, ZDES simulation set-up

. .. , Choice of geometry and mesh construction

, 112 6.2.2 Comparison of the mean ows obtained with dierent forcing types, p.115

. .. Investigation, 120 6.3.3 Assessment of the signal-to-noise ratio

. .. Conclusions, 131 6.1.1 Choice of geometry and mesh construction

. .. Investigation, 120 6.3.3 Assessment of the signal-to-noise ratio

. .. Conclusions, 131 105 List of Figures 1.1 Sketch of the conguration considered in Cumpsty and Marble's [1977] compact model. This gure is adapted from Leyko et al, p.10, 2014.

, Contours of reduced entropy concentration, S, on a symmetry plane of z = 0, at four time instants after the impulsive entropy source injection into the timevarying combustor ow-eld, 2018.

. Ewg-[bake, and (b) HAT [Knobloch et al., 2015a] test-facilities at DLR, 2007.

, Snapshot showing the pressure eld generated as an entropy line pulse is convected through a xed turbine blade, p.24, 2014.

. Papadogiannis, ?? /? of an instantaneous solution at mid-span without (a) and with (b) an entropy pulse, p.26, 2016.

, Classication of ow problems associated with ZDES modes. I: separation xed by the geometry, II: separation induced by a pressure gradient on a curved surface, III: separation strongly inuenced by the dynamics of the incoming boundary layer

, Amplitude and phase error over the distance convected and for a range of entropy wave spatial discretisation, p.36

, Diagrams of the waves involved for each forcing type in the subsonic case, p.43

, Contour of the Mach number in the nozzle

. .. Of-streamlines, , p.45

, CAA geometry and structured mesh. (b) and (c) have dierent scales, p.46

, Density uctuations injected into the computational domain in the entropyforced case

, Pressure uctuations measured in the entropic forcing case on the nozzle axis in the exit plane

, Contour of normalised entropy uctuations in an axial plane in the multiharmonic entropy-forced case

, 8 RMS values of pressure and velocity uctuations p and u in upstream and downstream ducts

, ) computed using CAA with a baseline 2.2 million node mesh, a mesh 1.5 times ner in all directions with 7.5 million elements and harmonic simulations

, + /? 1 ] (right) computed using CAA and both the Riemann invariants and DMM for acoustic wave separation

, Pressure and velocity uctuations computed as part of the multi-harmonic entropy-forced simulation both at a point of the nozzle axis in the middle of the downstream duct and averaged over the plane, and acoustic velocity uctuations computed from the impedance relation for a plane progressive wave u = p /(? 0 c 0 )

, Reection coecients obtained for the (a) entropy-forced simulation and (b) the two computations with acoustic forcing. Excitation types are given in brackets

. .. Without-non-reective-post-processing, , vol.54

, Amplitude and phase of the transfer functions resulting from entropic forcing

. Giauque, Huet et al., 2016] and Marble and Candel's compact solution, p.56, 1977.

. Giauque, Mean axial velocity (left) and pressure (right) proles used by the 2D model CHEOPS-Nozzle after sectional averaging and by the 1D model MARCAN, Huet, 2012.

. .. Hz, Contour of the normalised entropy uctuations ? inside the nozzle in 1D (top) and 2D (bottom) for harmonic forcing at 1000, p.57

, Amplitude and phase of the transfer functions resulting from upstream acoustic forcing

. Giauque, Huet et al., 2016] and Marble and Candel's compact solution, + /P + 1 ] (bottom) computed using CAA, CHEOPS-Nozzle, the 1D model MARCAN, p.58, 1977.

, Amplitude and phase of the transfer functions resulting from downstream acoustic forcing

+. Giauque, ? 2 ] (bottom) computed using CAA, CHEOPS-Nozzle, the 1D model MARCAN, p.60, 1977.

, Pressure uctuation contours in an axial plane with entropic excitation at 100 and 1,000 Hz respectively. Figures (b) and (d) show the convergent regions of the nozzle with reduced colour contour range, p.61

, Axial velocity uctuation contours in an axial plane of the nozzle with entropic excitation at 100 and 1,000 Hz. Dierent colour contour ranges are used upstream and downstream, p.61

, Unit vectors of the Cartesian coordinate system ( e x , e y ) and the mobile coordinate system ( e s, p.65

, 2 Sketch showing the radii of curvature R s and R n in the s and n directions respectively

. .. Cheops-stator,

, Normalised mass ow rate error (left) and pressure signal (right) close to the pressure side of the blade. The signals are computed for the baseline case towards the end of the transient simulations, after the switch to a second order time scheme

