3-d large eddy simulation for jet aeroacousticslyrintzi/aiaa-2003-3322.pdf · 3-d large eddy...

3-D Large Eddy Simulation for JetAeroacoustics ∗

A. Uzun†, G. A. Blaisdell‡, and A. S. Lyrintzis§

School of Aeronautics and AstronauticsPurdue University

West Lafayette, IN 47907

We present 3-D Large Eddy Simulation (LES) results for a turbulent isothermal roundjet at a Reynolds number of 100,000. Our recently developed LES code is part of aComputational Aeroacoustics (CAA) methodology that couples surface integral acousticsmethods with LES for the far field noise estimation of turbulent jets. The LES codeemploys high-order accurate compact differencing together with implicit spatial filteringand state-of-the-art non-reflecting boundary conditions. A localized dynamic Smagorin-sky subgrid-scale (SGS) model is used for representing the effects of the unresolved scaleson the resolved scales. A computational grid consisting of 12 million points was used inthe present simulation. Mean flow results obtained in our simulation are found to bein excellent agreement with the available experimental data of jets at similar flow con-ditions. Furthermore, the near field data provided by LES is coupled with the FfowcsWilliams-Hawkings method to compute the far field noise. Far field aeroacoustics resultsare also presented and comparisons are made with another computational study.

Introduction

THE turbulent jet noise problem still remains oneof the most complicated and difficult problems in

aeroacoustics. There is a need for substantial amountof research in this area that will lead to improved jetnoise prediction methodologies and aid in the designprocess of aircraft engines with low jet noise emissions.With the recent improvements in the processing speedof computers, the application of Direct Numerical Sim-ulation(DNS) and Large Eddy Simulation(LES) to jetnoise prediction methodologies is becoming more fea-sible. The first DNS of a turbulent jet was done for aReynolds number 2,000, supersonic jet at Mach 1.92by Freund et al.1 The computed sound pressure levelswere compared with experimental data and found tobe in good agreement with jets at similar convectiveMach numbers. Freund2 also simulated a Reynoldsnumber 3,600, Mach 0.9 turbulent jet using 25 milliongrid points, matching the parameters of the experi-mental jet studied by Stromberg et al.3 Excellentagreement with the experimental data was obtainedfor both the mean flow field and the radiated sound.Such results clearly show the attractiveness of DNS tothe jet noise problem. However, due to a wide rangeof length and time scales present in turbulent flows,DNS is still restricted to low-Reynolds-number flows inrelatively simple geometries. DNS of high-Reynolds-number jet flows of practical interest would necessitate

∗AIAA 2003-3322, Presented at the 9th AIAA/CEAS Aeroa-coustics Conference, May 2003, Hilton Head, SC.

†Graduate Research Assistant, Member AIAA.‡Associate Professor, Senior Member AIAA.§Professor, Associate Fellow AIAA.

tremendous resolution requirements that are far be-yond the reach of the capability of even the fastestsupercomputers available today.

Therefore, as frustrating as it might seem, turbu-lence still has to be modelled in some way to dosimulations for problems of practical interest. LES,with lower computational cost, is an attractive alter-native to DNS. In an LES, the large scales are directlysolved and the effect of the small scales or the sub-grid scales on the large scales are modelled. Thelarge scales are generally much more energetic thanthe small ones and are directly affected by the bound-ary conditions. The small scales, however, are usuallymuch weaker and they tend to have more or less auniversal character. Hence, it makes sense to directlysimulate the more energetic large scales and model theeffect of the small scales. LES methods are capableof simulating flows at higher Reynolds numbers andmany successful LES computations for different typesof flows have been performed to date. Since noise gen-eration is an unsteady process, LES will probably bethe most powerful computational tool to be used injet noise research in the foreseeable future since it isthe only way, other than DNS, to obtain time-accurateunsteady data.

One of the first attempts in using LES as a tool forjet noise prediction was carried out by Mankbadi etal.4 They employed a high-order numerical scheme toperform LES of a supersonic jet flow to capture thetime-dependent flow structure and applied Lighthill’stheory5 to calculate the far field noise. The first ap-plication of Kirchhoff’s method in combination withLES to far field jet noise was conducted by Lyrintzisand Mankbadi.6 LES has also been used together

1 of 12

American Institute of Aeronautics and Astronautics Paper 2003–3322

with Kirchhoff’s method7 for the noise prediction ofa Mach 2.0 jet by Gamet and Estivalezes8 as well asfor a Mach 1.2 jet with Mach 0.2 coflow by Choi etal.9 with encouraging results obtained in both stud-ies. Zhao et al.10 did LES for a Mach 0.9, Reynoldsnumber 3,600 jet obtaining mean flow results thatcompared well with Freund’s DNS and experimentaldata. They also studied the far field noise from aMach 0.4, Reynolds number 5,000 jet. They com-pared Kirchhoff’s method results with the directlycomputed sound and observed good agreement. Mor-ris et al. simulated high speed round jet flows us-ing the Nonlinear Disturbance Equations(NLDE).11–15

In their NLDE method, the instantaneous quantitiesare decomposed into a time-independent mean compo-nent, a large-scale perturbation and a small-scale per-turbation. The mean quantities are obtained using atraditional Reynolds Averaged Navier-Stokes(RANS)method. As in LES, they resolved the large-scale fluc-tuations directly and used a subgrid-scale model forthe small-scale fluctuations in their unsteady calcula-tions. In a recent paper,16 they also report the noisefrom a supersonic elliptic jet. Chyczewski and Long17

conducted a supersonic rectangular jet flow simulationand also did far field noise predictions with a Kirch-hoff method. Boersma and Lele18 did LES for a Mach0.9 jet at Reynolds numbers of 3,600 and 36,000 with-out any noise predictions. Bogey et al.19 simulateda Reynolds number 65,000, Mach 0.9 jet using LESand obtained very good mean flow results, turbulentintensities as well as sound levels and directivity. Con-stantinescu and Lele20 did simulations for a Mach 0.9jet at Reynolds numbers of 3,600 and 72,000. They di-rectly calculated the near field noise using LES. Theirmean flow parameters and turbulence statistics werein good agreement with experimental data and resultsfrom other simulations. The peak of the power spec-tra of the sound waves was also captured accurately intheir calculations. More recently, Bogey and Bailly’sLES21 for a Mach 0.9 round jet at Reynolds number,ReD = 400, 000 using 12.5 and 16.6 million grid pointsproduced mean flow and sound field results that are ingood agreement with the experimental measurementsavailable in the literature.

