Acoustic-hydrodynamic interaction in the entrainment regionof subsonic jet flow

Peter Jordan�

Francois Coiffet�

Joel Delville�

Yves Gervais�

Laboratoire d’Etudes Aerodynamiques UMR CNRS 6609, Poitiers, 86036, France

Fabienne Ricaud�

Renault iDVU, DPCVP sce 5070, 42 route de Beynes 78640 Villiers Saint Frederic, France.

The true global source of the farfield sound radiated from a subsonic jet is the entire dynamic found withinthe confines of the hydrodynamic field, a dynamic which comprises not only the unsteady compressive ex-citation of the medium resulting from turbulence mixing and unsteady temperature fluctuations, but alsosubsequent passage of the resultant perturbations across the jet flow. Interaction between the acoustic andhydrodynamic fields thus constitutes an integral part of the source mechanism, having a direct inluence on thecharacter of the farfield sound. Improved understanding, and subsequent modelling of the complete spatiotem-poral character of the global source term will be acheived through study of the different physical mechanismsimplicated in its dynamic. The phenomena which have to date been accepted as important in terms of the radi-ated sound are (i) sound generation due to turbulent mixing and shear, (ii) sound generation due to fluctuatingentropy, (iii) convective amplification and (iv) refraction and diffraction of sound by the mean and turbulentcomponents of the velocity field. The irrotational pressure field surrounding a jet also constitutes an integralcomponent of the global source mechanism however, and while it generally receives only very limited atten-tion, due in part to the absence of any significant sound generation mechanisms, it is an interesting region tostudy because of what it can tell us about the rest of the source structure. In addition, it sees the final passageof the sound field which will be radiated to the far field. In this work, measurements performed in the nearpressure field of an isothermal subsonic jet reveal a strong interaction mechanism between the hydrodynamicand acoustic fields, the essential features of which are captured by a simple model. On account of the successof the model a number of hypotheses concerning the relationship between the hydrodynamic and the acousticparts of the source term are confirmed, whence it is suggested that the global source constituted by a subsonicjet may be more deterministic than is generally accepted.

Nomenclature��� Speed of sound, m/s�

frequency, s-1�Jet exit diameter, m�Convection velocity, m/s�Wavenumber, m-1 Phase, rad�Pressure, Pa� Circular frequency, rad�

Researcher, Laboratoire d’Etudes Aerodynamiques UMR CNRS 6609, CEAT 86036, Poitiers, France.�PhD Student, Laboratoire d’Etudes Aerodynamiques UMR CNRS 6609, CEAT 86036, Poitiers, France.�Professor, Laboratoire d’Etudes Aerodynamiques UMR CNRS 6609, CEAT 86036, Poitiers, France.�Engineer, Renault iDVU, DPCVP sce 5070, 42 route de Beynes 78640 Villiers Saint Frederic, France.

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American Institute of Aeronautics and Astronautics

10th AIAA/CEAS Aeroacoustics Conference AIAA 2004-3020

Copyright © 2004 by Laboratoire d'Etudes Aérodynamiques. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

� Source strength� � Source radius, m � Source velocity, m/s� Radial position relative to shear-layer axis, m� Axial position relative to jet exit, m

Subscript�Hydrodynamic� Acoustic

I. Introduction

Jet noise prediction methodologies which involve a statistical description of the source mechanism, coupled withan acoustic analogy, are limited by their neglect of flow-acoustic interactions. Such interactions present themselvesin various forms, the most significant (and best understood) of these being the acoustic-mean flow interaction whichleads to refraction of sound away from the jet axis. For this particular phenomenon two approaches can be applied.The first involves analogies such as proposed by Lilley,1 and the second, the linearised Euler equations. The acoustic-hydrodynamic interaction problem is however more complex than this, involving at least two other phenomena whichare neglected by all but the DNS/LES approach. The first of these is manifest in the interaction between the soundfield and the mixing layer turbulence, interaction which leads to non-linear energy transfer between the hydrodynamicand acoustic fields, while the second occurs in the irrotational nearfield of the jet. Arndt et al.3 demonstrate, bothanalytically and experimentally, how the pressure field in the latter region is comprised of both acoustic and hydrody-namic phenomena, and identify thus a critical non-dimensional parameter

� � , � ����� defining a demarcation betweenregions where hydrodynamic and acoustic phenomena dominate.

