APPLICATION OF THE FINITE-VOLUME METHOD TO FLUID …

APPLICATION OF THE

FINITE-VOLUME METHOD TO

FLUID-STRUCTURE INTERACTION

ANALYSIS

A Thesis Submitted to the University of Manchester

for the Degree of Doctor of Philosophy

in the Faculty of Engineering and Physical Sciences

2011

By

Matthew Yates

School of Mechanical, Aerospace and Civil Engineering

Contents

List of Figures 15

List of Tables 18

Nomenclature 19

Abstract 21

Declaration 23

Copyright 25

Acknowledgements 27

1 Introduction 29

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.2 Study Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

1.3 Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2 Literature Review 35

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.2 Fluid-Structure Interaction Phenomena . . . . . . . . . . . . . . . . . . 35

2.3 Numerical Simulation of Fluid-Structure Interaction . . . . . . . . . . . 36

2.4 Application of the Finite-Volume Method to Elastic Analysis . . . . . . 36

2.5 Applications of Fluid-Structure Interaction . . . . . . . . . . . . . . . . 39

2.5.1 Flow Through Compliant Tubes and Channels . . . . . . . . . . 39

2.5.2 Stenosed Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.5.3 Aneurysmal Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.6 Numerical Grid Generation . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.6.1 Elliptic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.6.2 Solid-Body Methods . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3

CONTENTS

3 Mathematical Model of Fluid Motion 55

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2 The Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . 55

3.3 Turbulence Modelling Strategies . . . . . . . . . . . . . . . . . . . . . . 56

3.3.1 DNS Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.3.2 LES Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.3.3 RANS Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.4 The RANS Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.4.1 Reynolds Stress Models . . . . . . . . . . . . . . . . . . . . . . . 60

3.4.2 Eddy Viscosity Models . . . . . . . . . . . . . . . . . . . . . . . . 61

3.4.3 Non-linear Eddy Viscosity Models . . . . . . . . . . . . . . . . . 65

3.5 The Launder-Sharma Low-Re Model . . . . . . . . . . . . . . . . . . . . 65

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4 Mathematical Model of Solid Deformation 69

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2 The Stress Equilibrium Equations . . . . . . . . . . . . . . . . . . . . . 69

4.3 The Linear-Elastic Constitutive Relations . . . . . . . . . . . . . . . . . 71

4.4 The Kinematic Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.5 The Navier Displacement Equations . . . . . . . . . . . . . . . . . . . . 75

4.6 Solution of the Displacement Equations . . . . . . . . . . . . . . . . . . 77

4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5 Fluid Solver Implementation 79

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.2 The Finite-Volume Method . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.3 Convective Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.3.1 First Order Upwind Scheme . . . . . . . . . . . . . . . . . . . . . 83

5.3.2 QUICK Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.3.3 MUSCL Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.4 Temporal Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.4.1 Explicit Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.4.2 Implicit Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.4.3 Crank-Nicolson Method . . . . . . . . . . . . . . . . . . . . . . . 87

5.5 Storage Arrangement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.6 Calculation of the Pressure . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.7 Rhie-Chow Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.8 Non-Orthogonal Coordinate System . . . . . . . . . . . . . . . . . . . . 91

5.9 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.9.1 Inlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4

CONTENTS

5.9.2 Outlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.9.3 Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.9.4 Pipe Centre Line . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.10 Solution of the Discretized Equations . . . . . . . . . . . . . . . . . . . . 96

5.10.1 Source Term Linearization . . . . . . . . . . . . . . . . . . . . . . 96

5.10.2 Under-Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.10.3 The TDMA Solver . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.10.4 Convergence Criteria . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6 Solid Solver Development 99

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.2 Discretization of the Displacement Equations . . . . . . . . . . . . . . . 100

6.3 Temporal Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.4 Storage Arrangement in Solid Solver . . . . . . . . . . . . . . . . . . . . 103

6.5 Delta Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.6 Boundary Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.6.1 Traction Boundary Conditions . . . . . . . . . . . . . . . . . . . 106

6.6.2 Displacement Boundary Conditions . . . . . . . . . . . . . . . . . 107

6.6.3 Symmetry Boundary Conditions . . . . . . . . . . . . . . . . . . 107

6.7 Solution of the Discretized Equations . . . . . . . . . . . . . . . . . . . . 108


6.7.2 The TDMA Solver . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.8 Calculation of the Stress Components . . . . . . . . . . . . . . . . . . . 108

6.9 Validation of the Solid Solver . . . . . . . . . . . . . . . . . . . . . . . . 110

6.9.1 Planar Validation Exercise . . . . . . . . . . . . . . . . . . . . . 110

6.9.2 Axisymmetric Validation Exercise . . . . . . . . . . . . . . . . . 113

6.9.3 Unsteady Validation Exercise . . . . . . . . . . . . . . . . . . . . 117

6.10 Application to Grid Generation . . . . . . . . . . . . . . . . . . . . . . . 118

6.11 Fluid-Structure Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.11.1 Storage Arrangement . . . . . . . . . . . . . . . . . . . . . . . . 120

6.11.2 Calculation of Forces on Interface . . . . . . . . . . . . . . . . . . 122

6.11.3 Adaptation of Fluid Mesh . . . . . . . . . . . . . . . . . . . . . . 123


6.11.5 Convergence Criteria . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.11.6 Unsteady Coupled Analysis . . . . . . . . . . . . . . . . . . . . . 125

6.12 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5

CONTENTS

7 Pipe Flows 127

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

7.2 Simulation Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

7.2.1 Physical Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 128

7.2.2 Numerical Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

7.2.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 129

7.2.4 Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 130

7.2.5 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . 130

7.2.6 Under-Relaxation Factors . . . . . . . . . . . . . . . . . . . . . . 130

7.3 Rigid Tube Results and Discussion . . . . . . . . . . . . . . . . . . . . . 131

7.3.1 Analytical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7.3.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7.4 Compliant Tube Results and Discussion . . . . . . . . . . . . . . . . . . 132

7.5 Improvements to Re-meshing Algorithm . . . . . . . . . . . . . . . . . . 135

7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

8 Stenosed Flows 151

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

8.2 Steady Flow Through a Rigid Stenosis . . . . . . . . . . . . . . . . . . . 151


8.2.2 Numerical Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . 152


8.2.4 Fluid Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

8.2.5 Under Relaxation Factors . . . . . . . . . . . . . . . . . . . . . . 153

8.2.6 Laminar Flow Results and Discussion . . . . . . . . . . . . . . . 155

8.2.7 Turbulent Flow Results and Discussion . . . . . . . . . . . . . . 159

8.3 Steady Flow Through a Compliant Stenosis . . . . . . . . . . . . . . . . 173


8.3.2 Numerical Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . 174


8.3.4 Under Relaxation Factors . . . . . . . . . . . . . . . . . . . . . . 174


8.3.6 Sensitivity Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

8.3.7 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 177

8.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

9 Aneurysmal Flows 195

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

9.2 Steady Laminar Flow Through a Rigid Aneurysm . . . . . . . . . . . . . 195


6

CONTENTS

9.2.2 Numerical Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . 196





9.3 Steady Turbulent Flow Through a Rigid Aneurysm . . . . . . . . . . . . 200


9.3.2 Numerical Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . 204





9.4 Unsteady Laminar Flow Through a Rigid Aneurysm . . . . . . . . . . . 208


9.4.2 Numerical Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . 212





9.5 Unsteady Flow Through a Compliant Aneurysm . . . . . . . . . . . . . 232


9.5.2 Numerical Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . 232




9.5.6 Compliant Wall Results and Discussion . . . . . . . . . . . . . . 236

9.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

10 Conclusions and Further Work 255

10.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

10.1.1 Finite-Volume Solid Solver . . . . . . . . . . . . . . . . . . . . . 255

10.1.2 Straight Pipe Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 256

10.1.3 Stenosed Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

10.1.4 Aneurysmal Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

10.2 Suggestions for Further Work . . . . . . . . . . . . . . . . . . . . . . . . 259

10.2.1 Validation for New Test Cases . . . . . . . . . . . . . . . . . . . 259

10.2.2 Extension to Three Dimensions . . . . . . . . . . . . . . . . . . . 259

10.2.3 Addition of Implicit Transient Schemes . . . . . . . . . . . . . . 259

10.2.4 Code Parallelisation . . . . . . . . . . . . . . . . . . . . . . . . . 260

7

CONTENTS

10.2.5 Non-Linear Elastic and Plastic Constitutive Relations . . . . . . 260

A Elasticity Equations in Cartesian Coordinates 261

A.1 The Equilibrium Equations . . . . . . . . . . . . . . . . . . . . . . . . . 261

A.2 The Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . 261

A.3 Discretization of the u-Equilibrium Equation . . . . . . . . . . . . . . . 261

A.4 Discretization of the v-Equilibrium Equation . . . . . . . . . . . . . . . 264

B Elasticity Equations in Cylindrical Coordinates 267

B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

B.2 Transformation of Vector Functions . . . . . . . . . . . . . . . . . . . . . 267

B.3 Transformation of the Equilibrium Equations . . . . . . . . . . . . . . . 267

B.4 Transformation of the Constitutive Equations . . . . . . . . . . . . . . . 268

B.5 Discretization of the vz-Equilibrium Equation . . . . . . . . . . . . . . . 269

B.6 Discretization of the vr-Equilibrium Equation . . . . . . . . . . . . . . . 270

C Treatment of the South Boundary Equations 273

C.1 The u-Equilibrium Equation . . . . . . . . . . . . . . . . . . . . . . . . . 273

C.2 The v-Equilibrium Equation . . . . . . . . . . . . . . . . . . . . . . . . . 275

C.3 The vz-Equilibrium Equation . . . . . . . . . . . . . . . . . . . . . . . . 277

C.4 The vr-Equilibrium Equation . . . . . . . . . . . . . . . . . . . . . . . . 279

D Treatment of the West Boundary Equations 283

D.1 The u-Equilibrium Equation . . . . . . . . . . . . . . . . . . . . . . . . . 283

D.2 The v-Equilibrium Equation . . . . . . . . . . . . . . . . . . . . . . . . . 285

D.3 The vz-Equilibrium Equation . . . . . . . . . . . . . . . . . . . . . . . . 287

D.4 The vr-Equilibrium Equation . . . . . . . . . . . . . . . . . . . . . . . . 289

E Elasticity Equations in Planar Non-Orthogonal Coordinates 293

E.1 Non-orthogonal Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 293

E.2 The u-Displacement Equation . . . . . . . . . . . . . . . . . . . . . . . . 294

E.2.1 Transformation of Equilibrium Equation . . . . . . . . . . . . . . 294

E.2.2 Transformation of Constitutive Relations . . . . . . . . . . . . . 294

E.2.3 Finite Volume Discretization . . . . . . . . . . . . . . . . . . . . 297

E.3 The v-Displacement Equation . . . . . . . . . . . . . . . . . . . . . . . . 301

E.3.1 Transformation of Equilibrium Equation . . . . . . . . . . . . . . 301

E.3.2 Transformation of Constitutive Relations . . . . . . . . . . . . . 301

E.3.3 Finite Volume Discretization . . . . . . . . . . . . . . . . . . . . 304

8

F Elasticity Equations in Cylindrical Non-Orthogonal Coordinates 309

F.1 Transformation of the Governing Equations . . . . . . . . . . . . . . . . 309

F.1.1 Transformation of the Axial Displacement Equation . . . . . . . 309

F.1.2 Transformation of the Radial Displacement Equation . . . . . . . 310

F.2 Transformation of the Constitutive Relations . . . . . . . . . . . . . . . 310

F.2.1 Transformation of the Radial Stress . . . . . . . . . . . . . . . . 310

F.2.2 Transformation of the Axial Stress . . . . . . . . . . . . . . . . . 311

F.2.3 Transformation of the Hoop Stress . . . . . . . . . . . . . . . . . 311

F.2.4 Transformation of the Shear Stress . . . . . . . . . . . . . . . . . 311

F.3 Discretization of the Axial Displacement Equation . . . . . . . . . . . . 312

F.4 Discretization of the Radial Displacement Equation . . . . . . . . . . . . 317

Bibliography 324

Word count: 53,000 (approximate)

9

List of Figures

4.1 Two-dimensional stress element. . . . . . . . . . . . . . . . . . . . . . . 70

5.1 Typical finite-volume cell. . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.2 Fluid storage arrangement. . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.3 Chequer-board pressure field. . . . . . . . . . . . . . . . . . . . . . . . . 90

6.1 Solid storage arrangement. . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.2 Solid half-cell along south boundary. . . . . . . . . . . . . . . . . . . . . 106

6.3 Numerical mesh used for the planar validation exercise. . . . . . . . . . 112

6.4 Schematic of planar test case. . . . . . . . . . . . . . . . . . . . . . . . . 113

6.5 Results from the plate with a hole test case. . . . . . . . . . . . . . . . . 114

6.6 Schematic of axisymmetric test case. . . . . . . . . . . . . . . . . . . . . 115

6.7 Radial profiles of the radial stress and displacement in the cylinder under

internal pressure test case. . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.8 Schematic of unsteady test case. . . . . . . . . . . . . . . . . . . . . . . 118

6.9 Beam tip displacement - grid refinement . . . . . . . . . . . . . . . . . . 119

6.10 Beam tip displacement - time step refinement . . . . . . . . . . . . . . . 119

6.11 Coupled FSI Procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.12 Schematic of FSI storage arrangement. . . . . . . . . . . . . . . . . . . . 122

7.1 Schematic of straight compliant tube geometry. . . . . . . . . . . . . . . 128

7.2 Initial numerical mesh around the downstream end of compliant wall

section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

7.3 Comparison with the analytical solution for the rigid wall case. . . . . . 132

7.4 Comparison with the numerical data of Shim and Kamm. . . . . . . . . 134

7.5 Fluid flow vectors within the tube and radial stress contours in the com-

pliant wall. Pressure drop = 10 Pa . . . . . . . . . . . . . . . . . . . . . 138





11

LIST OF FIGURES





7.10 Axial u-displacement profiles for increasing pressure drop. . . . . . . . . 143

7.11 Radial v-displacement profiles for increasing pressure drop. . . . . . . . 143

7.12 Solid stress profiles for increasing pressure drop. . . . . . . . . . . . . . . 144

7.13 Wall shear stress profiles for increasing pressure drop. . . . . . . . . . . 145

7.14 Normalized wall shear stress profiles for increasing Reynolds number. . . 145

7.15 Wall pressure profiles for increasing pressure drop. . . . . . . . . . . . . 146

7.16 Normalized wall pressure profiles for increasing Reynolds number. . . . 146

7.17 Numerical mesh at downstream end of compliant wall section for increas-

ing pressure drop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

7.18 Detailed view of the numerical mesh at the downstream end of the com-

pliant wall section at pressure drop = 90 Pa. . . . . . . . . . . . . . . . 148

7.19 Effect of Poisson ratio upon the numerical mesh at the downstream end

of the compliant wall section at pressure drop = 90 Pa. . . . . . . . . . 149

8.1 Schematic of Young and Tsai geometry. . . . . . . . . . . . . . . . . . . 152

8.2 Mesh used during laminar, rigid walled stenosis simulations. . . . . . . . 153

8.3 Mesh used during turbulent, rigid walled stenosis simulations. . . . . . . 154

8.4 Laminar flow separation and re-attachment curves. . . . . . . . . . . . . 155

8.5 Laminar flow streamlines for rigid stenosis. . . . . . . . . . . . . . . . . 156

8.6 Laminar flow pressure drop across rigid walled stenosis . . . . . . . . . . 157

8.7 Wall shear stress plots for laminar flow through the rigid stenosis. . . . 158

8.8 Normalized wall shear stress plots for laminar flow through the rigid

stenosis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

8.9 Wall pressure plots for laminar flow through the rigid stenosis. . . . . . 159

8.10 Normalized wall pressure plots for laminar flow through the rigid stenosis.160

8.11 Laminar and turbulent flow pressure drop across rigid walled stenosis . 160

8.12 Turbulent kinetic energy and flow streamlines . . . . . . . . . . . . . . . 163

8.13 Pressure contours for turbulent flow through the rigid stenosis. Pressure

normalized according to (Pin − P )/(0.5ρU2in). . . . . . . . . . . . . . . . 165

8.14 Normalized wall pressure profiles for turbulent flow through the rigid

stenosis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

8.15 Normalized wall shear stress profiles for turbulent flow through the rigid

stenosis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

8.16 Turbulent flow mean velocity profiles in rigid stenosis at Re = 2000. . . 167


12

LIST OF FIGURES


8.19 Reynolds stress profiles in rigid stenosis at Re = 2000. . . . . . . . . . . 170



8.22 Schematic of compliant walled stenosis geometry. . . . . . . . . . . . . . 173

8.23 Numerical mesh used for the compliant walled stenosis simulations. . . . 175

8.24 Results from the compliant stenosis sensitivity tests. . . . . . . . . . . . 178

8.25 Flow rate comparison for flow through compliant stenosis. . . . . . . . . 179

8.26 Throat diameter comparison for flow through compliant stenosis. . . . . 179

8.27 Numerical results presented by Shim and Kamm . . . . . . . . . . . . . 180

8.28 Wall pressure profiles for increasing pressure drop for the compliant

stenosis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

8.29 Non-dimensionalized wall pressure profiles for increasing inlet Reynolds

number for the compliant stenosis. . . . . . . . . . . . . . . . . . . . . . 181

8.30 Wall shear stress profiles for increasing pressure drop for the compliant

stenosis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

8.31 Non-dimensionalized wall shear stress profiles for increasing inlet Reynolds

number for the compliant stenosis. . . . . . . . . . . . . . . . . . . . . . 182

8.32 u-displacement (axial) profiles for increasing pressure drop for the com-

pliant stenosis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

8.33 v-displacement (radial) profiles for increasing pressure drop for the com-

pliant stenosis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

8.34 Wall stress profiles for increasing pressure drop for the compliant stenosis.186

8.35 Velocity and turbulence field in the compliant stenosis at pressure drop

= 10 Pa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188


= 20 Pa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189


=30 Pa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190


= 40 Pa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191


= 50 Pa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

8.40 Deformation of the compliant stenosis at ∆P = 50Pa. The black line

represents the deformed stenosis geometry, whilst the grey line represents

the un-deformed geometry. . . . . . . . . . . . . . . . . . . . . . . . . . 193

9.1 Coarse, medium and fine meshes for the rigid walled aneurysm (Model

4, D/d = 2.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

13

LIST OF FIGURES

9.2 Comparison with the experimental data of Budwig et al. (1993). Aneurysm

centre velocity profiles, Model 3. . . . . . . . . . . . . . . . . . . . . . . 199

9.3 Comparison with the experimental data of Budwig et al. (1993). Aneurysm

centre velocity profiles, Model 4. . . . . . . . . . . . . . . . . . . . . . . 201

9.4 Velocity vectors and flow streamlines in aneurysm Model 3. . . . . . . . 202

9.5 Velocity vectors and flow streamlines in aneurysm Model 4. . . . . . . . 203

9.6 Mesh used for the steady state, rigid walled aneurysm simulations. . . . 204

9.7 Flow streamlines and velocity vectors from the rigid walled aneurysm

case (Model 5) at Re = 500 (above line of symmetry) and Re = 2600

(below line of symmetry). . . . . . . . . . . . . . . . . . . . . . . . . . . 206

9.8 Comparison with the experimental data of Asbury et al. (1995). Axial

velocity profiles for aneurysm Model 5 at Re = 2600. Numerical results

shown above line of symmetry, experimental data shown below. . . . . 207

9.9 Centre line turbulence intensity profiles at Re = 2600. . . . . . . . . . . 209

9.10 Normalized wall shear stress profile in aneurysm Model 2 at Re = 2600. 210

9.11 Flow streamlines and velocity vectors (upper figure); contours of turbu-

lent kinetic energy (central figure) and contours of the turbulent viscosity

ratio in aneurysm Model 7 at Re = 2600. . . . . . . . . . . . . . . . . . 211

9.12 Numerical mesh used for the unsteady, rigid walled aneurysm case. . . . 212

9.13 Time history of the inlet flow rate for Cases 1 and 2. . . . . . . . . . . . 213

9.14 Comparison between the experimental data of Yu (2000) and the nu-

merical results for the unsteady, rigid-walled case (Wormersley number

= 17). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

9.15 Comparison between the experimental data of Yu (2000) and the nu-

merical results for the unsteady, rigid-walled case (Wormersley number

= 22). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

9.16 Numerical mesh used for the unsteady, coupled FSI aneurysm simulations.233

9.17 Detailed view of the numerical mesh used for the unsteady, coupled FSI

aneurysm simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

9.18 Inlet velocity and aneurysm exit pressure waveforms used by Khanafer

et al. (2009). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

9.19 Velocity history of the unsteady, coupled FSI aneurysm simulation. . . . 237

9.20 Pressure history of the unsteady, coupled FSI aneurysm simulation. . . . 238

9.21 Flow streamlines and velocity vectors from the unsteady, compliant walled

aneurysm case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

9.22 Turbulent kinetic energy contours from the unsteady, compliant walled

aneurysm case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

9.23 Turbulent viscosity contours from the unsteady, compliant walled aneurysm

case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

14

9.24 Wall pressure profiles from the unsteady, compliant walled aneurysm case.245

9.25 Wall displacement profiles from the unsteady, compliant walled aneurysm

case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

9.26 Wall stress profiles from the unsteady, compliant walled aneurysm case. 249

9.27 Comparison of the temporal variation of the compliant aneurysm inlet

and outlet velocities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

9.28 Time history of the maximum radial displacement of the compliant

aneurysm wall. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

9.29 Comparison of the temporal variation of the maximum radial displace-

ment of the compliant aneurysm wall. . . . . . . . . . . . . . . . . . . . 251

9.30 Comparison of the temporal pressure difference for the compliant walled

aneurysm case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

15

List of Tables

3.1 Coefficients used in the standard k-ǫ model. . . . . . . . . . . . . . . . . 64

3.2 Coefficients used in the Launder-Sharma model. . . . . . . . . . . . . . . 67

4.1 Material property combinations. . . . . . . . . . . . . . . . . . . . . . . 77

7.1 Fluid solver boundary conditions. . . . . . . . . . . . . . . . . . . . . . . 129

7.2 Solid solver boundary conditions. . . . . . . . . . . . . . . . . . . . . . . 129

7.3 Fluid solver under-relaxation factors. . . . . . . . . . . . . . . . . . . . . 130

7.4 Solid solver under-relaxation factors. . . . . . . . . . . . . . . . . . . . . 130

7.5 Re-meshing algorithm under-relaxation factors. . . . . . . . . . . . . . . 130

8.1 Boundary conditions used in rigid walled stenosis simulations. . . . . . . 153

8.2 Under-relaxation factors used in the laminar, rigid walled stenosis sim-

ulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

8.3 Under-relaxation factors used in the turbulent, rigid walled stenosis sim-

ulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154



8.6 Under-relaxation factors used by the fluid solver. . . . . . . . . . . . . . 176

8.7 Under-relaxation factors used by the solid solver. . . . . . . . . . . . . . 176

8.8 Under-relaxation factors used by the re-meshing algorithm. . . . . . . . 176

9.1 Summary of models used in the steady state, rigid walled aneurysm

simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

9.2 Mesh configurations used in the steady state, rigid walled aneurysm sim-

ulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

9.3 Boundary conditions used for the steady state, rigid walled aneurysm

simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

9.4 Under-relaxation factors used for the steady state, rigid walled aneurysm

simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

17

9.5 Summary of models used in the steady state, rigid walled aneurysm

simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

9.6 Boundary conditions used in the rigid walled aneurysm simulations. . . 205

9.7 Under-relaxation factors used during the simulation. . . . . . . . . . . . 206

9.8 Summary of inlet conditions for unsteady, rigid walled cases. . . . . . . 212

9.9 Boundary conditions used for the unsteady, rigid walled aneurysm sim-

ulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

9.10 Under-relaxation factors used during the unsteady, rigid walled aneurysm

simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214



9.13 Under-relaxation factors used by the fluid solver during the unsteady,

coupled FSI aneurysm simulations. . . . . . . . . . . . . . . . . . . . . . 236

9.14 Under-relaxation factors used by the solid solver during the unsteady,

coupled FSI aneurysm simulations. . . . . . . . . . . . . . . . . . . . . . 236

9.15 Under-relaxation factors used by the re-meshing algorithm during the

unsteady, coupled FSI aneurysm simulations. . . . . . . . . . . . . . . . 236

18

Nomenclature

Symbol Description Unit

ǫ Solid strain m−1

ǫ Turbulent dissipation rate m2/s3

ǫ Isotropic turbulent dissipation rate m2/s3

E Solid elastic modulus Pa

k Turbulent kinetic energy m2/s2

µ, λ Solid Lame’s constants Pa

µ Fluid dynamic viscosity Pa s

µt Turbulent dynamic viscosity Pa s

ν Fluid kinematic viscosity (= µ/ρ) m2/s

ν Solid Poisson ratio -

P Mean fluid pressure Pa

P Instantaneous fluid pressure Pa

ρ Fluid density kg/m3

Rt Turbulent Reynolds number (= k2/ǫν) -

Re Reynolds number (= ρUL/µ) -

σ Solid normal stress Pa

τ Solid shear stress Pa

τwall Wall shear stress Pa

uiuj Kinematic Reynolds stress tensor m2/s2

Ui Instantaneous fluid velocity in xi-direction m/s

19

Acronym Description

CFD Computational Fluid Dynamics

DNS Direct Numerical Simulation

FE Finite-Element

FSI Fluid-Solid Interaction

FV Finite-Volume

LDA Laser Doppler Anemometry

LES Large Eddy Simulation

MUSCL Monotonic Upwind Scalar Transport Law

PDE Partial Differential Equations

PIV Particle Image Velocimetry

QUICK Quadratic Upwind Interpolation for Convective Kinematics

RANS Reynolds-Averaged Navier-Stokes

SIMPLE Semi-Implicit Method for Pressure-Linked Equations

STREAM Solution of the Turbulent Reynolds Equations for All Mach numbers

TDMA Tri-Diagonal Matrix Algorithm

Subscript Description

b Bulk value

n, s, e, w North, south, east and west cell faces

N , S, E, W North, south, east and west cell centres

P Cell centre

exit Exit value

in Inlet value

init Initial value

max Maximum value

min Minimum value

wall Wall value

Superscript Description

∗ Guessed value′ Corrected value

20

AbstractApplication of the Finite-Volume Method to FSI Analysis

Matthew Yates, November 2011

The University of Manchester, Doctor of Philosophy

This Thesis describes the numerical simulation of fluid-structure interaction (FSI)problems. A finite-volume based stress analysis code was developed and coupled toan existing in-house CFD code to form a general purpose FSI solver capable of beingused with the advanced turbulence and near-wall models developed within the researchgroup. The code has been used to study a number of physiological flows in the presentwork, although the general nature of the solver allows it to be used for other applicationsalso.

By using the same numerical method, implemented in a consistent manner, forboth fluid and solid domains, the inefficiencies associated with using separate packagesfor the fluid and solid were avoided. Separate packages typically store informationin different data structures; some form of software interface is required to transferinformation between the two packages. This additional software layer, which is calledduring each FSI iteration, causes a considerable overhead. By using a single numericalmesh across both domains, the inaccuracies associated with boundary interpolationwere also avoided. Typically, separate packages use meshes which do not conform attheir common boundary. In order to find nodal values of the fluid pressure, say, atthe solid nodes, some form of interpolation is necessary. The interpolation leads to theintroduction of truncation errors. These improvements allow for more accurate andefficient FSI simulations, particularly transient cases, to be performed.

The solid solver was verified against analytical solutions for a number of test cases,including: planar stress distribution in a square plate with a circular hole in the centre;axisymmetric stress in a thick walled cylinder under internal pressure, and unsteadydisplacement of a cantilevered beam under free vibration.

Before coupled analyses were performed, the flow solver was also validated througha number of rigid walled test cases, including steady flow through a stenosed tube andunsteady flow through an aneurysm. Many physiological flows are difficult to capturedue to flow separation and early transition to turbulence. The use of a low-Reynoldsnumber k-ǫ turbulence model was successful at capturing the flow field over a range ofphysiologically relevant flow rates.

Once the solid body and flow solvers had been validated in isolation, they werecoupled together and applied to a number of physiological flows, namely: steady flowthrough an initially straight tube with a compliant wall; steady flow through a com-plaint stenosis, and unsteady flow through a compliant aneurysm. The results from allthree test cases showed good agreement with the available experimental and numericaldata in terms of wall deformation.

The solid body solver also proved itself to be capable of producing high qualitynumerical meshes for use in other simulations. The fluid mesh was considered to be asolid body with arbitrary material properties; the required deformation was specifiedas prescribed displacement boundary conditions. The main benefit of this method,compared to simple elliptical grid generation methods, is that near-wall grid spacingwas preserved throughout the coupled simulation.

21

Declaration

No portion of the work referred to in this thesis has been submitted in support of an

application for another degree or qualification of this or any other university or other

institute of learning.

23

Copyright

i. The author of this thesis (including any appendices and/or schedules to this

thesis) owns certain copyright or related rights in it (the “Copyright”) and s/he

has given The University of Manchester certain rights to use such Copyright,

including for administrative purposes.

ii. Copies of this thesis, either in full or in extracts and whether in hard or electronic

copy, may be made only in accordance with the Copyright, Designs and Patents

Act 1988 (as amended) and regulations issued under it or, where appropriate,

in accordance with licensing agreements which the University has from time to

time. This page must form part of any such copies made.

iii. The ownership of certain Copyright, patents, designs, trade marks and other intel-

lectual property (the “Intellectual Property”) and any reproductions of copyright

works in the thesis, for example graphs and tables (“Reproductions”), which may

be described in this thesis, may not be owned by the author and may be owned by

third parties. Such Intellectual property and Reproductions cannot and must not

be made available for use without the prior written permission of the owner(s) of

the relevant Intellectual Property and/or Reproductions.

iv. Further information on the conditions under which disclosure, publication and

commercialisation of this theses, the Copyright and any Intellectual Property

and/or Reproductions described in it may take place is available in the Univer-

sity IP Policy (see http://www.campus.manchester.ac.uk/medialibrary/policies/

intellectual-property.pdf), in any relevant Thesis restriction declarations deposited

in the University Library, The University Library’s regulations (see http://www.

manchester.ac.uk/library/aboutus/regulation) and in The University’s policy on

presentation of Theses.

25

Acknowledgements

I would like to express my gratitude to my supervisors Prof. Hector Iacovides and Dr

Tim Craft for their advice and guidance throughout the course of this study. I am

grateful for their help and encouragement during my four years at the University of

Manchester. The financial support of the Douglas Prestwich Trust is greatly appreci-

ated.

Most importantly, I would like to thank my family, in particular my parents, Patricia

and David. Their constant support and encouragement have been vitally important to

me.

27

Chapter 1

Introduction

1.1 Background

The importance of numerical analysis as an engineering tool has increased greatly over

recent years. The ability to rapidly simulate engineering systems, without the associ-

ated cost of performing experiments, allows more informed decisions to be made during

the design phase. In many cases, for example in the simulation of turbulent flows, nu-

merical analysis can provide a level of detail that is unachievable using experimental

methods. Numerical analysis can also yield valuable results for cases which are im-

possible to study experimentally; for example, the failure of a nuclear reactor cooling

system.

A number of strategies for fluid modelling have been developed over the years, in-

cluding: direct numerical simulation (DNS); large eddy simulation (LES), and Reynolds

averaged Navier-Stokes (RANS). The most computationally expensive strategy is DNS,

which aims to resolve every feature of a flow. The wide range of length and time-scales

present in a turbulent flow requires special numerical methods and consumes large com-

putational resources. To accurately capture the entire range of length scales present in

a three dimensional turbulent flow would require approximately Re9/4 computational

nodes. The ratio of time scales present in a turbulent flow is of the order Re1/2, mean-

ing that fine temporal resolution is also necessary. These extremely high computational

requirements of DNS render it unsuitable for simulation of complex industrial flows;

instead, it can be used to simulate geometrically simple test cases for fundamental re-

search into turbulence physics. The LES approach aims to resolve only the larger scale

turbulent eddies in a flow, but is still regarded as too expensive for routine use in most

engineering applications.

The RANS approach requires much lower resources than DNS or LES by modelling

the effects of turbulence on the computed mean flow field. The drawback of the RANS

approach is that no single turbulence model is universally applicable to all turbulent

29

CHAPTER 1. INTRODUCTION

flows. The RANS approach has become the most widely used strategy for simulating

complex industrial flows. Many turbulence models have been extensively tested and

proven themselves to be robust and accurate for a range of industrially relevant flows.

The lower computational requirements allow more timely and efficient simulations than

with more complex modelling strategies, such as DNS and LES. The RANS modelling

framework is used for the turbulent fluid flow simulations in this work because of its

widespread use in industry and its computational efficiency.

As computational resources have grown over time, larger and increasingly complex

flows have been simulated. In many cases it may not only be the fluid flow in isolation

that is of interest; combustion, chemical reaction, heat transfer, radiation and many

other physical processes that interact with the fluid dynamics can be accounted for

within numerical simulations.

One of the more complex processes to be considered is that of fluid-structure inter-

action (FSI). Many, if not every, engineering system involves the interaction between

fluid flow and solid structures to a greater or lesser extent. The number of applications

which involve FSI effects is almost limitless; typical examples include: the wind induced

oscillations of bridges and tall buildings; the aeroelastic deformation of aerofoils and

rotor blades, and the expansion and contraction of physiological vessels.

Historically, due to insufficient computational resources, fluid-structure interaction

was either assumed to have minimal effect and was thus neglected or was treated in a

simplified one-way manner. A typical one-way coupling involves prescribing tractions

upon a solid structure based on the computed fluid loading. This allows the structural

engineer to design a structure strong enough to deform within allowable limits for a

given flow. It does not account for the effect of the structural deformation upon the fluid

flow field; something which becomes increasing important as the deformation becomes

larger. Whilst this may have been acceptable in the past, the current trend to design

lighter and more efficient structures, particularly in the aerospace industry, means that

FSI effects will have to be considered during the design of future engineering systems.

Typically, fluid mechanics and stress analysis problems have been simulated by

disparate groups, often using different numerical methods and with different objectives.

For example, a stress analyst may be satisfied to know that a component does not exceed

the yield strength of the material from which it is made, whilst a fluid mechanist

may be interested in the fine details of a complex turbulent flow. Typically, fluid

flow simulations are performed using the finite-volume method whilst stress analysis

simulations use the finite-element method. Both methods are relatively mature and well

understood within their respective areas. The principal advantage of the finite-volume

method for fluid flow problems is that variables are conserved across each finite-volume

cell. This is in contrast to the finite-element method, where variables are conserved

only across the global domain. In the past, the unstructured finite-element meshes were

30

1.1. BACKGROUND

better able to represent complex geometries - something which has only more recently

been possible with the finite-volume method.

The most basic strategy for simulating FSI problems is to simplify either the flow

field or deformation and concentrate simulation effort on the other. For example, a

stress analyst may approximate the wall shear stress imparted by a fluid upon a solid

body by prescribing a laminar boundary layer; alternatively, a fluid mechanist may

approximate the deformation of a solid body using a “spring law” equation based on

the calculated wall pressure and prescribed external pressure. Such an approximation

will only account for the wall normal deformation caused by the pressure difference,

the tangential deformation caused by the fluid shear stress is neglected.

A more accurate approach is to solve the differential equations governing fluid flow

and solid deformation in a coupled manner. The calculated pressure and wall shear

tractions acting to deform the solid in the wall normal and tangential directions re-

spectively are applied as boundary conditions to the solid solver, which then calculates

the resulting deformation within the solid domain. The deformation of the solid is

accounted for by adapting the fluid mesh and re-solving for the new pressure and wall

shear stresses. This process in continued until some form of convergence is achieved.

Perhaps the simplest method, in terms of computational implementation, for simulating

such fully coupled FSI problems is to use separate packages to simulate the fluid flow

and solid deformation; the necessary data (interface loading and deformation) being

manually, or semi-automatically, transferred between the packages. This becomes very

inefficient for all but one-way coupling, which is only suitable for problems involving

small deformation. Another issue is that the two packages may use different numer-

ical grids to cover their respective regions; this means that interpolation is required

to transfer the information between the two packages. This can lead to inaccuracy as

errors introduced through boundary interpolation affect the vitally important physics

governing the transfer of energy between the fluid and solid. The two packages will

also typically use differing numerical methods and store information in different data

structures; this means that some form of software interface is required to transfer the

information between the two packages. This can become very inefficient, particularly

for time-dependant problems, where the data transfer process must be repeated many

times.

The limitations described above can be overcome by using a single code to solve

the coupled fluid flow and solid deformation problem. Such a code would use the same

numerical method, implemented in a consistent manner, to solve the fluid and solid

equations. If a single numerical mesh were used to cover both fluid and solid regions,

the need to interpolate boundary tractions and deformations would be removed, leading

to increased accuracy. Consistent implementation also removes the need for a software

interface, allowing for more efficient computation of transient problems.

31


1.2 Study Objectives

The objective of this work was to develop a finite-volume based coupled FSI solver. This

was achieved by creating a finite-volume based stress analysis solver and coupling it to

an existing in-house CFD code. The motivation for this is twofold: firstly, by using the

same numerical method for both fluid and solid regions the inefficiency and inaccuracy

associated with solvers that use differing methods is avoided; secondly, the advanced

turbulence and near-wall models developed within the University of Manchester group,

and implemented in an existing finite-volume flow solver (the STREAM code), could

be applied to the study of fluid-structure interaction problems for the first time. The

STREAM code is an advanced RANS solver which has been extensively validated for

turbulent industrial flows.

The solid solver was developed by discretizing the elasticity equations, written in

terms of displacement, using the finite-volume method. To allow for calculations over

complex geometries, and to ensure consistency with the existing flow solver, the equa-

tions were discretized with respect to a boundary fitted, non-orthogonal coordinate

system. Initially, a stand-alone solid solver was developed to simplify the task of test-

ing and verification. The accuracy of the solver was proven through a number of test

cases for which analytical solutions were available. Planar, axisymmetric and unsteady

test cases all showed excellent agreement with experimental or analytical data.

Once the accuracy of the newly developed method had been proven it was coupled

to an existing CFD code. This task was greatly simplified by the consistent numerical

implementation used for both fluid and solid solvers. A single numerical mesh covered

both fluid and solid regions. The position of the fluid-solid interface is stored and used

to call the solvers over their respective domains. By using a novel storage arrangement,

whereby displacements are stored at the vertices of the fluid cells, the need to interpolate

data along the fluid-solid interface was removed.

The coupled solver has been applied to three cases: steady flow through an initially

straight tube with an elastic wall section; steady flow through a compliant walled

stenosis, and unsteady flow through a compliant walled aneurysm. The results were

compared to experimental and numerical data. In each case, rigid walled computations

were first carried out to validate the accuracy of the flow solver. Stenosed flows are

particularly difficult to compute as the large area constriction causes transition to

turbulence at a much lower Reynolds number than would be expected in an un-stenosed

tube. This difficulty was overcome by using the low-Reynolds-number Launder-Sharma

k–ǫ turbulence model and prescribing an initially high turbulence intensity over the

flow domain. Turbulence levels upstream and far downstream of the stenosis decayed

whilst the levels within the re-circulation bubble did not. The unsteady case featured a

physiological inlet wave form with both positive and negative inlet velocities at different

32

1.3. OUTLINE OF THESIS

stages of the flow cycle.

Whilst the newly developed FSI solver has been validated by simulating physiolog-

ical flows, the method is general purpose in nature and can thus be applied to many

other cases. The only limitation, owing to the choice of constitutive relation, is that

the deformation is within the linear-elastic region. By using non-orthogonal, boundary-

fitted grids, complex two-dimensional geometries, both planar and axisymmetric, can

be simulated.

A vitally important aspect of a FSI simulation is the adaption of the fluid region

mesh to account for the deformation of the solid region. The accurate calculation of

wall pressure and shear stress is heavily dependent upon having a high quality near-

wall mesh. This can be achieved by using the solid body solver as a grid adaption

algorithm. For the purposes of re-meshing, the fluid mesh is considered to be a solid

body with arbitrary material properties. The exact material properties, particularly

the Poisson ratio, can be varied to improve the grid adaption in certain cases. The

calculated displacements along the fluid-solid interface are applied as displacement

boundary conditions, with non-interface boundaries having zero displacement boundary

conditions. The displacement equations are solved over the fluid mesh region and

the grid coordinates are updated with the calculated displacements. The principal

advantage of such a method, over other grid generation algorithms based upon simple

elliptic equations, is the preservation of interior node distribution, such as near-wall

grid refinement. This is particularly important for the accurate simulation of turbulent

flows. The method is particularly advantageous in FSI calculations since the adaption

of an existing grid is more time efficient than complete re-meshing of the fluid domain

after each iteration of the solution.

1.3 Outline of Thesis

Chapter 2 provides a review of the most pertinent literature published in the field of

fluid-structure interaction. Following this, Chapters 3 and 4 describe the mathematical

models of turbulent fluid flow and elastic body deformation respectively. Chapter 5 de-

scribes the numerical implementation of the fluid flow solver; this includes a description

of the finite-volume method. The development of the coupled FSI solver is described

in Chapter 6. Initially, the solid body solver was developed and tested in isolation;

once this was completed, the solid solver was coupled to an existing CFD code. The

following three Chapters (7, 8 and 9) describe the application of the coupled FSI solver

to physiological flows: straight tube, stenosed tube and aneurysmal tube respectively.

Each chapter begins with the validation of the flow solver for the rigid walled case; once

the appropriate modelling strategies had been determined, the coupled solver was used

to simulate the compliant walled case. Results are compared to existing experimental

33


and numerical data from the literature. Finally, the conclusions drawn from this work

and recommendations for future work are presented in Chapter 10. The detailed deriva-

tion of the equations governing elastic deformation and their subsequent discretization

using the finite-volume method is provided in the Appendices.

34

Chapter 2

Literature Review

2.1 Introduction

The range of problems involving fluid-structure interaction, and hence the number

of published works on the topic, is too large to cover completely in this Chapter.

The purpose of this literature review is to give a brief description of the works of

most relevance to this Thesis. The first works to be considered are those relating

to the numerical simulation of FSI problems. In particular, papers describing the

application of the finite-volume method to the solution of the equations governing solid

body deformation are discussed. Following this, works describing the experimental

and numerical study of FSI problems are discussed. Attention is particularly focused

on physiological flows, including compliant stenoses and aneurysms, since these are

applications considered in the present study. In addition, this literature review also

contains a description of numerical grid generation techniques, since this is an important

topic in FSI calculations where grids need to be deformed or re-generated.

2.2 Fluid-Structure Interaction Phenomena

Almost every engineering system, be it a suspension bridge or a gas turbine, includes the

effects of fluid-structure interaction to some extent. Wind loading upon bridges and tall

buildings, aeroelastic deformation of aerofoils and rotor blades, and the expansion and

contraction of physiological vessels are just some examples of fluid-structure interaction.

The modern trend of designing lighter and more heavily loaded structures, particularly

within the aerospace industry, can only continue if engineers and designers are able to

accurately and efficiently determine the effects of FSI.

35

CHAPTER 2. LITERATURE REVIEW

2.3 Numerical Simulation of Fluid-Structure Interaction

Numerical simulation of separate fluid and solid systems is a relatively mature subject.

Typically, the finite-element method has been used to solve solid body problems and

the finite-volume method has been used to solve fluid flow problems. Some of the

reasons why different methods have historically been used were described in Chapter 1.

Also discussed was the need to use a single computer code, using the same numerical

method and a single numerical mesh for both fluid and solid regions, to obtain accurate

and efficient solutions.

Until recent advances in hardware, the computational requirements were too great

to perform fully-coupled simulations of fluid-structure problems. Instead, the effects

of FSI were approximated by describing either the fluid or solid with simple analytical

expressions, or by performing one-way coupled simulations. Such methods can only pro-

vide useful information for steady-state cases or those involving only small deflections;

if the deformation of the structure is large, the flow field and hence applied traction

upon the structure will change. This can only be correctly accounted for through fully

coupled FSI simulation.

2.4 Application of the Finite-Volume Method to Elastic

Analysis

The need to use a common discretization method for both the solid and the fluid

to efficiently solve FSI problems has already been mentioned. This leaves two main

options: to use the finite-element method for the fluid; or to use the finite-volume

(FV) method for the solid. Attention is here focused on attempts to discretize the

solid body equilibrium equations using FV methods. This option has been chosen for

two reasons: firstly, the FV method has proven itself capable of solving the complex

non-linear equations which govern turbulent fluid flow, and thus should be able to

cope with the linear elasticity equations; secondly, since the in-house codes used in this

study, which include some of the most advanced models for turbulence and near-wall

effects, are based on the FV method, it is convenient to use the same framework for

the solid solver, in order to avoid having to recode the fluid modelling.

The first use of the FV method to numerically solve solid body problems was by

Demirdzic et al. (1988), who used the method to simulate thermo-elastic problems.

Following on from this, Demirdzic and Martinovic (1993) simulated thermo-elasto-

plastic problems using the FV method. This method solved the unsteady form of

the equations governing thermal energy conservation and momentum balance, with

displacement components and temperature as dependant variables. The elasto-plastic

form of the solid body constitutive relations, which relate stresses to displacement

36

2.4. APPLICATION OF THE FINITE-VOLUME METHOD TO ELASTIC ANALYSIS

gradients, and Fourier’s law, which relates heat flux to temperature gradient, were used

to close the system of equations. In both works, a structured numerical grid was used.

To validate the method, results from a number of test cases were compared to either

analytical or experimental results. The test cases included: a long thin plate subjected

to a non-uniform temperature distribution; a cylinder subjected to a constant internal

pressure and a uniform temperature rise; compression of a thick plate; and arc-welding

of a thin plate. The authors concluded that the use of the FV method to solve solid

body problems had the following advantages: the method is simple and efficient; the

method is conservative on both the local and global scale; boundary conditions are

simple to prescribe; and non-linearities can be handled at little extra cost.

Demirdzic and Muzaferija (1994) then developed an FV solid solver which used

unstructured grids to improve the ability of the method to deal with complex geome-

tries. In this case the thermo-elastic constitutive relations were used. The method was

validated against a number of test cases. The most complex, in terms of geometry and

stress field, was a flat plate with a hole in the centre subjected to a uni-axial tension.

As a further development, Bijelonja et al. (2006) developed an FV based method to

solve incompressible elastic problems. In the incompressible limit, ν = 0.5, the Lame

constant, λ, tends to infinity. This can lead to a number of numerical issues such as

locking, where the calculated displacements are unrealistically small. To get around

this problem, they used the alternate constitutive relation:

[σ] = 2µ[ǫ] − p[I] (2.1)

where p is interpreted as the the “pressure”. Incompressibility is enforced by solving

an additional volumetric constraint equation:

∂u

∂x+

∂v

∂y+

∂w

∂z= 0 (2.2)

The “pressure” is solved for using the SIMPLE algorithm with Rhie and Chow

(1983) interpolation, as one would typically do in a fluid flow simulation, to prevent

unphysical pressure oscillations which can occur when using collocated grids. The

method was applied to a number of test cases. The results were accurate and locking

was not encountered.

Wheel (1996) presented an FV based method to calculate the stresses and displace-

ments within axisymmetric bodies. He stated that the finite-volume method would be

particularly useful in the fluid-structure interaction analysis of pressure vessels, pipes

and heat exchangers. The axisymmetric stress equilibrium equations were re-written

in terms of displacement gradients by substitution of the linear elastic constitutive re-

lations; the gradients were then replaced with linear difference approximations. This

resulted in a system of linear discretized equations in terms of displacements as shown

37


below:

AP uP +∑

(Ai ui) + BP vP +∑

(Bi vi) = 0 (2.3)

where the coefficients A and B are functions of the material properties and the cell ge-

ometry. Boundary conditions (in terms of traction and displacement) were prescribed

through the use of line cells along the domain boundary. These were essentially one di-

mensional cells over which the governing equations were discretized, in a similar manner

as for the internal cells, to produce one-sided difference equations which relate bound-

ary displacements to the internal displacements. The discretized boundary equations

were solved along with the internal equations using an iterative biconjugate gradient

method. This results in the displacement being calculated at every point within the

domain, including the boundary. The validity of the method was demonstrated through

its application to two test cases: a thick walled spherical pressure vessel, and a thin

walled cylinder-sphere intersection, as would be found at the two hemispherical ends

of a pressure vessel. The resulting stress distribution from the spherical test case was

shown to have good agreement with the analytical solution. The results were also com-

pared against finite-element results (obtained using the commercial package ANSYS),

and the FV method was shown to produce results which were as accurate as those from

the finite-element method. The results from the second case were also compared to an

analytical solution, where the FV method predictions were shown to be in good agree-

ment with the analytical solution, but slightly under-predicted the maximum stress. As

before, the results showed favourable comparison with equivalent finite-element results.

Wheel concluded that the method could be combined with an FV based CFD code to

simulate FSI problems.

Wheel (1997) presents an FV based method for predicting the displacement of trans-

versely loaded plates. There have been many reported problems associated with using

the finite-element method for the simulation of transversely loaded plates, including

locking of plate elements and prescription of boundary conditions. The shear forces

and bending moments appearing in the equilibrium were related to the displacements

and rotations via the Mindlin constitutive relation. As in the previous work (Wheel,

1996), the derivatives were approximated with linear difference relations, and line el-

ements were used to prescribe boundary conditions. The method was validated with

three test cases: a simply supported square plate; simply supported thin circular plate;

and a thin circular plate with clamped boundary conditions. The results from the first

test case showed good agreement to analytical results when a suitably refined mesh

(7 × 7 cells) was used. The accuracy of the method was also shown not to depend on

the thickness of the plate, something that can occur with the finite-element method.

The second test case was also compared to an analytical solution; as for the first case,

38

2.5. APPLICATIONS OF FLUID-STRUCTURE INTERACTION

the results were shown to be accurate and have good mesh refinement characteristics.

The third test was similar to the second but with different boundary conditions. The

circular test cases demonstrated the geometric versatility of the method.

Wheel (1999) proposed an FV based method to solve for the small strain defor-

mation of incompressible materials. Displacement and pressure were the dependent

variables, and a similar constitutive relation as before (Equation 2.1) was used. He

found that some restrictions had to be placed on the grid configuration to prevent

locking, namely that triangular based unstructured grids had to be used.

Wenke and Wheel (2003) also presented an FV based method for solving solid

body problems which used both displacements and rotations as independent variables.

This requires that an additional equilibrium equation has to be solved for the angular

moments. Unlike previous methods, the displacements were discretized using quadratic

approximations. A number of test cases were examined, all of which involved square

or rectangular membranes under different load configurations. The results proved that

the method could predict these stress distributions almost exactly, and was a marked

improvement on previous displacement-only based methods. The method was also

shown to be more accurate than finite-element methods which used a displacement and

rotation formulation.

Demirdzic and Muzaferija (1994) coupled a finite-volume CFD solver to a finite-

volume solid solver. The fluid solver included the SIMPLE pressure correction algo-

rithm and k–ǫ turbulence model, whilst the solid solver used the thermo-elastic form of

the constitutive relations. The solvers could be used either independently or in a cou-

pled manner to simulate FSI problems. Uncoupled test cases were presented for both

the fluid and solid solvers (in isolation) and the results were compared to experimental

and analytical results respectively. A coupled FSI simulation was performed for a sim-

plified air-cooled engine, however, only qualitative results were given to demonstrate

the possibilities of the method, rather than to test its accuracy.

2.5 Applications of Fluid-Structure Interaction

2.5.1 Flow Through Compliant Tubes and Channels

Perhaps the simplest (at least geometrically) case of fluid-structure interaction is that

of fluid flow through a tube or channel with compliant walls. This is of particular im-

portance to the study of physiological flows, where blood and other fluids are conveyed

through deformable vessels. Many theoretical, numerical and experimental investiga-

tions have been undertaken in this field.

Luo and Pedley (2000) performed numerical simulations of both steady and un-

steady flow through a two-dimensional compliant channel using the finite-element method.

39


Part of the upper wall of the channel was replaced with an elastic section. The elas-

tic section was modelled as a membrane subjected to constant longitudinal tension T

and external pressure pe. Steady Poiseuille flow was prescribed at the inlet boundary.

The fluid was modelled as water and the flow was laminar and incompressible. The

fluid flow was modelled with the Navier-Stokes equations whilst the normal displace-

ment of the elastic wall was modelled with the one-dimensional membrane equation,

in which the normal displacement is a function of the pressure difference across, and

the curvature along, the membrane. To simplify the analysis, the inertia of the wall

was neglected, i.e. the mass of the elastic wall was set to zero. With this treatment,

the wall shear stress exerted by the flowing fluid has no effect on the displacement.

Three different configurations were considered: first, the upstream and downstream

transmural pressure (which is defined as the difference between the internal pressure

acting upon the interior surface of the tube and the pressure acting upon the exter-

nal surface) was prescribed and the Reynolds number was obtained from the solution;

second, the upstream transmural pressure and Reynolds number were prescribed and

the downstream transmural pressure was obtained from the solution; finally, the down-

stream transmural pressure and Reynolds number were prescribed and the upstream

transmural pressure solved for.

Hazel and Heil (2003) carried out fully-coupled, three-dimensional simulations of

steady flow through compliant tubes using the finite-element method. The deformation

of the tube wall was governed by the Kirchoff-Love shell theory which is based on a

number of assumptions including low wall thickness to radius ratio. Symmetry was

assumed in both transverse planes so that only one quarter of the tube was modelled.

Heil (1997) performed both experimental and numerical investigations into flow

through compliant tubes. The numerical simulations consisted of very low Reynolds

number, steady-state and three-dimensional flow governed by the Stokes equations

through a compliant walled tube governed by shell theory. It was found that the

tube wall initially deformed axisymmetrically, until a critical transmural pressure was

reached, at which point the tube collapsed asymmetrically. Initially, the pressure dis-

tribution along the tube wall is due to viscous losses, and therefore varies linearly with

axial distance. However, as the tube deformation becomes more severe, the greatest

contribution to the transmural pressure difference comes from the Bernoulli effect, caus-

ing the point of maximum deformation to migrate towards the downstream end of the

elastic section. As the degree of tube collapse increases, the opposite walls nearly come

into contact, thus forming two outer passages through which the flow is accelerated.

Experiments were also conducted due to the lack of data corresponding to such low

Reynolds number flows. Numerical and experimental data were in good agreement in

terms of transmural pressure difference. Qualitative comparison between the shape of

the collapsed tube also showed broad agreement.

