Simulation and Optimization of Electromagnetohydrodynamic

FlowsBrian Dennis

Ph.D. Candidate

Aerospace Engineering Dept.

Penn State University

George DulikravichProfessor

Mechanical and Aerospace Dept.

University of Texas at Arlington

Overview

• Introduction• Governing equations for EMHD• General LSFEM formulation• h and p-type finite element solver• LSFEM for EMHD• Code validation cases • Applications• Conclusion and recommendations

Introduction

• EMHD is study of incompressible flow under the influence of electric and magnetic fields

• Various simplified analytical models exist and have been used for numerical simulation(EHD,MHD)

Introduction cont.• A fully consistent non-linear model for general

EMHD has been recently developed

• No numerical simulations of full EMHD has been report in open literature to date

• A few numerical simulations of simplified or inconsistent EMHD models have been report in open literature

• A computer code for numerical simulation for 2-D planar MHD/EMHD flows has been developed using LSFEM

Introduction cont.

• Numerical simulation is necessary for performing optimization involving EMHD flows

Applications of EMHD• Manufacturing(solidification,crystal

growth)

• Flow control

• Drag reduction/propulsion

• Pumps with no moving parts(artificial heart, liquid metal pumps)

• Compact heat exchangers

• Shock absorbers, active damping

Fully Consistent Model of EMHD

Conservation of Linear Momentum

30 i]ΤΤαρg[1Dt

vDρ me ppp

)vv(μ t

v )EE(σ 2

TTT

κ 2

sTEσT

κ5

5

Eq e BEσ1 BEdσ 2 BTσ 4

BTdσ 5 B)BE( σ 7 B)BT( T

κ10

EEε p BBμ1m

BEvε p

)BE(εDt

Dp

Conservation of Energy

EκΤdκΤκQDt

TDρC 421hp

BEκBΤκEdκ 1075

ΤdΤΤ

κΤEσEEσ 2

41

BΤEΤ

κΤdE

Τ

κ 105

Dt

BDEVε

μ

B

Dt

EεDE p

m

p

.

Conservation of Mass

0v

Maxwell’s Equations

ep qBvεEε ,

0B ,

t

BE

,

vqBvεEεt

)Evε(μ

Bepp

ΤdσΤσEdσEσ 5421

BΤκTBEσ 10-1

7 .

Sub-models

The LSFEM was applied to two sub-models of the fully nonlinear Electro-magneto-hydrodynamic system

•Magneto-hydrodynamics

•Electro-magneto-hydrodynamics with reduced number of source terms

Least-Squares Finite Element Method (LSFEM)

Advantages of the LSFEM

• Can use equal order basis functions for pressure and velocity

• can use the first order form of PDE’s

• can handle any type of equation and mixed types of equations

• Can discretize convection terms without upwinding or explicit artificial dissipation

• Stable and robust method

• Resulting system of equations is symmetric and positive definite

• Simple iterative techniques such as PCG and multigrid can be used to solve the system of equations

• Inclusion of divergence constraint for magnetic flux is straight forward

Least-squares finite element method (LSFEM)

h and p-Type Finite Element Solver

h and p-Type Methods• h-type finite element methods use large numbers of low order accurate elements

•p-type finite element methods use small numbers of high order accurate elements

•h/p-type finite element methods use a combination of both types together with an adaptive strategy

•p-type methods convergence to the exact solution is more rapid than in with h-type methods, if the underlying solution is smooth

p-Type Method• element approximation function are compose of p-type expansions. These expansions are composed of a summation of P polynomials

•p-type expansions can be categorized as nodal expansions or modal expansions

•the unknown coefficients in nodal expansions are the values of the function at the nodes of the element and therefor have some physical meaning. Nodal expansions based on Lagrange polynomials are typically used for low order finite elements

•the unknown coefficients in modal expansions are not associated with any nodes

p-Type Method•Modal expansions are hierarchical, that is, that all expansion sets less than P+1 are contained within the expansion set P+1

•Nodal expansions are not hierarchical

•Modal expansion typically produce finite element matrices that have a lower condition number than those produced with nodal expansions

•Expansions are usually developed in one dimension. Approximation functions for multidimensional elements are constructed through tensor products of the one dimensional expansions.

•Typically, the modal expansions are combined with the first order Lagrange polynomials so that continuity of the solution is satisfied between neighboring elements in a mesh

LSFEM CodeA serial code was developed in C/C++ to solve general systems with LSFEM

•Steady state problems only

•Mixed triangular and quadrilateral meshes for h-type elements.

