Effects of molecular diffusivity on shock-wave structures in monatomic gases

We present a full investigation into shock-wave profile description using hydrodynamics models. We identified constitutive equations that provide better agreement for all parameters involved in testing hydrodynamic equations for the prediction of shock structure in a monatomic gas in the Mach number range $1.0–11.0$. The constitutive equations are extracted from a previously derived thermomechanically consistent Burnett regime continuum flow model. The numerical computations of the resulting hydrodynamic equations along with classical ones are performed using a finite difference global solution (FDGS) scheme. Compared to previous studies that focused mainly on the density profile across the shock, here we also include temperature profiles as well as non-negativity of entropy production throughout the shock. The results obtained show an improvement upon those obtained previously in the bivelocity (or volume and mass diffusion) hydrodynamics and are more accurate than in the hydrodynamic models from expansions method solutions to the Boltzmann equation.

Figure 1

Variation of normalized velocity ($u_{\mathrm{N}}$) profiles in $\mathrm{Ar}$ shock layer: (a) ${\mathrm{Ma}}_1=1.55$, (b) ${\mathrm{Ma}}_1=3.38$

Figure 2

Variation of normalized density (${\rho }_{\mathrm{N}}$) profiles in $\mathrm{Ar}$ shock layer: (a) ${\mathrm{Ma}}_1=1.55$, (b) ${\mathrm{Ma}}_1=2.05$, (c) ${\mathrm{Ma}}_1=2.31,$ (d) ${\mathrm{Ma}}_1=3.38$. In each panel, the dashed line (black) represents the solution of classical Navier-Stokes equations (CNS), the solid line (red) represents the solution of modified Navier-Stokes equations (MNS), and the filled circles (blue) represent the experimental data of Alsmeyer [15]. The DSMC data from Alsmeyer [15] is superimposed in panels (a) and (b) as filled squares (green).

Figure 3

Variation of normalized density (${\rho }_{\mathrm{N}}$) profiles in $\mathrm{Ar}$ shock layer: (a) ${\mathrm{Ma}}_1=3.8$, (b) ${\mathrm{Ma}}_1=6.5$, (c) ${\mathrm{Ma}}_1=8,$ (d) ${\mathrm{Ma}}_1=9$. In each panel, the dashed line (black) represents the solution of classical Navier-Stokes equations (CNS), the solid line (red) represents the solution of modified Navier-Stokes equations (MNS), and the filled circles (blue) represent the experimental data of Alsmeyer [15]. The DSMC data from Alsmeyer [15] is superimposed in panels (a) and (d) as filled squares (green).

Figure 4

Variation of normalized temperature ($T_N$) profiles in $\mathrm{Ar}$ shock layer: (a) ${\mathrm{Ma}}_1=1.55$, (b) ${\mathrm{Ma}}_1=2.05$, (c) ${\mathrm{Ma}}_1=3.8,$ (d) ${\mathrm{Ma}}_1=9$. In each panel, the dashed line (black) represents the solution of classical Navier-Stokes equations (CNS), the solid line (red) represents the solution of modified Navier-Stokes equations (MNS), and the filled squares (green) represent the DSMC data taken from Alsmeyer [15].

Figure 5

Schematic of shock macroscopic parameters—shock thickness $\left(L_{\rho }\right)$, shock asymmetry $\left(Q_{\rho }\right),$ and temperature-density separation $\left({\delta }_{T\rho }\right)$.

Figure 6

(a, b) Variation of the reciprocal shock thickness, $\delta$, as a function of upstream Mach number, ${\mathrm{Ma}}_1$, for argon gas. Details on the description of plotted data is indicated in Table 1.

Figure 7

(a) Variation of density asymmetry or shape factor, $Q_{\rho }$, as a function of upstream Mach number, ${\mathrm{Ma}}_1$, for argon gas. (b) Variation of temperature-density spatial lag, ${\delta }_{T\rho }$, as a function of upstream Mach number, ${\mathrm{Ma}}_1$, for argon gas. Details on the description of plotted data is indicated in Table 1.

Figure 8

(a, b) Nondimensional entropy production rate $\left({\overline{\dot{s}}}_{\mathrm{gen}}\right)$ and (c, d) nondimensional specific entropy profiles within the shock: (a, c) ${\mathrm{Ma}}_1=1.75$, (b, d) ${\mathrm{Ma}}_1=6$. In each panel, the dashed line (black) and the solid line (red) refer to solutions from classical Navier-Stokes (CNS) and modified Navier-Stokes equations (MNS), respectively. The available DSMC data from Schrock [61] are superimposed for the specific entropy profiles in panels (c) and (d) as filled squares (green).