Bubble dynamics and related acoustics

The motion of a bubble of gas and vapour in an isochoric, inviscid liquid is numerically investigated in free space or in presence of a free surface and a gravitational force. The corresponding acoustic emission is evaluated and its relation with the bubble motion is discussed. The liquid is at rest at the initial time and, as a consequence, the flow is irrotational at any time. The bubble motion is described in terms of velocity potential. In order to evaluate the potential inside the flowfield, given its values and the normal velocities on the corresponding boundary, the classical integral representation is used. The normal velocity is then obtained by solving an integral equation. The pressure inside the bubble is assumed uniform and is related to the bubble volume $\cVs$ by the simplest state equation: $p_B(t) = p_V + p_{G0} \cVs(0)/\cVs(t)$, $p_V$ and $p_{G0}$ being the vapour and gas pressure at $t=0$. The corresponding pressure on the liquid face of the bubble boundary $\cBf$ follows as $p(\xv;t) = p_B(t) - 2 S/R_m(\xv;t)$, $S$ being the surface tension and $R_m$ the mean curvature radius. Once the pressure on the bubble boundary is known, the Bernoulli law is used in order to integrate in time the boundary values of the potential. The sound speed in the liquid being infinity, the acoustic pressure $p'$ at place $\xv_0$ and at time $t$ is evaluated by neglecting differences in the emission times. By indicating with $u_n$ the outward normal component of the velocity, $p'$ is computed as: \[ p'(\xv_0;t) = - \ds{\frac{\rho_L}{4\pi}} \ds{\frac{d}{dt}} \ds{\int_{\mbox{\fo $\cBf$}(t)}} dS(\xv;t) \hspace{1mm} \ds{\frac{u_n(\xv;t)}{|\xv-\xv_o|}} \hspace{1mm}, \] % $\rho_L$ being the liquid density. Comparisons with the acoustics induced by a spherical bubble are also discussed. As an interesting sample case, in the figures below the planar wave propagating at velocity $c$ in the $x$-direction: $p(x;t) = p_0 + \Delta p \hspace{.5mm} \{ 1-\tanh[2(x-x_0-ct)/\delta] \}/2$ crosses a spherical bubble with center on the origin (bubble radius $1$ mm, $p_V = 2338.474$ Pa, $S = 0.073$ N/m, $\rho_L = 998.207$ kg/m$^3$, $p_0 = \Delta p = 101325$ Pa, $x_0 = -2$, $\delta = 1$ mm, $c = 1$ m/s). Lengths are nondimensionalized with the bubble radius, times with the period ($0.364$ ms) of the free oscillations of the bubble and masses by means of $\rho_L$. Bubble shapes at times $0$ (green lines) and $1.17$ (red) are drawn in figure $a$, while the acoustic pressure in three positions ($(+4,0,0)$ blue line, $(-4,0,0)$ red and $(0,+4,0)$ green) is drawn vs. time in figure $b$. %Note that the curvature of the bubble boundary grows to infinity at a finite time, in absence of viscous dissipation. This singular behaviour leads to an intense acoustic emission.