This work experimentally investigates the flow structure around a rectangular cylinder with an aspect ratio of 2 under varying incidence angles to examine how acoustic perturbations modify and modulate the unsteady flow instabilities. In the absence of acoustic excitation, the angle of incidence is found to markedly influence the flow topology and the natural shedding pattern altering the vortex formation length and wake dynamics. With acoustic perturbations, it is observed that for incidence angles $\alpha = 0^{\circ }$ and $\alpha = 5^{\circ }$, the masked impinging leading-edge vortex (ILEV)/trailing-edge vortex shedding (TEVS) instability modes of $n= 1$ and $n= 2$ become evident when their frequencies coincide with the frequencies of the acoustic perturbations (i.e. resonant condition). Both trailing-edge and leading-edge vortices were found to be modulated by the acoustic pressure cycle. These instability modes, which are naturally present under non-resonant conditions for significantly higher aspect ratios, highlight the role of incidence angle and self-excited acoustic resonance in virtually augmenting the streamwise length of the cylinder, thereby facilitating the emergence and sustenance of the ILEV/TEVS shedding pattern.

Self-sustained thermoacoustic oscillations as observed in low-emission combustion- involved gas turbines and aero-engines involve complicated thermal fluid–acoustics interaction and rich nonlinear dynamics. Such pulsating oscillations are known as thermoacoustic instability. When it occurs, large-amplitude limit cycle oscillations (LCOs) of thermodynamic parameters are frequently observed. These LCOs could cause overheating, flame flashback, and even engine failures. Thus it is critical to understand and predict the generation mechanisms and nonlinear dynamics behaviours, and then develop corresponding control approaches to prevent or control the onset of such instabilities. In this work, we develop and extend the classical van der Pol oscillators by integrating a physics-informed neural networks (PINNs) algorithm with a modelled nonlinear Rijke-type thermoacoustic combustor. The theoretical Rijke tube system (with Galerkin expansion and modified King's law implemented) and a CFD simulation model are applied to provide ‘training/calibration data’ for the extended van der Pol (EVDP)-PINNs model. The optimized EVDP oscillators are confirmed to be capable of capturing the key nonlinear characteristics by comparing the transient growth behaviours of thermodynamic perturbations and LCO amplitude and frequency. Further investigations are conducted to obtain Hopf bifurcation and amplitude death (AD) characteristics. Comparison is then made to the coupled EVDP systems. Quite similar Hopf bifurcation features, but differences in regions of AD, are observed. In general, we demonstrate an applicable approach to intelligently ‘learn’ a nonlinear thermoacoustic system and to create reliable EVDP oscillator systems, which have great potential to contribute to the development and testing of control approaches, such as the coupling described in this work, which may replace costly experimental tests.

The gain in efficiency of the receptivity of jets to acoustic disturbances as the nozzle lip is thicker is investigated using numerical simulations. For that, axisymmetric acoustic pulses are introduced in jets with Blasius laminar boundary-layer profiles at Mach numbers $M=0.4$, 0.6, 0.9 and 1.3 for nozzle-lip thicknesses between 1 % and 93 % of the nozzle radius. They are located on the jet axis or outside the jet with incidence angles $\varphi$ between $5^{\circ }$ and $90^{\circ }$ with respect to the downstream direction. Instability waves develop in the jet shear layer after the acoustic disturbances hit the nozzle. In all cases except for $\varphi \geq 75^{\circ }$, their amplitudes and hence the efficiency of the jet receptivity to the disturbances increase with the nozzle-lip thickness. The gains in efficiency are greater for a pulse inside the jet, generating upstream-travelling pressure waves resembling guided jet waves, than for a pulse outside the jet, producing free-stream sound waves. In the second case, the gains are significant for $\varphi =5^{\circ }$ and decrease with the incidence angle, especially for $\varphi >30^{\circ }$. Moreover, the gains are stronger for a higher Mach number, and roughly double between $M=0.4$ and $1.3$, thus reaching, for a pulse inside the jet, values close to 6 between the thinnest and thickest lips. Finally, according to additional simulations for $M=0.9$, the gains in receptivity efficiency do not change appreciably when different azimuthal mode numbers of the acoustic disturbances, widths of the pulse and shapes of the boundary-layer profile are considered.

