An Implicit-Explicit Flow Solver for Complex Unsteady Flows

An Implicit-Explicit Flow Solver for Complex Unsteady FlowsFLOWS
ASTRONAUTICAL ENGINEERING
OF STANFORD UNIVERSITY
FOR THE DEGREE OF
All Rights Reserved
ii
I certify that I have read this dissertation and that, in my opinion, it is fully
adequate in scope and quality as a dissertation for the degree of Doctor of
Philosophy.
Antony Jameson Principal Adviser
Philosophy.
Robert MacCormack
Philosophy.
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iv
Abstract
Current calculations of complex unsteady flows are prohibitively expensive for use in real
engineering applications. Typical flow solvers for unsteady integration employ a fully im-
plicit time stepping scheme, in which the equations are solved by an inner iteration. In order
to achieve convergence within each physical time step, a substantial number of pseudo-time
steps (typically between 30-100, depending on the case) are required. Another unfavorable
characteristic of the dual time stepping method is that there are no available error esti-
mates for time accuracy available unless the inner iterations are fully converged, although
numerical experiments have demonstrated second order accuracy in time.
The approach in this thesis is to construct hybrid type schemes by combining implicit
and explicit schemes in a manner that guarantees second order accuracy in time. An initial
time accurate ADI step is introduced, followed by a small number of cycles of the dual-time
stepping scheme augmented by multigrid. The formal second order accuracy in time should
be retained without the need for large numbers of inner iterations. The number of inner
iterations required for convergence can thus be reduced while maintaining the same overall
error levels.
To investigate the effectiveness of the proposed scheme, several pitching airfoil test cases
were examined, offering a close look at possible reductions in computational cost by adopting
the present approach.
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Acknowledgments
I am grateful to all the people who have supported me during my graduate studies at
Stanford. Without their help, this research would never have been possible.
First, I would like to give my greatest appreciation to my advisor, Professor Antony
Jameson. Not only has he been open-minded to my ideas, very patient with my progress,
financially supportive to my research, his understanding to my struggles allowed me to
continue this research. His profound teachings and his re-known accomplishments definitely
drove my research to its end. In addition, I would like to thank my co-advisor Professor
Robert W. MacCormack whose mentorship helped my investigations in this field. Not
only did Professor MacCormack and Professor Jameson contributed, in every facet, to the
scientific content of my research and studies, they made me feel like I was part of a family
while at Stanford.
I would like to extend my, my most sincerest thanks to Professor Juan J. Alonso, and
Professor Thomas Pulliam. They have encouraged me throughout the entire process, were
model figures for me to accelerate. I thank them for their participation in my thesis com-
mittee.
I owe a special thanks to a select group of individuals: Peggy Huang, Peter Sturdza,
Joaquim Martins, Kihwan Lee, Nawee Butsuntorn and everyone in the Aerospace Com-
puting Laboratory. These people reinforced the importance of friendship in my life and
provided me with many fond memories during my stay at Stanford.
Finally, I would like to dedicate this work to my parents. I would be lost without their
unconditional acceptance of who I am.
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Contents
1.2 Background of CFD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Stability Analysis of Numerical Schemes . . . . . . . . . . . . . . . . . . . . 3
1.4 Levels of Approximations for Flow Simulations . . . . . . . . . . . . . . . . 5
1.5 Steady Versus Unsteady Fluid Flows Characteristics . . . . . . . . . . . . . 5
1.6 Current Unsteady Flow Solvers . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.7 Stanford’s ASCI Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.8 Objectives Of This Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.9 Preliminary Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.11 CFD Programming Style and Efficiency Considerations . . . . . . . . . . . 10
2 Governing Equations 12
2.2 Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Conservation of Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Closure relationships for Navier-Stoke’s Equation . . . . . . . . . . . . . . . 16
2.6 Navier-Stokes Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.7 Coordinate Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.7.2 Spatial Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.8 Transformed Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 Spatial Discretization 23
3.2 Inviscid Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Artificial Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4 JST Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.5 CUSP-Type Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4 Explicit Schemes 35
5 Implicit Schemes 37
5.2 Linearized Implicit Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.2.1 Numerical Dissipations for Implicit Operators . . . . . . . . . . . . . 38
5.3 Linearized Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.4 Alternate Direction Implicit (ADI) and ADI Scheme with 3-4-1 Backward
Difference Formula (ADI-BDF) . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.6 DDADI with BDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.7 Hybrid Implicit-Explicit Scheme . . . . . . . . . . . . . . . . . . . . . . . . 43
5.8 Iterative Dual-Time Stepping Scheme . . . . . . . . . . . . . . . . . . . . . 45
5.8.1 Explicit Unsteady Dual Time Stepping . . . . . . . . . . . . . . . . . 46
5.8.2 Implicit Unsteady Dual Time Stepping . . . . . . . . . . . . . . . . . 47
5.9 Point Implicit Treatment of Time Difference Source Term . . . . . . . . . . 48
5.9.1 Point Implicit Treatment of Source Term for Multi-stage Schemes . 48
5.9.2 Point Implicit Treatment for Implicit Pseudo-Time Iterations . . . . 49
5.10 Matrix Representation of ADI-BDF . . . . . . . . . . . . . . . . . . . . . . 50
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6.2 Internal Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.2.1 C-mesh Wake Implicit Boundary Implementation . . . . . . . . . . . 55
6.2.2 Implicit Periodic Boundary Condition for the O-mesh . . . . . . . . 57
6.3 Solid Wall Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 57
6.3.1 Inviscid Unsteady Wall Boundary Condition for Explicit Operators . 57
6.3.2 Inviscid Unsteady Wall Boundary Condition for Implicit Operators . 59
6.3.3 Viscous Flow Boundary Condition . . . . . . . . . . . . . . . . . . . 63
7 Convergence Acceleration Techniques 64
7.1 Implicit Residual Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
7.2 Multigrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
7.3 Variable Local Time Stepping . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7.3.1 Determining Time Step Size . . . . . . . . . . . . . . . . . . . . . . . 69
7.3.2 Calculating the Pseudo-Time Step Size . . . . . . . . . . . . . . . . 70
8 Accuracy Analysis 72
8.2 Accuracy of the ADI-BDF Scheme . . . . . . . . . . . . . . . . . . . . . . . 73
8.3 Accuracy of the DDADI-BDF Scheme . . . . . . . . . . . . . . . . . . . . . 74
9 Stability Analysis 75
9.1.1 General Form of Explicit and Implicit Stability Analysis . . . . . . . 75
9.1.2 Equivalence of the implicit and explicit schemes . . . . . . . . . . . . 77
9.2 Advantages of 3-4-1 BDF over Trapezoidal Rule . . . . . . . . . . . . . . . . 78
9.3 Choice of Model Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
9.3.1 Jameson’s MultiStage Scheme . . . . . . . . . . . . . . . . . . . . . . 80
9.3.2 ADI Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
9.3.3 ADI-BDF Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
9.3.5 DDADI with BDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
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10.2 A Note on Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . 105
10.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
10.5 Overview of the Hybrid Scheme for Unsteady Flow . . . . . . . . . . . . . . 108
10.6 Alternative Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
10.7.1 Inviscid Unsteady AGARD 702 Test Case 3 CT1 . . . . . . . . . . . 111
10.7.2 Inviscid Unsteady AGARD 702 Test Case 2 CT6 . . . . . . . . . . . 129
10.8 Viscous Unsteady Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
10.8.1 Viscous Unsteady AGARD 702 Test Case 3 CT1 . . . . . . . . . . . 141
10.8.2 Viscous Unsteady AGARD 702 Test Case 2 CT6 . . . . . . . . . . . 147
10.8.3 Viscous Unsteady NASA TP 1100 . . . . . . . . . . . . . . . . . . . 156
11 Conclusion 158
C CUSP Scheme Derivation 167
C.1 E-CUSP Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
C.2 H-CUSP Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
D.1 Two Dimensional Inviscid Euler Jacobian in UFLO82 . . . . . . . . . . . . 174
