Experimental and theoretical investigation of a polished-type single-mode fibers coupler with an intermediate planar waveguide [1973-10]

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Ultrasonics Sonochemistry 12 (2005) 79–84

Experimental and theoretical investigation of the meanacoustic pressure in the cavitation field

Cleofe Campos-Pozuelo a,*, Christian Granger b, Christian Vanhille c,Alexei Moussatov b,d, Bertrand Dubus b

a Instituto de Acustica, C.S.I.C., Serrano, 144, 28006 Madrid, Spainb IEMN, UMR CNRS 8520, Departement ISEN, 41 Boulevard Vauban, 59046 Lille Cedex, France

c E.S.C.E.T. Universidad Rey Juan Carlos. Tulipan, s/n. 28933 Mostoles, Madrid, Spaind Institut Meurice, 1 Avenue E. Gryzon, 1070 Bruxelles, Belgium

Received 23 February 2004; revised 22 June 2004; accepted 22 June 2004

Available online 13 August 2004

Abstract

The cavitation field radiated by a 20 kHz sonotrode-type transducer is experimentally and theoretically analyzed. Special interest

is paid to the origin of the strong fluid streaming appearing in low frequency sonoreactors. A new experimental procedure is pro-

posed to evaluate the mean acoustic pressure inside the fluid. This parameter has been quantified for different points and amplitudes.

The velocity of the radiating surface is controlled by a laser interferometer and is always sinusoidal. Train wave excitation is used.

The pressure wave and amplitude are measured in the tank with a calibrated hydrophone. The acoustic mean pressure is estimated

from the total pressure value at the end of the pulse after an adequate filtering. An analytical nonlinear second order model based on

the coupling of the equations of the fluid mechanics with the Rayleigh–Plesset equation is developed in order to relate the measured

acoustic parameters to the cavitation state of the fluid. The distributions of the fundamental amplitude and mean pressure are cal-

culated as a function of bubble density and bubble size. A qualitative theoretical description of the experimental data is presented.

Quantitative differences and model limitations are commented.

� 2004 Elsevier B.V. All rights reserved.

Keywords: Cavitation field characterization; Nonlinear acoustics; Ultrasonic reactors

1. Introduction

Although bubbly liquids are known to be highly non-

linear acoustic media [1], physical phenomena related to

the cavitation field, such as streaming, are usually stud-

ied considering linear acoustic waves [2,3]. According to

Mitome [7], two different types of acoustic streaming can

be considered. In the ‘‘genuine acoustic streaming’’, the

fluid motion is produced by the attenuation of theacoustic wave during its propagation. In the ‘‘quasi-

1350-4177/$ - see front matter � 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.ultsonch.2004.06.009

* Corresponding author.

E-mail address: [email protected] (C. Campos-Pozuelo).

acoustic streaming’’, fluid motion results from the gradi-ent of the mean acoustic pressure, as a consequence of

nonlinear phenomena in the ultrasonic wave propaga-

tion. The analysis of nonlinear waves in the cavitation

field is complicated by the stochastic nature of the bub-

ble and acoustic fields and by the complex interaction

between these fields. On the other hand, it is well known

that for a nonlinear wave propagating in a fluid the

mean acoustic pressure is not zero and its value stronglydepends on the nonlinearity parameter of the medium.

Thus, experimentally, the problem of analyzing the non-

linear acoustic behaviour of the cavitating fluid can be

handled by measuring the mean acoustic pressure which

is directly related to nonlinear acoustic parameters and

80 C. Campos-Pozuelo et al. / Ultrasonics Sonochemistry 12 (2005) 79–84

which naturally averages in time the physical quantities.

Measurements of mean acoustic pressure in the cavita-

tion field generated by a 20 kHz sonotrode are reported

in Section 2. A theoretical second order nonlinear model

is proposed to analyze these experimental results in

Section 3.

Fig. 2. Analyzed conelike bubble structure (CBS).

2. Measurement of mean acoustic pressure

The experimental set-up is schematized in Fig. 1. The

20 kHz source is made of a sandwich piezoelectric trans-

ducer, a mechanical amplifier and a 70 mm diameter

sonotrode. The source is driven by a pulsed sinusoidalexcitation with a typical pulse length of 6000 periods

at 20 kHz with 1 Hz repetition rate. Note that steady-

state in the tank is reached after a few hundred periods.

