Numerical Investigations of the Acoustics of a Coaxial

Nozzle

Mihai Mihaescu∗ Robert-Z. Szasz∗ Laszlo Fuchs†

Lund Institute of Technology, Department of Heat and Power Engineering, Lund, 22100, Sweden

Ephraim Gutmark‡

University of Cincinnati, 799 Rhodes Hall, P.O. Box 210070, Cincinnati, OH, 45221, USA

The acoustic field generated by two coaxial jets is studied numerically. The turbulentnon-isothermal flow is handled by Large Eddy Simulations (LES) and the acoustical per-turbations are treated by solving an inhomogeneous wave equation. The acoustical sourceis obtained from the instantaneous LES flow field. The computed flow and acoustical fieldsare compared to measured data. Good agreement between the computed and the measuredresults has been found.

Nomenclature

Acore core jet area, m2 Tcore core jet temperature, K

Afan fan jet area, m2 Tfan fan jet temperature, K

p pressure, Pa Tij Lighthill stress tensor, kg/(m · s2)

Pr Prandtl number ui velocity, m/s

Re Reynolds number Wcore core-stream velocity, m/s

SPL Sound Pressure Level, dB Wfan fan-stream velocity, m/s

St Strouhal number ρ density, kg/m3

t time, s ρ′ acoustic density fluctuation, kg/m3

T temperature, K

I. Introduction

The noise generated by jet engines is a major issue for airports and operators of jet airliners. Modern jetengines have usually a high by-pass ratio. Thus, the acoustic field generated by coaxial (non-isothermal) jetsis of major interest. Computationally, one could handle the flow and acoustics by solving the compressibleNavier-Stokes equations. However, such methods become expensive when far-field acoustic computations arerequired. Other strategies considered a computational domain that includes only the near-field region. Onthis limited domain, the flow field is solved providing the acoustic sources. Subsequently, an acoustic fieldis solved on the same or even a larger separate grid by using an acoustic approximation. In this way, thecombined system offers enhanced computational efficiency. This approach has been used in different forms.To compute the flow field, the simplest approach is based on the numerical solution of the Reynolds AveragedNavier-Stokes equations (RANS) with appropriate turbulence models. Bailly, Candel and Lafon1 obtainedthe aerodynamic field from a numerical solution of RANS associated with a k-ε closure. Computations havebeen carried out for cold supersonic free jets at Mach numbers of 1.67 and 2.0, respectively, and for a hotjet at Mach number 2.0. An expression for the acoustic spectral directivity is obtained after an estimationof the space-time correlation pressure combined with the free space Green function of Lighthill equation.

∗Ph.D Student LTH, Dept. of Heat and Power Engineering, 22100 Lund, Sweden.†Professor LTH, Dept. of Heat and Power Engineering, 22100 Lund, Sweden, AIAA Member.‡Professor UC, Dept. of Aerospace Engineering and Engineering Mechanics, Cincinnati OH, AIAA Member.

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AIAA 2005-420

Copyright © 2005 by Mihai Mihaescu. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

