failure analysis of fev

7/28/2019 Failure Analysis of FEV

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Abstract

This study was aimed towards improving the accuracy of Computational Fluid Dy-

namics (CFD) by developing methods for reliable estimation of the discretisation

error and its reduction.A new method for error estimation of the discretisation error for the second-

order accurate Finite Volume Method is presented, called the Face Residual Error

Estimator (FREE), which estimates the discretisation error on the cell faces. The es-

timator is tested on a set of cases with analytical solutions, ranging from convection-

dominated to diffusion-dominated dominated ones. Testing is also performed on a

set of cases of engineering interest and on polygonal meshes.

In order to automatically produce a solution of pre-determined accuracy an au-tomatic error-controlled adaptive mesh renement procedure is set up. It uses local

mesh renement to control the local error magnitude by rening hexahedral cells

parallel to the face with large discretisation error. The procedure is tested on four

cases with analytical solutions and on several laminar and turbulent ow cases of

engineering interest. It was found able to produce accurate solutions with savings

in computational resources.

In order to explore the possibilities of different mesh structures, a mesh generatorproducing polyhedral meshes based on the Delaunay technique is developed. An

adaptive mesh generation technique for polyhedral meshes is also developed and is

based on remeshing parts of the mesh which are selected for renement. The mesh

adaptation technique is tested on a case with an analytical solution. A comparison

of accuracy achieved on quadrilateral, triangular and polygonal meshes is also given,

where quadrilateral meshes perform best followed by polygonal meshes.


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Acknowledgements

I would like to express my gratitude to my supervisor Prof A. D. Gosman for his

interest and continuous guidance during this study.

I would like to use the opportunity to thank my colleagues from Prof Gosmans

CFD group, especially Dr. Hrvoje Jasak, Mr. Henry Weller and Mr Mattijs Janssens

for developing the FOAM C++ simulation code which made the implementation of

the ideas easier. Their support and suggestions were invaluable.

This study and the text of this thesis has beneted a lot from the numerous

suggestions and comments by Dr. Hrvoje Jasak.

It would be unfair not to thank Mrs Nicky Scott-Knight and Mrs Susan Clegg

for arranging administrative matters.Finally, the nancial support provided by the Computational Dynamics Ltd. is

gratefully acknowledged.


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Contents

1 Introduction 25

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.2 Present Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2 Governing Equations of Continuum Mechanics 31

2.1 Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2 Constitutive Relations for Newtonian Fluids . . . . . . . . . . . . . . 31

2.2.1 Turbulence Modelling . . . . . . . . . . . . . . . . . . . . . . . 32

2.3 General Form of a Transport Equation . . . . . . . . . . . . . . . . . 36

2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3 Finite Volume Discretisation 39

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 Measures of Mesh Quality . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3 Discretisation of Spatial Terms . . . . . . . . . . . . . . . . . . . . . 43

3.3.1 Convection Term . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.3.2 Diffusion Term . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.3.3 Source Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.4 Temporal Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.5 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.5.1 Boundary Conditions for the General Transport Equation . . . 56

3.5.2 Boundary Conditions for the Navier-Stokes Equations . . . . . 57

3.6 Discretisation Errors on different types of meshes . . . . . . . . . . . 58

3.6.1 Convection Term . . . . . . . . . . . . . . . . . . . . . . . . . 603.6.2 Diffusion Term . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.7 Solution of Linear Equation Systems . . . . . . . . . . . . . . . . . . 67

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3.8 Solution Algorithm for the Navier-Stokes System . . . . . . . . . . . 69

3.8.1 Pressure Equation . . . . . . . . . . . . . . . . . . . . . . . . 69

3.8.2 Algorithms for Pressure-Velocity Coupling . . . . . . . . . . . 71

3.9 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 73

4 Error Estimation 75

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.2 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.2.1 Methods Used in FEM Analysis . . . . . . . . . . . . . . . . . 76

4.2.2 Methods Used in FV Analysis . . . . . . . . . . . . . . . . . . 78

4.3 Error Transport Through a Face . . . . . . . . . . . . . . . . . . . . . 83

4.4 Face Residual Error Estimator . . . . . . . . . . . . . . . . . . . . . . 874.4.1 Analysis of the Normalisation Practice . . . . . . . . . . . . . 88

4.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.5.1 Planar Jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.5.2 Creeping Stagnation Flow . . . . . . . . . . . . . . . . . . . . 96

4.5.3 Convection Transport of Heat with a Distributed Heat Source 99


5 Mesh Adaptation 105

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.2 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.3 Adaptation Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.3.1 Selection of Cells and Mesh Renement . . . . . . . . . . . . . 109

5.3.2 Solution Mapping Between Meshes . . . . . . . . . . . . . . . 115

5.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.4.1 Planar Jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.4.2 Stokes Stagnation Flow . . . . . . . . . . . . . . . . . . . . . . 120

5.4.3 Convection Transport of Heat with a Distributed Heat Source 127

5.4.4 Convection and diffusion of a Temperature Prole without a

Heat Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129


6 Further Case Studies 137

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137


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6.2 Flow Over a Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.3 S-shaped Pipe Bend . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.4 Tube Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

6.5 Wall-Mounted Cube . . . . . . . . . . . . . . . . . . . . . . . . . . . 162


7 Adaptive Polyhedral Mesh Generation 179

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

7.2 Literature survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

7.3 Voronoi Polygons and Delaunay Triangulation . . . . . . . . . . . . . 185

7.3.1 Algorithm for calculation of the Dirichlet Tessellation . . . . . 187

7.4 Computational mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . 1907.5 Polyhedral Mesh Generation . . . . . . . . . . . . . . . . . . . . . . . 191

7.5.1 A comparison of accuracy on Quadrilateral, Polygonal and

Triangular Meshes . . . . . . . . . . . . . . . . . . . . . . . . 197

7.6 Mesh adaptation on Polyhedral Meshes . . . . . . . . . . . . . . . . . 205

7.6.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207


8 Conclusions and Future Work 213

8.1 Error Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

8.2 Mesh Adaptation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

8.3 Mesh Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

8.4 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216


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List of Figures

3.1 Computational cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2 Mesh non-orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3 Mesh skewness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.4 Variation of near the face . . . . . . . . . . . . . . . . . . . . . . . 49

3.5 Non-orthogonality treatment . . . . . . . . . . . . . . . . . . . . . . . 51

3.6 Boundary cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.7 Square mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.8 Triangular mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.9 Hexagonal mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.10 Split-hexahedron mesh . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.1 Inconsistency of the interpolated values on the face . . . . . . . . . . 84

4.2 Distance between points on non-orthogonal mesh . . . . . . . . . . . 90

4.3 Solution domain and boundary conditions for the jet case . . . . . . 92

4.4 Starting mesh for the jet case (10 x 4 cells) . . . . . . . . . . . . . . 93

4.5 Velocity eld for the jet case [m/s] (80 x 32 mesh) . . . . . . . . . . 94

4.6 Pressure isobars for the jet case [m2/s 2] (80 x 32 mesh) . . . . . . . . 94

4.7 Velocity error eld for the jet case [m/s] (80 x 32 mesh) . . . . . . . 95

4.8 Variation of errors with uniform mesh renement for the jet case

(|U norm | = 2 .474m/s ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.9 Solution domain and boundary conditions for the stagnation ow . . 97

4.10 Velocity and pressure for the stagnation ow (40 x 40 mesh) . . . . . 98

4.11 Velocity error elds for the stagnation ow [m/s] (40 x 40 mesh) . . 98

4.12 Variation of errors with uniform mesh renement for the stagnation

ow (|U norm | = 1 .107m/s ) . . . . . . . . . . . . . . . . . . . . . . . . 994.13 Solution domain and boundary conditions for the convection trans-por t case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

9


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10 List of Figures

4.14 Temperature and source elds for the convection transport case (40

x 40 mesh) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.15 Error elds for the convection transport case [ oC ] (40 x 40 mesh) . . 101

4.16 Uniform mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.17 Variation of errors with uniform mesh renement for the convection

transport case ( T norm = 1 oC) . . . . . . . . . . . . . . . . . . . . . . 102

5.1 A split-hexahedron cell with left face split in one direction shared

with two cells. The top face is cross-split and shared with four cells . 110

5.2 Directional splitting of cells . . . . . . . . . . . . . . . . . . . . . . . 111

5.3 Renement of split-hexahedron cells . . . . . . . . . . . . . . . . . . . 111

5.4 Node distances at a split face . . . . . . . . . . . . . . . . . . . . . . 1125.5 Additional splitting of cells . . . . . . . . . . . . . . . . . . . . . . . . 113

