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1 (eddy ) Navier-Stokes Navier-Stokes du dt u t +uru = 1 ; rp+r u (10.1) u p (kinematic viscosity) (convection term) 1 (viscous diusion term) d=dt =t+ur (substantial derivative) xt (interior derivative) (10.1) Bernoulli t Z t+t ; 1 u(x+x t+t) =u(x t)+ ; t rp+r u t (10.) x = R t+t ut (10.) t (uid particle) t x R t+t f;(1=)rp+r ugt t r u u r u u u (10.) u 1 (Reynolds number) Re = UL= 1 1
2 (10.1) [ ]/[ ] U L (hypersonic) Navier-Stokes r ru =0 (vorticity transport equation) d dt t +(ur) =(r)u + r (10.3) 1 (generation term ) (10.3) ( r)u 10.1: x (x) x 0 (x 0 ) a x 0 +a Taylor (x 0 +a) =(x 0 )+(ar)(x 0 )+ (1=!)(ar) (x 0 ) A u A B u+(r)u A u C +=t+(r) = +d=dt (10.3) d=dt =(r)u u+( r)u +d=dt D (frozen in) AB CD A B ( r)u AB (vortex laments) 3 AB (isotropic turburence) (anisotropy) [ ] U =L [ ] U=L LU 3 (constancy of circulation)
3 3 (r)u r u (boundary layer ow) TS (Tollmien-Schlichting wave) (hairpin vortex) (head) (leg) (bursting vortex) (ejection) (sweep) (horseshoe vorex) (kinetic theory) (eddy) (mixing length) (turbulence structure) Rosenhead(1931) Townsend(1949) 4 5 (DNS, direct numerical simulation) DNS (turbulent ow database) (helicity) 6 Stanford Illiac IV Navier-Stokes 198 NASA Ames Moin-Kim Illiac IV (turbulent shear ow) 4 interferogram( ), Schlieren method( ), shadowgraph( ) H = ju j=jujjj H =0 H 6= 0
4 4 (coherent) DNS (1984{ ) DNS DNS Re = Re =410 5 Gortler (free turbulent shear layer) (turbulent mixing layer) (roller) (rib) (shock waves) Benard (sophisticated model)
5 5 10. (DNS) NS NS NS (Reynolds stress) NS 7 (turbulence model) (modeling) NS u i t + u j u i = ; p + F i + n u i j +u o j i ; x j x i x j 3 iju k k (10.4) u i j = u i =x j Boussinesq F i u i u i +u 0 i (10.5) (10.4) (Reynolds equation) t u i + u j u i = ; p + F i + x j x i x j ; ui j +u j i ;u 0 i u 0 j (10.6) ;u 0u0 i j (Reynolds stress) u 0 i = u ju 0 i =0 u k k =0 (ensemble average) 8 (10.5) (Reynolds decomposition) (Reynolds average) (10.6) NS NS NS 7 close second-moment closure 8
6 6 ( ) (eddy viscosity model) (rate of strain tensor) S 9 (isotropic) (anisotropic) 3 u 1 1 u 1 +u 1 u 3 1 +u 1 3 S = 6 4u 1 +u 1 u u 3 +u u 1 3 +u 3 1 u 3 +u 3 u = u 1 1 ;(=3)u k k u 1 +u 1 u 3 1 +u u 6 k k u 1 +u 1 u ;(=3)u k k u 3 +u u 1 3 +u 3 1 u 3 +u 3 u 3 3 ;(=3)u k k (10.4) r r u =0 r u R ;u 0 1 ;u 0 u0 1 ;u 0 u R = 6 4;u 0 u0 1 ;u 0 ;u 0 u ;u 0 3 u0 1 ;u 0 3 u0 ;u 0 3 = ;u 0 ; 3 k (=3)k ;u0 1 u0 ;u 0 1 u0 3 ;u 0 u0 1 ;u 0 +(=3)k ;u0 u0 3 ;u 0 3 u0 1 ;u 0 3 u0 ;u 0 3 +(=3)k k =(1=)u 0 u0 k k (turbulent kinetic energy) Boussinesq ;u i u j = t (u j i +u i j ); 3 ijk (10.7) t (eddy viscosity) u k k =0 (10.6) r =(+ t ) ; (=3)kI I identity ij =(+ t ) u j i +u i j ; 3 ijk (i j =1 3) (10.8) ( ) t 9 U S u 1 1 u 1 u 1 3 u 1 u u 35 1 = 4 u 1 1 u 1 +u 1 u 1 3+u u u 1+u 1 u u 3+u ;u 1 u 1 3 ;u u 1 ;u 1 0 u 3 ;u 3 5 u 3 1 u 3 u 3 3 u 3 1+u 1 3 u 3 +u 3 u 3 3 u 3 1 ;u 1 3 u 3 ;u 3 0
