Investigation of Noise Sources in Turbulent Hot Jets using Large Eddy Simulation Data

45th AIAA Aerospace Sciences Meeting and Exhibit 8-11 January 2007, Reno, Nevada AIAA 2007-16 45th AIAA Aerospace Sciences Meeting and Exhibit, January 8-11, 2007, Reno, NV., USA. Investigation of Noise Sources in Turbulent Hot Jets using Large Eddy Simulation Data Phoi-Tack Lew, Gregory A. Blaisdell, and Anastasios S. Lyrintzis School of Aeronautics and Astronautics Purdue University West Lafayette, IN 477 Through the use of Lighthill s acoustic analogy, the aim of this paper is to investigate the noise sources of turbulent heated round jets using previously simulated Large Eddy Simulation (LES) data. Two heated and one unheated jet are considered to study the effects of heating on the noise source contributions to the far-field. Firstly, the computed overall sound pressure level (OASPL) and spectra are in good agreement with the prediction obtained from the porous Ffowcs Williams-Hawkings (FWH) surface integral method. Like the FWH prediction, however, the computed OASPL over-predicts the experiments by approximately 3dB but the trends agree reasonably well with the experimental results. Through decomposition of the Lighthill source term we obtain such sources as shear, self and entropy noise. An important finding is that when a high speed subsonic compressible jet is heated while keeping the ambient jet Mach number constant, significant cancellations occur in the far-field between the shear and entropy noise. In addition, heating a jet reduces the intensity of the nonlinear self noise terms compared to an unheated jet. For a low speed heated jet, the main contributing source is the entropy noise source while the shear and self noise sources hardly contribute to the far-field noise. Roman Symbols a j Jet centerline speed of sound a Ambient speed of sound D j Jet diameter M j Mach number = U j /a j M Acoustic Mach number = U j /a OASPL Overall sound pressure level p pressure r o Initial jet radius R Radial arc length from centerline jet exit Re D Jet Reynolds number = ρ j U j D j /µ j S r Strouhal number = f D j /U j T j Jet centerline temperature T i j Lighthill stress tensor, total noise Ti l j Shear noise Ti n j Self noise Ti s j Entropy noise T Ambient temperature Jet streamwise centerline velocity U j Graduate Research Assistant, Student Member AIAA. Associate Professor, Senior Member AIAA. Professor, Associate Fellow AIAA. Nomenclature Copyright c 2007 by P. Lew, G. A. Blaisdell and A. S. Lyrintzis. All other rights are reserved by the copyright owner. 1 of 32 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 variable x, y, z Cartesian coordinates x c /r o Potential core length Greek Symbols α Vortex ring forcing amplitude α f Filtering parameter t Time resolution Θ Observer angle relative to centerline jet axis κ Witze coefficient µ j Jet centerline viscosity ρ j Jet centerline density ρ Ambient density Viscous stress tensor σ i j Subscripts j Ambient condition Jet exit condition Superscripts ( ) Fluctuating component ( ) Time averaged value I. Introduction THERE is a current need to better understand the noise generation mechanisms in turbulent sub-sonic jets. This is because 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 communities surrounding the airport. Hence, a goal was introduced by NASA in 1997 aimed at eliminating community noise problems near airports. The goal is to reduce the perceived noise levels of future aircraft by a factor of two (10 EPNdB) from subsonic aircraft by 2007, and by a factor of four (20 EPNdB) by 2022. 1 This goal is not infeasible, but it is challenging nonetheless due to the fact that the underlying mechanisms/sources that cause jet noise are still not well understood and, therefore, cannot be fully controlled or optimized. Thus, the jet noise problem still remains one of the most elusive problems in aeroacoustics. With the advent of fast supercomputers, the application of direct numerical simulation (DNS) to jet noise prediction is becoming more feasible. 2, 3 DNS solves for the dynamics of all the relevant length scales of turbulence and thus no form of turbulence modeling is used. Unfortunately, due to the wide range of time and length scales present in turbulent flows and because of the limitations of current computational resources, DNS is still restricted to low Reynolds number flows. In contrast to DNS, large eddy simulation (LES), which computes the large scales directly and models the small scales or the subgrid scales, yields a cheaper alternative to DNS. It is assumed that the large scales in turbulence are generally more energetic compared to the small scales and are affected by the boundary conditions 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,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 of LES as an investigative tool for jet noise prediction was carried out by Mankbadi et al. 4 They performed a simulation of a low Reynolds number supersonic jet and applied Lighthill s analogy 5 to calculate the far-field noise. Lyrintzis and Mankbadi 6 were the first to use Kirchhoff s method with LES to compute the far-field noise. A string of other numerical calculations (e.g. References 7, 8, 9, 10, 11, 12, 13, 14, 15) were then carried out by investigators at higher Reynolds 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 hot exhaust conditions and at high Reynolds numbers. However, most of the current LES jet studies that have been carried out to date consists of either cold or isothermal jets. References [7,10,11,13,14,15] cited in the previous paragraph are such 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 2 of 32

