Flow Feature Aligned Mesh Generation and Adaptation

224 Pages·2013·11.58 MB·English

University of Sheffield

Department of Mechanical Engineering

Flow Feature Aligned Mesh Generation and
Adaptation

M. J. Harris

Supervised by Prof. Ning Qin

April 2013

This thesis is submitted to University of Sheffield in partial fulfilment of the requirement
for the degree of Doctor of Philosophy
ii
Abstract

Methods which allow for construction of flow feature aligned meshes in two- and
three-dimensions have been developed in this thesis to investigate their potential for
improvements in the numerical solution relative to globally refining the mesh. Of particular
interest in the work is the generation of high-quality quadrilateral and hexahedral elements
aligned with the dominant flow features. The two-dimensional techniques are applied on
unstructured quad-dominant meshes, whilst the three-dimensional problems involve
embedding high-quality hex-dominant mesh blocks into a hybrid volume mesh to improve
their ability to capture anisotropic flow features such as shock waves, trailing shear
layers/wakes and wing tip vortices.
A method involving the medial axis has been studied to provide a geometric
representation of two-dimensional flow features to allow feature-aligned meshes to be
generated. Due to the flexibility of the approach, a range of complex features can be
represented as simple geometric entities. These curves are embedded into the domain as
virtual geometries to force alignment of unstructured quad-dominant surface mesh
elements. The mesh locally mimics the attributes of a structured grid and provides high
quality numerical solutions due to the alignment of the cell interfaces with the flow
features.
To improve the capability of hybrid meshes to resolve anisotropic flow physics, a
method involving the extrusion of quad-dominant surface meshes has been developed.
Surface meshes are extruded in the direction of extracted flow features, yielding feature-
aligned semi-structured hex-dominant mesh blocks which can be embedded into the hybrid
volume mesh. The presence of feature-aligned hexahedra has been shown to greatly
enhance the resolution of anisotropic flow features compared with both isotropic and
anisotropic tetrahedral elements, due to a significant reduction in numerical diffusion.
Furthermore, improvements in the numerical solution have been also been obtained in a
more efficient manner than isotropically refining the hybrid mesh. The results indicate that
the type, orientation and size of the elements are significant contributing factors in the
resolution of the dominant flow features.
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Acknowledgements

Firstly I would like to thank Professor Ning Qin for acting as my supervisor for the
duration of this project. His advice, support and guidance have been invaluable during my
progression through the period of this thesis, and I have come to learn much from our
discussions on various CFD related topics.
During the project I frequently visited Aircraft Research Association (ARA) in
Bedford, England for a series of short work placements. I would like to express my
appreciation to all the members of the Computational Aerodynamics team who provided
me with helpful suggestions and technical expertise. Specifically, I would like to thank
David Martineau (now of ICON CFD) for serving as my initial industrial supervisor. I am
grateful for his assistance (and patience!) in the early stages of the project as I overcame the
steep learning curve involved with understanding and using the SOLAR source code. I feel
I have developed a significant level of programming capability within a short space of time
which would not have been possible without his initial guidance.
I would also like to thank Dr Craig Johnson of ARA for providing me with all the
software I required at the beginning of the project and generally being another source of
valuable technical expertise in both SOLAR and TAU. Other words of thanks are necessary
for Dr Andrew Peace, Head of Computational Aerodynamics at ARA, who (with Craig)
took the time to give several drafts of one of my papers a very thorough read. Their
detailed feedback was greatly appreciated and I hope greatly increased the quality of the
final paper.
Last but not least, I would like to take this opportunity to say thank you to all my
family and friends for the constant support given to me as I progressed through this work.
In particular I must say a massive thank you to my parents. Without them I would never
have got to where I am today. For all the support and encouragement they have given me
over the years (not just during the period of this thesis) I am eternally grateful. I hope that I
have made them proud through my achievements to date and that this goes some way in
repaying them for all they have done for me.

