Investigation of Noise Sources in Turbulent Hot Jets Using ...lyrintzi/Reno07Charlie.pdfThese high...

45th AIAA Aerospace Sciences Meeting and Exhibit, January 8-11, 2007, Reno, NV., USA.

Investigation of Noise Sources in Turbulent Hot Jets usingLarge Eddy Simulation Data

Phoi-Tack Lew∗, Gregory A. Blaisdell†, and Anastasios S. Lyrintzis‡

School of Aeronautics and AstronauticsPurdue University

West Lafayette, IN 47907

Through the use of Lighthill’s acoustic analogy, the aim of this paper is to investigate the noise sourcesof turbulent heated round jets using previously simulated Large Eddy Simulation (LES) data. Two heatedand one unheated jet are considered to study the effects of heating on the noise source contributions to thefar-field. Firstly, the computed overall sound pressure level (OASPL) and spectra are in good agreement withthe prediction obtained from the porous Ffowcs Williams-Hawkings (FWH) surface integral method. Likethe FWH prediction, however, the computed OASPL over-predicts the experiments by approximately 3dBbut the trends agree reasonably well with the experimental results. Through decomposition of the Lighthillsource term we obtain such sources as shear, self and entropy noise. An important finding is that when a highspeed subsonic compressible jet is heated while keeping the ambient jet Mach number constant, significantcancellations occur in the far-field between the shear and entropy noise. In addition, heating a jet reduces theintensity of the nonlinear self noise terms compared to an unheated jet. For a low speed heated jet, the maincontributing source is the entropy noise source while the shear and self noise sources hardly contribute to thefar-field noise.

Nomenclature

Roman Symbolsa j Jet centerline speed of sounda∞ Ambient speed of soundD j Jet diameterM j Mach number = U j/a j

M∞ Acoustic Mach number = U j/a∞OASPL Overall sound pressure levelp pressurero Initial jet radiusR Radial arc length from centerline jet exitReD Jet Reynolds number = ρ jU jD j/µ j

S r Strouhal number = f D j/U j

T j Jet centerline temperatureTi j Lighthill stress tensor, total noiseT l

i j Shear noiseT n

i j Self noiseT s

i j Entropy noiseT∞ Ambient temperatureU j Jet streamwise centerline velocity

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

Copyright c© 2007 by P. Lew, G. A. Blaisdell and A. S. Lyrintzis. All other rights are reserved by the copyright owner.

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American Institute of Aeronautics and Astronautics Paper 2007-0016

45th AIAA Aerospace Sciences Meeting and Exhibit8 - 11 January 2007, Reno, Nevada

AIAA 2007-16

Copyright © 2007 by P. Lew, G. A. Blaisdell and A. S. Lyrintzis. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

W Witze variablex, y, z Cartesian coordinatesxc/ro Potential core length

Greek Symbolsα Vortex ring forcing amplitudeα f Filtering parameter∆t Time resolutionΘ Observer angle relative to centerline jet axisκ Witze coefficientµ j Jet centerline viscosityρ j Jet centerline densityρ∞ Ambient densityσi j Viscous stress tensor

Subscripts∞ Ambient conditionj Jet exit condition

Superscripts(·)′ Fluctuating component(·) Time averaged value

I. Introduction

T is a current need to better understand the noise generation mechanisms in turbulent sub-sonic jets. This isbecause during the past several years, airports locally and abroad have implemented strict regulations on aircraft

with high jet noise emissions. These high jet noise levels can be unbearable and detrimental to the communitiessurrounding the airport. Hence, a goal was introduced by NASA in 1997 aimed at eliminating community noiseproblems near airports. The goal is to reduce the perceived noise levels of future aircraft by a factor of two (10EPNdB) from subsonic aircraft by 2007, and by a factor of four (20 EPNdB) by 2022.1 This goal is not infeasible, butit is challenging nonetheless due to the fact that the underlying mechanisms/sources that cause jet noise are still notwell understood and, therefore, cannot be fully controlled or optimized. Thus, the jet noise problem still remains oneof the most elusive problems in aeroacoustics.

With the advent of fast supercomputers, the application of direct numerical simulation (DNS) to jet noise predictionis becoming more feasible.2, 3 DNS solves for the dynamics of all the relevant length scales of turbulence and thusno form of turbulence modeling is used. Unfortunately, due to the wide range of time and length scales presentin turbulent flows and because of the limitations of current computational resources, DNS is still restricted to lowReynolds number flows. In contrast to DNS, large eddy simulation (LES), which computes the large scales directlyand models the small scales or the subgrid scales, yields a cheaper alternative to DNS. It is assumed that the largescales in turbulence are generally more energetic compared to the small scales and are affected by the boundaryconditions directly. In contrast, the small scales are more dissipative, weaker, and tend to be more universal in nature.Furthermore, most turbulent jet flows that occur in experimental or industrial settings are at high Reynolds numbers,usually greater than 100,000. With this idea in mind it is more appropriate to use LES as a tool for jet noise prediction,since it is capable of simulating high Reynolds number flows at a fraction of the cost of DNS. One of the first uses ofLES as an investigative tool for jet noise prediction was carried out by Mankbadi et al.4 They performed a simulationof a low Reynolds number supersonic jet and applied Lighthill’s analogy5 to calculate the far-field noise. Lyrintzisand Mankbadi6 were the first to use Kirchhoff’s method with LES to compute the far-field noise. A string of othernumerical calculations (e.g. References 7, 8, 9, 10, 11, 12, 13, 14, 15) were then carried out by investigators at higherReynolds numbers and were also found to be in good agreement with experimental results.

From a practical standpoint, it is desirable to study hot jets closely since jets fitted on all aircraft operate at hotexhaust conditions and at high Reynolds numbers. However, most of the current LES jet studies that have been carriedout to date consists of either cold or isothermal jets. References [7,10,11,13,14,15] cited in the previous paragraph aresuch examples. Only recently have LES simulations for hot jets been studied and compared to experiments. Bodony& Lele,16 for example, performed two LES simulations with different hot jet temperature ratios but at low Reynolds

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numbers of ReD = 13, 000 and ReD = 27, 000. Their results were consistent with the experimental observationsof Tanna17 and Bridges & Wernet.18 However, they found some discrepancies in their overall sound pressure level(OASPL) results due their limited grid resolution. Another example is the numerical simulation of Andersson et al.19

where they studied a M j = 0.75, ReD = 50, 000 hot jet and the results obtained were in good agreement with theexperimental data of Jordan et al. 20, 21 In addition, Shur et al.14, 15 studied several heated jets ranging from subsonicto supersonic Mach number and obtained excellent agreement with the experiments of Viswanathan22 and Tanna.17

Finally, Lew et al.23 performed two high Reynolds numbers (Re > 105) hot jet LES simulations and obtained goodturbulent flow results and satisfactory far-field noise predictions when compared to the hot jet experiments of Tanna17

and Viswanathan.22 Hence, the use of LES as an investigative tool to study the far-field noise for sub-sonic turbulenthot jets is certainly promising.

