Lattice Boltzmann Method without Invoking the M << 1 Assumption

SO Ronald

doi:10.61927/igmin223

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Biology Group Review Article 記事ID: igmin223

Lattice Boltzmann Method without Invoking the M << 1 Assumption

Environmental Sciences

SO Ronald ^*

Affiliation

DOI10.61927/igmin223 全文HTML 全文PDF この記事を引用する

要約

When a Maxwellian distribution is assumed for the distribution function in the BGK-type modelled BE, it will give rise to the Euler equations if it is the first-order approximation in the Chapman-Enskog method. Then the second-order equations will yield the N-S equations. Most LBM developed to date are formulated based on the second-order equations. Consequently, the assumption of a flow Mach number M << 1 is inherent in this formulation. This approach creates an unnecessary restriction on the LBM that should be avoided if possible. An alternative approach is to formulate a new LBM by considering an equilibrium distribution function where the first-order approximations give rise to the N-S equations. Adopting this approach, a new LBM has been formulated. This new LBM gives reliable results when applied to simulate aeroacoustics, incompressible flows, and compressible flows with and without shocks. Good agreement with measurements and numerical data derived from DAS/DNA calculations is obtained.

参考文献

Chapman S, Cowling TG. The Mathematical Theory of Non-uniform Gases. Cambridge University Press; 1939. Chapter 12.
Mott-Smith HM. The solution of the Boltzmann equation for a shock wave. Phys Rev. 1951;82:885-892.
Wang-Chang CS, Uhlenbeck GE, deBoer J, editors. Studies in Statistical Mechanics. Wiley; 1964; 2.
Morse TF. Kinetic model for gases with internal degrees of freedom. Phys Fluids. 1964;7:159-169.
Salwen H, Grousch CE, Ziering S. Extension of the Mott-Smith method for a one-dimensional shock wave. Phys Fluids. 1964;7:180-189.
Holway LH. New statistical models for kinetic theory: methods of construction. Phys Fluids. 1966;9:1658-1673.
Bhatnagar PL, Gross EP, Krook M. A model for collision processes in gases: I. Small amplitude processes in charged and neutral one-component systems. Phys Rev. 1954;94:511-525.
Broadwell J. Study of rarefied shear flow by the discrete velocity method. Phys Fluids. 1964;7:1243.
Cao NS, Chen S, Jin S, Martinez D. Physical symmetry and lattice symmetry in the lattice Boltzmann method. Phys Rev E. 1997;55:R21-24.
Mei R, Shyy W. On the finite difference-based lattice Boltzmann method in curvilinear coordinates. J Comput Phys. 1998;143:426-448.
Wolf-Gladrow DA. Lattice-Gas Cellular Automata and Lattice Boltzmann Models: An Introduction. Springer Verlag; 2000. Chapter 5.
Alexander FJ, Chen S, Sterling JD. Lattice Boltzmann thermohydrodynamics. Phys Rev E. 1993;47:R2249-R2252.
McNamara GR, Alder B. Analysis of the lattice Boltzmann treatment of hydrodynamics. Physica A. 1993;194:218-228.
Hu S, Yan G, Shi W. A lattice Boltzmann model for compressible perfect gas. Acta Mechanica Sinica (English Edition). 1997;13:218-226.
Kataoka T, Tsutahara M. Lattice Boltzmann model for the compressible Navier-Stokes equations with flexible specific-heat ratio. Phys Rev E Stat Nonlin Soft Matter Phys. 2004 Mar;69(3 Pt 2):035701. doi: 10.1103/PhysRevE.69.035701. Epub 2004 Mar 25. PMID: 15089354.
Xu K. Gas-Kinetic Scheme for Unsteady Compressible Flow Simulations. von Karman Institute for Fluid Dynamics Lecture Series, Vol. 1998-03. von Karman Institute; 1998.
Li XM, Leung RCK, So RMC. One-step aeroacoustics simulation using lattice Boltzmann method. AIAA J. 2006;44:78-89.
Li XM. Computational Aeroacoustics Using Lattice Boltzmann Model. PhD thesis. Mechanical Engineering Department, Hong Kong Polytechnic University; 2006.
Eucken A. Über das Wärmeleitvermögen, die spezifische Wärme und die innere Reibung der Gase. Physikalische Zeitschrift. 1913;14:324-332.