, Parameter f d simulated in the baseline case (a) at mid-span and (b) represented as isolines on a Mach number contour in an azimuthally constant plane downstream of the blade

. .. , Blade-to-blade contours of the mean Mach number computed at 5%, 50% and 95% of the blade's span with the baseline simulation, p.111

, Axial contours of the mean Mach number, static pressure and entropy elds, computed with the baseline simulation 32% (top) and 50% (bottom) of the axial chord downstream of the stator's trailing edge

, Vorticity in (a) the mid-span plane and (b) an azimuthally constant plane downstream of the stator in the baseline case

, PSD of velocity near the wake of the stator. The Welch estimator is used with one and ve blocks

, Eddy viscosity ratio µ t /µ in the mid-span plane

, They are computed one axial chord away from the stator blade upstream and downstream, from the baseline, entropy-forced and both upstream and downstream acoustically excited simulations, Radial proles of pitchwise-averaged mean ow variables: ow angle, static pressure

, Pressure and velocity uctuations computed both at a point of the upstream (top) and downstream (bottom) duct and averaged over the corresponding plane, and acoustic velocity uctuations computed from the impedance relation for a plane progressive or regressive wave u = ±p /(? 0 c 0 ). Entropy-forced simulation

, Normalised regressive and progressive waves P ? and P + computed in the upstream and downstream ducts in the entropy-forced case, using both the Riemann invariants and the DMM method

, Reection coecients obtained for the entropy forced ZDES simulation (left) and the two computations with acoustic perturbations (right), with both the Riemann invariants and DMM for post-processing

, Amplitude and phase of the transfer functions resulting from upstream acoustic forcing

?. .. , computed with and without non-reective post-processing, and using both the Riemann invariants and DMM, p.121

, Amplitude and phase of the transfer functions resulting from downstream acoustic forcing

?. , computed with and without nonreective post-processing, and using both the Riemann invariants and DMM, p.122

, Amplitude and phase of the transfer functions resulting from entropic forcing

. .. , computed with and without non-reective postprocessing, and using both the Riemann invariants and DMM, p.123

, A.1 Sketch of the coordinate system (e s

, Sketch of an hexahedral element

B. , Schémas des ondes présentes pour chacun des forçages étudiés en subsonique, p.147

, 3 Amplitude et phase des fonctions de transfert entropiques

. Caa, . Cheops-nozzle, and . Mar-can-[giauque, et la solution compacte de Marble and Candel [Marble and Candel, 1977.

, B.4 Fluctuations d'entropie normalisées ? dans la tuyère 1D (haut) et 2D (bas) pour un forçage harmonique à 1000 Hz

, Vecteurs unitaires du système de cordonnée cartésien ( e x , e y ) et du système de coordonnée mobile, p.151

, obtenues avec CHEOPS-Stator et la CAA avec des champs Euler et RANS, ainsi que la solution compacte de Cumpsty and Marble, p.153, 1977.

, Géométrie utilisée pour simuler le bruit dans un canal de stator isolé de turbine, p.154

, obtenues par des simulations ZDES post-traitées soit avec les invariants de Riemann soit avec la DMM, et par des calculs CAA avec des champs Euler ou RANS, Champs moyens simulés avec le calcul sans forçage et représentés dans des coupes axiales à 32% (haut) et 50% (bas) d'une corde axiale à l'aval du bord de fuite

A. Bibliography, Strategic Research and Innovation Agenda, p.145, 2017.

B. An, J. Liu, and H. Jiang, Numerical Investigation on Unsteady Eects of Hot Streak on Flow and Heat Transfer in Turbine Stage, Journal of Turbomachinery, vol.131, issue.3, p.27, 2009.

B. Aupoix and P. R. Spalart, Extensions of the Spalart-Allmaras Turbulence Model to Account for Wall Roughness, International Journal of Heat and Fluid Flow, vol.24, issue.4, p.113, 2003.

C. Bailly and G. Comte-bellot, Turbulence. CNRS éditions, vol.113, 2003.

F. Bake, U. Michel, and I. Roehle, Investigation of Entropy Noise in Aero-engine Combustors, Journal of Engineering for Gas Turbines and Power, vol.129, issue.2, p.159, 2007.

F. Bake, N. Kings, and I. Roehle, Fundamental Mechanism of Entropy Noise in Aero-engines: Experimental Investigation, Journal of Engineering for Gas Turbines and Power, vol.130, issue.1, p.93, 2008.

F. Bake, N. Kings, A. Fischer, and I. Röhle, Experimental Investigation of the Entropy Noise Mechanism in Aero-engines, International Journal of Aeroacoustics, vol.8, issue.1, p.22, 2009.

