AIAA 2003–3282 - Chalmerslada/postscript_files/billson_leuler_AIAA...Jet Noise Prediction Using...

AIAA 2003–3282Jet Noise Prediction UsingStochastic Turbulence ModelingMattias Billson

�, Lars-Erik Eriksson

��and Lars Davidson

�

�Department of Thermo and Fluid Dynamics, Chalmers

University of Technology, SE-412 96 Goteborg, Sweden�

Volvo Aero Corporation, Military Engines Division,SE-461 81 Trollhattan, Sweden

9th AIAA/CEAS Aeroacoustic Conferencemay 12–14, 2003/Hilton Head, South Carolina

For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics1801 Alexander Bell Drive, Suite 500, Reston, VA 20191–4344

Jet Noise Prediction Using StochasticTurbulence Modeling

Mattias Billson � , Lars-Erik Eriksson ��

and Lars Davidson �� Department of Thermo and Fluid Dynamics, ChalmersUniversity of Technology, SE-412 96 Goteborg, Sweden�

Volvo Aero Corporation, Military Engines Division,SE-461 81 Trollhattan, Sweden

This work is an attempt to capture the sound field of a high Reynolds number, high Machnumber subsonic jet without LES or DNS. The method used is called the SNGR (StochasticNoise Generation and Radiation) method and was first proposed by Bechara, Bailly, LafonCandel and Juve.1,2 The SNGR method is based on using the information given from a RANSsolution to generate a time dependent velocity source field which is used to evaluate sourceterms for the linearized Euler equations. The solution to the linearized Euler equationsprovide the acoustic information of the flow.

This method has in the present work been further developed with a new time depen-dence and convection of the generated velocity field.

The developed method has been applied to a Mach �� cold jet at �� and compared to recent LES computations and measurements of the same jet.

IntroductionAn aeroacoustic problem can be divided into ge-

neration of sound and propagation of the generatedsound. Solving the full compressible Navier-Stokesequations using LES or DNS captures both. Thecomputational cost however of the two methodsprohibits the use of them for most industrial ap-plications.

There has recently been some progress in a newmodeling approach called SNGR1,2 (Stochastic No-ise Generation and Radiation) model. It is basedon the idea that the linearized Euler equations area wave operator for acoustic perturbations. In-troducing suitable sources to the linearized Eulerequations result in accurate predictions of the ge-neration and propagation of acoustic perturbations.The objective is that this approach will cost lesscomputationally than performing a LES, especiallyat high Reynolds numbers.

In the SNGR method a RANS solution providestime averaged information about the flow field. Thechallenge is then to use the information given fromthe RANS solution to generate an instationary tur-bulent velocity field with the same local statisticalproperties as the RANS solution. This generatedturbulent field is used for evaluation source termsin the linearized Euler equations. Solving the li-nearized Euler equations with the source termsprovide the propagation of sound from the sourceregion to the far-field.

The present paper is organized as follows. Therole of the RANS solution is briefly discussed in

Copyright c�

2003 by M. Billson, L.-E. Eriksson and L. Davidson.Published by the American Institute of Aeronautics and Astronautics,Inc. with permission.

the following section followed by the presentationof the derived source terms for the linearized Eulerequations. The basic concept of how the synthesi-zed turbulence is generated is desribed and thenthe differences between the present method andthe ones proposed in Refs. (1,2) are discussed. Thedifferences concern how time dependence and con-vection is introduced in the generation of the synt-hesized turbulence.

Comparison of the synthesized turbulence withtheory for isentropic turbulence will first be presen-ted. Next a simulation of a 3D jet is presented be-ginning with the computational setup and followedby the results which are compared to both measu-rements, Jordan and Gervais3,4 and recent LargeEddy Simulations, Andersson et al.5 Differencesbetween the measurements and the present simu-lations are then investigated and this is followed bythe conclusions.

The SNGR methodThe SNGR model is performed in three steps.

These are:

step 1. A Reynolds-Averaged Navier-Stokes solu-tion of a compressible turbulent jet is calcu-lated using, for example, a �� turbulencemodel.

step 2. An instationary turbulent velocity fieldwith the same local turbulence kinetic energy,time scale and length scale as the RANS solu-tion is generated using random Fourier modes.

step 3. The linearized Euler equations are solvedusing the mean flow field computed in step (1)

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AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2003–3282

as mean flow solution. Source terms derived ina similar way as for Lighthill’s wave equationare evaluated using the turbulent field genera-ted in step (2). The linearized Euler equationsthen gives the propagation of sound from theturbulent field to the surrounding far-field.

RANS for Jet ComputationsThe Reynolds-Averaged Navier-Stokes solution

(RANS) of the flow field is computed using a stan-dard � � � turbulence model. The purpose of theRANS solution is to provide a mean flow solutionfor the linearized Euler computation. The RANSsolution is also used in the stochastic modeling ofthe turbulent field. The SNGR model needs in-put parameters in the form of turbulence kineticenergy, length scale and time scale. The turbulencekinetic energy is one of the solution variables fromthe RANS computation and the turbulence lengthscale and time scale are computed from the turbu-lence kinetic energy and the turbulence dissipationrate. For more details, see Billson.6

Linearized Euler Equations withSource Terms

A formal derivation of the source terms for thelinearized Euler equations is given in Billson.7 Gi-ven here is only the final set of equations.

��

��

��

�� "! � �#� �

��$� ��

��%�&��

�� ' & �� (�� (�� ' & � � � � � � ' & � (��

�� ' � �& � �� ' � �& � ��

(1)

The left-hand side of equations 1 is the linearizedEuler equations. The right-hand side contains allnon-linearities that emerged when the Euler equa-tions were rewritten into the linearized Euler equa-tions. Thus, equations 1 are still the full non-linearEuler equations. If the right-hand side of the equa-tions is treated as a source and is in some wayknown, then the left-hand side is a wave operatorresponding to the source.

