field variable and the phase field help variable that can serve as initial conditions for a transient simulation.




Enter initial values or expressions for the Phase field variable and the Phase field help

variable . The default values are 0.

2 – 2 1– 2

-----

f

+ +=

f

Additional Sources of Free Energy


Inlet

The Inlet feature node adds a boundary condition for inlets (inflow boundaries). At inlets a volume fraction Vf must be specified, typically either 0 or 1. Mathematically this boundary condition imposes

and



I N L E T

Specify a value of the Inlet volume fraction Vf. The value must be in the range from 0 to 1 and the default is 0.

Initial Interface

The Initial Interface node defines the boundary as the initial position of the interface = 0.



2Vf 1–= n

2------ 0=

If the Transient with Initialization ( ) study is being used, for the initialization to work it is crucial that there are two Initial Values nodes and one Initial Interface node. One of the Initial Values nodes is set to phipf = 1 and the other to phipf = -1. The Initial Interface node should have all interior boundaries where the interface is initially present as selection. If the selection of the Initial Interface node is empty, the initialization fails.


624 | C H A P T E

Wetted Wall

The Wetted Wall node is the default boundary condition representing wetted walls. Along a wetted wall the contact angle for the fluid, w, is specified, and across it the mass flow is zero. This is prescribed by

in combination with



WE T T E D WA L L

Enter a value or expression for the Contact angle w. The default value is /2 rad.

n 2 2 w cos =

n

2------ 0=


Th eo r y f o r t h e L e v e l S e t I n t e r f a c e

Fluid flow with moving interfaces or boundaries occur in a number of different applications, such as fluid-structure interaction, multiphase flows, and flexible membranes moving in a liquid. One way to track moving interfaces is to use a level set method. A certain contour line of the globally defined function, the level set function, then represents the interface between the phases. For The Level Set Interface the fluid-fluid interface can be advected with an arbitrary velocity field.

In this section:

• The Level Set Method

• Conservative and Non-Conservative Form

• Initializing the Level Set Function

• Variables For Geometric Properties of the Interface

• Reference for the Level Set Interface

The Level Set Method

The level set method is a technique to represent moving interfaces or boundaries using a fixed mesh. It is useful for problems where the computational domain can be divided into two domains separated by an interface. Each of the two domains can consist of several parts. Figure 10-1 shows an example where one of the domains consists of two separated parts. The interface is represented by a certain level set or isocontour of a globally defined function, the level set function . In COMSOL Multiphysics, is a smooth step function that equals zero (0) in one domain and one (1) in the other. Across the interface, there is a smooth transition from zero to one. The interface is defined by the 0.5 isocontour, or level set, of . Figure 10-2 shows the level set

T H E O R Y F O R T H E L E V E L S E T I N T E R F A C E | 625

626 | C H A P T E

representation of the interface in Figure 10-1.

Figure 10-1: An example of two domains divided by an interface. In this case, one of the domains consists of two parts. Figure 10-2 shows the corresponding level set representation.

Figure 10-2: A surface plot of the level set function corresponding to Figure 10-1.

The physics interface solves Equation 10-1 in order to move the interface with the velocity field u:

(10-1)

The terms on the left-hand side give the correct motion of the interface, while those on the right-hand side are necessary for numerical stability. The parameter, , determines the thickness of the region where varies smoothly from zero to one and is typically of the same order as the size of the elements of the mesh. By default, is constant within each domain and equals the largest value of the mesh size, h, within

t u + 1 –

----------–

=


the domain. The parameter determines the amount of reinitialization or stabilization of the level set function. It needs to be tuned for each specific problem. If is too small, the thickness of the interface might not remain constant and oscillations in can appear because of numerical instabilities. On the other hand, if is too large the interface moves incorrectly. A suitable value for is the maximum magnitude of the velocity field u.

Conservative and Non-Conservative Form

If the velocity is divergence free, that is, if

(10-2)

the volume (area for 2D problems) bounded by the interface should be conserved if there is no inflow or outflow through the boundaries. To obtain exact numerical conservation, switch to the conservative form

(10-3)

in The Level Set Interface settings window.

Using the conservative level set form, exact numerical conservation of the integral of is obtained. However, the non-conservative form is better suited for numerical

calculations and usually converges more easily. The non-conservative form, which is the default form, only conserves the integral of the level set function approximately, but this is sufficient for most applications.

Initializing the Level Set Function

If the study type Transient with Initialization is used in the model, the level set variable is first initialized so that it varies smoothly between zero and one over the interface. For that study, two study steps are created, Phase Initialization and Time Dependent. The Phase Initialization step solves for the distance to the initial interface, Dwi. The Time Dependent step then uses the initial condition for the level set function according to the following expression:

in domains initially outside the interface and

u 0=

t u + 1 –

----------–

=

01

1 eDwi –

+--------------------------=


628 | C H A P T E

in domains initially inside the interface. Here, inside refers to domains where <0.5 and outside refers to domains where >0.5.

Variables For Geometric Properties of the Interface

Geometric properties of the interface are often needed. The unit normal to the interface is given by

(10-4)

The curvature is defined as

(10-5)

These variables are available in the physics interface as the interface normal and mean curvature.

