Effects of variable deceleration periods on Rayleigh-Taylor instability with acceleration reversals

The dynamics of an interfacial flow that is initially Rayleigh-Taylor unstable but becomes statically stable for some intermediate period due to the reversal of the externally imposed acceleration field is studied. We discuss scenarios that consider both single and double-acceleration reversals. The accel-decel (AD) case consists of a single reversal imposed at an instant after the constant acceleration instability has entered a self-similar regime. The layer of mixed fluid ceases to grow upon acceleration reversal, and the dominant mechanics are due to internal wave oscillations. Variation of mass flux and the Reynolds stress anisotropy is observed due to the action of the internal waves. A second reversal of the AD case that is termed as accel-decel-accel, ADA is then explored; the response of the mixing layer is shown to depend strongly on the duration and the periodicity of the Reynolds stress anisotropy of the mixing layer during the deceleration period. We explore the effect of this variable deceleration period after the second acceleration reversal where the flow once again becomes Rayleigh-Taylor unstable based on metrics that include the integral mixing-layer width, bubble and spike amplitudes, mass flux, Reynolds stress anisotropy tensor, and the molecular mixing parameter.

Figure 1

(a) Schematic of the computational domain showing the heavy (red) and light (blue) fluids and their interface at $z=0$, (b) azimuthally averaged initial perturbations with energy in the modes 32–64, and (c) initial perturbations in physical space at $z=0$.

Figure 2

Representative accel-decel and accel-decel-accel profiles used for the current study. For the ADA simulations, the deceleration time is varied, as indicated in the figure.

Figure 3

(a) Oscillatory (internal wavelike) behavior observed in the vertical component of anisotropy tensor ($B_{33}$), its time derivative ${B^{\prime }}_{33}$, and the mass flux $\langle u_{3}c\rangle$, and (b) time evolution of total KE and total change in PE during the deceleration period for an RTI mixing layer undergoing a single acceleration reversal (AD case). The acceleration reversal occurs at $t=2\mathrm{s}$. The oscillations are used for selecting the reacceleration time, $t_2$, for the ADA test cases (see Table 1). Symbols denote the reacceleration times for I, P, D, and T cases.

Figure 4

(a) Vertical density contours for the CG and AD cases at $t=4\mathrm{s}$, $Z=32\mathrm{cm}$. (b) Kinetic energy is Fourier transformed and mapped onto transformed axes in which $N^2$ is a linear gradient. Points sampled in Fourier space are represented as black dots with radii that scale with the logarithm of energy. The red overlay is the region satisfying the dispersion relation for observed values of $N$ and thus admits resonant internal wave activity.

Figure 5

DMD analysis of RTI AD case. The plot of the left side shows a truncated distribution of eigenvalues ${\widehat{\mathrm{\Lambda }}}_{ii}$ and is plotted on a complex plane. The dashed gray line indicates a unit circle, and the dashed black lines indicate the buoyancy frequency. The zoomed subfigure on the right highlights the eigenvalues that correspond to the internal waves present during the RTI deceleration phase.

Figure 6

Visualization of modes: (a) a turbulent mode, (b) the zero-frequency mode, (c) the critical B33 mode, and (d) a propagating internal wave mode. The dashed lines on the internal wave mode correspond to the direction of propagation implied by its frequency given in Fig. 3.

Figure 7

Vertical and horizontal density contours for the CG at $Z=12\mathrm{cm}$. I, P, D, and T cases are shown for $Z+Z_{\mathrm{ADA}}=12\mathrm{cm}$. The deceleration period is subtracted for I, P, D, and T cases to evaluate RTI mix states at similar values of $Z+Z_{\mathrm{ADA}}$. The horizontal slices are taken from the height of the original heavy-light fluids interface.

Figure 8

The integral mixing width, $W$, scaled by $Z_{\mathrm{ADA}}$ for the (a) first-, (b) second-, and (c) third oscillatory period of $B_{33}$. (d) Average of integral mix widths over the three oscillatory periods during the RTI ADA reacceleration phase.

Figure 9

Bubble and spike widths ($h_b$ and $h_s$) averaged over the three oscillatory periods for ADA cases during reacceleration.

Figure 10

Scalar variance $\langle cc\rangle$, averaged for each D, P, I, and T case over the three oscillatory regions. The inset plot shows the scalar variance of the AD case with reacceleration times marked.

Figure 11

The normalized mass flux $\langle u_{3}c\rangle /h_{\mathrm{tot}}^{0.5}$, averaged for each D, P, I, and T case over the three oscillatory regions. The inset plot shows the mass flux of the AD case with reacceleration times marked.

Figure 12

The normalized vertical velocity fluctuation $\langle u_3{u}_3\rangle /h_{\mathrm{tot}}$, averaged for each D, P, I, and T case over the three oscillatory regions for $A=0.5$. The inset plot shows marked reacceleration times during the AD case.

Figure 13

θ averaged for each D, P, I, and T case over the three oscillatory regions. The inset shows the marked reacceleration times for the AD case.

Figure 14

Three-dimensional visualization of the density field where the behavior of the vertical mass flux $\langle u_{3}c\rangle$ during (a) classical RTI, and (b) RTI under deceleration at DPIT phases of the anisotropy tensor. As it is seen, in decreasing anisotropy (D) cases, mass flux behaves similarly to the classical RTI case, and the acceleration reversal at that point leads to the fastest regrowth. Meanwhile, the mass flux at the increasing anisotropy (I) point is just the opposite direction of the mass-flux behavior during the classical RTI; hence, these cases grow slowest. Besides, at anisotropy peaks (P) and trough (T) points, the behavior of the mass flux is almost neutral ($\langle u_{3}c\rangle \sim 0$), so their growth rates are between I and D cases. (Note that the horizontal slices are taken from the height of the original heavy-light fluids interface.)