Lallemand P, Luo LS. Theory of the lattice Boltzmann method: acoustic and thermal properties in two and three dimensions. Phys Rev E Stat Nonlin Soft Matter Phys. 2003 Sep;68(3 Pt 2):036706. doi: 10.1103/PhysRevE.68.036706. Epub 2003 Sep 23. PMID: 14524925.
Chen Y, Ohashi H, Akiyama M. Thermal lattice Bhatnagar-Gross-Krook model without nonlinear deviations in macrodynamic equations. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics. 1994 Oct;50(4):2776-2783. doi: 10.1103/physreve.50.2776. PMID: 9962315.
McNamara GR, Garcia AL, Alder BJ. Stabilization of thermal lattice Boltzmann models. J Statistical Phys. 1995;81:395-408.
Teixeira C, Chen H, Freed DM. Multi-speed thermal lattice Boltzmann method stabilization via equilibrium under-relaxation. Comput Phys Commun. 2000;129:207-226.
Shan X. Simulation of Rayleigh-Bénard convection using a lattice Boltzmann method. Phys Rev E. 1997;55:2780-2788.
Leung RCK, Kam EWS, So RMC. Recovery of the transport coefficients in the Navier-Stokes equations from the modeled Boltzmann equation. AIAA J. 2007;45:737-739.
Aristov VV. Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows. Kluwer Academic Publishers; 2001. Chapter 7.
So RMC, Leung RCK, Kam EWS, Fu SC. Progress in the development of a new lattice Boltzmann method. Computers & Fluids. 2019;190:440-469.
Fu SC, So RMC, Leung RCK. Modeled Boltzmann equation and its application to direct aeroacoustics simulation. AIAA J. 2008;46:1651-1662.
So RMC, Leung RCK, Fu SC. Modeled Boltzmann equation and its application to shock-capturing simulation. AIAA J. 2008;46:3038-3048.
So RMC, Fu SC, Leung RCK. Finite Difference Lattice Boltzmann Method for Compressible Thermal Fluids. AIAA J. 2010;48(6):1059-1071.
Lele SK. Direct numerical simulations of compressible turbulent flows: fundamentals and applications. In: Hanifi A, et al., editors. Transition, Turbulence and Combustion Modeling. Kluwer Academic Publishers; 1998. pp. 424-429.
Brenner H. Kinematics of volume transport. Physica A. 2005;349:11-59.
Brenner H. Navier-Stokes revisited. Physica A. 2005;349:60-132.
Greenshields CJ, Reese JM. The structure of shock waves as a test of Brenner’s modifications to the Navier-Stokes equations. J Fluid Mech. 2007;580:407-429.
Kam EWS, So RMC, Leung RCK. Lattice Boltzmann method simulation of aeroacoustics and nonreflecting boundary conditions. AIAA J. 2007;45:1703-1712.
Li XM, So RMC, Leung RCK. Propagation speed, internal energy and direct aeroacoustics simulation using lattice Boltzmann method. AIAA J. 2006;44:2896-2903.
Leung RCK, Li XM, So RMC. Comparative Study of Nonreflecting Boundary Condition for One-Step Duct Aeroacoustics Simulation. AIAA J. 2006;44:664-667.
Tam CKW, Webb JC. Dispersion-relation-preserving finite difference schemes for computational aeroacoustics. J Comput Phys. 1993;107:262-281.
Gilbarg D, Paolucci D. The structure of shock waves in the continuum theory of fluids. J Rat. Mech. Analy. 1953;2:617-642.
Ohwada T. Structure of normal shock waves: direct numerical analysis of the Boltzmann equation for Hard-sphere molecules. Phys Fluids A. 1993;5:217-234.
Alsmeyer H. Density profiles in argon and nitrogen shock waves measured by the absorption of an electron beam. J Fluid Mech. 1976;74:497-513.
Thomas LH. Note on Becker’s theory of the shock front. J Chem Phys. 1944;12:449-453.
Chu CK. Kinetic-theoretic description of the formation of a shock wave. Phys Fluids. 1965;8:12-22.
Bird GA. Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Clarendon Press; 1994.
Yang JY, Huang JC. Rarefied flow computations using nonlinear model Boltzmann equations. J Comput Phys. 1995;120:323-339.
Xu K, Tang L. Nonequilibrium Bhatnagar-Gross-Krook model for nitrogen shock structure. Phys Fluids. 2004;16:3824-3827.
Xu K, Josyula E. Gas-kinetic scheme for rarefied flow simulation. Math Comput Simul. 2006;72:253-256.
Weiss W. Continuous shock structure in extended thermodynamics. Phys Rev E. 1995;52:R5760-5763.

