Large vortex input in a noise of artificially excited subsonic jet

Victor Kopiev, Mikhail Zaitsev, Sergey Chernyshev and Nikolai Ostrikov

Central Aerohydrodynamics Institute (TsAGI), Acoustic Division 105005 Moscow, Radio str.17, Russia, vkopiev@mx.iki.rssi.ru

Abstract.

The main goal of the paper is to assess the contribution of large-scale vortices to the noise of the subsonic turbulent jet in the selected frequency bands (Sh~0.2-0.45). The mechanism of sound radiation according to which a separate vortex structures itself can be a significant sound source is considered. Vortex core eigen-modes are responsible for noise radiation according to this mechanism. The work is subdivided on three coupling items: (i) theoretical investigation of noise radiation of separate vortex ring (especially little-known octupole contribution to sound radiation), (ii) visualization of large vortices in excited jet and (iii) experimental investigation of turbulent jet noise (subsonic jet with velocity 120m/s excited at 2032Hz) decomposed on the separate azimuthal components with the help of azimuthal decomposition technique (ADT). The careful comparison of these results gives the quantitative assessment of the contribution of vortex rings to the real subsonic jet noise and gives the main source localization. The proposed approach for jet noise modelling was additionally checked by measurements with two co-axial microphone arrays. Good agreement of azimuthal noise modelling using the same parameter set for two radial distances was demonstrated.

1. Introduction

Turbulent jet noise is one of the topical problems in aero-acoustics. A notable advance has been made in jet noise research in terms of better understanding the nature of noise radiated by turbulence. The results leveraged primarily on Lighthill’s study in which the turbulent jet noise was replaced by the distribution of equivalent quadrupole sources moving in medium at rest [1, 2].Along with better understanding of small-scale turbulence as one of the main contributor to Lighthill’s tensor, it has been established with confidence that a turbulent jet contains coherent large-scale formations that can be associated either with separate vortex rings or instability wave packets depending on the jet velocity and the Reynolds number. For subsonic flows and low-speed jets, the large-scale structures look like vortex rings [3-9]. The importance of dynamic processes involving large coherent structures in relation to entrainment, mixing, and noise generation in free shear layers is well established. The phenomenon of large-scale vortex structures inspires a tempting idea of describing the noise source by a model of large quasi-determined vortices and a hope for finding a way to direct control the structures and the noise they generate. This approach, if successfully put into practice, would be a far more effective means of controlling the turbulent flow’s aerodynamics and acoustics than any attempt to control passive turbulence.

Quantitative estimation of coherent structure input in total noise should be based on some concrete mechanism. However, from the point of view of wavy-wall concept, the large scale vortex mechanism is not effective in subsonic jet due to the subsonic phase velocity of the objects. Therefore the common point of view [10-15] assumes that large-scale structures in the mixing layer of a jet do not radiate noise directly but only their interaction (merging, pairing, tearing etc.) can radiate sound.

We consider another mechanism of sound radiation by vortices according to which a separate vortex structures itself can be significant sound source. Vortex core eigen-modes are responsible for noise radiation according to this mechanism. It is not contradict to wavy wall interpretation of radiation efficiency: the mechanism is realized due to rotation of vortex core disturbances, which in its turn include supersonic wavy-wall disturbances in Fourier space.

13th AIAA/CEAS Aeroacoustics Conference (28th AIAA Aeroacoustics Conference) AIAA 2007-3647

Copyright © 2007 by Victor Kopiev. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

The measurements described in [16-21] indicate that the jet noise consists in fact of three quadrupoles with different azimuthal directivities – zero, first and second ones. To extract them using ADT it is enough to separate different azimuthal harmonics before spectrum averaging. Despite ADT gives more detailed information about sources [21] compared traditional techniques based on total noise measurements, the measurements of quadrupole components do not allow to extract vortex ring noise from total one, because the noise of any turbulent cluster always has a structure similar to a quadrupole. Therefore it has seemed virtually impossible until now to determine and assess the actual contribution of large-scale vortices to the total jet noise and one can ask if it is possible in principle to separate large-scale vortex noise and small-scale turbulence noise. To answer this principal question, on the one hand, we must determine some peculiarities distinguishing the sources of different types, on the other hand, these peculiarities should be noticeable enough to register them in experiment. Therefore if we could find such a peculiarity in radiation directivity then we can separate the noise sources in jet noise.

Thus, the main objective is to evaluate the contribution of large-scale vortices to the total jet noise. In this paper we give unambiguous and quantitative answer on this principal question. We have considered an excited subsonic jet (V=120m/s, jet excited at about 2 kHz on the Strouhal number St=0.68) in order to localize and intensify large-scale vortices in the initial part of the jet. The investigation is restricted to a frequency range from 600 to 1350Hz corresponded to the Strouhal number range (St=0.2– 0.45), which is quite representative from the standpoint of the jet noise. The frequency band was split into three typical regions for more accurate modelling: 600-800Hz, 800-1050Hz and 1050-1350Hz. Note that the refraction effect is not significant in this frequency range.

The realization is based on three coupling items: (i) theoretical results for vortex ring noise directivity (especially little-known part of octupole noise directivity), (ii) visualization of vortex rings and small-scale turbulence in excited jet, (iii) ADT experimental results for jet noise. A principle peculiarity in terms of main idea of the work is the transformation of noise directivity (i) due to octupole term. The octupole term for small-scale turbulence is small but for localize vortex with thin core is proportional to the ratio of two small parameters (Mach number and thickness parameter) and appears not to be small. Therefore the jet noise radiation includes the octupole term proportionally to the contribution of vortex ring input in the total noise. A visualization technique (ii) was used to make estimations of the vortex rings' velocities and geometric dimensions. These estimations helped make calculations of the vortex rings' acoustic radiation directivity distorted by the octupole term and built a set of basic functions for large-scale vortex noise directivity. In addition the visualization of excited jet helps to split off the small-scale turbulence zone and turbulent vortex ring zone in the whole turbulent region. Brüel & Kjær’s “Pulse 3560D” data acquisition and analysis system specially adapted by authors to ADT (iii) was used to measure the averaged spectra of the jet noise azimuthal harmonics which were then analyzed in different frequency bands. The sound radiation power of azimuthal harmonics was measured on a R=0.85m cylindrical surface around the jet along its axis. The measurements were compared with the directivity patterns of the convecting small-scale turbulence and oscillating vortex rings in order to make a quantitative evaluation of the vortex rings' contribution to the total subsonic jet noise in a considered specific case.

It is our luck that ADT data for excited jet noise directivity appears to be shifted to the upstream direction unlike non-excited jet [21, Fig.15] (in other manifestations these data are quite similar). This peculiarity helps to obtain unambiguous answer: for small-scale turbulence there are no any mechanisms to intensify the noise level in upstream direction.