The heated jet LES examples mentioned above deal only with predicting the far-field sound (via integral acous-tic methodologies) and not the root cause of it, i.e. the noise generating sources. In the literature, however, theinvestigation of noise sources in unheated jets using DNS and LES data itself is quite recent. Within the frame-work of Lighthill’s acoustic analogy,5, 24 Freund25 was the first to investigate noise sources in a low Reynolds numberReD = 3, 600, M j = 0.9 cold jet using his DNS data. One conclusion that Freund reported was that the contributionsfrom the shear, self and entropic noise sources are highly correlated at small angles to the jet axis, and not statisticallyindependent, as often assumed. However, Freund also noted that better parametrization of Reynolds-number effects onthe probable noise sources needed to be carried out since his DNS jet was set in the low-Reynolds-number limit. It is inthis regard that LES offers an attractive alternative whereby the jet can be simulated at higher Reynolds numbers with afraction of the cost compared to DNS, as mentioned earlier. Hence, in the realm of LES, Uzun et al.11 were the first touse LES data coupled with Lighthill’s acoustic analogy to investigate the noise sources in a M j = 0.9, ReD = 400, 000isothermal jet. In their investigation, they found that significant cancellations occur among the noise generated by theindividual components of the Lighthill stress tensor for a high Reynolds number isothermal jet. In addition, Bogey& Bailly26 used causality methods to study noise sources of several unheated jets with different Reynolds numbers.The investigation of noise sources in turbulent hot jets using LES data, however, is fairly limited. To the best of ourknowledge, the only use of LES data to study noise sources in turbulent heated jets is that of Bodony & Lele.27 Again,through the use of Lighthill’s acoustic analogy, their results indicate that when compared to an unheated jet, significantphase cancellation exists between the momentum and entropy sources in the near-field and that additional cancellationoccurs in the far-field for jets with Mach numbers within the range of 0.9 < M∞ < 1.95. Bodony & Lele then suggestthat the significant cancellations between these two sources in the far-field is a probable explanation as to why a heatedjet is quieter at some observer angles when compared to an unheated jet for the aforementioned jet Mach numbers.However, it must be noted that Bodony & Lele’s27 heated jets were run at rather high Mach numbers of M∞ ≥ 0.9.

With that in mind, our aim is to investigate the noise sources of two turbulent heated, high Reynolds numberjets (ReD > 105) using LES data. This work is a continuation of an earlier study of hot jets via LES by the sameauthors.23 The noise source investigation is conducted within the framework of Lighthill’s acoustic analogy.5, 24 Directcomparisons of hot and cold jet noise sources will be performed. In addition, we also use the acoustic analogy topredict the far-field sound and compare it to the far-field noise prediction based on the Ffowcs Williams-Hawkings(FWH) surface integral method performed in an earlier study.23 Hence, this paper is arranged as follows: Section IIprovides a brief description of the LES methodology used in this study; section III summarizes the grid setup and flowconditions used; sections IV and V summarize several turbulent flow physics and far-field noise results, respectively.Section VI briefly discusses the formulation and computational procedure and proceeds to the results of Lighthill’sacoustic analogy. Finally, some concluding remarks are given in section VII.

II. Brief Description of LES Methodology

The 3-D LES code used in this study was developed by Uzun et al.11, 12 and it uses either the classical28 or alocalized dynamic29 Smagorinsky (DSM) subgrid-scale model. However, the modeling of the subgrid-scale stresstensor still raises some fundamental issues as discussed by Bogey & Bailly30, 31 and Uzun et al.32 Eddy-viscositymodelings such as the classical Smagorinsky subgrid-scale model28 and the localized dynamic subgrid-scale model(DSM)33, 29 might dissipate the turbulent energy through a wide range of scales up to the larger ones, which should bedissipation free at sufficiently high Reynolds numbers.34 In addition, since the eddy-viscosity has the same functionalform as the molecular viscosity the effective Reynolds number is reduced in the simulated flows.35 See References[30] and [32] for a thorough analysis and discussion on the shortcomings of eddy viscosity subgrid-scale model onjet flows. An alternative to the use of an explicit eddy-viscosity model is the use of spatial filtering for modeling theeffects of the subgrid-scales. Using this alternative, the turbulent energy is only dissipated when it is transferred from

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the larger scales to the smaller scales discretized by the mesh grid.30 Hence, for the hot jets simulated here, we usespatial filtering as an implicit subgrid-scale model.36

Since we have a near sonic jet, the unsteady, Favre-filtered, compressible, non-dimensional LES equations aresolved. We transform from curvilinear coordinates to a uniform grid in computational space. The code uses thenon-dissipative sixth-order compact scheme developed by Lele37 to compute the solution on the internal points. Forthe points on the boundaries, however, a third-order one-sided compact scheme is used, and the points next to theboundaries are computed by a fourth-order compact central differencing technique. In order to eliminate numericalinstabilities that can arise from the boundary conditions, grid non-uniformities and unresolved scales the sixth-ordertri-diagonal spatial filter proposed by Visbal and Gaitonde38, 39 is employed with the filter parameter set to α f = 0.47.For time advancement, the explicit fourth-order Runge-Kutta scheme is used. Tam and Dong’s 3-D radiation andoutflow boundary conditions40 are implemented on the boundaries. In addition, a sponge zone41 is attached to the endof the computational domain to dissipate the vortices present in the flow field before they hit the outflow boundary.This is done so that unwanted reflections from the outflow boundary are suppressed. Figure 1(a) shows a schematic ofthe boundary conditions used in the 3-D LES code. A more in-depth discussion on the numerical methods used canbe found in Uzun.42

To excite the mean flow, randomized perturbations in the form of induced velocities from a vortex ring proposedby Bogey & Bailly43 are added to the velocity profile at a short distance (approximately one jet radius) downstreamfrom the inflow boundary (see Figure 1). This is done to ensure the break up of the potential core within a reasonabledistance. Studies regarding the effect of this inflow forcing on jet noise can be found in Lew et al.44 and Bogey &Bailly.13

III. Grid Setup and Test Cases

Table 1 summarizes the parameters for the heated and unheated jet test cases that are considered. The test cases areappropriately named according to the experimental test matrix of Tanna et al.45 Test case SP07 closely correspondsto Tanna’s set point SP07 which is a M j = 0.9 jet with a temperature ratio of T j/T∞ = 0.86 which is an unheated jet.Hence, we would like to see the effects of heating the jet, while keeping the ambient Mach number fixed. Hence, testcase SP46 is similar to SP07 in terms of the ambient Mach number but it is now heated with a temperature ratio ofT j/T∞ = 2.7. The case SP23 was chosen to examine jet flow physics and the far-field noise at a lower Mach numberwith the addition of heating. Furthermore, there are available LES16, 46 and experimental data45, 18, 47 for these testcases in the literature for us to compare with. Here, M j = U j/a j is the jet Mach number where U j is the centerline jet

Table 1. List of test cases. All physical domains correspond to (x, y, z) = (60ro,±20ro,±20ro).

