Study of the shock wave structure by regularized Grad’s set of equations

Abstract

In this work, we continue to study the possibility of applying moment equations for strongly nonequilibrium flows by an example of the problem of the shock wave structure in a monatomic gas in a wide range of Mach numbers for various models of molecular interaction. The object of the study is the so-called regularized 13-moment Grad's system (R13). First time, both linear and nonlinear versions of this system of equations were considered for the problem at such wide range of parameters. The Godunov method with increased accuracy is used as a numerical tool for solving the R13 system. The numerical results for the R13 system are analyzed by using data obtained by the Direct Simulation Monte Carlo (DSMC) method, experimental data, and analytical results. As a whole, the R13 system provides an adequate description of the shock wave structure in a wide range of Mach numbers. For Mach numbers around 2, good agreement with experimental and DSMC results is observed for both linear and nonlinear versions of the system. For high Mach numbers, the result strongly depends on the molecular interaction model used: shock wave structure predictions of the nonlinear R13 system are better for Maxwell molecules and worse for hard spheres as compared to the linear version. Particular attention in this work is paid to studying nonmonotonicity of the total temperature profile (temperature overshoot) in the structure of a strong shock wave. It is shown that the moment equations correctly predict the existence of the temperature overshoot. At the same time, the solution of the moment equations overpredicts the temperature overshoot at least two-fold for Mach number M = 8, and the nonlinear version of the R13 system yields a better result for this parameter than the linear version. C 2015 AIP Publishing LLC. [http://dx.

The analysis of different variants of R13 equations applied to the shock-wave structure

“…The terms omitted in the linear case form the so-called non-gradient transport mechanisms (NGTM) [5]. The linear variant (10) and the original nonlinear variant (7)-(9) were studied for shock structures in [6][7][8].…”

High-accuracy deterministic solution of the Boltzmann equation for the shock wave structure

“…The existence of this peak predicted by approximate analytical methods by several researchers, e.g., [14], is questioned in [1]. However, the overshoot was obtained in DSMC computations (see, e.g., [16][17][18][19]). Unfortunately, at the moment the existence of this peak cannot be confirmed by experiments because of the lack of temperature data at high Mach numbers.…”

R13 moment equations applied to supersonic flow with solid wall interaction

“…The variant of high resolution Godunov scheme with linear flow parameter reconstruction is chosen for numerical solution of the R13 set of equations [21]. The second-order accuracy in space for smooth solutions is achieved using essentially two-dimensional reconstruction procedures [12,21] for the primitive variables within each computational cell. The standard central difference approximation is used for elliptical part discretization of the fluxes (diffusion terms).…”

“…The difference between them lies in the relations for higher order moments [8,9]. It has been shown that all variants are well applicable for numerical modelling of the moderately rarefied slow gas flows and allow the modelling of main nonequilibrium effects in low-velocity micro-flow problems [10,11,12,13,14,15,16,17] and some moderately rarefacted supersonic flows [5,18,19,15,20,21,22]. The difficulty is that the solid wall boundary conditions for R13 equations were obtained from one of the variants [10] which is not applicable to near hypersonic flows [9].…”