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1 8 aldwn-lomax hen 8. +ru = t (8.a) u+r(uu+) =r +f t (8.b) e +rhu = r ( u);r q+f u t (8.c) u e = (+u 2 =2) H = h+u 2 =2=(e+)= q f (8.) (comressble Naver-Stokes equatons) T h = += = RT d = c v dt dh = c dt c v c R = c =c v R = c ;c v c = RT M = juj=c (DNS, drect numercal smulaton) (vscous stress tensor) (heat ux) q j = ; u j +u j ; 2 3 ju k k ( j = 2 3) (8.2a) q = ;T = ; c 2 ( = 2 3) (8.2b) ; P r x u j = u j =x j P r = c = (Prandtl number)

2 2 8.. Naver-Stokes (ensemble-averaed Naver-Stokes equatons) (Favre mean) u u = u+u u = (8.3) u u u u = eu+u eu = u= u = ; u = (8.4) eu u u = u u = eu;u = u = eu = eu = u = u+ u 2 (8.) + ru = t u + r(uu+) =r + f t (8.5b) E + r(e+) u = r u ;rq + f u t (8.5c) (8.5a) u = u;eu = u; u = (u+u );(+ )(u+u ) = (u ; u )=u ; u a = a ab = a b eab = ea b f ab = a e b 2 eu = (+ ) u = u+ u = u

3 3 u = eu uu = (eueu+2euu +u u )= eueu + u u E = (+u u=2) = e + 2 eu eu + ] 2 u u E e (E+)u = (h+u u=2)u = E e u+eu+ e h u + u u eu+ ^ 2 u u u (ensemble-averaed Naver- Stokes equatons) + reu = t t eu + r(eueu+) =r( f ; u u )+f t E e + r(e+) e ; eu = r (f ; u u )eu ; q ; h u + f u u^ u u =2 3. (8.6a) (8.6b) (8.6c) (8.5) (8.6) (8.) j q j = ; u j +u j ; 2 3 ju k k ; u u j ( j = 2 3) q = ; c 2 + h ; P u ( = 2 3) r ;u u j = ; u u j (Reynolds stress) h u = h u (turbulent heat ux) 2 (8.) j q ) u ) eu e ) e E ) H ) e E+= (8.6) (8.) e e ) e E = ; e + 2 eueu + 2 u u 3. ; e h = e+ = f = f ; = = =

4 4 = R e T de = c v d e T d e h = c de T (8.7a) =~e =~ e E; f 2 eh = e E+ = e E; f 2 = ~ ( f c2 +f 2 ) (8.7b) (8.7c) ~ = ; f 2 = 2 ~(eu eu+ ] u u ) f c2 = Re T ec = q fc 2 ` ' ` e ' 8..2 Naver-Stokes (8.) (cartesan coordnates) x q t + F +D+ = x q = u u 2 u 3 e D = ; x F = 2 3 u u u + u 2 u + 2 u 3 u + 3 Hu = ; j u j ;q f u ( = 2 3) f f 2 f 3 (8.8a) (8.8b) q F D (eneral curvlnear coordnates) (8.) J Jr = J(r ) U = ur t J u + J e U uu +r = (Jr ) HU + J u;q u

5 5 ^q t + ^F + ^D+^ = ^q = J u u 2 u 3 e ^D = ; J j ^F = J j j2 j3 U u U + u 2 U + 2 u 3 U + 3 HU ^ = ;J jk u k ;q j f k u k ( = 2 3) f f 2 f 3 (8.9a) (8.9b) ^q ^F ^D ^ 4 J = (x y z)=( ) j = =x j (8.9) (8.8) x (8.9) (curvlnear coordnate rd) J 8. J = x x x y y y z z z (x ;x )(x ;x )(x ;x ) J x x = x x 2 = y x 3 = z = 2 = 3 = Jr Jr = J x J y J z = y y (x ;x )(x ;x ) z z x x! z z x x y y 6 : H Hj H HH 6 H H HH W U H H : Hj V : V U H H Hj H 6 H H HH H W H 8.:

6 6 Jr = const. x = const. (8.9) J U =const. Jjr j x U l u r l (8.b) Jr l Jr l u = JU l Jr l (ruu+) =r l J(U u+r )= J(U U l + l ) ; J(U u+r ) r l ^q t + ^F + ^D+^ = ~q t + F ~ + R+ ~ D+~ ~ = ~q = J U U 2 U 3 e ~ F = J ~R = ;J(U u+r ) ~D = ;J j x k = k j 2 3 u U U U + U 2 U + 2 U 3 U HU r r 2 r 3 ; J j k T j ( = 2 3) ~ = ;Jf r r 2 r 3 u (8.a) (8.b) (8.c)

