Implementation of diffuse reflection boundary conditions in a thermal lattice Boltzmann model with flux limiters. (English) Zbl 1175.82048

Summary: We discuss three new implementation versions of diffuse reflection boundary conditions in a thermal lattice Boltzmann model. Their accuracy is investigated in the case of Couette flow by considering the slip regime. The best results are recovered with versions 2 and 3, which rely on outgoing fluxes to express the particle distribution functions in the ghost nodes outside the flow domain. Version 2 is found to be more economical since it involves no interpolation procedure. This version was thereafter used to investigate the temperature profile in Couette flow for various values of Prandtl number, as well as the capability of the thermal LB model to capture the Knudsen minimum in Poiseuille flow.


82C40 Kinetic theory of gases in time-dependent statistical mechanics
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
76M28 Particle methods and lattice-gas methods
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[1] Karniadakis, G.E.; Beskok, A.; Aluru, N., Microflows and nanoflows: fundamentals and simulation, (2005), Springer Berlin · Zbl 1115.76003
[2] Shen, C., Rarefied gas dynamics: fundamentals, simulations and micro flows, (2005), Springer Berlin
[3] Tabeling, P., Introduction to microfluidics, (2005), Oxford University Press Oxford
[4] M. Gad-el-Hak (Ed.), MEMS Handbook, Introduction and Fundamentals, vol. I, CRC Taylor and Francis, Boca Raton, 2006.
[5] Gad-el-Haq, M., The fluid mechanics of microdevices, Journal of fluids engineering, 121, 5, (1999)
[6] Reese, J.M.; Gallis, M.A.; Lockerby, D.A., New directions in fluid dynamics: non-equilibrium aerodynamic and microsystem flows, Philosophical transactions of the royal society series A, 361, 2967, (2003) · Zbl 1068.76044
[7] Benzi, R.; Succi, S.; Vergassola, M., The lattice Boltzmann equation: theory and applications, Physics reports, 222, 145, (1992)
[8] Rothman, D.H.; Zaleski, S., Lattice gas cellular automata: simple models of complex hydrodynamics, (1997), Cambridge University Press Cambridge · Zbl 0931.76004
[9] Chopard, B.; Droz, M., Cellular automata modeling of physical systems, (1998), Cambridge University Press Cambridge · Zbl 0973.82033
[10] Chen, S.; Doolen, G.D., Lattice Boltzmann method for fluid flow, Annual review of fluid mechanics, 30, 329, (1998) · Zbl 1398.76180
[11] Wolf-Gladrow, D.A., Lattice gas cellular automata and lattice Boltzmann models, (2000), Springer Berlin · Zbl 0999.82054
[12] Succi, S., The lattice Boltzmann equation for fluid dynamics and beyond, (2001), Clarendon Press Oxford · Zbl 0990.76001
[13] Sukop, M.C.; Thorne, D.T., Lattice Boltzmann modeling: an introduction for geoscientists and engineers, (2006), Springer Berlin
[14] Nie, X.; Doolen, G.D.; Chen, S., Lattice Boltzmann simulation of fluid flows in MEMS, Journal of statistical physics, 107, 279, (2002) · Zbl 1007.82007
[15] Lim, C.Y.; Shu, C.; Niu, X.D.; Chew, Y.T., Application of lattice Boltzmann method to simulate microchannel flows, Physics of fluids, 14, 2299, (2002) · Zbl 1185.76227
[16] Succi, S., Mesoscopic modeling of slip motion at fluid-solid interfaces with heterogeneous catalysis, Physical review letters, 89, 064502, (2002)
[17] Ansumali, S.; Karlin, I.V., Kinetic boundary conditions in the lattice Boltzmann method, Physical review E, 66, 026311, (2002)
[18] Li, B.; Kwok, D., Discrete Boltzmann equation for microfluidics, Physical review letters, 90, 124502, (2003)
[19] Niu, X.D.; Shu, C.; Chew, Y.T., A lattice Boltzmann BGK model for simulation of micro flows, Europhysics letters, 67, 600, (2004)
