A study in the near pressure field of co-axial subsonic jets

Transcription

1 th AIAA/CEAS Aeroacoustics Conference (th AIAA Aeroacoustics Conference), 8- May, 6, Cambridge, MA A study in the near pressure field of co-axial subsonic jets Charles E. Tinney Peter Jordan Antoine Guitton Joel Delville Laboratoire d Etudes Aérodynamiques, CNRS UMR 66, University of Poitiers, France Francois Coiffet Florida State University, Tallahassee, Florida, USA An experimental investigation of the near pressure field of unbounded subsonic jets has been performed. The near-field pressure was sampled, using linear and azimuthal arrays, on conical surfaces surrounding free jets generated by () a single axisymmetric nozzle, () a co-axial short-cowl nozzle, and () a co-axial short-cowl nozzle with serrations (the co-axial experiments were performed as part of the EU program, CoJeN (AST-CT--), where velocity and temperature-ratios were varied). The objective of the study is to better understand differences in the structure of the flows in terms of their sound production mechanisms. A model representation of the source mechanism associated with coherentstructures in the flow is considered, using both the pressure fluctuations themselves and the pressure-derivative source term from Curle s acoustic analogy. A filtering operation is then applied in order to identify the structure of the radiating source field. I. Introduction The irrotational pressure field surrounding subsonic jets has been the object of a considerable number of experimental investigations over the past half-century, whence researchers have endeavoured to better understand the nature of the large-scale flow dynamic and its sound production mechanisms. While this kind of measurement approach presents a number of advantages - pressure is a scalar quantity; it acts as a natural wavenumber filter (George et al. ); near-field measurements are relatively non-intrusive - it should not be forgotten that certain difficulties exist concerning physical interpretation of the pressure field in this region of the flow, and in particular when we try to relate it to the underlying turbulence. The near pressure field is governed by a linear hyperbolic differential equation, while it is essentially driven by a non-linear hydrodynamic pressure field which is well approximated by elliptic equations. Some qualification is thus required if we want to interpret measurements performed in the linear region in terms of the underlying non-linear mechanisms. We consider therefore some characteristics of these very different regions of the flow: The non-linear hydrodynamic region (the rotational part of the flow) is characterised by kr / (George et al. ), the linear hydrodynamic region (irrotational near-field) by kr / (Arndt et al. ) and the linear acoustic region (acoustic far-field) by kr (Arndt et al. ) where kr is a non-dimensional wavenumber (k and r are wavenumber and distance respectively). We thus see that in the linear hydrodynamic region of the flow (interpreted as a reactive acoustic field by Coiffet et al. ), for a given wavenumber, there is an extremely rapid spatial decay, r /. We can therefore expect pressure signatures registered in this region of the flow to constitute relatively local information concerning the large turbulence structures which are predominant in driving this part of the pressure field. And so, by positioning pressure sensors just outside the rotational region of the flow, we consider that the fluctuations we measure present a reasonable footprint of the underlying large-scale flow structure. LEA/CEAT, UMR CNRS 66, 866 Poitiers, France. FAMU-FSU College of Engineering, Pottsdamer St., Tallahassee, FL, 6, USA Copyright c 6 by P. Jordan. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. of American Institute of Aeronautics and Astronautics Paper 6-8

2 Now the picture of a sound production mechanism comprised of distinctly different mechanisms, associated with coherent structures on one hand, and relatively incoherent, small-scale turbulence on the other, is becoming increasingly accepted,, ). However, the question as to the precise mechanisms by which each of these flow structures generate sound remains an open one. We have previously suggested that it may make sense to further separate the mechanism associated with the coherent structures into a quasi-irrotational component (associated with the strain-deformation part of the Lighthill source term), due to wavy-wall-type behaviour involving growth, saturation and decay phases as structures are convected downstream, and a rotational component (associated with the rotational deformation part of the Lighthill source), more intimately related to the internal dynamic of the large-scale structures. a By considering the near pressure field to constitute a convective footprint of the large-scale structures we use the nearfield measurements to effect an estimate of a quasi-irrotational source, using both the signature itself and the surface pressure term which appears in Curle s acoustic analogy. 6 In effect we replace the jet with a model problem, constructed using the nearfield pressure signatures. In this paper results are presented from measurements performed in the nearfield of single and co-axial jets (jet conditions are described in the next section). The pressure fields are studied by means of linear and azimuthal microphone arrays with a view to better understanding how changes in the jet exit conditions (velocity ratio, temperature ratio, nozzle serrations) modify the large-scale behaviour of the flow. The temporal, axial and azimuthal structure of the pressure field is studied by means of correlation and Fourier analysis in (x, θ, t) and Proper Orthgonal Decomposition in x. Furthermore, as stated above an attempt is made to access information which is more directly related to the sound source mechanisms of the flow. Using Curle s acoustic analogy, spatial derivatives of the pressure are identified as an appropriate quantity to study. Spatial interpolations and appropriate windowing functions are thus implemented in order to facilitate twodimensional Fourier transforms (from p(x, t) and p(x, t)/ x to ˆP (k, ω) and k x ˆP (k, ω) ), whence a filtering operation is used to identify the components of the pressure field which are implicated in the generation of radiating pressure disturbances (those components which satisfy ω > ka, where ω, k and a are circular frequency, wavenumber and speed of sound respectively). Finally a Spectral Linear Stochastic Estimation is used to effect a full space-time reconstruction of the pressure and its derivatives on a conical surface surrounding the jet. II. Experimental setup Measurements were performed in the nearfield of jets issuing from three different nozzles:. Single-stream nozzle.. Short-cowl, co-axial nozzle (hereafter SCN). Short-cowl co-axial nozzle with serrations (hereafter SN). The experiments were conducted at two different facilities. For the single-stream nozzle [], experiments were performed at the Laboratoire d Etudes Aérodynamiques (LEA) in France. The setup, shown in figure a, consists of a line array of microphones, inclined at degrees to the jet axis as to follow the expanding flow of a Mach. jet at ambient temperature, and an azimuthal array comprising 8 microphones. Details regarding the experimental arrangement are described by Ricaud. The jet diameter is mm and the first microphone is located at r/d =.8 and x/d =. The azimuthal array was displaced in the axial direction and measurements were acquired at each axial microphone station. The radial displacement of the azimuthally arranged microphones was adjusted to follow the expanding flow, according to the reference microphone on the line array. A global conditioning technique (see A-) was applied in order to correct for discrepancies along the line array measurements between individual experiments. These individual experiments comprised moving the azimuthal array of microphones to new positions on the line array. More detail on this is provided in the next section concerning measurements of the co-axial jets as the experimental procedures were identical. The SCN experiments [ & ] were performed at the QinetiQ Noise Test Facility (NTF) in Farnborough, England. The microphone arrays are shown in figure b, the nozzles in figure. For each of the co-axial a Of course this in no way suggests that the turbulence is irrotational, rather it poses the question as to whether some part of the sound production mechanism associated with the coherent turbulent structures behaves in a quasi-irrotational manner, by virtue of their convection and the way in which this interacts with the irrotational nearfield. of American Institute of Aeronautics and Astronautics Paper 6-8

