, 孙世玮 中国科学院大气物理研究所,LASG 2 南京大学, 大气科学学院

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1 傅豪 1,2, 林一骅 1, 孙世玮 2, 周博闻 2, 王元 2 1 中国科学院大气物理研究所,LASG 2 南京大学, 大气科学学院

2 The Vortical Atmosphere & Ocean e_zoe_2002.jpg

3 Lagrangian point of view: Particle Method ω t + u ω = u ω + ν 2 ω 2D: Point vortex method Rosenhead L.,Proc. Roy. Soc., A, D: vortex filament method Rehbach et al., AIAA paper, 1978, u = 1 4π ω (r r ) r r Summation: multipole expansion method Greengard et al., J. Comput. Phys., 1987

4 Eulerian point of view: Vorticity-Velocity Method ω + u ω = t u ω + ν 2 ω 2 V = u ω Incompressible flow 2 A = ω Complex boundary condition u = k ψ + χ 2 V = ω Gatski et al., Appl. Numer. Math,1991 Bertagnolio et al., J. Comput.Phys,1995 Atmospheric Modelling: Heikes et al., Mon. Wea. Rev,2013 Explicit vortex dynamics No pressure: accurate divergence-free velocity Expense: Vector-Poisson Equation

5 dynamic vorticity boundary condition L u Two math properties = u t + u u ν 2 u L u B = n p B Wu 1994, Int. J. Numer. methods fluids Div-Curl problem => Curl-Curl problem u = ω, in Ω ( u) = ω, in Ω u = 0, in Ω u n = ω, on Ω n u = u b, on Ω n u = u b, on Ω Determined by vorticity equation! + ω = 0, in Ω

6 Define discrete operator on MAC grid u 1dl 1 i,j+ i,j+ 2 2 u 1dl 1 i 1,j+ i 1,j+ 2 2 D = u dn = + v i 1 2,j+1dl i 1 2,j+1 v i 1 2,jdl i 1 2,j v i+ 1 ω k = u dl = 2,jdl i+ 1 u 2,j i,j+ 1 2 v 1 i 2,jdl i 1 + u 2,j i,j 1 2 dl 1 i,j+ 2 dl 1 i,j 2 The discrete analog of ( ω) = 0 is preserved u = V ω = 0 τ

7 Parabolize the elliptic problem u + ( u) = ω τ Two-level implicit Euler scheme Pascazio et al., Computers & Fluids, 1996 Alternative direction implicit (ADI) Bertagnolio et al., J. Comput.Phys, 1995 Bertagnolio et al., Int.J.Numer. Meth. Fluids, 1998 Lo et al., Int.J.Numer. Meth. Fluids, 2005 u = 0 is not preserved in the iteration process! Explicit time stepping Stability condition: t h u n+1 = u n + τ[ ( u n ) ω] Equivalent to weighted Jacobi iteration in cartesian grid

8 Why using multigrid? u n+1 = Ru n + C spectral raadius: ρ R = max (eig(r)) Jacobi iteration method is hard to damp long waves the smaller, the better!

9 Geometric multigrid method u n+1 = u n + τ[ ( u n ) ω] fine h => Coarse 2h: I h 2h nothing required coarse 2h => fine h: I h 2h divergence free interpolation

10 fine => Coarse : transfer smoth error (small k) u τ + y ( v x u y ) = ω y v τ x v x u y = ω x 1D 3D: tensor product A_I h 2h B_I h 2h C_I h 2h

11 Smooth mode Spectral property of I h 2h w k h = sin ( jkπ n ) A_I h 2h w k 2h = sin ( 2jkπ n ) A_I 2h h w h k = 1 4 sin (j 1)kπ + 1 n 2 = cos 2 kπ 2n sin 2jkπ n A_I 2h h w h k = sin 2 kπ 2n w j 2h w h k = sin sin jkπ n = cos 2 kπ 2n w j 2h j n k π n sin (j + 1)kπ n = ( 1) n+1 w k h oscillatory mode

12 Spectral property of I h 2h A_I h 2h w k h = cos 2 kπ 2n w k 2h A_I 2h h w h k = sin 2 kπ 2n w k 2h B_I h 2h w k h = cos kπ 2n w k 2h B_I 2h h w h k = sin kπ 2n w k 2h C_I h 2h w k h = cos 3 kπ 2n w k 2h C_I h 2h w k h = sin 3 kπ 2n w k 2h

13 After some coarse grid process coarse => fine : transfer smooth correction (small k) M_I h 2h w 2h k = cos kπ 2n w k h sin kπ 2n cos jπk n N_I h 2h w 2h k = (cos kπ 2n sin 2kπ n sin kπ 2n )w k h +( sin kπ 2n sin 2kπ n kπ jπk cos ) cos 2n n Extension to 3D, Li et al., J. Comput. Phys., 2004

14 Bulk property Define A h u h = f as fine grid discretized equation A 2h u 2h = f as coarse grid TG = (I I h 2h A 2h ) 1 I 2h h A h R ν h h w k I h 2h w k h = c k w k h I h 2h w k h = s k w k h TGw k h = I c k cos 2 kπ 2n I h 2h 2h w k dangerous! For B_I h 2h : s k = sin kπ 2n TGw h k = I s k sin 2 kπ 2n For A_I h 2h : s k = sin 2 kπ 2n I h 2h 2h w k For C_I h 2h : s k = sin 2 kπ 2n

15 Numerical test I : wave1 and wave8 error damping process

16 Evolution of Fourier modes The long wave is hard to damp

17 Test II: monople wave

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19 Beyond stability condition 0.5Δτ Δτ t h Special treatment on interface

20 ω x t + vω x y Vorticity equation ω t + ω a u + wω x z = Bk + ν ( ω) = uω y y + uω z z + F x ω t = 0 ω y t + uω y x + wω y z = vω x x + vω z z + F y ω z t + uω z x + vω z y = wω x x + wω y y + F z Degenerate to 2D problem! 3 th order QUICK advection scheme 5 th order WENO advection scheme

21 Periodic Flow Advection Test --- Periodic in x direction WENO 5 -FV QUICK

22 MPI parallel speedup (on Tianhe 1A cluster) 80 cores, X 25

23 Driven cavity flow test: Re=100 y component of vorticity on x-z cross-section This model Lo et al.,2005, IJNFM

24 Re=400

25 Re=1000

26 Re=2000: 100*100*100 This model Lo et al.,2005

27 2D- double shear test: 256*256 This model t=1.8 Zhang et al., 2004, JCP

28 2D- double shear test: 256*256 t=1.793 This model t=1.8 Zhang et al., 2004, JCP

29 2D Vortex Merge Re = 1000

30 Density current

31 Rayleigh-Benard Convection Ra=1e5, Pr=7

32 Rotating Rayleigh-Benard convection test

33 Experimental Setup Rotating table Controller Water chiller

34 Water tank - photo Rail for laser motor for rail Tank for Water-cooling platinum sensor Tubes connected to chiller Red copper plate Heating board of casting Al Tank for experiment

35 Velocity Measurement Particle Image Velocimetry (PIV) CCD CCD (another apparatus) CCD laser laser laser Pr / 7.34 Ra g TH / Ta H /

36 3D Vertical vorticity measurement Descending model Ascending

37 The rotating instrument