Multiple Temperature Gas Dynamic Equations for Non-Equilibrium Flows (Journal of Computational Mathematics, 2021)
In an early approach, a kinetic model with multiple translational temperature [K. Xu, H. Liu and J. Jiang, Phys. Fluids 19, 016101 (2007)] to simulate non-equilibrium flows was proposed. In this paper, instead of using three temperatures in the $x$, $y$ and $z$-directions, we define the translational temperature as a second-order symmetric tensor. Under the new framework, the differences between the temperature tensor and the pressure tensor will be explicitly pointed out. Based on a multiple stage BGK-type collision model and the Chapman-Enskog expansion, the corresponding macroscopic gas dynamics equations in three-dimensional space will be derived. The zeroth-order expansion gives 10 moment closure equations similar to that of Levermore [C.D. Levermore, J. Stat. Phys 83, pp.1021 (1996)]. The derived gas dynamic equations can be considered as a regularization of the 10 moments equations in the first-order expansion. The new gas dynamic equations have the same structure as the Navier-Stokes equations, but the stress-strain relationship in the Navier-Stokes equations is replaced by an algebraic equation with temperature differences. At the same time, the heat flux, which is absent in Levermore's 10 moment closure, is recovered. As a result, both the viscous and the heat conduction terms are unified under a single anisotropic temperature concept. In the continuum flow regime, the new gas dynamic equations automatically recover the standard Navier-Stokes equations. Our gas dynamic equations are natural extensions of the Navier-Stokes equations to the near continuum flow regime and can be used for microflow computations. Two examples, the force-driven Poiseuille flow and the Couette flow in the transition flow regime, are used to validate the model. Both analytical and numerical results are presented. Theoretically, the Boltzmann equation can be also applied to the current multiple stage gas evolution model to derive generalized macroscopic governing equations in the near continuum flow regime. Instead of using Maxwellian as an expansion point in the Chapman-Enskog method, the multiple temperature Gaussian can be used as an expansion point as well.