Jet motion in flute-like instruments: experimental investigation through flow visualization and image processing

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1 Jet motion in flute-like instruments: experimental investigation through flow visualization and image processing Christophe Vergez LMA, CNRS, 3 Chemin Joseph Aiguier, 342 Cedex 2 Marseille, France, vergez@lma.cnrs-mrs.fr Patricio de la Cuadra CCRMA, Stanford University, CA , USA, pdelac@ccrma.stanford.edu Benoit Fabre LAM, Paris 6 University, rue Lourmel, 7, Paris, France, fabreb@ccr.jussieu.fr Flow visualisations on transverse oscillations of jets submitted to an acoustic perturbation are analysed in this paper. Two methods for image processing are compared, based on complementary approaches : inter image and intra image analysis. The results obtained using both methods are very close, showing the reliability of both methods. A typical application is shown on a classical example of the litterature. Introduction In flute-like instruments, acoustic oscillation is generated by the coupling of the unstable modes of an air jet and the acoustic modes of a pipe resonator. The intimate details of this coupling remain an unsolved problem. It appears that for both laminar and turbulent jets, the lack of theoretical models make experimental data useful to fit, calibrate and evaluate semi-empirical models ([]). In this paper, the development of instability on a jet is experimentally investigated using a simplified device (see section 2): the pipe resonator is removed from the flute and replaced by an acoustic field generated by loudspeakers. At the moment, there is no analytical model available to describe this jet oscillation. Using the Schlieren technique, flow visualizations have been carried out for a variety of Reynolds and numbers. Two different image analysis techniques have been developed (see section 3) and they are discussed in detail in this paper. Their common objective is to extract quantitative data from the images. The data are analyzed in the framework of a harmonic perturbation that is convected downstream and grows in the shear layers of the jet (see section 4). Finally, the application to a problem inspired by a paper in the literature is presented in section. 2 Experiment description Experimental setup : a jet, created by blowing through a slit, is acoustically forced by two out-of-phase loudspeakers (see figure ). The speed of the jet is altered by varying the pressure in the cavity p f just before the jet formation. The central speed of the jet U b at the channel Figure : Experimental setup. acoustic velocity sensor. Jet exit, speakers and exit can be estimated using Bernoulli s equation: 2pf U b =, () ρ where p f is the pressure in the cavity before the channel, and ρ the fluid density. Jet structure : two Reynolds numbers R e = and R e = 3, (R e U b h/ν, with h the channel height and ν the kinematic viscosity) were chosen. They correspond to two different operating modes of the jet: Re = gives a velocity around 7. m/s which corresponds to a normal blowing condition for a recorder and assures a laminar behavior, while Re = 3 gives a velocity around 4 m/s producing a turbulent jet frequently observed in the high register of the flute. The acoustic excitation frequency f is ranging from 7 to 4 Hz to cover a (S t ) range from approxi- 4

2 Forum Acusticum 2 Budapest mately. to. (S t fh/u b ). A fixed amplitude of the acoustic velocity (measured with a Microflown velocity sensor as shown in figure ) is chosen for the whole experiment, and a typical value is % of the jet velocity. Schlieren jet images : Schlieren technique 2 ([2]) is used to observe the behavior of the oscillating jet. We use CO 2 jet traversing the air, whose mass density assures enough difference to allow Schlieren visualization while producing a similar behavior from that found in real flutelike instruments. Figure 2 shows the basic implementation of the method, and figure 3 an example of a Schlieren jet image. Sequential images of the jet are taken with a digital camera synchronized with a stroboscope. Frequency of the camera was set to 4fps, exposure time to µs. Approximately images are taken covering two cycles of jet oscillation. Images are captured in raw, black and white, bmp files with size 28 x 448, and 8 intensity bits. Point light source lens Test area S lens Knife edge scre Figure 2: Schlieren basic implementation: a subset S of the test area with different medium (here the CO 2 jet) deviates light rays. A knife edge, placed in the focus of the second lens blocks part of the rays having crossed S, thus producing intensity variations on the screen (see figure 3). rithms are proposed. 3. Cross-correlation method A first algorithm based on cross-correlation between successive images is proposed. A typical intensity graph of one column of the image is shown in figure 4 (left picture). For that particular column the shape of the graph does not change as much as time goes on, only the shape is shifted following the movement of the jet. The particular shape varies with the column considered since it is determined by the mass distribution of the jet as shown in figure 3. Cross-correlation is calcu- Light intensity Vertical position [pixels] Figure 4: Intensity level of a single column from a Schlieren image (left) and cross-correlation of two intensity graphs corresponding to the same column in two successive images of the sequence (right). lated between two intensity graphs corresponding to the same column of two successive images of the sequence (see right part of figure 4). The position of the peak allows to measure the displacement of the jet between these two successive images. This is repeated for each column of the image. Since changes on the jet position may be smaller than one pixel, a parabola is fitted to the three highest points to determine the position of the peak. When considering all the columns of all the images in the sequence, the jet position can be reconstructed as shown in figure (R e = and S t =.2). Some defects of the 2. x 6 2. Figure 3: Schlieren Jet image (see figure 2 for the optic principle). 3 Image processing The goal of the image processing algorithm is to detect transverse jet displacements for every image. Two algo- This is considerably lower that the % observed in real flutes but allows us to cover an interesting range of frequencies without over exciting the speakers. 2 Using an optic scheme, light phase shift crossing an inhomogeneous media is converted into light intensity. Figure : Result of the cross-correlation method in the case R e = and S t =.2 46

