AIAA 2012-2965

42nd AIAA Fluid Dynamics Conference and Exhibit 25 - 28 June 2012, New Orleans, Louisiana

Towards Best Practices for Jet Noise Predictions with Unstructured Large Eddy Simulations Guillaume A. Br`es∗, Cascade Technologies Inc., Palo Alto, CA 94303

Joseph W. Nichols†, Sanjiva K. Lele‡, Stanford University, Stanford, CA 94305 and

Frank E. Ham§ Cascade Technologies Inc., Palo Alto, CA 94303

Experience gained from previous jet noise studies with the unstructured large eddy simulation (LES) flow solver “Charles” are summarized and put to practice for the predictions of supersonic jets issued from a converging-diverging round nozzle. In this work, the nozzle geometry is explicitly included in the computational domain using an unstructured body-fitted mesh with 42 million cells. Three different operating conditions are considered: isothermal ideally-expanded, heated ideally-expanded and heated over-expanded. Blind comparisons with the currently available experimental measurements carried out at United Technologies Research Center for the same nozzle and operating conditions are presented. The initial results show good agreement for both flow and sound field. In particular, the spectra shape and levels are accurately captured in the simulations for both near-field and far-field noise. In these studies, sound radiation from the jet is computed using an efficient permeable formulation of the Ffowcs Williams–Hawkings equation in the frequency domain. Its implementation in Cascade’s massively-parallel unstructured LES framework is reviewed and additional parametric studies of the far-field noise predictions are presented. As an additional step towards best practices for jet aeroacoustics with unstructured LES, guidelines and suggestions for the mesh design, numerical setup and acoustic post-processing steps are discussed.

Nomenclature c D dt f H Ma Mj M∞ NPR NTR n ˆ

Speed of sound Nozzle exit diameter Simulation time step Frequency Heaviside function Acoustic Mach number Jet Mach number Wind-tunnel Mach number Nozzle pressure ratio p0 /p∞ Nozzle temperature ratio T0 /T∞ Local unit normal vector to the surface

p Rej r ˆ r S St Stlim Stmin Stmax T t

Static pressure Jet Reynolds number Distance between observer and source Unit vector in the radiation direction FW-H integration surface Strouhal number f D/Uj Grid cutoff Strouhal number Minimum Strouhal number of the post-processing Maximum Strouhal number of the post-processing Static temperature Time

∗ Senior

Research Scientist, gbres@cascadetechnologies.com Associate, Center for Turbulent Research ‡ Professor, Dept. of Mechanical Engineering and Dept. of Aeronautics & Astronautics § Chief Technical Officer † Research

1 of 21 Copyright © 2012 by by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

American Institute of Aeronautics and Astronautics

Uj u x y β ∆ ∆t ∆St δ δθ γ µ φ ρ

Equivalent fully-expanded jet velocity Fluid velocity Observer position Source position p 2 1 − M∞ Cell spacing Sampling period Sampling Strouhal Dirac function Boundary layer momentum thickness Specific heat ratio Dynamic viscosity Jet inlet angle Density

I.

τ Total duration of simulation Subscript ∞ Free-stream quantity 0 Stagnation (total) property j Fully-expanded jet conditions n Projection in the normal direction Superscript ¯ Time-averaged quantity ˜ Temporal Fourier-transformed quantity ∗ Dimensional quantity ′ Disturbance quantity (e.g., ρ′ = ρ − ρ∞ ) ˆ Unit vector † Complex conjugate

Introduction

Over the last 40 years, the noise of propulsive jets powering civilian aircraft has been reduced by approximately 20 dB, in large measure by the introduction of the turbofan engine with progressively higher bypass ratio. Yet, more stringent noise regulations, the long term vision of effectively “silent” aircraft, and environmental impact concerns continue to push the civilian (and military) aircraft towards design configurations that further reduce the noise, while maintaining other performance metrics (like nozzle thrust). Until recently, the development of such designs has relied largely on laboratory and full-scale testing. However, cost constraints and the high complexity of the flow often limit the range of the parametric investigation and the success of the design optimization. Flow and noise-prediction tools that are fundamentally rooted in the flow-physics are needed to help improve the understanding of the sources of jet noise and guide the analysis processes towards quieter designs. In this context, large eddy simulation (LES), along with advancement in high-performance computing, is emerging as an accurate yet cost-effective computational tool for prediction of turbulent flows and their acoustic fields. As reviewed by Bodony & Lele,1 reliable LES prediction of jet noise requires particular attention to details in many different aspects, e.g., inclusion of nozzle geometry, appropriate treatment of the nozzle boundary layers,2–5 sufficient grid resolution, low numerical dissipation and dispersion, appropriate inflow and outflow boundary conditions,6, 7 shock capturing schemes,8, 9 use of acoustic analogy methodologies10–12 best suited for jet aeroacoustics, etc. The present study is part of a broader ongoing effort to improve understanding and develop predictive capabilities of propulsive jet noise, through high-fidelity physics-based simulations with the unstructured compressible flow solver “Charles” developed at Cascade Technologies. As a complement to experimental studies, the LES provide access to complete flow fields, allowing in-depth probing of the physics of jet noise production. In previous work, Charles has been used to investigate a wide range of high-speed unsteady flow processes for various complex configurations, including impinging flows,13, 14 circular15, 16 and rectangular17, 18 jets, chevrons,19, 20 and faceted military-style nozzle.21 Experience gained from these studies is currently used for the mesh design, numerical setup and acoustic post-processing steps of ongoing work, to continue advancing existing methodologies towards best practices for jet noise predictions with unstructured LES. The paper is intended as a review of some of the lessons learned so far and as an additional step towards such best practices. As a demonstration case, LES of supersonic jets issued from a converging-diverging round nozzle are performed at three different operating conditions: isothermal ideally-expanded, heated ideally-expanded and heated over-expanded. First, the numerical methods are reviewed in section II. Details on the flow configuration and computational setup are presented in section III. Comparisons with the available experimental measurements carried out at United Technologies Research Center (UTRC) for the same nozzle and operating conditions are shown in section IV. It is important to note that, to put the predictive capabilities to practice, all the simulations and post-processing were performed without prior knowledge of the experimental data, such that the initial results presented in this paper correspond to “blind” comparisons. Finally, as one of the important aspect of the jet noise problem, the far-field acoustic formulations best-suited for jets are discussed in section V and Appendix. 2 of 21 American Institute of Aeronautics and Astronautics

II. A.

Numerical methods

Compressible flow solver “Charles”

The compressible flow solver “Charles” solves the spatially-filtered compressible Navier–Stokes equations on unstructured grids using a control-volume based finite volume method where the flux is computed at each control volume face using a blend of a non-dissipative central flux and a dissipative upwind flux. This blending approach is often the basis of implicit approaches to LES, where the blending parameter is selected as a global constant with a value large enough to provide all the necessary dissipation (and potentially much more). Here, “Charles” does not use the implicit LES approach, but a heuristic algorithm13 to compute a locally optimal (i.e. minimal) blending parameter, which is purely grid-based, and can be pre-computed based on the operators only. Because the underlying numerical method has minimal numerical dissipation, it is critical to employ a sub-grid model to account for the physical effects of the unresolved turbulence on the resolved flow. For the large eddy simulations reported in this work, the Vreman model22, 23 is used with constant coefficient set to the recommended value of c = 0.07, and constant turbulent Prandtl number P rt = 0.9 to close the energy equation. Shocks, like sub-grid scale turbulence, are also sub-grid phenomena and thus require modeling to account for their effect on the resolved flow. However, unlike sub-grid scale turbulence, they are localized in the flow and a surgical introduction of modeling is potentially more appropriate. “Charles” uses a hybrid CentralENO scheme to simulate flows involving shocks. The scheme has three pieces: a central scheme (described previously), a shock-appropriate scheme24, 25 for computing a flux across a shock and a hybrid switch26 to identify the presence of flow discontinuities. B.

