 | Lenz C. Time step size adaptation for unsteady aerodynamic flow simulations: application to gust encounter simulations: Diss. … Dr.-Ing. / Deutsches Zentrum für Luft- und Raumfahrt, Institut für Aerodynamik und Strömungstechnik, Braunschweig. - Köln: DLR, 2018. - vii, 144 p.: ill., tab. - (Forschungsbericht; 2018-25). - Res. also Germ. - Bibliogr.: p.139-144. - ISSN 1434-845
Шифр: (Pr 1120/2018-25) 02
|
1 Introduction ............................................... 1
1.1 Motivation ................................................. 1
1.2 Previous Work .............................................. 2
1.2.1 Embedded Runge-Kutta methods for error estimation ... 3
1.2.2 Step size control algorithm ......................... 4
1.2.3 Step size control for flow simulations .............. 4
1.3 Contributions .............................................. 6
1.4 Thesis Overview ............................................ 7
2 Governing equations ....................................... 11
2.1 Euier equations ........................................... 11
2.2 Reynolds-Averaged-Navier-Stokes equations ................. 13
2.3 Spatial discretization and TAU code ....................... 16
2.4 Moving grid and basis transformation ...................... 20
2.5 Disturbance velocity approach for gust encounter
simulations ............................................... 21
2.6 Boundary conditions ....................................... 24
2.7 Summary of discrete equation .............................. 28
3 Numerical methods for ordinary differential equations ..... 29
3.1 Introduction to Runge-Kutta methods ....................... 30
3.2 Stiff equations and stability of Runge-Kutta methods ...... 32
3.2.1 Linear stability ................................... 33
3.2.2 Nonlinear stability ................................ 34
3.2.3 Order reduction and stage order .................... 36
3.3 Multistep methods ......................................... 37
3.4 Comparison of Multistep and Runge-Kutta methods ........... 39
3.5 Implemented methods ....................................... 40
3.5.1 Embedded exphcit Runge-Kutta methods ............... 40
3.5.2 Embedded diagonally implicit Runge-Kutta methods ... 42
3.5.3 Backward Difference Formulae ....................... 44
4 Solution of nonlinear algebraic systems ................... 47
4.1 Solution methods .......................................... 47
4.2 Stage value predictor ..................................... 51
4.3 Truncation criteria ....................................... 52
5 Time step size adaptation algorithms ...................... 55
5.1 Time step estimator ....................................... 55
5.1.1 Elementary (integral) control ...................... 58
5.1.2 Advanced controllers ............................... 60
5.1.3 New time step size limiter ......................... 65
5.1.4 Summary of adaptation algorithms ................... 69
5.2 Error estimator ........................................... 69
5.3 Error norm or functional .................................. 72
6 Numerical test cases ...................................... 75
6.1 Ehrenfried vortex ......................................... 75
6.2 Kármán vortex street ...................................... 76
6.3 NACA0012 laminar steady ................................... 78
6.4 NACA0012 turbulent pitching oscillation ................... 78
6.5 Gust encounter ............................................ 81
6.5.1 NACA0012 inviscid .................................. 81
6.5.2 SDM fighter configuration inviscid ................. 81
6.5.3 Transport aircraft half model turbulent ............ 84
7 Numerical results ......................................... 87
7.1 Temporal order of convergence of implemented methods ...... 88
7.2 Grid refinement study for explicit and implicit methods ... 91
7.2.1 Application of explicit methods .................... 92
7.2.2 Application of implicit methods .................... 93
7.3 Adaptation algorithm ...................................... 95
7.3.1 Verification ....................................... 95
7.3.2 Comparison of algorithms ........................... 95
7.3.3 Error functional .................................. 106
7.4 Solution of nonlinear systems ............................ 110
7.4.1 Solution methods .................................. 110
7.4.2 Stage value predictors ............................ 112
7.5 Investigation of DIRK method properties .................. 114
7.6 Comparison of BDFs and (adaptive) DIRK methods ........... 120
8 Conclusion ............................................... 127
Acronyms ...................................................... 131
Bibliography .................................................. 139
|
|
