Evaluation of BGK-type models of the Boltzmann equation

Abstract

A BGK-type model is a simplified model to the Boltzmann equation where the complicated collision integral is approximated using a simple relaxation term. In this paper, a generalized BGK-type model is proposed by combining the original BGK, Shakhov and ellipsoidal statistical models. To verify the new model, theoretical and numerical analysis is performed. The Chapman-Enskog expansion shows that the generalized model can recover the Navier-Stokes-Fourier and Burnett equations by choosing properly the introduced parameters. In addition, the collision term of the model is evaluated for several designed relaxation problems. Results show that the new model can match the results of third-order moments obtained with DSMC method by adjusting the free parameters.

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.…”

“…Assume that the integral in (4.9) is already evaluated and the result is (4.13) we can directly write I l 2 n 2 n 2 (θ) defined in (4.2) as 14) and the above integral can still be evaluated using formula (4.4). Substituting the above equation into (3.42), one can see that for any l, n and n , the coefficient a lnn is always proportional to m −1 θ n−n + η−5 2(η−1) .…”

Approximation of the Boltzmann collision operator based on hermite spectral method

“…Numerical evaluations on these BGK-type models can be found in [19,10], where one can find that these approximations are not accurate enough when the non-equilibrium is strong. Hence the study on efficient numerical methods for the original Boltzmann equation with the quadratic collision operator is still necessary.…”

Discrete Boltzmann multi-scale modelling of non-equilibrium multiphase flows

“…Correspondingly, high-order kinetic models consider non-equilibrium effects via higher-order distribution functions deviating from equilibrium. As a result, they are expected to be applicable to flows in the slip and transition flow regimes where the NS equations perform poorly (Struchtrup 2005; Gao & Sun 2014). A popular strategy for deriving such models is to perform a Chapman–Enskog multi-scale expansion.…”

“…2015) and collision frequency (Struchtrup 1997); pseudo-equilibrium distribution functions that contain non-equilibrium information (Holway 1966; Shakhov 1968; Shan et al. 2006; Gao & Sun 2014; Watari 2016); internal degrees of freedom (Rykov 1975) and even quantum vibrational energy (Wu et al. 2021 a ); and appropriate kinetic boundary conditions (Wagner & Pagonabarraga 2002; Sbragaglia & Succi 2005; Sofonea & Sekerka 2005; Toschi & Succi 2005; Benzi et al.…”