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Inertial Frame Independent Forcing for Discrete Velocity Boltzmann Equation: Implications for Filtered Turbulence Simulation

Published online by Cambridge University Press:  20 August 2015

Kannan N. Premnath  and
Sanjoy Banerjee
Kannan N. Premnath*
Affiliation:
Department of Mechanical Engineering, University of Wyoming, Laramie, WY 82071, USA
Sanjoy Banerjee*
Affiliation:
Department of Chemical Engineering, City College of New York, City University of New York, New York, NY 10031, USA
*
Corresponding author.Email:knandhap@uwyo.edu
Email:banerjee@che.ccny.cuny.edu
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Abstract

We present a systematic derivation of a model based on the central moment lattice Boltzmann equation that rigorously maintains Galilean invariance of forces to simulate inertial frame independent flow fields. In this regard, the central moments, i.e. moments shifted by the local fluid velocity, of the discrete source terms of the lattice Boltzmann equation are obtained by matching those of the continuous full Boltzmann equation of various orders. This results in an exact hierarchical identity between the central moments of the source terms of a given order and the components of the central moments of the distribution functions and sources of lower orders. The corresponding source terms in velocity space are then obtained from an exact inverse transformation due to a suitable choice of orthogonal basis for moments. Furthermore, such a central moment based kinetic model is further extended by incorporating reduced compressibility effects to represent incompressible flow. Moreover, the description and simulation of fluid turbulence for full or any subset of scales or their averaged behavior should remain independent of any inertial frame of reference. Thus, based on the above formulation, a new approach in lattice Boltzmann framework to incorporate turbulence models for simulation of Galilean invariant statistical averaged or filtered turbulent fluid motion is discussed.

Keywords

05.20.Dd47.27.-i47.27.E-Lattice Boltzmann methodcentral momentsGalilean invarianceturbulencefiltering
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 3 , September 2012 , pp. 732 - 766
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Frisch, U., Hasslacher, B. and Pomeau, Y., Lattice-Gas Automata for the Navier-Stokes Equation, Phys. Rev. Lett. 56 (1986) 1505.Google Scholar
[2]McNamara, G.R. and Zanetti, G., Use of the Boltzmann Equation to Simulate Lattice-Gas Automata, Phys. Rev. Lett. 61 (1988) 2332.Google Scholar
[3]Benzi, R., Succi, S. and Vergassola, M., The Lattice Boltzmann Equation: Theory and Applications, Phys. Rep. 222 (1992) 145.Google Scholar
[4]Chen, S. and Doolen, G., Lattice Boltzmann Method for Fluid Flows, Annu. Rev. Fluid Mech. 8 (1998) 2527.Google Scholar
[5]Succi, S., The Lattice Boltzmann Equation for Fluid Dynamics and Beyond, Clarendon Press, Oxford, 2001.Google Scholar
[6]Yu, D., Mei, R., Luo, L.-S. and Shyy, W., Viscous Flow Computations with the Method of Lattice Boltzmann Equation, Prog. Aero. Sci. 39 (2003) 329.Google Scholar
