Nonequilibrium thermohydrodynamic effects on the Rayleigh-Taylor instability in compressible flows

The effects of compressibility on Rayleigh-Taylor instability (RTI) are investigated by inspecting the interplay between thermodynamic and hydrodynamic nonequilibrium phenomena (TNE, HNE, respectively) via a discrete Boltzmann model. Two effective approaches are presented, one tracking the evolution of the local TNE effects and the other focusing on the evolution of the mean temperature of the fluid, to track the complex interfaces separating the bubble and the spike regions of the flow. It is found that both the compressibility effects and the global TNE intensity show opposite trends in the initial and the later stages of the RTI. Compressibility delays the initial stage of RTI and accelerates the later stage. Meanwhile, the TNE characteristics are generally enhanced by the compressibility, especially in the later stage. The global or mean thermodynamic nonequilibrium indicators provide physical criteria to discriminate between the two stages of the RTI.

Figure 1

Density evolutions in the RTI simulated by the DBM at various times: (a) $t=0.0$; (b) $t=0.2$; (c) $t=0.6$; (d) $t=1.0$; (e) $t=1.3$; (f) $t=1.6$; (g) $t=1.9$; and (h) $t=2.2$. The density range is $[0.40,1.70]$, and from blue to red, the density increases. The same results can also be obtained from the Navier-Stokes model. The larger the density, the stronger the inertial effects, the more difficult to change the velocity. Consequently, the upward perturbation grows into a shape similar to a bubble, while the downward perturbation grows into a shape similar to a spike.

Figure 2

Temperature profiles averaged in the $x$ direction versus the $y$ coordinate at different times. The mixing layer becomes wider with time. The perturbation depth in the upper part is smaller than that in the bottom part.

Figure 3

Profiles of macroscopic quantities and nonequilibrium effects along the central line $x=N_x/2$ at $t=1.6$. The gradients of macroscopic quantities work as driving forces of the TNE effects. The TNE quantities show the specific TNE status of the system. The most pronounced TNE effects occur around the interface where the macroscopic quantities have large gradients. Because the gravity is in the $y$ direction, the “flux” in the $y$ direction is more pronounced. The nonzero value of ${\mathrm{\Delta }}_{2xy}^{*}$ is a typical TNE quantity.

Figure 4

Positions of the bubble and spike versus time. The velocity of the bubble is smaller than that of the spike. The lighter fluid is “softer” and thus it is easier for the spike to pass through and grow.

Figure 5

The perturbation amplitudes obtained by two different tracking approaches. The local TNE strength can be used to track interfaces. The good agreement shows that the two approaches validate each other.

Figure 6

The growth of the perturbation amplitude by the DBM. The good linear fit is consistent with the linear theory on the RTI. The result shows that the linear theory for incompressible flows works also for a period in the compressible flows. The DBM and the linear theory validate each other.

Figure 7

The growth of the dimensionless amplitude $A^{*}$ with different values of compressibility $H_1$. In the initial stage, the compressibility tends to inhibit the RTI, while in the later stage it tends to strengthen the RTI and the strengthening effect approaches saturation with increasing compressibility.

Figure 8

The dimensionless amplitude versus the compressibility at $t^{*}=0.6$ in (a), and $t^{*}=2.5$ in (b). This figure shows more neatly the specific compressibility effects at one given time in each of the two stages.

Figure 9

The time evolutions of the various energy rates at different values of compressibility $H_1$. The energy rates include the rates of compressive energy ${\dot{E}}_c$ in (a), internal dissipation energy ${\dot{E}}_d$ in (b), kinetic energy induced by gravity ${\dot{E}}_{kg}$ in (c), kinetic energy induced by dissipation ${\dot{E}}_{kd}$ in (d), and kinetic energy induced by pressure ${\dot{E}}_{kp}$ in (e). It is clear that a larger compressive energy rate is involved in the later stage. This figure highlights significant differences as compared to the incompressible scenario.

Figure 10

The time evolution of the global average TNE strength with different values of compressibility $H_1$. The compressibility first decreases in the first stage and then increases in the later stage the global average TNE strength. The higher the compressibility, the stronger the global average TNE effects.

Figure 11

The average values of various components of TNE quantities versus compressibility at four nondimensional times: $t^{*}=0.1$ in (a), $t^{*}=0.6$ in (b), $t^{*}=2.0$ in (c), and $t^{*}=2.5$ in (d). The figure shows more specific global or mean TNE effects than Fig. 10. The relative strengths of those dependencies on compressibility may change with time.

Figure 12

Sketch of the DVM used in the present paper. The numbers in the figure are the indexes of the discrete velocities.