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Temperature gradient effects on acoustic and streaming velocities in standing acoustic waves resonator
Experimental Thermal and Fluid Science 66 (2015) 1–6
Contents lists available at ScienceDirect
Experimental Thermal and Fluid Science journal homepage: www.elsevier.com/locate/etfs
Temperature gradient effects on acoustic and streaming velocities in standing acoustic waves resonator Emeline Saint Ellier ⇑, Yannick Bailly, Laurent Girardot, David Ramel, Philippe Nika Energie department, FEMTO-ST Institute – Franche-Comté Electronic Mechanical Thermal et Optical-Sciences et Technologies, 2 avenue Jean Moulin, 90000 Belfort, France
a r t i c l e
i n f o
Article history: Received 16 December 2014 Received in revised form 9 March 2015 Accepted 9 March 2015 Available online 19 March 2015 Keywords: Temperature gradient Particle image velocimetry Acoustic Non-linearities Experimental investigation
a b s t r a c t This paper experimentally investigates the effects of an imposed axial temperature gradient in a cylindrical resonant cavity. The consequences on the acoustic velocity and acoustic streaming velocity are explored. The synchronised PIV (stands for Particle Image Velocimetry) technique was used in a 7 m long-standing wave air-filled acoustic resonator for the measurements. This method enabled the quantification of the changes in the velocity amplitudes due to the temperature gradient and highlighted the formation of one sole vortex in place of two Rayleigh vortices usually observed when acoustic streaming occurs. Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction Nonlinear phenomena can occur in a standing wave acoustic resonator if the system delivers high amplitude acoustic waves. These phenomena may distort originally harmonic waves and transform acoustic energy into higher harmonic components, which increase the dissipation of acoustic energy. They may also lead to the formation of a system of rotational cells that originates from within the boundary layers [1]. In this particular case, the interaction of the high amplitude acoustic waves with the walls of the resonator creates this flow, which is called Rayleigh streaming [2]. It appears as two symmetrical torus-shaped cells extending generally along a length of a quarter wavelength in the central part of the acoustic guide. While the acoustic wave is an oscillating wave, the Rayleigh streaming is a second-order non-oscillatory mean flow induced by the nonlinearities of the acoustic propagation inside the resonator. It is quasi-stationary and will be superimposed on the main acoustic wave. In recent years, studies on acoustic streaming found a revival with works related to thermoacoustics (whose works from Rott [3] and Swift [4] are considered as reference). In thermoacoustic processes, the reverse conversion between the thermal and acoustic energy is featured within an acoustic resonator. The systems use an environmentally friendly working medium (noble gas) to carry out a thermodynamic cycle, which facilitates the generation ⇑ Corresponding author. Tel.: +33 384578236; fax: +33 384570032. E-mail address: [email protected] (E. Saint Ellier). http://dx.doi.org/10.1016/j.expthermflusci.2015.03.007 0894-1777/Ó 2015 Elsevier Inc. All rights reserved.
or spending of acoustic energy [5]. The interaction between a temperature gradient set locally in the resonator, inside a porous medium, and the acoustic waves allow us to distinguish two operating functions in thermoacoustic machines: the receiver (heat pump or refrigerator) or prime-mover (motor). Nevertheless, this new technology, which is actually booming requires further development. New architectures machines (whose size is still important), a more detailed description of the temperature field established along the porous medium or a description of heat transfer within the machine are all subjects of study and improvement, where experimentation remains important. This remains true for this next point which is the subject of much attention from the scientific community in the thermoacoustic field. Various sources of energy dissipation [6] due to nonlinear effects are presently the leading cause of degradation in the machine performances of thermoacoustic systems. An important class of these non-linear effects is streaming flow [7]. Although already known and studied by the acoustic community [8,9], these effects and their interaction with the temperature gradient yet need further analysis through various observations. Despite several theoretical studies in this area, they sometimes lack detailed information since the evaluations of these effects, which among other things are characterised by the appearance of Rayleigh streaming, is not elementary, as these second-order phenomena generally lead to tricky situations where interactions with a temperature gradient and couplings between the different effects encountered are prominent. Hence the importance of experimental studies.
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Numerous experimental studies of Rayleigh-type streaming have been performed recently. However few experimental studies on nonlinear acoustic phenomena and the influence of an imposed temperature gradient have been published. Thompson et al. [10] studied the influence of an axial thermal gradient imposed on the entire resonator via measurements made with Laser Doppler Velocimetry (LDV) in a standing wave resonator. Increasing the temperature gradient (the maximum value is up to 8 K/m) distorts the Rayleigh cells, although the results do not match any existing theories. These researchers had previously analysed the acoustic velocity and streaming velocity within the same setup and showed that changes in the viscosity as a result of temperature in the fluid influence the amplitude of the streaming [11]. Nabavi et al. [12,13] were one of the first to observe the effects of a locally set transverse temperature gradient on acoustic Rayleigh wind via PIV measurements. They followed the disappearance of one of two vortices of the Rayleigh cell and the dominance of the second as the gradient increased, which led to the formation of a single vortex. Aktas and Ozgumus [14] numerically investigated the same subject and particularly studied the effects of acoustic streaming on thermal convection in an acoustic enclosure with differentially heated horizontal walls. Once again, the transverse temperature gradient strongly affected the acoustic streaming structures and velocities. They found that acoustic streaming enhanced the heat transfer, a conclusion also reached by Tajik et al. [15] from their experiments on a closed cylindrical enclosure filled with water. Daru et al. [16] numerically investigated the mean temperature evolution associated with the streaming motion for high intensity waves in the nonlinear streaming regime. In this paper, the influence of an axial temperature gradient on Rayleigh streaming flows in a standing acoustic wave resonator was first considered and is investigated via synchronised PIV measurements. The objective was to observe the behaviour of streaming when a temperature gradient is locally imposed in the resonator, where a high level of sound wave propagates. The measurements were focused in the area in which the temperature gradient is set. Section 2 of this paper presents the experimental PIV setup along with the associated PIV methodology and processing. Section 3 shows results for the study of the evolution of streaming with the drive ratio and the consequences on the acoustic streaming of the establishment of a temperature gradient inside the standing wave resonator. 2. Experimental setup and methodology 2.1. Experimental apparatus