[20] Lee, T.; Lin, C., Rarefaction and compressibility effects of the lattice-Boltzmann-equation method in a gas microchannel, Physical review E, 71, 046706, (2005)
[21] Zhang, Y.; Qin, R.; Emerson, D.R., Lattice Boltzmann simulation of rarefied gas flows in microchannels, Physical review E, 71, 047702, (2005)
[22] Sofonea, V.; Sekerka, R.F., Boundary conditions for the upwind finite difference lattice Boltzmann model: evidence of slip velocity in micro-channel flow, Journal of computational physics, 207, 639, (2005) · Zbl 1213.76150
[23] Sbragaglia, M.; Succi, S., Analytical calculation of slip flow in lattice Boltzmann models with kinetic boundary conditions, Physics of fluids, 17, 093602, (2005) · Zbl 1187.76469
[24] Ansumali, S.; Karlin, I., Consistent lattice Boltzmann method, Physical review letters, 97, 260605, (2005)
[25] Toschi, F.; Succi, S., Lattice Boltzmann method at finite Knudsen numbers, Europhysics letters, 69, 549, (2005)
[26] Sofonea, V.; Sekerka, R.F., Diffuse-reflection boundary conditions for a thermal lattice Boltzmann model in two dimensions: evidence of temperature jump and slip velocity in microchannels, Physical review E, 71, 066709, (2005)
[27] Guo, Z.; Zhao, T.S.; Shi, Y., Physical symmetry, spatial accuracy and relaxation time of the lattice Boltzmann equation for microgas flows, Journal of applied physics, 99, 074903, (2006)
[28] Benzi, R.; Biferale, L.; Sbragaglia, M.; Succi, S.; Toschi, F., Mesoscopic two-phase model for describing apparent slip in micro-channel flows, Europhysics letters, 74, 651, (2006) · Zbl 1116.76463
[29] Sofonea, V., Lattice Boltzmann approach to thermal transpiration, Physical review E, 74, 056705, (2006)
[30] Sofonea, V., Finite-difference lattice Boltzmann approach to pressure-driven microchannel flow with variable temperature, Europhysics letters, 76, 829, (2006)
[31] Ansumali, S.; Karlin, I.V.; Arcidiacono, S.; Abbas, A.; Prasianakis, N., Hydrodynamics beyond navier – stokes: exact solution to the lattice Boltzmann hierarchy, Physical review letters, 98, 124502, (2007)
[32] Karlin, I.V.; Ansumali, S., Renormalization of the lattice Boltzmann hierarchy, Physical review E, 76, 025701, (2007), R
[33] Zheng, L.; Shi, B.C.; Cai, Z.H., Lattice Boltzmann method for simulating the temperature jump and velocity slip in microchannels, Communications in computational physics, 2, 1125, (2007)
[34] Kim, S.H.; Pitsch, H.; Boyd, I.D., Slip velocity and Knudsen layer in the lattice Boltzmann method for microscale flows, Physical review E, 77, 026704, (2008)
[35] Zheng, L.; Guo, Z.L.; Shi, B.C., Discrete effects on thermal boundary conditions for the thermal latttice Boltzmann method in simulating microscale gas flows, Europhysics letters, 82, 44002, (2008)
[36] Maxwell, J.C., On stresses in rarified gases arising from inequalities of temperature, Philosophical transactions of the royal society of London, 170, 231, (1879) · JFM 11.0777.01
[37] Smoluchowski, M., Über Wärmeleitung in verdünnten gasen, Annalen der physik und chemie, 64, 101, (1898) · JFM 29.0787.01
[38] Watari, M.; Tsutahara, M., Two-dimensional thermal model of the finite-difference lattice Boltzmann method with high spatial isotropy, Physical review E, 67, 036306, (2003)
[39] R.J. LeVeque, Numerical Methods for Conservation Laws, Birkhäuser, Basel, 1992. · Zbl 0847.65053
[40] Toro, E.F., Riemann solvers and numerical methods for fluid dynamics, (1999), Springer Berlin · Zbl 0923.76004
[41] Cristea, A.; Sofonea, V., Two component lattice Boltzmann model with flux limiters, Central European journal of physics, 2, 382, (2004)
