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A comprehensive probability density function formalism for multiphase flows

Published online by Cambridge University Press:  01 June 2009

MADHUSUDAN G. PAI  and
SHANKAR SUBRAMANIAM
MADHUSUDAN G. PAI*
Affiliation:
Department of Mechanical Engineering, Iowa State University, Ames, IA 50011, USA
SHANKAR SUBRAMANIAM*
Affiliation:
Department of Mechanical Engineering, Iowa State University, Ames, IA 50011, USA
*
†Present address: Center for Turbulence Research, Stanford University, Stanford, CA 94305, USA. E-mail address for correspondence: [email protected]
‡E-mail address: [email protected]
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Abstract

A theoretical foundation for two widely used statistical representations of multiphase flows, namely the Eulerian–Eulerian (EE) and Lagrangian–Eulerian (LE) representations, is established in the framework of the probability density function (p.d.f.) formalism. Consistency relationships between fundamental statistical quantities in the EE and LE representations are rigorously established. It is shown that fundamental quantities in the two statistical representations bear an exact relationship to each other only under conditions of spatial homogeneity. Transport equations for the probability densities in each statistical representation are derived. Exact governing equations for the mean mass, mean momentum and second moment of velocity corresponding to the two statistical representations are derived from these transport equations. In particular, for the EE representation, the p.d.f. formalism is shown to naturally lead to the widely used ensemble-averaged equations for two-phase flows. Galilean-invariant combinations of unclosed terms in the governing equations that need to be modelled are clearly identified. The correspondence between unclosed terms in each statistical representation is established. Hybrid EE–LE computations can benefit from this correspondence, which serves in consistently transferring information from one representation to the other. Advantages and limitations of each statistical representation are identified. The results of this work can also serve as a guiding framework for direct numerical simulations of two-phase flows, which can now be exploited to precisely quantify unclosed terms in the governing equations in the two statistical representations.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 628 , 10 June 2009 , pp. 181 - 228
Copyright
Copyright © Cambridge University Press 2009

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