Reconstruction of Acoustic Velocity Field Excited by Complex Tones by Means of Particle Image Velocimetry

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1 Reconstruction of Acoustic Velocity Field Excited by Complex Tones by Means of Particle Image Velocimetry Witold Mickiewicz West Pomeranian University of Technology, Szczecin Faculty of Electrical Engineering ul. Sikorskiego 31, Szczecin, Poland Abstract The paper addresses a problem of the measurement and reconstruction of an acoustic velocity field using the particle image velocimetry technique (PIV). Nowadays this technique has become more and more popular non-invasive method of the sound field investigation. Using a so called phase-locked PIV there is possible to visualize sound fields excited by pure tones. In the paper some introductory results of numerical experiments are shown, which address the visualization of sound field excited by more complex signals - the sum of several sinusoids with different amplitude sets and frequency spacing. The experiments show, that the visualization of such field is a not straightforward task but it seems to be possible to achieve competent results. The main problem, which should be overcome is the correction of systematic error introduced by PIV system due to non-zero time interval between acquisition of two adjacent images of the field under investigation seeded by probe particles. Without the correction the reconstructed field is distorted due to different amplitude and phase shifts of its spectral components. Some notes on the correction method has been presented in that paper. I. INTRODUCTION Particle Image Velocimetry (PIV) is a velocity measurement method based on the analysis of a seeding particles shift registered on two consecutive images taken by a digital camera. The method was originally developed for the mass flows observation in the fluid mechanics [1][2]. Due to imaging and laser technology developments it become possible to observe and visualize oscillating particles movements around their mean positions, which don t move in time the mass. Such phenomena are observed for example in spaces, where acoustic waves propagate. Observation of an acoustic vector field in a non-intrusive way can be a valuable complement of the existing sound intensity methods [3][4][5][6][7][8]. In the case of audible sounds the image pairs should be acquired with sampling rate of kilohertz and time interval between image acquisition in a pair should be very small - microseconds. Nowadays such high sampling rates are still difficult to obtain. It requires high speed digital cameras and high speed double-pulsed lasers. Such systems called Time Resolved PIV are available on the market, but their prices are high and spatial resolution (1Mpixel) is quite low comparing to the more widespread traditional systems with lower sampling rate (about 10Hz) but higher spatial resolution of 4Mpixel. That is why to observe the dynamics of sound energy flow many researchers use the Phase Locked PIV technique [9][10][11][12][13][14][15][16][17]. The high sampling rate is obtained by a multiple images acquisition in the following time (phase) points synchronised to cyclic excitation As is was mentioned before, the PIV is an indirect velocity measurement method based on a displacement observation in a time interval, which is known. From the theoretical point of view for the instantaneous velocity measurement the time interval should be as short as possible aiming zero. Technically it is realized with a such small time interval, for which the displacement can be measured with the satisfactory precision [18]. There are some research articles devoted to the optimum time interval selection for the observation of harmonic wave excited by pure tone [19]. One of the main conclusions is that the optimum value strictly depends on the frequency of observed wave, so for different frequencies the optimum value changes. It is worth to mention that this non-zero time interval value introduces the systematic error of the velocity measurements and the reconstructed sinusoidal signal holds an amplitude error and a phase shift error. The theoretical model and correction method author has presented in [20]. Concluding this discussion: in existing literature the Phase Locked PIV is used for observation of simple harmonic waves with one frequency. In this paper the author wants to show some simulation results concerning the reconstruction of acoustic velocity field excited by multi tonal harmonic signal using the model presented in [20]. From the theoretical point of view such reconstruction should be possible. Some problems can arise with desirable frequency resolution, what is connected with a collection of huge enough amount of data. This in turn is connected with long examination time. When we use the phase-locked PIV methodology with high-energy laser light pulses, it leads to the problems of ensuring stable environmental conditions during the whole experiment. So it is worth to predict the minimal data number (examination time) allowing to make a signal reconstruction on the acceptable level of accuracy. In the paper the author shows computer simulation results concerning sound field reconstruction in the function of an /14/$ IEEE 311