resonator and PIV instrumentation) was previously used in [17] where more details can be found. The experimental investigation was performed on an air-filled cylindrical resonator that was 7 m in length (L = 7.03 m), made of stainless steel closed at one of its ends and equipped with an acoustic driver at the other. The inner diameter of the resonator was d = 56.3 mm. A transparent square section, which fits the resonator without modifying the cross section area, facilitated the optical measurements (see again [17] for its description and how it fits the resonator). Two heat exchangers were inserted in the resonator on either side of the PIV cell (i.e., the transparent section), thereby establishing a temperature gradient in the measuring cell, as shown in Fig. 1. These liquid–gas heat exchangers (water/air) are based on the shell and tube heat exchanger technology. More specifically, they are 1-pass straight-tube heat exchangers. The temperature gradient set between both exchangers is 86 °C/m: the hot heat exchanger, within which water circulates at 63 °C, was situated approximately 1.4 m from the resonator’s end. The cold heat exchanger is cooled by cold water flowing at 3 °C. A shaker (LDS – Ling Dynamic Systems – V450/1 – PA 500L) equipped with a piston whose diameter matched the internal diameter of the resonator was used as the acoustic source. It was chosen because of its ability to reach high amplitudes. Besides its working conditions maximise the possible single-frequency and thus do not affect the measurements via parasitical acoustic effects (i.e., harmonic frequencies). The superior harmonics amplitude reaches only 6% of that of the resonance frequency and therefore can be neglected. To vary the drive ratio (Dr), which is defined as the ratio of the maximum amplitude of the acoustic pressure to the mean pressure, the piston stroke is changed while the frequency remains the same. The system is tuned at a frequency of f = 24.4 Hz, which is the resonance frequency, the first mode of the system. This frequency was calculated for our experimental conditions and experimentally verified. An Nd YAG laser of 200 mJ pulse (PIV 190 PS1/TwinsBSL Quantel) at a wavelength of 532 nm combined with spherical and cylindrical optical components was used to generate the laser sheet. A prism was used to deflect the laser sheet into the vertical xy plane of the PIV cell. The image was then acquired with a CCD (Charge Couple Device) camera (TSI PIvCam 13-8) of 1024 1248 pixels. The camera was connected to a synchroniser (LASERPULSE Synchroniser – TSI Model 610034), which allowed the image acquisition to be triggered with the control signal of the shaker at a rate of 3.63 Hz. An aerosol generator (TOPAS ATM 210) was used to generate the seeding mist. The passive tracers are particles of DEHS (Di-2-EthylhexylSebacat). These particles have relaxation time of about 0.3 ls, which set the Stokes number to about 2 105.
St ¼ In this section, the experimental setup used to investigate the influence of an axial temperature gradient on the acoustic Rayleigh streaming is developed. The whole system (acoustic
su d
ð1Þ
where s is the particle relaxation time; u is the fluid velocity; and d is the resonator diameter.
Fig. 1. Zoom in: measurements configurations and setup.
E. Saint Ellier et al. / Experimental Thermal and Fluid Science 66 (2015) 1–6
3
Fig. 2. Experimental apparatus and PIV instrumentation setup.
The size of the droplets produced was on the order of 0.3 m, and the droplets completely evaporated after 4 h, which prevented clogging. Several valves were used to control the seeding and its injection into the resonator. Fig. 2 shows an exploded view of the PIV instrumentation setup. Following works from Moreau et al. [8] and Debesse et al. [18], velocity measurements are performed 30 min after the acoustic source is switched on to ensure steady state is reached and 10 min after seeding to ensure that the particles are well homogenised.
To evaluate the acoustic streaming, u2, and to monitor the acoustic cycle as closely as possible, velocity data collected at only one phase u is not enough. One must analyse the entire acoustic cycle. To this end, the acoustic period is divided into time lapses whose width is carefully chosen. It has to be sufficiently short to minimize uncertainties resulting from – the PIV algorithm where the oscillating acoustic flow displays a velocity inversion, – the post processing calculation (integral calculus in Eq. (4)).
2.2. PIV methodology The synchronised PIV technique was chosen because it provides an overall mapping of the flow with good spatial resolution. The 2D field [19] makes it more appropriate for the detection of vortex cells, such as those observed in Rayleigh streaming. 1D data can still be extracted and used to compare data at a given location. Conventionally, the Particle Image Velocimetry measurement method is well suited for measuring acoustic velocities as well as non-linear effects, such as acoustic streaming. The PIV has a history of being used in acoustics; for example Debesse [18], Reyt [20], Nabavi [21] and Saint Ellier [22] used it to measure acoustic streaming in a closed enclosure. The streaming velocity is a second order quantity; it is about two orders of magnitude smaller than the acoustic velocity and therefore particularly difficult to measure. To overcome these difficulties a particular methodology was developed. This is what next paragraphs describe: the method followed to extract acoustic, streaming and free convection velocities. Originally two methods were considered. Only the chosen one is described here. (To have more details about the 2nd way, please refer to [17] from the same authors.) First, via PIV acquisition, to ensure convergence of the streaming velocity field calculation, a series of instantaneous pairs of images are recorded at a given phase u with a time separation between laser pulses of Dt. This phase synchronisation with the acoustic signal is achieved thanks to the synchroniser. The velocity field can be deduced from the particle displacement between images of a pair. Then the calculated velocity fields at phase u are averaged to minimize errors.
Conversely numerous phases with very short spaces would increase the post processing. When the number of phases is chosen, each phase ui is studied with the same applied PIV post-processing as before. The PIV technique is primarily a non-intrusive method. However, it is indirect because the velocity measured is that of the particles (which is assumed to be the flow velocity). Applied to acoustic flows with streaming, this velocity can be decomposed into a term related to the acoustic oscillation, ua, and another one related to the streaming, u2, as follows:
uðx; y; tÞ ¼ ua ðx; y; tÞ þ u2 ðx; yÞ
ð2Þ
With the additional presence of a temperature gradient, we first assume that the velocity related to the natural convection, uc, superimposes on u(x, y, t) such that
uðx; y; tÞ ¼ ua ðx; y; tÞ þ u2 ðx; yÞ þ uc ðx; yÞ
ð3Þ
The integral of u(x,y,ui) over one period of the acoustic signal using the formula (4) gives :
u2 ðx; yÞ þ uc ðx; yÞ ¼
n1 1 X uðx; y; ui Þ ðtuiþ1 t ui Þ T ac i¼1
ð4Þ
where n = 17 is the number of phases over which the acoustic period Tac is divided, and tui is the time corresponding to the phase ui. If uc is known, the acoustic velocity u2 can be plotted from Eq. (4). A temporal representation of the particle velocity on a time period can be deduced by averaging each fields ui using the formula
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(5) where X is the number of points along the x coordinate and Y the number of points along the y coordinate.
