## THERMAL SCIENCE

International Scientific Journal

### DIFFERENTIATED HEATED LID DRIVEN CAVITY INTERACTING WITH TUBE: A LATTICE BOLTZMANN STUDY

**ABSTRACT**

The multiple-relaxation-time (MRT) lattice-Boltzmann method is implemented to investigate combined natural and forced convection occurring in a two-dimensional square cavity. The top wall slides to the right at constant speed, while the other three remain stationary. The solution is performed for a left vertical wall at a constant temperature, which is higher than of the right wall. This yields a “cooperating” case, in which dynamic and buoyancy forces are added together. The enclosure is filled with air and contains a heat conducting circular cylinder, which is placed at various positions. The double distribution model used in lattice Boltzmann methods has been adopted to simulate the hydrodynamic and thermal fields, with the D2Q9 and D2Q5 lattices selected to perform the corresponding computations. Simulations have been conducted over a wide range of Rayleigh (Ra) and Reynolds (Re) numbers, and the features of dynamic and thermal fields are presented for the spectra of this mixed convection phenomenon. The flow and heat transfer characteristics of the cylinder position are described and analyzed in terms of the average Nusselt number (Nu). The computed results show the influence of the cylinder on the corresponding heat transfer in the enclosure. It has been found that the power (i.e. shear stress) needed to lid the upper surface will depend on the governing parameters.

