Nodal Analysis of Nonlinear Behavior of the Instability at a Fluid Interface

In interface instabilities, deformations first grow exponentially, then enter a nonlinear regime affecting amplitude and symmetry. Most extant studies have focused on amplitude alone. Here, we study a 2D Rayleigh-Taylor instability for an initial sinusoidal deformation, analyzing its amplitude and asymmetry over time. For the latter, we define a metric based on the zero crossings of the interface. We develop a weakly nonlinear model and compare it to experimental data. It shows that our asymmetry metric complements the amplitude for an improved description of the instabilities’ nonlinear phases.

Figure 1

Interface shapes for discrete times $t=[10,11,\dots ,23]\times 16.7\mathrm{ms}$ for an experimental run for which $\lambda =10.0\mathrm{mm}$ and the Atwood number is 0.29. Distances $\mathrm{\Delta }{x}_b$ and $\mathrm{\Delta }{x}_s$ are represented in the figure for time $t=10\times 16.7\mathrm{ms}$. The top inset shows a close up near the interface average position for two successive times, the solid line for $t=12\times 16.7\mathrm{ms}$ and the dashed line for $t=15\times 16.7\mathrm{ms}$, allowing us to follow the interface nodes evolution.

Figure 2

Comparison of experimental measurements of $M_{\mathrm{SE}}$ and $M_{\mathrm{AE}}$ and their second-order theoretical predictions for an experimental run with $\lambda =20.3\mathrm{mm}$. To second order, the theoretical asymptotic value of $M_{\mathrm{AE}}$ is zero, which is observed experimentally by the presence of an initial random noise during the linear regime (the gray zone, where the absolute value of $M_{\mathrm{AE}}$ is plotted). The time marker $t_1$ represents the time when a secondary instability is first visible in the experimental images. $t_2$ corresponds to the limiting time of validity of the weakly nonlinear model (e.g., $R=1$).