乱流の数値解析 - kz.tsukuba.ac.jp

1 12010 ~ ~ 22 11 11 1 62010 0- 1- 2- LES 72010 (Turbulence) (Turbulent Flow) Leiseur, Turbulence In Fluids, 2nd ed., Kluwer Academics Publishers, 1991) 82010 1 diffusion fluctuation dissipation interaction ~gulib/kanpo/No32 2 diffusion z z z fluctuation z / dissipation interaction 102010 (ensemble average) (time average) (spatial average) (phase average) (correlation) (spectrum) ()txfpuiii.

2 , , , = 112010 .. fFf+= ()()()() = ==TiTiNkikNidttxfTxFtxfNtxf01, 1lim, 1lim,122010 (). , ..0 ijjiijjijijjijijijjiiiiiiixuxuDDxpxfxuut uxupPpuUu + == + += + = +=+= ( )( ) 132010 ()()()().,.. + ==+ += ++ + = =+ += + + = ( )( )( )( ) 142010 yA y0y0 yB y0y0 + jiijjiijuuxUxU + = jijiRuuoruu =00> <vvuU+uU UllyxyAy0yBUuv < 0, u> 0v> 0, u< 0152010 162010 [1] [3] [4] -[2]=[7] [5]=-[11] [10] [9]=-[12] [8] [6] 172010 ()()()][]/[][2/14/14/13ssmum ===( )() DDCLuC3'=.

3 ( ) ( )( )182010 u, v, w, p x, y, z, t RT 192010 LES DNS DNS RANS RSM 0 ( )1 2 (k- ),(k-l), 202010 ()()..0 + =+ += + + = ijjijijijijijijijiiixUxUTTPxFuuUUxtUxU ( )( )( ) uiuj 6 212010 ()(). + += ++ + iijijijijijijijijixupxfuuuuuUUuxtu ui ( )( ) ( ) i uj ( ) j ui ().2kjkiijjiijjikijkkjikkjikijkjijikkjik kjixuxufufuxuxupxUuuxUuuxuupuuuuuxuuUxtu u ++ + + + ++ = + ( )[1][2][3][4][5][6][7][8][9]222010 Convection [1]local rate of change of uiuj[2]advectionDiffusion[3]kinetic diffusion[4]pressure diffusion[5]viscous diffusionSource / Sink [6]production by mean strain[7]pressure strain correlation[8]production by body force[9]viscous dissipation232010.

4 0,2= =iiiixuuuk i= j ..2kikiiikiikkkiikkkkxuxufuxUuuxkpuuuuxk Uxtk = + + = + [1][2][3][4][5][6][8] [9]( ) [7] 0 [7] 242010 xk .2222122222 + + = + kjikijkjijkikkjkijijikkjikkkikikikijjjjx xuxuxuxuxuxuxuxuxUxuuxxUxxxuxpxxuxuuxUxt [1][2][3][4][5][6][7][8][9]( )kixu [1]local rate of change of [2]advection[3]kinetic diffusion[4]pressure diffusion[5]viscous diffusion[6]production by mean strain[7]production by mean rate of strain[8]production by turbulent rate of strain[9]decay of 252010 jjiuuu ,()()(). ++ + + + ++ = + + += + + ( )( ) (Closure problem)() XuUxtjjj=+ + 262010 1ccuiij 2 2.

5 Cccppxuxujiijjiij 31,232 , 2= = + =ij eddy viscosity TTkkiTjiVLuukUuu2= ( )272010 ( ) 2.,32,232 TTTT jiijjiTjiTTTkkjiijjiTjiVcLkxUxUuuVLcuuxU xUuu== + = = + = c T (eddy viscosity) T (kinematic eddy viscosity)( )LT, VT LTVT0 1 2 Boussinesq 282010 292010 0- 00> <vvPrandtl uU+uU UlldydU/y0 UluyA ()dydUlUuUuyA/ + +=yB ()dydUlUuUuyB/ =()dydUlVvuT/= uv (Mixing length)yxyAy0yBUuv < 0, u> 0v> 0, u< 0dydUldydUdydUluvTT222= = lLT=( )302010 lcl=( ) cl= ( ) ( ) 222) (1) (11 +=lyl = = ( ) Van-Driest wu=* =++Ayylexp1 ( )

6 *yuy=+ w u* A+=25 26312010 minmaxUUVclTl == ( ) cl= cl= cl= 1- ( ) [6] [7] [3],[4] k 22/12/121,,~iTTTTukkCLclLkV=== l() + + = + kkTkTdkiikTkkxkxLkCxUxUkUxtk 2/3C, Cd, k k ( )[6][7][3],[4]332010 2- LT VT 22/12/121,~,~iTTTukClklLkV== lk (lnkm) n, m 2- nm 2 -21 -13/2 -11/2 lkLVflklkLVVLV lklklkLVklTTTTTTTTTT2/12/12/3322/3222/12 22~/~~~===== k- 342010 k- lu32- 2 .,~,~~22/12/3constCkCkVklLTTT== C , k 352010 k- + + = + + = kTkkikiikTkkTkkikiikTxxkCxUxUxUkCtxkxxUx UxUtk 221( )( ) C , C 1, C 2, k, C C C 2 C 1 C = , C 1= , C 2= , k= , = k- 2- - k- k =++ + = 31,2332ijmkkmmijjiijjijiRSkxUxUkCuu 372010 kijkkjikijkSkkijijkkijjiijijjixUuuxUuuPx CPPC uuuuCPDtuuD = + = 31313221[ ][ ][ - ][ ]( )

7 Uk, , k, 382010 Large Eddy Simulation (LES) Coherent structure ..N-S Large eddySub-grid scale eddy392010 LES ()()''''12jijjijijjijijiijijiuuxuuuuxuuu uxuxpuuxtu + = + ( )N-S ' iiiuuu+= 2'',32''jisgsijsgsijjisgsjiuukkxuxuuu= + = ( )sgssgsk Sub-grid scale Sub-grid scale Smagorinsky model Vorticity model Turbulent energy model402010 LES DNS DNS RANS RSM 0 ( )1 2 (k- ),(k-l), jjxUtdtd + = 2iiUUdtd 2iiuudtd dtdE || || || [10] jjxx [1] PUxij [3] ()ijijUuux [4] ()jiijDUx [5] jijixUD [11] jijixUD [2] jijixUuu [7] jijixUuu [9] jijixud [12] jijixud [8] jijjdux [6] + 2iijjuupux