Theory of Turbulent Flow Between Parallel Plates 



set, in which the derivatives of density and entropy have been eliminated by use 

 of the thermodynamic relationships: 



Poo 



Dt 



p,i + A^oo[Vi,jj +Vj i-; 



y/^oo- -^oo 



i . J J ' 



p c 



DT 



- a„„T 



Dp 



k„ T 



. + u [V. . + V. .1 V. . 



O '^oo oo 



(2) 



^oo Dp 



C^ Dt 



oo 



DT 



-J V. . 



oo J , ] 



The parameters with oo subscripts are now constant. It is this set involv- 

 ing five variables, three components of velocity, one of pressure, and one of 

 temperature, that will be explored subject to the boundary conditions for parallel- 

 plate flow. . ■ ^ 



Solutions for the linear (small-amplitude) set were explored earlier [5,6,7] 

 for flow in a tube. Their validity (the solutions representing both laminar flow 

 and all modes of propagation) over the entire frequency range of possible con- 

 vergence of the Navier-Stokes (NS) equations was quite sharply tested by Green- 

 span [3]. The question now arises whether a second nonlinear solution, other 

 than the small- amplitude set, can exist. 



First, transforming the equations into dimensionless form: 



Momentum equation 



DR. 



1 



"d7 



.np+ (Ri,jj + Rj,ij) 



Energy equation 



DJ DP 



(^oo-l) 



Dt 



Dt 



eoo(Ri,j+Rj,i) 



n'j 



(3) 



Continuity equation 



DJ 

 Dt 



DP 



'°° Dt 



Incompressibility may be invoked by letting fi^^ approach zero. This lowers 

 the order of the combined equation set. Such a procedure is quite dangerous in a 

 nonlinear set. It leads to defects that are already suggested by the small- 

 amplitude linear solution. For the NS equations to be valid, /3 must be small 

 (in tube or plate experiments in the laboratory with normal air it will be about 

 10"^). The parameter r = p'^co must also be small. (The parameter j is a 

 dimensionless frequency.) However, /3aj is not very restricted, and, in fact, the 

 small-amplitude equations show that low-loss acoustic resonances are possible. 

 (In the turbulent field, the magnitude of /3 > may range from to 10 or more.) 



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