Hostname: page-component-77c89778f8-vsgnj Total loading time: 0 Render date: 2024-07-21T08:21:29.348Z Has data issue: false hasContentIssue false
  • English
  • Français

Averaged equations for inviscid disperse two-phase flow

Published online by Cambridge University Press:  26 April 2006

D. Z. Zhang  and
A. Prosperetti
D. Z. Zhang
Affiliation:
Department of Mechanical Engineering, The Johns Hopkins University, Baltimore, MD 21218, USA
A. Prosperetti
Affiliation:
Department of Mechanical Engineering, The Johns Hopkins University, Baltimore, MD 21218, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Averaged equations governing the motion of equal rigid spheres suspended in a potential flow are derived from the equation for the probability distribution. A distinctive feature of this work is the derivation of the disperse-phase momentum equation by averaging the particle equation of motion directly, rather than the microscopic equation for the particle material. This approach is more flexible than the usual one and leads to a simpler and more fundamental description of the particle phase. The model is closed in a systematic way (i.e. with no ad hoc assumptions) in the dilute limit and in the linear limit. One of the closure quantities is related to the difference between the gradient of the average pressure and the average pressure gradient, a well-known problem in the widely used two-fluid engineering models. The present result for this quantity leads to the introduction of a modified added mass coefficient (related to Wallis's ‘exertia’) that remains very nearly constant with changes in the volume fraction and densities of the phases. Statistics of this coefficient are provided and exhibit a rather strong variability of up to 20% among different numerical simulations. A detailed comparison of the present results with those of other investigators is given in § 10.

As a further illustration of the flexibility of the techniques developed in the paper, in Appendix C they are applied to the calculation of the so-called ‘particle stress’ tensor. This derivation is considerably simpler than others available in the literature.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 267 , 25 May 1994 , pp. 185 - 219
Copyright
© 1994 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

With Appendix C by H. F. Bulthuis.

