182 



where toj, 102/ 1^3 are the components of vorticity 

 given by 



ui 



3q, 



dq2 



h23x2 3x3 



3qj_ _ 9q3 

 3X3 hiSxj ' 



3q2 _ 9qi 



h]3xi h23x2 



and they satisfy 



V'O) = 0. 



0)2 



0)3 



(34) 



+ K2qi - Kiq2. 



(35) 



In the reduction of Eqs. (31, 32, 33) from the 

 Navier-Stokes equation, conventional predictions 

 for turbulent components are used; the velocity is 

 assumed to consist of time-averaged terms and 

 fluctuating terms. 



On the other hand, if the constant eddy viscosity 

 can be assumed, the following equations are derived 

 directly from Navier-Stokes equation; 



Introducing non-dimensional curvilinear small 

 line segments 3C, 3fi, and 3^, we represent the dif- 

 ferentiations 



1 3 



1 3 



hi3xi Le 3r ' h23x2 Le 3f| 



_3 1_ 3_ 



3X3 " L^^ 3^ 



Here we assume the derivatives by new variables 

 are all 0(1) , i.e. , 



(40) 



3C 



0(1) 



3n 



= 0(1) 





(41) 



The origin of the new variables coincides with that 

 of O-X1X2X3 but 1=0 corresponds to the position 

 of separation. 



We tentatively assume that the asymptotic series 

 for velocity and vorticity of C-region have the 

 following forms; 



qi/Ufl = QotS'^i'^' + eui(5,n,?) + e2u2(C,n,C) + ... 



h23x2 



(wiq2~U2qi) + 



(Miq3-t03qi) 



,1 32 32 

 _ 2 3x2 3x3 



{- 



3(1) 



7 0013 

 3- + i) 



h23x2 



(K2tOj-Kia)2) - K2 



hjSxi h23x2 8x3 

 3aj2 3mi 



hjBxj 



+ K 



2 2 



"1 

 h23x2 



(36) 



2,-v 



qj/Uo = £Vi(S,n,?) + e^V2(C,r|,<;) + . 

 qj/Ug = e2wi(|,f|,C) + £3*2(1, n,U + 



^^^^ - - Hr il,n.O + i^r (C.n,d + .■ 



e 5l 52 



Uq/L 



Ci)2 



-,- di {?,n,0 + - u^,(C,r:,c) + 

 p^ ni e n2 



3 3 



7— r-— (M2qi-U)iq2) + ■^;^(w2q3-W3q2) 

 h^oxi 0X3 



0)_ i~ ^-- ~ ^-- 



— ^ =-0) (5,n,C) + CO (5,n,0 + 

 Un/L e 51 £.2 



(43) 



r 1 32 ^ 32^ 



( — T — T + :r^") W5 



3 3mi 5"3 



h2 3x2 3x2^ 2 h23x2 hjSxi 8x3 



3(1)1 



3(1)9 



7-^7 (Ki0J2-K2(Dl) - Ki , , + Kji-— — 



hi3xi ^ ^ ^ ^ ■^h23x2 ^^hi3xi 



(37) 



''^l " hi3xi ' <'^l'^3-"3qi' + (K2- ^ 3^ ) ((H2q3-"3q2) 



,J^ i_ , _L il_i _ 3 3(1)1 3(1)2 

 ^h2 3x2 "^ h2 3x2'"3 3x3*hi3xi h23x2' 

 112 2 1 1 z 2 



All the quantities appearing in Eqs. (42) and (43) 

 are assumed to be 0(1). 



Moreover, we introduce non-dimensional variables 

 ki,k2,kii,k22 by 



ki = L-Ki , 

 k22 — Ij'K22 



L-K2 



^U 



L'K 



11 



(44) 



whose orders are 0(1) for all the regions. 



Substituting Eqs. (40) , (42) , and (43) into 

 Eq. (30) and Eq. (35) , we get as leading terms. 



H 



= 



(45) 



^ _(Ki(i)i+K2(i)2) - (Ki-Kn)j^ 



(K2-K22) 



3(1)3 

 h23x2 



where 



(38) 



3(0. 



ni 



3(5 



SI 



= 



3n 3c 

 and into Eqs. (36, 37, 38) we get. 



(46) 



(47) 



1 3h, 1 3h, 



'^ll =-K2^ ' ^22--^^ . (39) 



1 2 



Voitxcity Diffusion Region (C-region) 



52,7, 



^e 3^%1 3 ,- ~ ^ n 

 ' - 3f (%l"0> = ° 



£3 UqL 3r2 



32^ 



e3 UqL 3^ 



'J'l 3 - - 

 iT^'^Jl^ 



(48) 



(49) 



For the vorticity diffusion region, the constant 

 eddy viscosity is assumed. 



In order for the viscous diffusion term to exist, 

 Vg/UgL should be at least 0{e^). 