[7]Abe, T., Derivation of the Lattice Boltzmann Method by means of the discrete ordinate method for the Boltzmann equation, J. Comp. Phys. 131 (1997) 241.Google Scholar
[8]He, X. and Luo, L.S., A priori derivation of the lattice Boltzmann equation, Phys. Rev. E 55 (1997) 115.Google Scholar
[9]Shan, X. and He, X., Discretization of the Velocity Space in the Solution of the Boltzmann Equation, Phys. Rev. Lett. 80 (1998) 65.Google Scholar
[10]Luo, L.-S., Theory of the lattice Boltzmann method: Lattice Boltzmann models for nonideal gases, Phys. Rev. E 62 (2000) 4982.Google Scholar
[11]He, X. and Doolen, G., Thermodynamic Foundations ofKinetic Theory and Lattice Boltzmann Models for Multiphase Flows, J. Stat. Phys. 107 (2002) 1572.Google Scholar
[12]Asinari, P. and Luo, L.-S., A Consistent Lattice Boltzmann Equation with Baroclinic Coupling for Mixtures, J. Comput. Phys. 227 (2008) 3878.Google Scholar
[13]Shan, X., Yuan, X.F. and Chen, H., Kinetic Theory Representation of Hydrodynamics: A Way Beyond the Navier-Stokes Equation, J. Fluid Mech. 550 (2006) 413.CrossRefGoogle Scholar
[14]Junk, M., Klar, A. and Luo, L.S., Asymptotic Analysis of the Lattice Boltzmann Equation, J. Comp. Phys. 210 (2005) 676.Google Scholar
[15]Qian, Y., d’Humières, D. and Lallemand, P., Lattice BGK Models for Navier-Stokes Equation, Europhys. Lett. 17 (1992) 479.Google Scholar
[16]Chen, H., Chen, S. and Matthaeus, W.H., Recover of the Navier-Stokes Eqauations using the Lattice-Gas Boltzmann Method, Phys. Rev. A 45 (1992) 5339.CrossRefGoogle Scholar
[17]Bhatnagar, P., Gross, E. and Krook, M., A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems, Phys. Rev. 94 (1954) 511.Google Scholar
[18]d’Humières, D., Generalized Lattice Boltzmann Equations, Progress in Aeronautics and Astronautics (Eds. Shigal, B.D. and Weaver, D.P) 159 (1992) 450.Google Scholar
[19]Higuera, F. J. and Jiménez, J., Boltzmann Approach to Lattice Gas Simulations, Europhys. Lett. 9 (1989) 663.Google Scholar
[20]Higuera, F. J., Succi, S. and Benzi, R., Lattice gas-dynamics with enhanced collisions, Europhys. Lett. 9 (1989) 345.Google Scholar
[21]Grad, H., On the Kinetic Theory of Rarefied Gases, Comm. Pure Appl. Math. 2 (1949) 331.Google Scholar
[22]Lallemand, P. and Luo, Li-Shi, Theory of the Lattice Boltzmann Method: Dispersion, Dissipation, Isotropy, Galilean Invariance, and Stability, Phys. Rev. E 61 (2000) 6546.Google Scholar
[23]d’Humières, D., Ginzburg, I., Krafczyk, M., Lallemand, P. and Luo, L.-S., Multiple-Relaxation-Time Lattice Boltzmann Models in Three Dimensions, Phil. Trans. R. Soc. Lond. A 360 (2002) 437.Google Scholar
[24]Premnath, K.N. and Abraham, J., Three-dimensional Multi-Relaxation Time (MRT) Lattice-Boltzmann Models for Multiphase Flow, J. Comp. Phys. 224 (2007) 539.Google Scholar
[25]Premnath, K.N. and Abraham, J., Simulations of Binary Drop Collisions with a Multiple-Relaxation-Time Lattice-Boltzmann Model, Phys. Fluids 17 (2005) 122105.Google Scholar
[26]Premnath, K.N., McCracken, M.E. and Abraham, J., A Review of Lattice Boltzmann Methods for Multiphase Flows Relevant to Engine Sprays, Trans. of the SAE: J. Engines 114 (2005) 929.Google Scholar
