Next let us consider the validity of the so-called universal logarithmic laws of 

 turbulent boundary layers which is assumed in even the most recent work of Clauser, 

 Coles and Townsend. These laws are based upon two similarity laws, the law of the 

 wall and the velocity-defect law. Of these, the former may be derived by dimensional 

 reasoning and has been well-confirmed by experiment; the latter can be derived 

 theoretically only for pipe and channel flows, and its adoption for the case of flat-plate 

 boundary layers is based upon its correlation of velocity profiles over the narrow range 

 of Reynolds numbers for which data are available. 



It has recently been shown by Townsend, on the basis of the concept of approxi- 

 mate self-preservation of the structure of the turbulence in the outer part of a boundary 

 layer on a flat plate, that the velocity defect law may be retained, provided the previous 

 history of the flow is taken into account by replacing the local velocity scale by that 

 corresponding to some upstream position. Townsend's subsequent analysis is approxi- 

 mate and retains the logarithmic relations by assumption. A more precise analysis of 

 Townsend's theory yields a family of possible universal laws of which the logarithmic 

 law is only one member, and among which that one which best fits the boundary-layer 

 data is to be selected. 



The theory assumes the validity of the law of the wall 



u yu T 



- =/(</*), y* = — (72) 



u T v 



where u T =: \/r/p, - is the shear stress at the wall, and p and v are the density and 

 kinematic viscosity of the fluid; and in the outer part of the boundary layer, the validity 

 of a modified velocity-defect law 



Uo - u F(r) y U 



= — ; f = -, * = — (73) 



u T g(a) L u T 



where U is the free stream velocity, L is a length scale proportional to the boundary- 

 layer thickness, and g(a) is an unspecified function. Townsend assumes a specific form 

 for g(<j), but if it is further assumed that there is a range of values of y in which the 

 inner and outer similarity laws overlap, it is found that there is no freedom of choice 

 concerning either the form of g{a) or the similarity laws. 



By an exact functional analysis of the foregoing assumptions, one obtains the 

 result 



/"(*/*) 

 y* = n - 1 (74) 



/'(</*) 



If n = 0, this yields the well-known logarithmic relations for part of the velocity dis- 

 tribution and the variation of the coefficient of shear stress with Reynolds number. If 

 n=jL0, an entirely new set of relations may be derived. The results for the two cases 

 are compared in the following table: 



402 



