An Approximation for the Twenty-One-Moment Maximum-Entropy Model of Rarefied Gas Dynamics

Abstract

The use of moment-closure methods to predict continuum and moderately rarefied flow offers many modelling and numerical advantages over traditional methods. The maximumentropy family of moment closures offers models described by hyperbolic systems of balance laws. In particular, the twenty-one moment model of the maximum-entropy hierarchy offers a hyperbolic treatment of viscous flows exhibiting heat transfer. This twenty-one moment model has the ability to provide accurate solutions where the Navier-Stokes equations lose physical validity due to the solution being too far from local equilibrium. Furthermore, its first-order hyperbolic nature offers the potential for improved numerical accuracy as well as a decreased sensitivity to mesh quality. Unfortunately, higher-order maximum-entropy closures cannot be expressed in closed form. The only known affordable option is to propose approximations. Previous approximations to the fourteen-moment maximum-entropy model have been proposed [M c Donald and Torrilhon, 2014]. Although this fourteen-moment model also predicts viscous flow with heat transfer, the necessary moments to close the system renders it more difficult to approximate accurately than the twenty-one moment model. The proposed approximation for the fourteen-moment model also has realizable states for which hyperbolicity is lost.Unfortunately, the velocity distribution function associated with the twenty-one moment model is an exponential of a fourth-order polynomial. Such a function cannot be integrated in closed form, resulting in closing fluxes that can only be obtained through complex numerical methods. The goal of this work is to present a new approximation to the closing fluxes that respect the maximum-entropy philosophy as closely as possible. Preliminary results show that a proposed approximation is able to provide shock predictions that are in good agreement with the Boltzmann equation and surpassing the prediction of the Navier-Stokes equations. Furthermore, Couette flow results as well as lid-driven cavity flows are computed using a novel approach to Knudsen layer boundary conditions. A dispersion analysis as ii well as an investigation of the hyperbolicity of the model is also shown. The Couette flow results are compared against Navier-Stokes and the free-molecular analytical solutions for a varying Knudsen number, for which the twenty-one moment model show good agreement over the domain. The shock-tube problem is also computed for different Knudsen numbers. The results are compared with the one obtained by directly solving the BGK equation. Finally, the lid-driven cavity flow computed with the twenty-one moment model shows good agreement with the direct simulation Monte-Carlo (DSMC) solution.

Modeling high-Mach-number rarefied crossflows past a flat plate using the maximum-entropy moment method

“…Future work suggestions include the formulation of diffusive, instead of specular-reflection, wall boundary conditions, and the extension of the presented results to the 21-moment maximum-entropy method, for which an approximated interpolative closure was recently developed. 50…”

Numerical simulation of rarefied supersonic flows using a fourth-order maximum-entropy moment method with interpolative closure

“…In the following, this is referred to as the approximated interpolative closure. This approach is not available for all maximum-entropy systems, but, to date, has been developed for the 14-moment 21 and the 21-moment 23 systems, which are fourth-order members of the maximum-entropy family of moment methods. The former is the focus of this work.…”

Stabilizing the Maximal Entropy Moment Method for Rarefied Gas Dynamics at Single-Precision

“…The ill-conditioned maximal entropy optimization problem causes numerical overflow and breakdown of the optimization when the numerical precision is insufficient [35,36,37,38]. Previous works tackling the ill-conditioned optimization problem using closed-form approximations [35,39], adaptive basis [40,36,37,38], or regularizations [37,41]. However, ill-conditionedness still leads to breakdowns for strong non-equilibrium flows [38,42].…”