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Buildup of acoustic streaming in focused beams
ELSEVIER
Ultrasonics 34 (1996) 763-765
Research
Note
Buildup of acoustic streaming Kazuhisa
Matsuda
aPhysics bDepartment
a, Tomoo
Division, The Koganei
of Electronic
Engineering,
Kamakura
Technical
in focused beams b**, Yoshiro Kumamoto
High School, Koganei,
b
Tokyo 184, Japun
The University of Electra-Communications,
Chofu,
Tokyo 182. Japan
Received 12 April 1996
Abstract Using a laser Doppler velocimeter, we measured focusing source with a circular aperture along and beam is switched on, and a maximum velocity of experiment suggests that the numerical calculation the present source conditions. Keywords:
Acoustic
streaming;
Focused
the buildup velocity of acoustic streaming in water generated from a 2.8 MHz across the acoustic axis. Steady streaming is established a few seconds after the about 4 cm/s is observed near the focus. The good agreement of theory and method previously developed by the authors is valid within the framework of
beam; KZK equation
1. Introduction The propagation of ultrasound in a viscous fluid induces mass flow in the beams. This phenomenon is known as acoustic streaming. Recently, Duck et al. have carried out some experiments to measure the streaming velocity along and across sound beams in a focused system [ 11. Unfortunately, they did not provide a theoretical comparison with their experiments. A theory does exist, however [2]. Velocity distributions in focused beams are calculated by applying the stream-function vorticity method to the solution of flow equations [2]_ The aim of the present report is to examine the validity of the theory by comparing it with refined experiments.
2. Sound beam profiles An ultrasonic transducer with a circular aperture and concave acoustic lens (diameter 2.1 cm, focal length 7.5 cm) was driven at 2.8 MHz in fresh water. To theoretically determine the velocity field of streaming, the spatial distribution of the sound pressure in the beams must be measured or predicted as exactly as possible. We used a needle type PVDF hydrophone, which has a diameter * Corresponding author. e-mail: [email protected]
of 0.6 mm and an almost uniform frequency response from 1 to 20 MHz, to detect the pressure locally. The hydrophone can be precisely and flexibly moved in the horizontal (axial), transverse, and vertical directions by a three-dimensional translation stage controlled by a personal computer. Figs. la and lb show measured axial pressure amplitude curves along and across the acoustic axis for the first three harmonic waves. The fundamental frequency is 2.8 MHz, hence the second and third harmonic frequencies are 5.6 MHz and 8.4 MHz. The symbols are all the measured data and the solid curves are the theoretical prediction based on the KZK equation [ 31. Fairly good agreement is observed throughout the propagation data. The fundamental pressure attains a maximum level of 239 dB (pressure amplitude corresponds to 1.26 x lo6 Pa) at 73 mm, just before the focus. The two harmonics also have maximum levels of 223 dB and 210 dB near the focus. The second harmonic level is 16 dB lower than the fundamental, and the third harmonic, 29 dB lower. The sound energy losses due to harmonic generation, which does not occur actively under the present source conditions, enhances the driving force of streaming. Since the force is almost proportional to the absorption coefficient (quadratic increase with frequency) and the intensity of sound [2], a simple estimation of (2 x lo- 16’20)2+ (3 x lo- 2g/20)2 predicts about 10% enhancement at the focus.
0041-624X/96/$15.00 Copyright 8 1996 Elsevier Science B.V. All rights reserved PI2 SOO41-624X(96)00074-1
K Matsuda et al. / Ultrasonics 34 (1996) 763-765
764
/
I
I
/
240 ndamental(2.8MHz)
z
220
s ti *
200
180 5 (4
10 15 Axial distance [cm] r
I
20 I
Fundamental
(b)
I
,
(2.8MHz
butyle rubber material at the end of the tube precludes sound reflections. The water around the cylinder is held at 23 “C by a thermostatic bath. Fig. 2 shows buildup profiles of streaming at various points on the axis. The velocity increases rapidly with time near the focus. In particular, the flow speeds up very quickly at the focus of 7.5 cm in a few seconds and attains an almost steady velocity of about 4 cm/s. As the authors pointed out earlier, the velocities at 12 and 15 cm in the postfocal region begin to increase after a certain time delay. The solid curves theoretically predicted by the previous numerical calculation method [2] agree satisfactorily with the experimental data indicated in symbols. On-axis velocity profiles are shown in Fig. 3 for different times. Since the measuring system cannot perfectly detect low flow velocity, experimental data in the early stage are not determined in the figure. Significant streaming is not found near the source. However, as the measurement point approaches the focus, the velocity increases abruptly and attains a maximum value somewhat outside the focus. The streaming profiles in the postfocal region change significantly with time due to mass flow out of the focal region.