numbers of Re D = 13, 000 and Re D = 27, 000. Their results were consistent with the experimental observations of Tanna 17 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., Re D =, 000 hot jet and the results obtained were in good agreement with the experimental data of Jordan et al. 20, 21 In addition, Shur et al. 14, 15 studied several heated jets ranging from subsonic to supersonic Mach number and obtained excellent agreement with the experiments of Viswanathan 22 and Tanna. 17 Finally, Lew et al. 23 performed two high Reynolds numbers (Re > 10 5 ) hot jet LES simulations and obtained good turbulent flow results and satisfactory far-field noise predictions when compared to the hot jet experiments of Tanna 17 and Viswanathan. 22 Hence, the use of LES as an investigative tool to study the far-field noise for sub-sonic turbulent hot jets is certainly promising. The heated jet LES examples mentioned above deal only with predicting the far-field sound (via integral acoustic methodologies) and not the root cause of it, i.e. the noise generating sources. In the literature, however, the investigation of noise sources in unheated jets using DNS and LES data itself is quite recent. Within the framework of Lighthill s acoustic analogy, 5, 24 Freund 25 was the first to investigate noise sources in a low Reynolds number Re D = 3, 0, M j = 0.9 cold jet using his DNS data. One conclusion that Freund reported was that the contributions from the shear, self and entropic noise sources are highly correlated at small angles to the jet axis, and not statistically independent, as often assumed. However, Freund also noted that better parametrization of Reynolds-number effects on the probable noise sources needed to be carried out since his DNS jet was set in the low-reynolds-number limit. It is in this regard that LES offers an attractive alternative whereby the jet can be simulated at higher Reynolds numbers with a fraction of the cost compared to DNS, as mentioned earlier. Hence, in the realm of LES, Uzun et al. 11 were the first to use LES data coupled with Lighthill s acoustic analogy to investigate the noise sources in a M j = 0.9, Re D = 400, 000 isothermal jet. In their investigation, they found that significant cancellations occur among the noise generated by the individual components of the Lighthill stress tensor for a high Reynolds number isothermal jet. In addition, Bogey & Bailly 26 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 our knowledge, 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, significant phase cancellation exists between the momentum and entropy sources in the near-field and that additional cancellation occurs in the far-field for jets with Mach numbers within the range of 0.9 < M < 1.. Bodony & Lele then suggest that the significant cancellations between these two sources in the far-field is a probable explanation as to why a heated jet 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 s 27 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 number jets (Re D > 10 5 ) using LES data. This work is a continuation of an earlier study of hot jets via LES by the same authors. 23 The noise source investigation is conducted within the framework of Lighthill s acoustic analogy. 5, 24 Direct comparisons of hot and cold jet noise sources will be performed. In addition, we also use the acoustic analogy to predict 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 II provides a brief description of the LES methodology used in this study; section III summarizes the grid setup and flow conditions 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 s acoustic analogy. Finally, some concluding remarks are given in section VII. II. Brief Description of LES Methodology 11, 12 The 3-D LES code used in this study was developed by Uzun et al. and it uses either the classical 28 or a localized dynamic 29 Smagorinsky (DSM) subgrid-scale model. However, the modeling of the subgrid-scale stress tensor still raises some fundamental issues as discussed by Bogey & Bailly 30, 31 and Uzun et al. 32 Eddy-viscosity modelings such as the classical Smagorinsky subgrid-scale model 28 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 be dissipation free at sufficiently high Reynolds numbers. 34 In addition, since the eddy-viscosity has the same functional form 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 on jet flows. An alternative to the use of an explicit eddy-viscosity model is the use of spatial filtering for modeling the effects of the subgrid-scales. Using this alternative, the turbulent energy is only dissipated when it is transferred from 3 of 32

the larger scales to the smaller scales discretized by the mesh grid. 30 Hence, for the hot jets simulated here, we use spatial filtering as an implicit subgrid-scale model. 36 Since we have a near sonic jet, the unsteady, Favre-filtered, compressible, non-dimensional LES equations are solved. We transform from curvilinear coordinates to a uniform grid in computational space. The code uses the non-dissipative sixth-order compact scheme developed by Lele 37 to compute the solution on the internal points. For the points on the boundaries, however, a third-order one-sided compact scheme is used, and the points next to the boundaries are computed by a fourth-order compact central differencing technique. In order to eliminate numerical instabilities that can arise from the boundary conditions, grid non-uniformities and unresolved scales the sixth-order tri-diagonal spatial filter proposed by Visbal and Gaitonde 38, 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 and outflow boundary conditions 40 are implemented on the boundaries. In addition, a sponge zone 41 is attached to the end of 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 of the boundary conditions used in the 3-D LES code. A more in-depth discussion on the numerical methods used can be found in Uzun. 42 To excite the mean flow, randomized perturbations in the form of induced velocities from a vortex ring proposed by Bogey & Bailly 43 are added to the velocity profile at a short distance (approximately one jet radius) downstream from the inflow boundary (see Figure 1). This is done to ensure the break up of the potential core within a reasonable distance. 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 are appropriately named according to the experimental test matrix of Tanna et al. 45 Test case SP07 closely corresponds to 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, test case SP46 is similar to SP07 in terms of the ambient Mach number but it is now heated with a temperature ratio of T j /T = 2.7. The case SP23 was chosen to examine jet flow physics and the far-field noise at a lower Mach number with the addition of heating. Furthermore, there are available LES 16, 46 and experimental data 45, 18, 47 for these test cases 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) = (r o, ±20r o, ±20r o ). Test Case M j M N x N y N z Re D T j /T SP07 0. 0. 292 128 128,000 1.00 SP46 0. 0. 292 128 128 200,000 2. SP23 0.38 0. 