Leveraging large eddy simulations to assess noise source imaging of a controlled supersonic jet

Abstract

Noise source imaging based on phased array measurements is an essential tool in the aeroacoustic analysis of new nozzle designs, especially at full-scale. This investigation aims to assess the capability of a deconvolution-based beamforming technique to accurately estimate the changes in noise sources for model-scale heated military jets when fluid inserts are used for noise control. This goal is achieved by performing well-validated Large Eddy Simulations (LES) to complement the experimental measurements. The LES data is segregated into its hydrodynamic, acoustic and thermal components using Doak’s Momentum Potential Theory (MPT). The near-field MPT-derived components are subjected to Spectral Proper Orthogonal Decomposition (SPOD) to compare with the frequency-dependent noise source maps obtained directly from experiments. It is shown that fluid inserts alter the naturally occurring Kelvin-Helmholtz (K-H) instability in the jet shear layer, which leads to a change in the directivity of the noise radiated in the near-field. The upstream shift in the noise source distribution resulting from the modified K-H instability is accurately captured by the deconvolution-based source imaging technique using just the far-field measurements. These changes in source locations as a function of frequency are documented.

Keywords

Beamforming fluid inserts large eddy simulations momentum potential theory instability

Introduction

Supersonic, high-temperature jets exhausting from tactical aircraft engines create hazardous noise environments that affect the health and performance of military personnel working in the vicinity of these aircraft. They are also a source of annoyance in communities close to naval airbases and military training routes. According to a report in 2010 by Doychak,¹ the U.S. Department of Veterans Affairs spends hundreds of millions of dollars each year for hearing loss cases and noise complaints. A decade later, this number is almost certainly significantly higher. This motivates investigations into various noise reduction technologies for existing and next-generation military aircraft.

An accurate estimation of noise source characteristics is essential in assessing novel nozzle designs and noise reduction concepts. However, direct measurement of flow parameters in the jet near-field is extremely challenging and cost-prohibitive in full-scale tests due to the heated, turbulent nature of the flow-field. Therefore, source localization techniques typically rely on microphone measurements in the acoustic field to model a presumed form of noise source distribution that best fits the microphone data. Notably, Tam et al.² performed auto- and cross-correlations of far-field pressure measurements from both subsonic and supersonic jets to argue in favor of the existence of the now widely-accepted two-source model of jet noise. These findings were further corroborated by using an elliptic mirror microphone³ to estimate noise source distributions from fine-scale and large-scale turbulence. More sophisticated noise imaging techniques involve the use of polar-correlations^4–6 and phased arrays^7–10 to estimate frequency-dependent noise source maps in both model-scale jets and full-scale engines.^8,11–14

The goal of the present investigation is to assess the capability of a deconvolution-based source imaging technique¹⁵ to accurately estimate the changes in the frequency-dependent noise source distributions in heated over-expanded military-style jets when fluid inserts¹⁶ are used for noise control. A fluid insert is a linear array of small diameter injectors that blow a small fraction of the bypass air into the diverging section of a converging-diverging nozzle. The use of fluid inserts has been shown to be successful in reducing supersonic jet noise in both upstream and downstream directions at different nozzle sizes¹⁷ and operating conditions.¹⁸

This paper describes model-scale experiments carried out in the Pennsylvania State University’s high-speed aeroacoustics facility, both with and without fluid inserts. Noise source maps are generated for each jet using far-field measurements. The use of the deconvolution-based algorithm is motivated by its demonstrated improvement over traditional delay-and-sum processing of the array output.^9,15 In order to assess the accuracy by which the noise source maps indicate the changes in the noise sources due to fluid inserts, a pair of well-validated, experimentally-anchored Large Eddy Simulations (LES) are used to complement the experiments and serve as truth models. The LES flow-field is segregated into its hydrodynamic, acoustic and thermal components using Doak’s Momentum Potential Theory (MPT).¹⁹ Doak’s MPT is a physics-based technique that uses the principle of mass conservation to exactly decompose the momentum density fluctuations in the entire jet flow-field to obtain independently the radiating and non-radiating components. Due to its effectiveness, this technique has been used successfully to understand the exchange mechanisms between hydrodynamic and acoustic components in both free²⁰ and impinging jets,²¹ to analyze noise control techniques,^22,23 and to develop low-order jet noise models,^24,25 among other applications. Additional details of Doak’s MPT are provided in a later section.

The resulting hydrodynamic and acoustic components from MPT are subjected to spectral proper orthogonal decomposition (SPOD)²⁶ to examine the effect of fluid inserts on the Kelvin-Helmholtz (K-H) instability and the resulting noise signature as a function of frequency. The capability of the noise imaging technique to estimate the changes in the acoustic SPOD modes using just the far-field measurements from experiments is assessed by a direct comparison with the SPOD modes from the LES data. In addition, the peak noise source location is documented for both the baseline and the controlled jet as a function of frequency.

