194 



at first, consider the solution of the longitudinal 

 attachment-line Eqs . (30) and (31) and then the 

 solution of the full three-dimensional flow equations 

 as given by (24) and (25) . We shall not discuss the 

 Cebeci-Stewartson procedure for computing three- 

 dimensional flows with the transverse velocity, w, 

 containing flow reversal since that procedure will 

 be fully described in a forthcoming paper. 



Difference Equations for the Longitudinal 

 Attachment-Line Equations 



According to the Box method, we first reduce the 

 Eqs. (30), (31), (32), and (26) into a system of 

 five first-order equations by introducing new depen- 

 dent variables u(x,z,ri), v(x,z,ri), w(x,z,ri), 

 t(x,z,n), and 9(x,z,n). Equations (30) and (31) 

 then can be written as 



n-1/2 

 n-1/2 1 



^[j'n + ^n-lj. ^ 



j-1/2 = li'^j + '^D-1 



s" + si^-^ 

 2 V 3 3 



^j-l/2 = 2^=j ^ Vl 



(49) 



The difference equations which are to approximate 

 (47) are formulated by considering one mesh rect- 

 angle as in Figure 4. We approximate (47a, b) using 

 centered difference quotients and average them 

 about the midpoint (Xj^ "11-1/2' °^ ^^^ segment P1P2- 



;^(u? 



: 1 i,.,n 



"j-l' 



^-1/2 



(w'? - w^ J = t? 



D-1' 



"j-1/2 



(50a) 

 (50b) 



Similarly, (47c, d,e) are approximated by centering 

 them about the midpoint ^n-i/? ''^i-l /2 ""^ ^^^ rect- 

 angle PiP2P3Pit. This gives 



(bv) 



2 , 3u 



3v - m2U^ + nil 1 = '"lO " '^~ 



(bt) ' + 6t - m^uw - m^w - mgu + mj2 



3w 



= rnio u — 



m^u + m5W + mjo "5" 



(47a) 

 (47b) 

 (47c) 



(47d) 

 (47e) 



The boundary conditions (26) and (32) become 



n=0: u=w=e=0 (48a) 



n = noo: u = l, w = w^g/Uj,^^ (48b) 



We next consider the net rectangle shown in Figure 

 4 and denote the net points by 



no 



'j-i 



+ kj^ n = l,2, ...,N 

 j = 1, 2, ..., J 



We approximate the quantities (u,v,g,t,S) at 

 points (Xjj,rij) of the net by functions denoted by 

 (u!',v?J,w^,t?',e^) . We also employ the notation for 

 points and quantities midway between net points and 

 for any net function s?. 



*''l'n-1/2 



(x,) 



1'n-1 



'"I'n 



FIGURE 4. Net rectangle for the longitudinal 

 attachment-line equations. 



h. ■ 

 3 



(bv) . 



(bv) 



J-1 



^ '«^':-i/2 



, n , , 9. n „n-l 



(-2 + "n'<" 'j-1/2 = ^j-1/2 



n 

 "11 



(50c) 



(bt)'. 



(bt) 



:-i 



i^^l^ + a ) (uw) 



j-1/2 



^ '-'^V2 



n T n n , ?i 



"•3'" '-i_i /o ~ m9(u^) 



'j-1/2 



n 

 j-1/2 



n-1 n 

 ^j-l/2"j-l/2 



n- 1 n 

 u , , w . 

 3-1/2 3-1/2 



,n-l 

 =j-l/2 



-l/„n 



n 



- nil2 



(50d) 



'U 



3 V 3 



=t"-' 



j-1/2 



Here 



o""l , 2, n-1 



^j-1/2 = -"n'" 'j-1/2 



, -1 „ \ n n n 



-(mi t 2aJu._j/2 - %"j_i/2 



(50e) 



I(bv)"-1 - (bv)""!] + (6v)"-' 



j-1/2 



n-1 , 2%n-l n-1 



-■"2 (" )3_i/2 + ^l 



(51a) 



-n-1 , ,n-l 



S . , = -a (uw) 

 3-1/2 n' j-1/2 



h.'[(bt)"-^ 

 3 3 



*^'3:;j ^ <«^';:!/2 



n-1 2,n-l n-1 , 9, n-1 



n, w . , - m (u^) . 

 3 3-1/2 9 j-1/2 



(51b) 



mn-1 ^ n-1 



T = -2a u, , ., 



3-1/2 n 3-1/2 



hT^e"-! 

 3 3 



,n-l n-1 n-1 



^j-l' - ■"! "j-1/2 



n-1 n-1 

 m, w. 

 6 3-1/2 



(51c) 