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Table of Contents
Abstract iii
Acknowledgments iv
Table of Contents v
Nomenclature x

1 Introduction 1
1.1 Background 1
1.2 Mesh Classification 2
1.3 Generation of Flow Feature Aligned Meshes 4
1.4 Importance of Mesh Alignment 6
1.5 Aims and Objectives 8
1.6 Thesis Outline 9

2 Literature Review 12
2.1 Introduction 12
2.2 Mesh Adaptation 13
2.2.1 Classifying Mesh Adaptation Techniques 13
2.2.2 A Remark on Error Estimators 14
2.3 Mesh Adaptation: Methods for Feature Alignment 18
2.3.1 Structured Meshes 18
2.3.2 Unstructured Meshes 19
2.3.2.1 Triangular and Tetrahedral Meshes 19
2.3.2.2 Quadrilateral and Hexahedral Meshes 22
2.3.2.3 Cartesian Meshes 26
2.4 Mesh Insertion, Regeneration and Specialized Regeneration Techniques 28
2.5 Miscellaneous Techniques: Overset Grid (‘Chimera’) Approach 32
2.6 Summary 33

3 Governing Equations and Numerical Methods 36
3.1 Introduction 36
v
3.2 The Unsteady Navier-Stokes Equations 37
3.3 Preconditioning 41
3.4 Turbulence Modelling 42
3.4.1 Reynolds-Averaged Navier-Stokes (RANS) Simulations 43
3.4.2 Spalart-Allmaras Turbulence Model 44
3.5 Discretisation and Numerical Schemes 46
3.5.1 Spatial Discretisation 46
3.5.2 The Primary Grid 47
3.5.3 The Secondary Grid 50
3.5.4 Flux Discretisation 50
3.5.5 Higher-Order Reconstruction 50
3.5.6 Evaluation of the Gradient 52
3.5.7 Temporal Discretisation 54
3.6 Flow Feature Extraction 54
3.6.1 Shock Waves 55
3.6.1.1 Transonic Flows 55
3.6.1.2 Supersonic Flows 56
3.6.2 Wakes and Wing Tip Vortices 57
3.6.2.1 Steady Wakes 57
3.6.2.2 Wing Tip Vortices 58
3.7 Mesh Adaptation Schemes 59
3.7.1 Construction of Anisotropic Metric for Adaptation 59
3.7.2 Nodal Movement 62
3.7.3 Adaptation in TAU 63

4 Using the Medial Axis to Represent Complex Flow Features for Feature-
Aligned Unstructured Quad-dominant Surface Mesh Generation 65
4.1 Introduction 65
4.2 Building the Medial Axis 68
4.2.1 Concave Hull Construction 68
4.2.2 Medial Axis Approximation 70
4.2.3 Constrained Delaunay Triangulations 71
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4.2.4 Cleaning the Medial Axis 72
4.2.4.1 False Junction Nodes 72
4.2.4.2 Small Branch Removal 73
4.2.5 Splitting the Medial Axis 75
4.2.6 Recombining Medial Axis Branches 76
4.2.7 Fitting Polynomials 78
4.3 Controlling Mesh Spacing 80
4.4 Global Mesh Refinement 83
4.5 Identifying Multiple Regions of Points 85
4.6 Mesh Adaptation 86
4.6.1 Determining Local Mesh Spacing 86
4.7 Splitting Bad Quadrilateral Elements 89
4.8 Test Cases 90
4.8.1 Double Wedge Supersonic Inviscid Flow, M = 3.0 90
∞
4.8.2 NACA0012 Aerofoil 93
4.8.2.1 Inviscid Transonic Flow, M = 0.75, α = 1.25o 93
∞
4.8.2.2 Inviscid Transonic Flow, M = 0.95, α = 0.0o 96
∞
4.8.2.3 Inviscid Supersonic Flow, M = 1.2, α = 10.0o 100
∞
4.8.3 RAE2822 Aerofoil, Transonic Viscous Flow:
M = 0.730, α = 1.25o , Re = 6.5 x 106 102
∞
4.8.4 L1T2 Multi-Element Aerofoil, Subsonic Viscous Flow:
M = 0.197, α = 4.01o , Re = 3.52 x 106 110
∞
4.8.5 Wind Tunnel With Forward Facing Step, Inviscid Supersonic Flow:
M = 3.0 115
∞
4.9 Conclusions 120