In this work results from measurements performed in the hydrodynamic nearfield of an isothermal subsonic jetshow evidence of strong interaction between the hydrodynamic and the acoustic pressure fields - analysis of the cross-spectral matrix of the nearfield pressure signals revealing highly localised nulls in the space-frequency coherencepattern. An hypothesis is presented to explain this phenomenon whence a simple model is proposed which capturesthe essential features of the interaction mechanism. Non-dimensional quantities are also identified, and these result ina good collapse of the data as a function of axial and radial position and frequency.

In fact such interaction must occur between two highly correlated scalar fields, and one is thus inclined to see insome respects a tautology in the result. However, identification of the mechanism does lead to a number of usefulinsights. (i) The near pressure field of a jet can be considered a superposition of hydrodynamic and acoustic fields,the hydrodynamic field comprising plane waves, perpendicular to the outer shear layer extremity and convected withthe flow, while acoustic waves propagate spherically from an axial distribution of sources. (ii) The hydrodynamic andacoustic fields have the same spectral make-up (in terms of their phase), are highly correlated and out of phase by ��� � .(iii) The acoustic sources appear to be concentrated on the shear-layer axis. (iv) Results hint at an underlying dynamicwhich is more deterministic than is generally believed.

II. Theory

A simplified description of the nearfield pressure of a jet is obtained by Arndt et al. by considering the pressureand velocity in this region related by the unsteady Bernoulli equation�� !�#"$ �&% %(' *)

,+ ) � - (1)

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where�#"

is the pressure far from the flow and

is the velocity potential. The source is assumed to have a quadrupolenature, the corresponding solution to the spherical wave equation thus being

� %/.% � .0 21 � � .�354 �7698�:<; � ' � �>=@? - (2)

whence, using the boundary condition for an axial quadrupole

% % �BAAA CEDGFGH� 1I �/JLK�M .ON 6L8�:P; 1 � ' = (3)

to eliminate � , a solution for the mean-square pressure can be written, through substitution into equation 1:

Q �R$S�9�S� .� ; � � �9= . 0 � �� ?>T AAA�VUW� 1 � �2U ; 1 � �>= .X !Y ; � � �L= . U 1[Z X � � � ; � � �9=@\L] AAA

.>^ (4)

Farfield and nearfield expressions are obtained by considering� ��_a` and

� �<ba` , giving respectivelyQdc $��L��� .� ; � �>=[e . (5)

and Qfc $S�9��� .� ; � �>= e T ^ (6)

By then approximating the source velocity as a typical turbulence intensity, with spectral character comprising a flatenergy-containing region and inertial subrange with

� e(gih \ dependence, the spectral character of the energy-containingregion and inertial subrange of the nearfield pressure are shown to be characterised byQ�j � e T ; � �lkLmonPp ' = (7)

and Qqj � e T9r Tts ; �<�RkLmonPp ' = (8)

Two distinct regimes are thus defined (equations 5 and 8), corresponding to regions dominated by either hydrodynamicor acoustic pressure fluctuations. This character was verified experimentally and a demarcation between hydrodynamicand acoustic regions, manifest in a change of spectral roll-off from

� � e T9r Tts to� � e . , identified at

� � =2.

The transition region between the “hydrodynamic” and “acoustic” zones, found in the vicinity of� � =2, is of

particular interest because it is here that hydrodynamic and acoustic pressure fluctuations have similar energy, andshould thus be mutually influential.

III. Experimental setup

An extensive series of pressure measurements were performed2 in the near field of an isothermal subsonic jet withexit Mach number of 0.3 and jet exit diameter of 50mm, giving a Reynolds number of 300,000. The jet apparatus wasrun in an open circuit configuration, and the air delivery system was treated acoustically to eliminate fan noise. Thesettling chamber included a honeycomb structure and two grids to settle the flow, and had a contraction ratio of 25.The boundary layer inside the nozzle was turbulent, and the turbulence intensity of the exit flow of the order 0.4 u

An inclined array (at � � relative to the jet axis in order to follow the expanding flow) of 31 microphones, contigu-ously arranged at 10mm intervals, was used to sample the pressure field of the jet at 9 radial positions, from � = � ^ v �to � = ` � in � ^ ` � increments and from � = ` � to � = 3q� in ` � increments a. This configuration is shown in figure 1.

aThe radial position of the array is defined by the position of the first microphone relative to the jet axis

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wixzy

{| }

~ � � � � � � � ����������� ��� ����������

(a) (b)

Figure 1. Measurement setup: (a) Microphone array at �L�[����� (position of first microphone relative to the jet lip-line, (b) Measurementgrid �L�[�!����� � to � in increments of ����� and from � to � in increments of �

IV. Results and general discussion

A sample of some of the results is shown in figures 2 and 3, which show spectra as a function of � for �5� � � � ^�v ,and coherence as a function of � and

�for �5� � =0.1, �5� � =0.5 and �>� � =3.5.

