A Comparison and Unification of Ellipsoidal Statistical and Shakhov BGK Models (Advances in Applied Mathematics and Mechanics, 2018)
The Ellipsoidal Statistical model (ES-model) and the Shakhov model (S-model)were constructed to correct the Prandtl number of the original BGK modelthrough the modification of stress and heat flux. With the introduction of a new parameterto combine the ES-model and S-model, a generalized kinetic model can bedeveloped. This new model can give the correct Navier-Stokes equations in the continuumflow regime. Through the adjustment of the new parameter, it provides abundantdynamic effect beyond the ES-model and S-model. Changing the free parameter,the physical performance of the new model has been tested numerically. The unifiedgas kinetic scheme (UGKS) is employed for the study of the new model. In transitionflow regime, many physical problems, i.e., the shock structure and micro-flows, havebeen studied using the generalized model. With a careful choice of the free parameter,good results can be achieved for most test cases. Due to the property of the Boltzmanncollision integral, the new parameter in the generalized kinetic model cannotbe fully determined. It depends on the specific problem. Generally speaking, the S-modelpredicts more accurate numerical solutions in most test cases presented in thispaper than the ES-model, while ES-model performs better in the cases where the flowis mostly driven by temperature gradient, such as a channel flow with large boundarytemperature variation at high Knudsen number.