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Lattice Boltzmann modeling with discontinuous collision components: Hydrodynamic and Advection-Diffusion Equations

  • Hydrodynamic and advection-diffusion equations
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Abstract

Irrespective of the nature of the modeled conservation laws, we establish first the microscopic interface continuity conditions for Lattice Boltzmann (LB) multiple-relaxation time, link-wise collision operators with discontinuous components (equilibrium functions and/or relaxation parameters). Effective macroscopic continuity conditions are derived for a planar implicit interface between two immiscible fluids, described by the simple two phase hydrodynamic model, and for an implicit interface boundary between two heterogeneous and anisotropic, variably saturated soils, described by Richard’s equation. Comparing the effective macroscopic conditions to the physical ones, we show that the range of the accessible parameters is restricted, e.g. a variation of fluid densities or a heterogeneity of the anisotropic soil properties. When the interface is explicitly tracked, the interface collision components are derived from the leading order continuity conditions. Among particular interface solutions, a harmonic mean value is found to be an exact LB solution, both for the interface kinematic viscosity and for the interface vertical hydraulic conductivity function. We construct simple problems with the explicit and implicit interfaces, matched exactly by the LB hydrodynamic and/or advection-diffusion schemes with the aid of special solutions for free collision parameters.

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References

  1. M. Bakker and K. Hemker, Analytic solutions for groundwater whirls in box-shaped, layered anisotropic aquifers, Adv. Water Resour. 27:1075–1086 (2004).

    Article  ADS  Google Scholar 

  2. M. Bouzidi, M. Firdaouss and P. Lallemand, Momentum transfer of a Boltzmann-lattice fluid with boundaries, Phys. Fluids 13:3452–3459 (2001).

    Article  ADS  Google Scholar 

  3. R.H. Brooks and A.T. Corey, Hydraulic properties of porous media. Hydrol. Paper No. 3, Colorado State University, Fort Collins, Colo. 1964.

    Google Scholar 

  4. J.M. Buick and C.A. Greated., Gravity in a lattice Boltzmann model. Phys. Rev. E 61:5307–5320 (2000).

    Article  ADS  Google Scholar 

  5. J.M. Buick and C. A. Greated, Lattice-Boltzmann modeling of interfacial gravity waves. Phys. Fluids 10(6):1490–1511 (1998).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  6. M. Celia, E. Bouloutas and R. Zabra, A general Mass-Conservative Numerical Solution for the Unsaturated Flow Equation. Water Resour. Res. 26:1483–1496 (1990).

    Article  ADS  Google Scholar 

  7. S. Chen and G.D. Doolen, Lattice Boltzmann method for fluid flows. Ann. Rev. J. Fluid Mechanics 30:329–364 (1998).

    Article  ADS  MathSciNet  Google Scholar 

  8. D.A. Edwards, H. Brenner and D.T. Wasan, Interfacial transport process and rheology. Butterworth-Heinemann Series in Chemical Engineering, 1991.

  9. U. Frisch, D. d’Humières, B. Hasslacher, P. Lallemand, Y. Pomeau and J.P. Rivet, Lattice gas hydrodynamics in two and three dimensions Complex Sys. 1:649–707 (1987).

    MATH  Google Scholar 

  10. E. Guyon, J.-P. Hulin and L. Petit, Hydrodynamique physique, Inter Editions/Editions du CNRS, Paris, 1991.

    Google Scholar 

  11. X. He and L.S. Luo, Lattice Boltzmann Model for the Incompressible Navier-Stokes Equation. J. Stat. Phys. 88:927–945 (1997).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  12. F. J. Higuera and J. Jimenez. Boltzmann approach to lattice gas simulations. Europhys. Lett., 9:663–668, 1989.

    Article  ADS  Google Scholar 

  13. D. d’Humières, Generalized Lattice-Boltzmann Equations. AIAA Rarefied Gas Dynamics: Theory and Simulations. Progress in Astronautics and Aeronautics 59:450–548 (1992).

    Google Scholar 

  14. D. d’Humières, M. Bouzidi and P. Lallemand, Thirteen-velocity three-dimen-sional lattice Boltzmann model, Phys. Rev. E 63:066702-1–7 (2001).

    Article  Google Scholar 

  15. D. d’Humières, I. Ginzburg, M. Krafczyk, P. Lallemand and L.-S. Luo, Multiple-relaxation-time lattice Boltzmann models in three dimensions. Phil. Trans. R. Soc. Lond. A 360:437–451 (2002).

    Article  MATH  ADS  Google Scholar 

  16. D. d’Humières and I. Ginzburg, Some analytical results about the stability of Lattice Boltzmann models, in preparation, 2006.

  17. D. d’Humières and I. Ginzburg, Knudsen layers in Lattice Boltzmann modeling, in preparation, 2006.

  18. I. Ginzbourg. Boundary conditions problems in lattice gas methods for single and multiple phases, PhD, University Paris VI, 1994.

