## Publications

117 results found

Han X, Laera D, Morgans AS,
et al., 2019, Inlet temperature driven supercritical bifurcation of combustion instabilities in a lean premixed prevaporized combustor, *Experimental Thermal and Fluid Science*, Vol: 109, ISSN: 0894-1777

The present article reports experimental observation and analyses of a supercritical bifurcation of combustion instabilities triggered by the air inlet temperature (). The studies are performed with a pressurised kerosene fuelled Lean Premixed Prevaporized (LPP) combustor operated under elevated temperature. Unlike some previous studies, starting from an unstable condition of the system, the amplitude of combustion instabilities suddenly decrease when exceeds a critical value of = 570 K. When the temperature is lowered back the system returns to being unstable without featuring any hysteresis behaviour, as expected in case of a supercritical bifurcation. The unstable flames feature a periodic axial motion of lift-off and re-ignition, characterized as Helmholtz mode. The phase difference between chemiluminescence and pressure signals is found to increase with , exceeding 90 degrees (out of phase) for temperatures higher than 570 K. A low order network framework is conducted, illustrating that when is increased a stability shift of this mode is predicted at near 570 K, in good agreement with the experimental observations. The impact of on the spray characteristics is also examined, finding that higher promotes fuel evaporation and reduces equivalence ratio fluctuation at the exit of the swirler.

Sogaro FM, Schmid PJ, Morgans AS, 2019, Thermoacoustic interplay between intrinsic thermoacoustic and acoustic modes: non-normality and high sensitivities, *Journal of Fluid Mechanics*, Vol: 878, Pages: 190-220, ISSN: 0022-1120

This study analyses the interplay between classical acoustic modes and intrinsic thermoacoustic (ITA) modes in a simple thermoacoustic system. The analysis is performed using a frequency-domain low-order network model as well as a time-domain spatially discretised model. Anti-correlated modal sensitivities are found to arise due to a pairwise interplay between acoustic and ITA modes. The magnitude of the sensitivities increases as the interplay between the modes grows stronger. The results show a global behaviour of the modes linked to the presence of exceptional points in the spectrum. The time-domain analysis results in a delay-differential equation and allows the investigation of non-normal behaviour and its consequences. Pseudospectral analysis reveals that energy amplification is crucially linked to an interplay between acoustic and ITA modes. While higher non-orthogonality between two modes is correlated with peaks in modal sensitivity, transient energy growth does not necessarily involve the most sensitive modes. In particular, growth estimates based on the Kreiss constant demonstrate that transient amplification relies critically on the proximity of the non-normal modes to the imaginary axis. The time scale for transient amplification is identified as the flame time delay, which is further corroborated by determining the optimal initial conditions responsible for the bulk of the non-modal energy growth. The flame is identified as an active and dominant contributor to energy gain. The frequency of the optimal perturbation matches the acoustic time scale, once more confirming an interplay between acoustic and ITA structures. Flame-based amplification factors of two to five are found, which are significant when feeding into the acoustic dynamics and eventually triggering nonlinear limit-cycle behaviour.

Yang D, GUZMÁN-IÑIGO J, Morgans A, Sound generation by entropy perturbations passing through short circular holes, The 23rd International Congress on Acoustics

Xia Y, Laera D, Jones WP,
et al., 2019, Numerical prediction of the Flame Describing Function and thermoacoustic limit cycle for a pressurised gas turbine combustor, *Combustion Science and Technology*, Vol: 191, Pages: 979-1002, ISSN: 0010-2202

© 2019, © 2019 Taylor & Francis Group, LLC. The forced flame responses in a pressurized gas turbine combustor are predicted using numerical reacting flow simulations. Two incompressible1 large eddy simulation solvers are used, applying two combustion models and two reaction schemes (4-step and 15-step) at two operating pressures (3 and 6 bar). Although the combustor flow field is little affected by these factors, the flame length and heat release rate are found to depend on combustion model, reaction scheme, and combustor pressure. The flame responses to an upstream velocity perturbation are used to construct the flame describing functions (FDFs). The FDFs exhibit smaller dependence on the combustion model and reaction chemistry than the flame shape and mean heat release rate. The FDFs are validated by predicting combustor thermoacoustic stability at 3 and 6 bar and, for the unstable 6 bar case, also by predicting the frequency and oscillation amplitude of the resulting limit cycle oscillation. All of these numerical predictions are in very good agreement with experimental measurements.