In general, the LES results in the literature are en-couraging and show the potential promise of LES ap-plication to jet noise prediction. Except for the studyof Bogey,21 the highest Reynolds numbers reachedin the LES simulations so far are still below thoseof practical interest. Simulations of jets at higherReynolds numbers would be very helpful for analyz-ing the broad-banded noise spectrum at such highReynolds numbers. On the other hand, none of theLES studies done so far have predicted the high-frequency noise associated with the unresolved scales.Obviously, a subgrid-scale model for noise is highly de-sirable. Piomelli et al.,22 Rubinstein and Zhou,23 Seror

et al.,24 and Zhao et al.25 did some initial work on theinvestigation of the contribution of small-scales to thenoise spectrum. More research in this area that willlead to development of subgrid-scale acoustic modelsis needed.

With this motivation behind the current studygiven, we now present the two main objectives of thisresearch:

1. Development and validation of a versatile 3-DLES code for turbulent jet simulations. A high-order accurate 3-D LES code utilizing a robustdynamic subgrid-scale (SGS) model has been de-veloped. Generalized curvilinear coordinates areused, so the code can be easily adapted for calcula-tions in several applications with complicated ge-ometries. Since the sound field is several orders ofmagnitude smaller than the aerodynamic field, wemake use of high-order accurate non-dissipativecompact schemes which satisfy the strict require-ments of CAA. Implicit spatial filtering is em-ployed to get rid of the high-frequency oscillationsresulting from unresolved scales. Non-reflectingboundary conditions are imposed on the bound-aries of the domain to let outgoing disturbancesexit the domain without spurious reflection. Asponge zone that is attached downstream of thephysical domain damps out the disturbances be-fore they reach the outflow boundary. In ourrecent studies,26 we focused on this first objec-tive. Previous simulations were performed withthe constant-coefficient Smagorinsky SGS model.However, the results were found to be sensitiveto the choice of the Smagorinsky constant. Thecurrent version of our code has the dynamic SGSmodel implemented. As will be shown in theresults section, an LES with the dynamic SGSmodel for a compressible round jet produced meanflow results in excellent agreement with experi-mental observations.

2. Accurate prediction of the far field noise. Eventhough the sound is generated by a nonlinear pro-cess, the sound field itself is known to be linearand irrotational. This implies that instead of solv-ing the full nonlinear flow equations out in the farfield for sound propagation, one can use a cheapermethod instead, such as Lighthill’s acoustic anal-ogy5 or surface integral acoustic methods suchas Kirchhoff’s method7 and the porous FfowcsWilliams-Hawkings (FW-H) equation.27,28 Inthis paper, we will couple the near field data pro-vided by LES with the Ffowcs Williams-Hawkingsmethod for computing noise propagation to thefar field.

2 of 12


Governing EquationsThe governing equations for LES are obtained by

applying a spatial filter to the Navier-Stokes equa-tions in order to remove the small scales. The ef-fect of the subgrid-scales is computed using the dy-namic Smagorinsky subgrid-scale model proposed byMoin et al.29 for compressible flows. Since we aredealing with compressible jet flows, the Favre-filteredunsteady, compressible, non-dimensionalized Navier-Stokes equations formulated in curvilinear coordinatesare solved in this study using the numerical methodsdescribed in the next section.

Numerical MethodsWe first transform a given non-uniformly spaced

curvilinear computational grid in physical space to auniform grid in computational space and solve the dis-cretized governing equations on the uniform grid. Tocompute the spatial derivatives at interior grid pointsaway from the boundaries, we employ the followingnon-dissipative sixth-order compact scheme of Lele30

13f′i−1 + f

′i +

13f′i+1 =

79∆ξ

(fi+1 − fi−1

)

+1

36∆ξ

(fi+2 − fi−2

)(1)

where f′i is the approximation of the first derivative of

f at point i along the ξ direction, fi denotes the valueof f at grid point i, and ∆ξ is the uniform grid spac-ing in the ξ direction. For the left boundary at i = 1and the right boundary at i = N , we apply the follow-ing third-order one-sided compact scheme equations,respectively

f′1 + 2f

′2 =

12∆ξ

(−5f1 + 4f2 + f3

)(2)

f′N + 2f

′N−1 =

12∆ξ

(5fN − 4fN−1 − fN−2

)(3)

For the points at i = 2 and i = N − 1 next to theboundaries, we use the following fourth-order centralcompact scheme formulations, respectively

14f′1 + f

′2 +

14f′3 =

34∆ξ

(f3 − f1

)(4)

14f′N−2 + f

′N−1 +

14f′N =

34∆ξ

(fN − fN−2

)(5)

Spatial filtering can be used as a means of suppress-ing unwanted numerical instabilities that can arisefrom the boundary conditions, unresolved scales andmesh non-uniformities. In our study, we consideredthe following sixth-order tri-diagonal filter used by Vis-bal and Gaitonde31

αf fi−1 + fi + αf fi+1 =3∑

n=0

an

2(fi+n + fi−n) (6)

Tam & Dong’s radiation boundary conditions

Tam & Dong’s radiation boundary conditions

Tam & Dong’soutflow boundaryconditions

Sponge zone

Tam &Dong’sradiationbcs

Vortex ring forcing

Fig. 1 Schematic of the boundary conditions.

where

a0 =1116

+5αf

8, a1 =

1532

+17αf

16,

a2 =−316

+3αf

8, a3 =

132− αf

16(7)

The parameter αf must satisfy the inequality −0.5 <αf < 0.5. A less dissipative filter is obtained withhigher values of αf within the given range. With αf =0.5, there is no filtering effect.

