Bubble Structures in Acoustic Cavitatio RMettin 2005

8/10/2019 Bubble Structures in Acoustic Cavitatio RMettin 2005

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Bubble structures in acoustic cavitation

Robert Mettin

Drittes Physikalisches Institut, Universitat Gottingen,

Friedrich-Hund-Platz 1, 37077 Gottingen, Germany

E-mail: [email protected]

RUNNING TITLE: Bubble structures in acoustic cavitation

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Abstract

This article is reporting on bubble structures that represent different manifestations

of acoustic cavitation. General aspects relevant for structure formation in acoustic

cavitation are discussed, and a classification scheme into prototypes is proposed.

Characteristics and distributions of bubbles as well as the sound field environments

are reviewed for the different cavitation patterns. The study is mainly based on op-

tical and high-speed imaging investigations and is confined to acoustic frequencies

of the lower ultrasonic range.

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1 Introduction

The rupture of a liquid leading to cavities of gas phase is encountered in different

situations. Usually the circumstances are classified into the ideal cases of isobaric

rupture due to heat deposition (boiling), and isothermal rupture caused by tension.

The latter phenomenon, along with the subsequent dynamics of the created bubbles,

is generally called cavitation [1]. It is furthermore subdivided according to the

physical origin of the tension: Rupture in a low or negative pressure zone of a bulk

liquid flow is termed hydrodynamic cavitation [2, 3, 4, 5] , while the rupture in

strong sound fields is named acoustic cavitation [6, 7, 8, 9, 10, 3, 11]. This article is

devoted to the visual appearance of cavitation in the latter case - the generation by

an acoustic field.The findings and results are restricted to a certain range of applied sound frequen-

cies: the band between about 20 kHz and 50 kHz. The reason for this limitation

is based upon both the importance of this range for technical applications, and the

good accessibility by optical sensors. Acoustic cavitation at higher ultrasonic fre-

quencies creates increasingly small bubbles, and direct imaging becomes more and

more difficult. Then, indirect indicators have to be employed like erosion or sono-

luminescence [2, 12]. Lower frequencies (in the audible domain and below) are not

commonly used, and the cavitation phenomena approach in a way hydrodynamic

cavitation (oscillation periods are in the range of typical transit times of flows).

Nevertheless, investigations of bubble dynamics in low frequency sound are as well

worthwhile and can reveal highly interesting results [13, 14, 15].

The fields of applications of acoustic cavitation are permanently growing, and just

as much growing is the demand for fundamental knowledge of the accompanying

phenomena. In most applications, like in ultrasonic cleaning [16, 17] and sono-

chemistry [18], the collapse (violent implosion) of bubbles is the crucial event

which causes the observed effects. Consequently, the collapse phase is well studied,

and much effort is undertaken to clarify the physical and chemical conditions of a

single collapsing bubble [9, 12]. For most kinds of experimental or technical setups,

however, the link between applied sound field and bubble effects contains as well

the distribution of bubbles in space and time. This distribution is usually found to

appear in a given manner, and up to now it is not clear how to manipulate or controlit. A step towards this aim - a controlled acoustic cavitation - is a more complete

description and characterization of the existing bubble structures and their acoustic

environment. However, hitherto no satisfactory catalog of acoustic cavitation phe-

nomena exists, and this article tries to provide a starting or nucleation point for

further works in this direction.

It should be mentioned that a categorization of phenomena on the side of hydrody-

namic cavitation is already rather evolved. Owing to its importance in fluid dynam-

ics engineering, the treatment of cavitation in flow problems has a long history (see,

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e.g., [9]). Starting from works related to ship propellers, researchers and engineers

developed an own vocabulary of the observed patterns. For instance, they refer totraveling cavitation, fixed or attached cavitation, supercavitation, vortex cavitation,

and other manifestations of liquid rupture in flow problems [2, 9, 3, 5]. The main

aspect of distinction between the various types of hydrodynamic cavitation is the

location of appearance and the visual impression.

Quite in contrast, the existing categories of acoustic cavitation are led by qualita-

tive features of the bubbles. Typical attributes are transient vs. stable or hard

vs. soft cavitation. They describe the type of bubble collapse (transient = hard =

violent collapse, stable = soft = collapse without high energy concentration [10]),

but they do not qualify visual or structural bubble field properties or the environment

where the acoustic cavitation is observed. While the established notion is useful fora local description of cavitation, it clearly lacks the global aspect of bubble distri-

bution in space and time. Vice versa, the categories of hydrodynamic cavitation do

not tell much on the properties of single bubbles in the structures.

The classical literature is using a relatively small vocabulary of the different visual

manifestations of acoustic cavitation. This seems to be partly owed to the sparse-

ness of investigations, but also to the lack of sufficient imaging techniques. It has

been observed before that bubbles can appear in filamentary lines and conglomer-

ates which have been termed streamers [8, 10], and also in cluster-like concen-

trations [19, 20]. Additionally, some properties like subharmonic oscillations of

bubbles in structures have been visualized directly [21]. Apart of long-term ex-

posures, however, the imaging by high-speed cameras or holographic systems has

been highly elaborate and expensive, and systematic investigations could hardly be

performed. This has changed considerably, and nowadays medium- and high-speed

camera systems are available at moderate cost, and fully digital image processing

facilitates subsequent analysis.

2 Prerequisites

This chapter contains general remarks on technical issues and on some fundamen-

tals of acoustic cavitation which are relevant and useful for the discussion of struc-ture formation.

2.1 Experimental equipment

All of the following pictures have been recorded by several investigators at the Third

Physical Institute (Drittes Physikalisches Institut, DPI) at Gottingen University. The

imaging systems that have been used are a low framerate progressive scan video

camera (Pulnix TM 6701 AN), high framerate CCD cameras (HiSIS 2002, Photron

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ultima APX-RS), and an ultra high-speed camera with image intensifier (Imacon

468). The cameras have been connected to suitable magnifying optics like zoomlenses or long distance microscopes.

The illumination of cavitation processes is either arranged in a way that the bubbles

scatter the light for recording (bright bubbles), or that they move in front of a ho-

mogeneous bright background (dark bubbles). While the former way is somehow

easier to set up, it is hard to judge the absolute bubble sizes from the amount of

scattered light. In contrast, the latter method is suitable to obtain size statistics, but

its resolution of very small bubbles can be worse: while the light scattered by a bub-

ble could be enough to highlight a camera pixel, the shadowing effects of the same

bubble in front of an illuminated background might not be sufficient for detection if

the bubble image is smaller than a pixel. Likewise, the contrast of bubble signals inscattered light is typically better.

The acoustic set-ups that have been used are mainly simple resonator systems,

i.e. rectangular cuvettes with transparent walls, ultrasonic transducers at the bot-

tom, and an open top. Then, the generated sound field is typically close to a stand-

ing wave and consists of nodal and antinodal regions. Details of the field depend

on the liquid filling height and other liquid properties, and on possible influence

of the emerging cavitation (shifting resonance frequencies by impedance changes).

Additionally, the free surface can start to move and generate capillary and larger

surface waves which can build up and cause swashing. This, in turn, can modulate

the whole sound field with low frequencies. The emitting transducer surfaces are

relatively large in our resonator baths, and the (average) intensity at the emitter is

low (up to some W/cm2).

Other types of set-ups contain sonotrode emitters (Mason horn transducers). Such

emitters typically have a small surface area, resulting in high intensity at the sono-

trode tip in the range of 100 W/cm2. Structures will be shown near a small horn tip

and others near a more extended one, which might be considered as a transition to

larger emitter surfaces like in the resonator baths. The sound field near a sonotrode

can have the characteristics of a running wave: due to strong scattering and dis-

sipation in the strong cavitating zone directly in front of it, a shielding effect can

take place and the reflected waves from the container walls have less pronounced

influence than in the resonator case.All the structures reported in the following have been observed in water. Either the

water has been taken from the tap, or 20 m micropore filtered water has been used.In some cases the water has been degassed before the experiment. If non-degassed

water is used, its gas content decreases automatically under strong cavitating condi-

tions, and during this initial process sheets of bubbly layers (gaseous cavitation)

can appear. Later, hard (or true) cavitation remains or sets in. We always refer

to conditions after having run the experiments for a considerable time. Therefore,

the actual gas content of the used water is typically undersaturated to a certain de-

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gree which, though, is not quantified.

