本文对具有不同温度竖壁的封闭容器中二维自然对流问题进行了直接数值模拟。控制方程在非均匀网格上使用空间二阶精度和时间一阶精度进行离散，程序是在国家高性能计算中心（武汉）的并行计算机曙光1000A上实现的。使用4个结点，获得的加速比和效率分别约为2.89和0.72。

A boundary treatment for curved walls in lattice Boltzmann method is proposed. The distribution frmction at a wall node who has a link across the physical boundary is decomposed into its equilibrium and nonequilibrium parts. The equilibrium part is then approximated with a fictitious one where the boundary condition is enforced, and the nonequilibrium part is approximated using a first-order extrapolation based on the nonequilibrium part of the distribution on the neighboring fluid node. Numerical results show that the present treatment is of second-order accuracy, and has well-behaved stability characteristics.

A second-order moment (SOM) gas-phase turbulence model, combined with a Monte-Carlo (MC) simulation of stochastic particle motion using Langevin equation to simulate the gas velocity seen by particles, is called an SOM-MC two-phase turbulence model. The SOM-MC model was applied to simulate swirling gas-particle flows with a swirl number of 0.47. The prediction results are compared with the PDPA measurement data and those predicted using the Langevin-closed unified second-order moment (LUSM) model. The comparison shows that both models give the predicted time-averaged flow field of particle phase in general agreement with those measured, and there is only slight difference between the prediction results using these two models. In the near-inlet region, the SOM-MC model gives a more reasonable distribution of particle axial velocity with reverse flows due to free of particle numerical diffusion, but it needs much longer computation time. Both models underpredict the gas and particle fluctuation velocities, compared with those measured. This is possibly caused by the particle-wall and particle-particle interaction in the near-wall region, and the effect of particles on dissipation of gas turbulence, which is not taken into account in both models.

We show discrcte lattice offects must be considered in the introduotion of a foree into the lattice Boltzmann equation, A representation of the forcing temp is then proposed. With the representation, the Navier-Stokes equation is derived from the lattice. Boltzmann equation through the Chapman-Enskng expansion. Several other exisling force treatments are also examined.