The standard fourth-order explicit Runge-Kuttascheme is used for time advancement. We apply Tamand Dong’s 3-D radiation and outflow boundary condi-tions32 on the boundaries of the computational domainas illustrated in figure 1. We additionally use thesponge zone method33 in which grid stretching andartificial damping are applied to dissipate the vorticespresent in the flowfield before they hit the outflowboundary. This way, unwanted reflections from theoutflow boundary are suppressed. Since the actualnozzle geometry is not included in the present cal-culations, randomized perturbations in the form of avortex ring are added to the velocity profile at a shortdistance downstream of the inflow boundary in orderto excite the 3-D instabilities in the jet and cause thepotential core of the jet to break up at a reasonable dis-tance downstream of the inflow boundary. This forcingprocedure has been adapted from Bogey et al.34

For subgrid-scale modelling, both the standardconstant-coefficient Smagorinsky35 as well as the dy-namic Smagorinsky29 models have been implementedinto the LES code. The main idea in the dynamicmodel is to compute the model coefficients as functionsof space and time by making use of the informationcontained within the smallest resolved scales of mo-tion. For the dynamic model, a fifteen-point explicitfilter developed by Bogey and Bailly36 is used as thetest-filter. The ratio of the test-filter width to the grid

3 of 12


spacing is taken as 2. The usual practice in LES cal-culations is to average certain dynamically computedquantities over statistically homogeneous directionsand then use these averaged quantities to computethe model coefficients. This is an ad hoc procedurethat is used to remove the very sharp fluctuations inthe dynamic model coefficients and to stabilize themodel. Obviously, such an approach is not useful forturbulent flows for which there is no homogeneousdirection. In our implementation, the dynamicallycomputed model coefficients are locally averaged inspace using a second-order three-point filter in order toavoid the sharp fluctuations in the model coefficient.No negative model coefficients are allowed. This pro-cedure works reasonably well for the jet flows we arestudying.

LES for a Round Jet at a ReynoldsNumber of 100,000

The previous LES studies that we carried out can befound in Uzun.26 Here, we will present results from anLES performed with the dynamic SGS model for a tur-bulent isothermal round jet at a Mach number of 0.9,and Reynolds number ReD = ρjUjDj/µj = 100, 000where ρj , Uj , µj are the jet centerline density, velocityand viscosity at the nozzle exit, respectively and Dj isthe jet diameter. The jet centerline temperature waschosen the same as the ambient temperature.

A fully curvilinear grid consisting of approximately12 million grid points was used in the simulation. Thephysical portion of the domain extended to 60ro inthe streamwise direction and from −20ro to 20ro inthe transverse y and z directions, where ro = Dj/2is the jet radius. The coarsest resolution in this sim-ulation is estimated to be about 170 times the localKolmogorov length scale. Figures 2 through 5 showhow the grid is stretched in all three directions. Ini-tially, the grid points are clustered around the shearlayer of the jet in order to accurately resolve the shearlayer. After the potential core of the jet breaks up, thegrid is also stretched in the y and z directions in orderto redistribute the grid points and achieve a more uni-form distribution. The y − z cross-section of the gridremains the same along the streamwise direction aftera distance of 35ro.

It has been observed by Bogey and Bailly21 thatthe jet spreading rate and the jet decay coefficient donot reach their asymptotic values if the computationaldomain size is kept relatively short. Furthermore, theyfound out that Reynolds stresses do not reach theirself-similar state either in a relatively short domain.Our aim is to fully validate the dynamic model, soit is important that the asymptotic jet spreading rateand the centerline velocity decay rate as well as theself-similar state of Reynolds stresses are reached inthe simulation. This will facilitate the comparison ofour numerical results with the experimental jet data at

x / ro

y/r

o

0 10 20 30 40 50 60 70-20

-10

0

10

20

30

40

Fig. 2 The x − y cross-section of the grid on thez = 0 plane. (Every 4th grid node is shown.)

z / ro

y/r

o

-20 -10 0 10 20-20

-10

0

10

20

Fig. 3 The y − z cross-section of the grid on thex = 5ro plane. (Every 4th grid node is shown.)

similar flow conditions. Hence we decided to choose along domain length of 60ro in the streamwise direction.

In our simulation, the initial transients exited thedomain over the first 20,000 time steps. We then col-lected the flow statistics over 150,000 time steps. Thissampling period corresponds to a time length in whichan ambient sound wave travels about 60 times the do-main length in the streamwise direction. A total ofabout 123,000 CPU hours was needed for this sim-ulation on an IBM-SP machine using 160 processorsin parallel. It should be noted that the mean flowconvection speeds in the far downstream region arerelatively low, hence the turbulence convects slowly inthe far downstream region resulting in slowly converg-ing statistics. As a result, we had to collect data overa period of 150,000 time steps to obtain reasonablyconverged statistics.

4 of 12


z / ro

y/r

o

-20 -10 0 10 20-20

-10

0

10

20


z / ro

y/r

o

-20 -10 0 10 20-20

-10

0

10

20


For code validation, we first compared our jet’s cen-terline decay rate in the far downstream region withthe experiments of Zaman.37 Zaman37 did a series ofexperiments for initially compressible jets and exam-ined the asymptotic jet centerline velocity decay andentrainment rates. He found out that the asymptoticjet centerline velocity decay rate decreased with in-creasing initial jet Mach number. In figure 6, we plotthe streamwise variation of the inverse of the meanjet centerline velocity normalized by the jet inflow ve-locity. In the far downstream region, we see that theslope of the linear fit to the curve is 0.161. From the ex-periments of Zaman,37 the corresponding experimentalslope for a Mach 0.9 jet is about 0.155 which is veryclose to our computed value. For incompressible jets,on the other hand, experimental values ranging from0.165 to 0.185 have been previously reported in the

x / Dj

Uj/U

c

0 10 20 300

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

slope = 0.161

From Zaman’sexperiments (1998):slope ≈ 0.155 for Mj = 0.9

Fig. 6 Streamwise variation of Uj/Uc, where Uc =mean jet centerline velocity and Uj = jet inflowvelocity.

