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24th Symposium on Naval Hydrodynamics
Fukuoka, JAPAN, 8-13 July 2002
Influence of acoustic interaction in noise
generating cavitation
Jan Haliander (SSPA Sweden AB, Sweden)
Goran Bark (Chalmers University of Technology, Sweden)
ABSTRACT
The primary aim was to conduct a preliminary study of
the role of interaction between a few cavities on the
generation of high frequency cavitation noise. Differ-
ent cavitation processes, statistical distributions of
parameters and initial conditions for interacting cavi-
ties were observed using high-speed films from experi-
ments. The numerical study concerns acoustic
interaction between two spherical cavities. Monte
Carlo simulations with random input data were carried
out.
Except at the highest frequencies, the interaction
between cavities is not critical for most predictions of
cavitation noise. The sizes, the number of cavities and
the pressure forcing the collapse must still be well sim-
ulated. However, the interaction is important at the
highest frequencies.
1 INTRODUCTION
High frequency cavitation noise generated during the
collapse of vapour cavities, caused for example by a
marine propeller (Fig. 1), is investigated in this paper.
Although the discussion in the paper has primarily to
do with the sheet cavity, it is valid in principle also for
vortex cavities. During part of the cavitation cycle, the
sheet cavity typically breaks up into cloud-like sub-
cavities, that may later collapse close to each other.
These "break-offs" usually start with the growth of dis-
turbances due to re-entrant jets in the sheet cavity.
These disturbances make parts of the sheet grow
thicker, while the visual appearance changes from
glossy to a white bubbly structure. The broken-off
white cloud cavities are usually believed to be clusters
of a vast number of small vapour bubbles. The final
part of the collapse is often very violent, generating
high and narrow pressure pulses (Fig. 1~. These pulses
have a broadband (continuous) spectrum and dominate
Figure 1: Model propeller behind a ship model in the SSPA
cavitation tunnel. Noise signals from three hydrophores in
the near field of the cavitating propeller. The sharp pulses are
generated by collapsing cavities.
the high frequency part of the cavitation noise spec-
trum. The highest pulses can also contribute to erosion
of the propeller.
The flow field and its characteristics determine the
break up of the sheet cavity and its disintegration. This
disintegration determines the initial conditions for the
following collapses. Important initial conditions are the
sizes of the resulting cavities, the space between them
and other cavities or boundaries. The disintegrated
parts are convected downstream by the flow, i.e. they
are transformed to travelling cavities with the potential
to be noisy. The local flow field partly determines the
surrounding pressure and the start of the collapses.
The collapse is forced by the difference between
the surrounding pressure (time dependent) and the
pressure of the vapour and gas inside the cavity. Hence,
the surrounding pressure is a key parameter; it is com-
posed of the pressure in the undisturbed flow, the pres-
sure disturbance associated with the local non-
cavitating flow, the possible pressure disturbance from
stationary cavities and, finally, the pressure disturbance
from the motion of neighbouring cavities. This last
contribution, approached here primarily as an acoustic
interaction arising from the motion of neighbouring
cavities, is the main subject of the present paper. That
OCR for page 882
such interaction between collapsing and expanding
cavities can result in significantly higher pressure
pulses than the collapse of independent cavities was
first demonstrated theoretically by van Wijngaarden
(1964) and others. It is generally believed today that
this interaction plays a key role in erosion. For the ero-
sion problem, it is thought that a large number of small
cavities close together (cloud cavitation) and close to a
solid surface is of relevance; however, for noise gener-
ation, a few rather large cavities may dominate.
The main engineering question underlying this
study is whether it is necessary to account for interac-
tion between individual cavities in a model of the type
suggested by Matusiak (1992) for prediction of cavita-
tion noise. A preliminary analysis of the main effects,
made by applying a simple model, was intended to
yield some guidelines for the possible need of develop-
ment and application of more elaborate methods. Simi-
larly, scale effects in the distribution of cavities on a
model scale propeller in a cavitation tunnel may, under
the influence of interaction, also lead to scale effects
that affect the prediction of full scale noise. This pre-
liminary analysis was conducted as a Ph.D. project at
Chalmers University of Technology and the results are
presented in Hallander (2002~.
2 SYNCHRONIZATION AND INTERACTION
IN COLLAPSE PROCESSES
Synchronization and interaction are key concepts used
frequently in the paper. Synchronization of a number of
cavities in a collapse means here the process by which
a pressure field of a given spatial extent and temporal
duration causes the cavities to collapse more or less
simultaneously. The synchronizing pressure field can
be generated by a global flow, for example, over a pro-
peller blade or by a significant local flow process such
as the collapse of a nearby large cavity. Although the
cavities can collapse independently of each other, they
can also begin to interact acoustically and hydrody-
namically during the collapse. When the interaction
becomes strong, the importance of synchronization by
the global flow gradually declines, and the collapse of
some cavities can finally be forced by a pressure aris-
Synchronisation and interaction are most probable
when cavities are generated by disintegration of a
mother cavity, so that they appear close together in
time and space. Typical behaviours in different flows
are discussed in Sections 5 and 7.
3 CHOICE OF METHODS, PRIORITIES,
APPROXIMATIONS AND LIMITATIONS
The choice of cavity configurations in this study corre-
sponds to characteristic configurations among the large
voids (but not the very largest) that take part in collapse
events where interaction between the cavities is likely.
To find representative configurations with noisy cavita-
tion, high speed films were studied. That significant
interaction between small and closely spaced cavities
within collapsing clouds (that generate the very highest
frequencies) can be expected has been shown in
numerical studies by March (1980) and Chahine and
Duraiswami ( 1992~. In this study it was instead a prior-
ity to study the influence of the interaction between the
larger sub-cavities taking part in fast collapses. By
studying this problem, it is possible to estimate typical
lower frequencies at which the influence of interaction
can be expected. However, it was not a goal to deter-
mine the lowest frequency limit for interaction effects.