E Implicit Operator for Navier-Stokes Equations 176
E.1 Viscous 2D Jacobians Derivation . . . . . . . . . . . . . . . . . . . . . . . . 176
E.1.1 Derivation for Spatial Derivatives . . . . . . . . . . . . . . . . . . . . 177
F Inviscid NACA0012 CT1 Test Case 182
F.1 CT1 Inviscid O-mesh Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
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G.1 CT6 Inviscid O-mesh Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
H Viscous NACA0012 CT1 Test Case 190
H.1 CT1 Viscous C-mesh Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
I Viscous NACA64A010 CT6 Test Case 193
I.1 CT6 Viscous C-mesh Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
Bibliography 196
10.1 Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
10.2 Combination Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
10.3 CFL for Different Time Step Sizes . . . . . . . . . . . . . . . . . . . . . . . 111
10.4 CFL for Different Time Step Sizes . . . . . . . . . . . . . . . . . . . . . . . 129
10.5 Meshes used for viscous unsteady calculations for the CT1 test case. . . . . 141
10.6 Meshes used for viscous unsteady calculations for the CT6 test case. . . . . 147
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3.1 Fluxes through a cell in the computational domain . . . . . . . . . . . . . . 25
6.1 Perform matrix inversion along the strips across the wake. . . . . . . . . . . 56
6.2 Explicit unsteady Euler boundary condition . . . . . . . . . . . . . . . . . . 58
6.3 explicit unsteady Euler boundary condition . . . . . . . . . . . . . . . . . . 59
6.4 indices at a solid wall boundary . . . . . . . . . . . . . . . . . . . . . . . . . 61
9.1 Stability Region of 341 Backward Difference Formula (BDF) . . . . . . . . . 79
9.2 Stability Region of Trapezoidal Rule . . . . . . . . . . . . . . . . . . . . . . 79
9.3 5 Stage RK scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
9.4 5 Stage RK scheme with Point Implicit Treatment of Time Difference. . . . 81
9.5 5 Stage RK scheme with Separate Evaluation of Convection and Dissipation
Terms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
9.6 5 Stage RK scheme with Point Implicit Treatment of Time Difference, Sep-
arate Evaluation of Convection and Dissipation Terms. . . . . . . . . . . . . 82
9.7 Stability region of RK scheme for steady state case, λt = 0. . . . . . . . . . 82
9.8 RK scheme for steady state case, contour plot of amplification vs. modified
frequencies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
9.9 Stability region of RK scheme without point implicit treatment, λt = 0.5. . 83
9.10 Stability region of RK scheme with point implicit treatment, λt = 0.5. . . . 83
9.11 Stability region of RK scheme with point implicit treatment and tweaked
stage coefficients. λt = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
9.12 RK scheme without point implicit treatment at different ωt, for λt = 0.5. . . 83
9.13 RK scheme with point implicit treatment at different ωt, for λt = 0.5. . . . 84
9.14 Amplification factor of RK scheme, point implicit treatment and tweaked
stage coefficients. λt = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
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9.15 λx = λy = 4, λt = 0, unsmoothed residuals . . . . . . . . . . . . . . . . . . . 88
9.16 λx = λy = 4, λt = 0, smoothed residuals (after ADI), εx = εy = 0.1 . . . . . 88
9.18 λx = λy = 4, λt = 0, smoothed residuals (after ADI), εx = εy = 0.14 . . . . 89
9.20 λx = λy = 4, λt = 0, smoothed residuals (after ADI-BDF step), εx = εy =
0.05 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
9.21 λx = λy = 10, λt = 0, unsmoothed residuals . . . . . . . . . . . . . . . . . . 94
0.05 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
0.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
0.05 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
0.05 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
0.05 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
0.05 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
0.05 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
0.05 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
xiv
0.05 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
0.05 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
10.1 CFL distribution on the airfoil surface for PSTEP = 9 . . . . . . . . . . . 112
10.2 CP distribution on the airfoil surface with CFL at T.E. of about 26, 230. . . 112
10.3 CL versus α for PSTEP = 36 and NCY C = 100 . . . . . . . . . . . . . . . 113
10.4 CM versus α for PSTEP = 36 and NCY C = 100 . . . . . . . . . . . . . . 113
10.5 CD versus α for PSTEP = 36 and NCY C = 100 . . . . . . . . . . . . . . . 113
10.6 CL versus time step for PSTEP = 36 and NCY C = 100 . . . . . . . . . . 113
10.7 CL versus oscillation period for fixed α with PSTEP = 36 and NCY C = 100 114
10.8 CL versus oscillation period for fixed α with PSTEP = 36 and NCY C = 100 114
10.9 CD versus oscillation period for fixed α with PSTEP = 36 and NCY C = 100114
10.10CM versus oscillation period for fixed α with PSTEP = 36 and NCY C = 100114
10.11CL versus oscillation period for fixed α with PSTEP = 36 and NCY C = 2 115
10.13CD versus oscillation period for fixed α with PSTEP = 36 and NCY C = 2 115
10.15convergence history during an inner iteration step NCY C = 100 . . . . . . 116
10.16CL history at each time step . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
10.17CD history at each time step . . . . . . . . . . . . . . . . . . . . . . . . . . 116
10.18CM history at each time step . . . . . . . . . . . . . . . . . . . . . . . . . . 116
10.19CD history of hybrid scheme at each time step . . . . . . . . . . . . . . . . 117
10.20CM history of hybrid scheme at each time step . . . . . . . . . . . . . . . . 117
10.21CD history of DTSS at each time step . . . . . . . . . . . . . . . . . . . . . 118
10.22CM history of DTSS at each time step . . . . . . . . . . . . . . . . . . . . . 118
10.23Averaged absolute CL error through one pitching cycle for hybrid and DTSS
multigrid scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
10.24Averaged absolute CM error through one pitching cycle for hybrid and DTSS
10.25RMS of pressure error in entire flow field for ADI-BDF without inner iterations.120
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10.26CL percentage error in entire flow field for ADI-BDF without inner iterations. 120
10.27RMS of pressure error in entire flow field for hybrid and DTSS multigrid
scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
10.28CL percentage error through one pitching cycle for hybrid and DTSS multi-
grid scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
10.29RMS of pressure error in entire flow field for ADI-BDF and DTSS multigrid
scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
10.30CL percentage error through one pitching cycle for ADI-BDF and DTSS
10.31Comparison of CL versus α of Different grid sizes for PSTEP = 36 and
NCY C = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
NCY C = 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
NCY C = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
NCY C = 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
10.35Comparison of CL error of Different PSTEP for the Inviscid CT1: (a)
PSTEP = 9, (b) PSTEP = 18, (c) PSTEP = 36 and (d) PSTEP = 72 . 126
10.36Comparison of CL error of Different NCY C for the Inviscid CT1: (a) NCY C =
1, (b) NCY C = 2, (c) NCY C = 3 and (d) NCY C = 4 . . . . . . . . . . . 128
10.37CFL distribution over NACA64A010 airfoil with leading edge at x = 0 . . . 129
10.38160× 32 O-mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
10.39Comparison of CP Distributions of Different Numerical Schemes for the In-
viscid Case for PSTEP = 72 . . . . . . . . . . . . . . . . . . . . . . . . . . 130
10.40Comparison of CP Distributions of Different Numerical Schemes for the In-
viscid Case for PSTEP = 216 . . . . . . . . . . . . . . . . . . . . . . . . . 130
10.41CL versus α of the Hybrid Scheme . . . . . . . . . . . . . . . . . . . . . . . 131
10.42CL versus time of the Hybrid Scheme . . . . . . . . . . . . . . . . . . . . . 131
10.43CD versus α of the Hybrid Scheme . . . . . . . . . . . . . . . . . . . . . . . 131
10.44CM versus α of the Hybrid Scheme . . . . . . . . . . . . . . . . . . . . . . . 131
xvi
10.47CD versus oscillation period for fixed α with PSTEP = 36 and NCY C = 100132
10.48CM versus oscillation period for fixed α with PSTEP = 36 and NCY C = 100132
10.51CD versus oscillation period for fixed α with PSTEP = 36 and NCY C = 2 133
10.53Averaged CL Error vs. NCY C for PSTEP = 9 . . . . . . . . . . . . . . . 134
10.56Averaged CL Error vs. NCY C for PSTEP = 216 . . . . . . . . . . . . . . 135
10.57Comparison of CL versus α of Different Numerical Schemes for the Inviscid
Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
10.58Comparison of CL versus α of the Hybrid Scheme with Different Number of
Inner Iterations for the Inviscid Case . . . . . . . . . . . . . . . . . . . . . . 136
10.59Pressure Distribution on the NACA64A010 Airfoil at T = 576, α = 0.0 and