F. Bake, C. Richter, B. Mühlbauer, N. Kings, I. Röhle et al., The Entropy Wave Generator (EWG): a Refence Case on Entropy Noise, Journal of Sound and Vibration, vol.326, issue.3-5, p.146, 2009.

F. Bake, P. Gaetani, G. Persico, L. Neuhaus, and K. Knobloch, Indirect Noise Generation in a High Pressure Turbine Stage, 22nd AIAA/CEAS Aeroacoustics Conference, number AIAA, vol.84, p.107, 2016.

M. Bauerheim, I. Durán, T. Livebardon, G. Wang, S. Moreau et al., Transmission and Reection of Acoustic and Entropy Waves through a Stator-rotor Stage, Journal of Sound and Vibration, vol.374, p.59, 2016.

P. T. Beard, T. Povey, and K. S. Chana, Turbine Eciency Measurement System for the Qinetiq Turbine Test Facility, Journal of Turbomachinery, vol.132, issue.1, p.146, 2009.

C. Aguirre, Simulation of noise emitted by a reactive ow, vol.59, p.146, 2017.

D. J. Bodony, Scattering of an Entropy Disturbance into Sound by a Symmetric Thin Body, Physics of Fluids, vol.21, issue.9, p.108, 2009.

M. S. Bohn, Response of a Subsonic Nozzle to Acoustic and Entropy Disturbances, Journal of Sound and Vibration, vol.52, issue.2, p.22, 1977.

M. S. Bohn, Noise Produced by the Interaction of Acoustic Waves and Entropy Waves with High-speed Nozzle Flows, p.22, 1979.

A. Cader, C. Polacsek, and T. L. Garrec, Numerical prediction of rotor-stator interaction noise using 3D CAA with synthetic turbulence injection, AIAA/CEAS Aeroacoustics Conference, p.99, 2018.
URL : https://hal.archives-ouvertes.fr/hal-02070555

S. Candel, D. Durox, S. Ducruix, A. L. Birbaud, N. Noiray et al., Flame dynamics and combustion noise: progress and challenges, International Journal of Aeroacoustics, vol.8, issue.1, p.145, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00497146

S. M. Candel, Analytical Studies of Some Acoustic Problems of Jet Engines, vol.8, p.20, 1972.

A. Ceci, R. Gojon, and M. Mihaescu, Large Eddy Simulations for Indirect Combustion Noise Assessment in a Nozzle Guide Vane Passage. Flow, Turbulence and Combustion, vol.102, p.146, 2019.
URL : https://hal.archives-ouvertes.fr/hal-01881298

S. Chandramouli, R. Gojon, J. Fridh, and M. Mihaescu, Numerical Characterization of Entropy Noise with a Density Based Solver, 12th European Conference on Turbomachinery Fluid dynamics & Thermodynamics, number ETC2017-143, p.27, 2017.

B. Chu and L. S. Kovásznay, Non-linear Interactions in a Viscous Heat-conducting Compressible Gas, Journal of Fluid Mechanics, vol.3, issue.5, p.24, 1958.

V. Clair, C. Polacsek, T. L. Garrec, and G. Reboul, Experimental and Numerical Investigation of Turbulence-Airfoil Noise Reduction Using Wavy Edges, AIAA Journal, vol.51, issue.11, p.99, 2013.
URL : https://hal.archives-ouvertes.fr/hal-02085960

L. Crocco, Supercritical Gaseous Discharge with High Frequency Oscillations, Aerotechnica, vol.33, issue.1, p.4653, 1953.

N. A. Cumpsty, Excess Noise from Gas Turbine Exhausts, ASME International Gas Turbine Conference and Products Show, number 75-GT-61 in 1A, 1975.

N. A. Cumpsty and F. E. Marble, The Interaction of Entropy Fluctuations with Turbine Blade Rows; A Mechanism of Turbojet Engine Noise, Proceedings of the Royal Society of London A, vol.357, p.164, 1977.

F. De-domenico, E. O. Rolland, and S. Hochgreb, Detection of Direct and Indirect Noise Generated by Synthetic Hot Spots in a Duct, Journal of Sound and Vibration, vol.394, issue.5, p.146, 2017.

F. De-domenico, S. M. Lowe, L. Fan, S. Hochgreb, P. Shah et al., High Frequency Measurement of Temperature and Composition Spots with LITGS, Journal of Engineering for Gas Turbines and Power, vol.141, issue.3, p.31003, 2018.

F. De-domenico, E. Rolland, and S. Hochgreb, A Generalised Model for Acoustic and Entropic Transfer Function of Nozzles with Losses, Journal of Sound and Vibration, vol.440, p.23, 2019.

