Lattice Boltzmann Method for Sprays

  • K. N. PremnathEmail author
  • J. Abraham


Among the noncontinuum-based computational techniques, the lattice Boltzman method (LBM) has received considerable attention recently. In this chapter, we will briefly present the main elements of the LBM, which has evolved as a minimal kinetic method for fluid dynamics, focusing in particular, on multiphase flow modeling. We will then discuss some of its recent developments based on the multiple-relaxation-time formulation and consistent discretization strategies for enhanced numerical stability, high viscosity contrasts, and density ratios for simulation of interfacial instabilities and multiphase flow problems. As examples, numerical investigations of drop collisions, jet break-up, and drop impact on walls will be presented. We will also outline some future directions for further development of the LBM for applications related to interfacial instabilities and sprays.


Interfacial instabilities Lattice Boltzmann method Sprays 


  1. 1.
    Ashgriz, N. and Y. Poo. Coalescence and Separation in Binary Collisions of Liquid Drops. J. Fluid Mech. 221: 183–204 (1990).CrossRefGoogle Scholar
  2. 2.
    Asinari, P. Viscous Coupling based Lattice Boltzmann Model for Binary Mixtures. Phys. Fluids 067102: 1–22 (2005).MathSciNetGoogle Scholar
  3. 3.
    Bhatnagar, P., E. Gross, and M. Krook. A Model for Collision Processes in Gases, I. Small Amplitude Processes in Charged and Neutral One-Component Systems. Phys. Rev. 94: 511–525 (1954).zbMATHCrossRefGoogle Scholar
  4. 4.
    Carnahan, N. and K. Starling. Equation of State for Nonattracting Rigid Spheres. J. Chem. Phys. 51: 635–636 (1969).CrossRefGoogle Scholar
  5. 5.
    Chapman, S. and T. Cowling. Mathematical Theory of Non-Uniform Gases. Cambridge University Press, London (1964).Google Scholar
  6. 6.
    Chen, S. and G. Doolen. Lattice Boltzmann Method for Fluid Flows. Annu. Rev. Fluid Mech. 30: 329–364 (1998).CrossRefMathSciNetGoogle Scholar
  7. 7.
    d’Humieres, D., I. Ginzburg, M. Krafczyk, P. Lallemand, and L.-S. Luo. Multiple-Relaxation-time Lattice Boltzmann Models in Three Dimensions. Phil. Trans. Roy. Soc. Lond. Ser. A 360: 437–351 (2002).zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Gunstensen, A.K., D.H. Rothman, S. Zaleski, and G. Zanetti. Lattice Boltzmann Model of Immiscible Fluids. Phys. Rev. A 43: 4320–4327 (1991).CrossRefGoogle Scholar
  9. 9.
    Harris, S. An Introduction to the Theory of the Boltzmann Equation. Dover Publications, New York (2004).Google Scholar
  10. 10.
    He, X. and G. Doolen. Thermodynamic Foundations of Kinetic Theory and Lattice Boltzmann Models for Multiphase Flows. J. Stat. Phys. 107: 1572–4996 (2002).CrossRefGoogle Scholar
  11. 11.
    He, X. and L.-S. Luo. Theory of the Lattice Boltzmann Method: From the Boltzmann Equation to the Lattice Boltzmann Equation. Phys. Rev. E 56: 6811–6817 (1997).CrossRefGoogle Scholar
  12. 12.
    He, X., X. Shan, and G. Doolen. Discrete Boltzmann Equation Model for Nonideal Gases. Phys. Rev. E 57: R13–R16 (1998).CrossRefGoogle Scholar
  13. 13.