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 ambient speed of sound. N x, N y and N z 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 plane of the computational domain. The physical part of the computational domain extends to approximately r o in the streamwise direction and 20r o to 20r o in the transverse y and z directions. Beyond the streamwise location of r o is the sponge zone. The physical domain length of r o was chosen for two reasons. Firstly, Uzun et al. 42 reported that the Reynolds stresses achieve their full asymptotic self-similar state if a domain length of at least 45r o is used. Secondly, in order to capture the Overall Sound Pressure Levels (OASPL) adequately at shallow angles, a domain length of at least r o is required based on the recommendations of Uzun et al. 42 and Shur et al. 14 Based on the minimum grid spacing and ambient Mach number, the time resolution was determined to be t = 0.015 r o /a for cases SP07 and SP46, whereas case SP23 was set to t = 0.011 r o /a, respectively. The Reynolds number is defined as Re D = ρ j U j D j /µ j where ρ j, U j and µ j are the jet centerline density, velocity and viscosity at the inflow, respectively. D j = 2r o is the jet diameter. The Reynolds numbers specified above for both jets correspond to the experimental conditions of Tanna et al. 45 In the previous section, we mentioned that a vortex ring is used to excite the mean flow. The vortex ring used here contains a total of 16 azimuthal jet modes of forcing. 4 of 32

Bogey and Bailly 13 performed a simulation with all modes present and later removed the first four modes and found that the jet was quieter with the latter case and matched experimental results better. Hence, based on their results, the first 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 Freund 48 on the inflow boundary given by u(r) = 1 2 U j [ 1 tanh [ b ( r r o r o r )]], (1) where r = y 2 + z 2, r o = 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 a comparison, Freund 48 used a value of b = 12.5 for his 3-D jet DNS. Hence, we have a thicker shear layer compared to Freund s jet. For laboratory jets however, the measured value of b is usally an order of magnitude or higher compared to 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 Freund 48 ρ(r) = (ρ j ρ )ū(r) U j + ρ, (2) where ū(r)/u j is the mean streamwise velocity on the inflow boundary normalized by the jet centerline velocity, ρ j is the density at the jet centerline and ρ is the freestream density. The ratio ρ /ρ j determines whether or not the jet is hot or cold. A value lower than unity implies a cold jet, whereas a value greater than unity implies a hot jet. The next two 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 in Reference [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 & Lele 16 whereby the axial coordinate, i.e. x/r o, is shifted to aid in the presentation of near-field data over a range of operating conditions so that differences in compressibility or Mach number which affect the length of the potential core can be accommodated. 16, 47, 49 The procedure adopted by Bodony & Lele is called the Witze 49 correlation and is given by W = κ(x x c )/r o where κ = 0.08(1 0.16M j )(ρ /ρ j ) 0.22. Thus x c /r o is computed first and then x/r o is shifted axially. Then the data is re-scaled using the factor κ. Here, x c /r o is defined when the jet centerline velocity reduces to % of the inflow jet velocity, U c (x c ) = 0.U j. For Figure 3, we use Uzun et al. results. This was done since we did not have statistically converged results for our unheated jet SP07 using the current number of grid points, i.e. 4.9 million. Uzun s SP07 velocity decay results has approximately 12 millions grid points and compares very well with the experimental results of Brideges & Wernet. 18 In Bridges & Wernet s 18 technical report, they mention that the data for the mean streamwise velocity decay along the centerline for SP46 shows some problems beyond x/r o = 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 is nearly good agreement within the range of 0 W 1. We also note that Bodony & Lele s SP46 jet decays faster compared to Bridges & Wernet s data. There is also good agreement between our SP23 jet and the experimental results of Bridges & Wernet and Jordan et al. 20, 21 though our jet decays slightly faster. Bodony & Lele s SP23 jet decays the fastest when compared to experiments. As a note, we could not find velocity centerline decay data for Tanna s hot jet experiments. In addition, we also computed the mean centerline velocity decay rates, half-velocity growth rates, potential core lengths, mass flux growth rates and mean streamwise turbulence intensities of our heated jets. The results obtained were is good agreement with the experimental correlations of Zaman. 51 In brief, we found that for the heated jet case of M = 0.9, the jet grows faster which was also observed in the experimental correlations of Zaman and from the experimental data of Bridges & Wernet. Again, the reader is referred to Reference [23] for further results and discussion for the heated and unheated jets. V. Summary of Far-Field Aeroacoustic Results using the Ffowcs Williams-Hawkings Methodology This section will only give a brief discussion on the far-field noise results for our heated jets. Likewise, further acoustic results can be found in Reference [23]. The porous Ffowcs Williams-Hawkings 52, 53 (FWH) surface integral 5 of 32

acoustic method is used to study the far-field noise of our hot jets. The integral method follows the description of Lyrintzis & Uzun 54 and Lyrintzis. For simplicity, a continuous stationary control surface around the turbulent jet is used. For details regarding the numerical implementation of the Ffowcs Williams-Hawkings method, the reader is referred to Uzun. 42 As a note, all data here is presented alongside the Lighthill computations. The far-field noise prediction 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 starts about one jet radii downstream and is situated at approximately 7.5r 0 above and below the jet at the inflow boundary in the y and z directions. It extends streamwise until the end near of the physical domain at which point the cross stream extent of the control surface is approximately 30r o. Hence, the total streamwise length of the control surface is 59r o. We show results for an open control surface. A open control surface here is defined where there is no surface at the end of the physical domain, i.e. x = r o. Based on our grid resolution around our control surface and assuming that with our numerical method 6 points per wavelength are needed to accurately resolve an acoustic wave, 42 the maximum frequency resolved corresponds to a Strouhal number of S r 1 for both test cases SP07 and SP46 and S r 1.6 for test case SP23, where the Strouhal number is defined as S r = f D j /U j. The overall sound pressure levels are computed along an arc with a distance of R = 144r o