3 (a) (b) Figure. Experimental arrangement of the (a) single-stream nozzle at LEA, and (b) co-axial nozzles at the Noise Test Facility (NTF), QinetiQ. nozzles, short-cowl (SCN in figure a) and short-cowl with serrations (SN in figure b), the temperature and bypass ratios were varied according to the three conditions shown in table. (a) (b) Figure. (a) Short-cowl nozzle (SCN ). (b) Short-cowl nozzle with serrations (SN ). A line array of 8 microphones was mounted so as to follow the expanding jet (estimated to be around 8. o based on preliminary PIV measurements), while a circular array comprising fifteen microphones (the sixteenth being the one on the line array) was used to capture the azimuthal pressure field at nine axial stations. The locations of these nine stations are shown in figure a ( ) and were the same for both nozzles and all three conditions. For each microphone on the azimuthal array, a dedicated servo-motor moved it radially to match the radial position of the reference microphone on the line array for each new axial station. Positioning of the azimuthally arranged microphones was effected by matching a block averaged measurement of the reference microphone s rms voltage to that of each microphone on the azimuthal array and adjusting the position of the azimuthal microphone accordingly. The differences in the radial positions of the azimuthal array of microphones was of order cm and the variance of all of the microphone s sensitivities relative to the mean, was less than %. The /inch microphones and matching preamplifiers were MI- of American Institute of Aeronautics and Astronautics Paper 6-8

4 condition U s [ms ] U p [ms ] U s /U p T so [K] T po [K] T so /T po Table. Table of jet exit conditions for the co-axial and serrated nozzles. CROTECH GEFELL b type MK and MK, respectively, and were sampled at 6Hz for seconds using bit accurate A/D converters over a ±v bipolar range. Based on the nozzle exit conditions illustrated in table, the largest time scales of the flow estimated from U s /D s, where D s is.mm, demonstrates that there were a minimum of statistically independent samples. A sample profile of the rms pressure from the line array of microphones is shown in figure a. As a small number of microphones failed during the experiment, Spectral Linear Stochastic Estimation 8 (SLSE) was employed in order to regenerate the time series for the missing points. The form for the spectral estimate is as follows: p(x, f) = B j (x ; f)p j (f) () p j (f)p k(f) B j (x ; f) = p(x, f)p k(f) () and comprises an ensemble average of cross spectral densities. However, in order to construct the conditionally averaged term on the right-hand side of (), a cubic spline interpolation was used based on the conditional average of the cross spectral densities between points surrounding the missing data. The SLSE is thus used to reconstruct the time series for the missing events based on an interpolation of its averaged cross spectra with neighboring events. The results of this are displayed in figure a to show the effectiveness of the estimation procedure. Profiles were then generated by fitting two rd order polynomial curves to the original data. Special attention was given to the region near the jet exit so as to preserve the original profile. The resultant profiles are shown in figure b for both co-axial nozzles under all three conditions. III. Results and discussion In figure b the influence of the nozzle geometry and jet exit condition can be seen. Near the nozzle exit the serrations have clearly increased the hydrodynamic pressure when compared to the SCN. The profile of the SN is more complicated over the first two jet diameters, an abrupt increase in energy being manifest at >, which most likely corresponds to the roll-up structure of the primary jet. Farther downstream however, the pressure field is less energetic for the SN, a considerable reduction of nearly db being observed. For a given nozzle, differences due to changes in the jet exit conditions are most pronounced farther downstream, where the pressure level in the near-field is seen to increase as the bypass and temperature ratios decrease, that is as U s /U p and T so /T po. This may be due to a fundamental change in the structure of the flow, or to changes in the relative position of the jet s shear layer as the exit conditions vary. In order to determine which effect is predominant, information is required from the velocity field. This data is currently being processed. The noise production mechanism associated with coherent structures is sometimes explained by comparison with an instability wave which amplifies, saturates and decays, this process being central in the production of acoustically matched source components (in particular, abrupt amplification and decay can render a subsonically travelling instability wave efficient in generating noise ). Interpreted in this way, the serrations can be seen to have an effect on the amplification process. By comparison the pressure field of the single nozzle jet (figure c,) demonstrates a difference in peak energy of about db compared to the SCN, and a much smoother growth near the nozzle exit. The amplification phase looks similar to the SCN. b Specific information regarding these instruments can be found at of American Institute of Aeronautics and Astronautics Paper 6-8

