Direct Simulations Of Rayleigh-Bernard Convection With High Rayleigh Number By Lattice Boltzmann Model

The paper constructs a double distribution function (DDF) thermal lattice Boltzmann model based on D2Q9to simulate the thermal convective flows with the Boussinesq approximation. The model utilizes two kinds of distribution functions: fa(x,t) and ga(x,t).One is used to simulate the density and velocity fields, and the other one is used for temperature field. Under the hypothesis of Boussinisq, they are coupled by the external force term in evolution equation for density and velocity fields.The evolution function of Lattice Boltzmann model is given by: Where fa is the density distribution function, fαeq is the equilibrium distribution function for fα, gα is the temperature distribution function and gαeq is the equilibrium distribution function for gα. eα is the discrete velocity, τf and Tg characterize the dimensionless relaxation time of evolution functions respectively, Ωα is the additional term.Here the equilibrium distribution functions and the external force are expressed as: Where c is the lattice streaming speed, ωα is the weight coefficient, u is the velocity vector.In this paper, we assume that the additional term is the second order term of the expansion parameter ε, namely Ωα=ε2Φα. The advantage of this assumption is that the external force term is easy in expression and programming.Applying the multiscale Chapman-Enskog expansion for lattice Boltzmann equation, we prove that the model can recover the macroscopic momentum equation with the second order accuracy. The motion is in a low speed and small Mach number, therefore, the distribution function which has a first order accuracy in velocity field is selected for the model. Selection of the temperature function of this form can reduce the quantity of calculation while the numerical simulation results are satisfying.According to the external force expression obtained from the theoretical derivation and equilibrium distribution function, we write a programming in Fortran, then simulate the Rayleigh-Bernard convection. Before that, the natural convection in a square cavity whose physical model is simpler (in the way of boundary treatment) is simulated. Compared with the published experimental and numerical results, the model is proved in validity and accuracy of numerical simulation. Within a given Prandtl number Pr=0.71, we simulate RB convection in different Rayleigh numbers, also in the situation with the same Rayleigh numbers while the iteration steps are different. Through the figures of streamline and isotherm resulted from the simulation above, some conclusions are obtained as follows:With the increasing of Rayleigh number, the mechanism of convective heat transfer enhances as well. In low Rayleigh number (on the order of magnitude from103to105), the system will reach a steady state after a period of convective heat transfer. While, in high Rayleigh number (on the order of magnitude of106and above), the flow of system will appear periodic oscillating phenomenon instead of stability. Secondary instability occurs, and turbulence characteristics appear.