The sonotrode surface velocity is monitored indirectly

by measuring the radial velocity at half-height of the

sonotrode and using a preliminary measurement of the

longitudinal (surface)/radial (half height) velocity ratio.

A picture of the conical bubble structure (CBS) gener-ated below the sonotrode under these conditions is dis-

played in Fig. 2 [4]. The pressure is measured in the

tank using a Bruel & Kjaer (B&K) 8103 hydrophone.

In the following, the acoustic pressure p is

pðtÞ ¼ ptðtÞ � p0 ð1Þwhere pt is the total pressure, p0 the ambient pressureand t the time. The mean acoustic pressure is Æp(t)æwhere Æ æ denotes the time average. The acoustic pres-

Fig. 1. Experimental set-up.

sure is recorded at different points in the cavitation field.

In Fig. 3, an example of acoustic pressure on the axis of

the transducer is shown. A strong nonlinear distortion isto be noted. In order to establish the origin of this

strongly nonlinear behaviour and to verify the hypothe-

ses of linear excitation, simultaneous acquisition of

transducer vibration and pressure signals was continu-

ously done. In Fig. 4a comparison of the normalized

waveforms and spectral decomposition is presented. A

sinusoidal wave is obtained for the excitation while an

0.00E+0 2.00E-4 4.00E-4 6.00E-4 8.00E-4 1.00E-3

Time (s)

-2.00E+5

0.00E+0

2.00E+5

4.00E+5

6.00E+5

Pres

sure

(Pa)

Fig. 3. Measured waveshape at the axis (x = 4 cm from the source).

0 2 4 6

x 10-4

x 10-4

-2

-1.5

-1

-0.5

0

0.5

1

Time (s)

Nor

mal

ized

uni

ties

(a)

(b)0 1 2 3 4 5 6 7 8 9 10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (Hz)

Nor

mal

ized

uni

ties

Velocity at the transducerPressure

Velocity at the transducerPressure

Fig. 4. Comparison of the vibration velocity at the transducer and the pressure field: (a) waveshape and (b) spectral decomposition.

C. Campos-Pozuelo et al. / Ultrasonics Sonochemistry 12 (2005) 79–84 81

important harmonic distortion is observed for the pres-

sure wave. The pressure amplitude at the fundamental

frequency is chosen as reference pressure, in order,

firstly, to have a stable amplitude parameter, and sec-

ondly, to compare with the second order model devel-

oped in the following section. The mean acousticpressure is extracted using an adequate low-pass filtering

0.00 0.40 0.80 1.20 1.60 2.00

Time (s)

-80000.00

-60000.00

-40000.00

-20000.00

0.00

20000.00

Pres

sure

(Pa)

<p>

Fig. 5. Filtered signal corresponding to the mean pressure.

of the hydrophone signal. An example of filtered signal

is given in Fig. 5. The mean pressure is identified to the

signal value after the end of the excitation. In Fig. 6, the

amplitude of the fundamental and the mean acoustic

pressures before and after reaching the cavitation

threshold, are compared. In the absence of cavitation,

0.00 0.04 0.08 0.12 0.16 0.20Spatial coordinate (m)

0.00E+0

5.00E+4

1.00E+5

1.50E+5

2.00E+5

2.50E+5

Pres

sure

(Pa)

v=0.0525m/s. Fundamental amplitude

v=1.3133m/s. Fundamental amplitude

v=0.0525m/s.Averagedpressure

v=1.3133m/s. Averaged pressure

Fig. 6. Comparison of the fundamental and mean acoustic pressure

amplitudes before and after reaching the cavitation threshold.

0.00 0.40 0.80 1.20 1.60 2.00Excitation (m/s)

190.00

200.00

210.00

220.00

230.00

240.00

Pres

sure

(dB)

Fig. 7. Pressure amplitude evolution with the excitation.

82 C. Campos-Pozuelo et al. / Ultrasonics Sonochemistry 12 (2005) 79–84

a number of reflections appear in the reactor. The valuesof the mean pressure remain small, of the order of 1% of

the fundamental pressure. When the cavitation regime is

established the conelike bubble structure (CBS) appears

and the pressure field distribution and values are com-

pletely changed. An important mean pressure (42% of

the fundamental at the origin) is observed. In Fig. 7

the evolution of the pressure amplitude with excitation

is plotted. The strong attenuation and pressure satura-tion in the cavitation regime are clear.

In Fig. 8 the mean acoustic pressure distribution in

the cavitating fluid is shown for two transversal sections.

