29 



-E 



JP 



and 



c'"^' =\i«n. / ' 





- ,^, _(2) 2if2t _(o) -(-2) -2if2t 



22 (t) = n e + n + T) e , 



(35b) 



(36a) 



(36b) 



where 



■(2) 



■(0) 



N' \ t 



j=i 



2in-Y„ 



(37a) 



((£ "^ 



C <-l'.E<-^'c,<l' 



(37b) 



VpJ /'-v 



-(-2) 



n 



N' 



j=i 



(-1) (-1) 



(37c) 



-2in-Y„ 



We note that the order A perturbation, aj, of 

 the eigenvalue X is zero so that the long-term 



only of order A 



However, the short-term effect 



is still of order A because the eigenf unction, 

 Z] (t) , appears in the term I(t) in the relative 

 amplification ratio, e^ /e„ , given by Eq. (9) . In 

 fact, the structure of the matrix, E, is such that 

 all values of o ■ with odd indices are zero and X 

 has an expansion in even powers of A about the 

 simple eigenvalue, Ap. This can be surmized easily 

 from the fact that the phase of the imposed oscil- 

 latory part of the boundary layer flow should not 

 play a role in the modifications of the eigenvalue. 



the instability of the oscillatory boundary layer 

 as a whole is being compared to the instability of 

 the underlying steady Blasius boundary layer. How- 

 ever, it is easier to describe this comparison in 

 the terminology of the oscillatory forcing of the 

 Blasius boundary layer instability. For example, 

 if the oscillatory boundary layer is less stable 

 than the steady boundary layer by itself, then we 

 describe this situation as one in which the imposed 

 oscillations tend to destabilize the steady flow. 



The first set of calculations were made to test 

 for resonant interactions at second order in A. By 

 consulting the solutions (35) and (35), it can be 

 seen that the mean effect of the imposed oscilla- 

 tions on the eigenvalue, X , is manifested by the 

 term, 02- There are two types of resonances pos- 

 sible. The first type is the "harmonic parametric 

 resonance" which corresponds to values of fl given 

 by (Up/f! = 1/2, 1, 2,... where cOp is the response 

 frequency of the disturbance, (Dp = SmXp. The 

 second type of resonance is the "combination reso- 

 nance" corresponding to values of H given by<fl m 

 (ifl+Y^) = (note the denominators of solution 35). 

 Figure 1 shows the computational results at certain 

 frequencies H in the range, l<ai /f2<3. It can be 

 seen that the imposed oscillations stabilize the 

 flow. Figure 1 shows that no resonance effects are 

 predicted at either cOp/O = 1,2,3, or at Up/H = 1.417 

 and 1.74, which correspond to the two possible com- 

 bination resonances in the frequency range shown. 

 This lack of resonance effect results mainly be- 

 cause the external free stream oscillations induce 

 a significant amount of oscillatory vorticity in- 

 side the boundary layer only in a region very close 

 to the wall. This can be seen by examination of 

 the Stolies layer profile (14) where the exponential 

 factor has a vertical decay constant, 6, which is 

 equal to about 5 in the range of frequencies con- 

 sidered. The main fluctuations of the disturbance 

 velocity are concentrated at the mean critical layer, 

 Dc = 0.5 [where n^ is given by Cj^ = fg(ric) ^i^d ^y- 

 is the mean phase velocity of the disturbance] . 



Thus, instead of the Stokes layer interacting 

 directly with the disturbance of the underlying 

 steady boundary layer at the level, ric/ where most 

 of the disturbance energy is being produced, it is 

 confined mainly to the wall region where it cannot 

 be very effective. Furthermore, the Stokes layer 

 lacks a spatial structure in the x-direction that 

 can match in some way the spatial structure of the 



and Z2(t), exhibit clearly the possible effects, at 

 second order, of certain resonant couplings. None 

 of the denominators in (35) and (37) are zero be- 

 cause Y-i 7^ ± ikS for any integer values of j or k; 

 hence these solutions are uniformly valid for any 

 positive value of the frequency, n. It is possible, 

 however, that at resonant frequencies such as at 

 fi=±»8m(Yp) , the value of 02 will have a relative 

 maximum. Of particular importance is that in the 

 low frequency limit, f! -»- , the Oj's may be singu- 

 lar. The lower values of Q will be an important 

 consideration and will be discussed in detail in the 

 next section. 



4. NUMERICAL RESULTS AND DISCUSSION 



Before describing the computational results that 

 have been obtained, we emphasize that in this work 



FIGURE 1. The growth rate perturbation Re 02 for a = 



0.15. R,^ = 1128 (on the neutral curve) (o, = u /R^* 

 0*1. r p o'^ 



= 0.43 X 10-t. 