Vertical distance [mm]
Fig. 1. Axial pressure curves (a) and beam patterns at 73 mm Tar the first three harmonics. Solid curves denote the theoretical prediction based on the KZK equation.
Overall good agreement of theory and experiment is again obtained for the transverse beam patterns of the three harmonics at 73 mm, as shown in Fig. lb. The pressure of the fundamental frequency component 4 mm off the axis decreases by about 30 dB relative to the axial pressure, and the second and third harmonics decrease more. The total -6 dB beamwidth of the fundamental is 1.8 mm, which is precisely three times broader than the hydrophone diameter. As the harmonic frequency increases, the beam becomes narrower. Consequent averaging in the hydrophone active area must be considered for accurate comparison of the measured harmonic pressures with theory.
10 Time [s] Fig. 2. Buildup characteristics of the streaming velocity at different observation points. Solid curves denote the theory previously developed by the authors [2].
4 3. Velocity profiles Flow velocity generated in the beams after radiating 2.8 MHz CW ultrasound was measured by using a laser Doppler velocimeter (LDV) under the same source conditions as the sound field measurements. We included polystyrene particles of mean diameter 1.6 urn in water as a tracer to enhance the Doppler shift signals. An acrylic cylinder of length 20 cm and inner diameter 2.1 cm is flush attached to the source surface to establish the Eckart-type streaming model. A sound absorber of
F E3 z z2 .s 0
Zl > 0 0 Fig. 3. On-axis
5 10 15 Axial distance [cm] velocity
profiles as a function
20 of time.
K. Matsuda et al. / Ultrasonics 34 (1996) 763-765
165
slight asymmetry is noticeable in measured profiles; velocities in the region above the axis are slightly higher than those below the axis. A similar trend is observed in the report of Duck et al. Further investigation is needed to clarify the reasons for such results.
4. Conclusions I
-3
I
I
I
I
I
-2
-1
0
1
2
I
3
Vertical distance [mm] Fig. 4. Velocity profiles in the focal plane curve is the acoustic intensity.
at 5 and 20 s. The dotted
Fig. 4 shows streaming profiles at 5 and 20 s in the focal plane. The small measuring volume formed by the interaction of two laser beams enables us to observe the flow velocity with a spatial resolution of 120 pm. This resolution seems to be sufficient to compare experiment with theory in the present focusing system. In contrast to the experimental studies by Starritt et al. [4], the overall flow profiles are broader than the acoustic intensity profile indicated by the dotted curve. Moreover, a
As shown in Fig. 3, the velocity slope changes abruptly just before the focus. At this distance the hydrodynamic nonlinearity must be included in a theoretical prediction. The present numerical calculation including nonlinearity in an appropriate manner agrees well the measured velocity profiles.
References Cl1 F.A.
Duck, S.A. MacGregor and D. Grennwell, in: Advances in Nonlinear Acoustics, Proc. 13th ISNA, Bergen, Ed. H. Hobrek, (World Scientific, Singapore, 1993) pp. 6077612. c21 T. Kamakura, K. Matsuda, Y. Kumamoto and M.A. Breazeale, J. Acoust. Sot. Am. 97 (1995) 2740. Nonlinear c31 B.K. Novikov, O.V. Rudenko and V.I. Timoshenko, Underwater Acoustics (AIP, New York, USA, 1987) Section 2.5. Ultrasound in c41 H.C. Starritt, F.A. Duck and V.F. Humphrey, Med. Biol. 15 (1989) 363.