[42] Sofonea, V.; Lamura, A.; Gonnella, G.; Cristea, A., Finite-difference lattice Boltzmann model with flux limiters for liquid – vapor system, Physical review E, 70, 046702, (2004)
[43] Gonnella, G.; Lamura, A.; Sofonea, V., Lattice Boltzmann simulation of thermal nonideal fluids, Physical review E, 76, 036703, (2007)
[44] He, X.; Chen, S.; Doolen, G.D., A novel thermal model for the lattice Boltzmann method in incompressible limit, Journal of computational physics, 146, 282, (1998) · Zbl 0919.76068
[45] Peng, Y.; Shu, C.; Chew, Y.T., Simplified thermal lattice Boltzmann model for incompressible thermal flows, Physical review E, 68, 026701, (2003)
[46] Peng, Y.; Shu, C.; Chew, Y.T., A 3D incompressible thermal lattice Boltzmann model and its application to simulate natural convection in a cubic cavity, Journal of computational physics, 193, 260, (2004) · Zbl 1047.76107
[47] Shi, Y.; Zhao, T.S.; Guo, Z.L., Thermal lattice bhatnagar – gross – krook model for flows with viscous heat dissipation in the incompressible limit, Physical review E, 70, 066310, (2004)
[48] Chen, H.; Shan, X., Fundamental conditions for N-th-order accurate lattice Boltzmann models, Physica D, 2003, (2008) · Zbl 1143.76549
[49] Cao, N.; Chen, S.; Jin, S.; Martinez, D., Physical symmetry and lattice symmetry in the lattice Boltzmann method, Physical review E, 55, R21, (1997)
[50] Chen, Y.; Ohashi, H.; Akiyama, M., Prandtl number of lattice bhatnagar – gross – krook fluid, Physics of fluids, 7, 2280, (1995) · Zbl 1025.76544
[51] Soe, M.; Vahala, G.; Pavlo, P.; Vahala, L.; Chen, H.D., Thermal lattice Boltzmann simulations of variable Prandtl number turbulent flows, Physical review E, 57, 4227, (1998)
[52] Prasianakis, N.I.; Karlin, I.V., Lattice Boltzmann method for thermal flow simulation on standard lattices, Physical review E, 76, 016702, (2007)
[53] Ansumali, S.; Arcidiacono, S.; Chikatamarla, S.S.; Prasianakis, N.I.; Gorban, A.N.; Karlin, I.V., Quasi-equilibrium lattice Boltzmann method, European physical journal B, 56, 135, (2007)
[54] Shan, X.W.; Chen, H., A general multiple-relaxation time Boltzmann collision model, International journal of modern physica C, 18, 635, (2007) · Zbl 1144.82341
[55] Prasianakis, N.I.; Boulouchos, K.B., Lattice Boltzmann method for simulation of weakly compressible flows at arbitrary Prandtl number, International journal of modern physica C, 18, 602, (2007) · Zbl 1140.76435
[56] Li, Q.; He, Y.L.; Wang, Y.; Tao, W.Q., Coupled double-distribution-function lattice Boltzmann method for the compressible navier – stokes equations, Physical review E, 76, 056705, (2007)
[57] Zheng, L.; Shi, B.C.; Guo, Z.L., Multiple-relaxation-time model for the correct thermohydrodynamic equations, Physical review E, 78, 026705, (2008)
[58] Knudsen, M., Die gesetze der molecular stromung und die inneren reibungstromung der gase durch rohren, Annalen der physik, 29, 75, (1909) · JFM 40.0825.02
[59] Zhou, Y.; Zhang, R.; Staroselsky, I.; Chen, H.; Kim, W.T.; Jhon, M.S., Simulation of micro- and nano-scale flows via the lattice Boltzmann method, Physica A, 362, 68, (2006)
[60] Zhang, R.; Shan, X.; Chen, H., Efficient kinetic method for fluid simulation beyond the navier – stokes equation, Physical review E, 74, 046703, (2006)
[61] Ansumali, S.; Karlin, I.V.; Frouzakis, C.E.; Boulouchos, K.B., Entropic lattice Boltzmann method for microflows, Physica A, 359, 289, (2006)
[62] Cercignani, C., Theory and application of the Boltzmann equation, (1975), Scottish Academic Press Edinburgh · Zbl 0403.76065
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