Therefore to approximate the measurement data in the context of small-scale turbulence model of noise sources we have to place them near the nozzle and even inside the nozzle, where they can not be located. Directivity transformation due to octupole for large vortex with thin core intensifies the noise level exactly upstream and we do not need to put these sources inside the nozzle to meet the experimental data in upstream direction. However using the vortex ring model only we cannot meet experimental data in downstream direction due to slow speed of vortex (near half of mean velocity) and therefore insufficient convective amplification for this type of sources. Hence a mixture of two types of sources: convectively moving quadrupoles (that represent sound radiation by small-scale turbulence) and the oscillating vortex rings (that represent sound radiation by large-scale structures), will meet the measurement data. The careful estimation shows that the energy input of the vortex rings in the frequency bands under review is 30 to 45% depending on the harmonic number and frequency band.

The proposed approach for jet noise modelling was additionally checked by measurements with two co-axial microphone arrays. Good agreement of azimuthal noise modelling using the same parameter set for two radial distances was demonstrated. The results of this experiment are the additional independent validation of theoretical approach formulated in the present paper.

The results obtained give much deeper insight into the structure of the aerodynamic noise sources in a turbulent jet and can form the basis for a fundamentally new technique of studying the radiating turbulence in a subsonic jet. Subsequently, a new noise control concept and new approaches to devising robust numerical methods for turbulent jet noise analysis can be developed on this basis.

2. Basic assumptions

This study is based on the following assumptions. The turbulent jet noise is generated by three primary sources: (a) small-scale turbulence (mixing noise), (b) dipole source located at the nozzle edge (lip noise) and (c) vortex rings in the initial part of the jet. The first two assumptions are generally recognized in aero-acoustics [2, 22-24]. The last assumption is new (it was formulated in author’s papers [25,26]) and is checked in this paper for the jet with a given velocity of 120m/s excited at =2032Hz. We assume further that in a narrow frequency range, each source is localized in a specific area and can be replaced with an equivalent point source at the jet axis characterized by z coordinate, directivity, amplitude and convective velocity (Fig. 1).

≈jetV excf

a) b) Fig. 1. a) Three primary sources in turbulent jet. Photograph of excited turbulent jet with vortex rings in the

initial part of the jet; b) the coordinates in the jet.

2.1. Multipole decomposition of sound field spectral density The sound source classification applied in this study is based on the solution of a wave equation for air at rest and a narrow frequency range. For each frequency harmonic of exp(-iωt) type the solution which meets the radiation condition at infinity has the form of multipole decomposition:

( ) ( ) ( )χθωω ,kr

r

ce n

l/ll

l

ln

lnti YHP1

210

+= −=

− ∑∞

∑= (1)

where r, θ, χ corresponds to the spherical coordinate system 0<θ<2π, 0<χ<π, cln are the constants determined by the source structure; Hn

(1) is the first kind Hankel function; Yln are

spherical functions in the form ( ) θχ innl

nl ePY cos= where Pl

n(cosχ) are associated Legendre polynomials (Legendre polynomials at n=0). Obviously, the near and intermediate fields will have the same structure in terms of θ and χ but different functions from r. In the Eq. (1), the terms with l=0 represent monopole radiation, the terms with l=1 - dipole radiation, the terms with l=2 - quadrupole radiation, etc. The monopole terms can be neglected for the turbulent jet noise. Since the terms ( )θinexp transform into constants when square pressure (radiated power) is averaged by the azimuthal angle, we will not consider the dependence from θ. 2.2. Convective sources If the quadrupole sources move in the jet, the sound field directivity depends largely on the convective velocity. Therefore, the Eq.(1) derived for the motionless sources should be corrected. A standard technique [2,27-28] takes into account the convective motion of the source and its location in limited space. The result shows that the radiation intensity of quadrupole sources must be multiplied by the convective factor 5)cos1(1 χM− where M is the Mach number of the moving sources. Consider the simple model problem to demonstrate the effect of the convective amplification and obtain the formula for octupoles. Let the sound sources which are harmonic in time, exp(-iω0t+ niξ ), move with velocity V and exist only in region (-z1,z1), nξ is the initial phase of nth source, considered to be stochastic value. The sources are supposed to appear one by one with intensity A. The statement of the problem is quite adequate to the real jet where the operation time of acoustic sources (vortex ring life from origination to disappearance) is limited. We also can imagine the finite time of source operation as if they pass by the window, - z1 < z < z1, transparent for sound. This model task (Fig.2) permits analysis of convective apmplification effect in exact statement and helps to obtain the final equations. The pressure field generated by the sources defined above

( ) ( ) 003

0000 dtrdt,rqtt,rrGp ⋅−−= ∫ (2) where source density

( ) ( ) ( )( ) ( ) ( )zzzznTtvzyxeAq

n

niti +Θ−Θ−−⋅=∑ +−11δδδξω (3)

⎟⎠⎞

⎜⎝⎛ −=

crt

rG δ

π41 (4)

δ - is Dirac functhe source numbethe window trans

F

In accordenominator it is

∑ ∫−

⋅≈n

z

z

Ar

p1

141π

Substituting in exintegral from δ-fuwithin it, we obta

(

([ Mz

cMA

rp

−Θ⋅

−=

π

1

141

1

Consider

where τ >>T. Suwith the same n stochastic phase in the window (-z

p =2

z

r

z1 -z1

χ

ig.2. Sources moving along infinite axis are audible only when they pass by the window (-z1,z1).

tion, Θ - is unit Heviside function which cut off the audible window, n - is r. The sources (3) move with velocity V along z axis. Θ - functions cut off

parent for sound. dance with χcoszrrr 00 −≈− , neglecting differences from r in easy to obtain from (2)-(4)

( ) ( )[ ]++−−⋅+−+− n dzVnTMrVtcosMzicosikzikrtiexp 000 1 χδξχω (5)

ponent the argument of δ-function and using the obvious property that nction equals to zero beyond the interval of integration and equals to 1 in

)

) ] ( )[ ]VnTMrVtcosMzVnTMrVtcos

iVnTcosM

cosikcosM

ikrcosMtiexp

os nn

−−+−Θ⋅++−

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛+

−+

−+

−−∑

χχ

ξχ

χχχ

ωχ

1

111

1

(6)

the squared pressure averaged for a long period of time dtppp ∫ ⋅=τ

τ 0

*2 1

bstituting Eq.(6) in this integral it is clear that only the products of terms give input in the integral. The products of other terms include the exp with

nξ and their input cancels after summation over n if the number of sources 1,z1) is quite large, VzT 12<< . After some algebra we obtain

( ) ( )( )

τχ

χπ VcosMz

cosMrA

n

−⋅

−∑ 12

141

22

2

The sum of identical terms in this Equation is equivalent to multiplying the number of sources by TN τ= . This result in

( ) ( ) VT

zcosMr

Ap 12

22 2

14⋅

−=

χπ (7)

where VTz12 is the number of sources located in the window in each moment. After multiplying by A2 we obtain the total source intensity in the zone (-z1,z1). We see that squared of convective multiplayer ( )θcosM−1 , which appears in denominator of p2, is partly cancelled with the same factor in numerator, which appears in due to effective extension of source zone in argument of Θ-function in (6).

