Simulation of Natural Convection by Multirelaxation Time Lattice Boltzmann Method in a Triangular Enclosure

Abstract

The natural convection of incompressible flow confined within an enclosed right-angled triangular and isosceles cavity was investigated numerically using the multirelaxation time lattice Boltzmann method (MRT-LBM). According to the left and inclined walls thermal boundary conditions, two cases were considered in this study. In the first case, the inclined side of the enclosure was adiabatic, and the horizontal wall was heated, while the left one was kept at a cold temperature. However, the states of the left and inclined walls were interchanged in the second case. As the flow is only transported under the convection force, this study was carried out for the Rayleigh number ranging from Ra=103 to 106. The effects of the Rayleigh number on velocity and temperature profiles, streamlines, isotherms, and average Nusselt number were investigated. The position of cold and adiabatic walls had a great effect on the results. The results obtained are in good agreement with those of the literature and show the robustness of the MRT-LBM approach. In both cases, the heat-transfer rate increases with the increase in the Rayleigh number.

Cavity inclination effect on convection flow in a triangular geometry

“…Note that the particularity of the MRT‐LBM model is that the streaming process take place in the discrete velocity $${{\bf{c}}}_{i}$$ space of microscopic scale, while, the collisions events occur in the macroscopic space of moments $$({\bf{m}},{\bf{n}})$$ corresponding to $$({\bf{f}},{\bf{g}})$$. The following matrices $${\bf{M}}$$ (for $$f$$) and $${\bf{N}}$$ (for $$g$$) are used to interconnect the two levels 39 $$$({\bf{m}},{\bf{n}})=\hspace{2.22144pt}({\bf{M}}\hspace{2.22144pt}{\bf{f}},{\bf{N}}\hspace{2.22144pt}{\bf{g}})\,\,\text{and}\,\,({\bf{f}},{\bf{g}})=\hspace{2.22144pt}({{\bf{M}}}^{-{\bf{1}}}{\bf{m}},{{\bf{N}}}^{-{\bf{1}}}{\bf{n}}),$$$ where $${\bf{f}}={({f}_{0},\ldots ,{f}_{8})}^{T}$$ and $${\bf{g}}={({g}_{0},\ldots ,{g}_{4})}^{T}$$.…”

“…The moment vectors $$({{\bf{m}}}^{{\bf{eq}}},{{\bf{n}}}^{{\bf{eq}}})$$at the equilibrium states are given by 39 $$${{\bf{m}}}^{{\bf{eq}}}=({m}_{0-8}^{eq})=\left[\begin{array}{c}\rho \\ e=-2\rho +3({j}_{x}^{2}+{j}_{y}^{2})\\ \varepsilon =\rho -3({j}_{x}^{2}+{j}_{y}^{2})\\ \rho u={j}_{x}\\ {q}_{x}=-{j}_{x}\\ \rho v={j}_{y}\\ {q}_{y}=-{j}_{y}\\ {p}_{xx}={j}_{x}^{2}-{j}_{y}^{2}\\ {p}_{xy}={j}_{x}{j}_{y}\\ \end{array}\right],\,\text{and}\,{{\bf{n}}}^{{\bf{eq}}}=({n}_{0-4}^{eq})=\left[\begin{array}{c}T\\ u\hspace{2.22144pt}T\\ v\hspace{2.22144pt}T\\ a\hspace{2.22144pt}T\\ 0\\ \end{array},],.\right]$$$…”

“…For varied values of Richardson, Grashof, and Prandtl numbers, Shahid et al 38 employed the MRT‐LBM to model convective flow in a heated triangular enclosure with a moving lid. Using the last method, Baliti et al 39 simulated the natural convection within an enclosed right‐angled triangular and isosceles cavity filled with water. The research was done for Rayleigh numbers ranging from $$Ra=1{0}^{3}$$ to $$Ra=1{0}^{6}$$.…”

Numerical investigation of natural convection heat transfer using TiO₂/Al₂O₃-water nanofluids

“…Hssikou et al [28]and Elguennouni et al [29] studied using the LBM for heat transfer of a gas in a square enclosure. Also, Baliti et al [30] simulated by the Multirelaxation time lattice Boltzmann method, natural convection in a triangular enclosure. Asha et al [31] conducted a magnetic field impact on heat transfer in a C-shaped enclosure using MRT-LBM.…”

Modeling of natural convection by lattice Boltzmann in a partially heated cavity

“…This method tends to describe the behavior of fluid molecules at the mesoscopic scale using the distribution function of the fluid particles. Macroscopic properties, density, temperature, and velocity vector can be calculated from the distribution functions means [2][3][4]. According to researchers who have utilized this, we find Mohamed [5] who compared results obtained by the LBM and the FDM methods.…”