A Comparison and Unification of Ellipsoidal Statistical and Shakhov BGK Models

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.

Approximation of the linearized Boltzmann collision operator for hard-sphere and inverse-power-law models

“…Roughly speaking, the ES-BGK model is similar as the Shakhov model in the sense that both models fit only the Prandtl number, and some comparisons between these two models are carried out in [14,10]. The ES-BGK model gives a nonlinear collision term since non-equilibrium quantities appear in the parameter of the exponential.…”

Particle-based fluid dynamics: Comparison of different Bhatnagar-Gross-Krook models and the direct simulation Monte Carlo method for hypersonic flows

“…Chen et al 11 proposed a unified BGK model (UBGK), which merges the ESBGK and SBGK model. The new target distribution is…”

“…Up to now, only the ellipsoidal statistical BGK (ESBGK) model has been used in this context. Unfortunately, the ESBGK model has difficulties to reproduce the shock structure correctly 11 . In this publication, methods to sample in an efficient way from other target distribution functions used in the Shakhov and unified BGK models are described.…”

Thermally induced rarefied gas flow in a three-dimensional enclosure with square cross-section

“…The Shakhov model equation is appropriate for thermally driven flow as has been demonstrated by numerous applications in the literature, such as Refs. [19,20,[32][33][34], to name a few. It should be noted that to match the realistic gas viscosity [35,36] in a wide temperature range, the power-law index ω will vary significantly.…”