01

1 eDwi

+------------------------=

For the initialization to work it is crucial that there are two Initial Values nodes and one Initial Interface node. One of the Initial Values nodes should use Domain initially: Inside interface and the other Domain initially:

Outside interface. The Initial Interface node should have all interior boundaries where the interface is initially present as selection. If the selection of the Initial interface node is empty, the initialization fails.

• The Level Set Interface

• Studies and Solvers and Transient with Initialization in the COMSOL Multiphysics Reference Manual

n ----------

0,5=

=

– n 0,5==

It is only possible to compute the curvature explicitly when using second-order or higher-order elements.


Reference for the Level Set Interface

1. E. Olsson and G. Kreiss, “A Conservative Level Set Method for Two Phase Flow,” J. Comput. Phys., vol. 210, pp. 225–246, 2005.


630 | C H A P T E

Th eo r y f o r t h e Pha s e F i e l d I n t e r f a c e

The Phase Field Interface theory is described in this section:

• About the Phase Field Method

• The Equations for the Phase Field Method

• Conservative and Non-Conservative Forms

• Additional Sources of Free Energy

• Initializing the Phase Field Function

• Variables and Expressions

• Reference for the Phase Field Interface

About the Phase Field Method

The phase field method offers an attractive alternative to more established methods for solving multiphase flow problems. Instead of directly tracking the interface between two fluids, the interfacial layer is governed by a phase field variable, . The surface tension force is added to the Navier-Stokes equations as a body force by multiplying the chemical potential of the system by the gradient of the phase field variable.

The evolution of the phase field variable is governed by the Cahn-Hilliard equation, which is a 4th-order PDE. The Phase Field interface decomposes the Cahn-Hilliard equation into two second-order PDEs.

For the level set method, the fluid interface is simply advected with the flow field. The Cahn-Hilliard equation, on the other hand, does not only convect the fluid interface, but it also ensures that the total energy of the system diminishes correctly. The phase field method thus includes more physics than the level set method.

The free energy of a system of two immiscible fluids consists of mixing, bulk distortion, and anchoring energy. For simple two-phase flows, only the mixing energy is retained, which results in a rather simple expression for the free energy.

The Equations for the Phase Field Method

The free energy is a functional of a dimensionless phase field parameter, :


where is a measure of the interface thickness. Equation 10-6 describes the evolution of the phase field parameter:

(10-6)

where ftot (SI unit: J/m3) is the total free energy density of the system, and u (SI unit: m/s) is the velocity field for the advection. The right-hand side of Equation 10-6 aims to minimize the total free energy with a relaxation time controlled by the mobility (SI unit: m3·s/kg).

The free energy density of an isothermal mixture of two immiscible fluids is the sum of the mixing energy and elastic energy. The mixing energy assumes the Ginzburg-Landau form:

where is the dimensionless phase field variable, defined such that the volume fraction of the components of the fluid are (1+ )/2 and (1 )/2. The quantity (SI unit: N) is the mixing energy density and (SI unit: m) is a capillary width that scales with the thickness of the interface. These two parameters are related to the surface tension coefficient, (SI unit: N/m), through the equation

(10-7)

The PDE governing the phase field variable is the Cahn-Hilliard equation:

(10-8)

where G (SI unit: Pa) is the chemical potential and (SI unit: m3·s/kg) is the mobility. The mobility determines the time scale of the Cahn-Hilliard diffusion and must be large enough to retain a constant interfacial thickness but small enough so that the convective terms are not overly damped. In COMSOL Multiphysics the mobility is determined by a mobility tuning parameter that is a function of the interface thickness 2. The chemical potential is:

F T 12---2 2 f T + Vd ftot Vd= =

t u +

ftot

ftot–

=

fmix 12--- 2

42--------- 2 1–

2+=

2 23

--------------=

t u + G=

T H E O R Y F O R T H E P H A S E F I E L D I N T E R F A C E | 631

632 | C H A P T E

(10-9)

The Cahn-Hilliard equation forces to take a value of 1 or 1 except in a very thin region on the fluid-fluid interface. The Phase Field interface breaks Equation 10-8 up into two second-order PDEs:

(10-10)

(10-11)

Conservative and Non-Conservative Forms

If the velocity field is divergence free, use the conservative formulation:

Using the conservative phase field form, exact numerical conservation of the integral of is obtained. However, the non-conservative form is better suited for numerical calculations and usually converges more easily. The non-conservative form, which is the default form, only conserves the integral of the phase field function approximately, but this is sufficient for most applications.

Additional Sources of Free Energy

In some cases, the expression for the free energy can include other sources. It is possible to incorporate these by modifying Equation 10-11:

(10-12)

where f is a user-defined free energy (SI unit: J/m3).

G 2– 2 1–

2-----------------------+=

t u +

2------=

2 – 2 1– +=

t u+

2------=

2 – 2 1– 2

-----

f

+ +=

The expression for the external free energy must be manually differentiated with respect to and then entered into the field. In most cases, the external free energy is zero.

f


Initializing the Phase Field Function

If the study type Transient with Initialization is used in the model, the phase field variable is first initialized so that it varies smoothly between zero and one over the interface. For this study, two study steps are created, Phase Initialization and Time Dependent. The Phase Initialization step solves for the distance to the initial interface, Dwi. The Time Dependent step then uses the initial condition for the phase field function according to the following expression:

in Fluid 1 and

in Fluid 2. These expressions are based on a steady, analytic solution to Equation 10-10 and Equation 10-11 for a straight, non-moving interface.