from the jet nozzle exit. This arc length corresponds to the distance used by Tanna 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 that all experimental and LES data have been scaled to a common distance of R = 144r o (using a 1/R scaling). In addition to the experimental data shown, we have also included the SAE ARP 876C 56 database prediction for a jet operating at similar conditions as ours, i.e. SP07. This database prediction consists of actual engine jet noise measurements and can be used to predict overall sound pressure levels within a few db at different jet operating conditions. As we can see the prediction agrees well with the experimental results of Tanna et al. 45 From the LES results, test case SP07 compares well with the experimental results of Tanna and the SAE prediction within the range of o Θ o. Below that range the LES over predicts the OASPL values. Our data also compares rather well with the acoustic results of Bodony & Lele. 16 Figure 8 shows the OASPL plot for heated jet test case SP46 computed at R = 144r o. We also included Tanna s and Viswanathan s 22 experimental data as well the SAE ARP876C prediction. In terms of the shape of the OASPL curve, we are in good agreement with the experimental results of Tanna and Viswanathan though approximately 3 db higher. The peak radiation angle reported by Tanna is located at Θ = 30 o, whereas ours is located at approximately 32.5 o. Bodony s OASPL prediction is slightly higher than ours, but also follows the trend predicted by Tanna. The SAE ARP876C prediction is able to predict the values reported by Tanna very well, as shown. Hence, overall, our predicted OASPL are in good agreement with the experimental data. Figure 9, on the other hand, shows the OASPL data for test case SP23 and is compared to the experimental data of Tanna et al. Again, on average we over-predict the sound levels by approximately 3 db when compared to the results of Tanna. But nonetheless, our predicted OASPL follows the trend measured by Tanna et al. 45 The predicted values from the SAE ARP 876C show good agreement with the measured data from Tanna as well. The computed spectra using the FWH method are discussed alongside the Lighthill 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 surface integral acoustic method at least captures the directivity pattern in the far-field, albeit at higher noise levels than in the experiments. One possible explanation of this over-prediction could come from the artificial vortex ring forcing used to excite the mean flow. Nevertheless, the satisfactory comparison warrants an investigation into the noise generation mechanisms of the two turbulent heated jets. To accomplish this, we use Lighthill s 5, 24 acoustic analogy to compute and study the noise sources within our turbulent heated jets. Hence, this section is devoted to the noise source computation via the acoustic analogy and can be viewed as a continuation of a study of heated jets via LES. A formulation albeit brief is given in the next subsection followed by results. A. Brief Formulation We begin by considering Lighthill s 5 equation which is written as 2 ρ t 2 a2 2 ρ = 2 T i j, (3) x j x j x i x j 6 of 32

where the Lighthill stress tensor is given by T i j = ρu i u j + (p a 2 ρ)δ i j σ i j. (4) Here, a is the ambient speed of sound, σ i j is the viscous stress tensor and ρ is the fluctuation density. For this study the viscous stress term is neglected. It is important to note that the double divergence of T i j on the right hand side of Equation (3) serves only as a nominal acoustic source term, and in no way should it be interpreted as a true acoustic source. In Lighthill s original formulation of his equation, all effects aside from propagation in a homogenous stationary medium, such as refraction, are lumped into the right hand side. More specifically, the acoustic analogy assumes that the sound generated by the turbulent flow is equivalent to what a quadrupole distribution T i j per unit volume would emit if placed in an acoustic medium at rest. Hence, the quadrupole noise sources replace the actual fluid flow. It is also understood that most of T i j,i j does not radiate into the far-field. 25 However, what the right hand side 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, T i j, can be split into T i j = T m i j + T l i j + T n i j + T s i j, (5) whereby each of the individual components are given as, T m i j = ρu i u j + (p a 2 ρ)δ i j, (6) T l i j = ρu iu j + ρu ju i, (7) T n i j = ρu i u j, (8) T s i j = (p a 2 ρ )δ i j. (9) Here, Ti m j is the mean component which, by definition, does not make noise. T i l j is a component that is linear in velocity fluctuations and is called the shear noise, since this component consists of turbulent fluctuations interacting with the sheared mean flow. Ti n j is a component that is quadratic in velocity fluctuations and is called self noise, since this component involves the turbulent fluctuations interacting with themselves. Finally, Ti s j is the entropy component and shows 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 a unbounded flow coupled with the free-space Green s function, the far-field pressure fluctuations can be computed by ( ) y, t dy, (10) p p = (ρ ρ )a 2 1 (x i y i )(x j y j ) 1 2 4π V x y 3 a 2 t T 2 i j where x and y are the observer and source locations, respectively. B. Setup and Computational Details x y a The noise sources from all three jets including the isothermal case were computed in this study. The shape of the integration volume is similar to that of the FWH surface shown in Figure 6 with the exception that it is smaller in size in the lateral direction. The crosswise extent of the integration volume is roughly 7r o and opens up to 22r o. This crosswise length was chosen since Uzun et al. 11 showed that the majority of the noise sources for a high speed jet are confined within a crosswise extent of roughly 5 to 6 jet radii along the entire streamwise domain. The streamwise length of the integration volume is 59r o, and this plays a crucial role in the ability to capture the effective quadrupole noise sources in the computational domain. When Uzun et al. 11 used a domain length of 32r o for their Lighthill computations, they reported spurious noise levels in their OASPL directivity for observer angles Θ > o (Θ measured relative to jet centerline downstream axis). They suggest that the sudden truncation of the domain creates spurious dipole noise sources as the quadrupole sources pass through the down stream surface. Bodony and Lele 27 used a domain length of approximately r o for their Lighthill analysis and reported no spurious noise levels in their OASPL directivity. For each test case, the five primitive variables, q = [ρ, u, p] T, were saved every ten time steps over a duration of 40,000 time steps during the simulation. This resulted in roughly 430 GB of data saved in double precision unformatted. Due to this large data size, a total of 1,140 processors was used to compute Lighthill s volume integral with a total run time of 5 hours on the Lemieux supercomputer at the Pittsburgh Supercomputing Center (PSC). Based on the spatial grid resolution of the Lighthill control volume, the highest resolvable Strouhal number for test cases SP07 7 of 32

