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McBride, Diane (2003) Vertex-based discretisation methods for thermo-fluid flow in a finite volume-unstructured mesh context. PhD thesis, University of Greenwich.

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32-

Vertex-Based Discretisation Methods for Thermo - Fluid Flow in a

Finite Volume-Unstructured MeshContext

Diane McBride

Centre for Numerical Modelling and Process Analysis

School of Computing and Mathematical Sciences

the University of Greenwich

London

A thesis submitted in partial fulfillment of the requirements of the

University of Greenwich for the Degree of Doctor of Philosophy

July 2003

Abstract

The main aim of this research project is to investigate techniques to improve the resolution of flow variables on unstructured skewed meshes whilst working within a Finite Volume (FV) context. A three-dimensional vertex-based FV algorithm for the solution of thermo - fluid flow problems has been developed and integrated within a multi-physics FV framework PHYSIC A. Currently PHYSIC A employs a cell-centred discretisation technique for fluid mechanics problems and a vertex-based discretisation technique for solid mechanics problems. The vertex-based discretisa- tion approach is validated for a variety of heat transfer problems and comparisons are made with cell-centred solutions. A coupled thermo-mechanical problem, in- cluding solidification and radiation, is simulated using vertex-based and cell-centred techniques. Results, run-time and memory requirements are compared.

Hybrid vertex-based/cell-centred discretisation of the hydrodynamic variables is also investigated. The components of velocity are solved vertex-based with pressure cell- centred or conversely pressure is solved vertex-based with velocity cell-centred. The methods are applied to flow in a lid-driven cavity and solutions are obtained on a number of distorted meshes. Comparisons are made with the benchmark solutions. The hybrid discretisation enables solutions on distorted meshes where purely cell- centred techniques fail. The hybrid methods produce final solutions containing errors due to mesh distortion.

The co-located vertex-based flow solutions obtained on the distorted meshes are com- parable to solutions obtained on a uniform Cartesian mesh. Having a good resolution of the flow field on distorted meshes enables the solution of other transported vari- ables using cell-centred techniques. Finally, this hybrid vertex-based/cell-centred technique is applied to thermally driven flow, turbulent flow, and three-dimensional flow over an aircraft wing.

Acknowlegements

I would like to thank my supervisors Prof. Mark Cross and Dr. Nick Croft for their invaluable support throughout this research project. It is their encouragement, con- structive advice and helpful discussions that have made completion of this research project possible. I would also like to acknowledge the advice given by Dr. Nick Croft through his extensive knowledge of PHYSIC A.

Thanks go to my research colleagues for their friendship, suggestions and enjoyable working environment. Thanks also to the academic and technical staff of the school for their support.

This research would not have been possible without the financial support of the En- gineering and Physical Sciences Research Council and the University of Greenwich.

Especial thanks to my family, my husband Conrad and sons Jamie and Simon for their patience and understanding throughout my academic studies.

11

Contents

1 Introduction 1

1.1 Overview of Finite Volume Methods . .................. 3

1.1.1 Development of the Finite Volume Method ........... 5

1.2 The Finite Element Method ....................... 10

1.3 Mixed Finite Volume and Finite ElementTechniques ................................. 12

1.4 Outline of Contents ............................ 14

2 Finite Volume Method 17

2.1 Cell-Centred Approach .......................... 18

2.1.1 Non-Orthogonal Mesh ...................... 19

2.1.1.1 Non-Orthogonality ................... 20

2.1.1.2 Non-Conjunctionality ................. 22

2.2 Vertex-Based Approach .......................... 24

2.3 Comparison of Approaches ........................ 26

2.3.1 System of Discretised Equations ................. 29

2.4 Discretisation of the General TransportEquation ................................. 30

iii

CONTENTS 1V

2.4.1 Transient Term .......................... 31

2.4.2 Source Term ............................ 33

2.4.3 Diffusion Term .......................... 35

2.4.4 Convection Term ......................... 38

2.5 Convection Discretisation Schemes ................... 39

2.5.1 Central Differencing ....................... 40

2.5.2 Upwind Formulation ....................... 41

2.5.3 Hybrid Scheme .......................... 43

2.5.4 Higher Order Schemes ...................... 44

2.5.5 Flow Oriented Schemes ...................... 46

2.6 Closure. .................................. 50

3 Convection-Diffusion Test Cases 51

3.1 Transient Test Case ............................ 52

3.2 Steady-State Diffusion .......................... 60

3.3 Convected Scalar ............................. 64

3.4 Closure. .................................. 65

4 Coupled Physical Phenomena 67

4.1 Casting of a Turbine Blade ........................ 67

4.1.1 The Geometry ........................... 69

4.1.2 Material Data ........................... 69

4.1.3 Boundary Conditions ....................... 72

CONTENTS

4.2 Models of Physical Processes ...................... 74

4.2.1 Solidification ........................... 74

4.2.2 Radiation across Mesh Gap ................... 76

4.2.3 Solid Mechanics Equations .................... 79

4.3 Simulation Results ............................ 80

4.3.1 Run-time - Memory Requirements ................ 81

4.4 Closure. .................................. 87

5 Solution of Fluid Flow 88

5.1 Pressure - Velocity Coupling ....................... 89

5.1.1 SIMPLE .............................. 90

5.1.1.1 SIMPLEC ........................ 92

5.1.2 SIMPLER ............................. 93

5.1.2.1 Revised SIMPLER ................... 95

5.2 Velocity and Pressure Checkerboarding ................ 95

5.3 Cell-Centred Procedure ......................... 99

5.3.1 Rhie-Chow Interpolation Method ................ 99

5.3.2 The Solution Procedure ..................... 100

5.4 Vertex-Based Procedure ......................... 102

5.4.1 Discussion of Solution Method .................. 102

5.4.2 The Solution Procedure ..................... 104

5.5 Hybrid Discretisation of HydrodynamicVariables .................................. 106

CONTENTS vl

5.5.1 Pressure - Vertex-Based, Velocity - Cell-Centred ........ 108

5.5.2 Pressure - Cell-Centred, Velocity - Vertex-Based ........ Ill

5.6 Test Case - Lid-Driven Cavity ...................... 113

5.6.1 Results ............................... 116

5.6.2 Convergence and Run Times ................... 122

5.6.3 Discussion ............................. 126

5.7 Closure ................................... 127

6 Combined Method: Vertex-Based-Cell-Centred 128

6.1 Thermally Driven Flow .......................... 130

6.2 Closure ................................... 143

7 Turbulent Flow 147

7.1 k-e Model ................................ 152

7.2 Backward Facing Step .......................... 157

7.3 Flow over an Aircraft Wing ....................... 164

7.3.1 Two-dimensional Flow ...................... 166

7.3.2 Three-dimensional Flow - Uniform C-Mesh ........... 178

7.3.3 Distorted Three-dimensional Mesh ............... 192

7.3.4 Wing at 10° angle of attack ................... 206

7.3.5 Run Time and Memory Requirements .............. 221

7.4 Closure ................................... 221

8 Conclusions 223

CONTENTS

k-e

A Local Co-ordinate Systems 230

B Shape Functions 238

CONTENTS

List of Figures

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List of Tables

Chapter 1

Introduction

1.1 Overview of Finite Volume Methods

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o Cell-Vertex

o Vert ex-Cent red

1.1.1 Development of the Finite Volume Method

Mesh

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1.2 The Finite Element Method

1.3 Mixed Finite Volume and Finite Element

Techniques

1.4 Outline of Contents

Chapter 2

Finite Volume Method

2.1 Cell-Centred Approach

2.1.1 Non-Orthogonal Mesh

20

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