[27]Premnath, K.N., Pattison, M.J. and Banerjee, S., Generalized Lattice Boltzmann Equation with Forcing Term for Computation of Wall Bounded Turbulent Flows, Phys. Rev. E 79 (2009) 026703.Google Scholar
[28]Premnath, K.N., Pattison, M.J. and Banerjee, S., Dynamic Subgrid Scale Modeling of Turbulent Flows using Lattice-Boltzmann Method, Physica A 388 (2009) 2640.Google Scholar
[29]Pattison, M.J., Premnath, K.N., Morley, N.B. and Abdou, M., Progress in Lattice Boltzmann Methods for Magnetohydrodynamic Flows Relevant to Fusion Applications, Fusion Engg. Des. 83 (2008) 557.Google Scholar
[30]Shan, X. and Chen, H., A General Multiple-Relaxation Boltzmann Collision Model, Int. J. Mod. Phys. C 18 (2007) 635.CrossRefGoogle Scholar
[31]Karlin, I.V., Ferrante, A.Öttinger, H.C., Perfect Entropy Functions of the Lattice Boltzmann Method, Euro. Phys. Lett 47 (1999) 182.Google Scholar
[32]Boghosian, B.M., Yepez, J., Coveney, P.V. and Wagner, A., Entropic Lattice Boltzmann Methods, Proc. R. Soc. Lond. A 457 (2001) 717.Google Scholar
[33]Ansumali, S. and Karlin, I.V., Single Relaxation Time Model for Entropic Lattice Boltzmann Methods, Phys. Rev. E 65 (2002) 056312.Google Scholar
[34]Succi, S., Karlin, I. and Chen, H., Role of the H theorem in Lattice Boltzmann Hydrodynamic Simulations, Rev. Mod. Phys. 74 (2002) 1203.Google Scholar
[35]Asinari, P. and Karlin, I.V., Generalized Maxwell State and H Theorem for Computing Fluid Flows using the Lattice Boltzmann Method, Phys. Rev. E 79 (2009) 036703.Google Scholar
[36]Asinari, P. and Karlin, I.V., Quasiequilibrium Lattice Boltzmann Models with Tunable Bulk Viscosity for Enhancing Stability, Phys. Rev. E 81 (2010) 016702.Google Scholar
[37]Chikatamarla, S.S. and Karlin, I.V., Lattices for the Lattice Boltzmann Method, Phys. Rev. E 79 (2009) 046701.CrossRefGoogle ScholarPubMed
[38]Dubois, F. and Lallemand, P., Towards Higher Order Lattice Boltzmann Schemes, J. Stat. Mech.: Theory and Experiment P06006 (2009) 1.Google Scholar
[39]Karlin, I.V. and Asinari, P., Factorization Symmetry in the Lattice Boltzmann Method, Physica A 389 (2010) 1530.Google Scholar
[40]Rubinstein, R. and Luo, L.-S., Theory of the Lattice Boltzmann Equation: Symmetry Properties of Discrete Velocity Sets, Phys. Rev. E 77 (2008) 036709.Google Scholar
[41]Chenand, H.Goldhirsch, I. and Orszag, S.A., Discrete Rotational Symmetry, Moment Isotropy and Higher Order Lattice Boltzmann Models, J. Sci. Comput. 34 (2008) 87.Google Scholar
[42]Bandaand, M.K.Yong, W.-A. and Klar, A., A Stability Notion for Lattice Boltzmann Equations, SIAM J. Sci. Comput. 27 (2006) 2098.Google Scholar
[43]Junk, M. and Yong, W.-A., Weighted l2-Stability of the Lattice Boltzmann Method, SIAM J. Numer. Anal. 47 (2009) 1651.Google Scholar
[44]Yong, W.-A., Onsager-like Relation for the Lattice Boltzmann Method, Comp. Math. Appl. 58 (2009) 862.Google Scholar
[45]Junk, M. and Yang, Z., Convergence of Lattice Boltzmann Methods for Navier-Stokes Flows in Periodic and Bounded Domains, Numer. Math. 112 (2009) 65.Google Scholar
[46]Truesdell, C. and Noll, W., The Non-Linear Field Theories of Mechanics, Springer, New York, 2004.Google Scholar
[47]Muller, I., On the Frame Dependence of Stress and Heat Flux, Arch. Rational Mech. Anal. 45 (1972) 45.CrossRefGoogle Scholar
