Spectral Proper Orthogonal Decomposition Modes Research Articles

Effect of rotation on wake vortices in stratified flow

Stratified wakes past an isolated conical seamount are simulated at a Froude number of $Fr = 0.15$ and Rossby numbers of $Ro = 0.15$ , 0.75 and $\infty$ . The wakes exhibit a Kármán vortex street, unlike their unstratified, non-rotating counterpart. Vortex structures are studied in terms of large-scale global modes, as well as spatially localised vortex evolution, with a focus on rotation effects. The global modes are extracted by spectral proper orthogonal decomposition (SPOD). For all three studied $Ro$ ranging from mesoscale, submesoscale and non-rotating cases, the frequency of the SPOD modes at different heights remains coupled as a global constant. However, the shape of the SPOD modes changes from slanted ‘tongues’ at zero rotation ( $Ro=\infty$ ) to tall hill-height columns at strong rotation ( $Ro=0.15$ ). A novel method for vortex centre tracking shows that, in all three cases, the vortices at different heights advect uniformly at approximately $0.9U_{\infty }$ beyond the near wake, consistent with the lack of variability of the global modes. Under system rotation, cyclonic vortices and anticyclonic vortices (AVs) present considerable asymmetry, especially at $Ro = 0.75$ . The vorticity distribution as well as the stability of AVs are tracked downstream using statistics conditioned to the identified vortex centres. At $Ro=0.75$ , intense AVs with relative vorticity up to $\omega _z/f_{c}=-2.4$ (where $\omega_z$ is the vertical vorticity and $f_c$ is the Coriolis frequency) are seen with small regions of instability and they maintain large $\omega _z/f_{c}$ magnitude in the far wake. Recent stability analysis that accounts for stratification and viscosity is found to improve on earlier criteria and show that these intense AVs are stable.

Exploration on flutter mechanism of a damaged transonic rotor blade using high-fidelity fluid–solid coupling method

Structural damage in turbomachinery is a primary origin of aeronautic accidents, which is receiving increased attention. This study is thus focused on the aeroelastic analysis of damaged blades, including the onset of flutter and underlying mechanisms. First, a high-fidelity fluid–solid coupling system is established with computational fluid dynamics (CFD) and computational structural dynamics (CSD) technologies, via which the dynamic aeroelastic analysis is conducted based on static aeroelastic deformation. Second, a damaged rotor blade is parametrically modelled with variable damage levels, extents, and positions. Finally, the modal identification method of spectral proper orthogonal decomposition (SPOD) is applied to observe flow details and provide physical insight into the flutter mechanism for damaged blades. Numerical analysis finds that there is a critical damage level below which the aeroelastic stability is positively improved with increasing damage level; otherwise, a significant loss of stability is induced. The damage location and extent further affect this critical damage level and the change rate crossing the threshold. The simulation with CFD/CSD finds that the high pressure near the trailing edge induced from boundary layer separation suppresses vibrations in stable conditions, but motivates vibrations during flutter, which is because of the high-pressure spread to nearing blades. SPOD modes reveal that high-frequency disturbances with large scale are primary factors inducing flutter, which is further stimulated by the high-order disturbances with small scale. This study provides a crucial foundation for the fatigue prediction for rotor blades in service and the optimisation design for high-performance turbomachinery in the near future.

Numerical investigation of serration angle on trailing edge noise reduction

The influence of serration angle on the three-dimensional flow characteristics and noise attenuation mechanisms of a NACA6512-63 airfoil is explored in this research using large eddy simulation (LES) and spectral proper orthogonal decomposition (SPOD) methods. Analysis of the mean flow field reveals the formation of counter-rotating vortex pairs, resulting from the outward flow towards the serration edges and the downwash flow in the valleys. This flow modification alters the effective angle of the serration, which in turn impacts its noise reduction performance. Furthermore, the noise reduction mechanism is investigated by comparing the general solution of Lyu's semi-analytical equation with the SPOD mode results, and the frequency range associated with noise attenuation is analyzed. The findings indicate that the case with λ/h=0.15 demonstrates the most significant noise reduction effect. This research offers valuable insights into the interplay between serration geometry and the resulting flow characteristics and noise reduction mechanisms.

