Finite speed of sound effects on asymmetry in multibubble cavitation

Three-dimensional direct numerical simulations are used to revisit the experiments on multibubble cavitation performed by Bremond et al., Phys. Fluids 18, 121505 (2006); Phys. Rev. Lett. 96, 224501 (2006). In particular, we aim at understanding the asymmetry observed therein during the expansion and collapse of bubble clusters subjected to a pressure pulse. Our numerical simulations suggest that the asymmetry is due to the force applied by the imposed pressure pulse and it is a consequence of the finite effective speed of sound in the liquid. By comparing our numerical results to the experiments, we found that the effective speed of sound under the experimental conditions was smaller than that of degassed water due to microbubbles in the system which resulted from prior cavitation experiments in the same setup. The estimated values of the effective speed of sound are consistent with those derived from the classical theory of wave propagation in liquids with small amounts of gas. To support this theory, we also present evidence of tiny bubbles remaining in the liquid bulk as a result of the fragmentation of large bubbles during the prior cavitation experiments. Furthermore, we find that this asymmetry also alters the direction of the liquid jet generated during the last stages of bubble collapse.

Figure 1

Top view of multiple cavitation bubbles nucleating from pits ($4\mu \text{m}$ diameter) and collapsing in contact with a rigid wall. (a) A pair of bubbles nucleating from two pits separated by distances of 200 and $400\mu \text{m}$ are shown in the left and right panels, respectively. (b) A cluster of 37 bubbles nucleating from the pits drilled in a hexagonal configuration having a pitch distance of $200\mu \text{m}$. This figure is adapted from the experiments of Bremond et al. [4, 5].

Figure 2

(a) The simplified schematic (top view) of the problem where the initial nuclei (shown with circles) of radii $R_0$ are separated by a distance $d$ and exposed to a 1D pressure pulse of amplitude $\mathrm{\Delta }p=1.5\text{MPa}$ and a characteristic length $L_p$ (defined from the pulse duration $T_p$ given from hydrophone measurements). The direction of wave propagation is shown by the red arrow. (b) The 3D computational setup used in the current study where the projected color map corresponds to the initial pressure field taken from the experiments [5]. A zoomed-in view of the initial nuclei and the projection of the numerical grid on the bottom boundary of the domain are shown.

Figure 3

(a) The dramatic decrease in the effective speed of sound for a mixture of liquid and dispersed gas is shown as a function of the volume fraction of the gas phase ${\alpha }_g$ given by Eq. (9). The point obtained by matching ${\mathcal{A}}_y$ from experiments and numerical simulations is shown by a black cross, and its coordinates are specified in parentheses. (b) The tiny fragmented bubbles seen during the cavitation experiments are highlighted with red circles along with the bigger bubbles nucleating from the pits for the cases of two pits (top) and five pits (bottom). The scales in the top left corners of the top and bottom panels is $40\mu \text{m}$.

Figure 4

Comparison of the numerical results for two different values of the effective speed of sound, i.e., $c_e=1500\text{m/s}$ and $c_e=500\text{m/s}$. The bubble pair evolves from hemispherical nuclei with radii of $20\mu \text{m}$. In (a) and (b), the top view of the bubble shape and pressure field are displayed at subsequent times shown. (a) Separation distance $d=200\mu \text{m}$ between the nuclei, with $c_e=1500\text{m/s}$ on the left and $c_e=500\text{m/s}$ on the right. (b) Separation distance $d=400\mu \text{m}$ between the nuclei, with $c_e=1500\text{m/s}$ on the left and $c_e=500\text{m/s}$ on the right. (c) A particular snapshot showing the definition of the asymmetry parameter ${\mathcal{A}}_y\left(t\right)$, which is defined as the shift of the bubble centroid in the $y$ direction. (d) The time evolution of ${\mathcal{A}}_y\left(t\right)$ obtained from the numerical simulations for all the cases discussed in (a) and (b) along with a comparison with experimental data. The error bars are equal to one pixel size ($1.53\mu \text{m}$) in the experiment snapshots.

Figure 5

Effect of the effective speed of sound on the asymmetry for the particular case where the bubbles are nucleating from a pair of hemispherical nuclei with radii of $20\mu \text{m}$ separated by a distance $d=200\mu \text{m}$. (a) The time evolution of ${\mathcal{A}}_y$ is shown for different values of the effective speed of sound (color map), alongside the experimentally obtained values (black crosses). The experimental bubble shapes overlaid with their numerical counterparts (red dots) at $t=22$ and $23\mu \text{s}$ are illustrated in the inset. (b) The evolution of the dimensionless ${\mathcal{A}}_y\left(t\right)$ as a function of dimensionless time ($t^{*}=t{U}_c/R_0$) obtained using the scaling predicted from Eq. (12) for different values of $c_e$. (c) The maximum value of the asymmetry parameter $\text{max}\left({\mathcal{A}}_y\right)$ is plotted on a log-log scale as a function of the effective speed of sound along with the experimental point where its $c_e$ is obtained by matching $\text{max}\left({\mathcal{A}}_y\right)$ with the numerical data. A fit of all data (dashed line) is also plotted to demonstrate that $\text{max}\left({\mathcal{A}}_y\right)$ decays like $c_e^{-1}$ as predicted by Eq. (12) for given $T_p$. (d) The bubble shapes at the instant of $\text{max}\left({\mathcal{A}}_y\right)$ are shown for different effective speeds of sound in the liquid. (e) Semitransparent bubble shapes are shown to visualize the jet generated during the last stages of collapse, whose direction is highlighted with red arrows.

Figure 6

The numerical results for the particular case where the effective speed of sound is assumed to be $333\text{m/s}$ and $d=200\mu \text{m}$ for varying $R_0$. (a) The time evolution of the asymmetry parameter ${\mathcal{A}}_y\left(t\right)$ is shown for different sizes of nuclei depicted with the color map. A zoomed-in view around $\text{max}\left({\mathcal{A}}_y\right)$ is shown in the inset. (b) The value of $\text{max}\left({\mathcal{A}}_y\right)$ corresponding to the maximum in (a) is plotted as a function of the sizes of the nuclei. The color code in (a) and (b) characterizes the radii of nuclei $R_0$.

Figure 7

(a) Top view of the five bubbles in a straight line configuration, as discussed in Ref. [4]. The numerical results are shown in the left panel, where the bubble expands from $20\mu \text{m}$ hemispherical nuclei and the effective speed of sound is set to 667 m/s. The experimental bubble shapes are shown in the right panel. (b) The evolution of the asymmetry parameter ${\mathcal{A}}_y\left(t\right)$ obtained from the numerical simulations (solid line) and the experiments (crosses). The error bar is equal to one pixel size (i.e., $1.53\mu \text{m}$) in the experiment snapshots.

Figure 8

(a) Top view of the cluster of 37 bubbles in a hexagonal arrangement expanding from $20\mu \text{m}$ hemispherical nuclei. The effective speed of sound in the liquid is $1480\text{m/s}$. Isobars are shown with a color map. (b) The evolution of the asymmetry parameter ${\mathcal{A}}_y\left(t\right)$ obtained from the numerical simulations (solid line) and the experiments (crosses).

Figure 9

The time evolution of parameter ${\mathcal{A}}_y\left(t\right)$ is shown for varying values of the numerical slip length for the bubble pair case. The plot demonstrates the robustness of the results.