A non-slip boundary condition at a wall for the lattice Boltzmann method is presented. In the present method unknown distribution functions at the wall are assumed to be an equilibrium distribution function with a counter slip velocity which is determined so that fluid velocity at the wall is equal to the wall velocity. Poiseuille flow and Couette flow are calculated with the nine-velocity model to demonstrate the accuracy of the present boundary condition.

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SUMMARYThe lattice Boltzmann method (LBM) for a binary miscible fluid mixture is applied to problems of transport phenomena in a three-dimensional porous structure. Boundary conditions for the particle distribution function of a diffusing component are described in detail. Flow characteristics and concentration profiles of diffusing species at a pore scale in the structure are obtained at various Reynolds numbers. At high Reynolds numbers, the concentration profiles are highly affected by the flow convection and become completely different from those at low Reynolds numbers. The Sherwood numbers are calculated and compared in good agreement with available experimental data. The results indicate that the present method is useful for the investigation of transport phenomena in porous structures.

A new numerical method for incompressible non-Newtonian fluid flows based on the lattice Boltzmann method (LBM) is proposed. The essence of the present method lies in the determination of shear-dependent viscosity of the fluid by using a variable parameter related to the local shear rate. Also, the relaxation time in the BGK collision term is kept at unity taking account of numerical stability. The method is applied to two representative test case problems, power-law fluid flows in a reentrant corner geometry and non-Newtonian fluid flows in a three-dimensional porous structure. These simulations indicate that the method can be useful for practical non-Newtonian fluid flows, such as shear-thickening (dilatant) and shearthinning (pseudoplastic) fluid flows.

The asymptotic theory proposed by Sone [in Rarefied Gas Dynamics, edited by D. Dini (Editrice Tecnico Scientifica, Pisa, 1971), p. 737] is applied to the investigation of the accuracy of the lattice Boltzmann method (LBM) for small Knudsen number with finite Reynolds number. The S-expansion procedure of the asymptotic theory is applied to LBM with the nine-velocity model and fluid-dynamic type equations are obtained. From the fluid-dynamic type equations it is found that by using the LBM we can obtain the macroscopic flow velocities and the pressure gradient for incompressible fluid with relative errors of O(ε′2) where ε′ is a modified Knudsen number which is of the same order as the lattice spacing and is related to a dimensionless relaxation time. In two problems, the Couette flow with flow injection and suction through porous walls and a three-dimensional flow through a square duct, the accuracy of LBM is examined for relaxation times between 0.8 and 1.7 and the validity of the asymptotic theory for LBM is shown.