Kinetic modeling of nonequilibrium flow of hard-sphere dense gases

A kinetic model is proposed for the nonequilibrium flow of dense gases composed of hard-sphere molecules, which significantly simplifies the collision integral of the Enskog equation using the relaxation-time approach. The model preserves the most important physical properties of high-density gas systems, including the Maxwellian at rest as the equilibrium solution and the equation of state for hard-sphere fluids; all the correct transport coefficients, namely, the shear viscosity, thermal conductivity, and bulk viscosity; and inhomogeneous density distribution in the presence of a solid boundary. The collision operator of the model contains a Shakhov model-like relaxation part and an excess part in low-order spatial derivatives of the macroscopic flow properties; this latter contribution is used to account for the effect arising from the finite size of gas molecules. The density inhomogeneity in the vicinity of a solid boundary in a confined flow is captured by a method based on the density-functional theory. Extensive benchmark tests are performed, including the normal shock structure and the Couette, Fourier, and Poiseuille flow at different reduced densities and Knudsen numbers, where the results are compared with the solutions from the Enskog equation and molecular dynamics simulations. It is shown that the proposed kinetic model provides a fairly accurate description of all these nonequilibrium dense gas flows. Finally, we apply our model to simulate forced wave propagation in a dense gas confined between two plates. The inhomogeneous density near the solid wall is found to enhance the oscillation amplitude, while the presence of bulk viscosity causes stronger attenuation of the sound wave. This shows the importance of a kinetic model to reproduce density inhomogeneity and correct transport coefficients, including bulk viscosity.

Figure 1

Profiles of density (first row) and transverse velocity (second row) of the shear driven flow: (a), (d) ${\eta }_0=0.1$, $R=10$, and $\mathrm{Kn}=0.09$; (b), (e) ${\eta }_0=0.1$, $R=5$, and $\mathrm{Kn}=0.181$; (c), (f) ${\eta }_0=0.2$, $R=5$, and $\mathrm{Kn}=0.067$. The solutions to the Enskog equation are represented by empty circles. The DVM solutions of the kinetic model are shown by the solid lines.

Figure 2

Profiles of density (first row) and temperature (second row) of the thermally driven flow: (a), (d) ${\eta }_0=0.1$, $R=10$, and $\mathrm{Kn}=0.09$; (b), (e) ${\eta }_0=0.2$, $R=10$, and $\mathrm{Kn}=0.034$; (c), (f) ${\eta }_0=0.2$, $R=5$, and $\mathrm{Kn}=0.067$. The solutions to the Enskog equation are represented by empty circles. The DVM solutions of the kinetic model are shown by the solid lines.

Figure 3

Profiles of density (first row) and transverse velocity (second row) of the force-driven flow: (a), (d) ${\eta }_0=0.1$, $R=10$, and $\mathrm{Kn}=0.09$; (b), (e) ${\eta }_0=0.1$, $R=5$, and $\mathrm{Kn}=0.181$; (c), (f) ${\eta }_0=0.1$, $R=2$, and $\mathrm{Kn}=0.452$. The results obtained from the event-driven MD simulations are denoted by empty circles. The DVM solutions of the linearized kinetic model are shown by the solid lines. The velocity is normalized by $V_0=n_{0}FH\sqrt{2m{k}_B{T}_0}/p_0$. Only half domain $[-H/2,0]$ is plotted since the flow is symmetric with respect to $x=0$.

Figure 4

The flows at large Knudsen number: ${\eta }_0=0.01$, $R=10$, and $\mathrm{Kn}=1.149$. (a), (c) Profiles of density and transverse velocity of the Couette flow. (b), (d) Profiles of density and temperature for the Fourier flow. The DSMC solutions to the Enskog equation are represented by empty circles. The DVM solutions of the kinetic model are shown by the solid lines.

Figure 5

Profiles of normalized flow properties for a shock wave of Mach number $\mathrm{Ma}=4$: (a), (c) ${\eta }_1=0.022$ and (b), (d) ${\eta }_1=0.077$. The normalized properties are density $\tilde{\rho }=\frac{\rho -{\rho }_1}{{\rho }_2-{\rho }_1}$, temperature $\tilde{T}=\frac{T-T_1}{T_2-T_1}$, velocity $\tilde{U}=\frac{U-U_1}{U_2-U_1}$, total normal stress ${\tilde{P}}_{xx}=\frac{P_{xx}-p}{{\rho }_1{k}_{\mathrm{B}}{T}_1/m}$, and total heat flux ${\tilde{Q}}_x=\frac{Q_x}{{\rho }_1{k}_{\mathrm{B}}{T}_1/m\sqrt{2k_{\mathrm{B}}{T}_1/m}}$.

Figure 6

Profiles of velocity amplitude for the sound wave propagation in the confinement of $R=5$. The first row shows the results at $\mathrm{St}=1$, and the second row shows those at $\mathrm{St}=5$: (a), (d) ${\eta }_0=0.01$ and $\mathrm{Kn}=2.299$; (b), (e) ${\eta }_0=0.05$ and $\mathrm{Kn}=0.415$; (c), (f) ${\eta }_0=0.1$ and $\mathrm{Kn}=0.181$. The solutions of the linearized kinetic model with and without bulk viscosity are obtained by the DVM. The results for the sound wave in a dilute gas at the same Knudsen number and Strouhal number are included in (a) and (d), which are obtained by solving the Shakhov model equation.

Figure 7

Profiles of velocity phase for the sound wave propagation in the confinement of $R=5$. The first row shows the results at $\mathrm{St}=1$, and the second row shows those at $\mathrm{St}=5$: (a), (d) ${\eta }_0=0.01$ and $\mathrm{Kn}=2.299$; (b), (e) ${\eta }_0=0.05$ and $\mathrm{Kn}=0.415$; (c), (f) ${\eta }_0=0.1$, $R=2$, and $\mathrm{Kn}=0.181$. The solutions of the linearized kinetic model with and without bulk viscosity are obtained by the DVM. The results for the sound wave in a dilute gas at the same Knudsen number and Strouhal number are included in (a) and (d), which are obtained by solving the Shakhov model equation.