A validation of the derived source terms has beenperformed in the case of a forced 2D mixing layerin Billson6 where the solution of the linearized Eu-ler equations with the derived source terms wascompared to a direct simulation of the same flow.The source terms were shown to be working welland given that they are evaluated from a physi-cal solution, the response from the linearized Eu-ler equations is in good agreement with the direct

numerical simulation used to evaluate the sourceterms.

Stochastic Modeling of TurbulenceThis section concerns the generation of an insta-

tionary turbulent velocity field, i.e. Step (2) in theSNGR method. A time-space turbulent velocity fi-eld can be simulated using random Fourier modes.This was proposed by Kraichnan8 and Karweit etal.9 and further developed by Bechara et al.1 andBailly and Juve.2 The velocity field is then given by

)+* �,-� �/.012436587 2:9<;�= �> 2 , �@? 2 ��A 2 (2)

where 7 2 , ? 2 and

A 2 are amplitude, phase anddirection of the B *�C Fourier mode. Figure 1 showsthe geometry of the B *�C mode in wave numberspace.

0

PSfrag replacements

D$EF

D�EF

D�EG

D$EH D$EH

D FD G

D H

D�I

J I

K I

L IM I

Fig. 1 Geometry for the N�OQP mode.

The vector> 2 is chosen randomly on a sphere

with radius � 2 . This to ensure isotropy of the ge-nerated velocity field. By the assumption of in-compressibility the continuity equation gives thefollowing relation

> 2SR A 2UT � V ;XWZYX[\[ B (3)

The wave number vector> 2 and the spatial direc-

tionA 2 of the B *�C mode are thus perpendicular. The

angle ] 2 is a free parameter chosen randomly, seefigure 1. The phase of each mode ? 2 is chosen withuniform probability between �_^ ? 2 ^`.ba . The pro-bability functions of all the random functions c 2 ,? 2 , d 2 and ] 2 are given in table 1.

� c 2 � �febg .ba � �_^ c 2 ^`.ba� ? 2 � �febg .ba � �_^ ? 2 ^`.ba� d 2 � � e#gX. � =ih\j d � �_^ d 2 ^ka� ] 2 � �febg .4a � �_^ ] 2 ^`.ba

Table 1 Probability distributions of random vari-ables.

The probability function of d , � d 2 � �le#g4. =�h"j d �is chosen such that the distribution of the direction

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PSfrag replacements

��

��

��

��

��

Fig. 2 The probability of a randomly selecteddirection of a wave in wave-space is the same forall �� on the shell of a sphere.

PSfrag replacements

D�I

� D I

D�� D��

�� ! "##

$ � D I �

Fig. 3 von Karman-Pao spectrum

of> 2 is uniform on the surface of a sphere, see fi-

gure 2, i.e. the probability of a randomly selecteddirection is the same for all surface elements %'& .

The amplitude 7 2 of each mode is computed

so that the turbulence energy spectrum function( � 2 � correspond to the energy spectrum for iso-tropic turbulence. This gives

7 2 �*) ( � 2 �,+ � 2 (4)

where+� 2 is a small interval in the spectrum loca-

ted at � 2 , see figure 3. A model spectrum is usedto simulate the shape of an energy spectrum forisotropic turbulence. In this way the sum of thesquares of 7

2 over all B is equal to the total turbu-lence kinetic energy

� �0124365 7 .-2 (5)

The spectrum( � 2 � is subdivided with a linear

distribution as

� 2 � � 5 �0/ � 2�1 52 V ;�W B � e43 .53768686.3:9 (6)

where

/ � 2 � � 0 � � 5 �9 � e (7)

It has been argued in previous studies2 that a lo-garithmic distribution compared to a linear distri-bution results in a better resolution of the spectrumfor low wave numbers corresponding to the mostenergy containing eddies. This is true but it hasthe side effect that the amplitudes of the highestwave numbers which then are poorly resolved areamplified by the increasing

+� 2 for higher wave

numbers in equation 4. A comparison of the energydistribution and the amplitude distribution for li-near and logarithmic wave number distributions ispresented in the result section. This comparisonshows this effect and a linear distribution of thewave numbers is therefore used in this work.

The energy spectrum for isotropic turbulence issimulated by a von Karman-Pao spectrum

( � � � ] �� ;��<� g � < �>=? e � � g ��< � -A@ 5CBED:F %�G 1 -8HJI D IEK7L ;>M (8)

where � is the wave number, ��N � � 5OD =QP 1.R:D = is theKolmogorov wave number,

Pis the molecular visco-

sity and � is the dissipation rate. � ;

is the r.m.s.value of the velocity fluctuations corresponding tothe turbulent kinetic energy,

� ; � . � g�S . There aretwo free parameters in equation 8. The numericalconstant ] which determines the kinetic energy ofthe spectrum and the wave number �5< correspon-ding to the most energy containing eddies at thepeak in the spectrum. The available informationfrom the RANS solution is the turbulence kineticenergy � or, equivalently S � ; g4. , and the dissipationrate � . These must be used in order to determine] and � < and thereby the shape of the spectrumand the distribution of energy over different wavenumbers. The numerical constant ] can be deter-mined by the requirement that the integral of theenergy spectrum, equation 8, over all wave num-bers should be equal to the total turbulent kineticenergy

� �UTWV& ( � � / � (9)

Since equation 8 is derived for infinite Reynoldsnumber ] can be found independently of ��< by in-tegrating equation 9 to get

] � XY a[Z e�\Xg^] �Z e#g�S �`_ e�6 X�a .�\^] (10)

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The turbulence length scale from the RANS solu-tion is defined as

� ��&�� B� �

R:D -� (11)

Assuming that the length scale from the RANS so-lution is the same as the integral length scale forisotropic turbulence gives the following relation

� � a. � ; T V& ( � �

�/ � (12)

which is used to determine the wave number ��< cor-responding to the most energetic length scales. Therelation of � < to

�and ] is

��< � � aa4a ]� (13)

where ] is given in equation 10 and�

is obtainedfrom the RANS solution.