Acoustic cavitation is a complex phenomenon which involves scaling in time andspace over several orders of magnitude. However, certain aspects occur on relatively

distinct scales. Typical durations of processes are 1 ns for light emission and chem-

istry at the end of a bubble collapse, 20 . . . 50 s for a bubble volume (and soundfield) oscillation cycle, 0.1 . . . 10 ms for the life and translation time of a bubble

within a structure, 0.1 . . . 10 s for the change of bubble structures and sound fields

by retroaction of cavitation or by influence of swashing. Typical dimensions are less

than 1 m for the collapsed bubble, between 1 and 10 m for the equilibrium radiusR0(the bubble size without sound field), 10 . . . 100 m for the expanded bubble,0.1 . . . 10 mm for bubble travel distances and up to some cm for structure sizes (the

range of the sound wavelengths; all measures refer to the indicated frequency rangebetween about 20 and 50 kHz.).

The separation of scales allows to investigate and describe different aspects individ-

ually, which is an advantage for theoretical considerations. On the other hand, the

big span renders an all-embracing theoretical treatment difficult, if not impossible.

Experimentally, it necessitates different recording equipment for capturing features

on different scales. The bubble patterns reported in this article typically live much

longer than a single bubble oscillation period (which at all allows the notion of a

structure). Their lifetime can be limited by slower rearrangements of the con-

figuration of bubbles or sound fields, or can be also unlimited. At the same time,

features like period-doubling can only be registered by recordings synchronous to

the rather short sound field period. The spatial dimensions of the structures in the

range of mm to cm are as well very distinct from the size of individual bubbles.

Therefore, the optical recordings presented in the following can refer to different

time scales and spatial scales.

2.2 Sound fields

The characterization of the acoustic environment in the presence of cavitation is

generally an involved issue. It is well known that the acoustic emission spectrum

of cavitating liquids shows features like broad band noise and harmonics and sub-

harmonics relative to the excitation frequency [22, 23, 9]. While these spectralfeatures are essential for detection and characterization of cavitation, they are cer-

tainly induced effects and of only secondary importance for the actual generation

and structure formation of cavitation. Therefore, it is often valid to assume for the-

oretical considerations that only the principal component of the driving frequency

f= /(2)is present in the sound pressure field which is varying sinusoidally intimet:

p(x, t) =pa(x)cos(t(x)) . (1)

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The general notion of amplitudepaand phasedepending on the spatial coordinate

x IR3

allows for standing or traveling wave components in the field. For example,

pa(x) = pacos(k x) , = const (2)

results in a plane standing wave, while

(x) = k x , pa = const (3)

yields a plane traveling wave, both in the direction of a wave vector kand with a

wavelength= 2/kwithk = |k|.As already mentioned, the assumption (1) is not perfectly valid in the case cavita-

tion occurs in the liquid. Several effects lead to a retroaction of bubbles onto thesound field:

1. The presence of bubbles changes the sound speed in the liquid (which is actu-

ally a two-phase liquid-gas medium then). Accordingly, acoustic impedance and

acoustic wavelength alter, and the radiation from the transducer as well as reso-

nance frequencies of the cuvette can shift. Indeed, already a small void fraction of

gas can reduce the sound speed significantly (see, e.g., [24]), and recently the in-

fluence of a single levitated bubble on the resonance frequency of a glass cube has

been measured directly [25]. Such changes can introduce temporal fluctuations or

modulations of amplitude and phase.

2. The bubbles are created in the fluid and perform volume oscillations. This leadsto the loss and redistribution of energy of the primary wave: damping and scatter-

ing. Therefore the amplitude of the pressure wave will be reduced when it traverses

bubbly zones, and the phase can be changed. Amplitude loss will result in a higher

share of traveling wave in the field (towards the regions of losses), and scattering to

a certain randomization of the phase, which in turn leads to speckles in the sound

field that can vary in time if the bubble distribution changes.

3. For stronger sound fields (and these are typically implied for cavitation), the

bubble oscillations are nonlinear. This transfers energy to higher harmonics of the

basic frequency. Additionally, as mentioned, subharmonics and noise are generated

by the bubbles. The details of how exactly this happens are still partly unclear, but

the origin is necessarily nonlinear. Subharmonic (higher-periodic) oscillations of

bubbles and of whole bubble structures have been observed in experiments [21, 26]

and simulations [27, 28], and they are certainly linked to subharmonic lines. Also

chaotic (aperiodic) oscillations are shown in models, and there are hints from obser-

vations that chaotic dynamics indeed can play a role. Furthermore, lifetime effects

of the bubbles broaden the spectral lines and are partly responsable for noisy back-

ground.

The effects listed above lead to a distortion or perturbation of the ideal monofre-

quent sound field proposed above. Nevertheless it is mostly valid (and convenient)

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to neglect first of all any influence on structure formation. One subtle exception is

the secondary Bjerknes force being discussed later.The absolute values of pressure amplitudes shall be roughly distinguished intolow

which means in the range of 50 to 100 kPa and often lies below the cavitation thresh-

old for the considered frequency range, mediumwhich denotes pressures between

100 and 200 kPa, and highfor pressure amplitudes beyond about 200 kPa. Note

that it is not possible to reach arbitrary high pressure values in the liquid. Cav-

itation results in a certain saturation of the pressure amplitude because the liquid

rupture impedes further tension being built up [19, 20]. Furthermore, shielding and

dissipation effects of the bubbles limit the penetration depth of the sound.

Part of the energy of the sound field is converted into acoustic streaming of the liquid

[29]. The velocity of the liquid induced by acoustic forces has been monitored byobservation of submerged dust particles and by injection of ink. It turned out that

(macro-)streaming reaches values in the range of 1 cm/s, which is often one order of

magnitude smaller than the speed of cavitation bubbles. In many cases, acoustically

induced liquid streaming does not affect the formation of bubble structures, which

is why we mainly disregard it in this context.

2.3 Bubble sources and cavitation thresholds

For the discussion of bubble distributions in acoustic cavitation it is important to

realize that bubbles can be rapidly moving relative to the liquid, and that as well the

sources of bubbles can migrate. The cause of bubble translation are acoustic forces

which are theoretically quite well understood. They are called Bjerknes forces and

are discussed briefly in the next chapter, and more elaborately in A. Doinikovs

Section of this Volume [30]. The behavior of bubble sources is theoretically less

clear, and much information has to be extracted and inferred from observations.

Typically, bubbles appear at so-called seeds(or nuclei). Only in very clean and

pre-treated liquids and environments, rupture of the bulk liquid can be reached, and

then the necessary tension is mostly orders of magnitude higher than the experimen-

tally observed cavitation threshold in real liquids [2, 1, 3]. Otherwise, cavitation

initiates at weak points in the liquid (the nuclei) which are given by pre-existing mi-

crobubbles (of characteristic sizes below a micrometer), contaminations, containerwalls, object or transducer surfaces, or the pathways of high-energetic particle ra-

diation [19, 31]. Some of these nuclei are fixed in space, but microbubbles and

contaminations can flow freely in the liquid. Furthermore, new populations of mi-

crobubbles can be generated by other (larger) bubbles that split or disintegrate. The

spatial distribution of suchfree nucleiis difficult to observe in experiments because

of their small size. Usually it is inferred indirectly by occurrence of larger, de-

tectable bubbles.