F. De-la-puente-cerezo, Aeroacoustic Simulations of Landing Gears with Unstructured Grids and a ZDES Turbulence Model, p.115, 2017.
URL : https://hal.archives-ouvertes.fr/tel-01682189

S. Deck, Recent Improvements in the Zonal Detached Eddy Simulation (ZDES) formulation. Theoretical and Computational Fluid Dynamics, vol.26, p.159, 2012.

A. P. Dowling and Y. Mahmoudi, Combustion Noise. Proceedings of the Combustion Noise Institute, vol.35, p.146, 2015.
URL : https://hal.archives-ouvertes.fr/hal-00811265

I. Durán and S. Moreau, Study of the Attenuation of Waves Propagating Through Fixed and Rotating Turbine Blades, 18th AIAA/CEAS Aeroacoustics Conference, number AIAA2012-2133, p.21, 2012.

I. Durán and S. Moreau, Numerical Simulation of Acoustic and Entropy Waves Propagating through Turbine Blades, 19th AIAA/CEAS Aeroacoustics Conference, pp.2013-2102, 2013.

I. Durán and S. Moreau, Solution of the Quasi-one-dimensional Linearized Euler Equations Using Flow Invariants and the Magnus Expansion, Journal of Fluid Mechanics, vol.723, p.146, 2013.

I. Durán and A. S. Morgans, On the Reection and Transmission of Circumferential Waves Through Nozzles, Journal of Fluid Mechanics, vol.773, p.146, 2015.

I. Durán, M. Leyko, S. Moreau, F. Nicoud, and T. Poinsot, Computing Combustion Noise by Combining Large Eddy Simulations with Analytical Models for the Propagation of Waves through Turbine Blades, Comptes Rendus Mécanique, vol.341, issue.1-2, p.131140, 2013.

L. Eca, M. Hoekstra, A. Hay, and D. Pelletier, A Manufactured Solution for a Two-Dimensional Steady Wall-Bounded Incompressible Turbulent Flow, International Journal of Computational Fluid Dynamics, vol.21, issue.3-4, p.113, 2007.

A. Emmanuelli, M. Huet, T. L. Garrec, and S. Ducruix, CAA Study of Entropy Noise in Nozzle Flow for the Validation of a 2D Semi-Analytical Model, ASME Turbo Expo, number GT201763640 in 2C, p.41, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01863960

A. Emmanuelli, M. Huet, T. L. Garrec, and S. Ducruix, Study of Entropy Noise through a 2D Stator using CAA, AIAA/CEAS Aeroacoustics Conference, p.86, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01864034

A. Emmanuelli, J. Zheng, M. Huet, A. Giauque, T. L. Garrec et al., Description and Application of a 2D-axisymmetric Model for Entropy Noise in Nozzle Flows, Journal of Sound and Vibration, vol.472, p.41, 2020.

P. Gaetani and G. Persico, Hot Streak Evolution in an Axial HP Turbine Stage, International Journal of Turbomachinery Propulsion and Power, vol.2, issue.2, p.27, 2017.

P. Gaetani, G. Persico, V. Dossena, and C. Osnaghi, Investigation of the Flow Field in a High-Pressure Turbine Stage for Two Stator-Rotor Axial Gaps Part I: Three-Dimensional Time-Averaged Flow Field, Journal of Turbomachinery, vol.129, issue.3, p.572579, 2007.

P. Gaetani, G. Persico, V. Dossena, and C. Osnaghi, Investigation of the Flow Field in a High-Pressure Turbine Stage for Two Stator-Rotor Axial Gaps Part II: Unsteady Flow Field, vol.129, p.580590, 2007.

P. Gaetani, G. Persico, and C. Osnaghi, Eects of Axial Gap on the Vane-Rotor Interaction in a Low Aspect Ratio Turbine Stage, Journal of Propulsion and Power, vol.26, issue.2, p.325334, 2010.

P. Gaetani, G. Persico, and A. Spinelli, Entropy Wave Generator for Indirect Combustion Noise Experiments in a High-Pressure Turbine, 11th European Turbomachinery Conference, number ETC2015-025, 2015.

A. Giauque, M. Huet, and F. Cléro, Analytical Analysis of Indirect Combustion Noise in Subcritical Nozzles, Journal of Engineering for Gas Turbines and Power, vol.134, issue.11, p.164, 2012.