References

Arnold, G. S., Drew, D. A. & Lahey, R. T. 1989 Derivation of constitutive equations for interfacial force and Reynolds stress for a suspension of spheres using ensemble cell averaging. Chem. Engng Commun. 86, 43–54.Google Scholar
Auton, T. R., Hunt, J. C. R. & Prud'homme, M. 1988 The force exerted on a body in inviscid unsteady non-uniform rotational flow. J. Fluid Mech. 197, 241–257.Google Scholar
Banerjee, S. & Chan, A. M. C. 1982 Separated flow model, I – analysis of the averaged and local instantaneous formulations. Intl J. Multiphase Flow 6, 1–24.Google Scholar
Batchelor, G. K. 1970 The stress system in a suspension of force-free particles. J. Fluid Mech. 41, 545–570.Google Scholar
Batchelor, G. K. 1972 Sedimentation in a dilute dispersion of spheres. J. Fluid Mech. 52, 245–268.Google Scholar
Batchelor, G. K. 1974 Transport properties of two-phase materials with random structure. Ann. Rev. Fluid Mech. 6, 227–255.Google Scholar
Biesheuvel, A. & Gorissen, W. C. M. 1990 Void fraction disturbances in a uniform bubbly fluid. Intl J. Multiphase Flow 16, 211–231.Google Scholar
Biesheuvel, A. & Spoelstra, S. 1989 The added mass coefficient of a dispersion of spherical gas bubbles in liquid. Intl J. Multiphase Flow 15, 911–924.Google Scholar
Biesheuvel, A. & Wijngaarden, L. van 1984 Two-phase flow equations for a dilute dispersion of gas bubbles in liquid. J. Fluid Mech. 148, 301–318.Google Scholar
Bulthuis, H. F., Prosperetti, A. & Sangani, A. S. 1994 ‘Particle stress’ in disperse two-phase flow. J. Fluid Mech. (submitted).Google Scholar
Cercignani, C. 1988 Kinetic Theory and Gas Dynamics. Springer.
Drew, D. A. 1983 Mathematical modeling of two-phase flow. Ann. Rev. Fluid Mech. 15, 261–291.Google Scholar
Drew, D. A. 1989 Effect of particle velocity fluctuations on the inertia coupling in two-phase flow. In Constitutive Relationships and Models in Continuum Theories of Multiphase Flows (ed. R. Decker), pp. 103–125. NASA Conf. Pub. 3047.
Drew, D. A. 1991 Effect of particle velocity fluctuations in particle fluid flows. Physica 179A, 69–80.Google Scholar
Drew, D. A., Cheng, L. & Lahey, R. T. 1979 The analysis of virtual mass effects in two-phase flow. Intl J. Multiphase Flow 5, 233–242.Google Scholar
Drew, D. A. & Wallis, G. B. 1992 Fundamentals of two-phase flow modeling. In Proc. 3rd Intl Workshop on Two-Phase Flow Fundamentals, Imperial College, London.
Hansen, J. P. & McDonald, I. R. 1986 Theory of Simple Liquids. Academic.
Hinch, E. J. 1977 An averaged-equation approach to particle interactions in a fluid suspension. J. Fluid Mech. 83, 695–720.Google Scholar
Irving, J. H. & Kirkwood, J. G. 1950 The statistical mechanical theory of transport processes. IV. The equations of hydrodynamics. J. Chem. Phys. 18, 817–829.Google Scholar
Jones, A. V. & Prosperetti, A. 1985 On the suitability of first-order differential models for two-phase flow prediction. Intl J. Multiphase Flow 11, 133–148.Google Scholar
Joseph, D. D. & Lundgren, T. S. 1990 Ensemble average and mixture theory equations for incompressible fluid—particle suspensions. Intl J. Multiphase Flow 16, 35–42.Google Scholar
Koch, D. L. 1990 Kinetic theory for a monodisperse gas-solid suspension. Phys. Fluids A 2, 1711–1723.Google Scholar
Kok, J. B. W. 1989 Dynamics of gas bubbles moving through liquid. Doctoral dissertation, University of Twente, Enschede, The Netherlands.
Landau, L. & Lifshitz, E. M. 1959 Fluid Mechanics, Chap. 1. Pergamon.
Park, J. W., Drew, D. A. & Lahey, R. T. 1992 The analysis of void wave propagation in adiabatic monodispersed bubbly two-phase flow using an ensemble-averaged two-fluid model. Intl J. Multiphase Flow. (submitted).Google Scholar
Pauchon, C. & Banerjee, S. 1986 Interphase momentum interaction effects in the averaged multifield model. Part I: Void propagation in bubbly flows. Intl J. Multiphase Flow 12, 559–573.Google Scholar
Pauchon, C. & Smereka, P. 1992 Momentum interactions in dispersed flow: An averaging and a variational approach. Intl J. Multiphase Flow 18, 65–87.Google Scholar
Prosperetti, A. & Jones, A. V. 1984 Pressure forces in disperse two-phase flows. Intl J. Multiphase Flow 10, 425–440.Google Scholar
Prosperetti, A. & Zhang, D. Z. 1993 Ensemble phase averaging for disperse two-phase flows. In Proc. Eleventh Symp. on Energy Engineering Sciences. Argonne National Laboratories (in press).
Sangani, A. S. & Didwania, A. K. 1993a Dispersed-phase stress tensor in flows of bubbly liquids at large Reynolds numbers. J. Fluid Mech. 248, 27–54.Google Scholar
Sangani, A. S. & Didwania, A. K. 1993b Dynamic simulations of flows of bubbly liquids at large Reynolds numbers. J. Fluid Mech. 250, 307–337.Google Scholar
Sangani, A. S. & Prosperetti, A. 1993 Numerical simulation of the motion of particles at large Reynolds numbers. In Particulate Two-Phase Flow (ed. M. Roco), pp. 971–998. Butterworth-Heinemann.
Sangani, A. S., Zhang, D. Z. & Prosperetti, A. 1991 The added mass, Basset, and viscous drag coefficients in nondilute bubbly liquids undergoing small-amplitude oscillatory motion. Phys. Fluids A 3, 2955–2970.Google Scholar
Smereka, P. & Milton, G. W. 1991 Bubbly flow and its relation to conduction in composites. J. Fluid Mech. 233, 65–81.Google Scholar
Wallis, G. B. 1991a Two-phase potential flow. In Proc. 9th Symp. on Energy Engineering Sciences. Argonne National Laboratory Rep. Conf.9105116, pp. 32–39.
Wallis, G. B. 1991b The averaged Bernoulli equation and macroscopic equations of motion for the potential flow of a two-phase dispersion. Intl J. Multiphase Flow 17, 683–695.Google Scholar
Wijngaarden, L. van 1976 Hydrodynamic interactions between gas bubbles in liquid. J. Fluid Mech. 77, 27–44.Google Scholar
Zhang, D. Z. 1993 Ensemble phase averaged equations for multiphase flows. Doctoral dissertation, Department of Mechanical Engineering, Johns Hopkins University.
Zhang, D. Z. & Prosperetti, A. 1994 Ensemble phasic averaged equations for bubbly flows. Phys Fluids (submitted).Google Scholar
Zuber, N. 1964 On the dispersed two-phase flow in the laminar regime. Chem. Engng Sci. 19, 897–917.Google Scholar