7 7 Franke-Rodi-Schonung(1989) t ( ) q ij =(+ t ) u j i +u i j ; 3 iju k k ; 3 ijk (i j =1 3) (10.9) q i = 1 ; ;1 Pr + t c (i =1 3) (10.10) Pr t x i c Prandtl Pr =0:7 Prandtl Pr t =0:9 (eddy kinematic viscosity) t (= =) [ ][ ] Prandtl (Prandtl's mixing length theory) ( ) t = l u (10.11) y u y l van Driest(1956) n o l = y 1;exp ; y+ A (10.1) =0:41 Karman y + = yu = (wall coordinate) u = p w = w A =6 Prandtle-van Driest f g (viscous sublayer) ( buer layer) van Driest ( inertial sublayer, logarithmic region) l y t q = p k l t = C pkl (10.13) k ; " l (dissipation rate) " = u 0 i k u0 i k t = C k =" (10.14) k " k " t (two-equation turbulence model) 1 (one-equation turbulence model) (algebraic model) k ; " (law of the wall) ( 10.) u + = 1 ln y+ +5:15 (10.15)
8 8 u + = u=u k ; " k ; " (low Reynolds number k ; " model) k " (by-pass transition) 1 (10.14) Karman (10.1) (10.15) 10.:
9 NS (10.4) (10.4) i u 0 j u 0 j (10.4) i u 0 j t u i + u 0 j x k u k u i = ;u 0 j p i + u 0 j F i + u 0 j (ui k +u k i ) x k u 0 j =0 u k k = u 0 k k =0 u 0 j t u0 i + u 0 j u0 k u i k + u k u 0 j u0 i k + u0 j u0 k u0 i k = ; x i p0 u 0 j + p0 u 0 j i + u 0 j f0 i + x k u 0 j u0 i k ; u0 i k u0 j k (10.4) j u 0 i u0 i (10.4) j (transport equation for Reynolds stress) u0u0 i j t + u k u 0u0 i j x = ;u0 u0 j k u i k ;u 0u0 i k u j k + u 0 f0+u0f0 j i i j + p 0 (u 0 j i +u 0 i j ) k [ ] c ij [ ] P ij F ij [ - ] ij ; u 0u0 u0 i j k x +p 0 (kj u 0 i + kiu 0 j ); u 0u0 i j k x k [ ] d ij ; u 0 i k u0 j k (10.16) [ ] ;" ij (10.16) (production, ) - (pressure-strain correlation) (diusion) (viscous dissipation) P ij (10.3) R n [ ]= nr = Rn ( r)u nr (nrr)u n u 0u0 i j u0 u0 j k u i k u 0u0 i k u j k - ij p 0 Poisson r p = ;r(uru) Green p = Z dv ;^um^u l m 4 x l r
10 10 r p dv \^" dv ij = p0 (u 0 j i +u 0 i j )= ij1 + ij 4 Z ^u 0 l^u0 m x l x m (u 0 j i+u 0 i j) dv r + Z ^ul m^u 0 (u0 +u0 m l j i i j 4 ) dv r (10.17) - (mathematical model) (energy cascade) Rotta(1951) ij1 (return-to-isotropy) ij1 = ;c 1 " k u 0u0 i j ; 3 ijk (10.18) c 1 "=k u 0u0 i j ;(=3) ijk (u 0u0 i i i =1 3) (=3)k = u 0 u0 k k =3 (u0u0 i j i 6= j) 0 (contraction) (10.18) U 3 L ij Naot (1970) (isotropizaton of production, IP) ij = ;c P ij ; 3 ijp (10.19) P = P kk = (10.18) ij P ij 1 - ij (redistribution term) Launder ij1 ij 0:3c 1 +c =1 ij1 return ij rapid distortion theory rapid term ij1 slow trem DNS LES d ij r a