7 7 (8.b) ~q F ~ R ~ D ~ ~ (8.c) u U j = k j k (a b)= = a b= (8.) (mass ux)ju JU x = const. 3 (8.9) 3 (8.) R ~ 8..3 aldwn-lomax hen 2 aldwn-lomax l ( ) hen k ; k aldwn-lomax (978) hen (982) 2 q j =(+ t ) u j +u j ; 2 3 ju k k ; 2 3 jk ( j = 2 3) (8.a) q = ; ; P r + t P rt c 2 x ( = 2 3) (8.b) t (eddy vscosty) k = u u =2 (turbulent knetc enery) P rt (turbulent Prandtl number) P r = :72 P rt =:9() =:5( ) (8.a) (8.b) (8.2a) (8.2b) t = (8.a) (Reynolds stress) t (8.b) (turbulent heat ux) ;T=x (thermal eddy vscosty) t = c t =P rt t 2

8 8 aldwn-lomax 2 t = ( (t ) nner (d d crossover ) ( t ) outer (d>d crossover ) (8.2a) d d crossover ( t ) nner =( t ) outer d Prandtlvan Drest ( t ) nner = l 2 jj (8.2b) l = df;ex(;y + = + ) = ru =:4 Karman + =26 y + = u d= w = w w d= w u = w = w w lauser ( t ) outer =:6KF wake F Kleb (d) (8.2c) K =:68 lauser F wake = mn ; d max F max :25d max u df 2 =F max F Kleb (d) = h :3d +5:5 d max 6 ; d max F (d) =d jjf;ex(;y + = + ) F max d ex(;y + = + )= u df = juj max ;juj mn juj mn = F Kleb Klebano ( t ) max < 4 (8.3) t = aldwn-lomax ( t ) max (8.3) (8.) aldwn-lomax (8.2) hen k ; y = const. hen t = c f k 2 = (8.4a) k ^q t t + ^F t + ^D t +^ t = (8.4b)

9 9 ^q t = J k ^ t = ;J! ^Ft = JU k! P ;;D k (c P ;c 2 f;f 3 D) ^Dt = ; J j (+ t ) k j + t j (8.4c) P = ;u u j u =x j ;u u j D =2k=d2 d c =:9 c =:35 c 2 =:8 =:3 f =;e ;c3y+ f =; :4 2 :8 e;(ret=6) f 3 = e ;c4y+ c 3 =:5 c 4 =:5 Re t = k 2 = Re t y + du = =4 k k k = = k ; k q 2 q 3 k P k k 3 k

10 8.2 k ; k k (lnearzaton) (daonalzaton) 8.2. k k (8.4) +ru = t (8.5a) u+r(uu+) = t (8.5b) e +rhu = t (8.5c) k +rku = t (8.5d) +ru = t (8.5e) q t + F x = ( = 2 3) (8.6a) 2 3 q F q = u u 2 u 3 e k F = u u u + u 2 u + 2 u 3 u + 3 Hu ku u 2 ( = 2 3) (8.6b)

11 F q df = F q dq = dq F = q d q = (8.7) (8.6) q t + q = x ( = 2 3) (8.8a) = F =q 5 = 2 3 ;u u + 2 D 2 u ; ~u 2 3 u ; ~u 3 ~ ;u 2 u u 2 ; 2 ~u D 2 3 u 2 ; 2 ~u 3 2 ~ ;u 3 u u 3 ; 3 ~u 2 u 3 ; 3 ~u 2 D 3 3 ~ ;Hu + 2 u H ;~u u 2 H ;~u 2 u 3 H ;~u 3 u u ;ku k 2 k 3 k u ;u 2 3 u (8.8b) D j = u + j (;~)u j ~ = ; 2 =~u 2 =2 H = e+ = c2 + 2 ~ = e ;2 (8.5) + ur + r u = t (8.9a) u t + uru+ r = (8.9b) t + ur + c2 r u = k + urk = t (8.9d) + ur = t (8.9e) (8.9c) q q t + = ( = 2 3) (8.2a) x 5 F q q

12 2 q = u u 2 u 3 k = u 2 3 u = u 2 = u 3 = c 2 2 c 2 3 c 2 u u u (8.2b) (8.2) dq = q dq = Ndq q (8.2) (8.8) N = N N ; q q q = Nq dnq = N = q =q N ; 6 N = N ; = ;u = = ;u 2 = = ;u 3 = = ~u 2 =2 ;~u ;~u 2 ;~u 3 ~ ;k= = ;= = u u 2 u 3 u 2 =2 u u 2 u 3 =~ k (8.22a) (8.22b) = N ; N = N ; R L N = R L (8.23) 6 N q q q