Spectral proper orthogonal decomposition analysis of a spatially developing underexpanded round jet at Re = 45 000

This study conducts a comprehensive analysis of the coherent structures within a spatially developing underexpanded axisymmetric jet at a Reynolds number of 45 000, utilizing high-fidelity implicit large eddy simulations (iLES) in conjunction with spectral proper orthogonal decomposition (SPOD). In the frequency–wavenumber space, the global SPOD analysis identifies three distinct coherent structures, corresponding to three different mechanisms, namely, Kelvin–Helmholtz (KH), Orr, and lift-up. Their salient characteristics are discussed in detail. Local SPOD analysis further explores the streamwise evolution of these coherent structures, revealing that the influence of KH mechanism is confined to the near field, while the lift-up mechanism persists and dominates the energy content beyond the potential core, with the streaks of azimuthal wavenumbers one and two being the most energetic. The reconstruction of turbulent kinetic energy (TKE) and Reynolds shear stress from SPOD modes is assessed, and the first few azimuthal modes with low wavenumbers and frequencies are found crucial for capturing the dominant features of the flow. It is found that only the m = 0 and m = 1 modes contribute to the TKE at the centerline. The Reynolds shear stress reconstruction quality is comparable to TKE, but with a negligible contribution from the m = 0 mode. The azimuthal mode m = 1 captures the slope of the actual Reynolds shear stress profile in the vicinity of centerline, while m = 2 and higher modes capture the peak location of the actual profile.

Nested travelling wave structures in elastoinertial turbulence

Elastoinertial turbulence (EIT) is a chaotic flow resulting from the interplay between inertia and viscoelasticity in wall-bounded shear flows. Understanding EIT is important because it is thought to set a limit on the effectiveness of turbulent drag reduction in polymer solutions. Here, we analyse simulations of two-dimensional EIT in channel flow using spectral proper orthogonal decomposition (SPOD), discovering a family of travelling wave structures that capture the sheetlike stress fluctuations that characterise EIT. The frequency-dependence of the leading SPOD mode contains distinct peaks and the mode structures corresponding to these peaks exhibit well-defined travelling structures. The structure of the dominant travelling mode exhibits shift–reflect symmetry similar to the viscoelasticity-modified Tollmien–Schlichting (TS) wave, where the velocity fluctuation in the travelling mode is characterised by large-scale regular structures spanning the channel and the polymer stress field is characterised by thin, inclined sheets of high polymer stress localised at the critical layers near the channel walls. The travelling structures corresponding to the higher-frequency modes have a very similar structure, but are nested in a region roughly bounded by the critical layer positions of the next-lower-frequency mode. A simple theory based on the idea that the critical layers of mode $\kappa$ form the ‘walls’ for the structure of mode $\kappa +1$ yields quantitative agreement with the observed wave speeds and critical layer positions, indicating self-similarity between the structures. The physical idea behind this theory is that the sheetlike localised stress fluctuations in the critical layer prevent velocity fluctuations from penetrating them.

Effects of serration angle on noise reduction mechanism of trailing edge serration

This study investigates the effects of serration angle on the three-dimensional flow field and noise reduction mechanism of a NACA6512-63 airfoil using large eddy simulation (LES) and spectral proper orthogonal decomposition (SPOD) techniques. The serration angle is defined as the ratio of serration wavelength (λ) to serration amplitude (h). The mean flow field analysis on suction side of serration reveals the outward flow towards the serration edges and the downwash flow between the valleys that generate the counter-rotating vortex pair. This flow alteration changes the serration effective angle of the serration, consequently affecting the noise reduction performance. Additionally, by comparing the general solution of Lyu’s semi-analytical equation with the SPOD mode results, the noise reduction mechanism is examined, and the frequency range in which noise reduction occurs is analyzed. The linearly distributed multipole-like pressure patterns, similar to those in the semi-analytical solution, are observed in the low-frequency range for all three cases, with the λ/h = 0.3 case exhibiting the most effective noise reduction. The findings suggest that the local deviation of the turbulence axis, caused by the strong outward flow near the serration edges, can hinder the formation of the optimal pressure field distribution required for effective noise reduction.