11 11 du=dt = r a V Z Z Z d du udv = dt V V dt ZV dv = a r adv = S n ds S V n ds V a u r a a = ru >0 d ij 3 Daly-Harlow(1970) (10.0) d ij = c s x k k " u0 k u0 l (u0 i u0 j ) l c s ; r (k=")r rr k=" R " ij 0 " ij = u 0 i k u0 j k = (=3) u 0 i k u 0 i k (i = j) 0 (i 6= j) = 3 ij" (10.1) " = u 0 u0 i k i k k = u0 i " ; c P ij ; 3 ijp u0u0 i j t + u k u 0u0 i j x = P ij + F ij ; c 1 " k k u 0 i u0 j ; 3 ijk + c s x k k " u0 k u0 l (u0 i u0 j ) l " NS u 0 i l (10.4) i l t " + u j " = ;(u 0 u0 i l j l x + u0 u0 l i l j )u i j ; p 0 u0 l i l j x + i ; 3 ij" (10.) " t " + u i " = " x i k (c "1P + c "3 F )+c " k u0 u0 i l x i " " l ; c " " k (10.3)
12 1 k DNS " " LRR(Launder, Reece and Rodi) 10 (10.) " (10.3) (coecient) c 1 c c "1 c " c " c s c "1 c " c 1 =:0 c =0:6 c s =0: c "1 =1:44 c " =0:18 c " =1:9 k ; " k " k (10.16) k t k + x i u i k = P ; x i 1 u0 i u0 j + p 0 u 0 i ;k i ; " (10.4) [ ] c k [ ] [ ] d k [ ] P = ; u 0 i u0 j u i j - (10.4) t k + x i u i k = P + x i t k k i ; " (10.5) " k t " + " u i " = C "1 x i k P + t " i ; C " " x i " k (10.6) k ; " Launder-Spalding(1974) C =0:09 C "1 =1:44 C " = 1:9 k =1:0 " =1:3 Navier-Stokes LLR k ; " NS Launder, B.E., Reece, G.J. and Rodi, W., Progress in the development of a Reynolds-stress turbulence closure, J. Fluid Mdch., Vol.68(1975), 537{66.
13 " LES 6 4 (LES) (DNS) LES DNS Reynolds (algebraic model) Cebeci-Smith 11 Baldwin-Lomax 1 t = min ; ( t ) in ( t ) out = (t ) in (y y cross ) ( t ) out (y>y cross ) (10.7) y y cross ( t ) in =( t ) out y Prandtl-van Driest (10.11) (10.1) Z ( t ) out = C Cl (u e ;u)dyf Kleb (y) (10.8) 0 C Cl =0:0168 Clauser u e u u(y) 0 u e =(1=u e ) R 0 (u e ;u)dy F Kleb (y) Klebano F Kleb (y) =[1+5:5(y=) 6 ] ;1 (10.9) u =0:995u e y F Kleb 1;F Kleb 11 Cebeci, T. and Smith, A.M.O., "Analysis of Turbulent Boundary Layers.\ 1974, Academic Press. 1 Baldwin,B.S. and Lomax, H., Thin layer approximation and algebraic model for separated trubulent ows. AIAA Paper, 78{75(1978).
14 14 CS Michel R > 1:174 (1+ 400=R x )R 0:46 x (10.30) R = u e = =(1=ue ) R 0 u(u e;u)dy R x = u e x= x (local equilibrium) CS van Driest A =6 (adverse pressure gradient) l (favourable pressure gradient) l t F tr (x) Clauser C Cl =0:0168 [ CS ] Baldwin-Lomax Prandtl-van Driest ( t ) in = l jj (10.31) l = yf1;exp(;y + =A + )g = ru =0:4 Karman y + A + =6 Clauser ( t ) out =1:6C Cl F wake F Kleb (y) (10.3) C Cl =0:0168 Clauser F wake = min ; y max F max 0:5y max u dif =F max F Kleb (y) =[1+5:5(0:3y=y max ) 6 ] ;1 (10.33a) (10.33b) y max F (y) =yjjf1;exp(;y + =A + )g F max y exp(;y + =A + )=0 u dif = juj max ;juj min juj min =0 F Kleb Klebano t ( t ) max < 14 1 (10.34) t =0 BL u=y jj Clauser u e y max F max u dif 0:5y max u dif =F max Johnson-King(1985) Johnson, D.A. and King, L.S., A mathematically simple turbulence closure model for attached and separated turbulent boundary layers. AIAA J., Vol.3(1985), 1684{9.