5.6 Consistency over a split face in 2D (dotted lines represent the selected

renement) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.7 Consistency over a cross-split face in 3D (dotted lines represent the

selected renement) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.8 Treatment of incompatible cell splitting directions in 3D (dotted lines

represent the selected renement) . . . . . . . . . . . . . . . . . . . . 115

5.9 Mesh after 6 cycles of renement for the jet case (209 cells) . . . . . . 117

5.10 Variation of velocity errors with adaptive renement for the jet case

(|U norm | = 2 .474m/s ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.11 Velocity errors after 6 cycles of renement for the jet case [m/s] (209

cells) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.12 Estimated errors on the faces of the nal mesh with 209 cells (given

as percentage of the maximum estimated error on that mesh) . . . . . 119

5.13 Meshes for the creeping stagnation ow . . . . . . . . . . . . . . . . . 121

5.14 Velocity errors after 4 cycles of renement for the creeping stagnation

ow [m/s] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.15 Velocity error scaling with adaptive renement for the creeping stag-

nation ow (|U norm | = 1 .107m/s ) . . . . . . . . . . . . . . . . . . . . 1235.16 Estimated errors on the faces of the nal mesh with 280 cells (given


5.17 Meshes for the creeping stagnation ow (second calculation) . . . . . 125


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List of Figures 11

5.18 Variation of velocity errors with adaptive mesh renement for the

stagnation ow (Second calculation) ( |U norm | = 1 .107m/s ) . . . . . . 1265.19 Velocity error elds after 2 cycles of adaptive renement for the stag-

nation ow (Second calculation) [m/s] . . . . . . . . . . . . . . . . . 126

5.20 Mesh after 3 cycles of renement for the convection transport case . . 127

5.21 Fields after 3 cycles of renement . . . . . . . . . . . . . . . . . . . . 128

5.22 Variation of temperature errors with adaptive renement for the con-

vection transport case ( T norm = 1 oC ) . . . . . . . . . . . . . . . . . . 129

5.23 Temperature eld for the internal layer case [ oC] . . . . . . . . . . . . 129

5.24 Solution domain and boundary conditions for the convection and dif-

fusion of heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1315.25 Variation of temperature errors with adaptive renement for the con-

vection and diffusion of heat (T norm = 1 oC ) . . . . . . . . . . . . . . . 132

5.26 Temperature errors after 5 cycles of renement for the convection and

diffusion of heat [oC] (1706 cells) . . . . . . . . . . . . . . . . . . . . 133

5.27 Temperature and its gradient for the convection and diffusion of heat 133

5.28 Mesh and errors for the calculation driven by the exact face errors . . 134

6.1 Geometry and boundary conditions for the ow over a cavity . . . . . 138

6.2 Starting mesh for the ow over a cavity (36 cells) . . . . . . . . . . . 138

6.3 Velocity eld for the ow over a cavity on the nal adapted mesh

with 3257 cells (normalised by U avg ) . . . . . . . . . . . . . . . . . . . 139

6.4 Pressure coefficient eld for the ow over a cavity on the nal adapted

mesh with 3257 cells . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.5 Mesh after 8 cycles of renement for the ow over a cavity (3257 cells)140

6.6 Errors for the ow over a cavity on the nal adapted mesh with 3257

cells (given as percentage of U avg ) . . . . . . . . . . . . . . . . . . . . 141


ow over a cavity (errors given as percentage of U avg ) . . . . . . . . . 141

6.8 Estimated velocity error on the faces of the nal mesh with 3257 cells

(given as percentage of the maximum estimated error on that mesh) . 142

6.9 Case setup for the S-bend case . . . . . . . . . . . . . . . . . . . . . . 144

6.10 Starting mesh for the S-bend case (270 cells) . . . . . . . . . . . . . . 145


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12 List of Figures

6.11 Velocity eld in for the S-bend case obtained on the nal adapted

mesh with 390787 cells (normalised by U avg ) . . . . . . . . . . . . . . 146

6.12 Pressure coefficient in the symmetry plane for the S-bend case ob-

tained on the nal adapted mesh with 390787 cells . . . . . . . . . . 147

6.13 Section at X = 2 D (390787 cells) . . . . . . . . . . . . . . . . . . . . 148


S-bend case (given as percentage of U max ) . . . . . . . . . . . . . . . 149

6.15 Mesh after 9 cycles of renement for the S-bend case (390787 cells) . 150

6.16 Velocity error in the symmetry plane after 9 cycles of renement for

the S-bend case (390787 cells) (given as percentage of U max ) . . . . . 151

6.17 Geometry and boundary conditions for the tube bundle case . . . . . 153

6.18 Starting mesh for the tube bundle case (640 cells) . . . . . . . . . . . 154

6.19 Velocity eld for the tube bundle case (17505 cells)(given as UU avg ) . . 155

6.20 Pressure coefficient for the tube bundle case (17505 cells) . . . . . . . 156

6.21 q eld for the tube bundle case (17505 cells)(given as qU avg ) . . . . . . 156

6.22 eld for the tube bundle case (17505 cells)(given as DUavg 2 ) . . . . . 156

6.23 Mesh after 7 cycles of renement for the tube bundle case (17505 cells)158

6.24 Variation of errors with adaptive mesh renement for the tube bundle

case (errors given as percentage of U max , q max and max respectively) . 159

6.25 Exact and estimated error elds after 7 cycles of renement (17505

cells)(errors are given as percentage of U max , q max and max respec-

tively). Exact errors are calculated as the difference from the bench-

mark solution. Estimated errors are plotted as a weighted average of

face errors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

6.26 A comparison of proles for the tube bundle case . . . . . . . . . . . 161

6.27 Geometry and boundary conditions for the wall mounted cube case . 162

6.28 Starting mesh for the wall-mounted cube case (3444 cells) . . . . . . . 163

6.29 Distribution of ow variables in the symmetry plane for the wall-

mounted cube case, obtained on the nal adapted mesh (1.16149e+06

cells) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

6.30 Distribution of ow variables in the plane y/H = 0 .5 for the wall-mounted cube case, obtained on the nal adapted mesh (1.16149e+06

cells) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166


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List of Figures 13

6.31 Distribution of ow variables in the plane x/H = 0 .5 for the wall-

mounted cube case, obtained on the nal adapted mesh (1.16149e+06

cells) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6.32 Streamlines in the plane x/H = 2 for the wall-mounted cube case,

obtained on the nal adapted mesh (1.16149e+06 cells) . . . . . . . . 168

6.33 Mesh after 7 cycles of renement for the wall-mounted cube case

(1.16149e+06 cells) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

6.34 Variation of estimated errors with adaptive mesh renement for the

wall-mounted cube case (errors are given as percentage of U avg , kmax

at inlet and max at inlet) . . . . . . . . . . . . . . . . . . . . . . . . 170

6.35 Remaining estimated errors after 7 cycles of renement (errors are

given as percentage of U avg , kmax at inlet and max at inlet, respec-

tively) Estimated errors are plotted as a weighted average of face

errors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

6.36 Regions of highest gradients after 7 cycles of renement . . . . . . . . 172

6.37 Proles of ow variables taken at xH = 0,zH = 0.5 (above the corner

at which the leading edges meet) . . . . . . . . . . . . . . . . . . . . 173

6.38 C p on the bottom wall . . . . . . . . . . . . . . . . . . . . . . . . . . 1756.39 Velocity and turbulent energy k proles . . . . . . . . . . . . . . . . . 176

