Discretisation of the pseudo-incompressible Equations with implicit LES for Gravity Wave Simulation DACH2010. Felix Rieper, Ulrich Achatz

Transcription

1 Discretisation of the pseudo-incompressible Equations with implicit LES for Gravity Wave Simulation DACH2010, Ulrich Achatz Institut für Atmosphäre und Umwelt

2 Aim of the project Analysis: Propagation and breaking of gravity waves up to the thermosphere Development: New parameterizations for weather models

3 Methods and Means Pseudo-incompressible Equations 1 : sound proof (FVM discretisation): conserve mass and momentum Implicit LES: modern turbulence model (Linear model): linear analysis / optimal perturbations First simple Tests: Hot & cold air bubble 1 Durran 1989

4 Governing Equations Anelastic (Pillips, Ogura) or pseudo-incompressible (Durran)?

5 Intro GovEq ILES Results End Appendix GW break test I results by R. Klein I GW test by Smolarkiewiecz, Margolin: Wind over "witch of Agnesi" I N = 0.01 s 1, u = 10 m s 1 top: t = 2.5 h mid: t = 3.0 h (anel.) bot: t = 3.0 h (ps.-inc.)

6 Cold bubble test results by R. Klein isentropic background: θ = 300 K left: anelastic right: pseudo-incomp. top: θ = 30 K mid: θ = 150 K bot: θ = 270 K

7 The pseudo-incompressible equations (scaled, conservative) Background: ρ(z), θ(z), p(z) p z = ρ, ρ θ = p 1/γ

8 The pseudo-incompressible equations (scaled, conservative) Background: ρ(z), θ(z), p(z) p z = ρ, ρ θ = p 1/γ Variables: ρ, u, π ρ θ := ρ θ ρ = f (θ, not π)

9 The pseudo-incompressible equations (scaled, conservative) Background: ρ(z), θ(z), p(z) Pressure function p z = ρ, ρ θ = p 1/γ Variables: ρ, u, π ρ θ := ρ θ ρ = f (θ, not π) π = pκ κ, κ = γ 1 γ, γ = cp c V

10 The pseudo-incompressible equations (scaled, conservative) Background: ρ(z), θ(z), p(z) p z = ρ, ρ θ = p 1/γ Variables: ρ, u, π Pressure function π = pκ κ, Fluctuations κ = γ 1 γ, γ = cp c V ρ θ := ρ θ ρ = f (θ, not π) π = π π, θ = θ θ

11 The pseudo-incompressible equations (scaled, conservative) Background: ρ(z), θ(z), p(z) p z = ρ, ρ θ = p 1/γ Variables: ρ, u, π Pressure function π = pκ κ, Fluctuations κ = γ 1 γ, γ = cp c V ρ θ := ρ θ ρ = f (θ, not π) π = π π, θ = θ θ Prognostic and diagnostic equations: ρ t + (ρ u) = 0 (ρ u) t + (ρ u u) + 1 Ma 2 ρ θ π = 1 Fr 2 ρ θ θ k + 1 Re Visc ( ρ θu) = Heating

12 implicit LES Exploiting the numerical viscosity 2 2 Hickel et al. JCP 2006

13 Simulation of large eddies (LES) Large eddies

14 A perfect marriage: LES and FVM LES: Filtered equations u t + F x = 0 ū N t + G F N(u N ) x = G SGS

15 A perfect marriage: LES and FVM u t + F x = 0 LES: Filtered equations FVM: Volume averaged equations ū N t + G F N(u N ) x = G SGS ū N t + F(ũ N ) x = 0

16 A perfect marriage: LES and FVM u t + F x = 0 LES: Filtered equations FVM: Volume averaged equations ū N t + G F N(u N ) x = G SGS ū N t + F(ũ N ) x = 0 ILES: Filter G box = Volume Average & Special Flux function F G box F N(u N ) x F(ũ N ) x G SGS

17 A perfect marriage: LES and FVM u t + F x = 0 LES: Filtered equations FVM: Volume averaged equations ū N t + G F N(u N ) x = G SGS ū N t + F(ũ N ) x = 0 ILES: Filter G box = Volume Average & Special Flux function F G box F N(u N ) x F(ũ N ) x G SGS Find appropriate F and ũ N

18 Reconstruction and blending of polynomials k=1 r=0 x i i+1/2 Note: k = 1, 2, 3 (order of polynomial) r = 0,... k 1 (stencil) ũ = f (6 interpolants)

19 Reconstruction and blending of polynomials k=2 r=0 k=1 x i i+1/2 r=0 i+1/2 x i 1 x i x i+1 r=1 Note: k = 1, 2, 3 (order of polynomial) r = 0,... k 1 (stencil) ũ = f (6 interpolants)