The difference between a (pre-existing) microbubble and a (sound-induced) cav-

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itation bubble is not extremely well defined. From a practical point of view, the

cavitation bubble shows certain dynamics (in particular a strong collapse) whichleads to cavitation effects, while the nucleus bubble per sedoes not. The distinction

is caused by the surface tensionwhich imposes the pressurep = 2/Rupon theinterior of a spherical bubble of radius R. Depending on the applied sound ampliudeand frequency, there exists a critical bubble radius below which the microbubbles

remain inactive - i.e., they merely oscillate with small volume expansion, because

the negative pressure phase of the sound field is not high enough to overcome the

surface tension [32, 3]. Beyond this threshold radius, bubbles can expand to a mul-

tiple of their initial size and collapse violently afterwards [33]. For the case of

static tensionp < 0in the liquid, and a liquid vapor pressurepv, the threshold was

given by Blake [32] to be RBl = 4/3(p pv). Similar to a threshold radius,we can define a threshold pressure amplitudepBlfor a given sizeRof the nucleus:pBl =pv 4/3R. (The dynamical case of an oscillating pressure is discussed, forinstance, in [34].) This notion shows that a microbubble can turn into a cavitation

bubble not only by increasing its equilibrium size, but also by translation when it en-

ters a region of higher sound pressure. Accordingly, a cavitation bubble can become

a mere seed bubble if it shrinks or if the pressure falls below the Blake threshold.

In addition, it can as well generate or transform itself into many microbubbles by

splitting or disintegrating into small enough fragments, as mentioned above.

Besides inactive microbubbles and stronger oscillating cavitation bubbles, there can

also exist larger, but only weakly oscillating bubbles in an irradiated liquid. Such

bubbles are typically near nodal zones of standing waves or close to container walls

where the excitation pressure is low. They could be characterized as inactive (with

respect to strong collapse) macrobubbles. Nevertheless, they can serve as sources

of migrating microbubbles by splitting them off [25]. Inactive macrobubbles can

also collect neighboring bubbles and grow large enough to feel a dominant buoancy

force which lets them rise to the surface. During degassing processes by ultrasound,

as mentioned before, many such bubbles appear and play a certain role.

The surface tension is not only causing the inactivity of very small bubbles. Even

more, theoretically it forces small bubbles to dissolvebecause of the higher gaspres-

sure inside [35, 9]. The experience shows, however, that in real liquids free nuclei

exist even after long rest times. Thus one concludes that the microbubles are stabi-lized against dissolution, for instance by liquid contaminations (surface active sub-

stances gathering at the bubbles surface) or by freely floating solids (e.g. bubbles in

microscopic cracks of tiny particles). When a sound field is acting, the oscillation

of the bubbles can as well counteract the dissolution process. This effect is called

rectified diffusion(RD) and means that the net flux of gas through the bubble wall

after one oscillation cycle is not zero. Increased surface area, nonlinear oscillation

and the so-called shell effect contribute to a greater influx during the expanded

bubble phase than there is outflux during the collapse phase [36, 9, 37]. Again, this

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process shows a threshold: For too small bubbles, the oscillation is too weak and

surface tension prevails: the bubble should dissolve. It has been speculated that therectified diffusion is the dominant process which turns inactive microbubbles into

strongly collapsing ones, and therefore the RD threshold is partly considered as a

threshold for cavitation at all. However, recent calculations show that for increased

pressure and for small bubbles the RD threshold more or less coincides with the

Blake threshold [38], at least for the lower frequency ultrasonic range. Thus, very

small bubbles are either dissolving, or active (strongly oscillating), and there is only

very few parameter space for (inactive) microbubblesgrowing by rectified diffusion.

Alternatively, the process of translation, collison and merging may be dominant for

microbubble growth in many parameter regimes [39]. A deeper discussion of this

idea which would imply that merging works faster than dissolution results in certainestimates of microbubble densities and will be given elsewhere.

Still another threshold relevant for acoustic cavitation phenomena is the transition

to aspherical bubble shape [40, 41, 9, 10, 3]. Depending on frequency and bub-

ble equilibrium size, there exists always a sound pressure amplitude beyond which

the spherical shape of an oscillating bubble is unstable with respect to small per-

turbations. This threshold is generally lower for larger bubbles, but it shows reso-

nances where eigenfrequencies of surface modes fit to rational multiples of the driv-

ing frequency. Then the instability is easier triggered. In essence, at fixed driving

frequency, this surface instabilitythreshold limits the allowed bubble equilibrium

radius, depending on the local pressure amplitude. Bubbles being larger than the

stable size are going to split or disintegrate.

2.4 Bjerknes forces

The acoustic forces acting on and between oscillating bubbles are called Bjerknes

forces [42, 10]. Their origin is a nonzero average of the instantaneously acting force

F(t) = V(t)p(x, t)which is felt by the bubble at any instant of time under themain assumption that the bubble at position xis small compared to the wavelength

of the sound field (see [30]). Supposing both bubble volume Vand pressurepareoscillating periodically, we can time average over the period Tto arrive at

FB = V(t)p(x, t)T . (4)

For simplicity we keep the indicated assumptions (more about the excluded cases

like larger and shape distorted bubbles can be found in [30] and the literature

therein). If the pressure fieldpis caused by an external source (typically the domi-nant field), the force is called primaryBjerknes force, and if the field origins from

neighboring bubbles, it is termedsecondaryBjerknes force.

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For the monofrequent pressure distribution Eq. (1) we find

FB1 = pa(x) V(t)cos(t(x)) T+pa(x)(x) V(t)sin(t(x)) T . (5)

The first term on the right hand side can be identified as contributing for standing

waves, and the second for traveling waves. Forces are exerted in the direction of the

pressure amplitude gradient, and the phase gradient, respectively. The sign of the

partial forces depend on the phase relation of the bubble volume oscillation to the

driving pressure (cosine) and its derivative (sine). For small oscillations (linearized

motion of the bubble wall) the common notion is found where the linear resonance

radiusRres [10] plays a significant role: In standing waves, bubbles smaller thanRrestravel to pressure antinodes, while bubbles larger thanRresgo to nodes. In atraveling wave, the force is always in direction of the wave propagation, vanishes for

very small and very large bubbles, and is largest forR = Rres. If nonlinear bubbleoscillations are taken into account, it is found that also small bubbles are repelled

by pressure antinodes for sufficiently high pressure [43, 44], and that bubbles in

traveling waves can also run backwards [45].

A detailed discussion of the secondary Bjerknes force can be found in the literature

[10, 30]. Here we want to note that, to a certain approximation which is sufficient

for many aspects of bubble structures, the force of bubble 1 at position x1exerted

on bubble 2 at position x2can be written as

F1,2B2=

4

(x2 x1)

|x2 x1|3

V1(t)V2(t)

T

. (6)

Hereis the liquid density, and the dot denotes time derivatives of the bubble vol-umesV1andV2. Important features are that the force deminishes with the squaredbubble distance and that it acts typically attractive for bubbles of similar size which

are close to each other. However, many complications and details can be found for

nonlinearly oscillating [46] and shape distorted bubbles, very close encounters etc.

(see again [30] for a review).

It should be remarked that, in a way, the inclusion of the secondary Bjerknes force

is inconsistent with our previous assumption of the (primary) sound field being

undisturbed by the presence of bubbles. This can partly be justified by the fact that

the contributions scattered by the bubbles decay over a short range and act only

locally.

Generally speaking, the Bjerknes forces lead to a spatial concentration of bubbles.

Primary forces drive the cavitation bubbles towards pressure antinodes in standing

waves and larger (degassing) bubbles to nodes. Neigboring bubbles tend to cluster

due to secondary forces. Furthermore, acoustically hard object surfaces can attract

bubbles due to secondary Bjerknes forces from reflections (virtual mirror bubbles).