[48]Wang, C.-C., On the Concept of Frame-Indifference in Continuum Mechanics and in the Kinetic Theory of Gases, Arch. Rational Mech. Anal. 58 (1975) 381.Google Scholar
[49]Speziale, C.G., On Frame-Indifference and Iterative Procedures in the Kinetic Theory of Gases, Int. J. Engg. Sci. 19 (1981) 63.Google Scholar
[50]Woods, L.C., Frame-Indifferent Kinetic Theory, J. Fluid Mech. 136 (1983) 423.Google Scholar
[51]Speziale, C.G., The Effect of the Reference Frame on the Thermophysical Properties of an Ideal Gas, Int. J. Thermophys. 7 (1986) 99.Google Scholar
[52]Speziale, C.G., Comments on the “Material Frame-Indifference” Controversy, Phys. Rev. A 36 (1987) 4522.Google Scholar
[53]Wagner, A.J., Derivation of a Non-Objective Oldroyd Model from the Boltzmann Equation, arXiv:cond-mat/0105067v1 (2001).Google Scholar
[54]Degond, P. and Lemou, M., Turbulence Models for Incompressible Fluids Derived from Kinetic Theory, J. Math. Fluid Mech. 4 (2002) 257.Google Scholar
[55]Cao, N., Chen, S., Jin, S., and Martnez, D., Physical Symmetry and Lattice Symmetry in the Lattice Boltzmann Method, Phys. Rev. E 55 (1997) R21.Google Scholar
[56]Sofonea, V. and Sekerka, R.F., Viscosity of Finite Difference Lattice Boltzmann Models, J.Com-put. Phys. 184 (2003) 422.Google Scholar
[57]Lee, T. and Lin, C.-L., An Eulerian Description of the Streaming Process in the Lattice Boltz-mann Equation, J. Comput. Phys. 185 (2003) 445.Google Scholar
[58]Min, M. and Lee, T., A Spectral-Element Discontinuous Galerkin Lattice Boltzmann Method for Nearly Incompressible Flows, J. Comput. Phys. 230 (2011) 245.Google Scholar
[59]Geier, M., Greiner, A. and Korvink, J.G., Cascaded Digital Lattice Boltzmann Automata for High Reynolds Number Flow, Phys. Rev. E 73 (2006) 066705.Google Scholar
[60]Struchtrup, H., Macroscopic Transport Equations for Rarefied Gas Flows: Approximation Methods in Kinetic Theory, Springer, New York, 2005.Google Scholar
[61]Asinari, P., Generalized Local Equilibrium in the Cascaded Lattice Boltzmann Method, Phys. Rev. E 78 (2008) 016701.CrossRefGoogle ScholarPubMed
[62]Geier, M., Greiner, A. and Korvink, J.G., A Factorized Central Moment Lattice Boltzmann Method, Eur. Phys. J. Special Topics 171 (2009) 55.CrossRefGoogle Scholar
[63]Premnath, K.N. and Banerjee, S., Incorporating Forcing Terms in Cascaded Lattice Boltzmann Approach by Method of Central Moments, Phys. Rev. E 80 (2009) 036702.Google Scholar
[64]Chapman, S. and Cowling, T.G., Mathematical Theory of Nonuniform Gases, Cambridge University Press, New York (1964).Google Scholar
[65]Premnath, K.N. and Banerjee, S., On the Three Dimensional Central Moment Lattice Boltz-mann Method, J. Stat. Phys. 143 (2011) 747.CrossRefGoogle Scholar
[66]Scovazzi, G., A Discourse on Galilean Invariance and SUPG-Type Stabilization, Comp. Methods Appl. Mech. Engg. 196 (2007) 1108.Google Scholar
[67]Scovazzi, G., Galilean Invariance and Stabilized Methods for Compressible Flows, Int. J. Num. Meth. Fluids 54 (2007) 757.Google Scholar
[68]Chen, H., Succi, S. and Orszag, S., Analysis of Subgrid Scale Turbulence using the Boltzmann Bhatnagar-Gross-Krook Kinetic Equation, Phys. Rev. E 59 (1998) R2527.Google Scholar
