Extension of the Shakhov Bhatnagar–Gross–Krook model for nonequilibrium gas flows

Abstract

Bhatnagar-Gross-Krook (BGK) models are widely used to study rarefied gas dynamics. However, as simplified versions of the Boltzmann collision model, their performances are uncertain and need to be carefully investigated in highly nonequilibrium flows. In this study, several common BGK models, such as the Ellipsoidal Statistical BGK (ES-BGK) and Shakhov BGK (S-BGK) models, are theoretically analyzed using their moment equations. Then, numerical comparisons are performed between the Boltzmann collision model and BGK models based on various benchmarks, such as Fourier flow, Couette flow, and shock wave. The prediction performance of the ES-BGK model is better than that of the S-BGK model in Fourier flow, while prediction performance of the S-BGK model is better than that of the ES-BGK model in Couette flow and shock wave. However, with increasing Knudsen number or Mach number, the results of both ES-BGK and S-BGK deviate from the Boltzmann solutions. These phenomena are attributed to the incorrect governing equations of high-order moments of BGK models. To improve the performance of the current BGK models, the S-BGK model is extended by adding more high-order moments into the target distribution function of the original one. Our analytical and numerical results demonstrate that the extended S-BGK (S-BGK+) model provides the same relaxation coefficients as the Boltzmann collision model for the production terms of high-order moment equations. Compared to the other BGK models, the proposed S-BGK+ model exhibits better performance for various flow regimes.

Effect of thermal fluctuations on spectra and predictability in compressible decaying isotropic turbulence

“…where the right-hand side of (3.2) describes the relaxation of VDF towards a target distribution function f t with the relaxation time τ r comparable to the molecular mean collision time τ mic . In the BGK and S-BGK models, the target distribution functions are given by the local macroscopic quantities as (Yao et al 2023)…”

“…where C = c − u is the molecular thermal velocity, R is the specific gas constant, Pr is the Prandtl number and q i is the heat flux. Compared with the original BGK model with a fixed Pr of 1, the S-BGK model can be applied to gas flows with arbitrary Pr (Yao et al 2023).…”

“…In the BGK and S-BGK models, the target distribution functions are given by the local macroscopic quantities as (Yao et al. 2023) where is the molecular thermal velocity, is the specific gas constant, is the Prandtl number and is the heat flux. Compared with the original BGK model with a fixed of 1, the S-BGK model can be applied to gas flows with arbitrary (Yao et al.…”

“…Compared with the original BGK model with a fixed of 1, the S-BGK model can be applied to gas flows with arbitrary (Yao et al. 2023).…”