Analytic approach to nonlinear hydrodynamic instabilities driven by time-dependent accelerations

We extend our earlier model for Rayleigh-Taylor and Richtmyer-Meshkov instabilities to the more general class of hydrodynamic instabilities driven by a time-dependent acceleration $g\left(t\right)$. Explicit analytic solutions for linear as well as nonlinear amplitudes are obtained for several $g\left(t\right)$s by solving a Schrödinger-like equation $d^2\eta /d{t}^2-g\left(t\right)kA\eta =0$, where $A$ is the Atwood number and $k$ is the wave number of the perturbation amplitude $\eta \left(t\right)$. In our model a simple transformation $k\to k_L$ and $A\to A_L$ connects the linear to the nonlinear amplitudes: ${\eta }^{nonlinear}\left(k,A\right)\sim \left(1/k_L\right)\text{ln}{\eta }^{linear}\left(k_L,A_L\right)$. The model is found to be in very good agreement with direct numerical simulations. Bubble amplitudes for a variety of accelerations are seen to scale with $s$ defined by $s=\int \sqrt{g\left(t\right)}dt$, while spike amplitudes prefer scaling with displacement $\Delta x=\int \left[\int g\left(t\right)dt\right]dt$.

Figure 1

Three acceleration profiles similar to the ones used in LEM experiments [5, 6] and used in CALE simulations in this work.

Figure 2

The bubble amplitude $\eta \left(t\right)$, starting from ${\eta }_0={\eta }^{\ast }/2=0.065\text{cm}$, for the three acceleration profiles shown in Fig. 1. The dashed lines correspond to CALE simulations which we label level 1. The two continuous lines, undistinguishable in these figures, correspond to level 2 and level 3 solutions. The level 2 solution is obtained from the coupled Eqs. (5, 6). The level 3 solution is obtained from Eq. (1) for $\eta \le {\eta }^{\ast }$, after which we use Eq. (10). Analytic level 3 solutions in terms of Airy functions and level 4 scaling solutions in terms of elementary functions can also be written down (see the Appendix).

Figure 3

Quasiconstant acceleration (in arbitrary units) given by Eq. (29), and the corresponding amplitudes as calculated by CALE for $g_{\infty }=35g_E$, $T=1.2\text{ms}$, and the analytic solution Eq. (30), starting with ${\eta }_0={\eta }^{\ast }=0.13\text{cm}$.

Figure 4

Same as Fig. 3 with ${\eta }_0=4{\eta }^{\ast }=0.52\text{cm}$. The level 2 model, based on Eqs. (5, 6), fails for amplitudes ${\eta }_0>{\left({\eta }_0\right)}_{\text{max}}$ where, from Eq. (9), ${\left({\eta }_0\right)}_{\text{max}}\approx 0.48\text{cm}$. The level 3 model, using Eq. (10) for all ${\eta }_0\ge {\eta }^{\ast }$, gives a reasonable result which, in this case, is equivalent to the analytic solution in Eq. (30).

Figure 5

The idealized acceleration given by Eq. (29) (thick line) and the code-calculated interface acceleration (thin dashed line) showing oscillations around an average value. Imposing deliberately large oscillations as in Eq. (31) with $\epsilon =1$ and $\omega =2\pi /10{\text{ms}}^{-1}$ varies $g\left(t\right)$ by $\pm 100\mathrm{\%}$ around its average value of $35g_E$.

Figure 6

The deviation $\delta \left(t;\epsilon \right)$, as defined by Eq. (32), in bubble amplitude with and without the large oscillations in acceleration shown in Fig. 5. The dashed curve uses CALE and the continuous line uses Eq. (10) to evaluate the amplitudes $\eta \left(t;0\right)$ and $\eta \left(t;1\right)$ needed to construct $\delta \left(t;1\right)$.

Figure 7

Acceleration profile for a possible double-impulse experiment. The impulses are equal and opposite, bringing the tank to rest at 30 ms.

Figure 8

The “bubble” amplitude for the double-impulse experiment with $g\left(t\right)$ shown in Fig. 7 (repeated here in dashed line and arbitrary units). The initial amplitude is 0.065 cm. The bubble turns into a spike after 30 ms, the point where Eqs. (5, 10) both fail. Snapshots in Fig. 9.

