Inverse Chapman-Enskog derivation of the thermohydrodynamic lattice-BGK model for the ideal gas

A thermohydrodynamic lattice-BGK model for the ideal gas was derived by Ale xander et al. in 1993, and generalized by McNamara et al, in the same year. In these works, particular forms for the equilibrium distribution function and the transport coefficients were posited and shown to work, thereby est ablishing the sufficiency of the model. In this paper, we rederive the mode l from a minimal set of assumptions, and thereby show that the forms assume d for the shear and bulk viscosities are also necessary, but that the form assumed for the thermal conductivity is not. We derive the most general for m allowable for the thermal conductivity, and the concomitant generalizatio n of the equilibrium distribution. In this way, we show that it is possible to achieve variable (albeit density-dependent) Prandtl number even within a single-relaxation-time lattice-BGK model. We accomplish this by demanding analyticity of the third moments and traces of the fourth moments of the e quilibrium distribution function. The method of derivation demonstrates tha t certain undesirable features of the model - such as the unphysical depend ence of the viscosity coefficients on temperature - cannot be corrected wit hin the scope of lattice-BGK models with constant relaxation time.