�q���

� �q¡£¢¥¤f¦�§�¨ª©o«@¡�¨�©¨ª©5¬i­®«®¨�¬i¦�¦�¬¯ §�©5¬¦±°³²¬i©�¬E­@´�¨�¬µ §o©o«®¬i©q¡£©¶«· ²¬i´�¨�§�©

¸ ° §o© ¸ ¬· ²¬i´�¨�§�©º¹�¡o»#©5§o¢V¹�­®¬

¼�½ ¾o¿�ÀtÁ À®Â

¾o¿�ÃÄÆÅÇÄÆÈÉÊ ËËÊÄ Ì Í ÎË ÏÐ

ÑLÒÓÔÕÑÒ�Ö ÓÒ5Ö ÔÒ�Ö ÕÒ5ÖªÑÒ5Ö Ò¶ÓÒ5Ö Ò�ÔÒ�Ö Ò5Ñ

Ñ ¬E× ÒoÓÑ ¬E× Ò�ØÑ ¬E× Ò¶ÙÑ ¬E× Ò�ÚÑ ¬E× Ò�ÛÑ ¬E× Ñ[ÒÑ ¬E× Ñ�Ñ

Figure 2. Nearfield pressure spectra at �L�[� =0.5, horizontal axis shows ܣ�A clear demarcation is manifest between the hydrodynamic and acoustic zones at

� �<�l� , in agreement with Arndtet al., the two fields showing very different behaviour which will be discussed in what follows. It can be seen fromfigure 2 that the characteristic slopes given by equations 5 and 8 for the hydrodynamic and acoustic regions fit themeasured spectra very well.

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A. Evolution of the hydrodynamic field

Axial evolution in the hydrodynamic zone (� �ÞÝß� ) comprises a global increase in energy while the peak frequency

decreases The reasons for this decrease of the peak frequency are twofold. On one hand the characteristic frequenciesdecrease with increasing axial distance, due to a corresponding growth of the turbulence scales. And secondly, thelarger radial distance of the downstream microphones from the centre of the shear layer means that they fall outsidethe hydrodynamic field for the higher frequencies - leaving the lower hydrodynamic frequencies to dominate.

B. Spatial coherence

Cross-spectral matrices were computed for the 31 microphones (Y `Þà Y ` cross-spectra). In this way the level of

coherence ( á . ) can be assessed as a function of position and frequency for both the hydrodynamic and the acous-tic pressure fields. The result of this operation is presented in figure 3. The three columns correspond to the radialstations �5� � =0.1, �5� � =0.5 and �5� � =3.5, while the â rows are for â different axial positions of the reference micro-phone (shown on the � -axis of the subplots). A demarcation between the hydrodynamic and acoustic regions is againobserved at

� �ã�ä� . As � increases (following the columns in figure 3 from left to right)� �å�æ� is correspondingly

representative of lower frequencies, and so the hydrodynamic frequency range is seen to decrease with increasing dis-tance from the jet - the lowest frequency energy exerting its influence over the largest radial extent. Noteworthy at allradial locations is the more extensive spatial coherence of the hydrodynamic field in comparison to the acoustic field.Over the entire length of the jet the hydrodynamic field remains highly correlated and so it can be considered to behaveas a single oscillating unit, hydrodynamic perturbations at a given axial position communicating with perturbationsat all other axial locations. There is nonetheless a more rapid axial decay of coherence at the higher hydrodynamicfrequencies, representative of a transfer of energy to the lower frequency motion as the jet flow evolves in the down-stream direction.