Time dependence and Convection – previousstudies

In earlier studies, two methods have been deve-loped to introduce time dependence in the synthe-sized velocity field.1,2 The first method by Becharaet al1 is based on independent generation of velo-city fields ) * �,-� (see equation 2) in every time stepin the same manner as described above. The inde-pendent solutions are then filtered in time in everypoint to give a desired time correlation. The secondmethod Bailly and Juve2 is based on introductionof time dependence in the actual generation by ad-ding a time dependent term in the Fourier modes.The generated time dependent turbulent velocity)-* �, 3 �� is then given by

) * �, 3 �� /. 01243+5 7 2 9 ;�= �> 2 �, � � )� � � ? 2 �� 2 ��A 2(14)

In this expression )� is the local convection velo-city computed in the RANS solution, and � 2 is theangular frequency of the B *�C generated mode. Theangular frequency � 2 is a random function given bya Gaussian probability function

� � 2 � � e� 2 Y .ba % H 1 H�� 1 ��^L ; D - � ;� L (15)

where � 2 is the mean angular frequency of the B *�Cmode determined by � 2 � �� 2 , Bailly.2

The second method to introduce time dependenceis attractive from computational point of view dueto the difference in required computational effort

and storage. In the first method a number of Fou-rier modes have to be generated in each time step.The resulting velocity fields need to be filtered aposteriori to get suitable statistical properties. Thisdata need to be stored before it is used in the sourceterms in the linearized Euler computation, step (3).In the second method the velocity field can be ge-nerated for each time step independently and doesnot need to be filtered. It is then possible to in-clude the generation of the turbulent velocity fieldin the solver. The only data that need to be sto-red in order to repeat a computation in this caseare the random functions given in table 1. The firstmethod is however attractive in the way that thetime correlation can be chosen to follow a specifiedbehavior. This is done through the filtering of theindependent samples. This type of control is notpossible in the second method. Also, choosing therelation between the angular frequency � 2 and thewave number � 2 is somewhat arbitrary but at thesame time an important parameter for the soundgeneration. The first method does not include anyconvection whereas the second does through theterm

> 2 �, � � ) � in the cosine argument.

Time dependence and Convection – present workA way to retain the control of the time correlation

and at the same time keep the required computa-tional effort at a reasonable level is to introducethe time filter directly as the velocity field is gene-rated. To include convectional effects a convectionequation is solved for the filtered velocity field. Theproposed method to introduce time dependence andconvection is as follows. First define a realizationof the generated turbulent velocity field as )��* �,-�where superscript denotes time step � . Each gene-rated field )��* �,-� for all e�� 9 is independentof the others and they have a zero statistical meanin time. In other words the generated velocity fieldis locally white noise. A new turbulent velocity fieldcan then be computed via the equation

� �* �,-� �� 1 5* �,-� � � ) �* �,-� (16)

where � �"!$#�% � + � g�& � and � � ) e � � - � . & isreferred to as the time scale and defines the timeseparation for which the autocorrelation functionis reduced to !'#�% � e � 6 The expression for � ensu-res that the root mean square of �(�* �,-� is the sameas for )(�* �,-� . The time scale is computed from theRANS solution as

&U�)��* �� (17)

where the factor � * is introduced for the possibilityto modify the time scale.

To account for convection of the generated tur-bulent field �+�* �,-� a simple convection equation issolved for � � 1 5* �,-�

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�� 1 5* ��

�� (� � � 1 5* �� `� (18)

before it is used in equation 16. Absorbingboundary conditions based on characteristic vari-ables are used when solving equation 18.

Using equation 16 to introduce time dependenceas compared to equation 14 has the advantage ofrequiring a smaller number of modes. In equa-tion 14 the velocity field is totally determined byeach realization. A large number of modes isthen required to accurately describe isotropic tur-bulence. In equation 16 the velocity field ��* �,-� ata certain time is the weighted sum of all previousindependent velocity fields )��* �,-� each with diffe-rent random parameters. Much fewer modes arethen required to ensure accurate statistics. Thisproperty is shared with the first method describedabove.

The second method presented above has a di-sadvantage of loosing the space correlation cross amean shear flow through the convective argument> 2 �, � � )� � in equation 14. This has been reportedby Batten.10 This de-correlation does not occur inthe present method, as will be shown in the results.

Correlations of GeneratedTurbulence – Homogeneous Case

To see if the generated turbulent velocity fieldhas the specified time and length scales, the au-tocorrelation and two-point correlations are compu-ted from a generated homogeneous velocity field.

Using a first-order filter as in equation 16, thespecified autocorrelation is an exponentially decay-ing function which in a discrete time separation is

� �* � � 1��* � �* � - �� (19)

where � is the number of time steps separating ��*and � � 1��* and the overline denotes an average overmany realizations � . The specified and the compu-ted autocorrelations are shown in figure 4.

The longitudinal two-point correlation � 6�� 3 ��3i� � g � - ��3 ��3i� � is compared to the � -functionfor isotropic turbulence11 and the transversal�� 3 ��3 � � g �� '3i�'3i� � is compared to the � -function.The results are shown in figure 5. The correlationsin time and space clearly follow the specifiedcorrelations.

3D Jet SimulationThe simulated jet is a Mach � � ��6 \ a , � !f�� g P � \ a ��3i��X� cold jet with a nozzle diameter

of � � ��6 � a ? � @ . Jet flow conditions are shown in

table 2.

0 5 10 15 20

0

0.2

0.4

0.6

0.8

1

PSfrag replacements

� + � g�&

��

Fig. 4 Autocorrelation ��O � ��O� � ��O

"!for genera-

ted turbulence as a function of normalized timeseparation #�$&% �(' . First-order filter in time. Solidline: analytical expression, equation 19; others: )+* ,) ! , )-, velocity autocorrelation.