Role of very large-scale motions in shock wave/turbulent boundary layer interactions

The present study investigates the cause of low-frequency unsteadiness in shock wave/turbulent boundary layer (TBL) interactions. A supersonic turbulent flow over a compression ramp is studied using wall-resolved large eddy simulation (LES) with a freestream Mach number of 2.95 and a Reynolds number (based on δ0: the thickness of the incoming TBL) of 63 560. From the view of stability analysis, the effect of intrinsic instability on such low-frequency unsteadiness is excluded from the flow system by designing a ramp angle of 15°, and our attention is paid to the convective instability contributed by the incoming TBL. The LES results are analyzed by linear and nonlinear disambiguation optimization (LANDO), spectral proper orthogonal decomposition (SPOD), and resolvent analysis. The LANDO results reveal a streamwise scale-frequency relation of coherent structures in a very long (around 60δ0) TBL, which indicates that the dynamics of very large-scale motions (VLSMs) in the TBL are featured by a low frequency. The SPOD results reveal that the most energetic SPOD mode features a low frequency that is identical to the dominant low frequency of the wall-pressure spectrum. Additionally, coherent structures of the mode resemble the VLSMs in the incoming TBL. These consistencies imply that the dynamics of VLSMs contribute to the low-frequency unsteadiness of the present flow. A resolvent analysis then further suggests that the origins of low-frequency dynamics of the present flow are from the VLSMs, which can be optimally amplified by the forcing in the turbulent flow.

Cross-spectrum method for acoustic source identification and visualization of airfoil noise

The identification and visualization of airfoil noise sources are critical for comprehending the interplay between flow and acoustics, and for understanding the generation and propagation of sound. In this paper, we introduce an innovative method, specifically designed to identify and visualize flow-induced noise sources associated with airfoil noise. This approach employs cross power spectral density analysis to distinctly identify flow-induced noise sources. The data for our study comes from large-eddy simulations of a NACA 0012 airfoil, characterized by a Reynolds number of 4×105, a Mach number of 0.058, and an angle of attack of 6.25∘. Different cross-spectrum formulations are examined and evaluated. The analysis includes a comparison of the cross-spectrum method's strengths and limitations relative to established data-driven approaches like dynamic mode decomposition (DMD) and spectral proper orthogonal decomposition (SPOD). We find that the cross-spectrum method provides both spectral magnitudes and phase topologies, allowing it to effectively compare sound intensities among various sources at specific frequencies, while retaining high-resolution spatiotemporal coherent flow and acoustic dynamics. One of the notable advantages of the cross-spectrum method over DMD or SPOD modes is its lesser reliance on extensive data manipulation in the form of large-size data matrix, making it a more efficient and user-friendly approach for practitioners, particularly when dealing with complex systems or high-dimensional datasets. This characteristic enhances its practicality and accessibility in the field of flow and acoustic visualization. Finally, the newly developed method is applied to three distinct flow transition scenarios to evaluate its proficiency in distinguishing acoustic generation and propagation mechanisms, depending on the specific transition case.

Eduction of coherent structures from schlieren images of twin jets using SPOD informed with momentum potential theory in the spectral domain