The drawback of the RANS approach is its limited accuracy in predicting the Reynolds stresses. Since onlyinformation about the local mean flow are used, RANS lacks also a description of the unsteadiness of the flowfield and hence cannot represent well the dynamics of the acoustic sources. Improved approaches are usingLES or Direct Numerical Simulation (DNS) for the flow field. These approaches can capture the dynamicsof the turbulent flow. Using LES, Bogey and Bailly3 and recently Bogey, Bailly and Juve4 investigated asubsonic circular jet with a Mach number of 0.9 and Reynolds number of 65000. The acoustic field and theacoustic sources are investigated from the perspective of the dilatation (divergence of velocity). This accountsonly for compressible fluctuations and is linked to the acoustic pressure outside the jet. In concordance withthe observations, predominant sound sources are found just after the end of the potential core region. Usinga DNS database and Lighthill’s acoustic analogy, Rembold, Freund and Wang21 compute the far-field noiseof a low Reynolds number rectangular jet at Mach number 0.5. In the second step LES of the same flow,using this time 1/33 the DNS mesh size, has been performed. The sound has been computed again in thesame manner. It has been found that the low frequency part of the far-field spectrum are well reproducedby LES and a correction for the subgrid-scale contribution to the acoustic source terms does not changethe LES prediction. Aiming to obtain the sound field, Boersma and Lele2 performed LES of compressiblejet at Mach number 0.9 and different Reynolds numbers. The divergence of velocity has been used as ameasure for sound waves emitted by the flow. Colonius et al.9 analyzed data from the DNS of a turbulentjet at Mach number 1.92 in order to explain the noise generation mechanisms. A coarse-grid DNS of a coldaxisymmetric jet at 104 Reynolds number and 0.9 Mach number, coupled with Ffowcs Williams-Hawkings(FW-H) technique for the computation of far-field sound, has been performed by Shur et al.22 The sourcesof sound in isotropic turbulence and corresponding acoustic power per unit volume radiated to the far field,for hot and cold round jets at low turbulent Mach numbers, have been investigated by Lilley17 based on DNSdatabase and Lighthill’s acoustic analogy in a form derived by himself. Brentner and Farassat5 investigatethe aerodynamically generated sound of helicopter rotors, giving also an excellent overview of the theoreticalbackground in the field, starting with the Ffowcs Williams-Hawkings equation. The mechanisms of soundgeneration in a Mach 0.9, Reynolds number 3600 turbulent jet have been investigated by Freund.12 In thiscase, the Lighthill source has been computed by Fourier methods to isolate the portion of the source that mayradiate to the far field. Experimental measurements have been performed by Guj, Carley and Camussi14 inorder to identify the fluid-dynamic events responsible for noise generation in a cold low Mach number jet withReynolds number of 1.4 x 105. Turbulent structures generating noise in a high speed axisymmetric jet withMach number of 1.3, have been analyzed experimentally by Hileman and Samimy.16 Acoustic investigationsof a dual stream non-isothermal jet in a large anechoic chamber have been performed by Callender, Gutmarkand Dimicco6 and further on by Rask, Gutmark and Martens.20 In these cases, different setups with andwithout chevron nozzles are analyzed.

This paper describes a combined computational technique for computing sound production and propa-gation in near- and far- fields. The main objectives are to use an efficient numerical algorithm to analyzethe flow and acoustical fields generated by a coaxial jet. The flow field is handled by well resolved LES. Theinstantaneous flow field provides the instantaneous acoustical sources which enter the non-homogeneous waveequation. This approach has been applied to the coaxial jet set-up that has been used in the measurementsof Gutmark and Callender.7, 8 The agreement between the computed and the measured results is found tobe good.

II. Numerical Approaches

A. The flow solver

The flow solver is a non-dimensional, semi-compressible solver, where semi-compressibility implies that thedensity is dependent only on temperature, not pressure. The flow is described by a system of partialdifferential equations that account for the conservation of mass, momentum and energy. The latter iswritten in terms of the temperature (T ):

∂ρ

∂t+

∂ρui

∂xi

= 0 (1)

∂ρui

∂t+

∂ρuiuj

∂xj

= −∂p

∂xi

+1

Re·

∂2ui

∂xj∂xj

(2)

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∂T

∂t+ ui

∂T

∂xi

=1

Pr · Re·

∂2T

∂xi∂xi

(3)

where Re and Pr are the Reynolds and Prandtl numbers, respectively. A second order polynomial dependencebetween density and temperature is assumed:

ρ = A1 + A2 · T + A3 · T2 (4)

In equation (4), the parameters A1, A2 and A3 are the coefficients of the polynomial function which can beadjusted, depending on the temperature range. The normalized temperature used in the computations isgiven by:

T =T − Tref

Tmax − Tref

(5)

where T is the instantaneous, local temperature, Tmax is the maximum temperature in the domain (inlettemperature) and Tref is the reference temperature.

Turbulence is modelled using LES without any explicit subgrid scale (SGS) model. The main role forthe SGS models is to account for the dissipative effects of the small scales. Thus, the only requirement onthe SGS term is that its dissipation rate equals the energy transfer from the larger scales to the smallerunresolved ones. If the dissipation is too low, the energy will be accumulated on the small scales. If thedissipation is too large, the inertial sub-range with -5/3 slope will be more limited than the one that isanticipated by considering the local spatial resolution. Thus, for a high enough grid resolution, the SGSmodels effects are small and the SGS term can be neglected. This approach has been successfully appliedamong others by Fureby and Grinstein,13 Gullbrand and Chow,15 Olsson and Fuchs.19

Large Eddy Simulation of coaxial jets is performed on a Cartesian, staggered grid with local grid refine-ments. High order finite differences (third and fourth order) are used for the spatial discretization of themomentum equations and second order schemes for the energy equation. The integration in time is donethrough an implicit scheme (using a multi-grid solver) and an explicit three step, low-storage Runge-Kuttascheme for the energy equation. The blocked-cell approach is used in order to handle the complex geometryat the inlet, the cells which have their center outside of the computational domain are treated as containingno fluid and are excluded from the computations.