5 [db] raw estimated fit azimuthal 6 8 (a) [db] SCN,c= SCN,c= SCN,c= SN,c= SN,c= SN,c= 6 8 (b) log (σ p /x ) [db] Single jet 6 8 x/d (c) Figure. Pressure rms profiles. (a) Comparing the original raw data to the estimated, and to the joined rd order polynomial curve fits. Axial positions where the azimuthal array was traversed is identified ( ). (b) Profiles of the curve fits for all conditions of the Short-Cowl (SCN) and Serrated (SN) Nozzles. (c) Profiles from the single stream jet. of American Institute of Aeronautics and Astronautics Paper 6-8

6 A. Nearfield spectra Pressure spectra from the co-axial jets were determined from the line array data and are shown in figure. A % bandwidth moving filter has been used. The results again show the SN to generate higher pressure levels close to the exit plane (figure a). However, immediately downstream, at =, the SCN flow has higher levels, though the spectral densities have very similar shapes. It is mostly in the higher frequencies that differences in spectral shape become noticeable for different nozzle exit conditions. It is argued in the next section that the spectra can be divided in two regions, one in which the dynamic is dominated by hydrodynamic pressure fluctuations - the signature of the coherent structures of the flow - and a second high frequency region where the pressure fluctuations are predominantly acoustic. Thus, the differences observed in the higher frequencies are related more to differences in the sound fields radiated by the different jets, and less to the signature of the coherent structures. With this in mind, it is noteworthy that the hydrodynamic shape, manifest in the frequency range < f < khz changes very little for changes in the velocity ratio, suggesting that the large-scale structure of the jet has not changed in a fundamental way. Based on these observations, the differences in the rms profiles in figure b may be related to increases in the propagating acoustic energy in the higher frequency regions of the pressure spectrum, the large-scale hydrodynamic events having less of an effect. Closer observation of figure b at = shows that the peak frequency for the SCN is higher than for the SN, whereas at = 6 in figure d, the peak frequency for the SCN is now lower than the SN. This phenomenon is more clearly visible in figure e where the axial evolution of the spectra are shown for the SN and SCN flows. The effect of the serrations can be seen to completely change the evolution of the time scales of the flow. While in the SCN flow the evolution involves a rapid change from high to low frequency, the SN flow exhibits a more gradual evolution. The spectra do tend to similar forms however, showing how the jet eventually recovers its preferred structure - the change in the initial conditions of the flow is evidently not as strong a factor as the subsequent influence of the mean velocity profiles where the large-scale evolution of the turbulence is concerned. B. Space-frequency coherence - hydrodynamic and acoustic signatures Coherence spectra from microphones on the line array are shown in figure for reference microphones at four axial stations. These are shown for the first nozzle exit condition only, as changes in the exit conditions were found to produce negligible changes. These plots furnish information concerning the nature of the pressure fluctuations at different frequencies. The spectra can be seen to comprise two fairly distinct regimes, < f < khz and f > khz, most marked in the range < <. It has been shown by Arndt et al. and Coiffet et al. how these regimes correspond to energy dominated by either hydrodynamic or acoustic pressure fluctuations. In the frequency range < f < khz the pressure field is characterized by extensive axial coherence, corresponding to a convective signature related to the coherent structures of the flow. Whereas at f > khz the axial coherence is considerably less extensive - the signature of a more chaotic sound field, comprising sound waves generated by more random fine-scale turbulence, and which propagate thus in many different directions, as opposed to the sound field from the more coherent events which tend to be more directional (peak towards degrees). The abruptness of the switch from hydrodynamic to acoustic dominance is striking, and similar to the results of Coiffet et al., where a correspondingly sudden change in phase velocity was observed (at the same frequency as the change in coherence pattern), from the convection velocity of the jet to speeds which far exceed the sound speed; these supersonic phase speeds are due to spherical sound waves propagating at large angles to the line array, thereby presenting extremely high phase velocities along the array axis. Like the spectral densities in figure, the most noticeable difference between the serrated and nonserrated co-axial jets occurs near the jet exit, whereas the shapes downstream look quite similar. This again suggests that while the energy of the structures which generate the near-field signature may have changed dramatically, their basic structure has not. C. The azimuthal structure A Fourier-azimuthal decomposition using data acquired by the azimuthally arranged microphones is shown in figure 6. The distribution of the modal energy is represented in terms of percentage of local total energy. These results are consistent with previous studies of the near pressure field of axisymmetric jets by Ko & Davies, Arndt et al., Jordan et al. and Tinney et al., where the most coherent, low-order turbulence 6 of American Institute of Aeronautics and Astronautics Paper 6-8