The paper is organized as follows. First, a brief description of the two nozzle geometries is provided. This is followed by a description of the experimental methodology and the deconvolution-based noise source imaging technique. A summary of the LES calculations is provided next, including the implementation of Doak’s MPT to segregate the LES flow-field into its radiating and non-radiating components. The penultimate section compares the noise source images with the near-field LES data and documents the change in noise sources with frequency when fluid inserts are used for control. Concluding remarks are made in the final section.

Description of nozzles and operating conditions

The nozzles used in the present investigation represent a model-scale variant of the military-style GE F404 nozzle with an equivalent exit nozzle diameter D_e = 22.5 mm. Aircraft equipped with these nozzles can vary their geometry to produce different exit to throat area ratios to adapt to different flight regimes. The expansion portion of these nozzles contains 12 large flat seals interleaved with 12 smaller flaps to facilitate area adjustment in operational nozzles. In the present investigation, an exit to throat area ratio of 1.295, which is typical of a takeoff configuration, is selected. This area ratio corresponds to a design Mach number, M_d = 1.65.

The model-scale nozzles are fabricated through additive manufacturing by StrataSys, Ltd using the PolyJet photocuring process with the Amber Clear material, with a standard layer thickness of 0.015 mm. Additional details on the manufacturing process can be found in Stratasys’ guide to PolyJet.²⁷ The baseline nozzle is free of any noise reduction devices and is operated at a nozzle pressure ratio (NPR) of 3.0 and a total temperature ratio (TTR) of 3.0. These nozzle operating conditions correspond to a fully-expanded jet Mach number, M_j = 1.36 and is considered an appropriate representation of the takeoff condition.²⁸

For control, fluid inserts are generated by distributed, steady blowing of unheated air within the divergent section of the nozzle using a line of injectors. Figure 1(a) presents a CAD rendering of the fluid insert nozzle used in this study. This nozzle consists of three fluid inserts, each generated using a line of two injectors, spaced uniformly around the azimuth. This fluid insert nozzle is referred to as the 3FC-2FI (3 fluid corrugations, two injectors per corrugation) nozzle in the present study. A schematic diagram detailing a single fluid insert is presented in Figure 1(b).

Figure 1.

Diagrams of the fluid insert nozzle used for experiments and CFD simulations in this study: CAD rendering of nozzle with 3 fluid inserts, each generated using 2 in-line injectors (left) and Schematic of a single fluid insert (right).

Both the injectors have an exit diameter of D_inj = 0.06D_e. The upstream injector is inclined at θ_inj = 45^o to the jet centerline and is placed at 20% of the distance of the diverging section from the throat to the jet exit. In contrast, the downstream injector introduces air at θ_inj = 90^o to the nozzle centerline at 70% of the distance of the diverging section from the throat to the jet exit. Since the individual injectors introduce air at different locations inside the nozzle, for consistency, the injector pressure ratio (IPR) is defined as the ratio of the total inlet injector pressure with respect to the ambient pressure. All the controlled results reported in this investigation use an IPR₁ = 1.89 for the upstream injection and an IPR₂ = 4.5 for the downstream injector. This is based on previous observations by Prasad and Morris²⁹ that show that the primary function of the downstream injector in this fluid insert configuration is to provide noise reduction at downstream observer angles, whereas the upstream injector aids primarily in noise reduction in the upstream and sideline directions.

Experimental approach

High speed jet facility description

The experiments described in this investigation were conducted in the Pennsylvania State University’s high-speed jet aeroacoustics facility. A top-down schematic view of the facility is shown in Figure 2. This facility consists of a high-pressure air supply that exhausts into a 5.02 m × 6.04 m × 2.79 m anechoic chamber with a theoretical cutoff frequency of 500 Hz. The downstream section contains an exhaust fan that captures the jet exhaust and minimizes air re-circulation in the anechoic chamber. This single jet facility has been used extensively in several prior jet noise studies.^30–32 Other technical details of the facility, including its development, upgrades and flow conditioning can be found in Veltin,³³ Doty³⁴ and Powers.³¹

Figure 2.

Top-down view schematic of the Pennsylvania State University high speed jet aeroacoustics facility.