A. Giauque, M. Huet, F. Cléro, S. Ducruix, and F. Richecoeur, Thermoacoustic Shape Optimization of a Subsonic Nozzle, Journal of Engineering for Gas Turbines and Power, vol.135, issue.10, p.43, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01863996

A. Giusti, N. A. Worth, E. Mastorakos, and A. P. Dowling, Experimental and Numerical Investigation into the Propagation of Entropy Waves, AIAA Journal, vol.55, issue.2, p.20, 2017.

C. S. Goh and A. S. Morgans, Phase Prediction of the Response of Choked Nozzles to Entropy and Acoustic Disturbances, Journal of Sound and Vibration, vol.330, issue.21, p.146, 2011.

P. Guillaume, J. Schoukens, R. Pintelon, and I. Kollár, Crest-Factor Minimisation using Nonlinear Chebyshev Approximation Methods, IEEE Transactions on Instrumentation and Measurement, vol.40, issue.6, p.108, 1991.

J. Guzmán-iñigo, A. Morgans, and I. Durán, A Model for the Sound Generated by Entropy Disturbances Interacting with Isolated Blades, AIAA/CEAS Aeroacoustics Conference, p.17, 2018.

H. A. Hassan, Scaling on Combustion-generated Noise, Journal of Fluid Mechanics, vol.66, issue.3, p.145, 1974.

A. Holewa, S. Lesnik, G. Ashcroft, and S. Guérin, CFD-Based Investigation of Turbine Tonal Noise Induced by Steady Hot Streaks, International Journal of Turbomachinery Propulsion and Power, vol.2, issue.1, p.27, 2017.

S. M. Hosseinalipour, A. Fattahi, H. Afshari, and N. Karimi, On the Eects of Convecting Entropy Waves on the Combustor Hydrodynamics, Applied Thermal Engineering, vol.110, p.21, 2017.

M. Howe, Indirect Combustion Noise, Journal of Fluid Mechanics, vol.659, p.59, 2010.

M. Huet, Extension du Code MarCan pour l'Évaluation des Fonctions de Transfert de Tuyère aux Écoulements Supersoniques sans et avec Choc, ONERA, issue.13, 2013.

M. Huet, Nonlinear Indirect Combustion Noise for Compact Supercritical Nozzle Flows, Journal of Sound and Vibration, vol.374, p.13, 2016.

M. Huet, Budgets of Disturbances Energy for Nozzle Flows at Subsonic and Choked Regimes, Journal of Engineering for Gas Turbines and Power, vol.140, issue.11, p.59, 2018.
URL : https://hal.archives-ouvertes.fr/hal-02181779

M. Huet and B. Courbet, Évaluation Analytique et Numérique de la Non-réexion des Conditions aux Limites dans le code CEDRE, ONERA, vol.36, p.37, 2016.

M. Huet and A. Giauque, A Nonlinear Model for Indirect Combustion Noise Through a Compact Nozzle, Journal of Fluid Mechanics, vol.733, p.146, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01725170

M. Huet and F. Vuillot, Évaluation de la Précision des Schémas Numériques du Code CEDRE pour la Propagation de Perturbations Entropiques en Maillage Tétrahédrique, ONERA, vol.35, p.159, 2017.

M. Huet, F. Vuillot, N. Bertier, M. Mazur, N. Kings et al., Recent Improvements in Combustion Noise Investigation: from the Combustion Chamber to Nozzle Flow. Aerospace Lab, vol.11, p.164, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01369749

. Icao, Report of the Council, vol.3, p.145, 2017.

M. Ihme, Combustion and Engine-Core Noise, Annual Review of Fluid Mechanics, vol.49, issue.3, pp.277-310, 2017.

N. Kings, Indirect Combustion Noise: Experimental Investigation of the Vortex Sound Generation in Nozzle Flows, vol.3, p.25, 2014.

N. Kings and F. Bake, Indirect Combustion Noise: Noise Generation by Accelerated Vorticity in a Nozzle Flow, International Journal of Spray and Combustion Dynamics, vol.2, issue.3, p.25, 2010.

N. Kings, W. Tao, P. Scouaire, F. Richecoeur, and S. Ducruix, Experimental and Numerical Investigation of Direct and Indirect Combustion Noise in a Lean Premixed Laboratory Swirled Combustor, ASME Turbo Expo, number GT201657848 in 4B, p.25, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01863986

K. Knobloch, T. Werner, and F. Bake, Entropy Noise Generation and Reduction in a Heated Nozzle Flow, 21st AIAA/CEAS Aeroacoustics Conference, AIAA AVIATION Forum , number AIAA 20152818, vol.24, p.159, 2015.

K. Knobloch, T. Werner, and F. Bake, Noise Generation in Hot Nozzle Flow, ASME Turbo Expo, number GT2015-43702, vol.22, p.24, 2015.