Variables and Expressions

Unlike the level set method, the phase field method does not require expressions for the unit normal to the interface or smoothed delta functions, so they are not available for analysis. Variables that are defined are the chemical potential, which can be rewritten in terms of the dependent variable ,

0Dwi

2---------- tanh–=

0Dwi

2---------- tanh=

If the Transient with Initialization ( ) study is being used, for the initialization to work it is crucial that there are two Initial Values nodes and one Initial Interface node. One of the Initial Values nodes is set to phipf = 1 and the other to phipf = -1. The Initial Interface node should have all interior boundaries where the interface is initially present as selection. If the selection of the Initial Interface node is empty, the initialization fails.


T H E O R Y F O R T H E P H A S E F I E L D I N T E R F A C E | 633

634 | C H A P T E

and the surface tension force F G .

The mean curvature (SI unit: 1/m) of the interface can be computed by entering the following expression:

Reference for the Phase Field Interface

1. P. Yue, C. Zhou, J.J. Feng, C.F. Ollivier-Gooch, and H.H. Hu, “Phase-field Simulations of Interfacial Dynamics in Viscoelastic Fluids Using Finite Elements with Adaptive Meshing,” J. Comp. Phys., vol. 219, pp. 47–67, 2006.

G

2-------=

2 1 + 1 – G----=


11

G l o s s a r y

This Glossary of Terms contains application-specific terms used in the CFD Module software and documentation. For finite element modeling terms, mathematical terms, and geometry and CAD terms, see the glossary in the COMSOL Multiphysics Reference Manual. For references to more information about a term, see the index.

635

636 | C H A P T E

G l o s s a r y o f T e rm sanisotropy Directional dependence. Is often obtained from homogenization of regular structures, for example, monolithic structures in tubular reactors.

Boussinesq approximation An approximate method to include buoyancy effects, for which the density variation is only taken into account in the buoyancy term.

Brinkman equations A set of equations extending Darcy’s law in order to include transport of momentum through shear in porous media flow.

boundary layer Region in a fluid close to a solid surface. This region is characterized by large gradients in velocity and other properties. In turbulent flow it is often treated with approximative methods because of the difficulty to resolve the large gradients.

bubbly flow Flow with gas bubbles dispersed in a liquid.

conjugate heat transfer heat transfer that takes place in both a solid and a fluid.

creeping flow Models the Navier-Stokes equations without the contribution of the inertial term. This is often referred to as Stokes flow and is applicable when viscous flow dominates, such as in very small channels or microfluidic devices.

crosswind diffusion A numerical technique for stabilization of convection-dominated PDEs by artificially adding diffusion perpendicular to the direction of the streamlines. It reduces oscillations near sharp gradients.

Darcy’s law Equation that gives the velocity vector as proportional to the pressure gradient. Often used to describe flow in porous media.

Euler flow Flow of an inviscid fluid. Often used to approximate high speed compressible flows.

Euler-Euler model A two-phase flow model that treats both phases as inter-penetrating continua.

Fick’s laws The first law states that the diffusive flux of a solute infinitely diluted in a solvent is proportional to its concentration gradient. The second law introduces the first law into a differential material balance for the temporal evolution of the solute.

R 1 1 : G L O S S A R Y

fluid-structure interaction (FSI) When a fluid flow affects the deformation of a solid object and vice versa.

fully developed laminar flow Laminar flow along a channel or pipe that only has velocity components in the streamwise direction. The velocity profile does not change downstream.

Hagen-Poiseuille equation See Poiseuille’s law.

heterogeneous reaction Reaction that takes place at the interface between two phases.

homogeneous reaction Reaction that takes place in the bulk of a solution.

intrinsic volume averages The physical properties of the fluid, such as density, viscosity, and pressure.

k- turbulence model A two-equation RANS model that solves for the turbulent kinetic energy, k, and the dissipation of turbulence kinetic energy, . Utilizes wall functions to describe the flow close to solid walls.

k- turbulence model A two-equation RANS model that solves for the turbulent kinetic energy, k, and the specific dissipation rate, . Utilizes wall functions to describe the flow close to solid walls.

law of the wall See wall function.

low-Reynolds k- turbulence model Two-equation RANS model that solves for the turbulence kinetic energy, k, and the dissipation of turbulence kinetic energy, . Includes damping functions to be able to describe regions with low Reynolds numbers, for example close to solid walls.

low Reynolds number The region close to the wall where viscous effects dominate.

Mach number Dimensionless number equal to the flow velocity over the speed of sound. Compressible effects because of the flow speed can be neglected for Mach number less than 0.3.

multiphase flow Flow with more than one phase.

G L O S S A R Y O F TE R M S | 637

638 | C H A P T E

Navier-Stokes equations The momentum balance equation for a Newtonian fluid coupled to the equation of continuity. The meaning of the term originally only referred to the momentum balance but it is here used in the more general context.

