Discrete Boltzmann equation model for nonideal gases

Computer simulations of fluid dynamical problems involving fluid interfaces and phase transitions are of both fundamental and practical importance. Traditional computational fluid dynamics ~CFD! methods for solving macroscopic hydrodynamic equations have many difficulties in this area. For instance, fluid interfaces often undergo topological change due to both coalescence and phase transitions. In problems where the capillary effect is important, high resolution is required for accurate computation of interface curvature. The treatment of these problems using a Navier-Stokes solver is cumbersome, if not impossible, in many situations. The macroscopic motion of a fluid can also be solved by computing motions of its constituent particles. Since the complexity of the nonideal-gas fluid systems is essentially due to the microscopic interparticle interaction, particle methods such as molecular dynamics can simulate complex fluid phenomena naturally by implementing the correct interparticle potential. However, these methods are very inefficient for fluid simulations. At the mesoscopic level, the lattice-Boltzmann-equation ~LBE! method simulates the motion of fluids by following the evolution of a lattice Boltzmann equation that governs the behavior of the single-particle distribution function. It was found that solving the LBE directly is an efficient and accurate method for simulating fluid motion @1#. More importantly, the interparticle interaction can be easily incorporated into the LBE method to form a model that can simulate macroscopic complex fluid phenomena at least as efficiently as the conventional CFD methods solve the hydrodynamic equations for simple fluids @2#. Although the LBE method has shown its ability to simulate complex fluids, a recent study @3# shows that this method can be greatly improved if one can establish the relationship between the LBE and the continuous Boltzmann equation @3#. Historically, the continuous Boltzmann equation has mainly been used to solve supersonic flows @4#. This is partially due to both the extreme complexity of the collision kernel when dealing with dense, interacting particles and the tremendous computer resources required to resolve the sixdimensional distribution function. In this paper, we propose a computational scheme for the simulation of nonideal gases based on the continuous Boltzmann equation using a singlerelaxation-time approximation, also known as the BhatnagarGross-Krook ~BGK! collision model @5#. The interparticle attraction is treated using a mean-field approximation in the same way that the Coulomb interaction among the charged particles of a plasma is treated in the Vlasov equation @5#. Following the work of Enskog, the effect of the exclusion volume is taken into account by an additional term in the collision operator. The final Boltzmann equation is then discretized in the velocity space in a way that guarantees that the Navier-Stokes equation is obtained at the macroscopic level. This discretization is similar to the truncation made in the well-known 13-moment method of Grad @6#. The previously proposed nonideal LBE model @2# can be obtained with only minor differences. The present derivation allows the LBE model to be implemented on nonuniform grids. The ‘‘interaction potential’’ introduced previously now has a clear connection with the interparticle pairwise potential in real fluids. Analysis of some other schemes @7,8# for incorporating interparticle forces into LBE models in the framework of the present derivation shows that anisotropy is a consequence of an inappropriate intermolecular interaction. We start from the following Boltzmann equation in which the collision term is replaced by the BGK collision model,