5 Feature-Aligned Hex-Dominant Mesh Block Insertion: Part I: Shock Waves 121
5.1 Introduction 121
5.2 Generating Shock-Aligned Surface Meshes 122
5.3 Volume Mesh Generation 124
5.3.1 Generating Shock-Aligned Mesh Blocks 125
5.3.1.1 Summary of Process 134
5.4 Test Cases 135
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5.4.1 ONERA-M6 Wing, Viscous Transonic Flow:
M = 0.8395, α = 3.06o , Re = 11.72 x 106 135
∞
5.4.2 DLR-F4, Viscous Transonic Flow:
M = 0.75, α = 2.0o , Re = 3.0 x 106 145
∞
5.5 Conclusions 148

6 Feature-Aligned Hex-Dominant Mesh Block Insertion: Part II: Wing Tip
Vortices and Wakes 149
6.1 Introduction 149
6.2 Surface Mesh Extrusion 150
6.3 Defining Source Surfaces 150
6.3.1 Wing Tip Vortices 150
6.3.2 Trailing Wakes 151
6.4 Determining the Extrusion Direction 152
6.5 Mesh Block Growth and Expansion 153
6.6 Merging of Mesh Block into Hybrid Volume Mesh 155
6.7 Test Cases 156
6.7.1 ONERA-M6 Wing, Viscous Transonic Flow:
M = 0.8395, α = 3.06o , Re = 11.72 x 106 156
∞
6.7.1.1 Wing Tip Vortex Extrusion 156
6.7.1.2 Wing Tip Vortex and Wake Extrusion 159
6.7.2 DLR-F4, Viscous Transonic Flow:
M = 0.75, α = 2.0, Re = 3.0 x 106 163
∞
6.7.3 Delta Wing, Viscous Transonic Flow:
M = 0.75, α = 15.0, Re = 2.0 x 105 167
∞
6.8 Conclusions 174

7 Feature-Aligned Hex-Dominant Mesh Block Insertion: Part III: A
Comparison with Anisotropic Adaptation Techniques 175
7.1 Introduction 175
7.2 Test Cases 175
7.2.1 Delta Wing, Viscous Transonic Flow:
viii
M = 0.75, α = 15.0, Re = 2.0 x 105 175
∞
7.2.2 ONERA-M6 Wing, Viscous Transonic Flow:
M = 0.8395, α = 3.06o , Re = 11.72 x 106 178
∞
7.2.3 DLR-F4, Viscous Transonic Flow:
M = 0.75, α = 2.0, Re = 3.0 x 106 183
∞
7.3 A General Comment on the Reliability of Results 187
7.4 A Discussion on Feature-Alignment and Adaptation Techniques 187
7.5 Conclusions 189

8 Conclusions 190
8.1 Achievements and Concluding Remarks 190
8.2 General Concluding Remarks 192
8.3 Suggestions for Future Work 192

List of Published/Submitted Papers 195

REFERENCES 196

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Nomenclature
Latin Letters
Symbol Quantity
A Area of face i
i
dS Surface boundary
c Gas specific heat at constant pressure
p
c Gas specific heat at constant volume
v
C Drag coefficient
D
C Skin friction coefficient
f
C Lift coefficient
L
C Surface pressure coefficient
P
D Dt Material derivative,  tu  u
i i
e Total energy per unit mass
E Total energy
F Inviscid flux vector
F Inviscid flux vector on face i
i
G Viscous flux vector
G Viscous flux vector on face i
i
h Total enthalpy per unit mass or thickness
Hessian matrix

H Total enthalpy
i Cell or face notation
 Turbulent kinetic energy (per unit mass) or thermal conductivity
k Stiffness of the spring connecting nodes i, j
ij
𝑀𝑛 Normal Mach number
M Freestream Mach number

n Outward unit normal vector
ni Unit normal vector of face i
x