A,26 • EFFECT OF HEAT TRANSFER ON TRANSITION 



The value of Ukk/v may be estimated as follows: In compressible flow 

 the Blasius formula for velocity gradient near the wall holds approxi- 

 mately, provided the kinematic viscosity near the wall is used in form- 

 ing the Reynolds number. Hence du/dy = 0.332 (tZ/a:) 's/Ux/vy, where 

 U is the free stream velocity and v^ is the kinematic viscosity at wall 

 temperature. For small roughness Uk = kdu/dy. Thus 



^ = 0.332 f^V^") /^ 



Vy, \Vy,J \xJ\Vy, 



This may be written in the form 



- = 1.735 



X 



i-Tiyrm' 



where v^ is the kinematic viscosity at free stream temperature. The term 

 (j'm/j'w)~* gives the Mach number effect on the assumption that the critical 

 value of Ukk/v is independent of Mach number. Luther's data on Mach 

 number effect are fairly well represented by this relationship. Since the 

 roughness extended from a; = to a: = 0.1875, the basis for comparison 

 of the Reynolds number effect is uncertain. If a fixed value of x is used, 

 the theoretical relation gives somewhat smaller variation with Ux/v^ than 

 shown by Luther's curves for the value of k at which transition reaches 

 the downstream end of the roughness. 



Detailed studies of surface temperature distributions near roughness 

 elements are given in [115]. Some effects of distributed roughness on tran- 

 sition at a Mach number of 3.5 are described in [111]. Some observations 

 on roughness effects at a Mach number of 5.8 are discussed in [105]. At 

 this Mach number the boundary layer was markedly insensitive to 

 roughness. 



In summary of this section, there is every evidence that transition at 

 supersonic speed is a phenomenon of widespread occurrence and that the 

 factors which affect transition at subsonic speeds also influence transition 

 at supersonic speeds. 



A, 26. Effect of Heat Transfer on Transition at Supersonic 

 Speeds. The theory of the effect of heat transfer on the stability of a 

 laminar boundary layer developed by Lees and Lin [99,100] is reviewed 

 elsewhere [IV,F] in this series. This theory yields the important result 

 that the boundary layer could be completely stabilized by sufficient cool- 

 ing of the body. There has been much controversy about the validity of 

 the assumptions on which this theory is based and about the accuracy 

 of the numerical computations. 



For purposes of comparison with experimental data, we shall use the 

 more recent results of Dunn and Lin [116], although even these compu- 

 tations are not free from criticism because restricted to small wave 



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