A • TRANSITION FROM LAMINAR TO TURBULENT FLOW 



be generated and spread throughout the boundary layer. Even if the 

 shear layer does not itself undergo transition, it will roll up into discrete 

 vortices of a very small scale which diffuse through the boundary layer. 

 The recent work of Schubauer and Klebanoff [5] shows clearly that 

 the concept of two-dimensional separation is not applicable to transition 

 in a boundary layer in an air stream of small turbulence, and raises serious 

 doubts as to the utility of the separation concept in describing the phe- 

 nomena involved. The periodic ToUmien-Schlichting waves rapidly lose 

 their two-dimensional character, their amplitude varying along a direc- 

 tion normal to the flow. These variations have been shown to be directly 

 coupled with very small variations of mean velocity across the air stream 

 on a line parallel to the leading edge of the plate. The boundary layer 

 parameters, including the amplification ratio, are sensitive to these small 

 changes, resulting in increasing variations of wave amplitude across the 

 flow as the wave travels downstream. Apparently no dynamic instability 

 of a three-dimensional character is involved, at least in the early stages. 

 Turbulence originates locally in the regions of maximum amplitude as an 

 essentially three-dimensional phenomenon. The first sign of turbulence in 

 a hot wire record is a sharp and momentary large increase in wire temper- 

 ature which is normally interpreted as a momentary large decrease in 

 velocity. This appears first well out in the boundary layer rather than 

 close to the wall as would be expected from separation. In the present 

 state of the experiments it is difficult to believe that the results indicate 

 a three-dimensional localized separation bubble, whose shear layer may 

 roll up into a horseshoe vortex as described by Theodorsen [39]. The alter- 

 nate theory is that vortices of the Gortler type with axes parallel to the 

 flow develop at the wave amplitude maxima. 



A, 11. Theory of the Influence of Turbulence on Transition. 



Taylor [40] assumed that transition due to turbulence resulted from 

 momentary separation of the boundary layer caused by the pressure 

 gradients within the layer resulting from the fluctuating pressure gradi- 

 ents of the turbulence. Separation is determined by the parameter X = 

 {8^/v){dUe/dx) which may also be written as — {d^/v){l/pUe)idp/dx) where 

 8p/dx is the instantaneous pressure gradient. From the theory of isotropic 

 turbulence the root-mean-square pressure gradient is proportional to 

 {u'^/L)i{p/p^). Hence, the root-mean-square value of X is (Uel/vyid^/l-) 

 {u'/ue)^{L/l)~i where I is a reference dimension. Hence if separation 

 occurs at a fixed value of X, we have, noting that 8/1 at the transition 

 is some function of the Reynolds number RCi = {lU/v)^,, 



Be, = F 





According to this theory transition depends only on this combination of 



(30 ) 



