Wave propagation methods for hyperbolic problems on mapped grids

Transcription

1 Wave propagation methods for hyperbolic problems on mapped grids A France-Taiwan Orchid Project Progress Report Keh-Ming Shyue Department of Mathematics National Taiwan University Taiwan ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 1/47

2 Talk objective Review a body-fitted mapped grid approach for numerical approximation of hyperbolic balance laws in multi-d with complex geometries Present numerical results for problems arising from Compressible inviscid gas flow Barotropic cavitating flow Shallow granular avalanches ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 2/47

3 Mathematical model Consider a hyperbolic balance laws of the form N t q ( x, t) + j=1 x j f j (q, x) = ψ(q) in a general multidimensional domain x = (x 1, x 2,...,x N ): spatial vector, q lr m : vector of m state quantities f j lr m : flux vector, t: time ψ lr m : source terms Model is assumed to be hyperbolic, where N j=1 α j ( f j / q) is diagonalizable with real e-values α j lr ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 3/47

4 Mathematical model (Cont.) In a body-fitted mapped grid approach, we introduce a coordinate change x ξ via ξ = (ξ 1, ξ 2,...,ξ N ), ξ j = ξ j ( x) that transform a physical domain Ω to a logical one ˆΩ, see below when N = 2, x physical domain Ω x 1 mapping ξ 1 = ξ 1 (x 1, x 2 ) ξ 2 = ξ 2 (x 1, x 2 ) ξ logical domain ˆΩ ξ 1 ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 4/47

5 Mathematical model (Cont.) In a body-fitted mapped grid approach, we introduce a coordinate change x ξ via ξ = (ξ 1, ξ 2,...,ξ N ), ξ j = ξ j ( x) that transform a physical domain Ω to a logical one ˆΩ, so equations into the form and N q t + j=1 f j ξ j = ψ(q) with f j = N k=1 f k ξ j x k ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 4/47

6 Mathematical model (Cont.) Basic coordinate mapping relations in N = 3 are J x1 ξ 1 x2 ξ 1 x3 ξ 1 = 1 0 J 11 J 21 J 31 0 x1 ξ 2 x2 ξ 2 x3 ξ 2 J 0 J 12 J 22 J 32 0 x1 ξ 3 x2 ξ 3 x3 ξ 3 0 J 13 J 23 J 33 where J = (x 1, x 2, x 3 )/ (ξ 1, ξ 2, ξ 3 ) = det( (x 1, x 2, x 3 )/ (ξ 1, ξ 2, ξ 3 )), J 11 = J 12 = J 13 = (x 2, x 3 ) (ξ 2, ξ 3 ) (x 2, x 3 ) (ξ 3, ξ 1 ) (x 2, x 3 ) (ξ 1, ξ 2 ), J 21 =, J 22 =, J 23 = (x 1, x 3 ) (ξ 3, ξ 2 ) (x 1, x 3 ) (ξ 1, ξ 3 ) (x 1, x 3 ) (ξ 2, ξ 1 ), J 31 =, J 32 =, J 33 = (x 1, x 2 ) (ξ 2, ξ 3 ) (x 1, x 2 ) (ξ 3, ξ 1 ) (x 1, x 2 ) (ξ 1, ξ 2 ),,. ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 5/47

7 Compressible Euler equations t Cartesian coordinate case ρ ρu i E N + j=1 x j ρu j ρu i u j + pδ ij Eu j + pu j = 0 ρ xi φ ρ u φ, i = 1,...,N Generalized coordinate case t ρ ρu i E N + j=1 ξ j ρu j ρu i U j + p xi ξ j EU j + pu j = 0 ρ xi φ ρ u φ ρ: density, p = p(ρ, e): pressure, e: internal energy E = ρe+ρ N j=1 u2 j /2: total energy, φ: gravitational potential U j = t ξ j + N i=1 u i xi ξ j : contravariant velocity in ξ j -direction ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 6/47

8 Finite volume approximation Employ finite volume formulation of numerical solution Q n ijk 1 ξ 1 ξ 2 ξ 3 C ijk q(ξ 1, ξ 2, ξ 3, t n ) dv that gives approximate value of cell average of solution q over cell C ijk at time t n (sample case in 2D shown below) physical domain logical domain x 2 j + 1 j i 1 Ĉ ij i mapping x 1 = x 1 (ξ 1, ξ 2 ) x 2 = x 2 (ξ 1, ξ 2 ) j + 1 j ξ 2 ξ 2 ξ 1 C ij x 1 ξ 1 i 1 i ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 7/47