Test Case M j M∞ Nx × Ny × Nz ReD T j/T∞SP07 0.90 0.90 292 × 128 × 128 100,000 1.00SP46 0.55 0.90 292 × 128 × 128 200,000 2.70SP23 0.38 0.50 292 × 128 × 128 223,000 1.76

velocity and a j is the centerline jet speed of sound. M∞ = U j/a∞ is the acoustic Mach number where a∞ is the ambientspeed of sound. Nx, Ny and Nz simply correspond to the number of grid points in the x, y and z directions respectively.Each test case has a total of approximately 4.8 million grid points. Figure 2 shows the x − y cross sectional planeof the computational domain. The physical part of the computational domain extends to approximately 60ro in thestreamwise direction and −20ro to 20ro in the transverse y and z directions. Beyond the streamwise location of 60ro

is the sponge zone. The physical domain length of 60ro was chosen for two reasons. Firstly, Uzun et al.42 reportedthat the Reynolds stresses achieve their full asymptotic self-similar state if a domain length of at least 45ro is used.Secondly, in order to capture the Overall Sound Pressure Levels (OASPL) adequately at shallow angles, a domainlength of at least 55ro is required based on the recommendations of Uzun et al.42 and Shur et al.14 Based on theminimum grid spacing and ambient Mach number, the time resolution was determined to be ∆t = 0.015 ro/a∞ forcases SP07 and SP46, whereas case SP23 was set to ∆t = 0.011 ro/a∞, respectively.

The Reynolds number is defined as ReD = ρ jU jD j/µ j where ρ j, U j and µ j are the jet centerline density, velocityand viscosity at the inflow, respectively. D j = 2ro is the jet diameter. The Reynolds numbers specified above for bothjets correspond to the experimental conditions of Tanna et al.45 In the previous section, we mentioned that a vortexring is used to excite the mean flow. The vortex ring used here contains a total of 16 azimuthal jet modes of forcing.

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Bogey and Bailly13 performed a simulation with all modes present and later removed the first four modes and foundthat the jet was quieter with the latter case and matched experimental results better. Hence, based on their results, thefirst four azimuthal modes of forcing are not included in the forcing. The forcing amplitude is set to α = 0.007.

We consider a hyperbolic tangent velocity profile used by Freund48 on the inflow boundary given by

u(r) =12

U j

[1 − tanh

[b(

rro− ro

r

)]], (1)

where r =√y2 + z2, ro = 1, and U j is the jet centerline velocity. The parameter that controls the thickness of the shear

layer is b. In our code we have set this parameter to b = 2.8. A higher value of b implies a thinner shear layer. As acomparison, Freund48 used a value of b = 12.5 for his 3-D jet DNS. Hence, we have a thicker shear layer compared toFreund’s jet. For laboratory jets however, the measured value of b is usally an order of magnitude or higher comparedto that used in LES and DNS of jets. For case SP46, there are approximately 10 grid points in the initial jet shear layer.The inlet density profile is also adopted from Freund48

ρ(r) = (ρ j − ρ∞)u(r)U j

+ ρ∞, (2)

where u(r)/U j is the mean streamwise velocity on the inflow boundary normalized by the jet centerline velocity, ρ j isthe density at the jet centerline and ρ∞ is the freestream density. The ratio ρ∞/ρ j determines whether or not the jet ishot or cold. A value lower than unity implies a cold jet, whereas a value greater than unity implies a hot jet. The nexttwo sections give results for both jet development and noise calculations for each test case.

IV. Summary of Turbulent Flow Results

This section will briefly discuss the results for the jet cases mentioned above. Further details can be found inReference [23]. Figures 3 through 5 show the mean streamwise velocity decay for jet test cases SP07, SP46 and SP23,respectively. We adopt the procedure used by Bodony & Lele16 whereby the axial coordinate, i.e. x/ro, is shifted to aidin the presentation of near-field data over a range of operating conditions so that differences in compressibility or Machnumber which affect the length of the potential core can be accommodated.16, 47, 49 The procedure adopted by Bodony& Lele is called the Witze49 correlation and is given by W = κ(x − xc)/ro where κ = 0.08(1 − 0.16M j)(ρ∞/ρ j)0.22.Thus xc/ro is computed first and then x/ro is shifted axially. Then the data is re-scaled using the factor κ. Here, xc/ro

is defined when the jet centerline velocity reduces to 95% of the inflow jet velocity, Uc(xc) = 0.95U j. For Figure 3,we use Uzun et al.50 results. This was done since we did not have statistically converged results for our unheated jetSP07 using the current number of grid points, i.e. 4.9 million. Uzun’s SP07 velocity decay results has approximately12 millions grid points and compares very well with the experimental results of Brideges & Wernet.18 In Bridges &Wernet’s18 technical report, they mention that the data for the mean streamwise velocity decay along the centerline forSP46 shows some problems beyond x/ro = 20 or W & 1. They were not able to find an explanation for this behavior.Hence, the good collapse of our data in Figure 2(a) from W = 1 onwards is only fortuitous. Nevertheless, there isnearly good agreement within the range of 0 . W . 1. We also note that Bodony & Lele’s SP46 jet decays fastercompared to Bridges & Wernet’s data. There is also good agreement between our SP23 jet and the experimental resultsof Bridges & Wernet and Jordan et al. 20, 21 though our jet decays slightly faster. Bodony & Lele’s SP23 jet decays thefastest when compared to experiments. As a note, we could not find velocity centerline decay data for Tanna’s hot jetexperiments.

In addition, we also computed the mean centerline velocity decay rates, half-velocity growth rates, potential corelengths, mass flux growth rates and mean streamwise turbulence intensities of our heated jets. The results obtainedwere is good agreement with the experimental correlations of Zaman.51 In brief, we found that for the heated jetcase of M∞ = 0.9, the jet grows faster which was also observed in the experimental correlations of Zaman and fromthe experimental data of Bridges & Wernet. Again, the reader is referred to Reference [23] for further results anddiscussion for the heated and unheated jets.

V. Summary of Far-Field Aeroacoustic Results using the Ffowcs Williams-HawkingsMethodology

This section will only give a brief discussion on the far-field noise results for our heated jets. Likewise, furtheracoustic results can be found in Reference [23]. The porous Ffowcs Williams-Hawkings52, 53 (FWH) surface integral

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acoustic method is used to study the far-field noise of our hot jets. The integral method follows the description ofLyrintzis & Uzun54 and Lyrintzis.55 For simplicity, a continuous stationary control surface around the turbulent jetis used. For details regarding the numerical implementation of the Ffowcs Williams-Hawkings method, the readeris referred to Uzun.42 As a note, all data here is presented alongside the Lighthill computations. The far-field noiseprediction using Lighthill’s acoustic analogy is discussed in detail in the next section.

