Mixed convection in inclined lid driven cavity by Lattice Boltzmann Method and heat flux boundary condition

Laminar mixed convective heat transfer in two-dimensional rectangular inclined driven cavity is studied numerically by means of a double population thermal Lattice Boltzmann method. Through the top moving lid the heat flux enters the cavity whereas it leaves the system through the bottom wall; side walls are adiabatic. The counter-slip internal energy density boundary condition, able to simulate an imposed non zero heat flux at the wall, is applied, in order to demonstrate that it can be effectively used to simulate heat transfer phenomena also in case of moving walls. Results are analyzed over a range of the Richardson numbers and tilting angles of the enclosure, encompassing the dominating forced convection, mixed convection, and dominating natural convection flow regimes. As expected, heat transfer rate increases as increases the inclination angle, but this effect is significant for higher Richardson numbers, when buoyancy forces dominate the problem; for horizontal cavity, average Nusselt number decreases with the increase of Richardson number because of the stratified field configuration.

where fi and gi are the discrete populations which evolve when a standard first order integration strategy is adopted, the term Z 3 = c 3   The terms enclosed by the square bracket, multiplied by the corresponding weights M , will be called corresponding form for equilibrium.
Finally, using f 3 ; and g 3 ; , hydrodynamic and thermal variables are calculated as follows: The kinematic viscosity and the thermal diffusivity in the two-dimensional geometry are given by: ^= KL _ , `= 2 ' KL a In this problem shear stress applied by moving lid on the fluid layers results in fluid motion, thus creating suitable temperature gradient that enhances buoyancy forces. Therefore, mixed convection is produced in the fluid confined in the cavity. With the Boussinesq approximation, all the fluid properties are considered as constant, except in the body force term in the Navier-Stokes equations, where the fluid density is assumed ρ = ρ _(1 − β T − T a ), in which β is volumetric expansion coefficient, ρ _ and T a are the average fluid density and temperature. In order to simulate the mixed convection of nearly incompressible flows, buoyancy force per unit mass is defined as G = βg T − T a and this is used to drive the flow. By considering inclination angle, coordinate axis and gravity acceleration direction, as shown in figure 2, all of the aforementioned relations are maintained and used except those that are modified below: In this case hydrodynamic macroscopic variables are calculated as follows: where c 3 = jc 3m , c 3n l denote discrete particle speeds.

Boundary conditions
With regard to hydrodynamic boundary condition, no slip boundary condition is applied; It is obtained by means of the non-equilibrium bounce back rule as applied to the population perpendicular to the boundary. With regard to thermal boundary conditions, the top cavity lid and bottom wall are heated and cooled respectively by an uniform and constant heat flux entering and leaving respectively, and the sidewalls are insulated. These boundary conditions are obtained by means a thermal counter-slip approach as proposed by [6,8,9], in which the incoming unknown thermal populations are assumed to be equilibrium distribution functions with a counter slip thermal energy density e', which is determined so that suitable constraints are verified. For the top wall of the cavity, named as "north wall", in which entering heat flux is constant and equal to q N , the unknown g & P , g & X and g & Y are chosen as follows.
By definition: and become

