Turbulence and capillary waves on bubbles

We present a link between the theory of deep water waves and that of bubble surface perturbations. Theory correspondence is shown analytically for small wavelengths in the linear regime and investigated numerically in the nonlinear regime. To do so, we develop the second-order spatial perturbation equations for the Rayleigh-Plesset equation and solve them numerically. Our code is publicly available. Studying capillary waves on stable bubbles, we recreate the Kolmogorov-Zakharov spectrum predicted by weak turbulence theory, putting wave turbulence theory to use for bubbles. In this investigation, it seems that curvature does not affect turbulent properties. The calculated bubble surface responds qualitatively to low gravity experiments. The link demonstrated opens new possibilities for studying several bubble phenomena, including sonoluminescence and cavitation, using the extensive tools developed in the wave turbulence framework.

Figure 1

An example of the mean-squared elevation. The entire domain appears in (a). In (b), only the interval predicted inertial, between pumping, and the onset of high mode dissipation. Dashed green lines denote fits and blue dots denote simulation data. Data are fitted in the range $l\in [20,35]$. This simulation was run with $\beta =2.1, \epsilon =0.1$, and $l_{\text{max}}=55$. Other parameters are as previously defined.

Figure 2

Fitted power-law as a function of time (blue dots) and the Zakharov-Filonenko power law (dashed green line). Simulation parameters were selected as in Fig. 1.

Figure 3

An example state of the bubble, projected on a globe (from both sides). Simulation parameters were selected as in Fig. 1.

Figure 4

The figures above present simulation results for the amplitude of the mode $l=30,m=0$ alongside the theoretical prediction by (E10). Simulated with $l_{\text{max}}=40, \beta =1.0, \alpha =P_0=P_{\infty }=0$, and no dissipation or pumping terms. Amplitude scale $\epsilon$ is ${10}^{-4}$ (${10}^{-3}$) for (a) (b). $y$-axis units are arbitrary for both figures.