B • TURBULENT FLOW 



situation and pressure distribution. The defect law is correspondingly 

 expressed as 



^ = .(.,!) (21-3) 



Coles concluded from his survey of existing data that the central 

 problem was not so much a study of the defect function F as a study of 

 the original function g{Tr, y/d) which gives the departure of the mean ve- 

 locity profile from the logarithmic law of the wall. Since the characteristic 

 departure was obviously not confined to equilibrium flows, the mean- 

 velocity profile was expressed in the form 



U _JUry\ , t(x) Jy 



c7.=n^j+THy ^''-'^ 



where K is a constant, ir{x) denotes that x is now in general a function 

 of X, and oiiy/8) is a universal wake function common to all two-dimen- 

 sional turbulent boundary layer flows. 

 The term 



xCx) fy' 



K \8 



in Eq. 21-4 gives the departure from the logarithmic law of the wall, i.e. 

 from 



H"^) 



^ = jlnl^)+c 



where, according to Coles, K = 0.4 and c = 5.10 (Eq. 19-2). 



From an analysis of experimental data. Coles found the form of co(y/5) 

 as given in Fig. B,21a, in which co(?//5) has been subjected to the normal- 

 izing conditions co(0) = 0, co(l) = 2, and jl{y/8)dc>3 = 1. When plotted 

 against y/8 these curves have a nearly symmetrical S shape; and, due to 

 the normalization, have the maximum value oi 2 at y/8 = 1. The curves 

 obtained from Clauser's equilibrium profiles and the one obtained from 

 Wieghardt's data, which Coles finds to be also an equilibrium flow, are 

 plotted against the parameter yUj/{8*U^, which is equal to y/A in 

 Clauser's notation. Included in this set are data from nonequilibrium 

 profiles and the data of Liepmann and Laufer [94] for a region of turbu- 

 lent mixing between a uniform flow and a fluid at rest. 



The general working form of Eq. 21-4 may be written 



J/ _ 



Ur~ K 



^>n(^^)+^ + ^"(|) (2«) 



where the constants K and c have the numerical values as given above. 

 In order to use this formula, 7r(a;) must be known. It follows from Eq. 



< 140 ) 



