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Assessing moment-based boundary conditions for the lattice Boltzmann equation: a study of dipole-wall collisions. (English) Zbl 1410.76368

Summary: The accuracy of moment-based boundary conditions for no slip walls in lattice Boltzmann simulations is examined numerically by using the dipole-wall collision benchmark test for both normal and oblique cases. In the normal case the dipole hits the wall perpendicularly while in the oblique case the dipole hits the wall at an angle of \(30^\circ\) to the horizontal. Boundary conditions are specified precisely at grid points by imposing constraints upon hydrodynamic moments only. These constraints are then translated into conditions for the unknown lattice Boltzmann distribution functions at boundaries. The two relaxation time (TRT) model is used with a judiciously chosen product of the two relaxation times. Stable results are achieved for higher Reynolds number up to 10,000 for the normal collision and up to 7500 for the oblique case. Excellent agreement with benchmark data is observed and the local boundary condition implementation is shown to be second order accurate.

MSC:

76M28 Particle methods and lattice-gas methods
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
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[1] Orlandi, P., Vortex dipole rebound from a wall, Phys Fluids A (1989-1993), 2, 1429-1436, (1990)
[2] Jamart, B.; Nihoul, J., Mesoscale/synoptic coherent structures in geophysical turbulence, (1989), Elsevier: Elsevier New York
[3] Coutsias, E.; Lynov, J.-P., Fundamental interactions of vortical structures with boundary layers in two-dimensional flows, Physica D, 51, 482-497, (1991) · Zbl 0735.76023
[4] Clercx, H.; van Heijst, G., Dissipation of kinetic energy in two-dimensional bounded flows, Phys Rev E, 65, 066305, (2002)
[5] Clercx, H.; Bruneau, C.-H., The normal and oblique collision of a dipole with a no-slip boundary, Comput Fluids, 35, 245-279, (2006) · Zbl 1160.76328
[6] Wells, M.; Clercx, H.; Van Heijst, G., Vortices in oscillating spin-up, J Fluid Mech, 573, 339-369, (2007) · Zbl 1108.76325
[7] Kramer, W., Dispersion of tracers in two-dimensional bounded turbulence, (2007), Eindhoven University of Technology, Eindhoven, Netherlands, Ph.D. thesis
[8] Cieślik, A.; Akkermans, R.; Kamp, L.; Clercx, H.; Van Heijst, G., Dipole-wall collision in a shallow fluid, Eur J Mech B/Fluids, 28, 397-404, (2009) · Zbl 1167.76301
[9] Guzmán, J.; Kamp, L.; Van Heijst, G., Vortex dipole collision with a sliding wall, Fluid Dyn Res, 45, 045501, (2013) · Zbl 1469.76038
[10] Latt, J.; Chopard, B., A benchmark case for lattice Boltzmann: turbulent dipole-wall collision, Int J Mod Phys C, 18, 619-626, (2007) · Zbl 1388.76300
[11] Hardy, J.; De Pazzis, O.; Pomeau, Y., Molecular dynamics of a classical lattice gas: transport properties and time correlation functions, Phys Rev A, 13, 1949, (1976)
[12] Luo, L., Lattice-gas automata and lattice Boltzmann equations for two-dimensional hydrodynamics, (1993), Georgiargia Institute of Technology, Ph.D. thesis
[13] Luo, L., The lattice-gas and lattice Boltzmann methods: past, present, and future, (2000)
[14] Wolf-Gladrow, D., Lattice-gas cellular automata and lattice Boltzmann models: an introduction, (2000), Springer Science & Business Media · Zbl 0999.82054
[15] Guo, Z.; Shu, C., Lattice Boltzmann method and its applications in engineering, (2013), World Scientific · Zbl 1278.76001
[16] He, X.; Zou, Q.; Luo, L.; Dembo, M., Analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice Boltzmann BGK model, J Stat Phys, 87, 115-136, (1997) · Zbl 0937.82043
[17] He, X.; Luo, L.-S., Theory of the lattice Boltzmann method: from the Boltzmann equation to the lattice Boltzmann equation, Phys Rev E, 56, 6, 6811, (1997)
[18] Shan, X.; Yuan, X.; Chen, H., Kinetic theory representation of hydrodynamics: a way beyond the Navier-Stokes equation, J Fluid Mech, 550, 413-441, (2006) · Zbl 1097.76061
[19] He, X.; Luo, L., A priori derivation of the lattice Boltzmann equation, Phys Rev E, 55, R6333, (1997)
[20] Dellar, P., Incompressible limits of lattice Boltzmann equations using multiple relaxation times, J Comput Phys, 190, 351-370, (2003) · Zbl 1076.76063
[21] Allen, R.; Reis, T., Moment-based boundary conditions for lattice Boltzmann simulations of natural convection in cavities, Prog Comp Fluid Dyn, 16, 4, 216-231, (2016)
[22] D’Humières, D., Generalized lattice-Boltzmann equations, Prog Astronaut Aeronaut, 450-458, (1992)
[23] Lallemand, P.; Luo, L., Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability, Phys Rev E, 61, 6546, (2000)
[24] Ladd, A., Numerical simulations of particulate suspensions via a discretized Boltzmann equation. part 1. theoretical foundation, J Fluid Mech, 271, 285-309, (1994) · Zbl 0815.76085
[25] Reis, T.; Phillips, T., Alternative approach to the solution of the dispersion relation for a generalized lattice Boltzmann equation, Phys Rev E, 77, 2, 026702, (2008)
