Application of Compact Schemes to Large Eddy …lyrintzi/jsc-Alipaper.pdfApplication of Compact...

Application of Compact Schemes to Large Eddy

Simulation of Turbulent Jets

Ali Uzun, Gregory A. Blaisdell and Anastasios S. LyrintzisSchool of Aeronautics and Astronautics, Purdue University, West Lafayette,

Indiana 47907, USA

Abstract. We present 3-D Large Eddy Simulation (LES) results for a turbulentMach 0.9 isothermal round jet at a Reynolds number of 100,000 (based on jet nozzleexit conditions and nozzle diameter). Our LES code is part of a ComputationalAeroacoustics (CAA) methodology that couples surface integral acoustics techniquessuch as Kirchhoff’s method and the Ffowcs Williams-Hawkings method with LESfor the far field noise estimation of turbulent jets. The LES code employs high-orderaccurate compact differencing together with implicit spatial filtering and state-of-the-art non-reflecting boundary conditions. A localized dynamic Smagorinskysubgrid-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 usedin the present simulation. Mean flow results obtained in our simulation are foundto be in excellent agreement with the available experimental data of jets at similarflow conditions. Furthermore, the near field data provided by the LES is coupledwith the Ffowcs Williams-Hawkings method to compute the far field noise. Far fieldaeroacoustics results are also presented and comparisons are made with experimentalmeasurements of jets at similar flow conditions and another computational study[5]. The aeroacoustics results are encouraging and suggest further investigation ofthe effects of inflow conditions as well as the SGS model on the jet acoustic field.

Keywords: Large Eddy Simulation, Turbulent Jets, Jet Noise, Dynamic Smagorin-sky Subgrid-Scale Model, Compact Finite Difference Schemes, Implicit Spatial Fil-tering.

Abbreviations: LES – Large Eddy Simulation; DNS – Direct Numerical Simula-tion; CAA – Computational Aeroacoustics; SGS – Subgrid-Scale; OASPL – OverallSound Pressure Level; SPL – Sound Pressure Level; FWH – Ffowcs Williams-Hawkings

1. Introduction

The turbulent jet noise problem still remains one of the most com-plicated and difficult problems in aeroacoustics. There is a need forsubstantial amount of research in this area that will lead to improvedjet noise prediction methodologies and aid in the design process ofaircraft engines with low jet noise emissions. With the recent improve-ments in the processing speed of computers, the application of DirectNumerical Simulation (DNS) and Large Eddy Simulation (LES) to jetnoise prediction methodologies is becoming more feasible. The first

c© 2003 Kluwer Academic Publishers. Printed in the Netherlands.

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2 Uzun, Blaisdell and Lyrintzis

DNS of a turbulent jet was done for a Reynolds number 2,000, su-personic jet at Mach 1.92 by Freund et al. [14]. The computed soundpressure levels were compared with experimental data and found tobe in good agreement with jets at similar convective Mach numbers.Freund [15] also simulated a Reynolds number 3,600, Mach 0.9 turbu-lent jet using 25.6 million grid points, matching the parameters of theexperimental jet studied by Stromberg et al. [31]. Excellent agreementwith the experimental data was obtained for both the mean flow fieldand the radiated sound. Such results clearly show the attractivenessof DNS to the jet noise problem. However, due to the wide range oflength and time scales present in turbulent flows, DNS is still restrictedto low-Reynolds-number flows in relatively simple geometries. DNS ofhigh-Reynolds-number jet flows of practical interest would necessitatetremendous resolution requirements that are far beyond the reach ofthe capability of even the fastest supercomputers available today.Therefore, turbulence still has to be modeled in some way to do

simulations for problems of practical interest. LES, with lower com-putational cost, is an attractive alternative to DNS. In an LES, thelarge scales are directly resolved and the effect of the small scales orthe subgrid scales on the large scales are modeled. The large scales aregenerally much more energetic than the small ones and are directlyaffected by the boundary conditions. The small scales, however, areusually much weaker and they tend to have more or less a universalcharacter. Hence, it makes sense to directly simulate the more energeticlarge scales and model the effect of the small scales. LES methods arecapable of simulating flows at higher Reynolds numbers and manysuccessful LES computations for different types of flows have beenperformed to date. Since noise generation is an unsteady process, LESwill probably be the most powerful computational tool to be used in jetnoise research in the foreseeable future since it is the only way, otherthan DNS, to obtain time-accurate unsteady data.One of the first attempts in using LES as a tool for jet noise pre-

diction was carried out by Mankbadi et al. [25]. They employed afourth-order accurate numerical scheme to perform LES of a super-sonic jet flow to capture the time-dependent flow structure and appliedLighthill’s theory [20] to calculate the far field noise. The first applica-tion of Kirchhoff’s method [18] in combination with LES to far field jetnoise was conducted by Lyrintzis and Mankbadi [22]. LES has also beenused together with Kirchhoff’s method [18] for the noise prediction ofa Mach 2.0 jet by Gamet and Estivalezes [16] as well as for a Mach 1.2jet with Mach 0.2 coflow by Choi et al. [9] with encouraging results ob-tained in both studies. Zhao et al. [36] did LES for a Mach 0.9, Reynoldsnumber 3,600 jet obtaining mean flow results that compared well with