K. Knobloch, A. Holewa, S. Guérin, Y. Mahmoudi, T. Hynes et al., Noise Transmission Characteristics of a High Pressure Turbine Stage, 22nd AIAA/CEAS Aeroacoustics Conference, number AIAA, vol.84, p.107, 2016.

K. Knobloch, L. Neuhaus, F. Bake, P. Gaetani, and G. Persico, Experimental Assessment of Noise Generation and Transmission in a High-pressure Transonic Turbine Stage, ASME Turbo Expo, number GT2016-57209, vol.84, p.112, 2017.

J. Kopitz, E. Bröcker, and W. Polifke, Charactiristics-based Filter for Identication of Planar Acoustic Waves in Numerical simulation of Turbulent Compressible Flows. International Congress of Sound and Vibration, p.50, 2005.

M. Leyko, F. Nicoud, and T. Poinsot, Comparison of Direct and Indirect Combustion Noise Mechanisms in a Model and Combustor, AIAA Journal, vol.47, issue.11, p.22, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00803808

M. Leyko, S. Moreau, F. Nicoud, and T. Poinsot, Waves Transmission and Generation in Turbine Stages in a Combustion-Noise Framework, 16th AIAA/CEAS AeroAcoustics Conference, vol.4, p.21, 2010.

M. Leyko, S. Moreau, F. Nicoud, and T. Poinsot, Numerical and Analytical Modelling of Entropy Noise in a Supersonic Nozzle with a Shock, Journal of Sound and Vibration, vol.330, issue.16, p.146, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00802478

M. Leyko, S. Moreau, F. Nicoud, and T. Poinsot, Simulation and Modelling of the Waves Transmission and Generation in a Stator Blade Row in a Combustion-Noise Framework, Journal of Sound and Vibration, vol.333, issue.23, p.159, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01101467

T. Livebardon, S. Moreau, L. Gicquel, T. Poinsot, and E. , Bouty Combustion, and Flame. Combining LES of Combustion Chamber and an Actuator Disk Theory to Predict Combustion Noise in a Helicopter Engine, Combustion and Flame, vol.165, p.21, 2016.

J. M. Lourier, A. Huber, B. Noll, and M. Aigner, Numerical analysis of indirect combustion noise generation within a subsonic nozzle, AIAA Journal, vol.52, p.22, 2014.

W. Magnus, On the Exponential Solution of Dierential Equations for a Linear Operator, Communications on pure and applied mathematics, vol.7, issue.4, p.649673, 1954.

L. Magri, J. O'brien, and M. Ihme, Compositional Inhomogeneities as a Source of Indirect Combustion Noise, Journal of Fluid Mechanics, vol.799, issue.R4

Y. Mahmoudi, A. Giusti, E. Mastorakos, and A. P. Dowling, Low-Order Modeling of Combustion Noise in an Aero-Engine: The Eect of Entropy Dispersion, Journal of Engineering for Gas Turbines and Power, vol.140, issue.1, p.21, 2018.

F. E. Marble, Acoustic Disturbance from Gas Non-uniformities Convecting through a Nozzle. Symposium on Transportation Noise, vol.8, p.20, 1973.

F. E. Marble and S. M. Candel, Acoustic Disturbance from Gas Non-uniformities Convected Through a Nozzle, Journal of Sound and Vibration, vol.55, issue.2, p.164, 1977.

F. R. Menter, Two-equation Eddy-viscosity Turbulence Models for Engineering Applications, AIAA Journal, vol.32, issue.8, p.32, 1994.

J. H. Miles, Time Delay Analysis of Turbofan Engine Direct and Indirect Combustion Noise Sources, Journal of Propulsion and Power, vol.25, issue.1, p.218227, 2009.

J. H. Miles, Separating Direct and Indirect Engine Combustion Noise using the Correlation Function, Journal of Propulsion and Power, vol.26, issue.5, p.23, 2010.

A. Mishra and D. J. Bodony, Evaluation of Actuator Disk Theory for Predicting Indirect Combustion Noise, Journal of Sound and Vibration, vol.332, issue.4, p.108, 2013.

W. Moase, M. J. Brear, and C. Manzie, The Forced Response of Choked Nozzles and Supersonic Diusers, Journal of Fluid Mechanics, vol.585, p.146, 2007.

S. Moreau, C. Aguirre, and L. Gicquel, Large-eddy-simulation Prediction of Indirect Combustion Noise in the Entropy Wave Generator Experiment, International Journal of Spray and Combustion Dynamics, vol.10, issue.2, p.146, 2018.