u1 ðui Þ ¼
Y X X 1 1X uðx; y; ui Þ Y X y¼1 x¼1
ð5Þ
3. Results and discussion 3.1. Experimental conditions The experiments were performed at atmospheric pressure and at the resonance frequency of 24.4 Hz of the resonator filled with air. The transparent section where the measurements were carried out was located between a node and an antinode of the acoustic wave at 4L/15 (i.e. 5.6 m from the acoustic source) whose location was determined in previous work [22] from a calculated acoustic model of the system. Drive ratios from 1% to 4% were obtained. Seven test conditions were carried out: measurements were performed with and without the temperature gradient for three different drive ratios. A final test (test 4) was carried out to study the natural convection inside the measured section: no flow was imposed within the resonator. The experimental conditions are summarised in Table 1. The drive ratio was obtained from the measurements of a pressure transducer located at the resonator’s plug (see Fig. 1). 3.2. Results The experimental tests results yielded two main observations. First, Fig. 3, which shows the results of tests 1 through tests 3, indicates that the thermal gradient has little influence on the temporal Table 1 Operating conditions.
Test Test Test Test Test Test Test
1-1 1-2 2-1 2-2 3-1 3-2 4
P0 – mean pressure (bar)
Pa – acoustic pressure (mbar)
Drive ratio (%)
T0 (°C)
Temperature gradient
1.014 1.014 1.016 1.016 1.026 1.026 1.020
29 29 37 37 65 65 0
1.4 ± 12 % 1.4 ± 12 % 1.8 ± 9 % 1.8 ± 9 % 3.3 ± 6 % 3.3 ± 6 % 0
24.4 24.4 24.4 24.4 25 25 24.6
86 °C/m No 86 °C/m No 86 °C/m No 86 °C/m
variation of the axial velocity. The uncertainties associated with the PIV method correspond to a fraction of pixel in the order of two hundredths [19]. Given these uncertainties (see the corresponding error bars), both curves plotted for each test (with and without thermal gradient) are almost identical. The PIV measurements are compared to the theoretical acoustic sine profile which was calculated from the measured acoustic amplitude. The curves show little distortion due to the nonlinearities and propagations of the superior harmonics. When an acoustic wave propagates within the resonator, Rayleigh streaming is assumed to superimpose on the natural convection phenomenon (Eq. (3)). The quasi stationary flow velocity defined in Eq. (4) can be plotted. The three profiles resulting from this superposition are shown in Fig. 4. The thermo-convective flow is superimposed on the acoustic flow for tests 1-1, 2-1 and 3-1. The fourth curve (Dr = 0%) is the velocity profile resulting from the horizontal convective mass transfer only observed between the heat exchangers within the measurement cell without the presence of an acoustic wave. The temperature gradient affects the density of the fluid leading to a bulk fluid motion where the heavier components (cold fluid) fall, while lighter components (warm fluid) rise. It creates a convective rotational movement. Fig. 5, which represents the corresponding velocity field shows this particular motion. The maximum longitudinal velocity is 3.5 cm/s. Interestingly here the increase of the drive ratio values shows two physical phenomena: (1) it modifies the velocity profile of the convective flow by reducing its velocity amplitude and (2) it distorts the convection cell until forming a new structure, one cell with rotational direction opposite to the previous one. This new structure is illustrated in Fig. 6. Vorticity values obtained from this 2D velocity field do not exceed 6 102 s1. However these values should be considered with caution because the flow is actually strongly 3D and these velocity fields are obtained after a heavy post processing. In test 1-1, the velocity profiles of the quasi stationary flow and natural convective flow are qualitatively identical. As the drive ratio increases, the profile deforms (test 2-1). At a higher drive ratio (here, 3.3%), the phenomenon of streaming becomes dominant in the resonator and the temperature gradient imposed around the measuring cell ratio carries little influence. The work of Nabavi et al. [12] presents similitudes in their results [12], even though they studied a transversal temperature gradient. Unlike the present work where the drive ratio varies and the temperature gradient is imposed, they imposed a drive
5
Axial acoustic velocity ua [m/s]
4 3 2 1 0 -1 -2 -3
Dr=1,4% Dr=1,8% Dr=3,3%
-4 -5
0
10
20
30
40
50
60
70
80
90
Fig. 3. Temporal variations of the axial acoustic velocity for three drive ratios. The 4 correspond to no gradient cases and s to gradient cases. The dashed lines are the theoretical acoustic results.
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Dr=0% Dr=1,4% Dr=1,8% Dr=3,3%
30 20
r [mm]
10
-0,06
-0,04
-0,02
0
0,02
0,04
0,06
0 0,08 -10 -20 -30
Mean flow velocity [m/s] Fig. 4. Profiles of the mean flow velocity of test 1-1 (blue), 2-1 (orange) and 3-1 (purple) compared to the velocity profile of natural convection. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 6. Streaming velocity field for test 3-1.
30 20
r [mm]
10
-0,08
-0,06
-0,04
-0,02
0
0,02
0,04
0,06
0,08
0 0,1 -10 -20
Streaming velocity u 2 [m/s]
-30
Fig. 7. Comparisons of the streaming velocity profiles with (s) and without (—) temperature gradient. Blue: Dr = 1.4%, orange: Dr = 1.8%, violet: Dr = 3.3%. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Fig. 5. Bulk fluid motion velocity field resulting from the convective heat mass.
ratio of 0.8% avec varied the temperature gradient. However similitudes can be found. Their temperature gradient values correlated with the dominance of the convection phenomenon and deformation of streaming, whereas in our present work the drive ratio directly correlates with the dominance of streaming and deformation of the convection cell. At low drive ratio (0.8% for [12] and until 1.8% here), the profiles show one velocity cell resulting from the strong influence of the imposed temperature gradient. Beyond a given drive ratio (which could be around 1.8% in this work) the streaming becomes dominant over the convective flow. In some references the distinction between the previous acoustic-thermo-convective flow and acoustic streaming is not made. What is called here ‘‘stationary flow’’ or ‘‘mean flow’’, is often referred as ‘‘acoustic streaming’’ only (example in [12]) although they study the consequences of a combination between acoustic streaming and convective mass transfer. If we want to study the strict contribution of the acoustic streaming, given the hypothesis expressed for Eqs. (3) and (4), by
subtracting uc to Eq. (4), it is possible to deduce the streaming velocity profiles. The acoustic streaming velocity profiles of tests 1-1, 2-1 and 3-1 are plotted against streaming velocity profiles of tests 1-2, 2-2 and 3-2 in Fig. 7. If the convective flow and the acoustic streaming were dissociated, identical curves would be expected for a given drive ratio since only the sole contribution of acoustic streaming is studied here. However, even taking the PIV uncertainties into account, differences in the amplitude ranging from 10% to 100% were observed. These differences inversely increased with the drive ratio. The temperature gradient is intricately linked with the establishment of the streaming. These results show that the assumption made for Eqs. (3) and (4) is not suitable for the description of Rayleigh streaming in the presence of a temperature gradient. Indeed coupling phenomena are not considered in this equation. The superimposition of streaming on the natural convection must be assumed with care. Results of Fig. 7 show that strong coupling exists between the temperature gradient and the acoustic streaming within the resonator.
