Then 



D(x) = A+B , (31) 



which displays an unfortunate singularity wherever the gross field happens to have an 

 extremum. 



The surprising fact is that some turbulent shear flows, notably the so-called 

 "free shear flows" without solid boundaries (wakes and jets), seem to have mean 

 velocity distributions rather close to those given by assuming that turbulent shear stress 

 is Newtonian in character. Clearly this cannot be correct in principle. 



Although boundary layers and channels cannot be approximated even 



roughly as quasi-Newtonian, it turns out that two-fluid models having a wall layer 



vt/* 

 ^ ^12 with the true viscosity and an outer fluid with a "turbulent viscosity" 



V 



can be instructive. I have used such a pedagogical device for channel flow; it was 

 independently introduced into turbulent boundary layer analysis by F. H. Clauser [20]. 



— , the "fraction velocity", T = ^( — ) y=0 , 

 P dy 



the mean skin friction stress. 



The superficial success of a "turbulent viscosity" in describing the mean velocity 

 pattern of jets and wakes suggests evaluation of the corresponding "effective Reynolds 



number" R e = , in order to see how it varies with the true Reynolds number, 



v T 



R = . U ± and L are characteristic velocity and shear zone width. 



V 



For jets and wakes, it turns out that R e is independent of R* For the round 

 jet entering fluid at rest, 



Urn ■ A 



R e = « 15, (32) 



vt 

 where U m is the maximum velocity at a cross section and A is the "momentum 

 diameter," defined by 



tta 2 _ r» 



Urrr = 2tt / U*(r) r dr, (33) 



4 Jo 



for isopycnic flow. For the plane wake, 



(U x — U min )h 



R e = » 12, (34) 



vt 



where £/oo is free stream velocity, U min is the velocity on the axis at a section and h 

 is the lateral coordinate at which (t/oo — U) — Vi (C/oo — 17 min ). 



It is noteworthy that these values of R e are both of the same order as the "lower 

 critical Reynolds numbers" for the stability of laminar free shear flows. 



When a two fluid model is used for turbulent flow through a round tube, for 

 example, the "effective Reynolds number" defined with the v T of the turbulent core 



This observation has also been made by Townsend [1]. 



388 