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LES of Turbulent Jets 3

Freund’s DNS and experimental data. They also studied the far fieldnoise from a Mach 0.4, Reynolds number 5,000 jet. They comparedKirchhoff’s method results with the directly computed sound and ob-served good agreement. Boersma and Lele [1] did LES for a Mach 0.9 jetat Reynolds numbers of 3,600 and 36,000 without any noise predictions.Bogey et al. [2] simulated a Reynolds number 65,000, Mach 0.9 jet usingLES and obtained very good mean flow results, turbulent intensitiesas well as sound levels and directivity. Constantinescu and Lele [11]did simulations for a Mach 0.9 jet at Reynolds numbers of 3,600 and72,000. They directly calculated the near field noise using LES. Theirmean flow parameters and turbulence statistics were in good agreementwith experimental data and results from other simulations. The peakof the power spectra of the sound waves was also captured accuratelyin their calculations. More recently, Bogey and Bailly’s LES [5], [7], [8]for a Mach 0.9 round jet at Reynolds number, ReD = 400, 000 using12.5 and 16.6 million grid points produced mean flow and sound fieldresults that are in good agreement with the experimental measurementsavailable in the literature.In general, the LES results in the literature to date are encouraging

and show the potential promise of LES application to jet noise predic-tion. Except for the recent studies of Bogey and Bailly [5], [7], [8], thehighest Reynolds numbers reached in the LES simulations so far arestill below those of practical interest. Well-resolved LES calculationsof jets at higher Reynolds numbers close enough to practical valuesof interest would be very helpful for analyzing the broad-band noisespectrum and looking into the mechanisms of noise generation at suchReynolds numbers.With this motivation behind the current study given, we now present

the two main objectives of this research:

1. Development and validation of a versatile 3-D LES code for turbu-lent jet simulations. A high-order accurate 3-D LES code utilizinga robust dynamic subgrid-scale (SGS) model has been developed.Generalized curvilinear coordinates are used, so the code can be eas-ily adapted for calculations in several applications with complicatedgeometries. Since the sound field is several orders of magnitudesmaller than the aerodynamic field, we make use of high-order ac-curate non-dissipative compact schemes which satisfy the strict re-quirements of CAA. Implicit spatial filtering is employed to get ridof the high-frequency oscillations resulting from unresolved scalesand mesh non-uniformities. Non-reflecting boundary conditions areimposed on the boundaries of the domain to let outgoing distur-bances exit the domain without spurious reflection. A sponge zone

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that is attached downstream of the physical domain damps outthe disturbances before they reach the outflow boundary. In ourrecent studies [32], we focused on this first objective. Previous sim-ulations were performed with the constant-coefficient SmagorinskySGS model. However, the results were found to be sensitive to thechoice of the Smagorinsky constant [33]. The current version of ourcode has the dynamic SGS model implemented. As will be shownin the results section, an LES with the dynamic SGS model fora compressible round jet produced mean flow results in excellentagreement with experimental observations.

2. Accurate prediction of the far field noise. Even though the sound isgenerated by a nonlinear process, the sound field itself is known tobe linear and irrotational. This implies that instead of solving thefull nonlinear flow equations out in the far field for sound propa-gation, one can use a cheaper method instead, such as Lighthill’sacoustic analogy [20] or surface integral acoustic methods such asKirchhoff’s method [18] and the Ffowcs Williams-Hawkings (FWH)equation [13], [12]. In this paper, we couple the near field data di-rectly provided by LES with the Ffowcs Williams-Hawkings methodfor computing noise propagation to the far field. Overall soundpressure levels in the far field as well acoustic pressure spectra arepresented.

2. Governing Equations

The governing equations for LES are obtained by applying a spa-tial filter to the Navier-Stokes equations in order to remove the smallscales. The effect of the subgrid-scales is computed using the dynamicSmagorinsky subgrid-scale model for compressible flows proposed byMoin et al. [26]. Since we are dealing with compressible jet flows,the Favre-filtered unsteady, compressible, non-dimensionalized Navier-Stokes equations formulated in curvilinear coordinates are solved inthis study using the numerical methods described in the next section.

