A Two-Phase Model of Bubbly Fluids

We use a continuum theory for multiphase immiscible mixtures whose individual components are separated by infinitesimally thin interfaces. The average balance equations for the different phases, as well as for the mixture, result from a systematic spatial averaging procedure. In addition to equations for mass, momentum, and energy, together with the entropy inequality, the balance equations also include equations for microinertia and microspin tensors. These equations, together with appropriate constitutive equations consistent with the entropy inequality, enable the modeling of immiscible multiphase materials where internal parameters are important. Here, we apply the results to a simple microstretch bubbly fluid. We show that the equations for microspin and microinertia, under a number of simplifying assumptions, combine to yield a general form of the Rayleigh-Plesset equation.

Direct Numerical Simulation of Bubbly Flows and Application to Cavitation Mitigation

The direct numerical simulation (DNS) method has been used to the study of the linear and shock wave propagation in bubbly fluids and the estimation of the efficiency of the cavitation mitigation in the container of the Spallation Neutron Source liquid mercury target. The DNS method for bubbly flows is based on the front tracking technique developed for free surface flows. Our front tracking hydrodynamic simulation code FronTier is capable of tracking and resolving topological changes of a large number of interfaces in two- and three-dimensional spaces. Both the bubbles and the fluid are compressible. In the application to the cavitation mitigation by bubble injection in the SNS, the collapse pressure of cavitation bubbles was calculated by solving the Keller equation with the liquid pressure obtained from the DNS of the bubbly flows. Simulations of the propagation of linear and shock waves in bubbly fluids have been performed, and a good agreement with theoretical predictions and experiments has been achieved. The validated DNS method for bubbly flows has been applied to the cavitation mitigation estimation in the SNS. The pressure wave propagation in the pure and the bubbly mercury has been simulated, and the collapse pressure of cavitation bubbles has been calculated. The efficiency of the cavitation mitigation by bubble injection has been estimated. The DNS method for bubbly flows has been validated through comparison of simulations with theory and experiments. The use of layers of nondissolvable gas bubbles as a pressure mitigation technique to reduce the cavitation erosion has been confirmed.

A Vlasov equation for pressure wave propagation in bubbly fluids

The derivation of effective equations for pressure wave propagation in a bubbly fluid at very low void fractions is examined. A Vlasov-type equation is derived for the probability distribution of the bubbles in phase space instead of computing effective equations in terms of averaged quantities. This provides a more general description of the bubble mixture and contains previously derived effective equations as a special case. This Vlasov equation allows for the possibility that locally bubbles may oscillate with different phases or amplitudes or may have different sizes. The linearization of this equation recovers the dispersion relation derived by Carstensen & Foldy. The initial value problem is examined for both ideal bubbly flows and situations where the bubble dynamics have damping mechanisms. In the ideal case, it is found that the pressure waves will damp to zero whereas the bubbles continue to oscillate but with the oscillations becoming incoherent. This damping mechanism is similar to Landau damping in plasmas. Nonlinear effects are considered by using the Hamiltonian structure. It is proven that there is a damping mechanism due to the nonlinearity of single-bubble motion. The Vlasov equation is modified to include effects of liquid viscosity and heat transfer. It is shown that the pressure waves have two damping mechanisms, one from the effects of size distribution and the other from single-bubble damping effects. Consequently, the pressure waves can damp faster than bubble oscillations.