•Quadrilateral elements for p-type elements. Element approximation functions are constructed from a modal basis derived from Jacobi polynomials

•Support for multiple material domains such as in conjugate heat transfer problems for h-version

•Nonlinear equations are linearized with Newton or Picard method

LSFEM Code•Integrals are evaluated numerically using Legendre-Gauss quadrature. A P+2 rule was used for p-type elements.

•Static condensation is used to remove the interior degrees of freedom from p-type elements.

Sparse Linear SolversDirect Solver:

•Sparse LU factorization from PETSc library

Iterative Solvers:

•Jacobi preconditioned conjugate gradient method

•Multi-p multilevel method

LSFEM for EMHD

Nondimensional Form for EMHD Equations

Nondimensional Form for EMHD Equations

First Order Form for EMHD Equations

Iterative Solution Algorithm

General First Order System for 2-D

General First Order System for 2-D

General First Order System for 2-D

Code Validation

Verification of Accuracy

• Few analytic solutions for EMHD exist

• Some analytic solutions for MHD exist

• NSE portion of code was validated using analytic solutions for NSE and with experiment data from driven cavity flows and backward facing step

• Heat transfer/Electric/Magnetic field portions were verified with analytic solutions

Test Cases for NSE

• Test against analytical solutions for NSE

• Test against driven cavity numerical benchmark solutions

• Test against experiment data for flow over backward facing step

Mesh for analytic test cases

Comparison between computed and analytic solution for Couette-Poiseuille

flow

Mesh for driven cavity benchmark

Comparison with benchmark results for driven cavity at

Re = 1000

Computed streamlinesRe = 1000

Computed pressureRe = 1000

Computed vorticityRe = 1000

Comparison with benchmark results for driven cavity at

Re = 10000

Computed streamlinesRe = 10000

Computed pressureRe = 10000

Computed vorticityRe = 10000

Convergence history for driven cavity benchmark

Backward Facing Step Problem

Mesh for Backward Facing Step Problem

Computed Streamlines

Re = 100

Re = 400

Re = 500

Comparison of Computed and Measured Reattachment Lengths

Comparison of Present Results with Results from Commercial

CFD Codes

Mesh for Buoyancy-Driven Cavity

Computed Static Temperature Distribution

Computed Heat Flux on Vertical Walls

Computed Streamlines

Computed Streamline in Upper Left Corner

Comparison of Present Computations with Other Published Results

Test Cases for MHD and EMHD

• Test against analytical solutions for MHD and EMHD

LSFEM for MHD

First order system for MHD

First order system for 2-D MHD

First order system for 2-D MHD cont.

Mesh for Hartmann Flow Test Case

Hartmann FlowMHD LSFEM code was compared with the analytic solution to Poisuille-Hartmann flow

Comparison of LSFEM and analytic solution for Hartmann flow

Comparison of LSFEM and analytic solution for Hartmann flow

Comparison of LSFEM and analytic solution for Hartmann flow

Computed and analytic velocity profile

Comparison of LSFEM and analytic solution for Hartmann flow

Computed and analytic induced magnetic field

Comparison of h and p-type methods for Hartmann flow

Comparison of LSFEM and analytic solution for Hartmann flow with applied electric and

magnetic field

Simulation and Optimization of Magneto-hydrodynamic Flows

with LSFEM

Optimization of Magneto-Hydrodynamic Control of Diffuser

Flows Using Micro-Genetic Algorithms and Least-

Squares Finite Elements

Goal

• Given a fixed diffuser shape, use micro-GA and LSFEM MHD analysis to design a magnetic field distribution on the diffuser wall that will increase static pressure rise

Flow Solver

• LSFEM solver for 2-D steady incompressible Navier-Stokes together with Maxwell’s equations for steady magnetic field

• Uses hybrid quadrilateral/triangular grid

• One analysis takes around 22 min. on a single Pentium II CPU

BC’s and Parameterization

• Parabolic velocity specified at inlet• Static pressure specified at outlet• no-slip conditions on wall• magnetic field component along wall are

specified. They were parameterized with b-spline

• perfectly conducting wall bc used on all other solid surfaces

Hybrid mesh for diffuser analysis

Given diffuser shape

Convergence History for a Typical Design

Parallel Genetic Algorithm• GA is a naturally coarse grained parallel algorithm• One node maintains the population(master) and

distributes jobs to the slave nodes• Only simple synchronous message passing is

needed to implement on distibuted memory• Population size need not match the number of

slave nodes• Asynchronous models are also being developed

for use when function analysis computation times vary dramatically.