7.1 Delaunay Triangulation (solid lines) and Voronoi Polygons (dashed

lines) for a set of points P n (dots) . . . . . . . . . . . . . . . . . . . 186

7.2 Delaunay vs other triangulations . . . . . . . . . . . . . . . . . . . . 186

7.3 Initial hull for the Delaunay triangulation . . . . . . . . . . . . . . . 188

7.4 Dirichlet Tessellation and polyhedral mesh . . . . . . . . . . . . . . . 191

7.5 Generation of an internal face of the polyhedral mesh (section in the

plane of the face) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

7.6 Tetrahedra sharing a face. The edge of the polyhedral mesh (thick

lines) is perpendicular to the shared triangular face (red). There exist

a polygonal face for every edge of the triangular face. . . . . . . . . . 194

7.7 Generation of an internal face including the intersection of a boundary

edge (section in the plane of the face) . . . . . . . . . . . . . . . . . . 1957.8 Generation of boundary faces (coloured) including intersections with

boundary edges (thick lines) and a corner point . . . . . . . . . . . . 195


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14 List of Figures

7.9 An example of a 3D mesh for a cube with side length l = 1 m . . . . . 196

7.10 Meshes used for the jet case . . . . . . . . . . . . . . . . . . . . . . . 198

7.11 Exact velocity errors for the jet case [m/s] . . . . . . . . . . . . . . . 199

7.12 Quadrilateral, polygonal and triangular meshes used for comparison . 202

7.13 Variation of the exact velocity error on different types of meshes for

the cavity case. Errors are given as percentage of average inlet velocity

U avg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

7.14 Magnitude of the exact velocity error (given as percentage of average

inlet velocity U avg ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

7.15 The exact pressure error (given as percentage of pmax = 1.642 U 2avg ) . . 204

7.16 Scaling of the pressure drop for different types of meshes . . . . . . . 205

7.17 Adaptation of polyhedral meshes . . . . . . . . . . . . . . . . . . . . 206

7.18 Staring polygonal mesh for the convection and diffusion of heat . . . 208

7.19 Mesh after 9 cycles for the convection and diffusion of heat (5925 cells)209

7.20 Polygonal mesh from nearly degenerate Delaunay Triangulation . . . 209

7.21 Errors after 9 cycles of renement (5925 cells) . . . . . . . . . . . . . 210

7.22 Uniform mesh and the exact error (6769 cells) . . . . . . . . . . . . . 210

7.23 Variations of errors with adaptive renement for the convection and

diffusion of heat (T norm = 1 oC ) . . . . . . . . . . . . . . . . . . . . . 211


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List of Tables

5.1 Estimated errors on the faces of the nal mesh with 209 cells (given


6.1 Distribution of face errors for the cavity case . . . . . . . . . . . . . . 142

6.2 Pressure drop coefficients for the ow over a cavity . . . . . . . . . . 1436.3 Pressure drop coefficients C p, force coefficients C F and average vor-

ticity coefficient C for the S-bend case . . . . . . . . . . . . . . . . . 152

6.4 Maximum of , velocity gradient, k eld gradient and pressure gradi-

en t e lds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

6.5 Magnitude of the force acting on the cube and the lengths of vortices

downstream and upstream of the cube . . . . . . . . . . . . . . . . . 174

6.6 Distribution of estimated velocity error on the faces . . . . . . . . . . 177

7.1 Data structure for the triangulation . . . . . . . . . . . . . . . . . . . 187

7.2 Relations between objects forming Delaunay Triangulation and Dirich-

let Tessellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

7.3 Number of cells for different types of meshes (Jet case) . . . . . . . . 198

7.4 Number of cells for different types of meshes . . . . . . . . . . . . . . 201

7.5 A comparison of pressure drop for different types of meshes . . . . . . 2047.6 Measures of mesh quality . . . . . . . . . . . . . . . . . . . . . . . . . 208

15


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16 List of Tables


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Nomenclature

Latin Characters

a general vector property

aN matrix coefficient corresponding to the neighbour N

aP central coefficient

C constant dependent on the scheme for temporal discretisation

C F force coefficient

C p pressure coefficient

Co Courant number

d vector between P and N

d n vector between the cell centre and the boundary face

E exact error, required error tolerance

e total specic energy, solution error, truncation error

ef error on the face

F mass ux through the face

F conv convection transport coefficient

F diff diffusion transport coefficient

F norm normalisation factor for the residual

f face, point in the centre of the face

17


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18 List of Tables

f i point of interpolation on the face

f x interpolation factor

gb boundary condition on the xed gradient boundary

g acceleration of the gravity force

G matrix for Least-Squares Fit

H transport part, Hessian matrix

h mesh size

I unit tensor

k non-orthogonal part of the face area vector

k turbulent kinetic energy

L functional, set of edges

m skewness correction vector, second moment

M geometric moment of inertia, momentum

N point in the centre of the neighbouring cell, number of cells

P pressure, point in the centre of the cell, set of points

P atmospheric pressure

xdist position difference vector

p kinematic pressure, order of accuracy

q q in the q turbulence modelQP source for the system of linear equations

QV body forces

Q S surface forces

Re Reynolds number

r ratio


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List of Tables 19

Res f face residual

Res P f face residual from the cell owner

Res N F face residual from the cell neighbour

Res P cell residual

S outward-pointing face area vector

Sf face area vector

s parametric curve

S source term

S e error source term

Sp linear part of the source term

Su constant part of the source term

S CV area of a control volume

T temperature, time-scale

t time

U velocity

U b velocity of the arbitrary volumes face

V volume

V M material volume

V CV control volume

V i Voronoi Polygon

V P volume of the cell

x x component

x position vector

y y component

z z component


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20 List of Tables

Greek Characters

under-relaxation factor

N non-orthogonality angle

diffusivity

blending factor

orthogonal part of the face area vector

dissipation rate of turbulent kinetic energy

effectivity index, -eld in the q

turbulence model

heat conduction coefficient

dynamic viscosity

kinematic viscosity

T turbulent kinematic viscosity

density

turbulent Prandtl number

stress tensor

exact solution

general scalar property

measure of mesh skewness

Superscripts

qT transpose

q mean

q uctuation around the mean value, shadow points

q n new time-level


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22 List of Tables


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Abbreviations

ADT Alternating Digital Tree

AFT Advancing Front Method

Bi-CGSTAB Bi-Conjugate Gradient Stabilised

CAD Computer-Aided Design

CD Central Differencing

CFD Computational Fluid Dynamics

CG Conjugate Gradient

CV Control VolumeDNS Direct Numerical Simulation

FD Finite Difference Method

FEM Finite Element Method

FV Finite Volume

FVM Finite Volume Method

FREE Face Residual Error Estimator

ICCG Incomplete Cholesky Conjugate Gradient

LES Large Eddy Simulation

LSF Least Squares Fit

LU Lower-Upper

NVA Normalised Variable Approach

23


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24 List of Tables

NS Navier-Stokes equations

PISO Pressure-Implicit with Splitting of Operators

RANS Reynolds Averaged Navier-Stokes

RE Richardson Extrapolation

SIMPLE Semi-Implicit Method for Pressure-Linked Equations

TDMA Thomas algorithm

UD Upwind Differencing

2D Two-dimensional space

3D Three-dimensional space


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Chapter 1

Introduction

1.1 Background

Computational Fluid Dynamics (CFD) provides solutions to uid ow problems by

solving the governing equations on a computer. CFD has undergone rapid develop-

ment in the last two decades and problems which can be solved with it range from

simple laminar ows to very complicated multi-phase ows, including heat exchange.

Many of the existing CFD codes are coupled with the CAD systems to make the

design process easier and less expensive. As CFD is becoming an engineering tool its

accuracy is gaining more and more importance, introducing the need for a reliable

method for assessing and controlling accuracy.

The governing equations are of partial differential form, coupled in most cases.

Closed form solutions cannot be found except for some simple problems, which are

not of much practical interest. The numerical methods used for CFD provide solu-

tions by dividing the domain into smaller domains and assuming a certain variation

of the dependent elds over each subdomain. This, together with the conditions

specied at the boundary of the original domain, generates a system of N algebraic

equations with N unknowns for each dependent variable, N representing the number

of subdomains, which can be solved using a computer. The process of converting

a differential equation into a system of algebraic equations is called discretisation.

This process may introduce errors which can have a great inuence on the quality

of the results obtained.

There are many different discretisation practices. The most widely used ones

are [47]: Finite Difference Method (FD), Finite Element Method (FEM) and Finite

25


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1.1 Background 27

Iteration errors are a group of errors which arise if the governing equationsare solved using an iterative procedure. They are dened as the difference

between the exact solution of the FV equations and the solution obtained by

using an iterative procedure iteration , thus:

E iteration = F V iteration . (1.3)These errors can be reduced to the level of computer truncation error for any

given problem at the expense of time needed to complete the calculation.

Programming and User errors are a result of the incorrect implementationor use of the CFD methodology in a computer code.

Error estimators are tools for estimation of the discretisation error in the nu-merical solution by using properties of the discretisation practice and the governing

equations. They provide information about the discretisation error distribution and

its magnitude in some norm and therefore measure the quality of the results in this

respect.

The required discretisation accuracy is known before the analysis is performed.

It depends on the objective of the calculations and on the accuracy of the differential

equations which are used to describe the physics. Error estimators can be used asindicators of where and how to modify the mesh to achieve solutions of the required

discretisation accuracy. This can be achieved by locally rening the mesh where

the error is large and coarsening the mesh in the regions where the error is small,

in order to maintain the error at the required level and equidistribute it over the

computational domain. An adaptive procedure, used for achieving the required

accuracy, should be composed of a number of cycles, each cycle consisting of solving

equations using a current mesh and the discretisation practice, followed by error

estimation and nally modication of the mesh.