9 Finite volume (Cont.) In three dimensions N = 3, equations to be solved take q N t + j=1 f j ξ j = ψ(q) A simple dimensional-splitting method based on wave propagation approach of LeVeque et al. is used, i.e., Solve one-dimensional Riemann problem normal at each cell interfaces Use resulting jumps of fluxes (decomposed into each wave family) of Riemann solution to update cell averages Introduce limited jumps of fluxes to achieve high resolution ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 8/47

10 Finite volume (Cont.) Basic steps of a dimensional-splitting scheme ξ 1 -sweeps: solve q t + f 1 ξ 1 = 0 updating Q n ijk to Q ijk ξ 2 -sweeps: solve q t + f 2 ξ 2 = 0 updating Q ijk to Q ijk ξ 3 -sweeps: solve q t + f 3 ξ 3 = 0, updating Q ijk to Qn+1 ijk ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 9/47

11 Finite volume (Cont.) Consider ξ 1 -sweeps, for example, First order update is Q ijk = Q n ijk t [ (A + ξ 1 Q ) n + ( ] A i 1/2,jk 1 Q) n i+1/2,jk 1 with the fluctuations (A + 1 Q)n i 1/2,jk = m:(λ 1,m ) n i 1/2,jk >0 (Z 1,m ) n i 1/2,jk and (A 1 Q)n i+1/2,jk = m:(λ 1,m ) n i+1/2,jk <0 (Z 1,m ) n i+1/2,jk (λ 1,m ) n ι 1/2,jk & (Z 1,m) n ι 1/2,jk are in turn wave speed and f-waves for the mth family of the 1D Riemann problem solutions ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 10/47

12 Finite volume (Cont.) High resolution correction is with ( F 1 ) n i 1/2,jk = 1 2 Q ijk := Q ijk t ξ 1 m w m=1 [ ( F1 ) n [ sign (λ 1,m ) ( ) n F1 i+1/2,jk i 1/2,jk ( 1 t ) ] n λ 1,m Z 1,m ξ 1 i 1/2,jk ] Z ι,m is a limited value of Z ι,m It is clear that this method belongs to a class of upwind schemes, and is stable when the typical CFL (Courant-Friedrichs-Lewy) condition: ν = t max m (λ 1,m, λ 2,m, λ 3,m ) min( ξ 1, ξ 2, ξ 3 ) 1, ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 11/47

13 Smooth vortex flow: accuracy test Compressible Euler equations with ideal gas law Initial condition for vortex is set by ρ = ( 1 p = ρ γ, ) 1/(γ 1) 25(γ 1) 8γπ 2 exp(1 r 2 ), u 1 = 1 5 2π exp((1 r2 )/2) (x 2 5), u 2 = π exp((1 r2 )/2) (x 1 5), r = (x 1 5) 2 + (x 2 5) 2 E z 1, = z comput z exact 1, : discrete 1- or maximum-norm for state variable z ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 12/47

14 Smooth vortex flow: accuracy test Density contours on a grid at time t = Cartesian grid 10 Quadrilateral grid x2 x x x 1 ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 13/47

15 Smooth vortex flow: accuracy test Cartesian grid results Mesh size Order E ρ ( 1) ( 1) ( 2) ( 2) 1.97 E p ( 1) ( 1) ( 2) ( 2) 1.96 E u (00) ( 1) ( 1) ( 2) 1.99 E u (00) ( 1) ( 1) ( 2) 2.02 E ρ ( 1) ( 2) ( 3) ( 3) 2.00 E p ( 1) ( 2) ( 2) ( 3) 1.98 E u ( 1) ( 1) ( 2) ( 3) 2.01 E u ( 1) ( 2) ( 2) ( 3) 2.00 ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 14/47

16 Smooth vortex flow: accuracy test Quadrilateral grid results Mesh size Order E ρ (00) ( 1) ( 1) ( 2) 1.36 E p (00) ( 1) ( 1) ( 2) 1.33 E u (00) ( 1) ( 1) ( 1) 1.56 E u (00) ( 1) ( 1) ( 1) 1.44 E ρ ( 1) ( 2) ( 2) ( 3) 1.59 E p ( 1) ( 2) ( 2) ( 3) 1.63 E u ( 1) ( 1) ( 2) ( 2) 1.57 E u ( 1) ( 2) ( 2) ( 2) 1.50 ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 15/47

17 Shock waves over circular array A Mach 1.42 shock wave in water over a circular array t=0 shock ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 16/47