40


Rast (1994) presents numerical results obtained by simultaneously solving the Navier-

Stokes equations and the elastic membrane equation with the finite-element method for

steady flow through a channel with a compliant section on the upper wall. This is in

contrast to other methods in which the fluid and structure equations are decoupled and

solved iteratively until a solution is obtained which satisfies both sets of equations. The

position of the fluid nodes under the elastic wall section was defined with respect to a

number of radial spines which extend outwards to an origin, the coordinates of which

are related to the membrane deformation.

Bertram and Tscherry (2006) performed experiments of flow through a compliant

tube to investigate the phenomena of flow-rate limitation and tube wall collapse. The

experimental apparatus consisted of a so called ‘Starling resistor’, which is a compliant

tube mounted between two rigid tubes at either end and subject to uniform external

pressure. Measurements were made of the pressure difference across the length of

the compliant tube section, the cross-sectional area of the compliant tube and the

flow-rate through the tube. Flow was induced through the tube by reducing the exit

pressure, whilst maintaining the inlet and external pressure constant and equal. As the

downstream pressure was lowered, the flow-rate through the compliant tube initially

increased. At a critical pressure, the flow-rate reaches a maximum, and begins to

decrease with further reduction of the downstream pressure until reaching a constant

value shortly after. After this flow-rate limitation, self-excited oscillation of the tube

walls may be initiated. Tube wall oscillation was observed at Reynolds numbers as low

as 290–300. Both the points of peak flow-rate and the onset of oscillation occurred at

a lower streamwise pressure drop as the fluid viscosity was increased; the amplitude of

oscillation decreased as the viscosity was increased.

2.5.2 Stenosed Flow

Arterial stenoses are a commonly occurring problem in humans and a proper under-

standing of their effects is important to develop future treatments. Whilst their cause

is not completely understood, the most commonly given reason is the accumulation of

atherosclerotic plaque on the inside of artery walls (Young, 1979). The reduction in

cross sectional area caused by the stenosis can have significant impact on the local hy-

drodynamics, which in turn can cause additional stenosis growth or other physiological

problems.

Young and Tsai (1973) performed experiments of steady flow through rigid walled

tubes with both axisymmetric and non-symmetric stenoses of varying lengths and area

constrictions. Reynolds numbers ranged between 100 and 5000. Measurements were

made of the pressure drop across the stenosis and of the re-circulation region size. The

main finding of this work was that for tubes with severe stenosis, area constriction of

89%, the transition to turbulence occurs at much lower Reynolds numbers than would

41


be observed in straight tubes.

Ahmed and Giddens (1983) also conducted experiments of steady flow through

stenosed tubes with rigid walls. Detailed mean and fluctuating velocity profiles were

measured by laser Doppler anemometry, whilst qualitative information was found through

hydrogen bubble visualisation. The Reynolds number was in the range 500 to 2000,

whilst stenosis severities of 20, 50 and 75% area reduction were used. The flow visuali-

sation for the most severe stenosis indicated the presence of periodic oscillations in the

shear layer between the core flow jet and the re-circulation region at Re = 500; as the

flow rate increased these oscillations were replaced with random fluctuations. Radial

velocity profiles show that the re-attachment point moves downstream with increasing

Reynolds number, until transition to turbulence occurs, after which it moves back up-

stream. Measurements show that the axial velocity increases by a factor of 4 distal

to the throat of the stenosis due to the area reduction, before decaying back to the

proximal level after 2 diameters. The peak centre line disturbance intensity occurred

3.5 diameters from the throat. The main finding of the work is that for severe stenosis,

discrete oscillations or even turbulence can occur at relatively low Reynolds numbers;

something which must be taken into account when numerically simulating such a flow.

Ghalichi et al. (1998) numerically simulated steady flow through a number of rigid

stenosis models using a low-Reynolds-number form of the k–ω model of Wilcox (1998).

Stenosis severity varied between 50 and 86% area reduction whilst the Reynolds number

was in the physiological range. The numerical results were compared to the experimen-

tal data of Ahmed & Giddens. At lower flow rates, the size of the re-circulation region

was found to increase linearly with Reynolds number, until a critical value was reached,

at which point the re-circulation region reduced in size. The critical Reynolds num-

ber indicates the point at which the flow distal to the stenosis becomes transitional.

For a stenosis of 89% area reduction, the critical Reynolds number was 260. Some

of the results, namely streamwise variation of pressure and turbulence intensity, were

compared to numerical results obtained with the standard k–ǫ model. Whilst the k–ω

model was shown to better predict the pressure recovery and increase in k downstream

of the throat, this was most likely due to the fact that the k–ǫ model employed did

not have any additional low-Reynolds-number terms, rather than due to any inherent

limitation of the k–ǫ modelling strategy. Comparison was also made between the down-

stream velocity profiles and experimental LDA measurements. Whilst the agreement

was good, there did appear to be a discrepancy between the predicted and measured

mass flow rate around the point of flow re-attachment. For example, a velocity profile

2.5 diameters downstream from the throat of a 75% stenosis matched the experimental

data almost exactly, whilst a profile 4 diameters downstream under-predicted the ve-

locity across the entire profile. This suggests that either the numerical simulations were

failing to conserve mass, or, more likely, that the experimental data contain errors.

42


Varghese et al. (2007) performed direct numerical simulation of steady flow through

a rigid stenosis of 75% area reduction. The majority of the work focused on an eccentric

stenosis, though a stenosis similar to that used in the experiments of Ahmed & Giddens

was also used. Velocity profiles were shown to be in reasonable agreement with the

experimental data at a Reynolds number of 500. Results were also presented for stenoses

of 70 and 73% area reduction to highlight how a small change in the geometry can have

a significant effect on the flow.

In another work, Varghese and Frankel (2003) simulated pulsatile flow through a

rigid tube with an axisymmetric stenosis of 75% severity using four different RANS

turbulence models. As part of an initial validation exercise, they simulated steady

flow through the stenosis with Re = 500 and compared the velocity profiles with the

experimental data of Ahmed & Giddens. The simulations were performed using the

commercial CFD solver FLUENT. The turbulence models used included low-Re k–

ω, high and low-Re variants of the RNG k–ǫ and the standard k–ǫ. At locations

downstream of the stenosis, the numerical profiles underestimate the experimental data.

The discrepancies seem to increase with distance from the throat. Some of the models

failed to predict any re-circulation, or predicted premature re-attachment. As in the

results presented by Ghalichi, there also seemed to be a mass flow rate discrepancy.

However, the authors report that the k–ω model more accurately predicts the jet-like

core flow immediately downstream of the throat.

In a similar work, Ryval et al. (2004) numerically simulated pulsatile flow through

a stenosed rigid tube using Wilcox’s k–ω, both standard and transitional, turbulence

model. Steady simulations were performed initially for validation and grid indepen-

dence tests. The authors point out a major discrepancy between the predicted and

measured velocity profiles at the throat of the stenosis; they put the difference down to

either large uncertainty or error in the experimental data. Both the standard and tran-

sitional model performed well close to the throat of the stenosis, but the standard model

predicted premature re-attachment and recovery. The standard model predicted the

highest turbulence intensity to be at the throat of the stenosis, whilst the transitional

model was in agreement with the experimental data in predicting the peak around 4-5

diameters downstream of the throat. However, neither model could accurately predict

the flow over the entire domain.

The fact that there is some doubt about the validity of the Ahmed & Giddens data,

and the fact that earlier numerical studies report discrepancies in their comparisons,

points to the need for further computations and for the use of alternative validation

data. Here, we therefore concentrate on the cases experimentally investigated by Young

& Tsai. The main objective is to establish which turbulence modelling strategies are

necessary for the reliable computation of flow through stenoses with rigid walls, before

embarking on the analysis of fluid structure interactions in stenoses with compliant

43


walls.

Whilst the study of flow through rigid walled tubes is important to develop our

understanding of the fundamental processes involved, it is only a very simple approxi-

mation of the real physiological flow. The effects of wall compliance have to be included

to accurately simulate flow through human arteries. The pressure and wall shear stress

which the fluid imparts upon the solid wall cause it to deform; this deformation causes

a change in the cross-sectional area of the tube which, according to the law of mass

conservation, will have an effect on the fluid velocity and pressure. In the simplest case,

the interaction between fluid and solid can lead to a single, steady state solution; in

more complex cases it can result in complex time-dependant oscillations and even tube

wall buckling and other highly non-linear behaviour. Initially, only the simple case of

steady-state, axisymmetric deformation will be considered.

Stergiopulos et al. (1993) experimentally studied steady flow through a severe ax-

isymmetric stenosis with compliant walls. The stenosis model was made of silicone

rubber and had a sinusoidal shape with a maximum area reduction at the throat of

90%. The model was connected to straight rigid tubes at either end and was sub-

merged in a water bath to provide an external pressure. The flow rate through the

stenosis and the compliance of the stenosis walls were both in the physiologically rele-

vant range. The working fluid was water at 25. Upstream and downstream pressure

was controlled by means of two adjustable valves. Flow through the tube was initiated

by fixing the upstream pressure at 74 mmHg and reducing the downstream pressure

by 1 mmHg increments from 74 mmHg. After each reduction in pressure, both the

steady flow rate and the internal diameter of the throat of the stenosis were measured.

At low pressure differences, the flow rate was found to increase with pressure differ-

ence, whilst as a consequence of Bernoulli’s equation, the diameter of the stenosis was

found to decrease. However, at a pressure difference of 60 mmHg, the flow rate reached

a maximum value and the tube collapsed. Low frequency oscillations of the stenosis

walls were observed prior to collapse. At higher pressure differences, after the tube

had collapsed, the flow rate tended to a constant value and hence became independent

of downstream pressure. The collapsed tube had a region downstream of the stenosis

throat in which the opposite sides of the tube wall were in contact. Small channels

formed in the outer part of the collapsed region through which the fluid flowed. The

flow rate at which collapse occurred was shown to be reasonably well approximated by

the one-dimensional speed index criterion of Shapiro (1977).

Shapiro (1977) developed a 1-D theoretical model for steady flow through a thin

walled, compliant tube which is partially collapsed due to a negative transmural pres-

sure difference, p − pext. The model makes use of a so called ‘tube law’, which is a

relationship between the cross-sectional area of the tube and the transmural pressure

difference, and is analogous to the equation of state in a gas flow. The model has many

44


analogies to gas and free-surface flows, and has a controlling parameter, the speed index

(S = u/c; where u is the flow velocity and c is the wave speed), which is analogous

to the Mach and Froude numbers. As with the Mach and Froude numbers, S < 1

implies subcritical flow and S > 1 implies supercritical flow. The flow is said to be

critical when S = 1; at this point the flow can become choked, similar to the choking

in a converging-diverging nozzle flow in gas dynamics. The transition from subcritical

to supercritical flow can be either continuous, or by means of a discontinuous step or

‘shock’. The fluid-structure interaction mechanism is described as:

1. The fluid velocity is related to the pressure by means of the equation of motion.

2. The fluid pressure is related to the deformation of the tube (in terms of cross-

sectional area) by the tube law.

3. The cross-sectional area of the tube affects the velocity of the fluid by means of

the continuity equation.

The tube law is highly non-linear in nature for the case of negative transmural

pressure (tube collapse) because of the complex manner in which the tube buckles.

This means that no simple tube law relationship can be applicable over the entire

range of negative transmural pressure. In the case of negative transmural pressure

(tube collapse), the pressure difference is sustained by the bending stiffness of the tube;

in the case of positive transmural pressure (tube inflation), the pressure is sustained

by hoop tension. The model generally requires numerical integration of the governing

equations, however, a number of simple cases are presented for which the governing

equations can be integrated analytically. The cases which are analytically solved are:

friction flow, which is a steady flow through a horizontal tube with constant external

pressure where only the wall friction affects the flow variables; pressure-gravity flows, in

which only pressure and elevation affect the flow; a flow in which the neutral area (the

area of the tube under zero transmural pressure) has an effect on the flow; and finally,

flow in which friction and gravity have an effect of the flow variables. The results were

presented and discussed, but no comparison to experimental data was made.

Shim and Kamm (2002) performed numerical simulations of steady flow through

compliant walled tubes and channels. Of most relevance to the present work was a

validation exercise which replicated the experimental investigation performed by Ster-

giopulos. The fully coupled FSI simulations were performed with the commercial finite-

element package ADINA. The fluid was considered laminar and incompressible and was

modelled with the full Navier-Stokes equations. The finite-element mesh for the fluid

region consisted of 4848 3-node axisymmetric elements with a linear interpolation func-

tion. The solid was considered to be incompressible and was modelled using elasticity

equations with the Mooney-Rivlin constitutive relation. The finite-element mesh for

45


the solid region consisted of 2639 9-node axisymmetric elements with quadratic ele-

ments. The maximum Reynolds number, based on outlet velocity and unobstructed

diameter, was 2478 - this makes the laminar flow assumption seem questionable given

that transition to turbulence can occur at Re = 300 for such a flow. Converged solu-

tions could only be achieved for a pressure drop less than 33 mmHg. The numerical

results were in good agreement with the experimental data at low pressure differences.

However, at higher pressure differences the flow rate was under-predicted and thus the

diameter of the stenosis over-predicted.

Bathe and Kamm (1999) simulated the fully coupled, pulsatile flow of blood through

an elastic walled tube with an axisymmetric stenosis. They modelled the solid using

large displacement theory, closed with the Ogden model. The Ogden model is a con-

stitutive relationship used to represent incompressible, isotropic, nonlinear materials;

it contains a number of constants which were found from an experimental stress-strain

curve for a carotid artery. Blood was modelled as a homogeneous, Newtonian fluid

with constant properties. The material properties of the artery wall and the stenosis

were varied, to discover their effect on the flow field and the displacement of the wall.

The fluid was modelled with the Navier-Stokes equations with the assumptions of in-

compressibility and laminar flow. Both models also assumed axisymmetry, and both

were solved using the finite element method with the commercial package ADINA.

The geometry was described in terms of the percentage area reduction at the centre

of the stenosis. Simulations were performed using models with 51, 89 and 96% area

reduction. Results were given for the three geometries mentioned above, and at three

different times within the pulse cycle, in terms of pressure, wall shear stress and axial

velocity profiles for the fluid, and percentage area reduction, peak stenosal stress and

stenosal strain for the solid. The centre-line pressure profile, wall shear stress, and

axial velocity profiles are also presented for the 89% area reduction case. Minimum

pressure occurred slightly downstream of the stenosis centre, the pressure recovery oc-

curred across two separate stages further downstream. The increase in constriction

from 56% to 89% caused a 13 fold increase in peak wall shear stress, however, a further

increase to 96% constriction had little extra effect. The authors cite the axisymmetric

assumption as a major limitation in the model, as it does not allow the tube wall to

buckle or deform non axisymmetrically. In addition, the simplifying assumptions about

the properties of blood and artery walls mean that the results do not give an accurate

representation of actual physiological flows. The omission of turbulence modelling also

restricts the applicability of the model to low Reynolds number flows.

In summary, flow through stenosed tubes is of particular relevance to physiologi-

cal flows. Both experimental and numerical investigations have been conducted into

flows through both rigid and compliant walled tubes with stenoses. The flow is domi-

nated by a large pressure drop through the area constriction leading up to the stenosis

46


throat, followed by a re-circulation region downstream of the stenosis throat. This

re-circulation region creates large scale unsteadiness, which can lead to to turbulent

transition at much lower Reynolds numbers than would be expected in a straight tube.

In the compliant walled case, this pressure reduction leading up to the stenosis throat

can cause a negative transmural pressure difference, which causes a negative wall de-

formation in the radial direction - this can eventually lead to complete collapse of the

tube wall. The experimental investigation of laminar and turbulent flows through a

rigid walled stenosis conducted by Young and Tsai (1973) will be of particular use for

the validation of the numerical modelling strategies used for this class of flow. Simi-

larly, the experimental investigation of flow through a compliant stenosis conducted by

Stergiopulos et al. (1993) could be used the validate the fully coupled FSI solver.

2.5.3 Aneurysmal Flow

Another important category of physiological flow is the aneurysm. An aneurysm is

the dilation of an arterial vessel caused by localised weakening of the arterial wall. Of

particular interest in this work is the abdominal aortic aneurysm (AAA).

Asbury et al. (1995) performed an experimental study of steady flow through rigid

walled aneurysms. In total, seven symmetrical aneurysm models were used with a

uniform length of 4d and bulge diameters in the range 1.4d – 3.3d, where d is the di-

ameter of the undilated inlet. Quantitative measurements of the flow field were made

using laser Doppler velocimetry whilst qualitative visualisations were made using colour

Doppler flow imaging. Both laminar (Re = 500) and turbulent (Re = 2600) flow was

considered to cover the entire range of physiologically relevant conditions. The primary

objective of the research was to understand the effect of aneurysm diameter on turbu-

lence and wall shear stress patterns. Under laminar conditions a core flow through the

centre of the aneurysm was seen with a large re-circulation region filling the aneurysm;

the parabolic velocity profile within the core was maintained without any spreading

as it passed through the aneurysm. Under turbulent conditions, velocity fluctuations

and a flatter velocity profile were seen in the core which spread outwards as it flowed

through the aneurysm. This spreading caused extremely sharp velocity gradients at

the point where the downstream end of the aneurysm met the exit. It was found

that transition to turbulence occurs at lower Reynolds numbers in aneurysms with

greater cross-sectional area; this is ascribed to instabilities caused by the greater flow

deceleration occurring in aneurysms with greater cross-sectional areas. However, the

relationship between transition and aneurysm size was not strong, something that was

attributed to manufacturing variations (e.g. asymmetrical and/or eccentric aneurysm

geometries). Centre line turbulence intensity measurements showed higher levels in

the converging half of the aneurysm due to flow acceleration; larger aneurysm models

broadly display higher levels of turbulence intensity. Wall shear stress (estimated from

47


the gradient of the near-wall velocity measurements) was negative in the aneurysm due

to the flow re-circulation. Upon re-attachment, the laminar wall shear stress returned

to the fully developed levels found at the inlet; in the turbulent case, wall shear levels

rose to about twice the inlet level due to the sharp velocity gradients caused by the

velocity profile spreading.

Yu (2000) performed experiments of both steady and unsteady laminar flow through

rigid aneurysms. Particle image velocimetry (PIV) was used to quantitatively measure

the flow field. Only laminar flow was considered. The unsteady flow experiments

used a sinusoidal inlet velocity waveform, rather than a physiological waveform, to aid

comparison with CFD studies. Two elliptical aneurysm models of length 2.2d and bulge

diameters of 1.7d and 2d were considered. In the unsteady cases, the frequency of the

sinusoidal velocity waveform was prescribed to give Wormersley numbers of 17 and

22; the velocity waveform was always positive with mean Reynolds number of 1000.

In the case of steady flow at Re = 1000, a central core flow surrounded by a slow

moving re-circulation region was observed. The centre of the re-circulation bubble was

located nearer to the downstream end of the aneurysm and the core region showed no

spreading. The effect of the aneurysm size was found to be small; the only difference

being an increase in the re-circulation region strength with aneurysm radius. At the

peak flow rate of an unsteady flow cycle, a small re-circulation bubble formed at the

upstream end of the aneurysm (due to the adverse pressure gradient caused by the

area expansion in the first half of the aneurysm); as the flow rate reduced, this re-

circulation bubble grew in strength and was convected downstream. Slightly before

the minimum flow rate, the re-circulation region remained at the downstream end and

began to decay, and continued to do so as the flow rate increased. The strength of the

re-circulation region increased with the frequency of the inlet waveform because of the

greater deceleration (and hence adverse pressure gradient) present.

Yip and Yu (2002) performed experiments of unsteady turbulent flow through rigid

axisymmetric aneurysm models. The objective of their research was to study the transi-

tion to turbulence in complex geometries. Reynolds numbers, based on the Stokes layer

thickness, ranged from 405 to 806 whilst the Wormersley number of the sinusoidally

oscillating flow ranged from 7.2 to 12.3.

Peattie et al. (2004) performed experiments of unsteady flow through rigid aneurysm

models. Unlike previously mentioned unsteady experiments, in this study the inlet

velocity was negative during part of the cycle, thus providing a close approximation of

the physiological case.

Budwig et al. (1993) performed experiments and numerical simulations of steady

flow through rigid walled aneurysm models. Four axisymmetric models of length 4d and

bulge diameters in the range 1.3d to 2.1d were considered. Inlet Reynolds numbers were

in the range 500 to 2600. Numerical simulations were performed using the finite-element

48


flow solver, FIDAP. At Reynolds numbers below 1600, the calculated streamlines were

in reasonable agreement with the experimental measurements; the calculated centre of

the re-circulation bubble was slightly upstream of the experimentally measured one.

Wall shear stress profiles show that the length of the re-circulation bubble increases

as the bulge diameter is increased. At Reynolds numbers greater than 2500, the flow

was continually turbulent in both the aneurysm and the inlet sections; at Reynolds

numbers in the range 2000 < Re < 2500 the flow was intermittently turbulent in the

inlet tube and aneurysm region. In the smaller aneurysm models, the intermittency

factor was almost the same in both inlet and aneurysm regions; in larger models, the

turbulence was intensified in the aneurysm region.

Deplano et al. (2007) performed experiments of unsteady flow through both rigid

and compliant walled aneurysms. An idealized asymmetrical aneurysm geometry was

chosen to better replicate real aneurysms than the symmetrical geometries used in

other studies. The time variation of the inlet velocity was based upon physiological

measurements. The results showed that the energy stored in the compliant aneurysm

walls during the flow acceleration phase acts to push the re-circulation region further

downstream during the deceleration phase than occurs in the rigid walled case. The

elastic deformation of the wall due to the re-circulation region increases the wall stresses

and thus increases the likelihood of wall rupture.

Khanafer et al. (2009) performed numerical simulations of unsteady flow through

compliant walled, axisymmetric aneurysm models. A fully coupled, finite-element FSI

solver was used in the computations. The effects of turbulence were modelled using the

transitional form of the k–ω model of Wilcox. The wall of the aneurysm had a constant

thickness and was modelled with a non-linear, hyper-elastic constitutive relation. Time

varying, physiological inlet velocity and external pressure were prescribed whilst the

external pressure was fixed at zero. Two coupling strategies were employed: firstly, a

full FSI model in which the displacement of the tube wall was caused by both pressure

and shear stresses; and secondly, the so called CSS model, in which the displacement

was only dependent upon the wall pressure. For comparison, rigid walled calculations

were also performed. The CSS model under-predicted the peak wall stress by 8%

compared to the FSI model; this shows the extent of the fluid shear contribution to

the total wall stress. The CSS model showed oscillations in the time variation of the

peak wall deformation; these oscillations are damped by the viscous effect of the flow

and did not occur in the full FSI model. Comparison between the rigid and compliant

walled computations show that the re-circulation region is weaker and is convected

downstream faster in the compliant walled case. The initial vortex formed at the

upstream end of the aneurysm is weaker due to the energy dissipated by the expansion

of the elastic wall. The turbulence levels are higher and localized around the upstream

and downstream ends of the aneurysm in the rigid walled case; the compliant walled

49


case showed lower and more evenly spread turbulence levels.

Martino et al. (2001) performed fully coupled FSI simulations of pulsatile flow

through complex 3D models obtained from patient specific CT scans. The finite-element

method was used to solve both fluid and solid governing equations. Mesh adaption was

performed by treating the fluid mesh as a solid body subject to displacement boundary

conditions. The time variation of the inlet velocity profile and exit pressure were based

upon physiological measurements. The wall of the aneurysm model was assumed to

be linearly elastic. Whilst proving that patient specific FSI simulations are possible,

this study contributes little to our understanding of the fundamental principles of flow

through elastic aneurysms.

In a more complicated numerical analysis, Li and Kleinstreuer (2005) performed

the first fully-coupled simulations of pulsatile blood flow through an asymmetric, three-

dimensional stented aneurysm. A stent is an artificial vessel inserted into an aneurysm

to relieve the weakened artery wall from the stresses imparted by the blood flow. The

interaction occurs between the blood flow and the stent and between the stagnant

blood between the stent and the aneurysm wall. The incompressible fluid flow was

modelled with the Navier-Stokes equations with non-Newtonian constitutive relations

and the solid was modelled with the elasticity equations. Pulsatile inlet velocity and

exit pressure waveforms based on physiological measurements were employed. Cou-

pled simulations were carried out with the commercial finite-element solver ANSYS.

Although the stent significantly reduced the stresses acting on the diseased aneurysm

wall, they were not zero due to the complex fluid-structure interaction between the

stent, the stagnant blood and the aneurysm wall. The wall shear-stress and pressure

of the blood act to displace the stent in the streamwise direction. If the stent were to

migrate too far, blood may leak into the stagnant region and increase the stress acting

on the aneurysm wall to dangerous levels.

Miranda (2003) also studied the FSI problem of blood flow through an arterial

aneurysm. The axisymmetric blood flow was simulated using the CFD code, STREAM,

with the assumption that blood could be modelled as an incompressible, homogeneous,

Newtonian fluid. The deformation of the elastic wall was given by a simple spring

equation which related the pressure difference between the inside and outside of the

artery wall, ∆p, to the displacement of the wall, δ, via a spring constant k. The

pressure outside of the artery wall was specified and remained constant throughout

the simulation. The coupled FSI simulation began by solving the fluid equations for an

initially straight pipe. Next, the wall was displaced according to the calculated pressure

difference using the spring law and the grid was modified to account for this change.

Finally, the fluid equations were solved again for the new geometry. This process was

repeated until overall convergence was achieved, i.e. no further displacement of the wall

occurred with increasing iterations. The aim of the research was to test two different

50

2.6. NUMERICAL GRID GENERATION

methods of modifying the fluid grid as the wall of the aneurysm deformed. The first

method used a fluid grid with a constant number of nodes; the position of each node was

moved to reflect the displacement of the wall. The nodes were only moved in the radial

direction. As the wall of the artery deformed outward, the fluid grid became more and

more non-orthogonal. The second method used a fluid grid with a variable number of

nodes; more nodes were added as the wall of the artery deformed outwards. This was

achieved by defining a fixed orthogonal grid consisting of two regions: a fluid region

which covered the interior of the artery; and a solid region which covered the region

outside the artery wall. The artery wall was situated at the interface between the two

regions. As the artery wall deformed outwards, cells which were originally in the solid

region became part of the fluid region, and therefor took part in the next iteration of the

fluid solver. This meant that the majority of the fluid grid cells remained orthogonal,

except for the cells lying adjacent to the boundary, which followed the curvature of the

wall. The numerical results, whilst qualitatively similar, did show some dependence on

which grid adaption method was used.

In summary, flow through aneurysmal tubes is another important class of phys-

iological flow. As was the case for stenosed flow, the flow is dominated by a large

re-circulation region. However, unlike the stenosed flows, the size and position of the

re-circulation region is geometrically limited by the aneurysm bulge - this will tend

to increase the stability of any numerical simulations. The numerical results obtained

by Khanafer et al. (2009) for the case of unsteady, turbulent flow through a compli-

ant aneurysm will make and interesting a challenging test case for comparison. A

comparison with these results will allow for a comparison between finite-element and

finite-volume based elasticity solvers used within coupled FSI codes.

2.6 Numerical Grid Generation

The numerical grid is one of the most important aspects of any numerical simulation.

The grid has to have a fine enough resolution to be able to accurately capture all of the

information in the flow domain, for example near-wall velocity gradients. As the grid

is made finer, the accuracy of the simulation increases as the truncation errors in the

spatial approximations become smaller. However, an increased number of grid points

means an increase in computer storage and processor time. The compromise between

accuracy and efficiency makes grid generation a very difficult process.

The STREAM code (which is described later in the Thesis) uses structured, non-

orthogonal grids in which the grid lines are aligned with the boundary of the geometry,

rather than the Cartesian coordinate lines. This allows for more complex geometries to

be simulated and simplifies the prescription of boundary conditions. However, the gov-

erning equations have to be re-cast from their Cartesian form into the non-orthogonal

51


form. These equations contain more terms than the Cartesian form, as such the dis-

cretization and computer program are slightly more complex. Care has to be taken to

ensure that the grid is not too skewed, otherwise the non-orthogonal terms will domi-

nate and increase the source term in the discretized equations, which could lead to slow

convergence or even numerical instability.

2.6.1 Elliptic Methods

When simulating flow through a simple physical domain, such as a straight pipe or duct,

a Cartesian numerical grid can be generated with some simple algebraic expressions.

These expressions could involve terms which cluster grid points near walls or other

areas of importance. However, the vast majority industrial, or physiological, flows do

not have such simple geometries, and therefore cannot use simple Cartesian grids.

A more general way to generate boundary-fitted grids is to let grid points be the

solution to an elliptic system of PDE’s (Thompson et al., 1974). The simplest elliptic

PDE system is the Laplace equation, with Dirichlet boundary conditions, which can

be solved for the grid points:

∂2ξ

∂x2+

∂2ξ

∂y2= 0

∂2η

∂x2+

∂2η

∂y2= 0 (2.4)

where (ξ, η) are the non-orthogonal coordinates, and (x, y) are the Cartesian coordi-

nates. The dependant and independent variables can be exchanged to give:

α∂2x

∂ξ2− 2β

∂2x

∂ξ∂η+ γ

∂2x

∂η2= 0 α

∂2y

∂ξ2− 2β

∂2y

∂ξ∂η+ γ

∂2y

∂η2= 0 (2.5)

where the metric terms are given by:

α =

(∂x

∂η

)2

+

(∂y

∂η

)2

β =∂x

∂ξ

∂x

∂η+

∂y

∂ξ

∂y

∂ηγ =

(∂x

∂ξ

)2

+

(∂y

∂ξ

)2

(2.6)

When using such an approach to generate a structured grid, one coordinate (ξ

or η) is specified as constant along each of the boundaries, whilst the other varies

monotonically. Whilst this makes it very simple to control the grid spacing along the

boundary lines, it is impossible to control the distribution of the grid points within the

interior of the domain. To control the spacing of grid points within the domain, one

needs to alter the elliptic system which defines the grid points.

Thompson et al. (1977) proposed adding inhomogeneous terms to the right hand

side of the Laplace system which can be varied to control interior grid spacing. These

52

2.6. NUMERICAL GRID GENERATION

inhomogeneous terms can contain exponential decay terms to cluster grid points near

walls. The form of these equations is:

α∂2x

∂ξ2− 2β

∂2x

∂ξ∂η+ γ

∂2x

∂η2+ J2

[

P(ξ, η)∂x

∂ξ+ Q(ξ, η)

∂x

∂η

]

= 0 (2.7)

α∂2y

∂ξ2− 2β

∂2y

∂ξ∂η+ γ

∂2y

∂η2+ J2

[

P(ξ, η)∂y

∂ξ+ Q(ξ, η)

∂y

∂η

]

= 0 (2.8)

where J is the Jacobian of the transformation, P(ξ, η) and Q(ξ, η) are the functions

which control the interior grid spacing.

This approach is still limited in that it requires a number of parameters and con-

stants to be prescribed manually. This means that generating a grid requires a certain

amount of trial and error. Kaul (2003) devised a method for automatically calculating

the parameters in the inhomogeneous terms which give the best near wall grid. This

was based on the principle of conservation of thermal energy over the near boundary

region. Using this method, elliptic grids can be automatically generated with good

near wall grids. He applied this method to a number of test cases: a simple square; an

annulus; a convex geometry, and a gear tooth.

2.6.2 Solid-Body Methods

In a typical CFD simulation, the physical geometry of the domain does not change.

This means that a single grid can be generated before the simulation is carried out, and

then used for that, and subsequent simulations. Thus, a large amount of time, both

human and computer, can be dedicated to generating this grid. However, in a FSI

simulation, the physical domain does change. After the structure has displaced, the

fluid domain has changed slightly. This means that the original grid has to be altered,

or that the whole domain has to re-meshed. It is therefore vitally important for the

accurate simulation of FSI problems to have an efficient way of generating or adapting

grids.

Sheeta et al. (2006) developed a novel method of grid generation and adaption for

use in FSI problems based on the idea that the grid points are found as the finite-element

solution to the Navier equation which governs the displacements in a solid body. In this

method the CFD mesh is considered to be a solid, with prescribed material properties,

which is subjected to displacement boundary conditions, where the displacements on

the boundaries come from the displacement of the structure. This method is claimed

to preserve the quality of the original grid, i.e. near wall clustering, and minimise

grid distortion. Other than the prescription of suitable material properties, no other

parameters need to be specified, thus automating the process. The method is applicable

to both structured and unstructured grids. The method was tested for a number of

53


aeroelastic problems, including tail buffeting and wing flutter of simplified aircraft, with

positive results.

2.7 Summary

This Chapter has provided a summary of the literature published in the field of fluid-

structure interaction. As FSI is such a broad topic, covering many applications and

industries, only the most pertinent works were described here.

The finite-element method is the most commonly used numerical method within the

field of computational solid mechanics. Efficient simulation of FSI problems, particu-

larly time-dependant cases, requires the same numerical method to be used for both

fluid and solid equations. A number of papers describing the application of the finite-

volume method to solid mechanics were described. Some of the test cases presented

in these papers formed the basis of the validation of the newly developed solid body

solver.

Accurate simulation of many physiological flows can only be obtained by including

the effects of fluid-structure interaction. Many bio-fluid systems involve flow within a

compliant vessel. Experimental and numerical studies of a number of physiological flows

were presented, including: flow through an axisymmetric stenosis, or area reduction;

and flow through an axisymmetric aneurysm, or area expansion. In addition to the

compliant walled cases involving FSI, rigid walled cases were also considered to gain a

better understanding of the complex fluid mechanics present in such flows.

Finally, a description of numerical grid generation methods was given. The elliptic

generators are relatively simple and efficient to implement, but do not allow full control

over interior node distribution. During a coupled FSI simulation, the mesh covering

the fluid sub-domain is adapted according to the calculated displacement along the

interface. Grid generation methods in which the fluid mesh is considered to be a solid

body are relatively easy to implement within a coupled solver and allow control over the

vitally important near-wall node distribution in the vicinity of a fluid-solid interface.

54

Chapter 3

Mathematical Model of Fluid

Motion

3.1 Introduction

In this Chapter, the equations governing the unsteady flow of an incompressible fluid

are presented. The vast majority of industrially relevant flows are turbulent in nature.

To accurately and efficiently compute such flows, the effects of turbulence must be

accounted for. The most important computational methods and models which have

been developed for this purpose are presented in this Chapter.

The simulations presented later in this Thesis used the Reynolds Averaged Navier-

Stokes (RANS) turbulence modelling framework with a simple two-equation eddy-

viscosity model (EVM) which is also described in this Chapter.

3.2 The Navier-Stokes Equations

The Navier-Stokes equations are a system of non-linear, coupled partial differential

equations (PDEs) which are derived from the principles of mass and momentum con-

servation. The equation of mass conservation, or continuity equation, can be written

in Cartesian tensor notation as:

∂ρ

∂t+

∂(ρUi)

∂xi= 0 (3.1)

where ρ is the fluid density and Ui is the instantaneous fluid velocity in the xi-direction.

In the case of steady flow of an incompressible fluid, the above equation can be simplified

to:

∂Ui

∂xi= 0 (3.2)

55

CHAPTER 3. MATHEMATICAL MODEL OF FLUID MOTION

The equations of momentum conservation for a fluid are obtained from the applica-

tion of the force-momentum principle, and can be written in Cartesian tensor notation

as:

∂(ρUi)

∂t+

∂(ρUiUj)

∂xj= − ∂P

∂xi+

∂τij

∂xj(3.3)

where τij is the viscous stress tensor. In the case of a Newtonian fluid, the viscous stress

tensor is related to the mean strain rates through the following constitutive relation:

τij = µ

(

∂Ui

∂xj+

∂Uj

∂xi

)

(3.4)

where µ is the dynamic viscosity of the fluid. In the case of an incompressible flow,

the constitutive relation (Equation 3.4) can be substituted into the governing equation

(3.3) and simplified to give:

∂(ρUi)

∂t+

∂(ρUiUj)

∂xj= − ∂P

∂xi+

∂

∂xj

(

µ∂Ui

∂xj

)

(3.5)

where P is the instantaneous pressure. This describes a closed system of four coupled

PDEs written in terms of four unknown quantities (namely the instantaneous velocity

components, Ui, and the pressure P ) and two known fluid properties (density, ρ, and

dynamic viscosity, µ).

In the case of a non-isothermal flow, it is necessary to solve the energy equation,

subject to appropriate boundary conditions, in addition to the equations described

above. The energy equation can be written in Cartesian tensor notation:

∂T

∂t+

∂(UiT )

∂xi=

∂

∂xi

(

ν

Pr

∂T

∂xi

)

(3.6)

where T is the instantaneous fluid temperature, ν is the kinematic viscosity and Pr is

the Prandtl number.

Whilst none of the flows considered in this work are non-isothermal, the inclusion

of thermal strain terms in the solid constitutive relations (presented in the proceeding

Chapter) allow for the effects of thermally induced deformations upon the solid body.

3.3 Turbulence Modelling Strategies

The Navier-Stokes equations presented in the previous Section are applicable to all

flows. Unfortunately, the vast majority of flows of engineering, and many of physiologi-

cal relevance involve the effects of turbulence. The wide range of length and time scales

present in a turbulent flow cause all but the simplest of flows to be prohibitively expen-

56

3.3. TURBULENCE MODELLING STRATEGIES

sive to compute exactly. To overcome this obstacle, it is necessary to use some form

of turbulence model to account for the effects of turbulence. A brief description of the

most common turbulence modelling strategies is given in the proceeding Sub-sections.

3.3.1 DNS Modelling

The most accurate and complete method of computing turbulent flow is Direct Nu-

merical Simulation (DNS). DNS solves the Navier-Stokes equations on a fine enough

grid and with small enough time steps to resolve the entire range of length and time

scales present in the flow; the turbulent flow field is captured entirely. The results ob-

tained from DNS are so accurate and complete that they can be used for fundamental

research into turbulence physics and used for the development and validation of new

turbulence models. The principle disadvantage of DNS is its prohibitive computational

expense; resolution of small spatial and temporal variations requires a vast amount of

computer memory and time. DNS will not, therefore, be a viable option for computing

industrially relevant flows for the foreseeable future.

3.3.2 LES Modelling

The second most complete method of computing turbulent flow is Large Eddy Sim-

ulation (LES). LES computes the larger scale eddies exactly; models are introduced

to account for the effects of the smaller scale eddies. The Navier-Stokes equations

are spatially filtered to separate the large eddies from the small; the LES form of the

Navier-Stokes equations, for an incompressible flow, can be written:

∂(ρui)

∂t+

∂(ρuiuj)

∂xj= − ∂p

∂xi+

∂

∂xj

[

µ

(∂ui

∂xj+

∂uj

∂xi

)]

(3.7)

The continuity equation is unchanged by the filtering process:

∂(ρui)

∂xi= 0 (3.8)

where u and p are the filtered velocity and pressure respectively. The filtered velocity

field is given by:

ui(x) =

∫

G(x, x′)ui(x′) dx′ (3.9)

where G(x, x′) is called the filter kernel. Many filters exist, but typically they have

an associated cut-off length scale, ∆. The larger eddies (those greater than the filter

cut-off) are computed exactly because they interact with, and extract energy from,

the mean flow and are therefore mostly responsible for the transport of conserved

properties. The large eddies are more problem dependent and thus more difficult to

57


model accurately. The smaller eddies are more isotropic in nature and are thus less

problem dependent; this makes it simpler to find a model for them that is applicable

to many types of flow. The subgrid-scale stress is defined as:

τSij = −ρ(uiuj − uiuj) (3.10)

Perhaps the simplest and most widely used subgrid-scale model is that proposed

by Smagorinsky (1963). The subgrid-scale stress is represented by an eddy-viscosity

model of the form:

τSij −

1

3τSkkδij = µt

(∂ui

∂xj+

∂uj

∂xi

)

(3.11)

where µt is the sub-grid turbulent viscosity, which can be obtained through the relation

used by Deadroff (1970):

µt = ρ(Cs∆)2√

SijSij (3.12)

where Cs is the Smagorinsky constant and Sij is the filtered strain-rate tensor:

Sij =

(∂ui

∂xj+

∂uj

∂xi

)

(3.13)

and ∆ is the filter width, which is proportional to the cell volume:

∆ = 2(∆x∆y∆z)1/3 (3.14)

Despite advances in computer resources, computation of turbulent flows in complex

geometries using LES is still too costly for many industrial applications. This is mainly

due to the fact that all flows need to be computed as three-dimensional and time-

dependent, whilst resolving much of the turbulence structure.

3.3.3 RANS Modelling

The Reynolds Averaged Navier-Stokes (RANS) method computes the mean flow field

with models to account for the effects of turbulence. The governing Navier-Stokes

equations are ensemble averaged by decomposing the flow variables into mean and

time-varying components. The additional terms created by the averaging process are

called Reynolds stresses and represent the interaction between the time-varying fluc-

tuations. Models are introduced to relate the unknown Reynolds stresses with known

mean flow variables. The RANS approach is the most commonly used for industrial

flows as it requires the least computer resources and is not limited to simple geometries.

Many RANS models are robust, stable and have been extensively validated. The tur-

bulent flow calculations presented in this Thesis used the RANS modelling framework.

58

3.4. THE RANS EQUATIONS

Further details regarding the derivation of the time-averaged Navier-Stokes equations

and details of the most commonly used models used to represent the Reynolds stresses

are given in the following Section.

3.4 The RANS Equations

Often the intricate details of a complex turbulent flow are of little concern to an engineer

who is more interested in the time-averaged flow properties. Solution of the RANS

equations yields the mean flow field without the computational expense of resolving

the small scale features of the flow. The RANS equations are derived by decomposing

the flow variables into a mean and fluctuating component, as shown below:

Φ(xi, t) = Φ(xi) + φ(xi, t) (3.15)

where Φ is the instantaneous value of a scalar variable, Φ is the mean component and φ

is the fluctuating component of the variable. The conceptually simplest way of defining

the mean is a time-averaging where the value is of Φ is given by:

Φ(xi) = limT→∞

1

T

T∫

0

Φ(xi, t) dt (3.16)

The RANS approach can, however, be applied to both steady and unsteady flows;

the solution from an unsteady RANS simulation represents the ensemble averaged flow.

The ensemble average would be experimentally obtained by considering many realisa-

tions of the flow and averaging the values at any specific time over the realisations and

can be described mathematically as:

Φ(xi, t) = limN→∞

1

N

N∑

n = 1

Φ(xi, t) (3.17)

where N is the number of realisations in the ensemble and must be large enough to

avoid the effects of flow fluctuations.

However the averaging process is defined, the RANS form of the continuity equation

can be written as:

∂Ui

∂xi= 0 (3.18)

where Ui represents the mean fluid velocity in the xi-direction. It can be seen that

the time-averaging of the continuity equation has introduced no additional unknown

terms; this is because the continuity equation is linear.

The RANS form of the momentum equations can be written as:

59


∂(ρUi)

∂t+

∂(ρUiUj)

∂xj= − ∂P

∂xi+

∂

∂xj

(

µ∂Ui

∂xj− ρuiuj

)

(3.19)

where uiuj is called the Reynolds stress tensor and arises from the time averaging of

the non-linear convective term. The appearance of the stress tensor has added six

additional unknown variables to the system without adding a single extra equation.

Thus, to close the system of equations and allow for their solution, some form of

model for the Reynolds stress tensor is required. The above equations are an exact

representation of the turbulent flow; the effects of turbulence upon the mean flow are

entirely contained within the Reynolds stress tensor. Thus, the accuracy of a RANS

calculation depends upon the accuracy of the model used for the stress tensor. Many

models have been developed over the years; no single model has proven itself to be

capable of accurately computing the entire range of turbulent flows. It is therefore

essential for an engineer to understand the limitations and applicability of a particular

model before using it.

3.4.1 Reynolds Stress Models

The most complete, and computationally expensive, model for the Reynolds stresses

are the so called Reynolds Stress Models (RSMs). Transport equations for the indi-

vidual components of the Reynolds stress tensor are obtained from the Navier-Stokes

equations. The exact transport equation for the Reynolds stress tensor, Rij = uiuj,

can be written as:

∂Rij

∂t+

∂(UkRij)

∂xk︸︷︷︸

= Cij

= Pij + Dij − ǫij + Πij + Ωij (3.20)

where Cij is the convective term, Pij is the production term, Dij is the diffusion term,

ǫij is the dissipation rate, Πij is the pressure-strain interaction term, and Ωij is the

system rotation term. The convective, production and rotation terms can be treated

exactly:

Cij =∂(ρUkuiuj)

∂xk(3.21)

Pij = −(

Rim∂Uj

∂xm+ Rjm

∂Ui

∂xm

)

(3.22)

Ω = −2ωk (ujumeikm + uiumejkm) (3.23)

where ωk is the system rotation vector and eijk is the Levi-Civita symbol given by:

60


eijk =

1 if i, j and k are different and in cyclic order

−1 if i, j and k are different and in anti-cyclic order

0 if any two indices are the same

(3.24)

Appropriate models need to be introduced for all other terms. For example, the

dissipation term is often modelled as isotropic:

ǫij =2

3ǫδij (3.25)

where ǫ is obtained by solving a transport equation for the turbulent kinetic energy

dissipation rate, often similar to that used in the standard k-ǫ equation. The processes

represented by the modelled terms are of a more fundamental physical nature than

those modelled in two-equation schemes (to be described later), and are therefore less

problem dependent.

The principle advantage of the RSM models is that they account for transport effects

upon the Reynolds stresses. Reynolds stress models are able to accurately compute

flows involving anisotropic stress fields - unlike simpler models based upon the eddy-

viscosity model. Although the RSM models have been shown to perform better than

simpler models they are considerably more expensive as a total of seven transport

equations (in three-dimensions) have to be solved. Engineering judgment has therefore

to be exercised to determine whether this additional computational expense is deemed

necessary.

3.4.2 Eddy Viscosity Models

Within this class of model, instead of solving transport equations for the individual

Reynolds stress components, they are simply related to the mean strains according to

the Boussinesq relation:

ρuiuj = 23ρ k δij − µt

(∂Ui

∂xj+

∂Uj

∂xi

)

(3.26)

The above is also called the eddy-viscosity model for the Reynolds stresses. The

corresponding eddy-diffusivity model exists to represent the turbulent fluxes of a scalar

property:

ρujφ = −Γt∂Φ

∂xj(3.27)

where µt is the turbulent viscosity and Γt is the turbulent diffusivity. The effects of

turbulent mixing are approximated as additional diffusive transport of momentum.

The turbulent viscosity is a property of the flow; some of the more popular methods of

61


obtaining the turbulent viscosity are described in the following Sub-sections. Although

EVMs do not represent the physics of turbulent flow as accurately as RSMs, their

simple numerical numerical implementation make them more widely used - particularly

in commercial CFD solvers.

Mixing Length Models

From dimensional analysis it is known that the kinematic turbulent viscosity can be

expressed as the product of a representative velocity and length scale. The dynamic

turbulent viscosity can be written:

µt = C1ρvl (3.28)

where C1 is a dimensionless constant, ρ is the fluid density, v is a characteristic velocity

scale, and l is a characteristic length scale of the flow. The length scale is based on the

larger eddies as they interact with the mean flow. For simple two-dimensional shear

flows the velocity scale can therefore be written:

v = C2l

∣∣∣∣

∂U

∂y

∣∣∣∣

(3.29)

where C2 is a dimensionless constant, and l is the length scale. By combining Equa-

tions 3.28 and 3.29 the turbulent viscosity can be written:

µt = ρl2m

∣∣∣∣

∂U

∂y

∣∣∣∣

(3.30)

where the length scale lm includes the dimensionless constants C1 and C2. Equa-

tion 3.30 is called Prandtl’s mixing length model. The length scale has to be related to

the specific geometry under consideration; suitable algebraic relations exist for many

geometrically simple flows including pipes and channels. Prescription of the length

scale becomes more difficult for complex geometries. Mixing length models are not ca-

pable of capturing the transport and production/dissipation of turbulence properties;

for this reason, they can not be applied to many important categories of industrial flow

- particularly those involving separation.

One-Equation Models

The next category of models for the turbulent viscosity solve a single transport equation

for one of the turbulent properties of the flow. The equation for the turbulent kinetic

energy can be obtained from the exact Reynolds stress transport equation (3.20) by

summing the equations for the three normal stresses:

62


k =1

2(u2

1 + u22 + u2

3) (3.31)

which after expansion can be written:

∂(ρk)

∂t+

∂(ρUjk)

∂xj=

∂

∂xj

[(

µ +µt

σk

)∂k

∂xj

]

− ρuiuj∂Ui

∂xj− ρǫ (3.32)

where ǫ is the turbulent kinetic energy dissipation rate; in the one-equation model, it

is obtained from:

ǫ =k3/2

lǫ(3.33)

The turbulent viscosity is obtained from:

µt = ρ cµ lµ√

k (3.34)

where lǫ and lµ are prescribed length scales. The length scales can be related to the

near-wall distance according to:

lǫ = 2.55y[1 − exp(−0.235y∗)] (3.35)

and

lµ = 2.55y[1 − exp(−0.016y∗)] (3.36)

where the dimensionless near-wall distance is defined as:

y∗ =yk1/2

ν(3.37)

Whilst one-equation models allow for transport effects upon turbulent quantities,

they still suffer the limitation of requiring the prescription of a length scale.

Two-Equation Models

To overcome the difficulty of prescribing an algebraic length scale for a complex geom-

etry, and to account for transport effects upon turbulent properties, the two-equation

family of RANS models was developed. In these, transport equations are solved for

two turbulent properties, typically the turbulent kinetic energy, k, and its dissipation

rate, ǫ; a widely used alternative involves the inverse time scale, ω. Whilst an exact

equation for the turbulent kinetic energy dissipation rate can be obtained (see for ex-

ample Wilcox (1998)) the modelling required to close the equation is so great that it is

simpler to consider the entire equation as a model based on empirical observation.

63


In the case of the k-ǫ model, the velocity and length scale are related to the kinetic

energy and dissipation rate by:

v = k1/2 (3.38)

l =k3/2

ǫ(3.39)

The turbulent viscosity is obtained from:

µt = ρ cµk2

ǫ(3.40)

The standard k-ǫ model solves the following transport equations for turbulent ki-

netic energy and its dissipation rate respectively:

∂(ρk)

∂t+

∂(ρUik)

∂xi=

∂

∂xi

[(

µ +µt

σk

)∂k

∂xi

]

+ 2µtSijSij − ρǫ (3.41)

∂(ρǫ)

∂t+

∂(ρUiǫ)

∂xi=

∂

∂xi

[(

µ +µt

σǫ

)∂ǫ

∂xi

]

+ C1ǫǫ

k2µtSijSij − C2ǫρ

ǫ2

k(3.42)

where Sij is the strain tensor defined as:

Sij =

(∂Ui

∂xj+

∂Uj

∂xi

)

(3.43)

The values of the constants appearing in the standard k-ǫ equation are summarised

in Table 3.1. These constants are obtained from simple experiments, such as decay-

ing homogeneous turbulence, and analytical solutions, such as the limiting case of an

equilibrium boundary layer. An understanding of the assumptions used when tuning

the constants in the transport equations is necessary to produce accurate and valid

results. The limitations introduced through these approximations preclude the use of

these models, with these particular coefficients, in the low-Reynolds number flow region

near a wall.

cµ σk σǫ C1ǫ C2ǫ

0.09 1.0 1.3 1.44 1.92

Table 3.1: Coefficients used in the standard k-ǫ model.

Since two-equation models do not require the arbitrary prescription of either a

velocity or length scale they are more general in nature than simpler zero- and one-

equation models; this is the principle reason for their near universal use in commercial

CFD solvers.

64

3.5. THE LAUNDER-SHARMA LOW-RE MODEL

One of the main drawbacks of the two-equation models is that they have generally

been tuned to give broadly the correct shear stress in simple shear flows, and often

give incorrect normal stress values. Even if the turbulent kinetic energy is accurately

predicted, the stress field may be incorrect for flows involving complex strain fields

and strong body forces. Such flows include: swirling and rotating flows and flows with

anisotropic normal Reynolds stresses.

3.4.3 Non-linear Eddy Viscosity Models

The two-equation models discussed above used a linear stress-strain relation, some

failings of which were outlined in the previous Sub-section. To overcome the problem

of normal stress isotropy, but without introducing the computational expense of an

RSM, the non-linear EVM models were developed. These models aim to combine

the accuracy of the RSMs, particularly for flows with complex strain fields, with the

computational efficiency of two-equation models. Details of these schemes are not given

here, since they have not been employed in the present work, but the interested reader

might refer to Craft et al. (2000) and Craft et al. (1997), amongst others.

3.5 The Launder-Sharma Low-Re Model

Treatment of the near-wall region is one of the most important aspects of a turbulent

flow simulation. The difficulty lies in capturing the steep near-wall gradients of flow

variables. Mesh refinement in the near-wall region can become very expensive, par-

ticularly for complex three-dimensional geometries. To overcome this, wall functions

were developed which prescribe a log-law profile over a relatively coarse near-wall mesh.

Whilst wall functions can be very effective, they can be difficult to implement in a cou-

pled FSI solver as the wall displacement, and subsequent re-meshing, could move the

near-wall node outside of the log-law layer. Another consideration with relevance to

the simulations presented in this Thesis is that many physiological flows have Reynolds

numbers in the transitional regime; this means that parts of the flow domain remain

laminar whilst others are turbulent. It is therefore more convenient to employ a low-

Reynolds-number model, and there have been a number of such forms which have been

developed to account or these features.

In this work, all turbulent flow calculations have been performed using the Launder-

Sharma low-Reynolds-number model (Launder and Sharma, 1974). Although based on

the k-ǫ form given above, this model includes additional terms which account for near-

wall effects and requires a fine near-wall mesh to resolve the sharp gradients of flow

quantities. This particular model has been shown to perform well for a wide range of

cases. The Launder-Sharma low-Reynolds-number model can be written:

65


Dk

Dt= Pk − ǫ − 2ν

(

∂k1/2

∂xj

)2

+∂

∂xj

[

(ν + νt/σk)∂k

∂xj

]

(3.44)

Dǫ

Dt= cǫ1

Pk ǫ

k− cǫ2f2

ǫ2

k+ 2ννt

(∂2Ui

∂xj∂xk

)2

+∂

∂xj

[

(ν + νt/σǫ)∂ǫ

∂xj

]

+ Sǫ (3.45)

where k and ǫ are the turbulent kinetic energy and isotropic dissipation rate respectively.

