Improvement of acoustic theory of ultrasonic
waves in dilute bubbly liquids
Keita Ando, Tim Colonius, and Christopher E. Brennen
Division of Engineering and Applied Science, California Institute of Technology, Pasadena, California 91125
kando@caltech.edu, colonius@caltech.edu, brennen@caltech.edu
Abstract:
The theory of the acoustics of dilute bubbly liquids is reviewed,
and the dispersion relation is modified by including the effect of liquid com-
pressibility on the natural frequency of the bubbles. The modified theory is
shown to more accurately predict the trend in measured attenuation of ultra-
sonic waves. The model limitations associated with such high-frequency
waves are discussed.
© 2009 Acoustical Society of America
PACS numbers:
43.35.Bf, 43.30.Ft, 43.20.Hq [GD]
Date Received:
May 28, 2009
Date Accepted:
June 18, 2009
1. Introduction
The acoustics of bubbly liquids have been extensively studied for many years. The traditional
theory for a dilute bubbly mixture assumes that mutual interactions among the bubbles are
negligible. The bubble/bubble interactions can never be ignored at resonance even in the dilute
limit,
1
,
2
and the theory is known to overestimate attenuation under the resonant condition. The
theory is also known to be inaccurate in estimating the attenuation of ultrasonic waves. So far, to
the authors’ knowledge, the cause of the discrepancy under the ultrasonic condition has not
been revealed.
Herein, we briefly review the theory and modify the dispersion relation by including
the effect of liquid compressibility on the natural frequency of the bubbles and validate this
modified theory by comparing to experimental data. Finally, we discuss the model limitations
associated with ultrasonic waves.
2. Review of the theory
In the classic papers of Foldy
3
and Carstensen and Foldy,
4
wave propagation in a bubbly mixture
was treated as a problem of multiple scattering by randomly distributed isotropic scatterers
representing the spherical bubbles, and the dispersion relation for the mixture was derived. An
alternative approach is to treat the mixture as a single phase (continuum) medium. van
Wijngaarden
5
defined volume-averaged quantities in order to remove the local fluctuations due
to scattering and derived the averaged equations based on heuristic, physical reasoning. By
linearizing van Wijngaarden’s equations, Commander and Prosperetti
2
derived the dispersion
relation
1
c
m
2
=
1
c
l
2
+4
n
0
R
0
f
R
0
dR
0
N
2
−
2
+
i
2
,
1
where
c
m
is the complex sonic speed in the mixture,
c
l
is the sonic speed in the liquid alone,
n
is
the total bubble number density,
is the bubble-dynamic damping constant,
is the temporal
angular frequency
=2
f
,
N
is the natural frequency of the bubbles,
R
0
is the equilibrium
bubble radius, and
f
R
0
is the density function for the size distribution of the equilibrium
bubble radius, which satisfies
0
f
R
0
dR
0
=1. In the derivation of the dispersion relation
(1)
, the
void fraction,
Ando
etal.
: JASA Express Letters
DOI: 10.1121/1.3182858
Published Online 27 July 2009
J. Acoust. Soc. Am.
126
3
, September 2009
© 2009 Acoustical Society of America EL69
=
4
3
n
0
R
0
3
f
R
0
dR
0
,
2
is assumed to be small
1
and relative motion between the phases is ignored. The relative
motion has been shown to have minimal impact on the acoustic problem.
6
To complete the dispersion relation
(1)
, the bubble-dynamic constant and the natural
frequency need to be specified. Commander and Prosperetti
2
used the following expressions for
and
N
,
=
2
μ
l
l
R
0
2
+
p
g
0
2
l
R
0
2
I
+
2
R
0
2
c
l
,
3
N
2
=
p
g
0
l
R
0
2
R
−
2
S
p
g
0
R
0
.