2p

The same procedure can be used for multipoles of higher orders. Consider, at first, dipole sources in similar statement. In this case Eq. (3) could be rewritten in the form

( ) ( ) ( )( ) ( ) ( zzzznTtvzyxeAr

qn

nitii

i+Θ−Θ−−⋅

∂∂

= ∑ +−11δδδξω )

Substituting in (2), with standard manipulations: carrying the derivatives from q to Green function G under integral, altering the derivative argument from r0 to r and changing

( ) tcrrr ii ∂∂→∂∂ , we obtain the equation identical to (5) with derivative over time t:

( ) ( )[ ] ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛++−−⋅+−+−

∂∂

≈ ∑ ∫−n

z

zni

i dzVnTMrVtcosMzicosikzikrtiexpt

Acrr

rp

1

1000 1

41 χδξχωπ

Comparing this Eq. and Eq.(5) we can see that the space derivative in source q leads to time derivative in p. Differentiating two terms under integral over t, using the obvious

replacement ( ) ( ) ( ) ( ) (...zcosMV...t δ

χδ 01

∂∂−−

→∂∂ ) and replace the derivation of δ-function

to derivation of exponent under integral, after some algebra we easy obtain instead of (6)

( )

( ) ( )[ ] ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛++−−⋅+−+−⋅⋅

⋅−−

≈

∑ ∫−n

z

zni

i

dzVnTMrVtcosMzicosikzikrtiexpA

cosMik

rr

rp

1

1000 1

141

χδξχω

χπ

Thus we can conclude that multipole order increase leads to increase in the power of convection multiplier in sound pressure [23,27]. Using the same arguments for 2p as in (7)

we obtain the dipole convective multiplier as 311 )cosM( χ− . Repeating the same procedure for quadrupole

( ) ( ) ( )( ) ( ) ( zzzznTtvzyxeArr

qn

ntiij

ii+Θ−Θ−−⋅

∂∂∂

= ∑ +−11δδδξω )

and octupole

( ) ( ) ( )( ) ( ) ( zzzznTtvzyxeArrr

qn

ntiijm

mii+Θ−Θ−−⋅

∂∂∂∂

= ∑ +−11δδδξω )

we easy obtain for 2p the convective multiplier as 5)cos1(1 χM− and as ( )711 χcosM− correspondingly. Note that assumption that the sources fly one by one with the same period T is not obligatory but gives the Equation (5) in more simple form.

2.3. Dipole

The dipole is located on the nozzle and is normally taken into account for low Mach numbers [2,22] when the turbulent mixing noise fades and the weak dipole noise becomes noticeable. We will assume that the source is localized on the nozzle edge and, as suggested in (1), should take the form:

2

0022 r/cos~Pd χ , (8a)

for the zero azimuthal mode (n=0)

200

22 r/sin~Pd χ , (8b)

for the first mode (n=1). As it mentioned above, we assume the averaging of stochastic dipole directivities over azimuthal angle θ.

Clearly the dipole is zero in the second mode. The angle χ0 and the radius vector r0 are measured from the center of the nozzle exit (Fig.1b). The convective velocity of this source is obviously equal to zero.

2.4. Small-scale turbulence Accordingly to the Lighthill’s pioneer approach small-scale turbulence is a cluster of quadrupole sources moving convectively with velocity which is equal to the average flow velocity at the source point [1-2]. This source has been considered to be the principal generator of the free turbulent flow noise. It follows from (1) and convective motion of the quadrupole sources, briefly discussed in section 2.1, that

( ) ( )5222 131 χχ cosMr/cos~p tq −− , n=0 (9a)

( 52222 1 χχχ cosMr/cossin~p tq − )

)

, n=1 (9b)

( 5242 1 χχ cosMr/sin~p tq − , n=2 (9c)

where n is azimuthal harmonic number, Mt is convective Mach number based on mean velocity in the point of source location.

The angle χ and the radius vector r are measured from the source location on the axis (Fig. 1b). The visualization shows that a possible location of the sources is in the area z>12cm. The convective multiplier is determined by the jet velocity in the point of source location. This velocity varies from its maximum at the jet axis to zero at periphery outside the

shear layer. Since the radial position of the radiating small-scale vortices is to be determined, the value Mt remains a free parameter of the problem.

Fig. 3. Coordinate system in the vortex ring oscillation problem

3. Vortex ring

To obtain the directivity functions similar to (8)-(9) we briefly consider the properties

of the vortex ring as a noise source [25-26] and discuss the properties of vortex rings in the jet. 3.1. Vortex ring eigen modes The main source of vortex ring noise is supposed to be the vortex ring eigen-oscillations. The vortex ring oscillations are characterized by three numbers: the frequency number l, the azimuthal number n and the radial number j. Only fast oscillations with and have non-zero quadrupole moment and are interesting from the point of view of sound radiation. All the oscillations with have the frequency

1=l 2,1,0=n

1=l

( )µω O+Ω= 20 (10)

where is vorticity in the vortex ring core, 0Ω Ra=µ , R is the ring radius and is the vortex core section radius. Oscillations with different values have the form

an ( )θniexp

where θ is the angle around the ring axis (Fig. 3). Let us examine the key characteristics of the radiating vortex ring oscillations. At n=0, the oscillations are axisymmetric (about the ring axis) and representing elliptical deformation of the core section boundary in the main µ approximation: ( )ϕiexp 2 where ϕ is the polar angle in the core section. These oscillations are the only effective source of noise at n = 0.

A vortex ring has a more complex variety of eigen-oscillations at . Two types of oscillations can occur in this case (Fig.4). First, there can be isolated modes that are similar to axisymmetric modes described above, since both represent elliptical deformations of the core cross-section boundary in the absence of vorticity disturbances inside the vortex core.