7 SN SCN, c= SCN, c= SCN, c= SN, c= SN, c= SN, c= SCN S pp (x,f) S pp (x,f) SN = SCN = S pp (x,f) f [Hz] (a) S pp (x,f) f [Hz] (b) = =6 f [Hz] (c) f [Hz] (d).d.d.8d.d.8d S pp (x,f) SCN, c= 6 f [Hz] SN, c= 6 f [Hz] 6 f [Hz] (e) (f) Figure. Pressure spectra of the short-cowl nozzle jets (SCN and SN) at x = (a) D s, (b) D s, (c) D s and (d) 6D s. (e) Spectra contours of the short cowl nozzles, condition. (f) Pressure spectra of the single stream jet. of American Institute of Aeronautics and Astronautics Paper 6-8

8 . x=.ds x=.ds x=.d x=.d s s f [khz].. x /D s (a). x=.ds x=.d s x=.ds x=.d s f [khz].. x /D s (b) Figure. The coherence spectra at various axial positions from the (a) short-cowl nozzle, and (b) serrated nozzle. structures have been found to be the most efficient in driving the near-field pressure. Again we see that there are only small differences, in terms of azimuthal organization, of the near pressure fields generated by the nozzles with and without serrations. It is worth noting however that the very slight changes in the evolution of the amplitude of the axisymmetric and helical modes may produce important changes in the associated sound production mechanism. Information is required from flow measurements (PIV and LDV, currently being processed) before further conclusions can be drawn from these modal evolutions. D. Lagrangian correlations Space-time correlations have also been calculated, and the Lagrangian correlations, evaluated using four different reference microphones, are shown in figure. This provides information concerning changes to the pressure field as seen by an observer who moves with the convection speed of the flow. Here we see some structural differences related to changes in the nozzle geometry (mostly in the near nozzle region) and in the exit conditions (most marked further downstream). In the near nozzle region both nozzles show bi-modal behaviour, with two characteristic correlation shapes manifest (see the second bump at x/ds.). The different correlation function shapes in this region show the SCN to have significantly longer integral scales c. The serrations are found to reduce this integral scale by a factor of about two, indicating that they cause structures in the near-nozzle region to decay twice as fast as in the SCN flow. Further downstream we again see that the structural differences between the flows generated by the two nozzles are very slight, however more noticeable differences are manifest for changes in exit condition. c this integral scale corresponds to the product of the moving-frame time scale and the convection speed (the reader is referred 8 of American Institute of Aeronautics and Astronautics Paper 6-8

9 Short Cowl Nozzle 8 6 x/d= condition condition condition % Serrated Nozzle 8 x/d= % 6 Single Stream Nozzle x/d= % Figure 6. Fourier-azimuthal modes from the short-cowl, serrated and single stream jets at various axial locations in the flow. The zeroeth mode is identified as the start of each axial station. IV. Acoustically matched source components A necessary condition for the production of a propagating sound field, discussed by Crighton, implemented by Freund and Cabana et al., 6 and proposed recently by Goldstein as a means of identifying the true sources of aerodynamically generated sound, is that the source wavenumbers be acoustically matched. This amounts to requiring that the source phase velocities be supersonic, i.e. ω > ka. The line array measurements furnish a space-time signature of the near pressure field, which can be used to estimate the distribution of energy in wavenumber-frequency space by means of a two-dimensional Fourier transform (a windowing function W (x) is applied to the pressure field in the axial direction so that the Fourier transform is smooth at the boundaries well defined over this interval) ˆP (k, ω) = π k ˆP (k, ω) = π W (x)p(x, t)e ikx iωt dxdt. () W (x) p(x, t) e ikx iωt dxdt. () x In this way k ω spectra can be obtained, whence the dispersion inequality ω > ka can be used to identify how much energy is contained in regions of k ω space for which phase velocities are supersonic. It is only to Kerhervé et al. for discussion on this point) of American Institute of Aeronautics and Astronautics Paper 6-8

10 ρ pp (x,x ).8.6. SCN, c= SCN, c= SCN, c= SN, c= SN, c= SN, c= ρ pp (x,x ) x /D s 6 8 x /D s (a) (b).8.8 ρ pp (x,x ).6. ρ pp (x,x ) x /D s (c) x /D s (d) Figure. The correlations p(x, t)p(x = τu c, τ) at x = (a) D s, (b) D s, (c) D s and (d) 6D s. this energy which can be efficient in producing sound energy which will propagate from the flow. The question now arises however as to which quantity to consider as a source. As discussed earlier, in order to approximate the behaviour of a source system which is principally driven by convection of the most coherent structures of the flow, we can use the surface pressure term in Curle s acoustic analogy, where the radiated sound is expressed as p a (r, t) = ( π x i x j V [T ij ] r dv [p i ] ) x i S r ds. () Using the nearfield pressure in this context, we can model the sound production mechanism as a surface supporting fluctuating pressures generated by the coherent structures. The appropriate source quantity in this case is the spatial derivative of the pressure ( p/ x i ). On the other hand, there is evidence to suggest that the hydrodynamic pressure field itself may constitute a meaningful source term (Laurendeau et al. have recently found that these pressure signatures correlate very strongly with the radiated sound field 8 ). We have thus chosen to study k ω distributions of both the pressure and its spatial derivative. Results are displayed in figure 8 using all three jet exit conditions, and both co-axial nozzles. The first two columns pertain to the SCN, the last two are for the SN. For each of the nozzles, both ˆP (k, ω) and k ˆP (k, ω) are plotted. Before going on to discuss the significance of the results, we must first revisit the question of hydrodynamic and acoustic contributions to the measurements, first addressed in relation to the nearfield power of American Institute of Aeronautics and Astronautics Paper 6-8