4
Here,
μ
l
is the liquid viscosity,
S
is the surface tension, and
p
g
0
is the internal (gas) bubble
pressure (vapor pressure is typically negligible) given by
p
g
0
=
p
l
0
+
2
S
/
R
0
, where
p
l
0
is the
ambient pressure. The quantity
is a function of the Peclet number Pe=
R
0
2
/
th
, where
th
is
the thermal diffusivity of the gas,
=
3
1−
i
3
−1
Pe
−1
i
Pe coth
i
Pe − 1
,
5
where
is the ratio of specific heats. The effective polytropic index for thermal behavior of the
gas is then given by
=
R
/3. Since
→
1as
→
0orPe
→
0, the isothermal natural fre-
quency (defined below) is obtained in the quasistatic limit and is generally very close to the
resonant frequency.
N
2
=1
=
p
g
0
l
R
0
2
3−
2
S
p
g
0
R
0
.
6
It should be noted that the effect of liquid compressibility is ignored in Eq.
(4)
and is negligible
unless the frequency is extremely high compared to the resonant frequency.
7
We define the phase speed
V
and attenuation
A
(in decibels per unit length) as
V
=
R
1
c
m
−1
,
7
A
=−20
log
10
e
I
1
c
m
.
8
The estimated phase velocity
(7)
is known to yield quantitative agreement with experimental
data in a wide frequency range below and above the resonance.
2
,
8
However, the estimated at-
tenuation
(8)
under resonant and ultrasonic conditions appears to deviate from the experimental
values.
Before concluding this review, we examine the model limitations. In order that the
mixture be considered homogeneous and the wave structure be well resolved, we need to choose
a physically appropriate averaging volume,
V
, and presuppose the scale separation
9
l
=
n
−1/3
V
1/3
L
,
9
where
l
is the mean bubble spacing and
L
is the wavelength in the mixture. Note that
R
0
l
in
the dilute limit (i.e.,
→
0). Since the wavelength of ultrasonic waves may be comparable to or
shorter than the mean bubble spacing, the continuum model may be invalid. In addition, neglect
of the acoustic contribution to the bubble natural frequency
(4)
may also give rise to a discrep-
Ando
etal.
: JASA Express Letters
DOI: 10.1121/1.3182858
Published Online 27 July 2009
EL70 J. Acoust. Soc. Am.
126
3
, September 2009
Ando
etal.
: Acoustics of dilute bubbly liquids
ancy in theory and experiment between the attenuation of high-frequency waves. In Sec. 3, we
discuss the effect of liquid compressibility on the attenuation of ultrasonic waves.
3. Modification to the theory
3.1 Linearized dynamics of the spherical bubbles
To obtain the formulas for the bubble-dynamic damping constant and the bubble natural fre-
quency, we need to linearize the spherical bubble dynamics. It follows from Prosperetti
7
and
Prosperetti
et al.
10
that the linearized dynamics are described by
x
̈
+2
x
̇
+
N
2
x
=−
p
l
0
l
R
0
2
e
i
t
,
10
where
is the infinitesimal amplitude of sinusoidally oscillating (farfield) liquid pressure
1
and
x
is the corresponding perturbation in the bubble radius
x
1
:
p
l
=
p
l
0
1+
e
i
t
,
11
R
=
R
0
1+
x
.
12
Here, the damping constant and the natural frequency are
=
2
μ
l
l
R
0
2
+
p
g
0
2
l
R
0
2
I
+
2
R
0
2
c
l
1+
R
0
c
l
2
,
13
N
2
=
p
g
0
l
R
0
2
R
−
2
S
p
g
0
R
0
+
R
0
c
l
2
1+
R
0
c
l
2
2
,
14
where the last terms on the right-hand sides of the above equations represent the contributions
associated with liquid compressibility. To quantify the impact of liquid compressibility on the
ultrasonic waves, we compute the dispersion relation
(1)
based on Eqs.
(13)
and
(14)
instead of
Eqs.
(3)
and
(4)
and validate the modification by comparing to experimental data below.
For future use, we develop the asymptotic limits of Eqs.
(13)
and
(14)
. In the quasi-
static limit (i.e.,
→
0), it follows from Prosperetti
7
,
11
that
2
μ
l
l
R
0
2
+
−1
10
p
g
0
l
th
,
15
N
N
=1
,
16
where
N
=1
is the isothermal natural frequency
(6)
so that liquid compressibility is unimpor-
tant. On the other hand, in the limit of
N
=1
, it is readily shown that
c
l
2
R
0
,
17
N
.
