flow, the maximum velocity U m and the tube radius a, turns out to vary with the 



actual Reynolds number: _ 



U m • a 4 24V 2 



R e ^ pa — _ (35) 



vt Cf V c f 

 To U m a 

 where c f = — — — is a slowly varying function of R = . (36) 



As numerical example, we may take Laufer's two cases: 



R e tt 525 for R = 25,000, (37) 



R e « 900 for R = 250,000. 



It is possible that there exists a more suitable definition of R e for tube flow, 

 i.e. a definition which does not display this slow variation with R. However, in con- 

 trasting the actual Reynolds number dependence of turbulent jet and tube flows we 

 recall that the mean velocity profile of the former is virtually invariant, while that of 

 the latter is appreciably sensitive to Reynolds number. 



In terms of the turbulence parameters, we may regard the "turbulent diffusivity" 

 as behaving like , „ 



VT ~ v'J, (38) 



where v f is the r.m.s. velocity fluctuation in the transport direction, and 1 is a corre- 



v f 

 sponding Lagrangian integral length scale. It is found experimentally that — is 



U m 

 relatively insensitive to R in jets, but decreases with R in channels and tubes. The 

 R dependence of 2 is not established, but as a property of the large eddies it is likely 

 to be more or less proportional to shear zone width. The foregoing assumptions give 

 v T '— ' U m L for jets [.'. R e zz Const] and v T slowly decreasing with increasing R for 

 tubes. 



The low empirical values of R e for jet and wake, and the superficial agreement 

 of the corresponding constant exchange coefficient postulate, suggest that an engineer- 

 ing approximation to such flows may be gotten via a quasi-Oseen approximation to the 

 Reynolds equations, incorporating a constant v T as well.* 



Shear Stress Fluctuations 



For some applications it may be interesting to know the shear stress fluctuation 

 level in a turbulent flow. In a fully turbulent region the mean (Reynolds) turbulent 

 stress — pUiUj. dominates the viscous momentum transport, and we may begin by a 

 look at the fluctuation in turbulent stress. For simplicity, we restrict the discussion to 

 a boundary layer type flow, in which only one kind of mean shear component, 

 T T = — puv, is important. 



Since the so-called turbulent shear stress is actually the average y -convection of 

 x-momentum attributable to the turbulence, it seems plausible to identify 



r T = - p[U v +Vu+ {uv - m>)] (39) 



as the instantaneous fluctuation in turbulent shear stress. Then, for V <^ ~U and 

 turbulence levels not too high, 



r' T Uv' 



— •»— (40) 



T T uv 



a number considerably greater than unity. Figure 12 shows how t t varies across 

 the turbulent boundary layer. T is the mean shear stress at the wall. It should be 



* This has been applied to the wake by I. Imai (lecture at Fluid Mechanics Colloq., 

 The Johns Hopkins University, November, 1955). 



389 