The isotropic dissipation rate, ǫ, is related to the dissipation rate, ǫ, according to:

ǫ = ǫ − 2ν

(

∂k1/2

∂xj

)

(3.46)

The production term, Pk, is given as before by:

Pk = −uiuj∂Ui

∂xj(3.47)

The turbulent viscosity is given by:

µt = ρ fµ cµk2

ǫ(3.48)

The Launder-Sharma low-Reynolds-number k-ǫ model uses the following damping

functions to account for the effects of viscosity in the near-wall region:

fµ = exp

[ −3.4

(1 + Rt/50)2

]

(3.49)

f2 = 1 − 0.3 exp(−R2t ) (3.50)

where Rt is the turbulent Reynolds number, defined as:

Rt =k2

ǫν(3.51)

The additional source term, Sǫ, appearing on the right hand side of the ǫ equation is

the Yap length-scale correction (Yap, 1987) which was designed to reduce the predicted

turbulent length-scale in regions of flow re-attachment; it is given by:

Sǫ = cw(ǫ2/k) max

[(l

le− 1

)(l

le

)2

, 0

]

(3.52)

where l is the turbulent length scale (l = k3

2 /ǫ), le the equilibrium length scale (le = cly),

the constant cl = 2.55 and y is the near wall distance. The constant cw has been

empirically determined as 0.83 by Yap. The values of the other constants appearing in

66

3.6. SUMMARY

the above equations are listed in Table 3.2.

cǫ1 cǫ2 cµ σk σǫ

1.44 1.92 0.09 1.0 1.3

Table 3.2: Coefficients used in the Launder-Sharma model.

3.6 Summary

In this Chapter, the equations governing fluid flow have been presented. Various tur-

bulence modelling strategies have been discussed, including DNS, LES and RANS.

Most flows of practical importance include the effects of turbulence which have to be

modelled in order to allow for their efficient numerical simulation.

In the RANS framework, the governing equations are time averaged; a consequence

of which is the appearance of the Reynolds stresses which have to be modelled to close

the system of governing equations. The simplest form of model is the eddy-viscosity

model in which the turbulent stresses are related to the mean strain-rates through the

eddy viscosity. The computations presented in this Thesis use a relatively simple two-

equation model, whereby the eddy viscosity is obtained from transport equations which

are solved for the turbulent kinetic energy and its dissipation rate. The model includes

additional terms to account for near-wall effects. Although more accurate models exist,

the low-Re Launder-Sharma model was chosen for its efficiency, robustness and simple

implementation.

67


68

Chapter 4

Mathematical Model of Solid

Deformation

4.1 Introduction

In this Chapter, a full derivation of the governing equations of elasticity is presented.

By starting from first principles, the simplifications and assumptions inherent in a linear

elastic analysis will be highlighted. The difficulties associated with obtaining analytical

solutions to the elasticity equations necessitates the use of some numerical method for

all but the most trivial of problems; details of the numerical solution of the elasticity

equations will be deferred to a later Chapter. The equations in this Chapter have

been derived with reference to a Cartesian coordinate system; transformation of the

equations into polar and non-orthogonal coordinates is presented in the Appendices.

4.2 The Stress Equilibrium Equations

Consider an infinitesimal cube of side length dx, dy and dz. Each face of the cube has

three stresses acting upon it, where stress is defined as force per unit area. One stress

acts in a direction normal to the face and two stresses act parallel to the face. For

example, the stresses acting on a plane perpendicular to the x-axis are σxx, τxy and

τxz. In this notation, σ refers to a normal stress and τ refers to a shear stress. The

first subscript indicates the face on which the stress acts, whilst the second subscript

indicates the direction in which the stress acts. The stresses acting on a two-dimensional

element are shown schematically in Figure 4.1.

The stresses acting on the left hand x-plane are σxx, τxy and τxz. The stresses

acting on the right hand x-plane are (σxx + ∆σxx), (τxy + ∆τxy) and (τxz + ∆τxz). By

making use of the small deformation assumption, these can be approximated with a

Taylor series expansion of the form:

69

CHAPTER 4. MATHEMATICAL MODEL OF SOLID DEFORMATION

σxx + ∆σxx = σxx +∂σxx

∂xdx (4.1)

τxy + ∆τxy = τxy +∂τxy

∂xdx (4.2)

τxz + ∆τxz = τxz +∂τxz

∂xdx (4.3)

Similar expressions can be written for the stresses acting upon the y- and z-planes.

In addition to these surface stresses, the cube is also subjected to the three components

of the body force vector fx, fy and fz. Body forces are defined as forces per unit volume,

examples of which include gravity and inertia.

y6

-fx

fy

?

σyy

τyx

-6

σxx + ∆σxx

τxy + ∆τxy

6-

σyy + ∆σyy

τyx + ∆τyx

?

σxx

τxy

Figure 4.1: Two-dimensional stress element.

According to Newton’s Second Law of motion, the sum of the forces acting in the

x-direction is equal to zero; thus, the force equilibrium condition in the x-direction can

be written:

∑

Fx = 0 (4.4)

Thus, by summation of forces acting in the x-direction the equilibrium condition can

be written:

Inertial term︷︸︸︷

ρ∂2u

∂t2dx dy dz + [−σxx + (σxx + ∆σxx)] dy dz + [−τxy + (τxy + ∆τxy)] dx dz

+ [−τxz + (τxz + ∆τxz)] dx dy + fx dx dy dz = 0 (4.5)

70

4.3. THE LINEAR-ELASTIC CONSTITUTIVE RELATIONS

where the over-braced term (the second derivative of the displacement in the x-direction,

u, with respect to time, multiplied by the material density, ρ) represents the inertial

force due to the acceleration of the element in the x-direction. After substitution of

Equations 4.1 - 4.3 the above condition becomes:

ρ∂2u

∂t2dx dy dz +

[

−σxx +

(

σxx +∂σxx

∂xdx

)]

dy dz

+

[

−τxy +

(

τxy +∂τxy

∂ydy

)]

dx dz +

[

−τxz +

(

τxz +∂τxz

∂zdz

)]

dx dy

+ fx dx dy dz = 0 (4.6)

which can be simplified to give:

ρ∂2u

∂t2+

∂σxx

∂x+

∂τxy

∂y+

∂τxz

∂z+ fx = 0 (4.7)

Similar expressions can be derived by considering force equilibrium in the y- and

z-directions:

ρ∂2v

∂t2+

∂τxy

∂x+

∂σyy

∂y+

∂τyz

∂z+ fy = 0 (4.8)

ρ∂2w

∂t2+

∂τxz

∂x+

∂τyz

∂y+

∂σzz

∂z+ fz = 0 (4.9)

This system of coupled partial differential equations, called the stress equilibrium

equations, can be more compactly written using Cartesian tensor notation:

ρ∂2ui

∂t2+

∂σij

∂xj+ fi = 0 (4.10)

4.3 The Linear-Elastic Constitutive Relations

It has been proven experimentally that for a linear elastic material, the measured strain

in a body is proportional to the applied stress. This can be described mathematically

by the generalised form of Hooke’s law. For an isotropic material, the stress-strain

relationship can be written:

71


σxx

σyy

σzz

σxy

σxz

σyz

=

λ + 2µ λ λ 0 0 0

λ λ + 2µ λ 0 0 0

λ λ λ + 2µ 0 0 0

0 0 0 2µ 0 0

0 0 0 0 2µ 0

0 0 0 0 0 2µ

ǫxx

ǫyy

ǫzz

ǫxy

ǫxz

ǫyz

(4.11)

where ǫij is the strain (the same notation for the subscripts is used as for the stress,

σij), λ and µ are called Lame’s constants, which are related to the elastic modulus, E,

and the Poisson ratio, ν, according to:

λ =νE

(1 + ν)(1 − 2ν)(4.12)

µ =E

2(1 + ν)(4.13)

Equation 4.11 can also be written in Cartesian tensor notation:

σij = 2µǫij + λδijǫkk (4.14)

where δij is the Kronecker delta symbol, defined as:

δij =

1 if i = j

0 if i 6= j(4.15)

Summation occurs over repeated indices; for example:

ǫkk = ǫxx + ǫyy + ǫzz (4.16)

In the case of a thermo-elastic material, the stress-strain relation is given by the

Duhamel-Neumann law and includes an additional term to account for thermally in-

duced strains:

σij = 2µǫij + λδijǫkk − (3λ + 2µ)δijα(∆T ) (4.17)

which can also be written in vector notation:

[σ] = 2µ[ǫ] + λdiv u I − (3λ + 2µ)α(∆T )I (4.18)

where I is the identity matrix:

72

4.4. THE KINEMATIC RELATIONS

I =

1 0 0

0 1 0

0 0 1

(4.19)

4.4 The Kinematic Relations

The displacement field (u, v,w) describes the position of a point in an elastic body,

relative to its original position, after the body has been deformed due to the action of

some force. Strain describes the relative change in length of a body due to an applied

force. It is the purpose of this Section to derive relations between the displacements

and the strains. Since we are assuming linear elasticity, only small displacements will

be considered in what follows.

Consider a line segment of length ds which has its two endpoints at positions

(x, y, z) and (x+ dx, y + dy, z + dz), respectively. Under the action of some force, this

line segment deforms by amount (u, v,w) to have a new length dS, with endpoints at

(X,Y,Z) and (X + dX,Y + dY,Z + dZ), where:

X

Y

Z

=

x

y

z

+

u

v

w

(4.20)

This can be differentiated to give:

dX

dY

dZ

=

dx

dy

dz

+

du

dv

dw

(4.21)

The chain rule of differentiation can be used to replace the displacement terms

appearing in Equation 4.21 with:

du =∂u

∂xdx +

∂u

∂ydy +

∂u

∂zdz (4.22)

dv =∂v

∂xdx +

∂v

∂ydy +

∂v

∂zdz (4.23)

dw =∂w

∂xdx +

∂w

∂ydy +

∂w

∂zdz (4.24)

these can now be substituted into Equation 4.21 to give:

73


dS =

dX

dY

dZ

=

dx

dy

dz

+

∂u∂x

∂u∂y

∂u∂z

∂v∂x

∂v∂y

∂v∂z

∂w∂x

∂w∂y

∂w∂z

dx

dy

dz

(4.25)

or, more simply:

dS = ds + [G]ds (4.26)

where [G] is called the displacement gradient matrix. The next step is to find the differ-

ence between the square of the length of the segment before and after the deformation:

∆ = dS2 − ds2

= dST dS − dsT ds= (ds + [G]ds)T (ds + [G]ds) − dsT ds= dsT ds + [G]dsT ds + [G]T dsT ds +

[G]T [G]dsT ds − dsT ds= dsT ([G] + [G]T + [G]T [G])ds= dsT ([G] + [G]T )ds (4.27)

where the term [G]T [G] can be neglected due to the small displacement assumption.

Strain is defined to relate the change in length, ∆, with the original length, ds, in

the following way:

∆ = 2dsT [ǫ]ds (4.28)

thus:

[ǫ] =

ǫxx ǫxy ǫxz

ǫyx ǫyy ǫyz

ǫzx ǫzy ǫzz

=

1

2([G]T + [G]) (4.29)

finally:

ǫxx ǫxy ǫxz

ǫyx ǫyy ǫyz

ǫzx ǫzy ǫzz

=

∂u∂x

12

(∂u∂y + ∂v

∂x

)12

(∂u∂z + ∂w

∂x

)

12

(∂u∂y + ∂v

∂x

)∂v∂y

12

(∂v∂z + ∂w

∂y

)

12

(∂u∂z + ∂w

∂x

)12

(∂v∂z + ∂w

∂y

)∂w∂z

(4.30)

this can be written in the more compact Cartesian tensor notation:

74

4.5. THE NAVIER DISPLACEMENT EQUATIONS

ǫij =1

2

(∂ui

∂xj+

∂uj

∂xi

)

(4.31)

or in vector notation:

[ǫ] =1

2[gradu + (grad u)T ] (4.32)

4.5 The Navier Displacement Equations

It is possible to combine the equilibrium equations (4.7-4.9) with the constitutive re-

lations (4.11) and the strain-displacement relations (4.30) to give a set of three equa-

tions in terms of the three displacement components (u, v,w). First, consider the

x-equilibrium equation (4.7):

ρ∂2u

∂t2+

∂σxx

∂x+

∂τxy

∂y+

∂τxz

∂z+ fx = 0 (4.33)

From the stress-strain relation (4.17) we know that the stresses appearing in the

above equation can be replaced by:

σxx = 2µǫxx + λǫkk − (3λ + 2µ)α(∆T ) (4.34)

τxy = 2µǫxy (4.35)

τxz = 2µǫxz (4.36)

Using the strain-displacement equation (4.31) the above can be re-written as:

σxx = 2µ∂u

∂x+ λ

(∂u

∂x+

∂v

∂y+

∂w

∂z

)

− (3λ + 2µ)α(∆T ) (4.37)

τxy = µ

(∂u

∂y+

∂v

∂x

)

(4.38)

τxz = µ

(∂u

∂z+

∂w

∂x

)

(4.39)

These are then substituted back into the equilibrium equation (4.7) to give:

ρ∂2u

∂t2+

∂

∂x

[

2µ∂u

∂x+ λ

(∂u

∂x+

∂v

∂y+

∂w

∂z

)

− (3λ + 2µ)α(∆T )

]

+∂

∂y

[

µ

(∂u

∂y+

∂v

∂x

)]

+∂

∂z

[

µ

(∂u

∂z+

∂w

∂x

)]

+ fx = 0 (4.40)

75


which upon expansion becomes:

ρ∂2u

∂t2+ 2µ

∂2u

∂x2+ λ

∂2u

∂x2+ λ

∂2v

∂x∂y+ λ

∂2w

∂x∂z+ µ

∂2u

∂y2+ µ

∂2v

∂x∂y+ µ

∂2u

∂z2+ µ

∂2w

∂x∂z

− ∂

∂x[(3λ + 2µ)α(∆T )] + fx = 0 (4.41)

which can be simplified to give:

ρ∂2u

∂t2+ (λ + µ)

∂

∂x

[∂u

∂x+

∂v

∂y+

∂w

∂z

]

+ µ

[∂2u

∂x2+

∂2u

∂y2+

∂2u

∂z2

]

− ∂

∂x[(3λ + 2µ)α(∆T )] + fx = 0 (4.42)

Similar expressions can be found by considering the y- and z-equilibrium equations:

ρ∂2v

∂t2+ (λ + µ)

∂

∂y

[∂u

∂x+

∂v

∂y+

∂w

∂z

]

+ µ

[∂2v

∂x2+

∂2v

∂y2+

∂2v

∂z2

]

− ∂

∂y[(3λ + 2µ)α(∆T )] + fy = 0 (4.43)

ρ∂2w

∂t2+ (λ + µ)

∂

∂z

[∂u

∂x+

∂v

∂y+

∂w

∂z

]

+ µ

[∂2w

∂x2+

∂2w

∂y2+

∂2w

∂z2

]

− ∂

∂z[(3λ + 2µ)α(∆T )] + fz = 0 (4.44)

Equations 4.42-4.44 are collectively known as Navier’s equations of elasticity. They

can be condensed by writing them in Cartesian tensor notation:

ρ∂2ui

∂t2+ (λ + µ)

∂

∂xi

(∂uj

∂xj

)

+ µ∂2ui

∂x2j

− ∂

∂xi[(3λ + 2µ)α(∆T )] + fi = 0 (4.45)

or in vector notation:

ρ∂2u

∂t2+ (λ + µ)grad(divu) + µ∇2u− (3λ + 2µ)α grad(∆T ) + F = 0 (4.46)

where u is the displacement vector (u, v,w) and F is the body force vector (Fx, Fy, Fz).

76

4.6. SOLUTION OF THE DISPLACEMENT EQUATIONS

4.6 Solution of the Displacement Equations

To obtain a solution - analytical or numerical - to the displacement equations, the

following steps are performed:

1. Define the physical geometry of the solution domain.

2. Define the mechanical and thermal (if necessary) properties of the material.

3. Prescribe boundary conditions on all domain boundaries. Boundary conditions

can be prescribed either in terms of displacement (Dirichlet condition) or dis-

placement gradient (Neumann condition), typically corresponding to prescription

of the stress.

4. In the case of a transient problem, an initial condition must also be specified.

This might typically be a zero displacement condition.

5. Solve the system of PDEs using an appropriate technique to yield displacements

within the solution domain.

6. The corresponding stress field can be obtained from the calculated displacement

field through the constitutive relations.

Step 2 of the solution procedure states that the properties of the material need to be

prescribed. The behaviour of a linear elastic material is described by two independent

properties; Table 4.1 shows the possible combinations of material properties that may

be prescribed, where E is the Young’s modulus, G is the shear modulus, ν is the

Poisson ratio and λ is Lame’s first parameter. In the cases presented in this Thesis,

the material was defined by the Young’s modulus, E, and the Poisson ratio, ν (column

5 in Table 4.1).

1 2 3 4 5

(λ & G) (E & G) (λ & ν) (G & ν) (E & ν)

Table 4.1: Material property combinations.

4.7 Summary

This Chapter has presented the governing equations of linear elastic deformation writ-

ten in terms of displacement. The assumption of linear elasticity will not limit the

generality of the solid body solver too greatly as the majority of engineering systems

fall within this category; only systems with very large displacements or those involving

certain materials require an elasto-plastic or plastic constitutive relation. By using the

77


thermo-elastic constitutive relation, the effects of thermally induced strains may be

included in later simulations. The methodology of obtaining a solution to the displace-

ment equations was presented; this methodology is relevant for both analytical and

numerical solution. Details on the present approach of obtaining a numerical solution

to the displacement equations will be described in a later Chapter.

78

Chapter 5

Fluid Solver Implementation

5.1 Introduction

The system of non-linear PDEs which govern fluid flow was presented in Chapter 3. For

all but the simplest cases, an exact analytical solution to the Navier-Stokes equations

is not possible. Therefore, some form of numerical method is required to obtain an ap-

proximate solution. The numerical simulations presented in this Thesis were performed

using the finite-volume based STREAM code of Lien and Leschziner (1994).

The purpose of this Chapter is to give a thorough description of the finite-volume

method. Also, the details of how the method is implemented into the STREAM code

are given. As alluded to in the introductory Chapter, the efficient simulation of fluid-

structure interaction problems can only be achieved through the consistent use of the

same numerical method across both fluid and solid sub-domains; it is thus necessary

to give a full description of the finite-volume method before a description of the solid

solver is given in the proceeding Chapter.

5.2 The Finite-Volume Method

The finite-volume method involves the decomposition of the physical domain into a

number of contiguous cells defined by a numerical grid. The governing equations are

integrated over each of these cells; the terms appearing after integration are then ap-

proximated with finite-difference type relations. The result of this discretization process

is a system of N algebraic equations, where N is the number of discrete cells, which

are solved using an appropriate method, such as the Tri-Diagonal Matrix Algorithm

(TDMA) solver. The principle advantage of the finite-volume method is that it ensures

conservation of variables over each discrete cell, rather than over the global domain -

as is the case with the finite-element method, in which a functional form of the gov-

erning equations is minimised over the entire domain. This means that conservative

79

CHAPTER 5. FLUID SOLVER IMPLEMENTATION

solutions can be obtained cheaply on relatively coarse numerical grids. The use of

coarse grids will however introduce numerical errors which will reduce the accuracy of

the simulation.

~~ ~

~

~

PW E

S

N

mm m

m

m

w e

s

n

sw

nw

se

ne

- -

?

6?

6

?

6

?

6

-

-

∆xw ∆xe

∆ys

∆yn

∆xs

∆xn

∆yw ∆ye

Figure 5.1: Typical finite-volume cell.

A typical finite-volume cell is shown in Figure 5.1. The central node is denoted with

P ; neighbouring nodes are labelled with uppercase letters according to their compass

position relative to P ; cell face centres are labelled with lowercase letters. By way of

example, consider the governing equation of two-dimensional, steady-state transport of

a scalar quantity, φ:

∂(ρUφ)

∂x+

∂(ρV φ)

∂y=

∂

∂x

(

µ∂φ

∂x

)

+∂

∂y

(

µ∂φ

∂y

)

+ Sφ (5.1)

where ρ and µ are the fluid density and dynamic viscosity respectively; (U, V ) is the

velocity vector, which is assumed to be a known function of position; Sφ represents any

source or sink terms and φ is the transported scalar. This equation is integrated over

the cell shown in Figure 5.1:

n∫

s

e∫

w

[∂(ρUφ)

∂x+

∂(ρV φ)

∂y

]

dx dy =

n∫

s

e∫

w

[∂

∂x

(

µ∂φ

∂x

)

+∂

∂y

(

µ∂φ

∂y

)]

dx dy +

n∫

s

e∫

w

Sφ dx dy (5.2)

80

5.2. THE FINITE-VOLUME METHOD

which after integration becomes:

n∫

s

ρUφ dy

e

w

+

e∫

w

ρV φ dx

n

s

=

n∫

s

(

µ∂φ

∂x

)

dy

e

w

+

e∫

w

(

µ∂φ

∂y

)

dx

n

s

+

n∫

s

e∫

w

Sφ dx dy (5.3)

The above expression is an exact representation of the original transport equation

as no approximations have yet been introduced. However, approximations have to be

introduced to represent the cell face value of φ and its first derivative. The mid-point

rule is used to approximate the line integrals appearing in Equation 5.3. For example,

the x-direction convective term can be approximated:

n∫

s

ρUφ dy

e

w

≈ [ρUφ∆y]ew (5.4)

where ∆y is the height of the finite-volume cell. Similar expressions can be substituted

into Equation 5.3 to give:

(ρU∆y)e φe − (ρU∆y)w φw + (ρV ∆x)n φn − (ρV ∆x)s φs =

(µ∆y)e∂φ

∂x

∣∣∣∣e

− (µ∆y)w∂φ

∂x

∣∣∣∣w

+ (µ∆x)n∂φ

∂y

∣∣∣∣n

− (µ∆x)s∂φ

∂y

∣∣∣∣s

+ (∆x∆y)P Sφ (5.5)

The cell face values of φ and its first derivative are currently unknown, as φ is stored

at the cell centres. Thus, approximations for the unknown cell face values of φ and

its derivative have to be found in terms of the known values of φ at the cell centres.

The face values of the derivatives which appear from the discretization of the diffusive

terms can be approximated by central difference expressions of the form:

∂φ

∂x

∣∣∣∣e

≈ φE − φP

∆xe,

∂φ

∂x

∣∣∣∣w

≈ φP − φW

∆xw,

∂φ

∂y

∣∣∣∣n

≈ φN − φP

∆yn,

∂φ

∂y

∣∣∣∣s

≈ φP − φS

∆ys(5.6)

Interpolation of this form gives equal weighting to the computational nodes at either

side of the face. Whilst this is appropriate for the process of diffusion, it is not for the

process of convection, where information travels in the direction of the flow. For this

latter process, the simplest approximation for the face value of φ is obtained from the

upwind scheme, in which the face value is taken to be equal to the nearest upstream

nodal value. Assuming a positive velocity vector, the face values of φ can then be

81


approximated as:

φe = φP , φw = φW , φn = φP , φs = φS (5.7)

The above approximations for the convective (Equation 5.7) and diffusive (Equa-

tion 5.6) terms can be substituted into the integrated transport equation (5.5) to give:

(ρU∆y)e φE − (ρU∆y)w φW + (ρV ∆x)n φN − (ρV ∆x)s φS =

(µ∆y)e

(φE − φP

∆xe

)

− (µ∆y)w

(φP − φW

∆xw

)

+ (µ∆x)n

(φN − φP

∆yn

)

−

(µ∆x)s

(φP − φS

∆ys

)

+ (∆x∆y)P Sφ (5.8)

where Sφ is the value of the source term evaluated at the cell centre. The above equation

can be simplified to be written as:

AP φP =∑

i=N,S,E,W

(Ai φi) + Sφ (5.9)

where the coefficients are given by:

AP =∑

i=N,S,E,W

Ai (5.10)

AE = (ρU∆y)e +

(µ∆y

∆x

)

e

(5.11)

AW = (ρU∆y)w +

(µ∆y

∆x

)

w

(5.12)

AN = (ρV ∆x)n +

(µ∆x

∆y

)

n

(5.13)

AS = (ρV ∆x)s +

(µ∆x

∆y

)

s

(5.14)

Equation 5.8 is the discretized form of the original transport equation for a single

cell. Similar discretized equations are obtained for each cell within the computational

domain. This system of discretized equations can then be solved using a suitable

technique to yield an approximate value of φ at the centre of each cell.

82

5.3. CONVECTIVE DISCRETIZATION

5.3 Convective Discretization

The simplest method of approximating the convective fluxes at cell faces, namely up-

winding, was introduced in the previous Section. Whilst upwinding has the favourable

qualities of boundedness, conservativeness and transportiveness, is does lack the accu-

racy of higher order approximations. More detail on upwinding and some higher order

convective discretization schemes used in the present work is given in the proceeding

Sub-sections.

Every simulation presented in this Thesis used the MUSCL convective discretization

scheme for all flow variables. Like many higher order methods, this scheme aims to

preserve the boundedness of upwinding whilst improving upon its accuracy.

5.3.1 First Order Upwind Scheme

As alluded to earlier, the simplest form of convective discretization is the first order

upwind scheme. This scheme approximates the face value of a convected variable by

setting it equal to the value stored at the upstream node. The scheme thus takes into

account the direction of the flow and transfers information in the streamwise direction

only. The upstream value of a scalar variable, φ, at the east face of a cell is given by

the following expression:

φe = φP for Ue > 0 (5.15)

φe = φE for Ue < 0 (5.16)

Similar expressions exist for the other faces of the cell. These expressions are used to

replace the face values of the variable appearing in the discretized governing equation.

The principle advantage of the upwind scheme is universal boundedness. This means

that the face value of the variable will always lie within the bounds of the neighbouring

nodal values. The main disadvantage of the upwind scheme is that it introduces a large

amount of numerical diffusion due to the first order truncation error.

5.3.2 QUICK Scheme

The Quadratic Upwind Interpolation for Convective Kinematics (QUICK) scheme of

Leonard (1979) attempts to increase the accuracy of the discretization by including

higher order terms. Quadratic interpolation is used to approximate the face value of

the scalar variable; the quadratic curve is fitted across two upstream nodes and one

downstream node. On a uniform grid the QUICK value of a scalar variable, φ, at the

east face of a cell is given by the following expression:

83


φe =3

8φE +

6

8φP − 1

8φW for Ue > 0 (5.17)

φe =3

8φP +

6

8φE − 1

8φEE for Ue < 0 (5.18)

The QUICK scheme displays increased accuracy when compared to the upwind

scheme due to the higher order truncation error. The main disadvantage of the QUICK

scheme, however, is that it can become unbounded - particularly when the source term

variation, and hence the variation of φ, is large.

5.3.3 MUSCL Scheme

The Monotonic Upwind Scalar Conservation Law (MUSCL) of van Leer (1979) aims to

combine the accuracy of the higher order schemes, such as QUICK, with the favourable

numerical qualities of upwind, namely boundedness. The MUSCL scheme is one of a

family of convective discretization schemes called TVD (Total Variation Diminishing).

In the case of positive, one-dimensional flow the upwind approximation of a scalar

variable at the east face of a cell is given by:

φe = φP (5.19)

Higher order convective schemes can be thought of as an extension to the above

approximation; for example, the QUICK value of a scalar variable at the east face of a

cell can be written:

φe = φP +1

8(3φE − 2φP − φW )

︸︷︷︸

High−order correction

(5.20)

where the under-braced term represents the high-order correction. It is this correction

term which introduces not only the additional accuracy but also the numerical insta-

bility and unboundedness into the scheme. The TVD family of schemes use a limiter

function to effectively switch the high-order terms on and off as appropriate. The TVD

value of a scalar variable at the east face of a cell can be written:

φe = φP +1

2Ψ(r)(φE − φP ) (5.21)

where Ψ(r) is the limiter function and r is given by:

r =

(φP − φW

φE − φp

)

(5.22)

The equivalent limiter function for the QUICK scheme could be written:

84

5.4. TEMPORAL DISCRETIZATION

Ψ(r) =(3 + r)

4(5.23)

The MUSCL convective discretization scheme uses the limiter function proposed by

van Leer (1979):

Ψ(r) =r + |r|1 + r

(5.24)

5.4 Temporal Discretization

The transport equation presented earlier (Equation 5.1) did not include a time deriva-

tive. For unsteady flow, the temporal term must be included in the governing equation,

for example:

∂(ρφ)

∂t+

∂(ρUφ)

∂x+

∂(ρV φ)

∂y=

∂

∂x

(

µ∂φ

∂x

)

+∂

∂y

(

µ∂φ

∂y

)

+ Sφ (5.25)

The discretization process must now include an additional integration in time, from

t to (t + ∆t):

t+∆t∫

t

∫∫

V

∂(ρφ)

∂tdV dt +

t+∆t∫

t

∫∫

V

(∂(ρUφ)

∂x+

∂(ρV φ)

∂y

)

dV dt =

t+∆t∫

t

∫∫

V

[∂

∂x

(

µ∂φ

∂x

)

+∂

∂y

(

µ∂φ

∂y

)]

dV dt +

t+∆t∫

t

∫∫

V

Sφ dV dt (5.26)

The transient term is integrated in time, to give:

t+∆t∫

t

∂(ρφ)

∂tdt = [ρφ]t+∆t

t (5.27)

The first term on the left hand side of Equation 5.26 can be approximated by:

t+∆t∫

t

∫∫

V

∂(ρφ)

∂tdV dt ≈ ρ(φP − φo

P )∆V (5.28)

where φP represents the cell centre value of φ at the new time, (t + ∆t), φoP represents

the cell centre value of φ at the previous time, t, and ∆V is the cell volume. The

convective, diffusive and source terms appearing in Equation 5.26 are discretized in a

similar manner as in the steady-state case. However, the time integrals are approxi-

mated by evaluating the spatial integrals at some point within the interval (t, t + ∆t)

85


and multiplied by the time step size, ∆t. The three most commonly employed methods

of approximating these integrals are described in the following Sub-sections.

5.4.1 Explicit Method

The unsteady simulations performed in this work used the explicit time discretization,

in which the spatial derivative and source terms are evaluated at the previous time step,

t. The explicit form of the discretized governing equation for the unsteady transport

of a scalar can then be written:

(

AP +ρ∆xP ∆yP

∆t

)

φP =∑

i=N,S,E,W

(Ai φoi ) +

(

Soφ + φo

P

ρ∆xP ∆yP

∆t

)

(5.29)

The main advantage of the explicit method is its simplicity; the method is imple-

mented by simply adding an additional source term to the discretized equations based

on values of flow variables from the previous time step. The main limitations are that

the method is only first-order accurate in time and at least for a simplified problem has

a very restrictive stability limit on the time step size given by the following relation:

c =Ux∆t

∆x< 1 (5.30)

where c is the Courant number. It can be seen that mesh refinement requires a corre-

sponding reduction in the time step size, which greatly increases the computer resources

required for the simulation.

5.4.2 Implicit Method

An alternative is the implicit method in which the spatial derivatives are evaluated at

the current time step, (t+∆t). The implicit form of the discretized governing equation

for the unsteady transport of a scalar can be written:

(

AP +ρ∆xP ∆yP

∆t

)

φP =∑

i=N,S,E,W

(Ai φi) +

(

Sφ + φoP

ρ∆xP ∆yP

∆t

)

(5.31)

Unlike the explicit method, the implicit method is robust and unconditionally stable

for all time step sizes. It is also first-order accurate in time, so a small time step is

required if temporal accuracy is required. The main disadvantage of the implicit method

is that the right hand side of the discretized equations contain values of the variable

at the new time step; therefore, a system of algebraic equations must be solved at

each time step. This increases the computer resources required to obtain a solution.

86

5.5. STORAGE ARRANGEMENT

The implicit method can be more efficient than the explicit method when the flow is

dominated by large-scale unsteadiness or the solution tends towards the steady-state.

5.4.3 Crank-Nicolson Method

The Crank-Nicolson method evaluates the spatial derivatives mid-way between the

current and next time step. The method is based on a central difference approximation

so is second-order accurate in time; it is therefore possible to achieve greater temporal

accuracy than with simpler first-order schemes for a given time step size. The method

is implicit so requires a system of algebraic equations to be solved each time time step.

Although the method is unconditionally stable for all time step sizes, a condition on

the time step size is necessary to ensure that the solution is bounded.

5.5 Storage Arrangement

The STREAM code uses a collocated storage arrangement in which a computational

node, at which the flow variables and fluid properties are stored, lies at the geometric

centre of each internal cell. Additional nodes lie along the domain boundaries to aid the

prescription of suitable boundary conditions. A schematic of the storage arrangement

used in the STREAM code is shown in Figure 5.2.

The finite-volume cells are constructed from an initial numerical grid; the grid points

are shown in the schematic as crosses. These grid points define the vertices of the finite-

volume cells; for an internal cell with vertices at (xne, yne), (xse, yse), (xnw, ynw) and

(xsw, ysw) the coordinates of the computational node would be (xP , yP ) where:

xP = (xne + xse + xnw + xsw)/4 (5.32)

yP = (yne + yse + ynw + ysw)/4 (5.33)

It is often necessary to evaluate flow variables and material properties at cell faces;

for example, evaluation of φ at the east face of the cell can be obtained using linear

interpolation of the form:

φe = φP + fx (φE − φP ) (5.34)

where φP and φE are the known values of φ at the central and eastern nodes respectively

and fx is the interpolation factor defined by:

fx =xe − xP

xE − xP(5.35)

87


The advantages of the collocated arrangement are simple implementation and its

applicability to complex geometries. The principle drawback is the weak coupling

between the pressure and velocity which, without taking additional measure, can yield

non-physical solutions.

The discretized equations are assembled and solved for the internal nodes only (i.e.

within the range 2 ≤ i ≤ NI − 1 and 2 ≤ j ≤ NJ − 1). The boundary nodes either

maintain constant values or are related to the values at the internal nodes according to

one-sided difference equations, depending on the precise physical boundary conditions.

In the STREAM code, a Cartesian velocity decomposition is used whereby the

Cartesian components of the velocity vector are stored at the cell centres. Cartesian

components of the convective flux are stored at the cell faces.

PW E

S

N

mm m

m

m

ew

n

s

- -?

6

6

?∆ys

∆yn

∆xw ∆xe

i = 1 i = 2 i = NI−1 i = NI

j = 1

j = 2

j = NJ−1

j = NJ

Figure 5.2: Fluid storage arrangement.

5.6 Calculation of the Pressure

The discretization of the governing equation for a transported scalar was greatly sim-

plified by assuming a known velocity field. When solving the Navier-Stokes equations,

this is not the case as the velocity field is obtained as part of the solution. Another

complication is that the momentum equations include a pressure gradient term but, in

the incompressible case, there is no independent equation governing the pressure. To

overcome these problems, a pressure-velocity coupling has to be used to ensure that

the velocity and pressure obtained from the momentum equations satisfy the continu-

ity equation. The STREAM code uses the SIMPLE method of Patankar and Spalding

(1972) for pressure-velocity linkage. The discretized U - and V -momentum equations

can be written in the form:

88

5.6. CALCULATION OF THE PRESSURE

AP UP =∑

i=N,S,E,W

(Ai Ui) + (Pw − Pe)∆y + SU (5.36)

AP VP =∑

i=N,S,E,W

(Ai Vi) + (Ps − Pn)∆x + SV (5.37)

where the pressure gradient term has been discretized using a central difference centred

around P . The first step of the SIMPLE method is to solve the momentum equations

based upon a guessed pressure field:

AP U∗

P =∑

i=N,S,E,W

(Ai U∗

i ) + (P ∗

w − P ∗

e )∆y + SU (5.38)

AP V ∗

P =∑

i=N,S,E,W

(Ai V ∗

i ) + (P ∗

s − P ∗

n)∆x + SV (5.39)

where P ∗ represents this guessed pressure field and U∗, V ∗ the corresponding velocities

that are obtained by solving these momentum equations. In general, the velocity field

obtained from the guessed pressure will not satisfy continuity; thus corrections to the

pressure and velocity fields are defined as:

P = P ∗ + P ′ (5.40)

U = U∗ + U ′ (5.41)

V = V ∗ + V ′ (5.42)

where primed quantities are the required correction to the guessed fields to ensure that

continuity is observed. Equations relating the velocity corrections to those of pressure

can be obtained by subtraction of Equations 5.38-5.39 from Equations 5.36-5.37 to give:

U ′

P =∑

i=N,S,E,W

(Ai U ′

i

AP

)

︸︷︷︸

= 0

+(P ′

w − P ′

e)∆y

AP(5.43)

V ′

P =∑

i=N,S,E,W

(Ai V ′

i

AP

)

︸︷︷︸

= 0

+(P ′

s − P ′

n)∆x

AP(5.44)

To simplify the analysis, the under braced terms are neglected in the SIMPLE pro-

cedure. This is valid because the corrections are obtained within an iterative procedure

which is complete when continuity is satisfied, at which stage the corrections will be

89


zero.

The next stage of the procedure is to obtain the discretized form of the continuity

equation, which in two-dimensions can be written:

(ρU∆y)e − (ρU∆y)w + (ρV ∆x)n − (ρV ∆x)s = 0 (5.45)

By substituting the expressions for velocity corrections into the above, an equation

for the pressure correction can be obtained of the form:

AP P ′ =∑

i=N,S,E,W

(AiP

′

i

)+ Smass (5.46)

where the source term, Smass, represents the mass imbalance across the cell given by:

Smass = (ρU∆y)e − (ρU∆y)w + (ρV ∆x)n − (ρV ∆x)s (5.47)

The pressure correction equation (5.46) can be solved, with suitable boundary condi-

tions, to yield the pressure corrections; these corrections are then added to the pressure

at each cell within the domain. The velocities are then also corrected using Equa-

tions 5.43-5.44. To increase the rate of convergence, the convective fluxes stored at the

cell faces can also be corrected.

5.7 Rhie-Chow Interpolation

When using the collocated storage arrangement, care has to be taken when discretizing

the pressure gradient term appearing in the momentum equations. Suppose that the

pressure gradient in the x-direction were discretized on a uniform grid using a central

difference of the form:

∂P

∂x

∣∣∣∣P

=Pe − Pw

∆x=

PE − PW

2∆x(5.48)

This discretization would result in a zero source term if applied to a “chequer-board”

pressure field as shown in Figure 5.3. This is obviously non-physical and steps must be

taken to ensure its prevention.

WW W P E EE

0 1 0 1 0

Figure 5.3: Chequer-board pressure field.

Recall that the discretized equation for U at the node P can be written:

90

5.8. NON-ORTHOGONAL COORDINATE SYSTEM

UP =∑

i=N,S,E,W

(Ai Ui

AP

)

+SU

AP+ (Pw − Pe)

∆y

AP(5.49)

= HP + DUP (Pw − Pe) (5.50)

A similar expression can be written for the eastern neighbour cell:

UE = HE + DUE(Pe − Pee) (5.51)

where Pee is the pressure at the east face of the cell centred around the node E. Rhie and

Chow (1983) interpolation approximates the convective flux at the cell faces according

to:

Ue = He + DUe (PP − PE) (5.52)

=1

2(HP + HE) +

1

2(DU

P + DUE)(PP − PE) (5.53)

where HP and HE are obtained from Equations 5.50 and 5.51. Hence:

Ue =1

2(UP + UE) +

1

2(DU

P + DUE)(PP − PE)−

1

2DU

P (Pw − Pe) −1

2DU

E(Pe − Pee) (5.54)

Similar expressions for the other cell faces can be obtained in the same manner.

5.8 Non-Orthogonal Coordinate System

In order to perform simulations of flow through or around complex geometries it is

necessary to use more complex grid arrangements than the simple uniformly spaced,

rectangular grids presented thus far. The STREAM code uses boundary-fitted non-

orthogonal grids. To perform calculations on this type of grid, the governing equations

have to be transformed from Cartesian (x, y) into non-orthogonal coordinates (ξ, η)

whose directions are defined to follow the local grid lines which are not necessarily at

right angles to one another:

ξ = ξ(x, y) (5.55)

η = η(x, y) (5.56)

91


The chain rule of differentiation can be used to express derivatives with respect to

the non-orthogonal coordinates as:

∂φ

∂ξ=

∂φ

∂x

∂x

∂ξ+

∂φ

∂y

∂y

∂ξ(5.57)

∂φ

∂η=

∂φ

∂x

∂x

∂η+

∂φ

∂y

∂y

∂η(5.58)

This can be more conveniently written in matrix form:

∂φ∂ξ∂φ∂η

=

[∂x∂ξ

∂y∂ξ

∂x∂η

∂y∂η

]∂φ∂x∂φ∂y

(5.59)

By taking the inverse of this, the derivatives with respect to the Cartesian coordi-

nates can be obtained:

∂φ∂x∂φ∂y

=1

|J |

[∂y∂η −∂y

∂ξ

−∂x∂η

∂x∂ξ

]∂φ∂ξ∂φ∂η

(5.60)

where J is the Jacobian of the transformation given by:

J =∂x

∂ξ

∂y

∂η− ∂x

∂η

∂y

∂ξ(5.61)

For example, the convective term in the x-direction appearing in the U -momentum

equation would be transformed as:

∂(ρUφ)

∂x=

1

|J |

∂(ρUφ)

∂ξ

∂y

∂η− ∂(ρUφ)

∂η

∂y

∂ξ

(5.62)

In the STREAM code, the terms appearing on the right hand side of the above

equation are discretized in the following manner:

∂(ρUφ)

∂x=

1

|J |

[ρUφ]ew

∆ξ

∆yη

∆η− [ρUφ]ns

∆η

∆yξ

∆ξ

(5.63)

=1

Vol∆yη[(ρUφ)e − (ρUφ)w] − ∆yξ[(ρUφ)n − (ρUφ)s] (5.64)

where Vol is the volume of the cell and the geometrical terms, ∆yξ and ∆yη, are

calculated as:

∆yξ = 0.5(yne + yse − ynw − ysw) (5.65)

∆yη = 0.5(yne + ynw − yse − ysw) (5.66)

92

5.9. BOUNDARY CONDITIONS

where yne, ynw, yse and ysw is the y-coordinate of the north east, north west, south

east and south west corners of the cell respectively. In the case of an orthogonal cell,

the term yξ is zero and the term yη is simply the height of the cell.

Similar transformations can be performed on all terms appearing in the governing

equations. The finite-volume integration now occurs over the non-orthogonal cell of

volume ∆ξ∆η.

5.9 Boundary Conditions

To obtain a unique solution to a system of elliptic partial differential equations it is

necessary to prescribe boundary conditions for each dependant variable along each of

the domain boundaries.

In the following Sub-sections a presentation of the basic boundary conditions used

in the present simulations is given. Case specific details are deferred to their respective

Chapters.

5.9.1 Inlet

At the inlet, fully developed flow is typically prescribed in the present work, which in

the case of laminar pipe flow can be written:

Uin = 2 UB

[

1 −( r

R

)2]

(5.67)

Vin = 0 (5.68)

where UB is the prescribed inlet bulk velocity; R is the radius of the pipe; and r is

the radial coordinate. In the case of turbulent pipe flow, the inlet velocity can be

approximated by the 1/7th power law profile which can be written:

Uin = 1.2245 UB

(

1 − r

R

) 1

7

(5.69)

Vin = 0 (5.70)

Uniform turbulence levels can be specified on the inlet:

kin = (αUin)2 (5.71)

ǫin =cµk2

µRν(5.72)

93


where α is the inlet turbulence intensity; and µR is the prescribed inlet turbulent

viscosity ratio defined as:

µR =νt

ν

∣∣∣in

(5.73)

The inlet pressure is extrapolated from the interior of the domain, so for a boundary

on the west face of the domain:

Pin = Pi=2 − fx(Pi=3 − Pi=2) (5.74)

5.9.2 Outlet

At the outlet, zero gradient conditions are usually prescribed on all variables. In some

internal flows it can also be beneficial to apply a bulk mass flow correction to ensure

global mass conservation. This correction is defined as:

m′ = min − mout (5.75)

where m′ is the mass flow rate correction; min is the inlet mass flow rate; and mout is

the mass flow rate calculated at the penultimate row of nodes to the outlet plane. The

mass flow rate correction is related to corresponding velocity corrections according to:

m′ =

∫

ρ U ′ dA (5.76)

where integration is over the exit plane.

Recall that, within the SIMPLE algorithm, the velocity correction is related to the

pressure correction according to:

U ′ =∆y

AP∆P ′ = DU∆P ′ (5.77)

where ∆P ′ is the difference in the pressure correction between the west and east faces

of the cell. The two equations above can be combined to give a correlation between

the mass flow rate correction and the pressure drop correction necessary at the outlet

of the domain:

m′ =

∫

ρ DU ∆P ′ dA (5.78)

This can be rearranged to give:

∆P ′ =

∫ρ DU dA

m′(5.79)

This pressure drop correction is calculated within each iteration. The required

94

5.9. BOUNDARY CONDITIONS

pressure drop across the final cell to ensure global mass conservation is given by the

sum of the pressure drop corrections, which tends to zero as mass flow rate is conserved:

∆P =∑

∆P ′ (5.80)

The outlet pressure and velocities are then prescribed according to the calculated

pressure drop:

Pout =1

2(Pout−2 + Pout−1) − ∆P (5.81)

5.9.3 Wall

Along a wall, the no slip condition is prescribed, whereby the wall tangential velocity

is set to zero at the wall:

Uwall = 0 (5.82)

In the non-porous wall case, the wall normal velocity is also set to zero at the wall:

Vwall = 0 (5.83)

The wall normal gradient of the pressure is set to zero at the wall:

∂P

∂xn

∣∣∣∣wall

= 0 (5.84)

In a turbulent flow calculation, the kinetic energy and isotropic dissipation rate are

set to zero at the wall:

kwall = ǫwall = 0 (5.85)

5.9.4 Pipe Centre Line

Along the centre line of a pipe the radial velocity is set to zero:

Vr=0 = 0 (5.86)

The radial gradients of all other dependant variables are set to zero at the centre line:

∂U

∂r

∣∣∣∣r=0

=∂P

∂r

∣∣∣∣r=0

=∂k

∂r

∣∣∣∣r=0

=∂ǫ

∂r

∣∣∣∣r=0

= 0 (5.87)

95


5.10 Solution of the Discretized Equations

Once the discretized equations presented in the preceding Sections have been assembled

into a system of linear equations, they are solved using a suitable numerical method.

To improve the stability and efficiency of the solution procedure a number of additional

steps can be taken, including: linearization and under-relaxation. Since the solution

procedure is iterative in nature, a suitable criterion has to be used to decide when to

terminate. These considerations are the subject of this Section.

5.10.1 Source Term Linearization

If the source term on the right-hand side of the discrete equation is a function of the

dependant variable, it can be linearized and transferred to the left-hand side to increase

the diagonal dominance of the resulting system and thus improve the efficiency and

stability of the solver. The source term can be linearized by writing it in the form:

Sφ = S∗

φ + SP φP (5.88)

When SP is negative it can be transferred to the left-hand side of the discretized

equations:

[AP − min(SP , 0)] φP =∑

i=N,S,E,W

(Ai φoi ) + S∗

φ + max(SP , 0)φP (5.89)

thus increasing the diagonal term in the coefficient matrix and enhancing stability.

5.10.2 Under-Relaxation

Under-relaxation reduces the rate at which variables change within the iterative solution

procedure and can be used to prevent instabilities from developing when solving highly

non-linear and closely coupled systems of equations. Choice of appropriate under-

relaxation factors is largely based on trial and error; typical factors lie within the range

0.1 − 0.5. The value of the variable φ after an iteration is chosen to lie somewhere

between the value from the previous iteration, φold, and that just calculated, φnew:

φ = φold + α(φnew − φold) (5.90)

where α is the under-relaxation factor. The above can be substituted into the dis-

cretized transport equation to give:

AP

αφP =

∑

i=N,S,E,W

(Aiφi) + Sφ +(1 − α)

αφold

P (5.91)

The pressure correction equation uses a slightly different form of under-relaxation:

96

5.10. SOLUTION OF THE DISCRETIZED EQUATIONS

P = P old + αP ′ (5.92)

Large variations in P are avoided by adding only a small fraction of the calculated

correction.

5.10.3 The TDMA Solver

The system of discretized equations can be solved using any appropriate technique.

However, in the case of structured grids, such as those employed in the present work,

the most efficient method is to take advantage of the diagonal nature of the equa-

tions and solve them iteratively using the Tri-Diagonal Matrix Algorithm (TDMA).

In the one-dimensional case, where only three diagonal coefficients are non-zero, the

TDMA solver can be applied directly; in other cases, the off-diagonal terms have to

be treated explicitly as source terms to reduce the system to the required tri-diagonal

structure. Off-diagonal terms arising from the use of higher-order convective discretiza-

tion schemes can also be transferred to the source term in a process known as deferred

correction - this method is implemented in the STREAM code. The TDMA is the pre-

ferred solver because of its efficiency and simple implementation. Details of the TDMA

solver and its implementation are given in Anderson (1995).

5.10.4 Convergence Criteria

The residual is a measure of the error level in a numerical simulation and is defined as

the difference between the two sides of the discretized equation, for example:

rk =∑

i=N,S,E,W

(Aiφki ) + Sφ − AP φk

P (5.93)

where r is the residual and k is the iteration number. After each successive iteration,

the residual should decrease and the accuracy of the approximation to φ is improved.

The residuals are typically normalized with respect to inlet quantities; for example,

the residuals of the momentum equations are normalized with respect to the inlet

momentum. In the case of the pressure correction equation, the residual is defined as

the sum of the mass imbalance across each cell normalized with respect to the inlet

mass flow rate. When the simulation has reached a converged state, and hence the flow

field satisfies continuity, the mass imbalance, and hence pressure residual will tend to

zero.

The calculation is said to be converged, and hence complete, when the residual has

met a specified criteria. Typical convergence criteria are of the order ∼ 10−4. Care has

to be taken to ensure than the solution is fully converged and hence independent of the

chosen convergence criterion.

97


5.11 Summary

This Chapter has provided a description of the numerical methods used to solve the

Navier-Stokes equations. It has specifically described how these methods are imple-

mented into the STREAM code. A description of the finite-volume method was given by

way of example of its application to the discretization of the convection-diffusion equa-

tion (both steady and unsteady). Methods of evaluating the convective and diffusive

fluxes at the cell faces were presented. The various methods used for unsteady analysis

were also described. The collocated storage arrangement used in the STREAM code

has been explained. The SIMPLE pressure-velocity coupling is used by the STREAM

code as no independent equation for the pressure exists in an incompressible flow; also,

the use of Rhie-Chow interpolation scheme to avoid numerical instabilities was de-

scribed. The method of solving the system of algebraic equations resulting from the

discretization process, namely the TDMA solver, was described; in addition, methods

to improve the stability and efficiency of the solver have been mentioned.

In the following Chapter, the development and implementation of a numerical

method to solve the elasticity equations is presented.

98

Chapter 6

Solid Solver Development

6.1 Introduction

This Chapter describes the development and initial testing of a finite-volume based

solid body solver. The discretization and implementation was carried out in a manner

consistent with the fluid solver described in the previous Chapter. The greatest differ-

ence between the fluid and solid solvers lies in the storage arrangement; the solid solver

implementation was chosen to obviate the need to interpolate the calculated displace-

ments onto the boundary of the solid body. This was done to increase the accuracy of

the vitally important boundary displacement.

The newly developed method solves the unsteady, two-dimensional, linear-elastic

displacement equations for both planar and axi-symmetric geometries. The same

boundary-fitted non-orthogonal coordinate system is used as in the fluid solver to allow

complex geometries to be considered.

Before the solid solver could be coupled to the existing fluid solver to allow fully

coupled FSI simulations to be performed, the accuracy of the method had to be verified;

where verification involves the comparison of numerical results to an analytical solution.

This type of test is appropriate at this stage as it allows for detailed comparisons

to be made for fundamental cases. Validation of the solid body solver with respect

to experimental data will occur in later Chapters. In total, three verification test

cases were considered The steady state planar and axisymmetric terms were tested by

simulating a plate with a hole subject to uni-axial tension and a thick walled cylinder

under internal pressure respectively. The unsteady terms were tested by simulating

the free vibration of a thick beam. In each of the verification test cases, the numerical

results showed excellent agreement with the analytical solution.

This Chapter concludes with a description of the steps taken to couple the two

solvers together to form an unsteady FSI solver. These steps include: transfer of

boundary information; grid-regeneration, and the arbitrary Lagrangian-Eulerian (ALE)

99

CHAPTER 6. SOLID SOLVER DEVELOPMENT

method to account for movement of the fluid mesh.

6.2 Discretization of the Displacement Equations

The finite-volume discretization of the displacement equations is carried out in broadly

the same manner as described for the fluid flow equations in the previous Chapter.

Additional terms appear in the displacement equations as the divergence of the dis-

placement vector is non-zero; unlike the fluid equations, where the assumption of in-

compressibility was made.