In order to accurately simulate the acoustics of exhaust jets from full-scale aircraft engines, the TTR of the jet must be replicated. The Penn State facility uses helium-air jet mixtures to simulate heated air jets: in the present case, TTR = 3.0. The partial pressures of both helium and air are regulated manually in a piping control cabinet to produce helium-air mixture jets that replicate the lowered density and increased acoustic velocity of heated jets. Three helium cylinders, pressurized to approximately 2100 psi, supply helium into the pressure control cabinet. Accurate mixing of air and helium is achieved via a regulator and gate valve, while the mixture pressure transducer voltage is monitored manually. The mixture’s total pressure is controlled to within 5% of the target value during data acquisition. This methodology, developed by Kinzie³⁵ and Doty and McLaughlin,³⁶ has been demonstrated to accurately replicate the noise³⁷ and velocity profiles³⁸ of heated air jets. A comparison between experimental data from model-scale heat-simulated jets in the high-speed jet aeroacoustics facility at Penn State and the moderate-scale heated jet noise facility at NASA Glenn Research Center, which show excellent agreement, can be found in McLaughlin et al.³⁹

Acoustic measurements and data processing

Acoustic measurements are performed with 1/8 in. pressure field, model 40DP, GRAS microphones. These microphones are selected due to their high-frequency response (up to 120 kHz), high peak sound pressure levels (SPLs), and high signal-to-noise ratio. Up to 23 microphones can be mounted on a semi-permanent rotating microphone boom in grazing incidence. The average radial position of the microphones is 85D_e, measured from the center of the microphone grid cap to the nozzle exit.

Microphone calibrations are performed with a B&K acoustic calibrator, model 4231. The analog time-domain signals from the microphones are routed through a GRAS model 12AG power module. These signals are amplified and filtered for anti-aliasing, thus enabling their accurate digital conversion in the following data processing. A high-pass filter, set to 500 Hz, is used to remove any undesirable low frequency noise that could contaminate the data. Each microphone signal consists of 409,600 samples acquired at a rate of 300,000 samples per second. The measured data are converted into SPL using Welch’s algorithm. The data is split sequentially into 1024 point segments and a Hanning window function is applied with a 50% overlap between each window. The power spectral density (PSD) is calculated in each window and averaged from all segments. This PSD is then converted to decibels using a reference pressure of 20 µPa.

For each microphone, the SPL is converted to a non-dimensional, lossless value conforming to the ANSI Military Aircraft Measurement Standard established in 2014.⁴⁰ Data is corrected for microphone spectral response characteristics based on the manufacturer’s descriptions of each microphone obtained during factory calibration. These include the actuator correction, ΔC_act(f), and the appropriate free field response, ΔC_ff(f). The spectra are also corrected for the daily variations in atmospheric quantities during propagation by calculating the atmospheric attenuation (ISO 9613–2:1996) for each microphone using measured ambient pressures, humidities, and temperatures, ΔC_atm(f), and adding back the sound lost due to the atmospheric attenuation from the jet to the microphone. Finally, the spectra are non-dimensionalized to SPL, SPL per unit Strouhal number, St (= fU_j/D_e). Following Kuo et al.⁴¹ the different steps that lead to the SPL per unit St are given by

S P L (S t) = S P L_{r a w} (f) - Δ C_{a c t} (f) - Δ C_{f f} (f) + Δ C_{a t m} (f) + 10 \log_{10} f_{c} .

(1)

The microphones are assumed to be in the geometric far-field. Under this assumption, spherical spreading is applied to propagate the acoustic data to a uniform radius of 100D_e, measured from the nozzle exit plane.

Noise source imaging

As stated previously, noise source maps are determined from far-field acoustic measurements through a deconvolution of the coherence-based beamformer output. This method was developed and applied to far-field acoustic measurements of an unheated, Mach 0.9 jet by Papamoschou.¹⁵ The present investigation extends the application of this technique to far-field acoustic measurements of heat-simulated, imperfectly expanded jets. A brief summary of the deconvolution-based beamforming method is provided next. Details on the derivation of this method can be found in Papamoschou.¹⁵

The jet noise source is approximated as a linear distribution of spatially incoherent sources, ζ(x, ω), where x denotes the streamwise location and ω is the angular frequency. In the original formulation, this line of sources is typically placed along the jet centerline. However, in order to provide a direct one-to-one comparison with the LES data, we position ζ(x, ω) along the jet shear layer, coinciding at the nozzle lip (x/D_e = 0, r/D_e = 0.5) with a 5° spreading angle downstream of the nozzle exit. For a given microphone array, the coherence-based beamforming output, Φ(x, ω), can be defined as

Φ (x, ω) = \sum_{m = 1}^{N} \sum_{n = 1}^{N} T_{m n}^{*} (x, ω) γ_{m n} (ω),

(2)

where m and n denote the microphone numbers, T_mn is the array response matrix given by

T_{m n} (x, ω) = e^{i ω [τ_{m} (x) - τ_{n} (x)]},

(3)

and γ_mn(ω) is the complex coherence of the microphone signals defined as

γ_{m n} (ω) = \frac{G_{m n} (ω)}{\sqrt{G_{m m} (ω) G_{n n} (ω)}} .