M = 0.796 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
10.60Pressure Distribution on the NACA64A010 Airfoil at T = 619, α = −0.883
and M = 0.796 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
and M = 0.796 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
M = 0.796 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
10.63Comparison of CL versus α of the Hybrid Schemes for Different Grid Sizes . 138
10.64Comparison of CM versus α of the Hybrid Scheme with Different Grid Sizes 138
Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
10.69Averaged CL Error vs. NCY C for various PSTEP s . . . . . . . . . . . . . 140
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10.70Averaged Total Pressure Error vs. NCY C for various PSTEP s . . . . . . 140
10.71CFL distribution over airfoil surface for 192× 64 c-mesh . . . . . . . . . . . 141
10.73CL versus α for Hybrid scheme. . . . . . . . . . . . . . . . . . . . . . . . . 142
10.74CM versus α for Hybrid scheme. . . . . . . . . . . . . . . . . . . . . . . . . 142
10.75CD versus α for Hybrid scheme. . . . . . . . . . . . . . . . . . . . . . . . . 142
10.76RTRMS versus STEP for Hybrid scheme. . . . . . . . . . . . . . . . . . . 142
10.77CL history for fixed α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
10.79CD history for fixed α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
10.80CM history for fixed α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
10.83CD history for fixed α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
10.85 CL Convergence history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
10.86 CM Convergence history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
10.87CD Convergence history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
10.91CFL distribution over airfoil surface for 192× 64 c-mesh . . . . . . . . . . . 147
10.93CL versus α for hybrid scheme and DTSS . . . . . . . . . . . . . . . . . . . 148
10.94CM versus α for hybrid scheme and DTSS . . . . . . . . . . . . . . . . . . . 148
10.99CL Convergence history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
xviii
M = 0.796 for the Viscous Calculation with the Hybrid Scheme . . . . . . . 152
M = 0.796 for the Viscous Calculation . . . . . . . . . . . . . . . . . . . . . 152
10.105Demonstration of Temporal Accuracy for the Viscous Case . . . . . . . . . 153
M = 0.796 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
and M = 0.796 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
and M = 0.796 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
M = 0.796 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
10.110Comparison of CL versus α of Different Numerical Schemes for the Viscous
Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Inner Iterations for the Viscous Case . . . . . . . . . . . . . . . . . . . . . . 155
10.112Mach Contour of Dynamic Stall/Vortex Shedding Frame 1 . . . . . . . . . . 156
10.113Mach Contour of Dynamic Stall/Vortex Shedding Frame 10 . . . . . . . . . 156
10.118Several Force Coefficients History . . . . . . . . . . . . . . . . . . . . . . . . 157
E.1 two dimensional compact stencil for viscous calculation. . . . . . . . . . . . 177
F.1 O-mesh Grid 160X32 Cells Inviscid Calculation . . . . . . . . . . . . . . . . 182
G.1 O-mesh Grid 160X32 Cells Inviscid Calculation . . . . . . . . . . . . . . . . 185
xix
G.4 CL of CT6 072 1 1 100 ADIBDF fv2=1.3 4.0 . . . . . . . . . . . . . . . . . 188
G.5 CM of CT6 072 1 1 100 ADIBDF fv2=1.3 4.0 . . . . . . . . . . . . . . . . 188
G.10 CL of CT6 540 1 1 100 ADIBDF fv2=0.48 1.0 . . . . . . . . . . . . . . . . 189
H.1 C-mesh Grid 192× 64 Cells Viscous Calculation . . . . . . . . . . . . . . . 190
I.1 C-mesh Grid 192× 64 Cells Viscous Calculation . . . . . . . . . . . . . . . 193
xx
1.1 Practical Considerations and Motivations
Computational fluid dynamics (CFD) has been a field of intense research in the past twenty
years. CFD can be defined as a computer simulation of the flow of any fluid or gas. With
the advent of today’s computer technologies in terms of speed and data storage capacity,
not only have CFD increased its efficiency and accuracy, but the goal of CFD has expanded.
In the beginning of the computational era, CFD was restricted to only simplified versions of
the conservation laws, e.g. potential equations and small disturbance theory utilized CFD as
a tool. Today, with the exponential growth of the computing power in computer processors
and the desire to simulate more complicated phenomenon in fluid flows, CFD algorithms
are continuously improving so as to achieve greater capabilities, efficiency and robustness.
The author of this dissertation believes that presently, numerical algorithms for steady state
Euler calculations have been developed near the crux of their maturity. Some of the possible
short term goals for further improvements of CFD lie in the areas including: addition of the
viscous terms and/or turbulence modeling, grid generation techniques, parallel computation
techniques due to the advent of parallel computers and unsteady simulations.
Despite the ever developing relationship between physics of fluid and computer data
processing, there are a few essential breakdowns of fundamentals behind the operations of
CFD that have remained applicable. First, a mathematically well-posed problem must be
defined if the solution is expected to make any sense. Second, a grid must be generated along
with the proper boundary conditions defined. Third, a discrete formula constructed from
the laws of physics must be applied to each data point. Lastly, the overall flow solution is
1
obtained asymptotically by repeatedly running some preferred numerical algorithm -making
educated guesses of the flow field at each grid point based on the discretized equations. All
of these areas have been subjects of intense research in the past two and a half decades as
they are interrelated and critical to the performance of CFD as a design tool for industrial
applications.
Depending on the ability of the numerical algorithm, CFD researchers strive to reduce
CPU hour requirement for obtaining a reasonable solution of the flow field. During the
1980’s, the introduction and efficient implementation of concepts such as Total Variation
Diminishing (TVD) Scheme, multigrid and local preconditioning, have resulted in major
gains in efficiency over a wide range of numerical disciplines. For CFD in particular, the
CPU requirement for steady state Euler calculations have been reduced significantly. For
example, one can obtain the steady state solution about a two dimensional inviscid tran-
sonic airfoil in a matter of seconds on a personal computer. A three dimensional inviscid
transonic flow solution around an entire wing body configuration can be obtained in a mat-
ter of minutes on the same processor. Unsteady Euler calculations have been able to take
advantage of this improvement through techniques such as Dual Time Stepping. As a result,
phenomenon such as flutter and dynamic stall prediction for transonic airfoils can be easily
simulated on a personal computer.
In the aeronautical engineering design process, dynamics of unsteady fluid behavior can
lead to unforeseen instabilities in gas or fluid flows. The instability coupled with elastic
properties of the structure can trigger undesirable or even catastrophic behaviors in the
flight vehicle. The ability to predict and avoid unwanted events such as rotating stalls in
jet engines, buffeting or wing flutter should all depend on the ability of the flow solver to
simulate unsteady flows with sufficient resolution and accuracy. Unsteady flow simulation
has been historically challenging to the computational communities. The consideration of
time dependence adds an extra dimension to the already taxing problem of computational
fluid simulations. With the rapid advent of algorithms and computer hardware, the two
subjects must grow in congruency to best achieve and utilize the numerical tools available
today. As this thesis sets out to investigate the algorithmic side of the topic concerning
numerical simulations, a more efficient and robust unsteady flow solver is proposed and
analyzed in the subsequent text.
1.2. BACKGROUND OF CFD 3
1.2 Background of CFD
Computational fluid dynamics (CFD) has become a required field of study for all aero-
nautical engineers. CFD is a numerical analysis of fluid or gaseous flows. In order to
avoid expensive and time consuming wind tunnel experiments, aeronautical engineers need
efficient CFD algorithms to predict steady or unsteady flows about any aerodynamic body.
1.3 Stability Analysis of Numerical Schemes
One of the most important concepts in practical numerical analysis is the CFL number, first
introduced by Courant, Frederichs and Lewy. It was shown that the physics of the govern-
ing equations can be overshadowed by the instabilities of a numerical algorithm. Not all
consistent numerical algorithms are able to yield reasonable solutions that approximate the
exact solution of the governing Ordinary Differential Equation (ODE) or Partial Differential
Equation (PDE) with arbitrary degree of accuracy.