3. Numerical Methods

We first transform a given non-uniformly spaced curvilinear computa-tional grid in physical space to a uniform grid in computational spaceand solve the discretized governing equations on the uniform grid.To compute the spatial derivatives at interior grid points away from

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the boundaries, we employ the following non-dissipative sixth-ordercompact scheme of Lele [19]

1

3f

′

i−1 + f′

i +1

3f

′

i+1 =7

9∆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 alongthe ξ direction, fi denotes the value of f at grid point i, and ∆ξ is theuniform grid spacing in the ξ direction. For the left boundary at i = 1and the right boundary at i = N , we apply the following third-orderone-sided compact scheme equations, respectively

f′

1 + 2f′

2 =1

2∆ξ(−5f1 + 4f2 + f3) (2)

f′

N + 2f′

N−1 =1

2∆ξ(5fN − 4fN−1 − fN−2) (3)

For the points at i = 2 and i = N − 1 next to the boundaries, weuse the following fourth-order central compact scheme formulations,respectively

1

4f

′

1 + f′

2 +1

4f

′

3 =3

4∆ξ(f3 − f1) (4)

1

4f

′

N−2 + f′

N−1 +1

4f

′

N =3

4∆ξ(fN − fN−2) (5)

Spatial filtering can be used as a means of suppressing unwantednumerical instabilities that can arise from the boundary conditions,unresolved scales and mesh non-uniformities. In our study, we consid-ered the following sixth-order tri-diagonal filter used by Visbal andGaitonde [34]

αf fi−1 + fi + αf fi+1 =3∑

n=0

an2(fi+n + fi−n) (6)

where

a0 =11

16+5αf

8, a1 =

15

32+17αf

16, a2 =

−3

16+3αf

8, a3 =

1

32−

αf

16(7)

The parameter αf must satisfy the inequality −0.5 < αf < 0.5. A lessdissipative filter is obtained with higher values of αf within the givenrange. With αf = 0.5, there is no filtering effect.Since this filter has a 7-point right-hand side stencil, it obviously

cannot be used at near-boundary points. Instead, the following sixth-order equation with a one-sided right-hand side stencil is used at grid

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Tam & Dong’s radiation boundary conditions

Tam & Dong’s radiation boundary conditions

Tam & Dong’soutflow boundaryconditions

Sponge zone

Tam &Dong’sradiationbcs

Vortex ring forcing

Figure 1. Schematic of the boundary conditions. Vorticity magnitude contours areshown within the jet while divergence of velocity contours are shown in the outerpart of the computational domain.

points near the left boundary point i = 1 [34]

αf fi−1 + fi + αf fi+1 =7∑

n=1

an,ifn i = 2, 3 (8)

where the coefficients, an,i are given in Reference [34].For the points near the right boundary point i = N , we apply a

similar formulation

αf fi−1 + fi + αf fi+1 =6∑

n=0

aN−n,ifN−n i = N − 2, N − 1 (9)

where

aN−n,i = an+1,N−i+1 i = N − 2, N − 1 n = 0, 6 (10)

and the coefficients, aN−n,i are also given in Reference [34]. The bound-ary points, i = 1 and i = N are left unfiltered.The standard fourth-order explicit Runge-Kutta scheme is used for

time advancement. We apply Tam and Dong’s 3-D radiation and out-flow boundary conditions [3] on the boundaries of the computational

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domain as illustrated in figure 1. We additionally use the sponge zonemethod [10] in which grid stretching and artificial damping are appliedto dissipate the vortices present in the flowfield before they hit theoutflow boundary. This way, unwanted reflections from the outflowboundary are suppressed. Since the actual nozzle geometry is not in-cluded in the present calculations, randomized velocity perturbationsin the form of a vortex ring are added into the jet shear layers at a shortdistance downstream of the inflow boundary in order to excite the 3-Dinstabilities in the jet and cause the potential core of the jet to breakup at a reasonable distance downstream of the inflow boundary. Thisforcing procedure has been adapted from Bogey et al. [6]. A vortex ringwhose radius is equal to the initial jet radius, ro is considered in thistype of forcing. The vortex ring has a mean axial and radial velocitydistribution which can be computed using the analytical expressionsgiven in Bogey et al. [6] but it has no azimuthal velocity. There are 10azimuthal modes involved in the forcing. The amplitude and phaseof each forcing mode are randomly picked within a specified rangeof values by using a random number generator. Combining the basicvortex ring’s mean velocity distribution with the 10 azimuthal forcingmodes produces randomized axial and radial velocity disturbances inthe form of a vortex ring which are added into the initial jet shearlayers. The reader is referred to Reference [6] for more details.For subgrid-scale modelling, both the standard constant-coefficient

Smagorinsky [29] as well as the dynamic Smagorinsky [26] models havebeen implemented into the LES code. The main idea in the dynamicmodel is to compute the model coefficients as functions of space andtime by making use of the information contained within the smallest re-solved scales of motion. For the dynamic model, a fifteen-point explicitfilter developed by Bogey and Bailly [4] is used as the test-filter. Theratio of the test-filter width to the grid spacing is taken as 2. The usualpractice in LES calculations is to average certain dynamically computedquantities over statistically homogeneous directions and then use theseaveraged quantities to compute the model coefficients. This is an ad

hoc procedure that is used to remove very sharp fluctuations in thedynamic model coefficients and to stabilize the model. Obviously, suchan approach is not useful for turbulent flows for which there is no homo-geneous direction. In our implementation, the dynamically computedmodel coefficients are locally averaged in space using a second-orderthree-point filter along every spatial direction in order to avoid thesharp fluctuations in the model coefficient. No negative model coef-ficients are allowed. The upper limit for the model coefficients is setto 0.5. This procedure works reasonably well for the jet flows we arestudying.