0 0.5 1 1.5 2 2.5 3

0

0.2

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0.8

1

PSfrag replacements

�a) ./.�021�� (3 � ./.40 �� (3

0 0.5 1 1.5 2 2.5 3

0

0.2

0.4

0.6

0.8

1

PSfrag replacements

�b) 5(5�021�� (3 � 5(560 �� (3

Fig. 5 Spatial two-point correlations. (a) longi-tudinal two-point correlation, (b) transversal two-point correlation. Solid lines: analytical expres-sions; dashed lines: from generated turbulent ve-locity fields.

Jet exit conditionsDiameter,

�: ��6 � a m

Mach number, M : �'6�\ aPressure, � � : e �(e46 S kPaTemperature, 7 � : ./8�8 K

Ambient conditions��9 �;: : e �(e46 S kPa7 9 �;: : ./8�8 K

Table 2 Flow conditions

Source RegionThe region in which the SNGR is applied is

restricted to the region where the largest sourcesare expected. This region is a cylinder that startsfrom the nozzle and continues in the axial directiondown to

� g � � . X . Within this cylinder the sourceregion is the set of cells where

P * g P=< �'6 ��]�e . Thislimit is chosen for numerical reasons to ensure thatthe Kolmogorov wave number is larger than thepeak wave number � < in the model von Karman-Pao energy spectrum. This limit is quite arbitrarybut convenient since the turbulence kinetic energyturns out to be very small where this value of

P * g Pis reached.

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Numerical SolverThe code for the linearized Euler equations is ba-

sed on the finite volume method. The equationsare discretized on a structured curv-linear non-orthogonal boundary-fitted multi-block mesh. Theconvective terms are discretized with a six pointstencil. The coefficients of Tam’s12 fourth-order dis-persion relation preserving finite difference schemeis converted to the equivalent finite volume coeffi-cients. A fourth-order four step Runge-Kutta timemarching technique is used for the time stepping.Artificial selective damping is used to prevent spu-rious waves from the boundaries and regions withstretching to contaminate the solution. The man-ner in which the artificial selective damping is in-troduced is described in Eriksson.13

The boundary conditions are based on local one di-mensional analysis based on characteristic varia-bles, Billson.6

A buffer layer is applied in the outflow region toattenuate vorticity and entropy disturbances beforethey reach the outflow boundary. The attenuationis done by adding a damping term to the governingequations in the outflow region of the computatio-nal domain, see Refs. (6,14).

Computational setupThe block structure of the mesh is shown in fi-

gure 6. The spatial resolution is homogeneous+ � g � � �'6\e in the axial direction in the region� \k^ � g � ^ . � . The resolution in the �� -planeis chosen so as to resolve the core of the jet andinitial mixing layer with

+ � g �� 6 �4S with incre-ased mesh size further away from the nozzle exit.The mesh support acoustic waves up to a frequencyequivalent to Strouhal number � � � � � g � � e46 awith little dispersion and dissipation.

The buffer region is the part of the computationaldomain from

� g � � . � to the down-stream out-flow at

� g � � S�\ . Stretching of the mesh in thedown-stream direction in the buffer layer has theadditional effect that the artificial numerical dis-sipation at high wave numbers makes the bufferlayer more efficient.

The time step is+ � � 8'6 ] a � e�� 1�B = equivalent to

a maximum CFL-number based on spectral radiusof �� '6 a .For the base-line simulation (see results section)the length and time scale factors used are � � � e46 �and ��* � e�6 � . In the random field 9 � SX� modesare used and the lowest and highest wave numbersare � 5 � a and � 0 �l.X� � corresponding to � 5 �� B � < � g a and � 0 � .ba+g ] + � � respectively.

A linear distribution of the energy spectrum(equation 8) is used in the present work as oppo-sed to logarithmic in previous work.1,2,8,9 Whenusing a logarithmic distribution the amplitudes ofmodes at low wave numbers are suppressed andthe amplitudes of modes at high wave numbers are

a) � -plane through centerline.

b) � -plane atnozzle exit.Dashed lineindicates nozzleexit diameter.

c) � -plane at end ofcomputational domain.

Fig. 6 Slices of the block structure of the lineari-zed Euler mesh.

emphasized due to the increasing+� 2 in equation

4 for higher wave numbers. Using a linear distribu-tion gives a homogeneous distribution of the

+� 2 .

Figure 7 shows the energy distribution of thesynthesized velocity fields for each mode for diffe-rent down-stream positions in the center line (top)and in the jet shear layer (bottom). Figure 8 showsthe amplitude of the same modes when computedwith equation 4. A linear distribution is used to theleft and a logarithmic to the right. The shift of thepeak towards higher wave numbers between theenergy and the amplitude in the logarithmic caseis clear from figures 7 and 8. This shift in the peakis not present in the case with linear wave numberdistribution.

To validate the present method the far-field ac-oustic solution has been computed in two diffe-rent ways, the Kirchhoff integral method15 and thedouble time derivative formulation of the Lighthillacoustic analogy.16 The Lighthill analogy solutionis computed directly from the synthesized velocityfield, see figure 9 whereas the Kirchhoff solution isbased on the solution to the linearized Euler equa-tions and thus including all steps in the presentmethod.

Overall sound pressure levels (OASPL) andspectra of the acoustic data are compared with theLES of Andersson et al5 and the measurements ofthe same flow by Jordan and Gervais.4 The predic-ted acoustic data from the Kirchhoff integral met-hod have been high-pass filtered to attenuate theeffect of natural hydrodynamic instabilities in thelinearized Euler equations. The Kirchhoff surfaceis placed five cells from the outer boundary of the

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PSfrag replacementssynthesized

velocities, �+�*source termevaluation

solution toLEE

Kirchhoff methodLighthill method

Near-field solution

Far-field solution

Fig. 9 Near-field to far-field solutions using Lighthill and Kirchhoff methods.

