56 



where the matrix [bj^^] is the inverse of [A^-;]. 

 Setting ci, = C5 = Cg = in Eq. (24) leads to 



homogeneous problem corresponding to the eigenvalue 

 Oq. Thus, they are the solutions of 



^ b. .z = for j = 4,5, and 6 at y = y (25) 

 i=l ^^ 



where the b^^ are the elements of a 3 x 6 constant- 

 coefficient matrix. 



Using Eqs. (25) as the boundary condition at y 

 = y and guessing a value for ag, we use SUPORT to 

 integrate Eqs. (20) from y = y to y = and attempt 

 to satisfy the boundary conditions (21) . If the 

 guessed value for Oq is the correct eigenvalue, the 

 three boundary conditions will be satisfied. In 

 general, the guessed value is not the correct value 

 and the boundary conditions at the wall are not 

 satisfied. A Newton- Raphson procedure is used to 

 update the value of ag and the integration is re- 

 peated until the wall-boundary conditions are satis- 

 fied to within a prescribed accuracy. This leads 

 to a value for ag and a further integration of 

 Eqs. (20) leads to a solution that can be expressed 

 in the form 



^li " A(xi) C. (xi,y) for i = 1,2,. 



,6 



(26) 



where A is still an undetermined function at this 

 level of approximation. It is determined by im- 

 posing a solvability condition at the next level of 

 approximation . 



4. THE SECOND-ORDER PROBLEM 



With the solution of the first-order 'problem given 

 in Eq. (26) , the second-order problem becomes 



3z 



2i 



3y 



Z. a. .z., , 



dA 



= G . + F . A for i = 1 , 2 , . . . , 6 



1 dxj 1 



Z21 



Z23 = Z25 



at y 



(27) 



(28) 



3W. ° 



T-^ + y a. .W. = for i = 1,2, ... ,6 (31) 

 3y ^ Di D 



W2 = Wi^ = Wg = at y = 



W2 , Wi, , Wg ->■ as y 



(32) 



(33) 



Siibstituting for the G- and F. from Appendix II 

 into Eq. (30), we obtain the following equation for 

 the evolution of the amplitude A: 



i dA 



A dxj 



= iai (xi) 



(34) 



where 



/ Z F.W.dy/ Y. G.W.dy 

 j j=l 3D M .^j^ D D 

 / 



(35) 



The solution of Eq. (34) can be written as 



r 



A = Agexp[iE / ai(xi)dx] (36) 



where Ag is a constant of integration. 



To determine aj(xi), we need to evaluate dag/dxj 

 and the 3f;^/3xi. To accomplish this, we differen- 

 tiate Eqs. (20) -(22) with respect to Xj and obtain 



3C, 



/3?. 



3y \ 3x1 ) .^ ^ijVSxi 



j=l 



dag 



+ G. -— ^ + H. for i = 1,2,.. .,6 (37) 



1 dxi 1 



3Cl _ 3C3 

 3x1 3xi 



ac5 



3x1 



at y = 



(38) 



Z21'Z23' Z25 "^ 



as y -> 



(29) 



3^1 iy. lis 



3xi ' 3xi ' 3xi 



-> as y ^ 



(39) 



where the G^ and F. are known functions of the C^, 

 ag and the mean flow quantities. They are defined 

 in Appendix II. 



Since the homogeneous parts of Eqs. (27) -(29) 

 are the same as Eqs. (20) -(22) and since the latter 

 have a nontrivial solution, the inhomogeneous Eqs. 

 (27) -(29) have a solution if, and only if, a solva- 

 bility condition is satisfied. In this case the 

 solvability condition demands the inhomogeneities 

 to be orthogonal to every solution of the adjoint 

 homogeneous problem; that is. 



The initial conditions for the computational pro- 

 cedures are chosen to exclude any multiple of the 

 homogeneous solution. The R^ are known functions 

 of Ci, oig and the mean flow quantities and their 

 derivatives; they are given by 



3a. 



H. = Y C. T- 

 1 ^- ] 3x 



13 



and 



«0 



I 

 i=l 



r. dA 



G. + F.A 



1 dxi 1 



W.dy = 

 1 



(30) 



G. = V C. 11 



1 ^3 



j=l 



3xi 



for i = 1 , 2 , . . . , 6 



(40) 



where the Wi(xi,y) are the solutions of the adjoint 



Using the solvability condition of Eqs. (37)-(39), 

 we find that 