11 and coherence spectra in sections A and B. Because all sources of sound are to be found at a finite distance normal to the array axis, the line array will register sound waves which propagates across its extent at angles in the range < β < 8 (β is the angle between the propagation vector and the array axis). β can never be equal to or 8 degrees however, as this would imply a sound field issuing from a source lying on a line coincident with the array axis. Thus, the phase-speed of pressure perturbations which are truly acoustic in nature must always lie in the range c < U φ <, i.e. it can never be sonic or subsonic. This means that all energy on and above the line corresponding to the sound speed in the entrained flow in figure 8(red dash-dot line), which corresponds to pressure components with subsonic phase velocity, is devoid of any acoustic pressure fluctuations. A similar distinction is unfortunately not possible concerning the energy contained below the sonic line. In this region, pressure signatures with supersonic phase velocities can be generated both by hydrodynamic pressures, which will be efficient producers of sound, and sound waves whose propagation vectors are at angles which lie in the range < β < 8 (relative to the array axis). On the other hand, we can make an assumption that the majority of the sound sources are to be found on lines directly opposite, and approximately parallel to, the array axis. This being the case, the propagation vectors of the majority of the sound energy will lie closer to degrees (relative to the line array) than to or 8 degrees. This means that energy in the supersonic regime which is close to the sonic line (red dash-dot line) in figure 8 can be considered to be dominated by hydrodynamic fluctuations. Of course this distinction between hydrodynamic and acoustic phenomena is less clear-cut than in the subsonic regime. This should be kept in mind in what follows, where we use the criterion of supersonic phase velocity to identify hydrodynamic disturbances which are acoustically matched, and thus efficient producers of sound. In effect there will be a certain amount of bias in the identification procedure, related to the fact that in the supersonic k ω regime, a certain amount of acoustic energy is also present. In order to use the dispersion inequality ω > ka to identify acoustically matched components of these fields, a representative sound speed is required. However, as there are three different characteristic temperatures in this flow system (those of the primary jet, the secondary jet and the entrained flow) an appropriate choice of sound speed is not so straightforward. For each of the jet exit conditions, lines of ω = ka have therefore been plotted for the three characteristic sound speeds. These can be seen in figure 8, where a line corresponding to a convection velocity given by U conv =.6( U s + U p ) (6) has also been plotted in figure 8a. The red lines correspond to sound speeds calculated using the temperatures of the primary jet (dashed line), secondary jet (solid line) and entrained flow (dash-dot line). These lines give a qualitative picture of the efficiency of these convected structures as a source of sound with respect to radial position (as the sound speed is a function of radial position). It is clear that if the pressure fluctuations we measure are primarily driven by structures found towards the center of the jet, where the temperatures are highest, then the amount of energy contained in the supersonic regime (below the lines in figure 8) is relatively small and the structures are therefore not efficient producers of sound. If on the other hand these structures are active towards the outside of the jet then there is considerably more energy contained in the supersonic regime and the corresponding source mechanism is a good deal more efficient. The truth is probably somewhere between the two, and we will have to wait for further flow data in order to get a more definite answer to this question. Synchronous pressure-velocity measurements have also been performed and these should allow us to better understand which regions of the flow are most highly correlated with the nearfield pressure measurements. A further remark which can be made at this point concerns the two characteristic slopes which are manifest in the k ω energy distributions. It can be seen how one of these, where the majority of the energy is concentrated, corresponds to a characteristic convection velocity. As argued above this corresponds to purely hydrodynamic energy, and it gives information as to how frozen this pressure field can be considered to be. Were it perfectly so, the energy would be entirely contained on the line ω = ku conv. The amount of spread about this line is thus representative of the degree to which the field deviates from this condition (or how dispersive it is). The second characteristic slope is very close to the ambient sound speed, and so, as argued above, it corresponds primarily to acoustically matched hydrodynamic energy, but with some contributions from purely acoustic perturbations. The effect of the serrations can here be seen a little more clearly. In addition to a reduction in energy (the same colour scale is used for SCN and SN pressure, and another used for SCN and SN pressure derivatives), the k ω spectra of the flows generated by the SN show a greater spread of energy around the line ω = ku conv, of American Institute of Aeronautics and Astronautics Paper 6-8