The complexity of the cavitation process is a
major problem in this work: for example, it was hard to
select a model simple enough to be manageable but
still useful in describing the effects. In the present
study it was important to vary several parameters of
interacting cavities. A code based on the most
advanced model for the interaction of many cavities of
general shape can be very time consuming, which is
why it was disregarded at this stage. After an initial
study (Hallander 1995), a model describing two spheri-
cal cavities only was selected. Although two cavities
are of course the very lowest limit, they can neverthe-
less reflect basic properties of the processes studied on
film (which normally have three to five sub-cavities of
significant size in addition to the main cavity). The
number of high pressure pulses observed are often rela-
tively few (see Fig. 1), which also supports the choice
of a model with a few cavities. For spherical cavities,
process by which the pressure radiated from one cavity the model is rather complete, which permits acceptable
adds to the pressure forcing the collapse of a neigh- studies of parameter influences and adaptation to
bouring cavity. When the interaction is strong, this experimental observations. A key aspect was also the
pressure can be the dominating pressure forcing the chance to make extensive Monte Carlo simulations
collapse. using random parameter values for the model.
ing from the interaction between nearby cavities.
In this context, (acoustic) interaction refers to the
/
2
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Representative terms from entire chapter:
acoustic interaction
4 THEORETICAL MODEL
4.1 Fujikawa's two bubble equations with modifi-
cations
The bubble model selected was originally used by
Fujikawa (1984) and Fujikawa and Takahira (1986~.
The interaction is simulated with coupled equations of
radial motion for two spherical cavities:
R (t)R (t) 1- " + 3R2`t' 1 _4R ~ ~ (1)
[ Cw ] 2 [ 3cw]
+ ~ Spa ~ (t) + Pw 7 (R2~2)R2~2) + 2R2~2~)
R.(t). ~
-P7~(R~(t) t)- Pl~(R~(t) t)] = 0
_ 2' ~ + 3R2(t) 1 -
[ Cw ] 2 [ 3cw ] (2)
+ ~ ~Pa2(t) + Pw ~ (R~)R~) + 2R~)
-Pr2(R2(t),t)- C Pt2(R2(t)'t)] = 0
where Rifts is the radius of cavity number i (Ci), CW is
the velocity of sound in water, Pw is the density of
water and l is the distance between the cavity centres.
The main geometry is illustrated in Fig. 2. The exter-
nally applied pressure Pai~t) is further discussed in sec-
tion Section 4.3. The retarded time (i is
(i = t - (I - Ri(
cavities can be justified, at least as a first approxima-
tion and when the primary purpose is to study interac-
tion and radiation dominated by the monopole
behaviour.
In fact, it should be kept in mind that the volume
variation corresponding to spherically collapsing cavi-
ties (monopoles) is a more effective source of noise
than deformation motions (multipoles), so the pressure
generated is usually over-estimated in the model used
here. An effect of deformation is that it can be expected
to result in more viscous dissipation and thus less vol-
ume acceleration than for spherical cavities. This latter
effect is simulated in the computations by the artificial
damping term, Eqn. (5~. However, the selected value of
the damping parameter in this study tends to result in
too much damping of the motion during the final col-
lapses; the deformation losses occur earlier in the proc-
ess and should be compensated for in a different way.
Relative motion between the two cavities
Although the translational motions of the centres
of the cavities sometimes are critical, they are not taken
into account in the present study. This may be an
acceptable approximation for interaction where the dis-
tance between the cavities is relatively large. However,
when the forced cavities are generated at the boundary
of the forcing (main) cavity, as shown in Schoon and
Bark (1998b), this approximation can be expected to
lead to a significant underestimation of the interaction.
During a collapse, the flow induced by one cavity
usually results in a translation of the neighbouring cav-
ity. Experiments by Testud-Giovanneschi et al. (1990)
show that the cavities move towards each other, and
they deviate from their spherical shape. The closer the
cavities are to each other, the larger these effects
become. They also show that this influence is strongest
on the smallest cavity. Thus, the smaller cavity is
closer than the initial interdistance l to the larger one
when it collapses. However, support for the present
approximation was obtained from high speed films of
cavities on foils, e.g. Fig. 6, where no significant trans-
lation between the cavities is observed for distances
and diameters similar to the ones used in the present
simulations.
4.2 Gas and vapour pressures inside a cavity
It is often supposed that the vapour pressures inside the
cavities, PVift), increase during a very fast collapse,
because there is an upper limit of the rate of condensa-
tion of the vapour. The use of a non-equilibrium vapour
pressure allows a smaller amount of permanent (non-
condensable) gas, PgO, at the start of the collapse; thus,
the cavity will return to a smaller and more realistic
equilibrium radius after the collapse.
The equations of mass conservation and states for
the gas and vapour within the cavities give the rates of
change of gas pressure and vapour pressure within the
cavities (Tomita and Shima 1979~:
Pgi~t) = Pgi (t) (Ti (t) Ri (t) )
Pvitt) = Pvitt)
{Time—R. `~:Rift)_ i ~v: Lei
Pvi(~) 42~/ ,4~)
PVitt))-
ft1
(6)
(7)
where A is the accommodation coefficient for conden-
sation and evaporation, Kv is the gas constant of the
vapour and Taint) is the temperature of the liquid at the
wall of the cavity Ci. The saturation vapour pressure
PLei~t) iS a function of TLift) and Ri~t). An expression
can be found in Sato et al. (1996) or Atkins (1995~:
PI,ei (t) = P e {2sw/[Ri(~)pwKvT~;(~)] } (8)
where PVe is the equilibrium vapour pressure of the liq-
uid far from the cavities.
The temperature Ti~t) of the gas-vapour mixture
within Ci is given by Tomita and Shima (1979~:
Titt) = R. (t)~(K —DPvi(~) + (Kv—1)Pgitt)]
{(KV - 1 )(Pgi ft) + PVi ft))Ritt) +
APi, e i tt) ( Ti f t) - TL i tt) ) ~ }
- (9)
where Kg and KV are the specific heat ratios of the non-
condensable gas and vapour, respectively. The temper-
ature in the liquid at the wall of Ci is approximated by
the temperature in the liquid far from the cavities, Sato
et al. (1996~:
T~ift)~ Too
4
(10)
This approximation assumes that Trif t) is much
smaller than Titt), meaning that the heat transfer is
negligible. Although still approximate, this procedure
yields a more realistic behaviour of the interacting cav-
ities than the use of a constant vapour pressure. To take
the variation of temperature in the liquid with time into
account requires solution of the energy equation (Tom-
ita and Shima 1979, Fujikawa and Akamatsu 1980~.