Open rotors can play a critical role towards transitioning to a more sustainable aviation by providing a fuel-efficient alternative. This paper considers the sensitivity of an open-rotor engine to variations of three operational parameters during take-off, focusing on both aerodynamics and aeroacoustics. Via a sensitivity analysis, insights to the complex interactions of aerodynamics and aeroacoustics can be gained. For both the aerodynamics and aeroacoustics of the engine, numerical methods have been implemented. Namely, the flowfield has been solved using unsteady Reynolds Averaged Navier Stokes and the acoustic footprint of the engine has been quantified through the Ffowcs Williams-Hawking equations. The analysis has concluded that the aerodynamic performance of the open rotor can decisively be impacted by small variations of the operational parameters. Specifically, blade loading increased by 9.8% for a 5% decrease in inlet total temperature with the uncertainty being amplified through the engine. In comparison, the aeroacoustic footprint of the engine had more moderate variations, with the overall sound pressure level increasing by up to 2.4dB for a microphone lying on the engine axis and aft of the inlet. The results signify that there is considerable sensitivity in the model and shall be systematically examined during the design or optimisation process.

Acoustic resonance is an important factor that contributes to aeroengine compressor failure. In this study, a plane cascade of compressor blades was designed to reproduce acoustic resonance via a low-speed wind tunnel test. A high-frequency hot-wire, microphone and strain gauge were used to synchronously measure the fluid, acoustic and structural parameters. We analysed the variation in the amplitude and frequency of the multi-field parameters with increasing mean flow velocity and explored the multi-field interaction mechanism that induces the acoustic resonance of the plane cascade. The plane cascade effectively reproduced the acoustic resonance phenomenon. The first-order acoustic-mode frequency of the plane cascade flow duct, second-order torsional vibration mode frequency of the blade and shedding mode frequency of the tip clearance leakage vortex were equal under acoustic resonance. The fluid, acoustic and structural fields showed a strong interaction effect, achieving the maximum blade vibration amplitude and causing fatigue cracks of torsional vibration at the blade root. The frequency lock-in region of the compressor plane cascade was divided into an ‘acoustic–structure’ interaction region, a ‘fluid–acoustic–structure’ interaction region and a first-order acoustic-mode dominant region with increasing mean flow velocity, which demonstrates an interesting phenomenon in which the fluid–acoustic–structure modes compete: acoustic mode > blade vibration mode > vortex shedding mode. The results demonstrate a unique approach to the study of acoustic resonance that provides insight into the acoustic resonance mechanism in a cascade of compressor blades.

Serrations are commonly employed to mitigate the turbulent boundary layer trailing-edge noise. However, significant discrepancies persist between model predictions and experimental observations. In this paper, we show that this results from the frozen turbulence assumption. A fully developed turbulent boundary layer over a flat plate is first simulated using the large-eddy simulation method, with the turbulence at the inlet generated using the digital filter method. The space–time correlations and spectral characteristics of wall pressure fluctuations are examined. The simulation results demonstrate that the coherence function decays in the streamwise direction, deviating from the constant value of unity assumed in the frozen turbulence assumption. By considering an exponential decay function, we relax the frozen turbulence assumption and develop a prediction model that accounts for the intrinsic non-frozen nature of turbulent boundary layers. To facilitate a direct comparison with frozen models, a correction coefficient is introduced to account for the influence of non-frozen turbulence. The comparison between the new and original models demonstrates that the new model predicts lower noise reductions, aligning more closely with the experimental observations. The physical mechanism underlying the overprediction of the noise model assuming frozen turbulence is discussed. The overprediction is due to the decoherence of the phase variation along the serrated trailing edge. Consequently, the ratio of the serration amplitude to the streamwise frequency-dependent correlation length is identified as a crucial parameter in determining the correct prediction of far-field noise.

