Simulation of natural convection under high magnetic field by means of the thermal lattice Boltzmann method

The thermal lattice Boltzmann method （TLBM）, which was proposed by J. G. M. Eggels and J. A. Somers previously, has been improved in this paper. The improved method has introduced a new equilibrium solution for the temperature distribution function on the assumption that flow is incompressible, and it can correct the effect of compressibility on the macroscopic temperature computed. Compared to the previous method, where the half- way bounce back boundary condition was used for non-slip velocity and temperature, a non-equilibrium extrapolation scheme has been adopted for both velocity and temperature boundary conditions in this paper. Its second-order accuracy coincides with the ensemble accuracy of lattice Boltzmann method. In order to validate the improved thermal scheme, the natural convection of air in a square cavity is simulated by using this method. The results obtained in the simulation agree very well with the data of other numerical methods and benchmark data. It is indicated that the improved TLBM is also successful for the simulations of non-isothermal flows. Moreover, this thermal scheme can be applied to simulate the natural convection in a non-uniform high magnetic field. The simulation has been completed in a square cavity filled with the aqueous solutions of KC1 （llwt%）, which is considered as a diamagnetic fluid with electrically low-conducting, with Grashof number Gr=4.64~104 and Prandtl number Pr----7.0. And three cases, with different cavity locations in the magnetic field, have been studied. In the presence of a high magnetic field, the natural convection is quenched by the body forces exerted on the electrically low-conducting fluids, such as the magnetization force and the Lorentz force. From the results obtained, it can be seen that the quenching efficiencies decrease with the variation of location from left, symmetrical line, to the right. These phenomena originate from the different distributions of the magnetic field strengths in the zones of the symmetrical central line of the magnetic fields. The results are also compared with those without a magnetic field. Finally, we can conclude that the improved TLBM will enable effective simulation of the natural convection under a high magnetic field.

Numerical Predictions of the Effective Thermal Conductivity of the Rigid Polyurethane Foam

A reconstruction method is proposed for the polyurethane foam and then a complete numerical method is developed to predict the effective thermal conductivity of the polyurethane foam. The finite volume method is applied to solve the 2D heterogeneous pure conduction. The lattice Boltzmann method is adopted to solve the 1D homogenous radiative transfer equation rather than Rosseland approximation equation. The lattice Boltzmann method is then adopted to solve 1D homogeneous conduction-radiation energy transport equation considering the combined effect of conduction and radiation. To validate the accuracy of the present method, the hot disk method is adopted to measure the effective thermal conductivity of the polyurethane foams at different temperature. The numerical results agree well with the experimental data. Then, the influences of temperature, porosity and cell size on the effective thermal conductivity of the polyurethane foam are investigated. The results show that the effective thermal conductivity of the polyurethane foams increases with temperature; and the effective thermal conductivity of the polyurethane foams decreases with increasing porosity while increases with the cell size.

A Novel Dynamic Quadrature Scheme for Solving Boltzmann Equation with Discrete Ordinate and Lattice Boltzmann Methods

The Boltzmann equation(BE)for gas flows is a time-dependent nonlinear differential-integral equation in 6 dimensions.The current simplified practice is to linearize the collision integral in BE by the BGK model using Maxwellian equilibrium distribution and to approximate the moment integrals by the discrete ordinatemethod(DOM)using a finite set of velocity quadrature points.Such simplification reduces the dimensions from 6 to 3,and leads to a set of linearized discrete BEs.The main difficulty of the currently used(conventional)numerical procedures occurs when the mean velocity and the variation of temperature are large that requires an extremely large number of quadrature points.In this paper,a novel dynamic scheme that requires only a small number of quadrature points is proposed.This is achieved by a velocity-coordinate transformation consisting of Galilean translation and thermal normalization so that the transformed velocity space is independent of mean velocity and temperature.This enables the efficient implementation of Gaussian-Hermite quadrature.The velocity quadrature points in the new velocity space are fixed while the correspondent quadrature points in the physical space change from time to time and from position to position.By this dynamic nature in the physical space,this new quadrature scheme is termed as the dynamic quadrature scheme(DQS).The DQS was implemented to the DOM and the lattice Boltzmann method(LBM).These new methods with DQS are therefore termed as the dynamic discrete ordinate method(DDOM)and the dynamic lattice Boltzmann method(DLBM),respectively.The new DDOM and DLBMhave been tested and validated with several testing problems.Of the same accuracy in numerical results,the proposed schemes are much faster than the conventional schemes.Furthermore,the new DLBM have effectively removed the incompressible and isothermal restrictions encountered by the conventional LBM.

Multiple Relaxation Time Lattice Boltzmann Simulation of 2D Natural Convection in a Square Cavity at High Rayleigh Numbers

Natural convection in a square cavity at high Rayleigh numbers is simulated by multiple relaxation time(MRT)lattice Boltzmann method(LBM)with a separate distribution function to solve the temperature.The Rayleigh numbers examined here range from Ra=103 to Ra=10^(8).For Rayleigh numbers below 10^(8),the flow remains stationary and transition occurs beyond Ra=2×10^(8).Unsteady results at higher Rayleigh numbers(Ra=10^(9) and Ra=10^(10))are also investigated.To the best of our knowledge,this is the first accurate study which involves the high Rayleigh numbers Ra=10^(9),10^(10).