Newtonian fluid A fluid for which the stress is proportional to the rate of strain. Many common fluids such as water and air are Newtonian.

non-Newtonian fluid A fluid for which the stress is not proportional to the rate of strain. Blood and suspensions of polymers are examples of non-Newtonian fluids.

Poiseuille’s law Equation stating that the mass rate of flow in a tube is proportional to the pressure difference per unit length and to the fourth power of the tube radius. The law is valid for fully developed laminar flow.

pressure work Describes the reversible conversion of work, performed by the pressure in a fluid, into heat.

RANS Reynolds-averaged Navier-Stokes; implying that a time-averaging operation has been performed on the equations of motion. The Reynolds’ stresses (correlations between fluctuating velocity components) obtained from this averaging operation have to be obtained from an additional set of equations - a closure. Turbulence models like the k- and Spalart-Allmaras models constitute closures to the RANS equations.

Reynolds number A dimensionless number that describes the relative importance between inertia and viscous effects. Flow at high Reynolds number have a tendency to undergo transition to turbulence.

Soret effect Mass diffusion due to temperature gradients in multicomponent mixtures.

Spalart-Allmaras turbulence model A one-equation turbulence model that solves for the undamped turbulent kinematic viscosity, .

SST turbulence model The Shear Stress Transport model is a two-equation turbulence model combining the k- model in the near-wall region with the k- model in the free stream. The SST model is a low-Reynolds number model requiring high resolution near walls. The dependent variables are the turbulent kinetic energy, k, and the turbulent dissipation rate, .

Stokes flow See creeping flow.

T

R 1 1 : G L O S S A R Y

streamline diffusion A numerical technique for stabilization of convection-dominated PDEs by artificially adding upwinding in the streamline direction.

superficial volume averages The flow velocities, which correspond to a unit volume of the medium including both pores and matrix. They are sometimes called Darcy velocities, defined as volume flow rates per unit cross section of the medium.

thin-film flow Flow in very thin regions where the it can be assumed to always have a fully developed profile.

viscous heating The heat irreversibly generated from work by viscous friction in a fluid.

wall function Semi-empirical expression for the boundary-layer flow used in turbulence models. Often based on the assumption of negligible variations in the pressure gradient tangential to the surface.

G L O S S A R Y O F TE R M S | 639

640 | C H A P T E
R 1 1 : G L O S S A R Y

I n d e x

2D axisymmetric models

laminar flow and 114

A absolute pressure 68, 195

acceleration of gravity 259

acoustic boundary condition 591

added mass force 429

adding species 483

advanced settings 41

AKN model 166

B Babuska-Brezzi condition 139

Basset force 429

Basset history term 146

boundary conditions

heat equation, and 245

heat transfer coefficients, and 257

inlet and outlet, theory 122

boundary heat source (node) 205

boundary heat source variable 245

boundary nodes

Brinkman equations 452

bubbly flow 363

Darcy’s law interface 442

Euler-Euler interface 398

free and porous media flow 459

heat transfer 189

heat transfer in porous media 222

hmnf interfaces 311

level set 615

mm interfaces 383

nitf interfaces 277

phase field 620

reacting flow 526

reacting flow in porous media (rfcs)

540

reacting flow in porous media (rfds)

547

rmspf interfaces 106

spf 66

tff interface 583

tffs interfaces 583

tpdl interface 465

tpf interfaces 343

transport of concentrated species 505

transport of diluted species 488

boundary selection 42

boundary stress (node) 85

Boussinesq approximation 118, 295


Brinkman equations interface 450

theory 473

bubble number density 414

bulk velocity 259

C Cahn-Hilliard equation 630

Carreau model 118, 284

CFL number

high Mach number flow 328

pseudo time stepping, and 104

settings 55

theory 143

change thickness (node) 220

characteristic length 259

chemical potential variable 633

coefficient of volumetric thermal expan-

sion 260

compensating heat power contributions

233

compressible flow 115

concentration (node) 494

concentration of species 483

conduction, defined 227

conductive heat flux variable 239

I N D E X | 641

642 | I N D E X

conductive heat flux vector 245

conjugate heat transfer interface

theory 293

conjugate heat transfer interfaces 265

conservation of energy 228

consistent stabilization

settings 42

constraint settings 42

constraints, Galerkin 201

continuity (node)

heat transfer 207

continuity equation, Darcy’s law 471

continuity equation, multiphase flow 420

convection and diffusion (node)



convection, defined 227

convection, natural and forced 258

convective heat flux (node) 214

convective heat flux variable, cflux 240

convective heat flux variable, chflux 243

convective out-of-plane heat flux varia-

ble, chflux 241

convective terms, diluted species 552

coordinate system selection 42

Couette flow 599

creeping flow 340

creeping flow (spf) interface 57

theory 111

crosswind diffusion

definition 251

fluid flow 138

curves, fan 131

D Damköhler number 570

damping, squeezed- film 594

Darcy velocity 471


theory 471

defining species concentration 483

dense flows 429

dev_in and dev_out variables 130

diffusion (node)