Due to the nature of our curvilinear grid, the control surface is shaped as in Figure 6. The control surface startsabout one jet radii downstream and is situated at approximately 7.5r0 above and below the jet at the inflow boundary inthe y and z directions. It extends streamwise until the end near of the physical domain at which point the cross streamextent of the control surface is approximately 30ro. Hence, the total streamwise length of the control surface is 59ro.We show results for an open control surface. A open control surface here is defined where there is no surface at theend of the physical domain, i.e. x = 60ro. Based on our grid resolution around our control surface and assuming thatwith our numerical method 6 points per wavelength are needed to accurately resolve an acoustic wave,42 the maximumfrequency resolved corresponds to a Strouhal number of S r ' 1 for both test cases SP07 and SP46 and S r ' 1.6 fortest case SP23, where the Strouhal number is defined as S r = f D j/U j. The overall sound pressure levels are computedalong an arc with a distance of R = 144ro from the jet nozzle exit. This arc length corresponds to the distance used byTanna et al.45 in their experiments. The angle, Θ, however, is measured relative to the centerline jet downstream axis.

Figure 7 shows the overall sound pressure levels (OASPL) for SP07 LES and experimental data. Please note thatall experimental and LES data have been scaled to a common distance of R = 144ro (using a 1/R scaling). In additionto the experimental data shown, we have also included the SAE ARP 876C56 database prediction for a jet operating atsimilar conditions as ours, i.e. SP07. This database prediction consists of actual engine jet noise measurements andcan be used to predict overall sound pressure levels within a few dB at different jet operating conditions. As we cansee the prediction agrees well with the experimental results of Tanna et al.45 From the LES results, test case SP07compares well with the experimental results of Tanna and the SAE prediction within the range of 50o . Θ . 90o.Below that range the LES over predicts the OASPL values. Our data also compares rather well with the acousticresults of Bodony & Lele.16

Figure 8 shows the OASPL plot for heated jet test case SP46 computed at R = 144ro. We also included Tanna’sand Viswanathan’s22 experimental data as well the SAE ARP876C prediction. In terms of the shape of the OASPLcurve, we are in good agreement with the experimental results of Tanna and Viswanathan though approximately 3 dBhigher. The peak radiation angle reported by Tanna is located at Θ = 30o, whereas ours is located at approximately32.5o. Bodony’s OASPL prediction is slightly higher than ours, but also follows the trend predicted by Tanna. TheSAE ARP876C prediction is able to predict the values reported by Tanna very well, as shown. Hence, overall, ourpredicted OASPL are in good agreement with the experimental data. Figure 9, on the other hand, shows the OASPLdata for test case SP23 and is compared to the experimental data of Tanna et al. Again, on average we over-predict thesound levels by approximately 3 dB when compared to the results of Tanna. But nonetheless, our predicted OASPLfollows the trend measured by Tanna et al.45 The predicted values from the SAE ARP 876C show good agreementwith the measured data from Tanna as well. The computed spectra using the FWH method are discussed alongside theLighthill results in the next section.

VI. Computation of Noise Sources via Lighthill’s Acoustic Analogy

It is evident in the previous section that the predicted noise using the porous Ffowcs Williams-Hawkings surfaceintegral acoustic method at least captures the directivity pattern in the far-field, albeit at higher noise levels than in theexperiments. One possible explanation of this over-prediction could come from the artificial vortex ring forcing usedto excite the mean flow. Nevertheless, the satisfactory comparison warrants an investigation into the noise generationmechanisms of the two turbulent heated jets. To accomplish this, we use Lighthill’s5, 24 acoustic analogy to computeand study the noise sources within our turbulent heated jets. Hence, this section is devoted to the noise source compu-tation via the acoustic analogy and can be viewed as a continuation of a study of heated jets via LES. A formulationalbeit brief is given in the next subsection followed by results.

A. Brief Formulation

We begin by considering Lighthill’s5 equation which is written as

∂2ρ′

∂t2 − a2∞

∂2ρ′

∂x j∂x j=

∂2Ti j

∂xi∂x j, (3)

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where the Lighthill stress tensor is given by

Ti j = ρuiu j + (p − a2∞ρ)δi j − σi j. (4)

Here, a∞ is the ambient speed of sound, σi j is the viscous stress tensor and ρ′ is the fluctuation density. For thisstudy the viscous stress term is neglected. It is important to note that the double divergence of Ti j on the right handside of Equation (3) serves only as a nominal acoustic source term, and in no way should it be interpreted as a trueacoustic source. In Lighthill’s original formulation of his equation, all effects aside from propagation in a homogenousstationary medium, such as refraction, are lumped into the right hand side. More specifically, the acoustic analogyassumes that the sound generated by the turbulent flow is equivalent to what a quadrupole distribution Ti j per unitvolume would emit if placed in an acoustic medium at rest. Hence, the quadrupole noise sources replace the actualfluid flow. It is also understood that most of Ti j,i j does not radiate into the far-field.25 However, what the right handside of Equation (3) does provide is an exact connection between the near field turbulence and the far-field noise.

Following the standard Reynolds decomposition employed by Freund,25 the Lighthill stress tensor, Ti j, can be splitinto

Ti j = T mi j + T l

i j + T ni j + T s

i j, (5)

whereby each of the individual components are given as,

T mi j = ρuiu j + (p − a2

∞ρ)δi j, (6)

T li j = ρuiu′j + ρu ju′i , (7)

T ni j = ρu′iu

′j, (8)

T si j = (p′ − a2

∞ρ′)δi j. (9)

Here, T mi j is the mean component which, by definition, does not make noise. T l

i j is a component that is linear in velocityfluctuations and is called the shear noise, since this component consists of turbulent fluctuations interacting with thesheared mean flow. T n

i j is a component that is quadratic in velocity fluctuations and is called self noise, since thiscomponent involves the turbulent fluctuations interacting with themselves. Finally, T s

i j is the entropy component andshows the degree to which the pressure and density deviate from the isentropic relation in the turbulent flow.

To compute the far-field sound, Lighthill assumed that the source generating mechanism is compact and in aunbounded flow coupled with the free-space Green’s function, the far-field pressure fluctuations can be computed by

p − p∞ = (ρ − ρ∞)a2∞ ≈

14π

∫∫∫

V

(xi − yi)(x j − y j)|x − y|3

1a2∞

∂2

∂t2 Ti j

(y, t − |x − y|

a∞

)dy, (10)

where x and y are the observer and source locations, respectively.

