 | Li Dian. Application of point- and line-Implicit preconditioning in unsteady flow computations: Diss. … Dr.-Ing. / Deutsches Zentrum für Luft- und Raumfahrt, Institut für Aerodynamik und Strömungstechnik, Braunschweig. - Köln: DLR, 2015. - xix, 155 p.: ill. - (Forschungsbericht; 2015-18). - Res. also Germ. - Bibliogr.: p.149-155. - ISSN 1434-8454
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1 Introduction ................................................. 1
1.1 Motivation .............................................. 1
1.2 Implicit Preconditioning ................................ 4
1.3 Present Approach ....................................... 10
1.4 Outline ................................................ 10
2 Governing Equations ......................................... 13
2.1 Unsteady Navier-Stokes Equations ....................... 14
2.2 Favre- and Reynolds-averaged Navier-Stokes Equations ... 16
2.3 The Integral Form of the Unsteady Reynolds Averaged
Navier-Stokes Equations in a Moving Frame of
Reference .............................................. 19
2.4 Spalart-Allmaras Turbulence Model ...................... 23
3 Discretization .............................................. 27
3.1 Spatial Discretization ................................. 28
3.1.1 Convective Flux ................................. 30
3.1.2 Diffusive Flux .................................. 31
3.1.3 Treatment of Boundary Conditions and Fluxes ..... 33
3.1.4 Source Term ..................................... 35
3.1.5 Semi-discrete System ............................ 35
3.2 Temporal Discretization ................................ 36
3.2.1 Implicit Linear Multistep Methods ............... 37
3.2.2 Implicit Runge-Kutta Methods .................... 38
3.2.3 Fully-discrete System ........................... 39
4 Nonlinear Solver ............................................ 41
4.1 Newton's Method ........................................ 42
4.2 Basic Agglomeration Multigrid Solver ................... 43
4.2.1 Two Grid Method ................................. 44
4.2.2 Basic Multigrid Strategy ........................ 46
4.3 Preconditioned Explicit Multistage Runge-Kutta
Smoother ............................................... 50
4.3.1 Derivation of Preconditioner .................... 53
4.3.2 Construction of First-order Point- and Line-
Jacobi Preconditioner ........................... 58
5 Test Results ................................................ 73
5.1 Influence of Grid Density .............................. 74
5.1.1 NACA0012 Inviscid Pitching Oscillation .......... 74
5.1.2 RAE2822 Viscous Pitching Oscillation ............ 80
5.1.3 Summary of Grid Refinement ...................... 86
5.2 Influence of Stepsize of Physical-time ................. 90
5.3 Influence of Stopping Criteria ......................... 95
5.3.1 Relative Reduction of Unsteady Density
Residual ........................................ 95
5.3.2 Convergence of Lift and Drag Coefficient ........ 97
5.3.3 Further Comparison on Stopping Criteria ......... 98
5.4 Influence of Temporal Discretization Order ............ 101
5.5 Influence of Reynolds Number .......................... 104
5.6 Influence of Reduced Frequency ........................ 106
5.7 Three-dimensional Flows ............................... 107
5.7.1 LANN Wing Pitching Oscillation ................. 107
5.7.2 DLR-F12 Pitching Oscillation ................... 111
5.8 Influence Semi-coarsening Strategies .................. 114
5.8.1 Two-dimensional Hybrid Grid - NACA64A010
Airfoil Case ................................... 115
5.8.2 Three-dimensional Structured Grid - LANN Wing
Case ........................................... 120
5.8.3 Three-dimensional Unstructured Grid - DLR-F12
Case ........................................... 123
5.8.4 Summary of Influence Semi-coarsening
Strategies ..................................... 125
6 Conclusion and Outlook ..................................... 127
6.1 Conclusion ............................................ 127
6.2 Outlook ............................................... 128
A Local Fourier analysis ...................................... 131
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