0 5 10 15 200

50

100

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200

0 5 10 15 200

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� g �

wav

enu

mbe

r

a) linear

0 5 10 15 200

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100

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200

0 5 10 15 200

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� g �b) logarithmic

Fig. 7 Energy distribution

0 5 10 15 200

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� g �

wav

enu

mbe

r

a) linear

0 5 10 15 200

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� g �b) logarithmic

Fig. 8 Amplitude distribution

computational domain except in the outflow regionwhere the surface is not closed over the turbulentregion (see Ref. 17) and the Lighthill source regionis chosen as the same region in space as the SNGRsource region.

Choosing SNGR parametersThe SNGR parameters have been chosen in the

following way. The near-field is calibrated using thelength and time scale factors � � and ��* , see equa-tions 11 and 17. The basis is that the near-fieldshould be well calibrated before one can expect thesolutions in the far-field to be in good agreementwith measurements or the LES solutions. Thus thefar-field pressure spectra and amplitude is deter-mined by �� and � * .

ResultsThe RANS solution which governs the statistic

property of the proposed SNGR methodology has

been validated against the LES data of Anderssonet al.5 and measurements, Jordan et al.18 This ispresented in Billson.6

Near-Field ResultsFigure 10 shows the computed two-point space-

time correlation function,

� 5�� 5 �, 3�� 3 & � � � �, 3 �� , � � 3 � � & �� - �, 3 �� - �, � � 3 � � & � (20)

where,

and � are the position and the space sepa-ration in

� 5 direction respectively, overline 6 � deno-

tes time average and & is time separation. In figure10(a) the correlation is based on the LES data ofAndersson et al.5 and in figure 10(b) the correlationis based on the synthesized turbulent velocity field.Each curve represents the correlation in time of twopoints separated by a constant distance in space(axial direction). The envelope of the maxima ofthe correlation curves corresponds approximately19

to the Lagrangian autocorrelation. The correlationfor each space separation at zero time separationcorresponds to the two-point space correlation andconsequently the integral length scale.

Figure 11(a) shows the Lagrangian autocorrela-tion based on the space-time correlation for theLES data and the synthesized SNGR velocity fieldand figure 11(b) shows the two-point space correla-tion for the same sets of data. The correlations arein good agreement.

Since the spatial grid resolution is limited by theupper wave number constraint � 0 � ] + � , the ki-netic energy of the synthesized velocity field willbe lower than that of the RANS solution. The tur-bulence kinetic energy of the RANS solution, theresolved part (i.e. � 5 ^ � ^ � 0 ) of the energyspectrum (equation 3) and the synthesized turbu-lent velocity field in the shear layer are shown infigure 12. There is a large difference in the energyfrom the RANS solution compared with the resol-ved energy spectrum. This difference is due to thepoor resolution in the

�-direction which determines

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0 1 2 3 4−0.2

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& � g �

��

a) LES by Andersson

0 1 2 3 4−0.2

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& � g �

� ��

b) Synthesized velocity field

Fig. 10 Two-point space-time correlation (equa-tion 20 in the axial direction at

� ) � ��

the upper wave number, � 0 , in the syntetisation.The energy which is lost is at small scales, seefigure 13, and as such they are very compact assound sources. The energy of the synthesized tur-bulence is however close to that of the resolved partof the energy spectrum which would be expected. Asimulation where the convection operator has beenturned off (see below) is also included in figure 12.Comparing the two curves for the simulations onecan see that there is a small decrease in energywhen the convection operator is used which is at-tributed to the fact that the time filter, equation 16,only has a limited time to feed the synthesized tur-bulence with new energy as the convection operatoris transporting the velocity field down-stream.

The space and time correlations in figures 10 to11 show good agreement between the LES and thesynthesized velocity field. The turbulence kineticenergy levels in the simulations are lower than inthe RANS solution, and this should have the effectof producing too low far-field sound levels. Evenso, the proposed method to convect (equation 18)and filter (equation 16) the random velocity field(equation 2) can model the space-time statistics ofan inhomogeneous turbulent velocity field.

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& � g �a) Lagrangian autocorrelation

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Space separation � g �b) two-point space-space correlation

Fig. 11 Correlations at� ) � � �� . So-

lid (black) line: synthesized velocity field, dash-dotted (blue) line: LES by Andersson

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� g �

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Fig. 12 Turbulence kinetic energy on a line in theaxial direction at ��

. Dash-dotted (blue)line: RANS; dotted (red) line: resolved spectrum(equation 8); solid (black) line: synthesized veloci-ties (base-line case); dashed (magenta) line: synt-hesized velocities (no-convection case)

Far-Field ResultsDue to growing hydrodynamic instabilities in the

linearized Euler solution the far-field acoustic solu-tion is high-pass filtered. This is done by applyinga two-point Butterworth low-pass filter and sub-tracting the resulting signal. In this way, all lowfrequency information, below � � � �'6\e is dampedby the filter. Figure 14 shows time series of the un-filtered pressure disturbance, the low-pass filtered

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� g �

��

Fig. 13 Solid (black) line: highest synthesizedwave

�� ; dashed (blue) line: peak wave num-

ber��

in the shear layer, � � � ��

and the resulting high-pass filtered signal.

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� � g �

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Fig. 14 Time series of pressure fluctuation at� �� . Top: unfiltered; middle: low-pass filtered; bottom: high-pass filtered

In figure 15 the OASPL is computed at constantradius from the nozzle exit as a function of anglefrom the axial direction. Except for the offset ofe � %�� the directivity is the same for the measure-ments and the present method.

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da) 1 ��

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db) 1 ��

Fig. 15 OASPL ( �� ) at �� and � � � ��

for different angles � from jet axis direction. Solidline (diamonds): present method (base-line case);dashed line (circles): measurements

The e#g�S -octave band-pass filtered power spect-rum of pressure from the observation point at d �SX� degrees in figure 15(a) is shown in figure 16.One can immediately see that there is a large dif-

ference in the frequency-content between the me-asurements and the simulation. The peak in themeasured spectra is located at � � �f�'6 . as opposedto � � � e�6 � � e�6 a in the simulation. Also as shownin figure 15 and evident in figure 16, the amplitudeof the far-field signal is too high in the simulation.The OASPL is about e � %�� larger than the measu-red data corresponding to a far-field pressure amp-litude that is about S times over-estimated.