Unstructured grid and localized adaptive refinement

In high-fidelity LES, grid stretching, in particular in the downstream region, has been linked to numerical humps in the high frequency far-field spectra.13, 17 This numerical artifact is reminiscent of the hump found in far-field spectra of a round Mach 0.9 jet in the work of Bogey et al..5 To mitigate such spurious noise, the current meshing strategy promotes grid isotropy in the acoustic source-containing region, either directly in the initial mesh topology or through the use of adaptive refinement with “Adapt”. “Adapt” is a massively parallel tool developed in Cascade’s solver infrastructure that can refine elements locally to match a target length scale. This target length scale can vary throughout the domain, and can even be different in each direction for non-aeroacoustic applications. Specification of the target length scale can come from expert knowledge of the problem, a desired mesh size limit based on compute resources, or even from a solution on an unadapted or partially adapted mesh. The directional refinement capability dramatically reduces the overall mesh size and also prevents the addition of stiffness to the problem due to excessively small elements. The “Adapt” tool also provides a surface projection algorithm to respect nonplanar mesh boundaries during refinement, ensuring accurate representation of the underlying geometry, and allowing the use of very coarse grids as starting point. To illustrate the effect of grid stretching on acoustics and the use of adaptive refinement, LES of the hot supersonic jet issued from a SMC015 nozzle27 were performed. The converging-diverging nozzle is designed to obtained ideally expanded (shock-free) jet with exit Mach number Mj = 1.4. The ratio of the jet temperature to ambient temperature is Tj = 1.765. The original coarse mesh corresponds to the topology used by Mendez et al.28–30 The mesh consists of approximately 17 million purely hexahedral control volumes, with clustering of cells near the nozzle surface and the jet shear layers, and gradual stretching in the downstream and lateral directions. A refined mesh 137M is obtained by doubling the resolution in the axial, radial and azimuthal directions, for a total of 137 million control volumes,13 as shown in figure 1(a). Both meshes contain a significant amount of grid stretching, especially in the downstream regions. The mesh 137M is then adapted so that the resolution near the nozzle lip is extended isotropically throughout the volume inside, up to, and just outside the FW-H surface. The resulting mesh Iso217M contains 217 million control volumes and is presented in figure 1(b). Here, it is clear that the localized adaptive refinement introduces occasional hanging nodes throughout the domain. However, the numerical schemes in “Charles” are designed to handle transition type elements and produce accurate results on such unstructured grid. A qualitative comparison of the flow field results between the LES performed on the stretched grid 137M and the adapted grid Iso217M are presented in figure 2. High frequency sound waves originating from

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(a) Stretched mesh 137M

(b) Adapted mesh Iso217M

Figure 1. Planar cuts of the unstructured grids downstream of the nozzle exit. Only part of the computational domain is shown.

far downstream of the jet can be identified in figure 2(a). This spurious high frequency noise has been eliminated completely with the adapted mesh in figure 2(b). The spurious waves may not be apparent to a casual observer at this image scale but a magnified view of the sound field makes them evident. The same trend is observed on much coarse grids, though the spurious noise in these cases has lower frequency (i.e., larger wavelength, directly related to the larger grid spacing). Overall, the results show that this numerical artifact is likely associated with the grid stretching in the far downstream region and that mesh isotropy is desired.

(a) Stretched mesh 137M

(b) Adapted mesh Iso217M

Figure 2. Instantaneous flow field for SMC015 nozzle at Mj = 1.4 and temperature ratio Tj = 1.765, showing the reduction of spurious waves with the adapted mesh. Pressure and temperature contours are shown in gray scale and red scale, respectively.

C.

Far-field noise prediction

For far-field noise predictions, the accurate propagation of the small amplitude acoustic fluctuations from the near-field source region to the far-field microphones within the computational domain would be prohibitively expensive. The Ffowcs Williamsâ€“Hawkings (FW-H) equation31 is one of the most commonly used methods to overcome this difficulty. Sound at a far-field location can be computed from flow information on an arbitrarily-shaped surface S and the volume-distributed sources outside of S. If S corresponds to a physical solid surface (e.g., an helicopter blade or aircraft landing gear) the FW-H formulation is referred to as solid (impenetrable), and as permeable (porous) otherwise. One advantage of the permeable formulation is that it allows for the acoustic sources in the volume V outside the solid surface but inside the data surface S to be taken into account. The method has been successfully applied to a wide range of problems, including aircraft fly-over and rotorcraft noise (for instance, see the review by Brentner & Farassat32 ). Unlike these applications which involve arbitrary moving noise sources in a quiescent environment, jet flow configurations are typically considered in a fixed laboratory frame and, potentially in the presence of a uniform coflow to account for

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flight effects. Since the distance between the noise sources (i.e., stationary surface S enclosing the jet) and the observers (i.e., far-field microphones) is fixed and time-independent, there is no Doppler effect and an efficient frequency-domain permeable formulation33, 34 can be used. The formulation and its implementation for the prediction of jet noise in Cascade’s massively-parallel unstructured LES framework are discussed in Appendix.

III.

Flow configuration

The simulation setups A1, A2 and A3 reproduce the unheated ideally-expanded jet (B118), heated ideally-expanded jet (B122) and heated over-expanded jet (B122 OD, where OD stands for “Off-Design”) conditions tested at UTRC’s acoustic research tunnel facility,35, 36 respectively. Additional simulations of these conditions are currently ongoing, but only the initial LES corresponding to the blind comparisons are reported in this paper. A.

Numerical setup

The usual compressible formulation is used to nondimensionalize the Navier–Stokes equations, i.e., ρ=

ρ∗ , ρ∗∞

p=

p∗ p∗ , = γp∗∞ ρ∗∞ c∗∞ 2

T =

T∗ , ∗ T∞

u∗i , c∗∞

ui =

xi =

x∗i , D∗

t=

t∗ c∗∞ , D∗

f=

f ∗ D∗ , c∗∞

where the superscript ∗ refers to the dimensional quantity, and the subscripts ∞ and i denote the ambient (free-stream) property and the ith cartesian coordinate, respectively. p This nondimensionalization is based on an ambient speed of sound c∞ = γp∞ /ρ∞ , where γ = 1.4. The resulting form of the ideal gas law is p = ρT /γ. The Strouhal number typically defined as St = f D/Uj is related to the nondimensional frequency (i.e., the Helmholtz number) by the acoustic Mach number Ma = Uj /c∞ , namely St = f /Ma . The subscripts j refer to the fully-expanded jet properties and Uj is the equivalent fully-expanded jet velocity. The temperature dependence of viscosity is assumed to follow the power-law µ ∝ T m with constant coefficient m = 0.7. The round nozzle geometry is included in the computational domain and corresponds to UTRC convergingdiverging (CD) nozzle designed using a method-of-characteristics to provide ideally-expanded (i.e., shockfree) flow at the nozzle exit, with design Mach number of 1.5. The operating conditions are defined by the nozzle pressure ratio (NPR) and the nozzle temperature ratio (NTR). For off-design configurations, the definition of the operating conditions can be ambiguous. In this work, the nozzle pressure and temperature ratios, the jet Mach number Mj and temperature Tj , and the Reynolds number Rej are defined as NPR =

p0 , p∞

NTR =

T0 , T∞

Mj =

Uj , cj

Tj =

Tj , T∞

Rej =

ρj Uj D , µj

(1)

where the subscript 0 refers to the stagnation (total) properties. For all the cases, the nozzle exit diameter D is retained as reference scale. Table 1 summarizes the operating conditions and parameters of the simulations. Table 1. Summary of operating conditions and simulation parameters. The subscripts j refer to the fullyexpanded jet properties and δθ /D is the momentum thickness of the boundary layer at the nozzle exit.