The quality of a computational mesh is an important factor in minimising dis-

cretisation error [47]. The quality is inuenced by spatial resolution, skewness and

non-orthogonality of the mesh and also by the type of cells used.

The aim of the present study is to develop:

1. An accurate method for estimation of the discretisation error in the FV so-lution which is applicable to different types of differential equations and for

problems ranging from convection to diffusion dominated ones.


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1.3 Thesis Outline 29

Chapter 3 presents the Finite Volume Method used in this study. It is a second-

order accurate method for arbitrary unstructured meshes. Discretisation of spatial

terms in the transport equation is described term by term along with the errors

which may arise. Temporal discretisation and the errors which may results from

temporal discretisation are briey discussed. An analysis of the discretisation error

on different shapes of computational cells is performed. A solution algorithm for

Navier-Stokes equations is presented at the end of the chapter.

Developments in the eld of a posteriori error estimation made during thisstudy are presented in Chapter 4. A literature survey of the existing methods is

given rst. A new method for error estimation is proposed. The performance of

the proposed error estimator is tested on a set of cases with analytical solutions,

including convection and diffusion-dominated ones.

In Chapter 5 a mesh renement procedure is proposed. A literature survey of

mesh adaptation methods is presented rst, followed by the proposed mesh rene-

ment procedure based on directional cell-by-cell renement of hexahedral cells. The

performance of the renement procedure is examined on a set of test for which

analytical solutions are available.

In Chapter 6, the mesh renement procedure proposed in Chapter 5 is further

tested on four cases of engineering interest, involving laminar and turbulent ows.

Chapter 7 presents an algorithm for polyhedral mesh generation developed during

this study. A survey of mesh generation methods is given at the beginning of the

chapter. It is followed by an algorithm assembled for calculating polyhedral meshes

from the Delaunay Triangulation and the Voronoi Polygons which is described step

by step. A mesh adaptation technique for polyhedral meshes is also presented. A

comparison of relative accuracy which can be achieved on triangular, quadrilateral

and polygonal meshes is performed on cases introduced in earlier chapters. An

example of the adaptive mesh generation is also given.

Finally, a summary of the Thesis with some conclusions and suggestions for

future work are given in Chapter 8.


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30 Introduction


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Chapter 2

Governing Equations of

Continuum Mechanics

2.1 Navier-Stokes Equations

Governing equations of Fluid Mechanics so-called Navier-Stokes equations are a set

of partial differential equations which read [111, 112]:

Continuity equation:t

+ (U ) = 0 , (2.1)

Momentum equation:U

t+ (UU ) = g + , (2.2)

where g is the gravity acceleration and is a surface stress tensor.

Energy equation:

et

+ (eU ) = g.U + ( U ) q + Q. (2.3)

Here, q is the heat ux through the control volume surface and Q is the heat

source within the CV.

2.2 Constitutive Relations for Newtonian Fluids

The uids treated in this study are assumed to obey the following constitutive

relations [112]:

31


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32 Governing Equations of Continuum Mechanics

Newtons law of viscosity

= P +23

U I + U + (U )T (2.4)

where P is the pressure, is dynamic viscosity and I is the unit tensor.

The equation of state for the ideal gas

P = R T, (2.5)

where R is a universal gas constant.

Fourier law of heat conduction

q = T, (2.6) being a heat conduction coefficient.

When the above relations are inserted into Eqs. (2.2) and (2.3) a closed system

of equations is obtained, as follows [112]:

Continuity equation: t

+

(U ) = 0 , (2.7)

Momentum equation:U

t+ (UU ) = g P +

23

U+ U + (U )T ,

(2.8)

Energy equation:et

+ (eU ) = g U (P U ) 23 ( U ) U+ U + (U )T U + (T ) + Q,

(2.9)

2.2.1 Turbulence Modelling

Turbulent ows occur in most engineering applications and there are many methods

developed for prediction of such ows which differ in the level of detail the ow is

resolved [123].Direct Numerical Simulation ( DNS ) is the most detailed approach to turbulence

modelling and it numerically solves the governing equations over the whole range


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2.2 Constitutive Relations for Newtonian Fluids 33

of turbulent scales. This approach requires high spatial and temporal resolution,

demanding large computational resources and long simulation times, making DNS

unsuitable for most engineering applications. Some examples can be found in [19,

85, 123, 130].

Large Eddy Simulation ( LES ) ( Smagorinsky [137], Haworth and Jansen [58],

a review by Piomelli [121]) is an approach where the large scale eddies are resolved

and the small eddies are modelled. Therefore, this approach requires a spatial lter

separating the large scales from the small ones. As the small eddies are usually much

weaker and more isotropic than the large ones it makes sense to model them and

to fully resolve the large ones, as they are the main transporters of the conserved

properties. When the mesh size tends to zero such that it can resolve the smallest

eddies LES tends to DNS.

Reynolds-averaged method ( RANS ), originally proposed by Osborne Reynolds,

is a statistical approach to turbulence modelling. The rationale behind this approach

is that the instantaneous quantity (x, t ) in a certain point in the domain can be

written as the sum of an averaged value and a uctuation about that value, thus:

(x , t ) = (x , t ) + (x, t ), (2.10)

where (x , t ) denotes turbulent uctuations and (x , t ) is the averaged value. There

are three main techniques for calculating the averaged value namely time averaging,

space averaging and ensemble averaging Hinze [60].

Depending on whether the ow is incompressible or compressible, averaging can

be unweighted namely Reynolds averaging, or density weighted named Favre aver-

aging (eg. Favre [45], Cebeci and Smith [34]).

When the above averaging is applied to the momentum and the continuity equa-tions for incompressible isothermal ow without body forces Eqs. (2.1) and (2.8)

there results:

U = 0 , (2.11) Ut

+ (U U + U U ) U = p. (2.12)

where the term U U , called Reynolds stress tensor, is the only term containingU . In order to link the Reynolds stress with the mean ow variables, modelling

approximations have to be introduced and they are usually called turbulence models .


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The turbulence models used in this study are based on the Boussinesq approx-

imation [27] which assumes that turbulent stresses are linked to the averaged ow

variables as follows:

U U = t U + (U )T +

23

k I , (2.13)

where k stands for the kinetic energy of turbulence dened as:

k =12

U . U . (2.14)

The turbulent eddy viscosity can be calculated in many ways but the most popular

one is a two-equation approach where the t is dened as:

t = C k2

, (2.15)

where is the turbulence dissipation rate, dened as:= U U : U . (2.16)

The variables k and are calculated as the solution of their own transport equations.

The equation for the turbulent dissipation has the following form [123]:

(U ) (( T

+ )) = C 1Pk C 2

2

k, (2.17)

and the equation for the turbulent energy k reads [123]:

(Uk) (( T

k+ )k) = P . (2.18)

The production term P in the above equation has the following form:

P = 2 T (U + U

T )

2:

(U + UT

)2

. (2.19)

The values of the coefficients are: C = 0 .09, C 1 = 1 .44, C 2 = 1 .92, k = 1 .0 and

= 1 .3.

In the vicinity of the impermeable no-slip walls physics of the turbulence isdominated by the presence of the wall. The most general treatment for resolving

the ow near the wall is by solving the transport equations in the near wall region,

eg. Launder and Sharma [83]. However, as the large variations of ow variables exist

in the near-wall region, the computational mesh has to be very ne there. A model

developed to alleviate these problems is q [53] whereq and vary linearly nextto the wall. q and are dened as [53]:

q = k and (2.20) =

2q . (2.21)


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( = 0 .41), B is an empirical constant which depends on the thickness of the viscous

sublayer ( B 5.2 in a at plate boundary layer) and Y + is the dimensional distancefrom the wall dened as:

Y + =U d

. (2.31)

Here, d is the distance from the wall. The wall-function is valid when the near-wall

node is within the logarithmic region, i.e. Y + > 15. This imposes a limitation

on mesh resolution in the near-wall region characterised by high gradients of all

elds which need ne mesh resolution to achieve accurate solutions of the governing

equations. On the other hand, if the mesh becomes too ne ( Y + < 15) the wall-

function becomes invalid. It this is present over a large portion of wall boundaries

it may result in serious modelling errors.

2.3 General Form of a Transport Equation

All equations described above can be written in the form of a general transport equa-

tion, given below and used throughout this study to present the FV discretisation

practices and error analysis.

V CV t dV temporal derivative+ V CV (U ) dV convection term

V CV () dV diffusion term= V CV S () dV. source term (2.32)

Here is a tensorial property considered continuous in space, is the diffusion

coefficient and S () is the source term.