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3 From the zoo of bubble structures

This chapter provides a catalogue of observations and, in some cases, simulations

of bubble structures. The classification has been guided mainly by visual aspects,

and it has to be read as a suggestion to group and term the various manifestations of

acoustic cavitation.

Figure 1: One-dimensional streamer filament: Bubbles originate near the top at a needletip (arrow), and they run downwards in a straight streak to stop in a small cluster (at the

broadened end). Recording in scattered light, exposure time 16 ms (from [25]). All white

spots aside the central vertical strip are dust particles in the liquid and no bubbles.

3.1 Streamer and filament

An important building block of acoustic cavitation structures is astreamerwhich is

shown in its pure form in Fig. 1. It consists of a linear streak of bubbles that travel

rapidly from one end to the other, forming a directed bubble stream. The figure

shows a streamer in specially prepared environment where the bubbles are artifi-

cially seeded at the tip of a needle [25]. Often the sources of the streaming bubbles

are more obscure because the bubbles are first too small to be visible. Then the ap-

parent starting point of the streamer is located somewhere in the liquid. Typically,

however, a closer microscopic inspection reveals that many microbubbles already

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(a) (b)Figure 2: Two examples of filamentary streamer configurations (Acoustic LichtenbergFigures). Recordings at 25 kHz in scattered light, but with inverted grey scales (therefore

bubbles appear dark). (a): longterm exposure (20 ms) in a cylindrical resonator cell, picture

height 2 cm; (b): short exposure (5s) in a cubical cell, dimension 1 cm, from [47]).

form a stream to the apparent source point. The actual sources can be container

boundaries or really free floating microbubbles gathering at the virtual streamer ori-

gin. The end point in Fig. 1 is a small cluster of bubbles. In more natural cases

streamers merge with other streamers, and many of them form an agglomerate offilaments. This can be seen in Fig. 2 where typical filamentary bubble structures

are given. Bubbles move from outer regions towards the center and finally reach

a central point of high bubble concentration. The whole structure resembles roots

of a plant, neurons, or electrical discharge patterns. For the latter reason, filamen-

tary structures like in Fig. 2 have also been termed Acoustic Lichtenberg Figures

(ALFs) [48]. For only inward traveling bubbles in a filament, the question of con-

servation of gas mass arises. It seems that bubbles at the central cluster finally

disintegrate into microbubbles and dissolve or are taken away by liquid flow. The

latter is possible for small enough bubbles as they dont feel a strong Bjerknes force.

Indication for this scenario is a fuzzy mist or halo around the center in some record-ings like Fig. 2(a).

The acoustic environment typical for streamers are standing waves. Bubble source

points are close to lower pressure amplitude regions (nodes or walls), and the di-

rection of bubble translation is towards higher amplitudes, in particular towards the

pressure antinodes. The absolute pressure values at the antinode typically do not ex-

ceed 180 kPa, otherwise it repells the bubbles [43] and they cluster farther outside.

Therefore, streamers are mainly found for weakly cavitating systems just beyond

the cavitation threshold or in zones of low to medium pressure amplitudes. If a

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Figure 3: Comparison of measured and reconstructed three-dimensional bubble path data(lines) and the results of a particle model calculation (crosses). The model creates for each

detected bubble a particle that subsequently follows the equations of motion according

to Bjerknes forces. The generated tracks are relatively close to the real ones (from [52],

simulation parameters: 30 bubbles ofR0= 5m, driving 150 kPa at 20 kHz).

central filament is driven too strong, it rearranges to an off-center filament [48, 49].The main mechanism behind streamers and filaments can be explained by the Bjerk-

nes forces. Small bubbles which are generated near nodal zones feel attraction to-

wards the antinodes by the primary Bjerknes force. Clustering of microbubbles near

the (virtual) origins of streamers is mediated by the secondary Bjerknes force. The

same is true for longitudinal interaction along the path, and for merging of filament

branches. This process is relatively well understood up to quantitative consistency.

Simulations by a particle model [50, 51] which take into account Bjerknes forces

and a suitable viscous drag force could reproduce three-dimensionally reconstructed

bubble tracks very well [52, 53, 54]. An example from Ref. [52] is shown in Fig. 3.

Streamer filaments have been investgated quite well. From holographic and stereo-scopic recordings [21, 55, 52] it has been possible to reconstruct three-dimensional

pictures and bubble motions. Furthermore, bubble sizes and velocties have been

evaluated [56]. The main results are that the equilibrium radii of the bubbles stay

below about 10 m, and that their speeds reach up to about 1m/s (with a distribu-tion peak around 0.3m/s [57, 55]). Larger and faster bubbles are seldomly found.

Bubble densities within the streamers are smaller than one expects from a long-

term exposure like Fig. 2(a). Typically bubbles dont have lateral neighbors, and

the distance between subsequently traveling bubbles amounts about 0.5 to 2 mm.

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1 cm

Figure 4:Streamer of curved shape (bow). Recording in scattered light at 25 kHz, exposure0.3 ms, inverted grey scales.

It is important to notice, however, that the bubble stream can be rather discontinu-

ous and that bubbles can merge and split within the track. Consequently, there is a

longitudinal interaction between bubbles, and bubbles can temporarily stop or run

backwards to merge with another one.

A striking feature of streamers is their filamentary arrangement to ALFs. The spe-cific filament branches possess a remarkable high stability and long lifetime, which

is one reason for inhomogeneous spatial bubble densities even for longer time aver-

ages. In simulations, this could be reproduced by the assumption of an inhomoge-

neous distribution of bubble sources fixed in space like at walls or objects [48, 49],

together with secondary Bjkerknes force effects. However, it stays unclear why the

filaments are rather stable in the free liquid as well where the nuclei are supposed

to float arbitrarily in space. One can speculate that the microbubbles gather in a

self-amplifying way based on the instability of a homogeneous distribution [48].

Furthermore, self-stabilization of the bubble branches could be possible by the bub-

bles serving as sources themselves (by splitting off microbubbles along their path,which in turn feeds the branch). Such details are still to be clarified.

3.2 Bow and ring

Streaming flows of bubbles can also appear in a curved fashion. Figure 4 shows an

example of a bow which is ending in a one-sided filament near the left border of the

picture. The structure is actually three-dimensional, and we can see more than one

arc terminating in the same cluster (due to a lack of depth of focus, the respective

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(a) (b)

Figure 5:Rings of streaming bubbles at 25 kHz (scattered light, exposure 0.8 ms). Pictureheights approx. 4 cm.

bubbles appear blurred). In some cases, the bubble sources could definitely iden-

tified as attached to the container walls, in other cases (like in Fig. 4) the sources

remain unclear as for free streamer sources. It seems even possible to have closed

rings when the end cluster (or filament) of one arc appears to feed another one

and vice versa. This case is presented in Fig. 5. However, it remains to be shown

that indeed bubbles can travel all around the circle and that it is not composed of

two half-circles.

Typical acoustical environment is a standing wave field at small to moderate pres-

sure amplitudes. Hydrophone measurements show that the curved bubble tracks

encircle pressure nodal zones and lie close to antinodal regions. It seems to be

essential that the standing wave is more extended, i.e. it contains more than oneantinode. The absolute pressure range is similar to that of other filaments. This

indicate also the pressure measurements of Hatanaka et al., who as well observed

a ring-like structures in their study [58] and report pressure node amplitudes below

100kPa.

For increased pressure the ring structure can tranform to the double layer structure

which is described in the next section. This can already be anticipated in Fig. 5(b).

There the excitation is higher than in Fig. 5(a), and two parts of each ring show

stronger bubble population.

The following mechanism is likely to explain the structure: Similar to straight

streamers, the bubbles run, driven by the primary Bjerknes force, from low-pressureto high-pressure regions. The standing wave field is forming something like moun-

tains where the bubbles try to reach the (pressure amplitude) peak. In contrast to a

simple central antinode and a direct way upward like before, here the wave pattern

is more involved and provides a curved path. Furthermore, only few bubble sources

seem to be contributing. From this picture, closed rings should not be possible,

unless there is a mechanism for a bubble to fall down from a peak. Again, disin-

tegration into microbubbles might be a way for the gas to escape from the antinode,

following an outward (downhill) stream of fluid.

