A, 7 • SHEAR LAYERS IN THE FREE FLUID 



stream which was probably quite high. This compares with an estimated 

 value of the order of 515 for a boundary layer under similar conditions 

 (Fig. A,6). 



At cylinder Reynolds numbers from about 20,000 to about 200,000 the 

 laminar boundary layer becomes turbulent immediately on separation. 

 Since the value of Re^* at any fixed point on the cylinder varies as the 

 square root of the cylinder Reynolds number, the critical Re^* for the 

 boundary layer is of the order of \/l0 or 3.2 times that for a shear layer. 

 This is an independent experimental determination of the relative values 

 of Rci* for transition. In similar layers with zero pressure gradient and 

 low turbulence, the distance from stagnation point to transition would be 

 ten times as great for a boundary layer as for a shear layer. 



It is well known that a surface of discontinuity in an incompressible 

 frictionless fluid is unstable in the sense that any small disturbance in- 

 creases exponentially in amplitude with time as originally discussed by 

 Helmholtz and by Rayleigh. Rosenhead [26] attempted to follow the 

 motion to a later stage and showed that the disturbance becomes un- 

 symmetrical and tends to concentrate the vorticity at points spaced at 

 equal intervals. Because of the nonlinear character of the equations, the 

 final stages cannot be computed by superposition of the effects of separate 

 wavelengths. Rosenhead believed that the determination of the wave- 

 length which ultimately dominates cannot be determined except by con- 

 sidering the effects of viscosity and diffusion. The rolling-up process was 

 well advanced in the time required for the fluid to travel a distance equal 

 to one third of the wavelength. 



The effect of viscosity is not only to convert a discontinuity into a 

 shear layer of finite thickness but also to provide a damping effect. 

 Lessen [27] attempted to compute the stabihty of a shear layer using the 

 Tollmien-Schlichting theory. His computations are not complete but they 

 indicate a critical Re^* for the beginning of amplification of about 15 and 

 a predominant wavelength of approximately 35 times 5*. 



Shear layers produced by separation of the flow from a flat plate 

 normal to the wind have been studied experimentally by Fage and 

 Johansen [28]. These were, however, turbulent from their origin at the 

 edge of the plate, the value of Re^* being of the order of 600 or more. 



Flachsbart [29] shows some smoke flow pictures of the transition of 

 free vortex layers behind a flat plate. Reynolds numbers based on the 

 observed width of the smoke trail at transition vary from 40 to 100, but 

 it is unlikely that the width of the smoke filament has anything to do 

 with the usual measures of shear layer width. The values of i^Ct-s based 

 on distance from separation point to transition are better defined and 

 from them we infer values of Re^* of the order of 95 to 100. These values 

 are somewhat less than those found by Schiller and Linke for a cylinder, 

 perhaps due to different turbulence of the air streams. 



(23 > 



