×

zbMATH — the first resource for mathematics

A critical evaluation of force term in lattice Boltzmann method, natural convection problem. (English) Zbl 1406.76073
Summary: Few methods have been introduced and used in simulation fluid flows using lattice Boltzmann method (LBM) with external forces, such as buoyancy, surface tension, magnetic, etc. In some problems, the external force is constant, for instance gravitational force with constant density flows, while for other problems the force may vary spatially and/or temporally with non-zero gradients, such as gravitational force with variable density flows. For problems with the variable force term, adding force term to LBM may not be trivial. The paper evaluates mainly three different schemes of adding force term to LBM with BGK method. In this work, natural convection in a closed and an open ended cavities were used as a test platform. The results for the differentially heated cavity are introduced first. For the open cavity, the vertical left hand wall of the cavity is heated and opposing side is opened to the ambient, with other connecting boundaries are assumed to be adiabatic. Prior to the solution, the boundary conditions at the opening are unknown. The results of predictions using LBM are compared with results predicted by using finite volume method (FVM). The results are presented for \(Ra=10^{6}\) and for \(Pr=0.71\). It is found that most methods suggested in the literature produces similar results, despite that some authors claim that their scheme is more accurate than the other schemes.

MSC:
76M28 Particle methods and lattice-gas methods
76R10 Free convection
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Mohamad, A. A.: Applied lattice Boltzmann method for transport phenomena, momentum, heat and mass transfer, (2007)
[2] Succi, S.: The lattice Boltzmann equation for fluid dynamics and beyond, (2001) · Zbl 0990.76001
[3] L.-S. Luo, Lattice-Gas Automata and Lattice Boltzmann Equations for Two-Dimensional Hydrodynamics, Ph.D. thesis, Georgia Institute of Technology, 1993.
[4] Frisch, U.; D’humieres, D.; Hasslacher, B.; Lallemand, P.; Pomeau, Y.; Rivet, J. P.: Lattice gas hydrodynamics in two and three dimensions, Compl. syst. 1, No. 4, 633-647 (1987) · Zbl 0662.76101
[5] Shan, X.; Chen, H.: Simulation of nonideal gases and gas – liquid phase transitions by the lattice Boltzmann equation, Phys. rev. E 49, No. 4, 2941-2948 (1994)
[6] Luo, L. -S.: Unified theory of lattice Boltzmann models for non-ideal gases, Phys. rev. Lett. 81, No. 8, 1618-1621 (1998)
[7] Buick, J. M.; Greated, C. A.: Gravity in a lattice Boltzmann model, Phys. rev. E 61, No. 5, 5307-5320 (2000)
[8] Guo, Z.; Zheng, C.; Shi, B.: Discrete lattice effects on the forcing term in the lattice Boltzmann method, Phys. rev. E 65, No. 046308, 1-6 (2002) · Zbl 1244.76102
[9] Martys, N. S.; Shan, X.; Chen, H.: Evaluation of the external force term in the discrete Boltzmann equation, Phys. rev. E 58, No. 5, 6855-6857 (1998)
[10] Luo, L. -S.: Theory of the lattice Boltzmann method: lattice Boltzmann models for nonideal gases, Phys. rev. E 62, No. 4, 4982-4996 (2000)
[11] Ladd, A. J. C.; Verberg, R.: Lattice-Boltzmann simulations of particle – fluid suspensions, J. stat. Phys. 104, No. 5/6, 1191-1251 (2001) · Zbl 1046.76037
[12] Mohamad, A. A.; El-Ganaoui, M.; Bennacer, R.: Lattice Boltzmann simulation of natural convection in an open ended cavity, Int. J. Therm. sci. 48, No. 10, 1870-1875 (2009)
[13] He, X.; Chen, S.; Doolen, G.: A novel thermal model for the lattice Boltzmann method in incompressible limit, J. comput. Phys. 146, 282-300 (1998) · Zbl 0919.76068
[14] D’orazio, A.; Corcione, M.; Celata, G. P.: Application to natural convection enclosed flows of a lattice Boltzmann BGK model coupled with a general purpose thermal boundary condition, Int. J. Therm. sci. 43, No. 6, 531-630 (2004)
[15] Dixit, H. N.; Babu, V.: Simulation of high Rayleigh natural convection in a square cavity using the lattice Boltzmann method, Int. J. Heat mass transfer 49, 727-739 (2006) · Zbl 1189.76529
[16] Zou, Q.; He, X.: On pressure and velocity boundary conditions for the lattice botlzmann BGK model, Phys. fluids 9, No. 6, 1591-1598 (1997) · Zbl 1185.76873
[17] Hortmann, M.; Peric, M.; Scheuerer, G.: Finite volume multigrid prediction of laminar natural convection: bench-mark solutions, Int. J. Numer. meth. Fluids 11, 189-207 (1990) · Zbl 0711.76072
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.