Prediction of strong-shock structure using the bimodal distribution function

A modified Mott-Smith method for predicting the one-dimensional shock wave solution at very high Mach numbers is constructed by developing a system of fluid dynamics equations. The predicted shock solutions in a gas of Maxwell molecules, a hard-sphere gas, and in argon using the newly proposed formalism are compared with the experimental data, direct-simulation Monte Carlo (DSMC) solution, and other solutions computed from some existing theories for Mach numbers $M<50$. In the limit of an infinitely large Mach number, the predicted shock profiles are also compared with the DSMC solution. The density, temperature and heat flux profiles calculated at different Mach numbers have been shown to have good agreement with the experimental and DSMC solutions.

Figure 1

Temperature profiles plotted as the function of distance. The currently predicted results are compared with those based on the theories of Navier-Stokes, Mott-Smith, and DSMC.

Figure 2

Comparison of the predicted values of the temperature-density separation, which are plotted against the Mach number. Squares: DSMC results of Pham-Van-Diep (Ref. [4]), Nanbu (Ref. [25]), and Fiscko (Ref. [21]).

Figure 3

Density profile plotted as the function of distance. Comparison of the currently predicted density profile with the DSMC and Navier-Stokes simulation results against $x$ at $M=35$; argon, $s=0.72$.

Figure 4

Temperature profile plotted as the function of distance. Comparison of the currently predicted temperature profile with the DSMC, and Burnett and Navier-Stokes simulation results against $x$ at $M=35$; argon, $s=0.72$.

Figure 5

Comparison of the predicted values of the temperature-density separation, which are plotted against the Mach number; argon, $s=0.72$.

Figure 6

Density profile plotted as the function of distance. Comparison of the currently predicted temperature profile with the nonequilibrium molecular dynamics (Refs. [23] and [26]) NSx $=$ NS with the $T$-dependence of the transport coefficients being replaced by $T_{\mathrm{XX}}$. HM (0,0.5) is the Holian-Mareschal result with the temperature relaxation only. HM (2,0.5) includes the nonlinear Burnett conductivity as well as the relaxation. Navier-Stokes simulation results against $x$ at $M=134$; a hard-sphere gas, $s=0.5$.

Figure 7

Comparison of the predicted temperature profiles plotted as the function of distance. $M=134$; a hard-sphere gas, $s=0.5$. Notation: see Fig. 5.

Figure 8

The predicted heat flux profile plotted as the function of distance. $M=134$; a hard-sphere gas, $s=0.5$. Notation: see Fig. 5.

Figure 9

Comparison of the computed inverse density thicknesses, which are plotted against the Mach number, for two monatomic gases, Maxwell molecules, and a hard-sphere gas. DSMC (Refs. [25] and [27]).