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��Po

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Fig. 16�(��

-octave power spectrum of pressure inan observation point at

� �� . Solid(black) line: present method (base-line case); dash-dotted (blue) line: measurements

Evaluation of Results

The near-field two-point statistics of the synthe-sized turbulence seems to be in good agreementwith both measurements3 and LES data.5 This isshown in figures 10 and 11. The energy of the synt-hesized velocity field is however too low. Yet, theemitted far-field sound is over-predicted and hasa different frequeny-content compared to the mea-surements. Possible sources for these inconsistentresults in the present method has been classified asrelated to:

I The numerical issues in the computations

II The use of linearized Euler equations withsource terms

III The synthesis of turbulence

The numerical issues account for numerical sche-mes, truncation of the source region and limitedspatial and spectral resolution. In the second cate-gory one can include the use of the derived sourceterms for the linearized Euler equations, instabi-lities related to solving the linearized Euler equa-tions in mean shear and the use of Kirchhoff ’s in-tegral method. The third category involve, for ex-ample the spectral model of isentropic turbulenceand the time filter (equation 16) and convectionoperator (equation 18).

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The numerical issues in the computationsThe numerical issues have been thoroughly

tested and are well validated. The synthesized tur-bulence is well resolved with a shortest wave lengthof � � � 2 � ] + � for which the numerical scheme inthe convection operator (equation 18) has little dis-persion error. The numerical method used for theconvection operator is the same as for the lineari-zed Euler equations but with a four point stencil.The truncated source region would cause unwanteddisturbances if the source term evaluation was per-formed all the way to the source region boundary.This is however avoided by evaluating the sourceterms only in the interior of the source region.

The use of linearized Euler equations with sourceterms

The far-field solution using the Lighthill analogyis computed using the synthesized velocity fields� �* �,-� , i.e. the modeled SNGR velocities, see figure9. The Kirchhoff far-field solutions are computedfrom the solution of the linearized Euler equationsusing all the steps in the present methodology.If the far-field solution is the same for both theLighthill analogy and the Kirchhoff integral met-hod then the conclusion must be that the secondcategory of possible reasons for inconsistent solu-tions above (the use of linearized Euler equationswith source terms) can be ruled out.

Figure 17 shows the power spectrum of the far-field pressure computed using the Lighthill analogyand the Kirchhoff method. The far-field spectraare almost identical except for very low frequencieswhere there is presence of disturbances in the Kir-chhoff method solution originating from hydrody-namic instabilities in the solution of the linearizedEuler equations. The spectral peaks are located atthe same place and the levels are the same in thetwo curves.

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� �

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Fig. 17�(��

-octave power spectrum of pressure inan observation point at

� �� . Solid(black) line: Kirchhoff method; dash-dotted (blue)line: Lighthill’s analogy

The conclusion is that the linearized Euler equa-tions with source terms presented here indeed pro-

duce the correct sound field associated with a spe-cified turbulent source field.

The synthesis of turbulenceThere must be some feature in the synthesis of

turbulence that gives the errors in the far-field. Toinvestigate the reason for this, some parametersin the SNGR model were changed in order to in-vestigate their impact on the near- and far-fieldsolutions. The following parameters in the presentSNGR method were investigated

� Length scale, � �� Time scale, � *� Convection set to zero� Divergence of source field� Neglection of energy in wavespace

Length scaleFigure 18 shows the two-point space correlation� 5�� 5 �, 3�� 3i� � in the shear layer for two different va-

lues of the length scale parameter and the corre-sponding far-field power spectra of pressure areshown in figure 19. As can be seen in figures 18and 19 the increasing length scale results in far-field spectra with increasing energy and also a shiftin the peak in the spectral energy. The compactnessof the source field is decreased with the increasedlength scale and through the source terms in equa-tion 1, this changes the peak in the far-field spectraand the energy of the emitted sound.

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Space separation � g �Fig. 18 Two-point space correlation located at� ) � � �� . Solid (black) line: ��

;dash-dotted (blue) line: ��

Time scaleChanging the factor for the time scale from � * �

e to � * � ] result in the autocorrelations in figure20. The far-field power spectrum of pressure forthe increased time scale case is included in figure19. The effect of increasing the time scale of thesource field clearly increases the amplitude of theemitted sound. However, contrary to what might

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Fig. 19 Power spectrum of far-field pressure (1/3octave band pass filtered. Solid (black): base-linecase; dashed (blue): increased length scale; dash-dotted (red): increased time scale; dotted (ma-genta): no-convection case; diamonds (green): ho-mogeneous source field

be expected, the location of the spectral peak is notdrastically modified by the increased time scale.The time scale in the present SNGR model seemsto primarily have an effect on the compactness ofthe source field but that the effect is the same forall scales and thus not changing the spectral peakin the far-field sound.

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& � g �Fig. 20 Lagrangian autocorrelation (time correla-tion in the convected reference field). Solid (black)line: ��

; dash-dotted (blue) line: ��

Convection set to zeroThere was some concern that the convection

operator (equation 18) would cause the same de-correlation that has been observed in the pre-viously proposed method described above (equation14), see Ref. (10). The time dependence and con-vectional operator is not the same in the presentwork however and the issue needs to be investi-gated. Figure 19 shows the far-field spectrum fora computation where the convection in the SNGRmodel has been turned off. The result is almostidentical to the base-line case suggesting that theconvection operator does not greatly change the na-ture of the emitted sound.