C. L. Morfey, Amplication of Aerodynamic Noise by Convected Flow Inhomogeneities, Journal of Sound and Vibration, vol.31, issue.4, p.20, 1973.

A. S. Morgans and I. Durán, Entropy Noise: A review of Theory, Progress and Challenges, International Journal of Spray and Combustion Dynamics, vol.8, issue.4, p.20, 2016.

A. S. Morgans, C. S. Goh, and J. A. Dahan, The Dissipation and Shear Dispersion of Entropy Waves in Combustor Thermodynamics, Journal of Fluid Mechanics, vol.733, issue.R2

B. Mühlbauer, A. Wiedenhorn, M. Liu, B. Noll, and M. Aigner, Numerical Investigation of the Fundamental Mechanism of Entropy Noise Generation in Aero-engines, Acta Acustica united with Acustica, vol.95, issue.3, p.146, 2009.

M. Muthukrishnan, An Experimental Study of the Separation of Combustion Entropy Noise, p.22, 1977.

D. Papadogiannis, G. Wang, S. Moreau, F. Duchaine, L. Gicquel et al., Assessment of the Indirect Combustion Noise Generated in a Transonic High-pressure Turbine Stage, Journal of Engineering for Gas Turbines and Power, vol.138, issue.4, p.159, 2016.
URL : https://hal.archives-ouvertes.fr/hal-02050147

B. Pardowitz, U. Tapken, K. Knobloch, F. Bake, E. Bouty et al., Core Noise Identication of Broadband Noise Sources of a Turbo-shaft Engine, 20th AIAA/CEAS Aeroacoustics Conference, 2014.

G. Persico, A. Mora, P. Gaetani, and M. Savini, Unsteady aerodynamics of a low aspect ratio turbine stage: Modeling Issues and Flow Physics, Journal of Turbomachinery, vol.134, issue.6, p.146, 2012.

L. Pinelli, F. Poli, A. Arnone, S. Guérin, A. Holewa et al., On the Numerical Evaluation of Tone Noise Emissions Generated by a Turbine Stage: An In-Depth Comparison Among Dierent Computational Methods, ASME Turbo Expo, vol.5, p.23, 2015.

W. Polifke, C. Wall, and P. Moin, Partially Reecting and Non-reecting Boundary Conditions for Simulation of Compressible Viscous Flow, Journal of Computational Physics, vol.213, issue.1, p.49, 2006.

G. Reboul, A. Cader, C. Polacsek, and T. L. Garrec, CAA Prediction of Rotor-Stator Interaction Using Synthetic Turbulence: Application to a Low-Noise Serrated OGV, 23rd AIAA/CEAS Aeroacoustics Conference, AIAA AVIATION Forum, p.99, 2017.

S. Redonnet, Numerical Study of Acoustic Installation Eects with a Computational Aeroacoustics Method, AIAA Journal, vol.48, issue.5, p.147, 2010.

S. Redonnet, E. Manoha, and P. Sagaut, Numerical Simulations of Propagation of Small Perturbations Interacting with Flows and Solid Bodies, 7th AIAA/CEAS Aeroacoustics Conference, Aeroacoustics Conferences, vol.37, p.147, 2001.

A. Reoch, B. Courbet, A. Murrone, P. Villedieu, C. Laurent et al., Cedre software, CFD Platforms and coupling, vol.2, p.147, 2011.

N. Renard and S. Deck, Improvements in Zonal Detached Eddy Simulation for Wall Modeled Large Eddy Simulation, AIAA Journal, vol.53, issue.11, p.35, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01232564

E. Rolland, F. D. Domenico, and S. Hochgreb, Theory and Application of Reverberated Direct and Indirect Noise, Journal of Fluid Mechanics, vol.819, issue.23, p.435464, 2017.

E. Rolland, F. D. Domenico, and S. Hochgreb, Direct and Indirect Noise Generated by Entropic and Compositional Inhomogeneities, Journal of Engineering for Gas Turbines and Power, vol.140, issue.8, p.23, 2018.

C. L. Rumsey, D. O. Allison, R. T. Biedron, P. G. Buning, T. G. Gainer et al., CFD Sensitivity Analysis of a Modern Civil Transport Near Buet-Onset Conditions, p.115, 2001.

P. Sagaut, S. Deck, and M. Terracol, Multiscale and Multires-solution Approaches in Turbulence, p.33, 2013.

T. Sattelmayer, Inuence of the Combustor Aerodynamics on Combustion Instabilities from Equivalence Ratio Fluctuations, Journal of Engineering for Gas Turbines and Power, vol.125, p.20, 2003.