(4)

Here G_mn(ω) is denotes the cross-spectral density between the microphones m and n. τ(x) represents the propagation time for a sound wave to travel from any source at position (x, y) along the jet spreading angle to a far-field microphone (x_n, y_n) and is calculated using

τ_{n} (x) = \frac{\sqrt{{(x - x_{n})}^{2} + (y_{n}^{2})}}{c},

(5)

where c is the ambient speed of sound calculated based on atmospheric conditions within the anechoic chamber during the time of experiments.

The coherence-based beamforming output, Φ(x, ω), can be shown to be the convolution of the noise source distribution with the point spread function (PSF) over the streamwise extent of the noise source distribution, L, as follows

Φ (x, ω) = \int_{L} ζ (x, ω) V (x, ξ, ω) d ξ,

(6)

where V (x, ξ, ω) is the PSF defined as

V (x, ξ, ω) = \sum_{m = 1}^{N} \sum_{n = 1}^{N} T_{m n} (ξ, ω) T_{m n}^{*} (x, ω) .

(7)

The goal of the deconvolution-based beamforming method is to invert equation (6) in order to solve for the coherence-based noise source distribution, ζ(x, ω). For any frequency, the integral in equation (6) can be expressed as the summation over a finite number of discrete sources as follows

Φ_{i} = \sum_{q = 1}^{Q} V_{i q} ζ_{n} Δ ξ .

(8)

Following Papamoschou,¹⁵ we use Richardson-Lucy deconvolution to solve for the discrete source distribution, ζ_n. The inversion of equation (8) converges to a residual on the order of 10⁻⁵ or less within 75 iterations. The source distribution is calculated over the extent −10 ≤ x/D_e ≤ 30 (L = 40D_e), which is larger than the presumed physical extent of the jet noise source. Following the recommendation of Papamoschou,¹⁵ as a compromise between avoiding an unnecessarily coarse spatial resolution at low frequencies and the fine spatial resolution required at higher frequencies, a spatial resolution of Δξ = min (0.01 L, 0.25λ) is selected. Here λ = c/f represents the wavelength of the sound wave.

The positions of the far-field microphones are plotted in Figure 3. The origin is located at the center of the nozzle exit plane. The positions of the microphones on the boom, which were used for far-field acoustic measurements, are plotted as open, blue circles in Figure 3. Deconvolution-based beamforming of the measured far-field data is performed using a more spatially-resolved microphone array, plotted as the red dots in Figure 3.

Figure 3.

Position of microphones for far-field measurements. The beamforming array is focused on the peak noise emission angle: θ = 135°. Center of nozzle exit plane is located at (x, y) = (0, 0).

The design of the beamforming microphone array follows the guidance of Papamoschou.¹⁵ All the polar angles are measured from the jet upstream axis. The microphones are concentrated near θ = 135°, the peak noise emission direction. From the θ = 135° microphone, and expanding in either direction to polar angles of 120° and 150°, the microphone spacing is increased logarithmically. Logarithmic spacing is designed to mitigate the effects of spatial aliasing. The angular spacing between the microphones on either side of the θ = 135° microphone is Δθ = 0.4°, with a center-to-center spacing of 12.7 mm. This spacing was chosen as it is approximately half the wavelength of a St = 0.50 sound wave traveling through room-temperature air. This allows a majority of the sources contributing to the high-amplitude turbulent mixing noise to be mapped.

Large eddy simulation database

Formulation and setup

The LES calculations presented in this work are predicated on the previous numerical studies of fluid insert jets conducted by Prasad and Morris^18,22,29,42 for moderately larger-scale nozzles than those considered in the present study. The Favre-filtered Navier Stokes equations are solved in their dimensional form using the Wall-Adapting Local-Eddy (WALE) viscosity subgrid scale model⁴³ on trimmed hexahedral meshes using STAR-CCM+. The mesh distribution is identical to the finest mesh used in Prasad and Morris^22,29 and consists of 18.1 M cells for the baseline and 24M cells for the 3FC-2FI nozzle. Prismatic cells with a y⁺ < 1 are generated at the nozzle walls using the prism layer meshing feature in STAR-CCM+, no wall models are used. The convective fluxes are discretized using a hybrid third-order MUSCL scheme with a normalized variable diagram (NVD) approach⁴⁴ for numerical stability near shocks, whereas a second-order central difference scheme is used for the viscous terms. Time integration is performed using a second-order implicit dual time-stepping scheme with 10 sub-iterations.