LES case

Experiment

Description

NPR

NTR

Mj

Ma

Tj

Rej

δθ /D

A1 A2 A3

B118 B122 B122 OD

cold ideally-expanded hot ideally-expanded hot over-expanded

3.67 3.67 2.97

1.45 2.53 2.53

1.5 1.5 1.35

1.5 1.98 1.84

1.0 1.74 1.85

300,000 155,000 130,000

0.0017 0.0022 0.0029

The jet configuration is presented in figures 3(a) and (b), including the default FW-H surface used to compute the far-field sound (see section V for further analysis), and the sets of near-field probes (D2P 1 − D2P 17) and far-field microphones (D2M 1−D2M 12) in the UTRC experimental configuration, which match the LES microphone stations. Note that the more recent experimental results used in this paper correspond to a nozzle size of 2 inches, while previous comparisons15, 16 were done for a 3 inch nozzle35, 36 at the near-field 5 of 21 American Institute of Aeronautics and Astronautics

(a) Computational domain

(b) Far-field microphones

Figure 3. Schematics of the flow configuration and numerical setup.

microphones (P 1 − P 8) and far-field microphones (M 1 − M 9). The new experimental setup has more probes and therefore better resolution of the near-field flow, and the microphone distance is increased from 40D to 60D for a better approximation of the far-field conditions. B.

Boundary conditions

The round nozzle exit is at x = 0, and the axisymmetric computational domain increases with the jet diameter, extending to approximately 45D and 20D in the streamwise and radial directions, respectively. The jet exhausts into an anechoic chamber which is subject to a wind tunnel coflow with Mach number M∞ = U∞ /c∞ = 0.1, to avoid overheating the chamber. The same M∞ = 0.1 free-stream flow condition is used in the simulations. At the downstream outlet of the computational domain, a damping function7 is applied as a source term in the governing equations, such that the turbulent structures and sound waves will be damped in the outflow buffer before reaching the outlet boundary. Similarly, a sponge layer is applied at the upstream and radial outlets of computational domain by switching the numerical operators to lower-order dissipative discretization. A constant plug-flow is applied to the inlet of the nozzle such that the desired Mach number and the temperature ratio are achieved at the nozzle exit plane. It should be noted that we assume that the flow issued from the nozzle is laminar.a Consequently, the grid resolution inside the nozzle is only adequate for laminar flow and the value of the Reynolds number is reduced in the computations since there is no wall model inside the nozzle. For all cases, the experimental Reynolds number is about 5.5 times higher than in the computations. The momentum thickness of the boundary layer at the nozzle exit δθ /D is estimated and reported in the table 1. The use of such thin laminar boundary layers is a convenient and popular approach,4, 10, 11, 30 as it leads to rapid transition to turbulence near the nozzle exit while enabling coarser resolution inside the nozzle. C.

Computational grid

Based on previous grid resolution studies,15, 16 the refined mesh used in all the simulations contains approximately 42 million unstructured control volumes, mostly hexahedral. In general, hexahedral grids30 are found less dissipative than tetrahedral grids37, 38 for unstructured LES of jet noise. With the current standard in high-performance computing and unstructured LES, the present grid would be considered of medium size, compared to larger grids20, 21 used in recent studies. a In

the experiment, it is not known whether the flow issued immediately after nozzle is laminar or turbulent.

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The mesh consists of a fully unstructured core which transitions to a purely axisymmetric grid with 160 points in the azimuthal direction within the FW-H surface, and further stretching towards to domain boundaries. The cell spacing D/∆ at different streamwise locations is reported in table 2. While higher resolution is used near the nozzle exit and in the shear-layers, the ratio of radial and axial mesh-size is close to unity in the bulk of the mesh (i.e., 1 ≤ x/D ≤ 25) within the FW-H surface, with limited grid stretching, as discussed in section IIB. Table 2. Cell spacing at different streamwise locations along the FW-H surface, and estimated Strouhal number Stlim of an acoustic wave discretized with 8 points per wavelength.

Location x/D Cell spacing D/∆ Stlim case A1 case A2 case A3

0 300 25 18.9 20.4

1 71 5.9 4.5 4.8

2 65 5.4 4.1 4.4

5 62 5.2 3.9 4.2

10 60 5 3.8 4.1

20 54 4.5 3.4 3.7

25 52 4.3 3.3 3.5

Following the analysis of Mendez et al.,30 a limit Strouhal number Stlim of acceptable resolution, corresponding to waves resolved with 8 grid points per wavelength, can be estimated based on the resolution along the FW-H surface. Since the high-frequency noise sources are typically expected between the nozzle exit and the end of the potential core (i.e., 0 < x/D . 10), the present mesh is designed to accurately resolve far-field noise spectra up to St ≈ 4 to 5, with grid cutoff expected around St ≈ 8 to 10 (i.e., 4 points per wavelength), depending on the simulation (assuming sufficiently high sampling frequency of the input data, as discussed in the next section). D.

Simulation and post-processing procedures

In general, the LES are restarted from a statistically steady solution obtained from a coarser mesh and interpolated on the finer mesh. To remove the initial transient after interpolation, the simulations are typically ran for 60 to 90 time units, corresponding to the time of propagation of 2 to 3 acoustics waves from the nozzle exit to the start of the downstream outflow buffer. Then, snapshots of the whole transient flow field are collected every ∆tc∞ /D (i.e., sampling Strouhal ∆St = ∆f D/Uj ), rather than every time step because of file size constraints, for a total duration τ c∞ /D. Table 3 summarized the characteristics time parameters of the simulation and post-processing. The simulation time step is reduced for the heated case A2 because of CFL constraints due to the higher jet velocity. Note that the total duration corresponds to 220 to 320 convective time units D/Uj (depending on the simulation), which can be considered as a long time sample for high-fidelity LES. Table 3. Characteristics time parameters of the simulation and post-processing. The simulation time step, sampling period and total duration are dtc∞ /D, ∆tc∞ /D and τ c∞ /D, respectively. The Strouhal numbers Stmin and Stmax are the minimum and maximum frequency accessible by the post-processing.

LES case

dtc∞ /D

∆tc∞ /D

τ c∞ /D

Stmin

Stmax

A1 A2 A3

0.0002 0.0001 0.0002

0.02 0.02 0.02

215 112 142

0.0031 0.0045 0.0038

16.7 12.6 13.6

The maximum frequency accessible is the Nyquist Strouhal number Stmax = ∆St/2. Post-processing of the same LES database with different sampling frequencies (i.e., far-field noise prediction using every snapshots, one snapshot out of two, etc. ) show that the cutoff at high frequency is usually related to the grid resolution rather than the sampling frequency. Since the FW-H solver is implemented in the frequency domain using spectral methods, the results are reliable almost up to the Nyquist frequency. In practice, this means that the spectra cutoff can be estimated as min(Stmax , 2Stlim ). Similarly, Stmin = D/(τ Uj ) is the narrowband width and minimum frequency accessible by this postprocessing. In practice, the relatively short time signal in computation can lead to rather “peaky” narrow-

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band spectra, in some cases with large variations in the low frequencies due to lack of statistical convergence. For comparison with the smoother spectra obtained in experiment, it is convenient to computed bin-averaged spectra. For all the noise spectra calculations in the paper, the same post-processing procedure is applied to both numerical results and measurement data: a dimensional bin-averaged PSD (in dB/St) is computed from the narrowband data, with bin size ∆Stbin = 0.05. The total duration of the simulation yields approximately 16, 11 and 13 periods at St = 0.05 (i.e., the lowest frequency considered), for the case A1, A2 and A3, respectively. Post-processing of the same LES database with different total duration (i.e., far-field noise prediction using every snapshots, half of snapshot, etc. ) show that at least 10 periods of the frequency of interest is recommended for reasonable statistical convergence of the low-frequency noise spectra. Finally, if the calculation is applied to a jet with sufficient symmetry (e.g., round nozzle, chevron nozzle with small penetration angles,19 etc. ), microphones distributed along the azimuthal angle are statistically identical. As a result, the calculations of the spectra can benefit from the symmetry in these individual microphones and are ensemble-averaged to achieve better statistical convergence. E.