Gas diffusion The effect of gas diffusion is dis-
cussed by Watanabe and Prosperetti (1994~. The effect
of this mass transfer is that the gas content of the cavi-
ties is higher when they reach their final equilibrium
radii after several collapses and rebounds than it was in
the original cavitation nuclei. A cavity may also cap-
ture cavitation nuclei or other cavities during its life-
time. However, mass transfer of gas is not included in
the model above.
4.3 The externally applied pressure on a hydrofoil
or propeller blade
The externally applied pressure Paints in equations (1)
and (2) can be an arbitrary function of time (a slight
modification of Fujikawa's original equations). The
spatial pressure distribution on a cavitating foil or pro-
peller blade section is not obvious; with cavitation the
pressure distribution changes. The pressure equals the
vapour pressure for the part covered by the sheet cav-
ity. After a transition zone, the pressure returns to that
of a non-cavitating body, according to van Oossanen
(19744. If there is a stagnation point behind the sheet
cavity, the pressure gradient can be very steep. The
steepness of the pressure gradient in this zone (Fig. 3)
has a large influence on the collapse times and, thus, on
the violence of the collapses. This is demonstrated in
Hallander (1999~. The time history of the externally
applied pressure is also affected by the time variation
of the inflow and, for an oscillating foil, also by the
oscillation of the foil (Schoon 2000~. Thus, the gradi-
ent of Part) may be somewhat steeper than estimated
above.
A preliminary way to study this effect is to imple-
ment the time history of the pressure increase as an
increase from PVe at the point of disintegration to the
undisturbed pressure POO at the end of the foil, as Matu-
siak (1992) did. If the cavity is assumed to move with
the undisturbed flow velocity, UOO' after disintegration
at time, to, this motion in the spatial pressure distribu-
tion on a foil or propeller blade gives the external pres-
sure as a function of time (Fig. 3~. An observation
made by Schoon (2000, Fig. 6.3, p. 54) supports the
theory that a bubble just outside the boundary layer
16
12
10
8
6
4
2
~ Poo
/ ' Adopted pressure increase
Lip — - Linear pressure increase
ve
tOi+tinci
Figure 3: The solid line shows the time history of the exter-
nally applied pressure (Pa(t)) used in the simulations. The
collapse of a cavity starts at toi. The dashed line shows a lin-
ear pressure gradient, which was used by Matusiak (1992) for
example.
moves with a velocity close to UOO- In some experimen-
tal observations, due to local flow variations (e.g. vorti-
ces), the cavities do not move during the collapse.
Typical collapse times of the cavities studied are
about 2 to 4 ms for the first collapse. This is shorter
than the time of the pressure increase from Pee to POO
assumed above (which is 8.4 ms). The cavities thus
collapse under a pressure of less than POOP and the
motion decays before they reach the trailing edge.
4.4 The numerical solution
Fujikawa's equations, (1) and (2), were solved together
with equations (5), and (6H9) for each cavity. It was
assumed that the cavities were initially at rest at a uni-
form pressure and that each cavity starts its collapse
from an initial radius, Roi at time toi when it is exposed
to the externally applied pressure, Paid). These initial
conditions could have been reached either by the
growth of a cavitation nucleus or by the disintegration
of a larger cavity into parts.
The system of coupled ordinary differential equa-
tions (ODE) was solved with a MATLAB -
SIMULINK model. A variable order solver of the
Adams-Bashforth-Moulton type ("odell3") was used.
The time-stepping is critical when a cavity reaches its
minimum; an adaptive solver which varies both step
size and order is strongly recommended. The retarded
time, Eqn. (3), was approximated by
{i = t - (I - Rift)~/cw (11)
in the SIMULINK system.
The pressure generated by the two cavities,
pity =pltrl,t) +p2(r2,t), was calculated at the distance
of ri = 1 m from the centre of each cavity, Fig. 2. This
distance is much greater than the distance between the
cavities (ri>>l) in the configurations studied. The
finite speed of propagation is also disregarded when
the generated pressures are computed, since it is
merely a comparison at the same distance. The gener-
ated pressure is then:
pi~ri,t) = p(R' (t)Ritt) + 2Ri~t)R' (t)~/ri . (12)
For each simulation, the peak pressure amplitude
defined as Pma¢~` = maximum~p~t99 was recorded.
The average power density spectrum G69 for each
example (in the Monte Carlo simulations) was esti-
mated by the following process.
The signal pity from each simulation was interpo-
lated with a time step of 2 10-8 s (fs = 50 MHz).
This time step was found small enough to give a
good representation of the signal.
The interpolated signal was decimated (i.e. low-
pass filtered and down-sampled) tofu= 1 MHz.
· A windowed periodogram G( )~ was computed
for the decimated signal from each simulation:
N- ~ _
G(k)¢f' = ~ ~ p~n~w~njej21lin (13)
n = 0
k= 0,1,...,K-1
where pan) is the decimated sample sequence of
length N from each simulation and K is the number
of simulations in the example. The sample
sequence was padded with zeros so N became the
next power of two. The window function won) was
a Hanning window of length N.
The average power spectral density (PSD) G69
was computed as the sum of all the periodograms
divided by the number of periodograms (K) and
the norm squared of the window function (U9:
K- ~
(f) KU ~ (A (14)
k = 0
where
N- ~
U = N ~ W2(n). (15)
n = 0
4.5 Choice of model constants
The physical constants used in the simulations where
chosen to correspond to cavitation tunnel conditions
(fresh water, T.= = 20 °C), and are shown in Table 1.
The values of the initial gas content, Pgoi, the accom-
modation coefficient, A, and the damping parameter,
Kd, are constants that strongly affect the model behav-
iour. They are selected (tuned) to give the model rea-
sonable behaviour in comparison with experimental
data. In Hallander (1995) it was shown that a lower
artificial damping or a lower initial gas pressure makes
the simulated pulses higher and narrower. An increase
of the accommodation coefficient also has a similar
effect. A systematic variation of the accommodation
coefficient, carried out by Fujikawa and Akamatsu
(1980, p. 507, Table 2), shows that a higher value of A
causes a cavity collapse to a smaller minimum radius
and generates higher pressure.