The streamwise length of the computational domain is 65D_e and the radial size varies from 25D_e at the inlet to 40D_e at the outlet boundaries. All the jets simulated in this work operate at an ambient temperature of 288.15 K and a reference pressure of 101,008 Pa. All external fluid boundaries, except the downstream face, are set as non-reflecting free-stream boundaries based on characteristic variables. The downstream boundary is set as a pressure-outlet boundary. The pressure-outlet boundary condition only requires the specification of pressure and temperature, which are set to ambient values. A sponge zone treatment⁴⁵ is also applied to avoid reflections from the outer boundaries. The nozzle inlet is set as a stagnation inlet corresponding to the appropriate NPR and TTR. For the 3FC-2FI jet, the flow through each injector is simulated by specifying the appropriate IPR at the injector inlet. The inside walls of the nozzle are set as adiabatic with no-slip, whereas the outside walls are chosen to be adiabatic slip walls.

Figure 4 shows the instantaneous temperature contours for both the baseline and the 3FC-2FI nozzles at an arbitrary time-step.

Figure 4.

Instantaneous temperature contours for the baseline (top) and the 3FC-2FI (bottom) jet using LES. The nozzle exit is located at x/D_e = 0, and the injector ports are located at x/D_e ∼ − 0.26 and x/D_e ∼ − 0.66.

Due to the heated nature of the two jets, these temperature contours highlight the turbulent structures generated due to jet-mixing and the shock-cells due to the over-expanded nature of the jets. For the baseline jet, the interaction between the shocks from the opposite sides of the nozzle wall form a Mach disk near the jet axis, as shown in the top inset in Figure 4(a). For the 3FC jet, the injected fluid, due to its unheated nature, can be clearly observed in the top inset in Figure 4(b). The use of fluid inserts results in the generation of bow shocks in the diverging section due to the blockage of the supersonic core jet flow by the incoming fluid. This alters the shock-cell structure of the jet owing to less expansion between each shock-cell downstream of the nozzle exit. As a result, the Mach disk is not visible for the 3FC-2FI jet. In addition, fluid inserts also result in the formation of counter-rotating vortex pairs (CVPs) downstream of the injection as shown in the bottom inset in Figure 4(b). These CVPs convect downstream and enhance mixing in the jet shear layer.

Far-field predictions

The near-field LES results are extrapolated to the far-field using the Ffowcs Williams and Hawkings (FWH) analogy⁴⁶ in the time domain with a retarded-time approach. All the far-field observers are located at 100D_e from the jet exit. Two approaches are considered in order to obtain the far-field noise signals. In the first approach, the near-field results are calculated directly at 100D_e using the FWH analogy, whereas, in the second approach, the experimental procedure for determining the far-field microphone signals is closely replicated. The near-field results are calculated at 85D_e using FWH and then propagated to 100D_e using spherical spreading.

The effect of choosing one method over the other is observed primarily at polar angles where the far-field spectrum changes its shape from a peaked shape to a broadband shape (100° ≤ θ ≤ 125°). This is shown in Figure 5 for the baseline jet at θ = 120°. Directly propagating the LES results to 100D_e overshoots the LES predictions at 85D_e that are then extrapolated to 100D_e by approximately 2 dB. However, there is a good agreement when the experimental methodology is followed.

Figure 5.

Effect of the two approaches in extrapolating near-field LES results to the far-field at θ = 120°.

This clearly demonstrates the importance of closely replicating the experimental process in validating numerical predictions. Figure 6 compares the far-field predictions of the LES compared with the experimental measurements at different polar angles for the baseline jet.

Figure 6.

Comparison of LES predictions with experimental measurements at different polar angles for the baseline jet.

The LES predictions are seen to be in excellent agreement with the experimental far-field measurements up to the mesh cut-off St ≈ 1. At upstream angles, the LES predictions accurately capture the peak due to the broadband shock-associated noise component whereas, at downstream angles, the LES accurately capture the shape of the noise spectrum.In the present work, the noise benefit obtained by using the 3FC-2FI jet is quantified by calculating the change in Overall Sound Pressure Level (ΔOASPL), which is defined as the difference in OASPL of the 3FC-2FI jet with respect to the baseline jet. Therefore, a negative value of ΔOASPL denotes noise reduction. For the 3FC-2FI jet, the plane containing one of the fluid inserts is labeled as the ϕ = 0° azimuthal plane. This makes the planes given by ϕ = 120° and ϕ = 240° as the other two planes of fluid injection. For both the LES and experiments, the maximum noise reduction is observed in the planes bisecting any two fluid-inserts (ϕ = 60°, 180° and 300°). Figure 7 compares the predicted ΔOASPL values of the LES in the ϕ = 180° plane with the experimental measurements at different observer polar angles.

Figure 7.

Noise reduction in terms of ΔOASPL for different polar angles with LES and experiments.

