A Numerical Investigation of Unsteady Bubbly Cavitating Nozzle Flows

The effects of unsteady bubbly dynamics on cavitating flow through a converging-diverging nozzle are investigated numerically. A continuum model that couples the Rayleigh-Plesset equation with the continuity and momentum equations is used to formulate unsteady, quasi-one-dimensional partial differential equations. Flow regimes studied include those where steady-state solutions exist, and those where steady-state solutions diverge at the so-called flashing instability. these latter flows consist of unsteady bubbly shock waves traveling downstream in the diverging section of the nozzle. An approximate analytical expression is developed to predict the critical backpressure for choked flow. The results agree with previous barotropic models for those flows where bubbly dynamics are not important, but show that in many instances the neglect of bubbly dynamics cannot be justified. Finally the computations show reasonable agreement with an experiment that measures the spatial variation of pressure, velocity and void fraction for steady shockfree flows, and good agreement with an experiment that measures the throat pressure and shock position for flows with bubbly shocks. In the model, damping of the bubbly raidal motion is restricted to a simple "effective" viscosity, but many features of the flow are shown to be independent of the specific damping mechanism

Void fraction disturbances in a uniform bubbly fluid

The paper is concerned with the flow of dispersions of gas bubbles in liquid, with bubble sizes such that the inertia forces on the bubbles are of importance to the dynamics. One-dimensional conservation equations are derived, which govern the flow when the deviations from the uniform state are small. These are used to describe the features of the propagation of void fraction disturbances, and to investigate the stability of uniform bubbly flows. The results are compared with what has been observed in experiments

Numerical Computation of Shock Waves in a Spherical Cloud of Cavitation Bubbles

The nonlinear dynamics of a spherical cloud of cavitation bubbles have been simulated numerically in order to learn more about the physical phenomena occurring in cloud cavitation. A finite cloud of nuclei is subject to a decrease in the ambient pressure which causes the cloud to cavitate. A subsequent pressure recovery then causes the cloud to collapse. This is typical of the transient behavior exhibited by a bubble cloud as it passes a body or the blade of a ship propeller. The simulations employ the fully nonlinear continuum mixture equations coupled with the Rayleigh-Plesset equation for the dynamics of bubbles. A Lagrangian integral method is developed to solve this set of equations. It was found that, with strong bubble interaction effects, the collapse of the cloud is accompanied by the formation of an inward propagating bubbly shock wave, a large pressure pulse is produced when this shock passes the bubbles and causes them to collapse. The focusing of the shock at the center of the cloud produces a very large pressure pulse which radiates a substantial impulse to the far field and provides an explanation for the severe noise and damage potential in cloud cavitation

Acoustic Saturation in Bubbly Cavitating Flow Adjacent to an Oscillating Wall

Bubbly cavitating flow generated by the normal oscillation of a wall bounding a semi-infinite domain of fluid is computed using a continuum two-phase flow model. Bubble dynamics are computed, on the microscale, using the Rayleigh-Plesset equation. A Lagrangian finite volume scheme and implicit adaptive time marching are employed to accurately resolve bubbly shock waves and other steep gradients in the flow. The one-dimensional, unsteady computations show that when the wall oscillation frequency is much smaller than the bubble natural frequency, the power radiated away from the wall is limited by an acoustic saturation effect (the radiated power becomes independent of the amplitude of vibration), which is similar to that found in a pure gas. That is, for large enough vibration amplitude, nonlinear steepening of the generated waves leads to shocking of the wave train, and the dissipation associated with the jump conditions across each shock limits the radiated power. In the model, damping of the bubble volume oscillations is restricted to a simple "effective" viscosity. For wall oscillation frequency less than the bubble natural frequency, the saturation amplitude of the radiated field is nearly independent of any specific damping mechanism. Finally, implications for noise radiation from cavitating flows are discussed