Table 1: Physical constants and others that affect the
model behaviour used in the simulations.
Parameter Value
Too 20 °C
cw
Pw
.
sw
V .
Kg
Pve
A
Pgoi
Kd
1483 m/s
.
998.2 kg/m3
.
0.07061 N/m
1 13
461.9 J/(kgK)
1.40
i.33
~-
2.337 kPa
_-
0.015
0.010 Pa
2.337 kPa
50 Pas/m
The damping parameter is an artificial constant
(see page 3), while the other two parameters have a
physical interpretation. However, it is not possible to
experimentally determine their values. Values of A as
low as 0.01 have been suggested for evaporation in
some studies, according to Brennen (1995~. Fujikawa
and Akamatsu (1980) refer to a study where 0.04 is
suggested. The comparison of an average measured
sequence at unsteady cavitation with one of the highest
simulated ones, Section6.2, indicates that the total
6
(a) 1~
\~ /
(b)
Figure 4: Schematic cavitation model explaining the
nomenclature. A sheet cavity CO disintegrates into a main
cloud formation (first generation sub-cavity) Cat. During the
disappearance of CO and Cat, a number of (second generation)
sub-cavities C2 are generated by farther disintegration of CO
and Cat.
3.
model damping chosen was somewhat too high. Con-
sidering this, a value slightly higher than 0.015 would
have been a more appropriate choice.
The initial gas pressure is assumed to be the same
within all cavities in the present study. This simplified
assumption may be justified by arguing that all of the
cavities in a simulation are generated by disintegration
of the same main cavity; thus, they have the same ini-
tial gas pressure. In practice, the gas pressure is
unknown for a cavity which, after a complex history
(see "Gas diffusion" on page 5), is about to start its
collapse. However, for a cavity growing from a nucleus
of known diameter, the initial gas pressure can be esti-
mated from the pressure balance expressed by Eqn. (4~.
5 EXPERIMENTAL OBSERVATIONS OF
CAVITY DISINTEGRATION AND
STATISTICAL PROPERTIES
Use of experimental data Experimental data were
used at different stages of the study.
1. Choice of bubble model: Observations of the
number of larger cavities and the number of sharp
pulses were used in the choice of theoretical
model, Section 3.
Choice of geometrical configurations: Observa-
tions of sizes of cavities and spacing between
them.
Statistical data for the Monte Carlo simulations
(Section 6.2) were estimated from the experi-
ments.
Observations from earlier experiments Based on
basic studies (Hallander 1995, Hallander and Bark
1997) and the examination of high speed films, the
time displacement, space between cavity centres, and
initial radii of the cavities at the start of the collapses
were believed to be key parameters; hence, they were
treated as stochastic variables. High speed film obser-
vations revealed that the statistical properties of the
cavitation process (especially the disintegration of
larger cavities into parts) depend on the flow condi-
tions.
In Hallander (1998), two (somewhat idealised)
types of processes were identified. The first type,
shown in Fig. 4, was to simulate unsteady cavitation
which occurs when, for example, the wake behind a
ship, has large gradients in time and space. This pro-
duces strongly correlated disintegrations that generate
closely situated cavities which are likely to interact
during a rather synchronized collapse. The second type
of process was to simulate almost steady cavitation
which can occur in a wake with small gradients in time
and space. This results in weakly correlated disintegra-
tions and less synchronized collapses. Estimates of dis-
tribution parameters for the stochastic variables were
made for these two model processes. These distribution
parameters were then used to generate random input
data to Monte Carlo simulations with the system of two
spherical cavities, Section 6.2.
An improved experimental study In cooperation
with another project (Schoon 2000), a new method for
the generation of unsteady cavitation on a stationary
foil was developed and tested in the SSPA cavitation
tunnel (Hallander and Bark 1998, Hallander 1999~.
The properties of cloud cavitation for four conditions,
ranging from steady to highly unsteady, were analysed.
Input parameters for numerical studies were then esti-
mated from two of these conditions.
An unsteady inflow was generated by an oscillat-
ing foil (pitching around its mid-chord axis) positioned
upstream from a stationary test foil. This has advan-
tages over a conventional arrangement with an oscillat-
ing test foil. Although an oscillating test foil could
generate approximately the desired motion, it tends to
overemphasize the synchronization of bubbles, as well
as the collapse-forcing pressure, both due to unwanted
pressure terms caused by the pitching motion (Kruppa
1986 and Schoon and Bark 1998a). The new arrange-
ment used was intended to generate a more propeller-
like cavitation development. As expected, the new
experimental set-up did not generate as violent and
coherent collapses as the earlier experiments with one
oscillating foil did.
7
(alt= 100.7 ms (b~t=213.5ms
(a) t= 450.1 ms (b) t = 466.0 ms
c) t = 216.6 ms (d) t = 240.8 ms
Figure 5: Quasi-steady cavitation on a foil with oscillating
inflow (2 Hz), Uoo=5.0 m/s and cavitation number
~=1.0. The re-en~ant jets have time to develop fitlly and to
disintegrate parts of the sheet cavity several times during one
oscillation cycle.
When the inflow to the test foil is stationary or
varying slowly, disintegrations of a steady or quasi-
steady character occur, as shown in Fig. 5. The re-
entrant jets have time to fully develop and are con-
stantly breaking off parts from the aft section of the
sheet cavity in a boiling-like pattern. This pattern is
influenced by the length and thickness of the sheet cav-
ity. As this cavity grows longer and thicker, the disinte-
grated cavities also become larger. The formation of
first generation sub-cavities appears to be dispersed
quite well in time, as well as in space, although the
times at which they form are somewhat correlated by
the re-entrant jet mechanism.
When the oscillation of the inflow to the test foil is
further increased, the disintegrations become unsteady,
as shown in Fig. 6. More coherent cloud formations are
broken off from the sheet. These main, first generation
sub-cavities (typically about four) disintegrate further
into second generation sub-cavities which are close
together. The collapse of these coherent clouds gener-
ated significantly higher pulses (about ten times) than
the more random collapses under steady and quasi-
steady conditions.