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(a) (b)

Figure 6: Double layer structure from the side (a) and from the top (b). Scattered light,25 kHz, long-term exposure, picture heights about 4 cm (from [59]).

The bubble sizes, velocities and densities are similar to straight streamers and other

filaments. The main difference appears to be the detailed form of the standing wave,

but apart of this the phenomenon is just another form of a streamer.

The necessary properties of an extended standing wave field for curved and ring-

shaped streamers remain to be shown.

3.3 Double layer (jellyfish)

This structure consists of two layers of bubbles which appear flat or slightly curved

from the side, but like filamentary stars from the top. Both perspectives are shown

in Fig. 6 as seen by a long-term exposure. The layers seem to be always existing in

pairs, are often of similar extension, but can as well occur unbalanced in size. It is

possible that a layer stretches perpendicular to its planar direction to form a rather

conical entity. The double layer structure has been termedjellyfish[59, 60] because

its fuzzy and slightly drifting appearance bears some similarity to these transparent

and floating animals. It has been shown that the structure has strong cleaning and

erosive potential at objects [61].

Closer inspection by microscopic high-speed recordings reveals a very dynamic

nature of the layers. Small bubbles stream into the layers from the region between

them, and larger bubbles or clusters form within the layers and move in the opposite

direction. At times, a large stable bubble is found between the layers.

The structure exists in well defined standing waves at high pressure amplitudes. The

two layers arrange symmetrically with respect to a nodal pressure plane. Accord-

ingly, their bubbles oscillate in antiphase: While the bubbles of one layer are near

their maximum extension, the bubbles of the other layer are at their collapse, and

vice versa. This type of alternating collapse can be seen in Fig. 7.

A jellyfish structure can move a considerable amount in parallel to the nodal plane,

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12.5 us 18.75 us

43.75 us37.5 us

6.25 us

31.25 us

0 us

25 us

Figure 7: Antiphase oscillation of a double layer structure (driving frequency 40 kHz,

backlit exposures of 1s, frame width 1 cm; from [60]).

but usually never traverses it. Therefore, structures do not interact directly with

perpendicular neighbor structures, but can and do interact with other jellyfishes in

the same plane; sometimes they split into two, or two neighbors merge into one

structure. A transfer of bubbles across antinode regions takes place as well, but

only in form of fast moving larger clusters of bubbles.

The lifetime of a jellyfish structure can be very long, but in many cases it is limited

by some interaction with neighbors, or by deterioration or alteration of the standing

wave. For example, a free surface of the liquid can influence the acoustic field con-

siderably by swashing or surface waves, in particular at higher frequencies wherethe wavelength is shorter. This in turn can change or destroy/rebuild the double

layers in the rhythm of the swashing. At the same time, the noise level produced

by the cavitation is changing. Indeed, it has been shown that the emergence of the

jellyfish structure comes along with the subharmonic [25, 62].

Most bubbles appear from the region of the nodal plane between the layers. Some-

times the bubble stream is partially fed by a larger bubble which is trapped at the

node, but usually it is not clear from where exactly the bubbles originate, similar

to free origins of streamers. Due to primary Bjerknes forces, the bubbles travel

towards the antinode. However, the agglomeration within the layers is not at the

antinode itself, but somewhere before: The high pressure amplitude at the antinode

leads to repulsion of the bubbles, and the bubble layers settle near the location of

zero primary Bjerknes force. On their way towards and into a layer, the bubbles

again tend to form filaments because of the secondary Bjerknes force attraction.

This process has been simulated qualitatively by a particle model [60] which shows

that basic features can be reproduced. As an example see Fig. 8 which has been

calculated by P. Koch (DPI).

From Ref. [60] we cite some statistical data for jellyfishes at 40 kHz driving: A layer

consists of around 1000 visible bubbles, bubble number densities reach 20 mm3

within the filaments, maximum bubble radii are in the range of 20 to 50 m, and

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Figure 8: Particle model simulation of a double layer structure: side view (left) and topview (right). Simulation parameters: up to 700 bubbles ofR0= 3...5m, driving 200 kPa at

25 kHz.

Figure 9:Subsequent frames from a synchronous high-speed recording of a jellyfish struc-ture (backlit, exposure 1s, 26 kHz driving frequency). The interframe time amounts ex-

actly one driving period. In the filaments, a period-2 breathing of the structure can be seen.

bubble velocites reach 2 m/s. The actual pressure amplitude at the layer should be

around 200 kPa for a Bjerknes force balance, while the antinode pressure can be

considerably higher. The respective equilibrium radii of the contributing bubbles

can be recalculated to amount a few micrometer or less.

An intriguing feature of the jellyfishes is the connection with subharmonic emis-

sion. By synchronous recordings of images and acoustic spectra for a slowly mod-

ulated pressure amplitude it has been shown in Refs. [25, 62] that apparently only

the structure formation leads to emission at half the excitation frequency, while un-

structured cavitation (appearing in small single streamers) did not. The limitations

of magnification and time resolution in this work prevented a direct observation of

subharmonic bubble oscillations, but the phenomenon of a collective higher peri-

odic oscillation of a whole structure has been observed before for a larger filament

[21] and recently for large bubble clusters that we will discuss later [26]. Now

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(a) (b)

Figure10:Two configurations of a sub-surface bubble layer (starfish). Recording at 25kHzin scattered light, longterm exposure, frame width about 2 cm.

we present a high speed recording sequence which demonstrates a period-doubling

for a jellyfish directly. The frames of Fig. 9 have been recorded exactly with a

time interval of the driving period at 26kHz. Clearly a period-two breathing of

the structure is visible. What exactly leads to this structural period-doubling is not

clear yet. Synchronization between bubbles [63, 64] and/or global interaction of the

structure with the sound field are possible options.

3.4 Surface bubble layer (starfish)

While bubble layers of a jellyfish always appear in pairs, it is possible to find a

single bubble layer just beneath a free surface of the liquid (i.e., under the liquid-air

interface). This configuration is calledstarfish[25], and an almost circular example

is shown in Fig. 10(a).

The structure appears at the same acoustic conditions as the jellyfishes, and indeedit corresponds mainly to one half of a jellyfish. The free surface builds a nodal

plane in the standing wave, and the bubbles form the highest possible layer without

a partner layer, which would be outside the liquid. Nevertheless, there are at least

two specialities of this structure which render it distinct and which justify an own

name for it. First, its stream of bubbles (which is again from the node towards the

antinode) can be fed from the free surface. Air can be entrained, apparently through

surface waves or capillary waves. Therefore bubble sources are abundant, and the

filamentary layer frequently has a larger extent than the double layers within the

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Figure 11: Examples of small cluster configurations at 25 kHz (courtesy S. Luther [47];

frame sizes about 1mm, backlit exposure about 2s).

bulk liquid. Second, the surface can show a pronounced outward bulge right above

the starfish. It is not clear what causes the bulge, but it appears similar to the acoustic

fountain effect [65]. Possibly there exists a self-amplifying interaction between the

structure and the convex surface bulge, which in turn deforms the standing wave

locally.

Depending on the conditions of the resonator bath, starfishes can live long and stay

more or less stationary, or show a dynamic emergence and disappearance with life-

times in the range of seconds. Similar to jellyfishes they can move parallel to the

surface and collide with neighbors, in which case sometimes an elongated layerfilament with a pronounced backbone forms. This is shown in Fig. 10(b).