Analysis of the acoustic zone in terms of this spatial coherence leads to some interesting observations. For anacoustic field radiating from a single source one would expect very high levels of coherence over large distances.Here however a rapid axial decay is observed in comparison with the hydrodynamic field - in particular for the twoclosest array positions where the acoustic coherence has decayed to zero in the space of a jet diameter or so. In thecase of the closest array (column 1) the most extensive spatial coherence is found when the reference microphone islocated at around �/� � =

Y, coherence downstream of the end of the potential core (length of potential core = 4.5

�)

being almost non-existent. This is because of the high frequencies which correspond to� �ç_�� when the array

is this close to the jet - only the upstream sources contain appreciable energy at these frequencies. In the case of thesecond and third radial stations (columns 2 and 3) the spatial extent of the acoustic coherence increases with increasingdownstream distance. There are two reasons for this. Firstly, the progressively lower frequencies associated with theregion

� �ã_è� correspond to the energy containing region of the jet, are associated with the larger turbulence scalesand thus correlated over a greater distance, and secondly, the sources further downstream behave more like a singlesource mechanism, or a distribution of correlated sources, as the column instability (a global mode) is here dominant.

1. Coherence nulls

A curious phenomenon is manifest in the vicinity of� � =2, visible at �5� � =0.1 and 0.5, but most marked at �5� � =0.5.

For each reference position there exist highly localised nulls in the space-frequency coherence pattern. These nullsare observed just upstream and downstream of the reference microphone, and at a frequency which corresponds to� �<�l� (where acoustic and hydrodynamic energies are of similar order). Such a sudden loss of coherence can only beexplained by the existence of a local interference node, indicative of a highly coherent mechanism (at least locally) -i.e. at the axial position and frequency characteristic of a given null there is no energy coherent with the correspondingspectral components of the pressure field at the reference microphone. In the following section an hypothesis ispresented to explain this phenomenon and a simple model developed to capture its essential features.

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(a) (b) (c)

Figure 3. Coherence. Columns - (a) �L�[� =0.1 (b) �[�[� =0.5 (c) �L�[� =3.5.

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V. The interference mechanism

The interaction phenomenon can be best understood via reference to the schematics shown in figures 4(a) and 4(b).The assumptions necessary for explanation of the interaction mechanism are as follows:

é The hydrodynamic pressure signature registered by a microphone corresponds to an event directly opposite themicrophone [Ricaud2],é Registration of that event is instantaneous, i.e. there is no propagation delay,é Acoustic pressures propagate at the speed of sound and so there is an associated phase lag,é Acoustic and hydrodynamic pressures are out of phase by ��� � [Arndt et al.3].

x

p(x,t)

MICROPHONE ARRAY

s

r

Outer extremity of the shear layercenterline

Shear layer

c

m mm

rr

U

u

d

ud

r

c

(a) (b)

Figure 4. (a) Spatial hydrodynamic pressure pattern. (b) Interference mechanism, hydrodynamic field in red, acoustic field in blue

As the hydrodynamic pressure registered by a given microphone comprises contributions from local events, both thewavelength and energy of the axial pattern will increase with downstream distance (figure 4(a)) - a consequence of thegrowth of the corresponding turbulence structures. The hydrodynamic field registered by the array is thus an instan-taneous footprint of the shear-layer flow structure, and as pointed out earlier this footprint constitutes a single highlycorrelated entity, the dynamic at different axial locations being aware of the dynamic everywhere else. Where acousticfluctuations associated with the same flow dynamic are concerned however pressure perturbations propagate, and theaxial distribution comprises a collection of relatively uncorrelated sources, the implications of which are twofold - (i)the source of the acoustic pressure fluctuations registered by a given microphone is non-local, contributions being feltfrom the said distribution of sources, and (ii) there is a phase delay associated with the time taken for a pressure dis-turbance to travel from the source to the microphone. These differences are instrumental in the aforesaid interferencemechanism.

