Abstract
A model for dispersed two-phase flow is derived based on a Boltzmann equation. This model is shown to be compatible with the two-fluid model, and includes the source of dispersion. In this model, dispersion is the result of the correlation of the liquid velocity fluctuations with the number density (perhaps more appropriately, with the trajectories of the individual dispersed units). Using this derived force, and a very simple assumption regarding the correlation of the presence of a dispersed unit and the carrier fluid velocity, a form for this force can be derived. This form gives a force which is proportional to the scalar (dot) product of the fluid Reynolds stress tensor with the gradient of bubble number density. For isotropic turbulence, the force is proportional to the gradient of number density. The constant of proportionality depends on the ratio of the dispersed unit relaxation time to the liquid turbulence time scale.
Similar content being viewed by others
References
S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases. Cambridge: Cambridge University Press (1970) 423 pp.
C. M. Tchen, Mean Value and Correlation Problems Connected with the Motion of Small Particles Suspended in a Turbulent Fluid. Delft: Ph.D. Thesis (1947).
J. O. Hinze, Turbulence. New York: McGraw-Hill (1959) 586 pp.
M. Reeks, On a kinetic equation for the transport of particles in turbulent flows. Phys. Fluids A 3 (1991) 446–456.
G. G. Stokes, On the effect of the internal friction of fluids on the motion of pendulums. Trans. Cambridge Phil. Soc. 9 (1851) 1–141.
A. B. Basset, Treatise on Hydrodynamics. London: Deighton Bell (1888).
M. R. Maxey and J. J. Riley, Equation of motion for a small rigid sphere in non-uniform fluid. Phys. Fluids 26 (1983) 883–889.
J. D. Murray, On the mathematics of fluidization. Part I. Fundamental equations and wave propagation J. Fluid Mech. 21 (1965) 465–493.
G. B. Wallis, One Dimensional Two-Phase Flow. New York: McGraw-Hill (1969) 408 pp.
F. E. Marble, Dynamics of Dusty Gases Ann. Rev. Fluid Mech. 2 (1970) 397–446.
D. A. Drew, Averaged field equations for two-phase media Stud. Appl. Math. 50 (1971) 133–155.
Yu. A. Buyevich, Statistical hydrodynamics of disperse systems, physical background and general equations. J. Fluid Mech. 49 (1971) 489–507.
M. Ishii, Thermo-Fluid Dynamic Theory of Two-Phase Flow. Paris: Eyrolles (1975) 248 pp.
R. I. Nigmatulin, Spatial averaging in the mechanics of heterogeneous and dispersed systems. Int. J. Multiphase Flow. 5 (1979) 353–385.
S. L. Passman, J. W. Nunziato and E. K. Walsh, A theory of multiphase mixtures. Appendix 5C of C. Truesdell (ed.), Rational Thermodynamics. New York: Springer-Verlag (1984) 286–325.
D. A. Drew and S. L. Passman, Theory of Dispersed Two-Component Flow. New York: Springer-Verlag (1998) 308 pp.
S. E. Elghobashi and T. W. Abou-Arab, A two equation turbulence model for two-phase flow. Phys. Fluids 26 (1983) 931–938.
M. Maxey, The gravitational settling of aerosol particles in homogeneous turbulence and random flow fields. J. Fluid Mech. 174 (1987) 441–465.
M. Lopez de Bertodano, Two-fluid model for two-phase turbulent jet. Nuclear Eng. Design. 179 (1998) 65–74.
A. E. Larrateguy, P. M. Carrica, D. A. Drew and R. T. Lahey Jr., CFDShipM: Multiphase code for ship hydrodynamics. Version 2.24 Users Manual (1999) 146 pp.
O. A. Druzhinin and S. E. Elghobashi, Direct numerical simulation of bubble-laden turbulent flow using the two-fluid formulation. Phys. Fluids 10 (1983) 685–697.
F. J. Moraga, A. E. Larrateguy, D. A. Drew and R. T. Lahey Jr., An assessment of turbulent dispersion models for bubbly flow. Submitted (2001).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Drew, D.A. A turbulent dispersion model for particles or bubbles. Journal of Engineering Mathematics 41, 259–274 (2001). https://doi.org/10.1023/A:1011901711594
Issue Date:
DOI: https://doi.org/10.1023/A:1011901711594