and SP46 is 2 whereas for test case SP23 it is 2.7. Similar to the FWH methodology, the far-field sound is calculated along an arc with a radial distance of R = 144r o with the observer angle, Θ, measured relative to the jet centerline downstream axis. Please refer to Reference 11 for details regarding the numerical methods used for the Lighthill analysis. C. Results Referring to Figure 7 once more, we see that the far-field noise predicted by the acoustic analogy for jet SP07 is in good agreement with the experimental results of Viswanathan and Tanna. In addition, the comparison between FWH and Lighthill is also very good. However, the Lighthill prediction seems to give a better prediction, i.e. closer to experiments, for observation angles, 20 o < Θ < 40 o, compared to the FWH results. It is also important to note that we do not see any spurious noise levels for observer angles Θ > o as was reported by Uzun et al. 11 due to a longer integration domain used in this study. Figure 8 shows the OASPL directivity for the first heated jet SP46. Again, here we note the reasonable comparison between our Lighthill results and the experiments of Tanna 45 and Viswanathan. 22 Like the FWH results, the Lighthill computations over predict the laboratory results by approximately 3 db. The possible 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 results agree well with laboratory experiments of Tanna but over predict again by approximately 3 db. Note that at the peak radiation level at Θ 30 o, the Lighthill results are slightly lower compared to the FWH prediction and are closer to experiments. Thus, it seems so far that for all three jets considered here, Lighthill s acoustic analogy does a slightly better 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 OASPL contribution from T i j and its individual components Ti l j, T i n j and T i s j to the far-field noise for isothermal jet SP07. Even for an isothermal jet, the entropic part of the noise source is significant near the jet axis where the observation angle is small, 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. show spurious levels in the entropy noise for angles Θ > o. The shear (Ti l j ) and self (T i n j ) noise are greater than the total noise for angles Θ < 45 o while the entropy noise is greater compared to the total noise for Θ < 15 o. The shear noise, however, is seen here to have a more bi-directional character with an extinction near Θ = o. The shear noise shape computed here confirms the theory proposed by Ribner, 57 i.e. the sound intensity should vary proportional to a factor of I cos 4 Θ + cos 2 Θ for shear noise. This bi-directional behavior was also reported by Freund 25 using DNS and by Uzun et al. 11 Figure 11 shows a similar plot similar to that of Figure 10 but for the first heated test case SP46. The most noticeable difference between Figures 11 and 10 is that for case SP46 the entropy noise is greater compared to the total noise and the self noise is significantly lower compared to the total noise. More specifically, the entropy noise is louder than the total noise for observation angles Θ < o as opposed to SP07 s Θ < 45 o. The shear noise again shows a similar trend as in SP07, i.e. the sudden extinction at Θ o and then an increase. Finally, Figure 12 shows the noise source components for the second heated jet SP23. In this particular case, there is a stark contrast in the directivity behavior compared to the previous two jets. For this set point, M j = 0.38 and T j /T = 1.76, the most dominant source is the entropy noise. The self and shear noise contributions appear to be insignificant for a low speed heated 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 large computational cost required for each case, we were only able to include the first three cases. A similar run could be done 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 test cases together. Figure 13 shows the total noise for all three jets. The corresponding experimental data are plotted as well. Comparing the SP07 and SP46 date from the experiments, the effect of heating the jet actually makes the jet quieter while keeping the jet acoustic Mach number fixed. Our simulations captures the same behavior albeit for angles Θ < 45 o. For angles greater than 45 o, however, test case SP46 is slightly noisier. For the heated low speed jet, SP23, we see that overall it is the quietest compared to SP07 and SP46. Figures 14 through 16 shows the effect of heating on each of the sources for all three jets. From Figures 14 and 15, we see that the effect of heating actually decreases the shear and self noise if the Mach number is kept constant. We must bear in mind that all noise source levels could probably be lower since as we have seen, the LES results over predict the experimental measurements by approximately 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 as no surprise since entropy fluctuations are related to temperature variations. Test case SP23 overall shows the lowest 8 of 32