The creation of the structure is similar to one jellyfish layer, but there are no definite

data about bubble sizes, densities and velocities up to now. Visually the structure

often appears larger and more pronounced than a jellyfish, and possibly bubbles are

bigger and more dense due to air entrainment. An interesting approach has been car-

ried out by Krefting and Parlitz [25] to characterize the distinction between starfish

configurations and streamer filaments (Sec. 3.1) by a fractal dimension. Aware of

the limitations of the applicability of the method to the 2d-image data, they find a

slightly larger dimension for the starfish, which reflects the more dense and more

filigrane filaments compared to the streamer conglomerations (compare Fig. 2).

3.5 Cluster

Compact groups or agglomerations of many bubbles will be termed clusterin the

following. Clusters can be isolated, i.e. without visible inflow of bubbles or fila-

ments, or can be located at the ending of one or more streamers. They can exist

free in the liquid or attached to surfaces, for instance at submerged objects. At

least two types of clusters, characterized as smalland large, can be distinguished

phenomenologically. This is expounded in the following.

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Figure 12: Variability of a small cluster: every exposure has been taken at fixed phase atsubsequent driving periods at 24 kHz (from [25]; frame sizes about 1 mm, backlit exposure

1s).

3.5.1 Small cluster

A small cluster consists of only a few up to some dozens of visible bubbles which

are localized within a region of about 0.2 to 0.5 mm diameter [47, 25]. With the

naked eye it can appear like a larger single, jiggling or dancing bubble, but with suf-ficient magnification and short exposure time, individual bubbles within the small

cluster are clearly identified, see Fig. 11 for examples. The bubbles show a strong

oscillation behavior, and during their collapse phase they can shrink below optical

resolution to disappear from the image. While near their maximum, the individual

bubbles have distances comparable to their own size, but they can also touch each

other and partly merge. The structure in the cluster is extremely variable, as can

be seen from Fig. 12: Not only the positions of the bubbles can change during one

oscillation cycle, but also their number. The cluster as an entity, in contrast, is quite

stable and long living. It shows certain features of a single bubble, like the reaction

to Bjerknes forces: small clusters can be levitated in a standing wave field, theycan move and they feel attraction from other clusters in the liquid. Their seeding

can take place via a streamer. For instance, the single streamer filament from Fig. 1

ends in a small cluster. When the bubble inflow from a streamer stops, the cluster

can disappear, but it can also persist isolated from (visible) bubble sources. The

example from Fig. 12 has been recorded from a levitated, long living and isolated

cluster.

Small clusters appear for elevated pressure amplitudes above about 190 kPa and

below about 300 kPa [25]. Too strong excitation destroys the cluster or makes it

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Figure 13:Motion of a spontaneously emergent large cluster towards a glass plate. Back-lit, non-equidistant frames from a high-speed series, standing wave at 40 kHz, timing and

spatial scale indicated (picture from [25]).

drifting to the walls. All recorded examples stem from standing wave fields in the

20 kHz range.

Measurements show maximum radii from 15 to 65 m, and equilibrium radii ofroughly 1 m can be deduced from this. Furthermore it was found that they do notemit subharmonic lines, and that with respect to Bjerknes forces they behave similar

to single bubbles of about 10 m equilibrium radius [25].It is unknown why the structure is stable, while at the same time being extremely

variable in its constituents. Naively thinking, the bubbles should attract each other

and coalesce, because they oscillate in phase which causes an attractive secondary

Bjerknes force. However, it is not definite how the secondary Bjerknes force acts for

more than two bubbles oscillating strongly on such small distances. Probably, some

complications from a near field interaction come into play [66, 30] which modify

the simple attraction scheme. Additionally, a larger (merged) bubble is likely to be

shape unstable and would probably split. It seems that the structure finds a compro-

mise between merging and splitting. From recordings it is evident that there is not

much translation of the bubbles during their phase of extended volume, but there

might be a rapid spatial motion during collapse, not visible on the pictures. Re-

cent particle simulations by Konovalova et al. including nonlinearity, coupling and

translation of a few bubbles [67] show that for small clusters stable (non-merging)

parameter regions can exist. On the other hand, there is a remarkable similarity of

what we call a small cluster to the early reports and pictures of surface unstablebubbles by Kornfeld and Suvorov from 1944 [40]. Their images, taken at 7.5kHz

sound frequency, show a distorted bubble that at times splits into smaller parts, but

at other instants seems intact. Possibly the transition from a highly deformed bubble

that rarely splits off parts to a more or less permanent small group of individuals is

gradual and parameter dependent, and the small cluster is one extreme case.

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0.5 mm

1/4 T

5/4 T

2/4 T

6/4 T

3/4 T

7/4 T

4/4 T

8/4 T

Figure 14: Time resolved oscillation of the bubbles inside a large cluster which has beenstabilized in its position by a needle tip. Recording at 40 kHz, timing given in units of

T=25s (backlit exposure 2s).

3.5.2 Large cluster

Large clusters are composed of up to hundreds of individual bubbles, much more

bubbles than in the small clusters. They show maximum radii and mutual distances

in the range of their small cluster counterparts, but the agglomeration itself extends

up to the millimeter range. Large clusters can appear spontaneously in the free liq-

uid where they usually move rapidly in space. Surfaces can attract large clusters

and afterwards stabilize them in position and time. The shape of the structure is at

times surprisingly spherically (or hemispherically on surfaces), but it is evident thatit is, unlike the small cluster, far from a single merging/splitting bubble. Neverthe-

less large clusters are subjected to Bjerknes-like forces as entities and show other

properties of larger single bubbles. An example of a large cluster and its motion

can be found in [25] and is reproduced in Fig. 13. Here, the individual bubbles can

be optically resolved, and the spherical shape in the free liquid is nicely seen. The

cluster reaches the surface of a glass plate and evolves into a rather hemispherical

shape. During this transition, it flattens first, as having an inertia, and then reshapes

into the hemisphere as if under the influence of surface tension.

Large clusters have been observed in standing wave fields at elevated pressures un-

der similar conditions where jellyfish structures appear. They can travel and trans-port bubbles perpendicular to the standing wave planes, other than jellyfish layers.

Indeed, this perpendicular direction seems to be preferred by large cluster motion,

at least according to traces of surface cleaning and erosion [61].

The mechanism of the structure is not clear. Similar to small clusters all bubbles

should attract each other and merge, because they oscillate more or less in phase.

This can be seen from very high-speed images like the one in Fig. 14. Two com-

plete driving periods have been covered by the eight frames of the recording, and

the periodicity and the strong collapses of the constituting bubbles are resolved.

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There is a quite high local variability in the structure. The lifetime of individual

bubbles in the large cluster seems to be very short as in the small cluster version,but sometimes individual bubbles can be observed for some subsequent cycles [26].

Another characteristic feature of large clusters is shown in Ref. [28]: often they

tend to period-doubled or higher periodic oscillation which is reflected in the visi-

ble extension of the whole structure. Accordingly, subharmonic emissions can be

generated, similar as for jellyfish structures. This collective period-doubling effect

seems to be a feature connected with the bubble number in the agglomeration, and

it has been investigated on basis of a simple cluster model [28]. One result is that

indeed period-doubling can be facilitated by increasing the bubble population, if the

bubbles are not too small.

Larger clusters on surfaces frequently look like attached hemishperes, and theyshow a high erosive potential. There are models of shock wave amplification in

such a cluster [68] by a subsequent collapse of concentric shells, and this would

contribute to strong erosion. However, to the authors knowledge this phenomenon

has not been observed or proven experimentally yet.

Earlier high-speed observations of clusters are reported in Rozenbergs book [7]

and by Crum and Nordling [69]. Rozenbergs images show sequences of a larger

cluster attached to a surface, and Crum and Nordling show stroboscopic images of

free spherical large clusters with high translating speed (which they call comets).

Although their work was done in focused fields, there are obvious similarities of

their pictures to our recordings of larger clusters, and it is likely that they observed

similar structures.

