Thermal lattice-Boltzmann method for non-ideal gases with potential energy

We present a thermal lattice-Boltzmann method for gases with potential ener gy. In addition to the single particle distribution function, additional di stribution functions for the potential energy and the non-ideal part of the pressure tensor are defined which contain information about the two-partic le distribution function. Guided by the BBGKY-hierarchy, a set of three cou pled kinetic equations for these distribution functions is proposed. By mea ns of a Chapman-Enskog expansion it is shown that the correct hydrodynamic equations, including the equation for energy transport, are obtained in the limit of large length and time scales. We discuss how the model can be dis cretized in order to achieve second-order accuracy and Galilean-invariance. A reduced version of the model, in which the pressure field is adiabatical ly eliminated, is implemented in two dimensions on a hexagonal lattice. Its stability is investigated numerically, and tests of the accuracy for the t ransversal and longitudinal modes of linear hydrodynamics, as well as tests of Galilean-invariance, are performed. Comparisons are also made with a hy brid model, in which the energy equation is solved by a finite-difference s cheme. The method was further simplified in two cases: (i) for a constant t emperature, and (ii) for a gas with only excluded-volume interactions. (C) 2000 Elsevier Science B.V. All rights reserved.