Abstract This work presents a methodology to extract coherent structures from high-speed schlieren images of turbulent twin jets which are more physically interpretable than those obtained with currently existing techniques. Recently, Prasad and Gaitonde (J Fluid Mech 940:1–11, 2022) introduced an approach which employs the momentum potential theory of Doak (J Sound Vib 131(1):67–90, 1989) to compute potential (acoustic and thermal) energy fluctuations from the schlieren images by solving a Poisson equation, and combines it with spectral proper orthogonal decomposition (SPOD) to educe coherent structures from the momentum potential field instead of the original schlieren field. While the latter field is dominated by a broad range of vortical fluctuations in the turbulent mixing region of unheated high-speed jets, the momentum potential field is governed by fluctuations which are intimately related to acoustic emission, and its spatial structure in the frequency domain is very organized. The proposed methodology in this paper improves the technique of Prasad and Gaitonde (J Fluid Mech 940:1–11, 2022) in three new ways. First, the solution of the Poisson equation is carried out in the frequency-wavenumber domain instead of the time-space domain, which simplifies and integrates the solution of the Poisson equation within the SPOD framework based on momentum potential fluctuations. Second, the issue of solving the Poisson equation on a finite domain with ad hoc boundary conditions is explicitly addressed, identifying and removing those unphysical harmonic components introduced in the solution process. Third, the solution of the SPOD problem in terms of momentum potential fluctuations is used to reconstruct schlieren SPOD fields associated with each mode, allowing the visualization of the obtained coherent structures also in terms of the density gradient. The method is applied here to schlieren images of a twin-jet configuration with a small jet separation at two supersonic operation conditions: a perfectly-expanded and an overexpanded one. The SPOD modes based on momentum potential fluctuations retain the wavepacket structure including the direct Mach-wave radiation, together with upstream- and downstream-traveling acoustic waves, similar to SPOD modes based on the schlieren images. However, for the same dataset, they result in a lower-rank decomposition than schlieren-based SPOD and provide an effective separation of twin-jet fluctuations into independent toroidal and flapping oscillations that are recovered as different SPOD modes. These coherent structures are more consistent with twin-jet wavepacket models available in the literature than those originally obtained with direct schlieren-based SPOD, facilitating their interpretation and comparison against theoretical analyses. Graphical abstract

A closure mechanism for screech coupling in rectangular twin jets

The twin-jet configuration allows two different scenarios to close the screech feedback. For each jet, there is one loop involving disturbances which originate in that jet and arrive at its own receptivity point in phase (self-excitation). The other loop is associated with free-stream acoustic waves that radiate from the other jet, reinforcing the self-excited screech (cross-excitation). In this work, the role of the free-stream acoustic mode and the guided-jet mode as a closure mechanism for twin rectangular jet screech is explored by identifying eligible points of return for each path, where upstream waves propagating from such a point arrive at the receptivity location with an appropriate phase relation. Screech tones generated by these jets are found to be intermittent with an out-of-phase coupling as a dominant coupling mode. The instantaneous phase difference between the twin jets computed by the Hilbert transform suggests that a competition between out-of-phase and in-phase coupling is responsible for the intermittency. To model wave components of the screech feedback while ensuring perfect phase locking, an ensemble average of leading spectral proper orthogonal decomposition modes is obtained from several segments of large-eddy simulation data that correspond to periods of invariant phase difference between the two jets. Each mode is then extracted by retaining relevant wavenumber components produced via a streamwise Fourier transform. Spatial cross-correlation analysis of the resulting modes shows that most of the identified points of return for the cross-excitation are synchronised with the guided jet mode self-excitation, supporting that it is preferred in closing rectangular twin-jet screech coupling.

Identifying dominant flow structures in a bubbling gas-particle fluidized bed using the spectral proper orthogonal decomposition

This study applies the spectral proper orthogonal decomposition (SPOD) to analyze the spatio-temporal characteristics associated with the flow fields of a bubbling fluidized bed. The results suggest there is no singular dominant frequency linked to the highest energy levels; rather, there is a spectrum of frequencies. These frequencies are consistent with the natural frequency from published correlations. SPOD analysis allows for the capture of quasi-periodicity associated with the flow that arises from the semi-periodic evolution of bubbles in the bed and reveals the presence of multiple co-existing spatio-temporal patterns in the particle volume fraction and velocity fields. The dominant SPOD modes can be used for prediction, control, design, optimization, and reduced-order system representation. The capability of a few dominant modes to yield a low-rank reconstruction of the flow fields is demonstrated. The intermittent behavior of the bed is examined through frequency-time analysis of the gas pressure field.

