Discrete Boltzmann modeling of Rayleigh-Taylor instability in two-component compressible flows

A discrete Boltzmann model (DBM) is proposed to probe the Rayleigh-Taylor instability (RTI) in two-component compressible flows. Each species has a flexible specific-heat ratio and is described by one discrete Boltzmann equation (DBE). Independent discrete velocities are adopted for the two DBEs. The collision and force terms in the DBE account for the molecular collision and external force, respectively. Two types of force terms are exploited. In addition to recovering the modified Navier-Stokes equations in the hydrodynamic limit, the DBM has the capability of capturing detailed nonequilibrium effects. Furthermore, we use the DBM to investigate the dynamic process of the RTI. The invariants of tensors for nonequilibrium effects are presented and studied. For low Reynolds numbers, both global nonequilibrium manifestations and the growth rate of the entropy of mixing show three stages (i.e., the reducing, increasing, and then decreasing trends) in the evolution of the RTI. On the other hand, the early reducing tendency is suppressed and even eliminated for high Reynolds numbers. Relevant physical mechanisms are analyzed and discussed.

Figure 1

Mole fractions ($Y^A$ and $Y^B$) vs $x$ in the binary diffusion. The squares, circles, and triangles denote simulation results at instants $t=0.004$, 0.02, and 0.1, respectively. The lines stand for corresponding analytic solutions.

Figure 2

Initial configuration for the free fall case (a) and the height of material interface (b): type I (squares), type II (triangles), and the exact solution (line).

Figure 3

Comparison among the simulation results with various space steps. (a) Mole fraction $Y^A$ vs height $y^{*}$ along the grid $x^{*}=0.64$ at time $t^{*}=3.79$ in the evolution of RTI with $\text{Re}=2000$. The simulation is conducted with five different space steps: $\mathrm{\Delta }x=\mathrm{\Delta }y=1.0\times {10}^{-4}$ (solid), $2.0\times {10}^{-4}$ (dot), $4.0\times {10}^{-4}$ (short dashed), $8.0\times {10}^{-4}$ (dash dotted), and $1.6\times {10}^{-3}$ (long dashed). (b) The relation between space steps and numerical errors. The squares stand for the DBM results and the line represents the fitting function.

Figure 4

Contours of mole fraction $Y^A$ in the evolution of RTI with various Reynolds numbers at times $t^{*}=0.0$, 1.26, 3.79, and 6.32, respectively. Parts (a)–(d) are for $\text{Re}=2000$, (e)–(h) are for $\text{Re}=500$, and (i)–(l) are for $\text{Re}=125$.

Figure 5

The height $y^{*}$ (a) and speed $u^{*}$ (b) of the bubble front vs time $t^{*}$. The solid lines denote the analytic solutions and the symbols denote simulation results with various Reynolds numbers: $\text{Re}=2000$ (squares), 500 (upper triangles), and 125 (lower triangles).

Figure 6

Contours of invariant ${\mathrm{\Lambda }}_1^{\sigma *}$ of the tensor ${\mathbf{\Delta }}_2^{\sigma *}$ in the case of $\text{Re}=2000$ at times $t^{*}=0.126$, 1.26, 3.79, and 6.32, from left to right, respectively. The first row (a)–(d) is for ${\mathrm{\Lambda }}_1^{A*}$ and the second row (e)–(h) is for ${\mathrm{\Lambda }}_1^{B*}$.

Figure 7

The evolution of $\int \int |{\mathrm{\Lambda }}_1^{\sigma *}|dxdy$ and $\int \int |{\mathrm{\Lambda }}^{\sigma *}|dxdy$. Here ${\mathrm{\Lambda }}_1^{\sigma *}$ is the invariant of the tensor ${\mathbf{\Delta }}_2^{\sigma *}$, and ${\mathrm{\Lambda }}^{\sigma *}$ is the invariant of the tensor ${\mathbf{\Delta }}_{3,1}^{\sigma *}$. Panels (a)–(c) are for $\int \int |{\mathrm{\Lambda }}_1^{\sigma *}|dxdy$ with $\text{Re}=2000$, 500, and 125, respectively. Panels (d)–(f) are for $\int \int |{\mathrm{\Lambda }}^{\sigma *}|dxdy$ with $\text{Re}=2000$, 500, and 125, respectively. Species $A$ and $B$ are denoted by lines with squares and triangles, respectively.

Figure 8

Snapshots of entropy of mixing in the evolution of RTI with $\text{Re}=2000$ at times $t^{*}=0.126$, 1.26, 3.79, and 6.32, from left to right, respectively.

Figure 9

The integral of the entropy of mixing (a) and its growth rate (b) in the evolution of the RTI with various Reynolds numbers: $\text{Re}=2000$, 500, and 125, respectively.