Oscillations, Waves, and Interactions - GWDG

Acoustic cavitation 191
the bubble and get a system of equations, where N is the total bubble number:
F i M + F i D + F i B1 + �
F j,i
B2 = 0 , 1 ≤ i ≤ N . (8)
j�=i
The bubble sizes are defined by the rest radii R i 0. To describe their variation in time,
a gas diffusion law like Eq. (1) can be used, augmented by rules for merging with
other bubbles and splitting after surface instability.
This proposed N-body problem has some particularities, like a changing number
of particles 12 and forces that depend on the absolute position in space (i. e., on the
local sound pressure). It can be solved numerically, and the effort is considerably
reduced if approximations of the forces are taken. In particular a temporal averaging
over the oscillation period can be used, although some (typically small) errors are
introduced in added mass and drag (compare Sect. 4.3).
Hinsch employed an early version of this type of particle model already in 1975
to calculate the merging process of a few bubbles [51]. Yet, the idea was further
advanced only in the 1990s [43,52], probably due to the increased possibilities of
computer power. In the following we present some results from recent simulations of
bubble structures.
6.1 Streamer filaments
In many cases filamentary or “dendritic” arrangements of acoustic cavitation bubbles
are encountered [4,5,53]. The bubbles move relatively fast along branches that can
unite to form a new branch, like rivers form junctions. In a standing wave, a typical
picture is that many branches arrange around a pressure antinode, and bubbles
therefore stream inward the filamentary conglomerate [43]. At the center they may
form a larger bubble or bubble cluster where microbubbles are shed off, sometimes
visible as a “mist”. The microbubbles dissolve and thus compensate for the gas
arriving from the filaments.
The filamentary structure in a cubic resonator, driven at 25 kHz, has been analyzed
by stereoscopic high-speed recordings in Ref. [54]. Some of the reconstructed bubble
paths have afterwards been simulated by the particle approach described above. For
every bubble, only the first position and velocity have been used as an input for the
model, and the bubble sizes have been given a starting value of 5 µm. The pressure
value at the antinode and its position have been slightly fitted within the error
tolerance of a measurement. A direct comparison of experimental and calculated
bubble tracks is shown in Fig. 15. The correspondence is quite reasonable, and
the conclusion can be drawn that the dominant forces causing the bubble motion
have been taken into account by the particle model. Still, not every detail is met
perfectly by the simulation. This is probably due to variations in bubble sizes and
the fine structure of the filaments: they may contain smaller bubbles, overlooked by
the image processing of the experimental data, and travelling bubbles can undergo
repeated split-and-merge processes. Such phenomena add “noise” to the N-body
system, and because of the nonlinear bubble-bubble interaction, it is rather sensitive
12 N is not fixed in time because of nucleation, merging, splitting, and dissolution.