Discrete Boltzmann methods have emerged as a powerful technique for the computational modeling of a wide variety of complex fluid flow problems including single and multiphase flow in complex geometries [1,2,3,4,5,6,7]. While much progress has been made in the construction of discrete Boltzmann based models that describe single phase fluid flow with heat transport, isothermal fluids that undergo liquid/vapor phase transitions, and the phase separation of binary mixtures, until now no self-consistent model exists that conserves energy when a molecular interaction potential is included. Some computational fluid dynamics (CFD) based algorithms [8], which describe non-ideal gas systems, can be explicitly constructed to conserve mass, momentum, and energy. However, in this case, information about the effects of molecular interactions are incorporated into a modified stress tensor that is derived from a phenomenological Cahn Hilliard free energy which is typically expressed in terms of an expansion of density and density gradients.
In this paper it will be shown how energy conservation for a non ideal system can be incorporated into a discrete Boltzmann model. The basic idea is to properly construct the collision operator, used in the discrete Boltzmann equations, so as to account for momentum and energy transfer due to a molecular interactions. Information about momentum and energy transfer can be directly obtained from the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) [9,10] formalism. This method should serve as an ideal mesoscopic approach that bridges microscopic phenomena with the continuum macroscopic equations. Further, it can be directly implemented as a numerical method to model the time evolution of such systems.
An outline of this paper goes as follows. First, pertinent results obtained from the BBGKY equations will be reviewed including construction of moments of the collision operator and demonstration of energy conservation. A modified Bhatnagar, Gross and Krook [11] (BGK) representation of the collision operator that is consistent with energy conservation will then be described. An earlier model of Shan and Chen [4] describing an isothermal non-ideal gas as well as other mean-field approaches to modeling non-ideal systems, will be shown to be consistent with this formalism. A more general Hermite polynomial [12] representation of the collision operator will be presented and shown to be consistent with the modified BGK form up to 2nd order. An example of a numerical implementation of the modified BGK form will be given. The paper will conclude with a discussion concerning methods for further improvement of the numerical implementation and of other potential applications.