Coarse-grained Equations Consistent with Boltzmann Equation for Strong Non-Equilibrium Hydrodynamics

Predicting non-equilibrium hydrodynamics, in rarefied and strong shock-heated flows, requires computing solutions of the Boltzmann equation. A set of coarse-grained hydrodynamic equations, more accurate than the conventional Navier-Stokes equations, to model translational non-equilibrium is derived. The derivation does not rely on conventional techniques such as the Chapman-Enskog like perturbation expansion or Grad-like higher order moment methods. The proposed approach decomposes the velocity space into a number of discrete groups. Next, the maximum entropy principle is applied to obtain the probability density function for each group yielding a set of conservation equations for group-specific moments (mass, momentum and energy) using the Boltzmann equation. The accuracy of the proposed equations is investigated by simulations of homogeneous relaxation of non-equilibrium gas and a 1-D standing shock-wave for a range of Mach numbers. Non-equilibrium velocity distribution functions and macroscopic quantities obtained from the proposed coarse-grained model are shown to be in excellent agreement to the solutions of the BGK-Boltzmann equation. Shock thickness is demonstrated to be in excellent agreement between the coarse-grained model and the Boltzmann equation solution.