An Eulerian-Lagrangian Study of Cloud Dynamics near a Rigid Wall

Bubble cloud collapse is one of the most destructive forms of cavitation. However, numerical modeling of this phenomenon is challenging and involves strong bubble dynamics, bubble-bubble, and bubble-wall interactions. Here we consider an initially spherical cloud near a rigid wall and treat the cloud bubbly mixture as a continuum and the bubbles in the cloud as discrete singularities. These two components are coupled through the local and instantaneous void fractions associated with the bubble volumes and locations. Simulations with sinusoidal pressure excitation of various frequencies and amplitudes for the same initial bubble cloud enable determination of the cloud resonance frequency. This condition results in wall pressures orders of magnitudes higher than the excitation pressure. As the amplitude of the driving pressure increases, the resonance frequency deviates significantly from the classical value obtained from linearized theory [1]. It becomes gradually smaller as the excitation amplitude increases then reaches a limit value. For high amplitude excitations, the peak wall pressure generated at the wall reaches a maximum when the ratio of the initial bubble radius to the maximum bubble radius is minimum. Too weak or too strong bubble interactions in the cloud inhibit strong collective effects. The effects of other parameters such as the initial distance between the cloud center and the wall are also discussed. INTRODUCTION The collapse of a cloud of bubbles near a rigid boundary is known as one of the most destructive forms of cavitation. It occurs in practical applications such as in cavitation on propellers [2–4], cavitating jets for cutting, cleaning, oxidation, disinfection [5,6], Shock Wave Lithotripsy (SWL) for kidney stone fragmentation [7–9], cavitation erosion testing [10] etc. Numerical modeling of such a problem involves multi-scale physics ranging from the micro scale of the individual bubbles to the overall scale of the cloud, and involves bubble-bubble and bubble-wall interactions [10–17]. Two-phase bubbly flows are usually modeled using approaches such as equivalent homogeneous continuum models, Eulerian two-fluid models, or Eulerian-Lagrangian approaches. Homogeneous models are useful for low void fractions, whereas Eulerian-Lagrangian approaches are more appropriate for higher void fractions [12–18]. In recent studies a continuum homogeneous model, an Eulerian multicomponent model, and an Eulerian-Lagrangian model were compared for the simulation of a bubble cloud interacting with a large bubble [29-32]. It was found that although all the models capture the average low-frequency dynamics, modeling the discrete bubbles is essential to a good understanding of the physics with a reasonable computational cost. Using a 3D two-way coupled Eulerian-Lagrangian model [22–24], we study here the dynamics near a rigid wall of a bubble cloud excited by a sinusoidal pressure. To do so, we treat the two-phase medium as a continuum and solve the corresponding Navier-Stokes equations with time and space varying density using a fixed grid. The bubbles in the cloud are modeled as singularities. The source terms (bubble volume variations) are obtained from modified Rayleigh-Plesset-KellerHerring equations, while the dipole terms are obtained from bubble equations of motion. The two-way coupling between the Euler and Lagrange components is realized through the mixture