Intermittent sound generation in a free-shear flow

Transcription

1 16th AIAA/CEAS Aeroacoustic Conference and Exhibit, 7-9 June 21, Stockholm Intermittent sound generation in a free-shear flow André V. G. Cavalieri, Peter Jordan and Yves Gervais Institut Pprime, CNRS - University of Poitiers - Ensma, UPR 3346, Poitiers, France Mingjun Wei New Mexico State University, Las Cruces, NM, USA Jonathan B. Freund University of Illinois at Urbana-Champaign, Urbana, IL, USA Comparisons are made between direct numerical simulations of uncontrolled and optimally controlled mixing layers in order to understand what it is about the controlled flows that makes them substantially quieter. Special attention is paid to the possibility that the essential details of the source mechanism may be spatially and/or temporally localised: such features are hidden when second-order statistics such as spectra are considered; and indeed these are almost identical for the two flows. Analysis is thus performed in the time domain, in order to search for intermittent sound-producing events. The results show that a large-amplitude pressure wave associated with a triple vortex merger in the uncontrolled mixing layer contributes significantly to the farfield, and that this event has been eliminated in the controlled flow. The large amplitude pressure wave associated with this event appears to be due to two things: the axial concentration of a low-pressure zone associated with the merging of the three vortical structures on one hand, and an axially-extended high-pressure region which opens up in the low-vorticity region immediately upstream of the three said structures. These pressure distributions can be mechanistically understood in terms of centripetal forces associated with the vortex dynamics, and the sound production associated with this can be mechanistically understood in terms of the axial imbalance that occurs between the spatially-localised low pressure and the spatially extended highpressure. Having understood the above, we proceed to analyse a longer time-run simulation of the uncontrolled flow, to see if we can objectively extract similar events. We apply a wavelet transform to the radiated pressure field, and by means of this we identify a collection of similar signatures. In each case we find that these correspond to a similar mechanism. The results highlight the importance of considering sound-producing flows in the time domain, and using appropriately adapted signal processing. The implications for noisesource modelling, which are often based on second-order statistics, are also discussed. I. Introduction Although research in aeroacoustics has made considerable progress since the pioneering work of Lighthill, 1 it remains unclear how free turbulence generates sound, and it is thus difficult to propose technical solutions which might reduce the sound power radiated by propulsive jets. In this context, the direct numerical simulation (hereafter DNS) constitutes a valuable tool for studying the underpinning physics of the noise production by such flows. In our DNS the equations governing compressible, viscous flow equations are solved without modelling approximations, and this solution includes the sound field radiated by the flow. PhD student, Institut Pprime, 43 rue de l Aérodrome, 8636 Poitiers, France. Assistant Professor, ITA, Praça Mal. Eduardo Gomes, 5, Vila das Acácias, , São José dos Campos - SP, Brazil. Research Scientist, Institut Pprime, 43 rue de l Aérodrome, 8636 Poitiers, France. Professor, Institut Pprime, 4 avenue du Recteur Pineau, 8622 Poitiers, France. Assistant Professor, Mechanical and Aerospace Engineering, POBox 31/Dept 345 Las Cruces, NM Associate Professor, 211B Mechanical Engineering Laboratory, 126 West Green Street, MC-244, Urbana, IL of 1