A maximum noise reduction of 5.5 dB is observed in the peak noise direction (θ = 135°) in the experiments, whereas at the upstream locations, the change in shock-cell structure of the jet results in a 2–3 dB decrease in the broadband shock-associated noise. Overall, the noise reductions predicted by the LES are in good agreement with the experimental measurements at both upstream and downstream observer angles. This demonstrates the ability of LES (with the current mesh resolution) to accurately predict far-field noise from both baseline and fluid insert jets.

Segregation of radiating and non-radiating components

In order to uncover the mechanisms responsible for the observed noise reductions, the LES flow-field is segregated into its hydrodynamic, acoustic and thermal components using Doak’s MPT.¹⁹ The key step involves a Helmholtz decomposition of the momentum density into its solenoidal and irrotational components, each of which are calculated by manipulating the continuity equation. Following this approach, the momentum density of the flow can be written as

ρ u = \bar{B} + B^{'} - \nabla ψ^{'},

(9)

where

\nabla \cdot \bar{B} = 0.

(10)

Here primes denote a fluctuation about the mean, which is denoted with an overbar. B and ψ^′ represent the solenoidal component and irrotational scalar potential of ρu. By substituting equations (9) and (10) in the continuity equation, one can obtain the corresponding Poisson equation for the irrotational component of the flow

\nabla^{2} ψ^{'} = \frac{\partial ρ^{'}}{\partial t} .

(11)

The irrotational scalar potential can then be split into its acoustic $(ψ_{a}^{'})$ and thermal $(ψ_{t}^{'})$ components which are governed by their independent Poisson equations

\nabla^{2} ψ_{a}^{'} = \frac{1}{c^{2}} \frac{\partial p^{'}}{\partial t}, \nabla^{2} ψ_{t}^{'} = \frac{\partial ρ^{'}}{\partial S} \frac{\partial S^{'}}{\partial t},

(12)

where the entropy S is given by,

S = C_{p} [\log (T / T_{\infty}) - R \log (p / p_{\infty})] .

(13)

Here the subscript ∞ denotes ambient values, C_p is the specific heat at constant pressure, and R is the ideal gas constant.

In the solution procedure, the mean solenoidal component $(\bar{B})$ is assumed to be equal to the mean momentum density $(\bar{ρ u})$ . The outer boundaries of the LES domain are assumed to be purely acoustic. This results in $\nabla ψ^{'} = \nabla ψ_{a}^{'} = - (ρ u)^{'}$ which is integrated along the outer boundaries to provide a Dirichlet boundary condition to the Poisson solver. The Poisson equations for ψ^′ and $ψ_{a}^{'}$ are then discretized to second-order accuracy in the full 3D domain and solved using the BiCGSTAB algorithm. Once, ψ^′ is known, the hydrodynamic component is calculated using equation (9). This solution procedure has been validated extensively in literature for both external and internal flows, including free²⁰ and impinging jets,²¹ and supersonic boundary layers in a channel.⁴⁷

Comparison of noise source imaging with LES

Figure 8 presents the iso-surfaces of the streamwise components of the solenoidal and acoustic components for the two jets at an arbitrarily chosen time-step.

Figure 8.

Iso-contours of the streamwise component of the hydrodynamic component (top) and the acoustic component (bottom) for the baseline jet (left) and the 3FC-2FI jet (right). Contours of $- \partial ψ_{a}^{'} / \partial x$ in a 2D streamwise plane are also shown. For the 3FC-2FI jet, the top half of these contours represent the ϕ = 0° plane, whereas the bottom half represents the ϕ = 180° plane.

The thermal component does not play a major role in the dynamics of sound generation¹⁸ and is therefore not shown. The solenoidal component

(B_{x}^{'})

is associated with shear layer instabilities and represents the hydrodynamic fluctuations of the flow. These hydrodynamic fluctuations attenuate rapidly away from the shear layer. The use of fluid inserts results in the formation of streak-like hydrodynamic structures near the nozzle exit in the plane of fluid injection as shown by the inset in Figure 8(b). Similar structures have been observed in circular jets equipped with chevrons using proper orthogonal decomposition of the streamwise velocity in Rigas et al.⁴⁸ This existence of hydrodynamic streaks in the 3FC-2FI jet is not surprising as fluid inserts, similar to chevrons, generate streamwise CVPs as seen previously in Figure 4(b).