15 15 l k t uv Spalart-Allmaras(199) 14 FEM k ; " k ; " k =(1=)u 0 i u0 i " = u0 i k u0 i k k ; " Launder-Spalding(1974) 15 k " k " k " q = p k l k q " q 3 l ;1 k " 3 k " q m l n T l=q! q=l (specic dissipation rate) Wilcox(1988) k ;! 16 Hutton-Smith-Hickmott(1987) q ; f Mohammadi(1990) ; f q=l! l=q 1=l q k ; " k " (10.4) (10.6) k ; " k ; " t = C f k =" t k + t " + " = ";( x i u i k = P + " u i " = C "1 f 1 x i k P + p k=y) (10.35) n + o t k i ; " ; D (10.36) x i k n + o t " i ; C " f " + E (10.37) x i " k (10.38) t t =0 f " 14 Spalart, P.R. and Allmaras, S.R., A one-equation turbulence model for aerodynamic ows. AIAA Paper, (199). 15 Launder, B.E. and Spalding, D.B., The numerical computation of turbulent ow. Comp. Meth. Appl. Mech. Engng., Vol.3(1974), 69{ Wilcox, D.C., Multiscale model for turbulent ows. AIAA J., Vol.6(1988), 1311{0.
16 16 f 1 f C C "1 C " k " D E " (non-slip) f = f 1 = f =1 D = E =0 " = " k ; " k ; " Jones-Launder(197) 17 f = exp :5 ; f 1 =1:0 f = ;0:3 exp(;rt ) 1+R t =50 C =0:09 C "1 =1:45 C " =:0 k =1:0 " =1:3 D =( p k=y) E = t ( u=y ) R t (= ql=) =k =" (turbulent Reynolds number) k =" =0 Myong-Kasagi(1990) 18 " D = E =0 f = ; 1;exp(;y + =70) 1+ 3:45 p Rt f 1 =1:0 f = ; 1;exp(;y + =5) n 1; 9 exp ; R o t 6 C =0:09 C "1 =1:4 C " =1:8 k =1:4 " =1:3 " = k=y Chien ^q t t + ^Fti i + ^Dt +^g t =0 (10.39)!! 0 1 ^q t = J k k ^Fti = Ju i ^Dt = ; Jg ij (+ t)k= j " " i + A t "= j 0 1 P ;";D ^g t = ;J " A (10.40) k (C 1P ;C f";f 3 D) i J x g ij = i k j k P = ;u 0 i u0 j u i=x j D =k=d d C =0:09 C 1 =1:35 C =1:8 =1:3 f =1;e ;C3y+ f 0:4 =1; 1:8 e;(ret=6) f3 = e ;C4y+ C 3 =0:0115 C 4 =0:5 y + =1 4 k " ( k = " =0 ) k " k =" =0 k " Navier-Stokes k " 17 Jones, W.P. and Launder, B.E., The prediction of laminarization with a two-equation model of turbulence. Int. J. Heat Mass Transfer, Vol.15(197), 301{ Myoung, H.K. and Kasagi, N., A new approach to the improvement ofk ; " turbulence model for wall bounded shear ow, JSME Int. J. Fluids Eng., Vol.109(1990), 156{60.
