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Boundary conditions for the upwind finite difference lattice Boltzmann model: Evidence of slip velocity in micro-channel flow. (English) Zbl 1213.76150
Summary: We conduct a systematic study of the effect of various boundary conditions (bounce back and three versions of diffuse reflection) for the two-dimensional first-order upwind finite difference lattice Boltzmann model. Simulation of Couette flow in a micro-channel using the diffuse reflection boundary condition reveals the existence of a slip velocity that depends on the Knudsen number \(\varepsilon = \lambda/L\), where \(\lambda\) is the mean free path and \(L\) is the channel width. For walls moving in opposite directions with speeds \(\pm u_w\), the slip velocity satisfies \(u_{\text{slip}} = 2\varepsilon u_{\text{wall}}/(1 + 2\varepsilon)\). In the case of Poiseuille flow in a micro-channel, the slip velocity is found to depend on the lattice spacing \(\delta s\) and Knudsen number \(\varepsilon\) to both first and second order. The best results are obtained for diffuse reflection boundary conditions that allow thermal mixing at a wall located at half lattice spacing outside the boundary nodes.

MSC:
76M28 Particle methods and lattice-gas methods
Software:
HE-E1GODF
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[1] Rothman, D.H.; Zaleski, S., Lattice gas cellular automata: simple models of complex hydrodynamics, (1997), Cambridge University Press Cambridge · Zbl 0931.76004
[2] Chopard, B.; Droz, M., Cellular automata modeling of physical systems, (1999), Cambridge University Press Cambridge
[3] Wolf-Gladrow, D.A., Lattice gas cellular automata and lattice Boltzmann models, (2000), Springer Berlin · Zbl 0999.82054
[4] Succi, S., The lattice Boltzmann equation for fluid dynamics and beyond, (2001), Oxford University Press Oxford · Zbl 0990.76001
[5] Karniadakis, G.E.; Beskok, A., Micro flows: fundamentals and simulation, (2002), Springer New York · Zbl 0998.76002
[6] Nie, X.; Doolen, G.D.; Chen, S., Lattice Boltzmann simulations of fluid flows in MEMS, J. statist. phys., 107, 279-289, (2002) · Zbl 1007.82007
[7] Lim, C.Y.; Shu, C.; Niu, X.D.; Chew, Y.T., Application of lattice Boltzmann method to simulate microchannel flows, Phys. fluids, 14, 2299-2308, (2002) · Zbl 1185.76227
[8] Cao, N.; Chen, S.; Jin, S.; Martinez, D., Physical symmetry and lattice symmetry in the lattice Boltzmann method, Phys. rev. E, 55, R21-R24, (1997)
[9] Mei, R.; Shyy, W., On the finite difference-based lattice Boltzmann method in curvilinear coordinates, J. comput. phys., 143, 426-448, (1998) · Zbl 0934.76074
[10] Seta, T.; Kono, K.; Martinez, D.; Chen, S., Lattice Boltzmann scheme for simulating two-phase flows, JSME int. J. ser. B, 43, 305-313, (2000)
[11] Lee, T.H.; Lin, C.L., A characteristic Galerkin method for discrete Boltzmann equation, J. comput. phys., 171, 336-356, (2001) · Zbl 1017.76043
[12] Shi, W.; Shyy, W.; Mei, R., Finite-difference-based lattice Boltzmann method for inviscid compressible flows, Numer. heat transfer, part B, 40, 1-21, (2001)
[13] Sofonea, V.; Sekerka, R.F., Viscosity of finite difference lattice Boltzmann models, J. comput. phys., 184, 422-434, (2003) · Zbl 1062.76556
[14] Qian, Y.H.; D’Humières, D.; Lallemand, P., Lattice BGK models for Navier-Stokes equation, Europhys. lett., 17, 479-484, (1992) · Zbl 1116.76419
[15] He, X.; Luo, L.S., A priori derivation of the lattice Boltzmann equation, Phys. rev. E, 55, R6333-R6336, (1997)
[16] He, X.; Luo, L.S., Theory of the lattice Boltzmann method: from the Boltzmann equation to the lattice Boltzmann equation, Phys. rev. E, 56, 6811-6817, (1997)
[17] Luo, L.S., Theory of the lattice Boltzmann method: lattice Boltzmann models for nonideal gases, Phys. rev. E, 62, 4982-4996, (2000)