In this study, the asymptotic solutions of the pressure variations induced by two trains passing each other in a tunnel are theoretically investigated. The one-dimensional inviscid compressible airflow is analysed, and two methods to obtain numerically exact solutions and $M_{H}$ expansion formulas for approximate equations are presented, where $M_{H}$ is the Mach number of the high-speed train. The pressure coefficient, corresponding to the maximum value of the magnitude of the pressure, is expressed as $|c_{p}|_{max}=|c_{p,min}|=[({R}/({1-R}))$ $(1+\alpha )^{2}+({R(1-R)}/{(1-2R)^{2}})(1-\alpha )^{2}]+O[M_{H}]$, where $c_{p,min}<0$, $\alpha =U_{L}/U_{H}$ and $U_{L}$ and $U_{H}$ denote the speeds of the low- and high-speed trains, respectively, and $R$ is the cross-sectional area ratio of the train to the tunnel. The theoretical results indicate the dependence of the speeds of the two trains on the pressure distribution and that the maximum magnitude of the asymptotic pressure for a fixed value of $M_{H}$ is obtained for $\alpha =1$ and $\alpha =0$ when $R< R_{c}$ and $R>R_{c}$, respectively, where $R_{c}$ denotes the critical blockage ratio. Because the airflow along the side of the low-speed train, induced by the low-speed train, is along the running direction of the high-speed train and reduces the relative velocity of the high-speed train as the two trains pass each other, $|c_{p}|_{max}$ for $\alpha =0$ is larger than $|c_{p}|_{max}$ for $\alpha =1$ when $R>R_{c}$. It is theoretically demonstrated that, as conventional high-speed railway systems satisfy $R< R_{c}$, a conservative pressure estimation can be established assuming $\alpha =1$.

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.

The influence of the boundary layer (BL) thickness on the tonal noise generated by subsonic impinging jets is investigated. For that, initially laminar jets at Mach numbers $0.6$ and $0.9$ with BL thicknesses $0.05r_0$, $0.1r_0$ and $0.2r_0$, where $r_0$ is the pipe-nozzle radius, impinging on a plate at $6r_0$ from the nozzle, are simulated. For Mach number $0.9$, acoustic tones due to feedback loops of axisymmetric nature between the nozzle and the plate are generated at frequencies that do not vary with the BL thickness. The two dominant tones are, however, $17$ and $26\ \mathrm {dB}$ stronger for the thickest BL compared with the thinnest one. For Mach number $0.6$, for the thinnest BL no acoustic peaks appear, as observed in the experiments of the literature, but narrow peaks resulting from axisymmetric feedback loops emerge for thicker BLs. Therefore, low subsonic impinging jets can be resonant for specific nozzle-exit conditions. The increase in tone amplitude for Mach number $0.9$, and the establishment of feedback loops for Mach number $0.6$ with increasing BL thickness, are found to result from two changes in the jet flow. The first change is that the shear-layer laminar–turbulent transition occurs farther downstream for a thicker BL, leading to a greater predominance of the axisymmetric aerodynamic fluctuations near the plate. The second change is that the amplification of the flow fluctuations between the nozzle and the plate at the tone frequencies is stronger for thicker BLs.

The aeroacoustics of a boundary layer ingesting (BLI) ducted fan is investigated experimentally. The study examines a ducted fan immersed in an adverse streamwise pressure gradient turbulent boundary layer developed over a curved wall. Aeroacoustics measurements indicate that the noise from the BLI ducted fan results from a complex interaction among the fan, duct and the incoming boundary layer. The fundamental mechanisms of noise generation are explained using a general source separation strategy. A detailed noise comparison is made at varying fan rotational speeds and across a wide range of axial inflow velocities. In a low thrust regime, the noise is found to be driven by the fan loading, coupled with duct acoustics and the haystacking phenomenon. In a high thrust regime, the contribution from duct acoustics diminishes, and the noise is predominantly driven by the fan loading coupled with the haystacking phenomenon.

