A gas-kinetic model with multiple translational temperature for the continuum and near continuum flow simulations is proposed. The main purpose for this work is to derive the generalized Navier-Stokes equations with multiple temperature. It is well recognized that for increasingly rarefied flowfields, the predictions from continuum formulation, such as the Navier-Stokes equations lose accuracy. These inaccuracies may be partially due to the single temperature assumption in the standard Navier-Stokes equations. Here, based on an extended Bhatnagar-Gross-Krook (BGK) model with multiple translational temperature, the numerical scheme for its corresponding Navier-Stokes equations is also constructed. In the current approach, the energy exchange between x, y, and z directions is modeled through the particle collision, and individual energy equation in different direction is obtained. The kinetic model, newly constructed is an enlarged system in comparison with Holway’s ellipsoid statistical BGK model (ES-BGK). The detailed difference is presented in this paper. In the newly derived “Navier-Stokes” equations from the current model, all viscous terms are replaced by the temperature relaxation terms. The relation between the stress and strain in the standard Navier-Stokes equations is recovered only in the limiting case when the flow is close to the equilibrium, such as small temperature differences in different directions. In order to validate the generalized Navier-Stokes equations, we apply them to the study of Couette and Poiseuille flows with a wide range of Knudsen numbers. In the continuum flow regime, the standard Navier-Stokes solutions are precisely recovered. In the near continuum flow regime, the simulation results are compared with the direct simulation Monte Carlo solutions. The anomalous phenomena in the pressure and temperature distributions from the standard Navier-Stokes equations in the Poiseuille flow case at Kn = 0.1 are well resolved by the generalized Navier-Stokes equations. This paper clearly shows that many thermal nonequilibrium phenomena in the near continuum flow regime can be well captured by modifying some assumptions in the standard Navier-Stokes equations.
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