12 Short Cowl Nozzle p/ x Serrated Nozzle p/ x 6 U conv k [m ] c= c= c= c= 6 k [m ] c= c= c= c= 6 k [m ] c=.. f [khz] c= c= c= f [khz] f [khz] f [khz] (a) (b) (c) (d) Figure 8. Wavenumber-frequency spectra. Solid black line shows the convection velocity defined by equation 6. Red lines show ω = ka for sound speeds a corresponding to the core flow temperature (dashed line), the secondary flow temperature (solid line) and the ambient temperature (dash-dot line). Short-cowl nozzle: (a) ˆP (k, ω), (b) ˆP x (k, ω). Serrated nozzle: (c) ˆP (k, ω), (d) ˆP (k, ω). x indicating that their effect has been to reduce the frozen, convective character of the large-scale structures. This is consistent with the observations concerning the Lagrangian correlation presented earlier. Another effect can be seen for high k and low ω (i.e. small scales characterised by low frequency). There is a concentration of energy in this region of the spectra which dissappears under the influence of the serrations. The reason for this is not clear at present. On the other hand, where the supersonic regime is concerned we do not see major changes in the structure of the field, i.e. here it seems that the serrations simply lower the energy levels, whilst preserving the structure of the radiating components. The influence of exit condition is mostly manifest close to the sonic line, indicating that energy moves into the acoustically matched k ω regime as bypass and temperature ratios are decreased. This is consistent with the farfield acoustic spectra which show SPL c > SPL c > SPL c. By filtering the spectra such that only energy in the supersonic regime is retained, an inverse transform allows the radiating structure of the source field to be recovered. The result of this operation is shown in figure. It can be seen how the unfiltered field looks very similar to the field filtered using ω < ka, showing how the pressure field is predominantly comprised of fluctuations which are not acoustically matched (this is one of the reasons for the acoustic inefficiency of subsonic jets). The field filtered using ω > ka shows spatial structures with much longer wavelengths, oscillating at considerably higher frequency d. We see how the supersonic components of the field show high levels in the near-nozzle region, related to the intense, high-strain events which are characteristic in this part of the flow. In the region < < 6. activity is reduced, and then in the vicinity of = activity of the supersonic components increases again, due most likely to the collapse of the potential core and further high-strain events which this incurs. In order to better assess the radiating structures of the different flows the second-order statistics are recomputed using the filtered fields. Results of this operation are shown in figure. The result is shown in figure, where subsonic and supersonic rms profiles are plotted with their unfiltered counterparts. We see again how the pressure field is primarily contributed to by hydrodynamic pressures with subsonic phase velocity along the array. Examination of the supersonic profiles show three regions of elevated sound production activity, consistent with the observations just made regarding the space-time evolution. The two regions of d The colour scales are not comparable due to the large amplitude disparities between the subsonic and supersonic components. of American Institute of Aeronautics and Astronautics Paper 6-8

13 Original, SCN, c= t [s] x Subsonic, SCN, c= x Supersonic, SCN, c= x (a) (b) (c) Figure. (a) Reconstruction of original pressure data (space-time) for the SCN, c= measurements, (b) filtered reconstruction using subsonic components only ω < ka, (c) filtered reconstruction using supersonic components only ω ka (components with supersonic phase-speed) increase level in the near-nozzle region look to be related to the initial stages of the primary and secondary jets. Changes in exit condition can be seen to increase the source levels in the downstream region. And case SN,c= shows a second region of elevated source energy in the vicinity of 6, which may corresponds to a more intense transition region between the end of the primary and secondary jets as the velocity ratio decreases. Interestingly the source activity in the near nozzle region is relatively insensitive to changes in the exit conditions. On a final note, we must state that the accuracy of this interpretation depends on the validity of the assumption made concerning acoustic contributions in the supersonic regime. Further work is required in order to more rigorously address the problem of how departure from this assumption will bias our interpretation of the result. V. Farfield results The far field acoustics were measured using an arc array of microphones centered on the jet. A sample of these measurements are shown in figure, using the microphones positioned at o, 6 o, and o relative the jet axis. It can be seen that the serrations only become efficient in reducing the radiated sound for conditions and, i.e. when the velocity ratio decreases. In fact for a velocity ratio of. the serrations are found to increase the levels of the radiated sound. VI. Proper Orthogonal Decomposition of the line array A decomposition of the line array was performed using Lumley s Proper Orthogonal Decomposition. The technique is described elsewhere (Berkooz et al ) and was performed here using two different kernels, one comprising a time aceraged cross-correlation, the other an ensemble average of cross spectral densities (Citriniti & George ). The kernel used in the maximisation is constructed using Hilbert Schmidt s theory for integral equations with symmetric kernels, and the problem (using the time averaged kernel) results in of American Institute of Aeronautics and Astronautics Paper 6-8

14 log (σ p /x ) [db] SCN,c= SCN,c= SCN,c= SN,c= SN,c= SN,c= 6 8 (a) log (σ p /x ) [db] 6 8 (b) Figure. Pressure rms profiles of (a) subsonic filtered data, (b) supersonic filtered data for all conditions of the Short-Cowl (SCN) and Serrated (SN). Solid lines show original rms profiles Nozzles log (p/p ref )[db] SCN, o SCN, 6 o SCN, o SN, o SN, 6 o SN, o log (p/p ref )[db] log (p/p ref )[db] c= c= c= f [Hz] f [Hz] f [Hz] (a) (b) (c) Figure. c=. Far-field acoustics at three microphone positions and jet exit conditions, (a) c=, (b) c= and (c) of American Institute of Aeronautics and Astronautics Paper 6-8