2 Although this kind of simulation is limited to flows with low Reynolds numbers, it does allow a study of the fundamental vortex dynamics observed in turbulent flows; for reviews of these applications, see Colonius and Lele 2 and Wang et al. 3 The work of Wei and Freund 4 provides a valuable opportunity for studying sound-production mechanisms, and, in particular, for understanding what can be done to an unsteady vortical flow in order to make it significantly quieter. A two-dimensional, spatially-evolving mixing-layer was computed, and by means of an adjoint-based formulation a series of optimally-controlled flows were produced and compared with the uncontrolled baseline flow. Reductions in sound intensity of up to 6dB in the far acoustic field were achieved; however, the differences between the controlled and uncontrolled flows were found to be subtle: comparison of the statistics of the various flows showed only very slight differences, despite the one-order-of-magnitude reduction in radiated sound power. An analysis based on a Proper Orthogonal Decomposition (POD), where these modes served a surrogates for streamwise Fourier modes in this streamwise inhomogeneous flow, showed that there was an underlying organisation of the large structures, by the control, consistent with their more uniform streamwise advection. Some subsequent work addressed the differences between the noisy and quiet mixing-layers from other perspectives. Eschricht et al. 5 showed that there are differences in the wavenumber frequency spectra of the hydrodynamic pressure fields of the uncontrolled and controlled flows, less energy being found in the radiating sector of controlled-flow spectrum. In the same work an analysis based on causality-correlations with the Lighthill source term was presented. The approach showed the two-point, two-time correlations of the Lighthill source term to have been degenerated, the controlled flow comprising a space-time structure with more effective source cancellation, making it a less efficient generator of sound. While these analyses provide a statistical perspective on the differences between the noisy and quiet mixing-layers, the precise changes in the flow remained unclear. These approaches, being based on statistics, potentially mask the importance of local space-time events. They preclude the evaluation of intermittent events in the flow, and the extent to which these may be important in the production of sound. Such intermittency has been found to be important in previous experimental and numerical jet noise studies. For a turbulent jet high levels of intermittency are observed at the end of the potential core. Juvé et al. 6 showed for a Mach.9 jet, using causality correlations, that in this region, for the radiated sound at an angle of 3 to the jet axis, 5% of the sound is generated in 1 2% of the time. Guj et al. 7 performed conditional averages of turbulent velocity signals, acquired in a low-subsonic jet, using peaks in the far-field pressure as a trigger for event selection. The flow structures identified are characterised by exponentially-decaying functions for the separation-time probability density, which is an indication of their intermittency. Hileman et al. 8 performed a similar study on an ideally-expanded Mach 1.28 jet. Large amplitude events in the far-field pressure signals, which were analysed in the time domain, were used to select corresponding images from flow visualisations. A subsequent POD analysis of these noise-producing events allowed a characteristic flow signature to be obtained, and this identified the intermittent intrusion of turbulent structures into the potential region of the flow as the event associated with the high-amplitude farfield signatures. The same techniques were used by Kastner et al. 9 to the DNS of a Mach.9 jet of Freund; 1 similar intermittent bursts were detected in the acoustic field, and the turbulence was shown to comprise a truncation of a wave-packet structure, which is consistent with the conclusions of the experimental Mach 1.28 jet. Similar results have been reported by Bogey and Bailly 11 using Large Eddy Simulations of subsonic jets. At the end of the potential core intermittent vorticity bursts were observed, and these were correlated with positive pressure peaks at an angle of 4 in the far-field. In this paper, we analyse temporal data from the uncontrolled and controlled mixing layers of Wei and Freund 4 with a view to identifying intermittent events: we look for spatially- and/or temporally-localized signatures associated with strong sound production. In section II.A we show that most of the sound in the far acoustic field of the uncontrolled flow is associated with a single event. In section II.B we look at the vortex dynamics of both controlled and uncontrolled flows, and we identify the essential difference between these the loud flow event a triple vortex interaction which leaves behind it an extended region of quasiirrotational flow. This localised event momentarily disrupts the source cancellation mechanism and a strong pressure wave is emitted. In the controlled-flow the triple merger is prevented, and the source interference persists with much the same efficiency for the entire duration of the simulation. Finally, in section III, we use a continous wavelet transform to analyse the pressure data taken from a longer-run simulation of an uncontrolled mixing layer. We use the wavelet transform to objectively identify noisy events. Events of the same nature as those identified by comparing the controlled and uncontrolled flows are found. 2 of 1