20 Reconstruction and blending of polynomials k=1 r=0 k=2 r=0 k=2 r=1 x i i+1/2 i+1/2 x i 1 x i x i+1 r=1 r=2 x x x x x i 2 i 1 i i+1 i+2 r=0 Note: k = 1, 2, 3 (order of polynomial) r = 0,... k 1 (stencil) ũ = f (6 interpolants)

21 The flux function u u i F N i+1 R L u i u i+1 ) F N (x i+1/2 ) = F (ũr i + ũ L i+1 2 σ i+1/2 (ũ R i ũ L i+1) σ i+1/2 = σ ρu ū i+1 ū i σ ρu 1 free parameter

22 Numerical Tests Behaviour of the model with standard test cases: hot & cold bubble

23 Hot Bubble 3 Background: θ = 300 K Profile: θ = θ K cos 2 r Time: t = min Grid: CFL: ν = Mendez-Nunez, Carroll, MWR1994

24 Cold Bubble Background: θ = 300 K Profile: θ = θ 6.5 K cos 2 r Time: t = min Grid: CFL: ν = 1.0

25 Summary and Outlook Summary: 3D model of the pseudo-incompressible equations FVM conservative for mass and momentum ILES using WENO-type reconstructions Short-term Outlook: Linear and adjoint model Linear gravity wave test / optimal perturbations Examine wave packet propagation

26 Summary and Outlook Summary: 3D model of the pseudo-incompressible equations FVM conservative for mass and momentum ILES using WENO-type reconstructions Short-term Outlook: Linear and adjoint model Linear gravity wave test / optimal perturbations Examine wave packet propagation Thank you for your attention!

27 Appendix Details for the curious

28 FVM FVM discretisation Conserving mass and momentum

29 FVM Scaling of the equations Explicit references ρ ref = kg m 3, p ref = kpa, g = 10 m s 2, R sp = 287 J kg K

30 FVM Scaling of the equations Explicit references ρ ref = kg m 3, p ref = kpa, g = 10 m s 2, R sp = 287 J kg K Derived references a ref = p ref /ρ ref u ref = a ref l ref = p ref gρ ref = h sc t ref = l ref a ref θ ref = T ref = a2 ref R sp

31 FVM Scaling of the equations Explicit references ρ ref = kg m 3, p ref = kpa, g = 10 m s 2, R sp = 287 J kg K Derived references a ref = p ref /ρ ref u ref = a ref Dimensionless numbers l ref = p ref gρ ref = h sc t ref = l ref a ref θ ref = T ref = a2 ref R sp Ma = u ref a ref = 1 Fr 2 = p ref/ρ ref gl ref = 1

32 FVM Spatial Discretisation Data structure: C-grid Conservative treatement of mass and momentum transport Flux function > ILES w v cell i : f i+1/2 f i 1/2 x ρ,θ,π u cell i+1 : f i+3/2 f i+1/2 x

33 FVM Algorithm An overview Low storage Runge-Kutta 3rd order (Williamson, 1980) Single stage: ρ m+1 u from continuity from the momentum eq. using: (ρu) = ρ m+1 u π, u from the divergence constraint: ( ρ θu) = 0 u m+1, π m Corrector step: π m = π m 1 + π

34 ILES implicit LES Exploiting the numerical viscosity 4 4 Hickel et al. JCP 2006

35 ILES Turning Smogorinsky s LES into ILES (Burgers equation) (1) G box F N(u N ) x = ū N + 1 ū N 2 ū N 12 x x 2 x (2) G Smag = 2C S ū N x 2 ū N x 2 x2 Construct F(ũ N )/ x such that G N G Smag : Find appropriate flux function F(ũ i+1/2 ) and appropriate reconstruction ũ i+1/2 = reconstruct(ū N )

36 ILES Reconstruction process 1. Reconstruct / Deconvolve φ(x i+1/2 ) from φ i i Φ(x i+1/2 ) = j= xj+1/2 x j 1/2 φ(ξ) dξ = i φ j x j= 2. Find primitive interpolation polynomial Φ(x) 3. Differentiate: φ(x) = Φ (x) 4. Evaluate: φ(x i+1/2 )

37 ILES Polynomial blending 1. Smoothness measure (TV, different from WENO) β k,r = 1 (ε + 2. Blending weight stencil, k = 1, 2, 3 r = 0,... k 1 φ 2 2 ) ω k,r ( φ, x i ) 1 3 γ k,r β k,r ( φ, x i ) 3. Blending 3 k 1 φ(x i+1/2 ) = ω k,r ( φ, x i ) φ k,r (x i+1/2 ) k=1 r=0

38 Open Questions Open Questions Things we would like to know...

39 Open Questions Problem 1: Density oscillations Consistent? ρ θu and ρ t + ILES ρu Solid wall boundary conditions ok? Interpolation problems? (ρ, u, π ) (ρ, m, π )