Parallel Computer

• based on commodity hardware components and public domain software

• 16 dual Pentium II 400 MHz based PC’s

• 100 Megabits/second switched ethernet

• total of 32 processors and 8.2 GB of main memory

• Compressible NSE solver achieved 1.5 Gflop/sec with a LU SSOR solver on a 100x100x100 structured grid

Parallel Computer• based on commodity hardware components and public

domain software

• 16 dual Pentium II 400 MHz based PC’s

• 100 Megabits/second switched ethernet

Parallel Computer cont.• total of 32 processors and 8.2 GB of main memory

• Compressible NSE solver achieved 1.5 Gflop/sec with a LU SSOR solver on a 100x100x100 structured grid

• GA optimization of a MHD diffuser completed in 30 hours. Same problem would take 14 days on a single CPU

Genetic Algorithm

• Population size of 15

• 100 generations

• 9-bit strings for each design variable

• elitism

• tournament selection

• uniform crossover

• parallel micro-GA

GA Convergence History

Run 1 Run 2

ResultsTwo optimizations were run simultaneously with 16 processors each.

Generation 100 was reached by both in about 30 hours

run 1 achieved a pressure increase of .207 Pa run 2 achieved a pressure increase of .228 Pa

diffuser without an applied magnetic field achieved a pressure increase of .05 Pa.

With no applied magnetic field

With optimized applied magnetic field

Simulation of Magneto-hydrodynamic Flows with Conjugate Heat Transfer

with LSFEM

Objective

• Simulation of flow through channel with an applied magnetic field

• Simulation of heat transfer from the flow to a solid cold wall

• Observe the effect of applied magnetic field on flow patterns and heat transfer characteristics

Boundary Conditions• Inlet temperature of 2000 K

• Specified outlet pressure

• Specified parabolic velocity profile at inlet

• No-slip on walls

• Symmetry boundary condition on top

• Temperature of 300 K on bottom wall

• Perfectly conducting walls except in the region 7 < x < 8 where sinusoidal magnetic field components were specified. Magnitude was varied from 0 to 5 Tesla.

Results

• Presence of magnetic field induces a large separation in the flow field close to the wall

• Size and complexity are proportional to the strength of the magnetic field

• A drop in fluid/solid interface temperature was observed in the region where the magnetic field was applied

Effect of flow on magnetic field

Solidification with MHD

1.49E-04

2.97E-04

5.94E-04

1.34E-032.

97E-04

8.91E-04

1.34E-037.43E-04

2.23E-03

2.08E-03

1.04E-03

X

Y

0 0.05 0.10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

X

Y

0 0.05 0.10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Figure 14: Velocity magnitude for no applied magnetic field.

Figure 15: Streamlines for no applied magnetic field.

Optimized MHD Solidification

1.49E-04

5.94E-041.19E-03

8.91E-04

1.34E-03

2.23E-031.34E-03

7.43E-04

1.34E-03

2.97E-04

X

Y

0 0.05 0.10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

X

Y0 0.05 0.10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Figure 16: Velocity magnitude for optimized case 1.

Figure 17: Streamlines for optimized case 1.

Optimizer MHD Solidification

1.87E-05

3.11E-06

9.33E-06

2.18E-05

4.67E-05

2.80E-05

2.80E-05

2.18E-05

1.87E-05

3.11E-06

1.56E-05

1.56E-05

X

Y

0 0.05 0.10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

X

Y

0 0.05 0.10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Figure 21: Velocity magnitude for optimized case 2.

Figure 22: Streamlines for optimized case 2.

Solidification with Optimized Magnetic Boundary Conditions

 Figure 10. Isotherms and streamlines without and with an optimized magnetic field (test case 2 with six sensors)

(2D) 24 Oct 2002 MHD SOLIDIFICATION PROBLEM

-0.44-0.4

4

-0.37-0.3

7-0.37

-0.31-0.31

-0.25

-0.25

-0.19

-0.19

-0.12

-0.12

-0.12

-0.06

-0.06

0.00

0.00

0.06

0.06

0.13

0.13

0.19

0.19

0.19

0.25

0.25

0.25

0.31

0.31

0.31

0.38

0.38

0.38

0.44

0.44

0.44

(2D) 24 Oct 2002 MHD SOLIDIFICATION PROBLEM

(2D) 24 Oct 2002 MHD SOLIDIFICATION PROBLEM

- 0. 44

-0.44

-0. 44

-0.38

-0.38

-0.31

-0.31

-0.25

-0. 25

-0.25

-0.19

-0.19

-0.19

-0.13

-0.13

-0.13

-0.06

-0.06

0.00

0.00

0.06

0.06

0.13

0.13

0.13

0.19

0.19

0.19

0.25

0.25

0.25

0.31

0.31

0.38

0.38

0.44

0.44

(2D) 24 Oct 2002 MHD SOLIDIFICATION PROBLEM

Simulation of Electro-magneto-hydrodynamic Channel Flows

Objective

• Simulation of a steady state EMHD blood pump

• Both electric and magnetic fields are required to produce the driving force

Governing Equations

Boundary conditions and geometry

•Rectangular domain with height of 4 cm and length of 40 cm

•Triangular mesh:7021 nodes

3422 elements

parabolic triangles

•Specified parabolic inlet velocity profile and temperature of 310.15 K

•No slip on walls

•Wall temperature was 298.15 K

•Specified exit pressure of 1 Pa

•Positive electrode on bottom wall, negative electrode on top with 50 volts applied across them