18
In this limit, the damping due to liquid compressibility dominates over the viscous and thermal
contributions and the natural frequency is independent of the bubble size.
Ando
etal.
: JASA Express Letters
DOI: 10.1121/1.3182858
Published Online 27 July 2009
J. Acoust. Soc. Am.
126
3
, September 2009
Ando
etal.
: Acoustics of dilute bubbly liquids EL71
3.2 Validation of the modification
We compare the dispersion relation
(1)
to the experiment of Kol’tsova
et al.
12
who measured the
attenuation in a high-frequency range up to 30 MHz. In those experiments, hydrogen bubbles
were produced using electrolysis and had a size distribution with a mean radius of 15–20
μ
m.
The histogram of the size distribution for
=0.03% (probable size,
R
0
ref
=20
μ
m) is plotted in
Fig.
1
. We assume that the size distribution for different values of
is similar to that in Fig.
1
.
The actual distribution may be smooth, as shown in Fig.
1
, and we model it using a lognormal
density function with standard deviation
,
f
R
0
*
=
1
2
R
0
*
exp
−
ln
2
R
0
*
2
2
,
19
where
R
0
*
=
R
0
/
R
0
ref
.
Using the size distributions in Fig.
1
, the phase velocity
(7)
and attenuation
(8)
are
computed using Eqs.
(3)
and
(4)
or Eqs.
(13)
and
(14)
and are plotted in Fig.
2
. The attenuation
of Kol’tsova
et al.
12
is also plotted (
=0.004%,
N
=1
/2
=0.142 MHz for
R
0
ref
). It follows
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
R
∗
0
(
=
R
0
/R
ref
0
)
f
(
R
∗
0
)
histogram (Kol
′
tsova
etal.
, 1979)
lognormal (
σ
= 0.2)
lognormal (
σ
= 0.4)
Fig. 1. Normalized histogram of the bubble size distribution of Kol’tsova
et al.
Ref.
12
and lognormal distributions
with a standard deviation
. The probable size,
R
0
ref
, is set to be 20
m. The measured values are based on a
hydrogen/water mixture of
=0.03% at 15 °C and 1 atm.
10
2
10
1
10
0
10
1
10
1
10
0
10
1
10
2
f
[
MHz]
A
[dB
/
cm]
°
Kol
′
tsova
etal.
(1979)
10
2
10
1
10
0
10
1
1000
1500
2000
2500
f
[MHz]
V
[m
/
s]
}
histogram
}
lognormal
(
σ
=0.2)
}
lognormal
(
σ
=0.4)
Fig. 2. Phase velocity
left
and attenuation
right
for a hydrogen/water mixture of
=0.004% and
R
0
ref
=20
mat
15 °C and 1 atm. The lines and symbols
plus, cross, and asterisk
are computed using the dispersion relation
1
with Eqs.
3
and
4
and with Eqs.
13
and
14
, respectively. The symbols
circle
denote the experimental data of
Kol’tsova
et al.
Ref.
12
.
Ando
etal.
: JASA Express Letters
DOI: 10.1121/1.3182858
Published Online 27 July 2009
EL72 J. Acoust. Soc. Am.
126
3
, September 2009
Ando
etal.
: Acoustics of dilute bubbly liquids
from the phase velocity plot that the present modifications to
and
N
have negligible impact
on
V
. It is also found that the size distribution tends to smooth the transition in
V
at the resonant
frequency.
However, the present modification does lead to a striking change in the attenuation for
N
=1
. The dispersion relation
(1)
with the present modification predicts attenuations at
high frequencies that agree with the experimental measurements.That is, liquid compressibility
has major impact on the attenuation of the ultrasonic waves. As a result of Eqs.
(17)
and
(18)
,
the phase velocity and the attenuation for
N
=1
asymptote to the constant values,
V
c
l
,
20
A
20
log
10
e
3
2
R
0
ref
C
1
,
21
where the constant
C
1
is
C
1
=
0
R
0
*
2
f
R
0
*
dR
0
*
0
R
0
*
3
f
R
0
*
dR
0
*
.
