Solid-Boundary Treatment for Moment Systems.

Abstract

Hyperbolic-relaxation systems for describing fluid flow, such as Grad's 13-moment (13M) system, are an alternative to the systems generated by the Chapman-Enskog expansion. The former have a number of numerical advantages stemming from the compactness of their spatial operator. They have been demonstrated to be suited for subsonic and transonic flows.

In this dissertation we explore the boundary conditions that have to supplement these moment systems. Starting from the treatment proposed by Grad, we first remove an inconsistency in his theory that would lead to ambiguous results. We formulate two conditions, 1-C and 2-C, containing one and two parameters, respectively; these are used to satisfy the non-penetration condition in 1-C and 2-C, and an additional normalization condition in 2-C.

In order to study the accuracy of the new boundary condition we also develop a benchmark code based on the Boltzmann equation with Bhatnagar-Gross-Krook collision model (BGK equation), and implement the same conditions in there. Both codes include a Finite-Volume (FV) and a Discontinuous Galerkin (DG) option (piecewise-linear discretization). These codes are used to solve three types of flow problems: low-speed Couette flow with zero difference in wall temperatures, zero-speed Couette flow with appreciable wall-temperature difference, and low-speed Poiseuille flow. The Knudsen number ranged from 0.01 to 10. For these problems, incompressible Navier-Stokes-Fourier (NSF) solutions are available for comparison (except for the temperature-driven flow); here the boundary condition used is the velocity-slip boundary conditions.

The numerical experiments indicate that the 13M/1-C system consistently predicts a flow velocity at the wall that is closer to the BGK solution than NSF does. The 13M profiles of some quantities show a difficulty with relaxing toward the mid-channel value; the profiles are not monotone and become increasingly oscillatory with increasing Knudsen number. The use of the 1-C or 2-C condition with the 13M system makes little difference. A surprising finding is that using DG in velocity space with BGK was not superior to using FV. However, using a DG discretization in physical space is advantageous, as it removes an anomaly at the wall showing up in FV solutions due to one-sided reconstruction.