195 



(i-1,n-1,i-l) 



h. 



3 



(bv) . - (bv) . , 



^ <«-'j-i/2 -^ (-ii)-:!/' 



, ,n-l/2- 

 ("l0'i-i/2"j-i/2 



u - u 

 n n-x 



, , ,n-l/2- i 

 + (n'7)i_i/2Wj_i/2— 



Here, for example, 



x-l 



^^v"'^ + v"'^-' + v"-''^-' + v"-''^ 



(56) 



FIGURE 5. Net cube for the difference equations 



l/n,i n,i-l 

 a = — u. + u . 

 n 4 V D 3 



n.i n, i-1 

 u. ' + u. ' 



n-l/2 



n-1 



(51d) 



Difference Equations for the Full Three-Dimensional 

 Equations 



The difference equations for the full three- 

 dimensional equations, as given by (24) and (25), 

 are again expressed in terms of a first-order sys- 

 tem. With the definitions given by (47a) and (47b) , 

 they are written as 



(bv) ' + 9v - m2U^ - m5uw - mgw^ + ™ll 



3u 3u 



(bt) ' + Gt - mi^uw - m^v^ - mgu^ + mj2 



3w 3w 

 = mio "^+^7W^ 



3u , 3w 

 miu + mgw + '"lO -^ + my — 



Their boundary conditions, (26) become: 



11 = 0: u = w=e = 



n = n„: u = 1, w = Wg/Uj,gf 



(52a) 



(52b) 

 {52c) 



(53a) 

 (53b) 



The difference equations for (47a) and (47b) are 

 the same as those given by (50a) and (50b) : they 

 are written for the midpoint [ (x) , (z) j^ , n-|_, /^ ] of 

 the net cube shown in Figure 5; that is, 



hTVu"'^ - u^-i 



3 V 3 



^"'1 



j-lJ j-1/2' 



4 V 3 3 



n,i n-1 ,i 

 a. + u. 

 3-1 3-1 



and 



<"ll'i-l/2 



<'"ll'i ■*- ''"ll'i-l ■" '™ll' 



n-1 , ,n-l 



+ (m, , ) . , 



1 1 1 1-1 



(57) 



z„ = 



i = 1, 2, ... , I (58) 



Solution of the Difference Equations 



The difference equations (50) for the longitudinal 

 attachment-line flow and the difference equations 

 for (52) are nonlinear algebraic equations. We use 

 Newton's method to linearize them and then solve the 

 resulting linear system by the block-elimination 

 method discussed by Keller (1974) . A brief 

 description of it will be given for the streamwise 

 attachment-line equations. 



Using Newton's method, the linearized difference 

 equations for the system given by (50) are: 



h. 



6u^ - 6u. - -^ (6v. + 6v. ,) = (r,) . (59a) 

 3 :-i 2 J ]-l 1 J 



Sw. - 5w._ - ji (6t. + 6t._j) = ir^) . (59b) 



(Ci)j6v. + (C2)6v^_i + (C3)j'5e. + (5:t).6e._^ 



+ (<;5)j6u. + (C6).6u._^ = (rs). 



(59c) 



hT ( w'?'^ - wn.i 1 = t"'i 

 3 V 3 D-lJ D-1/2 



(54) 



(6l).6t. + (B2)j6t._j + (63)j6e. + {e4)j69j_j 



The difference equations which are to approximate 

 (52a, b,c) are rather lengthy. To illustrate the 

 difference equations for these three equations, we 

 consider the following model equation 



+ (65)j<5"3 + (B6)jfi"j_^ + (67)jSu. + (Bs'jS^j.j 



= (r.) . (59d) 



(bv) ' + ev + mil ="'10 u V^ + myw tt^ 



ox oz 



The difference equations for (55) are: 



(55) 



(ai) . 66. + (02).: fie. 



(03)^<5u. + (04)j6u._^ 



3 3 



+ (as).&v^ + {af,)-Sw._^ = (rj) 



(59e) 



