Towards physically realizable and hyperbolic moment closures for kinetic theory

摘要

The numerical prediction of continuum and non-equilibrium flows by using fully hyperbolic and realizable mathematical descriptions that follow from moment closures of gas kinetic theory is reviewed. A brief review is first given of some of the current capabilities and limitations of moment closures for predicting a range of continuum and non-equilibrium flow. Next, an extended but hyperbolic Gaussian closure for diatomic gases that does not account for heat-transfer effects, as well as a regularized version of this closure that incorporates anisotropic thermal-diffusion effects via the inclusion of higher order terms having an elliptic nature, are both reviewed and applied to a number of canonical flow problems. The numerical results for the Gaussian closures clearly demonstrate the capabilities and potential of moment closures and purely hyperbolic treatments. Following these reviews, a somewhat novel hierarchy of physically realizable and hyperbolic moment closures is considered and described. This alternative hierarchy is based on modifications to the more common maximum-entropy hierarchies, so as to ensure the validity and integrability of the approximate distribution function for all values of the velocity moments that are physically realizable. The predictive capabilities of this new closure hierarchy are then explored by considering numerical solutions of the closures for a one-dimensional kinetic equation with a relaxation-time collision operator and by comparing the closure solutions to discrete numerical solutions of this simplified kinetic equation. The study concludes with a brief summary of the findings and a discussion of the potential of the physically realizable and hyperbolic moment closures for application to fully three-dimensional physics.