[18] Watari, M.; Tsutahara, M., Two-dimensional thermal model of the finite-difference lattice Boltzmann method with high spatial isotropy, Phys. rev. E, 67, 036306, (2003)
[19] Toro, E.F., Riemann solvers and numerical methods for fluid dynamics, (1999), Springer Berlin · Zbl 0923.76004
[20] Bhatnagar, P.; Gross, E.P.; Krook, M., A model for collision processes in gases i: small amplitude processes in charged and neutral one-component systems, Phys. rev., 94, 511-525, (1954) · Zbl 0055.23609
[21] Welander, P., On the temperature jump in a rarefied gas, Ark. fys., 7, 507-553, (1954) · Zbl 0057.23301
[22] Chen, S.; Doolen, G.D., Lattice Boltzmann method for fluid flows, Ann. rev. fluid mech., 30, 329-364, (1998) · Zbl 1398.76180
[23] Martys, N.S.; Shan, X.; Chen, H., Evaluation of the external force term in the discrete Boltzmann equation, Phys. rev. E, 58, 6855-6857, (1998) · Zbl 1073.81571
[24] Grad, H., On the kinetic theory of rarefied gases, Commun. pure appl. math., 2, 331-407, (1949) · Zbl 0037.13104
[25] Burgers, J.M., Flow equations for composite gases, (1969), Academic Press New York · Zbl 0214.25207
[26] Sofonea, V.; Sekerka, R.F., BGK models for diffusion in isothermal binary fluid systems, Physica A, 299, 494-520, (2001) · Zbl 0972.82079
[27] Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P., Numerical recipes in Fortran: the art of scientific computing, (1992), Cambridge University Press Cambridge · Zbl 0778.65002
[28] Hoffmann, K.A.; Chiang, S.T., Computational fluid dynamics, (1998), Engineering Education System Wichita, KS
[29] Vincenti, W.G.; Kruger, Ch.H., Introduction to physical gas dynamics, (1965), Wiley New York
[30] Cercignani, C., Rarefied gas dynamics, from basic concepts to actual calculations, (2000), Cambridge University Press Cambridge · Zbl 0961.76002
[31] Y. Sone, Theoretical and Numerical Analyses of the Boltzmann Equation - Theory and Analysis of Rarefied Gas Flows, Part I, Lecture Notes, Department of Aeronautics and Astronautics, Graduate School of Engineering, Kyoto University, 1998. Available from: <http://www.users.kudpc.kyoto-u.ac.jp/a50077>
[32] Gad-el-Haq, M., The fluid mechanics of microdevices, J. fluids eng. - trans. ASME, 121, 5-33, (1999)
[33] K. Aoki, Dynamics of Rarefied Gas Flows: Asymptotic and Numerical Analyses of the Boltzmann Equation, Proc. of the 39th AIAA Aerospace Sciences Meeting & Exhibit, 8-11 January 2001, Reno, Nevada, American Institute of Aeronautics and Astronautics Meeting Paper AIAA 2001-0874
[34] Mo, G.; Rosenberger, F., Molecular dynamics simulations of flow with binary diffusion in a two-dimensional channel with atomically rough walls, Phys. rev. A, 44, 4978-4985, (1991)
[35] Koplik, J.; Banavar, J.R., No slip condition for a mixture of two liquids, Phys. rev. lett., 80, 5125-5128, (1998)
[36] Brenner, H.; Ganesan, V., Molecular wall effects: are conditions at a boundary “boundary conditions”?, Phys. rev. E, 61, 6879-6897, (2000)
[37] R.W. Barber, D.R. Emerson, A numerical study of low Reynolds number slip flow in the hydrodynamic development region of circular and parallel plate ducts, Daresbury Laboratory Technical Report DL-TR-01-001, 2001
[38] Marques, W.; Kremer, G.M.; Sharipov, F.M., Couette flow with slip and jump boundary conditions, Continuum mech. thermodyn., 12, 379-386, (2000) · Zbl 0970.76086
[39] Maxwell, J.C., On stresses in rarified gases arising from inequalities of temperature, Philos. trans. roy. soc. London, 170, 231-256, (1879) · JFM 11.0777.01
[40] Inamuro, T.; Yoshino, M.; Ogino, F., A non-slip boundary condition for lattice Boltzmann simulations, Phys. fluids, 7, 2928-2930, (1995) · Zbl 1027.76631
[41] Cornubert, R.; d’Humières, D.; Levermore, D., A Knudsen layer theory for lattice gases, Physica D, 47, 241-259, (1991) · Zbl 0717.76100
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