A Comparison and Unification of Ellipsoidal Statistical and Shakhov BGK Models

TL;DR

This paper introduces a generalized kinetic model combining ES and S models, validated through numerical simulations, which offers improved accuracy in non-equilibrium flow regimes by adjusting a free parameter.

Contribution

A new unified kinetic model blending ES and S models with a free parameter, enabling better accuracy in transition flows compared to existing models.

Findings

The generalized model can recover ES and S models as special cases.

The S-model generally predicts more accurate solutions in tough test cases.

The ES-model performs better in heat-driven flows with high Knudsen numbers.

Abstract

The Ellipsoidal Statistical model (ES-model) and the Shakhov model (S-model) are constructed for the correction of Prandtl number of the original BGK model through the modification of stress and heat flux. Even though in the continuum flow regime, both models can give the same Navier-Stokes equations with correct Prandtl number, their modification of the collision term may have different dynamic effect in the non-equilibrium transition flow regimes. With the introduction of one free parameter, a generalized kinetic model with the combination of the ES-model and S-model can be developed, and this new model can get the correct Navier-Stokes equations in the continuum flow regime as well, but with abundant dynamic effect through the adjustment of the new degree of freedom. In order to validate the generalized model, a numerical method based on the unified gas kinetic scheme (UGKS) has been developed for the new model. The physical performance of the new model with the variation of the free parameter has been tested, where the ES-model and S-model become the limiting cases. In transition flow regime, many physical problems, i.e., the shock structure and micro-flows, have been studied using the generalized model. With a careful choice of the free parameter, good results can be achieved for most test cases. The overall conclusion is that the S-model predicts more accurate numerical solutions in most tough test cases presented in this paper than the ES-model, while ES-model performs better in the cases when the flow is mostly driven by heat, such as a channel flow with large boundary temperature variations at high Knudsen number. The numerical study demonstrates the necessity of developing such a generalized model. With the inclusion of one more freedom, in the transition regime the new kinetic model may provide more accurate solution than the ES and Shakhov models.