This study presents the interplay of flow and acoustics within tandem deep cavities, focusing on the resonance mechanism occurring between turbulent shear layers and acoustic eigenmodes. The arrangement inside the tandem deep cavities includes both close and remote configurations. A combined fully coupled and decoupled aeroacoustic simulation strategy was devised. Employing an advanced high-order spectral/hp element method in conjunction with implicit large eddy simulation, the nonlinear compressible Navier–Stokes equations were solved to acquire internal flow–acoustic resonant field. In parallel, the linearized Navier–Stokes equations were tackled to determine coherent shear layer perturbations with external acoustic forcing. Based on acoustic measurements, the mainstream Reynolds number approaches approximately $R{e_{in}} = {O}({10^5})$, where we identified the presence of frequency lock-in and a resonance range. Aeroacoustic noise sources were examined by implementing spectral proper orthogonal decomposition to decompose the pressure fields into hydrodynamic and acoustic components. As feedback intensified, the flow characteristics by the acoustic forcing effect and the flow-interactive effect were categorized according to the development of concurrent turbulent shear layers. Subsequently, the alternating and synchronous behaviours of concurrent shear layers resonated with the out-of-phase and in-phase acoustic eigenmodes were identified, and the corresponding large-scale counter-rotating vortex pairs and co-rotating vortex structures at the cavity entrances were extracted. The acoustic power generated by the Coriolis force was calculated using Howe's vortex-sound analogy, and the aeroacoustic energy transfer mechanism between large-scale shear layer vortices with acoustic eigenmodes was further explored. Finally, a linear response of coherent perturbations of the concurrent shear layers by external acoustic forcing was established. The amplification of flow in the streamwise direction toward the main duct led to the formation of coherent vortex structures, accompanied by separation bubbles into the main duct.

Heated supersonic rectangular twin jets (SRTJ) with a total temperature ratio of 2, using nozzles of design Mach number 1.5 and aspect ratio 2, were investigated in flow regimes from overexpanded to the design condition (Mj = 1.3–1.5). This work complements our recently published work in unheated SRTJ using the same experimental facility (Samimy et al., J. Fluid Mech, vol. 959, 2023, A13). Localized arc filament plasma actuators (LAFPAs) were used to excite the natural instabilities in the jets, thereby controlling the flow and acoustics. The results show that the jets were coupled primarily out-of-phase in overexpanded cases, that the coupling had significant effects on the near-field (NF) pressure fluctuations, and that these fluctuations were considerably higher for in-phase than for out-of-phase coupled cases. The results also revealed that the far-field (FF) overall sound pressure level is significantly higher on the minor axis plane of the SRTJ and that the onset of Mach wave radiation contributes to the increased acoustic radiation at the peak noise direction. The LAFPAs successfully controlled the coupling and were able to reduce the NF pressure fluctuations by 10 dB. However, only 1 to 2 dB FF noise reduction at the peak noise radiation direction was achieved. The overall trends of the baseline results and response of the flow to excitation are qualitatively similar in unheated and heated cases, but the details are significantly different.

Compressible direct numerical simulations are employed to elucidate the low-wavenumber behaviour of wall-pressure fluctuations in turbulent channel flow and the effect of flow Mach number in the nearly incompressible regime. Simulations are conducted at bulk Mach numbers 0.4, 0.2 and 0.1, and friction Reynolds number 180. In addition to the convective ridge that is virtually Mach-number-independent, acoustic ridges, whose magnitudes are orders of magnitude lower, are identified in the two-dimensional wavenumber–frequency spectrum. At lower frequencies, the acoustic ridges represent propagating longitudinal and oblique waves that match the theoretical predictions of two-dimensional duct modes with a uniform mean flow. They decay with decreasing Mach number but remain distinctly identifiable even at Mach 0.1. At high frequencies, in contrast, no propagating waves are found, and the spectral level in the supersonic wavenumber range is broadly elevated and increases with decreasing Mach number.

