71 



TABLE 2. Properties of three-dimensional Falkner-Skan-Cooke boundary layers. 



6jj, and, unlike f ' (n) , never has an inflection 

 point even for an adverse pressure gradient. Indeed 

 it remains close to the Blasius profile in shape, 

 as underlined by a shape factor H (ratio of dis- 

 placement to momentum thickness) that only changes 

 from 2.703 to 2.539 as ^ goes from -0.1988377 (sep- 

 aration) to 1.0 (stagnation) . The weak dependence 

 of g(ri) on fi was first pointed out by Rott and 

 Crabtree (1952) , and made the basis of an approximate 

 method for calculating boundary layers on yawed 

 cylinders. For our purposes, it allows some of the 

 results of the stability calculations to be antici- 

 pated. For waves with the wavenumber vector aligned 

 with the local potential flow, we can expect the 

 amplification rate to vary smoothly from its value 

 for a two-dimensional Falkner-Skan flow to a value 

 not too far from Blasius as 6 goes from zero to 90°. 



The stability results in the next section will 

 be presented in terms of the Reynolds number R and 

 the similarity length scale L*. In order that the 

 results may be converted to the length scales of 

 the boundary-layer thickness, displacement thick- 

 ness and momentum thickness. Table 2 lists the 

 dimensionless quantities ng = S/L* , ng* = <5*/L* and 

 H = rii5*/n9 of the mainflow profile for several com- 

 binations of gjj and 6. Also listed are W^ax' the 

 average crossflow velocity W = (/wdri)/Tl5, the 

 deflection angle of the streamline at the inflection 

 point, i-infr an<i the location of the inflection 

 point, ninf- The quantity ng is defined as the 

 point where U = 0.999. 



STABILITY OF FALKNER-SKAN-COOKE BOUNDARY LAYERS 



Boundary Leyers with Small Crossflow 



In a two-dimensional boundary layer, the most un- 

 stable wave is two dimensional. Therefore, we can 

 expect that in three-dimensional boundary layers 

 with small crossflow the most unstable wave will 



have its wavenumber vector nearly aligned with the 

 local potential flow, and we can restrict ourselves 

 to waves with ij; = 0° for the purpose of determining 

 the maximum amplification rate. With the temporal 

 stability theory, this procedure is equivalent to 

 studying the two-dimensional instability of the 

 mainflow profile, but is only approximately so in 

 the spatial theory_unless iigr = 0°. As ii„^ is 

 usually small for ip = 0°, even with large cross- 

 flow, we may also view the ip = 0° spatial results 

 as a measure of the instability of the mainflow 

 profile. 



In order to place the three-dimensional effects 

 in context, it is helpful to first consider a small 

 deviation in the assumed pressure gradient on the 

 maximum amplification rate of two-dimensional 

 Falkner-Skan profiles. Figure 6 shows the maximum 

 spatial amplification rate (with respect to frequency) 

 as a function of Reynolds number for Blasius flow 

 and for B^ = - 0.02. What is noteworthy about 

 these results is the magnitude of the shift in Omax 

 for what are quite small pressure gradients. It 

 is evident that an experiment intended to measure 

 amplification rates in a Blasius boundary layer to 

 within an accuracy of 10% is required to maintain 

 an exceptional uniformity in the flow. 



The effect of the flow angle 9 on the maximum 

 spatial amplification rate of the waves with ^ = 0° 

 is shown in Figure 7 for Bh = - 0.02 and two Rey- 

 nolds numbers . In these calculations , ij/gr and i|)gi 

 were both taken equal to zero. The amplification 

 rate Oj^ax ^^ expressed as a ratio to the Blasius 

 value (Ob^max shown in Figure 6. It will be re- 

 called that with Bh = 0, g(n) = f (n) , and the 

 velocity profile remains the Blasius function for 

 all flow angles. The effect of a non-zero flow 

 angle with S , ?^ is destabilizing for a favorable 

 pressure gradient, and stabilizing for an adverse 

 pressure gradient. Consequently, it reduces the 

 pressure-gradient effect shown in Figure 6. The 

 reason for this result is easy to understand by 

 reference to (24) . We have already pointed out in 

 Section 4 that the spanwise velocity profile g(n) 