542



diffusion models 502

dilute flows 430

dimensionless distance to cell center var-

iable 168

discontinuous Galerkin

fluid flow 145

discontinuous Galerkin constraints 201

discretization 41

dispersed liquid droplets 428

dispersed phase boundary conditions

421

dispersed phase particles 426

dispersed solid particles 428

dispersivities, porous media 226

dissipation, turbulent 415

documentation 22

domain heat source variable 245

domain nodes


bubbly flow 363

Darcy’s law 442



heat transfer 189


hmnf interfaces 311

level set 615

mm interfaces 383

nitf interfaces 277

phase field 620

reacting flow 526


540


547


spf interfaces 66

tff interface 583

tffs interfaces 583

tpdl interface 465

tpf interfaces 343



domain selection 42

double dot product 113

drag force 429

drag law, Hadamard-Rybczynski 423

E eddy dissipation concept (EDC) 571

eddy viscosity 151

edge nodes


heat transfer 189


hmnf interfaces 311

nitf interfaces 277

tff interface 583

tffs interface 583

edge selection 42

effective volumetric heat capacity 193

elastic contribution to entropy 211

emailing COMSOL 24

energy powers 234

entropy waves 326

Eötvös number 415

equation view 41

equivalent thermal conductivity 255

equivalent volumetric heat capacity 255

Ergun packed bed expression 429

Ettehadieh solid pressure model 431

Euler-Euler equations 426, 431

bubbly flow 411

mixture model 418

Euler-Euler model, laminar flow (ee) in-

terface 395

theory 426

exit length 83

expanding sections 41

F fan (node) 90

fan curves

inlet boundary condition 91

theory 131

Favre average 152, 295

Fick’s law approximation diffusion 560

Fick’s law diffusion model 503

first law of thermodynamics 228

flow continuity (node) 96

fluid (node)

hmnf interfaces 313

nitf interfaces 279

rspf interfaces 528

fluid and matrix properties (node)


Darcy’s law 443

fluid flow

approaches to analysis 30

Brinkman equations theory 473

Darcy’s law theory 471

Mach number 114

particle tracing 146

single-phase theory 111

turbulent flow theory 149

fluid properties 31

fluid properties (node)

bf interfaces 364


spf interfaces 67

I N D E X | 643

644 | I N D E X

tpf interfaces 347

fluid-film properties (node) 583

fluids and matrix properties (node) 466

fluid-solid mixtures 427

flux (node)



flux discontinuity (node)

Darcy’s law 448



Forchheimer drag (node)



Fourier’s law 229

free and porous media flow interface 457

theory 476

frozen rotor 176

Fukui and Kaneko 606

G Galerkin constraints, heat transfer 201

Galerkin formulation, fluid flow 145

Galerkin least-squares (GLS), stabiliza-

tion 138

gas boundary conditions 375

gas constant 566

general stress (boundary stress condi-

tion) 86

geometric entity selection 42

geometry, simplifying 31

Gidaspow and Ettehadieh solid pressure

model 431

Gidaspow model 429

Gidaspow solid pressure model 431

Ginzburg-Landau equation 631

Grashof number 259

gravity 259

gravity (node)

bf interfaces 367

mm interfaces 388

tpf interfaces 349

grille (node) 95

H Hadamard-Rybczynski drag law 423, 430

Haider-Levenspiel model 430

Haider-Levenspiel slip model 423

heat flux (node) 202

heat flux, theory 230

heat source (node) 198

heat sources

defining as total power 198

line and point 209

heat transfer coefficients

out-of-plane heat transfer 218

theory 258

heat transfer in fluids (node) 194

heat transfer in porous media (node) 222

heat transfer in porous media interface

221

theory 255

heat transfer in solids (node) 190

heat transfer interfaces 186

theory 227

Henry’s law 368

hide (button) 41

high Mach number flow (hmnf) interfaces

theory 321

high Schmidt numbers 568

hybrid outlet 327

I implementing, Euler-Euler equations 431

incompressible flow theory 115

inconsistent stabilization

settings 42

inflow (node)

transport of concentrated species 513,

532


inflow heat flux (node) 212

initial interface (node)

level set 618

phase field 623

tpf interfaces 351

initial values (node)

bf interfaces 369


Darcy’s law 445

ee interface 401


heat transfer 197

hmnf interfaces 313

level set 617

mm interfaces 389

nitf interfaces 288

phase field 622


542


549


rspf interfaces 533

spf interfaces 71

thin-film flow 589

tpdl interface 468

tpf interfaces 350



initializing functions 627, 633

inlet (boundary stress condition) 100

inlet (node)

bf interfaces 371

Darcy’s law 447

ee interface 402

hmnf interfaces 317

level set 617

mm interfaces 392

phase field 623

single-phase flow 77

thin-film flow 589

tpdl interface 470

inlet boundary condition, theory 122

interface normal variable 628

interior fan (node) 92

interior wall (node)

nitf interfaces 287

spf interfaces 93

internal boundary heat flux variables 243

Internet resources 22

interphase momentum transfer 429

intrinsic volume averages 473

inward heat flux 246

isotropic diffusion

fluid flow 138

inconsistent stabilization methods 251

K Karman constant 299, 570

Kays-Crawford models 298

k-epsilon turbulence model 58, 153

bubbly flow 415

mixture models 422

Khan-Richardson model 146

knowledge base, COMSOL 25

Knudsen number 606

k-omega turbulence model 62

Krieger type model 429

Krieger type viscosity model 386

L laminar bubbly flow (bf) interface 357

theory 411

laminar flow

Reynolds number, and 116

laminar flow (hmnf) interface 303

laminar flow (nitf) interface, conjugate

heat transfer 269

laminar flow (nitf) interface, non-isother-

mal flow 265

laminar flow interface 52

I N D E X | 645

646 | I N D E X

theory 111

laminar inflow (inlet boundary condition)