First, the differential equations and the representative difference equations have zonal
dependencies due to the fact the fluid properties in the flow field such as density, pres-
sure, convection velocities and viscosity travel at finite speeds (characteristic speeds). The
numerical method must be carefully constructed to reflect the true zones of dependence
for each characteristic speed. The CFL condition states that the zone of dependence of
the numerical scheme must contain the zone of dependence of the flow. Obviously, if this
statement is not satisfied, no realistic flow solution can come of the numerical computation.
Upwinding is done by taking all the information necessary for evolving the solution
from upstream of the flow. This is in disregard of the fact that some of the flow properties
such as pressure in subsonic flow propagates in all directions rather than simply in the
direction of the fluid flow. For supersonic flows, using flow variables from upstream of the
flow leads to the correct zone of dependence. For transonic and subsonic flows, the term
upstream depends on the characteristic speed of the flow variable in question. Upwinding
of all flow variables in transonic flow provides stable results but leads to over-smoothing of
flow features. To prevent the problem of over-smoothing of the solution, partial upwinding
can be used. Some of the methods of partial upwinding are flux difference/vector splitting
and the addition of artificial dissipation. These approaches appear to be quite different at
first, but they can be shown to be rather similar.
Some successful numerical schemes for fluid/gas dynamics require active modification
4 CHAPTER 1. INTRODUCTION
of the difference method depending on the local flow characteristics. Implicit flow solvers
are typically accompanied by some kind of characteristic splitting of the flux Jacobians. By
choosing the zone of dependence from the numerical scheme to match that of the governing
equation, shocks and discontinuities are solved in the numerical solution. More recently,
artificial dissipation were used to indirectly pick the correct zone of dependence. For exam-
ple, Jameson’s JST scheme[?] reduces the central differencing in space to upwinding near
regions of strong flow gradients. Jameson’s explicit scheme uses a compact stencil and re-
quires low computer storage needs. Explicit schemes tend to be locally optimized and they
disregard anything that is too ’far’ away from a meshing perspective. Implicit schemes, on
the other hand, seek to take into consideration the values of the majority of the domain of
computation before updating each individual computational cell. This approach results in
dramatically increased computational cost for each update, but has the potential of taking
larger steps in time rather than an explicit scheme due to higher stability margins.
The most successful convergence acceleration technique during the present study is un-
doubtedly the multigrid method. Extending the numerical domain of an explicit scheme
through the use of coarser grids results in phenomenal convergence improvements. The
multigrid approach gives explicit schemes the advantage of increased time step size previ-
ously possible only to implicit schemes, while keeping the algorithm compact and efficient.
Multigrid has also been applied to implicit schemes, and has been demonstrated to have
similar improvements in performance of the implicit algorithms.
In general, implicit schemes are better behaved than explicit schemes while implicit
schemes consume more computational time and power. The advantages of implicit over
explicit schemes for stiff problems are largely motivated by the ability of the scheme to
handle larger CFL numbers; leading to reduced number of iterations. For most of the time,
the fluid dynamicist would choose the computational tool based on the flow physics, and
the computer hardware resources available. It would seem that a switched implicit/explicit
scheme can try and exploit the advantages of both worlds. To do so, one must first un-
derstand the advantages and disadvantages of explicit and implicit schemes. To perform a
study on the differences, a few sample explicit methods and implicit methods were consid-
ered. Numerical experiments were performed for the selected schemes. While the example
schemes used in this study may not be deemed the most ideal representation of explicit or
implicit numerical scheme for the study, an attempt is made to compare the fundamental
differences between the leading categories of schemes.
1.4. LEVELS OF APPROXIMATIONS FOR FLOW SIMULATIONS 5
1.4 Levels of Approximations for Flow Simulations
When CFD is used as a design tool for calculating flows, the degree of accuracy desired
varies depending on the specific application. The details of the flow are not always computed
exactly. Below is a list of the common approximations applied to the flow equations:
1. Small Disturbance Theory,
2. Potential Flow Calculation,
4. Thin Layer Navier Stokes Equations (TLNS),
5. Reynolds Averaged Navier Stokes Equations (RANS), and
6. Direct Numerical Simulation (DNS).
The scope of analysis in this work ranges from inviscid Euler’s equations up to fully
viscous Reynolds averaged Navier Stokes equations.
1.5 Steady Versus Unsteady Fluid Flows Characteristics
Fluid flows for the majority of aerodynamics engineering applications are unsteady due to
the inherent instabilities of the flow. Turbulence with increasing Reynolds number, shear
layer, vortex shedding, flow separation and shock-wave boundary layer interactions are some
of the examples of the instabilities of the fluid flow. It is important to study unsteady flows
since an unsteady solution can have drastically different characteristics when compared with
its quasi-steady counterpart. The non-trivial gradients of the state variables with respect
to time cannot be ignored because the transient solutions carry important information such
as the hysteresis effects.
The complexity behind unsteady flows can often be captured by a good numerical flow
solver. Several papers on studies of wing flutter, for example, were presented with experi-
mental validation of the numerical solvers proposed [60]. The accuracy of flow solvers is vital
to obtaining realistic flow solutions. Typical A-stable and O(t2) accurate flow solvers are
capable of resolving most of the unsteady effects. Due to limited computational power and
due to non-ideal convergence rates of numerical algorithms, the accuracy of flow solution is
often compromised to achieve reasonable computation times. Ideally, iterating the residual
errors to machine zero at every time step of an unsteady calculation will be preferred if no
time constraints were imposed. In reality however, in order to save computational costs,
only enough iterations are performed to obtain an approximate ensemble average of the
flow that captures the key components of physics.
Methods of unsteady flow analysis also depend on the time scales of the unsteady flow
structures and the amount of computational power allotted. Generally, in order to save
CPU hours, if the influence of the unsteady portion of the flow is not significant, then an
approximation technique is used to generalize the unsteady effect into steady flow. For
example, in designing an airplane’s cruise conditions, turbulence within the boundary lay-
ers is usually generalized by an ensemble averaging technique such as turbulence modeling.
Turbulence modeling is used to avoid the large number of mesh points needed near regions
of high gradients in the flow solution. Resolution of turbulent flow within the boundary
layer is usually not necessary from the perspective of an engineer. A turbulence model that
predicts the flow separation is adequate in most engineering applications. Although it is
doubtful whether a universally valid turbulence model can be constructed [52], the turbu-
lence modeling techniques may or may never reach a stage where a robust closure model
can be developed. The alternative to turbulence modeling is to perform direct numerical
simulations in order to resolve the details of the flow within the boundary layer. Given
that the time scales of the flow structures being simulated is not unreasonably small, the
unsteady time accurate flow solver should reproduce predominant characteristics of the flow
field.
1.6 Current Unsteady Flow Solvers
Typical flow solvers for unsteady integration employ a fully implicit time stepping scheme
in which the equations are solved by inner iterations. Some of the issues that need to be
addressed in the development of an effective unsteady flow solver are:
1. consistency, stability, convergence criteria,
2. spatial and temporal discretization,
3. conservativeness of the schemes,
4. oscillation control: LED/ELED, ENO/WENO, TVD,
1.7. STANFORD’S ASCI PROJECT 7
5. implicit and explicit schemes,
6. convergence acceleration (residual averaging, multigrid, etc.),
7. parallel computing, and
8. turbulence modeling.
Despite all the advances that have been made in increasing the efficiency of numerical
algorithms for CFD, computation time required remains a major roadblock for making
CFD a cost-efficient tool for aeronautical design.
1.7 Stanford’s ASCI Project
The ultimate goal of the ASCI project is to fully resolve the unsteady flow inside a jet turbine
engine using the most advanced CFD simulation tools available on a platform consisting of
hundreds of thousands of CPUs working in parallel. A key objective is to test the envelope
of CFD using the state of the art hardware architecture. Since CFD’s best performance
depends heavily on the type of unsteady flow solver used, it is necessary that a robust
and efficient numerical algorithm be implemented so as to speed up the design process.
The ASCI turbomachinery unsteady flow solver TFLO uses Jameson’ Dual-Time-Stepping-
Scheme with Multistage Runge-Kutta (RK) time integration, implicit residual averaging,
variable local time stepping and multigrid convergence acceleration techniques. The inner
iterations advance to a steady state in fictitious time at every real time step.