2,1=n

Fig.4

Secthe elliptidisturbanccenter line 3.2. VorteThe expreterms of M

where determinedthe time dcompared term for lo(Mach num

ijC

Subexpression

p

where θ,r

the sound

geometricadirectivitydimension

( )θχ ,Aq +

a) b) . Radiating modes of vortex ring l=1, n=2, accompanied by mean line oscillations and rotation of

vortex core deformations. a) isolated mode, b) bessel mode.

ond, the vortex ring also has a family of Bessel oscillations at in which cal deformation of the core boundary

2,1=n( )ϕiexp 2 is accompanied by vorticity

es inside the core ( )ϕiexp and displacement of the core boundary from the ring’s .

x ring noise ssion for the far sound field (pressure disturbance P) in two approximations in ach number M is as follows:

,r

crtD

r

rrrr

crtC

r

rrP ijkkjiijji⎟⎠⎞

⎜⎝⎛ −

−⎟⎠⎞

⎜⎝⎛ −

= 320

4ρπ

(11)

and are quadrupole and octupole densities localized in the vorticity region, by the dynamic of the vorticity field in the vortex core [29-30], top dots denote

erivatives. The octupole term is proportional to the next power of Mach number with quadrupole one and usually is expected to be small. However, the octupole calize vortex with thin core is proportional to the ratio of two small parameters ber and thickness parameter) and appears not to be small.

ijkD

stituting the forms of radiating eigen-oscillations [25] into (11) and using the s for and from [28] we get the sound field in the following form: ijC ijkD

( ) ([ θχβθχπ )]πρ ,A,ARk

re octq

rik+−= 220

4 (12)

χ, are the spherical coordinates (Fig.4), cωκ = , ω is the angular frequency, c is

velocity. For different azimuthal modes of vortex ring directivity has its characteristic form [30] depending on the frequency range,

l and other parameters of the vortex ring. Thus, the difference of the radiation from pure quadrupole directivity (Fig.5) is determined by the value of less coefficient

,,,n 210=

( θχβ ,Aoct )

β which has the value:

µβ M

cR 20 =Ω

= , ca

M2

0Ω= ,

Ra

=µ (13)

The directivity is a quadrupole one at ( ) ( )[ 2θχβθχ ,A,A~F octq + ] β =0. The value of β

for thin vortex rings is a ratio of two small parameters: Mach number M and parameter of ring thickness µ and even at small Mach numbers it appears to be not small. Exactly this fact indicates a substantial difference between the pure quadrupole noise produced by small-scale turbulence and the noise of localized vortices with thin cores which has a directivity with significant octupole contribution. Diagnostics of this difference offers a unique possibility of quantitative separation of the noise produced by localized vortices out of the total jet noise. 3.3. Evaluation of vortex ring mean parameters in the jet As it was noted above, the diagnostics of the difference between the experimentally measured jet noise directivity and the noise directivity of convecting quadrupoles distributed over the jet volume, gives a possibility of quantitative estimation of the contribution of the localized large-scale vortices to the total noise. For this purpose it is necessary not only to identify the vortex rings in the turbulent jet, but also to measure those parameters of vortex rings which determine the main characteristics of their acoustic directivity (velocity, diameter, thickness parameter µ ). For this purpose we make the frame-by-frame processing of video-fragments (Fig.1) to measure vortex ring velocities. The technique of the vortex ring visualization in the excited jet is presented in [31].

The results of vortex ring radius measurement give 52,R ≈ cm, the vortex core radius cm and subsequently the thickness parameter 70,a ≈ 30,Ra ≈=µ . This value is an upper

Fiinc

g.5. Transformation of radiation directivity F for 3D eigen-modes of vortex ring accordingly to Eq. (12) as βrease (the first column corresponds to pure quadrupole directivity, β =0). Vortex ring moves along positive z

direction; directivities for n=1 and 2 are averaged over the azimuthal angle θ.

estimate. This parameter directly enters the integrals for quadrupole and octupole momentums in (11). The ring velocity value lfV excring ⋅= was determined from the analysis of video-fragments, according to the measuring of the distance l between two rings and the time scale excf1=τ , where is the acoustic excitation frequency. The measured values of vortex ring velocity in the initial part of the jet are presented in Table 1 for the jet issue velocities in the range of 55–130 m/s.

Hzfexc 2032=

Table 1

Jet velocity, m/s

Distance between ring 1 and ring 2, cm

Vortex ring velocity, m/s

jetVringV=α

55 1.5 29.5 0.54 63 1.7 33.4 0.53 77 1.9 37.3 0.48 90 2.1 41.3 0.46 102 2.5 49.1 0.48 109 2.5 49.1 0.45 115 2.6 51.1 0.44 120 2.7 53 0.44 126 2.8 55 0.43 130 2.9 56 0.43

3.4. Evaluation of the characteristic frequencies of sound-generating oscillations of the vortex rings in jet Using the estimation of geometric parameters and mean velocity of vortex rings in excited jet, obtained through visualization, one can evaluate the mean value of vorticity in the vortex core and the eigen-oscillation frequency of radiating modes 20Ω=ω (10). Vortex ring structure assumes to be the simplest one, corresponding to uniform distribution of vorticity,

= const, in the core. The value in the thin core (vortex core radius is equal to ) can be estimated using the assumption, that all the vorticity contained in the jet shear layer shedding from the nozzle lip during the time

0Ω 0Ω a

excf/1=τ converts into the single vortex ring. In this case, the following relation is valid:

02Ω=⋅=Γ alV jet π

where is the circulation, Γexc

ring

fV

l = is the distance between the rings in the initial part of the

jet. Expressing the vorticity through the measured values, we get:

exc

jet

exc

ringjet

fR

V

fa

VV22

2

20 µπ

α

π

⋅≈

⋅=Ω

, (14)

In the right part of Eq. (14) all the values are estimated experimentally: 440,≈α (Table 1), parameter of vortex ring thickness s/mV,cm,R,, jet 1205230 =≈≤µ . Note, that it is rather difficult to evaluate the part of vorticity that converts into the vortex core, and

Eq. (14) gives only an upper estimation of the typical frequency of vortex radiation. Hence the typical frequency of the radiating vortex ring eigen-oscillations will be as follows:

HzfR

Vf

exc

jet 180044 222

20

0 ≈⋅

<Ω

=µπ

απ

The vortex rings having identical circulation Γ have the vortex core radius a distributed in some range. Therefore the mean vorticity 2

0 aπΓ=Ω is distributed too and, correspondingly, the vortex ring noise frequencies could cover the region from 200 to 2000Hz. Thus, the frequency range of 600-1350Hz considered in the paper is a quite representative spectrum part in which the vortex ring contribution into the overall jet noise is to be expected. 3.5. Vortex ring noise directivity The dimensionless coefficient β for the vortex rings in the frequency bands under study is within the range:

315010249 40 ,,f,c

Rrad −≈⋅=

Ω= −β (15)

where f is the frequency in Hz. In particular, - 8,05,0~ −β with mean value of 7,01 =β for the first frequency range Hzfrad 800600~ −- 0,17,0~ −β with mean value of 9,02 =β for the second frequency range

Hzfrad 1050800~ −- 3,10,1~ −β with mean value of 2,13 =β for the third frequency range

. Hzfrad 13501050~ −Thus the directivity A in the expression (12) takes the following form for vortex ring

oscillations: ( )χβ octq AAA 000 += , ( )χ2

0 31 cosAq −= , n=0 (16a)

( )χβ octq AAA 111 += , , n=1 (16b)

χχ cossinAq =1

( )χβ octq AAA 022 += , . n=2 (16c)

χ22 sinAq =

where, n is azimuthal harmonic number, β is taken from (15) for each frequency range under consideration

The first terms in these expressions correspond to the quadrupole’s contribution, it coincides with the small-scale turbulence directivity (2a-c). The second term represents the input of the octupole, which was calculated in [30]. Transformation of directivity of vortex ring noise due to addition of octupole term as β increase is presented on Fig.5. Taking into account the convective velocity of the vortex rings (Table 1), the total directivity should be transformed using the convective multipliers for quadrupole and octupole from section 2.1. It will result in the expression:

( )[ ] ( )7

222 11 χβθ cosMrAcosMA~P rocti

qii −+− , (17)

where is defined for each azimuthal harmonic by the expressions (16a)-(16c), the convective Mach number M

iA

r =0.16-0.17. The angle χ and the radius vector r are measured from the radiating vortex ring position z which has to be determined too. The visualization (Fig.1) shows that a possible location of the source in the region 4cm<z<12cm.

