Christopher E. Brennen

Cavitation and bubble dynamics

Cavitation and Bubble Dynamics deals with the fundamental physical processes of bubble dynamics and the phenomenon of cavitation. It is ideal for graduate students and research engineers and scientists; a basic knowledge of fluid flow and heat transfer is assumed. The analytical methods presented are developed from basic principles. The book begins with a chapter on nucleation and describes both the theory and observations in flowing and non-flowing systems. Three chapters provide a systematic treatment of the dynamics and growth, collapse, or oscillation of individual bubbles in otherwise quiescent fluids. The following chapters summarize the motion of bubbles in liquids; describe some of the phenomena that occur in homogeneous bubbly flows, with emphasis on cloud cavitation; and summarize some of the experimental observations of cavitating flows. The last chapter provides a review of free streamline methods used to treat separated cavity flows with large attached cavities.

Measurements of Solid Spheres Bouncing Off Flat Plates

Recent years have seen a substantial increase of interest in the flows of granular materials whose rheology is dominated by the physical contact between particles and between particles and the containing walls. Considerable advances in the theoretical understanding of rapid granular material flows have been made by the application of the statistical methods of molecular gas dynamics (e.g., Jenkins and Savage (1983), Lun et al. (1984)) and by the use of computers simulations of these flows (e.g., Campbell and Brennen (1985), Walton (1984)). Experimental studies aimed at measurements of the fundamental rheology properties are much less numerous and are understandably limited by the great difficulties involved in trying to measure velocity profiles, solid fraction profiles, and fluctuating velocities within a flowing granular material. Nevertheless, it has become clear that one of the most severe problems encountered when trying to compare experimental data with the theoretical models is the uncertainty in the material properties governing particle/particle or particle/wall collisions. Many of the theoretical models and computer simulations assume a constant coefficient of restitution (and, in some cases, a coefficient of friction). The purpose of the present project was to provide some documentation for particle/wall collisions by means of a set of relatively simple experiments in which solid spheres of various diameters and materials were bounced off plates of various thickness and material. The objective was to provide the kind of information on individual particle/wall collisions needed for the theoretical rheological models and computer simulations of granular material flows: in particular, to help resolve some of the issues associated with the boundary condition at a solid wall. For discussion of the complex issues associated with dynamic elastic or inelastic impact, reference is made to Goldsmith (1960) and the recent text by Johnson (1985).

FISSION OF COLLAPSING CAVITATION BUBBLES

High-speed observations (for example, Lauterborn and Bolle 1975, Tomita and Shima 1990, Frost and Sturtevant 1986) clearly show that though a collapsing cavitation bubble approaches its minimum size as a coherent single volume, it usually reappears in the first rebounding frame as a cloud of much smaller bubbles or as a highly distorted single volume (see, for example, figure 2). This paper explores two mechanisms that may be responsible for that bubble fission process, one invoking a Rayleigh-Taylor stability analysis and the other utilizing the so-called microjet mechanism. Both approaches are shown to lead to qualitatively similar values for the number of fission fragments and the paper investigates the flow parameters that effect that number. Finally, we explore the effective damping of the Rayleigh-Plesset single bubble calculation which that fission process implies and show that it is consistent with the number of collapses and rebounds which are observed to occur in experiments.

OBSERVATIONS OF THE DYNAMICS AND ACOUSTICS OF TRAVELLING BUBBLE CAVITATION

Ceccio and Brennen (1991 and 1989) recently examined the interaction between individual cavitation bubbles and the structure of the boundary layer and flow field in which the bubble is growing and collapsing. They were able to show that individual bubbles are often fissioned by the fluid shear and that this process can significantly effect the acoustic signal produced by the collapse. More recently Kumar and Brennen (1991-1992) have closely examined further statistical properties of the acoustical signals from individual cavitation bubbles on two different headforms in order to learn more about the bubble/flow interactions. All of these experiments were, however, conducted in the same facility with the same size of headform (5.08cm in diameter) and over a fairly narrow range of flow velocities (around 9m/s). Clearly this raises the issue of how the phenomena identified change with speed, scale and facility. The present paper describes experiments conducted in order to try to answer some of these important questions regarding the scaling of the cavitation phenomena. The experiments were conducted in the Large Cavitation Channel of the David Taylor Research Center in Memphis Tennessee, on similar Schiebe headforms which are 5.08, 25.4 and 50.8cm in diameter for speeds ranging up to 15m/s and for a range of cavitation numbers.

