A Monolithic Mass Tracking Formulation for Bubbles in Incompressible Flow

Abstract: This dissertation presents a novel method for treating bubbles in free surface incompressible flow that relies on the conservative advection of bubble mass and an associated equation of state in order to determine pressure boundary conditions inside each bubble. It is shown that executing this algorithm in a traditional manner leads to stability issues similar to those seen for partitioned methods for solid-fluid coupling. Therefore, the problem is reformulated monolithically. This is accomplished by first proposing a new fully monolithic approach to coupling incompressible flow to fully nonlinear compressible flow including the effects of shocks and rarefactions, and then subsequently making a number of simplifying assumptions on the air flow removing not only the nonlinearities but also the spatial variations of both the density and the pressure. The resulting algorithm is quite robust, has been shown to converge to known solutions for test problems, and has been shown to be quite effective on more realistic problems including those with multiple bubbles, merging and pinching, etc. Notably, this approach departs from a standard two-phase incompressible flow model where the air flow preserves its volume despite potentially large forces and pressure differentials in the surrounding incompressible fluid that should change its volume. The proposed method allows bubbles to readily change volume according to an isothermal equation of state.

This method is then extended to model both large and small scale bubble dynamics. Small under-resolved bubbles are evolved using Lagrangian particles that are monolithically two-way coupled to the surrounding flow in a manner that closely approximates the analytic bubble oscillation frequency while converging to the analytic volume as predicted by the well-known Rayleigh-Plesset equation. A novel scheme is presented for interconverting between these under-resolved Lagrangian bubbles and the larger well-resolved Eulerian bubbles. A novel seeding mechanism is also presented to realistically generate bubbles when simulating fluid-structure interaction with complex objects such as ship propellers. The proposed framework for bubble generation is general enough to be incorporated into all grid-based as well as particle-based fluid simulation methods.