Ensemble Averaging Techniques for Disperse Flows

  • A. Prosperetti
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 98)


In a number of recent papers by the author and co-workers, a new formalism for the derivation of averaged equations for disperse multiphase flow was developed. The techniques used in those studies are reviewed here in greater detail than was possible in the original publications. Some examples of the application of the formalism to the dilute case are then shown. Considerations on the numerical implementation of the method and on the incorporation of turbulence in the general framework are also given.


Disperse Phase Momentum Equation Reynolds Stress Multiphase Flow Ensemble Average 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    T.B. Anderson and R. Jackson, A fluid mechanical description of fluidized beds, I and EC Fundamentals, 6: 527–539, 1967.CrossRefGoogle Scholar
  2. [2]
    G.K. Batchelor, Sedimentation in a dilute dispersion of spheres, J. Fluid Mech., 52: 245–268, 1972.MATHCrossRefGoogle Scholar
  3. [3]
    G.K. Batchelor., Transport properties of two-phase materials with random structure, Ann. Rev. Fluid Mech., 6: 227–255, 1974.CrossRefGoogle Scholar
  4. [4]
    G.K. Batchelor, A new theory of the instability of a uniform fluidized bed, J. Fluid Mech., 193: 75–110, 1988.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    A. Biesheuvel and S. Spoelstra., The added mass coefficient of a dispersion of spherical gas bubbles in liquid, Int. J. Multiphase Flow, 15: 911–924, 1989.MATHCrossRefGoogle Scholar
  6. [6]
    A. Celmins, Representation of two-phase flows by volume averaging, Int. J. Multiphase Flow, 14: 81–90, 1988.CrossRefGoogle Scholar
  7. [7]
    D.A. Drew, Mathematical modeling of two-phase flow, Ann. Rev. Fluid Mech., 15: 261–291, 1983.CrossRefGoogle Scholar
  8. [8]
    D.A. Drew and R.T. JR. Lahey, Analytical modeling of multiphase flow, In Roco M.C., editor, Particulate Two-Phase Flow, pages 509–566, Butterworth-Heinemann, Boston, 1993.Google Scholar
  9. [9]
    G. Evans, Practical Numerical Integration, Wiley, Chichester, 1993.MATHGoogle Scholar
  10. [10]
    R.C. Givler, An interpretation of the solid phase pressure in slow fluid-particle flows, Int. J. Multiphase flow, 13: 717–722, 1993.CrossRefGoogle Scholar
  11. [11]
    E.J. Hinch, An averaged-equation approach to particle interactions in fluid suspension, J. Fluid Mech., 83: 695–720, 1977.MATHCrossRefGoogle Scholar
  12. [12]
    M. Ishii, Thermo-Fluid Dynamic Theory of Two-Phase Flow, Eyrolles, Paris, 1975.MATHGoogle Scholar
  13. [13]
    T.S. Lundgren, Slow flow through stationary random beds and suspensions of spheres, J. Fluid Mech., 51: 273–299, 1972.MATHCrossRefGoogle Scholar
  14. [14]
    H. Niederreiter, Random Number Generation and Monte Carlo Methods, Society for Industrial and Applied Mathematics, Philadelphia, 1992.MATHCrossRefGoogle Scholar
  15. [15]
    R.I. Nigmatulin, Spatial averaging in the mechanics of heterogeneous and dispersed systems, Int. J. Multiphase Flow, 5: 353–385, 1979.MATHCrossRefGoogle Scholar
  16. [16]
    A. Prosperetti and M. Marchioro, Conduction in non-uniform composites,In O. Manley and R. Goulard, editors, Proceedings of the 14th Symposium on Energy Engineering Sciences,Argonne National Laboratory, 1996, Report CONF-9605186.Google Scholar
  17. [17]
    A. Prosperetti and D.Z. Zhang, Finite-particle-size effects in disperse two-phase flows, Theor. Comput. Fluid Dynamics, 7: 429–440, 1995.MATHCrossRefGoogle Scholar
  18. [18]
    A. Prosperetti And D.Z. Zhang, Disperse phase stress in two-phase flow, Chem. Eng. Comm., 141–142: 387–398, 1996.Google Scholar
  19. [19]
    A.S. Sangani, A pairwise interaction theory for determining the linear acoustic properties of dilute bubbly liquids, J. Fluid Mech., 232: 221–284, 1991.MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    A.S. Sangani And A.K. Didwania, Dispersed-phase stress tensor in flows of bubbly liquids at large Reynolds numbers, J. Fluid Mech., 248: 27–54, 1993.MATHCrossRefGoogle Scholar
  21. [21]
    G.B. Wallis, The averaged Bernoulli equation and macroscopic equations of motion for the potential flow of a two-phase dispersion, Int. J. Multiphase Flow, 17: 683–695, 1991.MATHCrossRefGoogle Scholar
  22. [22]
    D.Z. Zhang, Renormalization in the closure relations for particulate flows, in preparation, 1997.Google Scholar
  23. [23]
    D.Z. Zhang And A. Prosperetti, Averaged equations for inviscid disperse two-phase flow, J. Fluid Mech., 267: 185–219, 1994a.MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    D.Z. Zhang And A. Prosperetti, Ensemble phase-averaged equations for bubbly flows, Phys. Fluids, 6: 2956–2970, 1994b.MathSciNetMATHCrossRefGoogle Scholar
  25. [25]
    D.Z. Zhang And A. Prosperetti, Momentum and energy equations for disperse two-phase flows and their closure for dilute suspensions, Int. J. Multiphase Flow, 23: 425–453, 1997.MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1998

Authors and Affiliations

  • A. Prosperetti
    • 1
  1. 1.Department of Mechanical EngineeringThe Johns Hopkins UniversityBaltimoreUSA

Personalised recommendations