x / Dj

Q/Q

e

10 15 20 25 304

5

6

7

8

9

10

11

slope = 0.267

From Zaman’sexperiments (1998):slope ≈ 0.26 for Mj = 0.9

Fig. 7 Streamwise variation of the mass flux nor-malized by the mass flux at the nozzle.

literature.Furthermore, in figure 7, we show the streamwise

variation of the mass flux normalized by the massflux through the jet nozzle. In the far downstreamregion, we again see a linear growth. This time theslope of the line is 0.267 showing very good agreementwith the experimental value of 0.26 from Zaman.37

The jet spreading rate, which is the slope of the half-velocity radius growth in the far downstream regiondepicted in figure 8, has been found to be 0.092 inour simulation. The half-velocity radius, r1/2 at agiven downstream location is defined as the radial lo-cation where the mean streamwise velocity is one halfof the jet mean centerline velocity at that location. Tothe best of our knowledge, no experimental value hasbeen reported from an initially compressible jet in the

5 of 12


slope= A = 0.092

experimental valuesof A : 0.086 - 0.096

x / ro

r 1/2/r

o

0 5 10 15 20 25 30 35 40 45 50 55 600

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Fig. 8 Streamwise variation of the half-velocityradius normalized by the jet radius.

literature for the jet spreading parameter. However,experimental values ranging from 0.086 to 0.096 havebeen previously found for incompressible jets. Hence,our computed value is within the experimental rangeobserved for incompressible jets.

Properly scaled mean streamwise velocity profilesat three downstream locations are plotted in figure 9.The self-similarity coordinate is taken as r/r1/2 wherer is the radial location and r1/2 is the half-velocity ra-dius. The vertical axis of the figure is normalized bythe mean jet centerline velocity. From this figure, itcan be observed that the profiles of our jet at the threedownstream locations collapse onto each other fairlywell and exhibit self-similarity which is consistent withexperimental observations. Our results are also in verygood agreement with the experimental profiles of Hus-sein et al.38 for an incompressible jet at ReD = 95, 500as well as that of Panchapakesan et al.39 for an incom-pressible jet at ReD = 11, 000. The convective Machnumber of the jet flow is low in the far downstreamregion, hence the compressibility effects are negligible.Therefore, it is safe to compare our profiles in the fardownstream region with incompressible experimentaldata.

We also compared our Reynolds stress profiles withthe experimental profiles of Hussein et al.38 andPanchapakesan et al.39 Figures 10 through 13 plotour properly scaled Reynolds stress profiles at threedownstream locations and compare them with the twoexperiments. Again, the self-similarity coordinate istaken as r/r1/2. The vertical axis of the figures isnormalized by the square of the mean jet centerlinevelocity. As can be observed from the plots, the prop-erly scaled profiles collapse onto each other exhibitingself-similarity. Overall, the agreement is very good.It should be noted that our Reynolds stresses com-puted by LES are based on filtered velocities, while

r / r1/2

u/U

c

0 0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x = 45ro

x = 50ro

x = 55ro

exp. data of Hussein et. al.exp. data of Panchapakesan et. al.

Fig. 9 Normalized mean streamwise velocity pro-files and comparison with experimental data.

r / r1/2

σ xx

0 0.5 1 1.5 2 2.50

0.025

0.05

0.075

0.1

x = 45ro

x = 50ro

x = 55ro


Fig. 10 Normalized Reynolds stress σxx profilesand comparison with experimental data.

the actual experimental profiles are based on unfilteredvelocities. Hence, some differences should be expectedbetween LES and experiments when the Reynoldsstresses are compared. As can be seen from the fig-ures, the Reynolds normal stress σxx profiles matchthe experiment of Panchapakesan et al.39 better, whilethe other Reynolds stresses are in between the twoexperimental profiles. It should also be noted thatthe difference between the initial conditions imposedin our simulation and the actual initial conditions inexperimental jets could be another possible reason forthe differences between LES and experimental profiles.We have imposed randomized velocity fluctuations inthe form of a vortex ring34 in our simulation sincethe actual nozzle geometry was not included in thecalculations. Furthermore, it can be argued that theexperimental Reynolds stress profiles have been mea-

6 of 12


r / r1/2

σ rr

0 0.5 1 1.5 2 2.50

0.01

0.02

0.03

0.04

0.05

0.06

x = 45ro

x = 50ro

x = 55ro


Fig. 11 Normalized Reynolds stress σrr profilesand comparison with experimental data.

r / r1/2

σ θθ

0 0.5 1 1.5 2 2.50

0.01

0.02

0.03

0.04

0.05

0.06

x = 45ro

x = 50ro

x = 55ro


Fig. 12 Normalized Reynolds stress σθθ profilesand comparison with experimental data.

sured in the very far downstream region, usually atdistances of 100 jet radii or more. It has also beenobserved experimentally that the different Reynoldsstress components reach asymptotic self-similarity atdifferent downstream locations, depending on initialconditions. Since our domain length of 60 jet radii isstill relatively short compared to experiments, the nor-malized Reynolds normal stresses σxx may not havereached their true asymptotic values. In a previoussimulation that we carried out using the same dynamicSGS model and a shorter domain length of 35ro, wefound out that the Reynolds stresses in the far down-stream region were lower than the ones in the presentsimulation. This clearly means that the Reynoldsstresses in the short domain simulation did not reachtheir asymptotic self-similar states. If we did a newsimulation using a domain longer than 60ro in the

r / r1/2

σ rx

0 0.5 1 1.5 2 2.50

0.005

0.01

0.015

0.02

0.025

x = 45ro

x = 50ro

x = 55ro


Fig. 13 Normalized Reynolds shear stress σrx pro-files and comparison with experimental data.

streamwise direction, it is possible that the Reynoldsnormal stress σxx profiles in the far downstream regionmight shift upwards and come in between the two ex-periments. However, such a simulation would requireeven more grid points and many more time steps torun since the very slowly convecting mean flow in thefar downstream region would cause very slowly con-verging flow statistics. Hence such a simulation wouldnot be very practical.