2 amount of processed data and a complexness of exciting signal - multi tonal signals with different amplitudes sets and frequency spacing. II. THEORETICAL BACKGROUND OF ACOUSTIC VELOCITY MEASUREMENT USING PIV The values of the medium instantaneous velocity are evaluated by the PIV system basing of the seeding particles spread analysis recorded on two images taken in two consecutive instants of time. The seeding particles are added to the transparent medium to make visible its movements. The physical parameters of the seeding tracers can be selected to exactly follow the medium movements. Using mathematical notation PIV system realizes approximation of the derivative of a displacement function with a difference quotient with finite non-zero time interval t. v i = d r i dt = lim r i (t + t) r i (t) r i(t + t) r i (t) t 0 t t (1) where v i is a velocity vector in the i-th subarea of measurement plane and r i is a position vector evaluated from seeding particles image analysis of this sub area for the given instant of time. In the case of sound field excited by pure tone the phaselocked PIV is simply to apply. Laser action and camera acquisition have to be synchronized to the excitation signal to collect the data sample by sample not in real time but in proper relation to each phase of the exciting If the synchronization is ensured, following samples can be acquired at lower real sampling frequency (for example 10 Hz). For such experiment 3 main parameters should be established: the number of phases in one exciting signal cycle, which will be collected, the time interval t for every image pair acquisition and the number of pairs acquired for one chosen phase to achieve the required signal-to-noise ratio in the resulting acoustic velocity field visualization. From own experience for pure tone excitation the signal cycle is divided into phases, t is about 1/8 of pure tone period and the number of pairs acquired for one chosen phase is connected with signal amplitude and is in range from fifty to hundreds. Selected values of these parameters influence the systematic and statistical errors in the final field reconstruction. These errors are functions of frequency, so if the exciting signal is no more one frequency pure tone, but complex tone, every harmonic component is influenced by different errors values, so the reconstructed field is distorted. In the following part of the paper some computer simulation are shown, which give us some insight in the nature of this problem. If we want to use the same methodology together with non intrusive PIV technique, the errors, if possible, should have to be eliminated or compensated. III. TEST SIGNALS FOR COMPUTER EXPERIMENTS The methodology of the research concerning acoustic energy flows in ducts and channels and around various obstacles bases on the frequency analysis of measured or calculated acoustic field parameter (pressure, velocity or sound intensity) in interesting area of investigated subject, which is excited by wideband noise (white or pink noise). The higher frequency resolution required, the narrower frequency bands are used. The acoustical phenomena are analysed in frequency domain using classical methods of Fourier analysis. For the analysis concerning audible frequencies the results are more often calculated and presented not for single frequency values, but for frequency bands. The frequency domain is divided into bands using non-linear logarithmic scale. The whole audible spectrum of frequencies between 20Hz and 20kHz is divided in octave and sub-octave bands. Most often used are oneoctave, one-third-octave, one-twelve-octave and one-twentyfour-octave bands, where fmax to f min is 2, 2 1 3, , consequently. The distance between the central band frequencies can be calculated using these bandwidth coefficients starting from normalized 1kHz frequency: f i = 1kHz ( f max f min ) i 12 ; i = 0, 1, 2, (2) Such frequencies values with non-constant frequency spacing could be used for creating artificial exciting signals besides signal with equidistant frequency spread. For numerical experiments in presented research a multitonal signals were used containing 13 sinusoidal components from one octave band with the non-equidistant frequencies calculated as follows: f MAX = 2; f i = f MIN 2 i 12 ; i = 0, 1, 2,...12, (3) f MIN and equidistant frequencies calculated as follows: f MAX f MIN = 2; f i = f MIN (1 + i ); i = 0, 1, 2, (4) 12 For both frequency sets 4 test signals were generated: 12 T EST i = A ij sin(2πf i + φ i ), (5) j=1 where initial phase values were random and constant for all experiments and amplitudes of harmonics were: A 1j = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]; (6) A 2j = [10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10]; (7) A 3j = 1 ; j = 0, 1, 2,...12 (8) 2j A 4j = 2j ; j = 0, 1, 2,...12; (9) 212 Figures 1 and 2 show the autocorrelation functions of signal T EST 1 for both frequency sets. As we can see, signal with non-equidistant frequency set lacks of the periodicity, so it will be difficult to calculate 312

