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

We present a thermal lattice-Boltzmann method for gases with potential energy. In addition to the single particle distribution function, additional distribution functions for the potential energy and the non-ideal part of the pressure tensor are defined which contain information about the two-particle distribution function. Guided by the BBGKY-hierarchy, a set of three coupled kinetic equations for these distribution functions is proposed. By means 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 discretized in order to achieve second-order accuracy and Galilean-invariance. A reduced version of the model, in which the pressure field is adiabatically eliminated, is implemented in two dimensions on a hexagonal lattice. Its stability is investigated numerically, and tests of the accuracy for the transversal and longitudinal modes of linear hydrodynamics, as well as tests of Galilean-invariance, are performed. Comparisons are also made with a hybrid model, in which the energy equation is solved by a finite-difference scheme. The method was further simplified in two cases: (i) for a constant temperature, and (ii) for a gas with only excluded-volume interactions.