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4. LES for a Round Jet at a Reynolds Number of 100,000

The previous LES studies that we carried out can be found in Uzun [32].Here, we will present results from an LES performed with the dynamicSGS model for a turbulent isothermal round jet at a Mach number of0.9, and Reynolds number ReD = ρjUjDj/µj = 100, 000 where ρj , Uj ,µj are the jet centerline density, velocity and viscosity at the nozzle exit,respectively and Dj is the jet diameter. The jet centerline temperaturewas chosen the same as the ambient temperature.A fully curvilinear grid consisting of approximately 12 million grid

points was used in the simulation. The physical portion of the domainextended to 60ro in the streamwise direction and from −20ro to 20roin the transverse y and z directions, where ro = Dj/2 is the jet radius.The filtering parameter was set to αf = 0.47.An animation was created using data from a previous similar jet

simulation performed at the same Reynolds number. The animationshows the convection of vortices in the jet near field together withsound radiation from the jet and can be downloaded at the followingURL: http://roger.ecn.purdue.edu/∼uzun/jetnoise.aviIt has been observed by Bogey and Bailly [5] that the jet spreading

rate and the jet decay coefficient do not reach their asymptotic valuesif the computational domain size is kept relatively short. Furthermore,they found out that Reynolds stresses do not reach their self-similarstate either in a relatively short domain. Our aim is to fully validate thedynamic model, so it is important that the asymptotic jet spreadingrate and the centerline velocity decay rate as well as the self-similarstate of Reynolds stresses are reached in the simulation. This will fa-cilitate the comparison of our numerical results with the experimentaljet data at similar flow conditions. Hence we decided to choose a longdomain length of 60ro in the streamwise direction.In our simulation, the initial transients exited the domain over the

first 20,000 time steps. We then collected the flow statistics over 150,000time steps. This sampling period corresponds to a time length in whichan ambient sound wave travels about 60 times the domain length in thestreamwise direction. A total of about 123,000 CPU hours was neededfor this simulation on an IBM-SP machine using 160 Power3 processorsin parallel. It should be noted that the mean flow convection speedsin the far downstream region are relatively low, hence the turbulenceconvects slowly in the far downstream region resulting in slowly con-verging statistics. As a result, we had to collect data over a period of150,000 time steps to obtain reasonably converged statistics.In figure 2, the one-dimensional spectrum of the streamwise ve-

locity fluctuations at the x = 20ro location on the jet centerline is

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k1

Eu(1

)(k

1)

5 10 15 2010-7

10-6

10-5

10-4

10-3

10-2

10-1

100

k1-5/3

Grid cutoff

Figure 2. One-dimensional spectrum Eu(1)(k1) of the streamwise velocity fluctua-

tions at the x = 20ro location on the jet centerline.

shown. The temporal spectrum of the streamwise velocity fluctuationsat the given location was coupled with Taylor’s hypothesis of frozenturbulence to compute this one-dimensional spectrum. As expected, theone-dimensional spectrum becomes almost flat as the axial wavenumberapproaches zero. We also show the grid cutoff wavenumber correspond-ing to our grid resolution at the given axial location. The spectralenergy at the grid cutoff wavenumber is about 2 orders of magnitudesmaller than that at the lowest wavenumbers. Before the grid cutoffwavenumber, the spectrum exhibits a decay which is quite similar toKolmogorov’s −5/3 decay rate prediction in the inertial range of tur-bulence. This means that our grid resolution is fine enough to resolve aportion of the inertial range of wavenumbers at the given axial location.For code validation, we first compared our jet’s centerline velocity

decay rate in the far downstream region with the experiments of Zaman[35]. Zaman [35] did a series of experiments for initially compressiblejets and examined the asymptotic jet centerline velocity decay andentrainment rates. He found out that the asymptotic jet centerlinevelocity decay rate decreased with increasing initial jet Mach number.In figure 3, we plot the streamwise variation of the inverse of the meanjet centerline velocity normalized by the jet inflow velocity. In thefar downstream region, we see that the slope of the linear fit to the

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x / Dj

Uj/U

c(x)

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

Figure 3. Streamwise variation of Uj/Uc(x), where Uc(x) = mean jet centerlinevelocity and Uj = jet inflow velocity.

