B • TURBULENT FLOW 



posed for connecting the outer and inner portions is left to the original 

 article. 



A treatment of the same character was applied to equilibrium flows 

 involving adverse pressure gradients. Again a good fit was obtained by 

 assuming a constant eddy viscosity given by Eq. 22-4 even for near-sepa- 

 ration profiles. Some of the more significant results of this work were: 

 (1) that {8*/T^)dp/dx proved to be the proper pressure gradient param- 

 eter which must be constant throughout an equilibrium layer, (2) that 

 a turned out to be practically independent of pressure gradient (inde- 

 pendent of the parameter {h*/r^)d'p/dx) and to have the value of approxi- 

 mately 0.018 in all cases tested. 



An interesting outcome of a constant a is a constant eddy Reynolds 

 number. If such a Reynolds number is defined by 



Re, = 



pUJ* 



we find from Eq. 22-4 that Re, = 1/a. Taking a = 0.018, Re, = 56. A 

 constant eddy Reynolds number is just another way of expressing the 

 behavior trend of all turbulent shear flows, namely a tendency for the 

 transferring agents to be proportional to the length and velocity scales 

 of the flow. 



Most important of all is the evidence from these sources that e^ be- 

 haves in equilibrium flows toward mean-velocity distributions beyond the 

 range of the logarithmic law as though it were constant. This cannot be 

 taken as a sweeping generalization, but it furnishes good evidence that 

 Cf, is likely to have a strong leaning in this direction generally and there- 

 fore will have only a weak dependence on local conditions. This being so, 

 there is little foundation for a mixing length theory in such regions, and it 

 renders of little significance the various arguments about how mixing 

 length should be expressed. The degree to which e^ is constant and the 

 exactness with which a gradient type of diffusion is obeyed for coarse 

 mixing are probably not sufficient to represent more sensitive quantities 

 like shear stress distributions. 



Near the wall the mixing length theory may be applied, and we see 

 that a valid procedure starts with an expression for e^ that has a striking 

 resemblance to that for the outer flow. The comparison is : 



Inner flow — = CiijUj; Ci — 0.4 



P 



Outer flow ^ = aC7,A; a = 0.018 



P 



In the first case the mixing scale is proportional to the distance from the 

 wall; in the second case it is proportional to the thickness of the shear 

 layer. 



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