Compressible Navier-Stokes (Euler) Solver based on Deal.II Library

Transcription

1 Compressible Navier-Stokes (Euler) Solver based on Deal.II Library Lei Qiao Northwestern Polytechnical University Xi an, China Texas A&M University College Station, Texas Fifth deal.ii Users and Developers Workshop Texas A&M University, College Station, TX, USA August 5,

2 Outline Motivation and Goal Current work Conclusion and TODOs 2

3 Motivation This is what I typically need to compute 3

4 The pain point Mesh generation: Time consuming Boring Experience depending Solution: Mesh adaptation let solver tell what is good mesh 4

5 Deal.II the library Mesh adaptation with tree data structure FEM discretization Parallelization and excellent scalability 5

6 Goal A N-S solver based on deal.ii that: Starts from initial value on coarse mesh Converges to solution on a reasonable fine mesh adaptively With High-order capability 6

7 Current work Governing Equation Solving technique Solver implementation Solver verification 7

8 Governing Equation Compressible + r F(w) =S(w) Use primitive variables as working var. w =(u j,,p) T 8

9 Governing Equation The Flux F = F c F v F c (w) = u u + Ip u 1 A = u i u j + ij p u i 1 A (E + p)u (E + p)u i 9

10 Governing Equation The Flux F = F c F v F v (w) = 1 Re 1 ij 0 ij u i j 1 A With T = p R ij = j i )+ k 10

11 Governing Equation Weak form: On domain Z v l + 8v = v l 2 T w =(u j,,p) T, Find w 2 S that satisfy Z v l,i i = Z v l S l (w) 11

12 Governing Equation Weak form: Integrate by part to get boundary flux Z l (w) v l Z l F l,i (w) i v l F l,i (w)n i Z v l S l (w) 12

13 Governing Equation Treat boundary conditions and discontinuities on hanging node with this boundary flux Compute numerical flux with Roe scheme[1] 13

14 Governing Equation Boundary conditions: Far field: Riemann invariant Slip wall: Non-penetration u n =0 out = in p out = p in 14

15 Solving technique BDF-1 (Implicit Euler) time integration Newton linearization Direct linear solver from Trilinos* * Iterative solver doesn t stable enough, only used for scalability test For steady run, time step size is determined according to norm of Newton update[2] 15

16 Solver implementation Starts from step-33 of deal.ii tutorial Change working vars. from conservative ones to primitive ones viscous flux Riemann boundary condition Roe flux[2] to replace the too viscous Lax flux Parallelize: learned from step-40 Version control with Git CMake project Regression test suit 16

17 Solver verification Scalability Manufactured solution[3] Subsonic Supersonic Adaptive simulation over circle Flow over foil NACA

18 Scalability 22,288 cells 91,312 DoFs GMRES solver from Trilinos with DD+ILUT precond. 18

19 Solver verification Manufactured solution[3] 8 u(x, y)= u 0 +u x sin(a ux x)+u y cos(a uy y)+u xy cos(a uxy xy) >< v(x, y)= v w = J(x, y) = 0 +v x cos(a vx x)+v y sin(a vy y) +v xy cos(a vxy xy) (x, y)= 0 + x sin(a x x)+ y cos(a y y)+ xy cos(a xy xy) >: p(x, y)= p 0 +p x cos(a px x)+p y sin(a py y) +p xy sin(a pxy xy) J h (x, + r F(w) =S(w) S(x, + r F(J) 19

20 Solver verification Manufactured solution[3] 8 u(x, y)= u 0 +u x sin(a ux x)+u y cos(a uy y)+u xy cos(a uxy xy) >< v(x, y)= v w = J(x, y) = 0 +v x cos(a vx x)+v y sin(a vy y) +v xy cos(a vxy xy) (x, y)= 0 + x sin(a x x)+ y cos(a y y)+ xy cos(a xy xy) >: p(x, y)= p 0 +p x cos(a px x)+p y sin(a py y) +p xy sin(a pxy xy) Choosing different Constants to get subsonic or supersonic case 20

21 Solver verification Manufactured solution Subsonic convergence order 21

22 Solver verification Manufactured solution Supersonic convergence order 22

23 Solver verification Adaptive simulation over circle C1 mapping on wall boundary Subsonic: Mach = 0.1 Almost inviscid: Cv=1e-6 just for stabilization Converged Cd= which ideal value is zero 23

24 Solver verification Adaptive simulation over cylinder Converged density contour 24

25 Solver verification Adaptive simulation over cylinder Converge history of density contour and mesh adaptation 25

26 Solver verification Flow over NACA2412 foil Subsonic: Mach=0.3 Converged pressure contour 26

27 Solver verification Flow over NACA2412 foil Subsonic: Mach=0.3 Oscillation may be caused by geometry non-smoothness 27

28 Solver verification Flow over NACA2412 foil Subsonic: Mach=0.3 Almost inviscid: Cv=3e-6 just for stabilization Converged Cd= which ideal value is zero Converged Cl=0.2576, Cm= As a reference, xfoil[4] gives Cd= Cl= Cm=

29 Conclusion A prototype NS solver based on deal.ii is constructed The solver could run in parallel but the solver doesn t scale perfectly High order convergence is confirmed by MMS Solving process could start from very coarsen mesh and go on with mesh adaptation The solver can give out reasonable aerodynamics data 29

30 ToDos Stable and efficient preconditioner for iterative solver (Now my hope is on MDF ordering of ILU) Describe wall boundary with NURBS geometry Non-slip boundary condition Anisotropic adaptation for boundary layer 30

31 Reference 1. J. Blazek. Computational Fluid Dynamics: Principles and Applications (Second Edition). Elsevier Science, Oxford, second edition edition, ISBN J. Gatsis. Preconditioning Techniques for a Newton Krylov Algorithm for the Compressible Navier Stokes Equations. PhD thesis, University of Toronto, C. J. Roy, C. C. Nelson, T. M. Smith, and C. C. Ober. Verification of Euler/Navier- stokes codes using the method of manufactured solutions. International Journal for Numerical Methods in Fluids, 44(6): , M. Drela, H. Youngren. xfoil/ 31

32 Thanks 32