3 Forum Acusticum 2 Budapest experimental setup are clearly visible in left part of figure 4 : non symetrical contrast appears as non symetrical amplitude peaks, and unhomogenous background intensity represented as a non horizontal line. However, these features of the images are useful for the cross-correlation since they help to produce the well shaped peak in the cross-correlation curve observed in the right part of figure 4. complemented image. The result of this method on one image is presented in figure 6. Bringing together the median line found for each image allows to construct 3D plots similar to figure. 3.2 Morphological method The idea of this method is to apply morphologic functions on binarized images, in order to identify large scales in this image (corresponding to the jet). This is done by the five following steps : Contrast homogenisation (between all the images of a sequence) and contrast enhancement by histogram equalization. Conversion from greyscale to black and white images : a statistical method based on the Otsu principle is used which estimates the threshold which best separates the histogram within two classes. The threshold for the binarization is therefore different for each image. Extraction of largest spatial scales in the image : a morphological opening is performed to clean all small structures. Then a morphological closure is applied in order to highlight large scales in the image. The size of the structuring element, found empirically, is the same for all the images of a given sequence. Identification of the jet among the large scales found at previous step : the image is first analysed in terms of different regions, i.e. groups of contiguous pixels. A discriminating criterion has been constructed according to geometrical characteristics calculated for each region. It involves the area of the region corresponding to the jet and the localization of each region in the image. Jet edges estimation and median line calculus : jet edges detection is trivial and is performed with Sobel algorithm which uses gradient information. Finally, the image is scanned column after column and the median line is constructed point after point, including empirical criteria to select, among several paths, the one to follow. This guarantees that the resulting median line is single-valued, but possibly discontinuous in the case of vortices. Moreover since we are interested in both half-jets (darker and brighter in figure 3), all the operations described above are applied on the image and on the Figure 6: Original Schlieren image superimposed with edge detected (black line) and calculated median line (white line). 4 Results comparison 4. Data analysis Every column of the images oscillates in time at the same frequency of the excitation. It is therefore possible to fit a sinusoid of that frequency to each column, and obtain the amplitude Y (f) and phase Y (f) of the fitted curve: Y (f) = N N k= ( X k exp jk 2πf ), (2) f s where f s is the sampling frequency, N is the number of images in the sequence, X i is the jet deflection for that column at image number k. These values Y (f) are used to recreate a fitted version of the position as shown in figure 7, and to analyze how the perturbation behaves. In a linear description of the jet [3], the perturbation is amplified while being convected downstream. The perturbation travels at a velocity about one half of the jet speed. Inspired from this theory, the experimental data can be analyzed assuming a jet transverse displacement η following : η(x, t) = η e γx e iω(t x/cp) (3) where η represents a complex initial amplitude of the oscillation, γ is the spatial growing rate, c p is the convection velocity and x is the distance from the flue exit. 4.2 Methods comparison The growing rate γ can be estimated by fitting an exponential to the detected amplitude curve. The convection 47