curve is 0.161. From the experiments of Zaman [35], the correspondingexperimental slope for a Mach 0.9 jet is about 0.155 which is very closeto our computed value. For incompressible jets, on the other hand,experimental values ranging from 0.165 to 0.185 have been previouslyreported in the literature.Furthermore, in figure 4, we show the streamwise variation of the

mass flux normalized by the mass flux through the jet nozzle. In the fardownstream region, we again see a linear growth. This time the slopeof the line is 0.267 showing very good agreement with the experimentalvalue of 0.26 from Zaman [35]. The jet spreading rate, which is theslope of the half-velocity radius growth in the far downstream regiondepicted in figure 5, has been found to be 0.092 in our simulation.The half-velocity radius, r1/2 at a given downstream location is definedas the radial location where the mean streamwise velocity is one halfof the jet mean centerline velocity at that location. To the best ofour knowledge, no experimental value has been reported from an ini-tially compressible jet in the literature for the jet spreading parameter.However, experimental values ranging from 0.086 to 0.096 have beenpreviously found for incompressible jets. Hence, our computed value iswithin the experimental range observed for incompressible jets.

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x / Dj

Q(x

)/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

Figure 4. Streamwise variation of the mass flux normalized by the mass flux at thenozzle.

slope= A = 0.092

experimental valuesof A : 0.086 - 0.096

x / ro

r 1/2(

x)/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

Figure 5. Streamwise variation of the half-velocity radius normalized by the jetradius.

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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 and Lumley

Figure 6. Normalized mean streamwise velocity profiles and comparison withexperimental data.

Properly scaled mean streamwise velocity profiles at three down-stream locations are plotted in figure 6. The self-similarity coordinateis taken as r/r1/2 where r is the radial location and r1/2 is the half-velocity radius. The vertical axis of the figure is normalized by themean jet centerline velocity. From this figure, it can be observed thatthe profiles of our jet at the three downstream locations collapse ontoeach other fairly well and exhibit self-similarity which is consistent withexperimental observations. Our results are also in very good agreementwith the experimental profiles of Hussein et al. [17] for an incompress-ible jet at ReD = 95, 500 as well as that of Panchapakesan and Lumley[28] for an incompressible jet at ReD = 11, 000. The convective Machnumber of the jet flow is low in the far downstream region (as implied byfigure 3), hence the compressibility effects are negligible there. There-fore, it is safe to compare our profiles in the far downstream regionwith incompressible experimental data.We also compared our Reynolds stress profiles with the experimental

profiles of Hussein et al. [17] and Panchapakesan and Lumley [28]. Thenormalized Reynolds stresses in cylindrical coordinates are defined asfollows

σxx =vx

′vx′

Uc2

σrr =vr

′vr′

Uc2

σθθ =vθ

′vθ′

Uc2

σrx =vr

′vx′

Uc2

(11)

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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


Figure 7. Normalized Reynolds stress σxx profiles and comparison with experimentaldata.

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


Figure 8. Normalized Reynolds stress σrr profiles and comparison with experimentaldata.

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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


Figure 9. Normalized Reynolds stress σθθ profiles and comparison with experimentaldata.

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


Figure 10. Normalized Reynolds shear stress σrx profiles and comparison withexperimental data.

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where vx′

, vr′

, vθ′

are the axial, radial and azimuthal components of thefluctuating velocity, respectively, Uc is the mean jet centerline velocityat a given axial location, and the overbar denotes time-averaging. Inorder to compare our Reynolds stresses with the experimental data, wetransformed our Reynolds stresses from the Cartesian coordinate sys-tem to the cylindrical coordinate system. The transformation processwas trivial and the details of it will be skipped here. Figures 7 through10 plot our properly scaled Reynolds stress profiles at three downstreamlocations and compare them with the two experiments. Again, the self-similarity coordinate is taken as r/r1/2. The vertical axis of the figuresis normalized by the square of the mean jet centerline velocity. Ascan be observed from the plots, the properly scaled profiles collapseonto each other exhibiting self-similarity. Overall, the agreement is verygood. It should be noted that our Reynolds stresses computed by LESare based on filtered velocities, while the actual experimental profilesare based on unfiltered velocities. Hence, some differences should beexpected between LES and experiments when the Reynolds stressesare compared. As can be seen from the figures, the Reynolds normalstress σxx profiles match the experiment of Panchapakesan and Lumley[28] better, while the other Reynolds stresses are in between the twoexperimental profiles. It should also be noted that the difference be-tween the initial conditions imposed in our simulation and the actualinitial conditions in experimental jets could be another possible reasonfor the differences between the LES and experimental profiles. We haveimposed randomized velocity fluctuations in the form of a vortex ring[6] in our simulation since the actual nozzle geometry was not includedin the calculations. Furthermore, it can be argued that the experimentalReynolds stress profiles have been measured in the very far downstreamregion, usually at distances of 100 jet radii or more. It has also beenobserved experimentally that the different Reynolds stress componentsreach asymptotic self-similarity at different downstream locations, de-pending on initial conditions. Since our domain length of 60 jet radii isstill relatively short compared to experiments, the normalized Reynoldsnormal stresses σxx may not have reached their true asymptotic values.If we did a new simulation using a domain longer than 60ro in thestreamwise direction, it is possible that the Reynolds normal stress σxx

