The average Nusselt number with the use of nanoﬂuid is higher than the use of water under the same Rayleigh number. However, the heat transfer rate of the nanoﬂuid takes on a lower value than water at a ﬁxed temperature difference across the enclosure mainly due to the signiﬁcant enhancement of dynamic viscosity. Furthermore, great deviations of computed Nusselt numbers using different models associated with the physical properties of a nanoﬂuid are revealed. A thermal lattice BGK model with doubled populations, together with a new boundary condition for temperature and heat ﬂux, is proposed to simulate the two-dimensional natural convection ﬂow in a cavity. In this framework, main aim of the present paper is to propose, discuss and validate a general purpose thermal boundary condition (GPTBC). The GPTBC proposed here, successfully demonstrated for the Dirichlet-type constraint in the case of thermal Couette and Poiseuille ﬂows. In this paper, a coupling lattice Boltzmann (LB) model for simulating thermal ﬂows on the standard two-dimensional nine-velocity (D2Q9) lattice is developed in the framework of the double-distribution-function (DDF) approach in which the viscous heat dissipation and compression work are considered. The discrete equilibrium density and total energy distribution functions are obtained from the Hermite expansions of the corresponding continuous equilibrium distribution functions. The pressure given by the equation of state of perfect gases is recovered in the macroscopic momentum and energy equations. The coupling between the momentum and energy transports makes the model applicable for general thermal ﬂows such as non-Boussinesq ﬂows, while the existing DDF LB models on standard lattices are usually limited to Boussinesq ﬂows in which the temperature variation is small. Extensive parametric studies on mixed convection in a driven cavity ﬁlled with porous medium are carried out for various values of Reynolds number, Richardson number and Darcy number. It is found that the ﬂow and temperature patterns change greatly with variations of these parameters. The effects of solid volume fraction of nanoﬂuids on hydrodynamic and thermal characteristics are investigated and discussed. It is found that the effects of solid volume fraction for Cu are stronger than Al2O3. This study presents a numerical investigation of laminar natural convection heat transfer in an inclined enclosure, differentially heated, with inclined partitions attached to its hot wall. Effects of partition length, partition inclination angle, partitions number and aspect ratio cavity are studied for Prandtl number 0.71 and cavity inclination angles of 45 and 90. Rayleigh numbers considered are 5 105 for a square cavity and 10 4 for a cavity with an aspect ratio equal to 6. The investigation is performed for various inclination angles and gap widths W. At lower Ra (Ra 6 10 5),the average hot wall Nusselt number Num is higher in inclined cavities than in vertical ones; while at larger Ra (Ra = 10 6 ), the opposite occurs. In vertical cavities, Num decreases with decreasing W. However in inclined ones, this trend is found only at lower Ra whereas at higher Ra, Num shows a maximum at about W = 0.5. Num also decreases when the number of partitions (Np) attached to the cold wall of the enclosure increases; however beyond a certain value of (Np), which depends of Ra, Num becomes almost constant. . The numerical approach adopted uses a hybrid thermal lattice Boltzmann method (HTLBM) in which the mass and momentum conservation equations are solved by using multiple relaxation time (MRT) model and the energy equation is solved separately by using the ﬁnite difference method (FDM). In addition, the radiative transfer equation (RTE) is treated by the discrete ordinates method (DOM) using the S8 quadrature to evaluate the source term of the energy equation. The results presented in terms of isotherms, streamlines and averaged Nusselt number, show that in the absence of the radiation, the temperature and the ﬂow ﬁelds are centro-symmetrics and the cavity core is thermally stratiﬁed. However, radiation causes an overall increase in temperature and velocity gradients along both thermally active walls. We adopted the double-distribution model to simulate heat transfer process. To simplify the calculation, a simpler model D2Q5 is applied to simulate temperature while D2Q9 is for velocity simulation. Brownian motion and the interaction potential between nanoparticles are considered in the simulation. This paper reports a numerical study on natural convection from a protruding heater located at the bottom of a square cavity ﬁlled with a copper-water nanoﬂuid. The vertical walls of the cavity are cooled isothermally; the horizontal ones are adiabatic, and the heater is attached to the bottom wall. The heat source is assumed either to be isothermal or to have a constant heat ﬂux. The eﬀective viscosity and thermal conductivity of the nanoﬂuid are modeled according to Brinkman and Patel, respectively. This work focuses on the study of natural convection heat transfer characteristics in a differentially-heated enclosure ﬁlled with a CuOeEGeWater nanoﬂuid for different published variable thermal conductivity and variable viscosity models. The problem is given in terms of the vorticityestream function formulation and the resulting governing equations are solved numerically using an efﬁcient ﬁnite-volume method. Different behaviors (enhancement or deterioration) are predicted in the average Nusselt number as the volume fraction of nanoparticles increases depending on the combination of CuOeEGeWater variable thermal conductivity and viscosity models employed. In general, the effects the viscosity models are predicted to be more predominant on the behavior of the average Nusselt number than the inﬂuence of the thermal conductivity models. The enclosure aspect ratio is predicted to have signiﬁcant effects on the behavior of the average Nusselt number which decreases as the enclosure aspect ratio increases. Transient natural convection in a low-Prandtl-number ¯uid was computed for a shallow rectangular enclosure of 5 5 1 heated from below and cooled from above with four dragless, adiabatic, vertical walls. The model consists of three-directional mass, momentum and energy balances. These equations were approximated by a HSMAC ®nite-difference method with a hybrid scheme for the inertial terms. The transient response after a step increase of the lower-wall temperature was at ®rst a regular oscillation, which then evolved to a series of long rolls with their axes in one direction. The present study aims to identify eﬀects due to uncertainties in eﬀective dynamic viscosity and thermal conductivity of nanoﬂuid on laminar natural convection heat transfer in a square enclosure. Numerical simulations have been undertaken incorporating a homogeneous solid– liquid mixture formulation for the two-dimensional buoyancy-driven convection in the enclosure ﬁlled with alumina–water nanoﬂuid. Signiﬁcant diﬀerence in the eﬀective dynamic viscosity enhancement of the nanoﬂuid calculated from the two adopted formulas, other than that in the thermal conductivity enhancement, was found to play as a major factor, thereby leading to contradictory results concerning the heat transfer eﬃcacy of using nanoﬂuid in the enclosure. A thermal lattice BGK model with doubled populations is proposed to simulate the twodimensional natural convection ﬂow in porous media (porous metals). The accuracy of this method is validated by the benchmark solutions. The detailed ﬂow and heat transfer at the pore level are revealed. The effects of pore density (cell size) and porosity on the natural convection are examined. Also the effect of porous media conﬁguration (shape) on natural convection is investigated. The results showed that the overall heat transfer will be enhanced by lowering the porosity and cell size. The square porous medium canhave a higher heat transfer performance than spheres due to the strong ﬂow mixing and more surface area. The Taylor series expansion- and least squares-based lattice Boltzmann method (TLLBM) was used in this paper to extend the current thermal model to an arbitrary geometry so that it can be used to solve practical thermo-hydrodynamics in the incompressible limit. The new explicit method is based on the standard lattice Boltzmann method (LBM), Taylor series expansion and the least squares approach. The ﬁnal formulation is an algebraic form and essentially has no limitation on the mesh structure and lattice model. Numerical simulations of natural convection in a square cavity on both uniform and nonuniform grids have been carried out. Favorable results were obtained and compared well with the benchmark data. It was found that, to get the same order of accuracy, the number of mesh points used on the nonuniform grid is much less than that used on the uniform grid.