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. Freund 25 suggests that the cancellations among the noise generating components must be correlated and defined the following correlating coefficients, C ln = p l p n, C p l rms p n ls = rms p l p s p l rms p s rms, C ns = pn p s p n rms prms s, (11) where the pressure terms are the fluctuating pressure history from each source and the superscripts l, n and s indicate the shear, self and entropy noise components, respectively. Figures 17 through 19 show how each of the correlation coefficients behave for an unheated and heated jets along an arc in the far-field at R = 144r o. Looking at the first correlation coefficient, C ln, we see that by heating the jet there appear to be less cancellations among the shear and self noise terms for angles Θ < o and then again after Θ = o. In other words, a high speed unheated jet has strong cancellations among the shear and self noise terms. From Figure 17 we can deduce that cancellations dominate the shear and self terms at nearly all observation angles. It is also important to note that Uzun et al. 11 reported the same observation for their unheated jet SP07. Figure 18 shows the correlation of C ls and from here we observe that over all observation angles the shear and entropy noise terms contain significant cancellations in the far-field when a M = 0.9 jet is heated. The cancellation is strongest at Θ = 5 o with C ls 0.9 for SP46 as opposed to C ls 0.3 for SP07 at the same angle. This observation could probably explain why a heated jet is quieter compared to an unheated jet from high Mach numbers, i.e. M > 0.7. Bodony & Lele 27 performed a similar analysis and also reported significant cancellations (both in noise amplitude and phase) between the momentum (shear) and entropy terms in their heated jet compared to a similar unheated case. For SP23 entropy noise dominates, so C ls and C ns are small. The correlation between self and entropy noise, C ns however, does not show as much cancellations as in C ls. Here in Figure 19 there is slightly more cancellation among the self and entropy noise for SP07 compared to SP46 for angles Θ < o but then SP46 registers more cancellations compared to SP23 for angles greater than Θ < o. Next we focus on spectra. Figures 20 through 22 show the 1/3-Octave pressure spectra for the unheated jet SP07 for observation angles Θ = 30 o, Θ = o and Θ = o, respectively. The experimental data by Tanna et al. 45 and Viswanathan 22 are also plotted as a reference. All spectra presented herein are in 1/3-Octave band format to facilitate the comparisons with experiments. At Θ = 30 o (Figure 20), the entropy noise registers the lowest energy across the spectrum compared to shear and self noise. However, the entropy noise surpasses the total noise at the higher frequency spectrum, i.e. for S r > 1.6. In addition, the shear noise is more intense than the total noise for all frequencies. The self noise, however, is lower than the total noise in the low frequency region, i.e. for S r < 0.3. An interesting observation is that 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 the spectral noise components, as we have seen in Figures 17 through 19. At the observation angle of Θ = o (Figure 21), the entropy noise is now lower compared to total and other noise components suggesting that the entropy noise source is negligible at this angle. In addition, the shear noise spectra overall is now lower compared to the total noise. An interesting note is that the frequency where the maximum SPL occurs shifts from S r = 0.3 to S r = 0.6 indicating that there is more high frequency content as an observer moves toward the near nozzle region. Moving on to the near nozzle region of Θ = o, i.e. Figure 22, we now see that the shear and entropy noise are lower compared to the total noise suggesting that the majority of the noise in the near nozzle region is due to the entropy term. Comparing the total noise spectra to the FWH results, we see good agreement as well. In addition, for all three observation angles the computed spectra 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 corresponding experimental data are plotted as well. At Θ = 30 o, all the noise source components are louder than the total noise for S r > 0.5. At this angle also, the dominant noise source is the entropy term. It is interesting to note that the comparison of the total noise with experiments are in good agreement especially in the low frequency portion of the spectra. The least dominant term here is the self noise. For the spectra at observation angle Θ = o, we still observe that the entropy noise term is the dominant source but compared to Θ = 30 o, its intensity is lowered. In addition, this time the shear and self noise terms are lower compared to the total noise spectra. Finally, at the Θ = o observation angle, we see that the shear noise term does not contribute much and again the self and entropy sources are probably the main contributors of noise at this angle. Again, we observe some of the noise sources being more intense and some lower than the total noise suggest the presence of cancellations amongst the spectra. In brief, we note that as we progress from the shallow angles to the near nozzle near region, the shear noise contribution decreases but the entropy noise 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) 9 of 32

for observation angles Θ = 30 o, Θ = o and Θ = o, respectively. The prevailing theme we see is that for this low speed heated jet, the entropy noise dominates across the frequency spectrum. The shear and self noise sources hardly contribute 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 and SP46. With the exception of Θ = 30 o, the total noise spectra of SP46 is slightly louder compared to SP07. This is no surprise because if we look closely at Figure 13, the OASPL of SP46 is also slightly higher compared to its unheated counterpart. 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 observation angle Θ = 30 o in the far-field distance of R = 144r o. For the shear noise term, Ti l j, the unheated jet spectra is louder compared to its heated jet counterpart for S r < 0.6 but is then less intense after that. For the self noise term, i.e. Figure 33, 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 = 0.9 jet. In other words, the intensity of the turbulent fluctuations interacting among themselves is lessened when the jet is heated. Hence, in addition to the added cancellations among the noise sources for heated jets (see Figure 18), the reduction 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 increased intensity level across the spectrum for heated jets. Figures 35 through 37 again show a similar comparison but at Θ = o. This time, the shear and self noise terms for SP46 are consistently lower compared to SP07 signifying that the shear noise source is now more intense for an unheated jet as we progress towards the near nozzle region. The entropy noise source terms for the heated jet are still higher for SP46 compared to SP07. For all plots here the spectral shape of the noise sources follow the experimental results reasonably well. Finally, Figures 38 through 40 show the spectral characteristics at Θ = o. The shear noise terms are reduced significantly by approximately 15 db for both SP07 and SP46 but the heated jet shear noise term is still lower compared to the unheated jet. Again, the self noise