Thus associated with each azimuthal component is an array of basic functions (8), (9) and (16) with a set of unknown parameters that should be found (or refined) on the basis of the experimental results.

4. Experimental investigation of noise azimuthal structure

The azimuthal decomposition technique (ADT) applied to the sound field of aerodynamic noise sources is based on analysis of far field power directivity for azimuthal noise components in narrow frequency bands. For measurements we use a circular array of six microphones and synchronous multi-channel analysis of acoustic data. The method was first used by V.F. Kopiev et al [18,19] to analyse the vortex ring noise and subsonic jet noise. In this study we analyse an axisymmetric subsonic jet issued from a conic dс=4cm nozzle at

120m/s. A loudspeaker located at the settling chamber was used as a longitudinal acoustic excitation source. The exact value of acoustic excitation frequency was 2032 Hz.

≈Vjet

The jet noise modal structure was measured in the acoustic anechoic chamber using a circular D = 1.7m array with 6 microphones (Brüel & Kjær's pre-polarized 1/4'' ICP® type 4935) that provide good phase performances at 0-5 KHz which is highly important for the azimuthal decomposition technique. The array moved along the jet axis by 0 to 290cm with a step of 5cm sweeping over the cylindrical surfaces at 85cm from the jet axis. The nozzle exit corresponded to the coordinate z= 65cm. The experiment set-up is shown in the diagram below (Fig.6).

Fig. 6. Microphone array in anechoic chamber

Averaged spectra of the azimuthal harmonics were measured in each section and analysed in different frequency bands. Brüel & Kjær’s “Pulse 3560D” data acquisition and Pulse LabShop X software adapted to ADT analysis system was used.

We represent the sound field of the jet in the form of a Fourier expansion in azimuthal θ-harmonics:

( )t,,P χθ = ...sinBcosAsinBcosAA +++++ θθθθ 22 22110 (18)

Accordingly ADT we measure the coefficients which are estimations

with some accuracy of the coefficients in (16) (see, for example, [20,21]). Fig.7 presents typical power spectra of the azimuthal harmonics

. The radiation directivity in terms of distance along the jet axis (corresponding to

23

22

22

21

21

20 ,,,,, ababaa

23

22

22

21

21

20 A,B,A,B,A,A

23

22

22

21

21

20 ,,,,, ababaa

χ angle by simplest algebra) of the three azimuthal modes averaged over the frequency bands is shown in Fig. 8. The smoothness and shape of the curves and their min and max positions are the main peculiarities for modelling. One can see that as the Strouhal number grows, the peaks on the curves shift steadily in the direction showed by the arrows. We notice also that the second azimuthal harmonic is less sensitive to frequency than the other harmonics. The frequency band was split into three typical subbands for the modelling purposes: 600-800Hz, 800-1050Hz and 1050-1350Hz. The refraction effect is not significant in this frequency range.

22

21

20 ,, aaa

4. Modeling

As was stated above, this study is conducted on the basis of several assumptions. The turbulent jet noise is generated by three sources: a) small-scale turbulence, b) a dipole localized at the nozzle exit and c) vortex rings in the initial part of the jet. The first two assumptions are generally recognized in acoustics. The third one is new and is checked in the current work for jet 120m/s excited at 2032Hz. We assume also that in a narrow frequency band, each source is localized in a specific area and can be replaced with an equivalent point source at the jet axis characterized by z coordinate, directivity, amplitude and convective velocity. The visualization results can be used to estimate the location regions for different types of sources:

(i) The turbulent vortex rings are significant in the region 4cm<z<12cm and their convective velocity is known to be М=0.16-0.17.

(ii) The small-scale turbulence is insignificant at 0-4cm. Turbulence is packed into separate large-scale vortices at 4-12cm. Thus, the small-scale turbulence is located in the region z>12cm. In this region the small-scale turbulent vortices can be found at any distance from the axis where the average velocity varies from Mt=0 to Mt=0.35, so the full range of possible velocities cannot be clearly identified beforehand.

(iii) The dipoles at the nozzle exit have zero velocity and are located at z=0. Although the contribution of these sources to the total noise is insignificant, they should be added to adjust the approximation to the measurement data at the angles with minimal radiation level. This minimal level doesn’t drop to zero and this fact cannot be modelled without dipole contribution (as an example, see the experimental curves for the first mode near the origin of coordinates, Fig.9).

Let us consider the modeling of each mode separately.

Fig

. 9. Location of dipoles (a), vortex rings (b) and small-scaleturbulence (c). Measurement line with 4cm scaling.

4.1. Axisymmetric mode (n=0) First, let us check if the zero mode can be modelled by conventional sources: convectively moving quadrupoles and an immobile dipole at the nozzle lip (it is not specifically indicated whether more dipoles should be added, but their presence becomes evident from the relevant Figures). In each frequency band (Fig.7a), the experimental curve has two distinct maximums and looks like a directivity of axisymmetric quadrupole source (Fig.5a, n=0). Let us try to approximate the measurement data in each frequency band using the selected set of modelling functions. Fig. 10a shows the experimental results (curve consisting of circles) in frequency band 1050-1350Hz and the curves for the averaged amplitudes of pressure squared generated by small-scale turbulent sources located at the smallest permissible distance from the nozzle edge (12cm) and for the varying Mach number of the sources’ convective motion. One can readily see that the ratio of levels in the curve’s peaks can become satisfactory at M=0.28. However, all the modelling curves are shifted to the right of the experimental curve. To meet the experimental data we have to place the sources in the region inside the nozzle (z<0)! The same behaviour is observed in the other frequency bands. Thus the convectively moving quadrupoles that represent the sound radiation by small-scale turbulence are not sufficient for describing the experimental data and we are to suggest that sources of some other kind exist in the jet which transform the directivity into upstream direction (see directivity transformation on Fig.5).