Divergence of source fieldOne of the assumptions made in the development

of the SNGR method is that the isotropic turbu-lence is indeed isotropic. The homogeneity assump-tion implied in this is used to ensure that the resul-ting velocity fields are solenoidal and thus having azero divergence. A velocity field that is not solenoi-dal will have a non-zero divergence which may be asource of sound when the velocity fields are used toevaluate the source terms. Some of the parametersin the SNGR model are evaluated locally (in space)from the RANS solution. These are the turbulencekinetic energy � , the time scale & and the lengthscale

�of the synthesized turbulence. This local

evaluation of SNGR parameters will introduce di-vergence and the role of this divergence as a sourceof sound must be investigated.

The divergence evaluated from the LES and thesynthesized velocity field in the base-line case isshown in figure 21. Two differences between thesynthesized velocities and the LES velocities areevident in figure 21. One is the difference in ampli-tude and the other is the shift in the peak regionof the divergence. In the LES the divergence ismost pronounced at

� ^ a , i.e. close to the endof the potential core whereas the divergence in thesynthesized velocity field continues to grow down-stream from this region.

The same property is shown in figures 22 and 23for the case of no convection and also a case witha homogeneous source term field. The state of thesource field in the homogeneous case was chosento that in RANS solution at the spatial location�� 3 �4� � a � 3i�'6 a � � , where the turbulence kineticenergy � is large. The divergence of the synthe-sized velocities in the no-convection case is clearlylower than in the base-line case. The convectionoperator in equation 18 does apparently introducedivergence in this inhomogeneous mean flow. Thedecrease in the divergence is even more evident inthe homogeneous case in figure 23. This is expectedsince the homogeneous case satisfies the assump-tions made for the flow to be solenoidal. The re-sulting far-field spectrum is plotted in figure 19 asthe diamonds (green). The divergence introducedby not satisfying the assumptions for the flow to besolenoidal is clearly not the source of the inconsi-stent results between the near-field (figures 10 and11) and the far-field (figures 15 and 16).

Neglection of energy in wavespaceThe last parameter tested was the effect of the

poor resolution in the axial direction on the energycontent of the synthesized velocity field. It is clearfrom figure 12 that a large portion of the energyis neglected in the synthesized velocity field, espe-cially in the beginnig of the shear layer wherethe length scales are smaller than further down-stream.

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Fig. 21 Divergence of velocity evaluated on a linein the axial direction at a radius � �� . So-lid (black): synthesized velocities (base-line case);dash-dotted (blue): LES by Andersson

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Fig. 22 Divergence of velocity evaluated on a linein the axial direction at a radius � �� . So-lid (black): synthesized velocities (no-convectioncase); dash-dotted (blue): LES by Andersson

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Fig. 23 Divergence of velocity evaluated on a linein the axial direction at a radius � �� . So-lid (black): synthesized velocities (homogeneouscase); dash-dotted (blue): LES by Andersson

The effect of truncating the spectrum at the highwave numbers is that the resulting two-point corre-lation and the length scale become too large. Thiscan be seen in figure 24 showing the � -function(longitudinal two-point correlation coefficient) forisotropic turbulence (see Hinze11) based on the

wavespace-truncated energy spectrum, i.e. the re-solved part of the von Karman-Pao spectrum andbased on the same spectra but for all wave num-bers.

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� g �

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Fig. 24 � -function evaluated from the energyspectrum (equation 8) located at

� ) �� . Solid line: for all wave numbers; dashedline: for resolved wave numbers.

In the base-line case above the parameter � � wascalibrated such that the two-point correlation in fi-gure 11(b) would match the LES data. In doing so,the value of the length scale parameter � � � e46 �was determined using a velocity field computedfrom a truncated spectrum. A length scale parame-ter of � � � e46 � is consequently calibrated to a toosmall value to compensate for the over-predictedtwo-point correlation resulting from the truncationof the energy at higher wave numbers.

The conclusion is that one should expect the two-point correlations from the synthesized velocity fi-eld to be larger than those of the LES data when alarge portion of the energy at the high wave num-bers has been omitted.

The effect of an increased length scale factor canbe seen in figure 18 and figure 19 where the lengthscale factor was increased to � �f� ] as comparedto the base-line case. Increasing the length scalemakes the synthesized turbulence less compact bymoving the energy to the lower wave numbers. Thisalso shifts the peak in the power spectrum of thefar-field sound towards lower frequencies, see fi-gure 19. The amplitude of the far-field sound isat the same time increased by the decreased com-pactness of the synthesized velocity field. Also theenergy of the synthesized velocity field is increaseddue to a better resolution of the synthesized veloci-ties.

By using �� ] as in the length scale test aboveand at the same time lowering the amplitude in thetime filter , i.e. � � ) � e � � - � in equation 16 using� � ��6"e , the far-field solution shown in figures 25and 26 is obtained. Observe that the calibration ofthe amplitude

�was performed so that the OASPL

would be in agreement with the measurements for�� g � 3 d � � SX��3:SX� � . The far-field spectrum is in

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quite good agreement with the measured data aswell as the directivity which is within X %�� fromthe measured OASPL.

The same arguments which were used for thelength scale factor and the two-point correlationsabove can also be used for the time scale factor andthe Lagrangian autocorrelation. It is thus likelythat the time scale factor � * � e is too small andshould be increased to compensate for the trun-cation of turbulence at high wave numbers. Theresult in this case would be a decrease of the far-field sound as shown in the time scale test above,see figure 19.

A simulation using the present method withincreased time- and length-scale factors would re-sult in an improved jet noise prediction comparedto the presented base-line case, without adjustingthe amplitude

�in the time filter. Due to time

constraints this has not yet been performed.

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Fig. 25 OASPL at ) �� and ) ��

for ang-les from jet axis direction. Solid line (diamonds):far-field calibrated simulation; dashed line (cir-cles): measurements

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Fig. 26 Power spectrum of pressure in an observa-tion point at

� �� . Solid line: far-fieldcalibrated simulation; dashed line: measurements

ConclusionsA method has been presented in which a RANS

solution of a jet is the basis of synthesized tur-bulence which in turn is used to evaluate sourceterms for the linearized Euler equations. The li-nearized Euler equations produce, when excited by

the source terms, the acoustic field connected to thesynthesized turbulence.

