Introducing a modified exact difference method for incorporating unsteady and non-uniform force terms in lattice kinetic models

摘要

The exact difference method (EDM) of Kupershtokh (2004), in this work, is modified to accurately incorporate unsteady non-uniform force terms into lattice kinetic models. We use the characteristic scheme for integrating the Boltzmann equation along with the trapezoidal rule to estimate the integration results. The method is based on discretizing the total derivative of the equilibrium distribution function (EDF) in the EDM, using a central second-order scheme, instead of a first order method. It is shown that the assumption of constant density in calculating the total derivative of the EDF is not necessary in the limit of incompressible fluid flows. Using the Chapman-Enskog analysis, it is shown that the present method can be applied to both the standard lattice Boltzmann method and the recently introduced constant speed kinetic model (CSKM) of Zadehgol and Ashrafizaadeh (2014). The following benchmark flows are simulated: (i) Womersley flow, (ii) Hartmann flow, (iii) unsteady Taylor-Green vortex flow, and (iv) the circular interface, using the CSKM, and the results are in excellent agreement with the analytical solutions. (C) 2019 Elsevier Ltd. All rights reserved.