3 II.A. Pressure data at the target line II. Results The data used in this work is the same presented by Wei and Freund; 4 we will therefore only briefly describe the flow. A two-dimensional mixing layer was computed by direct numerical solution of the equations governing compressible, viscous flow. The calculations were performed for a flow with Reynolds number ρ Uδ ω /µ = 5, where ρ is the ambient density common to both streams, U is the velocity difference between the streams, δ ω is the inflow vorticity thickness, and µ is the constant viscosity of the fluid. The Mach numbers for the free streams are.9 and.2, and the Prandtl number is equal to.7. The physical domain is rectangular, extending horizontally from x = to x = 1δ ω and from y = 8δ ω to y = 8δ ω. Wei and Freund 4 have already pointed out that although there are significant differences in the radiated noise between the uncontrolled and controlled flows, the changes in the flow are very slight. In order to identify and understand the differences between the mixing layers, we first evaluate how the radiated pressure field is changed by the control. Since the control objective is the noise reduction on a horizontal target line located at y = 7δ ω on the M =.2 side of the mixing layer, we use this line to identify the space-time intervals where the sound reduction is most effective. Figure 1 shows pressure signals for three points on the target line: one upstream point, one centered and one downstream. By means of these figures the spatial and temporal locality of the sound reduction can be assessed: sound reduction is more effective at the downstream point, and a significant portion of the noise reduction at this point occurs over a limited time-interval. (p p)/p (c) Figure 1. Pressure temporal signals for the ( ) uncontrolled and ( ) controlled mixing layers at the points = 2, = 5 and (c) = 8 of the y = 7δ ω target line. The most notable reduction in figure 1(c) happens between ta /δ ω = 3 and ta /δ ω = 42. In this interval the uncontrolled flow presents a large positive pressure peak, followed by a negative peak of similar level; the controlled flow presents much smaller amplitude presssure waves during the same period. We thus see that most of the sound reduction at the considered point is achieved by eliminating this one prominent peak; in fact, 7% of the noise reduction at this point happens during the period 3 < ta /δ ω < 42. This indicates that the noise reduction is due to the elimination of a single, acoustically-important, intermittent event in the flow. II.B. Flow dynamics for the uncontrolled and controlled flows Figure 2 shows visualisations of the pressure and vorticity fields for the uncontrolled mixing layer at six different times, and in figure 3 visualisations for the controlled case are shown at the same times. The time ta /δ ω = corresponds to the arrival of the large positive pressure peak at the point (8δ ω, 7δ ω ) for the uncontrolled mixing layer, as shown in figure 1. In figure 2(d) this can be seen as a group of positive contours that pass by the considered point. For the controlled mixing layer, we also verify positive contours in the figure 3(d) around the point (8δ ω, 7δ ω ), but with a much smaller amplitude. The instant ta /δ ω = was chosen by calculating the propagation time of a wave in a uniform flow at M =.2 between the points (8δ ω, 2δ ω ) and (8δ ω, 7δ ω ). For the uncontrolled mixing layer, we can see in figure 2 that there is a high pressure region around the point (8δ ω, 2δ ω ), close to the zero-vorticity region lying between the pair of vortices at x = 6δ ω and a zone of vorticity which has just crossed the downstream boundary of the computational domain. By following in succession figures 2 and 2(c) we see how this high pressure propagates to the points we considered on the target line. We see also that there is propagation of a similar high-amplitude pressure wave to the M =.9 side of the mixing layer. 3 of 1

4 8 8 (c) (d) (e) (f ) y/δω y/δω Figure 2. Visualisations of the uncontrolled mixing layer at times = 312.5, 341., (c) 354.5, (d) 367.9, (e) 383. and (f ) Center: vorticity module, with contour levels from.7 U/δ ω to 1.4 U/δ ω; outer regions: pressure field, with contour levels from.2p to.2p. Full lines correspond to positive contours, and dashed lines to negative ones. 8 8 (c) y/δω y/δω (d) (e) (f ) Figure 3. Visualisations of the controlled mixing layer. Center: vorticity module; outer regions: pressure field. Same contours and times of figure 2. 4 of 1

