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 maximum-entropy family of moment closures offers models described by hyperbolic systems of equations. In particular, the twenty-one moment model of the maximum-entropy hierarchy offers a hyperbolic treatment of viscous flows exhibiting heat transfer. This model has the ability to provide accurate solutions where the Navier–Stokes equations lose physical validity. Furthermore, its first-order hyperbolic nature offers the potential for improved numerical accuracy as well as a decreased sensitivity to mesh quality. Unfortunately, the distribution function associated with the 21 moment model is an exponential of a fourth-order polynomial. Such a function cannot be integrated in closed form, resulting in unobtainable closing fluxes. This work presents an approximation to the closing fluxes that respects the maximum-entropy philosophy as closely as possible. The proposed approximation is able to provide shock predictions in good agreement with the Boltzmann equation and surpassing the prediction of the Navier–Stokes equations. A dispersion analysis as well as an investigation of the hyperbolicity of the model is also shown.