Computational resources and performances

The LES code “Charles” is designed and implemented using Message Passing Interface (MPI) to function well in a massively-parallel, distributed memory environment. Because the numerical method is purely hyperbolic and fully explicit, the code requires relatively little communication at each time-step, as compared to other methodologies. Scalability tests on different systems showed that the code exhibits perfect scaling at the 20,000 core level39 and nearly linear scaling up 160,000 cores.20 Here, the majority of calculations were carried out on CRAY XE6 machines at DoD supercomputer facilities in ERDC and AFRL. Each simulation ran for a week on 1024 cores, for a computational cost of approximately 172 Kcore-hours. Note that even thought the LES cases A1 and A3 were performed for the same mesh, same number of processor and same number of core-hours, the total simulation time τ of the isothermal ideally-expanded case is almost 35% greater than for the heated over-expanded case, as shown in table 3. In the latter case, shocks are presents in the flow and the shock-capturing schemes require additional operations. However, the shock sensor, which identify control volumes containing shocks, is active for less than 1% of the cells. Consequently, these additional operations are only performed by the few processors which have shock-actived control volumes assigned to them, while the other processors are idle. One of the lesson learned from these simulations and from the earlier work of Nichols et al.18 is that improved performance can be obtained by enforcing processor load-balancing based on the shock sensor (i.e., the active cells have a computational “weight” greater than cells without shocks). This change in implementation is critical to obtained linear scaling for refined jet simulations with shocks on very large number of cores.20, 21 Since the whole transient flow field is stored, these simulations yield large databases, between 10TB and 18TB each. These databases reside in archival storage at ERDC and are currently continuing to be used for post-processing and analysis, probing the physics of jet noise production. For practical use where only a few specific measurements are of interest (e.g., thrust loss, near-field and far-field noise, etc. ), only a subset of the flow field needs to be recorded (e.g., mean measurements, transient data on FW-H surface, etc. ), significantly reducing the data storage requirement.

IV.

Blind comparisons with experiment: cold jet

This section focuses on the unheated ideally-expanded jet case A1 and the blind comparison with the corresponding experiment B118 (see table 1 for details). The experimental post-processing for the comparison with cases A2 and A3 is currently ongoing and will be presented in future publications. However, some LES results for these cases are reported in section V. A.

Visualization and qualitative description of the flow

An instantaneous snapshot of the vorticity field is shown in figure 4 and highlights the detailed turbulent structures resolved in the simulation. In this figure, the computational mesh is partially shown in planes normal to the axis of the jet, to illustrate the unstructured mesh capabilities. As expected, the thin laminar shear layer issued from the nozzle quickly transitions to turbulence, within one diameter from the nozzle exit. The turbulent flow mixes as it develops farther downstream of the nozzle, without significant grid transition effects. 8 of 21 American Institute of Aeronautics and Astronautics

Figure 4. Instantaneous vorticity magnitude for the isothermal ideally-expanded jet case A1. The cross-flow cuts of the vorticity magnitude and the mesh are shown in top and bottom subfigures. The dashed circular line corresponds to the nozzle exit outline.

A more global view of the transient flow field is presented in figure 5, showing the flow (visualized by the contours of temperature in red scale) and sound (visualized by contours of pressure in gray scale). The dominant noise radiation towards aft angles is clearly visible. Sound waves are emitted mainly in the vicinity of the nozzle exit, as well as downstream of the jet. Note that the visualized sound waves far from the jet plume do not represent the actual sound that can be computed using the FW-H solver. As discussed in section II, the coarser resolution and lower-order dissipative discretization outside of the acoustic sourcecontaining region (i.e., the zone enclosing the FW-H surface) leads to the intended smooth but quick damping of the pressure waves in the computational domain. Similar to laboratory experiments, exact pressure matched condition cannot be achieved in simulation, and residual shock cells are formed in the vicinity of the nozzle. The shock cell structures are visible in figure 5, and their presence suggests that the nozzle is operating slightly off design. As discussed in Ref. 15, the shocks were minimized in the experimental studies by varying the nozzle pressure ratio over a small range centered at the design value and determining when the minimum sound level was observed at the 90â—Ś far-field microphone. For the 2-inch nozzle, this corresponded to N P R = 3.55, N T R = 1.49 and Mj = 1.477 (rather than Mj = 1.5) for the unheated condition which was considered to be the shock free operating condition. For both simulations A1 and A2, only residual and weak shocks are present and the shock-capturing scheme discussed in section IIA is not active.

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Figure 5. Instantaneous pressure and temperature field for the isothermal ideally-expanded jet case A1. The colored contours show T from 1 to 1.15, while the gray scale contours show P from 0.704 to 0.724.

B.

Flow statistics

The measurements of the time-averaged streamwise velocity along the jet axis and in different cross-flow planes are compared to the azimuthal-averaged LES predictions in figure 6. The results show good agreement with experiments, as the centerline decay and length of the potential core are accurately captured in the simulation. The potential core length, taken as the distance up to which the flow velocity is greater than 95% of the jet exit velocity, is approximately 9.7D in the simulation and 10.2D in the experiment. In general, this quantity is challenging to predict numerically. Under-prediction is often observed in simulations and is typically associated with lack of resolution in the jet plume15, 40 and with initially laminar jets transitioning to turbulence outside of the nozzle.3, 4, 30 At the farther downstream station x/D = 15, the maximum velocity is predicted within 5% difference. This mismatch is essentially due to the slight under prediction of the potential core-length in figure 6(a) . C.

Near-field noise

Two different methods are used to compute the noise at the experimental near-field probes. The first method corresponds to the computation and propagation of the pressure fluctuations directly in the simulation (i.e., recording the pressure signal from simulation at specified locations). The LES pressure time histories are recorded over 48 near-field probes equally-spaced in the azimuthal direction and the spectra are azimuthalaveraged. The second method makes use of the FW-H solver mentioned in section IIC and described in more details in Appendix. Since there is no “far-field” approximation in the current algorithm, aside from the quadrupole terms being neglected, the far-field noise solver can be used to predict the noise at any location outside of the FW-H surface. Here, all the near-field probes are located outside that surface, as shown in the schematic in figure 3(a). To reproduce the LES setup and the presence of the coflow M∞ = 0.1 in the computation, the FW-H predictions are performed with the same coflow and at the same 48 near-field probes, with spectra averaging in the azimuthal direction. Figure 7 presents the near-field sound spectra for the LES case A1 and the blind comparison to experimental measurements (B118) at nine different probe stations. Overall, the agreement between both predictions and the experimental data is excellent over the whole frequency range, from St = 0.05 up to the experimental cutoff frequency of St ≈ 4. Similar agreement is obtained for the other probes locations.