B. Setup and Computational Details

The noise sources from all three jets including the isothermal case were computed in this study. The shape of theintegration volume is similar to that of the FWH surface shown in Figure 6 with the exception that it is smaller insize in the lateral direction. The crosswise extent of the integration volume is roughly 7ro and opens up to 22ro. Thiscrosswise length was chosen since Uzun et al.11 showed that the majority of the noise sources for a high speed jetare confined within a crosswise extent of roughly 5 to 6 jet radii along the entire streamwise domain. The streamwiselength of the integration volume is 59ro, and this plays a crucial role in the ability to capture the effective quadrupolenoise sources in the computational domain. When Uzun et al.11 used a domain length of 32ro for their Lighthillcomputations, they reported spurious noise levels in their OASPL directivity for observer angles Θ > 80o (Θ measuredrelative to jet centerline downstream axis). They suggest that the sudden truncation of the domain creates spuriousdipole noise sources as the quadrupole sources pass through the down stream surface. Bodony and Lele27 used adomain length of approximately 55ro for their Lighthill analysis and reported no spurious noise levels in their OASPLdirectivity.

For each test case, the five primitive variables, q = [ρ,u, p]T , were saved every ten time steps over a duration of40,000 time steps during the simulation. This resulted in roughly 430 GB of data saved in double precision unfor-matted. Due to this large data size, a total of 1,140 processors was used to compute Lighthill’s volume integral witha total run time of 5 hours on the Lemieux supercomputer at the Pittsburgh Supercomputing Center (PSC). Based onthe spatial grid resolution of the Lighthill control volume, the highest resolvable Strouhal number for test cases SP07

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and SP46 is 2 whereas for test case SP23 it is 2.7. Similar to the FWH methodology, the far-field sound is calculatedalong an arc with a radial distance of R = 144ro with the observer angle, Θ, measured relative to the jet centerlinedownstream axis. Please refer to Reference 11 for details regarding the numerical methods used for the Lighthillanalysis.

C. Results

Referring to Figure 7 once more, we see that the far-field noise predicted by the acoustic analogy for jet SP07 is ingood agreement with the experimental results of Viswanathan and Tanna. In addition, the comparison between FWHand Lighthill is also very good. However, the Lighthill prediction seems to give a better prediction, i.e. closer toexperiments, for observation angles, 20o < Θ < 40o, compared to the FWH results. It is also important to note thatwe do not see any spurious noise levels for observer angles Θ > 80o as was reported by Uzun et al.11 due to a longerintegration domain used in this study. Figure 8 shows the OASPL directivity for the first heated jet SP46. Again, herewe note the reasonable comparison between our Lighthill results and the experiments of Tanna45 and Viswanathan.22

Like the FWH results, the Lighthill computations over predict the laboratory results by approximately 3 dB. Thepossible spurious noise (as discussed in section IV B) is not substantial and the overall results are acceptable. Finally,Figure 9 shows the OASPL values for the second heated jet test case SP23. Trend wise, the computed Lighthill resultsagree well with laboratory experiments of Tanna but over predict again by approximately 3 dB. Note that at the peakradiation level at Θ ' 30o, the Lighthill results are slightly lower compared to the FWH prediction and are closer toexperiments. Thus, it seems so far that for all three jets considered here, Lighthill’s acoustic analogy does a slightlybetter job in predicting the far-field noise at the peak radiation angle compared to the FWH method.

Next we look at the individual noise source components of the Lighthill stress tensor. Figure 10 shows the OASPLcontribution from Ti j and its individual components T l

i j, T ni j and T s

i j to the far-field noise for isothermal jet SP07. Evenfor an isothermal jet, the entropic part of the noise source is significant near the jet axis where the observation angle issmall, but becomes insignificant in the nozzle region, i.e. large angles. Our observation follows that of Uzun et al.11

for the low angles but differs for the near nozzle region. Our results show a continuous decay but Uzun et al. showspurious levels in the entropy noise for angles Θ > 100o. The shear (T l

i j) and self (T ni j) noise are greater than the total

noise for angles Θ < 45o while the entropy noise is greater compared to the total noise for Θ < 15o. The shear noise,however, is seen here to have a more bi-directional character with an extinction near Θ = 80o. The shear noise shapecomputed here confirms the theory proposed by Ribner,57 i.e. the sound intensity should vary proportional to a factorof I ∼ cos4 Θ + cos2 Θ for shear noise. This bi-directional behavior was also reported by Freund25 using DNS and byUzun et al.11

Figure 11 shows a similar plot similar to that of Figure 10 but for the first heated test case SP46. The mostnoticeable difference between Figures 11 and 10 is that for case SP46 the entropy noise is greater compared to thetotal noise and the self noise is significantly lower compared to the total noise. More specifically, the entropy noiseis louder than the total noise for observation angles Θ < 60o as opposed to SP07’s Θ < 45o. The shear noise againshows a similar trend as in SP07, i.e. the sudden extinction at Θ ' 80o and then an increase. Finally, Figure 12 showsthe noise source components for the second heated jet SP23. In this particular case, there is a stark contrast in thedirectivity behavior compared to the previous two jets. For this set point, M j = 0.38 and T j/T∞ = 1.76, the mostdominant source is the entropy noise. The self and shear noise contributions appear to be insignificant for a low speedheated jet. To further study the effect of heating on this low Mach number case, it would be desirable to run this case,i.e SP23 with no heating. (This would correspond to set point SP03 in Tanna’s test matrix) However, due to the largecomputational cost required for each case, we were only able to include the first three cases. A similar run could bedone in the near future. Nonetheless, the directivity pattern of the shear noise follows that of the previous two jets.

To obtain a clearer representation of the effect of heating, we re-plot all the noise sources but with all the jet testcases together. Figure 13 shows the total noise for all three jets. The corresponding experimental data are plottedas well. Comparing the SP07 and SP46 date from the experiments, the effect of heating the jet actually makes thejet quieter while keeping the jet acoustic Mach number fixed. Our simulations captures the same behavior albeit forangles Θ < 45o. For angles greater than 45o, however, test case SP46 is slightly noisier. For the heated low speedjet, SP23, we see that overall it is the quietest compared to SP07 and SP46. Figures 14 through 16 shows the effectof heating on each of the sources for all three jets. From Figures 14 and 15, we see that the effect of heating actuallydecreases the shear and self noise if the Mach number is kept constant. We must bear in mind that all noise sourcelevels could probably be lower since as we have seen, the LES results over predict the experimental measurements byapproximately 3dB. Nonetheless, the trends are captured well here. The entropy noise source on the other hand, i.e.Figure 16, is amplified when the jet is heated. The increase in the entropy noise and density comes probably comes asno surprise since entropy fluctuations are related to temperature variations. Test case SP23 overall shows the lowest

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levels of noise where the entropy noise dominates the directivity.From the OASPL plots, it is clear that some of the noise components are louder and more intense than the total noise

at some observation angles, which suggests that cancellations are taking place. Freund25 suggests that the cancellationsamong the noise generating components must be correlated and defined the following correlating coefficients,