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Gaudron R, Morgans AS, The acoustic absorption coefficient of short circular holes sustaining a high Reynolds number bias flow, 23rd International Congress on Acoustics (ICA 2019)

Guzman Inigo J, Baddoo PJ, Ayton LJ, et al., 2019, Noise generated by entropic and compositional inhomogeneities interacting with a cascade of airfoils, 25th AIAA/CEAS Aeroacoustics Conference, Publisher: American Institute of Aeronautics and Astronautics

Dalla Longa L, Evstafyeva O, Morgans A, 2019, Simulations of the bi-modal wake past three-dimensional blunt bluff bodies, *Journal of Fluid Mechanics*, Vol: 866, Pages: 791-809, ISSN: 0022-1120

The bi-modal behaviour of the turbulent flow past three-dimensional blunt bluff bodies is simulated using wall-resolved large eddy simulations. Bi-modality (also called bi-stability) is a phenomenon that occurs in the wakes of three-dimensional bluff bodies. It manifests as a random displacement of the wake between preferred off-centre locations. Two bluff bodies are considered in this work: a conventional square-back Ahmed body representative of road cars, and a simplified lorry, which is taller than it is wide, with its aspect ratio corresponding to a 15 % European lorry scale model. To our knowledge, this is the first time that the asymmetric bi-modal switching behaviour of the wake, observed experimentally, has been captured in simulations. The resulting unsteady flow fields are then analysed, revealing instantaneous topological changes in the wake experiencing bi-modal switching. The best-resolved case, the simplified lorry geometry, is then studied in greater detail using modal decomposition to gain insights into the energy content and the dominant frequencies of the wake flow structures associated with the asymmetric states. High-frequency snapshots of the switching sequence allow us to propose that large hairpin vortices are responsible for the triggering of the switching.

Yang D, Laera D, Morgans AS, 2019, A systematic study of nonlinear coupling of thermoacoustic modes in annular combustors, *Journal of Sound and Vibration*, ISSN: 0022-460X

Thermoacoustic instabilities in annular gas turbine combustors often involve modes which vary in both the longitudinal and circumferential directions. Recent experimental studies show that during limit cycle oscillations, different thermoacoustic modes may be uncoupled, as is the case in purely longitudinal or circumferential spinning modes. They may also be coupled, for example two counter-rotating circumferential modes combining to give standing or mixed modes, and coupling between circumferential and longitudinal modes giving rise to the slanted mode. Accurately predicting such modal couplings and the resulting spatial pattern of limit cycle oscillations remains an open challenge. This work uses a 2-D low-order network model based on modal expansions, validated against a full 3-D Helmholtz solver, to systematically investigate these couplings. For the first time, low-order network modelling is shown to capture limit cycle oscillations exhibiting both uncoupled and nonlinearly coupled modes, the latter including coupling between counter-rotating circumferential modes and between longitudinal and circumferential modes. It is shown that limit cycle solutions with totally different mode patterns (longitudinal, circumferential spinning, circumferential standing and slanted) can all exist in a given thermoacoustic system, with switches between modal patterns arising from slight changes in parameters such as the flame time delay.

Yang D, Sogaro FM, Morgans AS,
et al., 2019, Optimising the acoustic damping of multiple Helmholtz resonators attached to a thin annular duct, *Journal of Sound and Vibration*, Vol: 444, Pages: 69-84, ISSN: 0022-460X

Helmholtz resonators (HRs) are widely used to damp acoustic oscillations, including in the combustors of aero-engines and power gas turbines where they damp thermoacoustic oscillations. The geometries of such combustors are often annular in shape, which means that low frequency acoustic modes exhibit both longitudinal and circumferential modeshapes, the latter across different circumferential wave numbers. For linear acoustic disturbances downstream of the flame, the presence of HRs leads to modal coupling and mode shape shifts, which makes design and placement of multiple HRs very complicated. A procedure which ensures that the design and placement of the HRs can be optimised for good acoustic damping performance would be very valuable, and such a procedure is presented in this work. A simplified linear, low-dimensional model for the acoustic behaviour in a hot annular duct sustaining a mean flow is extended to account for the attachment of multiple HRs. The HRs are assumed to sustain a cooling mean bias flow through them, towards the combustor, such that they can be modelled using linear, lumped element Rayleigh conductivity models. An optimisation method based on the gradient derived from adjoint sensitivity analysis is then applied to the low order network acoustic modelling framework for hot annular ducts incorporating HR models, for the first time. It is used to optimise over multiple HR geometry and placement parameters, to obtain optimum acoustic damping over all acoustic modes in a given frequency range. These optimisation procedures are validated via multi-dimensional parameter sweep results. Thus a novel and efficient tool for HR optimisation for thin annular ducts is presented.