In the ASCI Project, an attempt has been made to compute the unsteady flow within an
entire jet engine starting at the compressor through the combustion chamber and turbine,
and then exiting the exhaust nozzle. To capture the inherent unsteadiness and the physical
phenomenon that exist under such environments, from setting up the grid generation to
converging the numerical results, the entire process takes much longer than desired. In
one of the preliminary runs in the ASCI project, the unsteady flow through a complete
turbine with 9 blade rows is computed using a mesh with approximately ninety-four million
cells. Using the TFLO code, approximately twenty-five hundred time steps and two million
CPU hours are required to reach a stationary periodic state. Using five hundred and twelve
processors, the calculation requires approximately 8 months. With the present flow solver,
there are no available error bound estimates for accuracy of the flow solution unless the
inner iterations are fully converged. However, numerical experiments have demonstrated
second order accuracy in time with approximately twenty-five iterations per real time step.
The turn around time is too long, leaving design engineers the ability to run only a few
design cycles in a year.
1.8 Objectives Of This Dissertation
This dissertation is motivated by the clear need to solve the time-accurate Navier-Stokes
equations with greatly decreased computational cost compared to the state-of-the art solver
used in the ASCI project. An attempt is made to facilitate the unsteady calculations while
analyzing the differences between the various types of numerical schemes for the gas dy-
namics equations in two dimensions. The proposed new numerical method involves the
introduction of an initial Alternating Direction Implicit (ADI) step, guaranteeing second
order accuracy in time, followed by a small number of cycles of the dual time stepping
algorithm for reduction of the factorization and linearization error. The second-order accu-
racy in time should be retained with arbitrary (and hopefully fewer than typically required
for convergence of iterative methods) numbers of inner iterations following the initial ADI
step. Details of this new scheme will be presented along with examples that demonstrate
the second order accuracy and the convergence properties. Properties such as consistency,
stability, conservativeness and convergence are also considered.
1.9 Preliminary Considerations
The large time requirements for fully unsteady simulations in turbomachinery signify the
need to improve the efficiency of the numerical algorithms. Some approaches to be consid-
ered are:
1. To search for a more rapidly convergent inner iteration method; for this purpose, a Pre-
conditioned Symmetric Gauss-Seidel Relaxation Method is being studied [19,53,108],
2. To reformulate the scheme so that it yields sufficient accuracy without the need for
full convergence of the inner iterations,
3. To consider the alternative approach of representing the solution in the frequency
domain; studies of a nonlinear frequency domain approach are described by McMullen
1.10. CFD CODE DESIGN ASPECTS 9
et al [72], and
4. To increase the parallel efficiency of numerical algorithms across larger number of
interconnected memories and CPUs.
In this dissertation, the second approach is followed. The idea is to formulate a hybrid
scheme which introduces an initial linearized ADI step. When compared with commonly
used iterative schemes, the hybrid scheme does not need to iterate to full convergence at
each real time step to ensure 2nd order accuracy. However, if the ADI scheme is used
alone, the factorization error restricts the time step size for which sufficient stability and
accuracy can be attained. In addition, the three point backward time difference formula
is shown to be unconditionally unstable[84] without additional numerical dissipation. In
order to overcome these deficiencies in ADI-type schemes, a number of dual-time-stepping
iterations are added. The additional iteration reduces the errors due to both linearization
and factorization and thus increases the upper bound of the feasible time step of the ADI
scheme.
1.10 CFD Code Design Aspects
In general, the formulation of numerical algorithms includes a few main areas of study:
1. Derivation of the governing equations; this is generally adapted from one of the levels
of approximation from the previous section,
2. Construction of the grid/mesh; the present analysis uses O-meshes generated from
Karmen-Treftz transformation with stretched grid in the normal direction for inviscid
calculations. Hyperbolic mesh generators were used to create C-mesh for viscous
calculations,
3. Numerical discretization of the governing equations; the main efforts in writing a new
solver is consumed by this step.
4. Application of linear algebra; system of linearized difference equations can be solved
using conjugate gradient solvers. In the present approach, approximated factorization
is used to reduce the bandwidth of the implicit operator, resulting in inversion of block
tridiagonal and pentadiagonal systems,
in the present analysis, and
6. Software programming issues; briefly discussed the implementation issues.
1.11 CFD Programming Style and Efficiency Considerations
In the era of parallel programming and the ever changing architecture of computer hardware,
it is hard to decide what kind of software will last through more versions of compilers and
CPUs. Although the CFD software largely depends on the core technique implemented
in the flow solver, the programming methodology also plays a large role in the overall
performance and life expectancy of the code. While different factors to be considered drive
the software in different directions, the optimal balance of the main design issues are not
trivial. Sometimes, individual subroutines, if designed simply and precisely, may outlast
the main code.
There are several predominant issues for developing software, including data structure
of the solver for maximum operations speed and efficiency, clarity in arrangement of the
data structure, modularity of the subroutines, and platform dependence of the computer
code. Arrangement of the data structure is very important in terms of the resulting FLOPS.
Typical large arrays of the CFD code must be sequenced so that simultaneously accessed
elements are located closely in the hardware memory address for shortest data retrieval time
during the calculations. Examining the access of data by the CFD solver on average reveals
that in most cases, different state and geometric variables at adjacent nodes are accessed by
finite differencing operators simultaneously; thus, we would like to have the data grouped
in the hardware memory correspondingly. Access frequency of each set of variables need to
be considered in order to design the source code accordingly. The higher the frequency of
simultaneous access, (i.e. accessed in one pass of a loop), the closer the data should reside
in the memory. This usually leads to an ordering of the hierarchy of the flow variables:
1. different state and geometric variables,
2. adjacent mesh nodes,
4. multigrid levels.
1.11. CFD PROGRAMMING STYLE AND EFFICIENCY CONSIDERATIONS 11
Besides data storage, modularity of subroutines is also a key factor to consider in de-
signing a CFD code. Subroutines have modular dependency, i.e. they only operate on
certain variables and can be loaded when the corresponding variables are accessed. In a
single processor case, the most heavily used subroutines such as flux calculations, time
step calculations, prolongation and restriction, implicit Jacobian setup and implicit Jaco-
bian inversions should remain static in memory for shortest execution times. In practice,
modularizing subroutines are necessary only if the computer source code needs to be clearly
organized for auditing by more than one programmer. The source code organization remains
a lower priority than the speed and efficiency of the executable.
The decision on whether intermediate variables should be calculated on the fly or com-
puted in advance and stored in the computer memory is not so obvious. In order to un-
derstand the trade-off, the data size and hardware performance are some of the crucial
factors to consider. When the data size of the transient variables is large, storing the value
needed may increase the range of memory access and hence, dramatically deteriorate the
computational efficiency as the CPU must reload different memory pages into the cache
repeatedly. On the other hand, calculating the transient variables on the fly will require
a few extra FLOPS per element, resulting in increased calculation time by a factor of five
to fifty percent depending on the algorithm. If the system cache is large enough, then it is
desirable to store the needed variables. Otherwise, it is better to calculate them as needed.
Chapter 2
Governing Equations
The flow equations are a hyperbolic system of conservation laws. A brief derivation of the
governing equations is given in the following sections. In this dissertation, the bar notation
is used to denote vector variables while the double bar is used to denote matrices, i.e.
u = (ui)
x = (xi)
¯σ = (σij)
where i = 1, 2, ..N denoting N -dimensional space. In addition, we’ll let σj denote the jth
column of the matrix ¯σ.
2.1 Lagrangian and Eulerian Frames of Reference
In Lagrangian frame of reference, the control volume follows the fluid mass. In Eulerian
frame of reference, the control volume stays stationary in the inertial frame. The following
governing equations are derived under the Eulerian frame of reference; conservation of mass,
momentum and energy.
2.2 Conservation of Mass
Assuming the flow field of interest exists in the N -dimensional space then the coordinate
system is denoted by x ∈ <N and the flow velocity vector u (t, x) : < × <N → <N is a
12
2.3. CONSERVATION OF MOMENTUM 13
function of both time and space. Also, density of the continuous medium in question is
denoted by ρ (t, x) : <× <N → <.
In the Eulerian frame of reference, if we define a fixed control volume and take into
account the mass flow in and out of the control volume through its boundary denoted by
Γ, then ∫
where n is the outward surface unit normal vector.
∫
Γ ρur · ndΓ = 0 (2.2)
where ur = u− umesh is the convection velocity relative to the moving control volume.
