55 



To determine an approximate solution to Eqs. (2) 

 -(10), we use the method of multiple scales [Nayfeh 

 (1973) ] and seek a first-order expansion for the 

 eight dependent disturbance variables u, v, p, T, 

 p, c , y and k in the form of a traveling harmonic 

 wave; that is, we expand each disturbance flow 

 quantity in the form 



q(xi,y,t,) = [qi{xi,y) 



+ £q2(xi,y) + ...]exp(ie) (11) 



Li4(ui,vi,pi ,Ti 

 = iPo"o(uo - 



^Polfv, 



"0 



RPr c 



RPr c 

 e Po 



<0 



3y^ 



2 B^KQ 



<o«6 - -^ 



3ko 3Ti 



RPr c ay 3y 

 



(16) 



where 



36 , , 36 

 ^= "O'^l'' 3l 



(12) 



For the case of spatial stability, ap is the complex 

 wavenumber for the quasi-parallel flow problem and 

 10 is the disturbance frequency which is taken to be 

 real. 



Substituting Eqs. (11) and (12) into Eqs. (2)- 

 (10) , transforming the time and the spatial deriv- 

 atives from t and x to 6 and xj , and equating the 

 coefficients of £ and £ on both sides, we obtain 

 problems describing the qi and q2 flow quantities. 

 These problems are referred to as the first- and 

 second-order problems and they are solved in the 

 next two sections. 



The boundary conditions are : 



ui = vi = Tj = at y 

 uj , vj , Tj ->• as y 



(17) 

 (18) 



Equations (13) - (18) constitute an eigenvalue 

 problem, which is solved numerically. It is 

 convenient to express it as a set of six first-order 

 equations by introducing the new variables z^^ de- 

 fined by 



3ui 



3y ' 



zil 



ui, 



Z12 



Z13 = VI, 



zm 



Pi' 



-15 



Tl, 



Z16 



3Ti 



3y 



(19) 



3. THE FIRST-ORDER PROBLEM 



Substituting Eqs. (11) and (12) into Eqs. (2)-(10) 

 and equating the coefficients of e'' on both sides, 

 we obtain the following 



Lj (ui,Vi,pi,Ti) = iao[PoUl + (Uq 



-)Pl] 



Then, Eqs. (13) -(18) can be rewritten in the compact 

 form 



3z. 



li 



3y 



y a. .2 

 3=1 '' 



Ij 



for i = 1,2. . ,6 



Zll = Zl3 = Z15 = 3t y 

 Zll' Z13' Z15 ^0 as y 



(20) 



(21) 

 (22) 



+ 3^ (PQVl) 



(13) 



L2(ui'Vi ,pi ,Ti) 



IPO^OIUO - — 



i dVQ 



U0"0 



"1 



Po 



3U0 

 3y 



R 3y 



ag vj + laoPi 



Tl_ 3_ /duo. 3Uo\ 

 ■ R 3y VdTo 3y J 



1^ 3uo 3ui ih 

 R 3y 3y 



3vi 



R ^0"0 3y 



1^ dyp 3Uo 3Ti _ 1 3^ui 



R dTg 3y 3y 

 L3(ui,vi,pi ,Ti) 



VO 



3y2 



= 



1P0"0 Uo 



10 \ 1 2 



i^ r. 3P0 



^ "0 3?- "1 



ih 



3ui 



R ^0"0 3y 



i dyn 3Uo 



r_ 3yo 3vi 

 R 3y 3y 



(14) 



where the a- ■ are the elements of a 6 x 6 variable- 

 coefficient matrix. The nineteen nonzero elements 

 of this matrix are listed in Appendix I. 



We solve this eigenvalue problem by using SUPORT 

 [Scott and Watts ' (1977) ] . To set up the numerical 

 problem, we first replace the boundary conditions 

 (22) by a new set at y = y where y is a convenient 

 location outside the boundary layer. Outside the 

 boundary layer, the mean flow is independent of y 

 and the coefficients a.j_^ are constants. Hence, the 

 general solution of Eqs. (20) can be expressed in 

 the form 



■.. = f A. .c. 

 11 jt-i ^3 D 



exp(A.y) for i = 1,2,. ..,6 



(23) 



where the Aj are the eigenvalues of the matrix 

 [a^-:], the Aij are the corresponding eigenvectors 

 and the c: are arbitrary constants. The real parts 

 of three of the Aj are negative, while the real 

 parts of the remaining Aj are positive. Let us 

 order these eignevalues so that the real parts of 

 Ai,A2, and A3 are negative. Then, the boundary 

 condition (22) demands that 04,05 and eg are zero. 

 To set up this condition for SUPORT, we first solve 

 Eqs. (23) for the c-exp(A.y) and obtain 



3pi 

 3y 



r 3^vi 

 i ^° 3P~ 



(15) 



:.exp(A,y) = V b..z . for j = 1,2,.. .,6 (24) 

 3 3 A^ 13 11 



