Lattice Boltzmann method applied to the laminar natural convection in an enclosure with a heat-generating cylinder conducting body

In this paper, a numerical investigation of laminar convective flows in a differentially heated, square enclosure with a heat-conducting cylinder at its center, is carried out. The flow and the temperature are computed using respectively the lattice Boltzmann equation and finite-difference with suitable coupling to take natural convection into account. The investigation is performed for $\mathit{Pr}=0.71$, Rayleigh numbers of $\mathit{Ra}={10}^3\text{–}{10}^6$ and temperature-difference ratio of $\mathrm{\Delta }{T}^1=0\text{–}50$. The average hot and cold walls Nusselt numbers, the flow and temperature fields are presented and discussed. For a constant Ra, the average Nusselt number at the hot and cold walls (${\mathit{Nu}}_h$ and ${\mathit{Nu}}_c$) vary linearly with $\mathrm{\Delta }{T}^1$: ${\mathit{Nu}}_h$ decreases with $\mathrm{\Delta }{T}^1$ while ${\mathit{Nu}}_c$ increases with $\mathrm{\Delta }{T}^1$.     

Natural convection flows in porous trapezoidal enclosures with various inclination angles

Simulations were carried out using penalty finite element analysis with bi-quadratic elements to investigate the influence of uniform and non-uniform heating of bottom wall within a trapezoidal enclosure of various inclination angles $(\phi )$. Parametric study has been carried out for a wide range of Rayleigh number $(\mathit{Ra})({10}^3?\mathit{Ra}?{10}^6)$, Prandtl number $(\mathit{Pr})(0.026?\mathit{Pr}?988.24)$ and Darcy number $(\mathit{Da})({10}^{-3}?\mathit{Da}?{10}^{-5})$. Numerical results are presented in terms of stream functions, isotherm contours and Nusselt numbers. The heat transfer is primarily due to conduction at lower values of Darcy number $(\mathit{Da})$ and convection dominant heat transfer is observed at higher Da values. The intensity of circulation increases with increase in Darcy number. Increase in the intensity of circulations and larger temperature gradient are also observed with increase in $\phi$ from 0° to 45° especially at larger Pr and Ra. Non-uniform heating of the bottom wall produces greater heat transfer rate at the center of the bottom wall than uniform heating at all Rayleigh and Darcy numbers, but average Nusselt number is lower for non-uniform heating. Local heat transfer rates are found to be relatively greater for $\phi =0°$. It is observed that the local heat transfer rate at the central portion of bottom wall is larger for non-uniform heating case. Average Nusselt number plots show higher heat transfer rates at the bottom wall for $\phi =0°$ as compared to $\phi =45°$ and $\phi =30°$. It is observed that the average heat transfer rate at the bottom wall is found to be invariant with respect to $\phi$ at higher Ra for non-uniform heating. Critical Rayleigh numbers for conduction dominant heat transfer cases have been obtained and the power law correlations between average Nusselt number and Rayleigh numbers are presented for convection dominated regimes.