In the present computations inlet, outlet and wall boundary conditions are applied for solving the flowfield. The fluid velocity and temperature are specified at the inlet. No-slip boundary conditions for thevelocity and constant temperature are set on the solid walls. At the outlet, flux conserving zero gradientboundary condition is applied. Additionally, initial boundary conditions must be specified at the beginning ofthe computations. The flow solver, used to account only for the near-field region, provides the instantaneousdata necessary to compute the acoustic sources. The source terms are discretized in space using second orderfinite differences.

B. The acoustic solver

1. Governing equation

The acoustic solver is based on an inhomogeneous wave equation. Starting from the fundamental equationsof motion, namely continuity and momentum for a compressible fluid, one can obtain an inhomogeneous waveequation in terms of acoustic density fluctuation (ρ′). In dimensionless form the equation can be written as:

∂2ρ′

∂t2−

1

M2·

∂2ρ′

∂xi∂xi

=∂2Tij

∂xi∂xj

(6)

The non-dimensional Lighthill’s stress tensor (Tij) is given by:

Tij = ρuiuj + δij · (p −1

M2· ρ) (7)

where ρ, ui, p are the dimensionless density, velocity and pressure, respectively. M denotes the Mach numberand δij is the Kronecker delta function. The acoustic source is due to the velocity variations and due to the

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zy

x [1 x 1 x 4]

Figure 1. The geometry for the sine-bump channel.

semi-compressibility effects. Thus, the total acoustic source term can be decomposed into velocity variationsrelated acoustic source (I) and entropy related acoustic source (II):

∂2Tij

∂xi∂xj

=∂2 (ρuiuj)

∂xi∂xj︸ ︷︷ ︸

I

+∂2

[(p − 1

M2 ρ)δij

]

∂xi∂xj︸ ︷︷ ︸

II

(8)

The inhomogeneous wave equation is discretized on a Cartesian grid, which can be a separate grid,independent of the flow calculations. The biggest advantage is that the radiated far-field sound is found bysolving only one equation and not all the equations governing the flow (full compressible solver). Additionalgain is related to the numerical efficiency and the need for lower spatial resolution with the inhomogeneouswave equation. Second order finite differences are used for the spatial discretization of the derivatives in thewave equation. The near-field acoustic sources provided by instantaneous LES are transferred from the flowgrid to the acoustical one and constitute a bounded source region for far-field acoustic calculations. Thedata transfer between the grids is done by spatial averaging. In time, either explicit or implicit second orderfinite differences can be used.18

2. Boundary conditions

For the acoustic field, wall boundary conditions (totally reflected waves) and absorbing boundary conditions(non-reflective condition for acoustic waves) can be applied. A wall boundary condition requires a vanishinggradient of the acoustical density fluctuation (ρ′) in the direction normal to the wall (n):

∂ρ′

∂n= 0 (9)

In this way the acoustic waves will be totally reflected from the boundary into the interior of the compu-tational domain. The absorbing boundary conditions do not reflect the acoustic waves. The informationreaching such a boundary should be allowed to leave the computational domain. The condition is imple-mented by using a first order hyperbolic equation at the boundary. The outgoing characteristic at the solidboundary satisfies:

∂ρ′

∂t+

1

M

∂ρ′

∂n= 0 (10)

This relation is used to set up the non-reflective condition for the acoustic waves. Setting the characteristicpropagation speed positive (the sign in front of 1/M), will imply extrapolation of the information frominterior of the domain outside of it. Using upwind finite difference schemes in both time and space, one canobtain from equation (10) the expression for the acoustic density fluctuation at the time step m + 1, on theboundary:

ρ′m+1B = ρ′mB −

∆t

M·ρ′mB − ρ′mB−1

h(11)

where h is the cell size in the direction normal to the boundary. This condition gives a complete absorptionof the waves that are propagating normally towards a specified boundary.

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3. Wave propagation

A computational domain, sketched in Figure 1, is used in one of the test-cases. In order to test the imple-mented boundary conditions. The geometry of the channel consists of a rectangular domain [1× 1× 4] witha straight upper wall and a curved lower wall given by the function:

y(Z) =

0 , if Z ≤ 1

0.1 · {1 − cos [(x − 1)π]} , if 1 < Z < 3

0 , if Z ≥ 3

(12)

The 3D homogeneous wave equation is solved using totally reflecting boundary conditions for acousticwaves on the curved lower wall and on the straight upper wall. Non-reflecting boundary conditions foracoustic waves are employed at all other walls. The initial acoustic perturbation is given by function (13)and is imposed in the center of the domain (0.5 · Xmax; 0.5 · Ymax; 0.5 · Zmax).

