30 



disturbance mode. It is notable that the imposed 

 oscillations have an increased stabilizing effect 

 as il decreases (Up/fl increases) . This increased 

 stabilizing effect can be expected for two reasons. 

 The first reason is found in the solution (35) 

 which shows that the terms. 



^pj'''''*i«-^p'' 



may become unbounded if 



,(1) 

 "PP 



or E 



(-1) 

 PP 



remains bounded as fl -^ , because Yp = ■ Secondly, 

 it can be seen in the Stokes layer profile (14) that 

 the oscillations of the boundary layer become more 

 effective in penetrating up to the critical layer 

 when Q decreases (i.e., 6 also decreases since 



6 = /fiR^ /2) 



0* 



However, we cannot use the present parallel flow- 

 model at very low frequencies because in one period, 

 2Tr/fi, of the imposed oscillation, a disturbance 

 will propogate down. the boundary layer a distance, 

 6x, that is too large for the parallel flow assump- 

 tion to hold (i.e., constant boundary layer proper- 

 ties in the x-direction) . For example, the change, 

 6Rr^, in the displacement thickness Reynolds number, 

 ^S^i over the distance, 6x, (near the values of a = 

 0.15 and Rfi^ = 1200) is given approximately by 



6R, : 3.5N/U 

 0* p 



(38) 



where N = iDp/fl. Thus, in the range of values of a 

 and Kg* of our calculations (SRg* : 70 N so that for 

 N = 3, tSRjjt is nearly 20 percent of the value of 

 Ri5*. Under the circumstance, the parallel flow 

 approximation is only roughly valid. Nevertheless, 

 the values of SR^^ as a fraction of Rg^ decrease as 

 one goes downstream of the neutral curve for fixed 

 values of the frequency ratio, Up/fl. Thus, the 

 parallel flow approximation improves as one follows 

 a constant frequency disturbance downstream of the 

 neutral point. 



The second set of calculations that were per- 

 formed was for the amplification of a fixed frequency 

 disturbance propogating down the oscillatory bound- 

 ary layer. Two values of U)p/f2, equal to 2 and 3, 

 were chosen for illustration. The disturbance ex- 

 amined is an unstable Tollmien-Schlichting wave of 

 constant absolute frequency toj = Up/R^j* = 0.43 x lO""* 

 along the constant frequency line a = 0.00133 Rg^. 

 This disturbance first begins to grow in the steady 

 boundary layer at the values of a = 0.15, R.^ = 

 1128, and ceases to grow at about the values of a = 

 0.3 and R,;* = 2255. The disturbance trajectory a = 

 0.00133 Rg* passes nearly through the point in the 

 "' ^S* Pl^ne of maximum rate of amplification. 



Figure 2 shows the values of Re02 obtained for 

 the growing Tollmien-Schlichting wave along the 

 trajectory, a = 0.00133 Rj*, at the two different 

 values cop/n = 2 and 3. An interesting feature of 

 the results in Figure 2 is that |Rea2| increases 

 with R|5* although the quantity 6 also increases 

 which would seem to indicate further decoupling of 

 the oscillatory Stokes layer (14) from the distur- 

 bance oscillations.. Presumably, the values of 



FIGURE 2. Growth rate perturbation Re a^ along a = 

 0.00133 R^^. 



0* 



I Rea2 I decrease as Rr* becomes sufficiently large 

 for then 6 also becomes so large that the Stokes 

 layer will almost completely disappear. It can be 

 seen in Figure 2 that the stabilization of the 

 boundary layer can be substantial for the value of 

 Up/fi = 3 and at the larger values of R^*. 



Figure 3 shows the values of ReA = ReXp + A^Rea2 

 for the value of A = 0.1 and the three values of 

 Up/n = 0,2, and 3. (cOp/n = is equivalent to A = 

 0) . The total effect of the imposed oscillations 

 with A = 0.1 is not very substantial at the value, 

 cijp/n = 2, but at the value of ujp/Q = 3, the sta- 

 bilization of the flow is significant. We note that 

 an oscillation amplitude of A = 0.1 is a rather 

 large value at the frequencies considered here and 

 would require a large amount of power to achieve 

 in an experimental test facility such as a wind 

 tunnel unless the mean flow is very slow. 



The rates of amplification shown in Figure 3 can 

 be summed according to formula (9) to obtain the 



relative amplification ratio, e^ /e^ 

 show that 



One can 



X 2 



1600 1800 



FIGURE 3. The amplification rate Xj, along the trajec- 



tory a = 0.00133 R 



6*- 