5 A similar propagation occurs in the controlled mixing layer, but with a much smaller amplitude, as seen in figures 3 (d). The clearest difference in the vortex dynamics between this flow and the uncontrolled mixing-layer is the presence of a vortical region near x = 9δ ω for the controlled mixing layer at ta /δ ω = This last vortex is not visible in figure 2. The negative pressure peak for the uncontrolled mixing layer at the point (8δ ω, 7δ ω ), shown in figure 1(c), corresponds to the instant ta /δ ω = shown in figure 2(f ). As was done for the positive pressure wave, we calculate the propagation time of this wave to search for its origin, and the resulting time is ta /δ ω = 341., shown in figure 2. We can see in the uncontrolled mixing layer that a vortical structure enters the long quasi-irrotational, high-pressure region of flow. As this vortex is convected to the end of the computational domain we see, in figures 2(c) (f ), the formation of a low-pressure wave, which propagates to the far field and arrives at (8δ ω, 7δ ω ) with a high amplitude. The same instants are also portrayed for the controlled mixing layer, where again we see the propagation of a negative pressure wave to the coordinate (8δ ω, 7δ ω ) (figures 3 (f )), but with a considerably reduced amplitude. Although the vorticity-contours of the mixing layers at ta /δ ω = 341. the instant when the low pressure wave originates are similar, for the uncontrolled flow we see a pocket of vorticity enter an extended region of irrotational flow; for the controlled mixing layer, the passage of a similar concentration of vorticity occurs, but in a situation where the distances between succesive vortical regions are more balanced. If we consider the generation of sound-waves from a free shear flow to be the result of incomplete interference between regions of positive and negative stress, or pressure, as retarded-potential type solutions for the radiated pressure would have us believe, then we see that the occurence of an intermittent event can significantly disrupt the interference for a brief period of time. During this period, the mutual cancellations which occur between neighbouring vortices hold to a much lesser degree, making possible the propagation of a large amplitude sound wave. This assertion is consistent with the conclusions drawn by Wei and Freund 4 and Eschricht et al, 5 but in this case we can pinpoint a particular flow event and thus propose a local explanation, free from the cloudiness of averaging. It is interesting that the high-amplitude sound-wave comprises a compression followed by a depression, and not vica-versa. The observation may be of some use for mechanism identification and modelling; similar waveforms are observed in the two-dimensional, temporal simulation of Fortuné et al. 12 The difference in the evolution of the vorticity of the uncontrolled and controlled mixing layers is due to a triple vortex interaction which occurs in the uncontrolled flow. Figure 4 shows the temporal evolution of this triple merger up to the instant ta /δ ω = 312.5; this time corresponds to figure 2. We see that this triple interaction, which happens only once in the simulation of the uncontrolled mixing layer, is the cause of the extended irrotational region in figure 2. A large vortical structure is created by the merger. The two less intense vortices rotating at high speed around the more intense structure, causing the conglomeration to exit the flow domain earlier than their controlled counterparts, shown in figure 4. The controller modifies the flow dynamics such that the triple interaction is eliminated; this reduces the extent and level of the high-pressure irrotational region, and, as we have seen, the amplitude of the propagated sound wave Figure 4. Instantaneous vorticity module for the uncontrolled and controlled mixing layers at times = 245.3, 262.1, 278.9, and 312.5, from top to down. Contours range from 4 U/δ ω to.7 U/δ ω. So far we have only showed results for the y-direction body force actuation. But similar results are 5 of 1

6 observed for the other types of control. In all cases the triple vortex interaction is eliminated, and with it the extended region of high pressure. Figure 5 shows, for each of the control cases, the vorticity field at ta /δ ω = 312.5, which corresponds to the streamwise extensive high pressure region previously identified in the uncontrolled flow. Although there are slight differences between the vortex patterns, all of them present a smaller zero-vorticity region than the one shown for the uncontrolled flow in figure 2; the extent of this region is much closer to what is found at other times. Thus, all of the control formulations prevent the formation of the extended high pressure zone, demonstrating how this event constitutes the essential difference between the controlled and uncontrolled flows. (c) (d) (e) Figure 5. Vorticity contours at = for the flows with no control, mass source control, (c) x-direction body-force control, (d) y-direction body-force control and (e) internal-energy source control. Same contours of figure 4. III. Detection of intermittent radiation With a view to making the analysis procedure more objective, we analysed the pressure signal at the same point (8δ ω, 7δ ω ) by means of a continuous wavelet transform. Instead of projecting the time signal onto a set of Fourier basis functions, which are localised in frequency but infinitely extended in time, a projection is made onto a set of wavelet basis functions, which are localised in both time and time-scale (or pseudo-frequency). Further information and mathematical foundations of the wavelet transform can be found in the review article of Farge 13 and the references therein. The numerical formulation of Torrence and Compo 14 was used for the numerical calculation of the transform. We used Paul s wavelet, defined for s = 1 with an order m as ψ(1,t τ) = 2m i m m! π(2m)! [1 i(t τ)] (m+1). (1) This is a complex-valued wavelet function; for m = 4 its real and imaginary parts are shown in figure 6. The choice of this wavelet function with m = 4 is due in particular to its imaginary part, which approximates the shape of the noisy signature of the uncontrolled mixing layer seen in figure 1(c), with a negative sign. We expect thus that similar signatures will have a continuous wavelet transform with high energy content in a small-scale range and in a reduced interval of the simulation. A longer DNS simulation of the uncontrolled flow was used (1 times that of Wei and Freund 4 ). The same code was used for the simulation, with an identical grid and the same simulation parameters. The results are obviously identical over the time interval considered in that paper. The continous wavelet transform was applied to the pressure signal at the point (8δ ω, 7δ ω ) of this long simulation, the same point used in the analysis of section II.A. The resulting scalogram is shown in figures 7 and. In figure 7, which corresponds to the original simulation up to ta /δ ω = 626.6, we see the scalogram for the first half of the long simulation. There is a clear peak identified at ta /δ ω 38, which corresponds to that analysed in section II.A. In figure 7, we see that in the second half of the simulation there are similar peaks centered at about ta /δ ω 4, ta /δ ω 44, and ta /δ ω 61. To isolate the events which correspond to the peaks in the scalogram, we performed a filtering operation. This filter is based on the selection of a threshold α: 6 of 1