78

laminar mixture model (mm) interface

theory 418

laminar outflow (outlet boundary condi-

tion) 82

laminar two-phase flow, level set (tpf) in-

terface 338

laminar two-phase flow, phase field (tpf)

interface 341

leaking wall, wall boundary condition 74

level set functions, initializing 627, 633

level set interface 614

theory 405, 625

level set model (node) 616

lift force 429

line heat source (node) 209–210

line heat source variable 245

line mass source (node)

fluid flow 98

species transport 492

line source

fluid flow 137


local

CFL number 55, 143

local CFL number 104, 173

high Mach number flow 328

low re k-epsilon turbulence model 60

low Reynolds number

k-epsilon turbulence theory 166

neglect inertial term 266

lubrication 593

lumped curves 128

M Mach number

definition, single-phase flow 114

mass action law, laminar flow 566

mass balance 426

mass conservation, level set equations

408

mass flow inlet, theory 125

mass flow rate, theory 125

mass flux (node)

Darcy’s law 446

tpdl interface 469

mass fraction (node) 511

mass fractions 483

mass source (node)


Darcy’s law 445

mass sources

fluid flow 136

mass transfer (node)

bf interfaces 368

mm interfaces 387

mass transport 558

mathematics, moving interfaces

level set 614

phase field 619

theory 625, 630

mean curvature variable 628

mean effective thermal conductivity 192

mean effective thermal diffusivity 192

mean value closure 571

mechanisms of heat transfer 227

MEMS Module 463

meshing 32

microfluidic wall conditions (node) 462

mixture model, laminar flow (mm) inter-

face 377

mixture model, turbulent flow (mm) in-

terface 381

mixture properties (node) 385

mixture viscosity 386

mixture-averaged diffusion model 503,

559

Model Libraries window 23

model library examples

conjugate heat transfer, laminar flow

185

Euler-Euler model, laminar flow 397

heat transfer in fluids 194

heat transfer in solids 193

inlet (laminar flow) 78

laminar flow 57

laminar two-phase flow, level set (tpf)

341

laminar two-phase flow, phase field

(tpf) 343

mixture model, laminar flow 381

nitf interface, laminar flow 269

outlet (laminar flow) 78

phase field 620


thermodynamics 193

thin-film flow, shell 581


turbulent bubbly flow 363

turbulent flow (hmnf) interface 311

turbulent flow, k-epsilon (spf) 60

turbulent flow, k-omega (spf) 63

modeling strategy, physics 30

modified Reynolds equation, gas flow 605

modified Reynolds equation, thin-film

flow 593

momentum balance equations 427

moving interfaces 630

moving wall (wall functions), boundary

condition 76

moving wall, interior wall boundary con-

dition 95

moving wall, wall boundary condition 74

MPH-files 23

multiphase flow theory 405, 411, 418,

625, 630

N natural and forced convection 258

Navier-Stokes equations 112, 405

Neumann condition 157

Newtonian model 117

niterCMP variable 143

no flow (node)

Darcy’s law 448

level set 618

no flux (node)

tpdl interface 468



no slip, interior wall boundary condition

94

no slip, wall boundary condition 73

no viscous stress (open boundary) 85

non-conservative formulations 408, 553

non-isothermal flow interface

theory 293

non-isothermal flow interfaces 265

non-Newtonian fluids 113

non-Newtonian power law and Carreau

model 283

normal conductive heat flux variable 242

normal convective heat flux variable 242

normal stress (boundary condition) 78

normal stress, normal flow (boundary

stress condition) 86

normal total energy flux variable 243

normal translational heat flux variable

243

Nusselt number 259

O open boundary (boundary stress condi-

tion) 100

open boundary (node) 288

heat transfer 214

I N D E X | 647

648 | I N D E X

rspf interfaces 535

spf interfaces 84



outflow (node)

heat transfer 202



outlet (boundary stress condition) 100

outlet (node)

bf interfaces 373

Darcy’s law 449

ee interface 404

hmnf interfaces 319

mm interfaces 393

spf interfaces 81

thin-film flow 590

tpdl interface 470

outlet boundary condition 128

outlet boundary condition, theory 122

out-of-plane convective heat flux (node)

217

out-of-plane heat flux (node) 219

out-of-plane heat transfer

change thickness 220

general theory 253

shallow channel approximation 267

out-of-plane inward heat flux variable

242

out-of-plane radiation (node) 218

override and contribution 41–42

P pair boundary heat source (node) 205

pair nodes





heat transfer 189


hmnf interfaces 311

level set 615

nitf interfaces 277

phase field 620

reacting flow 526


540

reacting flow in porous media, (rfds)

547


spf interfaces 66

tff interface 583

tffs interfaces 583

tpf interfaces 343



pair selection 43

pair thin thermally resistive layer (node)

