and introduce a two-component vector potential 

 such that 



uhosine + -r^, wh,sin„ 



8y ' 8y 



d± 



vhjh2sin6 = - 



3^ ^ 8(j) 

 3x 3x 



where ip and ^ are defined as 



tij = (vsjUg) h2f {x,z, n)sine 



(fi = (vsiUg)ii{Uj.gj/Ug)hig(x,z,n)sine 



(22a) 

 (22b) 



(23a) 

 (23b) 



and Uj-gf is some reference velocity. 



Using these transformations and the relations 

 given by (9), (10), and (11), we can write the x- 

 momentum and z-momentuin equations for the general 

 case as: 



x-Momentum 



(bf")' + mjff" - m2(f')2 - msf'g' 



+ mgf "g - mg (g' )^ + mj i 



mio f 



z-Momentum 



3f 

 3x 



9z dz 



(24) 



(bg") ' + mifg" - mi+f g' - m3(g') + msgg" 

 - mg(f') + mi2 



= mio ( f 3^ - g' 3^ + n,7 ( g' g^ " g" g^ 



and their boundary conditions as 



n = 0: f=f'=g = g'=0; 

 ri = nco, f'=l»g'= We/U;^gf 



(25) 



(26) 



Here primes denote differentiation with respect to 

 n, and 



f = — , g' = ^^, 

 Ug Uj-ef 



b = 1 + e+ £+ = - 

 m' m ^ 



(27) 



The coefficients mj^ to mj2 si^e given by 



; 1 + ; — ; — ' ~ — (h^sinfl) 



hju 3x J hjh2sine 3x 2 



tlo = :; s,K,cotg, m, = -S,Ko 



2 hjUg 3x 11 o' 3 1 2 u 



sj Uj-ef 3Ug 



ref 



COtf 



e 

 "ref 



m^ - S1K21, ni5 - K7 ^I^ "^ ■*■ ^1*^12 "TT 

 ^ e e 



si 



^ hihosine 



/UgSl 



my 



^1 "ref 

 h2 Ug 



, mg = S1K2 



/ ., "»^ef . 



/UgS J hj sml 



'^ref 



(28) 



e S] 



mq = siKi csc9 , min = :; — 



ref '■ 



191 



e / "e 



■"11 = "2 + n>5 + mg 



"ref V"ref 



mi2 - mi. + mq 



u i ^ 



ref 



■'ref 



+ mg 



mjQ awg myWg 3wg 

 u „c 3x u2 3z 



^ref 



ref 



To transform the longitudinal attachment-line 

 flow equations and the boundary conditions , we use 

 the transformed coordinates given by (21) and de- 

 fine the two-component vector potential by 



uh2Sine = -r-^, w^hisinB = -r^ 



dy Z i jy 



vhjh2sine 



3i(j 



(29) 



with 41 and ip still given by (23) . With these vari- 

 ables, the longitudinal attachment-line equations 

 in the transformed coordinates can be written as 



(bf") ' + mjff" - m2 (f ')2 + mgf'g + m 



11 



hi V 8x 



f„ 3f 



(30) 



(bg") ' + mig"f - m^f'g' - m3(g')'^ + mggg" - m9(f')2 



+ "II2 = 7— If 



; ^ 3x 



(31) 



The boundary conditions and the coefficients mj to 

 mi2 are the same as in (26) and in (28) except now 



loo: 



g' 



si "ref 



ze 



^ref 



mg = m3 



mg = 0, 



m^ 2 ~ "13 



mil ~ ^2 

 2 



"ref 



3w_ 



+ mi^ 



"ref "^1 "ref ^^ 



(32) 



In terms of the transformed variables, the alge- 

 braic eddy-viscosity formulas as given by (17) to 

 (20) become 



(^m'i 



(33) 



(Ej„)q = 0.0168 /r^ 



(34) 



Here R = u si/v and 