  19. I. Ginzbourg and P. M. Adler, Boundary flow condition analysis for the three-dimensional lattice Bolzmann model, J. Phys. II France 4:191–214 (1994).

    Article  Google Scholar 

  20. GA95 I. Ginzbourg and P. Adler, Surface Tension Models with Different Viscosities, Transport Porous Media 20:37–76 (1995).

    Article  Google Scholar 

  21. I. Ginzburg and K. Steiner, A free surface Lattice Boltzmann method for modelling the filling of expanding cavities by Bingham fluids, Phil. Trans. R. Soc. Lond. A 360:453–466 (2002).

    Article  MATH  ADS  Google Scholar 

  22. I. Ginzburg and K. Steiner, Lattice Boltzmann model for free-surface flow and its application to filling process in casting. J. Comp. Phys. 185:61–99 (2003).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  23. I. Ginzburg and D. d’Humières, Multi-reflection boundary conditions for lattice Boltzmann models. Phys. Rev. E 68:066614-1-30 (2003).

    Article  ADS  Google Scholar 

  24. I. Ginzburg, Equilibrium-type and Link-type Lattice Boltzmann models for generic advection and anisotropic-dispersion equation, Adv. Water Resour. 28:1171–1195 (2005).

    Article  ADS  Google Scholar 

  25. I. Ginzburg, Generic boundary conditions for Lattice Boltzmann models and their application to advection and anisotropic-dispersion equations. Adv. Water Resour. 28:1196–1216 (2005).

    Article  ADS  Google Scholar 

  26. I. Ginzburg and J.-P. Carlier and C. Kao, Lattice Boltzmann approach to Richards’ equation, in: Computational methods in water resources. Proc. of the CMWR XV, Chapen Hill, USA, 2004, eds. C. T. Miller, Elsevier, pp. 583–597, 2004.

    Google Scholar 

  27. I. Ginzburg, Variably saturated flow described with the anisotropic Lattice Boltzmann methods. J. Comput. Fluids 35(8/9):831–848 (2006).available online:doi:10.1016/j.compfluid.2005.11.001

    Article  MATH  Google Scholar 

  28. I. Ginzburg and D. d’Humières, Lattice Boltzmann and analytical solutions for flow process in anisotropic heterogeneous stratified aquifers, submitted to Advances in Water Resour., 2006.

  29. A.K. Gunstensen, D.H. Rothmann, S. Zaleski and G. Zanetti, Lattice Boltzmann model of immiscible fluids. Phys. Rev. A 43:4320–4327 (1991).

    Article  ADS  Google Scholar 

  30. Z. Guo, C. Zheng, and B. Shi, Discrete lattice effects on the forcing term in the lattice Boltzmann method, Phys. Rev. E 65:066308-1-6 (2003).

    Google Scholar 

  31. D. Grunau, S. Chen and K. Eggert, A lattice Boltzmann model for multiphase fluid flows. Phys. Fluids A5 2557–2562 1993.

    ADS  Google Scholar 

  32. X. He, 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).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  33. HSZDS96 S. Hou, X. Shan, Q. Zou, G.D. Doolen and W.E. Soll, Evaluation of Two Lattice Boltzmann Models for Multiphase Flows. J. Comput. Phys. 138:695–713 (1996).

    Article  MathSciNet  ADS  Google Scholar 

  34. T. Inamuro, 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).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  35. A.J.C. Ladd, Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 2. Numerical results. J. Fluid Mech. 271:311–339 (1994).

    Article  ADS  MathSciNet  Google Scholar 

  36. D. Kehrwald, Numerical analysis of immiscible lattice BGK, PhD. diss, UNI Kaiserslautern, Germany, 2002.

  37. C. Körner, M. Thies, T. Hoffmann, N. Thürey and U. Rüde, Lattice Boltzmann Model for Free Surface flow for modeling Foaming, J. Stat. Phys. 121:179–197 (2005).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  38. P. Lallemand and L.-S. Luo, Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability. Phys. Rev. E 61:6546–6562 (2000).

    Article  ADS  MathSciNet  Google Scholar 

  39. semi98 F. Marinelli and D.S. Durnford, Semianalytical solution to Richards’ Equation for layered porous media. J. Irrigat. Drainage Eng. 124(6):290–299 (1998).

    Article  Google Scholar 

  40. C.T. Miller, C. Abhishek, A.B. Sallerson, J.F. Prins and M. W. Farthing, A comparison of computational and algorithmic advances for solving Richards’ equation, “Computational methods in water resources”. C.T. Miller (ed.), Proc. of the CMWR XV, June 13–17, Chapel Hill, NC, USA, vol. 2, pp. 1131–1145, Elsevier, 2004.