To summarize, the mean flow properties computedusing the dynamic SGS model compared very well withthe experimental data. Furthermore, the comparisonof Reynolds stresses in the present simulation with ex-periments is found to be reasonable. These findingsprovide us with valuable evidence towards the valid-ity of the numerical schemes as well as the dynamicsubgrid-scale model used in our LES code.

Far Field AeroacousticsNext, we will look at some far field aeroacoustics

results obtained by coupling the near field LES datawith a Ffowcs Williams-Hawkings code which has thecapability to work on any general control surface ge-ometry.

The Ffowcs Williams-Hawkings formulation for astationary control surface, S is given as follows

p′(~x, t) = p′T (~x, t) + p′L(~x, t) + p′Q(~x, t) (8)

where

4πp′T (~x, t) =∫

S

[ρoUn

r

]

ret

dS (9)

4πp′L(~x, t) =1a◦

∫

S

[Lr

r

]

ret

dS +∫

S

[Lr

r2

]

ret

dS

(10)

7 of 12


andUi =

ρui

ρo(11)

Li = Pij nj + ρuiun (12)

(~x, t) are the observer coordinates and time, r is thedistance from the source (surface) to the observer,subscript o implies ambient conditions, superscript ′

implies disturbances (e.g. ρ = ρ′ + ρo), ρ is the den-sity, ui is the velocity vector, a◦ is the ambient soundspeed, and Pij is the compressive stress tensor withthe constant poδij subtracted. The dot over a variableindicates a time derivative, a subscript r or n indicatesa dot product of the vector with the unit vector in theradiation direction r or the unit vector in the surfacenormal direction n, respectively. The surface integralsare over the control surface S, and the subscript retindicates evaluation of the integrands at the emission(retarded) time τ = t − r/a◦. The quadrupole noisepressure p′Q(~x, t) denotes the quadrupoles outside thecontrol surface and has been neglected in the cur-rent study. A more detailed description of the FfowcsWilliams-Hawkings method can be found in Lyrintzisand Uzun.40

We put a control surface around our jet as illus-trated in figure 14 and we gathered flow field data onthe control surface at every 10 time steps over a pe-riod of 23,000 time steps during our LES run. Thetotal acoustic sampling period corresponds to a timescale in which an ambient sound wave travels about9.4 times the domain length in the streamwise direc-tion. Assuming at least 6 points per wavelength areneeded to accurately resolve an acoustic wave, we seethat the maximum frequency resolved with our gridspacing around the control surface corresponds to aStrouhal number of approximately 1.0. The Nyquistfrequency, on the other hand, which is the maximumfrequency that can be resolved with the time incrementof our data sampling rate, corresponds to a Strouhalnumber of about 4.5. However, we choose the maxi-mum frequency that is based on spatial resolution asour grid cutoff frequency.

We computed the overall sound pressure levels alongan arc of radius 60ro from the jet nozzle. The centerof the arc is chosen as the jet nozzle exit and the an-gle θ is measured from the jet axis as illustrated infigure 15. We applied the Ffowcs Williams-Hawkingsmethod to compute the acoustic pressure signal at 36equally spaced azimuthal points on a full circle at agiven θ location on the arc. In order to confine spuriousspectral contributions to low frequencies, we multi-plied the pressure history at every azimuthal locationby a windowing function similar to that used by Fre-und.2 We ran the windowed pressure history througha 2048 point Fast Fourier Transform, and convertedto polar notation to obtain the power spectral den-sity at each frequency. We then averaged the acousticpressure spectra over the equally spaced 36 azimuthal

Control Surface

Control Surface

Jet Flow

Fig. 14 Schematic of the control surface surround-ing the jet flow.

x / ro

y/r

o

0 10 20-5

0

5

10

15

R

θ

Fig. 15 Schematic showing the center of the arcand how the angle θ is measured from the jet axis.

points and finally integrated the averaged spectra tocompute the overall sound pressure level at the givenθ location. In our spectra, we observed very ener-getic low frequencies corresponding to St ≤ 0.05. It isnot clear where these spurious energetic low frequen-cies are coming from. As pointed out by Bogey etal.19,34 who also made similar observations, these en-ergetic spurious low frequencies might be related tothe very low frequency reflections resulting from theoutflow boundary which may not be damped out ef-fectively by the sponge zone. Only 35 grid points wereused to construct the sponge zone in our grid. Be-cause of these observations, we did not include theStrouhal numbers less than 0.05 in the overall soundpressure level calculations. Only the resolved frequen-cies (0.05 ≤ St ≤ 1.0) were included in the integration.

Figure 16 shows our overall sound pressure lev-els computed along the arc and compares them withsome experimental data. Table 1 summarizes the

8 of 12


θ (deg)

OA

SPL

(dB

)

10 20 30 40 50 60 70 80 90 100 110 120100

102

104

106

108

110

112

114

116

118

120

LES+ FWH (isothermal jet)SAE ARP 876C predictionexp. of Mollo-Christensen et al. (cold jet)exp. of Lush (cold jet)exp. of Stromberg et al. (cold jet)

Fig. 16 Overall sound pressure levels at 60ro fromthe nozzle exit.

M ReD Reference0.9 540,000 Mollo-Christensen et al.41

0.88 500,000 Lush42

0.9 3,600 Stromberg et al.3

0.9 100,000 Our 3-D LES

Table 1 Some jet noise experiments with Machnumbers similar to that of our LES.

Mach numbers and Reynolds numbers of the exper-iments that we are doing comparisons with. It shouldbe noted that the experimental jets were cold jets,whereas our jet is an isothermal jet. We also plot theSAE ARP 876C43 database prediction for an isother-mal jet at the same operating conditions as ours. Thisdatabase consists of actual engine jet noise measure-ments and can be used to predict overall sound pres-sure levels within a few dB at various jet operatingconditions. The same database was also used to pre-dict the overall sound pressure levels for a cold Mach0.9 jet and the difference between the isothermal andcold jet predictions were found to be only 0.3 dB.