Fig. 10b shows an attempt to approximate the experimental curve on the base of the vortex rings and an immobile dipole at the nozzle lip (without small-scale turbulence). The difference from the previous modelling case is that the location of the rings is varied in the range 4cm<z<12cm with the Mach number fixed at M=0.17. This estimation of the rings’ motion was obtained in section 3.3. It can be easily seen that the modeling curve is now located to the left of the experimental curve. The ratio of the peaks’ amplitudes proves to be insufficient. The same behaviour is observed in the other frequency bands.

Hence an obvious conclusion could be made that the experimental data correspond to a mixture of two types of sources: convectively moving quadrupoles that represent sound radiation by small-scale turbulence and the vortex rings. The required source locations and relative amplitudes can be easily found by minimization of the root mean square deviation of the experimental and modelling curves with the restrictions on model parameters defined above. The obtained approximating curves are presented in Fig.11a,b,c in different frequency bands where they are marked by the same colours as the source types depicted by a colour point in the photos. The optimum model parameters corresponding to the approximating curves in Fig. 11a,b,c are listed in Table 2.

Table 2. Optimum parameters of the jet noise model for the axisymmetric (n=0) azimuthal harmonic.

Frequency band 600-800Hz 800-1050Hz 1050-1350Hz β parameter 0.74 0.97 1.11 Vortex ring position ringx 7cm 7cm 10cm

Convective Mach number of the vortex ring ringM 0.17 0.17 0.17

Small-scale turbulence position turbx 17cm 12cm 13cm Convective Mach number of the small-scale turbulence

turbM 0.33 0.35 0.34

Relative contribution of the vortex ring to the total acoustic power 0.40 0.30 0.45 Relative contribution of the small-scale turbulence to the total acoustic power 0.53 0.61 0.50

The energy input of different sources to the total noise was defined by means of numerical integration of for each source along the element of the cylinder on which the measurements were made (obviously this value is proportional to the energy flow

across the cylindrical surface around the jet). The values of energy contributions of various sources are given for the optimal parameters in Table 2. The energy contribution of the vortex rings to the total jet noise is about 34-45%.

χsinP2

VP

4.2. First azimuthal mode (n=1) The experimental curve for the first azimuthal mode of the jet noise (Fig.8b) has the main peak like the quadrupole directivity n=1 and the second peak in the back half-sphere.

Let us approximate the measurement data using the selected set of modeling functions. The procedure is basically the same as in the case of the zero mode. The only difference is that in contrast to zero harmonic, there are two types of vortex ring eigen-oscillations for n=1: the isolated and Bessel modes (see section 3.1).

Like for the zero mode, the curves modelling the input of the small-scale turbulence into the first azimuthal mode are located to the right from the experimental curve. When the convective Mach number increases, the curve becomes more similar to the experimental curve in terms of the inclination over the entire measurement area. However, the peak location moves further downstream. Varying the source position is not helpful either in approximating the experimental curve (Fig.12b). So, the sources of other kind should be taken into account.

Both the isolated and Bessel modes have been tried to approximate the measurement data. Fig.12b shows the evolution of the curves representing the contributions of these two modes to the first azimuthal mode, with the varied vortex ring position. The Bessel mode curves are located further to the left of the experimental curve which makes them hardly to be a candidate for modelling the total curve.

Thus, the experimental curve can be modelled by both the small-scale turbulence, isolated oscillations of the vortex ring like in the previous case (n=0) and immobile dipole at the nozzle lip. The approximating curves obtained by the minimization of the root mean square deviation in different frequency bands are displayed in Fig.13a,b,c. The optimum model parameters are listed in Table 3. The contribution of the vortex rings is about 20-30%.

Table 3. Optimum parameters of the jet noise model for the azimuthal harmonic n=1.

Frequency band 600-800Hz 800-1050Hz 1050-1350Hz β parameter 0.74 0.97 1.11 Vortex ring position ringx 9cm 9cm 9cm

Convective Mach number of the vortex ring ringM 0.17 0.17 0.17

Small-scale turbulence position turbx 15cm 19cm 16 cm Convective Mach number of the small-scale turbulence Mturb 0.31 0.33 0.31 Relative contribution of the vortex ring to the total acoustic power 0.20 0.23 0.31 Relative contribution of the small-scale turbulence to the total acoustic power 0.66 0.66 0.59

4.3. Second azimuthal mode (n=2) The experimental curve for the second mode (Fig.8c) is distinguished by a single peak slightly shifting as the frequency changes.

Let us consider the approximation in terms of three main sources. Like in the previous case, in the case n=2 there are two types of vortex ring eigen-oscillations: the isolated and Bessel modes. The convecting turbulence and both types of oscillations of the vortex ring have the required shapes although the convecting turbulence curve has a significantly different left slope and is much narrower than the experimental curve at low Mach numbers (Fig.14a). As the Mach number increases, the curve becomes wider, shifts entirely to the left and breaks away from the experimental curve. The vortex ring directivity is shifted to the left (Fig.14b,c), so to approximate the experimental curve by the vortex rings only, the rings must be placed in an area having no rings at all. Like in the first two cases, the experimental curve is modeled here by both types of sources (Fig. 15 a,b,c) – small-scale turbulence and isolated oscillations of vortex rings. The relative contribution of the vortex rings to the total jet noise is about 40-45%, i.e. almost half in the 1050-1350Hz frequency band (see Table 4).

Table 4. Optimum parameters of the jet noise model for the azimuthal harmonic n=2

Frequency band 600-800Hz 800-1050Hz 1050-1350Hz β parameter 0.74 0.97 1.11 Vortex ring position ringx 10cm 10cm 11cm

Convective Mach number of the vortex ring ringM 0.17 0.17 0.17

Small-scale turbulence position turbx 18cm 25cm 25cm Convective Mach number of the small-scale turbulence Mturb 0.32 0.26 0.26 Relative contribution of the vortex ring to the total acoustic power 0.41 0.40 0.45 Relative contribution of the small-scale turbulence to the total acoustic power 0.59 0.60 0.55

Thus the main conclusion is that the vortex rings observed in the excited V=120m/s jet visualization make a significant contribution to the total noise radiation in the frequency band 600-1350Hz.

5. Additional experiments with two coaxial microphone arrays (twofold ADT)

The proposed approach for jet noise modelling was additionally checked by measurements with two co-axial microphone arrays. These arrays were located at different radial distances from jet axis, 75,6sR cm= and 85,6lR cm= (Fig. 16).

The main peculiarity of experimental technique is the possibility of simultaneous acoustic measurements and azimuthal decomposition of jet noise at the same axial position and different radial position (Fig. 17). In other respects the present technique is similar to the one described in section 3.5. Averaged spectra of the azimuthal harmonics were measured in each section and analysed in different frequency bands. B&K Pulse 3560 data acquisition and Pulse LabShop X software adapted to double ADT analysis system was used. It was expected that azimuthal harmonic directivity of jet noise for two radial distances can be approximated using the same parameter set. The measurement data and modelling curves are presented in Fig. 18. One can see a good agreement of the results obtained with the same parameters for different radial positions in the range of axial distances up to 40 calibers from the nozzle. The discrepancy of measurement data and modelling curves at large distance can be possibly

related with the effect of refraction. Thus the results of present experiment are the additional independent validation of theoretical approach formulated in the present paper.