Referring then to the schematic shown in figure 4(b), the hydrodynamic pressure field is represented in red, theacoustic field in blue. ê C is the reference microphone, êìë and êìí are, respectively, the upstream and downstreamlocations of microphones which register the nulls in the coherence pattern, and p is a position directly opposite thereference microphone on the shear layer axis, considered the location of a pressure disturbance î ; ' =�� î ï ; ' =PU î(ð ; ' = .This disturbance has a certain frequency content, for which there exists a corresponding set of hydrodynamic and

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acoustic scales (wavelengthsñ � � and k£� � respectively). The reference microphone registers the hydrodynamic part

of the disturbance î ð ; ' = instantaneously, and the hydrodynamic phase at another axial location ( � ë or � í for example),relative to the phase at ê C , depends on (i) the frequency of the perturbation, (ii) the convection velocity of the hydro-dynamic field, (iii) the expansion rate of the mixing layer (cf. figure 4(a)), and (iv) the axial coordinate. The phase ofthe acoustic part of the perturbation on the other hand, î ï ; ' = will be a function of (i) the perturbation frequency, (ii)the speed of sound and (iii) the source-microphone distance (due to propagation of the acoustic field). î ð ; ' = and î ï ; ' =are considered to be highly correlated (this hypothesis is later verified), and so interference will occur between similarfrequencies (and therefore different scales because

ò Ýßk ) at locations where two criteria are satisfied - (i) there isa phase difference of

4between îOð ; ' = and î/ï ; ' = , and (ii) their energy magnitudes are of similar order. Using these

criteria a model can be obtained which will predict the space-frequency coordinates of the said nulls.

The axial position of the interference will always be found at one hydrodynamic wavelength from the referencemicrophone (modified to reflect the expansion of the mixing-layer), corresponding to the frequency for which

� �<�ó� ,as it is at one modified wavelength that the hydrodynamic phase is equal to � relative to the reference microphone andtherefore perfectly out of phase with an acoustic perturbation which lags by

4.

A. Modelling the interference

The phase of an acoustic field propagating from p (see figure 4(b)), as a function of � , � and�

is given byb

ï ��� 4 �2ô õ � . U�� .k U 4 � � �Oö õ � . U�� . U4 � - (9)

where4 �>� takes into account the inherent phase difference between hydrodynamic and acoustic pressures.3 The cri-

teria of opposite phase and equal energy for a given � are then represented by the intersection of the isocontour � 4

(function of � and�

) and the line� �Þ�÷� , which will give the space-frequency coordinates where destructive inter-

ference can be expected to occur, and a coherence of zero thus found between a microphone at this location and thereference microphone, provided of course that the aforesaid hypotheses are all correct. A result of this operation isshown in figure 5, interaction locations being identified both upstream and downstream of the reference microphone( ê C � õ �ø� � ). Figure 6 shows the same plots for a selection of axial and radial locations, with the experimentallyidentified nulls indicated by their coherence isocontour � ^ � (shown by a green ellipse). The model is found to correctlypredict the space-frequency coordinates of the coherence nulls for all locations.

Superposition of the model criteria on the experimental data reveals another interesting result (figure 7). A secondripple in the coherence pattern, farther from the reference microphone and at lower frequency, coincides with the iso-contour

� 4 , possibly a second interference zone where cancellation is incomplete because only one of the aforesaidcriteria are satisfied. The fields are in antiphase, but their energies are not the same.

Given the globality of the result, the space-frequency coordinates can be non-dimensionalised, the result of whichis shown in figure 8 for �5� � =0.3, 0.4, 0.5 and 1.5, where a good collapse of the data is obtained. Both the null loca-tions and the wavenumber are non-dimensionalised by the propagation distances ( � í and � ë ).

bthe Doppler factor has been ignored as it was found to have a negligible effect

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Figure 5. Identifying the nulls

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(a) (b) (c)

Figure 6. Phase - comparison with experiment. Columns - (a) �L�[� =0.3 (b) �L�[� =0.4 (c) �L�[� =0.5. Rows, from top to bottom - referencemicrophone positions ù5�[���ûú�� � , ù5�[�ü��ý�� þ and ù5�[���Þ�o� ÿ

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Figure 7. Coherence - comparison with experiment. Columns - (a) �L�[� =0.3 (b) �L�[� =0.4 (c) �L�[� =0.5. Rows, from top to bottom - referencemicrophone positions ù5�[���ûú�� � , ù5�[�ü��ý�� þ and ù5�[���Þ�o� ÿ