Results and discussion
Laminar mixed convection of a fluid inside a rectangular cavity with moving top lid and aspect ratio †K = ‡ˆ= 3 ⁄ , in which L and H are shown in Figure 2, is studied numerically using Lattice Boltzmann Method previously described. The bottom wall is cooled and the top lid is heated, and side walls are assumed insulated. Top lid moves with constant velocity U I . Characteristic dimensionless number in the analysis of mixed convection problems is Richardson number defined as Ri = Ra PrRe J ⁄ . As stated before, lattice Boltzmann method is used for near-incompressible flows and therefore Mach number is assumed as Ma ≪ 1. More specifically, the characteristic velocity of the flow for both natural regime, U * = •βgHΔT, and forced regime U * = νRe H ⁄ must be small compared The top lid is the hot moving wall at Y = 1, the bottom wall is the cold wall at Y=0 and two insulated side walls are at X = 0 and X = 3.
In order to obtain grid independent solution, a grid refinement study is performed for a horizontal cavity (i = 0). Grid independence of the results has been established in term of average Nusselt number on the lid and dimensionless values of x-velocity |, y-velocity w, and temperature Θ at X = 1.5 and Y = 0.5 (cavity center) for three different grid size, namely 300 × 100, 450 × 150 and 600 × 200 lattice nodes; due to small difference between the results of the last two grid sizes, the 450 × 150 grid is chosen as a suitable one in this work.
To validate the computer code, the comparison with the analytical solution given by Kimura and Bejan [10] and the value obtained by D'Orazio et al. [6] is examined for the Rayleigh number equal to 10 5 and 10 6        The motion of the cavity lid, with positive lid velocity, causes the fluid motion in the cavity and produces a strong clockwise rotational cell in it. This motion transfers hot fluid to the lower parts and enhances favourable pressure gradient along the vertical direction, leading to the generating buoyancy motions and transferring hot fluid to the upper parts. Therefore, the combination of free and forced convection, called "mixed convection", is made. In Figure 3, the Richardson number is K@ = 0.1, with forced convection dominant with respect to natural convection, and it implies that by increasing i, the rotational power of the cell in the center of the cavity increases slightly and it does not affect significantly the other moving and thermal behaviour of the fluid. When buoyancy forces dominate the problem, as for K@ = 10 reported in Figure 5, inclination angle has significant effect on the flow field and heat transfer. For i = 0, a strong cell in the upper half and a weak cell in the lower half of the cavity, are generated due to the forcing of moving wall. In addition, for i = 0, isotherms in the lower half of the cavity are straight lines and perpendicular to the side walls, indicating that conduction heat transfer is dominant in this region and that, without the forced convection contribution, no motion occurs, due the stratified fluid. As the inclination angle increases, the two cells merge, so that for i = 90 a large rotational cell covers the whole space of the cavity, with the central part practically motionless.
When the buoyancy effect is predominant (K@ = 10) in case of negative velocity, the effect of the direction of the moving wall, in term of modified flow field, can be detected. While for | I < 0 the two rotational cells lose their individuality already for low inclination angles, for | I < 0 they remain distinct because the hot moving wall has an opposite effect with respect to the buoyancy.
When K@ = 1, it becomes evident as the contribution of the moving wall with negative velocity and the buoyancy due to gravity acceleration have opposite effects on the fluid field; it gives rise to the formation of two rotational cell into the cavity.
For K@ = 0.1, when forced convection is the dominant effect on the flow field, streamlines and isotherms for negative velocity, have the same behaviour shown in case of positive wall velocity, although as in a mirror, but the result of the two cited opposing effects is the persistence of the two rotational cells.
In Figure 9, the average Nusselt number on the lid surface is reported as a function of inclination angle for different Richardson number in both case of positive and negative moving wall velocity. It has to be noted that for i = 90, Nu © decreases when K@ increases, since it implies to go to a stratified field configuration. With regard to γ ≠ 0, it is observed that for positive lid velocity when K@ = 0.1, Nu © increases slightly with the increase of γ. At Ri ≥ 1, Nu © is increased more intensively. In fact at K@ = 10, it is increased by a factor of 5 when γ varies from 0 to 90, indicating that in this case free convection effect is enhanced and its contribution can be added to the forced convection effect. When i = 0 (horizontal cavity) Nu © is maximum at K@ = 0.1 (significant forced convection), but for inclined cavity, the maximum Nu © occurs at K@ = 10 (significant natural convection). For negative velocity of moving wall, the opposite effect of buoyancy and forced convection contribution can be observed since the Nusselt number results ever lower than the value corresponding to the case of positive velocity. For low inclination angle, the contribution of natural convection is less important than the hindering effect of the negative moving wall velocity, whereas for increasing γ angle the importance of natural convection becomes predominant.

Conclusions
A thermal lattice Boltzmann BGK model with a dedicated boundary condition was used to study numerically laminar two-dimensional mixed convection heat transfer inside an inclined rectangular cavity when he heat transfer rate is imposed at the boundaries. Since the inclination of the cavity enhances the buoyancy force, which affects the velocity components of the flow, the forcing term simulating the buoyancy effect in the lattice Boltzmann equation were modified. The results show that, as expected, heat transfer rate increases as increases the inclination angle, but this effect is significant for the higher Richardson numbers, when buoyancy forces dominate the problem; for horizontal cavity, average Nusselt number decreases with the increase of the Richardson number because of the stratified field configuration. The effects of forced convection and natural convection can be considered as cooperating for positive velocity of the moving lid, while on the contrary they can be considered as opposite for negative velocity. This study shows that lattice Boltzmann method together with the counter-slip thermal energy density boundary condition can be effectively used to simulate heat transfer phenomena also in case of moving walls. The method can be successfully applied to simulate a wide class of cooling process where a given thermal power must be removed.