ρ′(r, 0) =

sin[0.5 · π

(1 − r

R

)], if r ≤ R

0 , otherwise

(13)

where R is the initial radius of the sphere and r is the radial position in the domain. This disturbance isused to excite acoustic oscillations inside the channel. Figure 2 depicts the wave propagation within the

z

y

z

y

(a) (b)

z

y y

z

(c) (d)

y

z

y

z

(e) (f)

Figure 2. Wave propagation in a channel with a curved lower wall; wall BC on the straight upper wall and onthe curved lower one, non-reflecting BC on all other walls; [98 × 98 × 386] cells.

channel (mid-plane y-z), at different time instances. The test shows the proper behavior of the boundaryconditions. The non-reflecting boundary conditions at the sides of the channel allow the waves to propagateundistorted through these boundaries.

III. Problem Formulation - Jet Engine Exhaust System

Flow and acoustical numerical investigations are performed for a model of a high by-pass turbofan engineexhaust. The geometry of the coaxial nozzle used in the simulations (Figure 3), is based on the dimensionsof the baseline separate flow nozzle test facility at the University of Cincinnati.6, 7

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Wcore

fanW

y

z

H1H2

H3

H4

H5

H6

Rc1R1iR

1oRc2R

2iR2o

Rc3

α2

α1

α3

(a) (b)

Figure 3. Separate flow exhaust nozzle design (a) and its generated geometry (b).

The core jet temperature (Tcore) is considered to be 400K, while the fan jet temperature (Tfan) is keptat 289K. The ratio between the core-stream axial velocity (Wcore) and the fan-stream axial velocity (Wfan)is taken to be 1.686 and the ratio between the core exit area (Acore) and the fan exit area (Afan) is 0.296,corresponding to the experimental case of Callender, Gutmark and Martens.7 The Reynolds number is 10000.The turbulent non-isothermal field is computed only in the near-field region. The near-field computationaldomain has a rectangular cross section of 7.5× 7.5 equivalent diameters (Deq) and a length of 15 equivalentdiameters in the flow direction (Z), the equivalent diameter being:

Deq = 2 ·

√

Acore + Afan

π(14)

The coarsest grid has 5×5×10 cells. The grid is then refined by halving the mesh spacing by a factor two ineach direction in each of the refinement steps. Four global levels (on the entire domain) and three additionallocal refinements (in the inlet region) are applied to obtain high resolution and in the same time to keep thecomputational effort on reasonable limits. On this restricted grid (roughly 4 × 106 computational cells) theflow dynamics and the mixing of the hot gas with the cold surrounding air form the acoustic sources whichare found by solving the flow.

The acoustic near-field is solved on the same mesh as the flow field. The acoustic source terms used in thewave equation are computed from the flow field at each time step. Near-field acoustic results are presentedin the next section. The acoustic field is also of great interest at large distances from the source. Theinstantaneous acoustic source term, provided by the near-field instantaneous LES data, can be transferredfrom the near-field grid to the far-field acoustical one, much larger, (8Deq × 40Deq × 32.5Deq) where thewave equation is solved. In order to eliminate the need for small time steps that is imposed by the explicitscheme, the wave equation is integrated in time using a second order implicit scheme. Non-reflecting acousticboundary conditions are set on the all the sides of the computational domain.

IV. Results and Discussions

A comparison between the normalized mean axial velocity profiles as computed with LES and experimen-tal data provided by Callender, Gutmark and Martens8 is depicted in Figure 4. The data has been extractedalong two different radial lines at 2.5 and 5.3 equivalent diameters downstream from the fan nozzle exitplane, respectively. As can be seen, the mean axial velocity component agrees well with the experimentalmeasurements. The discrepancy at the axis is not yet understood, but it might be due to the inaccuracyin modelling the geometry used in the experiments. This issue is to be further explored. The ”shoulder”seen at a W/Wexit value of roughly 0.57− 0.59, and a radial position of approximately 0.2 − 0.3 equivalentdiameters from the jet axis (Figure 4 (a)), indicates that the fan stream has not yet merged with the corestream. Similar behavior has also been observed in the experiments. At the downstream location of 5.3equivalent diameters (Figure 4 (b)), the fan and the core streams have completely merged into a single jet.