207

particle tracing in fluid flow 146

perfectly stirred reactor 571

periodic condition (node)


periodic flow condition (node) 89

periodic heat condition (node) 205

phase field interface 619

theory 405, 630

phase field model (node) 621

phase properties (node) 399

point heat source (node) 210

point heat source on axis (node) 211

point heat source variable 245

point mass source (node)

fluid flow 97


point nodes





heat transfer 189


hmnf interfaces 311

nitf interfaces 277

reacting flow 526


540


547


spf interfaces 66

tff interface 583

tffs interfaces 583

tpf interfaces 343

point selection 42

point source

fluid flow 136


pointwise mass flux, theory 125

Poiseuille flow 599

porous matrix properties (node) 460

porous media and subsurface flow

Brinkman equations interface 450


free and porous media flow interface

457

theory, Brinkman equations 473

theory, Darcy’s law 471

theory, free and porous media flow

476

power law, non-Newtonian 283

power law, single-phase flow theory 117

Prandtl number 259, 298

pressure (node) 446

pressure (outlet boundary condition) 82

pressure and saturation (node) 469

pressure point constraint (node) 99

pressure work (node)

heat transfer 211

pressure, no viscous stress (inlet and

outlet boundary conditions) 78

pressure-correction method 144

projection method, Navier-Stokes equa-

tions 144

pseudo time step 564

pseudo time stepping

hmnf interfaces 328

laminar flow theory 142

settings 55

theory 143

pseudoplastic fluids 117

pumps, lumped curves and 131

R radiation, out-of-plane 218

radiative heat flux variable 243

radiative heat, theory 248

radiative out-of-plane heat flux variable

241

RANS

mixture model interface 382

rotating machinery interface 105

theory, single-phase flow 150

rarefied gas 606

rarefied-general accommodation 607

rarefied-total accommodation 607

ratio of specific heats 194

Rayleigh number 259

reacting boundary (node) 534


543

reacting flow in porous media (rfcs) in-

terface 537

theory 574

reacting flow in porous media (rfds) in-

I N D E X | 649

650 | I N D E X

terface 544

theory 575

reacting flow interfaces

theory 563

turbulent flow k-epsilon 522

turbulent flow k-omega 524

turbulent flow low re k-epsilon 525

reacting flow laminar flow (rspf) inter-

face 518

reacting flow, turbulent flow k-epsilon

interface 522

reacting flow, turbulent flow k-omega in-

terface 524

reacting flow, turbulent flow low re

k-epsilon interface 525

reaction rate, turbulence and 570

reactions (node) 531



regularized mass fractions 539

removing species 483

Reynolds number 259

extended Kays-Crawford 298

low, turbulence theory 166

slip velocity models 423


Reynolds number definition 116

Reynolds particle number 146

Reynolds stress tensor 151, 154

Reynolds-averaged Navier-Stokes. See

RANS.

rotating domain (node) 108

rotating interior wall (node) 110

rotating machinery (rmspf) interfaces

theory 175

rotating machinery, laminar flow (rmspf)

interface 103

rotating machinery, turbulent flow (rm-

spf) interface 105

rotating wall (node) 109

S Schiller-Naumann model 430

Schiller-Naumann slip model 423

Schmidt number 423, 567

screen (node) 87

selecting

heat transfer interfaces 182, 186

high Mach number flow (hmnf) inter-

faces 302

multiphase flow interfaces 332

porous media and subsurface flow in-

terfaces 436

single-phase flow interfaces 46

SEMI standard E12-0303 80

settings windows 41

shallow channel approximation 267

shear rate magnitude variable 69

shear thickening fluids 117

show (button) 41

single-phase flow


single-phase flow interface

laminar flow 52

theory 111

SIPG method 145

slide-film damping 594

sliding wall (wall functions), boundary

condition 75

sliding wall, wall boundary condition 73

theory 120

slip length 606

slip model

Hadamard-Rybczynski 423

Schiller-Naumann 423

slip model theory 414

Haider-Levenspiel 423

Reynolds number 423

slip velocity, wall boundary condition 74

theory 120

slip, interior wall boundary condition 95

slip, wall boundary condition 73

theory 119

solid pressure and particle interaction

431

Soret effect 561

sound waves 326

Spalart-Allmaras turbulence model 64

specific heat capacity, definition 229

spf.cellRe variable 116

spf.sr variable 69, 529

squeezed-film damping 594

SST turbulence model 63

stabilization settings 42

stabilization techniques

crosswind diffusion 251

standard flow rate, theory 125

static pressure curves 88, 91

Stokes equations 57

Stokes flow. see Creeping Flow interface

strain-rate tensors 212

streamline diffusion

fluid flow 138

heat transfer, and 251

superficial volume average, porous me-

dia 474

supersonic inlet 327

supersonic outlet 328

surface tension force variable 634

surface-to-ambient radiation (node) 204

Sutherland’s law 315

swirl flow 114, 155

symmetric interior penalty Galerkin

method (SIPG) 145

symmetry (node)

