Two-fluid Model Formulation and Closure Limitations

For undisturbed turbulent flows the local instantaneous velocity of the continuous phase have been decomposed in various ways, not necessarily in accordance with the familiar Reynolds—and Favre averaging procedures. To explain the basic problem we use the Reynolds decomposition and averaging procedure, as an example. Introducing the peculiar velocity for the dispersed phase (4.184) can be re-arranged as  

The interpretations of the velocity variables in the first and second terms on the RHS represent a severe problem due to a general lack of knowledge about the physics they represent. Besides, when the F quantity in the Maxwell equation is substituted by this relation and Maxwellian averaged, the resulting covariances consist of velocity quantities which are averaged in an inconsistent manner (Maxwellian- vs. time-, volume-, or ensemble averaging). For this reason other somewhat ad hoc decomposition procedures have been defined intending to achieve more consistent closures for these terms. 

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