0.00 0.04 0.08 0.12Transversal spatial coordinate (m)

0.00

10000.00

20000.00

30000.00

40000.00

Pres

sure

(Pa)

v=1.4 m/sx=0.006 mx=0.044 m

Fig. 8. Mean acoustic pressure distribution for two transversal

sections.

The large mean acoustic pressure values, obtained only

in fluid areas where a large bubble concentration occurs,

indicate that this nonlinear phenomenon is associated to

the nonlinear vibration of bubbles. In fact, it is well

known that a bubbly liquid presents a strong nonlinear

acoustic behaviour and that a nonlinear acoustic waveproduces a mean pressure different to zero [1,5].

3. Theoretical interpretation

A simple model for linear and nonlinear wave propa-

gation is developed, on the basis of the Rayleigh–Plesset

expression for the nonlinear vibration of the bubble andthe conservation equations for the fluid. Linear behav-

iour of the fluid in the absence of bubbles is assumed,

i.e., the nonlinearity is only due to the nonlinear vibra-

tions of bubbles. Plane wave propagation is considered.

Then, the following equations describe the bubbly liquid

[5]:

o2pox2

� 1

c20

o2pot2

¼ �q0No2vot2

ð2Þ

€vþ dx0 _vþ x20vþ gp ¼ av2 þ bð2v€vþ _v2Þ ð3Þ

where p is acoustic pressure, c0 and q0 are, respectively,the sound speed and density of the liquid, N is the num-

ber of bubbles per unit volume, v the bubble volume, d isthe viscous damping coefficient, x2

0 ¼3cP0q0R

20

, R0 is the bub-

ble radius at the equilibrium, g ¼ 4pR0q0, a ¼ ðcþ1Þx2

0

2V 0and

b ¼ 16V 0, V 0 ¼ 4

3pR30. Eq. (2) results from the combination

of the moment and mass conservation laws. Eq. (3) de-

scribes the bubble dynamic in the Rayleigh–Plesset

approximation. Only second order terms are considered

in the bubble volume variations as well as in the pres-

sure. A perturbative technique is used to find the mean

acoustic pressure from Eqs. (2) and (3). Thus thesolution is assumed to be the linear solution pl (vl) plus

a second order perturbation p2 (v2):

p ¼ pl þ p2v ¼ vl þ v2

ð4Þ

Assuming also harmonic excitation the complex wavenumber k is easily obtained from Eqs. (2) and (3):

k ¼ k0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� q0Ngc20

ðx2 � x20 � jdxx0 � q0Ngc20Þ

s,ð5Þ

Assuming linear excitation, we obtain

hpi ¼ hp2i ¼Nða� bx2Þjv0j2e�2ax2ðx2

0 þ q0c20NgÞ 1þ x2

0

ðx20 þ 2q0c20NgÞ

� �

v0 ¼jgqc20ku0

ðx2 � x20 � jdxx0 � gNq0c

20Þ

ð6Þ

0.00 0.04 0.08 0.12Spatial coordinate (m)

0.00E+0

4.00E+4

8.00E+4

1.20E+5

1.60E+5

Pres

sure

(Pa)

Fundamental amplitude. Experimental

Mean pressure. Experimental

Theory

Fig. 9. Comparison between theoretical and experimental results.

Void fraction, NV0 = 0.175.

0.00 0.04 0.08 0.12 0.16Spatial coordinate (m)

0.00E+0

4.00E+5

8.00E+5

1.20E+6

1.60E+6

2.00E+6

Volu

met

ric F

orce

(N/m

3 ) Quasi-acoustic streaming

Genuine acoustic streaming

Fig. 10. Amplitude of the forces corresponding to genuine acoustic

streaming and quasi-acoustic streaming.

C. Campos-Pozuelo et al. / Ultrasonics Sonochemistry 12 (2005) 79–84 83

a being the imaginary part of the wave number in Eq.(5). Physically a corresponds to the acoustic attenuationof the cavitating medium.