Spectral feature extraction of rocket exhaust plume using spectral proper orthogonal decomposition

The spectral characterization of flow-field parameters provides a new perspective for understanding the spatiotemporal evolution of unsteady supersonic exhaust plumes and for extracting typical structures. In this study, a large-eddy simulation is performed to calculate the three-dimensional unsteady supersonic plume flow field of rocket engines, and a spectral proper orthogonal decomposition (SPOD) method with a spatiotemporal separation is established. This approach is used to extract the coherent structural features of the unsteady exhaust plume flow field and analyze the mode space structure at different frequencies. The three-dimensional reconstruction and denoising of the exhaust plume flow-field parameters can be achieved via the frequency- and time-domain reconstructions of the SPOD algorithm and oblique projection method, respectively. The ground rocket exhaust plume of ballistic evaluation motor-II is analyzed. The results indicate that the SPOD method can effectively extract the single-frequency mode structure of the reactive supersonic flow field, and that low-order behavior appears in the m = 0 and m = 1 azimuth modes. The potential core exhibits a high-frequency wave-packet structure that is affected by shock waves and shear layers. Time-domain reconstruction based on the oblique projection method facilitates the capture of the dynamic characteristics of the flow field. For the first-order SPOD mode, the frequency- and time-domain reconstruction errors are 3.3% and 1.5%, respectively. The frequency-domain reconstruction method exhibits a 4% improvement in denoising ability compared to low-pass filters. This study provides a novel method for the spectral characterization and spatiotemporal feature extraction of supersonic exhaust plume flow fields.

Implicit large-eddy simulation of an over-expanded screeching rectangular jet

Implicit large-eddy simulation of an over-expanded screeching rectangular jet is performed with a seventh-order finite difference scheme. Good agreement is found between the predicted flow- and acoustic fields with the experimental observations. Fourier decomposition, phase-averaging analysis and Spectral Proper Orthogonal Decomposition (SPOD) are used to investigate the origin of the screech, the shock leakage during the shear-layer flapping, and the distinguishing fluctuating characteristics in the minor- and major-axis plane of the rectangular jet. It finds that the screech is radiated from the end of the forth shock cell, where the interaction of the shock waves with the shear layer causes periodic leakages of shock-wave tips in the minor-axis plane, resulting in the generation of intense acoustic waves in the surrounding air. An obvious flapping mode at the same frequency of the screech is captured in the minor-axis plane and dominates the dynamic motions of the rectangular jet. The SPOD modes of pressure and velocity fluctuations at the screech frequency help to reveal the relationship between the screech generation and the coherent structures.

Connecting transonic buffet with incompressible low-frequency oscillations on aerofoils

Self-sustained, low-frequency, coherent flow unsteadiness over rigid, stationary aerofoils in the transonic regime is referred to as transonic buffet. This study examines the role of shock waves in sustaining this transonic phenomenon and its relation to low-frequency oscillations (LFO) that occur in flow over aerofoils in the incompressible regime (Zaman et al., J. Fluid Mech., vol. 202, 1989, pp. 403–442). This is investigated by performing large-eddy simulations of the flow over a NACA0012 profile for a wide range of flow conditions under free-transition conditions. At low Reynolds numbers, zero incidence angle and sufficiently high free-stream Mach numbers, $M$ , transonic buffet occurs with shock waves present in the flow. However, when $M$ alone is lowered, self-sustained, periodic oscillations at a low frequency are observed even though shock waves are absent and the entire flow field remains subsonic at all times. At higher incidence angles, the oscillations are sustained at progressively lower $M$ and are present even at $M=0.3$ , where compressibility effects are low. A spectral proper orthogonal decomposition (SPOD) shows that the spatial structure of these oscillations is consistent for all cases. The SPOD modes are topologically similar, suggesting a connection between transonic buffet and LFO in the incompressible regime. Comparisons with other studies examining transonic buffet on various aerofoils, under forced-transition and fully turbulent conditions support this hypothesis. Future studies using tools of global linear stability analysis, especially at high free-stream Reynolds numbers are required to examine whether the underlying mechanisms of transonic buffet and incompressible LFO are the same.