From the overall sound pressure level plot, we seethat our jet is louder than the experimental cold jetsfor values of θ smaller than 40o. It is also seen thatthere is as much as 3 dB difference between our predic-tion and the ARP 876C prediction at the downstreamangles. For larger values of θ, the overall sound pres-sure levels are comparable to the experimental val-ues of Mollo-Christensen et al.41 as well as those ofLush.42 The agreement with ARP 876C is also verygood for values of θ greater than 50o.

It is known that the large scale turbulent structuresin the jet flow are responsible for the noise radiationin the downstream direction while the high-frequencynoise generated by the fine scale turbulence becomesdominant in the sideline and upstream direction. Thedifferences in the overall sound pressure levels at the

downstream angles could be related to the differencein the nozzle exit conditions. This is because the jetnozzle exit conditions probably have a strong influ-ence on the generation of large scale structures in theflow, whereas the fine scale turbulence is much lessaffected by initial conditions and has a more univer-sal character. Our vortex ring forcing which is clearlyartificial and different than the initial conditions inthe actual experiments, is apparently generating largescale structures that result in somewhat higher overallsound pressure levels in the downstream direction.

Howe44 provides the following equation for estimat-ing the overall sound pressure level at the θ = 90o

location〈p2(r, 90o)〉

p2ref

≈ 9× 1013A

r2

(ρJ/ρo)wM7.5

[1− 0.1M2.5 + 0.015M4.5](13)

where r is the observer distance from the nozzle exit,〈p2(r, 90o)〉 is the mean square acoustic pressure at the(r, θ = 90o) observation point, pref is the standard ref-erence pressure, A is the nozzle cross-sectional area, Mis the jet Mach number, w is the so-called jet densityexponent, and ρJ/ρo is the ratio of the jet density tothe ambient density. Howe44 also provides an equa-tion in terms of the jet Mach number to estimate thejet density exponent, w. Since we have an isother-mal jet, our ρJ/ρo = 1 and therefore there is no needto estimate the jet density exponent for our jet. Theabove equation predicts an overall sound pressure levelof 105.82 dB at the R = 60ro, θ = 90o location. Thisprediction is very close to our computed value of 105.57dB as well as the ARP 876C prediction of 105.30 dBat the same location. From these findings, we see thatwe have captured most of the energetic frequencies inour simulation and the unresolved higher frequenciesare not likely to alter the overall sound pressure levelssignificantly.

Finally, we compare our acoustic pressure spectra atthree observation locations with the recent results ofBogey and Bailly21 for an isothermal round jet at aReynolds number of 400,000. It should be noted herethat Bogey and Bailly21 computed the spectra in thenear field of the jet only, using the data directly pro-vided by LES. The first observation point of Bogey andBailly21 is located inside of our control surface whilethe other two are located outside, hence we computethe near field spectra at the latter two points usingthe Ffowcs Williams-Hawkings method. On the otherhand, to compare our far field spectra with their nearfield spectra, we adjust the θ location along the arcsuch that an acoustic ray drawn between the end ofthe potential core (x = 11ro) and the θ location onthe arc crosses their observation point.

Figure 17 plots the spectra at the R = 60ro, θ = 25o

location over our resolved frequency range and com-pares it with the corresponding near field spectra ofBogey and Bailly.21 We see that our spectra has a

9 of 12


St

SPL

(dB

/St)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 170

80

90

100

110

120

130

Bogey’s spectra at x = 29ro and r = 12ro

Our spectra at R = 60ro and θ = 25o

4th order polynomial fit

Fig. 17 Acoustic pressure spectra at R = 60ro,θ = 25o in the far field and comparison with thecorresponding spectra of Bogey and Bailly.21

St

SPL

(dB

/St)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 170

80

90

100

110

120

130


Our spectra at x = 20ro and r = 15ro

4th order polynomial fitOur spectra at R = 60ro and θ = 50o


Fig. 18 Acoustic pressure spectra at R = 60ro, θ =50o in the far field, acoustic pressure spectra at x =20ro, r = 15ro in the near field and comparison withthe corresponding spectra of Bogey and Bailly.21

peak at around St = 0.25 while that of Bogey andBailly21 is at St = 0.3. Other than this difference, thetwo spectra look qualitatively similar. The spectra ofBogey and Bailly21 has higher sound pressure levelsthan ours since it was computed in the near field ofthe jet. We also show a 4th order polynomial fit toour spectra in the figure. From this polynomial fit, itis observed that there is a drop of about 15 dB/St inthe sound pressure level from the peak Strouhal num-ber of 0.25 to the grid cutoff Strouhal number of 1.0.Figures 18 and 19 show our near field spectra at theother two observation points, respectively and makecomparisons with the spectra obtained by Bogey andBailly.21 The acoustic rays that originate from the

St

SPL

(dB

/St)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 170

80

90

100

110

120

130


Our spectra at x = 11ro and r = 15ro

4th order polynomial fitOur spectra at R = 60ro and θ = 80o


Fig. 19 Acoustic pressure spectra at at R = 60ro,θ = 80o in the far field, acoustic pressure spectraat x = 11ro, r = 15ro in the near field and compar-ison with the corresponding spectra of Bogey andBailly.21

end of the potential core and pass through these twonear field observation points hit the far field arc atthe θ = 50o and θ = 80o locations, respectively. Wealso show our far field spectra corresponding to thesetwo near field observation points. From these figures,we see that our near and far field spectra basicallyhave the same shape. Our spectra also become broad-banded at these observation points demonstrating thefact that higher frequencies are becoming more dom-inant. It is also interesting to note that our spectrademonstrate a slow decay after they reach their peakwhereas the spectra of Bogey and Bailly21 stay rela-tively flat after the peak. A possible reason for thisdifference could be the different Reynolds numbers ofthe two simulations. There is almost 10 dB/St differ-ence between our near and far field spectra in figure 18.The distance between the far field point and the endof the potential core is about 53.60ro, whereas the dis-tance between the near field point and the end of thepotential core is 17.49ro. The ratio of these two dis-tances is about 3. A factor of 3 means about 10 dB/Stdifference according to the r−1 decay assumption ofacoustic waves. In this assumption, the acoustic waveamplitudes drop by a factor of 2 for every doubling ofthe distance from the source region. Similarly, the dif-ference between our near and far field spectra in figure19 is about 12 dB/St. In this case, the ratio of the farfield observation point distance to the near field obser-vation point distance is about 4 which also translatesinto 12 dB/St difference according to the r−1 decayassumption.