Cln =pl pn

plrms pn

rms, Cls =

pl ps

plrms ps

rms, Cns =

pn ps

pnrms ps

rms, (11)

where the pressure terms are the fluctuating pressure history from each source and the superscripts l, n and s indicatethe shear, self and entropy noise components, respectively. Figures 17 through 19 show how each of the correlationcoefficients behave for an unheated and heated jets along an arc in the far-field at R = 144ro. Looking at the firstcorrelation coefficient, Cln, we see that by heating the jet there appear to be less cancellations among the shear andself noise terms for angles Θ < 90o and then again after Θ = 90o. In other words, a high speed unheated jet has strongcancellations among the shear and self noise terms. From Figure 17 we can deduce that cancellations dominate theshear and self terms at nearly all observation angles. It is also important to note that Uzun et al.11 reported the sameobservation for their unheated jet SP07. Figure 18 shows the correlation of Cls and from here we observe that over allobservation angles the shear and entropy noise terms contain significant cancellations in the far-field when a M∞ = 0.9jet is heated. The cancellation is strongest at Θ = 5o with Cls ' −0.9 for SP46 as opposed to Cls ' −0.3 for SP07at the same angle. This observation could probably explain why a heated jet is quieter compared to an unheated jetfrom high Mach numbers, i.e. M∞ > 0.7. Bodony & Lele27 performed a similar analysis and also reported significantcancellations (both in noise amplitude and phase) between the momentum (shear) and entropy terms in their heatedjet compared to a similar unheated case. For SP23 entropy noise dominates, so Cls and Cns are small. The correlationbetween self and entropy noise, Cns however, does not show as much cancellations as in Cls. Here in Figure 19 there isslightly more cancellation among the self and entropy noise for SP07 compared to SP46 for angles Θ < 65o but thenSP46 registers more cancellations compared to SP23 for angles greater than Θ < 65o.

Next we focus on spectra. Figures 20 through 22 show the 1/3-Octave pressure spectra for the unheated jet SP07for observation angles Θ = 30o, Θ = 60o and Θ = 90o, respectively. The experimental data by Tanna et al.45 andViswanathan22 are also plotted as a reference. All spectra presented herein are in 1/3-Octave band format to facilitatethe comparisons with experiments. At Θ = 30o (Figure 20), the entropy noise registers the lowest energy across thespectrum compared to shear and self noise. However, the entropy noise surpasses the total noise at the higher frequencyspectrum, i.e. for S r > 1.6. In addition, the shear noise is more intense than the total noise for all frequencies. The selfnoise, however, is lower than the total noise in the low frequency region, i.e. for S r < 0.3. An interesting observation isthat at high frequencies, all noise components register higher sound pressure levels compared to the experimental data.The fact that the total noise is lower than the individual noise sources suggests that there are cancellations amongst thespectral noise components, as we have seen in Figures 17 through 19. At the observation angle of Θ = 60o (Figure 21),the entropy noise is now lower compared to total and other noise components suggesting that the entropy noise sourceis negligible at this angle. In addition, the shear noise spectra overall is now lower compared to the total noise. Aninteresting note is that the frequency where the maximum SPL occurs shifts from S r = 0.3 to S r = 0.6 indicating thatthere is more high frequency content as an observer moves toward the near nozzle region. Moving on to the near nozzleregion of Θ = 90o, i.e. Figure 22, we now see that the shear and entropy noise are lower compared to the total noisesuggesting that the majority of the noise in the near nozzle region is due to the entropy term. Comparing the total noisespectra to the FWH results, we see good agreement as well. In addition, for all three observation angles the computedspectra are in reasonable agreement with the two experimental measurements of Tanna et al. and Viswanathan.

Figures 23 through 25 show spectra for the first heated jet SP46, i.e. M∞ = 0.9 with T j/T∞ = 2.7. The corre-sponding experimental data are plotted as well. At Θ = 30o, all the noise source components are louder than the totalnoise for S r > 0.5. At this angle also, the dominant noise source is the entropy term. It is interesting to note that thecomparison of the total noise with experiments are in good agreement especially in the low frequency portion of thespectra. The least dominant term here is the self noise. For the spectra at observation angle Θ = 60o, we still observethat the entropy noise term is the dominant source but compared to Θ = 30o, its intensity is lowered. In addition, thistime the shear and self noise terms are lower compared to the total noise spectra. Finally, at the Θ = 90o observationangle, we see that the shear noise term does not contribute much and again the self and entropy sources are probablythe main contributors of noise at this angle. Again, we observe some of the noise sources being more intense andsome lower than the total noise suggest the presence of cancellations amongst the spectra. In brief, we note that as weprogress from the shallow angles to the near nozzle near region, the shear noise contribution decreases but the entropynoise source dominates for this heated jet.

Figures 26 through 28 show the spectral characteristics for the second heated jet SP23 (M∞ = 0.5, T j/T∞ = 1.76)

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for observation angles Θ = 30o, Θ = 60o and Θ = 90o, respectively. The prevailing theme we see is that for this lowspeed heated jet, the entropy noise dominates across the frequency spectrum. The shear and self noise sources hardlycontribute and this has already been seen in the OASPL plots in Figures 12, 17, 18 and 19.

To observe the effects of heating, Figures 29 through 31 show the spectral comparisons between SP07 and SP46.SP23 has been left out since this jet condition is different in terms of ambient jet Mach number compared to SP07 andSP46. With the exception of Θ = 30o, the total noise spectra of SP46 is slightly louder compared to SP07. This is nosurprise because if we look closely at Figure 13, the OASPL of SP46 is also slightly higher compared to its unheatedcounterpart. The experiments however show that at all observation angles SP46 is quieter compared to SP07. Hence,Figures 32 through 34 show the comparison between SP07 and SP46 for each noise source component at observationangle Θ = 30o in the far-field distance of R = 144ro. For the shear noise term, T l

i j, the unheated jet spectra is loudercompared to its heated jet counterpart for S r < 0.6 but is then less intense after that. For the self noise term, i.e. Figure33, the heated jet noise is consistently lower compared to SP07 by approximately 8 dB across the frequency spectrum.Thus the effects of heating while keeping the ambient Mach number fixed lowers the self noise source for a M∞ = 0.9jet. In other words, the intensity of the turbulent fluctuations interacting among themselves is lessened when the jetis heated. Hence, in addition to the added cancellations among the noise sources for heated jets (see Figure 18), thereduction in the self noise source could explain why a high speed heated jet is less noisy compared to an unheated jet.This observation is also supported by the findings of Bodony & Lele.27 The entropy source term shows an increasedintensity level across the spectrum for heated jets. Figures 35 through 37 again show a similar comparison but atΘ = 60o. This time, the shear and self noise terms for SP46 are consistently lower compared to SP07 signifying thatthe shear noise source is now more intense for an unheated jet as we progress towards the near nozzle region. Theentropy noise source terms for the heated jet are still higher for SP46 compared to SP07. For all plots here the spectralshape of the noise sources follow the experimental results reasonably well. Finally, Figures 38 through 40 show thespectral characteristics at Θ = 90o. The shear noise terms are reduced significantly by approximately 15 dB for bothSP07 and SP46 but the heated jet shear noise term is still lower compared to the unheated jet. Again, the self noisesource for the heated jet is lower across the frequency spectrum compared to when it is unheated. The entropy noisesource for the heated jet SP46 again shows an increased level compared to SP07 throughout the spectrum.

