70 



= ("c? 



+ wth'i, 



SI' 



with the local potential velocity, Up 

 as the reference velocity, the dimensionless main- 

 flow and cross flow velocity components are 



u(n) = f (n) cos e + g(n) sin 

 w(n) = pf (n) + g(n) cose si 



(24a) 

 (24b) 



These velocity profiles are defined by gjj, which 

 fixes f (n) and g(n), and the angle 9. We note 

 from (24b) that for a given pressure gradient all 

 crossflow profiles have the same shape; only the 

 magnitude of the crossflow velocity changes with 

 the flow direction. In contrast, according to 

 (24a), the mainflow profiles change shape as 9 varies. 

 For 9 = 0, u(n) = f'(n); for 9 = 90°, u(n) = g(n) ; 

 for 9= 45°, the two functions make an equal con- 

 tribution . 



When the velocity profiles (24) are used directly 

 in the stability relations, (2), the velocity and 

 length scales of the equations must be the same as 

 in (24) . This identifies the velocity scale as U*, 

 the length scale as 



v*x*/U* (x*) 

 c c, c 



and the Reynolds number U*L*/v* as 



P 



R /cos 9 

 c 



where 



R = Tu* (x*)x*)v* ^ 



C I CJ c c _J 



is the square root of 



the Reynolds number along the chord. For positive 

 pressure gradients (m > 0) , 9 =90° at x = and 

 9->-0°asx^«'; for adverse pressure gradients 

 (m < 0) , 9 = 90° at X = and 9 ^ 0° a's x ^ »; for 

 adverse pressure gradients (m<0),9=0°atx= 

 and 9 -»■ 90° as X ^ ». The Reynolds number R^ is 

 zero at X = for all pressure gradients, as is 

 R with one important exception. The exception is 

 where m = 1 (Sh = D is the stagnation-point solution; 

 here it is the attachment-line solution. In the 

 vicinity of x = 0, the chordwise velocity is 



U* = X* (dU* /dx*) „. 

 Cj c c c x=0 



The potential velocity along the attachment line is 

 W* , and the Reynolds number is 



SI 



R(x=0) = W* / 



SI 



V*(dU*^/dX*)^, 



»]■ 



a non-zero value. 



For the purposes of this paper, 9 may be regarded 

 as a free parameter, and the velocity profiles (24) 

 used at any Reynolds number. However, for the flow 

 over a given wedge, 9 can be set arbitrarily at only 



one Reynolds number. If 



the 9 at any other R 



ref 

 is given by 



is 9 at R = (R 



ref' 



tan 9 



tan 9 



ref 



(R 



JR 

 ref c 



m/(m+l) , 



For m << 1, the dependence on R is so weak that 9 

 is constant almost everywhere. One way of choosing 

 (Re) ref within the context of Figure 3 is to make 

 it the chord Reynolds number where i|jp = 0; i.e., 

 the local potential flow is in the direction of the 

 undisturbed freestream. 

 yaw angle ip 



Then 9 , is equal to the 

 ref 





 W xioo 



16 



FIGURE 4. Four crossflow velocity profiles, Falkner- 

 Skan-Cooke boundary layers. INF, inflection point; 

 MAX, maximum crossflow; SEP, separation pressure 

 gradient (6jj = -0.1988377). 



for 



45° and four values of i 



The inflection 



point and point of maximum crossflow velocity (Wj^ax) 

 are also noted on the figure . In Figure 5 , Wjj,ax f °^ 

 9 = 45° is given as a function of 3 j^ from near sep- 

 aration to Bj^ = 1.0. The crossflow velocity for 

 any other flow angle is obtained by multiplying the 

 Wmax of the figure by cos9 sin9. The maximum cross- 

 flow velocity of 0.133 is generated by the separa- 

 tion profiles rather than by the stagnation profiles, 

 where W = 0.120. However, W varies rapidly 

 with 6 , in the neighborhood of separation, as do 

 all other boundary-layer parameters, and for B j^ = 

 -0.190, W is only 0.102. 



The function g(ri) is only weakly dependent on 



Figure 4 shows the crossflow velocity profiles 



0.8 1.0 



FIGURE 5. Effect of pressure gradient on maximum cross- 

 flow, Falkner-Skan-Cooke boundary layers . 