    He, X., S. Chen, and R. Zhang. A Lattice Boltzmann Scheme for Incompressible Multiphase Flow and its Application in Simulation of Rayleigh–Taylor Instability. J. Comput. Phys. 152: 642–663 (1999).zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Holdych, D.J., D. Rovas, J.G. Georgiadis and R.O. Buckius. An Improved Hydrodynamic Formulation for Multiphase Flow Lattice Boltzmann Models. Int. J. Modern Phys. C 9: 1393–1404 (1998).CrossRefGoogle Scholar
  15. 15.
    Inamuro, T., N. Konishi, and F. Ogino. A Galilean Invariant Model of the Lattice Boltzmann Method for Multiphase Fluid Flows Using Free-Energy Approach. Comput. Phys. Commun. 129: 32–45 (2000).zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Inamuro, T., T. Ogata, S. Tajima, and N. Konishi. A Lattice Boltzmann Method for Incompressible Two-phase Flows with Large Density Ratios. J. Comput. Phys. 198: 628–644 (2004).zbMATHCrossRefGoogle Scholar
  17. 17.
    Junk, M., A. Klar, and L.-S. Luo. Asymptotic Analysis of the Lattice Boltzmann Equation. J. Comput. Phys. 210: 676–704 (2005).zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Kikkinides, E.S., A.G. Yiotis, M.E. Kainourgiakis, and A.K. Stubos. Thermodynamic Consistency of Liquid-Gas Lattice Boltzmann Methods. Phys. Rev. E 78: 036702 (2008).Google Scholar
  19. 19.
    Lallemand, P. and L.-S. Luo, Theory of the Lattice Boltzmann Method: Dispersion, Isotropy, Galilean Invariance, and Stability. Phys. Rev. E 61: 6546–6562 (2000).CrossRefMathSciNetGoogle Scholar
  20. 20.
    Lee, T. and C.-L. Lin. A Stable Discretization of the Lattice Boltzmann Equation for Simulation of Incompressible Two-Phase Flows at High Density Ratio. J. Comput. Phys. 206: 16 (2005).zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Luo, L.-S. Theory of the Lattice Boltzmann Method: Lattice Boltzmann Models for Nonideal Gases. Phys. Rev. E 62: 4982–4996 (2000).CrossRefMathSciNetGoogle Scholar
  22. 22.
    McCracken, M.E. and J. Abraham. Multiple-Relaxation-Time Lattice-Boltzmann Model for Multiphase Flow. Phys. Rev. E 71: 036701 (2005a).Google Scholar
  23. 23.
    McCracken, M.E. and J. Abraham. Simulations of Liquid Break up with an Axisymmetric, Multiple Relaxation Time, Index-Function Lattice Boltzmann Model. Int. J. Mod. Phys. C 16: 1671–1682 (2005b).zbMATHCrossRefGoogle Scholar
  24. 24.
    Mukherjee, S. and J. Abraham. A Pressure-Evolution-Based Multi-Relaxation-Time High-Density-Ratio Two-Phase Lattice-Boltzmann Model. Comput. Fluids 36: 1149–1158 (2007a).zbMATHCrossRefGoogle Scholar
  25. 25.
    Mukherjee, S. and J. Abraham. Lattice Boltzmann Simulations of Two-Phase Flow with High Density Ratio in Axially Symmetric Geometry. Phys. Rev. E. 75: 026701 (2007b).Google Scholar
  26. 26.
    Mukherjee, S. and J. Abraham. Investigations of Drop Impact on Dry Walls with a Lattice Boltzmann Model. J. Colloid Interface Sci. 312: 341–354 (2007c).CrossRefGoogle Scholar
  27. 27.
    Mukherjee, S. and J. Abraham. Crown Behavior in Drop Impact on Wet Walls. Phys. Fluids 19: 052103 (2007d).Google Scholar
  28. 28.
    Nourgaliev, R., T.N. Dinh, T.G. Theofanous, and D. Joseph. The Lattice Boltzmann Equation Method: Theoretical Interpretation, Numerics and Implications. Int. J. Multiphase Flow 29: 117–169 (2003).zbMATHCrossRefGoogle Scholar
  29. 29.
    Premnath, K.N. and J. Abraham. Lattice Boltzmann Model for Axisymmetric Multiphase Flows. Phys. Rev. E, 71: 056706 (2005a).Google Scholar
  30. 30.
