The Lattice Boltzmann Simulations Of Thermal Convections

In this paper, we use the double multiple-relaxation-time thermal lattice Boltzmann(MRT-TLB) model to simulate the thermal convective flows in two dimensions with theBoussinesq approximation, and the variables in the evolution equations are the particledistribution functions. And the double MRT-TLB model use two sets of distribution functions:one for mass and momentum conservations and the other for temperature, and two fields arecoupled by the buoyancy term in evolution equation of{f i}and the equilibrium moments inevolution equation of{g i}. Because we use the MRT collision model combined with somerestrains of relaxation rates, the imposed boundaries are all realized coincidentally at the locationδ x/2beyond the last fluid nodes, and the location of the boundary is independent of transportcoefficients or relaxation rates. But, for single-relaxation-time collision model, the boundarylocation depends on the relaxation time by error. And by using Chapman-Enskog expansion, themacroscopic equations with Boussinesq approximation could be derived form the above doubleMRT-TLB model, and the macroscopic density and velocity is the zeroth-order and first-ordermoment of discrete velocities corresponded to distribution functions{f i}, respectively, and thetemperature is the zeroth-order moment of discrete velocities corresponded to distributionfunctions{g i}. Then, we use the model to simulate cavity flow with differentially heatedvertical walls and Rayleigh-Benard convection heated from below. The flow parameters forcavity flow are the Prandtl number Pr=0.71and Rayleigh number Ra=103-106, and forRayleigh-Benard convection are Pr=0.71and7.0, and Ra=103-105. The results are as follows:(1) For cavity flow and Rayleigh-Benard convection, we use the results of the simulationsusing different meshes to calculate the L2-normed errors of velocity, pressure and temperaturefield. And we can see that, the errors of all variables monotonically decrease with increasingresolutions, and this is a solid evidence of the mesh convergence of our model. Also, theconvergence rates of errors are all around2.0, so our model is second-order accurate.(2) We study the influence of the Mach number on velocity, pressure and temperature field,and also study the effects on Nusselt numbers and hydrodynamic variables. And we find out thatthe Mach number has little effect on the accuracy of the double MRT-TLB simulations.(3) In simulations of cavity flow, we calculate the number of iterations to achieve steadystate using different Rayleigh numbers, Mach numbers and resolutions. Because Mach numberis equivalent to the CFL number, the resolution is inversely proportional to the time step, and theincreasing Rayleigh number will cause increasing temperature gradient near the boundaries, we can approximately derive that: Nt∝N·Ralg2·Ma-1. So we can make Mach number larger toenhance the computational efficiency, while the Mach number is still under a certain criticalvalue.(4) We compare our results for cavity flow with existing data obtained by usingmacroscopic simulations in different forms, and by using other LB models, and we can see thatour results agree well with the best existing data. With the simulation results of pressure,temperature and flow field, we can see that with the increasing Rayleigh number, thetemperature gradients near the vertical boundaries are increasing, and the isotherms in the centerbecome more straightness, which indicates that the dominant mechanism of heat transportchanges from the heat conduction in low Rayleigh number to heat convection in high Rayleighnumber. And the center of streamlines is moving form the center of the square cavity in lowRayleigh number to vertical walls in high Rayleigh number.(5) For Rayleigh-Benard convection with Pr=0.71and7.0and Ra no more than5×104, theevolvements of pressure, temperature and flow field with increasing Rayleigh number are almostsame: When Ra> Rac, the disturbances develop and self-organize to convection cycles, and withthe increasing Rayleigh number, the thermal convection become the dominant heat transportmechanism. And when Rayleigh number is large, the Nusselt number of Pr=7.0is lower thanPr=0.71, which is indicated by the smaller temperature difference in horizontal direction for Pr=7.0. But when Ra grows more, for example [6×104-8×104], the bifurcation comes up, that thesystem with Pr=0.71can still reach the steady state, when the streamlines and isotherms of thesystem with Pr=7.0will finally show periodic oscillations with period two fold of oscillatingperiod of Nusselt number.(6) We use different meshes to calculate the critical Rayleigh number Rac, and get theasymptotic value by using the least-square fit. The result of Racin our paper is1707.968375,differing form the theoretical value by about0.012%.(7) We investigate the effects of lateral boundary conditions on Rayleigh-Benardconvection at Pr=0.71and Ra=104, by using periodic lateral boundary conditions and adiabaticno-slip lateral boundary conditions. We can see that the adiabatic no-slip lateral walls restrict thedevelopment of thermal convection through viscosity friction. And the Nusselt number of thesimulation using adiabatic no-slip lateral walls is lower than using periodic lateral walls, in whichthe Nusselt number is2.4049and2.6551, respectively.