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1.0

0.8

0.8

0.6

0.6

u ¯/Uj

u ¯/Uj

1.0

0.4 0.2

0.2 0

5

10

20 15 x/D (a) centerline

25

0

30

1.0

1.0

0.8

0.8

0.6

0.6

u ¯/Uj

u ¯/Uj

0

0.4

0.4 0.2 0

-2

-1

1 0 r/D (b) at x/D = 0

2

-2

-1

0 1 r/D (d) at x/D = 15

2

0.4 0.2

-2

0 r/D (c) at x/D = 5

-1

1

0

2

Figure 6. Time-averaged streamwise velocity for the isothermal ideally-expanded jet case A1: horizontal ( ◦ ) and vertical ( ) experimental measurements; ( ) azimuthal-averaged LES.

For probes closer the nozzle, the LES results match exactly the FW-H predictions up to St ≈ 2. The under-prediction of the FW-H results at the higher frequencies could be due to the contributions from the quadrupole (i.e., volume terms) which are not negligible in the near-field but not computed with the current formulation. Overall, the agreement between the spectra computed within the LES solver and predicted with the FW-H solver provides further evidence that the different methods accurately capture the noise generation and propagation. Similar comparisons were done for a chevron nozzle in Ref. 19, with the same conclusions. For the probes farther downstream, such as probe D2P 16 at x/D = 18 in figure 7(i), some over-prediction is observed in the LES results for St > 0.7. Higher spectra levels at high frequencies directly predicted by LES indicates lack of resolution at these regions. The computational grid cannot support the propagation of small structures at such high frequencies. Because of the low-dissipative numerical scheme, the energy associated with these structures appears as numerical noise and contaminates the solution at high frequencies. D.

Far-field noise

Figure 8 shows the blind comparisons of the far-field sound for the unheated jet case A1 as a function of the experimental microphone stations in figure 3(b). For each microphone location, the FW-H calculation is performed at 72 stations equally-spaced in the azimuthal direction, and the far-field spectra are azimuthalaveraged. Two sets of far-field experimental results are presented in this figure. The first set ( ) is the spectra “as measured” in the experiment. The second set ( ◦ ) corresponds to the approximated ”lossless” spectra, in which a correction is applied to compensate for the atmospheric attenuation. The correction is a function of the distance of propagation and frequency. As shown in figure 8, its main effect is to increase the high 11 of 21 American Institute of Aeronautics and Astronautics

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Figure 7. Blind comparison of the sound spectra at different near-field probe locations: ( ◦ ) experiment; ( ) prediction from FW-H solver, coflow M∞ = 0.1; ( ) direct calculation in LES.

frequency levels, for St > 2. Based on the nozzle diameter size D = 2 inches and the jet velocity Uj = 1.5c∞ m/s, the experimental Strouhal number St = 2 corresponds to a dimensional frequency of about 20,000 Hz. At such high frequency, atmospheric attenuation is not negligible and needs to be accounted for. It is also important to note that the experimental data has no shear-layer correction at the moment: the noise radiated from the primary supersonic jet propagates across the entire coflow jet and through its mixing region toward the far-field microphones in the ambient fluid at rest (see figure 3(b)). This scenario cannot be reproduced easily with computations. Here, the FW-H predictions are performed without coflow (M∞ = 0), i.e., uniform fluid at rest assumed between the noise sources (i.e., the FW-H surface) and the microphones. However, the input to the far-field noise solver is the transient flow-field on the FW-H surface recorded for the simulation with coflow M∞ = 0.1 as free-stream flow condition, matching the experimental condition. The influence of coflow and forward flight effects will be investigated in future work. Overall the agreement is very good for these aft angles, in particular for microphones D2M 9 (φ = 140◦ ), D2M 10 (φ = 145◦ ), and D2M 11 (φ = 150◦ ), which correspond to the loudest angles. For the high frequencies, a better match is obtained with the approximated ”lossless” experimental spectra since the atmospheric attenuation is not taken into account in the FW-H solver. For most angles, the agreement extends from the low frequency of St = 0.05 up to St ≈ 10, which approximately corresponds to the cutoff frequency of both the experimental microphones and numerical predictions.

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Figure 8. Blind comparison of the sound spectra at different far-field microphone locations: ( ) experiment, “as measured”; ( ◦ ) experiment, lossless correction; ( ) prediction from FW-H solver, coflow M∞ = 0.

V.

Parametric studies of far-field noise predictions

Much like most jet noise studies in the literature, the current implementation of the FW-H solver makes use of a permeable surface S enclosing flow-generating sound sources, with the volume terms neglected (see Appendix). There is however no clear consensus in the literature on optimal location of the FW-H surface, treatment of the outflow disk and formulation details. Several parametric studies have been done in the past, by Rahier et al.,41 Uzun et al.,42 Shur et al.,10, 11 Spalart & Shur12 and Mendez et al.,28 among others. The purpose of this section is to document some of the results, analysis and conclusions obtained on these topics with high-fidelity unstructured LES. A.

Pressure-based FW-H formulation

Spalart & Shur12 argued that neglecting the volume term can be erroneous, in particular for simulation of hot jets where the surface S fails to entirely enclose the region of turbulence and important entropy fluctuations can therefore be expected to cross the FW-H surface. Starting from the study in Ref. 10, their work showed a reduction of this error with the use of a pressure-based variant of the original formulation, in which the value of density ρ = ρ∞ + ρ′ on the FW-H surface is not directly extracted from the simulation but instead computed using the pressure fluctuations, i.e., ρ = ρ∞ + p′ /c2∞ . Morfey & Wright43 also presented 13 of 21 American Institute of Aeronautics and Astronautics

similar modifications of the FW-H equation, derived from earlier work.44 The recent parametric studies by Mendez et al.28 yielded the same conclusion that the pressure-based formulation seems better suited for jet aeroacoustics, in particular for heated jets. Its added benefit is that only four variables (i.e., three components of velocity and pressure) need to be saved on the FW-H surface since the density is no longer required, consequently reducing storage requirements. The pressure-based FW-H formulation is therefore used is all the calculations in the present work. B.

Location of FW-H surfaces

For all three LES cases, calculations of the far-field noise directivity at 100D from the nozzle exit were performed for three different FW-H surfaces: S0 (tight), S1 (default) and S2 (loose), as shown in figure 9. The surfaces first follow the external nozzle shape then radially flare starting at (x/D = 0, r/D = 0.6), with slope 0.95, 0.11 and 0.13 for S0 , S1 and S2 , respectively. Here, the slopes are chosen based on rough estimate of the jet spreading rate.45–47 The distance 100D is a typical choice48 for the approximation of the far-field conditions.

Figure 9. Outline of the different FW-H surfaces considered: surfaces S0 ( )(tight), S1 ( ) (default), and S2 ( )(loose), with 16 end-caps; surface S3 ( ) extending in the outflow buffer and with open outflow disk.

In general, the choice of location of the FW-H surface is a trade-off between numerical resolution (i.e., how close to the jet should the surface be to ensure accurate propagation of the acoustic waves within the flow solver ?) and noise source containment (i.e., how far from the jet should the surface be to enclose all relevant acoustic sources ?). While the present mesh topology was designed a priori to provide sufficient numerical resolution, the question of acoustic source containment remained. Figure 10 shows azimuthalaveraged time statistics along the different FW-H surfaces, as an indication of the turbulence crossing these surfaces. A more rigorous definition, for instance based on the turbulence associated momentum flux or vorticity fluctuation will be considered in future work. As expected, the surface S0 , which was placed more aggressively near the jet plume, exhibit higher mean and rms levels for both streamwise velocity and pressure. As the surfaces S1 and S2 move outwards from the jet, the levels of fluctuation decrease and the mean streamwise velocity converges towards the coflow velocity M∞ = 0.1 downstream of the jet. 0.12 0.03 0.03 urms /c∞

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Figure 10. Flow statistics along the different FW-H surfaces (see details in figure 9): surfaces S0 ( S1 ( ), S2 ( ) and S3 ( ).

