B • TURBULENT FLOW 



a free constant: yi = cv/Ut and Ui — cUr. If we now integrate Eq. 22-2 

 as follows: 



1 f''y-ir<iu 



C2 Jyi y Ur J Ui 



we obtain exactly the law of the wall 



U 1 , Ury , 

 T^ = — m — - + const 



Ur Cl V 



where ci has the same numerical value as K. 



If we assume at the outset, as Prandtl did also, that v = Uj and again 

 take I = c^y, we again obtain 



This is identical to Eq. 22-2 and again yields exactly the law of the wall. 

 It is time to examine the consequences of these results. Since the law 

 of the wall is well founded and is one of the most universal features of 

 turbulent flow, we cannot escape the conclusion that the above assump- 

 tions are valid for the region in which the logarithmic law is obeyed. 

 We know, of course, that we must stay near the wall, if for no other 

 reason than that r changes with y. More specifically we may express the 

 eddy viscosity 



^ = C2yUr (22-3) 



P 



in the region where the logarithmic law of the wall is valid. 



Turning our attention to the outer 80 to 90 per cent of the layer, 

 we find that both Townsend [1] and Clauser [72] have explored the possi- 

 bility that e^j is constant in this region. Townsend employed the rather 

 straightforward procedure of solving the boundary layer approximation 

 of the equation of mean motion, considering both constant pressure flow 

 and equilibrium flow with pressure gradient. We call attention here only 

 to his treatment of the constant pressure case. When the constants in- 

 volving ep were chosen for the best fit of experimental results, fair agree- 

 ment was found for y/8 > 0.05. The principal defect was the usual one, 

 namely that a constant e^ yielded too slow an approach to the free stream 

 velocity. Evidently e^ effectively decreases near the outer edge, due no 

 doubt to intermittency of turbulent flow. The extent and quality of the 

 over-all agreement was, however, sufficiently good to show that an essen- 

 tially constant and valid e^ is a physical reality in the turbulent parts of 

 the flow beyond the logarithmic region. 



Clauser employed the novel approach of making laminar profiles re- 

 semble the outer portion of the constant pressure turbulent profile when 

 the laminar profiles were reduced to the basis oi (U — Ue)/Ur vs. y/8. 



( 144 ) 



