Theory of Turbulent Flow Between Parallel Plates 



(1) Estimate from ^>j = 100: 



g(^ 0)5/4) = 625 . 



This result, based on the low-frequency cutoff, emerges as a pressure 

 gradient condition (g = fRe^ where f = the friction factor). Since the friction 

 factor is the same function of Reynolds number in incompressible and com- 

 pressible flow, in nonsupersonic flow, then the gradient similarly is the same 

 function of Reynolds number. Thus, by this argument there is no difference 

 between the results for compressible and incompressible flow. The result did 

 not depend on experimental data. 



More fundamentally, where does the w^ = 100 criterion come from ? The 

 parameter uj itself made its appearance in the small-amplitude theory as a 

 damping parameter , even though proportional to frequency. As one attempts 

 to push the fluid back and forth at increasing frequency (or rate), one finds a 

 propagation parameter that is at first attenuative. It depends on viscosity. At 

 sufficiently high rate, an elastic "resonance" can come into existence. This 

 is true whether for gas or liquid. There is a critical value of oj (= 100, where 

 CO = h^Q/v depends on geometry -h-, viscosity -^ -, and frequency -0-, but 

 not on the velocity of propagation) at which the propagation is elastic. It does 

 not matter how high the propagation velocity is, as long as it is finite. 



(2) Crude estimate from the Strouhal number Sa As an approximation, 

 we wrote 



O) 



Re = , 



2S 



where the number 2 depends on the method of modelling equivalence among 

 fields of different geometries. We assume that the sudden appearance of a sus- 

 tained Strouhal number as Reynolds number is increased is associated with the 

 appearance of a von Karman-like vortex street shedding patchily from wall to 

 wall into the core. By assigning a numerical value (from wind tunnel data — 

 assuming that the same Strouhal number would be found for water tunnels) 

 then the critical Reynolds number at which a>^ = 100 would occur simultane- 

 ously gives the critical Reynolds number. 



The assumption here is that the critical Strouhal number does not depend 

 on the velocity of propagation. 



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