15 the following integral eigenvalue equation of the Fredholm type, R pp (x, x )φ (n) (x )dx = λ (n) φ (n) (x) () D where the kernel is constructed from the single transformation of the pressure time series, % R (+) (x,x ) 8 6 cumulative Individual SCN,c= SCN,c= SCN,c= SN,c= SN,c= SN,c= mode number..... x SCN,c= SCN,c= SCN,c= SN,c= SN,c= SN,c= (a) 6 8 (c) R (n) (x,x ) φ (n) (x,t)..... x n= n= n= (b) n=, original n=, extrapolated n=, original n=, extrapolated window. 6 8 (d) Figure. (a) Cumulative and relative energies of the eigenvalues λ (n). (b) Low-dimensional reconstruction of the kernel from the Short-Cowl nozzle, condition. (c) Cumulative reconstructions of the kernels using POD modes & for both co-axial nozzles and all three exit conditions. (d) Selective deconvolution of POD modes & from the Short-Cowl measurements and condition. R pp (x, x ) = p(x, t)p(x, t) (8) The convergence of the eigenvalues is shown in figure a for the different nozzle geometries and exit conditions. The first ten eigenvalues are seen to be organised in pairs, a sign of strongly convective phenomena. For the SN and condition three, the convergence is found to be slightly slower, indicative of a less organised field. It is also noteworthy, for this condition, that the difference in energy between the first and second modes is greater than for the other flow conditions. This is a sign that the pressure field for this flow condition is not as strongly convective as the other flows, in agreement with the observations made regarding the of American Institute of Aeronautics and Astronautics Paper 6-8

16 Lagrangian correlations and the wavenumber-frequency spectra. Indeed the SN,c= case shows the greatest spread about the ω = ku conv line in k ω space, in addition to the shortest Lagrangian integral scale. Figure b shows the first two eigenfunctions and their sum for condition SCN,c=. These modes can be considered the most representative low-order feature of the pressure field. An amplification-saturation-decay envelope is again manifest, and the change to this envelope, produced by changes in the exit condition, is shown in figure c. It can be seen how roughly the same evolution occurs with exit condition for both the SN and SCN. however, an effect of the serrations is to move this peak further downstream. POD was also performed on the fields filtered using ω < ka and ω > ka. The result is shown in figure. The eigenvalues of the supersonic field converge faster than those of the subsonic field, and percent of the energy is contained in the first mode. This indicates that the radiating component of the field is a ggod deal more structured than the acoustically redundant component. For the subsonic components the cumulative reconstructions of the kernel using POD modes mode and are very similar to those of the unfiltered field ( c), whereas the envelopes associated with the supersonic components show marked differences between the SCN and SN flows. This is in strong contrast to the differences observed in the rms profiles. These envelopes show much stronger source activity in the near-nozzle region of the SN flows, and reduced activity further downstream. This is consistent with the well-known effect of serrations; the source mechanisms of the jet are rendered less efficient, but a high frequency penalty is incurred due to additional sources generated by the serrations in the near-nozzle region. The envelopes from the SCN flows show interesting trends related to changes in the exit condition. For conditions c= and c=, in addition to the near-nozzle activity, a peak is manifest at., whereas for condition c=, the region at the close of the potential core has become considerably more important. Again this may be related to the additional mixing region which exists when the bypass ratio is reduced. Again, results from the PIV and the synchronous LDV-pressure measurements will help to further these interpretations. VII. Extrapolation by selective deconvolution As the axial extent of the measurements was limited to D, we have no information regarding the structure of the pressure field downstream of this point. From the point of view of noise modelling this will pose a problem, as the sudden truncation in the noise model may generate more noise than the sources we seek to estimate. To this end we have made an attempt to extrapolate the pressure data in the downstream direction. An exponential decay has been fitted to the nearfield pressure rms in order to model the energy decay downstream of the measurement region. This is shown in figure a, and can be seen to give a smooth continuation of the measured data. In order to perform a more complete estimate of the downstream evolution, in terms of its space-time structure, a combination of POD and selective deconvolution has been implemented (the reader is referred to Perret for a more involved discussion of this technique). The procedure involves performing a POD on the pressure field using a time averaged kernel: R pp (x, x )φ (n) (x )dx = λ (n) φ (n) (x) () D and then applying the selective deconvolution to each of the eigenvectors. This involves the following steps. () Substitute Φ (n) (X) with φ (n) (x) where X = x, and with zero where X x. () Then compute its wavenumber spectrum Φ (n) (k) = Φ (n) (X)e ikx dx () X followed by an inverse transform which only includes the first most energetic wavenumber, k (determined from the modulus of the spectrum) Φ (n) (X) = k Φ (n) (k )e ik x dk. () () Substitute Φ (n) (X) with Φ (n) (X) where X x, and with φ(n) (x) where X = x. This procedure is repeated according to Φ (n) (k) = Φ (n) (X)e ikx dx () X 6 of American Institute of Aeronautics and Astronautics Paper 6-8