7 ψ t τ Figure 6. Paul s wavelet: ( ) real and ( ) imaginary parts sa /δω sa /δω Figure 7. Wavelet spectrum. Levels are uniformely distribued between and.3(p δ ω/a ) 2 in steps of.3 (p δ ω/a ) 2. p f (s,t) = { p(s,t) if p(s,t) 2 > α if p(s,t) 2 < α The filtered pressure in the wavelet basis is then transformed back to the time domain by means of an inverse continuous wavelet transform. We chose as a filter α =.9(p δ ω /a ) 2, since above this value only the high energy concentrations shown in figures 7 and, analogous to that found at the beginning of the simulation, are thus retained. The results of this filtering operation applied to the beginning of the simulation are presented in figure 8. We see that the filtered pressure is zero in all but a reduced interval, where the triple vortex merger signature analysed in section II.A is captured. (2) (p p)/p Figure 8. Wavelet filtering: ( ) original and ( ) filtered pressures. For the remainder of the simulation, the reconstructed pressure signal is mostly zero, but at the intervals 7 of 1

8 where there are energy concentrations in both time and pseudo-frequency we see clear peaks for the pressure. We see in figure 9 that the filtered pressure around ta /δ ω = 4 has a pattern similar to that observed in figure 8, and which was related to the triple vortex interaction. In figure 9-(e), we see vorticity contours at the center of the mixing layer at instants prior to the arrival of the mentioned pressure peaks; we see a triple vortex merger similar to that shown in figure 4. (c) (p p)/p - (d) (e) Figure 9. Wavelet filtering, ( ) original and ( ) filtered pressures; -(e) vorticity module contours at = , = , = and = Contours range from 4 U/δ ω to.7 U/δ ω Figures 1 and 11 show results for the other intervals of the simulation where there are non-zero values for the filtered pressure. We see again in figures 1 and 11 a similar pressure signature as that of figures 8 and 9; and the corresponding vorticity contours, shown in figures 1-(e) and 11-(e), show that triple vortex mergers occur in each case, just prior to the arrival of the high-amplitude pressure wave at the considered point. (p p)/p (c) (d) (e) Figure 1. Wavelet filtering, ( ) original and ( ) filtered pressures; -(e) vorticity contours at = , = , = and = Same contours as figure 9 Although the reconstructed pressure signal is equal to zero for all the simulation time but the four events shown in figures 8, 9, 1 and 11, the filtered pressure has an RMS value equal to 23.6% of the value for the original pressure time series. We see thus that although the analysed events are rare in the dynamics of the mixing layer, they are responsible for a considerable portion of the sound power radiated by the flow. We can conjecture that an optimal control applied to the extended simulation would aim again to eliminate these noisy events. 8 of 1