Figure 5 shows the frequency spectrum of the turbulent kinetic energy, the monitoring point beingsituated on the jet axis at 8 equivalent diameters downstream from the fan nozzle exit plane. The spectrum

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5R / Deq

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

W /

Wex

it

LESExperiments

Normalized mean axial velocity profiles; Z =2.5 Deq

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5R / Deq

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

W /

Wex

it

LESExperiments

Normalized mean axial velocity profiles; Z =5.3 Deq

(a) (b)

Figure 4. Normalized mean axial velocity profiles along radial lines, jet mid-plane, at 2.5Deq and 5.3Deq

downstream from the fan nozzle exit plane; comparison with experimental data.

1 10St

0.0001

0.001

0.01

0.1

1

Am

plitu

de

Z = 8.00 Deq-5/3

Figure 5. The turbulent kinetic energy spectrum; jet axis; Z = 8Deq .

follows well the −5/3 line, the spatial resolution of the simulations being fine enough to resolve a large partof the inertial sub-range.

Figure 6 shows the instantaneous distributions of the velocity variations related acoustic source term(a) and semi-compressibility effects related acoustic source term (b), respectively. The acoustic sourcesare presented in the jet mid-plane section. Dominant sound sources are found in the shear layers, in theinteraction region of the two jets, in accordance with previous observations.10, 11 Figure 7 presents thefluctuating components (RMS) of both acoustic sources, along radial lines situated at different downstreamlocations from the fan nozzle exit plane. Before the fan and the core streams have completely merged into asingle jet, higher RMS values are found in the shear layers for the velocity variations related acoustic source(Figure 7 (a)). The RMS values of semi-compressibility related acoustic source term are also higher close tothe nozzle (Figure 7 (b)), but only the shear layer due to the heated core stream can be distinguished, sincelarger gradients in density are situated here.

The instantaneous acoustic density fluctuation is depicted in Figure 8, in a jet mid-plane section. Thepreferred direction of radiation, near 30 degrees angle from the downstream jet axis, is visible.

Figure 9 shows the frequency spectrum of the total acoustic source term (a) and the frequency spectrumof the acoustic density fluctuation (b). The monitoring point is situated in the core jet shear layer at 2.15equivalent diameters in the flow direction. Similarities can be observed in the frequency peaks between bothspectra. This is due to the fact that the acoustic density fluctuation is related with the acoustic sourceswhich are dependent of velocity variations and semi-compressible effects.

A comparison between the near field sound pressure levels computed with the previously described methodand the experimental data provided by Callender, Gutmark and Martens7 is presented in Figure 10. Thedata is extracted along axial lines, situated at different radial positions in the field, as shown in Figure 10(a).

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(a) (b)

Figure 6. Instantaneous acoustic source terms: velocity variations related acoustic source (a); semi-compressibility effects related acoustic source (b); (mid-plane sections on the second local refinement).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5R / Deq

0

100

200

300

400

500

600

700

800

900

RM

S of

vel

ocity

var

iatio

ns a

cous

tic s

ourc

e

Z = 2.15 DeqZ = 2.50 DeqZ = 5.30 DeqZ = 8.00 Deq

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5R / Deq

0

25

50

75

100

125

RM

S of

ent

ropy

aco

ustic

sou

rce

Z = 2.15 DeqZ = 2.50 DeqZ = 5.30 DeqZ = 8.00 Deq

(a) (b)

Figure 7. RMS profiles of the acoustic sources along radial lines, jet mid-plane, different downstream locations:velocity variations related acoustic source (a); semi-compressibility effects related acoustic source(b).

The differences between the computed and the experimental data close to the nozzle are probably due to theboundary conditions applied at the inlet, where in the computations a constant (top-hat shaped) velocityprofile is used, without turbulent fluctuations. Additionally, one may also expect some effects from theinaccuracy of describing the exhaust geometry itself (as stated above).

Far-field acoustic computations are performed on a much larger domain (8Deq × 40Deq × 32.5Deq). Thisinhomogeneous wave equation, with the acoustic terms provided by the near-field LES solution, is solvedon a coarser grid (as compared to the near-field region). Figure 11(b) shows a comparison between thepresent results and the experiments in terms of sound pressure level directivity. The monitoring points aredistributed in a radial fashion as can be seen in Figure 11(a). The computed sound pressure level directivityshows very similar behavior as the experimental data.