Vorticity forces of coherent structures on the NACA0012 aerofoil

This study examined the impact of coherent structures on the aerodynamic forces exerted on a NACA0012 aerofoil with angles of attack $7.5^{\circ }$ and $10^{\circ }$ , and a chord-based Reynolds number $50\,000$ . The study utilized the spectral proper orthogonal decomposition (SPOD) algorithm to identify the coherent structures, and vorticity force analysis to quantify their impact on lift and drag forces. Results showed that at $10^{\circ }$ , the zeroth frequency of the first SPOD mode had a significant impact on drag and lift forces due to a large vortex structure that caused a strong flow along the suction side of the aerofoil. The first and second frequencies of the first SPOD mode represented asymmetric vortex pairs and a series of vortex pairs that determined the leading-edge separation, respectively. At $7.5^{\circ }$ , the zeroth frequency of the first mode corresponded to an oscillating near-wall stream that followed the reattachment flow pattern, while the first frequency corresponded to a counter-rotating vortex pair that originated where the flow reattaches. Finally, the second frequency of the first mode corresponded to smaller counter-rotating vortex pairs at the shear layer originated near the reattachment point. These findings suggest that coherent structures have a significant impact on aerodynamic forces exerted on aerofoils, and can be identified and quantified using the SPOD algorithm and vorticity force analysis.

Wake characteristics of wall-mounted solid and foam-covered circular cylinders

The wake characteristics of wall-mounted finite submerged solid and foam-covered circular cylinders (height/diameter = 0.9) are explored using large-eddy simulation for a Reynolds number (based on the cylinder diameter) of 13 000, and the simulation results are validated using particle image velocimetry. The foam-covered cylinder consists of an inner cylinder wrapped with a highly porous open-cell foam and capped by circular plates at the top and bottom. Two recirculation regions are identified for both cylinders, based on time-averaged data, one downstream and another above the top surface of the cylinder. The presence of the foam induces changes in the characteristics of both recirculation regions. The foam covering also interrupts the formation of large periodic structures, the evidence of which can be seen in the temporal distribution and discrete Fourier analysis of the force coefficients and velocity fluctuations. The temporal lateral oscillations, represented by the coefficient of sway, are observed for the solid cylinder but are suppressed in the case of the foam-covered cylinder. Two dominating frequencies are consistently present for the solid cylinder, but no such frequencies can be identified for the foam-covered cylinder. Finally, spectral proper orthogonal decomposition modes are presented in order to further substantiate the findings.

Reduced-order models of aeroacoustic sources for sound radiated in twin subsonic jets

We construct reduced-order models of aeroacoustic sources for single and twin subsonic jets ( $M_j=0.9$ , $Re=3600$ ), with the goal of accurately recovering the far-field sound over a wide band of frequencies $St=[0.07,1.0]$ and directivity angles $\phi = [30^{\circ },120^{\circ }]$ within a subdecibel level accuracy. These models are realized via combining spatio-temporally coherent spectral proper orthogonal decomposition (SPOD) modes extracted directly from Lighthill's stress tensor, itself calculated using large-eddy simulation (LES). We consider two sets of twin subsonic jets of diameter $D$ each, with spacings of $0.1D$ and $1D$ , where the jets merge upstream and downstream of breakdown, respectively. The closely spaced twin jet decays the slowest due to reduced turbulent stresses which are, however, more broadband due to early merging. Such jets show strong shielding in the plane of jets, especially at shallow directivity angles where sound levels may drop below that of the single jet. The farther spaced twin jets have dynamics more akin to the constituent single jet with turbulent fluctuations peaking here at $St=0.34$ , but showing very little shielding, with their overall sound pressure level (OASPL) mostly linked to the nature of extra flow structures created during merging. Three-dimensional, energy-ranked, coherent structures for twin jets exhibit rather poor low-rank behaviour, especially at the far-field spectral peak $St=0.14$ . At $St \gtrsim 0.3$ , the SPOD wavepackets show strong visual coherence, resembling Kelvin–Helmholtz instability modes upstream of breakdown, while at the lower frequencies, there is very little spatial coherence with wavepackets peaking downstream of breakdown. Although the leading SPOD modes radiate poorly, reduced-order models using a subset of them, up to $45$ SPOD modes per frequency, show a remarkable match (within $1$ dB) against the LES-predicted sound over $0.1 \lesssim St \lesssim 0.5$ , at all angles investigated. At other frequencies, the closely spaced twin jet shows more error, due to its greater hierarchy of spatio-temporal structures, showing slower convergence at the shallower angles.

