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A study of two-fluid model equations

Published online by Cambridge University Press:  25 November 2011

K. Ueyama
K. Ueyama*
Affiliation:
Department of Environmental and Energy Chemistry, Kogakuin University, 1-24-2 Nishishinjyuku, Shinjyuku, Tokyo 163-8677, Japan
*
†Email address for correspondence: ueyama@cc.kogakuin.ac.jp
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Abstract

A theoretical study of the interaction term in the two-fluid model equations is presented. The relevant Navier–Stokes equation is volume-integrated in a control volume fixed in a field of dispersed two-phase flow; then it is time-integrated. An expression for the interaction term is obtained in the limit of infinitesimal control volume, which rigorously fits to the two-fluid model equation based on time averaging. The interaction term is then analysed for dispersed two-phase flow with homogeneous particle size. The mathematical expression of the resulting interaction term clearly shows its property, which has been overlooked for more than 40 years. It can be decomposed into the conventional interaction term and an additional ‘virtual force’ term. The virtual force term is evaluated approximately for two types of dispersed multiphase flow in order to demonstrate its effectiveness. The first is solid–liquid dispersed two-phase flow with spherical solid particles undergoing creeping flow, and the second is gas–liquid dispersed two-phase flow with large bubbles in a highly turbulent flow field. For solid spherical particles in a homogeneous creeping flow, the term vanishes, as was found numerically by Ten Cate and Sundaresan (Intl J. Multiphase Flow, vol. 32, 2006, pp. 106–131), although the term could be significant for a flow field with velocity gradient. For large bubbles in bubble columns in a recirculating turbulent flow regime, the term is significant. The two-fluid model equations are corrected by the introduction of these virtual force terms, which in some circumstances are important in simulating the macroscopic properties of dispersed two-phase flow with spatial variations.

JFM classification

Mathematical Foundations: Computational methods Multiphase and Particle-laden Flows: Multiphase flow
Type
Papers
Information
Journal of Fluid Mechanics , Volume 690 , 10 January 2012 , pp. 474 - 498
Copyright
Copyright © Cambridge University Press 2011

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