3.6 Sonotrode cavitation

Transducers of the form of a vertical bar dipping into the liquid are called sonotrodes

in the following. Typically they have a small emitting surface with accordingly high

intensity (many W/cm2). They are frequently used for applications in the 20 kHz

region like disintegration, homgenization or sonochemistry. The transducer oscilla-

tion is transformed to high amplitude by a mechanical amplifier in form of a horn

[70]. The resulting radiating surface is usually much smaller than the acoustic wave-

length, and a form of a propagating sound beam is created (piston source [71]).Further influence on the acoustic field is given by the shape and size of the liquid

container which causes standing wave contributions, but often these are negligible.

The sonotrode cavitation is specific for small and intense emitters of progressive

waves. Bubbles appear predominantly in front of and close to the sonotrode tip, and

the cavitating zone extends with increased transducer power. Detailed pictures of

bubbles at and near a sonotrode surface have been published already by Kornfeld

and Suvorov [40] and by Olson and Hammitt [72]. The authors were investigat-

ing the erosion mechanism of cavitation (see also [2]), and the pictures reveal an

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Figure 15: Examples of bubble configurations below a sonotrode of 1 cm diameter at24 kHz (backlit, exposure time 2s, picture height about 1 cm). The sonotrode tip is visible

at the top of the pictures, and three arbitrary, but increasing intensities are given from left to

right.

inhomogeneous coverage which sometimes appears star-like.

Examples of bubbles in the liquid from a side view are shown in Fig. 15. Increasing

the power stretches the bubbly region, but also raises the bubble density and seems

to enlarge the bubbles themselves. The sonotrode surface is more and more densely

covered with large bubbles.

The bubble distribution in the liquid looks more or less homogeneous, and con-

glomerations or filaments are not prominent. Sometimes, streaks of bubbles appear,

but overall the structure is relatively featureless and smooth. Also, the bubbly zone

can be bending to the side, deviating from a vertical extension of the sonotrode.

Bubbles seem to be mainly created close to or at the sonotrode tip, but some can

also appear from the liquid moving towards the center of the cavitation zone. The

lifetimes of recorded bubbles range from one or a few acoustic cycles (at locations

close to the sonotrode tip) to many cycles more far. The distant and longer living

bubbles can be tracked to find their motion velocity and direction. Often bubbles

move fast away from the tip (about 1 m/s), but also bubbles moving towards the tip

have been observed [73]. The bubble translation can be theoretically explained by

Bjerknes forces in traveling waves [53]. In this case, the phase gradient term in

Eq. (5) is of great importance. There is always a strong liquid streaming induced

by sonotrode transducers, and the liquid flow appears to be responsible for at least

part of the structure shape. When the flow is hindered artificially [74] or by larger

extension of the sonotrode surface, the bubbles form a conical zone, see the next

Section.

Statistics of bubble distributions below a sonotrode have been reported by Burdin et

al. [75]. They find mean bubble radii below about 10m and a homogeneous spatialdistribution below the tip. Additionally, void fractions have been determined.

It has been reported that cavitation effects can differ significantly between small,

high intensity sonotrodes and larger emitting surface setups [76]. This might be

connected with the bubble structures and in turn with the bubble densities, lifetimes

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(a) (b)

Figure 16: Bubbles below a large sonotrode in scattered light: (a) long-term exposure(width 15 cm), (b) frame from a high-speed recording (exposure time 0.88 ms, width 6 cm).

The sonotrode has a diameter of 120 mm which corresponds to 1.64for the driving fre-

quency of 20500 Hz (pictures from [78]).

or sizes. Much is still unknown with respect to this aspect, but it is a main problem in

upscaling acoustic cavitation processes. Some data has been published recently, but

more detailed investigations of sonotrode bubble distributions are still in demand.

3.7 Conical bubble structure

The conical bubble structure (CBS) appears at elevated intensitiesunder a sonotrode

of large emitting surface with a diameter in the range of the wavelength [77, 78]. Its

shape is shown in Fig. 16(a) as it appears to the naked eye. Short-term exposures

reveal the composition of streaming bubbles in filaments that origin at the transducer

surface and reach downwards to bundle near the symmetry axis. The stream of

bubbles is extended by some larger bubbles below the cone, but many smaller ones

stop and merge near the cone tip. Bubble sizes (R0of a few micrometer), velocities(about 1 m/s) and lifetimes (a few hundreds of driving periods or until collision with

other bubbles) seem to correspond more or less to the data of stronger streamers.

The sonotrode surface is partly covered by smoker and bubbly web structures (see

Sec. 3.9). For smaller sonotrodes with an artificially blocked liquid streaming at the

sonotrode edges, conical bubble structures on a smaller scale have been reported

[74].

The acoustic field is mainly a propagating wave with an amplitude maximum close

to (but not exactly at) the sonotrode (see [78] for field calculations). The radiated

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1 cm

Figure 17: Flare structure next to a submerged transducer of large area. The left part isattached to the vertical radiating surface, and the direction of bubble stream is to the right

(scattered light, 25 kHz, exposure time 333s).

intensities range from 1.5 to 8 W/cm2. Recordings with fixed phase relative to the

driving signal show mutual synchronous bubble oscillations, but drifting instants of

collapse [78]. This is an indication for a distorted wave propagation, probably due

to nonlinear effects induced by high bubble density [79].

The structure can be explained quite well by primary Bjerknes forces if the full

expression (5) is taken into account and if a traveling wave of decreasing ampli-

tude is assumed [80]. Following this explanation, the stagnation point at the conetip corresponds to a force equilibrium of the amplitude gradient part [first term in

Eq. (5)] and the phase gradient part (second term) of the primary Bjerknes force.

The amplitude decline is caused by geometrical spreading of the wave front and by

dissipation in the bubble field.

Some phenomena that have been recorded during the switching on of the transducer

and for a pulsed excitation [78] remain to be explained, in particular the emergence

of larger bubble clusters prior to the full development of the conical structure.

3.8 Flare

This acoustic cavitation structure has been observed at large radiating surfaces like

the bottom of cleaning baths or submerged transducers. From a limited part of the

surface (a few cm2), bubbles are ejected into the liquid, and they follow a broad

path which appears like a flare, see Fig. 17. The stream of bubbles wiggles and

develops bumbs similar to a Kelvin-Helmholtz instability which is shown by free

jet streams. Furthermore, the bubbly flare can detach from the surface and travel

a distance into the bulk liquid, whereupon it disappears. It is composed of a few

large and many more small bubbles which form small dense filaments. The part

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attached to the transducer appears reminiscent of the cone bubble structure, but the

flare is extended far into the liquid beyond the cone tip or stagnation point, as canbe seen on the figure. On its way, the width of the bubbly zone grows, being fed

from bubble sources at the side of its path. Bubble speeds of 1 m/s are common,

and the envelope of the whole structure is drifting with about 15cm/s away from

the transducer. Bubble sizes appear similar to streamer and jellyfish structures.

The sound field in front of the large emitter has not been determined experimen-

tally, but it probably possesses a complicated near field structure with shares of

both traveling and standing waves. The amplitude distribution of the oscillation on

the surface itself is likely to be complex, because the back is equipped with several

individual transducers. Nevertheless, the flare structure is universally found for dif-

ferent large emitters, e.g. in various cleaning bath setups. The pressure amplitudesare supposed to be quite high, because at the same time jellyfish structures appear

in other parts of the liquid container.

The mechanism of the flare is unclear. The part near the emitter is possibly created

in a similar way as the CBS, but the jet of bubbles needs further explanation. The

similarity to a free jet is striking, in particular because of the shear-like instabilities

at its boundaries. It remains to be explained at all why the boundaries of the bubbly

phase are relatively well defined, in particular because the actual motion of the

water, induced by acoustic streaming, appears to be much smaller than the front

velocity of the cavitating zone. Probably a significant interaction of propagating

sound wave and bubble density exists.