profiles in the far downstream region might shift upwards and come inbetween the two experiments. However, such a simulation would requireeven more grid points and many more time steps to run since the veryslowly convecting mean flow in the far downstream region would causevery slowly converging flow statistics. Hence such a simulation wouldnot be very practical.

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To summarize, the mean flow properties computed using the dy-namic SGS model compare very well with the experimental data. Fur-thermore, the comparison of Reynolds stresses in the present simulationwith experiments is found to be reasonable. These findings provide uswith valuable evidence towards the validity of the numerical schemesas well as the dynamic subgrid-scale model used in our LES code.

5. Far Field Aeroacoustics

Next, we will look at some far field aeroacoustics results obtained bycoupling the near field LES data with a Ffowcs Williams-Hawkingscode which has the capability to work on any general control surfacegeometry.The Ffowcs Williams-Hawkings formulation for a stationary control

surface, S is given as follows

p′(~x, t) = p′T (~x, t) + p′L(~x, t) + p′Q(~x, t) (12)

where

4πp′T (~x, t) =

∫

S

[

ρoUn

r

]

ret

dS (13)

4πp′L(~x, t) =1

a◦

∫

S

[

Lr

r

]

ret

dS +

∫

S

[

Lr

r2

]

retdS (14)

andUi =

ρui

ρo(15)

Li = Pijnj + ρuiun (16)

(~x, t) are the observer coordinates and time, r is the distance from thesource (surface) to the observer, subscript o implies ambient conditions,superscript ′ implies disturbances (e.g. ρ = ρ′ + ρo), ρ is the density,ui is the velocity vector, a◦ is the ambient sound speed, and Pij is thecompressive stress tensor with the constant poδij subtracted. The dotover a variable indicates a time derivative, a subscript r or n indicates adot product of the vector with the unit vector in the radiation directionr or the unit vector in the surface normal direction n, respectively.The surface integrals are over the control surface S, and the subscriptret indicates evaluation of the integrands at the emission (retarded)time τ = t − r/a◦. The quadrupole noise pressure p′Q(~x, t) denotesthe quadrupoles outside the control surface and has been neglected inthe current study. A more detailed description of the Ffowcs Williams-Hawkings method can be found in Lyrintzis and Uzun [23] as well asin the review paper by Lyrintzis [24].

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Control Surface

Control Surface

Jet Flow

Figure 11. Schematic of the control surface surrounding the jet flow. Vorticitymagnitude contours are shown within the jet.

We put a control surface around our jet as illustrated in figure 11and we gathered flow field data on the control surface at every 10time steps over a period of 23,000 time steps during our LES run. Thetotal acoustic sampling period corresponds to a time scale in whichan ambient sound wave travels about 9.4 times the domain length inthe streamwise direction. Assuming at least 6 points per wavelengthare needed to accurately resolve an acoustic wave, we see that themaximum frequency resolved with our grid spacing around the controlsurface corresponds to a Strouhal number of approximately 1.0. TheNyquist frequency, on the other hand, which is the maximum frequencythat can be resolved with the time increment of our data sampling rate,corresponds to a Strouhal number of about 4.5. However, we choose themaximum frequency that is based on spatial resolution as our cutofffrequency.We computed the overall sound pressure levels along an arc of radius

60ro from the jet nozzle. The center of the arc is chosen as the jet nozzleexit and the angle θ is measured from the jet axis as illustrated in figure12. We applied the Ffowcs Williams-Hawkings method to compute theacoustic pressure signal at 36 equally spaced azimuthal points on a fullcircle at a given θ location on the arc. In order to confine spuriousspectral contributions to low frequencies, we multiplied the pressure

jsc-paper.tex; 5/08/2003; 17:03; p.17


x / ro

y/r

o

0 10 20-5

0

5

10

15

R

θ

Figure 12. Schematic showing the center of the arc and how the angle θ is measuredfrom the jet axis. Contours shown are for the same variables as those in figure 1.