Gaudron R, Morgans AS, Acoustic absorption in a subsonic mean flow at a sudden cross section area change, 26th International Congress on Sound and Vibration (ICSV26)

Surendran A, Boakes C, Yang D, et al., 2019, Thermoacoustic response of heat exchanger tubes in future aero-propulsion engines

© Proceedings of the 26th International Congress on Sound and Vibration, ICSV 2019. All rights reserved. Cylindrical heat exchanger tubes can act as active or passive acoustic elements that influence the thermoacoustic stability of combustion systems. They can act as both unsteady heat sinks and acoustic dampers. The present work is motivated by heat exchangers in future aero-propulsion engines. One significant difference to heat exchanger tubes in domestic boiler systems is that rather than being tightly spaced, their spacing is likely to be of the same order as their diameter. The heat transfer behaviour can change depending on the tube spacing. The contributions of this work are twofold. Firstly, to build acoustic scattering models for heat sinks and simple area jump conditions, and secondly, to include them in 1D network models to study the influence of the spatial positioning of these models within the network. Preliminary results indicate that there is significant variation in the thermoacoustic behaviour if one were to consider an approximated model with a heat sink followed by an area jump, as compared to the one where an area jump is followed by a heat sink

Han X, Laera D, Morgans AS,
et al., 2018, The Effect of Stratification Ratio on the Macrostructure of Stratified Swirl Flames: Experimental and Numerical Study, *JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER-TRANSACTIONS OF THE ASME*, Vol: 140, ISSN: 0742-4795

Li J, Yang D, Morgans A, 2018, The effect of an axial mean temperature gradient on communication between one-dimensional acoustic and entropy waves, *International Journal of Spray and Combustion Dynamics*, Vol: 10, Pages: 131-153, ISSN: 1756-8277

This work performs a theoretical and numerical analysis of the communication between one-dimensional acoustic and entropy waves in a duct with a mean temperature gradient. Such a situation is highly relevant to combustor flows where the mean temperature drops axially due to heat losses. A duct containing a compact heating element followed by an axial temperature gradient and choked end is considered. The proposed jump conditions linking acoustic and entropy waves on either side of the flame show that the generated entropy wave is generally proportional to the mean temperature ratio across the flame and the ratio (F-1), where F is the flame transfer function. It is inversely proportional to the Mach number immediately downstream of the flame M2. The acoustic and entropy fields in the region of axial mean temperature gradient are calculated using four approaches: (1) using the full three linearised Euler equations as the reference; (2) using two linearised Euler equations in which the acoustic and entropy waves are assumed independent (thus allowing the extent of communication between the acoustic and entropy wave to be evaluated); (3) using a Helmholtz solver which neglects mean flow effects and (4) using a recently developed analytical solution. It is found that the communication between the acoustic and entropy waves is small at low Mach numbers; it rises with increasing Mach number and cannot be neglected when the mean Mach number downstream of the heating element exceeds 0.1. Predictions from the analytical method generally match those from the full three linearised Euler equations, and the Helmholtz solver accurately determines the acoustic field when M2≤0.1.

Yang D, Morgans A, LOW-ORDER NETWORK MODELING FOR ANNULAR COMBUSTORS EXHIBITING LONGITUDINAL AND CIRCUMFERENTIAL MODES, ASME Turbo Expo 2018: Turbomachinery Technical Conference and Exposition

Han X, Laera D, Morgans A,
et al., 2018, Flame macrostructures and thermoacoustic instabilities in strati fied swirling flames, *Proceedings of the Combustion Institute*, Vol: 37, Pages: 5377-5384, ISSN: 1540-7489

