On the accuracy of macroscopic equations for linearized rarefied gas flows

Abstract

Many macroscopic equations are proposed to describe the rarefied gas dynamics beyond the Navier-Stokes level, either from the mesoscopic Boltzmann equation or some physical arguments, including (i) Burnett, Woods, super-Burnett, augmented Burnett equations derived from the Chapman-Enskog expansion of the Boltzmann equation, (ii) Grad 13, regularized 13/26 moment equations, rational extended thermodynamics equations, and generalized hydrodynamic equations, where the velocity distribution function is expressed in terms of low-order moments and Hermite polynomials, and (iii) bi-velocity equations and "thermo-mechanically consistent" Burnett equations based on the argument of "volume diffusion". This paper is dedicated to assess the accuracy of these macroscopic equations. We first consider the Rayleigh-Brillouin scattering, where light is scattered by the density fluctuation in gas. In this specific problem macroscopic equations can be linearized and solutions can always be obtained, no matter whether they are stable or not. Moreover, the accuracy assessment is not contaminated by the gas-wall boundary condition in this periodic problem. Rayleigh-Brillouin spectra of the scattered light are calculated by solving the linearized macroscopic equations and compared to those from the linearized Boltzmann equation. We find that (i) the accuracy of Chapman-Enskog expansion does not always increase with the order of expansion, (ii) for the moment method, the more moments are included, the more accurate the results are, and (iii) macroscopic equations based on "volume diffusion" do not work well even when the Knudsen number is very small. Therefore, among about a dozen tested equations, the regularized 26 moment equations are the most accurate. However, for moderate and highly rarefied gas flows, huge number of moments should be included, as the convergence to true solutions is rather slow. The same conclusion is drawn from the problem of sound propagation between the transducer and receiver. This slow convergence of moment equations is due to the incapability of Hermite polynomials in the capturing of large discontinuities and rapid variations of the velocity distribution function. This study sheds some light on how to choose/develop macroscopic equations for rarefied gas dynamics.

Recasting Navier–Stokes equations

“…When Kn lies in the regime 0.001 Kn 0.1   contributions from the central Rayleigh part and the Brillouin side part become mixed and then the widths of both parts broaden due to the increased dissipation in sound propagation. Further increase of Kn results in vanishing contribution of the Brillouin part and the whole spectrum becomes nearly Gaussian [26][27][28]66]. In the typical spectra of the coherent RBS, one can notice the presence of Brillouin peaks only when the gas flow is in the hydrodynamic regime (Kn0.001).…”

“…In the typical spectra of the coherent RBS, one can notice the presence of Brillouin peaks only when the gas flow is in the hydrodynamic regime (Kn0.001). As Kn increases further, both peaks (the central Rayleigh and the two Brillouin side peaks) are present and the relative intensity of these peaks becomes large and later on the whole spectrum becomes Gaussian [26][27][28].…”

“…First, we compare the spontaneous RBS spectra solutions obtained from the re-casted Navier-Stokes with that of the classical Navier-Stokes solutions and also with the results calculated based on the Linearized Boltzmann equation (LBE) for Maxwellian gases and taken from Wu [28]. Figure 6 illustrates the spontaneous RBS spectra obtained from the re-casted Navier-Stokes-I model along with the spectra obtained from the classical Navier-Stokes and the LBE.…”

“…For the physical process and the experimental set-up of spontaneous RBS and coherent RBS, readers are referred to references [64,65]. The one-dimensional, linearized nature and lack of boundary conditions make the hydrodynamic equations simple and allow for analytical solutions for the Rayleigh-Brillouin scattering problem [24,25,28]. The spectrum of the scattered light is characterized by the Knudsen number (Kn), which is here the ratio of the mean free path of gas molecules to the characteristic length scale (L) of the system, identified as the scattering wave length k 2p .…”

“…Leaving aside non-linear configurations, conventional Navier-Stokes equations also fail to describe some of the linear flow problems accurately. For example, it is unsuccessful in describing the actual spectrum shapes in the Rayleigh-Brillouin scattering problem [23][24][25][26][27][28].…”