A,6 • APPLICATION OF DIMENSIONAL ANALYSIS 



of turbulence of less than 0.1 per cent. Favorable pressure gradients in- 

 crease the permissible roughness height considerably. 



A, 6. Application of Dimensional Analysis to Transition of a 

 Two-Dimensional Boundary Layer. So far we have considered the 

 study of transition on a flat plate, first without pressure gradient, then 

 with simple linear variation of the pressure, with uniform curvature, and 

 with simple roughness, each variable considered singly. However, in the 

 cases of technical interest, all of the variables may vary simultaneously 

 along the surface of the body generating the boundary layer. Further 

 attempts have been made to apply the knowledge gained in the simpler 

 cases by means of dimensional analysis. 



For a series of geometrically similar two-dimensional bodies the po- 

 sition at which transition occurs depends on the Reynolds number at 

 which the measurement is made and on the intensity and scale of the 

 turbulence of the main stream. Dimensional reasoning gives the result 

 that 



^ = F (—, —, ^^ 

 c \ V ' U' cj 



where x^ is the coordinate locating the transition point with relation to 

 a selected system of axes, c is the reference dimension, for example, the 

 chord of an airfoil, U is the free stream velocity, v the kinematic viscosity 

 of the fluid, u' the intensity, and L the scale of the turbulence. 



Since, as previously noted, transition occurs within the boundary 

 layer, one would like to relate its occurrence more directly to boundary 

 layer parameters rather than to the shape of the body. Boundary layers 

 on different two-dimensional bodies differ only in the variation of pres- 

 sure to which they are exposed by the flow around the body, and in the 

 curvature and roughness of the surface on which they are formed. Meas- 

 uring X along the surface from the stagnation point, x^ is dependent on 

 the three functions of x describing the variation of the stream velocity 

 just outside the boundary layer with x (which fixes the pressure vari- 

 ation), the variation of the curvature of the boundary with x, and the 

 variation of the roughness with x, as well as on the free stream turbulence. 

 Since the number of possible functions is indefinitely large, no simplifi- 

 cation results from this approach. It was for the reason of gaining some 

 insight into the problem that we considered in Art. 2 to 5 the simplest 

 types of functional variation, namely constant values of the parameters 

 independent of x, or linear variations. 



Since transition occurs suddenly, another approach is to relate tran- 

 sition to the local boundary layer parameters, eliminating a: as a variable. 

 The local situation is usually described by the boundary layer thickness 5, 

 the velocity u^ at the outer edge and its derivative du^/dx which is a 



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