An extreme example is that of downstream mov-
ing collapses at highly unsteady cavitation, see Fig. 7,
where a fast collapsing glossy sheet generates a large
number of surrounding cavities (Schoon and Bark
1998b). This results in a very violent, synchronized
and almost simultaneous collapse of all the cavities.
Summary of experimental observations From ear-
lier experiments, it is concluded that re-entrant jets can
significantly influence the disintegration of a sheet
8
c) t= 467.3 ms (d) t = 467.7 ms
Figure 6: Unsteady cavitation on a foil with oscillating
inflow (lOHz), Uoo=5.0 m/s and a=1.0. Disturbances
arise in the sheet cavity (a), which break up into cloud-like
sub-cavities that collapse (b - d).
(a) t= 231.7 ms (b) t = 234.9 ms
c~t=235.9ms (d~t=236.4ms
Figure 7: Highly unsteady cavitation (with a downstream
moving collapse) on a foil with oscillating inflow (15 Hz),
Uoo=5.0 m/s and ~=0.78.
cavity. From the present experiments with foils in
unsteady flow, it follows that the disintegration of the
sheet into parts can also be significantly influenced by
the variations of the unsteady inflow. Due to this
unsteady inflow, the sheet can disappear from the
upstream edge, which transforms a part of the sheet
into a travelling cavity. These two main mechanisms
and the balance between them influence the sub-cavity
distribution in time and space, and consequently, the
amount of acoustic interaction. Apart from re-entrant
jets and variation of inflow, the motion of a wavy cav-
ity surface towards the blade surface results, for rea-
sons of geometry, in further disintegration.
Some behaviours, which depend on the size and
spacing of the cavities, can be identified.
1. Classical cloud cavitation with a vast amount of
small bubbles close to each other can undergo a
highly synchronized collapse, as described by
March (1980) and others. This type of collapse is
not included in the present study. Instead, attention
is given to larger and more sparsely distributed
voids, which can be parts of a disintegrating cloud.
The aim was to investigate the significance of the
interaction between these larger voids.
2. When the variation of the inflow to a foil or pro-
peller blade is slow, the sheet cavity approaches a
steady state behaviour; re-entrant jets develop and
the sheet disintegrates into multiple, rather small
bubble formations, distributed over a relatively
large area. Consequently, the probability that
many bubbles would collapse simultaneously and
close together is rather limited. It is also signifi-
cant that most of the bubbles are of about the same
size. This behaviour is referred to as weakly syn-
chronized collapse, in which interaction can also
be expected to be rather weak, since the cavities
are relatively separated in time as well as in space.
When the inflow varies rapidly, the re-entrant jet
and the related cloud formation may not develop
completely; specifically, all the cloud formations it
generates may be almost simultaneously exposed
to the pressure forcing the collapse. In this case
interaction can be supposed to be more important.
6 NUMERICAL STUDIES
6.1 Basic studies
The behaviour of the equation system, describing two
spherical, interacting cavities (Section4.1), was stud-
ied by varying the input data. An example of time his-
tories of radii and pressure for such a system is shown
in Fig. 8. The second collapse of C2 is significantly
amplified here by the pressure generated from the first
collapse of Car. Figure 9 shows the time histories for
the same system without the interaction.
The effects of parameter variations on the interac-
tion between two cavities were demonstrated in Hal-
lander (1995~. Examples of parameters varied included
the space between the cavities, the initial gas pressure
and the artificial damping. The initial gas pressure
within the cavities was observed to strongly affect the
behaviour of the cavities and the pressure pulses gener-
ated. The results show that higher initial gas pressures
or higher artificial damping reduce the pressures gener-
ated, which results in less interaction. The maximum
was found to occur when Cat generates first a positive
pressure during the collapse of C2 followed by a nega-
tive one during the rebound. This implies that the time
displacement between the cavities is of great impor-
tance, as was also shown by Sato et al. (1994~.
The time displacement between the two collapsing
cavities was further investigated in Hallander and Bark
(1997~; the effect of a systematic variation of the time
displacement is given in Fig. 10. The maximum pres-
sure amplitudes generated by each cavity and the total
energy radiated by both cavities were studied as func-
tions of the time displacement. Although significantly
higher pressure pulses were found, the increase
occurred only within a quite narrow band of time dis-
placements, as shown in Fig. 10; this may reduce the
effective influence of interaction for statistically dis-
tributed cavities. The interaction influences the smaller
cavity much more than the larger one. According to
Fujikawa and Takahira (1986), this influence increases
when Ro2/Ro~ decreases.
Interaction and overlap were observed to increase
the total energy radiated at the highest frequencies. The
total energy radiated reaches its maximum when the
pressure peaks generated by the two cavities overlap; it
sinks to a minimum when they are in the opposite
phase. When the distance between the cavities is
decreased, the interaction becomes stronger, and both
the generated pressures and the total energy radiated
increase. Sometimes a "capture" of the smaller cavity
was observed. This means that the motion (expansion
and collapse) of the smaller cavity is strongly influ-
enced by the pressure from a larger neighbour cavity.
Such a capture is shown in Fig. 8. This capture gives a
small expansion of C2 and prolongs the time to its first
collapse.
The analysis of two interacting cavities was, by an
approximation, extended to the interaction of a few
more cavities in Hallander and Bark (2000~. This gave
an increase of approximately 3 dB to the energy spec-
tral density above 20 kHz.
6.2 Monte Carlo simulations
Basic studies (Section 6.1) showed that the interaction
between the cavities was sensitive to several parame-
ters. The following questions can then be raised: How
often does maximum interaction occur and what is its
average effect in more realistic cavitation processes?
To facilitate describing the generation and collapse of
cavity distributions, high speed films were analysed
and distribution parameters were estimated. Monte
7k
61
5 1
2
O
rat
0.12
0.10
0.08
0.06
-
0.04
0.02
O.
1 2
\
\
\
- 1
-0.02—
0 1 2 3
t [ms]
4 5 6
Figure 8: Time history of radii (upper) and generated pres-
sure (lower) of two interacting cavities: alto ~ 0.44 ms,
I ~ 11 mm, Rol ~ 6.4 mm and Ro2 ~ 3.0 mm. The second eol-
lapse of C2 is amplified significantly here by the pressure
generated from the first collapse of Cl.
