2. On Calculations of Cloud Cavitation

The experimental and practical observations of cloud cavitation exemplified by the preceding discussion have generated much interest in modelling the dynamics and acoustics of these flows. These efforts began with the work of van Wijngaarden (26) who first attempted to model the behavior of a collapsing layer of bubbly fluid next to a solid wall. The essence of his approach was to couple conventional continuity and momentum equations for the compressible mixture to a Rayleigh-Plesset equation for the bubble dynamics (see, for example, Brennen (27)) which would provide the neccessary relation connecting the local pressure with the bubble size and therefore the local mixture density. This is the basis for most of the computational investigations which have been carried out subsequently. Later investigators explored numerical methods which incorporate the individual bubbles (Chahine (28)) and continuum models which, for example, analyzed the behavior of shock waves in a bubbly liquid (Noordzij and van Wijngaarden (29), Kameda and Matsumoto (30)) and identified the natural frequencies of spherical clouds of bubbles (d’Agostino and Brennen (31)). Indeed the literature on the linearized dynamics of clouds of bubbles has grown rapidly (see, for example, d’Agostino et al. (32,33), Omta (34), Prosperetti (35)). However, apart from some weakly non-linear analyses (Kumar and Brennen (36,37,38)) only a few papers have addressed the highly non-linear processes which are an inevitable consequence of the non-linearities in the Rayleigh-Plesset equation and are most evident during the collapse of a cloud of bubbles. Chahine and Duraiswami (39) 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 and Kedrinskii and their co-workers (Mørch (2,3), Hanson et al. (4)). They surmised 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. We begin the review of computational efforts by describing studies of the dynamics of a spherical cloud of bubbles, a simple example which illustrates these processes of shock wave formation, propagation and focussing.

3. Computations: [1] Dynamics of a Spherical Cloud

Wang and Brennen (40,41,42) (see also Reisman et al. (1)) used a mixture model comprising the continuity, momentum and Rayleigh-Plesset equations (for specifics see, for example, d’Agostino et al. (31,32,33)) to study the non-linear growth and collapse of a spherical cloud of bubbles. A finite cloud of nuclei is subjected to an episode of low pressure which causes the cloud to cavitate; the pressure then returns to the original level causing the cloud to collapse. Wang and Brennen used the computational model to study the various cloud dynamics and acoustics exhibited in various parametric regimes. Key parameters are the cavitation number, σ, which characterizes the the initial pressure level, the magnitude of the low pressure episode characterized by a minimum pressure coefficient, Cpmin, and by a duration, D/U. The initial radius and void fraction of the cloud are denoted by A0 and α0 respectively and the initial radius of the bubbles within the cloud is denoted by R0.

This parametric exploration revealed that the dynamics and acoustics depended in an important way on the “cloud interaction parameter”, β, defined as

(1)

Note that, while the initial void fraction, α0, is very small, the ratio, A0/R0, may be very large so that β could be small or large compared with unity. Earlier linear and weakly nonlinear studies of cloud dynamics (d’Agostino & Brennen (31,32,33), Kumar & Brennen (36,37,38)) showed that the cloud natural frequency is strongly dependent on this parameter. If β is small, the natural frequency of the cloud is close to that of the individual bubbles in the cloud. In other words, the bubbles in the cloud tend to behave as individual units in an infinite fluid and the bubble/bubble interaction effects are minor. On the other hand bubble interaction effects dominate when the value of β is greater than order one. Then the collective oscillation of bubbles in the cloud results in a cloud natural frequency which is lower than the natural frequency of individual bubbles.

In all of the computations of Wang and Bren-



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