Beamforming with modified steering vectors for jet noise source location

Prof. Krish Ahuja has a longstanding interest in jet noise source location. His work in this area is grounded in the idea that if the assumed source location is correct, then the sound should obey the inverse square law relative to that point and the phase should be constant along lines originating at that point. He applied this with, conceptually, one microphone in 1985 and two microphones in 1998. In 2006 he commissioned a beamforming system, Array 48, from OptiNav, Inc. His student, Nick Breen, used this to measure subsonic jet noise source location in detail. The NASA-Glenn Research Center also purchased an Array 48. In the current work, a jet noise data set measured by Gary Podboy using Glenn’s array in 2008 is revisited with a new beamforming algorithm, Robust Functional Beamforming, to further support Tam’s two-source model and Breen’s source location. Beamforming with modified steering vectors is performed to measure the parameters of the wavepacket source model from the far field. This process suggested replacing the wavepacket spatial length parameter with a temporal lifetime parameter. Another steering vector modification aimed to measure modes with odd spinning order. It seems to have found them at an apparent location 10 jet diameters removed from the jet, laterally. This is tentatively interpreted as a Mach radius phenomenon like one observed by Csaba Horvath at NASA-Glenn, also using Array 48, to study counter-rotating propeller noise. In an observation unrelated to beamforming, the excess noise measured at 40° from the jet axis as compared with the 90° angle, is fully contained in the first few cross spectral matrix eigenvalues, or Spectral Proper Orthogonal Decomposition modes, in some cases.

An empirical model of noise sources in subsonic jets

Modelling the noise emitted by turbulent jets is made difficult by their acoustic inefficiency: only a tiny fraction of the near-field turbulent kinetic energy is propagated to the far field as acoustic waves. As a result, jet-noise models must accurately capture this small, acoustically efficient component hidden among comparatively inefficient fluctuations. In this paper, we identify this acoustically efficient near-field source from large-eddy simulation data and use it to inform a predictive model. Our approach uses the resolvent framework, in which the source takes the form of nonlinear fluctuation terms that act as a forcing on the linearised Navier–Stokes equations. First, we identify the forcing that, when acted on by the resolvent operator, produces the leading spectral proper orthogonal decomposition modes in the acoustic field for a Mach 0.4 jet. Second, the radiating components of this forcing are isolated by retaining only portions with a supersonic phase speed. This component makes up less than 0.05 % of the total forcing energy but generates most of the acoustic response, especially at peak (downstream) radiation angles. Finally, we propose an empirical model for the identified acoustically efficient forcing components. The model is tested at other Mach numbers and flight-stream conditions and predicts noise within 2 dB accuracy for a range of frequencies, downstream angles and flight conditions.

The source localization and dynamical evolution of axisymmetric screech modes in underexpanded supersonic jets

High order numerical simulations of underexpanded supersonic cold jets issuing from a circular sonic nozzle are conducted to investigate the effective source location of axisymmetric mode screech tones. The frequency and sound pressure level of screech tone agree well with the experimental results in literature. The detailed generation processes of axisymmetric screech modes in a complete cycle are exhibited through investigating the dynamic evolutions of the flow structures and acoustic waves. By implementing spectral proper orthogonal decomposition (SPOD) of jet velocity field and pressure field, the corresponding SPOD modes to the A1 and A2 screech modes are extracted respectively and their evolutions along streamwise direction are analyzed. The source location of A1 mode and A2 mode screech tones were revealed based on the dynamic evolution analysis and the Fourier mode decomposition. It is worth noting that the screech waves of each axisymmetric screech mode radiate from an isolated discrete source rather than the distributed sources at a given jet Mach number. Another interesting finding is that the effective source position of screech A2 mode varies with jet Mach number.