Rarefied Gas Dynamics conference 2022

On the Application of Maximum-Entropy Inspired Multi-Gaussian Moment Closure for Multi-Dimensional Non-Equilibrium Gas Kinetics

Co-Authors: Groth, Clinton P. T.; Frédérique Laurent

Maximum-entropy moment closures for predicting the behaviour of non-equilibrium rarefied gaseous flows have previously been shown to provide accurate and computationally efficient descriptions of transition-regime flows in which the usual continuum-regime Navier-Stokes-Fourier equations do not hold, and where Monte-Carlo and/or direct-discretization methods can be far too expensive for practical use [1]. Unfortunately, above second order in velocity space, there are no analytical closures for the systems of partial differential equations (PDEs) which govern the transport of the macroscopic moment quantities and instead approximate closures have been sought. In this study, a bi-Gaussian approximation to the number density function (NDF) is used to approximate the both the maximum-entriopy NDF and closing moment fluxes of fourth-order 14-moment maximum-entropy closure associated with fully multi-dimensional kinetic theory. Prior investigation by Laplante and Groth [2] of the bi-Gaussian approximation has shown that it can produce excellent results for one-dimensional univariate kinetic equations which compare very well with the actual maximum-entropy solutions as well as the interpolative-based maximum-entropy-based (IMEB) closure of McDonald and Torrilhon [3]. A potential benefit of the bi-Gaussian approach is that an essentially closed-form analytical solution results for the NDF. In this study, the extension of the bi-Gaussian closure to the multi-dimensional case is considered and compared to the equivalent multi-dimensional IMEB closure [3] [4]. It is shown that the bi-Gaussian closure suffers from several deficiencies: firstly, valid region of realizable space for the bi-Gaussian closure is a small subset of realizable 14-moment space; and secondly, the closure and moment equation eignestructure for solutions associated with zero heat flux become undefined.

Image: Number density function for (left) actual Maximum Entropy distribution, (middle) Bi-Gaussian distribution with small step in heat transfer in the X direction, and (right) Bi-Gaussian distribution with small step in heat transfer in the Y direction.