•Uniform magnetic field of .05 Tesla specified in Z direction

Physical parameters for EMHD blood pump

Density(kg m-3) = 1055.0

Inlet height(cm) = 4

Length(cm) = 40

Inlet temp.(K) = 310

Wall temp(K) = 298

heat conductivity(W kg-1 K-1) = .51

specific heat(J kg-1 K-1)= 4178

inlet velocity(m s-1) = .05

dynamic viscosity(kg m-1 s-1) = .004

electric conductivity (S m-1) = 1.4

outlet pressure (Pa) = 1

Computed Electric Potential

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35X

0

0.05

0.1

0.15

0.2

0.25

0.3

YPot21.87518.7515.62512.59.3756.253.1250-3.125-6.25-9.375-12.5-15.625-18.75-21.875

Computed Velocity Vectors

0.35 0.36 0.37 0.38 0.39 0.4 0.41X

0

0.01

0.02

0.03

0.04

0.05Y

Computed Static Pressure

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35X

0

0.05

0.1

0.15

0.2

0.25

0.3

Y

P-1.1625-3.32499-5.48749-7.64999-9.81249-11.975-14.1375-16.3-18.4625-20.625-22.7875-24.95-27.1125-29.275-31.4375

Computed Static Temperature

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35X

0

0.05

0.1

0.15

0.2

0.25

0.3

Y

T318.734317.525316.316315.107313.898312.689311.48310.271309.062307.853306.645305.436304.227303.018301.809

EMHD Channel Electrode Configuration

+

-

Mesh for EMHD channel problem

0 5 10 15 20X

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Y

Computed Electric Potential Distribution

0 5 10 15 20X

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Y

POT437.5375312.5250187.512562.50-62.5-125-187.5-250-312.5-375-437.5

Computed Electric Potential on Lower Wall

0 5 10 15 20X

0

100

200

300

400

500

PO

T

Compute Static Pressure

0 5 10 15 20X

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Y

P-67.4512-249.846-432.242-614.637-797.032-979.427-1161.82-1344.22-1526.61-1709.01-1891.4-2073.8-2256.19-2438.59-2620.98

Compute Static Pressure

0 5 10 15 20X

-2500

-2000

-1500

-1000

-500

0

P

P centerP wall

Computed Axial Velocity Distribution

0

50

U

0

5

10

15

20

X00.2

0.40.6

0.81

Y

X Y

Z

EMHD Channel Electrode Configuration

+ - + -

- + - +

Computed Electric Potential

1 2 3 4 5X

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Y

POT43.7537.531.252518.7512.56.250-6.25-12.5-18.75-25-31.25-37.5-43.75

Computed Electric Field Lines

1 2 3 4 5X

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Y

Computed Static Pressure

1 2 3 4 5X

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Y

P17.210815.017512.824110.63078.437396.244024.050661.8573-0.336063-2.52943-4.72279-6.91615-9.10951-11.3029-13.4962

Computed Streamlines

1 2 3 4 5X

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Y

Conclusions• A code for the simulation of MHD/EMHD was

developed based on the LSFEM• Code was tested against analytic solutions and

experimental data for separate disciplines• Code was applied to several MHD problems

including a MHD diffuser optimization problem• Code was used to simulate a EMHD pump and

several channel flows with finite length electrodes

Conclusions• LSFEM works well for NSE, MHD, and

EMHD when the problem are not very nonlinear.

• The ‘squaring’ effect inherent to the method appears to enhance any nonlinearity already present in the system. Thus, as Reynolds and Hartmann numbers increase, the required number of iterations increases rapidly

Conclusions• For LSFEM, first order forms for the PDE’s are

required in order to satisfy solution continuity requirements. This leads to a relatively large number of unknowns, especially in 3D, compared to a Galerkin type method

• The linear algebraic systems can become ill-conditioned. Better preconditioners are needed.

Future Work

• Better preconditioning of linear systems

• Closer look at scaling of the equations

• Extension to 3-D

• Better boundary conditions

• Unsteady