13 



gradients in damping and adverse pressure gradients in amplifying the Tollmien-Schlichting 

 waves. 



A rough empirical criterion for locating the point of self-excited transitions is ascribed 

 to Walz^^ who suggests a critical value of the Reynolds number of transition Rq ^^ &,s tiirice 

 that for neutral stability or 



^d,ts =^^9.N [40] 



There follov/s, however, a new empirical criterion which is based on more rational 

 grounds and which attempts to incorporate the effect of pressure gradients in determining the 

 position of self-excited transition. Since the transition point depends on the cumulative ef- 

 fect of the pressure gradients from neutral stability to transition, it would seem appropriate 

 to use as a first approximation the average pressure gradient for one of the significant parame- 

 ters. Accordingly two-dimensional data on transition points from tests on wing sections in 

 flight^"'^'* and in low-turbulence wind tunnels^ ^'^^'-^^ were analyzed on this basis. As shown 

 in Figure 4 the difference in Reynolds numbers from the point of neutral stability to transition 



Rq ts - Rq f^ is plotted against the average pressure gradient parameter -^ ^^ over the 

 same region where 



dx 



r^£s 6^ dV_ 

 J^ V d¥ 



lAH. IE „j, 



V dS 



r 



dx 



I 



The tildes over the sjmbols refer to two-dimensional flows. Examination of the plotted data 

 in Figure 4 indicates a reasonably consistent variation between the two parameters involved. 



The conversion of the preceeding two-dimensional data for use in axisymmetric flows 

 past bodies of revolution may be accomplished by means of Mangler's relations^' ^^ for trans- 

 forming the equation of n..>,:on of two-dimensional boundary layers to those of equivalent axi- 

 symmetric boundary layers on bodies of revolution. Mangler's transformation relations are 



- »■- "" 

 dx 



[42] 



= 



c 



2 _ 



w 

 L2 



dx 



y 



= 



C 



L 



y 



d 



= 



c 



L 



d 



U{x) = U{x) 