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[1] Chapman S, Cowling TG. The Mathematical Theory of Non-uniform Gases. Cambridge University Press; 1939. Chapter 12.

[2] Mott-Smith HM. The solution of the Boltzmann equation for a shock wave. Phys Rev. 1951;82:885-892.

[3] Wang-Chang CS, Uhlenbeck GE, deBoer J, editors. Studies in Statistical Mechanics. Wiley; 1964; 2.

[4] Morse TF. Kinetic model for gases with internal degrees of freedom. Phys Fluids. 1964;7:159-169.

[5] Salwen H, Grousch CE, Ziering S. Extension of the Mott-Smith method for a one-dimensional shock wave. Phys Fluids. 1964;7:180-189.

[6] Holway LH. New statistical models for kinetic theory: methods of construction. Phys Fluids. 1966;9:1658-1673.

[7] Bhatnagar PL, Gross EP, Krook M. A model for collision processes in gases: I. Small amplitude processes in charged and neutral one-component systems. Phys Rev. 1954;94:511-525.

[8] Broadwell J. Study of rarefied shear flow by the discrete velocity method. Phys Fluids. 1964;7:1243.

[9] Cao NS, Chen S, Jin S, Martinez D. Physical symmetry and lattice symmetry in the lattice Boltzmann method. Phys Rev E. 1997;55:R21-24.

[10] Mei R, Shyy W. On the finite difference-based lattice Boltzmann method in curvilinear coordinates. J Comput Phys. 1998;143:426-448.

[11] Wolf-Gladrow DA. Lattice-Gas Cellular Automata and Lattice Boltzmann Models: An Introduction. Springer Verlag; 2000. Chapter 5.

[12] Alexander FJ, Chen S, Sterling JD. Lattice Boltzmann thermohydrodynamics. Phys Rev E. 1993;47:R2249-R2252.

[13] McNamara GR, Alder B. Analysis of the lattice Boltzmann treatment of hydrodynamics. Physica A. 1993;194:218-228.

[14] Hu S, Yan G, Shi W. A lattice Boltzmann model for compressible perfect gas. Acta Mechanica Sinica (English Edition). 1997;13:218-226.

[15] Kataoka T, Tsutahara M. Lattice Boltzmann model for the compressible Navier-Stokes equations with flexible specific-heat ratio. Phys Rev E Stat Nonlin Soft Matter Phys. 2004 Mar;69(3 Pt 2):035701. doi: 10.1103/PhysRevE.69.035701. Epub 2004 Mar 25. PMID: 15089354.

[16] Xu K. Gas-Kinetic Scheme for Unsteady Compressible Flow Simulations. von Karman Institute for Fluid Dynamics Lecture Series, Vol. 1998-03. von Karman Institute; 1998.

[17] Li XM, Leung RCK, So RMC. One-step aeroacoustics simulation using lattice Boltzmann method. AIAA J. 2006;44:78-89.

[18] Li XM. Computational Aeroacoustics Using Lattice Boltzmann Model. PhD thesis. Mechanical Engineering Department, Hong Kong Polytechnic University; 2006.

[19] Eucken A. Über das Wärmeleitvermögen, die spezifische Wärme und die innere Reibung der Gase. Physikalische Zeitschrift. 1913;14:324-332.

[20] Lallemand P, Luo LS. Theory of the lattice Boltzmann method: acoustic and thermal properties in two and three dimensions. Phys Rev E Stat Nonlin Soft Matter Phys. 2003 Sep;68(3 Pt 2):036706. doi: 10.1103/PhysRevE.68.036706. Epub 2003 Sep 23. PMID: 14524925.

[21] Chen Y, Ohashi H, Akiyama M. Thermal lattice Bhatnagar-Gross-Krook model without nonlinear deviations in macrodynamic equations. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics. 1994 Oct;50(4):2776-2783. doi: 10.1103/physreve.50.2776. PMID: 9962315.

