Discrete unified gas kinetic scheme for all Knudsen number flows. III. Binary gas mixtures of Maxwell molecules

Recently a discrete unified gas kinetic scheme (DUGKS) in a finite-volume formulation based on the Boltzmann model equation has been developed for gas flows in all flow regimes. The original DUGKS is designed for flows of single-species gases. In this work, we extend the DUGKS to flows of binary gas mixtures of Maxwell molecules based on the Andries-Aoki-Perthame kinetic model [P. Andries et al., J. Stat. Phys. 106, 993 (2002). A particular feature of the method is that the flux at each cell interface is evaluated based on the characteristic solution of the kinetic equation itself; thus the numerical dissipation is low in comparison with that using direct reconstruction. Furthermore, the implicit treatment of the collision term enables the time step to be free from the restriction of the relaxation time. Unlike the DUGKS for single-species flows, a nonlinear system must be solved to determine the interaction parameters appearing in the equilibrium distribution function, which can be obtained analytically for Maxwell molecules. Several tests are performed to validate the scheme, including the shock structure problem under different Mach numbers and molar concentrations, the channel flow driven by a small gradient of pressure, temperature, or concentration, the plane Couette flow, and the shear driven cavity flow under different mass ratios and molar concentrations. The results are compared with those from other reliable numerical methods. The results show that the proposed scheme is an effective and reliable method for binary gas mixtures in all flow regimes.

Figure 1

Structure of a ${\text{Ma}}_1=1.5$ shock wave in a binary gas mixture with $m_A/m_B=0.5$ and $a_{BB}/a_{AA}=1$: (a) ${\chi }_1^A=0.1$; (b) ${\chi }_1^A=0.9$.

Figure 2

Structure of a ${\text{Ma}}_1=1.5$ shock wave in a binary gas mixture with $m_A/m_B=0.25$ and $a_{BB}/a_{AA}=1$: (a) ${\chi }_1^A=0.1$; (b) ${\chi }_1^A=0.9$.

Figure 3

Structure of a ${\text{Ma}}_1=3.0$ shock wave in a binary gas mixture with $m_A/m_B=0.5$ and $a_{BB}/a_{AA}=1$: (a) ${\chi }_1^A=0.1$; (b) ${\chi }_1^A=0.9$.

Figure 4

Schematic of the channel flow.

Figure 5

Particle flux versus Knudsen number of the pressure driven channel flow with ${\chi }_0^A=0.5$. $m_B/m_A$ in (a), (b), and (c) is 2, 4, and 10, respectively. The linearized solutions are from Ref. [51].

Figure 6

Same as Fig. 5 except that the flow is driven by a temperature gradient.

Figure 7

Same as Fig. 5 except that the flow is driven by a concentration gradient.

Figure 8

Schematic of the plane Couette flow.

Figure 9

Velocity profiles in the Couette flow for the Ne-Ar mixture with $C_0=0.5$ at (a) $\delta =0.1$, (b) $\delta =1$, and (c) $\delta =10$.

Figure 10

Same as Fig. 9 but for the He-Xe mixture.

Figure 11

The normalized stress of the Couette flow for gas mixtures (a) Ne-Ar and (b) He-Xe under different rarefaction parameters $\delta$ with $C_0=0.5$.

Figure 12

Velocity profiles across the cavity center at $\text{Re}=400$.

Figure 13

Velocity profiles along the center lines in the cavity flow for the Ne-Ar mixture with $C_0=0.5$ at (a) $\delta =0.1$, (b) $\delta =1$, and (c) $\delta =10$.

Figure 14

Same as Fig. 13 but for the He-Xe mixture.

Figure 15

Temperature contours in the lid driven cavity flow for the Ne-Ar mixture with $C_0=0.5$ at (a) $\delta =0.1$, (b) $\delta =1$, and (c) $\delta =10$. Black line: DSMC; white line with background: DUGKS.

Figure 16

Same as Fig. 15 but for the He-Xe mixture.