2.4 Summary

In this chapter the laws of the continuum mechanics have been presented. An intro-

duction into turbulence modelling is also given. Low- Re turbulence models solve the

turbulence equations in the near-wall region where they require ne mesh resolution

to resolve sharp gradients of solution variables, but they do not impose a limit on

mesh resolution there. High- Re turbulence models model the ow near wall bound-

aries by using wall-functions which reduce the number of cells required, but theyimpose a limit on mesh resolution there which may prevent the user from getting a

mesh-independent solution of the problem under consideration. The general trans-


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2.4 Summary 37

port equation which will be used for explaining FV discretisation and error analysis

is presented at the end of the chapter.


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Chapter 3

Finite Volume Discretisation

3.1 IntroductionThe Finite Volume discretisation used in this study will be described in this chapter

by using the general transport equation Eqn. (2.32), introduced in the previous

chapter, as the model. The FV discretisation of this equation will be performed term

by term and the resulting discretisation errors which can arise will be identied. The

boundary conditions and their inuence on the accuracy will also be discussed. A

solution algorithm for solving the Navier-Stokes equations will be presented at the

end of the chapter.

An important property required of a FV discretisation practice is that the ow

solution is sought at a certain number of nodes in space and time; and if the number

of nodes tends to innity then the solution should tend to the exact solution of

the governing equations. This will happen if the FV method satises the following

requirements [47]:

Consistency. The discretisation error in the numerical solutions must tendto zero as the grid spacing tends to zero. The discretisation can produce the

exact solution if the truncation error, dened as the difference between the

governing equation and its discrete approximation tends to zero when the grid

spacing tends to zero. The truncation error can be expressed as a power of

the grid size and/or time step where the power of the most important term

represents the order of the approximation. The order must be positive and if

possible equal or higher than the order of the differential equation [47].

Stability. The discretisation is considered stable if it does not magnify nu-39


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3.2 Measures of Mesh Quality 41

representation of the computational domain. For each cell the values of the elds

are stored at a node P , Fig. 3.1, located in the centroid of the cell x P , dened as:

V P (x xP )dV = 0 . (3.1)Every cell shares an internal face with a neighbouring cell, whose centroid is denoted

with N in Fig. 3.1. Faces which are not shared by two cells are boundary faces.

The values of the elds dened on the faces (i.e. face ux, surface normal gra-

dients) are stored in the node located in the centroid of the face x f , whose position

is given by:

f dS (x x f ) = 0 (3.2)The second step of the FV discretisation process is the approximation of the

governing equations over the typical cell here done in a second-order fashion by

assuming a linear variation of the property within each CV and during each time-

step. This can be expressed via the Taylor Series expansion:

(x) = P + ( x xP ) ()P + O(|(x xP )|2), (3.3)(t + t) = t + t

t

t

+ O( t2), (3.4)

where the subscript P relates to the node in which the solution is sought and the

superscript t denotes the current time step. O(|(x x P )|2) and O( T 2) are thetruncated terms in the full series, having the following form:

O(|(x xP )|2) =

i=2

1i!

(x xP )i ::: i(.. i

)P , (3.5)

O( t2) =

i=2

1i!

tn it i

(3.6)

where :::

iis a scalar product of ith rank tensors. ( x x P )i is a ith tensor product

of a vector with itself resulting in an ith rank tensor. The leading terms of the

truncation errors are proportional to ( x xP )2 and t2, so the approximations aresecond-order accurate.

3.2 Measures of Mesh Quality

The distribution of the nodes and the quality of the mesh inuence the accuracy

of results. The properties which determine mesh quality and their measures are


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42 Finite Volume Discretisation

presented here and they will be used when discussing different discretisation errors

later in this chapter. These properties are dened for mesh faces and their denition

is the same irrespective of the mesh type. Finally, the properties are:

Non-orthogonality is measured by the angle N between the vector d con-

necting nodes adjacent to a face and the face area vector S, as can be seen in

Fig. 3.2. The angle should be as small as possible. The reasons for this will

be given later in this chapter.

f

N

NPd

S

Figure 3.2: Mesh non-orthogonality

Mesh skewness. When the vector d does not intersect a face in its centre themesh is dened as skewed, Fig. 3.3. The degree of skewness can be measured

N P

Sf

f i

m

d

Figure 3.3: Mesh skewness

by:

= |m ||d |

. (3.7)

Here m and d are vectors dened in Fig. 3.7. Skewness affects the accuracyof the interpolation from the nodes onto the faces as will be shown in the

remainder of the chapter.


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3.3 Discretisation of Spatial Terms 43

Uniformity. A mesh is uniform when d intersects the face midway betweenthe nodes P and N , Fig. 3.3. Uniformity can be measured by:

f x = |x f i xN ||d |

, (3.8)

thus f x = 0 .5 on uniform meshes. The inuence of uniformity on accuracy willbe discussed later in the chapter.

3.3 Discretisation of Spatial Terms

The FV approximation of the spatial terms in Eqn. (2.32) will be given in this sec-

tion. Approximations of volume and surface integrals and interpolation techniques

which are needed for the FV discretisation of the spatial terms in Eqn. (2.32) willbe given rst.

Volume integrals of can be approximated by integrating Eqn. (3.3) over the

cell and using Eqn. (3.1) [68]:

V P (x) dV = P V P + O(|(x xP )|2) (3.9)where V P stands for the volume of the cell and P is the value of at the centroid.

Surface integrals can be evaluated in the similar fashion, thus [68]:

S f (x) dS = f S f + O(|(x x f )|2) (3.10) S f dS a(x) = Sf a f + O(|(x x f )|2) (3.11)

where f and a f are the values of tensorial property and a vector property a in

the centroid of the face dened in Eqn. (3.2). Values in the face centroids can be

interpolated or extrapolated from nodal values and are denoted by f and a f .

A second-order interpolation practice from the nodes onto internal faces can be

written as follows [20, 90]:

f = f i + m ()f i , (3.12)where f i and ()f i are the interpolated values of and at the point where the

vector d intersects the face, as shown in Fig. 3.3. f i and ()f i can be evaluated

by using linear interpolation:

f i = f x P + (1 f x )N , (3.13)()f i = f x ()P + (1 f x )()N . (3.14)


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The linear interpolation factor f x is dened as follows, Fig. 3.3:

f x = |x f i xN ||d |

. (3.15)

The truncation error for the interpolation practice dened in Eqn. (3.12) can be

estimated by using the following Taylor expansions:

P = f i + ( xP x f i ) ()f i +12

(xP x f i )2 : ()f i (3.16)N = f i + ( xN x f i ) ()f i +

12

(xN x f i )2 : ()f i (3.17)f = f i + m ()f i +

12

m 2 : ()f i (3.18)

By substituting P and N in Eqn. (3.13) with Eqn. (3.16) and Eqn. (3.17), respec-

tively, the truncation error for linear interpolation from Eqn. (3.13) can be obtainedas [47]:

el = f i f i=

12|xP x f i ||xN x f i |(d

2 : ()f i )

= 12

f x (1 f x )|d |2 (d 2 : ()f i ) (3.19)

d being an unit vector in the direction of d , Fig. 3.3.From Eqn. (3.19) it follows that the truncation error for the gradient interpolated

using Eqn. (3.14) has the form:

(e)l = ()f i ()f i = 12

f x (1 f x )|d |2 (d 2 : ()f i ) (3.20)

Taking the difference between the Eqn. (3.18) and Eqn. (3.12), the truncation error

for the linear interpolation scheme which is second-order accurate on every mesh

can be obtained in the following form:

einterpolation = 12|xP x f i ||xN x f i | (d

2 : ()f i ) + m (d 2 : ()f i )+

12|m |

2 m 2 : ()f i

= 12

f x (1 f x )|d |2 (d 2 : ()f i ) + m (d 2 : ()f i )+

12|m |

2 m 2 : ()f i

= 12|d |2 f x (1 f x )( d 2 : ()f i ) + |d |m (d 2 : ()f i )

+2

2 |d |2(m 2 : ()f i ). (3.21)


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Here, d and m are unit vectors in the directions of d and m , respectively. This

error reduces with the square of the distance between the neighbouring nodes and

is minimal when the mesh is not skewed ( = 0).

The gradient and divergence terms can be approximated by using the Gauss

theorem [68]:

Divergence of the vector property a can be approximated as follows:

V a dV = S CV dS a=

f

S

f

dS a

=f

Sf a f . (3.22)

Here, V represents the volume of the CV and S CV its surface area. dS is the

surface area vector pointing outwards and a f is evaluated using the interpola-

tion practice dened in Eqn. (3.12) for the vector property a .