[22] McNamara GR, Garcia AL, Alder BJ. Stabilization of thermal lattice Boltzmann models. J Statistical Phys. 1995;81:395-408.

[23] Teixeira C, Chen H, Freed DM. Multi-speed thermal lattice Boltzmann method stabilization via equilibrium under-relaxation. Comput Phys Commun. 2000;129:207-226.

[24] Shan X. Simulation of Rayleigh-Bénard convection using a lattice Boltzmann method. Phys Rev E. 1997;55:2780-2788.

[25] Leung RCK, Kam EWS, So RMC. Recovery of the transport coefficients in the Navier-Stokes equations from the modeled Boltzmann equation. AIAA J. 2007;45:737-739.

[26] Aristov VV. Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows. Kluwer Academic Publishers; 2001. Chapter 7.

[27] So RMC, Leung RCK, Kam EWS, Fu SC. Progress in the development of a new lattice Boltzmann method. Computers & Fluids. 2019;190:440-469.

[28] Fu SC, So RMC, Leung RCK. Modeled Boltzmann equation and its application to direct aeroacoustics simulation. AIAA J. 2008;46:1651-1662.

[29] So RMC, Leung RCK, Fu SC. Modeled Boltzmann equation and its application to shock-capturing simulation. AIAA J. 2008;46:3038-3048.

[30] So RMC, Fu SC, Leung RCK. Finite Difference Lattice Boltzmann Method for Compressible Thermal Fluids. AIAA J. 2010;48(6):1059-1071.

[31] Lele SK. Direct numerical simulations of compressible turbulent flows: fundamentals and applications. In: Hanifi A, et al., editors. Transition, Turbulence and Combustion Modeling. Kluwer Academic Publishers; 1998. pp. 424-429.

[32] Brenner H. Kinematics of volume transport. Physica A. 2005;349:11-59.

[33] Brenner H. Navier-Stokes revisited. Physica A. 2005;349:60-132.

[34] Greenshields CJ, Reese JM. The structure of shock waves as a test of Brenner’s modifications to the Navier-Stokes equations. J Fluid Mech. 2007;580:407-429.

[35] Kam EWS, So RMC, Leung RCK. Lattice Boltzmann method simulation of aeroacoustics and nonreflecting boundary conditions. AIAA J. 2007;45:1703-1712.

[36] Li XM, So RMC, Leung RCK. Propagation speed, internal energy and direct aeroacoustics simulation using lattice Boltzmann method. AIAA J. 2006;44:2896-2903.

[37] Leung RCK, Li XM, So RMC. Comparative Study of Nonreflecting Boundary Condition for One-Step Duct Aeroacoustics Simulation. AIAA J. 2006;44:664-667.

[38] Tam CKW, Webb JC. Dispersion-relation-preserving finite difference schemes for computational aeroacoustics. J Comput Phys. 1993;107:262-281.

[39] Gilbarg D, Paolucci D. The structure of shock waves in the continuum theory of fluids. J Rat. Mech. Analy. 1953;2:617-642.

[40] Ohwada T. Structure of normal shock waves: direct numerical analysis of the Boltzmann equation for Hard-sphere molecules. Phys Fluids A. 1993;5:217-234.

[41] Alsmeyer H. Density profiles in argon and nitrogen shock waves measured by the absorption of an electron beam. J Fluid Mech. 1976;74:497-513.

[42] Thomas LH. Note on Becker’s theory of the shock front. J Chem Phys. 1944;12:449-453.

[43] Chu CK. Kinetic-theoretic description of the formation of a shock wave. Phys Fluids. 1965;8:12-22.

[44] Bird GA. Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Clarendon Press; 1994.

[45] Yang JY, Huang JC. Rarefied flow computations using nonlinear model Boltzmann equations. J Comput Phys. 1995;120:323-339.

[46] Xu K, Tang L. Nonequilibrium Bhatnagar-Gross-Krook model for nitrogen shock structure. Phys Fluids. 2004;16:3824-3827.

[47] Xu K, Josyula E. Gas-kinetic scheme for rarefied flow simulation. Math Comput Simul. 2006;72:253-256.

[48] Weiss W. Continuous shock structure in extended thermodynamics. Phys Rev E. 1995;52:R5760-5763.

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