    Google Scholar 

  41. Y. Mualem, A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resour. Res. 12:513–522 (1976).

    Article  ADS  Google Scholar 

  42. X. Nie, Y.-H. Qian, G.D. Doolen and S. Chen, Lattice Boltzmann simulation of the two-dimensional Rayleigh-Taylor instability. Phys. Rev. E 58:6861–6864 (1998).

    Article  ADS  Google Scholar 

  43. Y. Qian, D. d’Humières and P. Lallemand, Lattice BGK models for Navier-Stokes equation. Europhys. Lett. 17:479–484 (1992).

    Article  MATH  ADS  Google Scholar 

  44. C. Pan, M. Hilpert and C.T. Miller, Lattice Boltzmann simulation of two-phase flow in porous media, Water Resour. Res. 40(1):W01501:1–14 (2004).

    Article  Google Scholar 

  45. C. Pan, L. Luo and C.T. Miller, An evaluation of lattice Boltzmann schemes for porous media simulations, J. Comput. Fluids 35(8/9):898–909 (2006).available online:doi:10.1016/j.compfluid.2005.03.008

    Article  MATH  Google Scholar 

  46. U. D’Ortona, D. Salin, M. Cieplak, R.B. Rybka and J.R. Banavar, Two-color nonlinear Boltzmann cellular automata: Surface tension and wetting, Phys. Rev. E 51:3718–3751 (1995).

    Article  ADS  Google Scholar 

  47. D. Raabe, Overview of the lattice Boltzmann method for nano-and microscale fluid dynamics in matherial science and engineering, Model. Simul. Mater. Sci. Eng. 12:R13–R46 (2004).

    Article  ADS  Google Scholar 

  48. L.O.E. dos Santos and P.C. Philippi, Lattice-gas model based on field mediators for immiscible fluids, Phys. Rev. E 65:046305-1-8 (2002).

    Article  ADS  Google Scholar 

  49. X. Shan and G.D. Doolen, Multi-component lattice-Boltzmann model with interparticle interaction, J. Stat. Phys. 81(1/2):379–393 (1995).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  50. X. Shan and G. Doolen, Diffusion in a multicomponent lattice Boltzmann equation model. Phys. Rev. E 54:3614–3620 (1996).

    Article  ADS  Google Scholar 

  51. M.R. Swift, E. Orlandini, W.R. Osborn and J.M. Yeomans, Lattice Boltzmann simulation of liquid-gas and binary fluid systems, Phys. Rev. E 54:5041–5052 (1996).

    Article  ADS  Google Scholar 

  52. D.H. Rothman and S. Zaleski, Lattice Gas Dynamics Automata - Simple Model for Complex Hydrodynamics, Cambridge University Press, ISBN: 0-521-55201-X, 1997.

  53. P.J. Ross, Modeling soil water and solute transport-Fast, simplified numerical solutions Agronomy J. 95:1352–1361 (2003).

    Article  Google Scholar 

  54. A. Tichonov and A. Samarsky, Equations of mathematical physics. Nauka, Moscow, 1977.

    Google Scholar 

  55. J. Tölke, Die Lattice Boltzmann Methode für Mehrphasenströmungen. PhD.diss LS Bauinformatik, TU, München, Germany, 2001.

  56. J. Tölke, M. Krafczyk, M. Schulz and E. Rank, Lattice Boltzmann simulations of binary fluid flow through porous media, Phil. Trans. R. Soc. Lond. A 360:535–545 (2002).

    Article  MATH  ADS  Google Scholar 

  57. H.-J. Vogel, J. Tölke, V. P. Schulz, M. Krafczyk and K. Roth, Comparison of Lattice-Boltzmann Model, a Full-Morphology Model, and a Pore Network Model for Determining Capillary Pressure-Saturation Relationships, Vadoze Zone J. 4:380–388 (2005).

    Article  Google Scholar 

  58. X. Zhang, A.G. Bengough, L.K. Deeks, J.W. Crawford and I. M. Young, A lattice BGK model for advection and anisotropic dispersion equation. Adv. Water Resour. 25:1–8 (2002a).

    Article  MATH  Google Scholar 

  59. X. Zhang, A.G. Bengough, L.K. Deeks, J.W. Crawford and I.M. Young, A novel three-dimensional lattice Boltzmann model for solute transport in variably saturated porous media, Water Resour. Res. 38:1–10 (2002b).

    Google Scholar 

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  1. Cemagref, Antony Regional Centre, HBAN, Parc de Tourvoie - BP 44, 92163, Antony cedex, France

    Irina Ginzburg PhD

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  1. Irina Ginzburg PhD
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Ginzburg, I. Lattice Boltzmann modeling with discontinuous collision components: Hydrodynamic and Advection-Diffusion Equations. J Stat Phys 126, 157–206 (2007). https://doi.org/10.1007/s10955-006-9234-4

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  • DOI: https://doi.org/10.1007/s10955-006-9234-4

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