The acoustic component $(- \partial ψ_{a}^{'} / \partial x)$ on the other hand, is an order of magnitude smaller than its hydrodynamic counterpart and consists of a highly organized wavepacket-like structure that is reminiscent of classical instability waves used to model the sound generation from jets.⁴⁹ Since the acoustic component radiates to the far-field, Figure 8(c) and (d) also show the $- \partial ψ_{a}^{'} / \partial x$ contours in a 2D streamwise plane along with the iso-contours. For the 3FC-2FI jet, the top half of these contours represent the ϕ = 0° plane, whereas the bottom half represents the ϕ = 180° plane. Due to the heated nature of the present jets, the acoustic component exhibits a strong Mach wave radiation signature in the peak noise radiation direction. For the 3FC-2FI jet, the length of the Mach wave radiating region is smaller, and the direction of the Mach wave radiation shifts to a shallower angle with respect to the downstream axis, especially in the ϕ = 180° plane, which shows the greatest noise reduction. As a result, the remainder of the analysis is focused on the ϕ = 180° plane for the 3FC-2FI jet.

Although the MPT procedure can separate the hydrodynamic and acoustic components independently, the acoustic component may be considered as an irrotational auditory signature that develops around the turbulent structures represented by the hydrodynamic component;²² This connection is developed further using SPOD. SPOD is a data-driven technique that decomposes the flow-field into a set of orthogonal modes in the frequency domain that optimally capture the flow’s energy based on a user-defined norm. In the present case, we select a norm consisting of correlated acoustic fluctuations integrated in the peak noise radiation region, as discussed later. The snapshots of hydrodynamic and acoustic components are arranged into a data-matrix $Q = [q^{(1)} q^{(2)} \cdot \cdot \cdot q^{(N)}]$ , where $q^{(k)} = {[B_{x}^{'}, - \partial ψ_{a}^{'} / \partial x]}^{T}$ is the state vector at the k^th snapshot. SPOD modes are then determined by solving the following Fredholm integral eigenvalue problem:

\int R (x, x^{'}, S t) W (x^{'}) ϕ_{n} (x^{'}) d x^{'} = λ_{n} (S t) ϕ_{n} (x, S t) .

(14)

Here $ϕ_{n} (x, S t)$ represents the n^th SPOD mode at a particular St value and λ_n is the energy associated with that mode. R is the cross-spectral density matrix given by $\hat{Q} (x, S t) {\hat{Q}}^{*} (x^{'}, S t)$ , where $\hat{Q}$ represents the Fourier modes of the data matrix Q. The Fourier transform is performed by arranging the q snapshots into blocks of 256 with a 75% overlap. A periodic Hann window is used to minimize spectral leakage. The weight matrix, W, is set equal to one for the acoustic component in the region given by 120° ≤ θ ≤ 150°, r/D_e > 1. The other elements of W are set to zero. This allows the $- \partial ψ_{a}^{'} / \partial x$ SPOD modes to be ranked optimally based on the peak downstream noise and permits the physical interpretation of $B_{x}^{'}$ SPOD modes that are correlated to these $- \partial ψ_{a}^{'} / \partial x$ modes without changing the SPOD spectrum.⁵⁰

Figure 9 shows the leading $- \partial ψ_{a}^{'} / \partial x$ SPOD modes for the two jets at St values in the vicinity of the far-field spectral peak in the downstream direction in Figure 6.

Figure 9.

Leading $- \partial ψ_{a}^{'} / \partial x$ SPOD modes for the baseline (left) and the 3FC-2FI jet (right) at St =0.09 (top),St =0.28 (middle) and St =0.45 (bottom).

For the baseline jet (Figure 9(a), (c) and (e)), the leading SPOD mode consists of a single coherent streamwise structure showing growth and decay, characteristic of the K-H instability.⁵¹ Moreover, due to the heated nature of both the jets, these leading SPOD modes exhibit a strong downstream radiation signature analogous to a supersonically traveling wavy wall.⁵² As expected, with an increase in the St value, the wavelength of the downstream radiation is reduced and the SPOD mode is confined closer to the jet exit. The leading SPOD modes for the 3FC-2FI jet (Figure 9(b), (d) and (f)) also consist of a single streamwise K-H type structure. However, as seen previously with the iso-contours, these SPOD modes radiate at a shallower angle with respect to the downstream axis when compared with the baseline. This is consistent with previous studies on fluid inserts^18,22 and indicates a decrease in the radiation efficiency of the acoustic mode. This decrease in radiation efficiency is responsible for the 5 dB decrease in the peak noise direction as shown previously.

Since these leading $- \partial ψ_{a}^{'} / \partial x$ SPOD modes exhibit a K-H type behavior, it is instructive to look at the corresponding $B_{x}^{'}$ SPOD modes to provide insights into the cause of this noise reduction. These are shown in Figure 10 for two frequencies.

Figure 10.

Leading $B_{x}^{'}$ SPOD modes for the baseline (left) and the 3FC-2FI jet (right) at St =0.09 (top) and St =0.45 (bottom).

