Cover Image

HARDBACK
$198.00



View/Hide Left Panel

R

Bubble radius

R0

Initial radius of the bubble

t*

Time (s)

t

Dimensionless time, Ut*/R0

tp

Duration of low pressure perturbation

T

Period of foil oscillation (s)

U

Reference velocity of the flow (m/s)

Vbmax

Maximum total volume of bubbles in the cloud

α

Void fraction of the bubbly mixture

α0

Initial void fraction of bubbly mixture

β

Cloud interaction parameter,

η

Bubble population per unit liquid volume

ρ

Density of the liquid

σ

Cavitation number,

µe

Effective dynamic viscosity of the liquid

ω

Foil oscillation frequency (rad/s)

ωn

Natural frequency of single bubbles

2.
INTRODUCTION

In many cavitating flows of practical interest one observes the periodic formation and collapse of a “cloud” of bubbles. This temporal periodicity may occur naturally as a result of bubble-filled vortical structures or it may be the response to a periodic disturbance imposed on the flow. Common examples of imposed fluctuations are (a) the interaction between rotor and stator blades in a pump or turbine or (b) the interaction between a ship's propeller and the non-uniform wake created by the hull. Much recent interest has focused on the dynamics and acoustics of finite clouds of cavitation bubbles because of the very destructive effects which are observed to occur when such clouds form and collapse in a flow (see, for example, Knapp (3), Bark and van Berlekom (4), Soyama et al. (5)). Many authors such as Wade and Acosta (6), Bark and van Berlekom (4), Shen and Peterson (7, 8), Bark (9), Franc and Michel (10), Le et al. (11), Kubota et al. (12, 13), Hart et al. (14), McKenney et al. (15), Reisman et al. (16) and de Lange et al. (17) have studied the complicated flow patterns involved in the production and collapse of a cavitating cloud on a hydrofoil.

Analytical studies of the dynamics of cavitation clouds can be traced to the work of van Wijngaarden (18) who first attempted to model the behavior of a collapsing layer of bubbly fluid next to a solid wall. Later investigators explored numerical methods which incorporate the individual bubbles (Chahine (19)) and continuum models which, for example, analyzed the behavior of shock waves in a bubbly liquid (Noordzij and van Wijngaarden ( 20), Kameda and Matsumoto (21)) and identified the natural frequencies of spherical cloud of bubbles (d'Agostino and Brennen (22))). Indeed the literature on the linearized dynamics of clouds of bubbles has grown rapidly (see, for example, Omta (23), d'Agostino et al. (24, 25), Prosperetti (26)). However, apart from some weakly non-linear analyses (27, 28, 29) only a few papers have addressed the highly non-linear processes involved during the collapse of a cloud of bubbles. Chahine and Duraiswami (30) have conducted numerical simulations using a number of discrete bubbles and demonstrated how the bubbles on the periphery of the cloud develop inwardly directed re-entrant jets. However, most clouds contain many thousands of bubbles and it therefore is advantageous to examine the non-linear behavior of continuum models.

Another perspective on the subject of collapsing clouds was that introduced by Mørch, Kedrinskii and Hanson (31, 32, 33). They speculated that the collapse of a cloud of bubbles involves the formation and inward propagation of a shock wave and that the geometric focusing of this shock at the center of cloud creates the enhancement of the noise and damage potential associated with cloud collapse. Recently Wang and Brennen (1, 2) have used the mixture models employed earlier by d'Agostino et al. (22, 24, 25) to study the non-linear growth and collapse of a spherical cloud of bubbles. A finite cloud of nuclei is subjected to a temporary decrease in the ambient pressure which causes the cloud to cavitate and then collapse. The calculations clearly confirm the view of cloud collapse put forward by Mørch and his co-workers. In the present paper, we present some further information from spherical cloud calculations.

How bubbly shocks are formed and propagate in the much more complex and non-spherical geometries associated with cavitating foils, propeller or pump blades is presently not very clear. The present experiments have allowed identification of some specific shock structures whose details remain to be resolved and modelled.

3.
EXPERIMENTS

This section describes an experimental investigation of the large unsteady and impulsive pressures which are experienced on the suction surface of a



The National Academies | 500 Fifth St. N.W. | Washington, D.C. 20001
Copyright © National Academy of Sciences. All rights reserved.
Terms of Use and Privacy Statement