The violent collapse of bubble clusters can be an important mechanism of damage to adjacent material surfaces in both engineering and biomedical applications. Because of their complexity, past theoretical studies have generally been restricted to significantly simplified models, such as homogenized continuum models based upon volume averages or arrays of strictly spherical bubbles, which neglect detailed bubble dynamics. However, the details of the bubble-scale dynamics are potentially important locally. For example, wall or tissue damage is expected to depend upon peak pressures rather than the average pressure that might be computed with a homogeneous model. Here, we simulate the expansion and subsequent collapse of hemispherical clusters of 50 bubbles adjacent to a planar rigid wall and viscous fluids as models for soft tissues in therapeutic ultrasound using a computationally efficient diffuse-interface numerical scheme for compressible multiphase flows. It represents in detail the coupled asymmetric dynamics of each bubble within the cluster. The development of this scheme and its application to simulate detailed bubble-cloud collapse are the principal contributions of this dissertation.
The numerical scheme represents multi-fluid interfaces using field variables (interface functions) with associated transport equations. In our formulation, these are augmented, with respect to an established formulation, to enforce a selected interface thickness. The resulting interface region can be set just thick enough to be resolved by the underlying mesh and numerical method, yet thin enough to provide an efficient model for dynamics of well-resolved scales. A key advance in our method is that the interface regularization is asymptotically compatible with the thermodynamic laws of the mixture model upon which it is constructed. It incorporates first-order pressure and velocity non-equilibrium effects while preserving interface conditions for equilibrium flows, even within the thin diffused mixture region. The finite-volume numerical solver is also integrated in a multi-resolution Adaptive Mesh Refinement (AMR) framework that allows efficient resolution of individual bubbles of the cluster in a sufficiently large domain. We first quantify the improved convergence of this formulation in an air-helium shock-tube problem and an air-water bubble-collapse problem, then show that it enables fundamentally better simulations of single-bubble dynamics. Demonstrations include both a spherical-bubble collapse, which facilitates comparison with a semi-analytic solution, and a jetting-bubble collapse adjacent a wall. For the spherical collapse, we show agreement with the semi-analytic solution, and the preservation of symmetry despite the Cartesian mesh. Comparisons for the near-wall case show that without the new formulation the re-entrant jet is suppressed by numerical diffusion leading to qualitatively incorrect results.
Next, the method is applied to simulate cluster dynamics adjacent to material surfaces. Simulations near the rigid wall show that collapse propagates inward, and a geometrical pressure focusing occurs, which generates impulsive pressures near the focus. The peak pressures depend strongly on the arrangement of the bubbles, particularly those near the focus. The initial acceleration of the bubbles that drives their expansion is identified as an important parameter governing the bubble interactions, and hence the pressure focusing. The simplified models we compare with provide good agreement for the gross cluster behavior, for example gas volume history, but fail to predict the same peak pressures seen in the detailed simulations during the collapse. Replacing the rigid wall with a viscous fluid, as a crude model for tissue, shows significantly different dynamics compared to the rigid wall. Simulations show weaker pressure focusing with substantially lower peak pressures
