A. S. Iberall 

 Equation of continuity 



Dt ^ J.J ' 



Thermodynamic relations (valid for near -equilibrium fields), one of 

 which is the relation of state 



7 

 dp = — dp - apdT , 



c 



dS = — dT - -^ dp , 



T P ^ ' ,.^, ,.^ 



A relation of compatibility 



y - 1 Cp 



a c2 



aT = 



Symbols are identified in the section on nomenclature. 



By examining their derivation within a modern statistical mechanical 

 framework [2], it is possible to determine the following limits [3,4] for their 

 applicability to continuous phenomena: 



/3[1 + Va^] < 0.001 « 1 , 



r [1+ \/u] < 0.1 << 1 , 



where 



/S = v/ch (a spatial continuum parameter — the ratio of mean free path 

 to dimensions), 



r = vO./c'^ (a temporal continuum parameter — the ratio of molecular re- 

 laxation time to shortest fluctuating period). 



In any hydrodynamic field, whether laminar or turbulent, in which these 

 conditions are met, the molecular ensemble will not manifest their fluctuations. 

 Any fluctuations that do exist must arise from the macroscopic dynamics that 

 are fully represented by Eq. (1). 



In this development for one elementary form of turbulent phenomena, that 

 induced by pressure gradients, attention will be restricted to small compressi- 

 bility effects by assuming that the square of the Mach number is not significantly 

 large compared to unity. Since there may still remain other sources of turbu- 

 lence, typically induced by heat transfer, rotation, or other relative wall mo- 

 tions, the set is specialized for fields that only show small density changes and 

 little temperature changes. This may be represented by the following nonlinear 



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