Figure 9

Snapshots of the interface as calculated by CALE for the double-impulse acceleration history shown in Fig. 7, starting with ${\eta }_0=0.13\text{cm}$. We have included the distance traveled in the first three snapshots. At 30 ms the tank comes to rest 6.2 cm below its initial position while the perturbation continues to evolve.

Figure 10

An acceleration history with a weaker reshock that can induce freeze-out, i.e., make $\eta \left(t\right)=\text{const}$. See Fig. 11.

Figure 11

The bubble amplitude calculated by CALE, by Eqs. (5, 6), and by Eq. (10), using the acceleration profile $g\left(t\right)$ shown in Fig. 10. CALE predicts complete freeze-out (we iterated on $g\left(t\right)$ to obtain this result). Equation (5) plus Eq. (6) or Eq. (10) predict slow increase or decrease. The calculations start with ${\eta }_0=0.065\text{cm}$. Without reshock $\eta$ would continue to grow, $\eta \sim \text{ln}t$, as in Eq. (15).

Figure 12

Bubble amplitudes as functions of $s$ defined by Eq. (19). The four cases A, B, C, and D correspond to Figs. 1a, 1b, 1c, and Eq. (29), respectively (see also Fig. 3 for the quasiconstant acceleration D). Initial amplitude ${\eta }_0=0.13\text{cm}$. Curve E is Eq. (22). Bubbles appear to scale with $s$ except after $g=0$ (curve C after $s\approx 5{\text{cm}}^{1/2}$).

Figure 13

Three acceleration profiles used for testing the scaling hypothesis with CALE simulations. A is a constant acceleration at $70g_E$. B has $g=g_3{t}^3$ with $g_3=0.06g_E/{\text{ms}}^3$. C is an impulsive acceleration reaching $500g_E$ in 1 ms and returning to zero at $t=2\tau =2\text{ms}$. Accelerations in units of $g_E$, displacements in cm, and scaling variables in ${\text{cm}}^{1/2}$ are plotted in diagrams a, b, and c respectively, all as functions of time in milliseconds.

Figure 14

Bubble $\left(\eta >0\right)$ and spike $\left(\eta <0\right)$ evolutions as calculated by CALE for the three acceleration profiles A, B, and C displayed in Fig. 13. They are plotted as functions of time in (a), $\Delta x$ in (b), and $s$ in (c). Bubbles appear to scale with $s$, while spikes prefer $\Delta x$. The impulsive acceleration C does not scale after $g=0$.

Figure 15

Same as Fig. 14 with the hexane $\left(\rho =0.66\text{g}/{\text{cm}}^3\right)$ in the tank replaced by air $\left(\rho =0.0012\text{g}/{\text{cm}}^3\right)$, hence $A\approx 1$.

Figure 16

Bubble and spike amplitudes calculated by CALE (thin dashed line) and by Eq. (36) (thick continuous lines). The acceleration is the negative of the impulse C in Fig. 13, and $A\approx 1$. The insets show the heavy fluid at $t=0$ and $t=24\text{ms}$.

Figure 17

Same as Fig. 16 with an equal and opposite reshock added at 10 ms, as shown by the thin dashed line in arbitrary units. CALE simulations starting with ${\eta }_0=0.13\text{cm}$. The spike reaches a magnitude of 9.5 cm by 24 ms, compared with 4.3 cm for the shock-only case.

Figure 18

Comparison of two tanks with a single shock only (left figure, $\eta$s in Fig. 16) and for $\text{shock}+\text{reshock}$ (right figure, $\eta$s in Fig. 17), both at 24 ms. The position $\left(\sim 15\text{cm}\right)$ and speed $\left(\sim 0.68\text{cm}/\text{ms}\right)$ of the tip of the jet in the laboratory are approximately the same in both cases. The tank at left is moving up at 0.49 cm/ms, hence ${\dot{\eta }}^s=0.49-0.68=-0.19\text{cm}/\text{ms}$. The tank at right is at rest, hence ${\dot{\eta }}^s=0-0.68=-0.68\text{cm}/\text{ms}$. Both tanks started with ${\eta }_0=0.13\text{cm}$, $y_0=0$ defining the laboratory position of the initial interface, carrying a reservoir of heavy fluid 4.4 cm thick.