6
5
2
O
0 1
0.12
0.10
0.08
0.06
0.04
0.02
OF
- 1
-0.02
o
. . 1 1 7
—- _ _ —- C
~" _C,
4
3 -
\\/ \
~_~
3 4 5 6
t [ms]
2
\
\
~-
\
\
/
\
\/ ~
~ . J. \'_
3 4 5
t [ms]
6
2
3
t [ms]
4
s
6
Figure 9: Time history of radii (upper) and generated pres-
sure (lower) for the same initial configuration as in Fig. 8 but
computed without interaction.
0.25
Carlo simulations were used to investigate the average
influence of interaction between statistically distrib- 0.20
uted cavities (Hallander 1998~.
In a new series of simulations (Hallander 1999),
distribution parameters estimated from two sets of new
experimental conditions were used to produce 2000
sets of random input data. These distributions of input
data are shown in Figures 1 1 and 13. The distributions
of peak pressure amplitude (resulting from the Monte
Carlo simulations) are shown in Figures 12 and 14.
Average power density spectra were also calculated,
see Fig. 15. The new estimates of distribution parame-
ters, in combination with a new model for the exter-
nally applied pressure (Section 4.3), gave smaller
effects of interaction on average than those used previ-
ously. Nevertheless, the simulations indicate that for
strongly coherent collapses, the interaction can result
in some very high pressure pulses.
10
0.15
0.10
0.05
o
-2 -1 0 1
Ato [ms]
Figure 10: Maximum pressure amplitude (at r = 1 m) gener-
ated by each cavity as a function of difference in starting
time: Rol = 8.0 mm, Ro2 = 4.0 mm and I = 32 mm.
0.8
, 0.6
0.4
0.2
c'
o
0.25
~ 0.2
5 0.15
, 0.1
0.05
O _
o
0.25
~ 0.2
`~5 0. 15
, 0.1
._
~ 0.05
, h . . 1 ~
( ) 2 4 6 8 0
let = t - t [ms]
J( _ .
_ 0.25
~ 0.2
`~= 0. 15
:> 0. 1
0.05
_ O ~
2 4 6 ~ 0
Rn1 [mm]
Figure 11: Distributions of random input data from simula-
tions of unsteady cavitation.
0.12,
0.1
c>
=3 0.08
0.06
c,
;>
._
V 0 04
0.02
O-
\J I /.
~ 1
C)
~ 0.~s
5
A: 0.06
._
V 0 04
0.02
,2 x10-3 1
Ill llm ~
0.04 0.06 ~ 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1 0.12
p lP
max 00
o
0 0.02 0.04 0.06 0.08 0.1 0.12
p lP
max of
Figure 12: Distribution of peak amplitudes from simulations
of unsteady cavitation with interaction (upper) and without
interaction (lower).
0.25
cat
0.2
0.15
;> 0.1
c, 0.05
O _
0.25
~ 0.2
Co.ls
, 0.1
~ 0.05
_ ~
O
20 40 60 -4 -2 0 2 4 no
I [mm] At =t -t [ms]
0.25
cat
~ 0.2
Co.ls
, 0.1
. -
0.05
O
20 40 60
I [mm]
0.25
~ 0.2
=0.15
, 0.1
_
0.05
O
2 4 6 8 0 2 4 6 8 0 2 4 6 8
Ro2 [mm] Ro1 [mm] Ro2 [mm]
Figure 13: Distributions of random input data from simula-
tions of quasi-steady cavitation.
0.1
t3 0.08
0.06
c'
._
c' 0 04
0.02
O
O.
O -
IL 004 0.064~0.08 0.1
0 0.02 0.04 0.06 0.08 0.1 0.12
p lP
manic on
0.12
c'
~ 0.08
3
G
0.06
<,, 0.04
0.02
O _
0 0.02 0.04 0.06 0.08 0.1 0.12
p lP
max of
Figure 14: Distribution of peak amplitudes from simulations
of quasi-steady cavitation with interaction (upper) and with-
out interaction (lower).
11
~ 100
m
80
-
60
$
c 40
o
O 20
120
=~
I— - without interaction I ~
. ~ . . . . . . . . . . . . . . ..... . . . . ....
105 lo6
103 104
f [Hz]
80
60
40
20
O
I with interaction I
I— - without interaction I
. . . . . . . . .
. . . ..... . . . . ....
lot lo2 103 104 lob lo6
f [Hz]
Figure 15: Average power density spectra from simulations
of unsteady cavitation (upper) and quasi-steady cavitation
(lower). The expectation of the initial radii are larger in the
quasi-steady ease, which results in higher spectral levels.
When comparing some time series of pressure sig-
nals, the simulated peak pressure amplitudes appear
higher than measured ones. However, after decimation
of the simulated signal (to a sampling frequency of
similar size), the order of magnitude is approximately
the same, as shown in Fig. 16. The upper graph shows
the same signal as in Fig. 8, but here it is decimated to
fs 250 kHz. The lower graph shows an average
sequence of a measured signal (from the unsteady
experimental condition, Fig. 6~. The decrease of the
simulated signal after decimation indicates that the
interaction has the most influence well above 100 kHz
in this example. Comparison of simulated spectra with
and without interaction, Fig. 15, shows that the average
increase of the PSD due to interaction is less than 3 dB
below 60 kHz in model scale (the difference increases
with frequency). The conclusion drawn from these
observations is that the interaction does not signifi-
cantly influence the radiated noise for lower frequen-
c~es.
0.030
0.025
0.020
0.015
0.010
0.005
Ot......
-1
-0.005-
2.5
0.03O
0.025
0.020
0.015
0.010
0.005
~~-
............. /
3 3.5 4
t [ms]
- 2 11
- 1: 1 ~
~1 ~
. .
-
-0.005
2275.5 2276 2276.5 2277
t [ms]
Figure 16: Comparison of a simulated signal of two inter-
aeting cavities (upper) with a measured signal where the
highest peaks are dominated by only a few cavities (lower).
The simulated signal is the same as in Fig. 8, but decimated
tof5 250 kHz. The measured signal is sampled atf5 262 kHz.