4 Forum Acusticum 2 Budapest the phase shift of the oscillation decreases roughly linearly downstream from flue exit, corresponding to the delay induced by the convection of the perturbation on the jet at constant velocity. Figure 7: Fitted position of jet, excitation frequency = Hz (St=.62), laminar jet (Re=) velocity c p of the perturbation can be estimated as the slope of the linear fitting to the phase as shown in figure 8 for the cross correlation method and the morphological method, in the case of a laminar jet (Re=). Oscilation amplitude [m] Oscilation phase [rad]. x 3 Amplitude of the oscillation, str =.36734,f = estim. ampl. (morpho) fit of the ampl. (morpho) estim. ampl.(cross cor) fit ampl. (cross cor)... Phase of the oscillation vs distance from the flue exit estim phase (morpho) fit phase (morpho) estim phase (cross cor) fit phase (cross cor) fit phase 2 (cross cor)... Figure 8: Amplitude and phase of the jet transverse displacement of the jet using the cross correlation method (pink line) and the morphological method (green line), excitation frequency = Hz (St=.3674), laminar jet (Re=). The two methods show very similar results : the oscillation amplitude follows roughly an exponential up to a distance of x/h 8 where h is the thickness of the slit. Looking closer at the upper part of figure 8 shows that the cross-correlation method allows to track the oscillation at distances larger (x/h 8.) than the morphological method x/h 7.. Both methods seems to allow to track the phase of the oscillation up to distances of x/h 9. where the amplitude curve clearly shows that the linear description of jet oscillation is no longer valid, due to non-linear roll-up of the jet [4] and triggering of turbulence []. Oscilation phase [rad] Oscilation amplitude [m] 6 x 4 Amplitude of the oscillation, str =.23624,f = 76.3 estim. ampl. (morpho) fit of the ampl. (morpho) estim. ampl.(cross cor) fit ampl. (cross cor) Phase of the oscillation vs distance from the flue exit estim phase (morpho) fit phase (morpho) estim phase (cross cor) fit phase (cross cor) fit phase 2 (cross cor) Figure 9: Amplitude and phase of the jet transverse displacement of the jet using the cross correlation method (pink line) and the morphological method (green line), excitation frequency = 76.3 Hz (St=.2362), Turbulent jet (Re=3). The two methods give similar results in the turbulent case, as shown in figure 9. It should be pointed out that, due to the lower relative acoustic excitation amplitude in the turbulent case (v ac /U j = 2.7e 3), the transverse jet displacement is much smaller on figure 9 than in the laminar case (v ac /U j = e 3). Analyses in figure 9 therefore correspond to a very severe case of a turbulent jet with very small transverse displacement (η max /h.3 compared to η max /h.8 for figure 8). Regarding the phase shifts estimated in figure 9, the slope of the curves increases with distance from flue-exit as expected in the case of a rapidly slowing jet velocity induced by turbulence. 48

5 Forum Acusticum 2 Budapest Discussion The results obtained with cross-correlation and morphological methods can be compared in a more general way by considering the evolution of the estimated parameters as function of the frequency in the case of a laminar jet as shown figure and in the case of a turbulent jet in figure. Both methods allow to estimate fairly well the.6.4 lumiere_droite cross corelation lumiere_droite morpholohical The application of the methods presented can be illustrated by the comparison of different flue channel geometries, as studied by Segoufin [] comparing the convection velocities of perturbation on jets issuing from a long channel compared to a short channel. In the case of the short channel with squared exit, the convection velocity is clearly faster than in the other cases, confirming the observations done by Ségoufin [] in the context of edge-tones. This can be interpreted, as suggested by Ségoufin, in terms of the boundary layers thickness of the jet, which becomes thinner in the case of short channel..9 short squared rounded 4 degrees Figure : Convection velocity of the perturbations on the jet as estimated through cross-correlation and morphological methods, laminar jet (Re=)..6 turbulent_3_n cross corelation turbulent_3_n morpholohical Figure 2: Dimensionless velocity, flue exit short (dashed), squared (dashdot), rounded (solid) and 4 degrees (dotted). References Figure : Convection velocity of the perturbations on the jet as estimated through cross-correlation and morphological methods, turbulent jet (Re=3). amplitude as well as phase evolution of the perturbation. The main difference is that the cross-correlation method performs an inter-image analysis while the morphological method is an intra-image technique. [] B. Fabre and A. Hirschberg. Physical modeling of flue instruments: a review of lumped models. Acta Acustica, 86:99:6, 2. [2] W. Merzkirch. Flow Visualization. Academic press Inc, 987. [3] J. W. S. Rayleigh. "The theory of sound". Dover, New York, 877. reprinted from the first edition in 94. [4] P. de la Cuadra and B. Fabre. Analysis of jet instability in flute-like instruments by means of image processing: effect of the excitation amplitude. ISMA, International Symposium on Musical Acoustics, 24. [] C. Ségoufin, B. Fabre, and L. de Lacombe. Experimental investigation of the flue channel geometry influence on edge-tone oscillations. Acustica/Acta Acustica, 9:966 97,