3 distortion for each frequency component, what leads to distorted shape of the velocity signal in time domain. It can be seen on figure 3. The ideal velocity signal (green) can t be easily reconstruct trough the gain and shift of the measured signal (red). The shape of both signals are different. Fig. 1. The autocorrelation function of signal T EST 1 for equidistant frequency set Fig. 3. The ideal (green) and PIV reconstructed (red) velocity signal calculated from test signal T EST 1 Fig. 2. The autocorrelation function of signal T EST 1 for non-equidistant frequency set precisely its spectrum basing of a not huge amount of measurement points. Also the synchronized data acquisition during PIV measurement will require a cyclic reset of the exciting This excitement discontinuity can generate artefacts in the investigated sound field and could destroy the proper image of phenomena occurring in the investigated region. In the case of equidistant frequency set the complex signal period is a multiplication of lowest frequency harmonic component period, so in presented case it should be observed through 12 cycles of lowest harmonic component. Together with the requirement of frequency resolution it give us the notions to calculate the minimum number of phases, which should be observed during the experiment. IV. EXPERIMENTS Presented numerical experiments were done in Matlab environment and consisted in comparable analysis of errors in acoustic velocity reconstruction for one observation point in space. It was assumed (eulerian approach), that displacement of seeding particles in thit point is described by test Reference (ideal) velocity signal was obtained by analytic differentiation of test signals. Reconstruction of the signal according to formula (1) is distorted due to non-zero time interval t. The PIV procedure introduces different amplitude and phase Fig. 4. Reconstructed signal spectrum for shortest time interval t = 0.05T MIN The shape-error is a function of non-zero time interval t and we can see its influence on particular spectral components on figure 4 and 5. The longer time interval t, the bigger amplitude differences, which increase with frequency. As it was mentioned before, the time interval t should be reasonable long, so the shape-error will occur practically always. So the question is, how strong it influences the measurement results? V. SIMULATION RESULTS For that stage of research only the cumulative energetic relative error for whole test signal was calculated and compared. The error was defined as: ε = W P IV W ideal W ideal, (10) 313

4 harmonic in the signal with frequency f MIN. The error is calculated in frequency domain for 1024 samples of the test Errors curves on figures 6 and 7 has the same pattern, although in second case only small portion of data were taken into account. So if the measurement data encompass the cycle of periodicity of the exciting signal, frequency analysis give the same quantitative information about the reconstructed Fig. 5. Reconstructed signal spectrum for longest time interval t= 0.25T MIN where W P IV is the energy of the reconstructed PIV signal and W ideal is the energy of the analytically differentiated test In figure 6 the reconstruction error of signal energy in time domain as a function of time interval t. The t is presented as a portion of the period of the lowest harmonic in the signal with frequency f MIN. The error is calculated in time domain for 0,5 s As we can see, the error increases with longer time interval t but also depends on the exciting signal spectrum. The cumulative error is a weighted sum of errors introduced by each harmonic and for given time interval t the compositional error increases with frequency. So if there is more energy concentrated in higher frequencies, the cumulative error will be bigger. Fig. 6. Reconstruction error of signal energy in time domain. Signal A, B, C, D corresponds to TEST signal 1,2,3,4. The same results were obtained in frequency domain. In figure 7 the reconstruction error of signal energy in frequency domain as a function of time interval t is shown. As before the t is presented as a portion of the period of the lowest Fig. 7. Reconstruction error of signal energy in frequency domain. Signal A, B, C, D corresponds to TEST signal 1,2,3,4. VI. CONCLUSION In the paper some experimental error analysis of PIV technique was presented. The