It is shown that the near-field statistics of a highMach number jet can be simulated using the pre-sent method. Two-point statistics as space corre-lations and Lagrangian time correlations based onthe synthesized turbulence are very close to thoseof the LES. The associated far-field solution doesnot agree well however, with those of the LES si-mulations or measurements. The directivity of thefar-field sound is well predicted but the amplitudeand spectral information differ from the measure-ments.

However, it is shown that the source terms to thelinearized Euler equations do make the equationsrespond correctly to a given turbulent velocity fieldand that the equations govern the associated soundfield. This confirms the results from Ref. (7).

It is further shown that the approach of calibra-ting the near-field synthesized velocity field to theLES solution will in fact cause the length scale andthe time scale of the synthesized velocity field tobe under-predicted. The reason for this lies in thelimitations of the SNGR method connected to thespatial grid resolution.

A simulation using a modified length scale andamplitude is shown to agree well with measuredfar-field data. Amplitude, directivity and spectraare close to those of the measurements, but the ge-nerality of the method is unclear since a far-fieldcalibration is the basis for the obtained results.

AcknowledgmentThis work was conducted as part of NFFP (Na-

tional Flight Research Program) as well as theEU 5th Framework Project JEAN (Jet ExhaustAerodynamics & Noise), contract number G4RD-CT2000-000313. Computer time at the Sun-cluster,provided by UNICC at Chalmers, is gratefully ack-nowledged.

References* Bechara, W., Bailly, C., Lafon, P., and Candel, S. M.,“Stochastic Approach to Noise Modeling for Free TurbulentFlows,” AIAA Journal, Vol. 32 , No. 3, 1994, pp. 455–463.!

Bailly, C. and Juve, D., “A Stochastic Approach To ComputeSubsonic Noise Using Linearized Euler’s Equations,” AIAA Pa-per 99-1872, 1999., P. Jordan and Y. Gervais, “Modelling self and shear no-ise mechanisms in anisotropic turbulence,” The 9th AIAA/CEASAeroacooustic Conference, AIAA 2003-8743, Hilton Head, SouthCarolina, 2003.

�

P. Jordan, Y. Gervais, J.-C. Valiere and H. Foulon, “Resultsfrom acoustic field measurements,” Project deliverable D3.6,JEAN - EU 5th Framework Programme, G4RD-CT2000-00313,Laboratoire d’Etude Aerodynamiques, Poitiers, 2002.

�

Andersson, N., Eriksson, L.-E., and Davidson, L., “Large–Eddy Simulation of a Mach 0.75 Jet,” The 9th AIAA/CEAS Ae-roacoustics Conference, AIAA 2003-3312, Hilton Head, SouthCarolina, 2003.

�

Billson, M., “Computational Techniques for Jet Noise Pre-dictions,” Lic. Thesis, Department of Thermo and Fluid Dyna-mics, Chalmers University of Technology, Gothenburg, 2002.

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�Billson, M., Eriksson, L.-E., and Davidson, L., “Acoustic

Source Terms for the Linear Euler Equations on Conservativeform,” The 8th AIAA/CEAS Aeroacoustics Conference, AIAA2002-2582, Breckenridge, Colorado, 2002.�

Kraichnan, R., “Diffusion by a random velocity field,” J.Comp. Physics, Vol. 13(1), 1970, pp. 22–31.

�Karweit, M., Blanc-Benon, P., Juve, D., and Comte-Bellot,

G., “Simulation of the propagation of an acoustic wave througha turbulent velocity field: A study of phase variance,” J. Acoust.Soc. Am., Vol. 89(1), 1991, pp. 52–62.*�� Batten, P., Goldberg, U., and Chakravarthy, S., “Recon-structed Sub-Grid Methods for Acoustics Predictions at all Rey-nolds Numbers,” The 8th AIAA/CEAS Aeroacoustics Confe-rence, AIAA 2002-2511, Breckenridge, Colorado, 2002.*"* Hinze, J., Turbulence, Second edition, McGraw Hill, 1975.* ! Tam, C. and Webb, J., “Dispersion-Relation-Preserving Fi-nite Difference Schemes for Computational Acoustics,” J. Comp.Physics, Vol. 107, 1993, pp. 262–281.* , Eriksson, L.-E., “Development and validation of highly mo-dular flow solver versions in G2DFLOW and G3DFLOW,” Inter-nal report 9970-1162, Volvo Aero Corporation, Sweden, 1995.* �

Colonius, T., Lele, S., and Moin, P., “Sound generation ina mixing layer,” Journal of Fluid Mechanics, Vol. 330, 1997,pp. 375 – 409.* �

Lyrintzis, A. S., “Review: The use of Kirchhoff ’s method incomputational aeroacoustics,” ASME: Journal of Fluids Engine-ering, Vol. 116, 1994, pp. 665 – 676.* �

Crighton, D., “Basic Principles of Aerodynamic Noise Ge-neration,” Progress in Aerospace Sciences, Vol. 16 No. 1, 1975,pp. 31–96.* � Freund, J. B., Lele, S. K., and Moin, P., “Calculation of theRadiated Sound Field Using An Open Kirchhoff Surface,” AIAAJournal, Vol. 34 No. 5, 1996, pp. 909 – 916.* � P. Jordan, Y. Gervais, J.-C. Valiere and H. Foulon, “Final re-sults from single point measurements,” Project deliverable D3.4,JEAN - EU 5th Framework Programme, G4RD-CT2000-00313,Laboratoire d’Etude Aerodynamiques, Poitiers, 2002.* � M. J. Fisher and P. O. A. L. Davies, “Correlation measure-ments in a non-frozen pattern of turbulence,” Journal of FluidMechanics, Vol. 18, 1963, pp. 97–116.

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