The present article investigates the correlation between flame macrostructures and thermoacoustic combustion instabilities in stratified swirling flames. Experiments are carried out in a laboratory scale longitudinal test rig equipped with the Beihang Axial Swirler Independently-Stratified (BASIS) burner, a novel double-swirled combustion system developed by adapting an industrial lean premixed prevaporized (LPP) combustor. At first, the flame macrostructures are investigated and discussed for various total equivalence ratios (ϕtotal) and stratification ratios (SRs). Depending on operating conditions, three different flame types are stabilized in the combustor: two attached flames comprising a stratified flame and a V-shaped flame (V-flame), as well as a lifted flame. Thermoacoustic instabilities are then investigated. The amplitude of the oscillations is found to be more sensitive to SR than the ϕtotal. Large amplitude limit cycles are found for low and high values of SR, for which the V-flame and the lifted flame are observed in the combustor, respectively. The flame dynamics are also investigated using local Rayleigh index maps. It is found that for both the lifted flame and V-flame, the major driving force comes from the flame-to-wall impingement region. Coherent structures associated with flame wrinkling are found along the flame brushes of the V-flame. On the contrary, the stratified flame is found to be more thermo-acoustically stable. Finally, incompressible Large Eddy Simulations is used to obtain the flame responses to forcing at 300 Hz, which is very close to the frequencies at which limit cycle oscillations occur. The results show that the global heat release rate response of the stratified flame exhibits a significant phase shift compared to the responses of the other two flame types, and this is the most likely cause of thermoacoustic stabilization.

Yang D, Guzman J, Johnson H, et al., 2018, On the sensitivity of the acoustics of short circular holes to inletedge geometries, AIAA/CEAS Aeroacoustics Conference, Publisher: AIAA

Short circular holes with a low Mach number mean bias flow passing through them can absorb or generate acoustics. A recent semi-analytical model (Yang & Morgans, Journal of Sound and Vibration, 384 (2016) pp. 294-311 & 393 (2017) pp. 41-61) based on the Green’s function method has been shown to be able to capture both the acoustic absorption and gen-eration. The model indicates the importance of accurately capturing the path of the shedvorticity from the hole inlet edge. In the present work, the effect of this vorticity path on thehole acoustics is first studied systematically using the semi-analytical model. As the vorticitypath is sensitive to the sharpness of the hole inlet edge, CFD tools are then used to obtain thevortex sheet shape after slightly changing the hole inlet edge shape. The acoustic responses ofthese modified holes are obtained using the semi-analytical model. A slightly chamfered edge case and a small ellipse edge case are considered to show that a very small change to the shape of the hole inlet edge can give a significant difference in the hole acoustic response.

Xia Y, Laera D, Morgans AS, et al., Thermoacoustic limit cycle predictions of a pressurised longitudinal industrial gas turbine combustor, ASME Turbo Expo 2018, Publisher: American Society of Mechanical Engineers

Morgans AS, 2018, Advective disturbances in combustor thermoacoustics, *International Journal of Spray and Combustion Dynamics*, Vol: 10, Pages: 101-102, ISSN: 1756-8277

Xia Y, Laera D, Morgans AS, Effect of flame-to-flame interaction on the flame describing function of a turbulent swirling flame, The 12th International ERCOFTAC Symposium on Engineering Turbulence Modelling and Measurements (ETMM12)

Xia Y, Laera D, Morgans A, Effect of flame-to-flame interaction on the flame describing function of a turbulent swirling flame in an annular combustor, The 25th International Congress on Sound and Vibration (ICSV25)

Xia Y, Laera D, Morgans A, Effect of wall heat transfer on the flame response to acoustic perturbation in a turbulent swirling combustor, The 25th International Congress on Sound and Vibration

Guzmán-Iñigo J, Morgans AS, Durán I, 2018, A model for the sound generated by entropy disturbances interacting with isolated blades

© 2018 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. This article presents a modelling approach to predict the low-frequency sound generated by entropy fluctuations interacting with isolated aerofoils. A model of the acoustic field is obtained based on a linearisation of the compressible Euler equations about a steady potential compressible mean flow. Neglecting the mean flow variations of velocity and density in the propagation of sound, but not in the source term, and using a Lorentz-type transformation, the problem is reduced to solving a Helmholtz equation. This equation is recast in integral form and a solution is obtained using a compact Green’s function method. This approach assumes that the acoustic wavelength is large compared to the profile chord and spacing, while no restrictions are made on the entropy wavelengths. The source term is further simplified by assuming that the steady flow is a small perturbation to an uniform flow. The model is illustrated using a symmetric aerofoil and its performance is assessed with numerical simulations of the compressible Euler equations.

