27 



2TT/a 



^T - §7 / / '"^ + v2)dxdn 



(8) 



Then the relative amplification ratio, erp /e-, , of 

 a disturbance as it grows during the time interval 

 from tg to ti can be shown to be 



-Tj I(t^) 



e„ I(t ) 



T 







exp 2 



/■ 



A dt 



(9) 



efficients, a , ...,aj,, are determined by the 

 boundary conditions once a is known. The matrices, 

 Q,P,J, and V, are the respective representations of 

 the operators ,£,£2, (fB£-32fg/3y2) ^^^ (.^-d^dy^) , 

 together with the boundary conditions (6) in the 

 space, Vj^, whose basis is the first N Chebyshev 



N' 

 polynomials, Tq 



T^_l[Orszag (1971)]. The 



function,^, is the Stokes layer profile 



jsf= Re (1-e 



-(l+i)l 



n, int 



)e 



(14) 



Note that the matrices, Q,P, and J, are real constant 

 matrices and V is real and time periodic and of the 

 form 



Kt) 



r 



3n 



a2|g|2 



an 



(10) 



V = V 



(1) int 



f-i) -int 



(15) 



and A, 



Re (A) 



Since the disturbance energy propogates down the 

 boundary layer at the group velocity, c [see 

 Gaster (1962)] one can compute the relative ampli- 

 fication ratio, e,p /erp , by calculating the integral 

 in the exponential function of Eq. (9) over the 

 spatial interval, Xq to x^, using the transformation. 



dx 



Cgdt. 



3. SOLUTION OF THE DISTRUBANCE EQUATION 



Solutions of Eq. (4) in the form (5) can be obtained 

 as a series in A, 



<p= (g^+Agj + . 



^(Ao+aiA+. ..) t 



(11) 



where V 



(1) 



and V 



(-1) 



are constant matrices. 



The matrix, Q, is invertable so that we can 

 multiply (13) by Q~^ to get 



da 

 dt 



(P'+iaJ')a+iaAV'a 



(16) 



■Ir 



where P' = Q~'-p etc.; henceforth we shall dispense 

 with the primes in (16) 



The perturbation procedure is most easily and 

 illuminatingly carried out by transforming (16) so 

 that the matrix, (P+iaJ)/R.^, is in diagonal form. 

 That is we will be working directly in the (approx- 

 imate) eigenspace of the steady Orr-Sommerfeld 

 equation for Blasius flow. Suppose that the in- 

 vertible matrix, B, transforms (P+iaJ)/R into 

 diagonal form. Then let 



where each term of (11) can be evaluated by solving 

 appropriate perturbation equations obtained by sub- 

 stituting (11) into (4) and (5) . Such perturbation 

 equations are basically inhomogeneous unsteady Orr- 

 Sommerfeld equations and must be solved numerically. 

 Our approach is equivalent to this except we reverse 

 the procedure by first executing a numerical pro- 

 cedure which reduces the Eqs. (4) and (5) to a sys- 

 tem of ordinary differential equations in time. 

 These are easily solved by perturbation theory to as 

 high an order as desired. 



Let us first expand the function <p in the Cheby- 

 shev series 



Bb 



(17) 



and substitute (17) into (16) and left-multiply by 



Then 



where 



db 

 dt 



Db + AEb 



(18) 



D = B ^ ^ (P+iaJ)B = rAi,...,A^_^J (19a) 

 6^ 



N 



f {y,t) 



n=l 



a„(t)T (y) 

 n n-1 



(12) 



where the Tj^(y) = cos~Mn cos y) , n = 0,1,... are the 

 Chebyshev polynomials of the first kind and where 

 we have mapped the interval, nE[0,no<>]/ onto ye [-1,1]. 

 Then we use the T-method as described by Orszag 

 (1971) to obtain the system of ordinary differential 

 equations 



da 

 dt 



(P+iaJ) a+iaAVa 



(13) 



where Q,P,J and V are (N-4)x(N-4) term matrices and 

 a = (aj , . . . ,aj^_^) ■ . The dagger (t) superscript de- 

 notes the transpose of a vector or matrix. The co- 



E = +iaB VB 



^(1) ^iS^t ^ ^(-1) ^-ifjt ^^g^J 



and the notation, fdi,..., d 1 , stands for a di- 

 agonal matrix of order n. 



The problem is now to find solutions of (18) in 

 the form 



b(t) 



z(t)e 



At 



(20) 



where z(t+2Tr/n) = z(t). We are mainly interested 

 in perturbations of magnitude A of the steady flat- 

 plate disturbance mode which becomes unstable far 

 downstream of the leading edge. This mode is as- 

 sociated with one of the eigenvalues of D, say A , 

 which for values of x between the two values , x < 

 xi, satisfies ReA > 0. It is known that Ap is a 

 simple eignevalue [see Mack (1976) ] so that a solu- 

 tion of the form (20) can be expanded as 



