75 



-0.2 



0.2 0.4 0.6 0.1 

 Fxio'' 



1.0 1.2 1.4 



FIGURE 12. Instability properties of Bj, = -0.10, 6 = 

 45° Falkner-Skan-Cooke boundary layer at R = 555. 

 (a) maximum amplification rate with respect to wave- 

 number and unstable k-F region; (b) unstable iji-F region. 



the most unstable, the unstable regions of Figure 

 12 bear more of a resemblance to those of a two- 

 dimensional boundary layer than to Figure 10. The 

 main differences from the two-dimensional case are 

 the asymmetry about i/ = 0° already noted in Figure 

 11, the one-sidedness of ^-^^y^, and, for F < 0.4 x 

 10~'t, the replacement of a lower cutoff frequency 

 for instability by a rapid shift with decreasing 

 frequency to waves oriented opposite to the cross- 

 flow direction and which are unstable down to zero 

 frequency. The instability shown in Figure 12 

 represents primarily an evolution of the small cross- 

 flow boundary layers of Figure 7 to larger cross- 

 flow. Only the frequencies, say |f| < 0.2 x 10"'^, 

 have to do with the pure crossflow instability of 

 Figure 10. For frequencies near 0.4 x 10"'* ,i|; 

 varies little with k in one part of the unstable 

 region, as with crossflow instability; in the other 

 part, as with mainflow instability, the opposite 

 is true. This behavior becomes more pronounced at 

 high Reynolds numbers . 



useful information concerning three-dimensional 

 boundary-layer stability can be obtained from par- 

 ticular pure spatial modes just as with two- 

 dimensional boundary layers . Arguments were given 

 to support using the modes whose growth direction 

 is determined from (17) or, more exactly, from 

 {19b) . A transformation (21a) , was derived to 

 enable the use of waves with an arbitrary growth 

 direction in calculating eigenvalues. The trans- 

 formation used in the temporal theory to reduce 

 the three-dimensional problem to a two-dimensional 

 probelm in the direction of the wavenumber vector 

 was shown to apply to spatial modes only when this 

 direction is close to the correct growth direction, 

 or the latter is the same as the potential-flow 

 direction (ij'qj- = 0°). 



The waves which have their wavenumber vector 

 aligned with the local potential flow (i|) = 0° when 

 the X axis of the mean-flow coordinate sytem is 

 also in the flow direction) always have their growth 

 direction very close to the potential-flow direction. 

 If the crossflow is small, the maximum amplification 

 rate of the ijj = 0° waves is almost identical to the 

 maximum amplification rate of the three-dimensional 

 boundary layer. Consequently, if we are only in- 

 terested in establishing the maximum amplification 

 rate of a small crossflow boundary layer, it can be 

 obtained from the mainflow profile alone. We used 

 this approach to obtain the effect of the flow (yaw) 

 angle on the instability of the Falkner-Skan-Cooke 

 yawed-wedge boundary layers for small pressure 

 gradients , and found that yaw reduces both the 

 stabilizing effect of a favorable pressure gradient 

 and the destabilizing effect of an adverse pressure 

 gradient. 



With moderate or large crossflow, crossflow in- 

 stability, which arises from the inflection point 

 of the crossflow velocity profile, is present and 

 can destabilize a boundary layer at low Reynolds 

 numbers which would otherwise be stable. As befits 

 the name, the unstable waves have their wavenumber 

 vectors oriented near the crossflow (or opposite) 

 direction. Also the instability covers a wide band 

 of unstable frequencies (including zero) and wave- 

 numbers. The growth direction of all unstable waves 

 is still near the potential-flow direction. If the 

 mainflow profile is also unstable, then the unstable 

 frequencies near zero act as with pure crossflow 

 instability and the higher frequencies as with pure 

 mainflow instability. Intermediate frequencies 

 have the latter behavior for small wavenumbers , and 

 the former for large wavenumbers . 



The results demonstrate why crossflow is more of 

 a problem for the maintenance of laminar flow with 

 strong favorable pressure gradients than with ad- 

 verse pressure gradients. In the former case, cross- 

 flow provides a powerful instability mechanism 

 even when the mainflow profile is stable; in the 

 latter, the crossflow only increases the amplifi- 

 cation rate over that of an already unstable main- 

 flow profile. This increase is about 50% for the 

 6 = 45° separation boundary layer. 



ACKNOWLEDGMENT 



CONCLUDING REMARKS 



All of the numerical results that have been presented 

 stem from the viewpoint adopted in Section 2 that 



This paper represents the results of one phase 

 of research carried out at the Jet Propulsion Lab- 

 oratory, California Institute of Technology under 

 Contract No. NAS7-100 sponsored by the National 

 Aeronautics and Space Administration. Financial 