���������� ��������� � ��������� � ��������� � �

����� � ���

� ��� � � � � � � � � � � � � � � � � � � � �� � �� � �� � �� � �� � �� � �� � �

�� ��� �

�� � �

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� � �

Figure 8. Non-dimensionalised space-frequency coordinates of coherence nulls

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VI. Discussion

The existence of such a marked interaction between the acoustic and hydrodynamic pressure fields has a numberof interesting implications, viz. verification of the hypotheses on which interpretation of the phenomenon depends. Afurther result, presented only very briefly here to lend support to the following arguments, is the correlation between thenearfield pressure and the axial and radial components of the turbulence velocity (obtained using two component LaserDoppler Velocimetry (LDV) performed synchronously with the nearfield array measurements). A more comprehensivepresentation and discussion of these results will be given in a later paper. It suffices for the present purposes to observethat through the identification of an ensemble of non-dimensional parameters it was possible to collapse the correlationdata. An example of this collapse is shown in figure 9, where the pressure-velocity correlations are shown for both theaxial and radial components of the turbulence velocity, non-dimensional time shown on the � -axis, non-dimensionalradial position (in the mixing-layer) shown on the � -axis. The surprising result is that for a given axial separationbetween the microphone and the LDV profile, regardless of the position of the reference microphone, the form ofthe pressure velocity correlation is approximately the same (hence the collapse of the data), and the characteristicscale is always that of the reference microphone. Thus while the dynamic of the velocity field at the location of theLDV measurement comprises all the local scales of the mixing-layer, its correlation with the nearfield pressure signalat a given axial location is characterised by the spatial hydrodynamic scale at that location, and the correlation isnon-negligible. This indicates that the velocity field of the jet is strongly correlated with the global hydrodynamic‘mode’ discussed earlier, this mode appearing thus to contain a large quantity of information concerning the overalljet dynamic.

(a) (b)

Figure 9. Pressure-velocity correlations (a) - !#" (b) - !%$

With this in mind the following observations can be made concerning the aforesaid hydrodynamic-acoustic inter-action, and the global dynamic of the jet-flow

A. Position of the maximum acoustic generation

In order to predict the location of the coherence nulls the model needs to assume a radial location of the source ofthe pressure disturbance î ; ' = (figure 4(b)). The radial position for which the correct null positions are identifiedcorresponds to the shear-layer axis - this implies that the main acoustic energy is generated at this location.

B. Correlation of hydrodynamic and acoustic pressure fields

For such an interaction to occur the hydrodynamic and acoustic pressure fields must be highly correlated, thus whiletheir spectral shapes at the source may be very different in terms of magnitude, the spectral shapec of their phase mustbe very similar, showing a global difference of

4 �>� .cThe term ‘spectral’ here refers to instantaneous spectra, and not power spectra.

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C. A deterministic underlying structure

It was deduced from the coherence plots that the acoustic sources are mutually uncorrelated while the hydrodynamicfield comprised a single highly correlated perturbation. However, given the high level of correlation necessary betweenthe hydrodynamic and the acoustic fields for the said interference to occur, the distributed acoustic “sources” must,by their correlation with the hydrodynamic field, be highly correlated amongst themselves. A similar observation wasmade above concerning the turbulence velocity field. The irrotational hydrodynamic pressure field appears thus to berich in information concerning the global dynamic of the jet.

It would appear, from the collapse of the pressure-pressure coherence and the pressure-velocity correlations, andfrom the high level of correlation implied between the hydrodynamic and acoustic fields, that the jet structure ishighly organised, from both a purely aerodynamic and an aeroacoustic point of view, organisation which hints at anunderlying determinism which could be exploited for example for the purposes of control. The precise nature of thisdeterminism must be better understood however, and this is the subject of ongoing study.

VII. Conclusion

A strong interaction mechanism, leading to destructive inteference between the hydrodynamic and acoustic pres-sure fields in the entrainment region of a subsonic jet has been identified and a simple model presented which capturesits essential features. The presence of such an interaction has some potentially far reaching implications concerningthe global structure of the jet, from a purely aerodynamic point of view, but also from an aeroacoustic point of view.The interaction mechanism and its implications are presented and discussed.

References&Lilley G.M., Theory of turbulence generated noise, noise radiated from upstream sources and combustion noise, AFAPL report, 72-53, 1972.'Ricaud, F., Etude de l’identification des sources acoustiques a partir du couplage de la pression en champ proche et de l’organisation

instantanee de la zone de melange de jet, Ph.D thesis, UniversitA c(

de Poitiers, 2003.)Arndt R.E.A., Long D.F., Glauser M.N., The proper orthogonal decomposition of pressure fluctuations surrounding a turbulent jet, JFM 340,

pp 1-33, 1997.

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