To illustrate how the discretization is carried out, recall that the steady state, two-

dimensional stress equilibrium equation in the x-direction can be written:

∂σxx

∂x+

∂τxy

∂y= 0 (6.1)

In this example, the body force, fx, is assumed to be zero. The derivation of the

discretized equations for a rectangular Cartesian cell begins with the integration of the

above equation over a finite-volume cell to give:

∫∫

V

(∂σxx

∂x+

∂τxy

∂y

)

dx dy = 0 (6.2)

∫

S

σxx dy

e

w

+

∫

S

τxy dx

n

s

= 0 (6.3)

σxx,e∆ye − σxx,w∆yw + τxy,n∆xn − τxy,s∆xs = 0 (6.4)

The stresses appearing in the above equation are related to the displacements ac-

cording to the linear-elastic constitutive relations:

σxx,e = (2µ + λ)e∂u

∂x

∣∣∣∣e

+ λe∂v

∂y

∣∣∣∣e

− (3λ + 2µ)eαe∆Te (6.5)

σxx,w = (2µ + λ)w∂u

∂x

∣∣∣∣w

+ λw∂v

∂y

∣∣∣∣w

− (3λ + 2µ)wαw∆Tw (6.6)

τxy,n = µn

(∂u

∂y

∣∣∣∣n

+∂v

∂x

∣∣∣∣n

)

(6.7)

τxy,s = µs

(∂u

∂y

∣∣∣∣s

+∂v

∂x

∣∣∣∣s

)

(6.8)

The displacement gradients at the cell faces are approximated using central differ-

ence formulae in an analogous manner to the diffusion terms in the fluid governing

equations:

100

6.2. DISCRETIZATION OF THE DISPLACEMENT EQUATIONS

σxx,e ≈ (2µ + λ)e

(uE − uP

∆xe

)

+ λe

(vne − vse

∆ye

)

−(3λ + 2µ)eαe∆Te (6.9)

σxx,w ≈ (2µ + λ)w

(uP − uW

∆xw

)

+ λw

(vnw − vsw

∆yw

)

−(3λ + 2µ)wαw∆Tw (6.10)

τxy,n ≈ µn

(uN − uP

∆yn

)

+ µn

(vne − vnw

∆xn

)

(6.11)

τxy,s ≈ µs

(uP − uS

∆ys

)

+ µs

(vse − vsw

∆xs

)

(6.12)

The subscripts ne, nw, se and sw refer to the four vertices of the cell; these terms

did not appear in the discretization of the diffusive terms of the fluid equations due to

the assumption of incompressibility, as discussed earlier. The relations above can be

substituted into the discretized equilibrium equation (Equation 6.4), and arranged into

the familiar form:

AP uP =∑

i=N,S,E,W

(Aiui) + Su (6.13)

where the coefficients are given by:

AP =∑

i=N,S,E,W

Ai (6.14)

AE = (2µ + λ)e∆ye

∆xe(6.15)

AW = (2µ + λ)w∆yw

∆xw(6.16)

AN = µn∆xn

∆yn(6.17)

AS = µs∆xs

∆ys(6.18)

and the source term is given by:

101


Su =∑

i=n,s,e,w

Si (6.19)

Se = λe(vne − vse) − ∆ye(3λ + 2µ)eαeTe (6.20)

Sw = −λw(vnw − vsw) + ∆yw(3λ + 2µ)wαwTw (6.21)

Sn = µn(vne − vnw) (6.22)

Ss = −µs(vse − vsw) (6.23)

The above decomposition of the source terms allows for easier prescription of bound-

ary conditions - this is described in more detail in Section 6.6. The discretized form of

the v-displacement equation is derived in exactly the same manner as the u-displacement

equation. Extension of the above to non-orthogonal grids is carried out in exactly the

same manner as for the discretization of the fluid equations.

6.3 Temporal Discretization

The unsteady displacement equations feature a second derivative with respect to time

(unlike the fluid equations, which have a first derivative with respect to time). To

account for transient effects, the transient term must be included in the governing

equations and an additional integration in time must be performed:

t+∆t∫

t

∫∫

V

ρ∂2u

∂t2dV dt +

t+∆t∫

t

∫∫

V

(∂σxx

∂x+

∂τxy

∂y

)

dV dt +

t+∆t∫

t

∫∫

V

fx dV dt = 0 (6.24)

The transient term appearing on the left hand side of the above equation can be

integrated in time to give:

t+∆t∫

t

ρ∂2u

∂t2dt =

[

ρ∂u

∂t

]t+∆t

t

(6.25)

The transient integral appearing in Equation 6.24 can be approximated by:

t+∆t∫

t

∫∫

V

ρ∂2u

∂t2dV dt ≈ ρ(uP − 2uo

P + uooP )∆V (6.26)

where uP represents the u-displacement at the new time, (t + ∆t); uoP represents the

displacement at the current time, t; uooP represents the displacement at the old time, (t−

∆t); and ∆V is the volume of the cell. This requires the storage of the displacement field

102

6.4. STORAGE ARRANGEMENT IN SOLID SOLVER

at three instances in time. The terms on the right hand side of the discretized equation

are treated explicitly, that is, evaluated at the current time, t. The explicit scheme was

chosen for its simplicity. The discretized form of the unsteady u-displacement equation

can be written:

(

AoP +

ρ∆xP ∆yP

(∆t)2

)

uoP =

∑

i=N,S,E,W

(Aiuoi )+

(

Su + uoP

2ρ∆xP ∆yP

(∆t)2− uoo

P

ρ∆xP ∆yP

(∆t)2

)

(6.27)

6.4 Storage Arrangement in Solid Solver

The solid solver uses a slightly different storage arrangement to the fluid solver. Fig-

ure 6.1 shows the storage arrangement used in the solid solver. The crosses represent the

grid points, whilst the circles represent computational nodes. Unlike the fluid solver,

which uses the grid points to define the vertices of the cell, the solid solver uses the

grid points to define the storage nodes. The vertices of the solid cell are located at the

geometric centre of the four neighbouring grid points. For example, the coordinates of

the north-east vertex of a solid cell are given by:

xne =1

4(xP + xE + xN + xNE) (6.28)

yne =1

4(yP + yE + yN + yNE) (6.29)

This means that the solid node, at which the displacement vector is stored, does

not necessarily lie at the geometric centre of the cell, if a non-uniform grid is used.

The above difference in storage location does not impact the grid generation process

for the coupled FSI solver. The simulations are still performed on a single numerical

grid which covers both the fluid and solid sub-domains, the appropriate finite-volumes

are simply constructed from the grid nodes according to whether a solid or fluid sub-

domain is being considered. Within each sub-domain, the finite-volume cells will be

constructed accordingly.

The reason for choosing a different storage arrangement within the solid sub-domain

is to locate nodes along the boundaries of the solid domain. This means that the

displacement of the solid boundaries is calculated directly, rather than extrapolated

from the interior nodes (as would be the case if the same storage arrangement as in

the fluid solver were used). This allows for a much more accurate evaluation of the

vitally important displacement of the fluid-solid interface in a coupled FSI simulation.

The overall accuracy of an FSI simulation depends upon the correct calculation of the

103


interface displacement.

Unlike the fluid storage arrangement, the solid solver does not store the material

properties and geometric quantities at the cell centre. Instead, material properties

are stored at the cell vertices. Recall that the vertices of the solid cells are in the

same location as the centres of the fluid cells (at which fluid material properties are

stored); this means that material properties are stored at consistent geometric locations

throughout the fluid-solid domain.

Interpolation of the displacement vector to the cell face centres and cell vertices is

greatly simplified by the above storage, for example:

ue =1

2(uP + uE) (6.30)

une =1

4(uP + uE + uN + uNE) (6.31)

Interpolation of the material properties to the cell face centres is equally simple:

ρe =1

2(ρne + ρse) (6.32)

i = 1 i = 2 i = NI−1 i = NI

j = 1

j = NJ

mPmW mE

mS

mN

ew

n

s

Figure 6.1: Solid storage arrangement.

6.5 Delta Formulation

The solid solver is implemented not by solving for the nodal displacements, but by

solving for the incremental changes to the displacement. The incremental changes to

the displacement, herein referred to as delta displacements, are defined by:

104

6.6. BOUNDARY TREATMENT

u = utot + ∆u (6.33)

v = vtot + ∆v (6.34)

where the subscript “tot” refers to the total displacement calculated at the previous

iteration. Once the equilibrium equations are satisfied, the calculated delta displace-

ments will be zero, and additional iterations will not alter the total displacement. At

this point the solution is said to have converged.

The decomposition of the displacement given above can be substituted into the

discretized displacement equation to give:

AP (uP,tot + ∆uP ) =∑

i=N,S,E,W

[Ai(ui,tot + ∆ui)] + Su (6.35)

which can be re-arranged to give:

AP ∆uP =∑

i=N,S,E,W

(Ai ∆ui) + S∆u (6.36)

where the new source term includes the additional terms arising from the total defor-

mation:

S∆u = Su +∑

i=N,S,E,W

(Ai ui,tot) − AP uP,tot (6.37)

When the displacement field satisfies the equilibrium equation, these additional

terms cause the source to become zero; thus, the calculated delta displacement will be

zero when the displacement field satisfies the equilibrium equation. It was considered

preferable to formulate a numerical method in the above form in which the calculated

variable is driven to zero, as it is likely to be more stable and easier to determine when

convergence is achieved.

At the conclusion of each solid solver iteration, the delta displacement is added

to the total displacement at each computational node within the domain. The cell

vertex values of the displacement arising from the cross derivatives, which arise during

the discretization of the displacement equations over non-orthogonal cells, are treated

implicitly by using the value of the total displacement at the previous iteration.

6.6 Boundary Treatment

The storage arrangement used by the solid solver was chosen so that computational

nodes were positioned along the solid domain boundaries. This means that along the

105


boundaries half-cells exist of the form shown in Figure 6.2:

mW mP

mN

mEw e

n

?

σ∗

yy,s

τ∗

xy,s

Figure 6.2: Solid half-cell along south boundary.

Similar half-cells exist along each of the domain boundaries. Quarter-cells exist at

the four corners of the domain.

Three types of boundary condition can be specified: traction, displacement and

symmetry. Implementation of these boundary conditions are described in detail in the

proceeding Sections. In the following examples, the boundary conditions are specified

along the south boundary.

6.6.1 Traction Boundary Conditions

Traction boundary conditions are specified by inserting the known boundary traction

(σ∗

yy,s and τ∗

xy,s) into the discretized form of the stress equilibrium equation (6.4):

σxx,e∆ye − σxx,w∆yw + τxy,n∆xn − τ∗

xy,s∆xs = 0 (6.38)

When discretizing this equation over the south boundary cell, the stresses on the

north, east and west faces are replaced by the displacements according to the consti-

tutive relations, as in the previously described case of an internal cell. Once this is

complete, the discretized equation can be written in the familiar form:

AP ∆uP =∑

i=N,E,W

(Ai∆ui) + S∆u (6.39)

To implement the traction boundary condition the south coefficient, AS , and the

south component of the source term, Ss, are now set to zero, and the desired trac-

tion force is added explicitly to the source term S∆u. This demonstrates the value of

decomposing the source term into cell face contributions. Traction is specified along

the other boundaries in exactly the same manner. The more general case of applying

106

6.6. BOUNDARY TREATMENT

traction boundary conditions on boundary-fitted non-orthogonal grids is presented in

the Appendix.

6.6.2 Displacement Boundary Conditions

Displacement boundary conditions are prescribed by setting the required value of the

displacement along the boundary and then ensuring that the calculated delta displace-

ment is zero along the boundary. This will ensure that the value of the displacement

along the boundary will remain unchanged throughout the simulation. The discretized

form of the delta u-displacement equation at the boundary cell is given by:

(AP − SP ) ∆uP =∑

i=N,S,E,W

(Ai ∆ui) + S∆u (6.40)

The delta displacement is set to zero by modifying the source terms as follows:

S∆u = 0 (6.41)

SP = −1030 (6.42)

The result of this is that the discretized equation effectively becomes:

(1030)∆u = 0 (6.43)

the solution to which is, of course, ∆u = 0.

6.6.3 Symmetry Boundary Conditions

Symmetry boundary conditions are prescribed by setting the wall normal displacement

to zero, and setting the wall normal gradient of the wall parallel displacement to zero.

For example, prescribing a symmetry boundary condition along the south boundary

would be done by:

∆v = 0∂u

∂y

∣∣∣∣s

= 0 (6.44)

The displacement gradient appears in the shear stress on the south boundary:

τxy,s = µs

(uP − uS

∆ys

)

︸︷︷︸

∂u∂y

˛

˛

˛

s=0

+ µs

(vse − vsw

∆xs

)

(6.45)

The result of setting the under braced term to zero is that the south coefficient of the

107


discretized equation, AS , is set to zero. All other coefficients and source terms remain

unchanged.

The corresponding symmetry conditions along a boundary normal to the x-axis are:

∆u = 0∂v

∂x

∣∣∣∣= 0 (6.46)

6.7 Solution of the Discretized Equations

The discretized displacement equations are solved in a very similar manner to the

discretized fluid equations. The source term is linearized in exactly the same manner

to increase the diagonal dominance of the coefficient matrix and thus increase the rate

of convergence. Due to its simplicity and efficiency, the discretized equations are solved

using the TDMA. Under-relaxation in used to improve the stability of the solver.


The stability of the iterative solution is increased through the use of under-relaxation.

The delta-form of the displacement equations are under-relaxed in a similar manner to

the pressure correction equation, updating the total displacement as:

u = uold + α ∆u (6.47)

where α is the under-relaxation factor. Since the displacement equations are linear,

higher under-relaxation factors can be used than are typically found in the fluid solver.

6.7.2 The TDMA Solver

The TDMA solver is used to solve the discretized displacement equations. A slight

modification had to be made to the implementation of this as used for the fluid solver,

as now the calculation occurs at both internal and boundary cells, whereas the fluid

TDMA only calculates at the internal cells.

Since the delta displacements are reset to zero at the beginning of each iteration,

it was found that faster convergence could be achieved by increasing the number of

TDMA sweeps per iteration; typically four sweeps are performed per iteration. This is

consistent with what is typically done for the pressure correction equation.

6.8 Calculation of the Stress Components

The governing equations were written in terms of displacements, thus their solution

yields a set of nodal displacement components ui. Whist displacements are useful

108

6.8. CALCULATION OF THE STRESS COMPONENTS

for describing the deformation of a body, it is also necessary to have the stress field.

The stress field is obtained from the calculated displacement field via the constitutive

relations; calculation of the stresses can be considered as part of the post-processing

stage. The two-dimensional, Cartesian, constitutive relations linking stresses at node

P to nodal displacement gradients are:

σxx,P = (2µ + λ)P∂u

∂x

∣∣∣∣P

+ λP∂v

∂y

∣∣∣∣P

−

(3λ + 2µ)P αP (∆T )P (6.48)

σyy,P = (2µ + λ)P∂v

∂y

∣∣∣∣P

+ λP∂u

∂x

∣∣∣∣P

−

(3λ + 2µ)P αP (∆T )P (6.49)

τxy,P = µP∂u

∂y

∣∣∣∣P

+ µP∂v

∂x

∣∣∣∣P

(6.50)

At the interior grid points, the displacement gradients appearing above are replaced

by central differences to give:

σxx,P = (2µ + λ)P

(ue − uw

∆xP

)

+ λP

(vn − vs

∆yP

)

−

(3λ + 2µ)P αP (∆T )P (6.51)

σyy,P = (2µ + λ)P

(vn − vs

∆yP

)

+ λP

(ue − uw

∆xP

)

−

(3λ + 2µ)P αP (∆T )P (6.52)

τxy.P = µP

(un − us

∆yP

)

+ µP

(ve − vw

∆xP

)

(6.53)

where the subscript P refers to nodal values, and the subscripts e, w, n and s refer to

the east, west, north and south face centres respectively; ∆xP and ∆yP refer to the

width and height of the solid cell.

Nodal values of material properties have to be interpolated since they are stored at

the corners of the cell; for example, the nodal value of µ is given by:

µP =1

4(µne + µse + µnw + µsw) (6.54)

The face centre values of displacements also have to be interpolated from the nodal

values. Since the solid arrangement has faces which lie exactly half way between the

nodes the interpolation is very simple; for example, the value of u on the east face is

given by:

ue =1

2(uP + uE) (6.55)

109


Along the boundaries, one-sided differences have to be used to represent the dis-

placement gradients; for example, along the south boundary, the discretized constitutive

relations are:

σxx,P = (2µ + λ)P

(ue − uw

∆xP

)

+ λP

(vn − vP

∆yP

)

−

(3λ + 2µ)P αP (∆T )P (6.56)

σyy,P = (2µ + λ)P

(vn − vP

∆yP

)

+ λP

(ue − uw

∆xP

)

−

(3λ + 2µ)P αP (∆T )P (6.57)

τxy.P = µP

(un − uP

∆yP

)

+ µP

(ve − vw

∆xP

)

(6.58)

where ∆yP is the height of the boundary half-cell and P refers to the node positioned

on the boundary.

6.9 Validation of the Solid Solver

Three test cases have been studied to validate the accuracy of the newly developed

solid solver. In each case, the numerical results have been compared with analytical

solutions. This series of tests involve: complex non-orthogonal geometries; planar and

axisymmetric geometries; all three types of boundary condition (traction, displacement

and symmetry), and transient displacements.

These tests do not include the effects of FSI, since it was necessary to validate the

accuracy of the method for purely solid body problems before the more complex FSI

simulations were undertaken.

6.9.1 Planar Validation Exercise

The first test case consists of a flat plate with a circular hole in the centre subject to

uni-axial tension. The purpose of this test is to evaluate the ability of the method to

correctly predict a complex stress field in a planar geometry. Additionally, the highly

non-orthogonal mesh in the vicinity of the hole will check that the non-orthogonal terms

have been correctly implemented into the solver.

Analytical Solution

This test case is the same as that performed by Demirdzic and Muzaferija (1994) who

give the analytical solution as:

110

6.9. VALIDATION OF THE SOLID SOLVER

σxx = fx

[

1 − a2

r2

(3

2cos 2θ + cos 4θ

)

+3

2

a4

r4cos 4θ

]

(6.59)

σyy = fx

[

−a2

r2

(1

2cos 2θ − cos 4θ

)

− 3

2

a4

r4cos 4θ

]

(6.60)

τxy = fx

[

−a2

r2

(1

2sin 2θ + sin 4θ

)

+3

2

a4

r4sin 4θ

]

(6.61)

where fx is the magnitude of the tensile force applied to the plate; σxx is the tensile

stress in the x-direction; σyy is the tensile stress in the y-direction; τxy is the in-plane

shear stress; a is the radius of the hole, and (r, θ) are polar coordinates with origin at

the centre fixed at the centre of the hole.

Physical Geometry and Material Properties

The geometry consists of a 4m by 4m plate with a circular hole of radius 0.5m located

in the centre. A tensile force of magnitude 10 kPa acts in the x-direction. Due to

the symmetrical nature of the problem, only one quarter of the geometry needs to be

simulated. This reduces the computational expense of the simulation and simplifies the

prescription of boundary conditions. In this simulation, the north-east quarter of the

plate was modelled. A schematic of the geometry and boundary conditions is shown in

Figure 6.4.

The plate was made of steel with elastic modulus, E = 107 Pa, and Poisson ratio,

ν = 0.3. For a steady-state simulation it is unnecessary to specify the density of the

material.

The elliptically generated mesh, which is shown in Figure 6.3, consisted of 45 nodes

in the x-direction and 45 nodes in the y-direction. The nodes were distributed uniformly

along each boundary.

Boundary and Initial Conditions

Symmetry conditions were prescribed along the west and south boundaries. Zero trac-

tion was prescribed along the circumference of the hole. To account for the finite

dimensions of the plate, the analytical solution for the stresses was applied as traction

boundary conditions along the north and east boundaries. This allows for a direct

comparison between the analytical solution and numerical result.

The displacement field was initialized to zero.

111


X (m)

Y(m

)

0 0.5 1 1.5 2

0

0.5

1

1.5

2

Figure 6.3: Numerical mesh used for the planar validation exercise.

Results and Discussion

The comparison between the numerical and analytical stress field is shown in Figure 6.5.

It can be seen that for all three components of stress there is good agreement. The

peak σxx stress occurs at the northernmost point of the hole. This peak is accurately

captured, despite the relative coarseness of the grid in this region. The σyy stress

field is slightly more complex; it consists of a compressive region at the easternmost

point of the hole, and a tensile region at the northernmost point. The tensile region

is not as accurately predicted as the compressive region; however, a similar finding is

shown in the results of Demirdzic and Muzaferija (1994). Another difference occurs at

the north-eastern boundary of the domain, where the numerical results show a small

region of high stress. The τxy field is similar to the σyy field, in that it has both tensile

and compressive regions and there is very good agreement between the numerical and

analytical solutions.

This test case has proven the ability of the method to accurately predict complex

stress fields when traction and symmetry boundary conditions are prescribed. The

method has also proven to be capable of handling highly non-orthogonal numerical

grids.

112


-

6

?

1.5 m

1.5

m

R=

0.5m

Symmetry

Zero Traction

Sym

met

ry

Applied

Tra

ctio

n

f x=

10kPa

-

-

-

-

-

-

-

-

-

Figure 6.4: Schematic of planar test case.

6.9.2 Axisymmetric Validation Exercise

The second test case consists of an infinitely long cylinder subject to internal pressure.

The purpose of this test was to evaluate the ability of the solver to predict the stress

field in an axisymmetric test. The solver was implemented in such a way as to allow a

simple switch between planar and axisymmetric geometries; this test will ensure that

implementation was correct.

Analytical Solution

The analytical solution for an infinitely long cylinder subject to internal pressure is

given by:

σrr = − pi a2

b2 − a2

[b2

r2− 1

]

(6.62)

σθθ =pi a

2

b2 − a2

[b2

r2+ 1

]

(6.63)

ur =pi a2

E(b2 − a2)

[

(1 + ν)b2

r+ (1 − ν)r

]

(6.64)

where σrr is the radial stress; σθθ is the hoop stress; ur is the radial displacement; a

113


6 67 7

8 89 910105 5

44

3 2 1321

LevelSXX:

11000

35000

59000

713000

917000

1121000

1325000

1529000

Analytical Numerical

(a) Normal stress in x-direction, σxx.

11

11

12

12

13 13

14

14

10

10

99

8 8

LevelSYY:

1-9500

3-7500

5-5500

7-3500

9-1500

11500

132500

154500


(b) Normal stress in y-direction, σyy.

10

10

9 9

99

8 8

77 6 6

LevelSXY:

1-8500

3-6500

5-4500

7-2500

9-500

111500


(c) In plane shear stress, τxy.

Figure 6.5: Results from the plate with a hole under uni-axial tension test case. Cal-culated stress field to the right of the vertical line of symmetry, analytical stress fieldshown to the left.

114


and b are in internal and external diameters of the cylinder respectively; and pi is the

difference between the internal and external pressure.


This test replicates that presented by Bijelonja et al. (2006) and consists of a cylinder

of inner radius 8m, outer radius 16m and length 20m. A schematic of the geometry

and boundary conditions is shown in Figure 6.6. A regular Cartesian grid with 40 and

40 nodes in the axial and radial directions respectively was used.

The cylinder material has elastic modulus, E = 1000 MPa, and Poisson ratio,

ν = 0.3. Prescription of the material density is not required in a steady-state simulation.

-20 m

ra = 8 m

rb = 16 m

Sym

met

ry

Sym

met

ry

Zero Traction

Applied Traction

pi = −10 MPa? ? ? ? ? ? ? ? ?

Figure 6.6: Schematic of axisymmetric test case.


Symmetry conditions were prescribed along the east and west boundaries to allow

comparison with the analytical solution for a cylinder of infinite length. Zero traction

was prescribed along the north boundary whilst a normal traction of magnitude, pi =

−10 MPa, was prescribed along the south boundary.

The displacement field was initialized to zero.


It can be seen from Figure 6.7 that both the radial stress and displacement profiles

have been accurately predicted through the thickness of the cylinder. This has proven

the method’s ability to accurately solve axisymmetric problems.

115


Radius (m)

Rad

ialS

tres

s,σ rr

(Pa)

8 9 10 11 12 13 14 15 16-1E+07

-8E+06

-6E+06

-4E+06

-2E+06

0

AnalyticalNumerical

(a) Radial stress, σrr.

Radius (m)

Rad

ialD

ispl

acem

ent,

u r(m

)

8 9 10 11 12 13 14 15 160.09

0.1

0.11

0.12

0.13

0.14

0.15

0.16

AnalyticalNumerical

(b) Radial displacement, ur.

Figure 6.7: Radial profiles of the radial stress and displacement in the cylinder underinternal pressure test case.

116


6.9.3 Unsteady Validation Exercise

The third test case consisted of the undamped vibrations of a short, two-dimensional

beam with an applied shear stress at its free end. Unlike the previous tests, this involves

time-dependant displacements. The purpose of this test is to ensure that the temporal

terms have been correctly implemented into the solver.

Analytical Solution

The analytical solution for the mean end displacement, δ, of the undamped vibration

of a short, fixed-free beam is given by Greenshields and Weller (2005) as:

δ =4τ

E

l3

h2(1 − ν2)

[

1 +3

4(1 + ν)

(h

l

)2]

(6.65)

and the frequency of oscillation, ωn, is:

ωn = µ2n

√

E

12ρs

h

l2(6.66)

where E, ρs and ν are the elastic modulus, density and Poisson ratio of the beam

material respectively; h and l and the beam height and length respectively; τ is the

applied shear stress on the free end of the beam, and µn is the eigenvalue for the nth

mode of bending. In the case of first mode bending µ1 = 1.875.


The beam was rectangular with length 20m and height 5m. The elastic modulus of

the beam material was E = 4 GPa, the Poisson ratio was ν = 0.32 and the density

was ρs = 1450 kg/m3. A schematic of the geometry and boundary conditions is shown

in Figure 6.8. A regular Cartesian grid was used, and grid refinement tests will be

discussed later.


Zero traction conditions were prescribed along the north and south boundaries. Zero

displacement conditions were prescribed along the west boundary to represent the fixed

end of the beam. At time t = 0 seconds, a shear stress, τ = 1 MPa, was applied to the

east boundary.

The displacement and velocity of the beam were initialized to zero.

117


?τ = 1 MPa

-20 m

?

6

5 m

Zero Traction

Zero Traction

Fix

ed

Applied

Tra

ctio

n

Figure 6.8: Schematic of unsteady test case.


Figure 6.9 shows the results from a grid dependency test. The results, in terms of time

history of the beam tip displacement, obtained from a 320 × 80 grid and a 160 × 40

grid are almost indistinguishable. In both cases, the time step was equal to 200× 10−6

seconds. The agreement between the numerical and analytical solutions is also good,

however, the frequency and amplitude of oscillation have been slightly under-predicted.

Figure 6.10 shows the results from a time step dependency test. In each case,

the 320 × 80 grid was used based upon the grid dependency test. The frequency of

oscillation was accurately predicted with each time step. However, the results obtained

from the largest time step (∆t = 2000× 10−6 seconds) show an under-prediction of the

amplitude due to artificial damping. For time steps smaller that 200×10−6 seconds no

change is seen in the results.

6.10 Application to Grid Generation

As indicated earlier, In addition to solving elasticity problems, the newly developed

method can be used to generate new, or adapt existing, numerical meshes. For FSI

problems, where frequent re-meshing of the domain may be required as the geometry

changes, this route of employing the solid solver methodology to adapt the grid is

considered particularly advantageous. To apply the method to grid generation, the

numerical mesh is considered to be a solid body with arbitrary material properties.

Displacements are prescribed along the boundaries to deform the solid body to the

desired geometry.

In practise, an initial mesh is generated algebraically, which broadly represents the

desired geometry. Grid points can be clustered around areas with expected large gra-

dients, such as the near-wall region in a turbulent flow calculation. Next, displacement

118

6.10. APPLICATION TO GRID GENERATION

Time (seconds)

End

Dis

plac

emen

t(m

)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8AnalyticalNumerical - 160x40Numerical - 320x80

Figure 6.9: Effect of grid refinement on time history of beam tip displacement.

Time (seconds)

End

Dis

plac

emen

t(m

)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8AnalyticalNumerical - ∆t = 20 msNumerical - ∆t = 200 msNumerical - ∆t = 2000 ms

Figure 6.10: Effect of time step refinement on time history of beam tip displacement.

119


boundary conditions are specified along all boundaries to deform the initial grid into

the desired final shape.

The principle advantage of this form of grid generation compared to other methods,

such as elliptic grid generators, is that it preserves the internal node distribution. For

example, near-wall node clustering in the initial mesh will be preserved in the final mesh.

Cells located near curved boundaries will be more orthogonal than those produced by

an elliptic generator; this will improve the accuracy of the simulation performed using

the grid.

6.11 Fluid-Structure Coupling

Once the newly developed solid solver had been verified in isolation, it was coupled to

the existing CFD code STREAM. This task was greatly simplified by the care taken

to implement the solid solver in a manner consistent with the fluid solver.

The details of how the two solvers are coupled together are presented in this Section.

The storage arrangement and transfer of data between the two solvers have been chosen

to maximise efficiency and accuracy of the coupled FSI solver. The additional steps

required to solve transient FSI problems are also described. The general procedure for

solving coupled FSI problems is shown in Figure 6.11.

6.11.1 Storage Arrangement

A key feature of the coupled FSI solver is that a single numerical mesh covers both fluid

and solid sub-domains. The storage arrangement is shown schematically in Figure 6.12.

The crosses represent the grid points from which the fluid and solid finite-volumes are

constructed over their respective sub-domains. The interface between the two regions

is indicated by the index j = JFSI. In the following, uppercase indices refer to solid

nodes whilst lowercase indices refer to fluid nodes.

Within the fluid sub-domain, the grid points are used to define the vertices of the

finite-volume cells; the computational node at which flow variables and fluid properties

are stored is located at the geometric centre of the cells. Convective fluxes are stored

at the face centres of the cell. Along the boundaries of the total domain (i = 1,

i = NI, j = 1), additional nodes are located to allow the prescription of the boundary

conditions. This means that, for regions having the same number of grid nodes, the

number of fluid nodes is one greater than the number of solid storage nodes. Fluid

nodes are not located along the fluid-solid interfaces within the interior total domain.

Within the solid sub-domain, the grid points are used to define the location of the

solid nodes. The vertices of the solid finite-volume cells are positioned at the geometric

centre of the four surrounding nodes (in two-dimensions). Computational nodes lie

along all boundaries of the solid sub-domain. This removes the need to interpolate the

120

6.11. FLUID-STRUCTURE COUPLING

Step 1: Read in mesh and construct fluid and solid volumes accord-ingly.

Step 2: Solve the Navier-Stokes equations over fluid sub-domain.This yields the stresses acting along the fluid-structure interface.

Step 3: Solve elasticity equations over solid sub-domain. This yieldsthe displacements in the solid sub-domain.

Step 4: Adjust the fluid mesh according to the calculated displace-ments along the fluid-structure interface.

Step 5: Check for FSI convergence within time step.

Step 6: Update time and check for steady-state solution.

Start

Stop

?

?

?

?

?

?

?

No

Yes

No

Yes

Figure 6.11: Coupled FSI Procedure.

121


i = 1 i = 2 i = ni−1 i = ni

I = 1 I = 2 I = NI−2 I = NI−1

j = 1

j = 2

j = jfsi−1

J = JFSI

J = NJ

yy=

=

=

Solid Node

Fluid Node

Grid Point

Figure 6.12: Schematic of FSI storage arrangement.

displacement from the interior to the boundary of the domain. Only the displacements

and stresses are stored at the solid nodes; solid material properties are stored at the

cell vertices. This means that material properties (both fluid and solid) are stored at

consistent locations throughout the entire domain.

6.11.2 Calculation of Forces on Interface

Solution of the fluid governing equations yields the fluid pressure and velocities at each

fluid node shown in Figure 6.12. The flow field is used to calculate the normal and

tangential tractions which act upon the fluid-solid interface. These are then applied as

traction boundary conditions to the solid solver which calculates the displacements at

each of the solid nodes.

Wall Pressure

The fluid pressure acts normal to the interface. Since a fluid node does not lie on

the interface, an assumption has to be made regarding the relationship between the

interface pressure and the pressure at the near-wall node. A zero pressure gradient is

assumed in the direction normal to the wall:

122

6.11. FLUID-STRUCTURE COUPLING

∂P

∂xn

∣∣∣∣wall

= 0 (6.67)

Therefore:

PJFSI = Pjfsi−1 (6.68)

The solid nodes are aligned with the faces of the fluid cells, thus interpolation is

needed to calculate the pressure at the location of the solid node of the form:

PI = Pi−1 + fx (Pi − Pi−1) (6.69)

These can be combined to give the pressure along the interface:

P(I,JFSI) = P(i−1,jfsi−1) + fx (P(i,jfsi−1) − P(i−1,jfsi−1)) (6.70)

where fx is the spatial interpolation factor.

Wall Shear Stress

The wall shear stress acts tangentially to the interface. It is calculated at the near-wall

fluid nodes (j = jfsi − 1) by assuming a zero velocity at the wall (no-slip condition):

τwall =∂Ut

∂xn

∣∣∣∣wall

(6.71)

where Ut represents the wall tangential velocity and xn the wall normal coordinate.

The use of the no-slip condition is valid in the case of a steady-state FSI simulation

because the velocity of the interface is zero once a steady state has been obtained; the

intermediate stages in a steady-state simulation are considered to be a number steady

simulations for which the velocity of the interface is zero. In the unsteady case, the

velocity of the interface is accounted for through the use of the Arbitrary Lagrangian-

Eulerian (ALE) method, which is described later. Interpolation is again required to

calculate the wall shear stress at the solid node location:

τI = τi−1 + fx (τi − τi−1) (6.72)

6.11.3 Adaptation of Fluid Mesh

Solution of the elasticity equations, with traction boundary conditions corresponding

to the viscous and pressure forced imparted by the fluid along the interface, leads to the

displacement vector being known at every node within the solid sub-domain. These

calculated displacements are then added to the nodal coordinates, to determine the

123


new geometry shape.

For the effect of the displacement along the interface to be felt by the fluid solver,

the numerical mesh covering the fluid sub-domain must be adapted. This is achieved

by exploiting the ability of the solid solver to adapt existing grids by treating them

as solid bodies as described earlier. The elasticity equations are therefore solved over

the fluid sub-domain with the calculated displacement along the interface applied as

a boundary condition; zero displacement boundary conditions are applied along the

other, outer, domain boundaries. Pseudo-material properties have to be specified for

the fluid sub-domain. Typically, these properties are chosen to be the same as those

in the solid sub-domain. However, the pseudo-material properties of the fluid sub-

domain can be varied in order to improve the quality of the fluid mesh after adaption.

The displacements calculated within the fluid sub-domain are then added to the nodal

coordinates to produce the deformed grid. This procedure allows for the fluid mesh to

be deformed throughout the course of an FSI simulation, whilst retaining the initial

internal node distribution, such as near-wall clustering. This greatly increases the

accuracy of the coupled FSI simulation.


In order to limit the deformation which occurs during each FSI iteration, and thus

prevent oscillatory or unstable behaviour, it is necessary to use under-relaxation. Only

a small fraction of the displacement calculated within an FSI iteration is added to the

total displacement. This fractional change in the displacement is used to move the grid

points in both solid and fluid sub-domains. The total displacement is thus updated at

the end of each FSI iteration according to:

utot = uold + α(u − uold) (6.73)

where utot represents the total u displacement; uold represents the u displacement at

the beginning of the particular FSI iteration, and α is the FSI under-relaxation factor.

Typically, the FSI under-relaxation factors employed lie in the range 0.3 < α < 0.6.

6.11.5 Convergence Criteria

Convergence is said to be achieved when additional FSI iterations cause no further

change to either fluid or solid solutions. This condition is implemented by counting

the number of fluid and solid inner-iterations within each FSI iteration; when they

are both zero, the coupled FSI simulation is said to have converged. In the case of a

transient simulation, the FSI simulation is said to have converged at the current time;

upon convergence, the proceeding time step is calculated.

124

6.12. SUMMARY

6.11.6 Unsteady Coupled Analysis

The discretization of the transient governing equations for both fluid flow and solid

deformation have been presented earlier. However, to perform transient coupled FSI

simulations, some additional steps need to be taken, other than simply including the

transient terms, because of the deforming mesh. The fluid equations were derived with

respect to a stationary, or Eulerian, reference frame; in the case of small deformation,

the elasticity equations are equivalent regardless of the reference frame. To account

for the movement of the fluid mesh during a coupled FSI simulation, the ALE method

has been used whereby the convective flux through the cell face in the fluid region is

calculated relative to the velocity of the cell faces:

∂(ρUi)

∂t+

∂

∂xj[ρ(Ui − U i

g)Uj ] = − ∂P

∂xi+

∂

∂xj

(

µ∂Ui

∂xj

)

(6.74)

where U ig is the velocity in the xi-direction of the finite-volume cell faces. The ALE

method is implemented by storing the relative convective flux at the cell faces. For

example, the convective mass flux in the x-direction at the east face of an orthogonal

cell is given by:

me = ρe(Ue − Ug,e)∆y (6.75)

where the face value of the fluid velocity is found by linear interpolation between the

neighbouring nodes:

Ue = UP + fx (UE − UP ) (6.76)

The velocity of the grid nodes (calculated during the mesh adaptation) is stored at

the vertices of the fluid-cells, and hence the cell face value of the grid velocity is:

Ug,e =1

2(Ug,ne + Ug,se) (6.77)

6.12 Summary

This Chapter has described the development and implementation of a finite-volume

based solid body solver. The use of the same numerical method, implemented in a

consistent manner, across both fluid and solid sub-domains, provides the most efficient

method of simulating FSI problems, especially those including transient effects, as no

software interface is required to transfer data between the different regions.

A novel storage arrangement was used which allows a single numerical grid to cover

both fluid and solid regions, without the need to interpolate to find the vitally important

125


displacement at the fluid-solid interface, as it is calculated directly. The solid storage

arrangement also greatly simplifies the implementation of the ALE method to allow

transient FSI simulations to be performed.

The solid solver is able to generate and adapt numerical grids by treating them

as solid bodies with arbitrary material properties. The benefit of this approach over

other, for example elliptic, methods is that the interior node distribution is preserved.

This means that the near-wall grid clustering of the fluid mesh is preserved throughout

the course of a FSI simulation.

A number of test cases were studied to verify the accuracy of the newly developed

solid solver before it was coupled to an existing flow solver. These tests proved the

ability of the method to correctly predict the complex stress field in a planar geometry,

the radial displacement of an axisymmetric geometry and the time variation of the end

displacement of an undamped beam.

126

Chapter 7

Pipe Flows

7.1 Introduction

Before more physiologically relevant flows can be simulated with the newly developed

FSI solver, it is necessary to validate the method for a much simpler flow. The case of

steady laminar flow through an initially straight tube with a compliant wall section was

simulated and the results were compared to the numerical data presented by Shim and

Kamm (2002). Since the fluid mechanics of this flow are relatively simple, particularly

when the wall deformation is small, the overall accuracy of the results will be dictated

mainly by the solid body solver and the fluid-solid coupling. The inlet and external

gauge pressure were held constant at 0 Pa; flow was induced through the tube by

reducing the outlet gauge pressure in 5 Pa increments. The use of gauge pressure

is most appropriate here because an incompressible flow is governed by the relative

pressure drop between the inlet and outlet; unlike the compressible flow case, the

absolute pressure has no effect upon the flow. Initially, a linear reduction in pressure

along the compliant wall section exists due to viscous losses in the fluid. Therefore, the

transmural pressure, and hence radial displacement, is greatest at the downstream end

of the compliant section. As the tube begins to deform at the downstream end of the

compliant section, the transmural pressure difference increases further in this region,

due to flow acceleration through the greater area reduction. As the outlet pressure is

reduced further, the tube deforms more, and eventually the flow separates downstream

of this area reduction.

7.2 Simulation Details

Information regarding geometry, material properties, numerical mesh and boundary

conditions is given in the proceeding Sub-sections.

127

CHAPTER 7. PIPE FLOWS

- - -

?

6 6

?

3.75 D 3.75 D 6.25 D

D = 8 mm

t = 0.4 mm

Figure 7.1: Schematic of straight compliant tube geometry.

7.2.1 Physical Geometry

The geometry consisted of an initially straight tube of diameter 8 mm and length 13.75

diameters. A compliant wall section of length 3.75 diameters begins 3.75 diameters

downstream from the inlet. The compliant wall section had a uniform thickness of 0.4

mm along its length. The geometry is shown schematically in Figure 7.1.

7.2.2 Numerical Mesh

To increase the numerical stability for the cases involving large wall deformation, and

therefore increased re-attachment length, the fluid domain was extended to 6.25 diam-

eters downstream from the end of the compliant wall section. This allowed sufficient

length for the flow to re-attach and undergo at least some re-development towards

fully-developed conditions before the outlet. The numerical mesh consists of 760 uni-

formly spaced nodes in the axial direction and 84 nodes in the radial direction. The

number of nodes in the streamwise direction could almost certainly be reduced from

this number by using non-uniform mesh spacing. This mesh had been carried forward

from an earlier validation simulation - it was considered to be more efficient to simply

re-use this mesh, rather than generate a new mesh. In these two-dimensional cases,

the use of an overly refined mesh does not have a significantly detrimental effect upon

the simulation time. The fluid sub-domain consists of 80 nodes in the radial direction

with expansion ratio Ry = 0.99. The sub-domain covering the compliant wall section

consists of 4 uniformly spaced nodes in the radial direction and 141 nodes in the ax-

ial direction. Although the mesh covering the solid sub-domain extends over the full

length of the fluid sub-domain, the solid solver is only called over the relevant portion

of the mesh. Figure 7.2 shows the numerical mesh around the downstream end of the

compliant wall section.

128

7.2. SIMULATION DETAILS

Outlet (East) Zero streamwise gradient U - and V -velocityPrescribed pressure

Inlet (West) Parabolic (laminar) U -velocity profileZero V -velocityZero pressure

Axis (South) Zero V -velocityZero normal gradient on all other variables

Wall (North) Zero U - and V -velocityZero normal gradient pressure

Table 7.1: Fluid solver boundary conditions.

Rigid Wall (East & West) Zero u- and v-displacementCompliant wall (South) Normal traction equal to fluid pressure

Tangential traction equal to fluid wall-shearCompliant wall (North) Zero normal and tangential traction

Table 7.2: Solid solver boundary conditions.

7.2.3 Boundary Conditions

The boundary conditions prescribed during the simulations are summarised in Ta-

bles 7.1 and 7.2. Further details regarding the implementation of the boundary condi-

tions is given in Chapter 5.

Fixed displacement boundary conditions were specified on all boundaries during the

re-meshing process. Along the West, South and East boundaries, zero displacement

boundary conditions were prescribed to ensure that the fixed boundaries were not

displaced. Along the North boundary, the displacement of the fluid region was equal

to the calculated displacement of the compliant wall section; this was done to ensure

that the fluid and solid sub-domains of the mesh remained in contact.

X / Diameter

R/D

iam

eter

5.6 5.8 6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8

0

0.1

0.2

0.3

0.4

0.5

0.6

Figure 7.2: Initial numerical mesh around the downstream end of compliant wall sec-tion.

129


U-Momentum 0.5V-Momentum 0.5

Pressure Correction 0.2

Table 7.3: Fluid solver under-relaxation factors.

U-Displacement 0.65V-Displacement 0.65

Table 7.4: Solid solver under-relaxation factors.

7.2.4 Material Properties

The working fluid was water with density, ρ = 1.0× 103 kg/m3, and dynamic viscosity,

µ = 1.0 × 10−3Pa s. The compliant wall material had Young’s modulus, E = 2.0 ×103 Pa, and Poisson ratio, ν = 0.3. For the purpose of re-meshing, the fluid mesh was

assigned the following material properties: Young’s modulus, E = 2.0 × 103 Pa, and

Poisson ratio, ν = 0.42. These material properties are arbitrary; values were chosen to

give the highest quality mesh after adaption.

7.2.5 Numerical Implementation

The fluid solver was modified to allow a fixed pressure drop between the pipe inlet and

outlet to be prescribed. This was done, instead of prescribing the mass flow rate, to

avoid numerical issues associated with flows that have multiple solutions for a given

flow rate; as can be the case for flow through a tube with compliant wall. The velocity

field is first calculated for the prescribed pressure drop with an arbitrary inlet flow

rate. At the end of each iteration, the mass flow rate at the outlet is calculated; the

inlet velocity profile is then scaled to ensure mass continuity and the flow field is re-

calculated. This procedure is continued until the convergence criteria are met for mass

and momentum conservation. The MUSCL convective discretization scheme was used

for all variables.

7.2.6 Under-Relaxation Factors

The under-relaxation factors used are summarised in Tables 7.3–7.5. The FSI under-

relaxation factor was α = 0.35.

U-Displacement 0.25V-Displacement 0.25

Table 7.5: Re-meshing algorithm under-relaxation factors.

130

7.3. RIGID TUBE RESULTS AND DISCUSSION

7.3 Rigid Tube Results and Discussion

Initially, the simple case of flow through a rigid tube was considered to ensure that the

CFD solver was correctly predicting the pressure and wall shear stress.

7.3.1 Analytical Solution

The analytical solution for the pressure drop in the case of fully developed, laminar

flow through a straight, rigid tube, is given by the following equation:

∆P = ρghf (7.1)

where ρ is the fluid density, g is gravitational acceleration and hf is the frictional head

loss, given by:

hf = f

(L

D

) (U2

B

2g

)

(7.2)

where L is the pipe length, D is the pipe diameter and UB is the bulk velocity. In the

case of fully developed laminar flow in a straight rigid tube, the friction factor, f , is

given by:

f =64

Re(7.3)

7.3.2 Numerical Results

Figure 7.3 shows a comparison of flow rate for increasing pressure drop between the

analytical solution, the presently computed numerical solution and the numerical data

presented by Shim and Kamm. The results have been normalized with respect to a

reference pressure (Pref = 200 Pa) and volumetric flow rate (Qref = 1.58 × 10−5 m3/s).

The reference values are those quoted by Shim and Kamm in presenting their compliant

tube results, defined as:

Pref = Eh1/R (7.4)

Qref = πR2(Eh1/ρD0)1/2 (7.5)

where E is the Young’s modulus of the compliant wall material, h1 is the thickness at

the upstream end of the compliant wall, R is the un-deformed radius of the tube, ρ is

the fluid density and D0 is the un-deformed diameter of the tube.

The computed results were obtained for the rigid wall case by setting the displace-

ments in the solid sub-domain equal to zero. The results show that the computed

131


∆ P / Pref

Q/Q

ref

0 0.05 0.1 0.15 0.20

0.5

1

1.5

2

2.5

Analytical SolutionNumerical ResultsShimm and Kamm Numerical Data

Figure 7.3: Comparison with the analytical solution for the rigid wall case.

numerical solution is in excellent agreement with the analytical solution over the entire

range of flow rates presented here. Whilst Shim and Kamm did not present any rigid

wall results, at the lowest flow rates shown in Figure 7.4 it can be seen that the wall

deformation is very small and the solution should therefore be close to that of the rigid

wall case. Data from the Shim and Kamm study is therefore presented in Figure 7.3 for

comparison. From this, it can be seen that a large discrepancy exists between the data

of Shim and Kamm and analytical and present rigid wall results. It can also be seen

that a large discrepancy exists between the data of Shim and Kamm and the analyti-

cal solution. Not only is the magnitude and gradient of the pressure-flow relationship

under-predicted, the line deviates from the linear at a much lower flow rate. A number

of explorations were performed in an attempt to ascertain the reason for these discrep-

ancies. After much testing, it was concluded that the discrepancy appeared to be due

to some error in the normalization of the results presented by Shim and Kamm. The

excellent agreement between the present results and the analytical solution confirms

the accuracy of the present approach for the fluid flow computation.

7.4 Compliant Tube Results and Discussion

Figure 7.4 shows the comparisons between the present results and the numerical data

of Shim and Kamm for the fully compliant tube case. Both flow rate and minimum

tube area are plotted against streamwise pressure drop. Pressure and flow rate have

132

7.4. COMPLIANT TUBE RESULTS AND DISCUSSION

been normalized using the quantities given in Equations 7.4 and 7.5; the area has been

normalized with respect to the un-deformed area of the tube:

Aref =πD2

4(7.6)

A large discrepancy exists between the computed and reported flow rate variation

with pressure difference. Both the computed maximum value and gradient of the flow

rate variation are much greater, by a factor of approximately two, than those reported

by Shim and Kamm. The maximum pressure drop achievable is approximately half

of that reported. It was not possible to obtain converged solutions at higher pressure

drops; the fluid portion of the numerical mesh around the downstream end of the

compliant section became highly skewed which lead to numerical instabilities in the

fluid solution. The tests conducted in the rigid wall case proved that the flow solver is

capable of correctly predicting the pressure-flow rate relationship in a laminar flow. The

discrepancy is again put down to incorrect normalization of the results by Shim and

Kamm. By choosing arbitrary normalization factors, rather than considering factors

which ensure dynamic similarity, it is difficult to check their results. Incorrectly set

material properties (such as density and dynamic viscosity) would not be highlighted

in their results.

The gradient of the minimum area with pressure relationship is predicted with

reasonable accuracy. However, the minimum area is over-predicted by a factor of ap-

proximately two. Both the present results and the computations published by Shim and

Kamm (2002) show the same physical trends, in terms of area and flow rate variation

with imposed pressure drop. It is most likely that the discrepancies described earlier

can be attributed to inconsistent post processing of the data in the study by Shim and

Kamm.

The effect of increasing the pressure drop upon the flow field and wall deformation

can be clearly seen from Figures 7.5 – 7.9. At the lowest pressure drop, ∆P = 10 Pa,

the wall deformation is very small; a fully developed laminar flow exists throughout the

tube. The linear pressure drop in the streamwise direction, caused by viscous losses,

means that the transmural pressure difference is greatest at the downstream end of the

compliant wall section. This large transmural pressure difference causes the greatest

negative radial displacement to occur at the downstream end of the compliant wall

section. This behaviour can be seen in Figures 7.10–7.11 which show the axial and

radial displacement profiles for increasing streamwise pressure difference respectively.

At the lowest pressure drop, ∆p = 10Pa, the radial, v, displacement is linear, owing to

the linear streamwise wall pressure variation. The axial, u, displacement is practically

zero, owing to the low wall shear stress. As the imposed pressure difference is increased,

the peak radial displacement remains at the downstream end of the compliant wall

133


∆P/Pref

Q/Q

ref

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

Shim and Kamm Numerical DataNumerical Results

(a) Flow rate comparison.

∆P/Pref

A/A

ref

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Shim and Kamm Numerical DataNumerical Results

(b) Minimum area comparison.

Figure 7.4: Comparison with the numerical data of Shim and Kamm.

134

7.5. IMPROVEMENTS TO RE-MESHING ALGORITHM

section for the reasons discussed above. Figure 7.11 shows that the radial displacement

profile remains almost linear through the central portion of the compliant wall section

even at higher pressure differences. The deformed shape of the compliant section means

that the fluid pressure exerts a force in both the streamwise and radial direction, unlike

the un-deformed shape, in which the fluid pressure exerts a force in the radial direction

only. This explains the increased axial displacements at higher pressure drops seen in

Figure 7.10.

The resulting solid body stress profiles, taken along the inner surface of the tube,

are shown in Figure 7.12. In each case, the radial stress is the largest component of

the stress tensor. At higher pressure drops, the stresses at the downstream end of the

compliant section become badly affected by the highly skewed mesh in this region.

The wall shear stress and pressure profiles responsible for the deformation are shown

in Figures 7.13 – 7.15. The wall pressure is significantly greater than the wall shear

stress, and therefore governs the deformation. Whilst the pressure is negative along

the entire length of the compliant wall section, the wall shear stress changes from

positive to negative at the downstream end of the compliant section at pressure drops

of 20 Pascals and greater. This is caused by the reversed flow direction within the

re-circulation region.

The increased deformation at the downstream end of the compliant wall section

causes an even greater pressure loss in the fluid as it flows through the area constriction,

owing to the Bernoulli effect; this in turn creates an even greater transmural pressure

difference. As a result of this, the initial increase in flow rate with imposed streamwise

pressure drop ceases once a critical streamwise pressure difference is reached, beyond

which, further increases in pressure drop to not increase the flow rate. Since the

downstream end of the tube is rigidly fixed, the maximum deformation occurs slightly

upstream of this point. A linear stress profile exists through the thickness of the

compliant wall. At the inner surface the radial stress is equal to the fluid pressure and

the tangential stress is equal to the fluid wall shear stress; the radial and tangential

stresses are zero at the outer surface.

As the deformation increases, the transmural pressure difference is governed by

pressure loss through the area reduction. When the pressure difference is increased to

50 Pa the deformation is significant enough to cause flow separation downstream of the

minimum area. Since the flow is laminar, the re-attachment length can be large.

7.5 Improvements to Re-meshing Algorithm

As the streamwise pressure drop increased, the maximum deformation - which occurred

just before the downstream end of the compliant wall - also increased. The abrupt

transition from the maximum deformation to the rigid attachment point can cause

135


difficulties for the re-meshing algorithm. Figure 7.17 shows the adapted numerical

mesh for increasing streamwise pressure drop. It can be seen at a pressure drop of 90

Pa (Figure 7.18) the fluid portion of the numerical mesh has become highly skewed

and the aspect ratio of the near-wall cells has increased. This can lead to spurious wall

shear stress and pressure predictions which introduce errors into the calculated wall

displacement. At streamwise pressure differences greater than 90 Pa these errors lead

to instabilities in the solution which lead to divergence.

The simplest method of improving the quality of the mesh adaption was to alter

the pseudo material properties of the fluid mesh. In particular, by varying the Poisson

ratio of the fluid mesh material, the quality of the near-wall mesh could be slightly

improved. Figure 7.19 shows the numerical mesh for increasing Poisson ratio at a

streamwise pressure difference of 90 Pa. It can be seen than an increase in the Poisson

ratio causes the radial grid lines to remain more normal to the wall and thus improve the

skewness of the near-wall cells. However, this is at the expense of more highly skewed

grid cells elsewhere in the flow domain. Although the near-wall skewness of the mesh is

reduced slightly, the improvement was not great enough to allow converged solutions to

be obtained at higher streamwise pressure drops. The near vertical fluid-solid interface

at the downstream end of the compliant wall section would cause problems for most grid

adaption algorithms. For cases involving such large deformation, it may be necessary

to generate a new mesh by completely re-meshing the fluid domain, rather than adapt

the existing mesh. The use of different meshing strategies, for cases which involve small

or large deformations, should be avoided as it reduces the generality of the method.

7.6 Summary

This Chapter has described the first application of the newly developed coupled fluid-

solid interaction solver. The case of flow through a compliant walled tube was simulated

and the results were compared to the numerical data of Shim and Kamm.

The simulations were performed by fixing the streamwise pressure difference be-

tween the inlet and the outlet. The prescribed external pressure was constant and

equal to zero.

The results were not in good agreement with the available data. However, this

was attributed to incorrect normalization of the results presented by Shim and Kamm.

This argument was strengthened by tests performed on a rigid walled tube which proved

that the flow solver was able to accurately predict the pressure-flow rate relationship

for laminar pipe flow.

The newly developed method was shown to be capable of predicting fluid-structure

interaction problems involving large displacements. Information regarding the displace-

ment and stress field is produced by the solver.

136

7.6. SUMMARY

Although the method is capable of predicting large displacements, additional steps

were required to ensure that the numerical mesh, particularly within the fluid sub-

domain, does not become skewed. The mesh skewness can lead to inaccuracy and

eventually numerical instability. To guarantee mesh quality when the deformation

becomes large, total re-meshing may be required.

137

CH

AP

TE

R7.

