72 



1.0 



1.5 

 R X 10" 



2.5 



3.0 



FIGURE 5. Effect of small pressure gradients on 

 the maximum amplification rate with respect to fre- 

 quency for two-dimensional Falkner-Skan boundary layers. 



is always close to the Blasius function. Thus as 

 the flow angle increases from zero the amplification 

 rate must change from the two-dimensional Falkner- 

 Skan value at 9 = 0° to a value not far from Blasius 

 at e = 90°. 



As discussed in Section 4, the only physically 

 meaningful flow with 6 = 90° and a non-zero Reynolds 

 number is the attachment- line flow ((5}^ = 1.0). For 

 all other values of Bj^, R at this flow angle must be 

 either zero (Bh > 0) or infinite (Bh ^ 0) • With 6j^ 

 = 1.0 and R = 1000 (R = 404.2, where Rg is the 

 momentijm-thickness Reynolds number), ^max/ '"^b ' max 

 = 0.765. The minimum critical Reynolds number of 

 this profile is (Rgj^r = 268 (the parallel-flow 

 Blasius value is 201) , yet turbulent bursts have 

 been observed as low as R^ = 250 for small distur- 

 bances by Poll (1977). 



FIGURE 7. Effect of flow angle on the maximum amplifi- 

 cation rate with respect to frequency of iJj = 0° waves 

 for two boundary layers with small crossflow at two 

 Reynolds numbers. 



We must still show that the waves with i|; = 0° 

 properly represent the maximum instability of three- 

 dimensional profiles with small crossflow. For this 

 purpose a calculation was made of a as a function 

 of ijj for Bh = -0-02, 6 = 45°, R = 1000 and F = 

 0.4256 X 10"'*, the most unstable frequency for "i/ = 

 0° at this Reynolds number. It was found that the 

 crossflow indeed introduces an asymmetry into the 

 distribution of a with i(), and the maximum of a is 

 located at t = -6.2° rather than at 0°. However, 

 this maximum value differs from the cjj^ax o^ Figure 

 7 by only 0^7%. It was also determined that ij^gi- = 

 -0.04° and ijjgi = -0.3° (approximately) for i|) = 0°, 

 which justifies taking both of these quantities 

 zero in all of the iji = 0° calculations. 



Crossflow Instability 



Minimum Critical Reynolds Number of Steady 

 Disturbances 



The instability that is unique to three-dimensional 

 boundary layers is called crossflow instability. 

 It was discovered experimentally by Gray (1952) and 

 later given a detailed theoretical explanation by 

 Stuart in Gregory et al . (1955). This instability 

 arises from the inflection point of the crossflow 

 velocity profile. As explained by Stuart, there 

 is a particular direction close to the crossflow 

 direction for which the mean velocity at the in- 

 flection point of the resultant velocity profile 

 is zero. Consequently, at sufficiently large 

 Reynolds numbers unstable steady disturbances exist 

 which have their constant phase lines nearly aligned 

 with the potential flow. 



Although crossflow instability is by no means 

 restricted to steady disturbances, these disturbances 

 do make a convenient starting point for our investi- 

 gation. The reason is that a suitable initial 

 guess for the angle \l) , which must be known rather 

 accurately for the eigenvalue search procedure to 

 converge , is given by 



t = (Bh/|Bhh (V2 - Ulinf) 



where Einf i^ ^^^ streamline deflection angle listed 

 in Table 2. It turns out that this value is with- 

 in a fraction of a degree of the angle of the most 

 unstable wavenumber. There is no such convenient 

 rule for the wavenumber itself, but the inverse of 



'mf ' 



the location of the inflection point in the 



similarity coordinate, or better still O.^/d^^^, is 

 usually an adequate enough initial guess to ensure 

 rapid convergence to an eigenvalue. 



As the crossflow is a maximum at 6 = 45° for a 

 given Bhf we can expect the crossflow instability 

 to also be a maximum near this angle. Figure 8 

 shows the minimum critical Reynolds number R^r 3t 

 9 = 45° for the zero-frequency disturbances as a 

 function of Bh- F°^ comparison, R^r of the two- 

 dimensional Falkner-Skan profiles, as computed by 

 Wazzan et al. (1968) , is also given. For adverse 

 pressure gradients, the steady disturbances become 

 unstable at Reynolds numbers well above the R^r of 

 the two-dimensional profiles. On the contrary, for 

 Bh > 0.07 the reverse is true, and for most pressure 

 gradients in this range the steady disturbances 

 become unstable at much lower Reynolds numbers than 

 the two-dimensional Rcr (foJ^ Bh = 1-0' the two- 