Ultrasonics 34 (1996) 763-765
Research
Note
Buildup of acoustic streaming Kazuhisa
Matsuda
aPhysics bDepartment
a, Tomoo
Division, The Koganei
of Electronic
Engineering,
Kamakura
Technical
in focused beams b**, Yoshiro Kumamoto
High School, Koganei,
b
Tokyo 184, Japun
The University of Electra-Communications,
Chofu,
Tokyo 182. Japan
Received 12 April 1996
Abstract Using a laser Doppler velocimeter, we measured focusing source with a circular aperture along and beam is switched on, and a maximum velocity of experiment suggests that the numerical calculation the present source conditions. Keywords:
Acoustic
streaming;
Focused
the buildup velocity of acoustic streaming in water generated from a 2.8 MHz across the acoustic axis. Steady streaming is established a few seconds after the about 4 cm/s is observed near the focus. The good agreement of theory and method previously developed by the authors is valid within the framework of
beam; KZK equation
1. Introduction The propagation of ultrasound in a viscous fluid induces mass flow in the beams. This phenomenon is known as acoustic streaming. Recently, Duck et al. have carried out some experiments to measure the streaming velocity along and across sound beams in a focused system [ 11. Unfortunately, they did not provide a theoretical comparison with their experiments. A theory does exist, however [2]. Velocity distributions in focused beams are calculated by applying the stream-function vorticity method to the solution of flow equations [2]_ The aim of the present report is to examine the validity of the theory by comparing it with refined experiments.
2. Sound beam profiles An ultrasonic transducer with a circular aperture and concave acoustic lens (diameter 2.1 cm, focal length 7.5 cm) was driven at 2.8 MHz in fresh water. To theoretically determine the velocity field of streaming, the spatial distribution of the sound pressure in the beams must be measured or predicted as exactly as possible. We used a needle type PVDF hydrophone, which has a diameter * Corresponding author. e-mail: [email protected]
of 0.6 mm and an almost uniform frequency response from 1 to 20 MHz, to detect the pressure locally. The hydrophone can be precisely and flexibly moved in the horizontal (axial), transverse, and vertical directions by a three-dimensional translation stage controlled by a personal computer. Figs. la and lb show measured axial pressure amplitude curves along and across the acoustic axis for the first three harmonic waves. The fundamental frequency is 2.8 MHz, hence the second and third harmonic frequencies are 5.6 MHz and 8.4 MHz. The symbols are all the measured data and the solid curves are the theoretical prediction based on the KZK equation [ 31. Fairly good agreement is observed throughout the propagation data. The fundamental pressure attains a maximum level of 239 dB (pressure amplitude corresponds to 1.26 x lo6 Pa) at 73 mm, just before the focus. The two harmonics also have maximum levels of 223 dB and 210 dB near the focus. The second harmonic level is 16 dB lower than the fundamental, and the third harmonic, 29 dB lower. The sound energy losses due to harmonic generation, which does not occur actively under the present source conditions, enhances the driving force of streaming. Since the force is almost proportional to the absorption coefficient (quadratic increase with frequency) and the intensity of sound [2], a simple estimation of (2 x lo- 16’20)2+ (3 x lo- 2g/20)2 predicts about 10% enhancement at the focus.
0041-624X/96/$15.00 Copyright 8 1996 Elsevier Science B.V. All rights reserved PI2 SOO41-624X(96)00074-1
K Matsuda et al. / Ultrasonics 34 (1996) 763-765
764
/
I
I
/
240 ndamental(2.8MHz)
z
220
s ti *
200
180 5 (4
10 15 Axial distance [cm] r
I
20 I
Fundamental
(b)
I
,
(2.8MHz
butyle rubber material at the end of the tube precludes sound reflections. The water around the cylinder is held at 23 “C by a thermostatic bath. Fig. 2 shows buildup profiles of streaming at various points on the axis. The velocity increases rapidly with time near the focus. In particular, the flow speeds up very quickly at the focus of 7.5 cm in a few seconds and attains an almost steady velocity of about 4 cm/s. As the authors pointed out earlier, the velocities at 12 and 15 cm in the postfocal region begin to increase after a certain time delay. The solid curves theoretically predicted by the previous numerical calculation method [2] agree satisfactorily with the experimental data indicated in symbols. On-axis velocity profiles are shown in Fig. 3 for different times. Since the measuring system cannot perfectly detect low flow velocity, experimental data in the early stage are not determined in the figure. Significant streaming is not found near the source. However, as the measurement point approaches the focus, the velocity increases abruptly and attains a maximum value somewhat outside the focus. The streaming profiles in the postfocal region change significantly with time due to mass flow out of the focal region.