9 (c) (p p)/p - (d) (e) Figure 11. Wavelet filtering, ( ) original and ( ) filtered pressures; -(e) vorticity contours at = 598.8, = , = and = Same contours as figure 9 IV. Conclusion An analysis was carried out to identify and understand the differences between the uncontrolled and controlled mixing layers of Wei and Freund. 4 The analysis of the pressure in the acoustic field showed that the noise reductions are especially significant at downstream radiation directions, and 7% of the sound reduction in this region is achieved by the suppression of a particular high amplitude pressure wave, comprising a compression followed by an expansion. A triple vortex merger, which occurs only once in the simulation, is shown to be indirectly responsible for these peaks in sound generation. In order to isolate similar noisy events, a longer DNS calculation for the uncontrolled mixing layer was performed, and the application of a continuous wavelet transform allowed the detection of similar peaks in the acoustic field; these again were found to be associated with triple vortex interactions. These are rare, intermittent events in the vortex dynamics of the mixing layer, but their contribution to the overall sound is considerable. This study demonstrates (confirming the results of previous findings) that such intermittency constitutes a major element in the production of sound by free shear-flows. It should thus be explicitly included in modelling strategies. In many statistical noise prediction schemes (typically Acoustic Analogy based) such intermittent events are not explicitly included, and indeed they are often explicitly excluded by certain modelling assumptions. It is possible that this may explain the poor robustness of nearly all current sound prediction schemes; these are mostly based on second-order turbulence statistics, which, as we have seen, can entirely miss the most important sound producing events. The uncontrolled and controlled mixing layers are superficially identical according to second-order statistics. In terms of the details of the mechanism by which the propagative wave is set up, our study shows that this can be explained in terms of the kind of multi-pole interference which retarded-potential type solutions imply. This supports the idea that the Acoustic Analogy models are conceptually correct, being made inaccurate only by the non-inclusion of intermittency effects. These points are the subject of ongoing modelling work (see companion paper Cavalieri et al. 15 ) Acknowledgments The present work is supported by CNPq, National Council of Scientific and Technological Development Brazil. JBF and MW acknowledge support from AFOSR for the original mixing layer control simulations. JBF acknowledges support from NASA for this work. References 1 Lighthill, M. J., On sound generated aerodynamically. I. General theory, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 1952, pp of 1

10 2 Colonius, T. and Lele, S., Computational aeroacoustics: progress on nonlinear problems of sound generation, Progress in Aerospace Sciences, Vol. 4, No. 6, 24, pp Wang, M., Freund, J., and Lele, S., Computational Prediction of Flow-Generated Sound, Annual Review of Fluid Mechanics, Vol. 38, 26, pp Wei, M. and Freund, J. B., A noise-controlled free shear flow, Journal of Fluid Mechanics, Vol. 546, 26, pp Eschricht, D., Jordan, P., Wei, M., Freund, J., and Thiele, F., Analysis of noise-controlled shear-layers, 13 th AIAA/CEAS Aeroacoustics Conference(28 th AIAA Aeroacoustics Conference), Vol. 211, American Institute of Aeronautics and Astronautics, 181 Alexander Bell Drive, Suite 5, Reston, VA, , USA,, 27, pp Juvé, D., Sunyach, M., and Comte-Bellot, G., Intermittency of the noise emission in subsonic cold jets, Journal of Sound and Vibration, Vol. 71, 198, pp Guj, G., Carley, M., Camussi, R., and Ragni, A., Acoustic identification of coherent structures in a turbulent jet, Journal of Sound and Vibration, Vol. 259, No. 5, 23, pp Hileman, J. I., Thurow, B. S., Caraballo, E. J., and Samimy, M., Large-scale structure evolution and sound emission in high-speed jets: real-time visualization with simultaneous acoustic measurements, Journal of Fluid Mechanics, Vol. 544, 25, pp Kastner, J., Samimy, M., Hileman, J., and Freund, J., Comparison of Noise Mechanisms in High and Low Reynolds Number High-Speed Jets, AIAA journal, Vol. 44, No. 1, 26, pp Freund, J. B., Noise sources in a low-reynolds-number turbulent jet at Mach.9, Journal of Fluid Mechanics, Vol. 438, 21, pp Bogey, C. and Bailly, C., Investigation of sound sources in subsonic jets using causality methods on LES data, AIAA Paper, Vol. 2885, 25, pp Fortune, V., Lamballais, E., and Gervais, Y., Noise radiated by a non-isothermal, temporal mixing layer. Part I: Direct computation and prediction using compressible DNS, Theoretical and Computational Fluid Dynamics, Vol. 18, No. 1, 24, pp Farge, M., Wavelet transforms and their applications to turbulence, Annual Review of Fluid Mechanics, Vol. 24, No. 1, 1992, pp Torrence, C. and Compo, G., A practical guide to wavelet analysis, Bulletin of the American Meteorological Society, Vol. 79, No. 1, 1998, pp Cavalieri, A. V. G., Jordan, P., Agarwal, A., and Gervais, Y., Jittering wave-packet models for subsonic jet noise, AIAA Paper , 16th AIAA/CEAS Aeroacoustic Conference and Exhibit, Stockholm, Sweden, June of 1