[69]Ansumali, S., Karlin, I. and Succi, S., Kinetic Theory of Turbulence Modelling: Smallness Parameter, Scaling and Microscopic Derivation of Smagorinsky Model, Physica A 338 (2004) 379.Google Scholar
[70]Chen, H., Orszag, S. and Staroselsky, I., Expanded Analogy between Boltzmann Kinetic Theory of Fluids and Turbulence, J. Fluid Mech. 519 (2004) 301.Google Scholar
[71]Teixeira, C., Incorporating Turbulence Models into the Lattice-Boltzmann Method, Int. J. Mod. Phys. C 9 (1998) 1159.Google Scholar
[72]Chen, H., Kandasamy, S., Orszag, S., Shock, R., Succi, S. and Yakhot, V., Extended Boltzmann Kinetic Equation for Turbulent Flows, Science 301 (2003) 633.Google Scholar
[73]Hou, S., Sterling, J., Chen, S. and Doolen, G.D., A Lattice Boltzmann Subgrid Model for High Reynolds Number Flows, Fields Inst. Comm. 6 (1996) 151.Google Scholar
[74]Yu, H., L.-S., and Girimaji, S., LES of Turbulent Square Jet Flow using an MRT Lattice Boltz-mann Model, Comput. Fluids 35 (2006) 957.Google Scholar
[75]Dong, Y.H., Sagaut, P. and Marie, S., Inertial Consistent Subgrid Model for Large-Eddy Simulation Based on the Lattice Boltzmann Method, Phys. Fluids 20 (2008) 035104.Google Scholar
[76]Sagaut, P., Toward Advanced Subgrid Models for Lattice-Boltzmann-Based Large-Eddy Simulation: Theoretical Formulations, Comp. Math. Appl. 59 (2010) 2194.Google Scholar
[77]Girimaji, S., Boltzmann Kinetic Equation for Filtered Fluid Turbulence, Phys. Rev. Lett. 99 (2007) 034501.Google Scholar
[78]Stolz, S. and Adams, N.A., An Approximate Deconvolution Procedure for Large-Eddy Simulation, Phys. Fluids 11 (1999) 1699.Google Scholar
[79]Speziale, C.G., Galilean Invariance of Subgrid-Scale Stress Models in the Large-Eddy Simulation of Turbulence, J. Fluid Mech. 156 (1985) 55.Google Scholar
[80]Germano, M., A Proposal for a Redefinition of the Turbulent Stresses in the Filtered Navier-Stokes Equations, Phys. Fluids 29 (1986) 2323.Google Scholar
[81]Fureby, C. and Tabor, G., Mathematical and Physical Constraints on Large-Eddy Simulations, Theoret. Comput. Fluid Dynamics 9 (1997) 85.Google Scholar
[82]Monin, A.S. and Yaglom, A.M., Statistical Fluid Mechanics, Vol I: Mechanics of Turbulence, Dover Publications, 2007.Google Scholar
[83]Germano, M., Turbulence: The Filtering Approach, J. Fluid Mech. 238 (1992) 325.Google Scholar
[84]Harris, S., An Introduction to the Theory of the Boltzmann Equation, Dover Publications, New York, 1999.Google Scholar
[85]Kogan, M.N., Rarefied Gas Dynamics, Plenum Press, New York, 1969.Google Scholar
[86]Müller, I. and Ruggeri, T., Extended Thermodynamics, Springer Tractsin Natural Philosophy, New York, 1993.Google Scholar
[87]He, X. and Luo, L.-S., Lattice Boltzmann Model for the Incompressible Navier-Stokes Equation, J. Stat. Phys. 88 (1997) 927.Google Scholar
[88]Maxwell, J.C., Illustrations of the Dynamical Theory of Gases, Phil. Mag. 19 (1860) 19.Google Scholar
[89]Reynolds, O., On the Dynamical Theory of Incompressible Viscous Fluids and the Determination of the Criterion, Phil. Trans. Roy. Soc. London A 186 (1895) 123.Google Scholar
[90]Durbin, P.A. and Reif, B. A. Pettersson, Statistical Theory and Modeling for Turbulent Flows, Wiley, West Sussex, 2010.Google Scholar