**KEYWORDS**

PAPER SUBMITTED: 2016-04-29

PAPER REVISED: 2016-05-30

PAPER ACCEPTED: 2016-06-29

PUBLISHED ONLINE: 2016-10-01

**THERMAL SCIENCE** YEAR

**2017**, VOLUME

**21**, ISSUE

**1**, PAGES [89 - 104]

- Guo Y., Bennacer R., Shen S., Ameziani D. E., and Bouzidi M., Simulation of mixed convection in slender rectangular cavity with lattice Boltzmann method, International Journal of Numerical Methods for Heat & Fluid Flow, vol. 20, (2010), 1, pp.130-148.
- McNamara, G. R. and Zanetti, G., Use of Boltzmann equation to simulate lattice-gas automata. Phys. Rev. L. 61, (1988), 20, :pp.2332-2335.
- Higuera F., Succi S., and Benzi R. Lattice gas dynamics with enhanced collisions Europhys Lett., 9, (1989),4, pp.345-349.
- Qian, Y., d'Humieres, D., and Lallemand, P., Lattice BGK Models for Navier-Stokes Equation. Europhys. Lett., 17 (1954),3, pp.479-484.
- Bhatnagar, P., Gross, E., and Krook. M., A model for collision processes in gases: small amplitude processes in charged and neutral one-component sys. Phys. Rev., 94 (1954),3, pp. 511-525.
- Shan, X. and He, X., Discretization of the Velocity Space in the Solution of the Boltzmann Equation. Phys. Rev. Lett., 80 (1998),1, pp.65-68.
- Abe, T., Derivation of the lattice Boltzmann method by means of the discrete ordinate method for the Boltzmann equation. J. Comp. Phys., 131, (1997),1, pp.241-246.
- He, X., and Luo, L.-S., Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation. Phys. Rev. E, 56 (1997), 6, pp.6811-6817.
- Von Neumann, J., Theory of Self-Reproducing Automata (edited and completed by Burks, A.), University of Illinois Press, 1966.
- Frisch, U., Hasslacher, B., and Pomeau, Y., Lattice-gas automata for the Navier-Stokes equation. Phys. Rev. Letter,56 (1986),14,pp.1505-1508.
- d'Humieres, D., Lallemand, P., and Frisch, U., Lattice gas models for 3D hydrodynamics. Europhy. Lett. 2 (1986), 4,pp. 291-297.
- Abe, T., Derivation of the lattice Boltzmann method by means of the discrete ordinate method for the Boltzmann equation. J. Comp. Phys., 131, (1938), pp.241-246.
- Higuera, F. and Jimenez, J., Boltzmann approach to lattice gas simulations. Europhys. Lett., 9 (1989a), 7, pp.663-668,
- Bhatnagar, P., Gross, E., and Krook. M., A model for collision processes in gases: small amplitude processes in charged and neutral one-component system. Phys. Rev., 94 (1954), 3, pp.511-525.
- D'Humieres, D., "Generalized lattice Boltzmann equation," in Rarefied Gas Dynamics: Theory and Simulations, Progress in Astronautics and Aeronautics, vol. 159, edited by Shizgal, B. D. and Weaver, D. P., AIAA, Washington, D.C., (1992), pp.450-458.
- Lallemand, P. and Luo, L.-S., "Theory of the lattice Boltzmann method: dispersion, dissipation, isotropy, Galilean invariance, and stability," Phys. Rev. E., vol. 61,(2000), 6,pp.6546-6562.
- Succi, S. The Lattice Boltzmann Equation for Fluid Dynamics and Beyond, Oxford University Press, 2001.
- Wolf-Gladrow, D. A. Lattice-Gas Cellular Automata and Lattice Boltzmann Models: an Introduction, Springer, Berlin, 2000.
- Chen, S., Chen, H., Martinez, D., and Matthaeus, W., Lattice Boltzmann model for simulation of magnetohydrodynamics, Phys. Rev. Lett. 67 (1991), 27, pp.3776-3779.
- Koelman, J. M. V. A., A simple Boltzmann scheme for Navier-Stokes fluid flow, Europhys. Lett. 15, (1991),6, pp.603-607.
- Lallemand, P. and Luo L. S., Theory of the lattice Boltzmann method: acoustic and thermal properties in two and three dimensions, Phys. Rev. E, 68 (2003), 3, pp. 036706.1-036706.25.
- d'Humières, D., Generalized lattice Boltzmann equations. in: B. . Shizgal, D. P. Weaver (Eds.), Rarefied Gas Dynamics: Theory and Simulations. Prog. Astro. Aero., AIAA, 159 (1992), pp. 450-458.
- Lallemand, P. and Luo L. S., Lattice Boltzmann method for moving boundaries, J. Comput. Phys., 184 (2003), 2,pp.406-421.
- Dubois, F., Une introduction au schéma de Boltzmann sur réseau, ESAIM: Proceedings 18, (2007), pp.181-215
- Moussaoui, M. A., Mezrah, A., and Naji, H., A computation of flow and heat transfer past three heated cylinders in a vee shape by a double distribution MRT thermal lattice Boltzmann model, International Journal of Thermal Sciences, 50 (2011),8, pp.1532-1542.
- Dubois, F., Une introduction au schéma de Boltzmann sur réseau, ESAIM: Proceedings 18, (2007), pp.181-215
- Dubois, F. and Lallemand, P., Towards higher order lattice Boltzmann schemes, J. Stat. Mech. (2009),6, pp. 1-40.
- Filippova, O. and Hänel, D., Grid refinement for lattice-BGK models J. Comput. Phys. 147 (1998), 1, pp.219-228.
- Mei, R., Yu, D., Shyy, W., and Luo L. Sh., Force evaluation in the lattice Boltzmann method involving curved geometry, Phys. Rev. E 65 (2002),4, pp.1-14.
- Guo, Z. L., Zheng, Ch., and Shi, B. C., An extrapolation method for boundary conditions in lattice Boltzmann method, Phys. Fluids 14 (2002),6, pp.2007-2010.
- Bouzidi, M., Firdaouss, M., and Lallemand, P., Momentum transfer of a Boltzmann lattice fluid with boundaries. Phys. Fluids 13, (2001), pp.3452-3459.
- Ameziani, D. E., Guo, Y., Bennacer, R., El Ganaoui, M., and Bouzidi, M., Competition between lid-driven and natural convection in square cavities investigated with a lattice Boltzmann method, Computational Thermal Sciences, 2 (2010),3, pp.269-282.