source for the heated jet is lower across the frequency spectrum compared to when it is unheated. The entropy noise source 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 individual sources of noise to the far-field sound for two heated jets and one isothermal/unheated jet. The individual noise sources are the shear, self and entropy noise. We found that overall sound pressure levels agree well in terms of trends with the experimental results of Tanna et al. and Viswanathan. In addition, the Lighthill results are also in good agreement with the results obtained using the Ffowcs Williams-Hawkings method. The heated jet results, however, over-predict the experimental results by about 3 db. It is found that when a high-speed subsonic compressible jet is heated while keeping the ambient jet Mach number constant, significant cancellations occur in the far-field between the shear and entropy noise sources. In addition, heating a jet reduces the intensity of the nonlinear self noise terms compared to an unheated jet. These observations could probably explain why a high-speed heated jet is quieter compared to an unheated jet when the jet ambient Mach number is fixed. For a low-speed heated jet, the main contributing source is the 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 and aeroacoustic post-processing codes. His assistance in understanding the inner workings of his code is also greatly appreciated. In addition, we would like to thank Dr. Loren Garrison from Rolls-Royce, Indianapolis in providing assistance in obtaining the SAE ARP 876C data for the jets presented above. The first author gratefully acknowledges the partial support of the Purdue Research Foundation (PRF) Special Incentive Research Grant (SIRG) and from Professor Luc Mongeau from McGill University s Department of Mechanical Engineering. This work was also partially supported by the National Computational Science Alliance under grant number ASC040044N and utilized the SGI Altix computing system at the University of Illinois, Urbana-Champaign and the Compaq Alpha Cluster at the Pittsburgh Supercomputing Center. Computational resources used at Purdue University include the 320 processor IBM-SP3 supercomputer and the 120 processor Sun F servers. 10 of 32

References 1 AeroAcoustics Research Consortium, http://www.oai.org/aeroacoustics/challenge.html. 2 Freund, J. B., Lele, S. K., and Moin, P., Direct Numerical Simulation of a Mach 1.92 Turbulent Jet and Its Sound Field, AIAA Journal, Vol. 38, No. 11, November 2000, pp. 2023 2031. 3 Freund, J. B., Noise Sources in a Low-Reynolds-number Turbulent Jet at Mach 0.9, Journal of Fluid Mechanics, Vol. 438, 2001, pp. 277 305. 4 Mankbadi, R. R., Hayder, M. E., and Povinelli, L. A., Structure of Supersonic Jet Flow and Its Radiated Sound, AIAA Journal, Vol. 32, No. 5, May 1994, pp. 897 6. 5 Lighthill, M. J., On the Sound Generated Aerodynamically: I, General Theory, Proc. Royal Soc. London A, Vol. 211, 12, pp. 564 587. 6 Lyrintzis, A. S. and Mankbadi, R. R., Prediction of the Far-field Jet Noise Using Kirchhoff s Formulation, AIAA Journal, Vol. 36, No. 12, 1996, pp. 1 4. 7 Choi, D., Barber, T. J., Chiappetta, L. M., and Nishimura, M., Large Eddy Simulation of High-Reynolds Number Jet Flows, AIAA Paper No. 99-0230, January 1999. 8 Boersma, B. J. and Lele, S. K., Large Eddy Simulation of a Mach 0.9 Turbulent Jet, AIAA Paper No. 99-1874, May 1999. 9 Zhao, W., A Numerical Investigation of Sound Radiated from Subsonic Jets with Application to Human Phonation, Ph.D. thesis, School of Mechanical Engineering, Purdue University, West Lafayette, IN, June 2000. 10 Zhao, W., Frankel, S. H., and Mongeau, L., Large Eddy Simulations of Sound Radiation from Subsonic Turbulent Jets, AIAA Journal, Vol. 39, No. 8, August 2001, pp. 1469 1477. 11 Uzun, A., Lyrintzis, A. S., and Blaisdell, G. A., Coupling of Integral Acoustic Methods with LES for Jet Noise Prediction, International Journal of Aeroacoustics, Vol. 3, No. 4, 2005, pp. 297 346. 12 Uzun, A., Blaisdell, G. A., and Lyrintzis, A. S., Application of Compact Schemes to Large Eddy Simulation of Turbulent Jets, Journal of Scientific Computing, Vol. 21, No. 3, December 2004, pp. 283 319. 13 Bogey, C. and Bailly, C., LES of High Reynolds, High Subsonic Jet: Effects of the Inflow Conditions on Flow and Noise, AIAA Paper No. 2003-31, May 2003. 14 Shur, M. L., Spalart, P. R., and Strelets, M. K., Noise Prediction for Increasingly Complex Jets, Part I: Methods and Tests, International Journal of Aeroacoustics, Vol. 4, No. 3 and 4, 2005, pp. 213 246. 15 Shur, M. L., Spalart, P. R., and Strelets, M. K., Noise Prediction for Increasingly Complex Jets, Part II: Applications, International Journal of Aeroacoustics, Vol. 4, No. 3 and 4, 2005, pp. 246 266. 16 Bodony, D. J. and Lele, S. K., Jet Noise Prediction of Cold and Hot Subnoic Jets Using Large-eddy Simulation, AIAA Paper No. 2004-3022, May 2004. 17 Tanna, H. K., An Experimental Study of Jet Noise Part 1: Turbulent Mixing Noise, Journal of Sound and Vibration, Vol., No. 3, 1977, pp. 405 428. 18 Bridges, J. and Wernet, M. P., Measurement of the Aeroacoustic Sound Source in Hot Jets, Tech. Rep. NASA/TM-2004-2128, NASA, Cleveland, OH, February 2004. 19 Andersson, N., Eriksson, L.-E., and Davidson, L., Large-Eddy Simulation of Subsonic Turbulent Jets and Their Radiated Sound, AIAA Journal, Vol. 43, No. 9, 2005, pp. 1899 1912. 20 Jordan, P., Gervais, Y., Valiére, J.-C., and Foulon, H., Final Results from Single Point Measurements, Tech. Rep. Project deliverable D3.4 JEAN-EU 5th Framework Programme, G4RD-CT2000-00313, Laboratoire d Etude Aeródynamiques, 2002. 21 Jordan, P., Gervais, Y., Valiére, J.-C., and Foulon, H., Results from Acoustic Field Measurements, Tech. Rep. Project deliverable D3.4 JEAN-EU 5th Framework Programme, G4RD-CT2000-00313, Laboratoire d Etude Aeródynamiques, 2002. 22 Viswanathan, K., Aeroacoustics of Hot Jets, Journal of Fluid Mechanics, Vol. 516, 2004, pp. 39 82. 23 Lew, P., Blaisdell, G., and Lyrintzis, A., Recent Progress of Hot Jet Aeroacoustics Using 3-D Large-Eddy Simulation, AIAA Paper 2005-3084, May 2005. 24 Lighthill, M. J., On the Sound Generated Aerodynamically: II, Turbulence as a Source of Sound, Proc. Royal Soc. London A, Vol. 222, 14, pp. 1 32. 25 Fruend, J. B., Noise Source Turbulence Statistics and the Noise from a Mach 0.9 Jet, Physics of Fluids, Vol. 15, 2003, pp. 1788 1799. 26 Bogey, C. and Bailly, C., Investigation of Sound Sources in Subsonic Jets using Causality Methods on LES data, AIAA Paper No. 2005-28, May 2005. 