40

),

Figure 11 shows the far-field noise spectra at inlet angle φ = 60◦ , 90◦ , and 150◦ , for all three LES cases. First, comparing A1 to A2 and A3, the main expected features tend to be captured in the simulations. The 14 of 21 American Institute of Aeronautics and Astronautics

P SD (dB/St)

φ = 60◦

heated case A2 has higher jet velocity Uj than the isothermal case A1, and therefore similar shape for the spectra but slightly shifted towards the lower frequency and with higher levels. For the over-expanded case A3, the presence of the shocks introduces new features in the far-field noise. The bump in the spectra around St ≈ 0.5 at the upstream angle φ = 60◦ corresponds to the broadband shock-associated noise (BBSAN). As predicted by theory and generally observed in experiment, the BBSAN shifts to higher frequency and decreases in amplitude as function of the direction of radiation,49–51 with lowest value for higher inlet angle. This is clearly visible for the sideline and downstream microphones at φ = 90◦ and 150◦ . Aside from these differences due to the operating conditions, the results showed nearly identical spectra over the whole frequency range for the three FW-H surfaces, despite the differences in the estimated turbulence crossing these surfaces shown in figure 10. This lack of sensitivity to the FW-H surface location is a good indication of the robustness of the methods. Here, the careful design of the mesh, with isotropic cells and limited grid stretching in the bulk of the acoustic source-containing region, was likely instrumental to these results. Except for the case A2 at φ = 90◦ in figure 11(b), there is no significant changes of the slope of the spectra at high frequency, up to St ≈ 8 − 10. These values are similar to the estimated grid cutoff 2Stlim discussed in section IIC. 140 140 140 120

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Figure 11. Sound spectra at 100D for different inlet angles and FW-H surfaces (see details in figure 9): surfaces )(tight), S1 ( )(default), and S2 ( )(loose), with 16 end-caps; surface S1 with open S0 ( outflow disk at x/D = 20 ( ), at x/D = 30 ( ), and closed with 1 end-cap at x/D = 30 ( ); surface S3 ( ) extending in the outflow buffer, with open outflow disk.

C.

Treatment of the outflow disk

Different school of thought exists when it comes to the treatment of the outflow disk for jet noise predictions. The first approach is to simply defined a closed permeable surface, as expected by the Ffowcs Williams & Hawkings equation. In many studies,41, 42 the FW-H surface is however left open at the outflow, to avoid spurious sound generated by vortices passing through the downstream face. In contrast, Shur et al.10 15 of 21 American Institute of Aeronautics and Astronautics

introduced a third option: averaging over outflow disk, also referred to as the method of “end-caps”. In this approach, the complex far-field pressure predicted from several FW-H surfaces with the same shape but outflow disks at different streamwise locations are phase-averaged, leading to the reduction or cancellation of the uncorrelated noise produced by the turbulent eddies crossing the successive outflow disks. The same concept can also be used to remove spurious high-frequency noise, by computing phase-average of the farfield sound over several embedded FW-H surfaces similar shape, as shown in Ref. 13. Another potential approach would be to account for the quadrupole noise associated with the flow in the downstream vicinity of the outflow disk(s), for instance with a parallel-flow Lilley-equation based acoustic analogy. Figure 11 illustrates the influence of the outflow disk on the noise spectra. While the predictions presented up to this point used the approach of Shur et al.,10 with 16 end-caps uniformly distributed from x/D = 20 to 30, additional FW-H calculations are performed with the default surface S1 where the outflow disk is left open at x/D = 20, at x/D = 30 and closed with a single end-cap at x/D = 30. For all three LES cases, these approaches lead to over-predictions in the low frequencies (up to 10 dB), in particular at the upstream and sideline angles. At the downstream angles (i.e., φ = 150◦ ), the physical noise becomes dominant and its amplitude is sufficiently high to mask this spurious noise. The same trend is observed when the strength of the noise sources increases through heating (i.e., higher jet exit velocity): the low-frequency spurious noise is more noticeable for the isothermal case A1 than for the hot case A2, in figure 11 (a) and (b), respectively. Here, these results are consistent with the analysis and conclusions reported by Shur et al.10 and Mendez et al.28 As a last parametric study, the surface S1 was extended in the outflow buffer to the end of the computational domain and left open at the outflow disk, resulting in surface S3 shown in figure 9. Comparing the thick solid red line with the black dashed line in figure 11, the predicted far-field noise spectra are essentially identical to the result from S1 with the end-caps. In the outflow buffer, the damping function and coarsening of the grid essentially drives the flow outwards steady state, such that the fluctuations on the corresponding parts of the FW-H surface can be expected to be small and their contribution to the total noise should be negligible. The flow statistics in figure 10 along the FW-H surface S3 at 30 ≤ x/D ≤ 45 would tend to confirm this trend. Likewise, the turbulent eddies are smoothly damped in the outflow buffer and their interactions with the FW-H surface should be minimal. This could be one potential reason (among other) why studies using open surfaces do not systematically report over-predictions in the low frequencies. Here, the hypothesis would be that, as long as interactions between the flow vortices are the FW-H surface are limited (because of coarser resolution, grid stretching, more dissipative schemes, etc. ), the spurious low frequency noise would likely be limited as well. Additional analysis with different LES databases and FW-H solvers would be needed to further investigate the cause of these discrepancies.

VI.

Conclusions and discussions

The paper is intended as a review of some of the lessons learned so far on jet noise predictions with unstructured large eddy simulation (LES), as well as a demonstration of the current predictive capabilities put to practice. The compressible flow solver “Charles” developed at Cascade Technologies is used to conduct LES of supersonic jets issued from a converging-diverging round nozzle at three operating conditions relevant to laboratory testing and tactical aircraft: isothermal ideally-expanded, heated ideally-expanded and heated over-expanded. In the numerical study, the nozzle geometry is explicitly included in the computational domain, using unstructured body-fitted mesh. Noise radiation from the jet is computed using an efficient frequency-domain implementation of the Ffowcs Williams–Hawkings equation, based on the LES data collected on a permeable surface located in the jet near-field. The initial comparisons between LES and experiments are very encouraging in terms of flow field as well as near-field and far-field acoustic spectra. It is important to note that all the simulations and postprocessing were performed without prior knowledge of the experimental data, such that the results presented in this paper correspond to “blind” comparisons. Additional analyses that leverage the large LES database recorded are also presented in the paper, in particular parametric studies of the far-field noise predictions. In future work, guidance from the experimental results will be used to further improve the comparisons and continue the development of methodologies towards best practices for LES-based jet noise predictions. From the results and analysis presented in this paper and in previous jet noise studies with high-fidelity unstructured LES, comments and guidelines can be suggested: (a) One of the first and more critical aspect of jet noise prediction is the careful design of the mesh for