[26] Ginzburg, I., Lattice Boltzmann modeling with discontinuous collision components: Hydrodynamic and advection-diffusion equations, J Stat Phys, 126, 1, 157-206, (2007) · Zbl 1198.82039
[27] Ginzburg, I., Equilibrium-type and link-type lattice Boltzmann models for generic advection and anisotropic-dispersion equation, Adv Water Resour, 28, 11, 1171-1195, (2005)
[28] Ginzburg, I.; d’Humières, D.; Kuzmin, A., Optimal stability of advection-diffusion lattice Boltzmann models with two relaxation times for positive/negative equilibrium, J Stat Phys, 139, 6, 1090-1143, (2010) · Zbl 1205.82049
[29] Chen, S.; Doolen, G., Lattice Boltzmann method for fluid flows, Annu Rev Fluid Mech, 30, 329-364, (1998) · Zbl 1398.76180
[30] Ginzbourg, I.; Adler, P., Boundary flow condition analysis for the three-dimensional lattice Boltzmann model, J Phys II France, 4, 191-214, (1994)
[31] Ginzburg, I.; d’Humiereśs, D., Multireflection boundary conditions for lattice Boltzmann models, Phys Rev E, 68, 066614, (2003)
[32] Zou, Q.; He, X., On pressure and velocity boundary conditions for the lattice Boltzmann BGK model, Phys Fluids, 9, 1591-1598, (1997) · Zbl 1185.76873
[33] Filippova, O.; Hänel, D., Grid refinement for lattice-bgk models, J Comp Phys, 147, 1, 219-228, (1998) · Zbl 0917.76061
[34] Dupuis, A.; Chopard, B., Theory and applications of an alternative lattice Boltzmann grid refinement algorithm, Phys Rev E, 67, 6, 066707, (2003)
[35] Bouzidi, M.; Firdaouss, M.; Lallemand, P., Momentum transfer of a Boltzmann-lattice fluid with boundaries, Phys Fluids, 13, 11, 3452-3459, (2001) · Zbl 1184.76068
[36] Yu, D.; Mei, R.; Shyy, W., A unified boundary treatment in lattice Boltzmann method, 41st aerospace sciences meeting and exhibit, 953, (2003)
[37] Zhao W., Yong W.. A family of single-node second-order boundary schemes for the lattice Boltzmann method. arXiv preprintarXiv:171208288; Zhao W., Yong W.. A family of single-node second-order boundary schemes for the lattice Boltzmann method. arXiv preprintarXiv:171208288
[38] Li, Z.; Favier, J.; D’Ortona, U.; Poncet, S., An immersed boundary lattice Boltzmann method for single-and multi-component fluid flows, J Comp Phys, 304, 424-440, (2016) · Zbl 1349.76708
[39] Noble, D.; Chen, S.; Georgiadis, G.; Buckius, R. O., A consistent hydrodynamic boundary condition for the lattice Boltzmann method, Phys Fluids, 7, 1, 203-209, (1995) · Zbl 0846.76086
[40] Ginzbourg, I.; d’Humières, D., Local second-order boundary methods for lattice Boltzmann models, J Stat Phys, 84, 927-971, (1996) · Zbl 1081.82617
[41] Chang, C.; Liu, C.-H.; Lin, C., Boundary conditions for lattice Boltzmann simulations with complex geometry flows, Comput Math Appl, 58, 940-949, (2009) · Zbl 1189.76402
[42] Ho, C.; Chang, C.; Lin, K.-H.; Lin, C., Consistent boundary conditions for 2D and 3D lattice Boltzmann simulations, CMES, 44, 137-155, (2009) · Zbl 1357.76068
[43] Schlaffer, M., Non-reflecting boundary conditions for the lattice Boltzmann method, (2009), Technischen Universität München, Ph.D. thesis
[44] Reis, T.; Dellar, P., Moment-based formulation of Navier-Maxwell slip boundary conditions for lattice Boltzmann simulations of rarefied flows in microchannels, Phys Fluids, (2012)
[45] Bennett, S., A lattice Boltzmann model for diffusion of binary gas mixtures, (2010), University of Cambridge, Ph.D. thesis
[46] Dellar, P., Moment-based boundary conditions for lattice Boltzmann magnetohydrodynamics, Numerical mathematics and advanced applications 2011, 83-90, (2013), Springer · Zbl 1273.76340
[47] Hantsch, A.; Reis, T.; Gross, U., Moment method boundary conditions for multiphase lattice Boltzmann simulations with partially-wetted walls, J Comput Multiphase Flows, 7, 1-14, (2015)
[48] Mohammed, S.; Reis, T., Using the lid-driven cavity flow to validate moment-based boundary conditions for the lattice Boltzmann equation, Arch Mech Eng, 64, 1, 57-74, (2017)
[49] He, X.; Shan, X.; Doolen, G., Discrete Boltzmann equation model for nonideal gases, Phys Rev E, 57, R13, (1998)
[50] Latt J., Chopard B.. Lattice Boltzmann method with regularized non-equilibrium distribution functions. arXiv preprintphysics/0506157; Latt J., Chopard B.. Lattice Boltzmann method with regularized non-equilibrium distribution functions. arXiv preprintphysics/0506157 · Zbl 1102.76056
[51] d’Humiereśs, D.; Ginzburg, I., Viscosity independent numerical errors for lattice Boltzmann models: from recurrence equations to magic collision numbers, Comput Math Appl, 58, 823-840, (2009) · Zbl 1189.76405
[52] Krastins I., Kao A., Pericleous K., Reis T.. Moment-based boundary conditions for three dimensional lattice Boltzmann simulations; Submitted to Int. J. Num. Meth. Fluids.; Krastins I., Kao A., Pericleous K., Reis T.. Moment-based boundary conditions for three dimensional lattice Boltzmann simulations; Submitted to Int. J. Num. Meth. Fluids.
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