27 Bodony, D. J. and Lele, S. K., Generation of Low Frequency Sound in Turbulent Jets, AIAA Paper No. 2005-3041, May 2005. 28 Smagorinsky, J. S., General Circulation Experiments with the Primitive Equations, Monthly Weather Review, Vol. 91, No. 3, March 1963, pp. 99 1. 29 Moin, P., Squires, K., Cabot, W., and Lee, S., A Dynamic Subgrid-Scale Model for Compressible Turbulence and Scalar Transport, Physics of Fluids A, Vol. 3, No. 11, November 1991, pp. 2746 27. 30 Bogey, C. and Bailly, C., Computation of a high Reynolds number jet and its radiated noise using LES based on explicit filtering, Computers and Fluids, Vol. To Appear, 2005. 31 Bogey, C. and Bailly, C., Decrease of the Effective Reynolds Number with Eddy-Viscosity Subgrid-Scale Modeling, AIAA Journal, Vol. 43, 2005, pp. 437 439. 32 Uzun, A., Blaisdell, G. A., and Lyrintzis, A. S., Sensitivity to the Smagorinsky constant in turbulent jet simulations, AIAA Journal, Vol. 41, No. 10, October 2003, pp. 2077 2079. 33 Germano, M., Piomelli, U., Moin, P., and Cabot, W., A Dynamic Subgrid-Scale Eddy Viscosity Model, Physics of Fluids A, Vol. 3, No. 7, July 1991, pp. 17 17. 34 Bogey, C. and Bailly, C., Selective Fitlering Versus Eddy Viscosity for Subgrid Scale Modelling in the LES of a High Reynolds Number Flow, Direct and Large-Eddy Simulation V, edited by H. Friedich, B. J. Geurts, and O. Métais, Kluwer Academic, Dordrecht, the Netherlands, 2004, pp. 23 30. 11 of 32

35 Domaradzki, J. A. and Adams, N. A., Direct modelling of subgrid scales of turbulence in Large Eddy Simulation, Journal of Turbulence, Vol. 3, No. 024, 2002, pp. 1. 36 Uzun, A., Blaisdell, G. A., and Lyrintzis, A. S., Impact of Subgrid-Scale Models on Jet Turbulence and Noise, AIAA Journal, Vol. 44, No. 6, 2006, pp. 13 1368. 37 Lele, S. K., Compact Finite Difference Schemes with Spectral-like Resolution, Journal of Computational Physics, Vol. 103, No. 1, November 1992, pp. 16 42. 38 Visbal, M. R. and Gaitonde, D. V., Very High-order Spatially Implicit Schemes for Computational Acoustics on Curvilinear Meshes, Journal of Computational Acoustics, Vol. 9, No. 4, 2001, pp. 1259 1286. 39 Koutsavdis, E. K., Blaisdell, G. A., and Lyrintzis, A. S., On the Use of Compact Schemes with Spatial Filtering in Computational Aeroacoustics, AIAA Paper No. 1999-03, May 1999. 40 Bogey, C. and Bailly, C., Three-dimensional Non-reflective Boundary Conditions for Acoustic Simulations: Far Field Formulation and Validation Test Cases, Acta Acustica, Vol. 88, No. 4, 2002, pp. 463 471. 41 Colonius, T., Lele, S. K., and Moin, P., Boundary Conditions for Direct Computation of Aerodynamic Sound Generation, AIAA Journal, Vol. 31, No. 9, September 1993, pp. 1574 1582. 42 Uzun, A., 3-D Large Eddy Simulation for Jet Aeroacoustics, Ph.D. thesis, School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN, December 2003, West Lafayette, IN, December 2003. 43 Bogey, C., Bailly, C., and Juvé, D., Noise Investigation of a High Subsonic, Moderate Reynolds Number Jet Using a Compressible LES, Theoretical and Computational Fluid Dynamics, Vol. 88, No. 4, 2003, pp. 463 471. 44 Lew, P., Uzun, A., Blaisdell, G., and Lyrintzis, A., Effects of Inflow Forcing on Jet Noise Using 3-D Large Eddy Simulation, AIAA Paper 2004-0516, January 2004. 45 Tanna, H. K., Dean, P. D., and Burrin, R. H., The Generation and Radiation of Supersonic Jet Noise, Volume III: Turbulent Mixing Noise Data, Tech. Rep. AFAPL-TR-74-24, Lockheed-Georgia Company, Marietta, GA, September 1976. 46 Bodony, D. J., Aeroacoustic Prediction of Free Shear Flows, Ph.D. thesis, Stanford University, Stanford, CA., June 2004. 47 Bridges, J. and Wernet, M. P., Measurements of the Aeroaocoustic Sound Source in Hot Jets, AIAA Paper No. 2003-3130, May 2003. 48 Freund, J. B., Direct Numerical Simulation of the Noise from a Mach 0.9 Jet, FEDSM Paper No. 99-7251, July 1999. 49 Witze, P. O., Centerline Velocity Decay of Compressible Jets, AIAA Journal, Vol. 12, No. 4, 1974, pp. 417 418. Uzun, A., Blaisdell, G. A., and Lyrintzis, A. S., 3-D Large Eddy Simulation for Jet Aeroacoustics, AIAA Paper No. 2003-3322, May 2003. 51 Zaman, K. B. M. Q., Asymptotic Spreading Rate of Initially Compressible Jets Experiment and Analysis, Physics of Fluids A, Vol. 10, No. 10, October 1998, pp. 22 26. 52 Ffowcs Williams, J. E. and Hawkings, D. L., Sound Generated by Turbulence and Surfaces in Arbitrary Motion, Philosophical Transactions of the Royal Society, Vol. A264, 1969, pp. 321 342. 53 Crighton, D. G., Dowling, A. P., Williams, J. E. F., Heckl, M., and Leppington, F. G., Modern Methods in Analytical Acoustics: Lecture Notes, Springer Verlag, London, 1992. 54 Lyrintzis, A. S. and Uzun, A., Integral Techniques for Aeroacoustics Calculations, AIAA Paper 2001-2253, May 2001. Lyrintzis, A. S., Surface Integral Methods in Computational Aeroacoustics - From the (CFD) Near-field to the (Acoustic) Far-Field, International Journal of Aeroacoustics, Vol. 2, No. 2, 2003. 56 SAE ARP 876C: Gas Turbine Jet Exhaust Noise Prediction, Society of Automotive Engineers, November 19. 57 Ribner, H. S., Quadrupole Correlations Governing the Pattern of Jet Noise, Journal of Fluid Mechanics, Vol. 38, 1969, pp. 1 24. 12 of 32

Tam & Dong s radiation boundary conditions Sponge Zone Tam & Dong s radiation bcs Tam & Dong s outflow boundary condition Vortex ring forcing Tam & Dong s radiation boundary conditions Figure 1. Boundary conditions used in the 3-D LES code. 40 30 Physical Domain Sponge Zone 20 y / r o 10 0-10 -20 0 20 40 x / r o Figure 2. The cross section of the computational grid on the z = 0 plane. (Every 3 rd grid point is shown). 13 of 32

1.1 1 0.9 0.8 0.U o Uzun s LES, SP07 Bodony-Lele s LES, SP07 Bogey & Bailly s LES, SP07 Freund s DNS, SP07 Bridges-Wernet experiment, SP07 Tanna s experiment, SP07 Jordan et al. experiment M j = 0. U c (x) 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0-0.5-0.25 0 0.25 0.5 0. 1 1.25 1.5 1. 2 W = k(x - x c )/r o Figure 3. Mean axial velocity centerline decay rate for isothermal jet SP07. 1.1 Current LES, SP46 Bodony s LES, SP46 Bridges & Wernet data, SP46 1 0.9 0.U o 0.8 U c (x) 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0-0.5-0.25 0 0.25 0.5 0. 1 1.25 1.5 1. 2 W = k(x - x c )/r o Figure 4. Mean axial velocity centerline variation for heated jet SP46. 14 of 32

1.1 1 0.9 Current LES, SP23 Bodony s LES, SP23 Bridges & Wernet s Exp., SP23 Jordan et al. Exp., M j = 0.53, T j / T a = 2 0.U o 0.8 U c (x) 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0-0.5-0.25 0 0.25 0.5 0. 1 1.25 1.5 1. 2 W = k(x - x c )/r o 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. 15 of 32