74 



lated in a single computer run with ijigj- = 0°, and 

 then corrected to an approximate ii)gj-(k) obtained 

 from (13) with constant wavenumber . A least-squares 

 curve fit to o (k) provided Oj^gx' to maximum spatial 

 amplification rate with respect to the vector wave- 

 number, and kpiax S"'^ '''max' the magnitude and direc- 

 tion of the wavenumber of Omax- 



Figure 10a gives o^^^ as a function of the di- 

 mensionless frequency F, and also shows the portion 

 of the iJi-F plane for which there is instability. 

 The unstable region is enclosed between the curves 

 marked i|/ and \li . These curves represent either 

 neutral stability points or extrema of ift. 



The corresponding wavenumber magnitudes are 

 shown in Figure 10b. The negative frequencies 

 signify that with \l) taken to be continuous through 

 F = 0, the phase velocity changes sign. If we 

 choose i|) so that the wavenumber and phase velocity 

 are both positive, then it is i|) that changes sign 

 at F = 0. Consequently, there are two groups of 

 positive unstable frequencies with quite different 

 phase orientations. The first group, which includes 

 the peak amplification rate, is oriented anywhere 

 from 5° to 31° (clockwise) from the direction opposite 

 to the crossflow direction. The second group is 

 oriented close to the crossflow direction itself. 



All of the unstable frequencies have in common 

 that the direction of growth is within a few degrees 

 of the potential-flow direction. The angle ijiqj- of 

 "{"max' ^s computed from (13) , is negative and has 

 its largest magnitude of just under 6° near F = 

 -0.60 X 10" . Orientations other than tmax "^^^ 

 have growth directions further removed from the 

 flow direction. 



Boundary Layers with both Crossflow and Mainflow 

 Instability 



As an example of a boundary layer which has both 

 crossflow and mainflow instability at low Reynolds 

 numbers, we selects, = -0.10 and 6 =45°. In con- 

 trast to the previous case, the steady disturbances 

 do not become unstable until a Reynolds number, R = 

 275, where the peak amplification rate is already 

 7.35 X 10"^. [For g^ = -0.10 and 6=0° Omax = 

 11.0 X 10-3 ^^ p ^ 2.2 X lO-"* according to Wazzan 

 et al. (1968)]. The distribution of a with i|) is 

 shown in Figure 11 for F = 2.2 x lO"**, a frequency 

 close to the most unstable frequency of F = 2 . 1 x 

 10" . We see that with a maximum crossflow velocity 

 of 0.0349 (cf . Table 2) , the distribution of a about 

 ij; = 0° is markedly asymmetric, and the maximum 

 cimplification rate of 7.31 x 10" 3 is located at iji = 

 -29.4° rather than near zero. This asymmetry was 

 barely perceptible for the small crossflow boundary 

 layers of Figure 7 where the crossflow is only one- 

 sixth as large. The a at i() = 0° of Figure 11 (5.82 

 X 10"^) is close to Oitiax with respect to frequency 

 of the 1^) = 0° waves (5.91 x 10~3) . since this value 

 is 20% below the peak amplification rate , the i|; = 

 0° waves are no longer adequate to represent the 

 maximum instability as with small crossflow boundary 

 layers. Figure 11 also gives the distribution with 

 iii of k and ij^gj.. The latter quantitiy was obtained 

 from (13) with constant wavenumber, and we see that 

 it remains within ± 7.5° of the potential-flow 

 direction throughout the unstable region. 



Because R = 276 is the minimum critical Reynolds 

 number of the steady disturbances, the unstable 

 region terminates in a neutral stability point at 



S 6 



1.2 



1.0 

 0.8 

 -^ 0.6 

 0.4 

 0.2 



-2.0 -1.5 -I.O -0.5 



0.5 1.0 1.5 2.0 2.5 3.0 

 FxlO^ 



FIGURE 10. Instability properties of Bj^ = 1.0, 9 = 45° 

 Falkner-Skan-Cooke boundary layer at R = 400. (a) maxi- 

 mum amplification rate with respect to vJavenxjmber and 

 unstable t|j - F region; (b) unstable k-F region. 



F = 0. We are particularly interested here in Rey- 

 nolds numbers where F = is also unstable, and as 

 an example. Figure 12 gives results for all unstable 

 frequencies at R = 555. Figure 12a shows a^ax ^^ 

 a function of F (here, as in Figure 10, Oj^ax ^^ the 

 maximum with respect to k) , as well as the unstable 

 region of the k-F plane ; the unstable region of the 

 \ii-F plane appears in Figure 12b. These two unstable 

 regions are quite different from those of Figure 10 

 where there is only crossflow instability. The 

 negative frequencies do resemble those of Figure 10 

 in that the unstable range of i(j is small, of k is 

 large, and with ip defined so that F > 0, the orien- 

 tations are close to the crossflow direction. How- 

 ever, for the higher frequencies, which are by far 



7 



6 



o 5 



X 

 ^_ 4 



CO 



o 

 X 3 



D 



2 

 1 



-70 -60 -50 -40 -30 -20 -10 

 iKdeg) 



10 20 30 



FIGURE 11. Effect of wavenumber angle on o, k and ijigj. 

 for 6h = -0.10, 6 = 45° Falkner-Skan-Cooke boundary 

 layer at R = 276. F = 2.2 x IQ-"*. 