Sogaro F, Schmid P, Morgans AS, 2018, Sensitivity analysis of thermoacoustic instabilities, Pages: 2063-2070

Copyright © (2018) by International Institute of Acoustics & Vibration.All rights reserved. Thermoacoustic instability is a phenomenon that occurs in numerous combustion systems, from rockets to land-based gas turbines. The thermoacoustic oscillations of these systems are of significant importance as they can result in severe vibrations, thrust oscillations, thermal stresses and mechanical loads that lead to fatigue or even failure. In this work we use a low-order network model representation of a combustor where linear acoustics are solved together with appropriate boundary conditions and flame jump conditions. Special emphasis is directed towards the interaction between instabilities associated with acoustic modes and flame-intrinsic modes. Adjoint methods are used to perform a receptivity and sensitivity analysis of the spectral properties of the system to changes in the parameters involved. To better analyse the extreme sensitivities that arise in the neighbourhood of a modal interaction, we compare the results with a time domain model that allows us to perform a pseudospectra analysis. The results provide key insights into the interplay between mode types.

Xia Y, Laera D, Morgans AS, et al., 2018, Thermoacoustic limit cycle predictions of a pressurised longitudinal industrial gas turbine combustor

Copyright © 2018 Siemens AG This article presents numerical prediction of a thermoacoustic limit cycle in an industrial gas turbine combustor. The case corresponds to an experimental high pressure test rig equipped with the full-scale Siemens SGT-100 combustor operated at two mean pressure levels of 3 bar and 6 bar. The Flame Transfer Function (FTF) characterising the global unsteady response of the flame to velocity perturbations is obtained for both operating pressures by means of incompressible Large Eddy Simulations (LES). A linear stability analysis is then performed by coupling the FTFs with a wave-based low order thermoacoustic network solver. All the thermoacoustic modes predicted at 3 bar pressure are stable; whereas one of the modes at 6 bar is found to be unstable at a frequency of 231 Hz, which agrees with the experiments. A weakly nonlinear stability analysis is carried out by combining the Flame Describing Function (FDF) predicted by LES with the low order thermoacoustic network solver. The frequency, mode shape and velocity amplitude corresponding to the predicted limit cycle at 209 Hz are used to compute the absolute pressure fluctuation amplitude in the combustor. The numerically reconstructed amplitude is found to be reasonably close to the measured dynamics.

Han X, Laera D, Morgans AS, et al., 2018, THE EFFECT OF STRATIFICATION RATIO ON THE MACROSTRUCTURE OF STRATIFIED SWIRL FLAMES: EXPERIMENTAL AND NUMERICAL STUDY, ASME Turbo Expo: Turbomachinery Technical Conference and Exposition, Publisher: AMER SOC MECHANICAL ENGINEERS

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Han X, Morgans A, 2017, Non-linear Interactions of Two Premixed Flames Explored by Large Eddy Simulation with External Acoustic Forcing, *Combustion Science and Technology*, Vol: 190, Pages: 424-435, ISSN: 0010-2202

This article describes a numerical study of the interactions between two lean premixed flames subjected to external acoustic forcing. This provides insights into the flame-to-flame interactions that may occur during combustion instability in annular combustors. Experimental measurements for comparison are available from the target combustor developed at Cambridge University (Worth and Dawson, 2012). Large eddy simulation is applied using the open source computational fluid dynamics toolbox, OpenFOAM, with the combustion modeled using the partially stirred reactor model with a four-step chemical reaction mechanism for methane/air. Harmonic velocity oscillations are imposed at the inlet; the flame responses are studied based on heat release rate signals in different combustion regions. The effect of the flame separation distance (Sd) on both the flame dynamics and unsteady heat release responses is analyzed. The results show that the flame-to-flame interactions are nonlinear for the flame separations studied. The spatial variation of the unsteady heat release rate demonstrates that flame-wall interactions play an important role, becoming even more important than flame-to-flame interactions for closely spaced flames (Sd < 2.00D). These findings imply that for the flame separation distances studied, any flame model used in the low-order annular combustion instability prediction should account for both nonlinearity and flame-to-flame interactions.