PIP

EFLO

WS

X / Diameter

R/D

iam

eter

5.75 6 6.25 6.5 6.75 7 7.25 7.5 7.75

0

0.1

0.2

0.3

0.4

0.5

0.6

R/D

iam

eter

3.5 3.75 4 4.25 4.5 4.75 5 5.25 5.5

0

0.1

0.2

0.3

0.4

0.5

0.6SRR: -2 -1 0 1 2 3 4 5 6

1

Figure 7.5: Fluid flow vectors within the tube and radial stress contours in the compliant wall. Pressure drop = 10 Pa

138

7.6

.SU

MM

ARY

X / Diameter

R/D

iam

eter

5.75 6 6.25 6.5 6.75 7 7.25 7.5 7.75

0

0.1

0.2

0.3

0.4

0.5

0.6

R/D

iam

eter

3.5 3.75 4 4.25 4.5 4.75 5 5.25 5.5

0

0.1

0.2

0.3

0.4

0.5

0.6SRR: -8 -4 0 4 8 12 16 20

1


139

CH

AP

TE

R7.

PIP

EFLO

WSR

/Dia

met

er

3.5 3.75 4 4.25 4.5 4.75 5 5.25 5.5

0

0.1

0.2

0.3

0.4

0.5

0.6SRR: -15 -5 5 15 25 35

1

X / Diameter

R/D

iam

eter

5.75 6 6.25 6.5 6.75 7 7.25 7.5 7.75

0

0.1

0.2

0.3

0.4

0.5

0.6


140

7.6

.SU

MM

ARY

X / Diameter

R/D

iam

eter

5.75 6 6.25 6.5 6.75 7 7.25 7.5 7.75

0

0.1

0.2

0.3

0.4

0.5

0.6

R/D

iam

eter

3.5 3.75 4 4.25 4.5 4.75 5 5.25 5.5

0

0.1

0.2

0.3

0.4

0.5

0.6SRR: -70 -50 -30 -10 10 30 50 70

1


141

CH

AP

TE

R7.

PIP

EFLO

WS

X / Diameter

R/D

iam

eter

5.75 6 6.25 6.5 6.75 7 7.25 7.5 7.75

0

0.1

0.2

0.3

0.4

0.5

0.6

R/D

iam

eter

3.5 3.75 4 4.25 4.5 4.75 5 5.25 5.5

0

0.1

0.2

0.3

0.4

0.5

0.6SRR: -350 -250 -150 -50 50

1


142

7.6. SUMMARY

X / Diameter

Udi

sp/D

iam

eter

3.5 4 4.5 5 5.5 6 6.5 7 7.5-3.0E-02

0.0E+00

3.0E-02

6.0E-02

9.0E-02

1.2E-01

1.5E-01

1.8E-01

Delp = 10 PaDelp = 30 PaDelp = 50 PaDelp = 70 PaDelp = 90 Pa

Figure 7.10: Axial u-displacement profiles for increasing pressure drop.

X / Diameter

Vdi

sp/D

iam

eter

3.5 4 4.5 5 5.5 6 6.5 7 7.5

-1.5E-01

-1.2E-01

-9.0E-02

-6.0E-02

-3.0E-02

0.0E+00


Figure 7.11: Radial v-displacement profiles for increasing pressure drop.

143


X / Diameter

Tra

ctio

n(P

a)

3.5 4 4.5 5 5.5 6 6.5 7 7.5

-4.0E+02

-3.0E+02

-2.0E+02

-1.0E+02

0.0E+00

1.0E+02

Radial StressAxial StressShear Stress

(a) Pressure drop = 10 Pa

X / Diameter

Tra

ctio

n(P

a)

3.5 4 4.5 5 5.5 6 6.5 7 7.5

-4.0E+02

-3.0E+02

-2.0E+02

-1.0E+02

0.0E+00

1.0E+02


(b) Pressure drop = 30 Pa

X / Diameter

Tra

ctio

n(P

a)

3.5 4 4.5 5 5.5 6 6.5 7 7.5

-4.0E+02

-3.0E+02

-2.0E+02

-1.0E+02

0.0E+00

1.0E+02


(c) Pressure drop = 50 Pa

X / Diameter

Tra

ctio

n(P

a)

3.5 4 4.5 5 5.5 6 6.5 7 7.5

-4.0E+02

-3.0E+02

-2.0E+02

-1.0E+02

0.0E+00

1.0E+02


(d) Pressure drop = 70 Pa

X / Diameter

Tra

ctio

n(P

a)

3.5 4 4.5 5 5.5 6 6.5 7 7.5

-4.0E+02

-3.0E+02

-2.0E+02

-1.0E+02

0.0E+00

1.0E+02


(e) Pressure drop = 90 Pa

Figure 7.12: Solid stress profiles for increasing pressure drop.

144

7.6. SUMMARY

X / Diameter

Wal

lShe

arS

tres

s(P

a)

4 5 6 7 8 9 10-1

0

1

2

3

4Delp = 10 PaDelp = 30 PaDelp = 50 PaDelp = 70 PaDelp = 90 Pa

Figure 7.13: Wall shear stress profiles for increasing pressure drop.

X / Diameter

Nor

mal

ized

Wal

lShe

arS

tres

s(τ

WA

LL/0

.5ρU

IN2 )

4 5 6 7 8 9 10-0.0002

0

0.0002

0.0004

0.0006

0.0008

Rey = 479.71Rey = 1112.80Rey = 1529.78Rey = 1697.70Rey = 1749.31

Figure 7.14: Normalized wall shear stress profiles for increasing Reynolds number.

145


X / Diameter

Wal

lPre

ssur

e(P

a)

4 5 6 7 8 9 10-140

-120

-100

-80

-60

-40

-20

0


Figure 7.15: Wall pressure profiles for increasing pressure drop.

X / Diameter

Nor

mal

ized

Wal

lPre

ssur

e(P

-PIN

)/(

0.5ρ

UIN2 )

4 5 6 7 8 9 10-0.025

-0.02

-0.015

-0.01

-0.005

0


Figure 7.16: Normalized wall pressure profiles for increasing Reynolds number.

146

7.6. SUMMARY

X / Diameter

R/D

iam

eter

5.6 5.8 6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8

0

0.1

0.2

0.3

0.4

0.5

0.6

(a) Pressure drop = 10 Pa

X / Diameter

R/D

iam

eter

5.6 5.8 6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8

0

0.1

0.2

0.3

0.4

0.5

0.6

(b) Pressure drop = 50 Pa

X / Diameter

R/D

iam

eter

5.6 5.8 6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8

0

0.1

0.2

0.3

0.4

0.5

0.6

(c) Pressure drop = 90 Pa

Figure 7.17: Numerical mesh at downstream end of compliant wall section for increasingpressure drop.

147

CH

AP

TE

R7.

PIP

EFLO

WS

X / Diameter

R/D

iam

eter

7 7.1 7.2 7.3 7.4 7.5 7.60.25

0.3

0.35

0.4

0.45

0.5

0.55

Figure 7.18: Detailed view of the numerical mesh at the downstream end of the compliant wall section at pressure drop = 90 Pa.

148

7.6. SUMMARY

X / Diameter

R/D

iam

eter

7 7.1 7.2 7.3 7.4 7.5 7.60.25

0.3

0.35

0.4

0.45

0.5

0.55

(a) Poisson ratio = 0.30

X / Diameter

R/D

iam

eter

7 7.1 7.2 7.3 7.4 7.5 7.60.25

0.3

0.35

0.4

0.45

0.5

0.55

(b) Poisson ratio = 0.42

X / Diameter

R/D

iam

eter

7 7.1 7.2 7.3 7.4 7.5 7.60.25

0.3

0.35

0.4

0.45

0.5

0.55

(c) Poisson ratio = 0.48

Figure 7.19: Effect of Poisson ratio upon the numerical mesh at the downstream endof the compliant wall section at pressure drop = 90 Pa.

149


150

Chapter 8

Stenosed Flows

8.1 Introduction

The first case involving physiological flows was that of flow through a stenosed tube.

Stenoses are localised constrictions within arteries caused by the accumulation of athero-

sclerotic plaque (Young, 1979). Accurate and efficient simulation of such flows could be

used to develop future treatments. The ability to simulate patient-specific cases could

be used to determine the most appropriate course of treatment.

Initially, rigid walled simulations were performed to validate the flow solver in iso-

lation, before cases involving fluid-structure interaction were considered. In both rigid

and compliant walled cases, the results were compared to experimental data from the

literature.

8.2 Steady Flow Through a Rigid Stenosis

Flow through a rigid walled stenosis was simulated and the results compared to the

experimental data of Young and Tsai (1973). Both laminar and turbulent simulations

were performed to cover the entire range of physiologically relevant flow rates; this

corresponds to Reynolds numbers in the range 30 – 5000. The purpose of this exercise

was to validate the flow solver and develop appropriate modelling strategies for this

category of flow.


The physical geometry consists of an axisymmetric tube with a sinusoidal stenosis. The

tube has internal diameter of 0.744 in. The stenosis has a length of 4 diameters and

maximum constriction of 89% (in terms of area reduction at the throat). The pressure

drop was measured between locations 5 diameters either side of the stenosis throat.

The geometry matched model M-2 used during the experiments of Young and Tsai. A

151

CHAPTER 8. STENOSED FLOWS

Ls = 4 Do

Do = 0.744 in (Ao-Amin)/Ao = 89%

X / Diameter

R/D

iam

eter

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3

-0.5

-0.25

0

0.25

0.5

0.75

Figure 8.1: Schematic of Young and Tsai geometry.

schematic of the geometry is given in Figure 8.1. In the laminar flow simulations, the

upstream and downstream lengths were 8.75 and 135 diameters respectively; the large

downstream length being required to ensure full re-attachment of the flow was achieved

in some cases. For the turbulent flow simulations, the upstream and downstream lengths

were 8.75 and 25 diameters respectively; it was possible to use a shorter downstream

length during the turbulent simulations due to the shorter re-circulation length found

in the turbulent case.


The numerical mesh used for the laminar simulations consisted of 2800 nodes in the

axial direction and 58 nodes in the radial direction. The nodes in the radial direction

had an expansion ratio of Ry = 0.95, in order to cluster nodes close to the wall. The

nodes were uniformly spaced in the axial direction. The numerical mesh used for the

turbulent simulations consisted of 1780 nodes in the axial direction and 120 in the radial

direction. The nodes in the radial direction had an expansion ratio of Ry = 0.975. In

the axial direction, nodes were clustered around the stenosis throat and in the region

downstream of the stenosis to accurately resolve the gradients within the separation

bubble. In every turbulent simulation reported in this Thesis, the grid refinement in

the near-wall region was great enough to ensure that the y+ value was always less than

one. This was checked by first running a simulation on a specific mesh and plotting

the value of y+ along the entire length of the wall. If the value of y+ was found to

be greater than one at any point, the mesh was refined in the near-wall region. This

process was continued until the near-wall node spacing was adequately refined to allow

the low-Reynolds-number turbulence model to accurately account for near-wall effects.

As discussed in the previous Chapter, the number of nodes in the axial direction could

be significantly reduced by using non-uniform mesh spacing. The meshes used during

these simulations were carried forward from previous validation simulations. The use

of such over-refined meshes for two-dimensional cases does not adversely affect the

simulation time as greatly as it would in a three-dimensional case. Portions of the

152

8.2. STEADY FLOW THROUGH A RIGID STENOSIS

X / Diameter

R/D

iam

eter

-2.2 -2.1 -2 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2

0

0.1

0.2

0.3

0.4

0.5

Figure 8.2: Mesh used during laminar, rigid walled stenosis simulations.

laminar and turbulent numerical meshes are shown in Figures 8.2 and 8.3 respectively.


The boundary conditions used during the rigid walled stenosis simulations are sum-

marised in Table 8.1.

8.2.4 Fluid Properties

The working fluid was water with density, ρ = 998.0 kg/m3, and dynamic viscosity,

µ = 1.003 × 10−3 Pa s.

8.2.5 Under Relaxation Factors

The under-relaxation factors used during the simulations are summarised in Tables 8.2 –

8.3. In the turbulent flow case, a low value of 0.05 was taken for the pressure correction

under-relaxation factor, to avoid numerical instabilities.

West (Inlet) Parabolic U -velocity profile (laminar)1/7th Power-law U -velocity profile (turbulent)

East (Outlet) Zero streamwise gradients for all variablesNorth (Wall) No-slip velocity

Zero wall normal pressure gradientSouth (Symmetry) Zero V -velocity

Zero normal gradient for all other variables

Table 8.1: Boundary conditions used in rigid walled stenosis simulations.

153


X / Diameter

R/D

iam

eter

-2.2 -2.1 -2 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2

0

0.1

0.2

0.3

0.4

0.5

Figure 8.3: Mesh used during turbulent, rigid walled stenosis simulations.

U -momentum equation 0.5V -momentum equation 0.5

Pressure correction equation 0.3

Table 8.2: Under-relaxation factors used in the laminar, rigid walled stenosis simula-tions.


Pressure correction equation 0.05k-equation 0.1ǫ-equation 0.1

Table 8.3: Under-relaxation factors used in the turbulent, rigid walled stenosis simula-tions.

154


X / Diameter

Rey

nold

sN

umbe

r

0 2 4 6 8 10 12 140

40

80

120

160

200

Experimental SeparationExperimental Re-attachmentNumerical SeparationNumerical Re-attachment

Figure 8.4: Laminar flow separation and re-attachment curves.

8.2.6 Laminar Flow Results and Discussion

It can be seen from the separation and re-attachment curve (Figure 8.4) that the

point of flow separation was accurately predicted at low Reynolds numbers by the

laminar solver. The point of separation moves slightly upstream towards the throat of

the stenosis with increasing Reynolds number. The re-attachment length, which was

identified by the point where the wall shear stress was equal to zero, is slightly under

predicted when compared with the experimental data. However, Young and Tsai report

that the location of the re-attachment point was more difficult to measure accurately

than the separation point. The re-attachment point shows a much stronger dependence

on the Reynolds number.

At Reynolds numbers greater than 200 the re-circulation bubble becomes very large,

as shown by the predicted streamlines in Figure 8.5. This large re-circulation bubble

is the reason the domain had to be extended to such a large downstream distance, in

order to ensure the flow had re-attached and was close to fully developed by the exit.

This is important to ensure that the outlet boundary conditions are valid and do not

introduce any significant error to the flow solution.

Figure 8.6 shows that the pressure drop across the stenosis is accurately predicted in

the laminar regime. The non-dimensional pressure drop is greatest at lower Reynolds

numbers where the viscous effects are dominant. Fully converged steady solutions

were only achievable for Reynolds numbers of less than 700, by which stage the re-

155

CH

AP

TE

R8.

ST

EN

OSE

DFLO

WS

X / DiameterR/D

iam

eter

-2 0 2 4 6 8 10 12 14 16 18 20

0

0.5

(a) Reynolds number = 50

X / DiameterR/D

iam

eter

-2 0 2 4 6 8 10 12 14 16 18 20

0

0.5

(b) Reynolds number = 100

X / DiameterR/D

iam

eter

-2 0 2 4 6 8 10 12 14 16 18 20

0

0.5

(c) Reynolds number = 150

X / DiameterR/D

iam

eter

-2 0 2 4 6 8 10 12 14 16 18 20

0

0.5

(d) Reynolds number = 200

X / DiameterR/D

iam

eter

-2 0 2 4 6 8 10 12 14 16 18 20

0

0.5

(e) Reynolds number = 250

Figure 8.5: Laminar flow streamlines for rigid stenosis.

156


Reynolds Number

Pre

ssur

eD

rop,

∆p

/ρU

2 in

101 102 103 1040

20

40

60

80

100

Experimental DataLaminar Results

Figure 8.6: Laminar flow pressure drop across rigid walled stenosis

attachment length was around 45 diameters. However, it can be seen from Figure 8.6

that the pressure drop is most accurately predicted at Reynolds numbers less than

300; this coincides with the experimentally observed point of transition to turbulence.

Even if the flow were to remain laminar beyond this point, the presence of the long

re-circulation bubble would most likely mean that it would no longer be steady, which

offers an explanation as to why the steady-state laminar computations fail to accurately

predict the pressure drop at higher Reynolds numbers.

It can be seen from the laminar flow wall shear stress plot (Figure 8.7) that the stress

increases to a maximum at the stenosis throat before rapidly decreasing downstream

of the throat. For all Reynolds numbers considered in this study, the flow separates

slightly downstream of the throat - this can be seen as a negative wall shear stress in the

plot. The minimum wall shear stress moves slightly upstream towards the throat with

increasing Reynolds number. It can also be seen that the wall shear stress is negative

far downstream of the throat at higher Reynolds numbers. The peak wall shear stress

increases linearly with Reynolds number.

Figure 8.9 shows that the laminar flow wall pressure decreases rapidly to a minimum

slightly downstream of the stenosis throat. The greater part of the pressure recovery

occurs in the downstream section of the stenosis, the remainder occurring in the straight

downstream section. It can also be seen that the magnitude of the change in pressure

is significantly greater than that of the wall shear stress. It is therefore expected that

157


X / Diameter

Wal

lShe

arS

tres

s(P

a)

-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Rey = 50Rey = 100Rey = 150Rey = 200Rey = 250

Figure 8.7: Wall shear stress plots for laminar flow through the rigid stenosis.

X / Diameter

Nor

mal

ized

Wal

lShe

arS

tres

s(τ

wal

l/0

.5ρU

B2 )

-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

12

14


Figure 8.8: Normalized wall shear stress plots for laminar flow through the rigid steno-sis.

158


X / Diameter

Wal

lPre

ssur

e(P

a)

-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-12

-10

-8

-6

-4

-2

0


Figure 8.9: Wall pressure plots for laminar flow through the rigid stenosis.

the greater part of the fluid-structure interaction in the compliant wall case will be

governed by the wall pressure traction.

8.2.7 Turbulent Flow Results and Discussion

To produce accurate predictions at higher Reynolds numbers, the effects of turbulence

must be accounted for. However, it was still found to be difficult to obtain a numerically

stable solution at moderate Reynolds numbers due to the transitional nature of the flow.

What makes these flows especially difficult to converge using the RANS approach is that

for these low transitional Reynolds numbers the flow is laminar upstream of the stenosis

and only undergoes transition to turbulence after the separation point. Some of these

difficulties were overcome by using the low-Reynolds-number Launder-Sharma model

and by prescribing an initial flow field with non-zero turbulence over the entire flow

domain. It can be seen from Figure 8.11 that the pressure drop is accurately predicted

at Reynolds numbers greater than 1500. Converged solutions were not achievable for

Reynolds numbers less than 1500.

The flow streamlines shown in Figure 8.12 indicate that the re-circulation bubble

reduces in size with increasing Reynolds number in the turbulent flow regime. Both the

separation and re-attachment point move upstream with increasing Re. The reduction

in size of the re-circulation bubble is due to the increased mixing caused by the turbu-

lent fluctuations. By comparing the turbulent flow streamlines (Figure 8.12) with the

159


X / Diameter

Nor

mal

ized

Wal

lPre

ssur

e(P

/0.5

ρUB2 )

-2 0 2 4 6 8 10-250

-200

-150

-100

-50

0Rey = 50Rey = 100Rey = 150Rey = 200Rey = 250

Figure 8.10: Normalized wall pressure plots for laminar flow through the rigid stenosis.

Reynolds Number

Pre

ssur

eD

rop,

∆p

/ρU

2 in

101 102 103 1040

20

40

60

80

100

Experimental DataLaminar ResultsTurbulent Results

Figure 8.11: Laminar and turbulent flow pressure drop across rigid walled stenosis

160


laminar flow streamlines (Figure 8.5), the dramatic reduction in re-circulation bubble

size due to the effects of turbulence can be seen.

Figure 8.12 also shows contours of the turbulent kinetic energy, k, with increasing

Reynolds number. It can be seen that a region of high k develops downstream of

the stenosis; this is due to the high shear rates in the separated shear layer. The

region of high k moves upstream with increasing Reynolds number. The region of

low k upstream of the throat indicates that the flow here is essentially laminar, before

transition to turbulence occurs slightly downstream of the throat. Further downstream,

beyond re-attachment, turbulence levels decay back towards the laminar state, since

the Reynolds number is still too low to sustain a fully developed turbulent pipe flow.

The pressure loss through the area constriction, assuming that viscous losses are

negligibly small, can be estimated from Bernoulli’s equation:

Pin + 0.5 ρU2in = P + 0.5 ρU2 (8.1)

where the subscript in refers to inlet values of the flow velocity, U , and pressure, P .

The above equation can be re-arranged to give:

P − Pin = 0.5 ρU2in − 0.5 ρU2 (8.2)

The continuity equation can be used to relate the flow velocity, U , to the tube

diameter, d:

U =

(din

d

)2

Uin (8.3)

Finally, the non-dimensional pressure at any point within the stenosed tube is given

by:

(P − Pin)

(0.5 ρU2in)

= 1 −(

din

d

)4

(8.4)

Figure 8.14 shows a comparison between the Bernoulli losses, calculated above,

and the computed pressure profiles for a range of turbulent Reynolds numbers. For

all Reynolds numbers considered, the computed minimum non-dimensional pressure,

which occurs at the stenosis throat, is approximately 20% less than that predicted by

Bernoulli’s equation. A small amount of pressure recovery occurs before the separation

point; however, within the re-circulation bubble, the pressure is relatively constant in

the axial direction. Beyond the re-circulation bubble, the pressure recovers further

towards the upstream level. At higher Reynolds numbers, the re-circulation region is

shorter, which allows for the second stage of pressure recovery to commence sooner.

The normalized turbulent flow pressure contours for increasing Reynolds number

161


9 8 7 6 5 4 3 2

X / Diameter

R/D

iam

eter

-2 -1 0 1 2 3 4 5 6 7-0.5

0

0.5

LevelTKE/U_IN**2:

10

20.4

30.8

41.2

51.6

62

72.4

82.8

93.2


9 8 7 6 5 4 3 2

X / Diameter

R/D

iam

eter

-2 -1 0 1 2 3 4 5 6 7-0.5

0

0.5

LevelTKE/U_IN**2:

10

20.4

30.8

41.2

51.6

62

72.4

82.8

93.2


9 8 7 6 5 4 3 2

X / Diameter

R/D

iam

eter

-2 -1 0 1 2 3 4 5 6 7-0.5

0

0.5

LevelTKE/U_IN**2:

10

20.4

30.8

41.2

51.6

62

72.4

82.8

93.2


9 8 7 6 5 4 3 2

X / Diameter

R/D

iam

eter

-2 -1 0 1 2 3 4 5 6 7-0.5

0

0.5

LevelTKE/U_IN**2:

10

20.4

30.8

41.2

51.6

62

72.4

82.8

93.2


8 7 6 5 4 3 2

X / Diameter

R/D

iam

eter

-2 -1 0 1 2 3 4 5 6 7-0.5

0

0.5

LevelTKE/U_IN**2:

10

20.4

30.8

41.2

51.6

62

72.4

82.8

93.2


8 7 6 5 4 3 2

X / Diameter

R/D

iam

eter

-2 -1 0 1 2 3 4 5 6 7-0.5

0

0.5

LevelTKE/U_IN**2:

10

20.4

30.8

41.2

51.6

62

72.4

82.8

93.2

(f) Reynolds number = 4000

162


7 6 5 4 3 2

X / Diameter

R/D

iam

eter

-2 -1 0 1 2 3 4 5 6 7-0.5

0

0.5

LevelTKE/U_IN**2:

10

20.4

30.8

41.2

51.6

62

72.4

82.8

93.2

(g) Reynolds number = 4500

7 6 5 4 3 2

X / Diameter

R/D

iam

eter

-2 -1 0 1 2 3 4 5 6 7-0.5

0

0.5

LevelTKE/U_IN**2:

10

20.4

30.8

41.2

51.6

62

72.4

82.8

93.2

(h) Reynolds number = 5000

Figure 8.12: Contours of turbulent kinetic energy (above line of symmetry) and flowstreamlines (below line of symmetry) for turbulent flow through the rigid stenosis.

are shown in Figure 8.13. It can be seen that the sharp pressure drop to a minimum

value at the throat is almost unaffected by the Reynolds number. The greatest effect

of the Reynolds number upon the pressure field can be seen in the downstream section.

The pressure recovery occurs further upstream as the Reynolds number is increased.

The normalized turbulent flow wall shear stress (Figure 8.15) is significantly smaller

in magnitude than that of the change in wall pressure - as was the case for laminar flow.

The peak wall shear stress occurs slightly upstream of the throat and the normalised

peak value reduces in magnitude with increasing Reynolds number. An effect of the

turbulent mixing is to reduce the size of the re-circulation bubble, which can be seen as

a shorter negative wall shear stress region. The re-attachment point, identified as the

point where the wall shear becomes positive again, moves upstream, towards the throat,

with increasing Reynolds number. The minimum value of the normalised wall shear

stress is relatively unaffected by the Reynolds number. This would suggest that the

peak negative wall shear stress is proportional to the square of the bulk velocity. The

wall shear stress begins to increase slightly downstream of the throat, before reducing

again in the straight downstream section.

Figures 8.16–8.18 show radial profiles of velocity (normalized with respect to bulk

inlet velocity) at Reynolds numbers of 2000, 3500 and 5000 respectively. The first

profile is at the stenosis throat; subsequent profiles are spaced one diameter further

downstream. The peak U -velocity, which occurs at the stenosis throat owing to mass

continuity through the severe area constriction, is relatively unaffected by the increase in

Reynolds number. For each case, the dimensionless U -velocity profile is approximately

ten times greater than the inlet bulk velocity. At a Reynolds number of 2000, the

163


1 2 3 8 9 9 8 7

X / Diameter

R/D

iam

eter

-2 -1 0 1 2 3 4 5 6 7-0.5

0

0.5

LevelP:

110

220

330

440

550

660

770

880

990


1 2 89 9 8 7

X / Diameter

R/D

iam

eter

-2 -1 0 1 2 3 4 5 6 7-0.5

0

0.5

LevelP:

110

220

330

440

550

660

770

880

990


1 2 3 89 9 8 7

X / Diameter

R/D

iam

eter

-2 -1 0 1 2 3 4 5 6 7-0.5

0

0.5

LevelP:

110

220

330

440

550

660

770

880

990


1 2 8 9 9 8 7

X / Diameter

R/D

iam

eter

-2 -1 0 1 2 3 4 5 6 7-0.5

0

0.5

LevelP:

110

220

330

440

550

660

770

880

990


1 2 8 9 9 8 7

X / Diameter

R/D

iam

eter

-2 -1 0 1 2 3 4 5 6 7-0.5

0

0.5

LevelP:

110

220

330

440

550

660

770

880

990


1 2 8 9 8 7 6

X / Diameter

R/D

iam

eter

-2 -1 0 1 2 3 4 5 6 7-0.5

0

0.5

LevelP:

110

220

330

440

550

660

770

880

990

(f) Reynolds number = 4000

164


1 2 8 9 8 7 6

X / Diameter

R/D

iam

eter

-2 -1 0 1 2 3 4 5 6 7-0.5

0

0.5

LevelP:

110

220

330

440

550

660

770

880

990

(g) Reynolds number = 4500

1 2 98 8 7 6

X / Diameter

R/D

iam

eter

-2 -1 0 1 2 3 4 5 6 7-0.5

0

0.5

LevelP:

110

220

330

440

550

660

770

880

990

(h) Reynolds number = 5000

Figure 8.13: Pressure contours for turbulent flow through the rigid stenosis. Pressurenormalized according to (Pin − P )/(0.5ρU2

in).

X / Diameter

Nor

mal

ized

Pre

ssur

eD

rop

(P-P

IN)

/(0.

5ρU

IN2 )

-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10-110

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

1-(dIN/d)4

Rey = 1500Rey = 2000Rey = 2500Rey = 3000Rey = 3500Rey = 4000Rey = 4500Rey = 5000

Figure 8.14: Normalized wall pressure profiles for turbulent flow through the rigidstenosis.

165


X / Diameter

τ wal

l/(0.

5ρ

Uin2 )

-4 -2 0 2 4 6 8 10-0.3

0

0.3

0.6

0.9

1.2

1.5

1.8

Re = 1500Re = 2000Re = 2500Re = 3000Re = 3500Re = 4000Re = 4500Re = 5000

Figure 8.15: Normalized wall shear stress profiles for turbulent flow through the rigidstenosis.

flow re-attaches at a point somewhere between 4 and 5 diameters downstream of the

throat. When the Reynolds number is increased to 5000, the re-attachment occurs

between 3 and 4 diameters downstream. In each case the flow is fully attached and has

almost achieved a fully developed turbulent velocity profile by 5 diameters downstream

of the throat. The negative U -velocity in the re-circulation region is accompanied by a

positive V -velocity in the core flow region.

Figures 8.19–8.21 show radial profiles of the Reynolds stress, normalized with re-

spect to the inlet bulk velocity squared, for increasing Reynolds number. The profiles

are at the same axial locations as the velocity profiles. In each case, the turbulence

levels are low at the throat owing to the laminar flow conditions upstream of the steno-

sis region. The turbulence levels begin to increase just downstream of the throat with

a sharp peak in the centre of the profile due to turbulence production in the shear

layer between the re-circulation bubble and the core flow. Downstream, the turbulence

levels increase in magnitude and grow to cover the entire radial profile. As the flow

re-attaches the turbulence levels begin to decay. The peak turbulence level moves up-

stream with increasing Reynolds number. The apparent zero value of the normal shear

stress, uiui, at the pipe centre line is a feature of the post-processing. The Reynolds

stress values are obtained from the calculated flow field during the post-processing stage

of the simulation. Since the flow field is not calculated along boundary nodes, neither

are the Reynolds stress values calculated at boundaries. In reality, a zero gradient for

166

8.2

.ST

EA

DY

FLO

WT

HR

OU

GH

AR

IGID

ST

EN

OSIS

Normalized Velocity

R/D

iam

eter

-2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5 U / Uin

V / Uin

(a) Velocity profile at X/D = 0.0

Normalized Velocity

R/D

iam

eter

-2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5 U / Uin

V / Uin

(b) Velocity profile at X/D = 1.0

Normalized Velocity

R/D

iam

eter

-2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5 U / Uin

V / Uin

(c) Velocity profile at X/D = 2.0

Normalized Velocity

R/D

iam

eter

-2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5 U / Uin

V / Uin

(d) Velocity profile at X/D = 3.0

Normalized Velocity

R/D

iam

eter

-2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5 U / Uin

V / Uin

(e) Velocity profile at X/D = 4.0

Normalized Velocity

R/D

iam

eter

-2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5 U / Uin

V / Uin

(f) Velocity profile at X/D = 5.0

Figure 8.16: Turbulent flow mean velocity profiles in rigid stenosis at Re = 2000.

167

CH

AP

TE

R8.

ST

EN

OSE

DFLO

WS

Normalized Velocity

R/D

iam

eter

-2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5 U / Uin

V / Uin


Normalized Velocity

R/D

iam

eter

-2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5 U / Uin

V / Uin


Normalized Velocity

R/D

iam

eter

-2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5 U / Uin

V / Uin


Normalized Velocity

R/D

iam

eter

-2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5 U / Uin

V / Uin


Normalized Velocity

R/D

iam

eter

-2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5 U / Uin

V / Uin


Normalized Velocity

R/D

iam

eter

-2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5 U / Uin

V / Uin



168

8.2

.ST

EA

DY

FLO

WT

HR

OU

GH

AR

IGID

ST

EN

OSIS

Normalized Velocity

R/D

iam

eter

-2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5 U / Uin

V / Uin


Normalized Velocity

R/D

iam

eter

-2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5 U / Uin

V / Uin


Normalized Velocity

R/D

iam

eter

-2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5 U / Uin

V / Uin


Normalized Velocity

R/D

iam

eter

-2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5 U / Uin

V / Uin


Normalized Velocity

R/D

iam

eter

-2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5 U / Uin

V / Uin


Normalized Velocity

R/D

iam

eter

-2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5 U / Uin

V / Uin



169

CH

AP

TE

R8.

ST

EN

OSE

DFLO

WS

Normalized Reynolds Stress

R/D

iam

eter

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5 uu / Uin2

vv / Uin2

ww / Uin2

uv / Uin2

(a) Reynolds stress profile at X/D = 0.0


R/D

iam

eter

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5 uu / Uin2

vv / Uin2

ww / Uin2

uv / Uin2

(b) Reynolds stress profile at X/D = 1.0


R/D

iam

eter

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5 uu / Uin2

vv / Uin2

ww / Uin2

uv / Uin2

(c) Reynolds stress profile at X/D = 2.0


R/D

iam

eter

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5 uu / Uin2

vv / Uin2

ww / Uin2

uv / Uin2

(d) Reynolds stress profile at X/D = 3.0


R/D

iam

eter

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5 uu / Uin2

vv / Uin2

ww / Uin2

uv / Uin2

(e) Reynolds stress profile at X/D = 4.0


R/D

iam

eter

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5 uu / Uin2

vv / Uin2

ww / Uin2

uv / Uin2

(f) Reynolds stress profile at X/D = 5.0

Figure 8.19: Reynolds stress profiles in rigid stenosis at Re = 2000.

170

8.2

.ST

EA

DY

FLO

WT

HR

OU

GH

AR

IGID

ST

EN

OSIS


R/D

iam

eter

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5 uu / Uin2

vv / Uin2

ww / Uin2

uv / Uin2



R/D

iam

eter

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5 uu / Uin2

vv / Uin2

ww / Uin2

uv / Uin2



R/D

iam

eter

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5 uu / Uin2

vv / Uin2

ww / Uin2

uv / Uin2



R/D

iam

eter

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5 uu / Uin2

vv / Uin2

ww / Uin2

uv / Uin2



R/D

iam

eter

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5 uu / Uin2

vv / Uin2

ww / Uin2

uv / Uin2



R/D

iam

eter

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5 uu / Uin2

vv / Uin2

ww / Uin2

uv / Uin2



171

CH

AP

TE

R8.

ST

EN

OSE

DFLO

WS


R/D

iam

eter

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5 uu / Uin2

vv / Uin2

ww / Uin2

uv / Uin2



R/D

iam

eter

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5 uu / Uin2

vv / Uin2

ww / Uin2

uv / Uin2



R/D

iam

eter

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5 uu / Uin2

vv / Uin2

ww / Uin2

uv / Uin2



R/D

iam

eter

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5 uu / Uin2

vv / Uin2

ww / Uin2

uv / Uin2



R/D

iam

eter

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5 uu / Uin2

vv / Uin2

ww / Uin2

uv / Uin2



R/D

iam

eter

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5 uu / Uin2

vv / Uin2

ww / Uin2

uv / Uin2



172

8.3. STEADY FLOW THROUGH A COMPLIANT STENOSIS

the Reynolds stresses would be expected along the pipe centre line.

The turbulent shear stress, uv, profiles show rather low turbulence levels within the

stenosis. At the downstream end of the stenosis, X/D = 2, the turbulent shear stress

increases and the profile shows a peak at a radial position of approximately R/D = 0.2.

This peak is caused by turbulence production within the separated shear layer, between

the re-circulation bubble and the core flow, where the mean velocity gradient, ∂U∂y , is

greatest. Moving further downstream, this peak begins to flatten to fill the entire radial

profile. Beyond X/D = 4, the velocity profile begins to return towards a fully developed

turbulent profile, as a result, the turbulent shear stress levels reduce in magnitude and

tend towards a fully developed profile.

As mentioned earlier, the components of the Reynolds stress tensor are obtained

from the mean flow field using the Boussinesq relation. The normal stress component

in the hoop direction, ww, is equal to two thirds of the turbulent kinetic energy, k; this

component exists despite the fact that there is no contribution from mean strain rate.

8.3 Steady Flow Through a Compliant Stenosis

Once the flow solver had been validated for the case of flow through a rigid walled

stenosis, the newly developed FSI solver was used to simulate the more complex, and

physiological realistic, case of flow through a compliant walled stenosis. Both laminar

and turbulent flow was considered. The results were compared to the experimental

data of Stergiopulos et al. (1993).


The physical geometry consists of a straight tube of diameter 8 mm and length 8.75

diameters, followed by a sinusoidal stenosis of length 4 diameters and area reduction

of 90%, followed by a straight tube of length 24.85 diameters. An elastic wall section

extends from 5 diameters upstream of the stenosis throat to 5 diameters downstream.

The wall thickness of the elastic material was not reported in the experimental paper,

therefore calculations were performed covering a range of wall thickness from 0.1 mm

to 0.2 mm, as will be shown later.

Diameter, Do = 8 mm

Lelastic = 10Do

(Ao-Amin)/Ao = 90%

Lstenosis = 3.75Do Thickness, t = 0.1 - 0.2 mm

Pin = 74 mmHg Pext = 3 mmHg

Figure 8.22: Schematic of compliant walled stenosis geometry.

173


Outlet (East) Zero streamwise gradient U - and V -velocityPrescribed pressure

Inlet (West) Parabolic (laminar) U -velocity profileZero V -velocityZero pressure

Axis (South) Zero V -velocityZero normal gradient on all other variables

Wall (North) Zero U - and V -velocityZero normal gradient pressure



Tangential traction equal to fluid wall-shearCompliant wall (North) Prescribed normal traction

Zero tangential traction



The fluid sub-domain of the numerical mesh consisted of 768 nodes in the axial di-

rection and 120 nodes in the radial direction. The nodes in the radial direction had

an expansion ratio of Ry = 0.975, in order to cluster nodes near the wall. The same

numerical mesh was used for both laminar and turbulent computations.

The solid sub-domain of the numerical mesh consisted of 258 uniformly spaced

nodes in the axial direction and 20 uniformly spaced nodes in the radial direction.

Figure 8.23 shows a portion of the numerical mesh around the upstream end of the

compliant stenosis section and a detailed view showing the solid sub-domain.


The boundary conditions are summarised in Tables 8.4 and 8.5.

Fixed displacement boundary conditions were specified on all boundaries during

re-meshing.

8.3.4 Under Relaxation Factors

The under-relaxation factors used during the simulations are summarised in Tables 8.6-

8.8. The FSI under-relaxation factor was 0.35.

174


X / Diameter

R/D

iam

eter

-2 -1.8 -1.6 -1.4 -1.2

0

0.1

0.2

0.3

0.4

0.5

X / Diameter

R/D

iam

eter

-1.56 -1.55 -1.54 -1.53 -1.52 -1.51 -1.5 -1.490.43

0.44

0.45

0.46

0.47

0.48

Figure 8.23: Numerical mesh used for the compliant walled stenosis simulations.

175




Table 8.6: Under-relaxation factors used by the fluid solver.

u-displacement equation 0.65v-displacement equation 0.65

Table 8.7: Under-relaxation factors used by the solid solver.


The working fluid was water with density, ρ = 998.0 kg/m3, and dynamic viscosity,

µ = 1.003 × 10−3 Pa s.

As already noted, the paper of Stergiopulos et al. (1993) did not specify the wall

thickness of the elastic material. Unfortunately, it also did not give the exact location

at which the reference pressure was measured, or values for the wall material properties,

but it reported that it was manufactured from Sylgard 184. These pieces of informa-

tion govern the displacement of the elastic tube wall. To obtain good estimates of the

required information, two tests were performed: firstly, a rigid walled CFD simulation

was performed at the flow rate for which the experiment reported zero wall displace-

ment. The reference pressure was then chosen so that the pressure difference between

the inside and outside of the wall at the throat of the stenosis was zero. The result of

this test was that the external pressure had to be increased from the reported 3 mmHg

to 42.1 mmHg. This pressure can easily be verified by application of Bernoulli’s equa-

tion between the tube inlet and the stenosis throat. Analytical calculations show that

pressure losses are negligibly small compared to Bernoulli losses over these tube lengths.

Secondly, sensitivity tests were conducted by varying the tube wall thickness and elastic

modulus, the objective being to accurately predict the throat diameter variation with

increasing flow rate. The elastic modulus was varied between 1.3 – 1.8 MPa, which is

slightly lower than those measured experimentally for Sylgard 184 by Schneider et al.

(2008). The wall thickness was varied between 0.1 – 0.2 mm. The results from these

sensitivity tests are presented in the following Section.

U -displacement equation 0.25V -displacement equation 0.25

Table 8.8: Under-relaxation factors used by the re-meshing algorithm.

176


8.3.6 Sensitivity Tests

The variation of the elastic modulus and wall thickness have only a small effect on the

pressure drop. This could be due to the fact that the pressure drop is measured over

the full length of the elastic region, after pressure recovery downstream of the throat,

and thus does not depend strongly on the exact deformation at the throat. Variation of

the elastic modulus and wall thickness did, however, have a large effect upon the wall

displacement as can be seen from Figure 8.24. Whilst the displacement profile remains

linear for all combinations of elastic modulus and wall thickness, the gradient increases

with reducing modulus and thickness.

8.3.7 Results and Discussion

Figures 8.25 and 8.26 show a comparison between the numerical and experimental re-

sults. Converged solutions were unobtainable between Reynolds numbers of 1000 and

1500, due to the difficulties in capturing transition described in earlier Sections. Simula-

tions were also not performed beyond the point at which the experiments reported tube

collapse as the current axisymmetric model is not able to capture such an asymmetric

phenomena.

The measured throat diameter decreases almost linearly with flow rate until a crit-

ical flow rate is reached at which point the tube collapses. As a result of the first test

described above, the key point of zero displacement is accurately predicted. The effect

of reducing the elastic modulus and the thickness of the tube wall is an increased rate at

which the throat diameter varies with flow rate. The experimental data is most closely

matched by setting the elastic modulus to 1.3 MPa and the wall thickness to 0.2 mm.

Whilst the results shown in Shim and Kamm (2002) show a similar level of agreement,

the absence of a turbulence model in their simulations limited them to low flow rates.

The numerical results from Shim and Kamm (2002) are reproduced in Figure 8.27.

The presently computed pressure drop across the stenosis, in Figure 8.25, shows

good agreement with the experimental data at low flow rates. However, at flow rates

greater than 15 ml/sec, the numerical results under-predict the pressure drop. The

results from the rigid walled CFD simulations presented earlier suggest that the flow

solver is able to accurately predict the pressure drop across an axisymmetric stenosis

at high flow rates. It is thought that this mismatch could be due to the fact that at

higher flow rates the elastic tube may have deformed into an asymmetric cross-section.

Figure 8.28 shows the wall pressure profiles for increasing axial pressure difference.

The profiles are similar in many respects to the rigid wall cases presented earlier. One

minor difference is that the pressure recovery within the stenosis region has a small

inflexion just downstream of the throat. The non-dimensionalized wall pressure pro-

files for increasing Reynolds number are shown in Figure 8.29. At the lowest Reynolds

177


Pressure Drop (mmHg)

Nor

mal

ized

Thr

oatD

iam

eter

,Dt-

Dt o/

Dt o

0 10 20 30 40 50 60 70-0.12

-0.09

-0.06

-0.03

0

0.03

0.06Experimental DataElastic Modulus = 1.8MPa, Thickness = 0.2mmElastic Modulus = 1.3MPa, Thickness = 0.2mmElastic Modulus = 1.8MPa, Thickness = 0.1mmElastic Modulus = 1.3MPa, Thickness = 0.1mm

(a) Throat displacement sensitivity.


Flo

wra

te(m

l/sec

)

0 10 20 30 40 50 60 700

5

10

15

20

25

Experimental DataElastic Modulus = 1.8MPa, Thickness = 0.2mmElastic Modulus = 1.3MPa, Thickness = 0.2mmElastic Modulus = 1.8MPa, Thickness = 0.1mmElastic Modulus = 1.3MPa, Thickness = 0.1mm

(b) Flow rate sensitivity.

Figure 8.24: Results from the compliant stenosis sensitivity tests.

178



Flo

wra

te(m

l/sec

)

0 10 20 30 40 50 60 700

5

10

15

20

Experimental DataNumerical Results

Figure 8.25: Flow rate comparison for flow through compliant stenosis.


Nor

mal

ized

Thr

oatD

iam

eter

,(D

T-D

T,o)

/DT

,o

0 10 20 30 40 50 60 70-0.12

-0.09

-0.06

-0.03

0

0.03

0.06

Experimental DataNumerical Results

Figure 8.26: Throat diameter comparison for flow through compliant stenosis.

179


Figure 8.27: Numerical results presented by Shim and Kamm (2002); (a) stenosis throatdiameter against pressure difference, (b) flow rate against pressure difference; solid linerepresents numerical prediction, symbols show experimental data of Stergiopulos et al.(1993).

180


X / Diameter

Wal

lPre

ssur

e(P

a)

-4 -2 0 2 4

-100

-80

-60

-40

-20

0


Figure 8.28: Wall pressure profiles for increasing pressure drop for the compliant steno-sis.

X / Diameter

Nor

mal

ized

Wal

lPre

ssur

e(P

/0.5

ρUB2 )

-4 -2 0 2 4-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0


Figure 8.29: Non-dimensionalized wall pressure profiles for increasing inlet Reynoldsnumber for the compliant stenosis.

181


X / Diameter

Wal

lShe

arS

tres

s(P

a)

-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10-40

-20

0

20

40

60

80

100

120

140


Figure 8.30: Wall shear stress profiles for increasing pressure drop for the compliantstenosis.

X / Diameter

Nor

mal

ized

Wal

lShe

arS

tres

s(τ

WA

LL/0

.5ρU

in2 )

-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10-0.5

0

0.5

1

1.5

2


Figure 8.31: Non-dimensionalized wall shear stress profiles for increasing inlet Reynoldsnumber for the compliant stenosis.

182


number (Re = 1408.71) the wall pressure profile shows a constant pressure region down-

stream of the stenosis within the re-circulation bubble. At higher Reynolds numbers,

the re-circulation bubble is much shorter and this constant pressure behaviour is not

observed. The pressure recovery is similar for all profiles for Reynolds numbers of

2539 and greater. The minimum pressure reduces at the stenosis throat for increasing

Reynolds number. This suggests that the flow upstream of the stenosis is still laminar

as viscous losses are still having a noticeable effect upon the total pressure drop.

Figure 8.30 shows the wall shear stress for increasing pressure difference. The

corresponding non-dimensional wall shear stress plot is shown in Figure 8.31. The

peak wall shear stress occurs just upstream of the stenosis throat. The wall shear

stress then changes in sign as the flow separated just downstream of the throat. From

the non-dimensional plot, it can be seen that at the lowest Reynolds number, the peak

shear stress value is slightly greater than values at higher Reynolds numbers - which are

all approximately equivalent. The behaviour downstream of the stenosis also shows a

difference between the lowest Reynolds number case and the others. This suggests that

the first case (Re = 1408.71) has not completely undergone transition to turbulence.

Figures 8.32 and 8.33 show the u-displacement (axial displacement) and v-displacement

(radial displacement) for increasing axial pressure drop respectively. The corresponding

wall tractions are shown in Figure 8.34. At low pressure differences, the v-displacement

is positive along the entire length of the compliant section; this is because the internal

pressure is greater than the external pressure along the entire length. As the pres-

sure drop between the inlet and outlet increases, and hence flow rate is increased, the

pressure at the stenosis throat becomes lower than the external pressure, causing a

negative v-displacement in the throat region. This transition from positive to negative

v-displacement at the throat occurs at approximately 20 Pa pressure difference. As the

axial pressure difference is further increased, the fluid pressure becomes lower than the

external pressure along the entire downstream compliant wall section. At these higher

pressure drops, the point of greatest (negative) v-displacement moves from the throat

to a point near the downstream end of the stenosis region. At the lowest axial pres-

sure drop, the u-displacement profile is negative along most of the compliant section

except for a region upstream of the throat. This is caused by a combination of the fluid

pressure acting on the wall of the converging section of the stenosis and the relatively

high wall shear stress upstream of the throat. As the axial pressure drop increases, the

downstream section of the tube displaces radially inwards; this causes a corresponding

displacement in the positive u-direction. The u-displacement upstream of the throat is

in the negative u-direction. The point of zero u-displacement lies just downstream of

the throat.

Figures 8.35–8.39 show the flow field (flow streamlines and velocity vectors above

the line of symmetry and contours of turbulent kinetic energy below) in the deformed

183


X / Diameter

Udi

sp/D

iam

eter

-4 -2 0 2 4-0.03

-0.02

-0.01

0

0.01

0.02

0.03

(a) Delp = 10 Pa

X / Diameter

Udi

sp/D

iam

eter

-4 -2 0 2 4-0.03

-0.02

-0.01

0

0.01

0.02

0.03

(b) Delp = 20 Pa

X / Diameter

Udi

sp/D

iam

eter

-4 -2 0 2 4-0.03

-0.02

-0.01

0

0.01

0.02

0.03

(c) Delp = 30 Pa

X / Diameter

Udi

sp/D

iam

eter

-4 -2 0 2 4-0.03

-0.02

-0.01

0

0.01

0.02

0.03

(d) Delp = 40 Pa

X / Diameter

Udi

sp/D

iam

eter

-4 -2 0 2 4-0.03

-0.02

-0.01

0

0.01

0.02

0.03

(e) Delp = 50 Pa

Figure 8.32: u-displacement (axial) profiles for increasing pressure drop for the com-pliant stenosis.

184


X / Diameter

Vdi

sp/D

iam

eter

-4 -2 0 2 4-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

(a) Delp = 10 Pa

X / Diameter

Vdi

sp/D

iam

eter

-4 -2 0 2 4-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

(b) Delp = 20 Pa

X / Diameter

Vdi

sp/D

iam

eter

-4 -2 0 2 4-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

(c) Delp = 30 Pa

X / Diameter

Vdi

sp/D

iam

eter

-4 -2 0 2 4-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

(d) Delp = 40 Pa

X / Diameter

Vdi

sp/D

iam

eter

-4 -2 0 2 4-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

(e) Delp = 50 Pa

Figure 8.33: v-displacement (radial) profiles for increasing pressure drop for the com-pliant stenosis.

185


X / Diameter

Tra

ctio

n(P

a)

-4 -2 0 2 4

-5.0E+04

0.0E+00

5.0E+04

1.0E+05


(a) Delp = 10 Pa

X / Diameter

Tra

ctio

n(P

a)

-4 -2 0 2 4

-5.0E+04

0.0E+00

5.0E+04

1.0E+05


(b) Delp = 20 Pa

X / Diameter

Tra

ctio

n(P

a)

-4 -2 0 2 4

-5.0E+04

0.0E+00

5.0E+04

1.0E+05


(c) Delp = 30 Pa

X / Diameter

Tra

ctio

n(P

a)

-4 -2 0 2 4

-5.0E+04

0.0E+00

5.0E+04

1.0E+05


(d) Delp = 40 Pa

X / Diameter

Tra

ctio

n(P

a)

-4 -2 0 2 4

-5.0E+04

0.0E+00

5.0E+04

1.0E+05


(e) Delp = 50 Pa

Figure 8.34: Wall stress profiles for increasing pressure drop for the compliant stenosis.

186

8.4. SUMMARY

compliant walled tube for increasing axial pressure drop. The upstream section of

the compliant tube is shown in the upper sub-figure whilst the downstream section

is continued in the lower sub-figure. It can be seen that the wall bulges outwards at

the upstream end of the compliant section at all axial pressure differences due to the

higher internal pressure. As the axial pressure difference is increased, the downstream

end of the compliant section is sucked downwards as described previously. The flow

field is qualitatively similar to the rigid wall case, however, the negative v-displacement

at the throat at higher pressure drops causes a slightly larger re-circulation region and

stronger core flow. At lower flow rates, corresponding to a pressure drop of 10 Pa,

the slight area reduction at the downstream end of the compliant section seems to

aid flow re-attachment and shorten the re-circulation bubble. The flow upstream of

the stenosis throat is laminar for all pressure differences. Figure 8.40 shows both the

original and deformed shape of the compliant wall at a pressure drop of 50 Pascals. The

positive transmural pressure difference upstream of the stenosis throat has caused the

compliant wall to deform radially outwards in this region; the deformed tube diameter

is constant throughout most of this upstream region, except in the vicinity of the fixed

displacement condition at the west boundary of the solid domain.

8.4 Summary

This Chapter has described the numerical simulation of stenosed flow. Initial rigid-

walled calculations were used to validate the flow solver. Comparisons with experimen-

tal data showed that the solver was capable of accurately predicting the pressure drop

across a severe axisymmetric stenosis for a wide range of physiologically relevant flow

rates.

Laminar simulations were only successful for Reynolds numbers up to 300. At these

Reynolds numbers a very large separation bubble existed downstream of the stenosis

throat. This caused numerical difficulties as the flow domain had to be extended to

allow the flow to fully re-attach upstream of the outlet; also, such a large separation

bubble would be unstable and prone to large scale unsteadiness - the physics of which

can not be captured with a steady-state flow solver.

Turbulent simulations using the Launder-Sharma low-Re model were successful for

Reynolds numbers greater than 1500. The flow was laminar upstream of the stenosis

before transitioning slightly downstream of the throat due to small scale instabilities

within the shear layer between the separation bubble and core flow. Turbulent levels

decayed downstream of the separation bubble. As the Reynolds number was increased,

the separation bubble reduced in size, and moved upstream towards the throat; this

caused a corresponding upstream migration of the high k region.

Fully coupled simulations showed good agreement with experimental data in terms

187

CH

AP

TE

R8.

ST

EN

OSE

DFLO

WS

R/D

iam

eter

-5.2 -4.8 -4.4 -4 -3.6 -3.2 -2.8 -2.4 -2 -1.6 -1.2 -0.8 -0.4 0-1

-0.5

0

0.5

1

SRR: -40000 -10000 20000 50000 80000 11000010

1 2 3 4 5 6

X / Diameter

R/D

iam

eter

0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4 4.4 4.8 5.2-1

-0.5

0

0.5

1

LevelTKE/UIN**2:

10.5

21

31.5

42

52.5

63

73.5

Figure 8.35: Flow streamlines and velocity vectors (above line of symmetry), and turbulent kinetic energy contours (below line ofsymmetry) for the compliant stenosis at pressure drop = 10 Pa. Compliant wall shows contours of radial stress (Pa).

188

8.4

.SU

MM

ARY

R/D

iam

eter

-5.2 -4.8 -4.4 -4 -3.6 -3.2 -2.8 -2.4 -2 -1.6 -1.2 -0.8 -0.4 0-1

-0.5

0

0.5

1

SRR: -40000 -10000 20000 50000 80000 11000010

6 5 4 3 2 1

X / Diameter

R/D

iam

eter

0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4 4.4 4.8 5.2-1

-0.5

0

0.5

1

LevelTKE/UIN**2:

10.5

21

31.5

42

52.5

63

73.5


189

CH

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TE

R8.