The aeroacoustic feedback loops in high-speed circular jets that impinge on a large flat plate are investigated via acoustic measurements and schlieren visualizations. In the present experiments, the nozzle pressure ratio ranges from 1.39 to 2.20, the corresponding ideally expanded jet Mach number $M_j$ is from 0.70 to 1.12 and the nozzle-to-plate distance ($H$) is from 4.0$D$ to 6.0$D$, where $D$ is the nozzle exit diameter. The results of acoustic measurements show that the strongest tones are generated in a limited frequency band. The empirical dispersion relations obtained from the fluctuating greyscales along the jet centreline of time-resolved schlieren images have good agreement with the dispersion relations from the vortex-sheet model. The coherent flow structures at tonal frequencies are extracted by spectral proper orthogonal decomposition and are analysed in detail. For the $M_j<0.82$ jets, the upstream-propagating guided jet mode is progressively confined to the potential core of jets with increasing tonal frequency, which provides the first direct experimental support for theoretical results. The evolution in the structures of acoustic resonance loops is studied along a single frequency stage of axisymmetric impinging tones. When the acoustic resonance between the upstream- and downstream-propagating guided jet modes is formed at tonal frequencies, the impinging tones are intenser. Slightly underexpanded impinging jets can simultaneously produce impingement tones and screech tones. Shock-cell structures have modulatory effects on the downstream-propagating Kelvin–Helmholtz wavepacket and the upstream- and downstream-propagating guided jet modes. Due to the interaction between the flow structures at the frequencies of impinging and screech tones, tones of axisymmetric modes can be produced outside the frequency ranges in which the axisymmetric upstream-propagating guided jet modes are supported by jets.

The acoustic response of a five-bladed rotor to an axisymmetric turbulent boundary layer at the tail end of a body of revolution (BOR) is investigated numerically to elucidate the physical sources of acoustics, particularly the role of coherent structures in sound generation. The BOR is at a length-based Reynolds number of $1.9 \times 10^6$ and free-stream Mach number of 0.059. Two rotor advance ratios, $1.44$ and $1.13$, are considered. The turbulent boundary layer on the nose and midsection of the BOR is computed using wall-modelled large-eddy simulation, whereas that in the acoustically important tail-cone section is wall-resolved. The radiated acoustic field is calculated using the Ffowcs Williams–Hawkings equation. The computed flow statistics and sound pressure spectra agree well with the experimental measurements at Virginia Tech. In addition to broadband turbulence-ingestion noise, spectral humps near multiples of the blade-passing frequency and accompanying valleys are captured. They are shown to be caused by correlated blade unsteady-loading dipole sources and their constructive and destructive interference as a result of successive blades cutting through the same coherent structures. The latter undergo rapid growth in the decelerating tail-cone boundary layer before their interaction with the rotor. The acoustic radiation is dominated by the outer region of the blade owing to a combination of larger blade chord-length, inflow turbulence intensity and blade speed. The numerical results also correctly predict the effect of the rotor advance ratio on the acoustic field. A mixed free-stream/convection Mach-number scaling successfully collapses the sound pressure spectra at the two advance ratios.

The interaction between ingested turbulence and rotating blades is a key source of broadband noise in engineering applications. In this paper, a far-field noise model accounting for source correlation across the span of the blade and between blades is developed and applied to the study of homogeneous isotropic turbulence ingestion by a model cooling fan, wind turbine and aircraft propeller. The widely used theory of Amiet is revisited and it is shown that previous works produce conflicting results when attempting to account for blade-to-blade correlation. Central to the model is the calculation of the time between blade chops of the same turbulent eddy as heard by the observer. In this paper it is shown that, when derived correctly, Amiet's theory accounts for correlated sources between blades and, thus, can predict haystacking tones. Comparisons with the new rotational formulation and with experimental data enable us to show that Amiet's theory can be used to accurately predict turbulence ingestion noise from open rotors. In particular, it is found that the infinite-span assumption in strip theory and the neglect of correlation effects across the span do not undermine the accuracy of this theory. This is of great importance because, unlike Amiet's theory, models which treat rotational effects and source correlation exactly are expensive to evaluate routinely at high frequencies due to the slow convergence of infinite series with Bessel functions.