V. Conclusions

A combined method for computing sound production and propagation have been described. The methodis based on LES to solve the flow field and an acoustic correction (i.e. the non-homogeneous wave equationwith appropriate boundary conditions) to handle the acoustics. The results using this approach have beencompared with experimental results of Gutmark et al, who studied the acoustic field generated by a coaxialturbofan jet engine exhaust model. Good agreement has been found between the computed results and theexperimental data, demonstrating the inherent advantages of combined semi-compressible LES approach,enhanced with a simple linear acoustic solver.

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Figure 8. Instantaneous acoustic density fluctuation field in mid-plane section.

0.1 1 10 100St

0

25

50

75

100

125

150

Am

plitu

de

Monitoring point location: core jet shear layer; Z = 2.15 Deq.

0.1 1 10 100St

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

Am

plitu

de

Monitoring point location: core jet shear layer; Z = 2.15 Deq.

(a) (b)

Figure 9. Frequency spectra of the total acoustic source term (a) and acoustic density fluctuation (b), in thecore jet shear layer, at 2.15Deq downstream from the fan nozzle exit plane.

Acknowledgments

This work has been financially supported by The Swedish National Energy Administration through the”Center of Competence of Combustion Processes”. Computational resources at the Center for Scientific andTechnical Computing at Lund University (LUNARC) are greatly acknowledged. Specifically, we wish tothank Bryan Callender for helping with the experimental data.

References

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Turbulence Closure, Journal of Sound and Vibration 194(2), pp. 219-242, 1996.2Boersma, B. J., Lele, S. K., Large Eddy Simulation of Mach 0.9 Compressible Jet, AIAA paper, 99-1874, 1999.3Bogey, C., Bailly, C., Downstream Subsonic Jet Noise: Link with Vortical Structures Intruding into Jet Core, C.R.

Mecanique, 330, pp. 527-533, 2002.4Bogey, C., Bailly, C., Juve, D., Noise Investigation of a High Subsonic, Moderate Reynolds Number Jet Using a Com-

pressible Large Eddy Simulation, Theoret. Comput. Fluid Dynamics, 16, pp. 273-297, 2003.5Brentner, K. S., Farassat, F., Modeling Aerodynamically Generated Sound of Helicopter Rotors, Progress in Aerospace

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paper, 2002-0369, 2002.7Callender, B., Gutmark, E., Martens, S., A Near-Field Investigation of Chevron Nozzle Mechanisms, AIAA paper,

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Jet, Center for Turbulence Research, Proc. of the Summer Program, pp. 159-167, 1998.10Fisher, M. J., Preston, G. A., Bryce, W. D., A Modelling of the Noise from Simple Coaxial Jets, Part I: with Unheated

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American Institute of Aeronautics and Astronautics

0 1 2 3 4 5 6 7 8 9 10Z / Deq

Soun

d Pr

essu

re L

evel

Exp; R = 0.000DeqExp; R = 0.378DeqExp; R = 0.757DeqExp; R = 1.136DeqExp; R = 1.514DeqPresent results; R=0.000DeqPresent results; R=0.378DeqPresent results; R=0.757DeqPresent results; R=1.136DeqPresent results; R=1.514Deq

Sound Pressure Levels along axial lines; different radial locationsNote*: the distance between two horizontal dotted lines is 2 dB

(a) (b)

Figure 10. Near-field sound pressure level distribution along axial lines at different radial locations, comparisonwith experimental data (b); the location of the lines (a).

4Deq

35 Deqr =

[8 x 40 x 32.5] Deq

90 100110

120130

140

150

z

y

x

.

80 90 100 110 120 130 140 150 160Directivity angle

SPL

ExpSimulation

Note*: the distance between two horizontal lines is 1 dB

(a) (b)

Figure 11. Sound pressure level directivity comparison; the radial distribution of the monitoring points (a)and the comparison with experimental data (b).

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15Gullbrand, J., Chow, F. K., The Effect of Numerical Errors and Turbulence Models in Large-Eddy Simulations of a

Channel Flow, with and without Explicit Filtering, Journal of Fluid Mechanics, Vol. 495, pp. 323-341, 2003.16Hileman, J., Samimy, M., An Attempt to Identify Noise Generating Turbulent Structures in a High Speed Axisymmetric

Jet, AIAA paper, 2000-2020, 2000.17Lilley, G. M., The Radiated Noise from Isotropic Turbulence with Applications to the Theory of Jet Noise, Journal of

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