13 3 k L l k R = L; r k k l k ( ) k 7 j ; Ij = u ; 2 3 u ; = u ; 2 = u ; 3 = c 2 2 c 2 3 c 2 u ; u ; u ; = = u (5 ) 2 = u +c 3 = u ;c l k = u l ( ; I) =(l k l2 k = u c l ( ; I)=(l k l 2 k l7 k ) l 7 k ) 2 3 = 2 = 3 = c 2 2 c 2 3 c 2 c 2 3 c = c 2 = c 3 = c 2 2 c 2 3 c 2 c c = c 7 L u u +c u u ;c u k u L R L L L = L N =

14 4 R = N ; R = L = R = N ; u u + c u + 2 c u + 3 c u ;c u u ;=c 2 =c 2 =c 3 =c 2 3 ;=c =2c 2 =2c 3 =2c ;=2c ; =2 =2 ; 2 =2 2 =2 ; 3 =2 3 =2 c=2 2 c=2 3 c=2 ;c=2 N (8.24a) (8.24b) (8.24c) (8.2) L (8.6) L q L t + F = L q x t + L q = (8.25) x ; t +u ; ; x c 2 t +u = x n t +(u + j c) o u j + n j x c t +(u +c) o = (j = 2 3) (8.26) x n t +(u ;c) ou ; n x c t +(u ;c) o = x ; t +u ; k = x t +u = x dx =dt = u dx =dt = u +c dx =dt = u ;c =t+u =x =t+(u + c)=x =t+(u ;c)=x dx =dt = u dx =dt = u dx =dt = u +c dx =dt = u ;c dx =dt = u k

15 k t J + JU = t Ju + J(uU +r )= t Je+ JHU = t Jk + JkU = t J + JU = (8.27a) (8.27b) (8.27c) (8.27d) (8.27e) ^q t + ^F = ( = 2 3) (8.28a) ^q = J u u 2 u 3 e k ^F = J U u U + u 2 U + 2 u 3 U + 3 HU ku U ( = 2 3) (8.28b) ^F ^q j ^F ^q Euler d ^F = ^F ^q d^q = ^ d^q ^F = ^ ^q d ^ ^q = (8.29) (8.28) ^q t + ^ ^q = ( = 2 3) (8.3a) ^ = ^F = ^q ^ = 2 3 ;u U + 2 ^D 2 u ;~ u 2 3 u ;~ u 3 ~ ;u 2 U u 2 ;~ 2 u ^D2 3 u 2 ;~ 2 u 3 ~ 2 ;u 3 U u 3 ;~ 3 u 2 u 3 ;~ 3 u 2 ^D3 ~ 3 ;HU + 2 U H ;~U u 2 H ;~U u 2 3 H ;~U u 3 U ;ku k 2 k 3 k U ;U 2 3 U (8.3b)

16 6 ^D j = U +(;~) j u j ~ = ; 2 =~u 2 =2 j = =x j (8.9) ur = U = r = r = t + U + r u = (8.3a) u t + U u + r = (8.3b) t + U + c 2 r u = (8.3c) k t + U k = (8.3d) t + U = (8.3e) ^q t + ^ ^q = ( = 2 3) (8.32a) ^q = q = u v w k ^ = U 2 3 U = U 2 = U 3 = c 2 2 c 2 3 c 2 U U U (8.32b) (8.32) (8.3) ^q = ^q = N=J d^q =( ^q = ^q)d^q =(N=J)d^q (8.32) ^ ^ ^ ^ = N ; ^ N = N ; ^R ^ ^L N = ^R ^ ^L (8.33) ^ ^ = U (5 ) ^ 2 = U + c ^3 = U ; c j = k j k ^ ^ = U U + c U c U c U ; c U U (8.34a)

17 7 ^L = ^R = N ; ;=c 2 2 ; ; 3 3 =c 2 2 ; ; =c 3 3 ; ; =c 2 3 ; =c 2 2 c 2 22 c D 3 2 2; ; ; D ; ; ; c ; 2 c N D c 2 c 3 c ; c D j = 3 2 j; j 2 +(; j ) (j = 2 3) (8.34b) (8.34c) (8.28) ^L J ^q t + ^F = ^L ^q t + ^ ^q ^L = (8.35) t +U ; c 2 t +U = n +; U + t o n c U + +; U + c t o c = ( j = ) t +U u j ; j t +U u = ( j 6= ) (8.36) n +; U ; t o n c U ; ;; U ; c t o c = t +U k = t +U = d =dt = U d =dt = U + c d =dt = U ; c =t+u = =t+(u + c) = =t+(u ; c) = k