17 subsonic supersonic % % SCN,c= SCN,c= SCN,c= SN,c= SN,c= SN,c= mode number (a) mode number (b) x R (+) (x,x )... R (+) (x,x ) (c) 6 8 (d) Figure. Cumulative and relative energies of the eigenvalues λ (n) for field filtered using (a) ω < ka, (b) ω > ka. Cumulative reconstructions of the kernels using POD modes & for fields filtered using (c) ω < ka, and (d) ω > ka, for both co-axial nozzles and all three exit conditions. Φ (n) j (X) = Φ (n) (k j )e ikjx dk j k j () where the index j does not necessarily correspond to an ordered sequence of wavenumbers (Fourier modes are rather sorted according to their energy). This procedure is repeated until the extrapolated result is satisfactory. Since the extrapolation is being performed on the empirical eigenfunctions of the POD, the orthogonality between the extrapolated regions modes must be retained. A Gram-Schmidt method is performed in order to ensure that this orthogonality is retained for the new eigenfunctions Φ (n) (X). Φ (n) (x) = Φ (n) (x) () ( Σn Φ (n) (x)φ (n) (x ) ) Φ (m) (x) () Σ n Φ (n) (x)φ (n) (x) Since the extrapolation assumes that the eigenfunctions are periodic in space, consideration must be given to the realistic decay of the pressure field in the extrapolated region. This is effected by adjusting the of American Institute of Aeronautics and Astronautics Paper 6-8

18 db=log (P/Pref) original extended SCN, c= (a) (b) Figure. (a)exponential decay (b) Original and low-dimensional (POD modes n=,,) reconstruction with extrapolation. averaged rms of the extrapolated field to follow the decay law P (x i ) = P (x m )e α(xm xi). In order to ensure continuity, the coefficient α is determined using the slope of the last two points on the rms profile. A typical result is shown in figure d. Once all of the POD modes have been extrapolated in this way, a low order estimate of the space-time structure of the pressure field can be reconstructed (using POD modes, and ). An example is shown in figure b. The continuity of the pressure field is good at the truncation point, and the extrapolated field shows a space-time evolution which is in good agreement with the measured region - energy continues to decay, the characteristic space-scales continue to increase and a similar convection velocity is maintained. This extrapolated data will allow more confident modelling of the source field, and at a later stage it will permit spatial Fourier transforms without the window functions used to date. 8 of American Institute of Aeronautics and Astronautics Paper 6-8

19 VIII. Conclusion Data from measurements performed in the near pressure fields of subsonic single and co-axial jets (serrations, bypass-ratio, temperature ratio) have been presented, where the pressure field is considered to constitute a reasonable footprint of the large-scale coherent structures (or low-order dynamic) of the flows. A comparative study shows how serrations produce considerable reductions in the nearfield pressure levels, however structural changes in the flows are more subtle. These changes are studied and discussed using a variety of analysis tools. A two-dimensional Fourier transform from (x, t) to (k, ω) is used to study the pressure fields in wavenumberfrequency space. Arguments are presented which permit a clearer separation of the measured pressure field into () purely hydrodynamic components, with subsonic phase velocity (and therefore redundant in the production of a progressive sound field) and () components with sonic and supersonic phase velocity which are predominantly hydrodynamic in nature (but which do contain some acoustic contribution), and thus active in the production of radiating pressures. A filtering operation, based on the dispersion inequality ω > ka, leads to a reconstruction of the sound-producing dynamic of the low-order flow structure and the statistics of this field are compared to those of the unfiltered flow dynamic. This allows us to access the true sound producing components of flow structure associate with the most coherent structures of the jet. A Proper Orthogonal Decomposition of the filtered pressure field allows some the subtleties of the structural changes induced by the serrations to be more clearly identified. The serrations are found to considerably increase source levels in the near-nozzle region, while reducing source activity further downstream. Work is ongoing, and in particular Spectral Linear Stochastic Estimation is being used to effect a full space-time reconstruction of the pressure field on a conical surface surrounding the jet. This will allow the analysis procedure presented in this paper to be applied to a more extensive spatial domain, whereby the azimuthal dependence of the flow structure can be studied. Finally, synchronous LDV-pressure measurements (currently being processed) will allow some of the observations made to be further developed and validated. IX. Acknowledgements This work was funded by the EU program CoJeN (AST-CT--). The authors would also like to acknowledge the support of the staff of the NTF (Noise Test Facility), QinetiQ, whose participation in the experimental campaign was invaluable, also DLR (Ulf Michel and Sébastien Guérin) both for supplying the microphones and for their participation in the experiment, and ISVR (Juan Battaner-Moro) for provision of an acquisition system and for assitance provided in setting up the experiment. Finally the authors are grateful to Kevin Britchford (Rolls Royce) and Craig Mead (QinetiQ) for co-ordinating the experimental campaign. A. X. Appendices In order to correct for small discrepancies between individual experiments, a global conditioning of the entire pressure field was performed using Proper Orthogonal Decomposition (POD). That is, several individual experiments (i) were performed based on the re-positioning of the azimuthal array along the axis of the jet. A POD of the line array was performed for each of the individual experiments, followed by a decomposition of the global field (averaged over the individual experiments). This allowed for a proper set of transfer functions to be generated in order to correct for differences between the coefficients from the decompositions of the individual experiments, and the averaged global coefficients. This global conditioning technique begins with a decomposition of the individual experiments, D R i (x, x ; f)φ (n) i (x ; f)dx = λ (n) i φ (n) i (x; f) (6) followed by a decomposition of the global field, R(x, x ; f)φ (n) (x ; f)dx = Λ (n) Φ (n) (x; f) () D of American Institute of Aeronautics and Astronautics Paper 6-8