On derivation and verification of a kinetic model for quantum vibrational energy of polyatomic gases in the gas-kinetic unified algorithm

Abstract

In this paper, a kinetic Boltzmann model equation involving internal degrees of freedom is constructed for thermodynamic non-equilibrium polyatomic gases, in which the continuous distribution mode of rotational energy and discrete quantum effect of the vibrational energy are both considered, respectively. By theoretically analyzing the “recurrence feature” of this model, it is found that the root of the quantum effect comes from the translational-rotational-vibrational relaxation term (G3t,r,v), because of the additional term relative to the continuous distribution mode of vibrational energy. Besides, some important properties of this model are also analyzed and proved, including the conservative property and entropy inequality. Then, the balance equations on each vibrational energy level and balance equations about macroscopic variables are derived by the moment method. The corresponding thermodynamic relationship in the hydrodynamic limit is also analyzed by the simplified version of Chapman-Enskog method, and the correct Prandtl number is obtained. Numerical methods and wall surface boundary condition to solve this model equation are presented here. Finally, planar flows past a cylinder with different Mach numbers are numerically simulated to test and verify the reliability of this model, and to analyze the influence rules of the excited quantum vibrational energy for polyatomic gas flows.