    Premnath, K.N. and J. Abraham. Simulations of Binary Drop Collisions with a Multiple-Relaxation-Time Lattice-Boltzmann Model. Phys. Fluids 17: 122105 (2005b).Google Scholar
  31. 31.
    Premnath, K.N. and J. Abraham. Three-Dimensional Multi-Relaxation Time (MRT) Lattice-Boltzmann Models for Multiphase Flow. J. Comput. Phys. 224: 539–559 (2007).zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Premnath, K.N., McCracken, M.E. and J. Abraham. A Review of Lattice Boltzmann Methods for Multiphase Flows Relevant to Engine Sprays. SAE Trans: J. Engines, 114: 929–940 (2005).Google Scholar
  33. 33.
    Qian, J. and C. Law. Regimes of Coalescence and Separation in Droplet Collision. J. Fluid Mech. 331: 59–80 (1997).CrossRefGoogle Scholar
  34. 34.
    Roisman, I. Dynamics of Inertia Dominated Binary Drop Collisions. Phys. Fluids 16: 3438–3449 (2004).CrossRefMathSciNetGoogle Scholar
  35. 35.
    Rowlinson, J. and B. Widom. Molecular Theory of Capillarity. Clarendon Press, Oxford (1982).Google Scholar
  36. 36.
    Sankaranarayanan, K., I.G. Kevrekidis, S. Sundaresan, J. Lu and G. Tryggvason. A Comparative Study of Lattice Boltzmann and Front-Tracking Finite-Difference Methods for Bubble Simulations. Int. J. Multiphase Flow 29: 109–116 (2003).zbMATHCrossRefGoogle Scholar
  37. 37.
    Sbragaglia, M., R. Benzi, L. Biferale, S. Succi, K. Sugiyama, and F. Toschi. Generalized Lattice Boltzmann Method with Multirange Pseudopotential. Phys. Rev. E 75: 026702 (2007).Google Scholar
  38. 38.
    Shan, X. and H. Chen. Lattice Boltzmann Model of Simulating Flows with Multiple Phases and Components. Phys. Rev. E 47: 1815–1819 (1993).CrossRefGoogle Scholar
  39. 39.
    Shan, X., X.-F. Yuan, and H. Chen. Kinetic Theory Representation of Hydrodynamics: A Way Beyond the Navier-Stokes Equation. J. Fluid Mech. 550: 413–441 (2006).zbMATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Stone, H. and L. Leal. Relaxation and Breakup of an Initially Extended Drop in an Otherwise Quiescent Fluid. J. Fluid Mech. 198: 399–427 (1989).CrossRefGoogle Scholar
  41. 41.
    Stone, H., B. Bentley, and L. Leal. An Experimental Study of Transient Effects in the Breakup of Viscous Drops. J. Fluid Mech. 173: 131–158 (1986).CrossRefGoogle Scholar
  42. 42.
    Succi, S. The Lattice Boltzmann Equation for Fluid Dynamics and Beyond. Oxford University Press, New York (2001).zbMATHGoogle Scholar
  43. 43.
    Swift, M., S. Orlandini, W. Osborn, and J. Yeomans. Lattice Boltzmann Simulations of Liquid-Gas Binary-fluid Systems. Phys. Rev. E 54: 5041–5042 (1996).CrossRefGoogle Scholar
  44. 44.
    Wagner, A.J. Thermodynamic Consistency of Liquid-Gas Lattice Boltzmann Simulations. Phys. Rev. E 74: 056703 (2006).Google Scholar
  45. 45.
    Wolf-Gladrow, D. Lattice-Gas Cellular Automata and Lattice Boltzmann Models, Lecture Notes in Mathematics, No. 1725. Springer, Berlin (2000).Google Scholar
  46. 46.
    Zheng, H.W., C. Shu, and Y.T. Chew. A Lattice Boltzmann Model for Multiphase Flows with Large Density Ratio. J. Comput. Phys. 218: 353–371 (2006).zbMATHCrossRefMathSciNetGoogle Scholar

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© Springer US 2011

Authors and Affiliations

  1. 1.Mechanical EngineeringUniversity of WyomingLaramieUSA

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