18 k t J + JU = t JU + t Je+ JHU = t Jk + JkU = t J + JU = J ; UU +(r r) = (8.37a) (8.37b) (8.37c) (8.37d) (8.37e) ~q t + ~ F = ( = 2 3) (8.38a) ~q = J U U 2 U 3 e k ~ F = J U U U + U 2 U + 2 U 3 U + 3 HU ku U ( = 2 3) (8.38b) ~F ~q d ~ F = ~ F ~q d~q = ~ d~q ~ F = ~ ~q d ~ ~q = (8.39) (8.38) ~q t + ~ ~q = ( = 2 3) (8.4a) ~ = ~ F = ~q ~ = 2 3 ;U U + 2 ~ D 2 U ;~ 2 3 U ;~ 3 ~ ;U 2 U U 2 ;~ 2 ~ D2 3 U 2 ;~ 2 3 ~ 2 ;U 3 U U 3 ;~ 3 2 U 3 ;~ 3 2 ~ D3 ~ 3 ;HU + 2 U H ;~ U 2 H ;~ 2 U 3 H ;~ 3 U U ;ku k 2 k 3 k U ;U 2 3 U (8.4b)

19 9 ~D j = U + j U j ;~ j j =(x k = )u k = j U j j =(x k = )(x k = j ) t + U + U = (8.4a) U t + U U + r r = (8.4b) t + U + c 2 U = (8.4c) k t + U k = (8.4d) t + U = (8.4e) ~q t + ~ ~q = (8.42a) ~q = U U 2 U 3 k ~ = U 2 3 U = U 2 = U 3 = c 2 2 c 2 3 c 2 U U U (8.42b) (8.42) (8.4) ~q = ~q = N=J ~ N ~ = J~q = ~q N ~ ; ~N = ~N ; = ;U = = ;U 2 = = ;U 3 = = 2 ;~ ;~ 2 ;~ 3 ~ ;k= = ;= = U U 2 U 3 u 2 =2 2 3 =~ k (8.43a) (8.43b)

20 2 d~q =( ~q = ~q)d~q =( N=J)d~q ~ ~ ~ ~ = ~ N ; ~ ~ N = ~ N ; ~ R ~ ~ L ~ N = ~ R ~ ~ L (8.44) ~ ~ L ~ R ~ = L ~ ; ~ = ^ ~L = ~R = ~ N ; ;=c 2 ; 2 2 = 22 ; 3 3 = 33 =c ; 2 = ; 3 23 = =c ; 3 = ; 2 32 = =c 2 3 ; =c =2 c 2 =2 22 c 3 =2 33 c ;=2 c ; =2 2 2 = =2 33 =2 2 =2 ; 2 = = =2 3 = =2 22 ; 3 =2 3 =2 c=2 2 c= c=2 33 ;c=2 (8.38) ~L J (8.45a) ~N (8.45b) (8.45c) ~q t + F ~ = L ~ ~q t + ~ L ~ ~q = (8.46) (8.36) k

21 2 8.3 Euler ourant FL =maxj t=x j = maxj^ t= j y + > 4 y + = 4 ourant (8.8) hen 2 (8.4) (delta-form mlct method) ni +t o ( n +D ) q n = rhs n (8.47a) x rhs = ;t F +D+ q n+ = q n +q n x ni +t o ( n x +D ) q (m) = ;(q (m;) ;q n )+ 2; rhs n +rhs (m;) (8.47b) q (m) = q (m;) +q (m) q = u u 2 u 3 e k D = ; x F = 2 3 (+ t ) k x + t u u u + u 2 u + 2 u 3 u + 3 x Hu ku u = ; ( = 2 3) f f 2 f 3 f u P ;;D k (c P ;c 2 f;f 3 D) (8.47c) (D q n )=x =x q n (8.47a) q n

22 22 (8.47b) t q n =(q (m;) ;q n )+q (m) q (m;) ;q n q (m) (8.) hen 2 (8.4) ni +t o ( ~ n + D ~ ) ~q n n = rhs (8.48a) rhs = ;t ^F + ^D+^ ni +t o ( ~ n + D ~ ) ~q (m) =~q (m;) +~q (m) ~q n+ =~q n +~q n ~q (m) = ;(~q (m;) ; ~q n )+ 2; rhs n + rhs (m;) (8.48b) ~q = J U U 2 U 3 e k ^D = ; J j ^F = J U u U + u 2 U + 2 u 3 U + 3 j j2 j3 jk u k ;q j HU ku U k j (+ t )k= k k j + t = k ( = 2 3) ^ = ;J f f 2 f 3 f k u k P ;;D k (c P ;c 2 f;f 3 D) (8.48c)

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