VII. Closing Remarks

In summary, through the use of Lighthill’s acoustic analogy we have examined the contribution of the individualsources of noise to the far-field sound for two heated jets and one isothermal/unheated jet. The individual noise sourcesare the shear, self and entropy noise. We found that overall sound pressure levels agree well in terms of trends withthe experimental results of Tanna et al. and Viswanathan. In addition, the Lighthill results are also in good agreementwith the results obtained using the Ffowcs Williams-Hawkings method. The heated jet results, however, over-predictthe experimental results by about 3 dB. It is found that when a high-speed subsonic compressible jet is heated whilekeeping the ambient jet Mach number constant, significant cancellations occur in the far-field between the shear andentropy noise sources. In addition, heating a jet reduces the intensity of the nonlinear self noise terms compared toan unheated jet. These observations could probably explain why a high-speed heated jet is quieter compared to anunheated jet when the jet ambient Mach number is fixed. For a low-speed heated jet, the main contributing source isthe entropy noise while the shear and self noise sources hardly contribute to the far-field sound.

Acknowledgments

We would first like to thank Dr. Ali Uzun from Florida State University who provided both his 3-D LES andaeroacoustic post-processing codes. His assistance in understanding the inner workings of his code is also greatlyappreciated. In addition, we would like to thank Dr. Loren Garrison from Rolls-Royce, Indianapolis in providingassistance in obtaining the SAE ARP 876C data for the jets presented above. The first author gratefully acknowl-edges the partial support of the Purdue Research Foundation (PRF) Special Incentive Research Grant (SIRG) andfrom Professor Luc Mongeau from McGill University’s Department of Mechanical Engineering. This work was alsopartially supported by the National Computational Science Alliance under grant number ASC040044N and utilizedthe SGI Altix computing system at the University of Illinois, Urbana-Champaign and the Compaq Alpha Cluster atthe Pittsburgh Supercomputing Center. Computational resources used at Purdue University include the 320 processorIBM-SP3 supercomputer and the 120 processor Sun F6800 servers.

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Tam & Dong’s radiation boundary conditions

Sponge Zone

Tam & Dong’s radiation boundary conditions

Tam &Dong’sradiationbcs

Tam &Dong’soutflowboundarycondition

Vortex ring forcing

Figure 1. Boundary conditions used in the 3-D LES code.

x / ro

y/r

o

0 20 40 60 80-20

-10

0

10

20

30

40

Physical Domain Sponge Zone

Figure 2. The cross section of the computational grid on the z = 0 plane. (Every 3rd grid point is shown).

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W = k(x - xc )/ro

Uc(x

)/U

j

-0.5 -0.25 0 0.25 0.5 0.75 1 1.25 1.5 1.75 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Uzun’s LES, SP07Bodony-Lele’s LES, SP07Bogey & Bailly’s LES, SP07Freund’s DNS, SP07Bridges-Wernet experiment, SP07Tanna’s experiment, SP07Jordan et al. experiment Mj = 0.75

0.95Uo

Figure 3. Mean axial velocity centerline decay rate for isothermal jet SP07.

W = k(x - xc )/ro

Uc(

x)/U

j

-0.5 -0.25 0 0.25 0.5 0.75 1 1.25 1.5 1.75 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Current LES, SP46Bodony’s LES, SP46Bridges & Wernet data, SP46

0.95Uo

Figure 4. Mean axial velocity centerline variation for heated jet SP46.

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W = k(x - xc )/ro

Uc(

x)/U

j

-0.5 -0.25 0 0.25 0.5 0.75 1 1.25 1.5 1.75 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Current LES, SP23Bodony’s LES, SP23Bridges & Wernet’s Exp., SP23Jordan et al. Exp., Mj = 0.53, Tj / T a = 2

0.95Uo

Figure 5. Mean axial velocity centerline decay rate for heated jet SP23.

Figure 6. The control surface used for the Ffowcs Williams-Hawkings surface integral method.

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Θ (deg)

OA

SP

L(d

B)

0 20 40 60 80 100 120 140 16090

95

100

105

110

115LES-Lighthill, SP07LES-FWH, SP07, Uzun et al.Bodony’s LES, SP07SAE ARP876C PredictionTanna’s Exp., SP07Viswanathan’s Exp. SP07

Figure 7. Overall sound pressure level variation for unheated jet SP07 at R = 144ro from the nozzle exit.

Θ (deg)

OA

SP

L(d

B)

0 20 40 60 80 100 120 140 16085

90

95

100

105

110

115

LES-Lighthill, SP46LES-FWH, SP46Bodony’s LES, SP46SAE ARP 876C predictionTanna’s Exp., SP46Viswanathan’s Exp, SP46

Figure 8. Overall sound pressure level variation for heated jet SP46 at R = 144ro from the nozzle exit.

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Θ (deg)

OA

SP

L(d

B)

0 20 40 60 80 100 120 140 16070

75

80

85

90

95

100

LES-Lighthill, SP23LES-FWH, SP23Bodony’s LES, SP23SAE ARP876C, SP23Tanna’s Exp., SP23

Figure 9. Overall sound pressure level variation for heated jet SP23 at R = 144ro from the nozzle exit.

Θ (deg)

OA

SPL

(dB

)

0 20 40 60 80 100 120 140 16070

75

80

85

90

95

100

105

110

115

120

total noise, Tij , SP07

shear noise, T lij , SP07

self noise, T nij , SP07

entropy noise, T sij , SP07

Figure 10. Overall sound pressure level variation of the noise from Ti j and its components for SP07 at R = 144ro from the nozzle exit.

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Θ (deg)

OA

SPL

(dB

)

0 20 40 60 80 100 120 140 16075

80

85

90

95

100

105

110

115

120

125






Θ (deg)

OA

SPL

(dB

)

0 20 40 60 80 100 120 140 16030

40

50

60

70

80

90

100

110

120






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Θ (deg)

OA

SPL

(dB

)

0 20 40 60 80 100 120 140 16070

75

80

85

90

95

100

105

110

115

total noise, Tij , SP07total noise, Tij , SP46total noise, Tij , SP23Tanna’s Exp., SP07Tanna’s Exp., SP46Tanna’s Exp., SP23

Figure 13. Overall sound pressure level variation of total noise, Ti j, for all jets at R = 144ro from the nozzle exit.

Θ (deg)

OA

SPL

(dB

)

0 20 40 60 80 100 120 140 16030

40

50

60

70

80

90

100

110

120

130




Figure 14. Overall sound pressure level variation of shear noise, T li j, for all jets at R = 144ro from the nozzle exit.

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Θ (deg)

OA

SPL

(dB

)

0 20 40 60 80 100 120 140 16030

40

50

60

70

80

90

100

110

120

130




Figure 15. Overall sound pressure level variation of self noise, T ni j, for all jets at R = 144ro from the nozzle exit.

