Volume 7, Issue 2
A Comparison and Unification of Ellipsoidal Statistical and Shakhov BGK Models

Songze Chen, Kun Xu, & Qingdong Cai

Adv. Appl. Math. Mech., 7 (2015), pp. 245-266.

Published online: 2018-05

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  • Abstract

The Ellipsoidal Statistical model (ES-model) and the Shakhov model (Smodel) were constructed to correct the Prandtl number of the original BGK model through the modification of stress and heat flux. With the introduction of a new parameter to combine the ES-model and S-model, a generalized kinetic model can be developed. This new model can give the correct Navier-Stokes equations in the continuum flow regime. Through the adjustment of the new parameter, it provides abundant dynamic effect beyond the ES-model and S-model. Changing the free parameter, the physical performance of the new model has been tested numerically. The unified gas kinetic scheme (UGKS) is employed for the study of the new model. 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. Due to the property of the Boltzmann collision integral, the new parameter in the generalized kinetic model cannot be fully determined. It depends on the specific problem. Generally speaking, the Smodel predicts more accurate numerical solutions in most test cases presented in this paper than the ES-model, while ES-model performs better in the cases where the flow is mostly driven by temperature gradient, such as a channel flow with large boundary temperature variation at high Knudsen number.

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@Article{AAMM-7-245, author = {Songze Chen, Kun Xu, and Qingdong Cai}, title = {A Comparison and Unification of Ellipsoidal Statistical and Shakhov BGK Models}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {7}, number = {2}, pages = {245--266}, abstract = {

The Ellipsoidal Statistical model (ES-model) and the Shakhov model (Smodel) were constructed to correct the Prandtl number of the original BGK model through the modification of stress and heat flux. With the introduction of a new parameter to combine the ES-model and S-model, a generalized kinetic model can be developed. This new model can give the correct Navier-Stokes equations in the continuum flow regime. Through the adjustment of the new parameter, it provides abundant dynamic effect beyond the ES-model and S-model. Changing the free parameter, the physical performance of the new model has been tested numerically. The unified gas kinetic scheme (UGKS) is employed for the study of the new model. 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. Due to the property of the Boltzmann collision integral, the new parameter in the generalized kinetic model cannot be fully determined. It depends on the specific problem. Generally speaking, the Smodel predicts more accurate numerical solutions in most test cases presented in this paper than the ES-model, while ES-model performs better in the cases where the flow is mostly driven by temperature gradient, such as a channel flow with large boundary temperature variation at high Knudsen number.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.2014.m559}, url = {http://global-sci.org/intro/article_detail/aamm/12047.html} }
TY - JOUR T1 - A Comparison and Unification of Ellipsoidal Statistical and Shakhov BGK Models AU - Songze Chen, Kun Xu, & Qingdong Cai JO - Advances in Applied Mathematics and Mechanics VL - 2 SP - 245 EP - 266 PY - 2018 DA - 2018/05 SN - 7 DO - http://doi.org/10.4208/aamm.2014.m559 UR - https://global-sci.org/intro/article_detail/aamm/12047.html KW - AB -

The Ellipsoidal Statistical model (ES-model) and the Shakhov model (Smodel) were constructed to correct the Prandtl number of the original BGK model through the modification of stress and heat flux. With the introduction of a new parameter to combine the ES-model and S-model, a generalized kinetic model can be developed. This new model can give the correct Navier-Stokes equations in the continuum flow regime. Through the adjustment of the new parameter, it provides abundant dynamic effect beyond the ES-model and S-model. Changing the free parameter, the physical performance of the new model has been tested numerically. The unified gas kinetic scheme (UGKS) is employed for the study of the new model. 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. Due to the property of the Boltzmann collision integral, the new parameter in the generalized kinetic model cannot be fully determined. It depends on the specific problem. Generally speaking, the Smodel predicts more accurate numerical solutions in most test cases presented in this paper than the ES-model, while ES-model performs better in the cases where the flow is mostly driven by temperature gradient, such as a channel flow with large boundary temperature variation at high Knudsen number.

Songze Chen, Kun Xu, & Qingdong Cai. (1970). A Comparison and Unification of Ellipsoidal Statistical and Shakhov BGK Models. Advances in Applied Mathematics and Mechanics. 7 (2). 245-266. doi:10.4208/aamm.2014.m559
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