69 



30 40 50 

 l// (deg) 



70 80 



FIGURE 2. Amplification rate as function of ^ for four 

 frequencies. Blasius boundary layer, R = 1600. 



of its two-dimensional value until ijj has increased 

 to 60°. With unstable waves for -79° < i|) < 79°, a 

 consideration of only the two-dimensional wave gives 

 an incomplete picture of the instability of the 

 boundary layer. 



THREE-DIMENSIONAL FALKNER-SKAN BOUNDARY LAYERS 



In order to study the influence of three dimension- 

 ality in the mean flow on boundary- layer stability, 

 it is necessary to have a family of boundary- layers 

 where the magnitude of the crossflow can be varied 

 in a systematic manner. The two-parameter yawed- 

 wedge flows introduced by Cooke (1950) are suitable 

 for this purpose. One parameter is the usual Falkner- 

 Skan dimensionless pressure gradient; the other 

 is the ratio of the spanwise and chordwise velocities. 

 A combination of the two parameters makes it possible 

 to simulate simple planar three-dimensional boundary 

 layers . 



The inviscid velocity in the plane of the wedge 

 and normal to the leading edge is 



U* = 

 c, 



C*(x* 



where the wedge angle is (Tr/2) B and 8 = 2m/(m+l). 

 We shall refer to this velocity as the chordwise 

 velocity. The velocity parallel to the leading 

 edge, or spanwise velocity is 



W* = const. 



SI 



The subscript 1 refers to the local freestream. For 

 this inviscid flow, the boundary- layer equations 

 in the x direction, as shown by Cooke (1950), 

 reduce to 



f " + ff " 



m+1 

 2 



This equation is the usual Falkner-Skan equation 

 for a two-dimensional boundary layer, and is inde- 



pendent of the spanwise flow. The dependent vari- 

 able f(n) is related to the dimensionless chordwise 

 velocity by 



U = 

 c 



U* 



2_\ f'(n) 

 ,m+l/ 



and the independent variable is the similarity 

 variable 



n = y 



v*x* 



where x^ is measured normal to the leading edge. 

 Once f (ri) is known, the flow in the spanwise di- 

 rection Zg is obtained from 



g^' + fg' 







where 



W* 

 = s 



W* 

 Sl 



g(Ti) 



Both f ' (n) and g(n) are zero at n = and approach 

 unity as n -> ". Tabulated values of g(ri) for a 

 few values of 6 , may be found in Rosenhead (1963, 

 p. 470) . 



The final step is to use f (n) and g(r)) to con- 

 struct the mainflow and crossflow velocity components 

 needed for the stability equations. A flow geometry 

 appropriate to a swept back wing is shown in Figure 

 3. There is no undisturbed freestream for a Falkner- 

 Skan flow, but such a direction is assumed and a 

 yaw, or sweep, angle ifi is defined with respect to 

 it. The local freestream, or potential flow, is at 

 an angle iip with respect to the undisturbed free- 

 stream. It is the potential flow that defines the 

 x,z coordinates of the stability equations. The 

 angle of the potential flow with respect to the 

 chord is 



tan 



W* 



U*~ 

 c. 



and 6 is related to i1j and Ui by 

 sw p 



UNDISTURBED 

 FREESTREAM 



FIGURE 3. Diagram of coordinate systems used for 

 Falkner-Skan-Cooke boundary layers . 