Vertical distance [mm]
Fig. 1. Axial pressure curves (a) and beam patterns at 73 mm Tar the first three harmonics. Solid curves denote the theoretical prediction based on the KZK equation.
Overall good agreement of theory and experiment is again obtained for the transverse beam patterns of the three harmonics at 73 mm, as shown in Fig. lb. The pressure of the fundamental frequency component 4 mm off the axis decreases by about 30 dB relative to the axial pressure, and the second and third harmonics decrease more. The total -6 dB beamwidth of the fundamental is 1.8 mm, which is precisely three times broader than the hydrophone diameter. As the harmonic frequency increases, the beam becomes narrower. Consequent averaging in the hydrophone active area must be considered for accurate comparison of the measured harmonic pressures with theory.
10 Time [s] Fig. 2. Buildup characteristics of the streaming velocity at different observation points. Solid curves denote the theory previously developed by the authors [2].
4 3. Velocity profiles Flow velocity generated in the beams after radiating 2.8 MHz CW ultrasound was measured by using a laser Doppler velocimeter (LDV) under the same source conditions as the sound field measurements. We included polystyrene particles of mean diameter 1.6 urn in water as a tracer to enhance the Doppler shift signals. An acrylic cylinder of length 20 cm and inner diameter 2.1 cm is flush attached to the source surface to establish the Eckart-type streaming model. A sound absorber of
F E3 z z2 .s 0
Zl > 0 0 Fig. 3. On-axis
5 10 15 Axial distance [cm] velocity
profiles as a function
20 of time.
K. Matsuda et al. / Ultrasonics 34 (1996) 763-765
165
slight asymmetry is noticeable in measured profiles; velocities in the region above the axis are slightly higher than those below the axis. A similar trend is observed in the report of Duck et al. Further investigation is needed to clarify the reasons for such results.
4. Conclusions I
-3
I
I
I
I
I
-2
-1
0
1
2
I
3
Vertical distance [mm] Fig. 4. Velocity profiles in the focal plane curve is the acoustic intensity.
at 5 and 20 s. The dotted
Fig. 4 shows streaming profiles at 5 and 20 s in the focal plane. The small measuring volume formed by the interaction of two laser beams enables us to observe the flow velocity with a spatial resolution of 120 pm. This resolution seems to be sufficient to compare experiment with theory in the present focusing system. In contrast to the experimental studies by Starritt et al. [4], the overall flow profiles are broader than the acoustic intensity profile indicated by the dotted curve. Moreover, a
As shown in Fig. 3, the velocity slope changes abruptly just before the focus. At this distance the hydrodynamic nonlinearity must be included in a theoretical prediction. The present numerical calculation including nonlinearity in an appropriate manner agrees well the measured velocity profiles.
References Cl1 F.A.
Duck, S.A. MacGregor and D. Grennwell, in: Advances in Nonlinear Acoustics, Proc. 13th ISNA, Bergen, Ed. H. Hobrek, (World Scientific, Singapore, 1993) pp. 6077612. c21 T. Kamakura, K. Matsuda, Y. Kumamoto and M.A. Breazeale, J. Acoust. Sot. Am. 97 (1995) 2740. Nonlinear c31 B.K. Novikov, O.V. Rudenko and V.I. Timoshenko, Underwater Acoustics (AIP, New York, USA, 1987) Section 2.5. Ultrasound in c41 H.C. Starritt, F.A. Duck and V.F. Humphrey, Med. Biol. 15 (1989) 363.