1 Introduction .............................................. 13
2 The RANS equations ........................................ 25
2.1 Governing equations ....................................... 25
2.2 Turbulence models ......................................... 30
2.2.1 Spalart-Allmaras model ............................. 32
2.2.2 kω-model ........................................... 34
2.3 Nondimensionalization ..................................... 37
2.4 Initial and boundary value problems ....................... 47
2.5 Turbulence modeling: An inverse view ...................... 49
3 Discretization ............................................ 55
3.1 Computational mesh and Graph Theory ....................... 57
3.2 Discretization: The invistid part ......................... 61
3.2.1 Derivative of convective flux ...................... 67
3.2.2 Eigendecomposition of the derivative of the
convective flux .................................... 68
3.2.3 Implementation of the Matrix Dissipation operator .. 71
3.3 Low Mach number modifications ............................. 81
3.3.1 Modification of Turkei ............................. 84
3.3.2 Definition of a low Mach number modification ....... 84
3.3.3 Eigendecomposition of Turkel's modification ........ 86
3.3.4 Extension of Roe matrix to the incompressible
limit .............................................. 89
3.4 Discretization: The viscous part .......................... 89
3.4.1 Approximation of gradients ......................... 89
3.4.2 Discretization of viscous flux terms ............... 93
3.5 Discretization of turbulence models ....................... 94
3.6 Discretization of boundary conditions ..................... 97
3.7 Discrete set of equations ................................ 104
4 Total derivative of Discretization ....................... 107
4.1 Global considerations .................................... 108
4.1.1 Graphs and sparse matrices ........................ 108
4.1.2 Construction of dR/dW ............................. 110
4.2 Derivative of inviscid terms ............................. 116
4.2.1 Derivative of compact inviscid flux ............... 116
4.2.2 Derivative of compact inviscid fiux and constant
ARoe ............................................. 122
4.2.3 Derivative of full inviscid flux .................. 123
4.3 Derivative of viscous terms .............................. 125
4.3.1 Derivative of viscous terms assuming TSL .......... 125
4.3.2 Eigendecomposition of viscous flux Jacobian ....... 134
4.4 Derivative of turbulence models .......................... 135
4.4.1 Derivative of Spalart-Allmaras model .............. 136
4.4.2 Derivative of kω-model ............................ 139
4.4.3 Structure of derivative for turbulence models ..... 141
4.4.4 Derivative of eddy viscosity ...................... 142
4.5 Derivative of boundary conditions ................... 142
5 Solution algorithms ...................................... 147
5.1 Solution methods for nonlinear ecjuations ................ 148
5.1.1 Determination of lines ............................ 148
5.1.2 Agglomeration techniques .......................... 151
5.1.3 Nonlinear multigrid ............................... 154
5.1.4 Construction of transfer operators ................ 158
5.1.5 Coarse grid equations ............................. 160
5.1.6 Construction of a multigrid smoother .............. 162
5.1.7 Globalization strategies .......................... 167
5.2 Linear solution methods .................................. 171
5.2.1 Krylov subspace methods ........................... 171
5.2.2 Construction of preconditioner .................... 172
5.2.3 Iterative solution methods for linear equations ... 174
5.3 Hierarchy of multigrid smoothers ......................... 178
5.3.1 Low cost smoothers ................................ 179
5.3.2 The solution algorithm: A castle in a sand pit .... 181
6 Examples ................................................. 185
6.1 Choice of suitable solution algorithm .................... 186
6.2 Characteristic values .................................... 188
6.3 Example 1: CASE 9, RAE 2822 .............................. 190
6.4 Example 2: MDA30P30N ..................................... 193
6.5 Example 3: DPW III, Case 2, Wing 1 ....................... 195
6.6 3D Transonic turbulent flow .............................. 199
6.6.1 Influence of choice of gradients .................. 200
6.6.2 Assessment of low cost smoother ................... 201
6.6.3 Assessment of Newton kind smoother ................ 203
6.7 Example 5: High-lift Prediction Workshop II. Case 2a ..... 204
6.8 Example 6: 3D three element high lift wing-body
configuration ............................................ 206
6.9 Investigations for the incompressible limit .............. 209
6.9.1 Example 7: Inviscid flow over NACA0012 airfoil ... 211
6.9.2 Example 8: Transonic turbulent flow over
a Common Research Model .......................... 218
6.9.3 Example 9: 3D high lift wing-body configuration .. 218
6.9.4 Example 10: NASA TRAP Wing ....................... 219
6.10 Example 11: Laminar flow over NACA0012 airfoil ........... 229
6.11 Apphcation of kω-model ................................... 231
6.11.1 Example 12: CASE 10, RAE 2822 ..................... 234
6.11.2 Example 13: CASE 9, RAE 2822 ...................... 235
6.11.3 Example 14: MDA30P30N ............................. 237
6.11.4 Example 15: Transonic turbulent flow over
a Conunon Research Model .......................... 237
6.11.5 Example 16: NASA TRAP Wing ........................ 240
6.12 Sunniiary of immerical examples .......................... 241
7 Assessment of algorithms ................................. 243
7.1 Scalability investigations ............................... 244
7.2 Computer aided analysis .................................. 248
7.3 Numerical results ........................................ 251
7.3.1 Laminar flow around NACA0012 airfoil .............. 252
7.3.2 Turbulent flow around DPW5 CRM .................... 253
7.4 Considerations for the kω-model .......................... 255
8 Conclusion and Summary ................................... 259
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