18 Shock waves over circular array Grid system time= ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 17/47

19 Shock waves over circular array Contours for density t= ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 18/47

20 Steady state shock tracking A Mach 3 compressible gas flow over a 20 o ramp Exact shock location (an oblique shock with 37.8 o ) is inserted into underlying grid for computation Grid Mach x ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 19/47

21 Compressible Multiphase Flow Homogeneous equilibrium pressure & velocity across material interfaces Volume-fraction based model equations (Shyue JCP 98, Allaire et al. JCP 02) t (α iρ i ) + 1 J t (ρu i) + 1 J E t + 1 J α i t + 1 J N d j=1 N d j=1 N d j=1 N d j=1 ξ j (α i ρ i U j ) = 0, i = 1, 2,..., m f ξ j (ρu i U j + pj ji ) = 0, i = 1, 2,...,N d, ξ j (EU j + pu j ) = 0, U j α i ξ j = 0, i = 1, 2,..., m f 1; ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 20/47

22 Moving cylindrical vessel Initial setup time=0 interface air helium ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 21/47

23 Moving cylindrical vessel Solution at time t = 0.25 ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 22/47

24 Moving cylindrical vessel Solution at time t = 0.5 ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 23/47

25 Moving cylindrical vessel Solution at time t = 0.75 ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 24/47

26 Moving cylindrical vessel Solution at time t = 1 ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 25/47

27 Moving cylindrical vessel cross-sectional plot along boundary 3 t = 0.25 t = 0.5 t = 0.75 t = ρ p θ(π) θ(π) θ(π) θ(π) ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 26/47

28 Barotropic cavitating flow A relaxation model of Saurel et al. (JCP 090 t (α 1ρ 1 ) + t (α 2ρ 2 ) + t (ρu i) + α 1 t + N j=1 N j=1 N j=1 N j=1 x j (α 1 ρ 1 u j ) = 0, x j (α 2 ρ 2 u j ) = 0, x j (ρu i u j + pδ ij ) = 0, u j α 1 x j = 1 µ (p 1 (ρ 1 ) p 2 (ρ 2 )) i = 1,...,N Each phasic pressure p ι satisfies p ι (ρ) = (p 0 + B) (ρ/ρ 0 ) γ B Mixture pressure p satisfies p = α 1 p 1 + α 2 p 2, µparameter ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 27/47

29 Mixture speed of sound Wood formula (stiffness in c vs. α) 1 ρ c 2 = α ρwc 2 w + 1 α ρgc 2 g ρw = 10 3 kg/m 3, cw = m/s, ρg = 1.0kg/m 3, cg = 374.2m/s c ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 28/47

30 Cavitation test: 1D Homogeneous gas-liquid mixture with αg = 10 2 u 1 = 100m/s u 1 = 100m/s ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 29/47

31 Cavitation test: 1D Formation of cavitating gas bubble 10 x p 5 cavitation u ρ 500 α x x ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 30/47

32 Cavitation test: 2D steady state Uniform gas-liquid mixture with speed 600m /s over a circular region ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 31/47

33 Cavitation test: 2D steady state Pseudo colors of volume fraction Formation of cavitation zone (Onset shock induced, diffusion,...?) ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 32/47

34 Cavitation test: 2D steady state Pseudo colors of pressure Smooth transition across liquid-gas phase boundary ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 33/47

35 Cavitation test: 2D steady state Pseudo colors of volume fraction: 2 circular case Convergence of solution as the mesh is refined? t=0.05 ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 34/47

36 Shallow granular avalanches Depth-average Savage-Hutter equations h t + N j=1 t (hu i) + x j (hu j ) = 0, N j=1 x j (hu i u j + 12 β xh 2 δ ij ) = hψ i, i = 1,...,N, where Mohr-Coulomb closure is used with β x = K x cosζ, Kx if u > 0 K x = K x + if u < 0, K ± x = 2 cos 2 φ ( 1 ± ψ i = sinζδ 1i u i u tanδ ( ) cosζ + κu 2 B 1 cosζ x i 1 cos2 φ cos 2 δ ) 1, ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 35/47

37 Earth pressure coefficients Jump discontinuity on K x, K + x K x 0 (see below where δ = 30 o is used as a reference) K x + K x K ± x φ (degree) ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 36/47

38 Avalanche on an inclined channel Hemispherical granular material Parameters: ζ = 35 o, φ = 30 o, δ = 30 o t = 0 ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 37/47

39 Avalanche on an inclined channel Contour plots for granular height (normal to channel) 6 t = ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 38/47

