31 



(39) 



J = 2 / ReAdt 

 t 







r &*i 



= 0.676 (ReX/c ) dR^. 



where R^^ and Rg^j are the values of R^^^ at the 

 locations of the disturbance at the times, t^ and 

 t] , respectively. The value of c-, the group ve- 

 locity, along the trajectory a = 0.00133 Rj^^, was 

 computed to be about 0.356. We neglected the 0(A ) 

 modification of c due to the imposed oscillations. 

 This modification of Cg is 0(10~^) and thus does 

 not affect the integral, J, in a substantial way. 

 From the results shown in Figure 3, one obtains (by 

 a trapezoidal rule integration) , the values of J = 

 11,8.7, and 6.3 for o) /JJ = 0,2, and 3 respectively. 

 The integral I(t) of (10) is evaluated by certain 

 sums and products of the vector components of the 

 solutions, (35) and (37). We omit the details. The 

 resulting expression for I(t) , to second order in A, 

 has the following form 



I(t) = Ai+A2A^+A(BiCOsC2t+B2Sinnt) 

 +A2(Cicos2nt+C2Sin2nt) 



(40) 



where Ai>0 ,A2 ,Bi ,62 ,Ci and C2 are real numbers that 

 depend on the Reynolds number, Rj^. These coeffi- 

 cients have been computed along the disturbance 

 trajectory, a = 0.00133 R^^ and are plotted in 

 Figure 4. By using the values of A2<Bj, B2, Cj and 

 C2 from Figure 4 (Aj = 1.0 by suitable normalization) 

 in Eq. (40) for the value of A = 0.1 one finds that 



0.5 



I(t) 

 I(t J 



< 2 



at all the values of Reynolds number, R 



&*' 



for which 



the disturbance grows. It is customary to assess 

 the overall growth of a disturbance by considering 

 the natural logarithm of the amplification ratio, 



e,,,/e„ 



From (9) we have 



S,n 



In 



I(t) 



+ J 



and one can see that although the term, JLn I(t)/I(tQ) 

 contributes an oscillatory factor to Jin s^/e^ (re- 



call that, by following the disturbance down the 

 plate, t = Rr ) this contribution is minor relative 

 to the maximum value attained by J. Thus it can be 

 seen that the major effect of the parallel free 

 stream oscillations is to reduce the mean growth 

 rate of the unstable disturbances. This effect is 

 small for small values of A but can be significantly 

 large at such large values of A as A > 0.1. We note 

 that typical free stream turbulence rarely has a 

 velocity magnitude as large as 10 percent of the 

 mean free stream speed. 



Experimental results on the effects of parallel 

 free stream oscillation on the instability and 

 transition of the flat plate boundary layer are re- 



FIGURE 4. The coefficients of I (t) along a = 0.00133 



R,., u /a = 3. 

 6* p 



viewed by Loehrke, Morkovin and Fejer (1975). How- 

 ever, we shall not make any comparison with their 

 experiments because these were for very low frequency 

 oscillations (to /fi ~ 10) for which our parallel 

 flow instability theory is of doubtful applicability. 

 An appropriate analytical instability theory for 

 comparison with these experiments is a quasi-steady 

 and parallel flow theory [see Obremski and Morkovin 

 (1969) ] . 



5. CONCLUDING REMARKS 



Our main result is that the parallel free stream 

 oscillations, which manifest themselves in the 

 Blasius boundary layer as a Stokes layer, lead to 

 a mean stabilization of the flow. This stabiliza- 

 tion is very weak except for oscillation amplitudes 

 that are at least near 10 percent of the mean free 

 stream speed. Precise experimental data on the 

 effects of such oscillations on Blasius boundary 

 layer instability is not available in the frequency 

 range considered in this work. However, the results 

 are in accord with transition data for oscillatory 

 pipe flows. Sarpkaya (1966) has shown experimen- 

 tally that transition is delayed substantially when 

 harmonic axial oscillations are superimposed on 

 steady pipe flow. Furthermore, von Kerczek and 

 Davis (1975) have shown that the oscillatory Stokes 

 layer by itself is very stable, probably at all 

 Reynolds numbers, so that one might conjecture that 

 if the Stokes layer begins to dominate the boundary 