The truncation error for the divergence term consists of the error in the inter-polation of a f , thus:

ediv =f

Sf (a f a f )

=f

Sf einterpolation

=f

1

2|d

|2Sf

f x (1

f x )( d 2 : (

a)f i ) +

|d

|m

(d 2 : (

a)f i )

+f

2

2 |d |2Sf (m 2 : (a)f i ). (3.23)

This error reduces with the square of d and is smallest in case when = 0.

The error is also dependent on the shape of the CV. This will be discussed in

Section 3.6.

Gradient term. Discretisation of the gradient term can be performed eitherusing the Gauss Theorem or the Least Squares Fit (LSF). Discretisation using


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3.3.1 Convection Term

The discretisation of the convection term is performed using Eqn. (3.22) in the

following fashion:

V P (U) dV = f S (U)f

=f

S (U )f f

=f

F f , (3.30)

where the volume integral is rst transformed into a sum over the faces and then

approximated. Here F represents the mass ux through the face:

F = S (U )f . (3.31)

These uxes have to satisfy continuity for every CV. They can be estimated using

the interpolated values of U and onto the face. This interpolation may introduce

an error into the mass ux which can then be written:

F = ( S (U )f ) + e flux . (3.32)

The procedure for obtaining conservative uxes and errors which can arise will be

described in Section 3.8.

The next issue is how to obtain the value of on the face. Many different inter-

polation techniques can be used to obtain f but some do not ensure boundedness.

Linear Interpolation (Central Differencing) (CD) is a natural second-order interpolation practice for obtaining the value of on the face. This

practice has already been described in Eqn. (3.12) and is:

f = ( f x P + (1 f x )N + m ()f i ) + econv . (3.33)

The truncation error, dened in Eqn. (3.21), is:

eCD = 12|d |

2 f x (1 f x )( d 2 : ()f i ) + |d |m (d 2 : ()f i )+

2

2 |d |2(m 2 : ()f i ). (3.34)

In [61, 114, 119, 148] it is shown that with CD the convective contribution to

the coefficients of downstream nodes is always negative, which may give rise to


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non-physical oscillations which violate boundedness and degrade the quality

of the results. On convection-diffusion problems this undesired behaviour may

occur if the value of face Peclet Number is greater than two. This undesirable

property may not become apparent if the gradients in the solution are not

large, but if boundedness of the solution is essential (as in solutions of the eg.

k and in turbulence equations) some other scheme may have to be used.

Upwind Differencing (UD) was introduced to overcome the problem of oscillatory solutions and make the convection term unconditionally positive.

Boundedness of the convection term is achieved by assuming that the value

on the face is determined by the upstream node, thus:

f i =f i = P for F > 0.

f i = N for F < 0.(3.35)

This discretisation practice ensures the boundedness of the solution by making

the matrix coefficients unconditionally positive.

The truncation error for the UD scheme can be obtained by using the following

Taylor series expansions:

f = P + ( x f xP ) ()P + 12 (x f xP )2 : ()P for F > 0.

N + ( x f xN ) ()N + 12 (x f xN )2 : ()N for F < 0.(3.36)

The truncation error can be found as a difference between Eqn. (3.36) and

Eqn. (3.35), thus:

eUD =(x f xP ) ()P + 12 (x f xP )2 : ()P for F > 0.

(x f xN ) ()N +12 (x f xN )

2

: ()N for F < 0.

(3.37)

The leading error term in the above equation is a function of ( x f xP ) ()P ,resembling a form of the diffusion term, and is therefore called numerical dif-

fusion [114]. This discretisation practice is rst-order accurate and it requires

high spatial resolution to achieve accurate solutions [114].

Gamma differencing scheme (Gamma) described in [68, 75] is a boundedscheme formed by blending CD with UD in the regions where CD would not

produce a bounded solution, thus:

f = () (f )CD + (1 ()) (f )UD + eGamma . (3.38)


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Here () is a blending factor, 0 () 1, dependent on the nature of the distribution around the face. The procedure used for evaluating () is based

on Normalised Variable Approach (NVA) of Leonard [87] and Gaskell et al.

[49]. A normalised variable is dened as [87]:

P =P U D U

, (3.39)

where P , U and D are the values in the node P , upwind node U and

downstream node D, as depicted in Fig. 3.4. The solution is bounded if the

P DUFlow direction

U

D

P

f

Figure 3.4: Variation of near the face

following conditions are satised:

U P D , (3.40)

or

U P D , (3.41)from where it follows that P should obey:

0 P 1. (3.42)

Jasak [68] has modied Eqn. (3.39) to be applicable to arbitrary meshes, thus:

P = 1 ()f d

2(

)P

d

(3.43)

where the node P must be an upwind node to the face f and d = x D xP connects the D and P nodes, Fig. 3.4. Depending on P , the blending factor


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() is determined as follows [68]:

P > 1 () = 0 ,

m P 1 () = 1 ,0 P < m () =

P

m,

P < 0 () = 0 ,

where m is a constant of the scheme which was introduced to ensure linear

transition between CD and UD when 0 P < m to improve convergencefor steady-state problems. The range of m recommended by Jasak [68] is

0.1

m

0.5.

The truncation error for this interpolation practice can be written as follows:

eGamma = ()eCD + (1 ())eUD , (3.44)

where eCD and eUD are dened in Eqs. (3.34) and (3.37). The scheme is

second-order accurate when () = 1 but it reduces down to rst order when

() < 1.

The total error resulting from the discretisation of the convection term has con-

tributions from the interpolation of mass ux and the interpolation of , so:

V P (U ) dV = f (F + eflux )(f + eint )=

f

F f + econv (3.45)

where econv has the following form:

econv =f

(F eint + eflux f + eint eflux )

C flux eint + C f eflux + eint eflux . (3.46)

The order of the approximation is therefore equal to the lowest order approximation

used in the process. If the procedures for interpolation of and F are second-order accurate then the approximation of the convection term is also second-order

accurate.


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3.3.2 Diffusion Term

The discrete approximation of the diffusion term is obtained by using Eqn. (3.22)

and taking the gradients on the faces to be constant due to the assumed linear

variation of the property . The result is:

V P () dV = f S ()f =

f

()f (S )f , (3.47)

where the terms ( S ) f and ()f need further treatment. The latter is interpo-lated onto the faces using Eqn. (3.12), where is substituted by . Approximation

k

P Nf d

S

N

Figure 3.5: Non-orthogonality treatment

of (S )f on a non-orthogonal mesh, Fig. 3.5, when vectors d and S are not par-allel, is performed using the following expression [68]:

(S )f = | |N P |d |+ k ()f (3.48)

where ()f can be evaluated using Eqn. (3.14). Here, is parallel with d where

and k have the property:

S = + k . (3.49)

In [68] Jasak has tested different treatments of and k . The one for which k is

orthogonal to S, Fig. 3.5, performed best in terms of accuracy and convergence and

is adopted here. The length of can be expressed as follows:

| | = |S|

cos N , (3.50)


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and the length of k can be calculated from:

|k| = |S| tan N . (3.51)The truncation error for the approximation of the ( S ) f term can be obtainedusing the following Taylor expansions:

P = f + ( xP x f ) ()f +12

(xP x f )2 : ()f +16

(xP x f )3 :: ()f (3.52)

N = f + ( xN x f ) ()f +12

(xN x f )2 : ()f +16

(xN x f )3 :: ()f (3.53)

By substituting Eqn. (3.52) and Eqn. (3.53) into Eqn. (3.48) and by adding the error

from the interpolation of () f , the truncation error for ( S ) f is obtained, thus:esnGrad = ( S )f (S )f

= 12| ||d |

(|xN x f |2 |xP x f |2)d 2 : ()f

16| ||d |

(|xN x f |3 + |xP x f |3)d 3 :: ()f

12|xN x f ||xP x f |k (d

2 : ()f )

= |S|cos N |d |2 (2f x 1)d

2 : ()f

|S|

6 cos N |d |2 (1 f x )3 + f x 3 d 3 :: ()f

|S| tan N |d |22

f x (1 f x )k (d 2 : ()f ), (3.54)where d and k are unit vectors in directions of d and k , respectively, and f x is the lin-

ear interpolation factor dened in Eqn. (3.15). From the dependence of Eqn. (3.54)

on the f x , it follows that the approximation is rst-order accurate except for f x = 0 .5,i.e. present when the mesh is uniform. It is therefore advisable to keep the mesh

as uniform as possible to obtain best accuracy. If the mesh is uniform, the approxi-

mation becomes second-order accurate. The error is also dependent on the angle of

non-orthogonality and is minimal when N = 0.

