The accuracy and efficiency of the lattice Boltzmann method (LBM) and the finite difference method (FDM) are numerically investigated. In the FDM for incompressible viscous flows, it is usually needed to solve a Poisson equation for the pressure by iteration or relaxation technique, while in the LBM, such special treatment is not required. Two-dimensional problems of incompressible viscous flows and thermal fluid flows are computed by using the LBM and FDM. In the problem of flows through a porous structure, the present results indicate that the LBM is more efficient than the FDM, because there is no need to relax the pressure fields in the LBM at relatively high Reynolds numbers. Therefore, it is found that the LBM is useful for the investigation of transport phenomena in complex geometries such as porous structures.

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SUMMARYThe generalized MSR-method (GMSR-method) is proposed as a ÿnite element uid analysis algorithm for arbitrarily deformed elements using the error analysis approach. The MSR-method was originally developed by one of the authors in our previous research works using a modiÿed Galerkin method (MGM) for a convection-di usion equation and the SIMPLER-approach. In this paper, this MGM is developed theoretically in the case of arbitrarily deformed elements using the error analysis approach. In the GMSR-method, since the inertia term and the pressure term are considered explicitly, only symmetrical matrices appear. Hence, it helps us reduce computational memory and computation time. Moreover, artiÿcial viscosity and di usivity are introduced through an error analysis approach to improve the accuracy and stability. This GMSR-method is applied for two-and three-dimensional natural convection problems in a cavity. In the computations at di erent Rayleigh numbers, it is shown that this method gives reasonable results compared to other research works. Thus, it is found that the GMSR-method is applicable to thermal-uid ow problems with complicated boundaries.

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