Euler-Lagrange Simulations of Bubble Cloud Dynamics Near a Wall

We present in this paper a two-way coupled EulerianLagrangian model to study the dynamics of microbubble clouds exposed to incoming pressure waves and the resulting pressure loads on a nearby rigid wall. The model simulates the twophase medium as a continuum and solves the N-S equations using Eulerian grids with a time and space varying density. The microbubbles are modeled as interacting spherical bubbles, which follow a modified Rayleigh-Plesset-Keller-Herring equation and are tracked in a Lagrangian fashion. A two-way coupling between the Euler and Lagrange components is realized through the local mixture density associated with the bubbles volume change and motion. Using this numerical framework, simulations involving a large number of bubbles were conducted under driving pressures of different frequencies. The results show that the frequency of the driving pressure is critical in determining the overall dynamics: either a collective strongly coupled cluster behavior or non-synchronized weaker multiple bubble oscillations. The former creates extremely high pressures with peak values orders of magnitudes higher than that of the excitation pressures. This occurs when the driving frequency matches the natural frequency of the bubble cloud. The initial distance between the bubble cloud and the wall is also critical on the resulting pressure loads. A bubble cloud collapsing very close to the wall exhibits a cascading collapse with the bubbles farthest from the wall collapsing first and the nearest ones collapsing last, thus the energy accumulates and then results in very violent pressure peaks at the wall. Farther from the wall, the bubble cloud collapses quasi spherically with the cloud center collapsing last. INTRODUCTION Collapse of clouds of bubbles near boundaries occur in various engineering applications such as in ultrasonic cavitation, cavitating jets, unsteady sheet cavities on propeller blades, etc. [1,2,3]. Numerical modeling of such a problem is very challenging, since it involves bubble-bubble, bubble-flow, and bubble-wall interactions. In addition the problem involves multi-scale physics ranging from micron-scale individual bubbles to meterscale flow field (e.g., propeller blade). In between, there are very rich and complex meso-scale phenomena at the scale of the bubble cloud, where the bubbles can act collectively [4-8]. The Boundary Element Method (BEM) or Direct Numerical Simulation (DNS) which can fully resolve individual bubbles’ behavior provide details at several scales of interest with impressive progress reported [e.g.,8-11]. However, due to the computational cost these methods are mostly limited to small scale problems addressed for fundamental study purposes, such as developing subgrid relationships for larger scale models. For real applications, twophase bubbly flows are usually modeled using one of several approaches: equivalent homogeneous continuum models, Eulerian two-fluid models, or Eulerian-Lagrangian approaches wherein the bubbles are treated as discrete particles.