history at every azimuthal location by a windowing function similarto that used by Freund [15]. We ran the windowed pressure historythrough a 2048 point Fast Fourier Transform, and converted to polarnotation to obtain the power spectral density at each frequency. Wethen averaged the acoustic pressure spectra over the equally spaced 36azimuthal points and finally integrated the averaged spectrum to com-pute the overall sound pressure level at the given θ location. Only theresolved range of frequencies in the LES was included while integratingthe acoustic pressure spectrum to obtain the overall sound pressurelevels.Figure 13 shows the overall sound pressure levels computed along

the arc and compares them with some experimental data. Table Isummarizes the Mach numbers and Reynolds numbers of the exper-iments that we are doing comparisons with. It should be noted thatthe experimental jets were cold jets, whereas our jet is an isothermaljet. We also plot the SAE ARP 876C database [30] prediction for anisothermal jet at the same operating conditions as ours. This databaseconsists of actual engine jet noise measurements and can be used topredict the overall sound pressure levels within a few dB at various jetoperating conditions. The same database was also used to predict theoverall sound pressure levels for a cold Mach 0.9 jet and the difference

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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)

Figure 13. Overall sound pressure levels at 60ro from the nozzle exit.

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

M ReD Reference

0.9 540,000 Mollo-Christensen et al. [27]

0.88 500,000 Lush [21]

0.9 3,600 Stromberg et al. [31]

0.9 100,000 Our 3-D LES

between the isothermal and cold jet predictions was found to be only0.3 dB.From the overall sound pressure level plot, we see that our jet is

louder than the experimental jets of Lush [21] and Stromberg et al.

[31] for all points along the arc. On the other hand, our prediction iswithin 1.5 dB of the measurements of Mollo-Christensen et al. [27]. Wealso see as much as 3 dB difference between our prediction and theSAE ARP 876C prediction at the downstream angles where θ < 40o.The missing higher frequencies in the LES are expected to increaseour OASPL predictions by a few more dB. The differences betweenthe current predictions and experimental measurements are believedto be related to the inflow forcing employed in the simulation. Since

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the actual nozzle geometry is not included in the present computations,randomized velocity fluctuations in the form of a vortex ring have beenimposed a short distance downstream of the inflow boundary in orderto trigger the instabilities in the flow. This vortex ring forcing is thesame type of forcing employed by Bogey and Bailly [6] in their LEScalculations. In a recent study, Bogey and Bailly [7] took a close lookat the effects of the inflow conditions on the jet flow and noise. Theyfound out that reducing the amplitude of the initial disturbances resultsin an increase in radiated noise. On the other hand, a thinner initialjet shear layer thickness was observed to cause increased noise levelsin the sideline direction and reduced noise levels in the downstreamdirection. The most important changes were obtained when the firstfour azimuthal modes of forcing were removed. In this case, the noiselevels were noticeably reduced. They concluded that further improve-ments are still needed to reduce the sideline noise levels which areoverestimated with respect to experimental data. The reader is referredto Reference [7] for more detailed information.Finally, we compare our acoustic pressure spectra at three obser-

vation locations with the recent results of Bogey and Bailly [5] for anisothermal round jet at a Reynolds number of 400,000. It should benoted here that Bogey and Bailly [5] computed the spectra in the nearfield of the jet only, using the data directly provided by LES. Thefirst observation point of Bogey and Bailly [5] is located inside of ourcontrol surface while the other two are located outside, hence we candirectly compute the near field spectra at the latter two points usingthe Ffowcs Williams-Hawkings method. On the other hand, in orderto make a comparison at their first near field observation point whichis located inside of our control surface, we first adjust our θ locationalong the far field arc at R = 60ro such that an acoustic ray drawnbetween the end of the potential core (x = 11ro) and the θ location onthe arc crosses their near field observation point. After we compute ourfar field spectrum at the corresponding θ location on the arc, we canthen transfer the far field spectrum to the near field observation pointusing the r−1 decay assumption of acoustic waves. In this assumption,the acoustic wave amplitudes drop by a factor of 2 for every doublingof the distance from the source region. The source region is assumedto be compact and taken at the end of the potential core of the jet.The acoustic pressure spectra are plotted as the sound pressure level

(SPL) versus the Strouhal number which is defined as St = fDj/Uj ,where f is the frequency, Dj and Uj are the jet nozzle diameter and thejet centerline velocity on the inflow boundary, respectively. It should bementioned here that since the computed spectra are noisy, the spectrathat are shown are polynomial fits to the actual computed spectra.

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St

SPL

(dB

/St)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 190

100

110

120

130

Bogey and Bailly’s spectrum at x = 29ro and r = 12ro

Our spectrum at x = 29ro and r = 12ro

Our spectrum at R = 60ro and θ = 25o

Figure 14. Acoustic pressure spectrum at R = 60ro, θ = 25o in the far field, acousticpressure spectrum at x = 29ro, r = 12ro in the near field and comparison with thecorresponding spectrum of Bogey and Bailly [5].