3.9 Smoker and web

This acoustic cavitation structure is bound to a surface in the liquid, and one exam-

ple is seen in Fig. 18(a). It consists of an active spot on the object which is the origin

of a cloud or plume of very small bubbles. Because this plume is directed, and be-

cause its constituents are hardly resolved, the optical appearance is reminiscent of

the smoke emitted from a localized fire and blown away by the wind, watched from

the top. Therefore the figure has been named smoker. It is always attached to a

solid object, but the emission spot can be either fixed or drifting on its surface. The

bubble plume often runs close or parallel to the surface, and its direction can appearstationary or fluctuating. If more smokers are close to each other, their plumes can

merge. In extreme cases, merged plumes form awebor matrix of connected bubbly

zones close to the object surface. Such a case is shown in Fig. 18(b).

Smokers appear on objects in strong acoustic fields. Preferred locations are trans-

ducer surfaces, but the recording of Fig. 18(a) has been done on a submerged alu-

minium block in a standing wave bath. Multiple smokers can appear on passive

objects, while really dense webs have been seen only directly on emitting surfaces

like in Fig. 18(b) which shows a large sonotrode surface generating a CBS (compare

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(b)(a)

Figure 18:Short exposure time recordings in reflected light of a smoker at 40 kHz (a) anda web at 20 kHz (b). Exposure times 2s, picture heights about 1.5 cm in (a) and 3 cm in

(b), material aluminium.

also [77]).

On high-speed recordings, parts of the bubbles forming the smoker or web can

visually disappear during collapse. Furthermore, it has been observed that the plume

can oscillate in doubled or higher period [81], which means that the structure can

emit subharmonics.The origin of the smoker is unclear. The smoker tips are strongly erosive points,

and their location, if fixed, might be attached to local cracks on the surface. Smok-

ers on materials other than aluminium, for example titanium, have been observed.

Statistics about size and speed of the bubbles in the plume do not exist, but individ-

ual bubbles have been measured to move with about 1 m/s away from the tip [81].

Many bubbles are too small for optical resolution, and therefore they appear nebu-

lous like a cloud. The boundary of such a cloud has been determined to move in the

range of 10 m/s [81], but it is not sure if individual bubbles reach this speed, or if it

is only a virtual speed of the front of bubbly liquid. The reason for the directivity of

the bubble plume is also obscure. Liquid streaming is likely to be too slow to have

any influence, but acoustic forces might be responsible. The merging of smoker

clouds to webs suggests that a certain attraction is acting between them.

Once a transducer surface is covered by a dense bubble web, its emission proper-

ties might change considerably and impedance mismatch, shielding and high local

dissipation can appear. Nevertheless, the region close to the transducer can be very

active and effective for certain processes [77].

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4 Summary

An introduction to the topic of acoustic cavitation and structure formation has been

given, addressing aspects of sound fields and retroaction by cavitation, nucleation

and bubble sources, cavitation thresholds, and Bjerknes forces.

A variety of acoustic cavitation structures in the low frequency ultrasonic range has

been described, illustrated by images, and characterized up to available knowledge.

Roughly they can be grouped into standing wave, traveling wave, and surface bound

figures, but this classifying scheme is not working perfectly. Structures connected

intimately with standing waves are streamer filaments, bowsand rings, and the

layer structures (jellyfishand starfish). On the other side, sonotrode, conicaland

flarebubble structures seem to rely strongly on traveling wave parts in the soundfield. Clustersof different sizes can appear in all kinds of acoustic fields, in the

bulk liquid and on object surfaces. Smokersand websare definitely adherent to

solids (transducers, walls, or submerged objects). More types of bubble structures

are likely to be found and described in the future, but a reasonable classification into

a limited and small number of prototypes seems feasible.

Distinctions of the proposed bubble structure candidates are obviously given in their

optical shape and their acoustic environment. On the level of bubble dynamics,

bubble sizes (equilibrium radii) of a few micrometer and below are quite common

and similar for all observed formations. As well, the measured translation speeds

are more or less generally limited at about 1m/s. Differences appear in the lifetime

of individual bubbles, the bubble number density, and the collapse strength. These

parameters are probably the most significant ones to explain different cavitation

effects under different conditions or setups.

Lifetimes range from only one or few expansion and collapse cycles in clusters

and below a sonotrode up to hundreds of cycles in streamers and filaments. Even

arbitrary many oscillations with strong collapse have been realized for levitated iso-

lated bubbles [12]. Limiting factors of the lifetimes are high bubble density (or

low distance to a neighboring bubble, dominant for instance in clusters), collision

with neighbors after spatial translation (dominant in streamer filaments), and sur-

face instability after growth by rectified diffusion [38]. Probably the lifetime of

the bubbles in different structures is explaining partly the variation of effects underdifferent acoustic conditions.

The bubble density for many of the proposed situations varies strongly in space,

which is the well known inhomogeneity of acoustic cavitation. Next to intensely

cavitating regions, like in clusters, jellyfish layers, or flares, no bubbles can be no-

ticed by optical means. Also the density within the populated regions of the struc-

tures differs. Higher number densities can be found in layers, clusters, sonotrode

structures and flares, and in smoker plumes. Lower densities appear in filaments

of streamers and conical structures. The density should have an influence on the

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sphericity of the bubbles, and therefore on both the bubble lifetime and collapse

strength. Also, the overall effect or yield of a cavitation application should scalewith the density.

In the considered frequency range and for the applied sound field intensities, pre-

sumably all active (i.e. strongly expanding and collapsing) bubbles are visible at

high optical resolution. Therefore, the observed bubbly regions should give an up-

per estimate of the active sites. Of course, not all visible bubbles need to be strongly

imploding, and also the real violence of the collapse is difficult to assess. Bubbles

disappearing from the image during collapse likely emit shock waves, but even then

the vehemence is unclear. It depends onR0and on the local pressure amplitude,which might be quite different for different structures. It can be speculated that in

standing waves the bubbles typically do not reach areas of pressure amplitude aboveabout 200kPa when their structures are Bjerknes force controled [82]. In contrast,

in traveling waves near sonotrodes the driving pressures are supposed to be signif-

icantly higher. Then, however, the stronger collapse is accompanied by a shorter

lifetime.

5 Outlook

This article has proposed classifications of bubble structures that represent different

manifestations of acoustic cavitation. This has been done on the basis that, with

respect to this topic, not much information is available from prior publications, thatmany aspects are still unclear and that a general scheme of bubble figures has not

been elaborated before. Being aware of the limitations of a review of this kind, it

is intended that the information given will serve as a reference for forthcoming re-

search. Future work needs to extend the results for a larger variety of acoustic and

geometric environments, as well as for higher and multiple excitation frequencies.

Many aspects of the differences of bubble dynamics within the different structures

remain to be highlighted, as well as the consequences on distinct applications of

acoustic cavitation like cleaning or chemistry. An increasing amount of such ap-

plications demands for a detailed knowledge of the link between acoustic field and

cavitation effects, which is mediated by the distribution and properties of the cavi-tation bubbles.

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Acknowledgement

Many thanks go to all actual and former members of the cavitation group at the DPI in

Gottingen. Without their contributions, this article would not have been possible. I am

indebted to Dagmar Krefting, Stefan Luther, Philipp Koch, Topi Tervo, Alexei Moussatov

and Bertrand Dubus for making available pictures from shared or their own work. Special

thanks go to Claus-Dieter Ohl and Reinhard Geisler. I thank Werner Lauterborn for strong

support, Juan Antonio Gallego-Juarez and Enrique Riera for their kind hospitality during

a stay at CSIC in Madrid, and Carsten Knie from VKT Company for making available

their high-speed camera. Parts of this work have been supported by the German Science

Foundation (DFG Graduiertenkolleg Stromungsinstabilitaten und Turbulenz), the Euro-

pean Union (INCO Copernicus IC15-CT98-0141), and the German Ministry of Research

and Technology (BMBF project Untersuchung von Kavitationsfeldern).

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