: LES Energy RANS Kawai & Lele JCP (2010) Wavenumber Accuracy LES, DNS RANS 1970 s Efficiency
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1 MHD (ISAS), (JAXA) kawai@flab.isas.jaxa.jp hp://flab.eng.isas.jaxa.jp/member/kawai JAXA
2 : LES Energy RANS Kawai & Lele JCP (2010) Wavenumber Accuracy LES, DNS RANS 1970 s Efficiency
3 FDS, FVS, AUSM: 1970 s Q n+1 j = Q n+1 j + x E j+1/2 E j 1/2 E j+1/2 = 1 2 [E j+1 + E j A ave (Q j+1 Q j )] Turbulence: K-H insabiliy w/u Exac FDS High-order cenral x/r c
4 : non-dissipaive : FDS, FVS, AUSM 5 4 Densiy Reference FDS x WENO (Jiang & Shu JCP 1996) WCNS (Deng & Maekawa JCP 1997) Universiy of Chicago LAD (Kawai & Lele JCP 2008,2010) Soshi Kawai, ISAS/JAXA
5 ESA NASA B =0 NASA
6 MHD B =0 Eigh-wave formulaion Powell NASA repor (1994) - Projecion Brackbill & Barnes JCP (1980) - B =0 Consrained Transpor - 2 Evans & Hawley APJ (1988) B =0
7 LAD Kawai e al. JCP (2008, 2010) u + ( uu + p ar ( u) ) =0 u u
8 LAD Kawai e al. JCP (2008, 2010) u + ( uu + p ar ( u) ) =0 u u Kawai & Lele AIAA J. (2010) B B B B
9 B B B B B x 0 A z A y B y = RHS phyb + A z 0 A x B z A y A x 0 A = A x A y = ar B = ar J A z
10 B B B B B x B y = RHS phyb + B z : : ( B) ( B) =0 0 A z A y A z 0 A x A y A x 0 2 (A z A z ) x y A = ar B = ar J + 2 (A y A y ) x z = J = RHS phyj + 2 (A) ( A) + 2 (A x A x ) y z J = RHS phyj + 2 ( ar J) Soshi Kawai, ISAS/JAXA
11 ar = C ar 1 a 4 J x 4 r J 2 x r x r+3 B x B y = RHS phyb + B z J A z A y A z 0 A x A y A x 0 A = ar B = ar J x J ar PDE ( B) =0 (D B) n+1 =(D B)n =0
12 B (n[i,j,k]) ( B) n = 1 x DL Bx,i [B x]+ 1 y DL By,j [B y]+ 1 z DL Bz,k [B z] B x (n[i,j,k]) B n+1 x, = Bn x, y DL RHS,j [RHS]+D L Az,j [A z] + z DL RHS,k [RHS]+D L Ay,k [A y] 1x DL Bx,i Inducion eqs.: B x B y = RHS phyb + B z 0 A z A y A z 0 A x A y A x 0 1x DL Bx,i B n+1 x, = 1 x DL Bx,i B n x, + x y DL Bx,i x z DL Bx,i D L RHS,j [RHS]+D L Az,j [A z] D L RHS,k [RHS]+D L Ay,k [A y]
13 B (n[i,j,k]) ( B) n+1 =( B)n + x y {F xy} + x z {F xz} + y z {F yz} if F xy = F xz = F yz =0 ( B) n+1 =( B)n =( B)0 F xy = D L Bx,i +D L By,j D L RHS,j [RHS]+D L Az,j [A z] D L RHS,i [RHS]+D L Az,i [A z] if D L Bx,i DL Az,j = DL By,j DL Az,i F xy = F xz = F yz =0
14 D L Bx,i DL Az,j = DL By,j DL Az,i D L Bx,i f = N a, p=m a, a p, f (i+p+1,j,k) f (i+p,j,k) D L Az,j f = D L By,j f = D L Az,i f = N b, q=m b, b q, f (i,j+q+1,k) f (i,j+q,k) N c, q=m c, c q, f (i,j+q+1,k) f (i,j+q,k) N d, p=m d, d p, f (i+p+1,j,k) f (i+p,j,k) D L Bx,i DL Az,j f = N a, p=m a, N b,(i+p+1,j,k) q=m b,(i+p+1,j,k) a p, b q,(i+p+1,j,k) f (i+p+1,j+q+1,k) f (i+p+1,j+q,k) D L By,j DL Az,i f = N c, q=m c, N d,(i,j+q+1,k) p=m d,(i,j+q+1,k) c q, d p,(i,j+q+1,k) f (i+p+1,j+q+1,k) f (i+p,j+q+1,k) a m, = d m,(i,j+n,k) k i f ( OK!) 6 j i Soshi Kawai, ISAS/JAXA
15 PDE B x B y = RHS phyb + B z 0 A z A y A z 0 A x A y A x 0 4 J A = ar B = ar J ar x 4 PDE OK 6 k i f j Soshi Kawai, ISAS/JAXA i
16 : C 1) ar = C 1 a r J 2 x r x r+3 B x B y = RHS phyb + B z 0 A z A y A z 0 A x A y A x 0 A = ar B = ar J 2) : Alfven 3) : 4) : Orszag-Tang
17 C 1D slow swich-off shock problem Falle e al. MNRAS (1998) ar = C 1 a r J 2 x r x r+3 J = RHS phyj + 2 ( ar J) By x C =0 C =1 C = 10 C = 100 Wiggles ampliude(%) Thickness Wiggles C Shock hickness (grid poins)
18 Alfven Exac N=4 N=8 N=16 = 45 2N B N x N=8 B = N=16 Soshi Kawai, ISAS/JAXA Toh JCP (2000)
19 h-order slope Error Y L error + L 2 error X
20 Dai-Woodward 1.6 = 45 Dai & Woodward JCP (1994) Lef condiions, u,u,u z,p,b,b,b z L =(1.08, 1.2, 0.01, 0.5, 0.95, Righ condiions 2/ 4, 3.6/ 4, 2/ 4 ), u,u,u z,p,b,b,b z R =(1, 0, 0, 0, 1, 2/ 4, 4/ 4, 2/ 4 ) B D (N=4000) 1D HLLD (N=800) Miyoshi & Kusano JCP (2005) 2D presen (N=800) x x B 0 Soshi Kawai, ISAS/JAXA
21 Orszag-Tang Orszag & Tang JFM (1979) Tempraure Soshi Kawai, ISAS/JAXA
22 Orszag-Tang Orszag & Tang JFM (1979) Temperaure Arificial diffusion A = ar J
23 Orszag-Tang Orszag & Tang JFM (1979) Temperaure Divergence B field B = 10 12
24 y =0.5 1 N=1600 N=200 N=400 N=800 Temperaure T = p/ ar Arificial diffusion x x Soshi Kawai, ISAS/JAXA
25 MHD - - PDE - =0 pressure u-velociy X1 X2 X3 X1 X2 X3 MHD =20 =40 =60 =80 =100 Soshi Kawai, ISAS/JAXA
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