Multiple-relaxation-time discrete Boltzmann modeling of multicomponent mixture with nonequilibrium effects

A multiple-relaxation-time discrete Boltzmann model (DBM) is proposed for multicomponent mixtures, where compressible, hydrodynamic, and thermodynamic nonequilibrium effects are taken into account. It allows the specific heat ratio and the Prandtl number to be adjustable, and is suitable for both low and high speed fluid flows. From the physical side, besides being consistent with the multicomponent Navier-Stokes equations, Fick's law, and Stefan-Maxwell diffusion equation in the hydrodynamic limit, the DBM provides more kinetic information about the nonequilibrium effects. The physical capability of DBM to describe the nonequilibrium flows, beyond the Navier-Stokes representation, enables the study of the entropy production mechanism in complex flows, especially in multicomponent mixtures. Moreover, the current kinetic model is employed to investigate nonequilibrium behaviors of the compressible Kelvin-Helmholtz instability (KHI). The entropy of mixing, the mixing area, the mixing width, the kinetic and internal energies, and the maximum and minimum temperatures are investigated during the dynamic KHI process. It is found that the mixing degree and fluid flow are similar in the KHI process for cases with various thermal conductivity and initial temperature configurations, while the maximum and minimum temperatures show different trends in cases with or without initial temperature gradients. Physically, both heat conduction and temperature exert slight influences on the formation and evolution of the KHI morphological structure.

Figure 1

Sketch of the discrete velocities and mesh grids.

Figure 2

Initial configuration of the three-component diffusion.

Figure 3

Grid convergence analysis: (a) the horizontal distribution of mole fractions $X^A$ at the time $t=0.05$, (b) relative errors under various spatial steps.

Figure 4

Molar fractions in the diffusion process: $X^A$ (top), $X^B$ (middle), and $X^C$ (bottom). Squares, circles, triangles, and diamonds denote DBM results at time instants $t=0.005$, 0.02, 0.06, and 0.2, respectively. Solid lines stand for the corresponding analytical solutions.

Figure 5

Nonequilibrium quantities at time $t=0.02$ in the diffusion process. Squares, circles, and triangles denote DBM results of ${\widehat{f}}_5^{A\mathrm{sneq}}, {\widehat{f}}_5^{B\mathrm{sneq}}$, and ${\widehat{f}}_5^{C\mathrm{sneq}}$, respectively. Solid lines stand for the corresponding analytical solutions.

Figure 6

Initial configuration of the thermal Couette flow.

Figure 7

Vertical distribution of the horizontal speed $u_x$ (a) and nonequilibrium quantity ${\widehat{f}}_6^{A\mathrm{sneq}}$ (b) in the thermal Couette flow. Squares, circles, triangles, and diamonds represent DBM results at time instants $t=0.1$, 0.4, 2.0, and 30, respectively. Solid lines stand for the corresponding analytical solutions.

Figure 8

Vertical distribution of the temperature in the steady Couette flow. (a) Cases with $Pr=1.0$ and $\gamma =1.3$, 1.4, and 1.5, respectively. (b) Cases with $\gamma =1.4$ and $Pr=0.5$, 1.0, and 2.0, respectively.

Figure 9

Initial configuration of the Sod shock tube.

Figure 10

Profiles of density at a time instant $t=0.2$ in the Sod shock tube with various spatial steps (a) and temporal steps (b).

Figure 11

Profiles of density (a), horizontal speed (b), internal energy (c), and pressure (d) at a time instant $t=0.2$ in the Sod shock tube. Symbols represent DBM results, and solid lines stand for Riemann solutions.

Figure 12

Initial configuration of the KHI.

Figure 13

Contours of the entropy of mixing at time instants $t=0.0$, 0.5, 1.0, 1.5, and 3.0 in the evolution of KHI.

Figure 14

Physical quantities in the evolution of KHI with $T_L=T_R$: (a) the entropy of mixing and its growth rate, (b) the mixing area and its growth rate, (c) the mixing width and its growth rate, (d) the kinetic energy, (e) the internal energy, (f) the total energy, (g) the maximum temperature, (h) the minimum temperature, and (i) the temperature difference. The inserts in (d) and (e) correspond to the rectangles, respectively. The lines with squares, circles, upper triangles, lower triangles, and diamonds indicate $Pr=0.25$, 0.5, 1.0, 2.0, and 4.0, respectively.

Figure 15

Physical quantities in the evolution of KHI with $T_L\ne T_R$: (a) the entropy of mixing and its growth rate, (b) the mixing area and its growth rate, (c) the mixing width and its growth rate, (d) the kinetic energy, (e) the internal energy, (f) the total energy, (g) the maximum temperature, (h) the minimum temperature, and (i) the temperature difference. The inserts in (d) and (e) correspond to the rectangles, res- pectively. The lines with squares, circles, upper triangles, lower triangles, and diamonds indicate $Pr=0.25$, 0.5, 1.0, 2.0, and 4.0, respectively.

Figure 16

Comparison of physical fields at the time $t=1.5$ in the KHI process. From top to bottom are the mass fraction of species $A$, vorticity, and temperature in the three rows, respectively. From left to right are the cases ($T_L=T_R$ and $Pr=0.25$), ($T_L=T_R$ and $Pr=4.0$), ($T_L\ne T_R$ and $Pr=0.25$), and ($T_L\ne T_R$ and $Pr=4.0$) in the four columns, respectively. Only a part of the horizontal range $0.5\le x\le 1.0$ is shown in each subfigure.