ST

EN

OSE

DFLO

WS

6 5 4 3 2 1

X / Diameter

R/D

iam

eter

0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4 4.4 4.8 5.2-1

-0.5

0

0.5

1

LevelTKE/UIN**2:

10.5

21

31.5

42

52.5

63

73.5

R/D

iam

eter

-5.2 -4.8 -4.4 -4 -3.6 -3.2 -2.8 -2.4 -2 -1.6 -1.2 -0.8 -0.4 0-1

-0.5

0

0.5

1

SRR: -40000 -10000 20000 50000 80000 11000010


190

8.4

.SU

MM

ARY

R/D

iam

eter

-5.2 -4.8 -4.4 -4 -3.6 -3.2 -2.8 -2.4 -2 -1.6 -1.2 -0.8 -0.4 0-1

-0.5

0

0.5

1

SRR: -40000 -10000 20000 50000 80000 11000010

6 5 4 3 2 1

X / Diameter

R/D

iam

eter

0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4 4.4 4.8 5.2-1

-0.5

0

0.5

1

LevelTKE/UIN**2:

10.5

21

31.5

42

52.5

63

73.5


191

CH

AP

TE

R8.

ST

EN

OSE

DFLO

WS

1

R/D

iam

eter

-5.2 -4.8 -4.4 -4 -3.6 -3.2 -2.8 -2.4 -2 -1.6 -1.2 -0.8 -0.4 0-1

-0.5

0

0.5

1

SRR: -40000 -10000 20000 50000 80000 11000010

6 5 4 3 2 1

X / Diameter

R/D

iam

eter

0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4 4.4 4.8 5.2-1

-0.5

0

0.5

1

LevelTKE/UIN**2:

10.5

21

31.5

42

52.5

63

73.5


192

8.4

.SU

MM

ARY

R/D

iam

eter

-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0

0

0.2

0.4

0.6

X / Diameter

R/D

iam

eter

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0

0.2

0.4

0.6

Figure 8.40: Deformation of the compliant stenosis at ∆P = 50Pa. The black line represents the deformed stenosis geometry, whilstthe grey line represents the un-deformed geometry.

193


of wall response to streamwise pressure difference. Due to the lack of detailed informa-

tion regarding material properties used during the experiments a number of parametric

tests were performed to obtain good estimates. The properties obtained from these

tests were broadly in agreement with experimentally measured values. As in the rigid

walled case, there existed a range of intermediate Reynolds numbers for which con-

verged solutions were not achievable due to the numerical difficulties associated with

capturing turbulent transition.

The finite-volume method has proven itself to be more than capable of accurately

predicting internal flows through stenosed passages with compliant walls. The use of

the same numerical method for both the fluid and solid domains to create a coupled

FSI solver is both efficient and accurate. The use of the finite-volume solid body solver

to adapt the fluid portion of the numerical mesh preserves the original mesh features,

such as near-wall node clustering.

Many physiological flows occur within the transitional Reynolds number regime.

The Launder-Sharma low-Reynolds-number turbulence model has been shown to be

accurate and efficient for such flows. In the case of stenosed flows, transition can occur

at much lower Reynolds numbers than would be expected for straight pipe flows. To

compute such flows, it is necessary to prescribe an initially non-zero turbulence field

throughout the fluid domain and allow turbulence levels to decay in the laminar regions

- such as those upstream of the stenosis throat at low Reynolds numbers.

194

Chapter 9

Aneurysmal Flows

9.1 Introduction

The second and final physiological flow to be considered in this work is that of flow

through an aneurysmal tube. An aneurysm is a localised expansion of an artery due to

a weakened wall material. Two types of aneurysm exist: fusiform, which are axisym-

metric expansions which commonly occur in the abdominal aorta; and saccular, which

are balloon like, non-symmetric expansions which commonly occur in the brain. This

Chapter considers only the fusiform type of aneurysm.

Initially, rigid walled simulations of steady and unsteady flow are considered to

validate the flow solver. Both laminar and turbulent flows are considered to cover the

entire range of physiologically relevant flow rates. Results are compared to experimental

data for a range of aneurysm geometries. Unsteady FSI simulations are then performed

and the results compared to numerical data from the literature.

9.2 Steady Laminar Flow Through a Rigid Aneurysm

The first case to be considered was that of steady-state, laminar flow through a rigid

aneurysm. In total, four different aneurysm models, in terms of aneurysm bulge di-

ameter, were considered for a range of Reynolds numbers extending from 400 to 2000.

Results were compared to the experimental data of Budwig et al. (1993), which con-

sists of LDV velocity measurements made across a radial profile at the centre of the

aneurysm.


The geometry consisted of a tube of diameter d = 1.9 cm with an axisymmetric

aneurysm of length 4 diameters and bulge diameter, D, measured at the centre of

the aneurysm. Four aneurysm models were considered with bulge diameters, D/d, of

195

CHAPTER 9. ANEURYSMAL FLOWS

Model 1 D/d = 1.3Model 2 D/d = 1.5Model 3 D/d = 1.8Model 4 D/d = 2.1

Table 9.1: Summary of models used in the steady state, rigid walled aneurysm simula-tions.

Coarse Grid NI = 310 NJ = 40Medium Grid NI = 388 NJ = 50Fine Grid NI = 465 NJ = 60

Table 9.2: Mesh configurations used in the steady state, rigid walled aneurysm simu-lations.

1.3, 1.5, 1.8 and 2.1; the bulge diameter of the various aneurysm models is summarised

in Table 9.1. Upstream of the aneurysm was a straight section of length 3.7 diameters;

downstream of the aneurysm was a straight section of length 11 diameters. The long

downstream sections used in the stenosis simulations are not necessary in this case as

the size of re-circulation bubble is constrained by the aneurysm.


Three meshes of various node densities were considered to ensure mesh independence;

the three densities are shown in Figure 9.1. The total number of nodes in both the

streamwise and radial directions for each of the three meshes is summarised in Table 9.2.

The nodes were uniformly spaced in the axial direction; in the radial direction, an

expansion ratio of Ry = 0.98 was used to cluster the nodes around the wall to better

resolve the near-wall gradients of the flow variables.


The boundary conditions used for the simulations are summarised in Table 9.3.


The working fluid was water with density, ρ = 1.0× 103 kg/m3, and dynamic viscosity,

µ = 1.0 × 10−3 Pa s.


The under-relaxation factors used for the simulations are summarised in Table 9.4.

196

9.2. STEADY LAMINAR FLOW THROUGH A RIGID ANEURYSM

X / Diameter

R/D

iam

eter

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

1

1.2

(a) Coarse mesh

X / Diameter

R/D

iam

eter

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

1

1.2

(b) Medium mesh

X / Diameter

R/D

iam

eter

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

1

1.2

(c) Fine mesh

Figure 9.1: Coarse, medium and fine meshes for the rigid walled aneurysm (Model 4,D/d = 2.1).

197



The U -velocity profiles across the centre (X/D = 0) of the aneurysm with D/d = 1.8

(Model 3) at Reynolds numbers of 800 and 2000 are shown in Figure 9.2. The profile

shows a central core flow which maintains the parabolic profile of the straight inlet

section with a large but slow moving re-circulation region in the aneurysm bulge. It

can be seen that the medium and fine meshes produce the same result; the coarse

mesh slightly under-predicts the peak velocity in comparison with the medium and fine

meshes. In Figure 9.2 the agreement with the experimental data is mostly good for all

three mesh densities. However, there is a slight under-prediction (particularly at the

higher Reynolds number) between r/R = 0.15 and 0.4; in the higher Reynolds number

case, the velocity magnitude in the recirculation region is slightly over-predicted. The

results obtained with the laminar flow solver at a Reynolds number of 800 are in much

closer agreement with the experimental data that those obtained at a Reynolds number

of 2000; this suggests that the flow may be close to turbulent transition at this higher

Reynolds number. It is quite possible that instabilities in the shear-layer within the

aneurysm bulge could lead to turbulent transition at a lower Reynolds number than

would be expected for straight pipe flow. At the higher Reynolds number, both the

experimental data and numerical results show that the velocity profile within the central

core flow is flatter than at the lower Reynolds number.

Figure 9.3 shows the central U -velocity profile in the aneurysm with the largest

bulge diameter (Model 4, D/d = 2.1) at Reynolds numbers of 400 and 2000. At the

lower Reynolds number, Re = 400, the numerical results are in good agreement with

the experimental data; the computed velocity at the centre-line is, however, slightly

under-predicted. The results obtained with the fine mesh are in best agreement with

the experimental data at the centre-line. At the higher Reynolds number, the numerical

results are not in such good agreement with the experimental data. The velocity in

Inlet (West) Parabolic (laminar) U -velocity profileZero V -velocityInlet pressure extrapolated from interior of domain(See Section 5.9.1 for details)

Outlet (East) Mass correction condition for pressure(See Section 5.9.2 for details)Zero streamwise gradient for all other variables

Wall (North) No-slip condition for velocity componentsZero wall normal gradient for pressure

Symmetry (South) Zero V -velocityZero normal gradient for all other variables

Table 9.3: Boundary conditions used for the steady state, rigid walled aneurysm simu-lations.

198

9.2. STEADY LAMINAR FLOW THROUGH A RIGID ANEURYSM

R / Radius

U/U

in

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

Coarse GridMedium GridFine GridExperimental Data


R / Radius

U/U

in

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1



Figure 9.2: Comparison with the experimental data of Budwig et al. (1993). Aneurysmcentre velocity profiles, Model 3.

199




Table 9.4: Under-relaxation factors used for the steady state, rigid walled aneurysmsimulations.

the central core region is under-predicted whilst the magnitude of the very flat profile

in the re-circulation region is over-predicted. This could again be due to the laminar

solver’s inability to accurately resolve this transitional flow.

The flow fields in aneurysm Model 3 (D/d = 1.8) at Reynolds numbers of 800 and

1500 are shown in Figure 9.4. As the flow enters the aneurysm at the lower Reynolds

number, the laminar velocity profile of the straight inlet section is preserved with only

a very small re-circulation region in the bulge region. Moving further downstream, as

the aneurysm bulge increases, the re-circulation region grows to fill the larger diameter.

The peak velocity at the centre-line of the tube increases accordingly to ensure mass

conservation. As the aneurysm bulge begins to reduce in size, the re-circulation region

reduces in size (in the radial direction) and the velocity increases which in turn further

increases the velocity in the core region. At the exit of the aneurysm, the velocity

profile has a steep near-wall gradient, as might be seen in a fully developed turbulent

pipe flow. Further downstream, the near-wall velocity gradient diminishes and the

flow returns to the fully developed laminar velocity profile. The flow at the higher

Reynolds number is broadly similar and is characterised by a central core region with

a parabolic profile with a large re-circulation region, the downstream end of which has

much stronger back-flow, filling the bulge region.

The flow fields in the largest aneurysm (Model 4, D/d = 2.1) show broadly the same

characteristics as those in the smaller model (Figure 9.5). The re-circulation region is

larger as it now fills a larger aneurysm. The velocity profile is slightly flatter in both

the core region and the re-circulation bubble.

9.3 Steady Turbulent Flow Through a Rigid Aneurysm

In a similar work to that of Budwig et al. (1993), Asbury et al. (1995) experimentally

investigated steady flow through a rigid aneurysm. Whilst the experimental study of

Budwig et al. (1993) provided detailed experimental data at a specific location within

the aneurysm, the study of Asbury et al. (1995) provides data for the entire flow

field. Centre line turbulence intensity profiles were also provided to allow appropri-

ate turbulence modelling strategies to be identified. Although the primary focus of

this investigation was to develop the most accurate turbulence modelling strategies for

aneurysmal flow, both laminar and turbulent flow simulations were performed to cover

200

9.3. STEADY TURBULENT FLOW THROUGH A RIGID ANEURYSM

R / Radius

U/U

in

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1



R / Radius

U/U

in

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1



Figure 9.3: Comparison with the experimental data of Budwig et al. (1993). Aneurysmcentre velocity profiles, Model 4.

201

CH

AP

TE

R9.

AN

EU

RY

SM

AL

FLO

WS

X / Diameter

R/D

iam

eter

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

-1

-0.5

0

0.5

1

1


X / Diameter

R/D

iam

eter

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

-1

-0.5

0

0.5

1

1


Figure 9.4: Velocity vectors and flow streamlines in aneurysm Model 3.

202

9.3

.ST

EA

DY

TU

RB

ULE

NT

FLO

WT

HR

OU

GH

AR

IGID

AN

EU

RY

SM

X / Diameter

R/D

iam

eter

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

-1

-0.5

0

0.5

1

1


X / Diameter

R/D

iam

eter

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

-1

-0.5

0

0.5

1

1


Figure 9.5: Velocity vectors and flow streamlines in aneurysm Model 4.

203


Model 1 D/d = 1.41Model 2 D/d = 1.88Model 3 D/d = 2.27Model 4 D/d = 2.56Model 5 D/d = 2.75Model 6 D/d = 2.97Model 7 D/d = 3.27

Table 9.5: Summary of models used in the steady state, rigid walled aneurysm simula-tions.

X / Diameter

R/D

iam

eter

-4 -3 -2 -1 0 1 2 30

0.5

1

1.5

Figure 9.6: Mesh used for the steady state, rigid walled aneurysm simulations.

a range of physiologically relevant Reynolds numbers. The laminar flow simulations

were performed to further validate the flow solver as the previous case only allowed for

comparisons with the experimental data at a single profile within the flow domain.


The geometry consisted of a straight tube of diameter, d = 1.3 cm and length 2.7 diam-

eters, followed by a rigid aneurysm of length 6 diameters. A total of 7 aneurysm models

were considered with bulge diameters varying from 1.4 cm to 3.3 cm. A straight tube

section of length 10 diameters lies downstream of the aneurysm. The bulge diameter

ratios of the various models are summarised in Table 9.5.


The numerical mesh consisted of 555 uniformly spaced nodes in the axial direction

and 60 nodes in the radial direction. The nodes in the radial direction were clustered

around the wall with an expansion ratio of Ry = 0.98. The numerical mesh within the

aneurysm region of Model 5 can be seen in Figure 9.6.

204

9.3. STEADY TURBULENT FLOW THROUGH A RIGID ANEURYSM


The boundary conditions used during both laminar and turbulent flow simulations are

summarised in Table 9.6. Reynolds numbers of 500 and 2600 were considered. The

parabolic inlet velocity profile was used for both laminar and turbulent flow calculations.

Whilst it is not common to use a parabolic inlet velocity profile for a turbulent flow, it

was felt that by initialising the flow field with a high turbulence intensity, the flow would

transition to turbulence - as happened in the turbulent stenosis case. In the turbulent

case, the flow field was initialized with a turbulence intensity of 20% throughout the flow

domain. Many simulations were performed for a range of inlet and initial conditions to

find values which gave the most accurate results. Despite appearing quite high for such

a flow, it was found that the closest agreement with the experimental data could be

achieved by prescribing an inlet turbulence intensity of 8% and turbulent viscosity ratio

of 1.5. Transitional flows are extremely sensitive to boundary and initial conditions;

small changes in the inlet values can lead to relatively large changes in the results - or

even divergence.


The working fluid was a mixture of distilled water and sodium thiocyanate with density,

ρ = 1.17 × 103 kg/m3, and dynamic viscosity, µ = 1.6497 × 10−3 Pa s.


The under-relaxation factors used during the simulations are summarised in Table 9.7.

The same under-relaxation factors were used for the momentum and pressure correction

equations for both laminar and turbulent flow calculations.


The difference between the laminar and turbulent flow field within the aneurysm can

be clearly seen in Figure 9.7. The normalized laminar velocity along the centre line

West (Inlet) Parabolic U -velocity profilekin = 8%µ/µt|in = 1.5

East (Outlet) Zero streamwise gradients for all variablesNorth (Wall) No-slip velocity

Zero wall normal pressure gradientSouth (Symmetry) Zero V -velocity

Zero normal gradient for all other variables

Table 9.6: Boundary conditions used in the rigid walled aneurysm simulations.

205




Table 9.7: Under-relaxation factors used during the simulation.

X / Diameter

R/D

iam

eter

-4 -3 -2 -1 0 1 2 3 4-1.5

-1

-0.5

0

0.5

1

1.5

1U/Uin

Figure 9.7: Flow streamlines and velocity vectors from the rigid walled aneurysm case(Model 5) at Re = 500 (above line of symmetry) and Re = 2600 (below line of symme-try).

is greater than the turbulent velocity throughout the aneurysm. The re-circulation

bubble is similarly sized in both cases as the separation and re-attachment points are

governed largely by geometry. The back flow is slightly stronger in the laminar case.

In both cases, the re-circulation is stronger at the downstream end of the aneurysm.

Figure 9.8 shows a comparison of the velocity field between the numerical results and

experimental data in aneurysm Model 5 at a Reynolds number of 2600. It can be seen

that each but the final velocity profile is in good agreement with the experimental data.

The numerical results seem to predict a more rapid return towards fully developed flow

after the exit of the aneurysm. This may be due to over-prediction of the turbulence

levels in this region.

The comparison of the centre line turbulence intensity at Reynolds number of 2600

is shown in Figure 9.9. Centre-line turbulence levels are low over the first half of

each aneurysm and then just before x/D = 0 (the maximum width location) there

is a sudden jump, signifying transition to highly turbulent conditions. This feature is

present in all models. This is due to turbulence production in the separated shear layer.

The predicted peak turbulence level increases from Model 1 to Model 3, beyond which

it remains constant until Model 7, after which the level reduces. The experimental data

show the same features as the computations and for most models there is also good

qualitative agreement between predictions and experimental data, with the exception

206

9.3

.ST

EA

DY

TU

RB

ULE

NT

FLO

WT

HR

OU

GH

AR

IGID

AN

EU

RY

SM

Figure 9.8: Comparison with the experimental data of Asbury et al. (1995). Axial velocity profiles for aneurysm Model 5 at Re = 2600.Numerical results shown above line of symmetry, experimental data shown below.

207


of Model 6, which has a much higher peak turbulence intensity, and Model 1 which

shows a lower peak turbulence level. In Model 1, the over-prediction of the turbulence

intensity could be a consequence of prescribing a high initial turbulence field; in this

case, the turbulence levels did not adequately decay. Moreover, the experimental data

for Model 6 seem to be out of line with those for Models 5 and 7, which suggests that

the disagreement between predictions and measurements for Model 6 may be due to

unreliable experimental data.

Measurements of the wall shear stress variation within the aneurysm for Model 2

show that after about four diameters upstream of the middle of the aneurysm the wall

shear stress reduces in the flow direction and at a distance of about two diameters

from the middle, the wall shear stress changes sign. It remains negative, but with

low magnitude, until about one diameter after the half-way point. Beyond the region

of negative values, the measured wall shear stress shows a steep rise. The wall shear

stress (Figure 9.10) is generally in reasonable agreement with the experimental data

for Model 2 at Reynolds number 2600. However, the peak shear stress, which occurs

at the downstream end of the aneurysm, appears to be over-predicted by a factor

of approximately two. This is consistent with the over-prediction of the turbulence

intensity for this model, downstream of the half-way point, shown in Figure 9.9(c).

Figure 9.11 shows the flow field in aneurysm Model 7 at a Reynolds number of

2600. The upper sub-figure shows the velocity vectors and flow streamlines; the middle

and lower sub-figures show contours of turbulent kinetic energy and turbulent viscosity

ratio respectively. The turbulence levels are relatively low in the straight inlet section

and the upstream portion of the aneurysm. Turbulent kinetic energy is produced

within the shear-layer between the core flow and the re-circulation bubble. The highest

turbulence levels are found at the downstream end of the aneurysm where the flow

leaves the aneurysm bulge. The near-wall velocity gradient is high at the downstream

end of the aneurysm as the core flow, which has spread slightly as it passed through

the aneurysm, flows into the straight outlet section.

From the above results, and a number of explorations of alternative treatments,

the best strategy for modelling turbulent flow within these axisymmetric aneurysms

at physiological Reynolds numbers is to prescribe a parabolic laminar velocity profile

at the inlet and specify an initially high turbulence intensity throughout the flow do-

main. However, it still proved difficult to accurately predict the centre-line turbulence

intensity for all of the aneurysm models.

9.4 Unsteady Laminar Flow Through a Rigid Aneurysm

The third case to be considered is that of laminar, unsteady flow through a rigid-

walled aneurysm. Results were compared to the experimental data obtained by the

208

9.4. UNSTEADY LAMINAR FLOW THROUGH A RIGID ANEURYSM

X/Diameter

Tu

rbu

len

ceIn

tens

ity

-10 -5 0 5 10 150

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

(a) Model 1

X/Diameter

Tu

rbu

len

ceIn

tens

ity

-10 -5 0 5 10 150

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

(b) Model 2

X/Diameter

Tu

rbu

len

ceIn

tens

ity

-10 -5 0 5 10 150

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

(c) Model 3

X/Diameter

Tu

rbu

len

ceIn

tens

ity

-10 -5 0 5 10 150

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

(d) Model 4

X/Diameter

Tu

rbu

len

ceIn

tens

ity

-10 -5 0 5 10 150

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

(e) Model 5

X/Diameter

Tu

rbu

len

ceIn

tens

ity

-10 -5 0 5 10 150

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

(f) Model 6

X/Diameter

Tu

rbu

len

ceIn

tens

ity

-10 -5 0 5 10 150

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

(g) Model 7

Figure 9.9: Centre line turbulence intensity profiles at Re = 2600. Symbols representthe experimental data of Asbury et al. (1995), solid line represents the present numericalresults.

209


X/Diameter

No

rmal

ized

WS

S

-10 -5 0 5 10 15-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Figure 9.10: Normalized wall shear stress profile in aneurysm Model 2 at Re = 2600.Symbols represent the experimental data of Asbury et al. (1995), solid line representsthe present numerical results.

PIV technique by Yu (2000). The purpose of this test was to ensure that the flow

solver can accurately simulate unsteady flow through an aneurysm before the more

complex case of unsteady flow through a compliant walled aneurysm was considered.

The inlet velocity waveform was sinusoidal in nature. Two cases were considered with

different inlet velocity frequency and magnitude.


The physical geometry consisted of a straight inlet section of diameter 40 mm and length

3.75 diameters. This was followed by an aneurysm section of length 2.2 diameters.

Finally, an outlet section of length 2.65 diameters followed downstream of the aneurysm.

The geometry of the aneurysm was elliptical in nature and defined by the following

algebraic equation:

(x2

(A/2)2

)

+

(y2

(D/2)2

)

= 1 (9.1)

where x and y are the axial and radial coordinates respectively; A = 100 mm is the

aneurysm length, and D = 80 mm is the aneurysm bulge diameter. Small fillet radii

were included at the upstream and downstream ends of the elliptic bulge to give a

smooth transition from the straight pipe sections to the aneurysm.

210

9.4

.U

NST

EA

DY

LA

MIN

AR

FLO

WT

HR

OU

GH

AR

IGID

AN

EU

RY

SM

X/D

Y/D

-4 -3 -2 -1 0 1 2 3 40

0.5

1

1.5LevelPSI:

11E-06

22E-06

33E-06

44E-06

55E-06

66E-06

77E-06

88E-06

99E-061

0.01

0.01

0.01

0.01

0.02

0.02

0.02

0.03

0.03

0.04

0.04

0.04

0.05

0.05 0.05

0.06

0.06

0.06

0.07

0.07

X/D

Y/D

-4 -3 -2 -1 0 1 2 3 40

0.5

1

1.5Level

TKE/UIN**2:1

0.012

0.023

0.034

0.045

0.056

0.067

0.07

10

10

10

20

2020

20

30

30

30

30

40

40

40

50

50

50

X/D

Y/D

-4 -3 -2 -1 0 1 2 3 40

0.5

1

1.5Level

TVIS/VIS:1

102

203

304

405

50

Figure 9.11: Flow streamlines and velocity vectors (upper figure); contours of turbulent kinetic energy (central figure) and contours ofthe turbulent viscosity ratio in aneurysm Model 7 at Re = 2600.

211


X / Diameter

R/D

iam

eter

-1.5 -1 -0.5 0 0.5 1 1.5

0

0.2

0.4

0.6

0.8

1

Figure 9.12: Numerical mesh used for the unsteady, rigid walled aneurysm case.


The numerical mesh consisted of 300 uniformly spaced nodes in the axial direction and

60 nodes in the radial direction. In the radial direction, an expansion ratio of Ry = 0.98

was used to cluster the nodes in the near-wall region. The mesh within the aneurysm

region is shown in Figure 9.12.


Two cases were considered with different inlet velocity amplitude and frequency. The

inlet velocity was non-dimensionalized using the Reynolds number and the frequency

using the Wormersley number, defined as:

α = 0.5D√

ω/ν (9.2)

where D is the aneurysm bulge diameter, ω is the inlet velocity frequency, and ν is

the fluid kinematic viscosity. The Reynolds number and Wormersley numbers for both

cases are summarised in Table 9.8.

The sinusoidal inlet flow rate waveforms of the form 1 + sin(ωt) are shown in Fig-

ure 9.13; where a and ω are the amplitude and frequency of oscillation respectively.

The parabolic inlet velocity profile was simply scaled according to the time dependent

Case 1 Case 2

Remax 1274 1162Reaverage 1000 1000Remin 726 838Wormersley number, α 17 22

Table 9.8: Summary of inlet conditions for unsteady, rigid walled cases.

212

9.4. UNSTEADY LAMINAR FLOW THROUGH A RIGID ANEURYSM

Time (secs)

Q(t

)/Q

mea

n

0 0.5 1 1.5 2 2.5 30.7

0.8

0.9

1

1.1

1.2

1.3

α = 17α = 22

Figure 9.13: Time history of the inlet flow rate for Cases 1 and 2.

waveform. The boundary conditions are summarised in Table 9.9.

For both Case 1 and 2 a total of 160 time steps were used per time period. In Case

1 (α = 17) this corresponded to a CFL number of approximately 0.64 and in Case 2 a

CFL number of approximately 0.38. In total, five flow cycles were simulated to ensure

that fully periodic conditions were arrived at.


The working fluid was water with density, ρ = 1121 kg/m3, and viscosity, µ = 3.5 ×10−3 Pa s.

Inlet (West) Parabolic (laminar) U -velocity profileZero V -velocityInlet pressure extrapolated from interior of domain(See Section 5.9.1 for details)


Wall (North) No-slip condition for velocity componentsZero wall normal gradient for pressure

Symmetry (South) Zero V -velocityZero normal gradient for all other variables

Table 9.9: Boundary conditions used for the unsteady, rigid walled aneurysm simula-tions.

213




Table 9.10: Under-relaxation factors used during the unsteady, rigid walled aneurysmsimulations.


The under-relaxation factors used during the simulations are summarised in Table 9.10.


The velocity vectors and flow streamlines obtained numerically for Case 1 can be com-

pared to the experimental data in Figure 9.14. It can be seen that a weak re-circulation

bubble begins to form in the centre of the aneurysm bulge at the start of the flow decel-

eration phase of the flow cycle (t/T = 0.375). As the flow continues to decelerate, this

re-circulation bubble begins to grow in strength (in terms of velocity magnitude and

radial dimension) and move downstream toward the downstream end of the aneurysm.

By the end of the deceleration phase (t/T = 0.813) the re-circulation bubble is strong

enough to influence the core flow region through the centre of the aneurysm; this is

a feature that was not found in the steady state flows considered earlier. As the flow

begins to accelerate once more, t/T = 0.0625 to 0.25, the re-circulation bubble begins

to reduce in strength and size. Until the flow once more begins to decelerate, the flow

through the aneurysm is characterised by a strong central core region surrounded by

two small, slow moving re-circulation regions at the upstream and downstream ends of

the aneurysm bulge. These relatively small re-circulation bubbles combine to form the

single, central bubble which is later convected downstream as described earlier.

The numerical and experimental flow fields for Case 2 can be seen in Figures 9.15. In

this case, the amplitude of the inlet velocity wave form is greater, whilst the frequency is

lower. The flow is broadly similar to that of the lower frequency case described earlier.

In the high frequency case, the two small re-circulation bubbles exist for a greater

proportion of the flow cycle before combining; as before, the combined re-circulation

bubble is convected to the downstream end of the aneurysm. The experimentally

measured re-circulation bubble is much stronger in the higher frequency case; whilst

the numerical results broadly agree with this observation, the difference in re-circulation

strength is not so pronounced between the two cases.

214

9.4

.U

NST

EA

DY

LA

MIN

AR

FLO

WT

HR

OU

GH

AR

IGID

AN

EU

RY

SM

X / Diameter

R/D

iam

eter

-1.5 -1 -0.5 0 0.5 1 1.5

0

0.2

0.4

0.6

0.8

1

1.2

1

(a) Numerical, t/T = 0.0625

(b) Experimental, t/T = 0.0625

215

CH

AP

TE

R9.

AN

EU

RY

SM

AL

FLO

WS

X / Diameter

R/D

iam

eter

-1.5 -1 -0.5 0 0.5 1 1.5

0

0.2

0.4

0.6

0.8

1

1.2

1

(c) Numerical, t/T = 0.1875

(d) Experimental, t/T = 0.1875

216

9.4

.U

NST

EA

DY

LA

MIN

AR

FLO

WT

HR

OU

GH

AR

IGID

AN

EU

RY

SM

X / Diameter

R/D

iam

eter

-1.5 -1 -0.5 0 0.5 1 1.5

0

0.2

0.4

0.6

0.8

1

1.2

1

(e) Numerical, t/T = 0.25

(f) Experimental, t/T = 0.25

217

CH

AP

TE

R9.

AN

EU

RY

SM

AL

FLO

WS

X / Diameter

R/D

iam

eter

-1.5 -1 -0.5 0 0.5 1 1.5

0

0.2

0.4

0.6

0.8

1

1.2

1

(g) Numerical, t/T = 0.375

(h) Experimental, t/T = 0.375

218

9.4

.U

NST

EA

DY

LA

MIN

AR

FLO

WT

HR

OU

GH

AR

IGID

AN

EU

RY

SM

X / Diameter

R/D

iam

eter

-1.5 -1 -0.5 0 0.5 1 1.5

0

0.2

0.4

0.6

0.8

1

1.2

1

(i) Numerical, t/T = 0.4375

(j) Experimental, t/T = 0.4375

219

CH

AP

TE

R9.

AN

EU

RY

SM

AL

FLO

WS

X / Diameter

R/D

iam

eter

-1.5 -1 -0.5 0 0.5 1 1.5

0

0.2

0.4

0.6

0.8

1

1.2

1

(k) Numerical, t/T = 0.5

(l) Experimental, t/T = 0.5

220

9.4

.U

NST

EA

DY

LA

MIN

AR

FLO

WT

HR

OU

GH

AR

IGID

AN

EU

RY

SM

X / Diameter

R/D

iam

eter

-1.5 -1 -0.5 0 0.5 1 1.5

0

0.2

0.4

0.6

0.8

1

1.2

1

(m) Numerical, t/T = 0.625

(n) Experimental, t/T = 0.625

221

CH

AP

TE

R9.

AN

EU

RY

SM

AL

FLO

WS

X / Diameter

R/D

iam

eter

-1.5 -1 -0.5 0 0.5 1 1.5

0

0.2

0.4

0.6

0.8

1

1.2

1

(o) Numerical, t/T = 0.6875

(p) Experimental, t/T = 0.6875

222

9.4

.U

NST

EA

DY

LA

MIN

AR

FLO

WT

HR

OU

GH

AR

IGID

AN

EU

RY

SM

X / Diameter

R/D

iam

eter

-1.5 -1 -0.5 0 0.5 1 1.5

0

0.2

0.4

0.6

0.8

1

1.2

1

(q) Numerical, t/T = 0.8125

(r) Experimental, t/T = 0.8125

223

CH

AP

TE

R9.

AN

EU

RY

SM

AL

FLO

WS

X / Diameter

R/D

iam

eter

-1.5 -1 -0.5 0 0.5 1 1.5

0

0.2

0.4

0.6

0.8

1

1.2

1

(s) Numerical, t/T = 0.875

(t) Experimental, t/T = 0.875

Figure 9.14: Comparison between the experimental data of Yu (2000) and the numerical results for the unsteady, rigid-walled case(Wormersley number = 17).

224

9.4

.U

NST

EA

DY

LA

MIN

AR

FLO

WT

HR

OU

GH

AR

IGID

AN

EU

RY

SM

X / Diameter

R/D

iam

eter

-1.5 -1 -0.5 0 0.5 1 1.5

0

0.2

0.4

0.6

0.8

1

1.2

1

(a) Numerical, t/T = 0.0625

(b) Experimental, t/T = 0.0625

225

CH

AP

TE

R9.

AN

EU

RY

SM

AL

FLO

WS

X / Diameter

R/D

iam

eter

-1.5 -1 -0.5 0 0.5 1 1.5

0

0.2

0.4

0.6

0.8

1

1.2

1

(c) Numerical, t/T = 0.1875

226

9.4

.U

NST

EA

DY

LA

MIN

AR

FLO

WT

HR

OU

GH

AR

IGID

AN

EU

RY

SM

X / Diameter

R/D

iam

eter

-1.5 -1 -0.5 0 0.5 1 1.5

0

0.2

0.4

0.6

0.8

1

1.2

1

(d) Numerical, t/T = 0.25

(e) Experimental, t/T = 0.25

227

CH

AP

TE

R9.

AN

EU

RY

SM

AL

FLO

WS

X / DiameterR

/Dia

met

er-1.5 -1 -0.5 0 0.5 1 1.5

0

0.2

0.4

0.6

0.8

1

1.2

1

(f) Numerical, t/T = 0.375

X / Diameter

R/D

iam

eter

-1.5 -1 -0.5 0 0.5 1 1.5

0

0.2

0.4

0.6

0.8

1

1.2

1

(g) Numerical, t/T = 0.4375

228

9.4

.U

NST

EA

DY

LA

MIN

AR

FLO

WT

HR

OU

GH

AR

IGID

AN

EU

RY

SM

X / Diameter

R/D

iam

eter

-1.5 -1 -0.5 0 0.5 1 1.5

0

0.2

0.4

0.6

0.8

1

1.2

1

(h) Numerical, t/T = 0.5

(i) Experimental, t/T = 0.5

229

CH

AP

TE

R9.

AN

EU

RY

SM

AL

FLO

WS

X / DiameterR

/Dia

met

er-1.5 -1 -0.5 0 0.5 1 1.5

0

0.2

0.4

0.6

0.8

1

1.2

1

(j) Numerical, t/T = 0.625

X / Diameter

R/D

iam

eter

-1.5 -1 -0.5 0 0.5 1 1.5

0

0.2

0.4

0.6

0.8

1

1.2

1

(k) Numerical, t/T = 0.6875

230

9.4

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NST

EA

DY

LA

MIN

AR

FLO

WT

HR

OU

GH

AR

IGID

AN

EU

RY

SM

X / Diameter

R/D

iam

eter

-1.5 -1 -0.5 0 0.5 1 1.5

0

0.2

0.4

0.6

0.8

1

1.2

1

(l) Numerical, t/T = 0.8125

X / Diameter

R/D

iam

eter

-1.5 -1 -0.5 0 0.5 1 1.5

0

0.2

0.4

0.6

0.8

1

1.2

1

(m) Numerical, t/T = 0.875

Figure 9.15: Comparison between the experimental data of Yu (2000) and the numerical results for the unsteady, rigid-walled case(Wormersley number = 22).

231


9.5 Unsteady Flow Through a Compliant Aneurysm

The final case to be considered is that of unsteady flow through a compliant aneurysm.

The results are compared to the numerical data of Khanafer et al. (2009), who reported

results from a RANS-based simulation using a low-Re-number form of the k–ω turbu-

lence model; unlike the present work, elastic wall deformation was computed using the

finite element method. Only turbulent flow is considered in this Section to match the

set-up of the simulations performed by Khanafer et al. (2009) and thus allow direct

comparisons to be made.


The physical geometry consists of an axisymmetric aneurysm of diameter, D = 2.5d,

where d is the diameter of the inlet tube (d = 2.5 cm). The aneurysm has length L = 4d

and is sinusoidal in nature; the profile of the aneurysm wall is given by the following

algebraic equation:

f(x) =D − d

4

[

1 + sin

(2πz

L− π

2

)]

+d

2(9.3)

where d is the diameter of the inlet tube; D is the diameter of the aneurysm; L is the

length of the aneurysm, and x is the axial coordinate with origin at the centre of the

aneurysm. Straight tube sections of length 3.5d and 6.5d lie either side of the aneurysm

section. The compliant wall section has thickness t = 1.5 mm and extends to cover the

aneurysm and part of the straight upstream and downstream sections (from x = −3.5d

to x = 3.5d, where x has its origin at the centre of the aneurysm).


The fluid portion of the numerical mesh consisted of 290 uniformly spaced nodes in

the streamwise direction and 80 nodes in the radial direction. An expansion ratio of

Ry = 0.975 was used in the radial direction to cluster nodes in the near-wall region. The

fluid portion of the numerical mesh around the aneurysm region is shown in Figure 9.16

and in more detail in Figure 9.17. The fluid portion of the mesh was denser than

necessary (sixty radial nodes were shown to be sufficient in the previous simulations),

however, in the case of two-dimensional simulations, using a slightly finer mesh does

not increase the simulation time too greatly.

The solid portion of the numerical mesh consisted of 150 and 10 nodes in the

streamwise and radial directions respectively. Nodes were uniformly distributed in

both directions.

232

9.5. UNSTEADY FLOW THROUGH A COMPLIANT ANEURYSM

X / Diameter

R/D

iam

eter

3 4 5 6 7 8

0

0.5

1

1.5

Figure 9.16: Numerical mesh used for the unsteady, coupled FSI aneurysm simulations.

X / Diameter

R/D

iam

eter

3 3.2 3.4 3.6 3.8 4 4.20.2

0.3

0.4

0.5

0.6

Figure 9.17: Detailed view of the numerical mesh used for the unsteady, coupled FSIaneurysm simulations.

233


Inlet (West) 1/7th power-law (turbulent) U -velocity profileZero V -velocityInlet pressure extrapolated from interior of domain(See Section 5.9.1 for details)


Axis (South) Zero V -velocityZero normal gradient for all other variables

Wall (North) Zero U - and V -velocityZero normal gradient for pressure



Tangential traction equal to fluid wall-shearCompliant wall (North) Prescribed normal traction

Zero tangential traction



The boundary conditions used during the coupled simulations are summarised in Ta-

bles 9.11 and 9.12. The time history of the inlet velocity is shown in Figure 9.18. The

flow cycle had a period, T , of 1.1 seconds; this time period was discretized with 440

time steps of size 2.5 × 10−3 seconds. The Reynolds number, based upon the inlet

diameter d, and peak inlet velocity, was 3348. In the simulations reported by Khanafer

et al. (2009) the external pressure was constant and set to zero, whilst the exit fluid

pressure was varied according to the waveform shown in Figure 9.18. In the present

simulations, it was deemed simpler to maintain the fluid exit pressure at zero, and vary

the external pressure to give the equivalent transmural pressure. In total, fifteen flow

cycles were simulated to ensure cyclic conditions had been achieved and that the effect

of the initial conditions had been removed. It was found tat the flow had reached a

cyclic state after four cycles. The simulation was initiated at the point of maximum

flow rate (t = 0.4 seconds).


The working fluid was water with density, ρ = 1050 kg/m3, and viscosity, µ =

0.00345 Pa s.

The elastic wall material had density, ρ = 2200 kg/m3, elastic modulus, E = 2.7 ×106 Pa, and Poisson ratio, ν = 0.45. The same material properties were used by the

234


Time (seconds)

Vel

ocity

(m/s

)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1-0.1

0

0.1

0.2

0.3

0.4

(a) Inlet velocity waveform

Time (seconds)

Pre

ssur

e(m

mH

g)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.160

70

80

90

100

110

(b) Outlet pressure waveform

Figure 9.18: Inlet velocity and aneurysm exit pressure waveforms used by Khanaferet al. (2009).

235




Table 9.13: Under-relaxation factors used by the fluid solver during the unsteady,coupled FSI aneurysm simulations.


Table 9.14: Under-relaxation factors used by the solid solver during the unsteady,coupled FSI aneurysm simulations.

re-meshing algorithm.


The under-relaxation factors used during the coupled FSI simulation are summarised

in Tables 9.13–9.15. The FSI under-relaxation factor was α = 0.75.

9.5.6 Compliant Wall Results and Discussion

The time history of the bulk fluid velocity at different sections within the compliant

walled aneurysm is shown in Figure 9.19. The bulk velocity is plotted at the inlet,

centre and outlet of the aneurysm. The inlet and outlet velocities are very similar. If

the aneurysm wall was rigid, the inlet and outlet velocities would have to match exactly

(due to mass conservation). However, in the compliant walled case, the flow volume

within the aneurysm is changing, and the difference between the inlet and exit bulk

velocities reflects this time-variation. The centre velocity is much lower due to the area

difference. The bulk velocities appear to have reached a periodic state after the second

flow cycle. All subsequent results presented in this Section are taken from the fourth

cycle.

The time history of the fluid wall pressure within the compliant walled aneurysm

is shown in Figure 9.20. Again, profiles are given at the inlet, centre and outlet of

the aneurysm. The pressure at the outlet of the aneurysm was set to zero to aid the


Table 9.15: Under-relaxation factors used by the re-meshing algorithm during the un-steady, coupled FSI aneurysm simulations.

236


Time (seconds)

Ubu

lk(m

/s)

0 1 2 3 4 5-0.1

0

0.1

0.2

0.3

0.4

0.5InletOutletCentre

Figure 9.19: Velocity history of the unsteady, coupled FSI aneurysm simulation.

specification of the boundary conditions as described earlier. The maximum pressure

occurs at the aneurysm inlet at the time when the velocity is highest (t = 4.8 seconds);

the minimum pressure occurs shortly afterwards (t = 4.9 seconds) as the flow begins

to decelerate. The pressure at the aneurysm centre is slightly lower in magnitude and

in phase with the inlet pressure.

Figure 9.21 shows the flow streamlines and velocity vectors at different times within

the fourth flow cycle. The times in this Figure correspond to times within the fourth

flow cycle which relate to the inlet cycle shown in Figure 9.18. Near the beginning of the

cycle, t/T = 0.18, the re-circulation region is slightly skewed in shape with the centre

moving further downstream. The strength of the back-flow is greatest along the wall

at the downstream end of the aneurysm. At the point of peak velocity, t/T = 0.36, the

flow remains completely attached throughout the aneurysm. The velocity profile at the

downstream end of the aneurysm has a greater near-wall gradient than at the upstream

end due to the flow acceleration near the wall as the flow exits the aneurysm bulge. At

the next instance in time, t/T = 0.64, the bulk flow has reversed in direction; a large

re-circulation bubble has formed towards the downstream end (with regard to the flow

direction) of the aneurysm. The magnitude of the velocity is greater near the wall and

almost zero in the central region of the tube. This differs from the cases previously

237


Time (seconds)

Pre

ssur

e(m

mH

g)

0 1 2 3 4 5-6

-4

-2

0

2

4

6

8InletOutletCentre

Figure 9.20: Pressure history of the unsteady, coupled FSI aneurysm simulation.

considered where the flow was strongest in the central core region. By t/T = 0.91,

the flow has again reversed in direction. The re-circulation bubble has now migrated

to the downstream end of the aneurysm and reduced in strength. The magnitude of

the velocity is relatively low throughout the aneurysm; however, the greatest velocity

is found within the central core region.

The corresponding plots of turbulent kinetic energy are shown in Figure 9.22. Near

the beginning of the cycle, t/T = 0.18, a region of high k moves to the downstream end

of the aneurysm and becomes quite skewed. At t/T = 0.36, the peak value of turbulence

kinetic energy occurs at the downstream end of the aneurysm. Turbulence levels in the

downstream end of the aneurysm are relatively low. At t/T = 0.64, the peak value of k

occurs at the inlet of the aneurysm; a secondary peak in turbulent kinetic energy exists

at the centre of the aneurysm. By t/T = 0.91, this peak has grown in magnitude and

migrated to the downstream end of the aneurysm. Turbulence levels are relatively low

in the upstream half of the aneurysm bulge.

The normalized turbulent viscosity contours (Figure 9.23) show a broadly similar

pattern to the turbulent kinetic energy profiles. The peak value of the normalized

turbulent viscosity was approximately 13 and occurred at the downstream end of the

aneurysm at the time corresponding to peak inlet velocity.

238

9.5

.U

NST

EA

DY

FLO

WT

HR

OU

GH

AC

OM

PLIA

NT

AN

EU

RY

SM

X / Diameter

R/D

iam

eter

3 4 5 6 7 8

0

0.5

1

1.5

1

(a) t/T = 0.18

X / Diameter

R/D

iam

eter

3 4 5 6 7 8

0

0.5

1

1.5

1

(b) t/T = 0.36

239

CH

AP

TE

R9.

AN

EU

RY

SM

AL

FLO

WS

X / Diameter

R/D

iam

eter

3 4 5 6 7 8

0

0.5

1

1.5

1

(c) t/T = 0.64

X / Diameter

R/D

iam

eter

3 4 5 6 7 8

0

0.5

1

1.5

1

(d) t/T = 0.91

Figure 9.21: Flow streamlines and velocity vectors from the unsteady, compliant walled aneurysm case.

240

9.5

.U

NST

EA

DY

FLO

WT

HR

OU

GH

AC

OM

PLIA

NT

AN

EU

RY

SM

1

1

1

2

2

2

3

3

3

4

4

45

5

56

6

7

7

X / Diameter

R/D

iam

eter

3 4 5 6 7 8

0

0.5

1

1.5Level

TKE/UIN**2:1

0.00054

0.0027

0.003510

0.00513

0.0065

(a) t/T = 0.18

1

1

1

2 22

22

3

3

3 3

4

4

4

5

5

6 6

7 8

89

910

10

11

1213

14

X / Diameter

R/D

iam

eter

3 4 5 6 7 8

0

0.5

1

1.5Level

TKE/UIN**2:1

0.00054

0.0027

0.003510

0.00513

0.0065

(b) t/T = 0.36

241

CH

AP

TE

R9.

AN

EU

RY

SM

AL

FLO

WS

1

1

1

12

2

2

2

3

33

3

4

4

44

4

5

5

6 7 10

X / Diameter

R/D

iam

eter

3 4 5 6 7 8

0

0.5

1

1.5Level

TKE/UIN**2:1

0.00054

0.0027

0.003510

0.00513

0.0065

(c) t/T = 0.64

1

1

1

11

2

2

2

2

3

3

3

3

4

4

4

55

6

6

7

7

X / Diameter

R/D

iam

eter

3 4 5 6 7 8

0

0.5

1

1.5Level

TKE/UIN**2:1

0.00054

0.0027

0.003510

0.00513

0.0065

(d) t/T = 0.91

Figure 9.22: Turbulent kinetic energy contours from the unsteady, compliant walled aneurysm case.

242

9.5

.U

NST

EA

DY

FLO

WT

HR

OU

GH

AC

OM

PLIA

NT

AN

EU

RY

SM

1

1 1

2

2

2

3

3

3

34

4

4

45

5

5

6

6

X / Diameter

R/D

iam

eter

3 4 5 6 7 8

0

0.5

1

1.5Level

TVIS/VIS:11

22

33

44

55

66

77

88

99

1010

1111

1212

1313

(a) t/T = 0.18

1

1

1

2

2

2

23

3

3

3

4

4

4

4 5

5

5

5

6

6

7

7

88

9

9

10

10

11

12 13

X / Diameter

R/D

iam

eter

3 4 5 6 7 8

0

0.5

1

1.5Level

TVIS/VIS:11

22

33

44

55

66

77

88

99

1010

1111

1212

1313

(b) t/T = 0.36

243

CH

AP

TE

R9.

AN

EU

RY

SM

AL

FLO

WS

1

1

1

1

2

2 2

3

3

3

4 4

4

4

5

5

5

5

6

6

6

7

7

8

889

10

10

X / Diameter

R/D

iam

eter

3 4 5 6 7 8

0

0.5

1

1.5Level

TVIS/VIS:11

22

33

44

55

66

77

88

99

1010

1111

1212

1313

(c) t/T = 0.64

1

1

1

1

1

22

2

22

3

3

3

3

4

4

4

4

5

5

5

56

6

67

7

7

8 89

9

10

X / Diameter

R/D

iam

eter

3 4 5 6 7 8

0

0.5

1

1.5Level

TVIS/VIS:11

22

33

44

55

66

77

88

99

1010

1111

1212

1313

(d) t/T = 0.91

Figure 9.23: Turbulent viscosity contours from the unsteady, compliant walled aneurysm case.

244


X / Diameter

Pre

ssur

e(P

a)

2 3 4 5 6 7 8 9 10-150

-100

-50

0

50

100

150

t / T= 0.18t / T = 0.36t / T = 0.64t / T = 0.91

Figure 9.24: Wall pressure profiles from the unsteady, compliant walled aneurysm case.

The wall pressure profiles at different times within the flow cycle are shown in

Figure 9.24. Initially, the pressure is positive throughout the aneurysm model. At

times t/T = 0.64 and t/T = 0.91, the pressure is approximately zero. When fluid is

injected at a high velocity from the upstream end of the aneurysm, the pressure in the

aneurysm is higher than the exit pressure at the downstream end. When the fluid is

sucked into the aneurysm from the downstream end, the pressure within the aneurysm

becomes lower than the exit pressure. At all times, the external pressure is lower than

the internal pressure, resulting is a positive radial displacement.

The displacement profiles along the compliant wall section are shown in Figure 9.25.

Both u- and v-displacement profiles are symmetric about the centre of the aneurysm

at every instance in time. The u-displacement is negative in the upstream half of the

aneurysm and positive in the downstream half. The displacement is largely governed by

the pressure force; the change of sign in the u-displacement is due to the change of sign

in the outward normal vector. The v-displacement is positive along the entire length of

the compliant section at all instances in time. Two points of peak displacement occur

at either side of the aneurysm centre. The maximum u-displacement is approximately

0.2% of the pipe diameter whilst the maximum v-displacement is 0.3% of the pipe

diameter. The v-displacement at the centre of the aneurysm is approximately 3 times

less than the peak value.

The wall stress profiles are shown in Figure 9.26. As with the displacement profiles,

the stress profiles are symmetrical about the centre of the aneurysm; this is unsurprising

as the stresses are obtained by post-processing the displacements. The radial stress

245


X / Diameter

Udi

sp/D

iam

eter

2 3 4 5 6 7 8 9

-0.002

-0.001

0

0.001

0.002

t / T = 0.18t / T = 0.36t / T = 0.64t / T = 0.91

(a) Axial u-displacement profiles

X / Diameter

Vdi

sp/D

iam

eter

2 3 4 5 6 7 8 90

0.0005

0.001

0.0015

0.002

0.0025

t / T = 0.18t / T = 0.36t / T = 0.64t / T = 0.91

(b) Radial v-displacement profiles

Figure 9.25: Wall displacement profiles from the unsteady, compliant walled aneurysmcase.

246


profiles show a similar pattern to the radial displacement. Peak values of radial stress

exist either side of the centre, at which point the stress is a minimum. The peak axial

stress is approximately 4 times greater than the peak radial stress. The peak axial

stress exists at the centre of the aneurysm. The axial stress becomes negative near the

rigid attachment points at either end of the compliant wall section.

It can be seen from Figure 9.27 that the velocity at the inlet to the compliant

aneurysm is in excellent agreement with the numerical data presented by Khanafer

et al. (2009). The outlet velocity predicted here, on the other hand, does not match

that of the Khanafer et al predictions. During the period of low inlet velocity (t = 1.0 –

1.4 and t = 2.0 – 2.2 seconds) the two velocity variations are in reasonable agreement.

During the steep acceleration and deceleration in the middle phase of the flow cycle,

the predicted outlet velocity is very similar to the inlet velocity. This could be due

to an under-prediction of the wall displacement. The data shows a rapidly oscillating

velocity during the middle phase of the cycle. This could be due to rapid changes in

the flow volume within the aneurysm caused by large wall displacements. The graphs

presented in this Section are plotted against non-dimensional time through the fourth

cycle period, with the start of the cycle chosen to match that defined by Khanafer et al.

(2009), in order to produce valid comparisons.

Figure 9.28 shows the time history of the maximum radial displacement over the

full fifteen flow cycles that were simulated. At first sight it might be concluded that

the results do not appear to be cyclic as the maximum displacement changes with each

successive flow cycle. However, closer investigation showed this to be a consequence of

the frequency at which the results were output from the solver, as this did not quite

result in an integer number of output intervals per flow cycle. As can be seen from

the figure, the maximum displacement appears to be the same every three cycles, and

this also corresponds to the number of cycles required for the output times to repeat.

To simplify comparison with the numerical data of Khanafer et al. (2009), the data

presented in Figure 9.29 has been taken from three successive flow cycles, in order to

use data at different non-dimensional times and thus avoid any inconsistencies caused

by the issue described above.

The findings from the velocity comparisons are consistent with the predicted radial

displacement comparison, as shown in Figure 9.29, in which comparison is made of the

temporal variation of the maximum radial displacement. To simplify the comparison,

the data from the literature is plotted on the left ordinate whilst the numerical data is

plotted on the right ordinate. The present results have been normalized with respect

to the inlet diameter of the tube (this is consistent with all other displacement plots

presented in this work); it is not made clear in the study of Khanafer et al. (2009) which

length units or normalization factor has been used when presenting their results. This

uncertainty makes it difficult to make valid quantitative comparisons with their data.

247


X / Diameter

Rad

ialS

tres

s(P

a)

2 3 4 5 6 7 8 9-400

0

400

800

1200

t / T = 0.18t / T = 0.36t / T = 0.64t / T = 0.91

(a) Radial stress profiles

X / Diameter

Axi

alS

tres

s(P

a)

2 3 4 5 6 7 8 9-2000

-1000

0

1000

2000

3000

4000

5000

t / T = 0.18t / T = 0.36t / T = 0.64t / T = 0.91

(b) Axial stress profiles

248


X / Diameter

She

arS

tres

s(P

a)

2 3 4 5 6 7 8 9-3000

-2000

-1000

0

1000

2000

t / T = 0.18t / T = 0.36t / T = 0.64t / T = 0.91

(c) Shear stress profiles

Figure 9.26: Wall stress profiles from the unsteady, compliant walled aneurysm case.

The overall behaviour of the maximum radial displacement of the elastic wall through a

flow cycle is in very close agreement with the data. The peak radial displacement occurs

at the time of peak velocity (approximately t/T = 0.4). As the flow begins to decelerate,

the displacement reduces, initially at a rapid rate, until approximately t/T = 0.8,

at which point the displacement reduces in an almost linear manner until the flow

begins to accelerate again, at which point the displacement rapidly increases back to

its maximum value. The close qualitative agreement with the displacement behaviour,

in conjunction with the velocity and pressure results presented earlier, suggests quite

strongly that the units of the maximum radial displacement presented by Khanafer

et al. (2009) are incorrect.

