Validation and analysis of gas-kinetic unified algorithm for solving Boltzmann model equation with vibrational energy excitation

Abstract

With the increase of temperature in flow field,gas molecules possess not only rotational degree of freedom,but also vibrational energy excitation.In order to simulate and study the influence of internal energy excitation on polyatomic gas flow with high temperature and high Mach number,according to the general Boltzmann equation,we consider the rotational and vibrational energy modes as the independent variables of gas molecular velocity distribution function.It is assumed that the rotational and vibrational energy modes are described by continuous distribution with degree of freedom and temperature.Based on the Borgnakke-Larsen collision model used in direct simulation Monte Carlo (DSMC) method, the collision term of Boltzmann equation with internal energy excitation is divided into elastic and inelastic collision terms.The inelastic collision is decomposed into translational-rotational energy relaxation and translational-rotationalvibrational energy relaxation according to a certain relaxation rate obtained from the reciprocalities of rotational and vibrational collisions numbers per one elastic collision.Then a kind of Boltzmann model equation considering the excitation of vibrational energy is constructed.For showing the consistency between the present model equation and Boltzmann equation,the conservation of summational invariants and the H-theorem of this model are proved.When solving the present model equation with numerical methods,because of the continuous energy modes,it is difficult to simulate this model equation directly.In this paper,three control equations are derived and solved by the LU-SGS (lower-upper symmetric Gauss-Seidel) method,and the cell-centered finite volume method with multi-block patched grid technique in physical space.As a result,these gas-kinetic unified algorithm (GKUA) with vibrational energy excitation has been developed.Results are presented for N2 with different Knudsen numbers around cylinder from continuum to rarefied gas flow by using the present Boltzmann model equation,GKUA with simple gas model,and DSMC method. Very good agreement between the present model and DSMC results is obtained,which shows that the accuracy and reliability of the present model.Comparing the translational,rotational,vibrational,and total temperatures computed by different methods,the effects of the rotational and vibrational degrees of freedom are demonstrated.For the simple gas model,the translational temperature is much higher than those for the other two models with internal energy excitation. At the same time,the distance from shock wave to wall for the simple gas model is about twice those for the other two models.On the other hand,the obtained aerodynamic force coefficients of the cylinder are increasing according to the sequence from the simple gas model to the rotational energy excitation model to the vibrational energy excitation model, but the variation range is very small.By reducing the gas characteristic vibrational temperature,the temperature after the shock wave is much lower,and the heat flux declines evidently at the stagnation point with the same temperature as the wall temperature.This implies that with the wall temperature increasing the heat flux declines.