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E. Saint Ellier et al. / Experimental Thermal and Fluid Science 66 (2015) 1–6
4. Conclusions This study is part of an experimental exploration phase of acoustic streaming in thermoacoustic systems. The study of an axial temperature gradient localised around the measured section showed that the interaction between acoustic streaming and the temperature gradient is complex and most likely leads to strong coupling phenomena that were not taken into account with the assumption made at the beginning of this paper. The simple superimposition of acoustic streaming and free convection was insufficient. However, the measurement first revealed the formation of a sole streaming vortex whose direction of rotation seemed to be opposite to that of the free convection. This difference should be verified by performing experiments with gradients established in the opposite direction. Interestingly, the drive ratio directly correlated with the dominance of streaming, as the velocity profiles of the given mean flow and natural convection were qualitatively identical. Experiments performed with a higher temperature gradient would yield additional results to pursue this avenue of research. This work will be continued, as this area is crucial to improving the efficiency of thermoacoustic machines (e.g., effect of secondary flows and harmonics). Acknowledgment The experimental work was partially supported by the European Union under the FP7. Contract No. 226415 – www.thatea.eu. References [1] [2] [3] [4]
N. Riley, Acoustic streaming, Theor. Comp. Fluid Dyn. 10 (1998) 349–356. Lord. Rayleigh, The Theory of Sound, MacMillan, London, 1896. N. Rott, Thermoacoustics, Adv. Appl. Mech. 20 (1980) 135–175. G.W. Swift, A unifying perspective for some engines and refrigerators, Acoust. Soc. Am. through the American Institute of Physics, 2002. [5] T.D. Rossing, F. Dunn, W.M. Hartmann, D.M. Campbell, N.H. Fletcher, Thermoacoustics, in: Handbook of Acoustics, Springer, New York, 2007. 239–255.
[6] S.-Y. Xiao, S. Tao, M.-C. Qiu, H. Ge, L. Fan, S.-Y. Zhang, H. Zhang, Acoustic non linearities in themoacoustic refrigerators driven by loudspeakers, in: 19th International Congress of Sound and Vibration (ICSV19), Vilnius, Lithuania, July 2012. [7] M.J. Lighthill, Waves in Fluids, Cambridge University Press, Cambridge, 1978. [8] S. Moreau, H. Bailliet, J.-C. Valière, Measurements of inner and outer streaming vortices in a standing waveguide using laser doppler velocimetry, J. Acoust. Soc. Am. 123 (2) (2008) 640–647. [9] J.P. Sharpe, C.A. Greated, C. Gray, D.M. Campbell, The measurements of acoustic streaming using particle image velocimetry, Acustica 68 (1989) 168–172. [10] M.W. Thompson, A.A. Atchley, M.-J. Maccarone, Influence of a temperature gradient and fluid inertia on acoustic streaming in a standing wave, J. Acoust. Soc. Am. 117 (4) (2005) 1839–1849. [11] M.W. Thompson, A.A. Atchley, Simultaneous measurements of acoustic and streaming velocities in a standing wave using laser Doppler anemometry, J. Acoust. Soc. Am. 117 (4) (2005) 1828–1838. [12] M. Nabavi, K. Siddiqui, J. Dargahi, Effects of transverse temperature gradient on acoustic and streaming velocity fields in a resonant cavity, Appl. Phys. Lett. 93 (2008) 5. [13] M. Nabavi, K. Siddiqui, J. Dargahi, Influence of differentially heated horizontal walls on the streaming shape and velocity in a standing wave resonator, Int. Commun. Heat Mass 35 (2008) 1061–1064. [14] M.K. Aktas, T. Ozgumus, The effects of acoustic streaming on thermal convection in an enclosure with differentially heated horizontal walls, Int. J. Heat Mass Transfer 53 (2010) 5289–5297. [15] B. Tajik, A. Abbassi, M. Saffar-Avval, A. Abdullah, H. Mohammad-Abadi, Heat transfer enhancement by acoustic streaming in a closed cylindrical enclosure filled with water, Int. J. Heat Mass Transfer 60 (2013) 230–235. [16] V. Daru, D. Baltean-Carlès, C. Weisman, P. Debesse, G. Gandikota, Twodimensional numerical simulations of nonlinear acoustic streaming in standing waves, Wave Motion 50 (2013) 955–963. [17] E. Saint Ellier, W. Kdous, L. Girardot, D. Ramel, P. Nika, Acoustic streaming measurements in standing wave resonator using particle image velocimetry, Wave Motion, http://dx.doi.org/10.1016/j.wavemoti.2014.07.011, 2014. [18] P. Debesse, D. Baltean-Carlès, F. Lusseyran, M-X. François, Experimental analysis of nonlinear phenomena in a thermoacoustic system, in: 18th International Symposium on Nonlinear Acoustics (ISNA), July 2008. pp. 355– 358, DOI: http://dx.doi.org/10.1063/1.2956229. [19] M. Raffel, C. Willert, J. Kompenhans, Particule Image Velocimetry – A practical guide, Springer. ISBN 3-540-63683-8. [20] I. Reyt, H. Baillet, J.-C. Valière, Experimental investigation of acoustic streaming in a cylindrical wave guide up to high streaming Reynolds numbers, J. Acoust. Soc. Am. 135 (1) (2014) 27–37. [21] M. Nabavi, K. Siddiqui, J. Dargahi, Measurement of the acoustic velocity field of non linear standing waves using the synchronized PIV technique, Exp. Therm. Fluid Sci. 33 (2008) 123–131. [22] E. Saint Ellier, Y. Bailly, L. Girardot, D. Ramel, P. Nika, Acoustic streaming measurements by PIV, in: 15th International Symposium on Flow Visualisation (ISFV15), Minsk, Bielorussia, June 2010.