Figure 9.30 shows comparisons of the fluid pressure (difference between the aneurysm

inlet and exit pressures) and the transmural pressure (difference between the internal

and external pressures at the exit of the aneurysm). It can be seen that the transmural

pressure is in good agreement with the numerical data of Khanafer et al. (2009). This

is slightly surprising given that the transmural pressure difference is largely responsible

for the radial displacement - which was shown to be in poor agreement with the nu-

merical data. This adds further weight to the suggestion that the radial displacement

data presented by Khanafer et al. (2009) has been inconsistently non-dimensionalized.

The fluid pressure difference is not in good agreement with the numerical data. The

magnitude of the pressure difference is reasonably wall captured, however the temporal

249


t / Time Period

Ubu

lk(m

/s)

0 0.2 0.4 0.6 0.8 1-0.2

-0.1

0

0.1

0.2

0.3

0.4

Khanafer et al - Inlet VelocityNumerical Results - Inlet Velocity

(a) Aneurysm Inlet Velocity

t / Time Period

Ubu

lk(m

/s)

0 0.2 0.4 0.6 0.8 1-0.2

-0.1

0

0.1

0.2

0.3

0.4

Khanafer et al - Outlet VelocityNumerical Results - Outlet Velocity

(b) Aneurysm Outlet Velocity

Figure 9.27: Comparison of the temporal variation of the compliant aneurysm inletand outlet velocities.

250


Time (seconds)

Max

.Rad

ialD

ispl

acem

ent(

Vdi

spm

ax/D

iam

eter

)

0.4 1.5 2.6 3.7 4.8 5.9 7 8.1 9.2 10.3 11.4 12.5 13.6 14.7 15.8 16.9

0.0015

0.002

0.0025

0.003

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Figure 9.28: Time history of the maximum radial displacement of the compliantaneurysm wall.

t / Time Period

Vdi

sp(K

hana

fer

etal

)

Vdi

sp/D

iam

eter

(Num

eric

alR

esul

ts)

0 0.2 0.4 0.6 0.8 1 1.2 1.42.1

2.3

2.5

2.7

2.9

0.04

0.05

0.06

0.07

Khanafer et alNumerical Results

Figure 9.29: Comparison of the temporal variation of the maximum radial displacementof the compliant aneurysm wall.

251


variation is poorly captured. This is probably an effect, rather than a cause, of the

poorly predicted wall deformation.

9.6 Summary

This Chapter has described the numerical simulation of aneurysmal flow. Before un-

steady, fully coupled FSI simulations were performed a number of simpler rigid walled

test cases were considered to validate the flow solver. These were followed by an un-

steady, rigid-walled case intended to validate the implementation of the unsteady terms

into the flow solver. Finally, after the flow solver had been shown to accurately sim-

ulate rigid-walled aneurysmal flow, the fully coupled case of unsteady flow through a

compliant walled aneurysm was considered.

The steady-state flow solver was shown to be in good agreement with the experi-

mental data for both laminar and turbulent flow. The flow field was characterised by a

central core which flowed through the centre of the aneurysm surrounded by a slowly

rotating re-circulation region. The strength of the re-circulating flow increases towards

the downstream end of the aneurysm bulge. At the exit to the aneurysm, the near-

wall velocity gradient was very steep as the core, which had spread radially outwards

slightly, re-enters the straight section. In the turbulent case, this shear-layer led to

the production of turbulence. The peak level of turbulent kinetic energy was relatively

unaffected by the bulge diameter of the aneurysm. Whilst the Reynolds numbers were

within the transitional regime, the difficulties encountered during the stenosis simula-

tions were not found in the aneurysm case. This is largely because the re-circulation

bubble is constrained by the geometry of the aneurysm bulge. Both separation and re-

attachment points are governed by geometry; this removes the need to have the large

downstream section used during the stenosis simulations.

Once the flow solver had been validated for the steady-state case, the logical pro-

gression was to investigate the effect of a time-varying inlet velocity profile. The flow

was characterised by a re-circulation bubble which formed during the acceleration phase

of the flow cycle and migrated downstream during the deceleration phase.

The final case to be considered in this Chapter was that of unsteady flow through

a compliant walled aneurysm. Unlike the rigid-walled, unsteady case, the time-varying

inlet waveforms were based on physiological measurements, rather than an idealised

sinusoidal waveform. The velocity waveform included both positive and negative flow.

Whilst the comparisons with numerical data from the literature showed that the wall

displacement was under-predicted, the pattern of the temporal variation was reason-

ably well captured. Since the displacements were very small, the flow field would not

have differed greatly from the rigid walled case. The displacement profiles, and the

corresponding stress profiles, were symmetric about the centre of the aneurysm centre.

252

9.6. SUMMARY

t / Time Period

Pre

ssur

e(m

mH

g)

0 0.2 0.4 0.6 0.8 1-15

-10

-5

0

5


(a) Fluid Pressure Difference Comparison

t / Time Period

Pre

ssur

e(m

mH

g)

0 0.2 0.4 0.6 0.8 160

70

80

90

100

110


(b) Transmural Pressure Difference Comparison

Figure 9.30: Comparison of the temporal pressure difference for the compliant walledaneurysm case.

253


The peak radial displacement occurred at the same time as the maximum aneurysm

inlet velocity. More work is necessary to understand the discrepancy with the data

from the literature.

254

Chapter 10

Conclusions and Further Work

10.1 Conclusions

This thesis has described the development and subsequent testing of a fully coupled

fluid-structure interaction solver. To improve the accuracy and efficiency of coupled

simulations, the solver had a number of novel features, including consistent use of the

finite-volume method to solve both the fluid and solid governing equations, and the use

of a single numerical mesh to cover both fluid and solid sub-domains.

The coupled solver was verified and validated through a number of test cases, includ-

ing: steady laminar flow through an initially straight tube with a compliant wall section;

steady flow (laminar and turbulent) through a compliant walled stenosis, and unsteady

turbulent flow through a compliant aneurysm. Comparison with available data were

made difficult by what are believed to be incorrectly presented results and limited infor-

mation regarding material properties. However, the newly developed method showed

itself to be capable of accurately predicting the fluid-structure interaction effects for a

range of complex flows.

10.1.1 Finite-Volume Solid Solver

Initially, the finite-volume solid body solver was developed and tested in isolation. Tests

were performed for a number of cases for which analytical solutions were available,

including: planar stress in a plate with a circular hole in the centre; axisymmetric

stress in a thick walled cylinder under internal pressure, and undamped vibration of a

fixed-free cantilever beam.

Once the solid solver had been tested, it was coupled to an existing in-house CFD

code to create a new FSI solver. This task was simplified by developing the solid solver

in a consistent manner to the CFD code. Although not fully exploited in the present

study, the newly developed FSI code allows for the advanced turbulence and near-wall

models developed within the Manchester research group to be applied to the study of

255

CHAPTER 10. CONCLUSIONS AND FURTHER WORK

FSI problems. This is a major improvement upon many existing FSI solvers which are

often limited to simple turbulence models.

10.1.2 Straight Pipe Flow

The first fully-coupled FSI case considered with the newly developed solver was steady

laminar flow through an initially straight tube with a compliant wall section. Initial

rigid-wall tests proved that the flow solver was capable of accurately predicting the

streamwise pressure variation in a laminar pipe flow.

When the effects of wall compliance were included in the simulation an initially

linear variation of wall deformation in the streamwise direction was found; this was

caused by the linear reduction in pressure, and hence transmural pressure difference,

due to laminar viscous losses. The deformation was thus greatest slightly before the

downstream end of the compliant wall section. As the streamwise pressure difference

was increased, the deformation also increased; however, the linear pressure variation

no longer existed as the area-reduction at the downstream end of the compliant section

dominated the pressure drop. As the streamwise difference was increased further, the

compliant wall began to collapse at the downstream end.

The re-meshing algorithm struggled to produce a good quality mesh in the fluid

sub-domain when there was a significant deformation towards the end of the com-

pliant section. Attempts were made to reduce the skewness of the fluid cells in the

region with little effect. One conclusion from this is that in FSI simulations with very

large displacements it may be necessary to completely re-mesh the fluid sub-domain.

For large displacements, the assumption of linear-elasticity inherent in the solid solver

development may also no longer be valid in such cases.

Although the coupled results were not in good agreement with the numerical data

from the literature in terms of area reduction and volumetric flow rate variation with

applied streamwise pressure drop, a number of consistency tests were performed. These

led to the conclusion that the discrepancies were likely to be a result of incorrect non-

dimensionalisation of the data presented by Shim and Kamm (2002).

10.1.3 Stenosed Flow

The second fully-coupled FSI test case was that of steady flow through a compliant

walled stenosis. Both laminar and turbulent flow was considered to cover the entire

range of physiologically relevant flow rates.

Firstly, flow through a rigid walled stenosis was computed and compared to the

experimental data of Young and Tsai (1973). Initial laminar simulations were shown

to be in excellent agreement with the experimental data in terms of streamwise pres-

sure drop across the stenosis up to a Reynolds number of 300. Beyond this point,

256

10.1. CONCLUSIONS

convergence issues were encountered; specifically the mass residual would remain fixed.

This was attributed to the extremely large re-circulation bubble which formed down-

stream of the stenosis. Simply increasing the downstream length of the domain, in

order to accommodate the recirculation bubble, was not successful. The size of the re-

circulation bubble meant that large-scale unsteadiness was likely, the physics of which

were unaccounted for with the steady-state solver. It was also noted that in these flows

transition to turbulence occurs at a much lower Reynolds number than that associated

with straight pipe flows. Use of the Launder-Sharma low-Re turbulence was successful

for Reynolds numbers greater than 1500. Unlike the laminar case, in the turbulent flow

calculations the re-circulation reduced in size with increasing Reynolds number. The

region of high turbulent kinetic energy produced by the high shear-rate in the separated

shear layer acted to reduce the re-circulation bubble size through increased momentum

mixing. The flow upstream of the stenosis was laminar, before transition occurred

slightly downstream of the throat; further downstream, the turbulence decayed. An

intermediate Reynolds number range (300 < Re < 1500) existed for which converged

solutions were not achievable due to the numerical difficulties associated with capturing

turbulence transition.

Once the flow solver had been validated for the rigid walled case, coupled simulations

were performed using the newly developed FSI solver for the case of flow through a

compliant walled stenosis. Simulations were performed to match the case examined

experimentally by Stergiopulos et al. (1993). The paper describing the experiment

was lacking some information regarding the vitally important compliant wall thickness

and material properties, but a series of parametric tests were conducted to find best

estimates for the necessary information.

Coupled simulations were shown to be in good agreement with the experimental

data of Stergiopulos et al. (1993) in terms of flow rate and wall deformation variation

with streamwise pressure difference. At low streamwise pressure differences, and hence

low flow rates, the internal pressure was everywhere greater than the external pressure;

this caused a positive radial displacement over the entire compliant wall section. As the

flow rate was increased, the pressure drop at the stenosis increased until the pressure

was lower than the external pressure, resulting in a negative radial displacement in the

region immediately downstream of the stenosis throat. Further increases of the flow

rate resulted in the pressure recovery downstream of the stenosis not being able to

raise the internal pressure above the external pressure, resulting in a negative radial

displacement over the entire downstream compliant section.

10.1.4 Aneurysmal Flow

The inlet velocity waveform was based on physiological measurements and included

both positive and negative flow.

257


The third and final fully-coupled FSI test case to be considered was that of unsteady,

turbulent flow through a compliant walled aneurysm. This case was chosen to test the

newly developed method for a realistic physiological flow.

Initial rigid-wall validation simulations were shown to be in good agreement with the

experimental data. The steady-state flow field was characterised by a core flow through

the centre of the aneurysm surrounded by a slowly rotating re-circulation region in the

outer part of the aneurysm bulge. In the turbulent case, the highest turbulence levels

occurred at the downstream end of the aneurysm where the flow exited the aneurysm.

The difficulties associated with simulating stenosed flows were not encountered as the

re-recirculation region is contained by the aneurysm geometry, thus removing the need

for a large downstream section to allow re-attachment and flow development before the

exit.

In the unsteady case, where the inlet velocity waveform had a sinusoidal variation

with time, the flow through a rigid walled aneurysm is characterised by the formation

of a large re-circulation bubble in the aneurysm bulge shortly after the flow deceleration

phase of the flow cycle. This re-circulation bubble is convected downstream toward the

downstream end of the aneurysm. As the flow begins to accelerate, this re-circulation

bubble begins to decreases in strength. Eventually, two smaller re-circulation bubbles

form at either end of the aneurysm bulge. These grow and eventually combine to form

the single, larger re-circulation bubble at the end of the acceleration phase.

Unlike the rigid-walled case, which used a sinusoidal inlet velocity profile, the com-

pliant walled case used a physiological flow waveform. Results from the unsteady, cou-

pled simulations were shown to be in poor agreement with numerical data, obtained

using the finite-element method, taken from the literature. It was thought that the

highly under-predicted wall displacement was caused by an incorrectly specified exter-

nal pressure. Investigations will be necessary to rectify this problem. The flow field

experienced relatively abrupt changes during the steep acceleration and deceleration

phases of the flow cycle. This is in contrast to the relatively gradual changes seen dur-

ing the rigid walled cases. The wall displacement profiles were symmetrical about the

centre of the aneurysm at all instances in time. The axial displacement was negative

in the upstream and positive in the downstream portion of the aneurysm; the change

in sign is attributed to the change in sign of the unit normal vector, which causes a

change in sign to the pressure force. The radial displacement was greatest at points

either side of the aneurysm centre. The peak displacement occurred at the same instant

in time as the peak aneurysm inlet velocity. Whilst the numerical results were in poor

agreement with the data from the literature, in terms of radial displacement of the

compliant wall, some of the main features of the temporal variation of the deformation

were reproduced. More work is necessary to investigate and rectify the discrepancies

with the data from the literature.

258

10.2. SUGGESTIONS FOR FURTHER WORK

10.2 Suggestions for Further Work

This thesis has described the development and testing of a new and complex solver.

Time constraints imposed a limit upon the amount of development, and the following

Sub-sections describe a number of avenues that could be explored to further improve

the new method.

It was found during the course of this work that a lack of detailed experimental data

exists for coupled FSI problems. Further experimental studies providing such details

would be highly valuable.

10.2.1 Validation for New Test Cases

The test cases considered in this work were drawn from physiological and bio-fluid

applications. To prove the generality of the new method, it is important to validate

the solver for new applications. Possible example include: external aeroelastic flows,

such as flow past a deformable aerofoil, or internal aeroelastic flows, such as flow past a

deformable gas turbine rotor blade. In addition to considering the elastic deformation

caused by the fluid flow upon the rotor blade, it would be interesting to consider the

non-isothermal case and investigate the effects of heat transfer upon the deformation of

the rotor blade. This type of analysis was made possible by the use of the thermo-elastic

constitutive relations when developing the solid body solver.

10.2.2 Extension to Three Dimensions

Currently, the FSI solver is capable of solving two-dimensional problems - both planar

and axisymmetric. The use of a non-orthogonal, boundary fitted coordinate system al-

lows a complex geometries to be considered. However, to further increase the geometric

flexibility, and make the solver more general in nature, it will be necessary to extend the

method to three dimensions. The existing three-dimensional version of the STREAM

code could be used as a framework to simplify the development of a three-dimensional

FSI solver.

The cases considered during this work, particularly the compliant tube and stenosis,

can only be considered axisymmetric for relatively small deformations. As the point of

tube collapse is approached, the assumption of axisymmetry becomes invalid. Simula-

tion of such tube collapse can only be simulated accurately with a three-dimensional

solver.

10.2.3 Addition of Implicit Transient Schemes

The explicit time discretization scheme was chosen because of its simple implementa-

tion. However, this scheme is limited by a very stringent stability condition on the

259


time step size. An improvement would be to include the implicit scheme, which has no

limit upon the time step size. This scheme is more difficult to implement as it requires

an additional set of equations to be solved at the end of each time step, owing to the

appearance of terms on the right hand side of the discretized equations.

10.2.4 Code Parallelisation

To allow the efficient and timely calculation of complex three-dimensional problems it

will be necessary to parallelize the coupled solver. This will allow large problems to

be divided and solved across a number of processors. This task will made simpler by

the existence of a parallel version of the STREAM flow solver; the parallel solid solver

could be implemented in a consistent manner. However, development and subsequent

testing of a parallel solver will not be a trivial task.

10.2.5 Non-Linear Elastic and Plastic Constitutive Relations

The inherent assumption of linear-elasticity limits the use of the new solver to cases

involving relatively small deformation and simple materials. To allow the simulation of

more complex cases, it will be necessary to use a different constitutive relation. This

is true for both solid and fluid; many physiological flows include blood which is best

described by a non-Newtonian constitutive relation.

260

Appendix A

Elasticity Equations in Cartesian

Coordinates

A.1 The Equilibrium Equations

The u-equilibrium equation in two-dimensional Cartesian coordinates can be written:

∂σxx

∂x+

∂τxy

∂y+ fx = 0 (A.1)

The v-equilibrium equation in two-dimensional Cartesian coordinates is:

∂τxy

∂x+

∂σyy

∂y+ fy = 0 (A.2)

A.2 The Constitutive Relations

The stresses are related to the displacements through the linear-elastic constitutive

relations:

σxx = (2µ + λ)∂u

∂x+ λ

∂v

∂y− (3λ + 2µ)α(∆T ) (A.3)

σyy = (2µ + λ)∂v

∂y+ λ

∂u

∂x− (3λ + 2µ)α(∆T ) (A.4)

τxy = µ

(∂u

∂y+

∂v

∂x

)

(A.5)

A.3 Discretization of the u-Equilibrium Equation

The first step in the finite-volume discretization process is the integration of the u-

equilibrium equation over an internal cell:

261

APPENDIX A. ELASTICITY EQUATIONS IN CARTESIAN COORDINATES

∫∫

V

(∂σxx

∂x+

∂τxy

∂y+ fx

)

dx dy = 0 (A.6)

∫

S

σxx dy

e

w

+

∫

S

τxy dx

n

s

+

∫∫

V

(

fx

)

dx dy = 0 (A.7)

(σxx,e∆ye) − (σxx,w∆yw) + (τxy,n∆xn) − (τxy,s∆xs) + ∆xP ∆yP fx,P = 0 (A.8)

Where the stresses on the cell faces are approximated by:


∂x

∣∣∣∣e

+ λe∂v

∂y

∣∣∣∣e

− (3λ + 2µ)eαe(∆T )e (A.9)

= (2µ + λ)e

(uE − uP

∆xe

)

+ λe

(vne − vse

∆ye

)

−

(3λ + 2µ)eαe(∆T )e (A.10)


∂x

∣∣∣∣w

+ λw∂v

∂y

∣∣∣∣w

− (3λ + 2µ)wαw(∆T )w (A.11)

= (2µ + λ)w

(uP − uW

∆xw

)

+ λw

(vnw − vsw

∆yw

)

−

(3λ + 2µ)wαw(∆T )w (A.12)

τxy,n = µn∂u

∂y

∣∣∣∣n

+ µn∂v

∂x

∣∣∣∣n

(A.13)

= µn

(uN − uP

∆yn

)

+ µn

(vne − vnw

∆xn

)

(A.14)

τxy,s = µs∂u

∂y

∣∣∣∣s

+ µs∂v

∂x

∣∣∣∣s

(A.15)

= µs

(uP − uS

∆ys

)

+ µs

(vse − vsw

∆xs

)

(A.16)

The above equations are then substituted into Equation A.8 to give the discretized

u-equilibrium equation:

262

A.3. DISCRETIZATION OF THE U -EQUILIBRIUM EQUATION

∆ye

[

(2µ + λ)e

(uE − uP

∆xe

)

+ λe

(vne − vse

∆ye

)

−

(3λ + 2µ)eαe(∆T )e

]

−

∆yw

[

(2µ + λ)w

(uP − uW

∆xw

)

+ λw

(vnw − vsw

∆yw

)

−

(3λ + 2µ)wαw(∆T )w

]

+

∆xn

[

µn

(uN − uP

∆yn

)

+ µn

(vne − vnw

∆xn

)]

−

∆xs

[

µs

(uP − uS

∆ys

)

+ µs

(vse − vsw

∆xs

)]

+ ∆xP ∆yP fx,P = 0 (A.17)

This can be arranged into the form:

AP uP = AE uE + AS uS + AN uN + AW uW + Su (A.18)

Where the coefficients are given by:


∆xe(A.19)


∆xw(A.20)

AN = µn∆xn

∆yn(A.21)

AS = µs∆xs

∆ys(A.22)

AP = AE + AW + AN + AS (A.23)

Su = Su,s + Su,n + Su,e + Su,w + Su,P (A.24)

The components of the source term are given by:

Su,s = −µs(vse − vsw) (A.25)

Su,n = µn(vne − vnw) (A.26)

Su,e = λe(vne − vse) − ∆ye(3λ + 2µ)α(∆T )e (A.27)

Su,w = −λw(vnw − vsw) + ∆yw(3λ + 2µ)α(∆T )w (A.28)

Su,P = ∆xP ∆yP fx,P (A.29)

263


A.4 Discretization of the v-Equilibrium Equation

The first step in the finite-volume discretization process is the integration of the u-

equilibrium equation over an internal cell:

∫∫

V

(∂τxy

∂x+

∂σyy

∂y+ fy

)

dx dy = 0 (A.30)

∫

S

τxy dy

e

w

+

∫

S

σyy dx

n

s

+

∫∫

V

(

fy

)

dx dy = 0 (A.31)

(τxy,e∆ye) − (τxy,w∆yw) + (σyy,n∆xn) − (σyy,s∆xs) + ∆xP ∆yP fy,P = 0 (A.32)

Where the stresses on the faces are approximated by:

τxy,e = µe∂u

∂y

∣∣∣∣e

+ µe∂v

∂x

∣∣∣∣e

(A.33)

= µe

(une − use

∆ye

)

+ µe

(vE − vP

∆xe

)

(A.34)

τxy,w = µw∂u

∂y

∣∣∣∣w

+ µw∂v

∂x

∣∣∣∣w

(A.35)

= µw

(unw − usw

∆yw

)

+ µw

(vP − vW

∆xw

)

(A.36)

σyy,n = (2µ + λ)n∂v

∂y

∣∣∣∣n

+ λn∂u

∂x

∣∣∣∣n

− (3λ + 2µ)nαn(∆T )n (A.37)

= (2µ + λ)n

(vN − vP

∆yn

)

+ λn

(une − unw

∆xn

)

−

(3λ + 2µ)nαn(∆T )n (A.38)

σyy,s = (2µ + λ)s∂v

∂y

∣∣∣∣s

+ λs∂u

∂x

∣∣∣∣s

− (3λ + 2µ)sαs(∆T )s (A.39)

= (2µ + λ)s

(vP − vS

∆ys

)

+ λs

(use − usw

∆xs

)

−

(3λ + 2µ)sαs(∆T )s (A.40)

264

A.4. DISCRETIZATION OF THE V -EQUILIBRIUM EQUATION

The above equations are then substituted into Equation A.32 to give the discretized

v-equilibrium equation:

∆ye

[

µe

(une − use

∆ye

)

+ µe

(vE − vP

∆xe

)]

−

∆yw

[

µw

(unw − usw

∆yw

)

+ µw

(vP − vW

∆xw

)]

+

∆xn

[

(2µ + λ)n

(vN − vP

∆yn

)

+ λn

(une − unw

∆xn

)

−

(3λ + 2µ)nαn(∆T )n

]

−

∆xs

[

(2µ + λ)s

(vP − vS

∆ys

)

+ λs

(use − usw

∆xs

)

−

(3λ + 2µ)sαs(∆T )s

]

+

∆xP ∆yP fy,P = 0 (A.41)


AP vP = AE vE + AS vS + AN vN + AW vW + Sv (A.42)


AE = µe∆ye

∆xe(A.43)

AW = µw∆yw

∆xw(A.44)

AN = (2µ + λ)n∆xn

∆yn(A.45)

AS = (2µ + λ)s∆xs

∆ys(A.46)

AP = AE + AW + AN + AS (A.47)

Sv = Sv,s + Sv,n + Sv,e + Sv,w + Sv,P (A.48)


265


Sv,s = −λs(use − usw) + (3λ + 2µ)sαs(∆T )s (A.49)

Sv,n = λn(une − unw) − (3λ + 2µ)nαn(∆T )n (A.50)

Sv,e = µe(une − use) (A.51)

Sv,w = −µw(unw − usw) (A.52)

Sv,P = ∆xP ∆yP fy,P (A.53)

266

Appendix B

Elasticity Equations in

Cylindrical Coordinates

B.1 Introduction

In the previous Appendix, the governing equations of elasticity were presented. These

equations were all derived with respect to a Cartesian coordinate system. In this

Appendix, the equations will be transformed into cylindrical polar coordinates.

B.2 Transformation of Vector Functions

The gradient of a vector field in polar coordinates is given by:

gradu =

∂vr

∂r∂vθ

∂r − vθ

r∂vz

∂r1r

∂vr

∂θ1r

∂vθ

∂θ + vr

r1r

∂vz

∂θ∂vr

∂z∂vθ

∂z∂vz

∂z

(B.1)

The divergence of a vector field in polar coordinates is given by:

divu =1

r

∂(r vr)

∂r+

1

r

∂vθ

∂θ+

∂vz

∂z(B.2)

B.3 Transformation of the Equilibrium Equations

The equilibrium equations in polar coordinates are:

∂σrr

∂r+

1

r

∂τrθ

∂θ+

∂τrz

∂z+

σrr − σθθ

r+ fr = 0 (B.3)

∂τθr

∂r+

1

r

∂σθθ

∂θ+

∂τθz

∂z+

2τrθ

r+ fθ = 0 (B.4)

267

APPENDIX B. ELASTICITY EQUATIONS IN CYLINDRICAL COORDINATES

∂τzr

∂r+

1

r

∂τzθ

∂θ+

∂σzz

∂z+

τzr

r+ fz = 0 (B.5)

In the case of axisymmetric coordinates all derivatives with respect to θ vanish, as

does the shear stress τrθ and the body force fθ.

∂σrr

∂r+

∂τrz

∂z+

σrr − σθθ

r+ fr = 0 (B.6)

∂τθz

∂z= 0 (B.7)

∂τzr

∂r+

∂σzz

∂z+

τzr

r+ fz = 0 (B.8)

B.4 Transformation of the Constitutive Equations

The strain-displacement relationship in polar coordinates is:

[ǫ] =

ǫrr ǫrθ ǫrz

ǫθr ǫθθ ǫθz

ǫzr ǫzθ ǫzz

=

1

2

[(gradu)T + (gradu)

]

=

∂vr

∂r12

(∂vθ

∂r − vθ

r + 1r

∂vr

∂θ

)12

(∂vz

∂r + ∂vr

∂z

)

12

(1r

∂vr

∂θ + ∂vθ

∂r − vθ

r

)1r

∂vθ

∂θ + vr

r12

(1r

∂vz

∂θ + ∂vθ

∂z

)

12

(∂vr

∂z + ∂vz

∂r

)12

(∂vθ

∂z + 1r

∂vz

∂θ

)∂vz

∂z

(B.9)

The stress-strain relationship in polar coordinates is:

[σ] =

σrr σrθ σrz

σθr σθθ σθz

σzr σzθ σzz

= 2µ[ǫ] + λdivu I − (3λ + 2µ)α(∆T )I (B.10)

These can be combined to give the stress-displacement relationship:

268

B.5. DISCRETIZATION OF THE VZ -EQUILIBRIUM EQUATION

σrr σrθ σrz

σθr σθθ σθz

σzr σzθ σzz

=

2µ

∂vr

∂r12

(∂vθ

∂r − vθ

r + 1r

∂vr

∂θ

)12

(∂vz

∂r + ∂vr

∂z

)

12

(1r

∂vr

∂θ + ∂vθ

∂r − vθ

r

)1r

∂vθ

∂θ + vr

r12

(1r

∂vz

∂θ + ∂vθ

∂z

)

12

(∂vr

∂z + ∂vz

∂r

)12

(∂vθ

∂z + 1r

∂vz

∂θ

)∂vz

∂z

+

λ

(1

r

∂rvr

∂r+

1

r

∂vθ

∂θ+

∂vz

∂z

)

1 0 0

0 1 0

0 0 1

− (3λ + 2µ)α(∆T )

1 0 0

0 1 0

0 0 1

(B.11)

In the case of cylindrical polar coordinates, the stress-displacement relationships

are given by:

σrr = 2µ∂vr

∂r+ λ

(1

r

∂(r vr)

∂r+

∂vz

∂z

)

− (3λ + 2µ)α(∆T ) (B.12)

= (2µ + λ)∂vr

∂r+ λ

∂vz

∂z+

λvr

r− (3λ + 2µ)α(∆T ) (B.13)

σθθ = 2µvr

r+ λ

(1

r

∂(r vr)

∂r+

∂vz

∂z

)

− (3λ + 2µ)α(∆T ) (B.14)

= (2µ + λ)vr

r+ λ

∂vr

∂r+ λvzz − (3λ + 2µ)α(∆T ) (B.15)

σzz = 2µ∂vz

∂z+ λ

(1

r

∂(r vr)

∂r+

∂vz

∂z

)

− (3λ + 2µ)α(∆T ) (B.16)

= (2µ + λ)∂vz

∂z+ λ

∂vr

∂r+

λvr

r− (3λ + 2µ)α(∆T ) (B.17)

τrz = µ

(∂vz

∂r+

∂vr

∂z

)

(B.18)

B.5 Discretization of the vz-Equilibrium Equation

The axial displacement equation is integrated:

2πr

∫∫

V

(∂τrz

∂r+

∂σzz

∂z+

τrz

r+ fz

)

dr dz = 0 (B.19)

269


2π

∫∫

V

(∂(r τrz)

∂r+

∂(r σzz)

∂z− σzz

∂r

∂z+ fz

)

dr dz = 0 (B.20)

Since:

r∂τrz

∂r=

∂(r τrz)

∂r− τrz (B.21)

r∂σzz

∂z=

∂(r σzz)

∂z− σzz

∂r

∂z(B.22)

2π

∫

S

(r τrz) dz

n

s

+ 2π

∫

S

(r σzz) dr

e

w

+ 2π

∫∫

V

(

rfz − σzz∂r

∂z

)

dr dz = 0 (B.23)

(rnτrz,n∆zn) − (rsτrz,s∆zs) + (reσzz,e∆re) − (rwσzz,w∆rw)+(

rP fz,P − σzz,P∂r

∂z

∣∣∣∣P

)

∆rP ∆zP = 0 (B.24)

B.6 Discretization of the vr-Equilibrium Equation

The radial displacement equation is integrated:

2πr

∫∫

V

(∂σrr

∂r+

∂τrz

∂z+

σrr − σθθ

r+ fr

)

dr dz = 0 (B.25)

2π

∫∫

V

(∂(r σrr)

∂r+

∂(r τrz)

∂z− σθθ − τrz

∂r

∂z+ rfr

)

dr dz = 0 (B.26)

Since:

r∂σrr

∂r=

∂(r σrr)

∂r− σrr (B.27)

r∂τrz

∂z=

∂(r τrz)

∂z− τrz

∂r

∂z(B.28)

∫

S

(rσrr) dz

n

s

+

∫

S

(rτrz) dr

e

w

+

∫∫

V

(

rfr − σθθ − τrz∂r

∂z

)

dr dz = 0 (B.29)

270

B.6. DISCRETIZATION OF THE VR-EQUILIBRIUM EQUATION

(rnσrr,n∆zn) − (rsσrr,s∆zs) + (reτrz,e∆re) − (rwτrz,w∆rw)+(

rP frP − σθθ,P − τrz,P∂r

∂z

∣∣∣∣P

)

∆rP ∆zP = 0 (B.30)

271


272

Appendix C

Treatment of the South

Boundary Equations

C.1 The u-Equilibrium Equation

Integration of the u-equilibrium equation over a half-cell on the south boundary leads

to:

∫∫

V

(∂σxx

∂x+

∂τxy

∂y+ fx

)

dx dy = 0 (C.1)

∫

S

σxx dy

e

w

+

∫

S

τxy dx

n

s

+

∫∫

V

(

fx

)

dx dy = 0 (C.2)

(σxx,e∆ye) − (σxx,w∆yw) + (τxy,n∆xn) − (τxy,s∆xs) + ∆xP ∆yP fx,P = 0 (C.3)

Where the stresses on the cell faces are approximated by:


∂x

∣∣∣∣e

+ λe∂v

∂y

∣∣∣∣e

− (3λ + 2µ)eαe(∆T )e (C.4)

= (2µ + λ)e

(uE − uP

∆xe

)

+ λe

(vne − vse

∆ye

)

−

(3λ + 2µ)eαe(∆T )e (C.5)

273

APPENDIX C. TREATMENT OF THE SOUTH BOUNDARY EQUATIONS


∂x

∣∣∣∣w

+ λw∂v

∂y

∣∣∣∣w

− (3λ + 2µ)wαw(∆T )w (C.6)

= (2µ + λ)w

(uP − uW

∆xw

)

+ λw

(vnw − vsw

∆yw

)

−

(3λ + 2µ)wαw(∆T )w (C.7)

τxy,n = µn∂u

∂y

∣∣∣∣n

+ µn∂v

∂x

∣∣∣∣n

(C.8)

= µn

(uN − uP

∆yn

)

+ µn

(vne − vnw

∆xn

)

(C.9)

τxy,s = τ∗

xy,south (C.10)

Where τ∗

xy,south is the prescribed traction on the south boundary.

∆ye

[

(2µ + λ)e

(uE − uP

∆xe

)

+ λe

(vne − vse

∆ye

)

−

(3λ + 2µ)eαe(∆T )e

]

−

∆yw

[

(2µ + λ)w

(uP − uW

∆xw

)

+ λw

(vnw − vsw

∆yw

)

−

(3λ + 2µ)wαw(∆T )w

]

+

∆xn

[

µn

(uN − uP

∆yn

)

+ µn

(vne − vnw

∆xn

)]

−

∆xs

[

τ∗

xy,south

]

+

∆xP ∆yP

[

fx,P

]

= 0 (C.11)

Which can be written in the form:

AP uP = AE uE + AS uS + AN uN + AW uW + Su (C.12)


274

C.2. THE V -EQUILIBRIUM EQUATION


∆xe(C.13)


∆xw(C.14)

AN = µn∆xn

∆yn(C.15)

AS = 0 (C.16)

AP = AE + AW + AN (C.17)

Su = Su,s + Su,n + Su,e + Su,w + Su,P (C.18)


Su,s = −∆xs τ∗

xy,south (C.19)

Su,n = µn(vne − vnw) (C.20)

Su,e = λe(vne − vse) − ∆ye(3λ + 2µ)eαe(∆T )e (C.21)

Su,w = −λw(vnw − vsw) + ∆yw(3λ + 2µ)wαw(∆T )w (C.22)

Su,P = ∆xP ∆yP fx,P (C.23)

C.2 The v-Equilibrium Equation

Integration of the v-equilibrium equation over a half-cell on the south boundary leads

to:

∫∫

V

(∂τxy

∂x+

∂σyy

∂y+ fy

)

dx dy = 0 (C.24)

∫

S

τxy dy

e

w

+

∫

S

σyy dx

n

s

+

∫∫

V

(

fy

)

dx dy = 0 (C.25)

(τxy,e∆ye) − (τxy,w∆yw) + (σyy,n∆xn) − (σyy,s∆xs) + ∆xP ∆yP fy,P = 0 (C.26)


275


τxy,e = µe∂u

∂y

∣∣∣∣e

+ µe∂v

∂x

∣∣∣∣e

(C.27)

= µe

(une − use

∆ye

)

+ µe

(vE − vP

∆xe

)

(C.28)

τxy,w = µw∂u

∂y

∣∣∣∣w

+ µw∂v

∂x

∣∣∣∣w

(C.29)

= µw

(unw − usw

∆yw

)

+ µw

(vP − vW

∆xw

)

(C.30)


∂y

∣∣∣∣n

+ λn∂u

∂x

∣∣∣∣n

− (3λ + 2µ)nαn(∆T )n (C.31)

= (2µ + λ)n

(vN − vP

∆yn

)

+ λn

(une − unw

∆xn

)

−

(3λ + 2µ)nαn(∆T )n (C.32)

σyy,s = σ∗

yy,south (C.33)

Where σ∗

yy,south is the prescribed traction on the south boundary.

∆ye

[

µe

(une − use

∆ye

)

+ µe

(vE − vP

∆xe

)]

−

∆yw

[

µw

(unw − usw

∆yw

)

+ µw

(vP − vW

∆xw

)]

+

∆xn

[

(2µ + λ)n

(vN − vP

∆yn

)

+ λn

(une − unw

∆xn

)

−

(3λ + 2µ)nαn(∆T )n

]

−

∆xs

[

σ∗

yy,south

]

+

∆xP ∆yP

[

fy,P

]

= 0 (C.34)


AP vP = AE vE + AS vS + AN vN + AW vW + Sv (C.35)


276

C.3. THE VZ -EQUILIBRIUM EQUATION

AE = µe∆ye

∆xe(C.36)

AW = µw∆yw

∆xw(C.37)


∆yn(C.38)

AS = 0 (C.39)

AP = AE + AW + AN (C.40)

Sv = Sv,s + Sv,n + Sv,e + Sv,w + Sv,P (C.41)


Sv,s = −∆xs σ∗

yy,south (C.42)

Sv,n = λn(une − unw) − ∆xn(3λ + 2µ)nαn(∆T )n (C.43)

Sv,e = µe(une − use) (C.44)

Sv,w = −µw(unw − usw) (C.45)

Sv,P = ∆xP ∆yP fv,P (C.46)

C.3 The vz-Equilibrium Equation

Integration of the vz-equation over a half-cell on the south boundary leads to:

∫∫

V

(∂(r τrz)

∂r+

∂(r σzz)

∂z+ fz

)

dr dz = 0 (C.47)

∫

S

(r τrz) dz

n

s

+

∫

S

(r σzz) dr

e

w

+

∫∫

V

(

rfz

)

dr dz = 0 (C.48)

(rnτrz,n∆zn) − (rsτrz,s∆zs) + (reσzz,e∆re) − (rwσzz,w∆rw)+

(rP fz,P ) ∆rP ∆zP = 0 (C.49)


277


τrz,n = µn∂vz

∂r

∣∣∣∣n

+ µn∂vr

∂z

∣∣∣∣n

(C.50)

= µn

(vz,N − vz,P

∆rn

)

+ µn

(vr,ne − vr,nw

∆zn

)

(C.51)

τrz,s = τ∗

rz,south (C.52)

σzz,e = (2µ + λ)e∂vz

∂z

∣∣∣∣e

+ λe∂vr

∂r

∣∣∣∣e

+λevr,e

re− (3λ + 2µ)eαe(∆T )e (C.53)

= (2µ + λ)e

(vz,E − vz,P

∆ze

)

+ λe

(vr,ne − vr,se

∆re

)

+λevr,e

re−

(3λ + 2µ)eαe(∆T )e (C.54)

σzz,w = (2µ + λ)w∂vz

∂z

∣∣∣∣w

+ λw∂vr

∂r

∣∣∣∣w

+λwvr,w

rw− (3λ + 2µ)wαw(∆T )w (C.55)

= (2µ + λ)w

(vz,P − vz,W

∆zw

)

+ λw

(vr,nw − vr,sw

∆rw

)

+λwvr,w

rw−

(3λ + 2µ)wαw(∆T )w (C.56)

Where τ∗

rz,south is the prescribed traction on the south boundary.

rn∆zn

[

µn

(vz,N − vz,P

∆rn

)

+ µn

(vr,ne − vr,nw

∆zn

)]

−

rs∆zs

[

τ∗

rz,south

]

+

re∆re

[

(2µ + λ)e

(vz,E − vz,P

∆ze

)

+ λe

(vr,ne − vr,se

∆re

)

+λevr,e

re−

(3λ + 2µ)eαe(∆T )e

]

−

rw∆rw

[

(2µ + λ)w

(vz,P − vz,W

∆zw

)

+ λw

(vr,nw − vr,sw

∆rw

)

+λwvr,w

rw−

(3λ + 2µ)wαw(∆T )w

]

+

∆rP ∆zP

[

rP fz,P

]

= 0 (C.57)


278

C.4. THE VR-EQUILIBRIUM EQUATION

AP vz,P = AE vz,E + AS vz,S + AN vz,N + AW vz,W + Svz (C.58)


AE = (2µ + λ)ere∆re

∆ze(C.59)

AS = 0 (C.60)

AN = µnrn∆zn

∆rn(C.61)

AW = (2µ + λ)wrw∆rw

∆zw(C.62)

AP = AE + AN + AW (C.63)

Svz = Svz ,s + Svz ,n + Svz ,e + Svz,w + Svz ,P (C.64)

The components of the source term are:

Svz ,s = −rs∆zs τ∗

rz,south (C.65)

Svz ,n = µnrn(vr,ne − vr,nw) (C.66)

Svz ,e = λere(vr,ne − vr,se) + λe∆revr,e − (3λ + 2µ)eαe(∆T )e (C.67)

Svz ,w = −λwrw(vr,nw − vr,sw) − λw∆rwvr,w + (3λ + 2µ)wαw(∆T )w (C.68)

Svz ,P = ∆rP ∆zP rP fz,P (C.69)

C.4 The vr-Equilibrium Equation

Integration of the vr-equation over a half-cell on the south boundary leads to:

∫∫

V

(∂(r σrr)

∂r+

∂(r τrz)

∂z− σθθ + rfr

)

dr dz = 0 (C.70)

∫

S

(r σrr) dz

n

s

+

∫

S

(r τrz) dr

e

w

+

∫∫

V

(

rfr − σθθ

)

dr dz = 0 (C.71)


rP frP − σθθ,P

)

∆rP ∆zP = 0 (C.72)

279



σrr,n = (2µ + λ)n∂vr

∂r

∣∣∣∣n

+ λn∂vz

∂z

∣∣∣∣n

+λnvr,n

rn− (3λ + 2µ)nαn(∆T )n (C.73)

= (2µ + λ)n

(vr,N − vr,P

∆rn

)

+ λn

(vz,ne − vz,nw

∆zn

)

+λnvr,n

rn−

(3λ + 2µ)nαn(∆T )n (C.74)

σrr,s = σ∗

rr,south (C.75)

τrz,e = µe∂vz

∂r

∣∣∣∣e

+ µe∂vr

∂z

∣∣∣∣e

(C.76)

= µe

(vz,ne − vz,se

∆re

)

+ µe

(vr,E − vr,P

∆ze

)

(C.77)

τrz,w = µw∂vz

∂r

∣∣∣∣w

+ µw∂vr

∂z

∣∣∣∣w

(C.78)

= µw

(vz,nw − vz,sw

∆rw

)

+ µw

(vr,P − vr,W

∆zw

)

(C.79)

The stress at the cell centre is given by:

σθθ,P = (2µ + λ)Pvr,P

rP+ λP

∂vr

∂r

∣∣∣∣P

+ λP∂vz

∂z

∣∣∣∣P

−

(3λ + 2µ)P αP (∆T )P (C.80)

= (2µ + λ)Pvr,P

rP+ λP

(vr,n − vr,s

∆rP

)

+ λP

(vz,e − vz,w

∆zP

)

−

(3λ + 2µ)P αP (∆T )P (C.81)

Where σ∗

rr,south is the prescribed traction on the south boundary.

280

C.4. THE VR-EQUILIBRIUM EQUATION

rn∆zn

[

(2µ + λ)n

(vr,N − vr,P

∆rn

)

+ λn

(vz,ne − vz,nw

∆zn

)

+λnvr,n

rn−

(3λ + 2µ)nαn(∆T )n

]

−

rs∆zs

[

σ∗

rr,south

]

+

re∆re

[

µe

(vz,ne − vz,se

∆re

)

+ µe

(vr,E − vr,P

∆ze

)]

−

rw∆rw

[

µw

(vz,nw − vz,sw

∆rw

)

+ µw

(vr,P − vr,W

∆zw

)]

+

∆rP ∆zP

[

rP fr,P − (2µ + λ)Pvr,P

rP− λP

(vr,n − vr,s

∆rP

)

−

λP

(vz,e − vz,w

∆zP

)

+ (3λ + 2µ)P αP (∆T )P

]

= 0 (C.82)


AP vr,P = AE vr,E + AS vr,S + AN vr,N + AW vr,W + Svr (C.83)


AE = µere∆re

∆ze(C.84)

AS = 0 (C.85)

AN = (2µ + λ)nrn∆zn

∆rn(C.86)

AW = µwrw∆rw

∆zw(C.87)

AP = AE + AN + AW (C.88)

Svr = Svr ,s + Svr ,n + Svr,e + Svr ,w + Svr ,P (C.89)


281


Svr ,s = −rs∆zsσ∗

rr,south (C.90)

Svr,n = λnrn(vz,ne − vz,nw) + λn∆znvr,n − (3λ + 2µ)nαn(∆T )n (C.91)

Svr,e = µere(vz,ne − vz,se) (C.92)

Svr ,w = −µwrw(vz,nw − vz,sw) (C.93)

Svr ,P = ∆rP ∆zP rP fr,P − ∆rP ∆zP (2µ + λ)Pvr,P

rP−

∆zP λP (vr,n − vr,s) − ∆rP λP (vz,e − vz,w +

∆rP ∆zP (3λ + 2µ)P αP (∆T )P (C.94)

282

Appendix D

Treatment of the West Boundary

Equations

D.1 The u-Equilibrium Equation

Integration of the u-equilibrium equation over a half-cell on the west leads to:

∫∫

V

(∂σxx

∂x+

∂τxy

∂y+ fx

)

dx dy = 0 (D.1)

∫

S

σxx dy

e

w

+

∫

S

τxy dx

n

s

+

∫∫

V

(

fx

)

dx dy = 0 (D.2)

(σxx,e∆ye) − (σxx,w∆yw) + (τxy,n∆xn) − (τxy,s∆xs) + ∆xP ∆yP fx,P = 0 (D.3)



∂x

∣∣∣∣e

+ λe∂v

∂y

∣∣∣∣e

− (3λ + 2µ)eαe(∆T )e (D.4)

= (2µ + λ)e

(uE − uP

∆xe

)

+ λe

(vne − vse

∆ye

)

−

(3λ + 2µ)eαe(∆T )e (D.5)

σxx,w = σ∗

xx,west (D.6)

283

APPENDIX D. TREATMENT OF THE WEST BOUNDARY EQUATIONS

τxy,n = µn∂u

∂y

∣∣∣∣n

+ µn∂v

∂x

∣∣∣∣n

(D.7)

= µn

(uN − uP

∆yn

)

+ µn

(vne − vnw

∆xn

)

(D.8)

τxy,s = µs∂u

∂y

∣∣∣∣s

+ µs∂v

∂x

∣∣∣∣s

(D.9)

= µs

(uP − uS

∆ys

)

+ µs

(vse − vsw

∆xs

)

(D.10)

Where σ∗

xx,west is the prescribed traction on the west boundary.

∆ye

[

(2µ + λ)e

(uE − uP

∆xe

)

+ λe

(vne − vse

∆ye

)

−

(3λ + 2µ)eαe(∆T )e

]

−

∆yw

[

σ∗

xx,west

]

+

∆xn

[

µn

(uN − uP

∆yn

)

+ µn

(vne − vnw

∆xn

)]

−

∆xs

[

µs

(uP − uS

∆ys

)

+ µs

(vse − vsw

∆xs

)]

+

∆xP ∆yP

[

fx,P

]

= 0 (D.11)


AP uP = AE uE + AS uS + AN uN + AW uW + Su (D.12)



∆xe(D.13)

AW = 0 (D.14)

AN = µn∆xn

∆yn(D.15)

AS = µs∆xn

∆ys(D.16)

AP = AE + AN + AS (D.17)

Su = Su,s + Su,n + Su,e + Su,w + Su,P (D.18)

284

D.2. THE V -EQUILIBRIUM EQUATION


Su,s = −µs(vse − vsw) (D.19)

Su,n = µn(vne − vnw) (D.20)

Su,e = λe(vne − vse) − ∆ye(3λ + 2µ)eαe(∆T )e (D.21)

Su,w = −∆ywσ∗

xx,west (D.22)

Su,P = ∆xP ∆ypfx,P (D.23)

D.2 The v-Equilibrium Equation

Integration of the v-equilibrium equation over a half-cell on the west boundary leads

to:

∫∫

V

(∂τxy

∂x+

∂σyy

∂y+ fy

)

dx dy = 0 (D.24)

∫

S

τxy dy

e

w

+

∫

S

σyy dx

n

s

+

∫∫

V

(

fy

)

dx dy = 0 (D.25)

(τxy,e∆ye) − (τxy,w∆yw) + (σyy,n∆xn) − (σyy,s∆xs) + ∆xP ∆yP fy,P = 0 (D.26)


τxy,e = µe∂u

∂y

∣∣∣∣e

+ µe∂v

∂x

∣∣∣∣e

(D.27)

= µe

(une − use

∆ye

)

+ µe

(vE − vP

∆xe

)

(D.28)

τxy,w = τ∗

xy,west (D.29)


∂y

∣∣∣∣n

+ λn∂u

∂x

∣∣∣∣n

− (3λ + 2µ)nαn(∆T )n (D.30)

= (2µ + λ)n

(vN − vP

∆yn

)

+ λn

(une − unw

∆xn

)

−

(3λ + 2µ)nαn(∆T )n (D.31)

285


σyy,s = (2µ + λ)s∂v

∂y

∣∣∣∣s

+ λs∂u

∂x

∣∣∣∣s

− (3λ + 2µ)sαs(∆T )s (D.32)

= (2µ + λ)s

(vP − vS

∆ys

)

+ λs

(use − usw

∆xs

)

−

(3λ + 2µ)sαs(∆T )s (D.33)

Where τ∗

xy,west is the prescribed traction on the west boundary.

∆ye

[

µe

(une − use

∆ye

)

+ µe

(vE − vP

∆xe

)]

−

∆yw

[

τ∗

xy,west

]

+

∆xn

[

(2µ + λ)n

(vN − vP

∆yn

)

+ λn

(une − unw

∆xn

)

−

(3λ + 2µ)nαn(∆T )n

]

−

∆xs

[

(2µ + λ)s

(vP − vS

∆ys

)

+ λs

(use − usw

∆xs

)

−

(3λ + 2µ)sαs(∆T )s

]

+

∆xP ∆yP

[

fy,P

]

= 0 (D.34)


AP vP = AE vE + AS vS + AN vN + AW vW + Sv (D.35)


AE = µe∆ye

∆xe(D.36)

AW = 0 (D.37)


∆yn(D.38)

AS = (2µ + λ)s∆xs

∆ys(D.39)

AP = AE + AN + AS (D.40)

Su = Sv,s + Sv,n + Sv,e + Sv,w + Sv,P (D.41)


286

D.3. THE VZ -EQUILIBRIUM EQUATION

Sv,s = −λs(use − usw) + ∆xs(3λ + 2µ)sαs(∆T )s (D.42)

Sv,n = λn(une − unw) − ∆xn(3λ + 2µ)nαn(∆T )n (D.43)

Sv,e = µe(une − use) (D.44)

Sv,w = −∆ywτ∗

xy,west (D.45)

Sv,P = ∆xP ∆yP fy,P (D.46)

D.3 The vz-Equilibrium Equation

Integration of the vz-equation over a half-cell on the west boundary leads to:

∫∫

V

(∂(r τrz)

∂r+

∂(r σzz)

∂z+ fz

)

dr dz = 0 (D.47)

∫

S

(r τrz) dz

n

s

+

∫

S

(r σzz) dr

e

w

+

∫∫

V

(

rfz

)

dr dz = 0 (D.48)

(rnτrz,n∆zn) − (rsτrz,s∆zs) + (reσzz,e∆re) − (rwσzz,w∆rw)+

(rP fz,P ) ∆rP ∆zP = 0 (D.49)


τrz,n = µn∂vz

∂r

∣∣∣∣n

+ µn∂vr

∂z

∣∣∣∣n

(D.50)

= µn

(vz,N − vz,P

∆rn

)

+ µn

(vr,ne − vr,nw

∆zn

)

(D.51)

τrz,s = µs∂vz

∂r

∣∣∣∣s

+ µn∂vr

∂z

∣∣∣∣s

(D.52)

= µs

(vz,P − vz,S

∆rs

)

+ µs

(vr,se − vr,sw

∆zs

)

(D.53)

287


σzz,e = (2µ + λ)e∂vz

∂z

∣∣∣∣e

+ λe∂vr

∂r

∣∣∣∣e

+λevr,e

re− (3λ + 2µ)eαe(∆T )e (D.54)

= (2µ + λ)e

(vz,E − vz,P

∆ze

)

+ λe

(vr,ne − vr,se

∆re

)

+λevr,e

re−

(3λ + 2µ)eαe(∆T )e (D.55)

σzz,w = σ∗

zz,west (D.56)

Where σ∗

zz,west is the prescribed stress on the west boundary.

rn∆zn

[

µn

(vz,N − vz,P

∆rn

)

+ µn

(vr,ne − vr,nw

∆zn

)]

−

rs∆zs

[

µs

(vz,P − vz,S

∆rs

)

+ µs

(vr,se − vr,sw

∆zs

)]

+

re∆re

[

(2µ + λ)e

(vz,E − vz,P

∆ze

)

+ λe

(vr,ne − vr,se

∆re

)

+λevr,e

re−

(3λ + 2µ)eαe(∆T )e

]

−

rw∆rw

[

σ∗

zz,west

]

+

∆rP ∆zP

[

rP fz,P

]

= 0 (D.57)


AP vz,P = AE vz,E + AS vz,S + AN vz,N + AW vz,W + Svz (D.58)


AE = (2µ + λ)ere∆re

∆ze(D.59)

AS = µsrs∆zs

∆rs(D.60)

AN = µnrn∆zn

∆rn(D.61)

AW = 0 (D.62)

AP = AE + AS + AN (D.63)

Svz = Svz ,s + Svz ,n + Svz ,e + Svz ,w + Svz ,P (D.64)

288

D.4. THE VR-EQUILIBRIUM EQUATION


Svz ,s = −µsrs(vr,se − vr,sw) (D.65)

Svz ,n = µnrn(vr,ne − vr,nw) (D.66)

Svz ,e = λere(vr,ne − vr,se) + λe∆revr,e − re∆re(3λ + 2µ)eαe(∆T )e (D.67)

Svz ,w = −rw∆rwσ∗

zz,west (D.68)

Svz ,P = ∆rP ∆zP rP fz,P (D.69)

D.4 The vr-Equilibrium Equation

Integration of the vr-equation over a half-cell on the west boundary leads to:

∫∫

V

(∂(r σrr)

∂r+

∂(r τrz)


)

dr dz = 0 (D.70)

∫

S

(r σrr) dz

n

s

+

∫

S

(r τrz) dr

e

w

+

∫∫

V

(

rfr − σθθ

)

dr dz = 0 (D.71)


rP frP − σθθ,P

)

∆rP ∆zP = 0 (D.72)


σrr,n = (2µ + λ)n∂vr

∂r

∣∣∣∣n

+ λn∂vz

∂z

∣∣∣∣n

+λnvr,n

rn− (3λ + 2µ)nαn(∆T )n (D.73)

= (2µ + λ)n

(vr,N − vr,P

∆rn

)

+ λn

(vz,ne − vz,nw

∆zn

)

+λnvr,n

rn−

(3λ + 2µ)nαn(∆T )n (D.74)

σrr,s = (2µ + λ)s∂vr

∂r

∣∣∣∣s

+ λs∂vz

∂z

∣∣∣∣s

+λsvr,s

rs− (3λ + 2µ)sαs(∆T )s (D.75)

= (2µ + λ)s

(vr,P − vr,S

∆rs

)

+ λs

(vz,se − vz,sw

∆zs

)

+λsvr,s

rs−

(3λ + 2µ)sαs(∆T )s (D.76)

289


τrz,e = µe∂vz

∂r

∣∣∣∣e

+ µe∂vr

∂z

∣∣∣∣e

(D.77)

= µe

(vz,ne − vz,se

∆re

)

+ µe

(vr,E − vr,P

∆ze

)

(D.78)

τrz,w = τ∗

rz,west (D.79)



rP+ λP

∂vr

∂r

∣∣∣∣P

+ λP∂vz

∂z

∣∣∣∣P

−

(3λ + 2µ)P αP (∆T )P (D.80)

= (2µ + λ)Pvr,P

rP+ λP

(vr,n − vr,s

∆rP

)

+ λP

(vz,e − vz,w

∆zP

)

−

(3λ + 2µ)P αP (∆T )P (D.81)

Where τ∗

rz,west is the prescribed stress on the west boundary.

rn∆zn

[

(2µ + λ)n

(vr,N − vr,P

∆rn

)

+ λn

(vz,ne − vz,nw

∆zn

)

+λnvr,n

rn−

(3λ + 2µ)nαn(∆T )n

]

−

rs∆zs

[

(2µ + λ)s

(vr,P − vr,S

∆rs

)

+ λs

(vz,se − vz,sw

∆zs

)

+λsvr,s

rs−

(3λ + 2µ)sαs(∆T )s

]

+

re∆re

[

µe

(vz,ne − vz,se

∆re

)

+ µe

(vr,E − vr,P

∆ze

)]

−

rw∆rw

[

τ∗

rz,west

]

+

∆rp∆zP

[

rP fr,P −

(2µ + λ)Pvr,P

rP− λP

(vr,n − vr,s

∆rP

)

− λP

(vz,e − vz,w

∆zP

)

+

(3λ + 2µ)P αP (∆T )P

]

= 0 (D.82)


AP vr,P = AE vr,E + AS vr,S + AN vr,N + AW vr,W + Svr (D.83)

290

D.4. THE VR-EQUILIBRIUM EQUATION


AE = µere∆re

∆ze(D.84)

AS = (2µ + λ)srs∆zs

∆rs(D.85)

AN = (2µ + λ)nrn∆zn

∆rn(D.86)

AW = 0 (D.87)

AP = AE + AS + AN (D.88)

Svr = Svr ,s + Svr ,n + Svr,e + Svr ,w + Svr ,P (D.89)


Svr ,s = −λsrs(vz,se − vz,sw) − λs∆zsvr,s +

rs∆zs(3λ + 2µ)sαs(∆T )s (D.90)

Svr ,n = λnrn(vz,ne − vz,nw) + λn∆znvr,n −rn∆zn(3λ + 2µ)nαn(∆T )n (D.91)

Svr ,e = µere(vz,ne − vz,se) (D.92)

Svr ,w = −rw∆rwτ∗

rz,west (D.93)

Svr ,P = ∆rP ∆zP rP fr,P − ∆rP ∆zP (2µ + λ)Pvr,P

rP−

∆zP λP (vr,n − vr,s) − ∆rP λP (vz,e − vz,w) +

∆rP ∆zP (3λ + 2µ)αP (∆T )P (D.94)

291


292

Appendix E

Elasticity Equations in Planar

Non-Orthogonal Coordinates

E.1 Non-orthogonal Coordinates

The equations in the previous Appendices are valid for a Cartesian coordinate system.