Fig. 16 Photograph of coaxial microphone array in anechoic chamber

Fig. 18 Modeling curves approximating the axisymmetricazimuthal mode directivity for two radial position (red line–

75,6sR cm= , green line – 85,6lR cm= ) measured in thеfrequency band 800-1050Hz. Experimental data are displayed ascircles. Modelling parameters: vortex ring position , small-

scale turbulence position , convective Mach number of the

vortex ring and small-scale turbulence , are thesame for both modelling curves

ringx

turbx

ringM turbM

Fig. 17 Scheme of experimental set-up with two microphone arrays

6. Conclusion

The main objective of the study is to assess quantitative contribution of large-scale vortices to the total jet noise for a specific parameter range.

The subject of investigation is a turbulent jet with a flow velocity of 120m/s excited at 2032Hz. Analysis of the problem helped select a frequency range for the investigation: 600-1350Hz - quite a representative part of the spectrum where the vortex rings should be expected to make a noticeable contribution to the total jet noise and where the refraction is negligibly small. A visualization technique developed earlier was used to obtain estimations of the vortex rings' velocities and geometric dimensions. They helped make the first calculations of the vortex rings' acoustic radiation directivity distorted by the octupole at 600-1350Hz. A set of basic functions was created for a vortex ring and used to model azimuthal noise components and assess the quantitative contribution of the vortex rings to the total jet noise.

The averaged spectra of the azimuthal harmonics of the jet noise were measured and analyzed by azimuthal decomposition technique (ADT) in different frequency bands. The sound radiation power of azimuthal harmonics was measured on a R=0.85m cylindrical surface around the jet along its axis. The frequency band was split into three typical subbands for more accurate modelling: 600-800Hz, 800-1050Hz and 1050-1350Hz.

A quantitative assessment of the vortex ring oscillation contribution to the total subsonic jet noise in a considered specific case was made for the first time based on the comparison of the measurements for each azimuthal harmonic with the directivity patterns of the convecting turbulence and vortex rings. It was obtained that the energy input of the vortex rings to the total jet noise in the frequency bands under consideration is 30 to 45% depending on the harmonic number and frequency band.

The proposed approach for jet noise modelling was additionally checked by measurements with two co-axial microphone arrays. Good agreement of azimuthal noise modelling using the same parameter set for two radial distances was demonstrated. The results of this experiment are the additional independent validation of theoretical approach formulated in the present paper.

The results obtained give much deeper insight into the structure of the aerodynamic noise sources in a turbulent jet and can form the basis for a fundamentally new technique of investigation the radiating turbulence in a subsonic jet. Subsequently, a new noise control concept and new approaches to devising robust numerical methods for turbulent jet noise analysis can be developed on this basis.

Acknowledgments The work is partly realized with the support of Russian Foundation of Basic Research

(05-01-00670) and AIRBUS.

References 1. Lighthill, M. J.: On Sound Generated Aerodynamically: 1. General Theory. Proc. Royal Soc.

London, ser. A, v. 211, No. 1107, Mar. 20, 1952, pp. 564-587. 2. Lighthill, M. J.: On Sound Generated Aerodynamically:11. Turbulence as a Source of Sound. Proc.

Royal Soc. London, ser A, 222, No. 1148, Feb. 23, 1954, pp.1-32. 3. Crow, S. C. and Champagne, F. H.: Orderly structure in Jet Turbulence. J. Fluid Mech., v. 48, pt.3,

Aug. 16, 1971, pp. 547-591.

4. Yule, A. J.: Large-Scale Structure in the Mixing Layer of a Round Jet. J. Fluid Mech., v. 89, pt. 3, Dec. 13, 1978, pp. 413-432.

5. Moore, C. J.: The role of shear-layer instability waves in jet exhaust. J. Fluid Mech., 1977, v. 80, pt. 2, pp. 321-367.

6. Hussain, A. K. M. F.: Coherent Structures - Reality and Myth. Phys. Fluids, v. 26, No. 10, Oct. 1983, pp. 2816-2850.

7. Vlasov, Ye. V. and Ginevsky, A. S.: Coherent structures in turbulent jets and wakes, “Science and mechanics”, v. 20, Moscow, VINITI, 1986, pp. 3-84.

8. Davies, P. O. A. L. and Yule, A. J.: Coherent structures in turbulence. J. Fluid Mech. 1975, v.69, pp. 513-537. 9. Browand, F. K. Laufer, J. The role of large scale structures in the initial development of circular jets.:

Proc. 4th Biennial symp. Turbulence in Liquids, Univ. Missouri-Rolla, 1975, pp. 333-344. Princeton, New Jersey: Science Press.

10. Armstrong, R. R., Michalke, A. and Fuchs, H. V.: Coherent structures in jet turbulence and noise. American Institute of Aeronautics and Astronautics Journal 15, pp. 1011-1017.

11. Tam, C. K. W. “Jet Noise Generated by Large-Scale Coherent Motion” in “Aeroacoustics of flight vehicles. Theory and practice” ed. H. Hubbard, v. 1, pp. 311-390 ASA/AIP.

12. M. Cabana, V. Fortune, P.Jordan. "A look Inside the Lighthill source term". AIAA Paper 2006-2484. 13. J. Freund. "Noise source in a low-Reynolds-number turbulent jet at Mach 0.9". J. Fluid Mech., v.438,

2001, p. 277-305. 14. M. Samimy, I. Adamovich, B. Webb, J. Kastner, J. Hileman, S. Keshav, and P. Palm “Development

and application of locolized arc filament plasma actuators for jet flow and noise control” AIAA 2004-0184, 2004, p.1-19

15. Ffowcs Williams, J.E., Kempton A.J., The noise from large-scale structure of a jet., JFM, v.84(4), 1978.

16. Juve, D., and Sunyach, M., 1981, “Near and Far Field Azimuthal Correlations for Excited Jets,” AIAA Paper 81-2011.

17. Maestrello, L., “Two point correlations of sound pressure in the far field of a jet: experiment”. NASA-TMX-72835, 1976.

18. Michalke A., Fuchs H.V., “On turbulence and noise of an axisymmetric shear flow”, J. of Fluid Mech., 1975, v. 70, pp. 179-205.

19. Fuchs H.V., Michel U., “Experimental evidence of turbulent source coherence effecting jet noise”, AIAA Paper 77-1348, 1977, pp. 1-10.

20. Kopiev V.F., Zaitsev M.Yu., Chernyshev S.A. and Kotova A.N. The role of large-scale vortex in a turbulent jet noise. AIAA Paper 99-1839, 1999, 13p.