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aeroacoustics: here, sufficient resolution in the shear-layer, as well as cell isotropy and limited grid stretching in the acoustic source-containing region, are strongly recommended. (b) Likewise, the importance of the computational methods used in the flow solver needs to be stressed: numerical schemes with low dispersion and dissipation are essential to aeroacoustics predictions. (c) Inclusion of the nozzle in the computational domain will only become more important as more complex geometries like chevrons and faceted nozzles are being investigated numerically. Unstructured solvers seem ideally suited to handle such complex cases. (d) For fair comparison with measurements, attention to the details of the experimental procedure is important (e.g., shear-layer correction, atmospheric attenuation, far-field assumption, consistent postprocessing of time signal for spectra computations, etc. ) (e) Special care is also required for the time signal processing: windowing of the time data should be used to avoid spectral leakage; the total (statistically converged) simulation time should be at least 10 periods of the lowest frequency of interest, for reasonable convergence of the acoustic spectra; the transient flow sampling frequency should be at least twice the grid cutoff frequency, to capture high frequencies. (f) To further improve statistical convergence of the numerical data, azimuthal averaging of the far-field spectra (over 18 to 36 positions) can be used for jet with sufficient level of azimuthal symmetry (e.g., round jets, chevron nozzle with small penetration angles, etc. ) (g) To limit spurious noise cause by vortices crossing the side parts of the FW-H surface, the pressurebased formulation,12, 43 rather than the original FW-H formulation, is recommended, in particular for heated jets. Its additional benefit is the reduction in file size, since the density on the FW-H surface is no longer stored. (h) While different alternative exist for the treatment for the outflow disk, a closed FW-H surface with the method of end-cap10 tends to lead to better acoustic results: it is a robust and reliable method, with limited performance penalty. (i) In terms of performance, processor load-balancing based on the shock sensor was found critical for the flow solver to retain linear scalability on large number of cores, when shocks are present in the flow. For the far-field noise solver, the frequency-domain formulation33, 34 of the FW-H equation is a robust and efficient method ideally suited for jet aeroacoustics (including forward flight effects). (j) Other important aspects of the jet noise problem which were not addressed in details in the present study include characterization of the nozzle boundary layers (e.g., evaluation of the momentum thickness at nozzle exit), appropriate inflow and outflow boundary conditions, shock capturing schemes, effect of subgrid scale model, etc.

Acknowledgments This work is supported by NAVAIR STTR program under grant number N68335-11-C-0026 managed by Dr. John Spyropoulos. The simulations were performed as part of the HPC Challenge Project NAWCP30952C5 entitled “Large Eddy Simulations of Supersonic Jet Noise and Combustor & Augmentor Spray Atomization”. The majority of calculations were carried out on CRAY XE6 machines at DoD supercomputer facilities in ERDC and AFRL. The authors would like to thank Robert Schlinker, John Simonich and Ramons Reba of United Technologies Research Center, for providing the nozzle geometry, the experimental results, and conducting the blind comparisons.

Appendix For prediction of jet noise, the acoustic analogy module was developed in Cascade’s massively-parallel unstructured LES framework based on the early work of Ffowcs Williams & Hawkings31 and its extension by Lockard.33, 34 For the specific case of a stationary source in a medium moving at constant subsonic speed U = (U1 , U2 , U3 ), the convective form of the Ffowcs Williams–Hawkings equation for a permeable surface S is given by33 2 ∂2 ∂2 ∂ ∂ ∂2 ∂ 2 − c + 2U + U U [(ρ − ρ∞ )H(S)] = [Qn δ(S)] − [Fi δ(S)] j i j ∞ ∂t2 ∂xj ∂xj ∂t∂xj ∂xi ∂xj ∂t ∂xi (2) ∂2 + [Tij H(S)] , ∂xi ∂xj 17 of 21 American Institute of Aeronautics and Astronautics

where the variables Qn and Fi are defined as Qn = Qi n ˆ i = [ρ(ui + Ui ) − ρ∞ Ui ] n ˆi,

(3)

Fi = [Pij + ρ(ui − Ui )(uj + Uj ) + ρ∞ Ui Uj )] n ˆj .

(4)

The bold font denotes vector quantities and the summation convention is used for repeated vector component indices i,j. The subscript ∞ indicates a ambient quantity and c∞ is the speed of sound. The unit outward normal to the surface is n ˆ , and ui are the local fluid velocities on S. The Heaviside function H(S) and the Dirac delta function δ(S) are present to account for the discontinuity of the functions at the surface interface. In equations (2) and (4), the Lighthill stress tensor and the compressive stress tensorb are defined as Tij = ρui uj + Pij − c2∞ (ρ − ρ∞ )δij , and Pij = (p − p∞ )δij − σij , respectively, where σij is the viscous stress tensor and δij is the Kronecker delta. In general, the viscous term in Pij is a negligible source of sound and is typically neglected, i.e., Pij = (p − p∞ )δij . Note that the specific form of the convective wave operator on the right hand side of equation 2 takes into account the presence of the mean flow. The terms Qn and Fi also include mean velocity terms, while the Lighthill stress tensor Tij is unchanged. When U = (0, 0, 0), the equations simplify to the classical FW-H formulation and source term definitions. Equation 2 is in a form suitable to perform Fourier analysis, and becomes h i 2 ∂2 ∂2 ∂2 2 e n δ(S) + ∂ Fei δ(S) + k − 2iMj k − Mi Mj c∞ ρ˜′ H(S) = − iwQ ∂xj ∂xj ∂xj ∂xi ∂xj ∂xi (5) i h 2 ∂ e Tij H(S) , − ∂xi ∂xj

after application of the Fourier transform pair defined as Z +∞ Z +∞ 1 g(t) exp(−iωt)dt, g(t) = g˜(ω) exp(iωt)dω, g˜(ω) = 2π −∞ −∞

(6)

where ω = 2πf is the angular frequency, k = ω/c∞ is the wavenumber and Mi = Ui /c∞ . Without loss of generality, it can be assumed that the mean flow velocity is in the +x1 directionc , i.e., U = (U∞ , 0, 0). For subsonic wind-tunnel Mach numbers M∞ = U∞ /c∞ < 1, the three-dimensional free-space Green’s function for equation 5 is G(x, y) = where the distance R is defined as R= with R∗ = and

p

−1 exp (−ikR) , 4πR∗

−M∞ (x1 − y1 ) + R∗ , β2

(x1 − y1 )2 + β 2 [(x2 − y2 )2 + (x3 − y3 )2 ], β=

p

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(8) (9) (10)

Much like in the time-domain formulation with convective effects,53–55 R represents the effective acoustic distance (rather than the geometric distance) between the source y and the observer x in terms of time delay between emission and reception, in the presence of coflow. The solution of equation 5 can be further simplified33 by moving the Green’s function inside the spatial derivative operator, applying Green’s theorem, and using the properties of the Dirac delta function. Also, outside of the source region, density perturbations are expected to be small, so the term c2∞ ρ˜′ is replaced by the acoustic pressure p˜′ . b While P = pδ −σ is the compressive stress tensor as defined by Lighthill,52 Ffowcs Williams & Hawkings31 reinterpreted ij ij ij the symbol as the difference of the tensor from its mean value, p∞ δij . Brentner & Farassat32 use the notation ∆Pij to avoid confusion. c Otherwise, the reference frame can be rotated to satisfy this condition

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Therefore, for jet noise predictions in uniform subsonic coflow, the complex acoustic pressure p˜′ at an observer x outside of S is given as a function of frequency by the following integral solutiond : Z Z ∂G(x, y) e n (y, ω)G(x, y)dS dS − iω Q p˜′ (x, ω) = − Fei (y, ω) ∂yi S S (11) Z ∂ 2 G(x, y) dV. − Teij (y, ω) ∂yi ∂yi V