The differential form of the equation of conservation of mass is obtained using the
Divergence Theorem, ∂ρ
2.3 Conservation of Momentum
The conservation of momentum can be derived in a similar manner to the derivation of the
conservation of mass. Some of the constants of the fluid properties are:
µ ≡ Coefficient of dynamic viscosity of fluid (= ρν)
λ ≡ Second viscosity coefficient (= −2
3 µ by Stoke’s relation.[6])
µv ≡ Bulk viscosity
where the bulk viscosity is zero by Stoke’s Hypothesis, except for acoustic motion and
interior of shocks; both phenomena are not addressed in the present analysis. This equation
also neglects body forces such as gravity or magnetism.
Again, balancing fluxes around a fixed control volume, the conservation of momentum
can be expressed as an integral equation,
∫
∫
14 CHAPTER 2. GOVERNING EQUATIONS
where ¯σ is the stress tensor and the dot product ¯σ · n can be written in tensor notation as
σijni summed over i following Einstein’s notation.
∫
∫
¯σ · ndΓ = 0 (2.5)
where ur = u− umesh is the convection velocity relative to the moving control volume.
The differential form of the conservation of momentum is then
∂ρu
For clarification, expanding the vector and matrix products for three dimensional flow
field into tensor notations,
∂x1 +
∂ρuu2
∂x2 +
∂ρuu3
∂x3 .
The components of the stress tensor ¯σ is defined in terms of velocity gradients,
σij = µ
+ λδij5 · u + µvδij5 · u
where the first subscript of σ denotes the direction normal to the plane that the force is
acting on, and the second subscript denotes the direction of the force.
The following is a matrix representation of the stress tensor in three dimensions.
¯σ =
2.4 Conservation of Energy
Some of the constants used in deriving the conservation of energy for fluid flow are
Pr ≡ Prandtl’s number
k ≡ Thermal conductivity = Cpµ
Pr .
Fourier’s Law of heat conduction states that the heat flux is proportional to local temper-
ature gradient according to the thermal conductivity coefficient
qx = −k ∂T
∂x . (2.7)
Balancing the heat flux around a fixed control volume, the conservation of energy can be
stated as follows:
∫
) · ndΓ = 0 (2.8)
where the dot product u · (¯σ · n) is defined as ujσijni summed over i and j. For a control
volume moving with velocity umesh
∫
∫
) · ndΓ = 0
where ur = u−umesh is the convection velocity relative to the moving control volume. Using
the divergence theorem, the conservation of energy can be cast into partial differential form,
∂ (ρE)
( 5 · ¯σ
u · ( 5 · ¯σ
Given the definition of total enthalpy,
H = E + p
∂ (ρE)
2.5 Closure relationships for Navier-Stoke’s Equation
A few more equations are needed to close the model. Other than conservation of mass,
momentum and energy, some description of the medium needs to be provided. For ideal
gas,
p = ρRT .
Given that the gas has a constant pressure specific heat of Cp = (
∂h ∂T
)
v , then the universal gas constant, R = Cp − Cv. The ratio of
specific heats is designated as γ,
γ = Cp
E = e + |u|2 2
.
In addition, we use Southerland’s law to approximate the dynamic viscosity coefficient,
µ = 1.461 · 10−6T
2.6 Navier-Stokes Equation
Combining the conservation laws and the properties of an ideal gas, the Navier-Stoke’s
equation can be written as a set of first order partial differential equation
∂w
∂t +
∂fi
∂xi +
∂fvi
w =
fi =
(2.12)
where δij is the Kronecker’s delta function. The viscous flux vectors are
fvi = −
(2.13)
where Einstein’s notation is used to sum over indices j.
2.7 Coordinate Transformation
The coordinate transformation is used to transform the nonuniform and curvilinear grid into
rectangular computational grid with the cell size of one. This results in unity difference
18 CHAPTER 2. GOVERNING EQUATIONS
quotients with the addition of metric terms.
(x1, x2, x3, t)→ (X1,X2,X3, τ)
therefore the transformation is defined as
Xi = Xi (x1, x2, x3, t)
τ = τ (x1, x2, x3, t)
with the reverse transformation
xi = xi (X1,X2,X3, τ)
t = t (X1,X2,X3, τ) .
For simplicity, t = τ for our calculations. By the chain rule, the temporal derivative
transformation is,
∂
∂Xi
∂Xi
∂xj .
Therefore, the PDE (2.10) is transformed to a new set of conservative equations in the new
coordinate system,
∂w
∂t +
∂Fi
Fi = fj ∂Xi
2.7.1 Time Related Unsteady Metrics
The metrics ∂xi
∂Xj and ∂xi
∂t can be calculated for any i and j.
First, try to find the temporal metrics. In reference to Anderson [7],
( ∂xi
∂t
− (
.
−

umesh i =

[ ∂X ∂x
∂X1
∂x1 =
1
J
( ∂x2
∂X2
∂x3
Substituting the unsteady metric terms (2.17) into equation(2.15) and (2.16),
∂w
∂t +
= 0.
Since the determinant of the spatial metric J is analogous to the volume of a computa-
tional cell, it is customary to multiply J throughout the entire equation
∂Jw
term in equations(2.26) are canceled out. Finally, analogous to (2.15),
∂Jw
where the transformed fluxes are now,
JFi = J ∂Xi
)
.
Note that the metric Jacobian J is analogous to the volume of the computational cells.
∂
dSj = 0 (2.28)
In the finite volume formulation, the values of w are assumed to be constant throughout
each computational cell while the fluxes at the boundaries are assumed to be constant at
respective cell walls. Also, J ∂Xi
∂xj is analogous to the surface areas of the cell.
Chapter 3
Spatial Discretization
In the present approach, the governing equations will be discretized using the finite volume
method. The finite volume approach creates numerical methods that conserve the total
mass, momentum and energy.
Given any type of spatial discretization, the resulting difference equation involves algebraic
terms representing approximations to the spatial partial derivatives. The partial differential
equation is reduced to an ordinary differential equation by collecting the terms correspond-
ing to the spatial derivatives and denoting them by R(w). The equation (2.27) is rewritten
in the semidiscretized form dJw
dt + JR(w) = 0 (3.1)
If the residual is evaluated at time step ≤ n, i.e.
dJnwn
dt + JR(wn, wn−1, wn−2, . . . ) = 0, (3.2)
then the resulting ODE can be solved explicitly. Otherwise, if the residual contains unknown
variables from the time step n + 1, then an implicit solver is needed. Nonlinear implicit
difference equations must be solved by iterative or linearization techniques.
Semidiscretization allows us to examine the spatial and temporal difference operators
23
separately. The spatial difference operators are constructed first, turning the partial dif-
ferential equations into ordinary differential equations. Next, a numerical time integration
technique is chosen to discretize the ordinary differential equation into simple algebraic
equations. Using the resulting difference equation to advance the flow solution in time
requires simple algebraic operations.
In the present dissertation, the spatial derivatives are discretized using compact second or-
der central differencing in space. Although central differencing is known to be oscillatory
as shown by its Fourier footprints being entirely on the imaginary axis, the present imple-
mentation discourages the formation of local extrema by the use of artificial dissipation. In
fact, at points of extrema, with the addition of first order artificial dissipation, the central
differencing operator in space is converted into an upwind operator[50].
Recall the semidiscretized equation, dJw dt
+ JR(w) = 0, where the term R(w) is the
discrete residual of the differential variables ∂Fi
∂Xi . Given that the transformed coordinate
system has unit mesh spacings, Xi = 1, the present numerical discretization scheme is
written as follows:
R(w) = Q(w)−D(w)
2 ,j,k −
2 ,j,k
2 ,k −
2 ,k
2 − (
2
2 ,j,k + F2i,j+ 1
2 ,k − F2i,j− 1
2 ,k + F3i,j,k+ 1
2 ,j,k
2 ,k
2
is the artificial numerical dissipation.
Implementation of the above discretization in the Fortran program is illustrated by the
following procedure. For simplicity, we’ll focus on the convective flux in the X1-direction.