17 17 k ; " t 1 ;u0 v 0 6 u 0 i u0 j " (algebraic stress model) 19 k ; " k ; " u 0 i u0 j k c ij = P ij + ij + d ij ; " ij (10.41) c k = P + d k ; " (10.4) (10.41) u 0 i u0 j =k r ij 0 [r ij ] c ij = D Dt u0 i u0 j = r ij D Dt k = r ijc k d ij = c s x k k " u0 k u0 l (u0 i u0 j ) l = r ij c s x k k " u0 k u0 l k l = r ij d k D=Dt =t+u i =x i c ij ;d ij = r ij (c k ;d k )=r ij (P ;") (10.41) u 0 i u0 j ; 3 ijk =(1;c ) k P ij ;(=3) ij P P ;"(1;c 1 ) (10.43) k " k ; " Navier-Stokes (ensemble average) (turbulent heat ux) 3 q ti = h0 u 0 i h 0 (= c p 0 ) ( ) 1 (thermal eddy diusivity model) 0 u 0 i = ;a t =x i (10.44) 19 ASM algebraic second-moment approach 0 Rodi, W., A new algebraic relation for calculating the Reynolds stresses. ZAMM, Bd.56(1976), T19{T1. 1 ( ) (Favre mean)
18 18 a t ( ) (thermal eddy diusivity) (algebraic heat transfer model) a t = t =P r t Pr t Pr t (two-equation heat transfer model) a t = C f k (k=") n (0 =" ) m (10.45) n+m =1 0 " n =1 m =0 Nagano-Kim(1988) n = m =1= (turbulent heat ux transport equation model) 3 q ti = h0 u 0 i ;" i 0 " 1 Baldwin-Lomax ANUT(J),J=1,JE FORTRAN (y =0) u = v=y =0 k y " y t y 3 k ; "
19 LES LES(large eddy simulation) LES DNS( ) DNS (unresolvable) Fourier (energy spectrum)e() Kolmogorov(1941) Kolmogorov (Kolmogorov local equilibrium) " E() =" m n F (= d ) m n Kolmogorov 1 E() =" =3 ;5=3 F (= d ) (10.46) d =("= 3 ) 1=4 Kolmogorov l d == d Kolmogorov (Kolmogorov microscale) (10.46) [ p.14, 199, ] 10.3: Kolmogorov
20 (10.46) (10.46) E " Kolmogorov ;5=3 (Kolmogorov ) E() =" =3 ;5=3 (10.47) =1:6 Kolmogorov ;5=3 ( ) Kolmogorov (inertial subrange) (Kolmogorov ) 1 Fourier Z f i (r i )= 1 u i (x i )u i (x i +r i )= 1 F i ( i ) cos( i r i )d i ui ui 0 f (x t) Z 1 f(x t) = G(r)f (x+r t) dr (10.48) 0 Fourier f( t) =G()f( t) (10.49) G(r) G() (spatial lter function) (Gaussian lter), (spectrum cuto lter, sharp cuto lter), (top hat lter ) r 6 Gaussian lter: G(r i )= ; 6r i G( i ) = exp ; ( i i ) 1 i exp i spectrum cuto lter: G(r i )= sin(r i= i ) r i G( i )= top hat lter: G(r i )= ( 1= i (jr i j i =) 0 (jr i j > i =) ( 4 1 (j i j= i ) 0 (j i j >= i ) (10.50) (10.51) G( i )= sin( i i =) i i (10.5) i f (x t) = f (x t)+f 0 (r t) f (x t) f (x t) f 0 =0 f = f fg 0 6=0 f=x = f=x Navier-Stokes (10.1) u i x i =0 (10.53) u i t +u u i j = ; 1 p x j x i + u i x j ; x j (u i u j ; u i u j ) (10.54)
21 1 10.4: SGS(subgrid-scale) ( ) u i u j ; u i u j =(u i u j ; u i u j )+(u i u 0 j +u0 i u j)+(u 0 i u0 j )=L ij + C ij + R ij L ij C ij R ij Leonard cross Reynolds Leonard L ij =0 Cross L ij +C ij Reynolds SGS Reynolds SGS SGS (10.7) u 0 i u0 j = ; t D ij iju 0 k u0 k D ij = u i j +u j i (10.55) t SGS t > 0 LES SGS Smagorinsky L ij +C ij Smagorinsky, J., General circulation experiments with the primeitive equations. I. The basic experiment. Mon. Weather Rev., Vol.91(1963), 99{164.
22 t t =(C S ) p D D =( D ij D ij ) 1= (10.56) C S Smagorinsky 0:1 0: =( 1 3 ) 1=3 t van Driest Smagorinsky Germano(1991) Dynamic SGS SGS GS(grid-scale) Smagorinsky Bardina(1983) ( )SGS GS SGS Kolmogorov LES 1 (10.46) (10.47) Kolmogorov Smagorinsky SGS (REY(I,J),I=1,3),J=1,3 FORTRAN
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