Due to challenges associated with the computation of the volume-distributed sources, the surface S is often chosen such that it encloses flow-generating sound sources and that the volume term can be assumed small and neglected. In the current far-field noise solver, the third integral in equation 11 is not computed. The time histories of the terms Fi and Qn are then computed and a Hanning window is applied after their mean values are removed. Note that windowing of the time data should be used to avoid spectral leakage, otherwise spurious noise contaminates the low frequencies, in particular at the quieter angles φ < 120◦ . The resulting quantities are then Fourier transformed. Since the terms Fi and Qn are not functions of the observer location x, the Fourier transforms only need to be computed once, at every element (i.e., face) for the surface S. The surface integrations in equation 11 are then performed for each observer and each frequency of interest. Note that analytical expressions for the spatial derivatives of the Green’s function G(x, y) can be derived and used instead of numerical differentiation. Finally, the narrowband (one-sided) Power Spectral Density (P SD) level in dB/St is calculated as ! † 2 p˜′ (x, St)p˜′ (x, St) , (12) P SD(x, St) = 10 log10 p2ref Stmin

and the Overall Sound Pressure Level (OASP L) in dB is computed as OASP L(x) = 10 log10

St max X Stmin

† 2 p˜′ (x, St)p˜′ (x, St) p2ref

!

,

(13)

where the superscript † denotes the complex conjugate, and pref is the reference pressure. Recall that Stmax and Stmin are the maximum and minimum frequency (i.e., the narrowband width) accessible by this post-processing, respectively (see section IID). While acoustic analogy calculations are typically very efficient compare to the CFD near-field simulations, the computational cost may become not negligible for predictions involving a large number of time samples, surface elements and/or observers. In particular for jet noise, a long time record is usually needed to ensure statistical convergence of the low frequencies, leading to large LES databases. In addition, directivity mapping in three dimensions can require several thousands of observers. Special care is therefore needed in terms of memory management, efficiency and performance. Fortunately, the method is well-suited for parallelization, since the calculations for each surface element, each frequency and each observers are independent. The current MPI implementation is done in the same massively parallel infrastructure than the flow solver “Charles”. It uses a standard load-balancing approach on the FW-H surface elements: each processor computes the noise contribution of only a portion of the surface, for all the observers and frequencies; the individual contributions are then recombined linearly as final output.

References 1 Bodony, D. J. and Lele, S. K., “Current Status of Jet Noise Predictions Using Large-Eddy Simulation,” AIAA J., Vol. 46, 2008, pp. 346–380. 2 Bogey, C., Bailly, C., and Juv´ e, D., “Noise investigation of a high subsonic, moderate Reynolds number jet using a compressible large eddy simulation,” Theor. Comput. Fluid Dyn., Vol. 16, 2003, pp. 273–297. 3 Bogey, C., Bailly, C., and Juv´ e, D., “Investigation of Downstream and Sideline Subsonic Jet Noise Using Large Eddy Simulation,” Theor. Comput. Fluid Dyn., Vol. 20(1), 2006, pp. 23–40. 4 Bogey, C. and Bailly, C., “Influence of nozzle-exit boundary-layer conditions on the flow and acoustic fields of initially laminar jets,” J. Fluid Mech., Vol. 663, 2010, pp. 507–538. d While equation (20) in Ref. 33 is correct, there is a typographical error in equation (17) of Ref. 34, the term ∂G(y; ξ)/∂y i should be replaced by ∂G(y; ξ)/∂ξi

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37 Liu, J., Kailasanath, K., Ramamurti, R., Munday, D., and Gutmark, E., “Large-Eddy Simulations of Imperfectly Expanded Jets from an Chevron Nozzle,” AIAA paper 2009-3192, 2009. 38 Pouangu´ e, A. F., Sanjos´ e, M., and Moreau, S., “Jet noise simulation with realistic nozzle geometries using fully unstructured LES solver,” AIAA paper 2012-2190, 2012. 39 Nichols, J. W., Ham, F., Lele, S. K., Khalighi, Y., and Spyropoulos, J., “High-fidelity LES of Asymmetric Supersonic Jets,” Proceedings of DoD High Performance Computing Modernization Program User Group Conference 2011, 2011. 40 Saxena, S. and Morris, P. J., “Noise Predictions for High Subsonic Single and Dual-Stream Jets in Flight,” AIAA paper 2012-2082, 2012. 41 Rahier, G., Prieur, J., Vuillot, F., Lupoglazoff, N., and Biancherin, A., “Investigation of integral surface formulations for acoustic predictions of hot jets starting from unsteady aerodynamic simulations,” AIAA paper 2003-3164, 2003. 42 Uzun, A., Lyrintsis, A. S., and Blaisdell, G. A., “Coupling of integral acoustics methods with LES for jet noise prediction,” International Journal of Aeroacoustics, Vol. 3(4), 2004. 43 Morfey, C. L. and Wright, M. C. M., “Extension of Lighthill’s Acoustic Analogy with Application to Computations Aeroacoustics,” Proceedings of the Royal Society A, Vol. 463, 2007, pp. 2101–2127. 44 Morfey, C. L., Hu, Z. W., and Wright, M. C. M., “Modelling of Turbulent Jets and Wall Layers: Extension of Lighthill’s Acoustic Analogy with Application to Computations Aeroacoustics,” Proceedings of European Conference on Computational Fluid Dynamics, ECCOMAS CFD-2006, 2006. 45 Richards, C. D. and Pitts, W. M., “Global density effects on the self-preservation behaviour of turbulent free jets,” J. Fluid Mech., Vol. 254, 1993, pp. 417–435. 46 Zaman, K. B. M. Q., “Asymptotic spreading rates of initially compressible jets – experiment and analysis,” Phys. Fluids, Vol. 10(10), 1998, pp. 2652–2660. 47 Zaman, K. B. M. Q., “Spreading characteristics of compressible jets from nozzles of various geometries,” J. Fluid Mech., Vol. 383, 199, pp. 197–228. 48 Viswanathan, K. and Czech, M. J., “Measurment and Modeling of Effect of Forward Flight on Jet Noise,” AIAA J., Vol. 49(1), 2011, pp. 216–234. 49 Harper-Bourne, M. and Fisher, M. J., “The noise from shock waves in supersonic jets,” Tech. Rep. CP 131, AGARD, 1974. 50 Tam, C. K. W. and Tanna, H. K., “Shock associated noise of supersonic jets from convergent-divergent nozzles,” J. Sound Vib., Vol. 81, 1982, pp. 337–358. 51 Tam, C. K. W., Seiner, J. M., and Yu, J. C., “Proposed Relationship Between Broadband Shock Associated Noise and Screech Tones,” J. Sound Vib., Vol. 110(2), 1986, pp. 309–321. 52 Lighthill, M. J., “On sound generated aerodynamically: I. General theory,” Proc. Royal Soc. Lond. A, Vol. 211, 1952, pp. 564–587. 53 Wells, V. L. and Han, A. Y., “Acoustics of a Moving Source in a Moving Medium with Application to Propeller Noise,” Journal of Sound and Vibration, Vol. 184(4), 1995, pp. 651–663. 54 Najafi-Yazdi, A., Br` es, G. A., and Mongeau, L., “An Acoustic Analogy Formulation for Moving Sources in Uniformly Moving Media,” Proceedings of the Royal Society A, Vol. 467(2125), 2011, pp. 144–165. 55 Br` es, G. A., Freed, D., Wessels, M., Noelting, S., and P´ erot, F., “Flow and Noise Predictions for the Tandem Cylinder Aeroacoustic Benchmark,” Phys. Fluids, Vol. 24(036101), 2012.

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