The other two directions can be derived analogously. Writing the discrete flux difference in
3.2. INVISCID FLUXES 25
i-direction, at the i + 1 2 and the i− 1
2 cell surfaces
2 ,j,k (3.3)
where the flux at the i + 1 2 surface is graphed below
i
j
k
Figure 3.1: Fluxes through a cell in the computational domain
F1i+ 1 2 ,j,k =
(3.4)
where δij is the Kronecker’s Delta and the equation is summed over m and n using Einstein’s
notation. If we used the flow variables from cells (i, j, k) and (i + 1, j, k) to compute the
average value at ( i + 1
2 , j, k ) , then
F1i+ 1 2 ,j,k =
+wi+ 1 2 ,j,k
. (3.8)
If we define the transformed velocities at the interfaces ( i + 1
2 , j, k )
( ∂X1
∂xn
2 ,j,k
summed over indices n, then equation (3.8) may be further simplified as
F1i+ 1 2 ,j,k =
(3.12)
summed over indices l and n, where metrics, q±1 and qmesh 1 are defined at the surface
3.3. ARTIFICIAL DISSIPATION 27
2 , j, k ) .
3.3 Artificial Dissipation
The purpose of artificial diffusion is to reduce gradients that exist in the flow solution.
The addition of numerical dissipation can help maintain stability of numerical schemes. For
example, the JST scheme uses artificial dissipation to maintain positivity and thus preserves
the LED property of the flow solution. In the present approach, centered differencing is
inherently non-dissipative, therefore we must provide additional numerical dissipation to
prevent generation of oscillations in the solution near local extrema. For pure transport
equations, the locus of the Fourier symbol for central differencing is along the imaginary
axis. The imaginary axis is included in the stability region of one step multistage schemes of
three or more stages with appropriate coefficients. The addition of numerical dissipation to
the flow equations bends the Fourier footprint of the spatial difference operator toward the
negative real axis and away from the pure imaginary axis, making the transient solutions
somewhat less oscillatory.
The amount of artificial diffusion necessary to keep a numerical evolution stable is critical
to the success of a numerical scheme. While insufficient numerical dissipation can result in
ringing effects near shocks and even instability, too much dissipation smears out gradients in
the solution masking real physical details of the flow. The JST scheme[51] achieves positivity
by implementing a gradient sensor switch. The sensor controls the amount of numerical
dissipation near local extrema so that the LED property is maintained. Since LED schemes
are TVD, the JST scheme applied to a scalar conservation law is a TVD scheme. According
to Harten’s theorem, any TVD scheme can be, at best, first order accurate. Therefore, first
order accuracy exists in the neighborhood of discontinuities or large numerical gradients.
In order to maintain higher order accuracy in the smooth parts of the flow solution, the
scheme must discern between a smooth varying function and a discontinuity. This is done
by a gradient sensor constructed from local pressure gradients. Ideally, numerical schemes
should be able to preserve sharp discontinuities occurring naturally in the flow with a single
interior point. The requirement of a single intermediate point in shocks is imposed by
discrete conservation of the global solution.
In the non-scalar case, if the dissipation is not removed according to the eigenvalues of
the convective flux Jacobians or have the wrong zone of dependence, then too much artificial
28 CHAPTER 3. SPATIAL DISCRETIZATION
dissipation results in a smoothed-out shock front. Jameson has shown that while the JST
scheme will not allow shocks with single interior point, both CUSP and HCUSP schemes
do admit solutions with sharp discontinuities satisfying the Rankine-Hugoniot conditions
containing a single interior point.
Assume the semidiscrete equation results in difference equation of the form:
dw
2
where the fluxes through the surface (i + 1 2 , j, k) are
hi+ 1 2 ,j,k = Fi+ 1
2 ,j,k − di+ 1
2 ,j,k. (3.13)
Following this formulation, the two main types of artificial dissipations in the form of flux
splitting are described next.
3.3.1 Flux Difference Splitting
If the flux Jacobian A = ∂F ∂X is diagonalizable, then the flux Jacobian can be decomposed
into its eigenvalues and eigenvectors
A = RΛR−1, (3.14)
where Λ is a diagonal matrix. As a result, upwinding of the difference equation 3.13 can be
achieved by separating the positive and negative eigenvalues,
hi+ 1 2 ,j,k =
1
2
)
= 1
2
) − 1
2
)
Flux difference splitting is used in the Roe method [91].
3.4. JST SCHEME 29
3.3.2 Flux Vector Splitting
An alternative to flux difference splitting is to split the flux vector by separating the plus
and minus components of the characteristics at cell centers,
hi+ 1 2 ,j,k = f+
i,j,k + f− i+1,j,k
)
)
)
hi+ 1 2 ,j,k = f+
i,j,k + f− i+1,j,k
2 (εi+1,j,kwi+1,j,k − εi,j,kwi,j,k) . (3.17)
The JST scheme, Steger-Warming [97] and the Van Leer MUSCL-TVD (Monotonic Upstream-
Centered Scheme for Conservation Laws-Total Variation Diminishing) Schemes all fall into
this category. Flux vector difference splitting (3.15) and flux vector splitting (3.16) differs
in where the flux Jacobian for the numerical dissipation is evaluated.
3.4 JST Scheme
Defining the numerical dissipation flux as a combination of first and third order differences,
di+ 1 2 ,j,k = ε
(2)
2 ,j,k − ε
2 ,j,k + wi− 1
(3.18)
where wi+ 1 2 ,j,k = (wi+1,j,k − wi,j,k) and λi+ 1
2 ,j,k is the spectral radius of the Jacobians
at ( i + 1
2 , j, k ) . Because central differencing of the inviscid flux is used, in order for the
residual to converge toward machine zero, third order diffusion is needed at smooth parts of
the solution. First order diffusion is needed at local extrema to maintain the LED property
of the scheme.
In order to determine when to turn on the first order diffusion and turn off the third
order diffusion, a shock sensor comprised of second order pressure gradients is used,
νi,j,k =
.
Thus,
ε (2) i+1,j,k = k(2)max ( νi−1,j,k, νi,j,k, νi+1,j,k, νi+2,j,k)
and
[
di+ 1 2 ,j,k = D2
ξ (w)−D4 ξ (w)
with
2 ,j,k
2 ,j,k + i− 1
Λi+ 1 2 ,j,k =
φξi,j,k
where λξi,j,k is the spectral radius of the Jacobian in the i-coordinate direction and φ is the
3.4. JST SCHEME 31
λξi,j,k = (
~n =
i+ 1 2 ,j,k
local pressures,

ε (4)
k(2) = 1
k(4) = 1
.
The parameter k(4) is determined by minimizing the high frequency portion of the ampli-
fication factor of the root locus of the Fourier symbol for the spatial difference operator.
The linear stability analysis is performed by using the transport equation with third order
numerical dissipation (equation 9.2 without first order diffusion) as the model PDE and
Jameson’s 5 stage RK scheme with separate evaluation of convective and dissipative terms
(equation 9.4).
3.5 CUSP-Type Schemes
JST scheme uses scalar diffusion, thus, it does not admit shock structures with single interior
point. In practice, JST scheme smooths shocks over several grid points. In order to capture
shocks with single interior point, a flux vector splitting technique is used. The Convective
Upwind Split Pressure (CUSP) scheme, developed by Jameson [51, 52], treats pressure
separately from convective variables so each part has the correct zones of dependence and
admits shocks structures with single interior point.
3.5. CUSP-TYPE SCHEMES 33
3.5.1 Comparison of E-CUSP and H-CUSP Schemes
A summary of E-CUSP and H-CUSP schemes is presented here in tabular form,
Scheme ECUSP
Flux Split
dj+ 1 2 =1
)
[ 1 2βwu
sign (M) |M |> 1
this leads to α∗= 0 at M = 1 (sonic line)
Eigenvalues λ± = u± c
2α∗c (wj+1 − wj) +1 2β (fj+1 − fj)
use limiter for diffusive flux
Scheme HCUSP
Flux Split
2α∗c (wj+1 − wj) +1 2β (fj+1 − fj)
dj+ 1 2 =1
[ 1 2βwhu
sign (M) |M | > 1
u± √ (
1 2βwhu
Chapter 4
Explicit Schemes
Various temporal discretizations are examined for this ordinary differential equation and
are presented in the following sections.
4.0.2 Multistage Runge-Kutta (RK) Schemes
Using the discrete residual at the beginning of the time step n results in an explicit equation
dJnwn
dt + JR (wn) = 0. (4.2)
An explicit m-stage multistage scheme can be used to solve for the solution at time step
n + 1,
w(0) = wn
36 CHAPTER 4. EXPLICIT SCHEMES
For time-accurate calculations, time step size is the same throughout the entire mesh.
