Asymptotic equivalence of forcing terms in the lattice Boltzmann method within second-order accuracy.

We show the asymptotic equivalence of two forcing schemes in the lattice Boltzmann method (LBM) within second-order accuracy through the asymptotic analysis instead of the Chapman-Enskog analysis. We consider the single relaxation time LBM with the following two forcing schemes: the simplest scheme by He et al. [J. Stat. Phys. 87, 115 (1997)10.1007/BF02181482] (referred to as He forcing); the most popular scheme by Guo et al. [Phys. Rev. E 65, 046308 (2002)10.1103/PhysRevE.65.046308] (referred to as Guo forcing). It has been shown by using the Chapman-Enskog analysis that the He forcing leads the unphysical terms in the macroscopic equations due to the spatial and time derivatives of the body force, whereas the Guo forcing does not lead such terms. However, we find by using the asymptotic analysis that the order of the unphysical terms is comparable to or less than (Δx)^{3} for the continuity equation and (Δx)^{4} for the Navier-Stokes equations (where Δx is the lattice spacing). Therefore, not only the Guo forcing but also the He forcing give the macroscopic flow velocity and pressure for incompressible viscous fluid with relative errors of O[(Δx)^{2}]. To verify the result of the asymptotic analysis, we simulate two benchmark problems in which the body force is changed in space and time: a generalized Taylor-Green problem and a natural convection problem. As a result, we find that the calculated results of macroscopic variables by the He forcing converge to those by the Guo forcing at the second-order convergence rate. Therefore, we can conclude that the He forcing and the Guo forcing are equivalent within the second-order accuracy even for the space- and time-dependent body force.

Numerical simulations of the behaviour of a drop in a square pipe flow using the two-phase lattice Boltzmann method.

The lattice Boltzmann method for multi-component immiscible fluids is applied to simulations of the behaviour of a drop in a square pipe flow for various Reynolds numbers of 10

A lattice kinetic scheme for bubble flows.

A lattice kinetic scheme for two-phase immiscible fluids with large density ratios is presented. The difficulty in the treatment of large density ratio is resolved by using the projection method. The method can simulate two-phase fluid flows with a density ratio up to 1000. The method is applied to the simulation of many bubbles rising in a long square duct. The complicated unsteady structures of the interface and the velocity field can be simulated in a stable manner.

A lattice kinetic scheme for incompressible viscous flows with heat transfer.

A lattice kinetic scheme for incompressible viscous flows with heat transfer is developed based on the lattice Boltzmann method. In the new scheme, macroscopic variables are calculated without velocity distribution functions. Thus, the scheme can save computer memory because there is no need to store the velocity distribution functions. Governing equations for the macroscopic variables are obtained by applying the asymptotic theory. The continuity equation, the Navier-Stokes equations, and the convection-diffusion equation for fluid temperature are obtained with relative errors of O(epsilon2), where epsilon is a small parameter that is of the same order as a lattice spacing and is related to a relaxation parameter. In order to verify the accuracy of the scheme, natural convection flows in a square cavity are simulated, and the calculated results are in good agreement with available standard results.