methodology is based on simulating non-ideal differentiation of the system. The increment of the measurement error is due to non-zero t time interval and is bigger for higher frequencies. For systematic error compensation reconstructed signals should be analysed in frequency domain. In figures 4 i 5 two spectrum are shown with different error levels, what is caused by different values of the chosen time interval t. To obtain better results in time domain signal reconstruction the systematic error can be corrected applying the following steps:. Phase-locked PIV measurement using t =1/2T, where T is the period of the highest frequency component in the excitation signal spectrum. The number of recorded phase steps should ensure sufficient resolution for the FFT analysis of the reconstructed signal and encompass the cycle of complex Estimation of amplitudes and phases of frequency components in measured signal using FFT analysis. Correction of amplitudes and phases using different coefficient values depending on the frequency calculating from model presented in [20]. Synthesis of the signal time form using corrected harmonic components. 314

5 The correction is especially needed, when the exciting signal has more energy concentrated in higher frequencies. The robustness of the procedure was proofed for 2 harmonic complex tone in [20] and now the next step of the presented research is to check the correction procedure for more complex signals. [19] S.Moreau, H.Bailliet, J.Ch.Valiere, R.Boucheron, G.Poignand, Development of Laser Techniques for Acoustic Boundary Layer Measurements. Part II: Comparison of LDV and PIV Measurements to Analytical Calculations, Acta Acoustica united with Acoustica, 95, pp , [20] W.Mickiewicz, Systematic error of acoustic particle image velocimetry and its correction, Metrology and Measurement Systems, 21(3), VII. ACKNOWLEDGEMENTS The author likes to express its gratitude to prof. Stefan Weyna for his scientific support and unconstraint access to his Image Laser Anemometry Laboratory at the Faculty of Maritime Technology and Transport of West Pomeranian University of Technology in Szczecin, Poland. REFERENCES [1] M.Raffel, C.Willert, J.Kompenhans, Particle image velocimetry: a practical guide, Springer, Berlin, New York, Heidelberg, [2] J.Westerweel, Digital particle image velocimetry - Theory and application, Ph.D. Thesis. Delft University, [3] F.J.Fahy, Measurement of acoustic intensity using the crossspectral density of two microphone signals, J. Acoust. Soc. Am., 62(L), pp , [4] F.J.Fahy, Sound Intensity, 2nd edition London, England: E&FN Spon, [5] G.Rasmussen, Measurement of vector sound fields, Proc. 2nd Int. Congr. Acoustic Intensity, pp , [6] F.Jacobsen, Sound Intensity and its measurement applications, Lyngby, Denmark: B&K [7] W.Mickiewicz, M.Jablonski, M.Pyla, Automatized system for 3D sound intensity field measurement, Proceedings of 16th Methods and Models in Automation and Robotics Conference, Miedzyzdroje, [8] S.Weyna, W.Mickiewicz, Multi-modal acoustic flow decomposition examined in a hard walled cylindrical duct, Archives of Acoustics, 39(2), [9] D.B.Hann, C.A.Greated, The measurement of flow velocity and acoustic particle velocity using particle image velocimetry, Meas. Sci. Technol. pp , [10] D.J.Skulina, R.MacDonald, D.M.Campbell, PIV Applied to the Measurement of the Acoustic Particle Velocity at the Side Hole of a Duct, Proceedings of Forum Acusticum 2005, Budapest. [11] A.Tonddast-Navæi, Acoustic Particle-Image Velocimetry Development and Applications, Ph.D. Thesis. Open University, Milton Keynes, UK, [12] R.MacDonald, D.J.Skulina, D.M.Campbell, J.Ch.Valiere, D.Marx, H.Bailliert, PIV and POD Applied to High Amplitude Acoustic Flow at a Tube Termination, Proceedings of 10-eme Congress Francais d Acoustique, Lyon, [13] M.Nabavi, K.Siddiqui, J.Dargahi, Measurement of the acoustic velocity field of nonlinear standing waves using the synchronized PIV technique, Experimental Thermal and Fluid Science, 33, pp , [14] A.Fischer, E.Sauvage, I.Röhle, Acoustic PIV: Measurements of the acoustic particle velocity using synchronized PIV-technique, Proceedings of 14th Int Symp on Applications of Laser Techniques to Fluid Mechanics, Lisbon, [15] D.J.Skulina, A study of Non-linear Acoustic Flows at the Open End of a Tube using Particle Image Velocimetry, Ph.D. Thesis. University of Edinburgh, [16] S.Weyna, W.Mickiewicz, M.Pyła, M.Jabłoński, Experimental acoustic flow analysis inside a section of an acoustic waveguide, Archives of Acoustics, 38(2), pp , [17] S.Weyna, W.Mickiewicz, Phase-Locked Particle Image Velocimetry Visualization of the Sound Field at the Outlet of a Circular Tube, Acta Physica Polonica A, 125(4A), pp. A , [18] J.Westerweel, Theoretical analysis of the measurement precision in particle image velocimetry, Experiments in Fluids, Suppl. S3-S12,

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