Xia Y, Duran I, Morgans AS,
et al., 2017, Dispersion of entropy perturbations transporting through an industrial gas turbine combustor, *Flow, Turbulence and Combustion*, Vol: 100, Pages: 481-502, ISSN: 1386-6184

In the context of combustion noise and combustion instabilities, the transport of entropy perturbations through highly simplified turbulent flows has received much recent attention. This work performs the first systematic study into the transport of entropy perturbations through a realistic gas turbine combustor flow-field, exhibiting large-scale hydrodynamic flow features in the form of swirl, separation, recirculation zones and vortex cores, these being ubiquitous in real combustor flows. The reacting flow-field is simulated using low Mach number large eddy simulations, with simulations validated by comparison to available experimental data. A generic artificial entropy source, impulsive in time and spatially localized at the flame-front location, is injected. The conservation equation describing entropy transport is simulated, superimposed on the underlying flow-field simulation. It is found that the transport of entropy perturbations is dominated by advection, with both thermal diffusion and viscous production being negligible. It is furthermore found that both the mean flow-field and the large-scale unsteady flow features contribute significantly to advective dispersion — neither can be neglected. The time-variation of entropy perturbation amplitude at combustor exit is well-modelled by a Gaussian profile, whose dispersion exceeds that corresponding to a fully-developed pipe mean flow profile roughly by a factor of three. Finally, despite the attenuation in entropy perturbation amplitude caused by advective dispersion, sufficient entropy perturbation strength is likely to remain at combustor exit for entropy noise to make a meaningful contribution at low frequencies.

Li J, Morgans A, 2017, Commentary on manuscript “Comment on ‘The one-dimensional acoustic field with arbitrary mean axial temperature gradient and mean flow’ (J. Li and A. S. Morgans, Journal of Sound and Vibration 400 (2017) 248–269)”, *Journal of Sound and Vibration*, Vol: 410, Pages: 488-494, ISSN: 0022-460X

Davide L, Morgans AS, 2017, Large eddy simulations for the flame describing functionof apremixed turbulent swirling flame, 24th International Congress on Sound and Vibration (ICSV24)

Computational prediction of thermoacoustic instabilities arising in gas turbine and aeroengine combustors is an ongoing challenge. Approaches which couple separate treatments for the acoustic waves and the flame all rely on a model for the response of the flame to oncoming acoustic perturbations. In the frequency domain, in the limit of small perturbations, this function is usually given in terms of the so-called Flame Transfer Function (FTF), i.e., a function that relates the heat release rate perturbation at the flame location with longitudinal velocity fluctuations taken at an upstream reference point. With the increase of the amplitude of oscillations, nonlinear combustion process controls the dynamics of the systems. A flame describing function (FDF) is so-defined introducing a dependence of the gain and phase of the FTF on velocity fluctuations amplitude |u'/ū|. The present work uses numerical simulations to obtain the FDF of a turbulent premixed swirling flame. The swirled burner developed at NTNU university is considered in a longitudinal combustor setup. Simulations are performed using Large Eddy Simulations (LES) via the open source Computational Fluid Dynamics (CFD) code, OpenFOAM. The predicted unperturbed flame structure is at first presented and discussed. Subsequently, the nonlinear flame response is characterised submitting the flame to a harmonically varying longitudinal velocity fluctuation, for which the forcing frequency is varied from 300 Hz to 1900 Hz considering two forcing amplitude levels, |u'/ū|=0.1 and |u'/ū|=0.2. The shape of the gain and the phase of the full FDF is discussed and the main characteristics are investigated.

Iñigo JG, Morgans A, Durán I, 2017, Sound generated by blades interacting with entropy waves, 24th International Congress on Sound and Vibration 2017 (ICSV 24)

This paper deals with the theoretical modelling of the sound generated by convected entropic perturbations interacting with blades. A model is obtained for inviscid small perturbations evolving around a potential steady flow past an aerofoil. Neglecting the mean flow variations of velocity and density in the propagation of sound and using a Lorentz-type transformation, the equation governing the acoustics is reduced to the Helmholtz equation with a modified source term. The simplified problem is recast in integral form and solved using compact Green's function method. This approach assumes that the acoustic wavelength is large compared to the profile chord and spacing, while no restrictions are made on the entropy wavelengths. The aerofoil thickness, camber and angle of attack are restricted such that the steady flow is a small perturbation to a uniform flow. For a symmetric profile, it is found that the main contribution to the radiated sound arises from the deceleration of the flow at the stagnation point of the leading-edge.

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