Linearized Dynamics of Spherical Bubble Clouds

The present work investigates the dynamics of the one-dimensional, steady flow of a spherical bubble cloud subject to harmonic far-field pressure excitation. Bubble dynamics effects and energy dissipation due to viscosity, heat transfer, liquid compressibility and relative motion of the two phases are included. The equations of motion for the average flow and the bubble radius are linearized and a closed-form solution is obtained. The results are then generalized by means of Fourier synthesis to the case of arbitrary far-field pressure excitation. The flow displays various regimes (sub-resonant, trans-resonant and super-resonant) with different properties depending on the value of the relevant flow parameters. Examples are discussed in order to show the effects of the inclusion of the various energy dissipation mechanisms. Finally the results for the case of Gaussian-shaped far-field pressure change are presented and the most important limitations of the theory are briefly discussed. The simple linearized dynamical analysis developed so far clearly deminstrates the importance of the complex phenomena connected to the interaction of the dynamics of the bubbles with the flow and provides an introduction to the more realistic study of the same flows with nonlinear bubble dynamics.

The Dynamic Transfer Function for a Cavitating Inducer

Knowledge of the dynamic performance of pumps is essential for the prediction of transient behavior and instabilities in hydraulic systems; the necessary information is in the form of a transfer function which relates the instantaneous or fluctuating pressure and mass flow rate at inlet to the same quantities in the discharge from the pump. The presence of cavitation within the pump can have a major effect on this transfer function since dynamical changes in the volume of cavitation contribute to the difference in the instantaneous inlet and discharge mass flow rates. The present paper utilizes results from free streamline cascade theory to evaluate the elements in the transfer function for a cavitating inducer and shows that the numerical results are consistent with the characteristics observed in some dynamic tests on rocket engine turbopumps.

Rotor-Stator Interaction in a Diffuser Pump

The interaction between impeller blades and diffuser vanes in a diffuser pump was investigated. Steady and unsteady pressure measurements were taken on the diffuser vanes, and the shroud wall of a vaned and a vane less diffuser. Steady, unsteady, and ensemble averaged unsteady data, as well as frequency spectra are presented. The measurements were made for different flow coefficients, shaft speeds, and radial gaps between impeller blade trailing and diffuser vane leading edge (1.5% and 4.5% based on impeller discharge radius). The resulting lift on the vane, both steady and unsteady, was computed from the pressure measurements at mid vane height. The magnitude of the fluctuating lift was found to be greater than the steady lift. The pressure fluctuations were larger on the suction side than on the pressure side attaining their maximum value, of the same order of magnitude as the total pressure rise across the pump, near the leading edge. Pressure fluctuations were also measured across the span of the vane. and those near the shroud were significantly smaller than those near the hub. The pressure fluctuations on the shroud wall itself were larger for the vaned diffuser than a vaneless diffuser. Lift, vane pressure, and shroud wall pressure fluctuations decreased strongly with increasing radial gap.

Heat transfer to flowing granular media

The convective heat transfer resulting from a granular flow over a heated surface is investigated. The specific type of flow considered is that in which adjacent material particles are in physical contact. The qualitative features of this type of flow are discussed, and the existing equations of motion are extended. With regard to the equations of motion, an exact solution is shown which has applications concerning the mass flow rate of granular materials through hoppers. The particular heat transfer problem investigated is convection from a flat plate with its long axis parallel to the flow field. An approximate analytical solution, which takes into account the particulate nature of the medium, is developed and experimental measurements obtained. The theory was found to correctly predict the trends of the experimental data. The results indicate that the Nusselt number for this configuration is influenced substantially, under certain conditions, by the noncontinuous nature of the medium. A semi-empirical correlation is presented, based on experimental results obtained with four different granular materials.

Vertical Vibration of a Deep Bed of Granular Material in a Container

A deep bed of granular material (more than six layers of particles) was subjected to sinusoidal, vertical vibrations. Several phenomena were observed depending on the amplitude of excitation. These included heaping, surface waves, and arching; the transitions from one state to another involved various dynamic instabilites and bifurcations. The paper includes a description of these phenomena and the characteristic properties associated with each in addition to measurements of the transitions from one phenomena to another.

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.