At St = 0.09, the

B_{x}^{'}

SPOD mode for the 3FC-2FI jet shows smaller structures than the baseline jet. This indicates that the mixing introduced by the fluid inserts results in a breakup of the large streamwise elongated hydrodynamic structures at this St value. At St = 0.45, the fluid inserts alter the K-H instability by contributing to the hydrodynamic component near the nozzle exit as highlighted by the rectangle in Figure 10(c) and (d). These factors are responsible for the decrease in radiation efficiency of the acoustic component as shown in Figure 9.

In order to assess the accuracy of the frequency-dependent noise source maps, ζ(x, ω), obtained from the far-field measurements, we treat the acoustic SPOD modes shown in Figure 9 as truth models. Figure 11 shows the noise source distributions at the three St values shown in Figure 9 for the two jets.

Figure 11.

Noise source maps at the three St values for the baseline (left) and the 3FC-2FI jet (right). The vertical dashed lines indicate the point of peak amplification along the shear layer in the $- \partial ψ_{a}^{'} / \partial x$ SPOD modes in Figure 9.

These noise source distributions are determined from measurements using the beamforming array shown in Figure 3. The noise source maps obtained from the original array were compromised with spatial aliasing effects for St > 0.2 and are therefore not shown. Based on these noise source maps, several important observations can be made. First, even though the beamforming procedure assumes the existence of a line of uncorrelated point sources, the noise source distributions consist of a Gaussian shape that closely follow the growth and decay of the K-H instability represented by the near-field SPOD modes. Second, similar to the SPOD modes, the noise source distribution moves upstream with an increase in frequency indicating that higher frequency waves are radiated from closer to the nozzle exit. Third, the vertical dashed lines on these figures represent the point of peak amplification in the SPOD modes along the line of the source distribution ζ(x, ω). For each of the three frequencies, the peak amplification location is predicted with reasonable accuracy; a maximum deviation of

\sim 0.35 D_{e}

is observed for the 3FC-2FI jet at St = 0.28. Finally, as shown in Figure 10, the use of fluid inserts results in the breaking up of large-scale hydrodynamic structures at lower frequencies and an early onset of K-H instability at others, resulting in a decrease in radiation efficiency of the acoustic mode. The upstream shift in the peak source location due to these phenomena is captured accurately by the ζ(x) distribution by using just the far-field measurements. This demonstrates that the deconvolution-based beamforming approach, although originally developed for a Mach 0.9 jet, can be used to provide accurate estimates even for military-style configurations at supersonic conditions, thus making it an important tool for full-scale testing.

Figure 12 shows the noise source peak locations as a function of frequency, determined from experimental data for both the jets.

Figure 12.

Comparison of peak noise source location as a function of St value for the baseline and the 3FC-2FI jet obtained from far-field measurements. The horizontal dashed lines correspond to the three St values compared in Figure 11.

A least-squares power fit is applied to each data set. In general, the source location for the 3FC-2FI jets in the plane bisecting any two fluid inserts peaks is

\sim 2 D_{e}

upstream of the baseline jet.

Conclusion

This investigation demonstrates that the effect of fluid inserts on the near acoustic field of an over-expanded supersonic jet can be predicted from the far-field measurements using a deconvolution based beamforming technique. This is achieved by performing two large eddy simulations (LES) corresponding to a baseline and a fluid insert jet to serve as truth models for the experiments. The LES flow-field is decomposed into its hydrodynamic, acoustic and thermal components using Doak’s MPT. When combined with spectral proper orthogonal decomposition (SPOD), the leading SPOD modes of the acoustic component represent a single streamwise structure exhibiting strong Mach wave radiation. These SPOD modes are compared directly with the noise source maps obtained from far-field measurements. It is observed that the use of fluid inserts results in a decrease in the radiation efficiency of the wavepackets which can be attributed to the breaking up of large-scale structures at low frequencies and a modification of the K-H instability at other frequencies. The upstream shift in the peak amplification due to this process is accurately estimated using the noise source maps obtained from far-field measurements. This makes the deconvolution-based beamforming technique a viable candidate for full-scale testing.

Final remarks

It is a privilege for the authors to provide this contribution in honor of Professor John (Shon) E. Ffowcs Williams. Shon was a powerful figure in the development of aeroacoustics beginning with his graduate student days at the University of Southampton. His contributions to the understanding and prediction of the sound generated by high-speed turbulence and by moving bodies remain the foundations for much research more than 50 years after their original publication. The Ffowcs Williams and Hawkings equation and its solution, also used in the present paper, is a key element of computational aeroacoustics prediction methods based on unsteady flow simulations. Professor Ffowcs Williams was a much larger than life character and knew how to enjoy himself. His presence will be greatly missed.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was funded in part by the Office of Naval Research.

ORCID iD

Chitrarth Prasad

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