40 Avalanche on an inclined channel Down-flow phase 6 t = ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 38/47

41 Avalanche on an inclined channel Down-flow phase 6 t = ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 38/47

42 Avalanche on an inclined channel Down-flow phase 6 t = ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 38/47

43 Avalanche on an inclined channel Deposit phase 6 t = ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 38/47

44 Avalanche on an inclined channel Deposit phase 6 t = ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 38/47

45 Avalanche on an inclined channel Deposit phase 6 t = ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 38/47

46 Avalanche on an inclined channel Deposit phase 6 t = ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 38/47

47 Avalanche on an inclined channel Deposit phase 6 t = ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 38/47

48 Avalanche on an inclined channel Cross-sectional plot along the channel t = 0 ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 39/47

49 Avalanche on an inclined channel Down-flow phase t = 3 ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 39/47

50 Avalanche on an inclined channel Down-flow phase t = 6 ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 39/47

51 Avalanche on an inclined channel Down-flow phase t = 9 ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 39/47

52 Avalanche on an inclined channel Deposit phase t = 12 ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 39/47

53 Avalanche on an inclined channel Deposit phase t = 15 ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 39/47

54 Avalanche on an inclined channel Deposit phase t = 18 ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 39/47

55 Avalanche on an inclined channel Deposit phase t = 21 ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 39/47

56 Avalanche on an inclined channel Deposit phase t = 24 ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 39/47

57 Avalanche on an inclined channel Hemispherical granular material Parameters: ζ = 35 o, φ = 40 o, δ = 30 o t = 0 ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 40/47

58 Avalanche on an inclined channel Contour plots for granular height (normal to channel) 6 t = ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 41/47

59 Avalanche on an inclined channel Down-flow phase 6 t = ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 41/47

60 Avalanche on an inclined channel Down-flow phase 6 t = ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 41/47

61 Avalanche on an inclined channel Down-flow phase 6 t = ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 41/47

62 Avalanche on an inclined channel Deposit phase 6 t = ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 41/47

63 Avalanche on an inclined channel Deposit phase 6 t = ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 41/47

64 Avalanche on an inclined channel Deposit phase 6 t = ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 41/47

65 Avalanche on an inclined channel Deposit phase 6 t = ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 41/47

66 Avalanche on an inclined channel Deposit phase 6 t = ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 41/47

67 Avalanche on an inclined channel Cross-sectional plot along the channel t = 0 ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 42/47

68 Avalanche on an inclined channel Down-flow phase t = 3 ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 42/47

69 Avalanche on an inclined channel Down-flow phase t = 6 ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 42/47

70 Avalanche on an inclined channel Down-flow phase t = 9 ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 42/47

71 Avalanche on an inclined channel Deposit phase t = 12 ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 42/47

72 Avalanche on an inclined channel Deposit phase t = 15 ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 42/47

73 Avalanche on an inclined channel Deposit phase t = 18 ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 42/47

74 Avalanche on an inclined channel Deposit phase t = 21 ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 42/47

75 Avalanche on an inclined channel Deposit phase t = 24 ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 42/47

76 Avalanche on an inclined channel Pseudo colors of velocity divergence: Down-flow phase ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 43/47

77 Avalanche on an inclined channel Pseudo colors of velocity divergence: Down-flow phase ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 43/47

78 Avalanche on an inclined channel Pseudo colors of velocity divergence: Deposit phase ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 43/47

79 Avalanche on an inclined channel Pseudo colors of velocity divergence: Deposit phase ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 43/47

80 Steady state ramp computation A Froude 7 shallow granular flow over a 24.9 o ramp Parameters: ζ = 32.6 o, φ = 38 o, δ = 31 o ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 44/47

81 Steady state ramp computation A Froude 7 shallow granular flow over a 24.9 o ramp Parameters: ζ = 32.6 o, φ = 38 o, δ = 31 o ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 45/47

82 Steady state ramp computation Cross-sectional plot along ramp with three different φ h 5 φ=31 φ=38 φ= p u x ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 46/47

83 Future direction Numerical methodology Vacuum (dry) state treatment Flux & source terms well-balanced Interface sharping by techniques such as Lagrange-like moving mesh or front tracking Applications Relaxation model as applied to more practical cavitation problems General depth-average models to granular flows ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 47/47

84 Future direction Numerical methodology Vacuum (dry) state treatment Flux & source terms well-balanced Interface sharping by techniques such as Lagrange-like moving mesh or front tracking Applications Relaxation model as applied to more practical cavitation problems General depth-average models to granular flows Thank You ISCM II & EPMESC XII, 29 November 3 December, 2009 p. 47/47