---

- Natural convection heat transfer in a square cavity induced by heated plate is investigated using the lattice Boltzmann method. A suitable forcing term is represented in the Boltzmann equation. With the representation, the Navier-Stokes equation can be derived from the lattice Boltzmann equation through the Chapman-Enskog expansion. Top and bottom of the cavity are adiabatic; the two vertical walls of the cavity have constant temperatures lower than the plate’s temperature. The flow is assumed to be two-dimensional. Air is chosen as a working fluid (Pr=0.71). The study is performed for different values of Grashof number ranging from 103 to 105 for different aspect ratios and position of heated plate. The effect of the position and aspect ratio of heated plate on heat transfer are discussed. With increase of the Grashof number, heat transfer rate is increased in both vertical and horizontal position of the plate. The obtained results of the lattice Boltzmann method are validated with those presented in the literature.

---

In this paper, natural convection heat transfer inside an enclosure which is partially filled with porous layer is reported using lattice Boltzmann method. Generalized equations in modeling flow in porous media have been employed which are coupled with the lattice Boltzmann formulation of the momentum and energy equations. The present study investigates the effect of position of porous layer on heat transfer rate for different dimensionless parameters, such as Rayleigh number, Darcy number and porosity of the porous layer. In addition, a modified Rayleigh number is presented as an effective parameter which affects the degree of penetration of the fluid into the porous layers. The obtained results showed that the heat transfer rate in the case of vertical layer is more than that of horizontal porous layers.

---

In the present paper,by performing a two-dimensional simulation,the heat transfer from a hot cylinder to a cold square enclosure has been studied parametrically and the consequent effect of changing in cylinder diameter has been investigated. The 2-D governing equations have been solved using the finite volume method and TDMA in an ADI procedure for different diameters of cylinder inside a square enclosure with a constant characteristic length for two different Rayleigh numbers of 104 and 105.Results showed that the patterns of streamlines, isotherms and the Nusselt number values depend strongly on the Rayleigh number and also ratio of cylinder diameter to characteristic length of enclosure (2R/H). In this case, the centers of vortices created around the cylinder appear in bottom half of enclosure in 2R/H=0.4 for Ra=104 and in 2R/H= 0.5 for Ra=105. Moreover, it is observed that increasing the Rayleigh number and 2R/H ratio, the heat transfer rate from the enclosure is also increased.For example,in 2R/H=0.5, by increasing the Rayleigh number from 104 to 105, the average Nusselt enhances about 30 percent of its initial value and in Ra=105, by changing the 2R/H ratio from 0.2 to 0.5, the average Nusselt climbs almost 35 percent of its initial value.

---

In this paper, natural convective heat transfer of nanofluids in a uniform magnetic field between the square cavity and inner cylinder, was simulated via Lattice Boltzmann Method. The inner cylinder in square shape, diamond, and circular has been examined. Square cavity walls and inner cylinder surfaces are at a constant cold and warm temperature, respectively. The flow, temperature, and magnetic field is calculated with solving flow, temperature, and magnetic distribution functions simultaneously. D2Q9 lattice arrangement for each distribution function is used. The results clearly show the behavior of fluid flow and heat transfer between the cavity and the cylinder. The results have been validated with available valid results showing relatively good agreement. The effects of Rayleigh number, Hartmann number, void fraction and type of nanoparticles on natural convective heat transfer are investigated. This study shows that for all three geometries used with the same void fraction, type of nanofluid, and Rayleigh number, natural convective heat transfer decreases with Hartmann number. Also, when Hartmann number was had fixed, natural convective heat transferwas increased with Rayleigh number. Thus, to select the right geometry for optimum natural convective heat transfer, our needs to pay special attention to Hartmann and Rayleigh numbers. In addition, viod fraction and type of nanofulid can affect heat transfer directly.

---