In Fig. 9, theoretical and experimental values of fun-

damental (i.e. at driving frequency) and mean acoustic

pressures are compared, showing a good qualitative cor-

respondance. Nevertheless, this agreement is obtained

for bubble density NV0 of 0.175, this value being clearly

overestimated by this simple model. The strong nonlin-ear distortion observed in Fig. 3 is not explained by a

perturbation model. Moreover this perturbation model

does not take into account nonlinear attenuation, which

is clearly important in the cavitation field. If stronger

nonlinear attenuation is considered, the experimental

pressure amplitude gradient and mean acoustic pressure

values (Fig. 9) could be fitted with smaller values of bub-

ble density. The high values of Æpæ (of the order of a halfof the fundamental) can be obtained for a value of

NV0 = 2 · 10�5, but in this case the attenuation ob-tained by the model is really small, and does not justify

the decrease of the fundamental and the mean pressure.

For this strong nonlinearity, a really important nonlin-

ear attenuation is expected, giving easily a pressure de-

crease as experimentally observed for small values of a(see Fig. 7). Other limiting hypotheses in the modelare plane wave approximation (the geometrical diver-

gence can also contribute to this amplitude decrease)

and homogeneous bubble distribution in the cone. More

sophisticated numerical models, based in the same phys-

ical approach, are then needed to have a realistic charac-

terization of the bubble density in the fluid. Nevertheless

the qualitative behaviour experimentally observed is

correctly described and the physical interpretation cor-rect. We use this idea to give a new interpretation of

the origin of the acoustic streaming in a cavitating fluid.

The second order model described above has also been

used to calculate the magnitude of the forces generatingthe streaming. In the ‘‘genuine acoustic streaming’’, the

fluid motion is produced by the attenuation of the

acoustic wave during its propagation. The well-known

solution proposed by Nyborg [6] is used to calculate

the volume force generated in a fluid when a plane

acoustic wave propagates. In this approximation, the

linear propagation of the wave in the bubbly liquid is

considered to calculate the effective attenuation of thewave and the magnitude of the force. In the ‘‘quasi

acoustic streaming’’, fluid motion results directly from

the gradient of the mean acoustic pressure [7]. In Fig.

10 the magnitude of the volume forces generating each

type of streaming (genuine acoustic streaming and qua-

si-acoustic streaming) are compared for the values in

Fig. 9. It appears clearly that the force associated to

the fluid nonlinearity (due to the nonlinear vibrationof bubbles and origin of the quasi-acoustic streaming)

is much larger than the force associated to attenuation

increase (origin of the genuine acoustic streaming) when

cavitation turns on. The strong fluid streaming observed

in the cavitation field mainly originates in acoustic wave

nonlinearity.

4. Conclusion

The cavitation field of a low frequency sonotrode is

analyzed from the nonlinear acoustic point of view. Be-

cause of the strong nonlinear distortion of the acoustic

field, it is not easy to have direct evaluation of the acous-

tic parameters. In this paper, it is proposed to take as

the reference the pressure amplitude at fundamentalfrequency for the linear part, and the mean acoustic

84 C. Campos-Pozuelo et al. / Ultrasonics Sonochemistry 12 (2005) 79–84

pressure for the nonlinear distortion. A new experimen-

tal procedure is presented to evaluate the mean acoustic

pressure inside the fluid. A simple perturbation model

based on the Rayleigh–Plesset equation for the bubbles

vibration and linear fluid mechanics is also developed.

The qualitative comparison between theoretical andexperimental data indicates that the proposed physical

approach is correct. Limitations of the model and need

for more sophisticated numerical models are discussed.

Acoustic streaming in a cavitating fluid is shown to re-

sult from bubbly fluid acoustic nonlinearity rather than

attenuation.

Acknowledgment

This work was supported by CNRS and CSIC

(French-Spanish cooperation project 2002FR0004),

URJC (Rey Juan Carlos University) No. PPR-2003-46

and the European Union (Feder 2).

References

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Acoustics, Cambridge University Press, Cambridge, 1998.

[2] N. Gondrexon, V. Renaudin, C. Petrier, M. Clement, P. Boldo, Y.

Gonthier, A. Bernis, Ultrason. Sonochem. 5 (1998) 1–6.

[3] O. Dahlem, J. Reisse, V. Halloin, Chem. Eng. Sci. 54 (1999) 2829–

2838.

[4] A. Moussatov, C. Granger, B. Dubus, Ultrason. Sonochem. 10

(2003) 191–195.

[5] M.F. Hamilton, D.T. Blackstock (Eds.), Nonlinear Acoustics,

Academic Press, 1998.

[6] W.L. Nyborg, Acoustic streaming, in: W.P. Mason (Ed.), Physical

Acoustics, vol. 2B, Academic Press, 1965.

[7] H. Mitome, Acoustic streaming and nonlinear acoustics––research

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