J. Smagorinsky, General Circulation Experiments with the Primitive Equations, Monthly Weather Review, vol.91, issue.3, p.33, 1963.

S. Spagnolo, Unsteady Aerodynamic Loads on Aircraft Landing Gear, p.115, 2016.

P. Spalart, W. Jou, M. Strelets, and S. Allmaras, Comments on the feasibility of LES for wings and on a hybrid RANS/LES approach, Proceedings of the 1st AFSOR International Conference on DNS/LES, p.33, 1998.

P. Spalart, S. Deck, M. Shur, K. Squires, M. Strelets et al., A New Version of Detached-eddy Simulation, Resistant to Ambiguous Grid Densities, Theory of Computational Fluid Dynamics, vol.20, p.33, 2006.

P. R. Spalart, Trends in Turbulence Treatments, Fluids 2000 Conference and Exhibit, number AIAA, p.115, 2000.

P. R. Spalart and S. R. Allmaras, A One-equation Turbulence Model for Aerodynamic Flows, La Recherche Aerospatiale, vol.1, p.32, 1994.

S. R. Stow, A. P. Dowling, and T. P. Hynes, Reection of circumferential modes in a choked nozzle, Journal of Fluid Mechanics, vol.467, p.146, 2002.

C. K. Tam, Advances in Numerical Boundary Conditions for Computational Aeroacoustics, Journal of Computational Aeroacoustics, p.39, 1998.

C. K. Tam and Z. Dong, Radiation and Outow Boundary Conditions for Direct Computation of Acoustic and Flow Disturbances in a Nonuniform Mean Flow, Journal of Computational Acoustics, vol.4, issue.2, p.152, 1996.

C. K. Tam and S. A. Parrish, On the Generation of Indirect Combustion Noise, 20th AIAA/CEAS Aeroacoustics Conference, AIAA Aviation, number AIAA 2014-3315, vol.17, p.159, 2014.

C. K. Tam and S. A. Parrish, Noise of High-performance Aircraft at Afterburner, Journal of Sound and Vibration, vol.352, issue.23, p.103128, 2015.

C. K. Tam and S. A. Parrish, The Physical Processes of Indirect Combustion Noise Generation, International Journal of Aeroacoustics, vol.17, issue.2, p.24, 2018.

C. K. Tam and J. Webb, Dispersion-Relation-Preserving Finite Dierence Schemes for Computational Acoustics, Journal of Computational Physics, vol.107, issue.2, p.152, 1993.

C. K. Tam, S. A. Parrish, J. Xu, and B. Schuster, Indirect Combustion Noise of Auxiliary Power Units, Journal of Sound and Vibration, vol.332, p.17, 2013.

H. S. Tsien, The Transfer Function of Rocket Nozzles, Journal of the American Rocket Society, vol.22, issue.8, p.139143, 1952.

W. C. Ullrich, J. Gikadi, C. Jörg, and T. Sattelmayer, Acoustic-entropy Coupling Behavior and Acoustic Scattering Properties of a Laval Nozzle, 20th AIAA/CEAS Aeroacoustics Conference, p.22, 2014.

W. C. Ullrich, M. Schulze, and T. Sattelmayer, Fundamental Indirect Noise Generation by Interactions between Entropy, Vorticity and Acoustic Waves in the Context of Aero Engine Applications, Inter.noise (43rd Int. Congress on Noise Control Engineering), p.25, 2014.

G. Wang, M. Sanjose, S. Moreau, D. Papadogiannis, F. Duchaine et al., Noise Mechanisms in a Transonic High-pressure Turbine Stage, International Journal of Aeroacoustics, vol.15, issue.1-2, p.159, 2016.

Y. Xia, I. Durán, A. Morgans, and X. Han, Dispersion of Entropy Perturbations Transporting through an Industrial Gas Turbine Combustor. Flow Turbulence and Combustion, vol.100, p.159, 1920.

J. Zheng, Analytical and Numerical Study of the Indirect Combustion Noise Generated by Entropy Disturbances in Nozzle Flows, vol.146, p.147, 2016.

J. Zheng, M. Huet, A. Giauque, F. Cléro, and S. Ducruix, A 2D-axisymmetric Analytical Model for the Estimation of Indirect Combustion Noise in Nozzle Flows, 21st AIAA/CEAS Aeroacoustics Conference, vol.133, p.146, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01864011

E. E. Zukoski and J. M. Auerbach, Experiments Concerning the Response of Supersonic Nozzles to Fluctuating Inlet Conditions, Journal of Engineering for Power, vol.98, issue.1, p.22, 1976.