Simulation of low Mach number flows

Transcription

1 Simulation of low Mach number flows Improvement of accuracy and convergence of the TAU code using time derivative preconditioning Ralf Heinrich, Braunschweig, 9th February 008 Simulation of low Mach number flows Improvement of accuracy and convergence of the TAU code using time derivative preconditioning Overview Motivation Numerical problems associated with low Machnumber flows Time-derivative preconditioning arameter settings

2 Simulation of low Mach number flows Improvement of accuracy and convergence of the TAU code using time derivative preconditioning Motivation Goal: Efficient and accurate simulation of high-lift flows Although the onflow Mach number is small, incompressible methods are not the appropriate tool for simulation of high lift flows Features of RANS method TAU Time stepping method High efficiency and accuracy in the trans- and supersonic regime BUT: Numerical problems for low Mach number flows! 3 Numerical problems for simulation of low Mach number flows Simple numerical experiment to find the symptoms: NACA00, α =.0 o, inviscid Mach number range from 0. down to 0.00 Expectation: Flow should behave incompressible Solution (c p, c L ) should be independent of the Mach number! c d should be 0 Ma=0. Ma=0.0 Ma=0.00 4

3 Ma=0. Ma=0.0 Ma= Ma=0. Ma=0.0 Ma=

4 Ma=0. Ma=0.0 Ma= What are the reasons?... for bad convergence: Condition of the system of equations D Euler: W t A W x = 0 Δt λ Δx λ o λ - stiffness Ma 0 λ / λ 0 min t x max Eigenvalues: λ = u, λ = u a, λ = u ~ Δx Δtmax λ O max Number of timesteps for convection of a disturbance Δx along particle path a Ma λ λλλ o - λ o λ O Δt λ - transonic λ / λ =O() Ma Ma N = Ma min min max convection per time step Ma λ / λ max 8 4

5 What are the reasons?... for bad accuracy: Unbalanced artificial dissipation D Euler equations in characteristic form; usage of scalar dissipation: ~ ~ 4 ~ W W W λ = K4 λmax 4 t x x λ = u ~ ~ 4 ~ W W W λ = K4 λmax 4 t x x = u a ~ W3 t ~ W λ3 x For steady solution time derivative vanishes 3 = K λ 4 max 4 ~ W3 4 x λ λ 3 = u a For Mach number approaching 0 λ is vanishing small compared to λ max : Excessive dissipation Detailed analysis based on perturbation expansion can be found in: S. Venkateswaran and C. L. Merkle Analysis of reconditioning Methods for the Euler and Navier-Stokes Equations Von Karman Institute Lecture Series, March How to bypass the problems? roblems seem to be closely related to discrepancy of the eigenvalues of the system Idea: Try to manipulate the eigenvalues of your system of equations in such a way, that all eigenvalues are clustered, like for transonic flow (have a similar order of magnitude) We are primarily interested in steady solutions. Multiply the time derivative with an appropriate matrix. This does not change the steady solution Now the relevant eigenvalues of our system of equations come from the matrix A 4 W W W A = K4 λmax 4 t x x 4 W W W A = K4 λ' max 4 λ t x x max of A Task: Construct the matrix in such a way, that our system of equations is still hyperbolic and the eigenvalues are clustered 0 5

6 6 reconditioner of Choi and Merkle AD z W C y W B x W A t W = or AD z G y F x E t W = ( ) T e t w v u W ρ ρ ρ ρ ρ = or AD z G y F x E t W = Γ Γ ( ) T T w v u p W = γ γρ ρ ρ ρ ρ ρ ρ = Γ w v u ) M /(a h ) M w /(a ) M v /(a ) M u /(a ) M /(a r t r r r r revent singularity near stagnation point )) min(,max(m,k M M r = Γ is local quantity! User parameter reconditioner of Choi and Merkle AD z W C y W B x W A t W = Eigenvalues of A in x-direction: ( ) a' u' a' u' u u u ( ) ( ) ( ) r r r M 4 M a u M a' ' u = m m Verhalten für, u = q, K= Ma Number of timesteps for convection of a disturbance Δx along particle path Ma Ma N = reconditioning

7 Subsonic flow around NACA risms Ma = Ma = 0.6 Study with respect to Convergence Accuracy 3 Subsonic flow around NACA risms Ma min = Ma max = 0.6 Study with respect to Convergence Accuracy 4 7

8 Subsonic flow around NACA risms Ma min = Ma max = 0.6 Study with respect to Convergence Accuracy 5 Subsonic flow around NACA risms Ma min = Ma max = 0.6 Study with respect to Convergence Accuracy 6 8

9 Subsonic flow around NACA risms Ma min = Ma max = 0.6 Study with respect to Convergence Accuracy 7 Subsonic flow around NACA risms Ma min = Ma max = 0.6 Study with respect to Convergence Accuracy 8 9

