Regularized extended-hydrodynamic equations for a rarefied granular gas and the plane shock waves

The regularized versions of extended-hydrodynamic equations for a dilute granular gas, in terms of 10-, 13-, and 14-moments, are derived from the inelastic Boltzmann equation. The regularization is achieved by adding higher-order gradient terms that are obtained following a Chapman-Enskog-like gradient-expansion [H. Struchtrup, Stable transport equations for rarefied gases at high orders in the Knudsen number, Phys. Fluids, 16, 3921 (2004)]. For both granular and molecular gases, the resulting moment equations are found to be free from the well-known finite Mach-number singularity (that occurs in the Riemann problem of planar shock waves) since the regularized gradient terms yield parabolic equations in contrast to the hyperbolic nature of original moment equations. In order to clarify the advantage of these regularized equations, the 10-moment model for the plane shock-wave problem is solved numerically for both molecular and granular gases; the calculated hydrodynamic profiles compare favorably with previous simulation results for molecular gases. For a granular gas, both regularized and nonregularized equations predict asymmetric density and temperature profiles, with the maxima of both density and temperature occurring within the shock layer, and the hydrodynamic fields are found to be smooth for the regularized equations for all Mach numbers studied. It is demonstrated that, unlike in the case of molecular gases, a “second” regularization of the regularized equations must be carried out in order to arrest the unbounded growth of density within the shock layer in a granular gas.

Figure 1

Dimensionless inverse shock width ($l_1/\delta$) versus upstream Mach number (${\mathrm{Ma}}_1$) for the stationary plane shock-wave problem for a hard-sphere gas. All data are extracted from Ref. [37]: NS and R13 refer to solutions from Navier-Stokes and “regularized” 13-moment equations, respectively; results from DSMC are also superimposed. The inset displays the variation of normalized density across the shock and the definition of shock width $\delta$.

Figure 2

Steady-state profiles of the solutions of the viscous Burgers equation (18) for the initial data given in Eq. (19), with $\nu$ equal to $0.025,0.05,0.1$ and the limit of $\nu \to 0$.

Figure 3

Schematic of the Riemann problem of a plane shock wave: the upstream and downstream properties of the shock are denoted by (${\rho }_1,u_1,{\theta }_1$) and (${\rho }_2,u_2,{\theta }_2$), respectively. The green line represents a typical density profile across the shock.

Figure 4

Normalized density profiles ${\rho }_N$ [Eq. (58)] in a molecular gas predicted by 10-moment model using two different numerical methods; see the text for details: (a) ${\mathrm{Ma}}_1=1.2$, (b) ${\mathrm{Ma}}_1=2$, and (c) ${\mathrm{Ma}}_1=3$.

Figure 5

Comparison of shock profiles for a molecular gas predicted by R10 (dot-dashed line) and 10-moment (solid line) model: (a) normalized density ${\rho }_N$ [Eq. (58)], (b) temperature ${\theta }_N$ [Eq. (59)], (c) velocity $u_N$ [Eq. (59)], (d) longitudinal stress $\sigma \equiv {\sigma }_{xx}$, (e) heat flux $q\equiv q_x$, and (f) deviatoric third moment $\mathcal{Q}\equiv {\mathcal{Q}}_{xxx}$ for an upstream Mach number of ${\mathrm{Ma}}_1=2>{\mathrm{Ma}}_{\mathrm{cr}}$. The DSMC data of Timokhin et al. [58] for density and temperature are superimposed in panels (a) and (b).

Figure 6

Early-time evolutions of granular shock-wave profiles as predicted by 10-moment (upper panels) and R10 model (lower panels) for an upstream Mach number ${\mathrm{Ma}}_1=2$ and for a restitution coefficient of $\alpha =0.9$: (a), (d) density, (b), (e) temperature, and (c), (f) velocity.

Figure 7

Profiles of (a) longitudinal stress ($\sigma \equiv {\sigma }_{xx}$), (b) heat flux ($q\equiv q_x$), and (c) deviator of third-moment ($\mathcal{Q}\equiv {\mathcal{Q}}_{xxx}$) predicted by R10 (red dashed curve) and 10-moment (black curve) models for an upstream Mach number ${\mathrm{Ma}}_1=2$ and at time $t=10$ for restitution coefficients of $\alpha =0.9$ (upper panels) and $\alpha =0.7$ (lower panels).

Figure 8

(a) Temporal evolution of the density overshoot, $▵\rho \equiv ({\rho }_{max}-{\rho }_2)$, for ${\mathrm{Ma}}_1=1.2$ with $\alpha =0.9$; the inset shows the variation of the spatial location of ${\rho }_{max}$ with time. (b, c) Evolution of normalized shock speed, ${\tilde{v}}_s=v_s/c$, where $v_s=x(\rho ={\rho }_{max})/t$ is the speed of the density peak and $c=\sqrt{\gamma {\theta }_1}$ is the adiabatic sound speed, for $\alpha =0.9$ (main panel) and $\alpha =0.7$ (inset) with (b) ${\mathrm{Ma}}_1=1.2$ and (c) ${\mathrm{Ma}}_1=2$. (d) Variation of asymptotic shock speed at large times $[{\tilde{v}}_s^{\infty }\equiv {\tilde{v}}_s(t\to \infty )]$ with initial upstream Mach number; the circles and squares denote data from R10 and Navier-Stokes [11] model, respectively.

Figure 9

Comparison of Haff's law (red curve) with the numerical solution of R10 equations for ${\mathrm{Ma}}_1=1.2$: (a) upstream temperature ${\theta }_1$ (main panel), downstream temperature ${\theta }_2$ (inset), and (b) maximum temperature within the shock layer ${\theta }_{max}$.

Figure 10

(a) Comparison of Haff's law (red curve) with the numerical solution of R10 equations for ${\mathrm{Ma}}_1=2$ and $\alpha =0.9$: upstream temperature ${\theta }_1\left(t\right)$ (main panel), downstream temperature ${\theta }_2\left(t\right)$ (inset). (b) Temporal evolution of maximum granular temperature ${\theta }_{max}\left(t\right)$ (main panel, with time axis on the top). The black and green curves in panel (b) represent the spatial evolution of granular temperature $\theta \left(x\right)$ at two different times; the inset of panel (b) is a zoomed version of the black curve, containing the temperature maxima and minima. (c) Spatial evolution of granular temperature (within shock layer, such as in the inset of panel (b) at late times $t\gg t_c$; the abscissa has been shifted with respect to the location of the temperature minima.

Figure 11

Temporal evolutions of (a) density $\rho (x,t)$, (b) pressure $p(x,t)$, and (c) longitudinal stress $\sigma (x,t)$, as predicted by R10 equations, for (a), (b), (c) ${\mathrm{Ma}}_1=1.2$ and $\alpha =0.7$ and (d), (e), (f) ${\mathrm{Ma}}_1=2$ and $\alpha =0.9$.

Figure 12

Arrest of maximum density ${\rho }_{max}$ via a regularization procedure for R10 equations: (a) ${\mathrm{Ma}}_1=1.2$ and $\alpha =0.7$ and (b) ${\mathrm{Ma}}_1=2$ and $\alpha =0.9$. For definitions of ${\theta }_{\text{cut}}$, see Sec. 6c.