Θ (deg)

OA

SPL

(dB

)

0 20 40 60 80 100 120 140 16070

75

80

85

90

95

100

105

110

115

120

125

130




Figure 16. Overall sound pressure level variation of entropy noise, T si j, for all jets at R = 144ro from the nozzle exit.

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Θ (deg)

Cor

rela

tion

Coe

ffici

ent,

Cln

0 20 40 60 80 100 120 140 160-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4Cln , SP07

Cln , SP46

Cln , SP23

Figure 17. Correlation amongst the shear and self noise compenents, Cln, for all jets in the far-field at R = 144ro from the nozzle exit.

Θ (deg)

Cor

rela

tion

Coe

ffici

ent,

Cls

0 20 40 60 80 100 120 140 160-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Cls , SP07

Cls , SP46

Cls , SP23

Figure 18. Correlation amongst the shear and entropy noise compenents, Cls, for all jets in the far-field at R = 144ro from the nozzle exit.

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Θ (deg)

Cor

rela

tion

Coe

ffici

ent,

Cns

0 20 40 60 80 100 120 140 160-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Cns , SP07

Cns , SP46

Cns , SP23

Figure 19. Correlation amongst the self and entropy noise compenents, Cns, for all jets in the far-field at R = 144ro from the nozzle exit.

Strouhal Number, Sr = f Dj / Uj

1/3

-Oct

ave

SP

L(d

B/S

r)

0 0.5 1 1.5 2 2.5 350

55

60

65

70

75

80

85

90

95

100

105

total noise, Tij , SP07,Θ = 30 o

shear noise, Tlij , SP07,Θ = 30 o

self noise, Tnij , SP07,Θ = 30 o

entropy noise, Tsij , SP07,Θ = 30 o

Tanna’s Exp., SP07,Θ = 30 o

Viswanathan’s Exp., SP07,Θ = 30 o

Cut-off, Sr = 2

Figure 20. Spectra of the noise from Ti j and its components for SP07 at Θ = 30o, R = 144ro from the nozzle exit.

22 of 32



1/3

-Oct

ave

SP

L(d

B/S

r)

0 0.5 1 1.5 2 2.5 350

55

60

65

70

75

80

85

90

95

100







Cutf-off, Sr = 2



1/3

-Oct

ave

SP

L(d

B/S

r)

0 0.5 1 1.5 2 2.5 325

30

35

40

45

50

55

60

65

70

75

80

85

90

95







Cut-off, Sr = 2


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1/3

-Oct

ave

SP

L(d

B/S

r)

0 0.5 1 1.5 2 2.5 335

40

45

50

55

60

65

70

75

80

85

90

95

100

105

110





LES-FWH, SP46,Θ = 30 o



Cut-off, Sr = 2



1/3

-Oct

ave

SP

L(d

B/S

r)

0 0.5 1 1.5 2 2.5 345

50

55

60

65

70

75

80

85

90

95

100








Cut-off, Sr = 2


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1/3

-Oct

ave

SP

L(d

B/S

r)

0 0.5 1 1.5 2 2.5 325

30

35

40

45

50

55

60

65

70

75

80

85

90

95








Cut-off, Sr = 2



1/3

-Oct

ave

SP

L(d

B/S

r)

0 0.5 1 1.5 2 2.5 30

10

20

30

40

50

60

70

80

90

100

110

120







Cut-off


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1/3

-Oct

ave

SP

L(d

B/S

r)

0 0.5 1 1.5 2 2.5 30

10

20

30

40

50

60

70

80

90

100

110

120







Cut-off



1/3

-Oct

ave

SP

L(d

B/S

r)

0 0.5 1 1.5 2 2.5 30

10

20

30

40

50

60

70

80

90

100

110

120

130







Cut-off


26 of 32



1/3

-Oct

ave

SP

L(d

B/S

r)

0.5 1 1.5 2 2.5 350

55

60

65

70

75

80

85

90

95

100

105







Cut-off, Sr = 2

Figure 29. Spectra of the total noise, Ti j, for SP07 and SP46 at Θ = 30o, R = 144ro from the nozzle exit.


1/3

-Oct

ave

SP

L(d

B/S

r)

0 0.5 1 1.5 2 2.5 350

55

60

65

70

75

80

85

90

95

100



Cut-off, Sr = 2


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1/3

-Oct

ave

SP

L(d

B/S

r)

0 0.5 1 1.5 2 2.5 345

50

55

60

65

70

75

80

85

90

95



Cut-off, Sr = 2



1/3

-Oct

ave

SP

L(d

B/S

r)

0 0.5 1 1.5 2 2.5 350

55

60

65

70

75

80

85

90

95

100

105



Cut-off, Sr = 2

Figure 32. Spectra of the shear noise, T li j, for SP07 and SP46 at Θ = 30o, R = 144ro from the nozzle exit.

28 of 32



1/3

-Oct

ave

SP

L(d

B/S

r)

0 0.5 1 1.5 2 2.5 350

55

60

65

70

75

80

85

90

95

100

105



Cut-off, Sr = 2

Figure 33. Spectra of the self noise, T ni j, for SP07 and SP46 at Θ = 30o, R = 144ro from the nozzle exit.


1/3

-Oct

ave

SP

L(d

B/S

r)

0 0.5 1 1.5 2 2.5 350

55

60

65

70

75

80

85

90

95

100

105

110



Cut-off, Sr = 2

Figure 34. Spectra of the entropy noise, T si j, for SP07 and SP46 at Θ = 30o, R = 144ro from the nozzle exit.

29 of 32



1/3

-Oct

ave

SP

L(d

B/S

r)

0 0.5 1 1.5 2 2.5 355

60

65

70

75

80

85

90

95

100



Cut-off, Sr = 2



1/3

-Oct

ave

SP

L(d

B/S

r)

0 0.5 1 1.5 2 2.5 355

60

65

70

75

80

85

90

95

100



Cut-off, Sr = 2


30 of 32



1/3

-Oct

ave

SP

L(d

B/S

r)

0 0.5 1 1.5 2 2.5 350

55

60

65

70

75

80

85

90

95

100



Cut-off, Sr = 2



1/3

-Oct

ave

SP

L(d

B/S

r)

0 0.5 1 1.5 2 2.5 330

35

40

45

50

55

60

65

70

75

80

85

90

95



Cut-off, Sr = 2


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1/3

-Oct

ave

SP

L(d

B/S

r)

0 0.5 1 1.5 2 2.5 335

40

45

50

55

60

65

70

75

80

85

90

95



Cut-off, Sr = 2



1/3

-Oct

ave

SP

L(d

B/S

r)

0 0.5 1 1.5 2 2.5 325

30

35

40

45

50

55

60

65

70

75

80

85

90

95



Cut-off, Sr = 2


32 of 32


Investigation of Noise Sources in Turbulent Hot Jets Using ...lyrintzi/Reno07Charlie.pdfThese high...

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Transcript of Investigation of Noise Sources in Turbulent Hot Jets Using ...lyrintzi/Reno07Charlie.pdfThese high...