Numerical investigation on the collapse of a bubble cluster near a solid wall

This paper studies numerically the collapse of a cluster of cavitation bubbles (as a primitive model for a bubble cloud) near a solid wall. The homogeneous two-phase mixture model is used, with the liquid-vapor interface resolved by volume of fluid method. The liquid is treated as compressible, allowing the propagation of pressure waves at the speeds determined by a state equation. This cluster consists of 27 identical bubbles, evenly distributed in a cubic region, with various bubble-wall and bubble-bubble distances considered. Our simulations suggest that the bubble-wall distance plays a more significant role. The maximum impulsive pressure of $41\mathrm{MPa}$ is achieved when the cluster is very close to the wall. The inward progress of collapse is observed by examining the evolutions of bubble shapes and flow fields, with two distinctly different sequences of collapse identified between the small and large bubble-wall distances. At a large bubble distance, the centermost bubble is the last to collapse, while at a small bubble distance, it is the central bubble nearest to the wall which collapses lastly. This difference can also explain the more intensive impulsive pressure for the smaller bubble-wall distances. The proposed numerical approach is of special interest because it can resolve the details of bubble-bubble and bubble-wall interactions, which are significant to the study of the collapse of a cavitation cloud, and its potential damage to hydraulic systems.

Figure 1

Outlines of the cavitation bubble at various stages of collapse for (a) the present simulations and (b) experimental results from Kling and Hammitt [46].

Figure 2

Relationships between the dimensionless bubble-wall distance and the time span of bubble collapse. Solid line: the analytic solution from formula (9). Symbols: the present numerical simulations.

Figure 3

Schematic of the geometric configuration for the simulations of 27 cavitation bubbles (side view).

Figure 4

Evolutions of (a) the vapour volumes and (b) the pressures at the center of the solid wall as the bubble cluster collapses, for different distances. The inset in (b) presents the pressure evolution at $\gamma =1.5$ for a single bubble.

Figure 5

Evolutions of the bubble shapes for $\gamma =3.5$ at a sequence of time instants of (a) $t=108\mu \mathrm{s}$, (b) $t=202\mu \mathrm{s}$, (c) $t=218\mu \mathrm{s}$, (d) $t=234\mu \mathrm{s}$, (e) $t=246\mu \mathrm{s}$, and (f) $t=258\mu \mathrm{s}$.

Figure 6

Velocity vectors within the symmetry plane for $\gamma =3.5$ at different time instants of (a) $t=218\mu \text{s}$, (b) $t=234\mu \text{s}$, (c) $t=246\mu \text{s}$, and (d) $t=258\mu \text{s}$.

Figure 7

Evolutions of the bubble shapes for $\gamma =1.5$ at different time instants of (a) $t=214\mu \text{s}$ or $t^{*}=2.32$, (b) $t=240\mu \text{s}$ or $t^{*}=2.60$, and (c) $t=274\mu \text{s}$ or $t^{*}=2.97$.

Figure 8

Pressure contours within the symmetry plane for $\gamma =1.5$ at different time instants of (a) $t=220\mu \text{s}$, (b) $t=242\mu \text{s}$, (c) $t=256\mu \text{s}$, (d) $t=266\mu \text{s}$, (e) $t=274\mu \text{s}$, and (f) $t=282\mu \text{s}$. Blue (dark gray) and red (light gray) colors denote low and high values of pressure, respectively.

Figure 9

Radial distributions of the pressure on the wall for a sequence of time instants ($\gamma =1.5$).

Figure 10

Evolutions of the pressures at the center of the solid wall as the bubbles collapse for different bubble-bubble spacings with a fixed bubble-wall distance of $\gamma =1.2$.