Theory of Turbulent Flow Between Parallel Plates 



(ii) We will assume that the temperature fluctuations are continuous across 

 the transition layer. Thus, 



= -ja(y-l)SgSe^ ^ + a C e^ ^ - (y - I ) E , 



from which C can be evaluated. Thus, the set remaining in the core is 



U = ja^bjSe^^''' - j (l-a)\bjSe^^i"- J ^ E 



V=-aa;(2e' + aco(l-a)%e ' +^E 



-, J b , X § 



W = Sw(l-cr)iBe ' + - E 



^ ' CO 



Mean- Flow Equations "* """ ' 



Except for eliminating two more constants of integration, we are up to a 

 critical point — how to satisfy the mean-flow equations, say, by determination 

 of a spectral density function. 



While there are five equations, two involve terms of lower order of magni- 

 tude than the others, and so may be dropped in a first-order approximation. 

 Instead they are replaced by the following lemmas. 



(iii) The mean flow at low Mach number essentially behaves as if it were 

 an incompressible, in the present instant, one-dimensional flow. 



In the fifth mean-flow equation, the left-hand side can contribute only 

 negligible residue, so that 



itself represents the fifth mean-flow equation. 



(iv) The isothermal injection of fluid with isothermal boundaries at the 

 same temperature, at low Mach number, creates a mean flow field which be- 

 haves essentially as if it were isothermal. 



Similarly in the fourth mean-flow equation, other terms contribute negli- 

 gibly, so that 



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