In this experimental study, the impact of symmetric local blowing on suppressing the vortex-induced noise of a circular cylinder was investigated. A highly instrumented cylinder with pressure taps and a series of blowing chambers was used to inject air along the span (seven times the cylinder diameter) at circumferential angles $\theta _{b}={\pm }41^{\circ }$, ${\pm }90^{\circ }$ and ${\pm }131^{\circ }$ corresponding to the boundary layer, shear layers on the cylinder and separated shear layers, respectively. The investigation aimed to understand the noise reduction mechanism of local blowing by conducting near-field pressure and far-field noise measurements in synchronisation with flow field velocity measurements. Near-field pressure was measured around the circumference of the cylinder using a remote-sensing technique and planar particle image velocimetry was implemented to measure the velocity of the wake flow field at a diameter-based Reynolds number of $Re=7\times 10^{4}$. The results revealed that the interaction of the rolling up separated shear layers, under the influence of high-momentum fluid travelling from the free stream to the wake, induced significant vertical flow movement in the vortex-formation region. This movement led to strong alternating surface pressure fluctuations at the cylinder's shoulders, contributing to the scattering of noise. It was demonstrated that local blowing delayed vortex shedding for all cases, except at $\theta _{b}={\pm }90^{\circ }$, which elongated the shear layers and pushed the high-momentum transfer area farther downstream. The application of local blowing at $\theta _{b}={\pm }41^{\circ }$ was particularly effective in increasing the vortex formation size due to reduced entrainment of fluid-bearing vorticity.

A computational analysis is performed to study the three-dimensional response of rectangular shear layers to plasma actuator-based control, in the context of sound mitigation of supersonic non-axisymmetric jets. A Mach $1.5$ rectangular jet with an aspect ratio $2:1$ is controlled using experimentally informed actuation patterns, referred to as M0, M1, M2, M3, M${\rm \pi}$ and M+/$-$1. While the first five progressively increase the phase difference between successive actuators thus enhancing three-dimensionality of the shear layer structures, the latter corresponds to the flapping mode of the jet. A preliminary linear analysis identifies that the frequency, $St\sim 1$, has a relatively high overall amplification within the baseline shear layer, and is hence utilized for control in the subsequent nonlinear simulations. Each actuation reveals unique near-field vortical and acoustic responses that have a profound impact on far-field noise levels. The M0 actuation induces circumferentially interconnected strong streamwise vortices, while M1 actuation enhances the circumferential variability in the coherent structures. The M2 actuation encompasses both these effects, and along with a very low tonal impact of forcing, produces the most desirable far-field noise mitigation (${\sim }2.6\,{\rm dB}$), contributed by a broadband reduction around the column-mode peak of the baseline jet. Beyond M2 actuation, effectiveness of control saturates, particularly along the direction of peak noise radiation. Through a near-field analysis of the acoustic component, the efficacy of M2 actuation is attributed to the attenuation of the radiative efficiency of the jet, including reduced energy in the supersonic phase speeds, and redistribution of energy into the higher helical modes. Further, it curtails the nonlinear difference interactions in the plume that energize column-mode frequencies, which often appear as strong intermittent sound-producing events. While the shear layer turbulent kinetic energy decreases with actuation, the controlled jets show minimal variations in mean flow properties, particularly under M2 actuation, suggesting this to be a promising small-perturbation-based noise control strategy.

We investigate self-excited vortex-acoustic lock-in in a bluff body combustor. The bluff body not only anchors the flame but also sheds vortices. Mutual interaction between vortex shedding and the combustor acoustic field leads to lock-in. A lower-order model for vortex shedding is coupled to the acoustic equations to obtain a set of discrete dynamical maps, which are then solved numerically. Lock-in is identified when a definite phase relationship between vortex shedding and acoustic field remains unaltered with time. The common frequency during (phase) lock-in is either close to the natural acoustic or vortex shedding frequencies, accordingly termed A- or V-lock-in, respectively. Instability and amplitude suppression occur with most parts of the A- and V-lock-in regions, respectively. Furthermore, we relate the lock-in and pre-lock-in regimes observed in our previous experiments (Singh & Mariappan, Combust. Sci. Technol., 2019, pp. 1–29) to A- and V-lock-in phenomena, respectively. Among the two lock-in phenomena, combustion instability is favoured by $1:1$ A-lock-in, whose boundaries in the parameter space can be obtained from a forced response of the vortex shedding process. Finally, we discuss the bifurcations leading to lock-in.

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.