Contents lists available at ScienceDirect
Experimental Thermal and Fluid Science journal homepage: www.elsevier.com/locate/etfs
Temperature gradient effects on acoustic and streaming velocities in standing acoustic waves resonator Emeline Saint Ellier ⇑, Yannick Bailly, Laurent Girardot, David Ramel, Philippe Nika Energie department, FEMTO-ST Institute – Franche-Comté Electronic Mechanical Thermal et Optical-Sciences et Technologies, 2 avenue Jean Moulin, 90000 Belfort, France
a r t i c l e
i n f o
Article history: Received 16 December 2014 Received in revised form 9 March 2015 Accepted 9 March 2015 Available online 19 March 2015 Keywords: Temperature gradient Particle image velocimetry Acoustic Non-linearities Experimental investigation
a b s t r a c t This paper experimentally investigates the effects of an imposed axial temperature gradient in a cylindrical resonant cavity. The consequences on the acoustic velocity and acoustic streaming velocity are explored. The synchronised PIV (stands for Particle Image Velocimetry) technique was used in a 7 m long-standing wave air-filled acoustic resonator for the measurements. This method enabled the quantification of the changes in the velocity amplitudes due to the temperature gradient and highlighted the formation of one sole vortex in place of two Rayleigh vortices usually observed when acoustic streaming occurs. Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction Nonlinear phenomena can occur in a standing wave acoustic resonator if the system delivers high amplitude acoustic waves. These phenomena may distort originally harmonic waves and transform acoustic energy into higher harmonic components, which increase the dissipation of acoustic energy. They may also lead to the formation of a system of rotational cells that originates from within the boundary layers [1]. In this particular case, the interaction of the high amplitude acoustic waves with the walls of the resonator creates this flow, which is called Rayleigh streaming [2]. It appears as two symmetrical torus-shaped cells extending generally along a length of a quarter wavelength in the central part of the acoustic guide. While the acoustic wave is an oscillating wave, the Rayleigh streaming is a second-order non-oscillatory mean flow induced by the nonlinearities of the acoustic propagation inside the resonator. It is quasi-stationary and will be superimposed on the main acoustic wave. In recent years, studies on acoustic streaming found a revival with works related to thermoacoustics (whose works from Rott [3] and Swift [4] are considered as reference). In thermoacoustic processes, the reverse conversion between the thermal and acoustic energy is featured within an acoustic resonator. The systems use an environmentally friendly working medium (noble gas) to carry out a thermodynamic cycle, which facilitates the generation ⇑ Corresponding author. Tel.: +33 384578236; fax: +33 384570032. E-mail address: [email protected] (E. Saint Ellier). http://dx.doi.org/10.1016/j.expthermflusci.2015.03.007 0894-1777/Ó 2015 Elsevier Inc. All rights reserved.
or spending of acoustic energy [5]. The interaction between a temperature gradient set locally in the resonator, inside a porous medium, and the acoustic waves allow us to distinguish two operating functions in thermoacoustic machines: the receiver (heat pump or refrigerator) or prime-mover (motor). Nevertheless, this new technology, which is actually booming requires further development. New architectures machines (whose size is still important), a more detailed description of the temperature field established along the porous medium or a description of heat transfer within the machine are all subjects of study and improvement, where experimentation remains important. This remains true for this next point which is the subject of much attention from the scientific community in the thermoacoustic field. Various sources of energy dissipation [6] due to nonlinear effects are presently the leading cause of degradation in the machine performances of thermoacoustic systems. An important class of these non-linear effects is streaming flow [7]. Although already known and studied by the acoustic community [8,9], these effects and their interaction with the temperature gradient yet need further analysis through various observations. Despite several theoretical studies in this area, they sometimes lack detailed information since the evaluations of these effects, which among other things are characterised by the appearance of Rayleigh streaming, is not elementary, as these second-order phenomena generally lead to tricky situations where interactions with a temperature gradient and couplings between the different effects encountered are prominent. Hence the importance of experimental studies.
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E. Saint Ellier et al. / Experimental Thermal and Fluid Science 66 (2015) 1–6
Numerous experimental studies of Rayleigh-type streaming have been performed recently. However few experimental studies on nonlinear acoustic phenomena and the influence of an imposed temperature gradient have been published. Thompson et al. [10] studied the influence of an axial thermal gradient imposed on the entire resonator via measurements made with Laser Doppler Velocimetry (LDV) in a standing wave resonator. Increasing the temperature gradient (the maximum value is up to 8 K/m) distorts the Rayleigh cells, although the results do not match any existing theories. These researchers had previously analysed the acoustic velocity and streaming velocity within the same setup and showed that changes in the viscosity as a result of temperature in the fluid influence the amplitude of the streaming [11]. Nabavi et al. [12,13] were one of the first to observe the effects of a locally set transverse temperature gradient on acoustic Rayleigh wind via PIV measurements. They followed the disappearance of one of two vortices of the Rayleigh cell and the dominance of the second as the gradient increased, which led to the formation of a single vortex. Aktas and Ozgumus [14] numerically investigated the same subject and particularly studied the effects of acoustic streaming on thermal convection in an acoustic enclosure with differentially heated horizontal walls. Once again, the transverse temperature gradient strongly affected the acoustic streaming structures and velocities. They found that acoustic streaming enhanced the heat transfer, a conclusion also reached by Tajik et al. [15] from their experiments on a closed cylindrical enclosure filled with water. Daru et al. [16] numerically investigated the mean temperature evolution associated with the streaming motion for high intensity waves in the nonlinear streaming regime. In this paper, the influence of an axial temperature gradient on Rayleigh streaming flows in a standing acoustic wave resonator was first considered and is investigated via synchronised PIV measurements. The objective was to observe the behaviour of streaming when a temperature gradient is locally imposed in the resonator, where a high level of sound wave propagates. The measurements were focused in the area in which the temperature gradient is set. Section 2 of this paper presents the experimental PIV setup along with the associated PIV methodology and processing. Section 3 shows results for the study of the evolution of streaming with the drive ratio and the consequences on the acoustic streaming of the establishment of a temperature gradient inside the standing wave resonator. 2. Experimental setup and methodology 2.1. Experimental apparatus