The expectation of the initial radii are larger in the
quasi-steady example, which results in higher spectral
levels, Fig. 15. This is so because only parts of the cav-
itation processes are taken into account when investi-
gating the importance of interaction between cavities
on different scales. In more complete models of cavita-
tion processes, the number of collapses of different
types involved in each process must be taken into con-
sideration.
Further comparison of the two signals in Fig. 16
shows that they behave similarly, but the simulated one
has wider pulses. Although the measured sequence
shown is a quite average one, the simulated one is
among the highest found in the Monte Carlo simula-
tions. The conclusion can be drawn from this that the
total model damping used in the simulation studies was
too high. The choice of these parameters was discussed
in Section 4.5. However, the two sequences show that
the model is able to generate reasonably realistic sig-
nals.
12
6.3 Influence of interaction on cavity motions, sig-
nals and spectra: Summary of observations
1.
From an acoustical point of view, the pressure
fields generated by each cavity are scattered by
neighbouring ones. In particular, if the time dis-
placement of the motions of the cavities is favour-
able, the pressure pulse from a large cavity
changes substantially the motion of a small cavity,
while the influence of the small one on the large
one can be quite limited. The collapse of the small
cavity often becomes more violent, while the col-
lapse of the large one is slowed a bit. (These ~en-
eral observations have been made by several
authors.)
2. The amplitude of the pulse from the large cavity
decreases a bit because of the interaction, while
the pulse from the small one becomes significantly
higher, although of very short duration. Here' the
energy spectral density of the sum of the pulses,
when the interaction is taken into account,
increases considerably at the highest frequencies,
while the levels at low frequencies mostly remain
relatively unchanged.
In real cavitation processes, involving many cavities
that behave partly randomly, the trends indicated above
still apply, however the details are much influenced by
the statistical properties of the cavitation processes.
FORMATION OF PRESSURE FIELDS THAT
LEAI) TO INTERACTION AND
SYNCHRONIZATION
The present foil experiments, with varying strengths
and periodicity of the gusts entering the test foil, con-
f~rm earlier indications that the variations of the global
flow in time and space can generate differing bubble
distributions and influence the synchronization of the
collapses. The synchronization of the collapses of bub-
bles increases, i.e. all of the bubbles tend to collapse
more nearly simultaneously, with increasing amplitude
and frequency of the gusts in the inflow. Hence, the
general conclusion can be drawn that the variation in
time and space of the global pressure that corresponds
to the inflow to the foil is significant; the degree
depends on the amplitude, duration and spatial extent
of the temporal variations of the pressure. These cir-
cumstances can significantly influence noise, vibra-
tions and erosion as well.
Synchronization can occur on multiple spatial
scales. In experiments with foils in unsteady flow, con-
ditions can be found in which collapses are undoubt-
edly synchronized by the global flow. However,
towards the end of the process, the collapses seem also
to be influenced by strong interaction which gives rise
to a second synchronization, this time within a smaller
region. Although these configurations are of primary
interest for erosion, they can also be expected to influ-
ence noise at the very highest frequencies. Due to this
complexity, it is concluded that simulation of cavita-
tion noise by model experiments is still the most relia-
ble method; this can be expected to remain so, possibly
for a long time. However, even experimental methods
can be tricky.
The processes involved in the disintegration of the
main cavity (a sheet, for example) determine the distri-
bution in time and space of large as well as small sub-
cavities. Large sub-cavities, which later on become the
forcing cavities in the interaction process, are specifi-
cally generated by two mechanisms: the early action of
break-off due to re-entrant jets or the possible increase
of the pressure (above the level of cavitation pressure)
at the upstream cavitation edge which can result in a
transient or "downstream moving collapse". The small
sub-cavities are formed by minor break-off processes
in the remaining sheet or by further disintegration of
the large sub-cavities mentioned above. All such cavi-
ties are then, more or less synchronously, forced to col-
lapse by the globally increasing pressure (generated by
the motion of the propeller in the wake).
The numerical interaction model predicts higher
individual pulses as well as higher mean spectra (PSD)
at high frequencies, when using statistical input from
the unsteady experimental condition in which the sub-
cavities collapse in the most synchronized way. This
trend could be anticipated and it should be noted that
its influence at the highest frequencies could make
some difference, although the model can be expected
to underestimate the interaction effect. An important
conclusion drawn from the comparison of the simu-
lated examples is that the simulations, by using the sta-
tistical distributions, demonstrate the effect of large
scale synchronization of sub-cavities. This synchroni-
sation emanates from the global flow field, which
determines the development of re-entrant jets, down-
stream moving collapses, etc.
In summary, two conceptual synchronization
mechanisms can be identified for a collective collapse
of cavities.
1. The first and, usually, most global synchronization
is due to the increase of the environmental pres-
sure, resulting from the decrease of the angle of
attack of a propeller blade or foil. This starts the
collapse of bubbles in a relatively large region.
13
Second, a large and violently collapsing cavity
can, by acoustic interaction, synchronize the col-
lapse of nearby small cavities. Supposing that the
pressure pulse from the large cavity has roughly
spherical spreading, it follows that this synchroni-
zation is usually more limited in space than the
first one. When the second type of synchronization
is added to the first one, small cavities close to the
forcing one can collapse very violently.
8 SOME FURTHER COMMENTS ON THE
INFLUENCE OF ACOUSTIC
INTERACTION ON THE NOISE SPECTRA
GENERATED
Some questions, about the contribution of acoustic
interaction to the noise spectra generated, which have
not been addressed elsewhere in this study, are dis-
cussed below.
The influence of small, non-interacting cavities It
is evident that cavities smaller than the ones studied in
the present simulations occur in real cavitation proc-
esses. Even if every pulse from the small cavities may
be weak, the number of small pulses can be high, and
overlapping of the pulses is possible. A general answer
to this question may not exist. Interacting as well as
non-interacting cavities of different sizes may exist in a
specific cavitation process. The two types of processes
identified in Section 6.2 indicate that the relation
between interacting and non-interacting collapses can
shift and is controlled by the large scale development
of the cavity in time and space. It is thus possible, at
least in cavitation processes where the interaction is
weak and random, that the high frequency levels can be
dominated by a large number of small cavities which
collapse without significant interaction. However, the
fact remains that noisy cavitation events in many
instances are dominated by relatively few (about three
to five) very high and sharp pulses per cavitation cycle.