10 of 12


Concluding RemarksAs part of a jet noise prediction methodology, we

have developed and fully tested a 3-D Large Eddy Sim-ulation code. The localized dynamic model was foundto perform very well. The mean flow results obtainedin our simulation for a Reynolds number 100,000 jetwere in excellent agreement with experimental data.We also coupled the near field data computed by ourLES code with the Ffowcs Williams-Hawkings (FW-H) formulation to compute the far field noise of thejet. Far-field aeroacoustics results were encouraging.

In our near future work, we plan to simulate thesame Reynolds number 400,000 jet test case whichBogey and Bailly21 studied. This will help us fullyvalidate our CAA methodology. We also plan to do amore detailed study of the far field acoustics of the jetusing both the Kirchhoff’s and the Ffowcs Williams-Hawkings (FW-H) formulations. Furthermore, we willdo noise calculations using the r−1 decay assumptionwhich is cheaper than integral acoustic methods. Wewill compare far field noise computations with the sur-face integral methods to the results obtained by usingthe r−1 decay assumption. We will also investigate thesensitivity of far field noise predictions to the positionof the control surface on which aeroacoustic data iscollected.

Our eventual goal in this research is to study theaeroacoustics of a high Reynolds number jet by doinga well-resolved LES. It is imperative that there existswell-documented experimental data for the LES testcase chosen so that comparisons can be made withthe experiment to ensure the validity of simulation re-sults. Only then the simulation database can be usedreliably to investigate the jet noise generation mecha-nisms and recover valuable information about the jetnoise physics.

AcknowledgmentsThis work is part of a joint project with Rolls-

Royce, Indianapolis and is sponsored by the Indiana21st Century Research & Technology Fund. It wasalso partially supported by the National Computa-tional Science Alliance under the grant CTS010032N,and utilized the SGI Origin 2000 computer systemsat the University of Illinois at Urbana-Champaign.Some of the computations were performed on the IBM-SP research computer of Indiana University. We aregrateful to Dr. Christophe Bogey for his help whilewe implemented the vortex ring forcing method andthe dynamic subgrid-scale model into our code. Wealso thank him for providing us with data from hisReynolds number 400,000 jet simulation.

References1Freund, J. B., Lele, S. K., and Moin, P., “Direct Numerical

Simulation of a Mach 1.92 Turbulent Jet and Its Sound Field,”AIAA Journal , Vol. 38, No. 11, November 2000, pp. 2023–2031.

2Freund, J. B., “Noise Sources in a Low-Reynolds-numberTurbulent Jet at Mach 0.9,” Journal of Fluid Mechanics,Vol. 438, 2001, pp. 277–305.

3Stromberg, J. L., McLaughlin, D. K., and Troutt, T. R.,“Flow Field and Acoustic Properties of a Mach Number 0.9 Jetat a Low Reynolds Number,” Journal of Sound and Vibration,Vol. 72, No. 2, 1980, pp. 159–176.

4Mankbadi, R. R., Hayder, M. E., and Povinelli, L. A.,“Structure of Supersonic Jet Flow and Its Radiated Sound,”AIAA Journal , Vol. 32, No. 5, May 1994, pp. 897–906.

5Lighthill, M. J., “On the Sound Generated Aerodynami-cally: I, General Theory,” Proc. Royal Soc. London A, Vol. 211,1952, pp. 564–587.

6Lyrintzis, A. S. and Mankbadi, R. R., “Prediction of theFar-field Jet Noise Using Kirchhoff’s Formulation,” AIAA Jour-nal , Vol. 1, No. 2, 1996, pp. 1–4.

7Kirchhoff, G. R., “Zur Theorie der Lichtstrahlen,” Annalender Physik und Chemie, Vol. 18, 1883, pp. 663–695.

8Gamet, L. and Estivalezes, J. L., “Application of Large-eddy Simulations and Kirchhoff Method to Jet Noise Predic-tion,” AIAA Journal , Vol. 36, No. 12, December 1998, pp. 2170–2178.

9Choi, D., Barber, T. J., Chiappetta, L. M., and Nishimura,M., “Large Eddy Simulation of High-Reynolds Number JetFlows,” AIAA Paper No. 99-0230, January 1999.

10Zhao, W., Frankel, S. H., and Mongeau, L., “Large EddySimulations of Sound Radiation from Subsonic Turbulent Jets,”AIAA Journal , Vol. 39, No. 8, August 2001, pp. 1469–1477.

11Morris, P. J., Wang, Q., Long, L. N., and Lockhard, D. P.,“Numerical Predictions of High Speed Jet Noise,” AIAA PaperNo. 97-1598, May 1997.

12Morris, P. J., Long, L. N., Bangalore, A., and Wang,Q., “A Parallel Three-Dimensional Computational Aeroacous-tics Method Using Nonlinear Disturbance Equations,” Journalof Computational Physics, Vol. 133, 1997, pp. 56–74.

13Morris, P. J., Long, L. N., Scheidegger, T. E., Wang, Q.,and Pilon, A. R., “High Speed Jet Noise Simulations,” AIAAPaper No. 98-2290, 1998.

14Morris, P. J., Long, L. N., and Scheidegger, T., “Parallelcomputations of High Speed Jet Noise,” AIAA Paper No. 99-1873, 1999.

15Morris, P. J., Scheidegger, T. E., and Long, L. N., “JetNoise Simulations for Circular Nozzles,” AIAA Paper No. 2000-2080, June 2000.

16Boluriaan, S., Morris, P. J., Long, L. N., and Scheidegger,T., “High Speed Jet Noise Simulations for Noncircular Nozzles,”AIAA Paper No. 2001-2272, May 2001.

17Chyczewski, T. S. and Long, L. N., “Numerical Predictionof the Noise Produced by a Perfectly Expanded RectangularJet,” AIAA Paper No. 96-1730, May 1996.