resonator and PIV instrumentation) was previously used in [17] where more details can be found. The experimental investigation was performed on an air-filled cylindrical resonator that was 7 m in length (L = 7.03 m), made of stainless steel closed at one of its ends and equipped with an acoustic driver at the other. The inner diameter of the resonator was d = 56.3 mm. A transparent square section, which fits the resonator without modifying the cross section area, facilitated the optical measurements (see again [17] for its description and how it fits the resonator). Two heat exchangers were inserted in the resonator on either side of the PIV cell (i.e., the transparent section), thereby establishing a temperature gradient in the measuring cell, as shown in Fig. 1. These liquid–gas heat exchangers (water/air) are based on the shell and tube heat exchanger technology. More specifically, they are 1-pass straight-tube heat exchangers. The temperature gradient set between both exchangers is 86 °C/m: the hot heat exchanger, within which water circulates at 63 °C, was situated approximately 1.4 m from the resonator’s end. The cold heat exchanger is cooled by cold water flowing at 3 °C. A shaker (LDS – Ling Dynamic Systems – V450/1 – PA 500L) equipped with a piston whose diameter matched the internal diameter of the resonator was used as the acoustic source. It was chosen because of its ability to reach high amplitudes. Besides its working conditions maximise the possible single-frequency and thus do not affect the measurements via parasitical acoustic effects (i.e., harmonic frequencies). The superior harmonics amplitude reaches only 6% of that of the resonance frequency and therefore can be neglected. To vary the drive ratio (Dr), which is defined as the ratio of the maximum amplitude of the acoustic pressure to the mean pressure, the piston stroke is changed while the frequency remains the same. The system is tuned at a frequency of f = 24.4 Hz, which is the resonance frequency, the first mode of the system. This frequency was calculated for our experimental conditions and experimentally verified. An Nd YAG laser of 200 mJ pulse (PIV 190 PS1/TwinsBSL Quantel) at a wavelength of 532 nm combined with spherical and cylindrical optical components was used to generate the laser sheet. A prism was used to deflect the laser sheet into the vertical xy plane of the PIV cell. The image was then acquired with a CCD (Charge Couple Device) camera (TSI PIvCam 13-8) of 1024 1248 pixels. The camera was connected to a synchroniser (LASERPULSE Synchroniser – TSI Model 610034), which allowed the image acquisition to be triggered with the control signal of the shaker at a rate of 3.63 Hz. An aerosol generator (TOPAS ATM 210) was used to generate the seeding mist. The passive tracers are particles of DEHS (Di-2-EthylhexylSebacat). These particles have relaxation time of about 0.3 ls, which set the Stokes number to about 2 105.
St ¼ In this section, the experimental setup used to investigate the influence of an axial temperature gradient on the acoustic Rayleigh streaming is developed. The whole system (acoustic
su d
ð1Þ
where s is the particle relaxation time; u is the fluid velocity; and d is the resonator diameter.
Fig. 1. Zoom in: measurements configurations and setup.
E. Saint Ellier et al. / Experimental Thermal and Fluid Science 66 (2015) 1–6
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Fig. 2. Experimental apparatus and PIV instrumentation setup.
The size of the droplets produced was on the order of 0.3 m, and the droplets completely evaporated after 4 h, which prevented clogging. Several valves were used to control the seeding and its injection into the resonator. Fig. 2 shows an exploded view of the PIV instrumentation setup. Following works from Moreau et al. [8] and Debesse et al. [18], velocity measurements are performed 30 min after the acoustic source is switched on to ensure steady state is reached and 10 min after seeding to ensure that the particles are well homogenised.
To evaluate the acoustic streaming, u2, and to monitor the acoustic cycle as closely as possible, velocity data collected at only one phase u is not enough. One must analyse the entire acoustic cycle. To this end, the acoustic period is divided into time lapses whose width is carefully chosen. It has to be sufficiently short to minimize uncertainties resulting from – the PIV algorithm where the oscillating acoustic flow displays a velocity inversion, – the post processing calculation (integral calculus in Eq. (4)).
2.2. PIV methodology The synchronised PIV technique was chosen because it provides an overall mapping of the flow with good spatial resolution. The 2D field [19] makes it more appropriate for the detection of vortex cells, such as those observed in Rayleigh streaming. 1D data can still be extracted and used to compare data at a given location. Conventionally, the Particle Image Velocimetry measurement method is well suited for measuring acoustic velocities as well as non-linear effects, such as acoustic streaming. The PIV has a history of being used in acoustics; for example Debesse [18], Reyt [20], Nabavi [21] and Saint Ellier [22] used it to measure acoustic streaming in a closed enclosure. The streaming velocity is a second order quantity; it is about two orders of magnitude smaller than the acoustic velocity and therefore particularly difficult to measure. To overcome these difficulties a particular methodology was developed. This is what next paragraphs describe: the method followed to extract acoustic, streaming and free convection velocities. Originally two methods were considered. Only the chosen one is described here. (To have more details about the 2nd way, please refer to [17] from the same authors.) First, via PIV acquisition, to ensure convergence of the streaming velocity field calculation, a series of instantaneous pairs of images are recorded at a given phase u with a time separation between laser pulses of Dt. This phase synchronisation with the acoustic signal is achieved thanks to the synchroniser. The velocity field can be deduced from the particle displacement between images of a pair. Then the calculated velocity fields at phase u are averaged to minimize errors.
Conversely numerous phases with very short spaces would increase the post processing. When the number of phases is chosen, each phase ui is studied with the same applied PIV post-processing as before. The PIV technique is primarily a non-intrusive method. However, it is indirect because the velocity measured is that of the particles (which is assumed to be the flow velocity). Applied to acoustic flows with streaming, this velocity can be decomposed into a term related to the acoustic oscillation, ua, and another one related to the streaming, u2, as follows:
uðx; y; tÞ ¼ ua ðx; y; tÞ þ u2 ðx; yÞ
ð2Þ
With the additional presence of a temperature gradient, we first assume that the velocity related to the natural convection, uc, superimposes on u(x, y, t) such that
uðx; y; tÞ ¼ ua ðx; y; tÞ þ u2 ðx; yÞ þ uc ðx; yÞ
ð3Þ
The integral of u(x,y,ui) over one period of the acoustic signal using the formula (4) gives :
u2 ðx; yÞ þ uc ðx; yÞ ¼
n1 1 X uðx; y; ui Þ ðtuiþ1 t ui Þ T ac i¼1
ð4Þ
where n = 17 is the number of phases over which the acoustic period Tac is divided, and tui is the time corresponding to the phase ui. If uc is known, the acoustic velocity u2 can be plotted from Eq. (4). A temporal representation of the particle velocity on a time period can be deduced by averaging each fields ui using the formula
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(5) where X is the number of points along the x coordinate and Y the number of points along the y coordinate.