Finally, the discrete form of the diffusion term can be written:

V P

(

) dV =f

(()f + einterpolation )(

|

|N P

|d |+ k

(

)f + esnGrad )

=f

( )f (| |N P

|d |+ k ()f ) + ediff (3.55)


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where the truncation error for the diffusion term ediff has the form:

ediff =f

( )f esnGrad

+f

(

|

|N P

|d |+ k

(

)f ) einterpolation

+f

einterpolation esnGrad

f

(C int esnGrad + C snGrad eint + esnGrad eint ). (3.56)

from which it follows that the order of the approximation is equal to the lowest order

found in Eqs. (3.54) and (3.21). Thus, the discretisation is second-order accurate on

uniform meshes and reduces to rst-order on the non-uniform ones. The behaviour

of the truncation error and the achievable accuracy on CVs of different shapes will

be compared in Section 3.6.

3.3.3 Source Terms

As it was previously mentioned, all the terms in the equations which cannot be

expressed as convection, diffusion or temporal terms are grouped into the so-called

source term. If the source term is dependent on , linearisation should be performed

[114], such as:

S (, x) = Su (x , ) + Sp(x , ) (x). (3.57)

When the Eqn. (3.57) is integrated over the control volume using Eqn. (3.9) the

discretised form of the source term is obtained, thus:

V P

S () dV = Su V P + Sp V P P + esource . (3.58)

The truncation error for the source term can be estimated by using the following

Taylor expansion:

V P S (x , ) dV = V P S (xP , P ) + ( x xP ) (S (x , ))P + S (x, ) dV + V P 12(x xP )2 :: (S (x , ))P dV +

V P

1

2(x

x

P )

(

S (x , ))P

S

dV

+ V P 12( )2 2S (x , )2 dV, (3.59)


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where can be substituted by:

= ( x xP ) ()P . (3.60)After substituting and S (xP , P )+ S (

x , ) in Eqn. (3.59) using Eqn. (3.60)

and Eqn. (3.57), respectively, the truncation error can be found as the differencebetween Eqn. (3.59) and Eqn. (3.58), thus:

esource =12

(x xP )2 : (S )P V P +12

(x xP )2 : (()P (S )P )S

V P

+12

(x xP )2 : (()P )2 2S (x, )

2V P . (3.61)

The error is a function of (x xP )2 but it also depends on the linearisation practiceused.

3.4 Temporal Discretisation

Temporal discretisation is performed on a semi-discretised form of the transport

equation, where the spatial terms have already been approximated using the prac-

tices described in the previous section. This form reads [61]:

t+ t

tt P

V P +f

F f f ( )f S ()f dt

= t+ tt (Su V P + Sp V P P ) dt.(3.62)

and can be written in a shorter form [47]:

V P t+ tt t dt = t+ tt f (t, (x, t )) dt, (3.63)where f (t, (x , t )) contains all spatial terms from Eqn. (3.62). After performing

integration of Eqn. (3.63) there results:

V P t+ tt t dt = V P (n o) = t+ tt f (t, (x , t )) dt, (3.64)where the subscripts o and n represent old and new time levels, respectively. The

second important part of the temporal discretisation process is to choose an approx-

imation for t+ tt f (t, (x, t )) dt, which cannot be evaluated exactly. Taylor seriesexpansion gives:f (t + t, (x , t + t)) = f (t o, (x , t o))+ t

df (to, (x , t o))dt

+12

( t)2d2f (to, (x , t o))

dt2,

(3.65)


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3.4 Temporal Discretisation 55

which will be used later to establish the order of temporal discretisation. Some of

the most common temporal discretisation practices are:

Euler Explicit method. This method approximates Eqn. (3.64) by assuming[47]:

t+ t

tf (t, (x , t )) dt = f (t, (x , t o)) t (3.66)

i.e. it integrates the spatial terms by using the values at the beginning of the

time interval. These spatial terms can be calculated because o is known and

the value at the end of the interval can be obtained directly for every node

without having to solve a system of equations. It is shown in [47, 61] that

this scheme is stable for Courant numbers Co = |U | th < 0.5. Here, U is the

transport velocity and h is the cell size.

The truncation error for this practice can be found as a difference between

Eqn. (3.65) and Eqn. (3.66) which yields:

eexpEuler = tdf (to, (x , t o))

dt+

12

( t)2d2f (to, (x , t o))

dt2(3.67)

This error is proportional to t, i.e. this method is rst-order accurate in

time.

The Crank-Nicholson discretisation practice assumes linear variation of f (t, (x , t )) in time [47, 61], giving:

t+ tt f (t, (x , t )) dt = 12 [f (t, (x , t o)) + f (t, (x, t n ))] t (3.68)Eqn. (3.68) requires evaluation of spatial terms for old and new time steps.

Because the values of n are not known at the new time level this method

requires a solution of a system of algebraic equations for each time step [61, 68].

The truncation error for this practice is:

eCN =12

( t)2d2f (to, (x, t o))

dt2(3.69)

Thus, it is second-order accurate in time.

Euler Implicit method. This method approximates Eqn. (3.64) by assuming

[47]:

t+ tt f (t, (x , t )) dt = f (t, (x , t n )) t (3.70)


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3.5 Boundary Conditions 57

P

b

dd

S

n

m = k

Figure 3.6: Boundary cell

which shows that the approximation Eqn. (3.72) is second-order accurate.

Diffusive uxes require the gradient of normal to the boundary face which is

known exactly in case of Von Neumann boundary condition. In case of Dirichlet

boundary condition it can be evaluated as follows:

(S )b = |S|

|d n

|(b P k ()P ), (3.75)

where k is a non-orthogonality vector, Fig. 3.6. The truncation error for ( S )bapproximated using Eqn. (3.75) can be derived by using the following Taylor series

expansion:

b = P + d ()P +12

d 2 : ()P . (3.76)

Substituting b in Eqn. (3.75) with Eqn. (3.76), after some algebra it gives:

ebouSnGrad = |S| |d

|cos N (d

2

: ()P ). (3.77)

d is a unit vector in the direction of d . The error is a function of |d | and is thereforerst-order accurate. This method corresponds to backward and forward differences

in Finite Difference Method which are rst-order accurate [47]. The error is minimal

when the mesh is orthogonal ( N = 0).

3.5.2 Boundary Conditions for the Navier-Stokes Equations

The most common boundary conditions which occur when solving the laminar and

turbulent ows are:


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Inlet. The velocity distribution is prescribed and the surface normal gradientis set to zero for the pressure. Values of turbulent properties ( eg. k, , q and

) are prescribed.

Outlet. The pressure distribution is prescribed there. Surface normal gradi-

ents of the velocity eld and turbulent properties are set to zero.

Symmetry plane. The surface normal gradients of all scalar elds are zeroat the symmetry plane. The convection ux through the symmetry plane is

zero. Viscous stress of the velocity component parallel to the boundary is zero

and it must be imposed directly [47] while the viscous stress of the velocity

component normal to the wall is evaluated using Eqn. (3.75).

Impermeable no-slip walls. The velocity eld is xed to the velocity of thewall and the normal gradient of the pressure is set to zero there. The convection

ux through the wall is zero. Viscous stress of the velocity component normal

to the wall is zero at the wall and it must be imposed as an additional condition

[47]. Viscous stress of the velocity component parallel to the wall is not zero

and its evaluation is dependent on the near-wall modelling employed [47].

Conditions imposed on the turbulence properties depend on the turbulence

model and the type of near-wall treatment. The values of q and elds,

dened in Chapter 2, are xed at the wall boundary while the zero normal

gradient is imposed on k and elds.

3.6 Discretisation Errors on different types of meshes

Discretisation errors are dependent both on the type of computational mesh ( i.e.

tetrahedral, hexahedral, polyhedral) and the approximations for different terms in

the governing equations. This section will present a comparison of accuracy achiev-

able on different mesh types. This is done by using the expressions for the truncation

errors presented in the preceeding sections of this chapter. The comparison will be

presented for convection and diffusion terms. Discretisation procedures for diver-

gence and gradient terms are very similar to the discretisation procedure for the

convection term and everything that will be said about error for the convectionterm can be applied to them.

Four types of mesh will be considered for the analysis:


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3.6 Discretisation Errors on different types of meshes 59

Square mesh. A representative cell with its neighbours which inuence itare shown in Fig. 3.7.

U = c

o n s t.

X

Y

P E

N

W

S

l

Figure 3.7: Square mesh

This mesh is orthogonal, it is not skewed and f x = 0 .5 because the centre of

each internal face lies midway between the neighbouring nodes.

Triangular mesh. A mesh consisting of equilateral triangles is uniform, or-thogonal and not skewed, see Fig. 3.8.

X

Y

U = c

o n s t.

