Boundary conditions for Grad's 13 moment equations

Abstract

A complete set of boundary conditions for Grad's 13 moment equations is derived from Maxwell's boundary conditions for the Boltzmann equation. The equations are solved for plane Couette flow. The results exhibit temperature jump and slip, and agree well with DSMC calculations for Knudsen numbers Kn ≤ 0.1. Nonlinear effects lead to unphysical results at larger Knudsen numbers, and for very fast flows. A simplified version of the Grad 13 equations, the so-called bulk equations, gives meaningful results in conditions where the full set of equations fails.

NR$xx$ Simulation of Microflows with Shakhov Model

“…The planar Couette flow is a classic benchmark test in the field of microflows. The moment method for this problem has been investigated in a lot of papers such as [25,18,26,10,28,11]. Here we consider the symmetric Couette flow.…”

“…It is numerically implemented by first constructing a set of moments satisfying the boundary conditions, and then approximating the flow state in the ghost cell with a first order extrapolation of each moment. Thus, boundary conditions for the NRxx method of all orders are collected into a uniform framework, which avoids separate and involved implementation for different systems with sophisticated expressions [26,28,11].…”

“…Boundary conditionsIn the moment methods, the boundary condition is always a complicated issue when simulating microflows. As discussed in[7,20,26,28,11] and the references therein, delicate derivations and careful numerical techniques are needed for a solid wall. In this section, a numerical way for dealing with boundary conditions in the NRxx method is introduced, which appears to be uniform for all orders of moment systems.…”

Application of the Method of Generating Functions to the Derivation of Grad’s N-Moment Equations for a Granular Gas

“…This question is far from trivial [28]. A very popular choice is the Maxwell boundary conditions (MBC) which is a boundary condition on the whole single particle distribution function and indirectly on its moments (e.g., [30]). The model is defined as follows.…”