However, the maximum time step size is limited by the smallest computational cell in th
mesh. This poses severe limitations on advancement in numerical time integration. By
using different time step sizes for different mesh sizes, variable time stepping, steady state
convergence is accelerated, but the flow solution is no longer time-accurate. Therefore,
variable time stepping with explicit schemes is only good for steady state calculations in
non-autonomous problems.
Chapter 5
Implicit Schemes
Various temporal discretization are examined for this ordinary differential equation. Several
implicit schemes are discussed here.
5.1 Fully Implicit Backward Difference Formula
Discretize the time difference by the fully implicit backward difference formula (BDF)
dJnwn
dt =
3
t Jnwn +
and thus,
t Jnwn +
( wn+1
) = 0.
Although this equation is second order accurate and A-stable (the third order accurate
scheme is stiffly stable), it is not directly solvable without iterations.
37
5.2 Linearized Implicit Operators
Many options of creating the implicit operator are available. The computationally most
expensive to most economical implicit operator are as follows:
1. Linearized Equations,
2. Approximate Factorizations – ADI and Diagonally Dominant ADI (DDADI), and
3. Diagonalized Approximate Factorizations.
The implicit drivers can be applied on the pseudo-time or the real time. When the implicit
operator is applied on the real time, a time accurate marching technique is created.
5.2.1 Numerical Dissipations for Implicit Operators
For linearized implicit methods, addition of implicit numerical dissipations can be treated
separately from the explicit counterparts[88, 101]. Strictly first order accurate dissipation
in the implicit operator without JST-type switching mechanisms produces error terms in
the smooth part of the solution that are O(tw); which produces global error of O(t2).
Therefore, it is not necessary to use second or higher order artificial dissipation on the left
hand side in order to maintain global second order accuracy in time. This concept can be
described briefly by the following equation[71]:
{Numerics}wn i,j = {Physics} . (5.2)
5.3 Linearized Scheme
One way to avoid iterations is to linearize the R ( wn+1
) term. Consider the semi-discrete
R ( wn+1
5.3. LINEARIZED SCHEME 39
where wn = wn+1−wn and wn−1 = wn−wn−1. Examining the spatial differences from
equation (5.3),
) (5.5)
n) ∂Xi
.
Using Backward Euler difference formula in time and assuming Jn+1 = Jn = Jn−1 ≡ J ,
Jwn+1 − Jwn + J t
2
(5.6)
with 0 ≤ θ ≤ 1 and the scheme is fully implicit if θ = 1 or fully explicit if θ = 0.
[
= Jt
. (5.7)
Alternatively, using the fully implicit (θ = 1) 3-4-1 Backward Euler difference formula
in time,
2t Jwn−1 + JR
( wn+1
) , then we arrive at the linearized 3-4-1 scheme,
[
. (5.10)
The problem with solving linearized equations directly is that the cost of inversion is
too large.
5.4 Alternate Direction Implicit (ADI) and ADI Scheme with
3-4-1 Backward Difference Formula (ADI-BDF)
Original contributors to the ADI scheme are Mitchell-Gourlay, Beam-Warming and Briley-
MacDonald. Using approximate factorization to reduce the complexity of the linearized
trapezoidal scheme,
= Jt
[
= Jt
. (5.14)
For the linearized 3-4-1 scheme, similar factorization can be applied, resulting in the
ADI-BDF scheme,
. (5.16)
The additional factorization error is t 4 DX1A1DX2A2w for the ADI with trapezoidal
rule, and 2t 3J
DX1A1DX2A2w for the ADI-BDF scheme. The problems with ADI type
schemes are:
1. The factorization error dominates at large CFL numbers, and
2. The scheme isn’t amenable to parallel processing, although this is presently being
remedied.
The ADI or ADI-BDF schemes are usually solved by sweeping in different directions.
[
followed by another sweep in the other coordinate direction,
[
Consider the upwind flux split,
R = D− X1
F+ 1 + D+
[
Setting θ = 1 and defining the shift operators E,
D+ X1
where
A+ i −A−
D = I + t(A+ 1 −A−
1 + A+ 2 −A−
Removing a factor of D from equation (5.18),
D [
and factorize
Jwn = −JtR. (5.19)
The two additional diagonal terms exist in DDADI when compared to traditional ADI
schemes, making the implicit matrix diagonally dominant. Solve (5.19) in two steps by
1. first solve
2. then solve
5.6 DDADI with BDF
A different approach is used to derive the DDADI with 3-4-1 Backward Differencing in time.
Starting with
3 R (wn)
[
3 R (wn) .
D
3
= J
3 R (wn)
D
3
3
3 R (wn) . (5.22)
Multiply out the factored form to see what the extra terms are
D
3V (
3 R (wn)
[
3
3 R (wn) (5.23)
3
5.7 Hybrid Implicit-Explicit Scheme
[
3 wn−1 = 0 (5.25)
44 CHAPTER 5. IMPLICIT SCHEMES
and then iterate with a multistage time stepping scheme augmented by multigrid to drive
the solution in the steady state limit towards the linearized equation (5.5)
dJw
dτ +
J
3 R(wn)− J
3 wn−1
w(2) = w(1) − β1τ
dwn
dτ +
3
t Jnwn +
by applying the Dual Time Stepping Scheme (5.28).
The initial ADI step is already formally O(t2), subtracting the product of β1 and
(5.14) from (5.26) we get
w(2) −w(1) = β1 4t2
9 DX1A1DX2A2w(1) = O(t2)
and subsequently, any w(k) −w(k−1) is also O(t2).
The advantages of this scheme are:
1. Ability to retain formal second order accuracy with any number of iterations. It
should not be necessary to iterate to convergence within each implicit time step, in
contrast to existing dual-time stepping schemes which are only second order accurate
if the inner iterations are fully converged, and
2. The additional iterations with multigrid should provide information exchange between
5.8. ITERATIVE DUAL-TIME STEPPING SCHEME 45
processors which is needed to stabilize the ADI scheme running in separately proces-
sors.
5.8 Iterative Dual-Time Stepping Scheme
Using an inner iteration, drive the equation to steady state in pseudo time τ ,
dw
dτ +
[ 3
t Jnwn +
]
w(0) = wn+1
using
4. multigrid.
The main snag with this scheme is that no error estimate for time accuracy can be found
unless the inner iteration is fully converged.
46 CHAPTER 5. IMPLICIT SCHEMES
5.8.1 Explicit Unsteady Dual Time Stepping
Given the governing difference equation,
Dt
) = 0
and the kth order accurate backward difference operator for the time difference is defined
by
−wn+1 = wn+1 − wn
then the governing equation can be written in the alternate form,
Dt
= 1
t
[ a0w
)] .
Next, consider the backward second order equation (3-4-1 Backward Difference Formula)
M = 2
a0 = 3
t wnJn +
or
E ( wnJn, wn−1Jn−1, . . . , wn+1−MJn+1−M
) = − 2
5.8. ITERATIVE DUAL-TIME STEPPING SCHEME 47
Finally, the Dual Time Stepping equation,
dw
dτ +
[ 3
t wnJn + JR (w)
]
5.8.2 Implicit Unsteady Dual Time Stepping
ADI with trapezoidal rule can be used to drive the pseudo time iteration. However, since
pseudo time accuracy is not as important as pseudo time convergence, first order ADI will
suffice,
. (5.29)
DDADI is a popular ’pseudo second order’ implicit scheme with excellent high frequency
damping properties,
. (5.30)
ADI-BDF can also be used for driving the pseudo time iterations, but at less efficiency
[
3
)
The difference between time accurate implicit schemes (equations (5.14),(5.16) and
(5.19) ) and dual time stepping implicit schemes (equations (5.29) ∼ 5.31) is the time
step increment t→ τ and the time difference term in the right hand side source term:
( 3wn+1 − 4wn + wn
5.9 Point Implicit Treatment of Time Difference Source Term
The stability of the numerical schemes are affected by the addition of the time-difference
terms in unsteady problems. As Fourier analysis have sh

An Implicit-Explicit Flow Solver for Complex Unsteady Flows

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