22
For the lognormal
f
R
0
*
,
C
1
=exp
−2.5
2
so that the attenuation decreases as
increases. It
should be pointed out that Kol’tsova
et al.
12
presented the different data sets of the attenuation
(with different void fractions) which remains almost constant under the ultrasonic condition. It
is therefore concluded that the modified theory is superior to the previous theory when it comes
to predicting this trend.
Furthermore, we notice that the size distribution increases the attenuation below the
resonant frequency. From Eqs.
(15)
and
(16)
, the asymptotic values at low frequency become
V
c
l
1+
l
c
l
2
p
l
0
,
23
A
20
log
10
e
l
2
V
2
3
p
l
0
2
C
2
,
24
where the constant
C
2
is
C
2
=
0
R
0
*
5
f
R
0
*
dR
0
*
0
R
0
*
3
f
R
0
*
dR
0
*
.
25
Here, we have neglected the viscous contribution in Eq.
(15)
since the thermal damping gener-
ally dominates over the viscous damping. In addition, it is assumed that the surface tension is
negligible in Eq.
(16)
. For the lognormal
f
R
0
*
,
C
2
=exp
8
2
so that the attenuation increases as
increases. To interpret this tendency, consider linear bubble oscillations under a sinusoidal
forcing
p
l
−
p
l
0
sin
t
of the farfield liquid pressure. The corresponding perturbation in the
bubble radius oscillates with the forcing frequency
and with a phase shift
such that
cos
=
N
2
R
0
−
2
N
2
R
0
−
2
2
+4
2
R
0
2
.
26
Therefore, phase cancellations due to the different phases among the different-sized bubbles
occur in the low-frequency regime since
N
N
=1
. The phase cancellations can be re-
garded as apparent damping of the wave propagation in the polydisperse mixture, and the damp-
ing mechanism becomes more effective as the bubble size distribution broadens.
13
,
14
As a result,
Ando
etal.
: JASA Express Letters
DOI: 10.1121/1.3182858
Published Online 27 July 2009
J. Acoust. Soc. Am.
126
3
, September 2009
Ando
etal.
: Acoustics of dilute bubbly liquids EL73
the size distribution increases the attenuation, as seen in Fig.
2
. However, in the ultrasonic limit,
all the different-sized bubbles oscillate with the same phase due to the fact that
N
(regard-
less of the bubble sizes); hence, in this case, the phase cancellations do not occur.
Finally, we check the model limitation
(9)
. The mean bubble spacing in Fig.
2
is
l
=
3
4
0
R
0
*
3
f
R
0
*
dR
0
*
3
R
0
ref
1100
μ
m,
27
where the histogram in Fig.
1
is used for
f
R
0
*
.At
f
=10 MHz, the wavelength is approximated
by
L
c
l
f
150
μ
m,
28
which is larger than the mean bubble radius but shorter than the mean bubble spacing. Hence,
the continuum assumption is invalidated, while bubble/bubble interactions may be ignored.
However, despite this violation, as seen in Fig.
2
, the continuum theory accurately predicts the
trend in the attenuation around
f
=10 MHz. This implies that the validity of the dispersion rela-
tion
(1)
may extend beyond the limitation
(9)
.
4. Conclusion
A modification to the traditional dispersion relation of linear waves in dilute bubbly liquids is
made to take into account the effect of liquid compressibility (which is very important far above
the resonant frequency) on linearized dynamics of spherical bubbles. The modified dispersion
relation is found to accurately predict the trend in measured attenuation of ultrasonic waves.
The agreement between the modified theory and experiment implies that the validity of the
dispersion relation
(1)
may extend beyond the continuum model limitation
(9)
.
Acknowledgments
The authors gratefully acknowledge the support by ONR Grant No. N00014-06-1-0730 and by
NIH Grant No. PO1 DK043881.
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Ando
etal.
: JASA Express Letters
DOI: 10.1121/1.3182858
Published Online 27 July 2009
EL74 J. Acoust. Soc. Am.
126
3
, September 2009
Ando
etal.
: Acoustics of dilute bubbly liquids