u1 ðui Þ ¼
Y X X 1 1X uðx; y; ui Þ Y X y¼1 x¼1
ð5Þ
3. Results and discussion 3.1. Experimental conditions The experiments were performed at atmospheric pressure and at the resonance frequency of 24.4 Hz of the resonator filled with air. The transparent section where the measurements were carried out was located between a node and an antinode of the acoustic wave at 4L/15 (i.e. 5.6 m from the acoustic source) whose location was determined in previous work [22] from a calculated acoustic model of the system. Drive ratios from 1% to 4% were obtained. Seven test conditions were carried out: measurements were performed with and without the temperature gradient for three different drive ratios. A final test (test 4) was carried out to study the natural convection inside the measured section: no flow was imposed within the resonator. The experimental conditions are summarised in Table 1. The drive ratio was obtained from the measurements of a pressure transducer located at the resonator’s plug (see Fig. 1). 3.2. Results The experimental tests results yielded two main observations. First, Fig. 3, which shows the results of tests 1 through tests 3, indicates that the thermal gradient has little influence on the temporal Table 1 Operating conditions.
Test Test Test Test Test Test Test
1-1 1-2 2-1 2-2 3-1 3-2 4
P0 – mean pressure (bar)
Pa – acoustic pressure (mbar)
Drive ratio (%)
T0 (°C)
Temperature gradient
1.014 1.014 1.016 1.016 1.026 1.026 1.020
29 29 37 37 65 65 0
1.4 ± 12 % 1.4 ± 12 % 1.8 ± 9 % 1.8 ± 9 % 3.3 ± 6 % 3.3 ± 6 % 0
24.4 24.4 24.4 24.4 25 25 24.6
86 °C/m No 86 °C/m No 86 °C/m No 86 °C/m
variation of the axial velocity. The uncertainties associated with the PIV method correspond to a fraction of pixel in the order of two hundredths [19]. Given these uncertainties (see the corresponding error bars), both curves plotted for each test (with and without thermal gradient) are almost identical. The PIV measurements are compared to the theoretical acoustic sine profile which was calculated from the measured acoustic amplitude. The curves show little distortion due to the nonlinearities and propagations of the superior harmonics. When an acoustic wave propagates within the resonator, Rayleigh streaming is assumed to superimpose on the natural convection phenomenon (Eq. (3)). The quasi stationary flow velocity defined in Eq. (4) can be plotted. The three profiles resulting from this superposition are shown in Fig. 4. The thermo-convective flow is superimposed on the acoustic flow for tests 1-1, 2-1 and 3-1. The fourth curve (Dr = 0%) is the velocity profile resulting from the horizontal convective mass transfer only observed between the heat exchangers within the measurement cell without the presence of an acoustic wave. The temperature gradient affects the density of the fluid leading to a bulk fluid motion where the heavier components (cold fluid) fall, while lighter components (warm fluid) rise. It creates a convective rotational movement. Fig. 5, which represents the corresponding velocity field shows this particular motion. The maximum longitudinal velocity is 3.5 cm/s. Interestingly here the increase of the drive ratio values shows two physical phenomena: (1) it modifies the velocity profile of the convective flow by reducing its velocity amplitude and (2) it distorts the convection cell until forming a new structure, one cell with rotational direction opposite to the previous one. This new structure is illustrated in Fig. 6. Vorticity values obtained from this 2D velocity field do not exceed 6 102 s1. However these values should be considered with caution because the flow is actually strongly 3D and these velocity fields are obtained after a heavy post processing. In test 1-1, the velocity profiles of the quasi stationary flow and natural convective flow are qualitatively identical. As the drive ratio increases, the profile deforms (test 2-1). At a higher drive ratio (here, 3.3%), the phenomenon of streaming becomes dominant in the resonator and the temperature gradient imposed around the measuring cell ratio carries little influence. The work of Nabavi et al. [12] presents similitudes in their results [12], even though they studied a transversal temperature gradient. Unlike the present work where the drive ratio varies and the temperature gradient is imposed, they imposed a drive
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Axial acoustic velocity ua [m/s]
4 3 2 1 0 -1 -2 -3
Dr=1,4% Dr=1,8% Dr=3,3%
-4 -5
0
10
20
30
40
50
60
70
80
90
Fig. 3. Temporal variations of the axial acoustic velocity for three drive ratios. The 4 correspond to no gradient cases and s to gradient cases. The dashed lines are the theoretical acoustic results.
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Dr=0% Dr=1,4% Dr=1,8% Dr=3,3%
30 20
r [mm]
10
-0,06
-0,04
-0,02
0
0,02
0,04
0,06
0 0,08 -10 -20 -30
Mean flow velocity [m/s] Fig. 4. Profiles of the mean flow velocity of test 1-1 (blue), 2-1 (orange) and 3-1 (purple) compared to the velocity profile of natural convection. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 6. Streaming velocity field for test 3-1.
30 20
r [mm]
10
-0,08
-0,06
-0,04
-0,02
0
0,02
0,04
0,06
0,08
0 0,1 -10 -20
Streaming velocity u 2 [m/s]
-30
Fig. 7. Comparisons of the streaming velocity profiles with (s) and without (—) temperature gradient. Blue: Dr = 1.4%, orange: Dr = 1.8%, violet: Dr = 3.3%. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Fig. 5. Bulk fluid motion velocity field resulting from the convective heat mass.
ratio of 0.8% avec varied the temperature gradient. However similitudes can be found. Their temperature gradient values correlated with the dominance of the convection phenomenon and deformation of streaming, whereas in our present work the drive ratio directly correlates with the dominance of streaming and deformation of the convection cell. At low drive ratio (0.8% for [12] and until 1.8% here), the profiles show one velocity cell resulting from the strong influence of the imposed temperature gradient. Beyond a given drive ratio (which could be around 1.8% in this work) the streaming becomes dominant over the convective flow. In some references the distinction between the previous acoustic-thermo-convective flow and acoustic streaming is not made. What is called here ‘‘stationary flow’’ or ‘‘mean flow’’, is often referred as ‘‘acoustic streaming’’ only (example in [12]) although they study the consequences of a combination between acoustic streaming and convective mass transfer. If we want to study the strict contribution of the acoustic streaming, given the hypothesis expressed for Eqs. (3) and (4), by
subtracting uc to Eq. (4), it is possible to deduce the streaming velocity profiles. The acoustic streaming velocity profiles of tests 1-1, 2-1 and 3-1 are plotted against streaming velocity profiles of tests 1-2, 2-2 and 3-2 in Fig. 7. If the convective flow and the acoustic streaming were dissociated, identical curves would be expected for a given drive ratio since only the sole contribution of acoustic streaming is studied here. However, even taking the PIV uncertainties into account, differences in the amplitude ranging from 10% to 100% were observed. These differences inversely increased with the drive ratio. The temperature gradient is intricately linked with the establishment of the streaming. These results show that the assumption made for Eqs. (3) and (4) is not suitable for the description of Rayleigh streaming in the presence of a temperature gradient. Indeed coupling phenomena are not considered in this equation. The superimposition of streaming on the natural convection must be assumed with care. Results of Fig. 7 show that strong coupling exists between the temperature gradient and the acoustic streaming within the resonator.