In order to make this method as geometrically flexible as possible, a general non-

orthogonal coordinate system will be used. The first step is to find the relationship

between spatial derivatives in the Cartesian system and the non-orthogonal system,

then to use these relations to replace the terms appearing the the Cartesian governing

equations. From the chain rule of differentiation we know that:

∂φ

∂ξ=

∂φ

∂x

∂x

∂ξ+

∂φ

∂y

∂y

∂ξ(E.1)

∂φ

∂η=

∂φ

∂x

∂x

∂η+

∂φ

∂y

∂y

∂η(E.2)

Or in matrix form:

∂φ∂ξ∂φ∂η

=

[∂x∂ξ

∂y∂ξ

∂x∂η

∂y∂η

][∂φ∂x∂φ∂y

]

(E.3)

By taking the inverse of this relation we get:

∂φ∂x∂φ∂y

=1

|J |

[∂y∂η −∂y

∂ξ

−∂x∂η

∂x∂ξ

] [∂φ∂ξ∂φ∂η

]

(E.4)

Or explicitly:

293

APPENDIX E. ELASTICITY EQUATIONS IN PLANAR NON-ORTHOGONAL COORDINATES

∂φ

∂x=

1

|J |

∂y

∂η

∂φ

∂ξ− ∂y

∂ξ

∂φ

∂η

(E.5)

∂φ

∂y=

1

|J |

−∂x

∂η

∂φ

∂ξ+

∂x

∂ξ

∂φ

∂η

(E.6)

Where |J | is the Jacobian of the transformation, and is given by:

|J | =∂x

∂ξ

∂y

∂η− ∂y

∂ξ

∂x

∂η(E.7)

E.2 The u-Displacement Equation

E.2.1 Transformation of Equilibrium Equation

The two-dimensional, Cartesian form of the x-equilibrium equation is:

∂σxx

∂x+

∂τxy

∂y+ fx = 0 (E.8)

This can be transformed into non-orthogonal coordinates using Equations E.5-E.6:

1

J

∂y

∂η

∂σxx

∂ξ− ∂y

∂ξ

∂σx

∂η

+1

J

−∂x

∂η

∂τxy

∂ξ+

∂x

∂ξ

∂τxy

∂η

+ fx = 0 (E.9)

Which can be simplified by moving the metric terms to give:

1

J

∂

∂ξ

[

σxx∂y

∂η

]

− ∂

∂η

[

σxx∂y

∂ξ

]

+1

J

− ∂

∂ξ

[

τxy∂x

∂η

]

+∂

∂η

[

τxy∂x

∂ξ

]

+fx = 0 (E.10)

E.2.2 Transformation of Constitutive Relations

The Cartesian form of the constitutive relations are:

σxx = 2µǫxx + λ(ǫxx + ǫyy) − (2µ + 3λ)α(∆T )

= (2µ + λ)ǫxx + λǫyy − (2µ + 3λ)α(∆T )

= (2µ + λ)∂u

∂x+ λ

∂v

∂y− (2µ + 3λ)α(∆T ) (E.11)

τxy = 2µǫxy

= µ

(∂u

∂y+

∂v

∂x

)

(E.12)

294

E.2. THE U -DISPLACEMENT EQUATION

Which are transformed in the same way as for the equilibrium equation to give:

σxx =(2µ + λ)

|J |

∂y

∂η

∂u

∂ξ− ∂y

∂ξ

∂u

∂η

+λ

|J |

−∂x

∂η

∂v

∂ξ+

∂x

∂ξ

∂v

∂η

−

(2µ + 3λ)α(∆T )

=(2µ + λ)

|J |

∂

∂ξ

[

u∂y

∂η

]

− ∂

∂η

[

u∂y

∂ξ

]

+

λ

|J |

− ∂

∂ξ

[

v∂x

∂η

]

+∂

∂η

[

v∂x

∂ξ

]

− (2µ + 3λ)α(∆T ) (E.13)

τxy =µ

|J |

−∂x

∂η

∂u

∂ξ+

∂x

∂ξ

∂u

∂η

+µ

|J |

∂y

∂η

∂v

∂ξ− ∂y

∂ξ

∂v

∂η

=µ

|J |

− ∂

∂ξ

[

u∂x

∂η

]

+∂

∂η

[

u∂x

∂ξ

]

+

µ

|J |

∂

∂ξ

[

v∂y

∂η

]

− ∂

∂η

[

v∂y

∂ξ

]

(E.14)

The cell face values of the stresses are given by:

σxx,e =(2µ + λ)e

|J |

[YYe

∆ηe

UE − UP

∆ξe− YX e

∆ξe

Une − Use

∆ηe

]

+

λe

|J |

[

−XYe

∆ηe

VE − VP

∆ξe+

XX e

∆ξe

Vne − Vse

∆ηe

]

−

(2µ + 3λ)eαe(∆T )e (E.15)

=(2µ + λ)e

Vole[YYe(UE − UP ) − YX e(Une − Use)] +

λe

Vole[−XYe(VE − VP ) + XX e(Vne − Vse)] −

(2µ + 3λ)eαe(∆T )e (E.16)

σxx,w =(2µ + λ)w

|J |

[YYw

∆ηw

UP − UW

∆ξw− YXw

∆ξw

Unw − Usw

∆ηw

]

+

λw

|J |

[

−XYw

∆ηw

VP − VW

∆ξw+

XXw

∆ξw

Vnw − Vsw

∆ηw

]

−

(2µ + 3λ)wαw(∆T )w (E.17)

=(2µ + λ)w

Volw[YYw(UP − UW ) − YXw(Unw − Usw)] +

λw

Volw[−XYw(VP − VW ) + XXw(Vnw − Vsw)] −

(2µ + 3λ)wαw(∆T )w (E.18)

295


σxx,n =(2µ + λ)n

|J |

[YYn

∆ηn

Une − Unw

∆ξn− YX n

∆ξn

UN − UP

∆ηn

]

+

λn

|J |

[

−XYn

∆ηn

Vne − Vnw

∆ξn+

XX n

∆ξn

VN − VP

∆ηn

]

−

(2µ + 3λ)nαn(∆T )n (E.19)

=(2µ + λ)n

Voln[YYn(Une − Unw) − YX n(UN − UP )] +

λn

Voln[−XYn(Vne − Vnw) + XX n(VN − VP )] −

(2µ + 3λ)nαn(∆T )n (E.20)

σxx,s =(2µ + λ)s

|J |

[YYs

∆ηs

Use − Usw

∆ξs− YX s

∆ξs

UP − US

∆ηs

]

+

λs

|J |

[

−XYs

∆ηs

Vse − Vsw

∆ξs+

XX s

∆ξs

VP − VS

∆ηs

]

−

(2µ + 3λ)sαs(∆T )s (E.21)

=(2µ + λ)s

Vols[YYs(Use − Usw) − YX s(UP − US)] +

λs

Vols[−XYs(Vse − Vsw) + XX s(VP − VS)] −

(2µ + 3λ)sαs(∆T )s (E.22)

τxy,e =µe

|J |

[

−XYe

∆ηe

UE − UP

∆ξe+

XX e

∆ξe

Une − Use

∆ηe

]

+

µe

|J |

[YYe

∆ηe

VE − VP

∆ξe− YX e

∆ξe

Vne − Vse

∆ηe

]

(E.23)

=µe

Vole[−XYe(UE − UP ) + XX e(Une − Use)] +

µe

Vole[YYe(VE − VP ) − YX e(Vne − Vse] (E.24)

τxy,w =µw

|J |

[

−XYw

∆ηw

UP − UW

∆ξw+

XXw

∆ξw

Unw − Usw

∆ηw

]

+

µw

|J |

[YYw

∆ηw

VP − VW

∆ξw− YXw

∆ξw

Vnw − Vsw

∆ηw

]

(E.25)

=µw

Volw[−XYw(UP − UW ) + XXw(Unw − Usw)] +

µw

Volw[YYw(VP − VW ) −YXw(Vnw − Vsw] (E.26)

296


τxy,n =µn

|J |

[

−XYn

∆ηn

Une − Unw

∆ξn+

XX n

∆ξn

UN − UP

∆ηn

]

+

µn

|J |

[YYn

∆ηn

Vne − Vnw

∆ξn− YX n

∆ξn

VN − VP

∆ηn

]

(E.27)

=µn

Voln[−XYn(Une − Unw) + XX n(UN − UP )] +

µn

Voln[YYn(Vne − Vnw) −YX n(VN − VP )] (E.28)

τxy,s =µs

|J |

[

−XYs

∆ηs

Use − Usw

∆ξs+

XX s

∆ξs

UP − US

∆ηs

]

+

µs

|J |

[YYs

∆ηs

Vse − Vsw

∆ξs− YX s

∆ξs

VP − VS

∆ηs

]

(E.29)

=µs

Vols[−XYs(Use − Usw) + XX s(UP − US)] +

µs

Vols[YYs(Vse − Vsw) − YX s(VP − VS)] (E.30)

E.2.3 Finite Volume Discretization

Now that the governing equations have been re-written in non-orthogonal coordinates,

we can perform the finite volume discretization process. This process begins by inte-

grating the the x-equilibrium equation over a finite-volume cell. The equation governing

u-displacement is given by:

∫∫

V

(∂σxx

∂x+

∂τxy

∂y+ fx

)

dx dy = 0 (E.31)

Which in non-orthogonal coordinates can be written:

∫∫

V

(1

|J |

∂

∂ξ

[

σxx∂y

∂η

]

− ∂

∂η

[

σxx∂y

∂ξ

]

+

1

|J |

− ∂

∂ξ

[

τxy∂x

∂η

]

+∂

∂η

[

τxy∂x

∂ξ

]

+ fx

)

|J | dξ dη = 0 (E.32)

Since, dx dy = |J | dξ dη.

297


∫∫

V

(∂

∂ξ

[

σxx∂y

∂η

]

− ∂

∂η

[

σxx∂y

∂ξ

]

−

∂

∂ξ

[

τxy∂x

∂η

]

+∂

∂η

[

τxy∂x

∂ξ

]

+ fx

)

dξ dη = 0 (E.33)

[ ∫

S

(

σxx∂y

∂η− τxy

∂x

∂η

)

dη

]e

w

+

[ ∫

S

(

τxy∂x

∂ξ− σxx

∂y

∂ξ

)

dξ

]n

s

+

∫∫

V

fx|J | dξ dη = 0 (E.34)

(

σxx,e∂y

∂η

∣∣∣∣e

− τxy,e∂x

∂η

∣∣∣∣e

)

∆ηe −(

σxx,w∂y

∂η

∣∣∣∣w

− τxy,w∂x

∂η

∣∣∣∣w

)

∆ηw +

(

τxy,n∂x

∂ξ

∣∣∣∣n

− σxx,n∂y

∂ξ

∣∣∣∣n

)

∆ξn −(

τxy,s∂x

∂ξ

∣∣∣∣s

− σxx,s∂y

∂ξ

∣∣∣∣s

)

∆ξs +

fx,P |J |∆ξ∆η = 0 (E.35)

The final form of the discretized u-displacement equation is:

YYeσxx,e −XYeτxy,e − YYwσxx,w + XYwτxy,w +

XX nτxy,n − YX nσxx,n −XX sτxy,s + YX sσxx,s + fx,P VolP = 0 (E.36)

The geometrical terms are given by:

XX =∂x

∂ξ∆ξ XY =

∂x

∂η∆η YX =

∂y

∂ξ∆ξ YY =

∂y

∂η∆η

When the transformed constitutive relations have be substituted into Equation E.36

we get:

298


YYe

[(2µ + λ)e

Vole[YYe(UE − UP ) − YX e(Une − Use)] +

λe

Vole[−XYe(VE − VP ) + XX e(Vne − Vse)] − (2µ + 3λ)eαe(∆T )e

]

−XYe

[µe


µe

Vole[YYe(VE − VP ) −YX e(Vne − Vse)]

]

−YYw

[(2µ + λ)w

Volw[YYw(UP − UW ) − YXw(Unw − Usw)] +

λw

Volw[−XYw(VP − VW ) + XXw(Vnw − Vsw)] − (2µ + 3λ)wαw(∆T )w

]

+XYw

[µw


µw

Volw[YYw(VP − VW ) − YXw(Vnw − Vsw)]

]

+XX n

[µn


µn

Voln[YYn(Vne − Vnw) − YX n(VN − VP )]

]

−YX n

[(2µ + λ)n

Voln[YYn(Une − Unw) − YX n(UN − UP )] +

λn

Voln[−XYn(Vne − Vnw) + XX n(VN − VP )] − (2µ + 3λ)nαn(∆T )n

]

−XX s

[µs


µs

Vols[YYs(Vse − Vsw) − YX s(VP − VS)]

]

+YX s

[(2µ + λ)s

Vols[YYs(Use − Usw) −YX s(UP − US)] +

λs

Vols[−XYs(Vse − Vsw) + XX s(VP − VS)] − (2µ + 3λ)sαs(∆T )s

]

+VolP fx = 0 (E.37)

Which can be written:

AP UP = AE UE + AW Uw + AN UN + AS US + Su (E.38)


299


AE =YY2

e(2µ + λ)eVole

+XY2

eµe

Vole(E.39)

AW =YY2

w(2µ + λ)wVolw

+XY2

wµw

Volw(E.40)

AN =XX 2

nµn

Voln+

YX 2n(2µ + λ)nVoln

(E.41)

AS =XX 2

sµs

Vols+

YX 2s(2µ + λ)sVols

(E.42)

AP = AE + AW + AN + AS (E.43)

Su = Su,e + Su,w + Su,n + Su,s + Su,P (E.44)


Su,e = −(2µ + λ)e YYe YX e

Vole(une − use) +

λe YYe XX e

Vole(vne − vse)

−YYe (2µ + 3λ)eαe(∆T )e

−µe XYe XX e

Vole(une − use) +

µe XYe YX e

Vole(vne − vse)

−λe YYe XYe

VolevE − µe XYe YYe

VolevE

(E.45)

Su,w =(2µ + λ)w YYw YXw

Volw(unw − usw) − λw YYw XXw

Volw(vnw − vsw)

+YYw (2µ + 3λ)wαw(∆T )w

+µw XYw XXw

Volw(unw − usw) − µw XYw YXw

Volw(vnw − vsw)

−λw YYw XYw

VolwvW − µw XYw YYw

VolwvW

(E.46)

Su,n = −µn XX n XYn

Voln(une − unw) +

µn XX n YYn

Voln(vne − vnw)

−(2µ + λ)n YX n YYn

Voln(une − unw) +

λn YX n XYn

Voln(vne − vnw)

+YX n (2µ + 3λ)nαn(∆T )n

−µn XX n YX n

VolnvN − λn YX n XX n

VolnvN (E.47)

300

E.3. THE V -DISPLACEMENT EQUATION

Su,s =µs XX s XYs

Vols(use − usw) − µs XX s YYs

Vols(vse − vsw)

+(2µ + λ)s YX s YYs

Vols(use − usw) − λs YX s XYs

Vols(vse − vsw)

−YX s (2µ + 3λ)sαs(∆T )s

−µs XX s YX s

VolsvS − λs YX s XX s

VolsvS

(E.48)

Su,P =λe YYe XYe

VolevP +

µe XYe YYe

VolevP +

λw YYw XYw

VolwvP

+µw XYw YYw

VolwvP +

µn XX n YX n

VolnvP +

λn YX n XX n

VolnvP

+µs XX s YX s

VolsvP +

λs YX s XX s

VolsvP + fx,P VolP (E.49)

E.3 The v-Displacement Equation

E.3.1 Transformation of Equilibrium Equation

The two-dimensional, Cartesian form of the y-equilibrium equation is:

∂τxy

∂x+

∂σyy

∂y+ fy = 0 (E.50)

After transformation into non-orthogonal coordinates this becomes:

1

J

∂

∂ξ

[

τxy∂y

∂η

]

− ∂

∂η

[

τxy∂y

∂ξ

]

+1

J

− ∂

∂ξ

[

σyy∂x

∂η

]

+∂

∂η

[

σyy∂x

∂ξ

]

+fy = 0 (E.51)

E.3.2 Transformation of Constitutive Relations

The Cartesian form of the constitutive relations are:

σyy = 2µǫyy + λ(ǫxx + ǫyy) − (2µ + 3λ)α(∆T )

= (2µ + λ)ǫyy + λǫxx − (2µ + 3λ)α(∆T )

= (2µ + λ)∂v

∂y+ λ

∂u

∂x− (2µ + 3λ)α(∆T ) (E.52)

τxy = 2µǫxy

= µ

(∂u

∂y+

∂v

∂x

)

(E.53)

301


Which are transformed to give:

σyy =(2µ + λ)

|J |

−∂x

∂η

∂v

∂ξ+

∂x

∂ξ

∂v

∂η

+λ

|J |

∂y

∂η

∂u

∂ξ− ∂y

∂ξ

∂u

∂η

−

(2µ + 3λ)α(∆T )

=(2µ + λ)

|J |

− ∂

∂ξ

[

v∂x

∂η

]

+∂

∂η

[

v∂x

∂ξ

]

+

λ

|J |

∂

∂ξ

[

u∂y

∂η

]

− ∂

∂η

[

u∂y

∂ξ

]

− (2µ + 3λ)α(∆T ) (E.54)

τxy =µ

|J |

−∂x

∂η

∂u

∂ξ+

∂x

∂ξ

∂u

∂η

+µ

|J |

∂y

∂η

∂v

∂ξ− ∂y

∂ξ

∂v

∂η

=µ

|J |

− ∂

∂ξ

[

u∂x

∂η

]

+∂

∂η

[

u∂x

∂ξ

]

+

µ

|J |

∂

∂ξ

[

v∂y

∂η

]

− ∂

∂η

[

v∂y

∂ξ

]

(E.55)

The cell face value of the stresses are:

σyy,e =(2µ + λ)e

|J |

[

−XYe

∆ηe

VE − VP

∆ξe+

XX e

∆ξe

Vne − Vse

∆ηe

]

+

λe

|J |

[YYe

∆ηe

UE − UP

∆ξe− YX e

∆ξe

Une − Use

∆ηe

]

−

(2µ + 3λ)eαe(∆T )e (E.56)

=(2µ + λ)e

Vole[−XYe(VE − VP ) + XX e(Vne − Vse)] +

λe

Vole[YYe(UE − UP ) − YX e(Une − Use)] −

(2µ + 3λ)eαe(∆T )e (E.57)

σyy,w =(2µ + λ)w

|J |

[

−XYw

∆ηw

VP − VW

∆ξw+

XXw

∆ξw

Vnw − Vsw

∆ηw

]

+

λw

|J |

[YYw

∆ηw

UP − UW

∆ξw− YXw

∆ξw

Unw − Usw

∆ηw

]

−

(2µ + 3λ)wαw(∆T )w (E.58)

=(2µ + λ)w

Volw[−XYw(VP − VW ) + XXw(Vnw − Vsw)] +

λw

Volw[YYw(UP − UW ) − YXw(Unw − Usw)] −

(2µ + 3λ)wαw(∆T )w (E.59)

302


σyy,n =(2µ + λ)n

|J |

[

−XYn

∆ηn

Vne − Vnw

∆ξn+

XX n

∆ξn

VN − VP

∆ηn

]

+

λn

|J |

[YYn

∆ηn

Une − Unw

∆ξn− YX n

∆ξn

UN − UP

∆ηn

]

−

(2µ + 3λ)nαn(∆T )n (E.60)

=(2µ + λ)n

Voln[−XYn(Vne − Vnw) + XX n(VN − VP )] +

λn

Voln[YYn(Une − Unw) − YX n(UN − UP )] −

(2µ + 3λ)nαn(∆T )n (E.61)

σyy,s =(2µ + λ)s

|J |

[

−XYs

∆ηs

Vse − Vsw

∆ξs+

XX s

∆ξs

VP − VS

∆ηs

]

+

λs

|J |

[YYs

∆ηs

Use − Usw

∆ξs− YX s

∆ξs

UP − US

∆ηs

]

−

(2µ + 3λ)sαs(∆T )s (E.62)

=(2µ + λ)s

Vols[−XYs(Vse − Vsw) + XX s(VP − VS)] +

λs

Vols[YYs(Use − Usw) −YX s(UP − US)] −

(2µ + 3λ)sαs(∆T )s (E.63)

τxy,e =µe

|J |

[

−XYe

∆ηe

UE − UP

∆ξe+

XX e

∆ξe

Une − Use

∆ηe

]

+

µe

|J |

[YYe

∆ηe

VE − VP

∆ξe− YX e

∆ξe

Vne − Vse

∆ηe

]

(E.64)

=µe


µe

Vole[YYe(VE − VP ) − YX e(Vne − Vse] (E.65)

τxy,w =µw

|J |

[

−XYw

∆ηw

UP − UW

∆ξw+

XXw

∆ξw

Unw − Usw

∆ηw

]

+

µw

|J |

[YYw

∆ηw

VP − VW

∆ξw− YXw

∆ξw

Vnw − Vsw

∆ηw

]

(E.66)

=µw


µw

Volw[YYw(VP − VW ) − YXw(Vnw − Vsw] (E.67)

303


τxy,n =µn

|J |

[

−XYn

∆ηn

Une − Unw

∆ξn+

XX n

∆ξn

UN − UP

∆ηn

]

+

µn

|J |

[YYn

∆ηn

Vne − Vnw

∆ξn− YX n

∆ξn

VN − VP

∆ηn

]

(E.68)

=µn


µn

Voln[YYn(Vne − Vnw) − YX n(VN − VP )] (E.69)

τxy,s =µs

|J |

[

−XYs

∆ηs

Use − Usw

∆ξs+

XX s

∆ξs

UP − US

∆ηs

]

+

µs

|J |

[YYs

∆ηs

Vse − Vsw

∆ξs− YX s

∆ξs

VP − VS

∆ηs

]

(E.70)

=µs


µs

Vols[YYs(Vse − Vsw) − YX s(VP − VS)] (E.71)

E.3.3 Finite Volume Discretization

Integration of the v-displacement equation cell gives:

∫∫

V

(∂τxy

∂x+

∂σyy

∂y+ fy

)

dxdy = 0 (E.72)

Or, in non-orthogonal coordinates:

∫∫

V

(1

|J |

∂

∂ξ

[

τxy∂y

∂η

]

− ∂

∂η

[

τxy∂y

∂ξ

]

+

1

|J |

− ∂

∂ξ

[

σyy∂x

∂η

]

+∂

∂η

[

σyy∂x

∂ξ

]

+ fy

)

|J |dξdη = 0 (E.73)

Which can be simplified to give:

∫∫

V

(∂

∂ξ

[

τxy∂y

∂η

]

− ∂

∂η

[

τxy∂y

∂ξ

]

−

∂

∂ξ

[

σyy∂x

∂η

]

+∂

∂η

[

σyy∂x

∂ξ

]

+ fy

)

dξdη = 0 (E.74)

304


[ ∫

S

(

τxy∂y

∂η− σyy

∂x

∂η

)

dη

]e

w

+

[ ∫

S

(

σyy∂x

∂ξ− τxy

∂y

∂ξ

)

dξ

]n

s

−

∫∫

V

fy|J |dξdη = 0 (E.75)

(

τxy,e∂y

∂η

∣∣∣∣e

− σyy,e∂x

∂η

∣∣∣∣e

)

∆ηe −(

τxy,w∂y

∂η

∣∣∣∣w

− σyy,w∂x

∂η

∣∣∣∣w

)

∆ηw +

(

σyy,n∂x

∂ξ

∣∣∣∣n

− τxy,n∂y

∂ξ

∣∣∣∣n

)

∆ξn −(

σyy,s∂x

∂ξ

∣∣∣∣s

− τxy,s∂y

∂ξ

∣∣∣∣s

)

∆ξs +

fy|J |∆ξ∆η = 0 (E.76)

Finally:

YYeτxy,e −XYeσyy,e − YYwτxy,w + XYwσyy,w +

XX nσyy,n − YX nτxy,n −XX sσyy,s + YX sτxy,s +

fyVolP = 0 (E.77)

The transformed constitutive relations are substituted into Equation E.77 to give:

305


YYe

[µe


µe

Vole[YYe(VE − VP ) − YX e(Vne − Vse)]

]

−XYe

[(2µ + λ)e

Vole[−XYe(VE − VP ) + XX e(Vne − Vse)] +

λe

Vole[YYe(UE − UP ) − YX e(Une − Use)] − (2µ + 3λ)eαe(∆T )e

]

−YYw

[µw


µw

Volw[YYw(VP − VW ) − YXw(Vnw − Vsw)]

]

+XYw

[(2µ + λ)w

Volw[−XYw(VP − VW ) + XXw(Vnw − Vsw)] +

λw

Volw[YYw(UP − UW ) − YXw(Unw − Usw)] − (2µ + 3λ)wαw(∆T )w

]

+XX n

[(2µ + λ)n

Voln[−XYn(Vne − Vnw) + XX n(VN − VP )] +

λn

Voln[YYn(Une − Unw) − YX n(UN − UP )] − (2µ + 3λ)nαn(∆T )n

]

−YX n

[µn


µn

Voln[YYn(Vne − Vnw) − YX n(VN − VP )]

]

−XX s

[(2µ + λ)s

Vols[−XYs(Vse − Vsw) + XX s(VP − VS)] +

λs

Vols[YYs(Use − Usw) − YX s(UP − US)] − (2µ + 3λ)sαs(∆T )s

]

+YX s

[µs


µs

Vols[YYs(Vse − Vsw) − YX s(VP − VS)]

]

+VolP fy = 0 (E.78)

Which can be written:

AP VP = AE VE + AW Vw + AN VN + AS VS + Sv (E.79)


306


AE =YY2

eµe

Vole+

XY2e(2µ + λ)eVole

(E.80)

AW =YY2

wµw

Volw+

XY2w(2µ + λ)wVolw

(E.81)

AN =XX 2

n(2µ + λ)nVoln

+YX 2

nµn

Voln(E.82)

AS =XX 2

s(2µ + λ)sVols

+YX 2

sµs

Vols(E.83)

AP = AE + AW + AN + AS (E.84)

Sv = Sv,e + Sv,w + Sv,n + Sv,s + Sv,P (E.85)


Sv,e =µe YYe XX e

Vole(une − use) −

µe YYe YX e

Vole(vne − vse)

−(2µ + λ)e XYe XX e

Vole(vne − vse) +

λe XYe YX e

Vole(une − use)

+XYe (2µ + 3λ)eαe(∆T )e

−µe YYe XYe

VoleuE − λe XYe YYe

VoleuE

(E.86)

Sv,w = −µw YYw XXw

Volw(unw − usw) +

µw YYw YXw

Volw(vnw − vsw)

+(2µ + λ)w XYw XXw

Volw(vnw − vsw) − λw XYw YXw

Volw(unw − usw)

−XYw (2µ + 3λ)wαw(∆T )w

−µw YYw XYw

VolwuW − λw XYw YYw

VolwuW

(E.87)

Sv,n = −(2µ + λ)n XX n XYn

Voln(vne − vnw) +

λn XX n YYn

Voln(une − unw)

−XX n (2µ + 3λ)nαn(∆T )n

+µn YX n XYn

Voln(une − unw) − µn YX n YYn

Voln(vne − vnw)

−λn XX n YX n

VolnuN − µn YX n XX n

VolnuN (E.88)

307


Sv,s =(2µ + λ)s XX s XYs

Vols(vse − vsw) − λs XX s YYs

Vols(use − usw)

+XX s (2µ + 3λ)sαs(∆T )s

−µs YX s XYs

Vols(use − usw) +

µs YX s YYs

Vols(vse − vsw)

−λs XX s YX s

VolsuS − µs YX s XX s

VolsuS

(E.89)

Sv,P =µe YYe XYe

VoleuP +

λe XYe YYe

VoleuP +

µw YYw XYw

VolwuP (E.90)

+λw XYw YYw

VolwuP +

λn XX n YX n

VolnuP +

µn YX n XX n

VolnuP (E.91)

+λs XX s YX s

VolsuP +

µs YX s XX s

VolsuP + fy,P VolP (E.92)

308

Appendix F

Elasticity Equations in

Cylindrical Non-Orthogonal

Coordinates

In this Chapter, the equations governing the elastic behaviour of cylindrical bodies will

be transformed into non-orthogonal coordinates.

F.1 Transformation of the Governing Equations

F.1.1 Transformation of the Axial Displacement Equation

The equation governing uz-displacement in polar coordinates is given by:

∫∫

V

2π

(∂τrz

∂r+

∂σzz

∂z+

τrz

r+ fz

)

r dr dz = 0 (F.1)


∫∫

V

(∂(r τrz)

∂r+

∂(r σzz)

∂z+ rfz

)

dr dz = 0 (F.2)

The above can be transformed into non-orthogonal coordinates using the above

relations:

∫∫

V

(1

|J |

− ∂z

∂η

∂(r τrz)

∂ξ+

∂z

∂ξ

∂(r τrz)

∂η

+

1

|J |

∂r

∂η

∂(r σzz)

∂ξ− ∂r

∂ξ

∂(r σzz)

∂η

+ r fz

)

|J | dξ dη = 0 (F.3)

309

APPENDIX F. ELASTICITY EQUATIONS IN CYLINDRICAL NON-ORTHOGONAL COORDINATES


∫∫

V

(

− ∂

∂ξ

[

r τrz∂z

∂η

]

+∂

∂η

[

r τrz∂z

∂ξ

]

+

∂

∂ξ

[

r σzz∂r

∂η

]

− ∂

∂η

[

r σzz∂r

∂ξ

])

dξ dη +

∫∫

V

(

r fz

)

|J | dξ dη = 0 (F.4)

F.1.2 Transformation of the Radial Displacement Equation

The equation governing ur-displacement in polar coordinates is given by:

∫∫

V

2π

(∂σrr

∂r+

∂τrz

∂z+

σrr − σθθ

r+ fr

)

r dr dz = 0 (F.5)


∫∫

V

(∂(r σrr)

∂r+

∂(r τrz)


)

dr dz = 0 (F.6)

Which can be transformed into non-orthogonal coordinates using the above relations

to give:

∫∫

V

(1

|J |

− ∂z

∂η

∂(r σrr)

∂ξ+

∂z

∂ξ

∂(r σrr)

∂η

+

1

|J |

∂r

∂η

∂(r τrz)

∂ξ− ∂r

∂ξ

∂(r τrz)

∂η

− σθθ + r fr

)

|J | dξ dη = 0 (F.7)


∫∫

V

(

− ∂

∂ξ

[

r σrr∂z

∂η

]

+∂

∂η

[

r σrr∂z

∂ξ

]

+

∂

∂ξ

[

r τrz∂r

∂η

]

− ∂

∂η

[

r τrz∂r

∂ξ

])

dξ dη +

∫∫

V

(

r fr − σθθ

)

|J | dξ dη = 0 (F.8)

F.2 Transformation of the Constitutive Relations

F.2.1 Transformation of the Radial Stress

The radial stress is given by:

310

F.2. TRANSFORMATION OF THE CONSTITUTIVE RELATIONS

σrr = (2µ + λ)∂vr

∂r+ λ

∂vz

∂z+

λvr

r− (3λ + 2µ)α(∆T ) (F.9)

Which can be transformed to give:

σrr =(2µ + λ)

|J |

− ∂

∂ξ

[

vr∂z

∂η

]

+∂

∂η

[

vr∂z

∂ξ

]

+

λ

|J |

∂

∂ξ

[

vz∂r

∂η

]

− ∂

∂η

[

vz∂r

∂ξ

]

+λvr

r− (3λ + 2µ)α(∆T ) (F.10)

F.2.2 Transformation of the Axial Stress

The axial stress is given by:

σzz = (2µ + λ)∂vz

∂z+ λ

∂vr

∂r+

λvr

r− (3λ + 2µ)α(∆T ) (F.11)


σzz =(2µ + λ)

|J |

∂

∂ξ

[

vz∂r

∂η

]

− ∂

∂η

[

vz∂r

∂ξ

]

+

λ

|J |

− ∂

∂ξ

[

vr∂z

∂η

]

+∂

∂η

[

vr∂z

∂ξ

]

+λvr

r− (3λ + 2µ)α(∆T ) (F.12)

F.2.3 Transformation of the Hoop Stress

The hoop stress is given by:

σθθ = (2µ + λ)vr

r+ λ

∂vr

∂r+ λ

∂vz

∂z− (3λ + 2µ)α(∆T ) (F.13)


σθθ = (2µ + λ)vr

r+

λ

|J |

− ∂

∂ξ

[

vr∂z

∂η

]

+∂

∂η

[

vr∂z

∂ξ

]

+

λ

|J |

∂

∂ξ

[

vz∂r

∂η

]

− ∂

∂η

[

vz∂r

∂ξ

]

− (3λ + 2µ)α(∆T ) (F.14)

F.2.4 Transformation of the Shear Stress

Finally, the shear stress is given by:

τrz = µ

(∂vz

∂r+

∂vr

∂z

)

(F.15)


311


τrz =µ

|J |

− ∂

∂ξ

[

vz∂z

∂η

]

+∂

∂η

[

vz∂z

∂ξ

]

+

µ

|J |

∂

∂ξ

[

vr∂r

∂η

]

− ∂

∂η

[

vr∂r

∂ξ

]

(F.16)

F.3 Discretization of the Axial Displacement Equation

The equation governing axial displacement is integrated over a non-orthogonal cell:

∫∫

V

(

− ∂

∂ξ

[

r τrz∂z

∂η

]

+∂

∂η

[

r τrz∂z

∂ξ

]

+

∂

∂ξ

[

r σzz∂r

∂η

]

− ∂

∂η

[

r σzz∂r

∂ξ

])

dξ dη +

∫∫

V

(

r fz

)

|J | dξ dη = 0 (F.17)

Which leads to:

[ ∫

S

(

− r τrz∂z

∂η+ r σzz

∂r

∂η

)

dη

]e

w

+

[ ∫

S

(

r τrz∂z

∂ξ− r σzz

∂r

∂ξ

)

dξ

]n

s

+

∫∫

V

(

r fz

)

|J | dξ dη = 0 (F.18)

Which can be simplified to give;

(

re σzz,e∂r

∂η

∣∣∣∣e

− re τrz,e∂z

∂η

∣∣∣∣e

)

∆ηe −(

rw σzz,w∂r

∂η

∣∣∣∣w

− rw τrz,w∂z

∂η

∣∣∣∣w

)

∆ηw

+

(

rn τrz,n∂z

∂ξ

∣∣∣∣n

− rn σzz,n∂r

∂ξ

∣∣∣∣n

)

∆ξn −(

rs τrz,s∂z

∂ξ

∣∣∣∣s

− rs σzz,s∂r

∂ξ

∣∣∣∣s

)

∆ξs

+

∫∫

V

(

r fz

)

|J | dξ dη = 0 (F.19)

Finally:

re σzz,e YYe − re τrz,e XYe − rw σzz,w YYw + rw τrz,w XYw +

rn τrz,n XX n − rn σzz,n YX n − rs τrz,s XX s + rs σzz,s YX s +

rP fz,P VolP (F.20)

312

F.3. DISCRETIZATION OF THE AXIAL DISPLACEMENT EQUATION

The metric terms appearing above are given by:

XX =∂z

∂ξ∆ξ XY =

∂z

∂η∆η YX =

∂r

∂ξ∆ξ YY =

∂r

∂η∆η

The normal stress on the cell faces are given by:

σzz,e =(2µ + λ)e

VoleYYe(vz,E − vz,P ) − YX e(vz,ne − vz,se) +

λe

Vole−XYe(vr,E − vr,P ) + XX e(vr,ne − vr,se) +

λe vr,e

re−

(3λ + 2µ)eαe(∆T )e (F.21)

σzz,w =(2µ + λ)w

VolwYYw(vz,P − vz,W ) − YXw(vz,nw − vz,sw) +

λw

Volw−XYw(vr,P − vr,W ) + XXw(vr,nw − vr,sw) +

λw vr,w

rw−

(3λ + 2µ)wαw(∆T )w (F.22)

σzz,n =(2µ + λ)n

VolnYYn(vz,ne − vz,nw) − YXw(vz,N − vz,P ) +

λn

Voln−XYn(vr,ne − vr,nw) + XX n(vr,N − vr,P ) +

λn vr,n

rn−

(3λ + 2µ)nαn(∆T )n (F.23)

σzz,s =(2µ + λ)s

VolsYYs(vz,se − vz,sw) − YX s(vz,P − vz,S) +

λs

Vols−XYn(vr,se − vr,sw) + XX s(vr,P − vr,S) +

λs vr,s

rs−

(3λ + 2µ)sαs(∆T )s (F.24)

The shear stress on the cell faces are given by:

τrz,e =µe

Vole−XYe(vz,E − vz,P ) + XX e(vz,ne − vz,se) +

µe

VoleYYe(vr,E − vr,P ) − YX e(vr,ne − vr,se) (F.25)

313


τrz,w =µw

Volw−XYw(vz,P − vz,W ) + XXw(vz,nw − vz,sw) +

µw

VolwYYw(vr,P − vr,W ) − YXw(vr,nw − vr,sw) (F.26)

τrz,n =µn

Voln−XYn(vz,ne − vz,nw) + XX n(vz,N − vz,P ) +

µn

VolnYYn(vr,ne − vr,nw) − YX n(vr,N − vr,P ) (F.27)

τrz,s =µs

Vols−XYs(vz,se − vz,sw) + XX s(vz,P − vz,S) +

µs

VolsYYs(vr,se − vr,sw) − YX s(vr,P − vr,S) (F.28)

Which can be substituted into Equation F.20 to give:

314

F.3. DISCRETIZATION OF THE AXIAL DISPLACEMENT EQUATION

re YYe

[(2µ + λ)e


λe


λe vr,e

re−

(3λ + 2µ)eαe(∆T )e

]

−re XYe

[µe


µe

VoleYYe(vr,E − vr,P ) − YX e(vr,ne − vr,se)

]

−rw YYw

[(2µ + λ)w


λw


λw vr,w

rw−

(3λ + 2µ)wαw(∆T )w

]

+rw XYw

[µw


µw

VolwYYw(vr,P − vr,W ) − YXw(vr,nw − vr,sw)

]

+rn XX n

[µn


µn

VolnYYn(vr,ne − vr,nw) − YX n(vr,N − vr,P )

]

−rn YX n

[(2µ + λ)n

VolnYYn(vz,ne − vz,nw) − YXw(vz,N − vz,P ) +

λn


λn vr,n

rn−

(3λ + 2µ)nαn(∆T )n

]

−rs XX s

[µs


µs

VolsYYs(vr,se − vr,sw) − YX s(vr,P − vr,S)

]

+rs YX s

[(2µ + λ)s

VolsYYs(vz,se − vz,sw) −YX s(vz,P − vz,S) +

λs

Vols−XYn(vr,se − vr,sw) + XX s(vr,P − vr,S) +

λs vr,s

rs−

(3λ + 2µ)sαs(∆T )s

]

+rP VolP fz,P = 0 (F.29)

315



AP vz,P = AE vz,E + AS vz,S + AN vz,N + AW vz,W + Svz (F.30)


AE =re YY2

e (2µ + λ)eVole

+re XY2

e µe

Vole(F.31)

AW =rw YY2

w (2µ + λ)wVolw

+rw XY2

w µw

Volw(F.32)

AN =rn XX 2

n µn

Voln+

rn YX 2n (2µ + λ)nVoln

(F.33)

AS =rs XX 2

s µs

Vols+

rs YX 2s (2µ + λ)sVols

(F.34)

AP = AE + AW + AN + AS (F.35)

Svz = Svz ,s + Svz ,n + Svz ,e + Svz ,w + Svz ,P (F.36)


Svz ,e = −re YYe YX e (2µ + λ)eVole

(vz,ne − vz,se) +re YYe XX e λe

Vole(vr,ne − vr,se)

+YYe λe vr,e − re YYe (3λ + 2µ)eαe(∆T )e

−re XYe XX e µe

Vole(vz,ne − vz,se) +

re XYe YX e µe


−re YYe XYe λe

Volevr,E − re XYe YYe µe

Volevr,E

(F.37)

Svz ,w =rw YYw YXw (2µ + λ)w

Volw(vz,nw − vz,sw) − rw YYw XXw λw

Volw(vr,nw − vr,sw)

−YYw λw vr,w + rw YYw (3λ + 2µ)wαw(∆T )w

+rw XYw XXw µw

Volw(vz,nw − vz,sw) − rw XYw YXw µw


−rw YYw XYw λw

Volwvr,W − rw XYw YYw µw

Volwvr,W

(F.38)

Svz ,n = −rn XX n XYn µn

Voln(vz,ne − vz,nw) +

rn XX n YYn µn

Voln(vr,ne − vr,nw)

−rn YX n YYn (2µ + λ)nVoln

(vz,ne − vz,nw) +rn YX n XYn λn


−YX n λn vr,n + rn YX n (3λ + 2µ)nαn(∆T )n

−rn XX n YX n µn

Volnvr,N − rn YX n XX n λn

Volnvr,N (F.39)

316

F.4. DISCRETIZATION OF THE RADIAL DISPLACEMENT EQUATION

Svz ,s =rs XX s XYs µs

Vols(vz,se − vz,sw) − rs XX s YYs µs

Vols(vr,se − vr,sw)

+rs YX s YYs (2µ + λ)s

Vols(vz,se − vz,sw) − rs YX s XYs λs


+YX s λs vr,s − rs YX s (3λ + 2µ)sαs(∆T )s

−rs XX s YX s µs

Volsvr,S − rs YX s XX s λs

Volsvr,S

(F.40)

Svz ,P =re YYe XYe λe

Volevr,P +

re XYe YYe µe

Volevr,P +

rw YYw XYw λw

Volwvr,P

+rw XYw YYw µw

Volwvr,P +

rn XX n YX n µn

Volnvr,P +

rn YX n XX n λn

Volnvr,P

+rs XX s YX s µs

Volsvr,P +

rs YX s XX s λs

Volsvr,P + rP VolP fz,P (F.41)

F.4 Discretization of the Radial Displacement Equation

The equation governing radial displacement is integrated over a non-orthogonal cell:

∫∫

V

(

− ∂

∂ξ

[

r σrr∂z

∂η

]

+∂

∂η

[

r σrr∂z

∂ξ

]

+

∂

∂ξ

[

r τrz∂r

∂η

]

− ∂

∂η

[

r τrz∂r

∂ξ

])

dξ dη

+

∫∫

V

(

r fr − σθθ

)

|J | dξ dη = 0 (F.42)

Which leads to:

∫

S

(

−r σrr∂z

∂η+ r τrz

∂r

∂η

)

dη

e

w

+

∫

S

(

r σrr∂z

∂ξ− r τrz

∂r

∂ξ

)

dξ

n

s

+

∫∫

V

(

r fr − σθθ

)

|J | dξ dη = 0 (F.43)


317


(

re τrz,e∂r

∂η

∣∣∣∣e

− re σrr,e∂z

∂η

∣∣∣∣e

)

∆ηe −(

rw τrz,w∂r

∂η

∣∣∣∣w

− rw σrr,w∂z

∂η

∣∣∣∣w

)

∆ηw +

(

rn σrr,n∂z

∂ξ

∣∣∣∣n

− rn τrz,n∂r

∂ξ

∣∣∣∣n

)

∆ξn −(

rs σrr,s∂z

∂ξ

∣∣∣∣s

− rs τrz,s∂r

∂ξ

∣∣∣∣s

)

∆ξs +

∫∫

V

(

r fr − σθθ

)

|J | dξ dη = 0 (F.44)

Finally:

re τrz,e YYe − re σrr,e XYe − rw τrz,w YYw + rw σrr,w XYw +

rn σrr,n XX n − rn τrz,n YX n − rs σrr,s XX s +

rs τrz,sYX s + (rP fr,P − σθθ,P )VolP (F.45)

Where the normal stress on the cell faces are given by:

σrr,e =(2µ + λ)e


λe


λe vr,e

re−

(3λ + 2µ)eαe(∆T )e (F.46)

σrr,w =(2µ + λ)w


λw


λw vr,w

rw−

(3λ + 2µ)wαw(∆T )w (F.47)

σrr,n =(2µ + λ)n


λn

VolnYYn(vz,ne − vz,nw) − YX n(vz,N − vz,P ) +

λn vr,n

rn−

(3λ + 2µ)nαn(∆T )n (F.48)

318


σrr,s =(2µ + λ)s

Vols−XYs(vr,se − vr,sw) + XX s(vr,P − vr,S) +

λs


λs vr,s

rs−

(3λ + 2µ)sαs(∆T )s (F.49)

Where the shear stress on the cell faces are given by:

τrz,e =µe


µe

VoleYYe(vr,E − vr,P ) − YX e(vr,ne − vr,se) (F.50)

τrz,w =µw


µw

VolwYYw(vr,P − vr,W ) − YXw(vr,nw − vr,sw) (F.51)

τrz,n =µn


µn

VolnYYn(vr,ne − vr,nw) − YX n(vr,N − vr,P ) (F.52)

τrz,s =µs


µs

VolsYYs(vr,se − vr,sw) − YX s(vr,P − vr,S) (F.53)



rP+

λP

VolP−XYP (vr,e − vr,w) + XXP (vr,n − vr,s) +

λP

VolPYYP (vz,e − vz,w) − YXP (vz,n − vz,s) −

(3λ + 2µ)P αP (∆T )P (F.54)

The above can be substituted into Equation F.45 to give:

319


re YYe

[µe


µe

VoleYYe(vr,E − vr,P ) −YX e(vr,ne − vr,se)

]

−re XYe

[(2µ + λ)e


λe


λe vr,e

re−

(3λ + 2µ)eαe(∆T )e

]

−rw YYw

[µw


µw

VolwYYw(vr,P − vr,W ) − YXw(vr,nw − vr,sw)

]

+rw XYw

[(2µ + λ)w


λw


λw vr,w

rw−

(3λ + 2µ)wαw(∆T )w

]

+rn XX n

[(2µ + λ)n


λn

VolnYYn(vz,ne − vz,nw) − YX n(vz,N − vz,P ) +

λn vr,n

rn−

(3λ + 2µ)nαn(∆T )n

]

−rn YX n

[µn


µn

VolnYYn(vr,ne − vr,nw) −YX n(vr,N − vr,P )

]

−rs XX s

[(2µ + λ)s

Vols−XYs(vr,se − vr,sw) + XX s(vr,P − vr,S) +

λs


λs vr,s

rs−

(3λ + 2µ)sαs(∆T )s

]

+rs YX s

[µs


µs

VolsYYs(vr,se − vr,sw) − YX s(vr,P − vr,S)

]

320


−VolP

[

(2µ + λ)Pvr,P

rP+

λP


λP

VolPYYP (vz,e − vz,w) −YX P (vz,n − vz,s) −

(3λ + 2µ)P αP (∆T )P

]

+rP VolP fr,P = 0 (F.55)


AP vr,P = AE vr,E + AS vr,S + AN vr,N + AW vr,W + Svr (F.56)


AE =re YY2

e µe

Vole+

re XY2e (2µ + λ)eVole

(F.57)

AW =rw YY2

w µw

Volw+

rw XY2w (2µ + λ)wVolw

(F.58)

AN =rn XX 2

n (2µ + λ)nVoln

+rn YX 2

n µn

Voln(F.59)

AS =rs XX 2

s (2µ + λ)sVols

+rs YX 2

s µs

Vols(F.60)

AP = AE + AS + AN + AW (F.61)

Svr = Svr ,s + Svr ,n + Svr ,e + Svr ,w + Svr ,P (F.62)


321


Svr ,e =re YYe XX e µe

Vole(vz,ne − vz,se) −

re YYe YX e µe


−re XYe XX e (2µ + λ)eVole

(vr,ne − vr,se) +re XYe YX e λe

Vole(vz,ne − vz,se)

−XYe λe vr,e + re XYe (3λ + 2µ)eαe(∆T )e

−re YYe XYe µe

Volevz,E − re XYe YYe λe

Volevz,E

(F.63)

Svr ,w = −rw YYw XXw µw

Volw(vz,nw − vz,sw) +

rw YYw YXw µw


+rw XYw XXw (2µ + λ)w

Volw(vr,nw − vr,sw) − rw XYw YXw λw

Volw(vz,nw − vz,sw)

+XYw λw vr,w − rw XYw (3λ + 2µ)wαw(∆T )w

−rw YYw XYw µw

Volwvz,W − rw XYw YYw λe

Volwvz,W

(F.64)

Svr ,n = −rn XX n XYn (2µ + λ)nVoln

(vr,ne − vr,nw) +rn XX n YYn λn

Voln(vz,ne − vz,nw)

+XX n λn vr,n − rn XX n (3λ + 2µ)nαn(∆T )n

+rn YX n XYn µn

Voln(vz,ne − vz,nw) − rn YX n YYn µn


−rn XX n YX n λn

Volnvz,N − rn YX n XX n µn

Volnvz,N (F.65)

322


Svr ,s =rs XX s XYs (2µ + λ)s

Vols(vr,se − vr,sw) − rs XX s YYs λs

Vols(vz,se − vz,sw)

−XX s λs vr,s + rs XX s (3λ + 2µ)sαs(∆T )s

−rs YX s XYs µs

Vols(vz,se − vz,sw) +

rs YX s YYs µs


−rs XX s YX s λs

Volsvz,S − rs YX s XX s µs

Volsvz,S

(F.66)

Svr ,P =re YYe XYe µe

Volevz,P +

re XYe YYe λe

Volevz,P +

rw YYw XYw µw

Volwvz,P

+rw XYw YYw λw

Volwvz,P +

rn XX n YX n λn

Volnvz,P +

rn YX n XX n µn

Volnvz,P

+rs XX s YX s λs

Volsvz,P +

rs YX s XX s µs

Volsvz,P + rP VolP fr,P

−VolP

[

(2µ + λ)Pvr,P

rP+

λP


λP

VolPYYP (vz,e − vz,w) − YXP (vz,n − vz,s) −

(3λ + 2µ)P αP (∆T )P

]

(F.67)

323


324

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