10 Subsonic flow around NACA risms Ma min = Ma max = 0.6 Study with respect to Convergence Accuracy 9 Subsonic flow around NACA risms Ma min = Ma max = 0.6 Study with respect to Convergence Accuracy 0 0

11 Subsonic flow around NACA risms Ma min = Ma max = 0.6 Study with respect to Convergence Accuracy Subsonic flow around NACA risms Ma min = Ma max = 0.6 Study with respect to Convergence Accuracy

12 Subsonic flow around NACA risms Ma min = Ma max = 0.6 Study with respect to Convergence Accuracy 3 Subsonic flow around NACA risms Ma min = Ma max = 0.6 Study with respect to Convergence Accuracy 4

13 Laminar flow around flat plate 5 Laminar flow around flat plate 6 3

14 Laminar flow around flat plate 7 Laminar flow around flat plate 8 4

15 Laminar flow around flat plate 9 Laminar flow around flat plate 30 5

16 Laminar flow around flat plate 3 Laminar flow around flat plate 3 6

17 Laminar flow around flat plate 33 Laminar flow around flat plate 34 7

18 Laminar flow around flat plate 35 3 element profile (LT) 5000 hexahedra 0000 prisms All computations for α = 0.8 and Re = 3.5x0 6 Ma = 0., 0.0, 0.00 Ma =

19 3 element profile (LT) 5000 hexahedra 0000 prisms All computations for α = 0.8 and Re = 3.5x0 6 Ma = 0., 0.0, 0.00 Ma = element profile (LT) 5000 hexahedra 0000 prisms All computations for α = 0.8 and Re = 3.5x0 6 Ma = 0., 0.0, 0.00 Ma =

20 3 element profile (LT) 5000 hexahedra 0000 prisms All computations for α = 0.8 and Re = 3.5x0 6 Ma = 0., 0.0, 0.00 Ma = element profile (LT) 5000 hexahedra 0000 prisms All computations for α = 0.8 and Re = 3.5x0 6 Ma = 0., 0.0, 0.00 Ma =

21 3 element profile (LT) 5000 hexahedra 0000 prisms All computations for α = 0.8 and Re = 3.5x0 6 Ma = 0., 0.0, 0.00 Ma = element profile (LT) 5000 hexahedra 0000 prisms All computations for α = 0.8 and Re = 3.5x0 6 Ma = 0., 0.0, 0.00 Ma =

22 reconditioning for vortical flows reconditioning is usually also an appropriate technique for subsonic vortical flows (e.g. if you are interested in the vortices behind an aircraft) 43 arameter settings Entries in control file of TAU solver: M r = min(, max( M, K M )) reconditioning : - reconditioning (0//): Cut-off value:.0 Default value is Default value is 0: reconditioning switched off : Artificial dissipation is computed based on primitive variables (p, u, v, w, T) : Artificial dissipation is computed based on conservative variables.0 (smaller values tend to make computation more unstable) 44

23 arameter settings, remarks and hints In general preconditioning helps to improve the accuracy properties in the low Machnumber regime due to a better scaling of the artificial dissipation In many cases preconditioning helps to improve the convergence properties (especially for very low Mach numbers < 0.)... but there is absolutely no guaranty for improvement of the convergence properties! Due to the lower amount of dissipation computations tend to be more unstable, especially in the transient phase. The following can be done, in order to solve this problem Start computation with a higher cut-off value, e.g. 4.0; after the st transient phase the cut-off value can then be reduced (very easy with TAU-HYTON) Start without preconditioning; after the st transient phase, preconditioning can be switched on, maybe with a larger cut-off value; then again the cut-off value can then be reduced 45 arameter settings, remarks and hints In some cases computations converge without preconditioning, but not with preconditioning In many cases an unsteady behavior of the flow is detected earlier with preconditiong, because of the lower amount of dissipation!! If you apply preconditioning in the transonic regime, in some cases computations become unstable A similar observation has been made for FLOWer Blending between subsonic and supersonic regimes have been improved by Frédéric le Chuiton in FLOWer The same procedure has now also been implemented in TAU and will be available in release

24 Further remarks Influence of Farfield Distance for D High Lift Applications A distance of 0 chord length might be too small for high lift applications original TC mesh comparison of results with and without vortex correction (v.c.) usage of extended mesh using chimera technique farfield distance = 00, 00, 500, 000 chord length 47 Further remarks Influence of Farfield Distance for High Lift Applications A distance of 0 chord length might be too small for high lift applications comparison of results with and without vortex correction (v.c.) usage of extended mesh using chimera technique farfield distance = 00, 00, 500, 000 chord length Chimera boundaries 48 4

25 Further remarks Influence of Farfield Distance for High Lift Applications A distance of 0 chord length might be too small for high lift applications comparison of results with and without vortex correction (v.c.) usage of extended mesh using chimera technique farfield distance = 00, 00, 500, 000 chord length New farfield bound. 49 Further remarks Influence of Farfield Distance for High Lift Applications C p on slat 50 5

26 Further remarks Influence of Farfield Distance for High Lift Applications 5 6