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4. Conclusions This study is part of an experimental exploration phase of acoustic streaming in thermoacoustic systems. The study of an axial temperature gradient localised around the measured section showed that the interaction between acoustic streaming and the temperature gradient is complex and most likely leads to strong coupling phenomena that were not taken into account with the assumption made at the beginning of this paper. The simple superimposition of acoustic streaming and free convection was insufficient. However, the measurement first revealed the formation of a sole streaming vortex whose direction of rotation seemed to be opposite to that of the free convection. This difference should be verified by performing experiments with gradients established in the opposite direction. Interestingly, the drive ratio directly correlated with the dominance of streaming, as the velocity profiles of the given mean flow and natural convection were qualitatively identical. Experiments performed with a higher temperature gradient would yield additional results to pursue this avenue of research. This work will be continued, as this area is crucial to improving the efficiency of thermoacoustic machines (e.g., effect of secondary flows and harmonics). Acknowledgment The experimental work was partially supported by the European Union under the FP7. Contract No. 226415 – www.thatea.eu. References [1] [2] [3] [4]
N. Riley, Acoustic streaming, Theor. Comp. Fluid Dyn. 10 (1998) 349–356. Lord. Rayleigh, The Theory of Sound, MacMillan, London, 1896. N. Rott, Thermoacoustics, Adv. Appl. Mech. 20 (1980) 135–175. G.W. Swift, A unifying perspective for some engines and refrigerators, Acoust. Soc. Am. through the American Institute of Physics, 2002. [5] T.D. Rossing, F. Dunn, W.M. Hartmann, D.M. Campbell, N.H. Fletcher, Thermoacoustics, in: Handbook of Acoustics, Springer, New York, 2007. 239–255.
[6] S.-Y. Xiao, S. Tao, M.-C. Qiu, H. Ge, L. Fan, S.-Y. Zhang, H. Zhang, Acoustic non linearities in themoacoustic refrigerators driven by loudspeakers, in: 19th International Congress of Sound and Vibration (ICSV19), Vilnius, Lithuania, July 2012. [7] M.J. Lighthill, Waves in Fluids, Cambridge University Press, Cambridge, 1978. [8] S. Moreau, H. Bailliet, J.-C. Valière, Measurements of inner and outer streaming vortices in a standing waveguide using laser doppler velocimetry, J. Acoust. Soc. Am. 123 (2) (2008) 640–647. [9] J.P. Sharpe, C.A. Greated, C. Gray, D.M. Campbell, The measurements of acoustic streaming using particle image velocimetry, Acustica 68 (1989) 168–172. [10] M.W. Thompson, A.A. Atchley, M.-J. Maccarone, Influence of a temperature gradient and fluid inertia on acoustic streaming in a standing wave, J. Acoust. Soc. Am. 117 (4) (2005) 1839–1849. [11] M.W. Thompson, A.A. Atchley, Simultaneous measurements of acoustic and streaming velocities in a standing wave using laser Doppler anemometry, J. Acoust. Soc. Am. 117 (4) (2005) 1828–1838. [12] M. Nabavi, K. Siddiqui, J. Dargahi, Effects of transverse temperature gradient on acoustic and streaming velocity fields in a resonant cavity, Appl. Phys. Lett. 93 (2008) 5. [13] M. Nabavi, K. Siddiqui, J. Dargahi, Influence of differentially heated horizontal walls on the streaming shape and velocity in a standing wave resonator, Int. Commun. Heat Mass 35 (2008) 1061–1064. [14] M.K. Aktas, T. Ozgumus, The effects of acoustic streaming on thermal convection in an enclosure with differentially heated horizontal walls, Int. J. Heat Mass Transfer 53 (2010) 5289–5297. [15] B. Tajik, A. Abbassi, M. Saffar-Avval, A. Abdullah, H. Mohammad-Abadi, Heat transfer enhancement by acoustic streaming in a closed cylindrical enclosure filled with water, Int. J. Heat Mass Transfer 60 (2013) 230–235. [16] V. Daru, D. Baltean-Carlès, C. Weisman, P. Debesse, G. Gandikota, Twodimensional numerical simulations of nonlinear acoustic streaming in standing waves, Wave Motion 50 (2013) 955–963. [17] E. Saint Ellier, W. Kdous, L. Girardot, D. Ramel, P. Nika, Acoustic streaming measurements in standing wave resonator using particle image velocimetry, Wave Motion, http://dx.doi.org/10.1016/j.wavemoti.2014.07.011, 2014. [18] P. Debesse, D. Baltean-Carlès, F. Lusseyran, M-X. François, Experimental analysis of nonlinear phenomena in a thermoacoustic system, in: 18th International Symposium on Nonlinear Acoustics (ISNA), July 2008. pp. 355– 358, DOI: http://dx.doi.org/10.1063/1.2956229. [19] M. Raffel, C. Willert, J. Kompenhans, Particule Image Velocimetry – A practical guide, Springer. ISBN 3-540-63683-8. [20] I. Reyt, H. Baillet, J.-C. Valière, Experimental investigation of acoustic streaming in a cylindrical wave guide up to high streaming Reynolds numbers, J. Acoust. Soc. Am. 135 (1) (2014) 27–37. [21] M. Nabavi, K. Siddiqui, J. Dargahi, Measurement of the acoustic velocity field of non linear standing waves using the synchronized PIV technique, Exp. Therm. Fluid Sci. 33 (2008) 123–131. [22] E. Saint Ellier, Y. Bailly, L. Girardot, D. Ramel, P. Nika, Acoustic streaming measurements by PIV, in: 15th International Symposium on Flow Visualisation (ISFV15), Minsk, Bielorussia, June 2010.