21. Kopiev, V. F. “Azimuthal decomposition of turbulent jet noise and its role for diagnostic of noise sources” VKI Lecture Series 2004-2005 "Advance in Aeroacoustics and Applications", pp.1-24.

22. J.E. Ffowcs-Williams, C.G. Gordon “Noise of highly turbulent jets at low exhaust speeds” AIAA Journal vol. 3, No. 4, 1965.

23. M.E. Goldstein, "Aeroacoustics" McGraw-Hill Int.Co. 1976. 24. D.G. Crighton, "Basic priciples of aerodynamic noise generation", Prog. Aerospace Sci., 1975, v.16, n1, 31-96. 25. Kopiev, V. F. and Chernyshev, S. A. “Vortex ring eigen-oscillation as a source of sound”. J. of Fluid

Mech. 1997, v. 341, pp. 19-57. 26. Kopiev, V. F. “Theoretical and Experimental Investigation of VortexRing Noise” VKI Lecture Series

2004-2005 "Advance in Aeroacoustics and Applications", pp.1-33 27. Ffowcs Williams J.E. “The noise from turbulence convected at high speed. Phil. Trans. Roy. Soc.,

A225, pp. 469-503, 1963. 28. H.S. Ribner, "The generation of sound by turbulent jets". Adv. Appl. Mech., 1964, v.8, 103-182. 29. Kop'ev, V.F. and Leont'ev, E.A., Some Comments on Lighthill's Theory in Connection with Sound

Emission by Compact Vortices, Akust. Zh., 1986, vol. 32, no. 2, pp. 184 - 189 [Sov. Phys. Acoust., vol. 32, no. 2, pp. 109-112].

30. Kopiev V.F., Chernyshev S.A. “Octupole radiation of localized vortex”, AIAA Paper 2005-2955, 2005, pp. 1-16.

31. Kopiev V.F., Zaitsev M.Yu., Inshakov S.I., Guriashkin L.P. (2003) “Visualization of the large-scale vortex structures in excited turbulent jets”, Journal of Visualisation, The Visualization Society of Japan and Oshma, 6 (3), pp. 303-311.

Figures

0 400 800 1.2k 1.6k 2k 2.4k 2.8k 3.2k[Hz]

20

30

40

50

60

70

80

90[dB/400p PaІ]

Signal 1Signal 2Signal 3Signal 4Signal 5Signal 6

a)

400 800 1.2k 1.6k 2k 2.4k 2.8k 3.2k[Hz]

0

10

20

30

40

50

60

70

80

90[dB/400p PaІ]

FFT Analyzer \ Signal 1FFT Analyzer \ Signal 2FFT Analyzer \ Signal 3FFT Analyzer \ Signal 4FFT Analyzer \ Signal 5FFT Analyzer \ Signal 6

b)

Fig. 7. Averaged spectra of the azimuthal harmonics of the turbulent jet noise . Distance from the nozzle exit: a) x=90cm, b) x=195cm

23

22

22

21

21

20 ,,,,, ababaa

0

0.00001

0.00002

0.00003

0.00004

0.00005

0.00006

0.00007

0 50 100 150 200 250 300

x,см

Pa^2

/Hz

350-600 600-800 800-1050 1050-1350

22af∆

a)

0

0.00001

0.00002

0.00003

0.00004

0.00005

0.00006

0.00007

0.00008

0 50 100 150 200 250 300

x, см

Pa^2

/Hz

350-600 600-800 800-1050 1050-1350b)

22af∆

0.00006

0.00007

2a

0

0.00001

0.00002

0.00003

0.00004

0.00005

0 50 100 150 200 250 300

x, cm

Pa^2

/Hz

350-600 600-800 800-1050 1050-1350c)

2

f∆

Fig. 8. Distribution of azimuthal harmonics power averaged over frequency bands along the jet axis. Distance from the microphone array to the jet axis: R=85cm. Nozzle position: x=65cm. a) n=0, b) n=1; c) n=2.

n=0

Fig. 10 a. Evolution of the directivity curves modeling the contribution of small-scale turbulenceto the azimuthal mode n=1 with the increasing Mach number =0.1; 0.17; 0.25; 0.34. Source position:

. Experimental curve frequency band: 1050-1350Hz. The arrow shows the curve displacements at higher M.

convMcmxturb 12=

n=0

Fig.10 b. Evolution of directivity curves modeling the contribution of the vortex ring to the azimuthal mode n=0 with a varying source position . Convecting Mach number: . Experimental

curve frequency band: 1050-1350Hz. The arrow shows the curve displacements at higher x. cmxring 12,8,5= 17.0≈convM

a) b)

c)

Fig. 11. “Vortex ring + small-scale turbulence” modeling curves approximating the axisymmetric azimuthal mode directivity measured in three subbands: a) 600-800Hz, b) 800-1050Hz and c) 1050-1350Hz. Total, vortex

ring and dipole noise. Measurements display as circles.

n=1 n=1

a) b)

Fig. 12 a ,b. Evolution of the directivity curves modeling the contribution of small-scale turbulence to the azimuthal mode n=1. Experimental curve frequency band: 1050-1350Hz. a) increased convective Mach number

=0.1; 0.2; 0.3, source position: convM cmxturb 12= ; b) changed source position ,

convective Mach number: =0.2. The arrow shows the curve displacement due to parameter increase.

cmxturb 30,20,12=

convM

n=1 n=1

Isolated oscillations Bessel oscillations Fig. 12 c. Evolution of the directivity curves modeling the contribution of two types of vortex ring oscillations

to the azimuthal mode n=1 with the changing source position cmxring 12,10,6= . Convective Mach number:

. Experimental curve frequency band: 1050-1350Hz. 17.0≈convM

n=1

a) b)

c)

Fig. 13. “Vortex ring + small-scale turbulence” modeling curves approximating the n=1 azimuthal mode directivity measured in three bands: a) 600-800Hz; b) 800-1050Hz; c) 1050-1350Hz. Total, vortex ring and

dipole noise. Measurements display as circles.

n=2

Fig.14 a. Evolution of the directivity curves modeling the contribution of small-scale turbulence to the

azimuthal mode n=2. Experimental curve frequency band: 600-800Hz. Source position increase: , convective Mach number =0.3. The arrow shows the curve displacement due to

parameter increase. cm,,xturb 302012= convM

b) Isolated oscillations c) Bessel oscillations

Fig. 14 b, c. Evolution of the directivity curves modeling the contribution of two types of vortex ring oscillations to the azimuthal mode n=2 with the changing source position cmxring 12,9,6= . Convective

Mach number: . Experimental curve frequency band: 600-800Hz. 17.0≈convM

n=2

a) b)

c)

Fig. 15. “Vortex ring + small-scale turbulence” modeling curves approximating the n=2 azimuthal mode directivity measured in three bands: a) 600-800Hz; b) 800-1050Hz; c) 1050-1350Hz. Total, vortex ring and

dipole noise. Measurements display as circles.