bf interfaces 375

Darcy’s law 447

heat transfer 202

mm interfaces 394

spf interfaces 83

thin-film flow 592



symmetry, flow (node) 291

symmetry, heat (node) 290

T technical support, COMSOL 24

temperature (node) 201

tensors

Reynolds stress 154

strain-rate 212

viscous stress theory 116

theory

bf interfaces 411


conjugate heat transfer interface 293

Darcy’s law 471

Euler-Euler model, laminar flow 426


heat equation definition 228

heat transfer 227

heat transfer coefficients 258


hmnf interfaces 321

inlet and outlet boundary conditions

122

laminar flow 111

level set 625

mm interfaces 418

non-isothermal flow 293

out-of-plane heat transfer 253

phase field 630

reacting flow 563


574

I N D E X | 651

652 | I N D E X


575



spf interfaces 111

thin-film flow 593

tpf interfaces 405

transport of concentrated species in-

terface 558

transport of diluted species interface

551

turbulent flow k-epsilon model 149

turbulent flow low re k-epsilon model

149

thermal conductivity, mean effective 192

thermal creep, wall boundary condition

75

thermal diffusion 561

thermal diffusivity 192

thermal dispersion (node) 226

thermal expansivity 259

thermal insulation 246

thermal insulation (node) 200

thin diffusion barrier (node) 499

thin impermeable barrier (node) 500

thin thermally resistive layer (node) 207

thin-film flow interfaces

theory 593

thin-film flow, domain (tffs) interface 581

thin-film flow, edge (tff) interface 582

thin-film flow, shell (tffs) interface 579

thin-film gas flows 605

TMAC, microfluidic wall conditions 463

Torczynski and Gallis 608

total accumulated energy power 234

total accumulated heat power 233

total energy flux variable 241

total fluid losses 233

total heat flux 203

total heat flux variable 239

total heat source 233

total net energy power 235

total net heat power 233

total normal heat flux variable 242

total power 198

total work source 235

traction boundary conditions 85

translational heat flux variable 241

translational motion (node) 193

transport mechanisms 489, 506

transport of concentrated species inter-

face 501

theory 558

transport of diluted species interface 485

theory 551

transport properties (node)


541


548

turbulence models

k-epsilon 58, 153

k-epsilon, bubbly flow 415

k-epsilon, mixture models 422

k-omega 62

low re k-epsilon 60

nitf interfaces 271


Spalart-Allmaras 64, 169

SST 63

turbulence models, reacting flow 566

turbulent bubbly flow (bf) interface 360

theory 411

turbulent compressible flow 152

turbulent conjugate heat transfer

theory 295

turbulent dissipation rate

multiphase flow 415

turbulent flow k-epsilon (nitf) interface

270

turbulent flow k-epsilon (spf) interface

58

turbulent flow k-epsilon interface

theory 149

turbulent flow k-omega (nitf) interface

276

turbulent flow k-omega interface 62

turbulent flow low re k-epsilon (nitf) in-

terface 270

turbulent flow low re k-epsilon (spf) in-

terface 60

turbulent flow low re k-epsilon interface

theory 149

turbulent flow Spalart-Allmaras (nitf) in-

terface 273

turbulent flow Spalart-Allmaras (spf) in-

terface 64

turbulent flow SST (nitf) interface 274

turbulent flow, k-epsilon (hmnf) inter-

face 306

turbulent flow, Spalart-Allmaras (hmnf)

interface 309

turbulent flow, SST (spf) interface 63

turbulent heat flux variable 240

turbulent kinetic energy theory 415

k-epsilon model 153

mixture model 422

RANS 153

turbulent length scale 171

turbulent mass transport flux 566

turbulent mixing 556, 562

turbulent mixing (node)



turbulent mixture model (mm) interface

theory 418

turbulent non-isothermal flow interfaces

theory 295

turbulent Prandtl number 298

turbulent Schmidt number 557, 562

turbulent two-phase flow, level set (tpf)

interface 352

turbulent two-phase flow, phase field

(tpf) interface 355

two-fluid Euler-Euler model

bubbly flow 411

mixture model 418

two-phase Darcy’s law (tpdl) interface

464

U undamped turbulent kinematic viscosity

169

user community, COMSOL 25

V vacuum pump (node) 88

vacuum pump theory 128

variable, shear rate magnitude 529

variables

dimensionless distance to cell center

168

level set interface 628

niterCMP 143

phase field interface 633

spf.cellRe 116

velocity (inlet and outlet boundary con-

ditions) 78

viscous drag, coefficient 414

viscous heating (node)

heat transfer 212

nitf interfaces 290

viscous slip, wall boundary condition 74

viscous stress tensors, theory 116

viscous stress, theory 126

volume averages 473

I N D E X | 653

654 | I N D E X

volume force (node)



spf interfaces 71

volumetric heat capacity 193

vorticity waves 326

W wall (node)

bf interfaces 369

ee interface 402

mm interfaces 390

nitf interfaces 285


thin-film flow 591

wall distance initialization study step 164,

167

wall functions, turbulent flow 156

wall functions, wall boundary condition

75

weak constraint settings 42

web sites, COMSOL 25

well posedness 324

Wen and Yu fluidized state expression

429

wetted wall (boundary condition) 345

wetted wall (node) 624

Z zero shear rate viscosity 118

Date post:	13-Apr-2018
Category:	Documents
Upload:	lydien
View:	247 times
Download:	5 times

The CFD Module User’s Guide - lost-contact.mit.edu home … · Visit the Contact COMSOL page at...

Documents