f/WuiMo 1 



(a) 



(b) 



Figure ?. From Shen (10), by peimission of the Journal of Aeronautical Sciences. 



must therefore be caused by viscosity. The viscous forces act as follows. Consider a 

 travelling wave over a solid boundary. It can be shown that the viscous forces at the 

 wall produce a phase shift of the oscillation which in turn produces a Reynolds stress. 

 The Reynolds stress r is found to be given by the formula [12] 



pv' 



1 



3a5o 



(1) 



where 



a = wave number 



5 = thickness of the Stokes boundary layer 



= W2«, 



V2u. 

 -' £ T = frequency 



v 2 = mean square of the component of oscillation perpendicular 

 to the wall. 



Thus, there is a Reynolds stress acting in the same direction as the shear of the mean 

 motion, if the wave is moving in the direction of the basic flow. At large distances, this 

 stress must be zero. The transition occurs at the critical layer where the wave speed is 

 equal to the flow speed and the effect of viscosity there is again dominant. In a region 

 essentially free from the effect of viscosity, the Reynolds stress remains constant [13]. 

 This Reynolds stress converts energy from the basic flow into the small oscillations. If 

 this energy conversion is sufficiently large to overcome the damping effect of viscosity, 

 the motion will be unstable. Thus, viscosity plays a dual role. It is at the same time 

 the indirect cause of instability and the suppressor of the oscillations. The relative 

 importance of these effects decides whether the flow is stable or not. This is why dis- 

 turbances of a given frequency are stable both for large and for small Reynolds num- 

 bers; because in the one case there is too much damping; and in the other case, there 

 is insufficient excitation. 



It has been found possible [12] to pursue this line of reasoning to derive, in a 



355 