Frequency range of acoustic interaction What the
present study shows is that interaction between cavities
can, for a given group of cavities, extend the frequency
range upwards by tenfold and generate some extremely
high pulses. However, analysis of the linear scaling of
cavitation noise, Hallander and Bark (2000), indicates
that the formulas based on linear acoustics can be
regarded as approximations for frequencies at which
the spectral levels are influenced by interaction.
As to the energy transferred to higher frequencies
due to interaction, it cannot be excluded that the total
energy radiated by the cavities is lowered by this. One
reason is that when a cavity collapses to a smaller min-
imum radii, the viscous dissipation increases. Another
1
reason is that the development of shock waves with
higher dissipation may increase due to the extremely
high and sharp pulses generated. This loss in shock
waves is indicated in Levkovskii's scaling theory (Lev-
kovskii 1968~.
As noted above, the choice of cavity sizes implies
that typical low frequencies (but not the very lowest)
influenced by interacting cavities have been estimated.
However, there are some other possible ways that inter-
action effects can be found at lower frequencies.
The behaviour of the main (forcing) cavity is dis-
turbed (slowed down) by the driven cavity.
The driven cavity is sometimes observed to
expand early in the process due to entrapment in
the low pressure generated by the forcing cavity.
The effect of this is probably weak due to low
radiation efficiency for the forced cavity.
3. There might be cavities present that are larger than
the ones being studied.
If the smallest bubbles take part in the interaction,
this will determine the levels at the highest frequencies.
The interaction increases by the ratio Ro~/Ro2, which
means that all cavities smaller than the Ro2 used in the
simulations are exposed to even more interaction; this
raises the high frequency range. An exception to this
may arise if the smaller cavities are much further away
from the main cavity than the one being studied. In this
situation it is possible that the high frequency level
could be dominated by many non-interacting cavities,
as discussed above.
The probability of synchronization becomes
higher when the ratio Ro~/Ro2 increases. In extreme
cases, such as the one in Schoon and Bark (1998b), the
probability of powerful synchronization of the col-
lapses is high and superposition of the pulses from the
driven cavities is also likely. Consequently, there are
reasons to believe that interaction often dominates the
high frequency spectrum, despite the possible excep-
tions mentioned in the discussion above.
9 CONCLUSIONS
1. A main engineering conclusion drawn from the
numerical simulations is that, provided the levels
at the very highest frequencies are disregarded,
the acoustic interaction between medium sized
dominating structures does not make much differ-
ence.
This implies that if a numerical method, for exam-
ple of the type presented by Matusiak (1992), can
generate an acceptable size and time distribution of
14
cavities and a realistic collapse forcing pressure, it
can also be expected to generate realistic noise lev-
els up to medium high frequencies (i.e. up to the
order of some hundred times the blade-rate). Since
these requirements are rather demanding, methods
of this type are not yet standard engineering proce-
dures.
When making predictions with model tests, the
implication is that, except for the very highest fre-
quencies, the distribution in space and time of
nearby cavities is not very critical. However, the
number and size of the cavities have to be reasona-
bly well simulated, particularly if the mean value
spectrum over all pulses is of interest.
When an accurate estimate of the noise at the very
highest frequencies is desired, the numerical simu-
lations indicate that an accurate simulation of both
the acoustic interaction and the statistical proper-
ties of cloud cavitation is required. As the time
displacements between cavities are very critical,
this problem is very difficult to solve in numerical
simulations. In model testing, there are still serious
problems, but some success can be expected with
a proper global flow. In extreme cases, for exam-
ple with small cavities close to a large one, a good
simulation of interaction can be anticipated
according to observations.
The statements in points 1 and 2 hold for interac-
tion between small to medium sub-cavities (clouds or
single voids) typically generated by a disintegrating
sheet. These sub-cavities can be expected to determine
the lower, if not the very lowest, end of the frequency
range at which interaction effects appear.
ACKNOWLEDGEMENTS
Most of the work for this paper was carried out within
the "Doctoral Student Programme" in underwater
acoustics and underwater technology. The programme
was funded by the Swedish Armed Forces (FM) and
managed by the Swedish Defence Research Agency
(FOI).
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16
DISCUSSION
J. Matusiak
Helsinki University of Technology, Finland
I would like to congratulate the author for a very
interesting paper. The author has successfully
demonstrated the effect of the interaction of
cavitation bubbles on noise. I was particularly
happy with the conclusion that although
interaction may result in high pressure pulses, it
does not significantly affect pressure spectrum
except at very high frequencies. This was my
simplifying assumption in the method of
evaluating the pressure generated by a cavitating
propeller (Matusiak 1 992a, 1 992b).
I found particularly interesting your
experimentally derived information on the
distributions of cavitation bubbles radii (Figures
1 1 and 13) used in the Monte Carlo simulations.
Did you find a correlation of the radii and total
volume of cavitation bubbles with the sheet
cavitation geometry and a disintegration of fixed
cavitation? I had a simple model (Matusiak
1 992a, 1 992b) relating these quantities. I am
curious whether your observations support this
model.
AUTHORS' REPLY
We did not try to find a correlation of the radii
and total volume of cavitation bubbles with the
sheet cavitation geometry. The estimates of
distribution parameters were made from high-
speed film recordings of two experimental
conditions, Figures S and 6. The geometry of
these sheet cavities are relatively similar, but the
cavity dynamics are very different (quasi-steady
and unsteady). In fact, all experimental
parameters except the oscillation frequency were
the same for these two conditions.
The time displacement, initial radii and distance
between the cavities were supposed to be
stochastic variables. Data for these variables
were sampled from the high-speed film
recordings by identifying start times of collapses
and by measuring the corresponding initial radii
and distances on the screen (Hallander 2002:
Paper IV). Distribution parameters were
estimated by fitting probability distributions to
the data. Since the high-speed film recordings
were limited to 1.3 s, the number of samples
used for the data fitting was small. The estimated
distribution parameters were finally used to
generate random input data to the Monte Carlo
simulations, Figures 11 and 12.
