Effect of direct bubble-bubble interactions on linear-wave propagation in bubbly liquids

We study the influence of bubble-bubble interactions on the propagation of linear acoustic waves in bubbly liquids. Using the full model proposed by Fuster and Colonius [J. Fluid Mech. 688, 253 (2011)], numerical simulations reveal that direct bubble-bubble interactions have an appreciable effect for frequencies above the natural resonance frequency of the average size bubble. Based on the new results, a modification of the classical wave propagation theory is proposed. The results obtained are in good agreement with previously reported experimental data where the classical linear theory systematically overpredicts the effective attenuation and phase velocity.

Figure 1

Histogram of the bubble size distribution used in the simulation for validation against the classical linear theory results (results presented in Fig. 2). $R_0=110\mu \mathrm{m}$.

Figure 2

Speed of sound and attenuation of a polydisperse bubble cloud of randomly distributed bubbles obtained from simulation results and comparison with the traditional linear theory neglecting bubble-bubble interactions [22].

Figure 3

Influence of the fitting parameter $L/R_0$ on the phase velocity and attenuation of a monodisperse bubble cloud of randomly distributed air bubbles in water obtained from simulation results for case A ($g=1$ in all cases). For reference, the predictions of the classical linear theory ($g=0$) and the results from simulations are included.

Figure 4

Phase velocity and attenuation of a monodisperse bubble cloud of randomly distributed air bubbles in water obtained from simulation results for cases A (top), B (middle), and C (bottom). The dashed horizontal line on the attenuation plots indicates the upper bound for which the numerical attenuation values are reliable, due to numerical dissipation. For reference, the predictions of the classical linear theory and the proposed extension of the theoretical results with constant $C=2.3$ are included.

Figure 5

Best fitting value of the nondimensional interaction distance, $L{\beta }^{1/3}/R_0$, for various bubble radius and concentrations for monodisperse bubble clouds. The number of bubbles range from $n/n_{\mathrm{ref}}=0.5$ to 2. The nondimensional value $C=\frac{L{\beta }^{1/3}}{R_0}$ remains approximately constant irrespective of the concentration and bubble radius.

Figure 6

Phase velocity and attenuation of a polydisperse bubble cloud of randomly distributed bubbles obtained from simulation results and comparison with the experimental results of Silberman [9], the linear theory (LT) predictions of Ref. [22] (LT $g=0$) and the modified linear theory accounting for direct bubble-bubble interactions in the limiting case of $g=1$. The dashed horizontal line on the attenuation plots indicates the upper bound for which the numerical attenuation values are reliable, due to numerical dissipation.