18Boersma, B. J. and Lele, S. K., “Large Eddy Simulationof a Mach 0.9 Turbulent Jet,” AIAA Paper No. 99-1874, May1999.

19Bogey, C., Bailly, C., and Juve, D., “Computation of theSound Radiated by a 3-D Jet Using Large Eddy Simulation,”AIAA Paper No. 2000-2009, June 2000.

20Constantinescu, G. S. and Lele, S. K., “Large Eddy Simu-lation of a Near Sonic Turbulent Jet and Its Radiated Noise,”AIAA Paper No. 2001-0376, January 2001.

21Bogey, C. and Bailly, C., “Direct Computation of theSound Radiated by a High-Reynolds-number, Subsonic RoundJet,” CEAS Workshop From CFD to CAA, Athens, Greece,November 2002.

22Piomelli, U., Streett, C. L., and Sarkar, S., “On the Com-putation of Sound by Large-eddy Simulations,” Journal of En-gineering Mathematics, Vol. 32, 1997, pp. 217–236.

23Rubinstein, R. and Zhou, Y., “Characterization of SoundRadiation by Unresolved Scales of Motion in Computational

11 of 12


Aeroacoustics,” Tech. Rep. NASA/CR-1999-209688, ICASE Re-port No. 99-39, ICASE, NASA Langley Research Center, Octo-ber 1999.

24Seror, C., Sagaut, P., Bailly, C., and Juve, D., “On theRadiated Noise Computed by Large-eddy Simulation,” Physicsof Fluids, Vol. 13, No. 2, February 2001, pp. 476–487.

25Zhao, W., Frankel, S. H., and Mongeau, L., “Effects ofSpatial Filtering on Sound Radiation from a Subsonic Axisym-metric Jet,” AIAA Journal , Vol. 38, No. 11, November 2000,pp. 2032–2039.

26Uzun, A., Blaisdell, G. A., and Lyrintzis, A. S., “RecentProgress Towards a Large Eddy Simulation Code for Jet Aeroa-coustics,” AIAA Paper No. 2002-2598, June 2002.

27Williams, J. E. F. and Hawkings, D. L., “Sound Generatedby Turbulence and Surfaces in Arbitrary Motion,” PhilosophicalTransactions of the Royal Society, Vol. A264, 1969, pp. 321–342.

28Crighton, D. G., Dowling, A. P., Williams, J. E. F., Heckl,M., and Leppington, F. G., Modern Methods in AnalyticalAcoustics: Lecture Notes, Springer–Verlag, London, 1992.

29Moin, P., Squires, K., Cabot, W., and Lee, S., “A DynamicSubgrid-Scale Model for Compressible Turbulence and ScalarTransport,” Physics of Fluids A, Vol. 3, No. 11, November 1991,pp. 2746–2757.

30Lele, S. K., “Compact Finite Difference Schemes withSpectral-like Resolution,” Journal of Computational Physics,Vol. 103, No. 1, November 1992, pp. 16–42.

31Visbal, M. R. and Gaitonde, D. V., “Very High-order Spa-tially Implicit Schemes for Computational Acoustics on Curvi-linear Meshes,” Journal of Computational Acoustics, Vol. 9,No. 4, 2001, pp. 1259–1286.

32Bogey, C. and Bailly, C., “Three-dimensional Non-reflective Boundary Conditions for Acoustic Simulations: FarField Formulation and Validation Test Cases,” Acta Acustica,Vol. 88, No. 4, 2002, pp. 463–471.

33Colonius, T., Lele, S. K., and Moin, P., “Boundary Con-ditions for Direct Computation of Aerodynamic Sound Genera-tion,” AIAA Journal , Vol. 31, No. 9, September 1993, pp. 1574–1582.

34Bogey, C., Bailly, C., and Juve, D., “Noise Investigationof a High Subsonic, Moderate Reynolds Number Jet Using aCompressible LES,” Theoretical and Computational Fluid Dy-namics, Vol. 16, No. 4, 2003, pp. 273–297.

35Smagorinsky, J. S., “General Circulation Experiments withthe Primitive Equations,” Monthly Weather Review , Vol. 91,No. 3, March 1963, pp. 99–165.

36Bogey, C. and Bailly, C., “A Family of Low Dispersive andLow Dissipative Explicit Schemes for Computing AerodynamicNoise,” AIAA Paper No. 2002-2509, June 2002.

37Zaman, K. B. M. Q., “Asymptotic Spreading Rate of Ini-tially Compressible Jets - Experiment and Analysis,” Physics ofFluids, Vol. 10, No. 10, 1998, pp. 2652–2660.

38Hussein, H. J., Capp, S. C., and George, W. K., “Ve-locity Measurements in a High-Reynolds-number, Momentum-conserving, Axisymmetric, Turbulent jet,” Journal of Fluid Me-chanics, Vol. 258, 1994, pp. 31–75.

39Panchapakesan, N. R. and Lumley, J. L., “Turbulence Mea-surements in Axisymmetric Jets of Air and Helium. Part 1. AirJets.” Journal of Fluid Mechanics, Vol. 246, 1993, pp. 197–223.

40Lyrintzis, A. S. and Uzun, A., “Integral Techniques forAeroacoustics Calculations,” AIAA Paper No. 2001-2253, May2001.

41Mollo-Christensen, E., Kolpin, M. A., and Martucelli,J. R., “Experiments on Jet Flows and Jet Noise Far-field Spectraand Directivity Patterns,” Journal of Fluid Mechanics, Vol. 18,1964, pp. 285–301.

42Lush, P. A., “Measurements of Subsonic Jet Noise andComparison with Theory,” Journal of Fluid Mechanics, Vol. 46,No. 3, 1971, pp. 477–500.

43SAE ARP 876C : Gas Turbine Jet Exhaust Noise Predic-tion, Society of Automotive Engineers, November 1985.

44Howe, M. S., Acoustics of Fluid-Structure Interactions,Cambridge University Press, 1998.

12 of 12


3-d large eddy simulation for jet aeroacousticslyrintzi/aiaa-2003-3322.pdf · 3-d large eddy...

Documents