Eulerian-Lagrangian method for simulation of cloud cavitation

We present a coupled Eulerian-Lagrangian method to simulate cloud cavitation in a compressible liquid. The method is designed to capture the strong, volumetric oscillations of each bubble and the bubble-scattered acoustics. The dynamics of the bubbly mixture is formulated using volume-averaged equations of motion. The continuous phase is discretized on an Eulerian grid and integrated using a high-order, finite-volume weighted essentially non-oscillatory (WENO) scheme, while the gas phase is modeled as spherical, Lagrangian point-bubbles at the sub-grid scale, each of whose radial evolution is tracked by solving the Keller-Miksis equation. The volume of bubbles is mapped onto the Eulerian grid as the void fraction by using a regularization (smearing) kernel. In the most general case, where the bubble distribution is arbitrary, three-dimensional Cartesian grids are used for spatial discretization. In order to reduce the computational cost for problems possessing translational or rotational homogeneities, we spatially average the governing equations along the direction of symmetry and discretize the continuous phase on two-dimensional or axi-symmetric grids, respectively. We specify a regularization kernel that maps the three-dimensional distribution of bubbles onto the field of an averaged two-dimensional or axi-symmetric void fraction. A closure is developed to model the pressure fluctuations at the sub-grid scale as synthetic noise. For the examples considered here, modeling the sub-grid pressure fluctuations as white noise agrees a priori with computed distributions from three-dimensional simulations, and suffices, a posteriori, to accurately reproduce the statistics of the bubble dynamics. The numerical method and its verification are described by considering test cases of the dynamics of a single bubble and cloud cavitaiton induced by ultrasound fields.Comment: 28 pages, 16 figure

A brief summary of L. van Wijngaarden's work up till his retirement

This paper attempts to provide an overview of Professor Leen van Wijngaarden's scientific work by briefly summarizing a number of his papers. The review is organized by topic and covers his work on pressure waves in bubbly liquids, bubble dynamics, two-phase flow, standing waves in resonant systems, and flow cavitation noise. A list of publications up till his retirement in March 1997 is provided in the Appendix

Stability of Parallel Bubbly and Cavitating Flows

This paper examines the bubble dynamic effects on the stability of parallel bubbly and cavitating flows of low void fraction. Inertial effects associated with the bubble response and energy dissipation due to the viscosity of the liquid, the heat transfer between the two phases, and the liquid compressibility are included. The equations of motion are linearized for small perturbations and a modified Rayleigh equation for the inviscid stability of the two-dimensional parallel flow is derived. Numerical solutions of the characteristic problem for the modified Rayleigh equation of a free shear layer are obtained by means of a multiple shooting method. Depending on the dispersion of the gaseous phase in the bubbly mixture, the ambient pressure and the free stream velocities, the pressure of air bubbles can induce significant departures from the classical solution for a single phase fluid. Results are presented to illustrate the influence of the relevant flow parameters

Statistical mechanics of bubbly liquids

The dynamics of bubbles at high Reynolds numbers is studied from the viewpoint of statistical mechanics. Individual bubbles are treated as dipoles in potential flow. A virtual mass matrix of the system of bubbles is introduced, which depends on the instantaneous positions of the bubbles, and is used to calculate the energy of the bubbly flow as a quadratic form of the bubbles' velocities. The energy is shown to be the system's Hamiltonian and is used to construct a canonical ensemble partition function, which explicitly includes the total impulse of the suspension along with its energy. The Hamiltonian is decomposed into an effective potential due to the bubbles' collective motion and a kinetic term due to the random motion about the mean. An effective bubble temperature-a measure of the relative importance of the bubbles' relative to collective motion-is derived with the help of the impulse-dependent partition function. Two effective potentials are shown to operate: one due to the mean motion of the bubbles, dominates at low bubble temperatures, where it leads to their grouping in flat clusters normal to the direction of the collective motion, while the other, temperature-invariant, is due to the bubbles' position-dependent virtual mass and results in their mutual repulsion. Numerical evidence is presented for the existence of the effective potentials, the condensed and dispersed phases, and a phase transition

Improvement of acoustic theory of ultrasonic waves in dilute bubbly liquids

The theory of the acoustics of dilute bubbly liquids is reviewed, and the dispersion relation is modified by including the effect of liquid compressibility on the natural frequency of the bubbles. The modified theory is shown to more accurately predict the trend in measured attenuation of ultrasonic waves. The model limitations associated with such high-frequency waves are discussed