P

EW

S

l

Figure 3.8: Triangular mesh

Regular hexagonal mesh. This type of mesh can be generated using aDelaunay algorithm, see Chapter 7. It consists of hexagons and is uniform,

orthogonal and not skewed, Fig. 3.9.

X

Y U

= c o n s

t.

P

N

NE

SE

NW

SW

S

l

Figure 3.9: Hexagonal mesh

Split-hexahedron mesh. This type of mesh is produced when local rene-


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ment of hexahedral cells is done by splitting cells, such that a cell face gets

divided into two or four faces. An example is shown in Fig. 3.10.

X

Y

d s

U = c

o n s t.

P E

N

S

l

SW

NWS

N

mk

Figure 3.10: Split-hexahedron mesh

Unlike the previous examples, a split-hexahedron mesh is skewed and not or-

thogonal but f x = 0 .5. The distances between nodes at the split face are also

larger than on other faces. Thus, for the example above:

d s = ls d =l

cos N d , (3.78)

where N can be calculated from:tan N =

0.25 ll

=14

. (3.79)

3.6.1 Convection Term

The analysis of errors in the convection term will be performed by using the Central

Differencing scheme and taking the uid velocity and density to be constant:

U = const.

= const.

in order not to include errors coming from evaluation of mass uxes, Eqn. (3.31).

The truncation error for the CD scheme was given in Section 3.3.1 and it is:

econv = 12

f

F |d |2 f x (1 f x )( d 2 : ()f i ) + |d |m (d 2 : ()f i )

+f

F 2

2 |d |2(m 2 : ()f i ). (3.80)

Here, f x is a linear interpolation factor dened in Eqn. (3.15). The second and the

third terms in Eqn. (3.80) are zero on meshes which are not skewed.


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Error on square mesh. The distance between the two nodes is equal to thelength of a side. Hence, the mass ux is:

F = |U | l (3.81)

because |S| = l; and f x = 0 .5. Replacing |d | in Eqn. (3.80) with l yields:

econvSquare = 18

F cos l 2(n 2e : ()e)

18

F sin l 2(n 2n : ()n )

+18

F cos l 2(n 2w : ()w )

+18F sin l 2(n 2s : ()

s )

= 18

l2 F cos (n 2e : (()e ()w ))

18

l2 F sin (n 2n : (()n ()s )) , (3.82)

where d 2 from Eqn. (3.80) is replaced by n which represents a unit face normal

vector pointing outwards from the cell P . The subscript n e denotes that the

face is shared with a neighbour E . is the angle between the velocity vectorand the x axis.

Equilateral Triangular Mesh. For this the mass ux is:

F = 3 |U | l. (3.83)

The truncation error can be found using Eqn. (3.80), yielding:

econvTri = 316

l2F cos ((n 2e : ()e) (n 2w : ()w ))

18

l2F sin (12

(n 2e : ()e) (n 2s : ()s ) 12

(n 2w : ()w )).

(3.84)

Hexagonal Mesh. Here the expression for the mass ux is:F =

33

|U | l, (3.85)


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and Eqn. (3.80) gives the truncation error as:

econvHexagon = 316

l2F cos n 2ne : (()ne ()sw )

316

l2F cos n 2se : (()se ()nw )

18l2F sin n 2n : (()s ()s )

116

l2F sin n 2ne : (()ne ()sw )

116

l2F sin n 2se : (()se ()nw ) . (3.86)

Split-hexahedron Mesh. The truncation error for the conguration shownin Fig. 3.10 is:

econvSplitHex = 18

l2F sin n 2n,s : (()n ()s )

18

F cos l2 n 2e : ()e 12

l2s d 2nw : ()nw

18

F cos 12

l2s d 2sw : ()sw

+12

F cos l 3s m nw (d 2nw :: ()nw )+

12

F cos l 3s m sw (d 2sw :: ()sw )

F cos 2

2l2s m 2nw : (()nw + ()sw ) (3.87)

where F is dened in Eqn. (3.81), ls can be calculated from Eqn. (3.78) and

can be obtained from Eqn. (3.7), thus:

= |m |ls

=l

4 ls. (3.88)

Evidently, the error in Eqs. (3.82), (3.84), (3.86) and (3.87) is dependent bothon the spatial resolution and the form of the solution itself as expressed through its

gradients. A comparison of accuracy can therefore only be performed by examining

different forms of solutions, as follows:

1. = constant. When the solution eld has a uniform and xed gradient the

error is zero on all types of meshes. This is consistent with the assumption

expressed in Eqn. (3.3).

2. = constant. This class of solutions with uniform curvature reveals differ-

ences in accuracy between different mesh types. The discretisation on square


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and hexagonal meshes still produces exact solutions Eqs. (3.82) and (3.86),

while triangular meshes and meshes with split-hexahedra produce errors. The

truncation error for triangular meshes, Eqn. (3.84), can be simplied to:

econvT ri = 316 l

2

F cos ((n2

e n2

w ) :)

18

l2F sin ((12

n 2e n 2s 12

n 2w ) :) (3.89)

while the truncation error for a split-hexahedron mesh, Eqn. (3.87), can be

reduced to:

econvSplitHex = 18

F cos (l2 n 2e 12

l2s d 2nw 12

l2s d 2sw ) :

F cos

2

2l2s

(m 2nw

+ m 2sw

) :

. (3.90)

The above equations show that the errors exist because these mesh types have

not got face pairs such that the error on each pair cancel. Two cell faces form

a face pair if the sum of their outward-pointing normal vectors is a zero vector.

From there it follows that the difference between the second tensors of the face

normals, i.e. (n 2e n 2w ) = 0, is a zero tensor for every face pair; which resultsin zero discretisation error. Triangular meshes have not got any face pairs

because (n 2e n 2w ) and ( 12 n 2e n 2s 12 n 2w ) in Eqn. (3.89) are not zero tensors.Skewness contributes to the error in Eqn. (3.90) because ( m 2nw + m 2sw ) is not

a zero tensor.

3. = constant. This type of problem cannot be resolved exactly on any

type of mesh. Hence, it should show which of the square or hexagon shapes

should be the most accurate. The variation of (x) within the cell is:

(x) = ()P + ( x xP ) (3.91)When in Eqn. (3.80) is substituted with Eqn. (3.91), the following ex-

pressions for the truncation errors result.

econvSquare = 18

l3F cos n 3e :: 18

l3F sin n 3n ::. (3.92)

econvHexagon = 18l3F cos

32 (n3ne + n 3nw ) ::

18

l3F sin (n 3n +12

n 3ne 12

n 3nw ) ::. (3.93)


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In order to determine which type of mesh is the most accurate the problem

is simplied such that 3

x 3 = constant while all other components of

are zero. The velocity U is at angle = 0. When the above is inserted in

Eqs. (3.92) and (3.93) there results:

econvSquare = 18

l3F 3x 3

, (3.94)

econvHexagon = 964

l3 3x 3

. (3.95)

For the same number of cells in the domain the area of the square cell is equal

to the area of the hexagon. From there it follows that:

lhexagon = 2 33 lsquare . (3.96)

By knowing the ratio between lhexagon and lsquare , Eqn. (3.96) the ratio between

the errors is:

econvHexagoneconvSquare

= 1 .155. (3.97)

Eqn. (3.97) shows that the error on the hexagonal mesh is expected to be

higher than for the square mesh. The reason for this lies in a fact that hexagons

always have more faces with non-zero mass uxes than the square such that

the sum of errors committed on those faces is larger than for the square.

3.6.2 Diffusion Term

The accuracy of the diffusion term discretisation on different types of meshes canbe performed by comparing the truncation errors dened by Eqn. (3.56). In order

to simplify the analysis, the uid density and the diffusion coefficient are assumed

to be constant, i.e. :

= constant.

= constant.

This makes the error arising from interpolation of onto the cell faces equal to

zero; and the only remaining error comes from the approximation of surface normal


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gradients, Eqn. (3.48):

ediff = f

() |S|cos N

|d |2

(2f x 1)d 2 : ()f

f () |S|

6 cos N |d

|2 (1

f x )3 + f x 3 d 3 :: (

)f

f

()|S| tan N |d |22

f x (1 f x )k (d 2 : ()f ), (3.98)

This will now be evaluated for the different mesh types.

Square Mesh. For the conguration depicted in, Fig. 3.7, the magnitude of the surface vector is:

|S| = l. (3.99)Noting that |d | = l, N = 0 and f x = 0 .5, the truncation error can be written:

ediffHex = 124

()|S|l2(n 3e :: ()e)

124

()|S|l2(n 3w :: ()w )

124

()|S|l2(n 3n :: ()

failure analysis of fev

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