St

SPL

(dB

/St)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 190

100

110

120

130





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St

SPL

(dB

/St)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 190

100

110

120

130





Figure 14 plots the spectrum at the x = 29ro and r = 12ro locationover our resolved frequency range and compares it with the correspond-ing near field spectrum of Bogey and Bailly [5]. An acoustic ray thatoriginates from the end of the potential core and passes through thisnear field observation point hits the far field arc at around the θ = 25o

location. The far field spectrum at this θ location on the arc is alsoshown in the figure. The distance between the far field observationpoint and the end of the potential core is about 50.25ro, whereas thedistance between the near field point and the end of the potential coreis 21.63ro. The ratio of these two distances is about 2.32. A factor of2.32 means about 7.32 dB/St difference according to the r−1 decayassumption of acoustic waves. Hence we shift our far field spectrumcomputed at R = 60ro and θ = 25o upwards by 7.32 dB/St to get ournear field spectrum at the first observation point of Bogey and Bailly[5]. We see that our spectrum has a peak at around St = 0.25 whilethat of Bogey and Bailly [5] is at St = 0.3. The peak of our spectrumhas a sound pressure level (SPL) slightly higher than that of Bogeyand Bailly [5]. We also see that after the peak, our spectrum decayssomewhat faster than that of Bogey and Bailly [5]. Finally, from ourspectrum, it is observed that there is a drop of about 15 dB/St in the

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sound pressure level from the peak Strouhal number of 0.25 to the gridcutoff Strouhal number of 1.0.Figures 15 and 16 show our near field spectra at the other two ob-

servation points, respectively and make comparisons with the spectraobtained by Bogey and Bailly [5]. The acoustic rays that originatefrom the end of the potential core and pass through these two nearfield observation points hit the far field arc at the θ = 50o and θ = 80o

locations, respectively. We also show our far field spectra correspondingto these two near field observation points. From these figures, we seethat our near and far field spectra basically have the same shape. Ourspectra also become broad-band at these observation points demon-strating the fact that higher frequencies are becoming more dominant.It is also interesting to note that our spectra demonstrate a decay afterthey reach their peak whereas the spectra of Bogey and Bailly [5] stayrelatively flat after the peak.One possible reason for the differences between our spectra and those

of Bogey and Bailly [5] could be the different Reynolds numbers of thetwo simulations. It should also be reminded here that Bogey and Bailly[5] did not employ any SGS model in their LES. Instead, they treatedthe spatial filter used in their simulation as the SGS model. In a recentstudy, Bogey and Bailly [7] also looked at the effect of SGS modelling onthe computed jet noise. Their comparisons of acoustic pressure spectraobtained with and without an SGS model for a round jet at Reynoldsnumber 400,000 are very similar to our observations in this study. Thesefindings imply that the eddy viscosity associated with the SGS modelis somehow affecting the well-resolved length scales of motion therebyresulting in a reduction of the amplitudes of the well-resolved acousticwaves.There is almost 10 dB/St difference between our near and far field

spectra in figure 15. The distance between the far field point and theend of the potential core is about 53.60ro, whereas the distance betweenthe near field point and the end of the potential core is 17.49ro. Theratio of these two distances is about 3. A factor of 3 means about10 dB/St difference according to the r−1 decay assumption of acousticwaves. Similarly, the difference between our near and far field spectrain figure 16 is about 12 dB/St. In this case, the ratio of the far fieldobservation point distance to the near field observation point distanceis about 4 which also translates into 12 dB/St difference according tothe r−1 decay assumption.

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6. Concluding Remarks

As part of a jet noise prediction methodology, we have developed andtested a 3-D Large Eddy Simulation code. High-order accurate compactfinite difference schemes were found to be very efficient and usefulin the computation of turbulent jet flows. Furthermore, the localizeddynamic model was found to perform very well. The mean flow resultsobtained in our simulation for a Reynolds number 100,000 jet werein excellent agreement with experimental data. We also coupled thenear field data computed by our LES code with the Ffowcs Williams-Hawkings (FWH) formulation to compute the far field noise of the jet.Far-field aeroacoustics results were encouraging and there is room forfurther improvement of the inflow conditions to get better agreementwith experimental measurements. The effect of the SGS model on thecomputed noise also deserves further investigation.

Acknowledgements

This work is sponsored by the Indiana 21st Century Research & Tech-nology Fund. It was also partially supported by the National Compu-tational Science Alliance under the grant CTS010032N, and utilizedthe SGI Origin 2000 computer systems at the University of Illinoisat Urbana-Champaign. The simulation presented in this paper wasperformed on the IBM-SP research computer of Indiana University. Weare grateful to Dr. Christophe Bogey for his help while we implementedthe vortex ring forcing method and the dynamic subgrid-scale modelinto our code. We also thank him for providing us with data from hisReynolds number 400,000 jet simulation.

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Address for Offprints:

Ali UzunSchool of Aeronautics and AstronauticsPurdue University315 N. Grant StreetWest Lafayette, Indiana 47907-2023USAe-mail: [email protected]: +1 765 494 5144Fax: +1 765 494 0307

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