domain: spi phys

text:
Équipe

M Éric

Lamballais Professeur

M Sofiane

Khelladi Professeur

Ensam -Rapporteur M Thomas

Mme Estelle

Ph.D Loris Casadei

Cesare, Clément, Eleonora Amelia Federica

Fran Gabi

Giulia

D J Guillaume

Juan

Ludovic, Majo, Manon, Manuel Lorenzo

Maria Morgana Marion

Natalia Nicky

Sergio Ramon

Tomek Un

E Rebecca

Agli Incredibili Giulio

ED MEGeP : Dynamique des fluides

Keywords: Time-domain impedance boundary condition, Oscillo-Diffusive Representation, Aircraft noise, Acoustic liner, Duct acoustics, Nonlinear propagation, Acoustic boundary condition, Buzz-Saw Noise, Computational Fluid Dynamics (CFD), Navier-Stokes Characteristic Boundary

Time-domain broadband impedance model for computational aeroacoustics : application to shock-wave propagation in lined intakes Modèle d'impédance temporel large bande pour les codes d'aéroacoustique numérique : application à la propagation du bruit de choc dans les entrées d'air de nacelle traitées

Modèle d'impédance temporel large bande pour les codes d'aéroacoustique numérique: application a la propagation du bruit de choc dans les entrées d'air de nacelle traitées

présentée par

Loris Casadei

Ph.D. dissertation soutenue le 23 Mars, 2022. ISAE-SUPAERO, Université de Toulouse, France.

Resumé

Le bruit de choc du rotor est aujourd'hui une des principales sources de bruit des moteurs d'avions dans les conditions de décollage et de montée. Le contrôle et la réduction de cette source de bruit sont d'une importance primordiale pour les avionneurs pour se conformer aux réglementations internationales et améliorer le confort des passagers. Des outils de simulation haute fidélité sont nécessaires pour son étude, avec prise en compte de tous les effets géométriques et d'écoulement 3D ainsi que la modélisation des traitements acoustiques incorporés dans les parois de l'entrée d'air de nacelle. Les solveurs Euler et Navier-Stokes proposent des solutions pour calculer la propagation non linéaire des fluctuations de pression de forte amplitude des chocs créés par les rotors dans l'entrée d'air. Cependant, la modélisation des revêtements acoustiques dans ces solveurs reste un défi de l'acoustique numérique moderne, en raison de leur appartenance naturelle au domaine fréquentiel. Le présent travail se concentre sur la validation et l'extension de la condition limite d'impédance dans le domaine temporel (TDIBC) basée sur la Représentation Oscillo-Diffusive (ODR) et son implémentation dans un solveur CFD industriel. L'ODR s'est déjà avéré être un outil mathématique performant pour traduire l'opérateur d'impédance (ou de reflection) dans le domaine temporel. Un développement numérique dans un formalisme Navier-Stokes Characteristic Boundary Condition (NSCBC) a permis l'implémentation de ce modèle temporel dans le solveur Navier-Stokes à volumes finis elsA. Les validations de cette méthodologie sont réalisées par rapport à des données acoustiques issus de la littérature et de mesures expérimentales industrielles. Ils ont tous démontré que la nouvelle TDIBC est correctement implémentée dans le solveur CFD et prouvé son efficacité en termes de temps de calcul et stabilité numérique. Enfin, une application à la propagation et à l'atténuation des ondes de choc créées par les rotors dans une entrée d'air traiteé est proposée. 3 Spectral Difference algorithm for a 2nd order polynomial in 1D -source from [START_REF] Fiévet | Numerical Study of Hypersonic Boundary-Layer Transition Delay through Second-Mode Absorption[END_REF].

ix

Notation

This list is intended to resume those variables that are recalled multiple times throughout the different chapters of this work. Many more were used temporarily or when citing the literature, and they will be explicitly defined in the text to not burden this list of fundamental properties.

Greek letters

β
Z impedance kg • m -2 • s -1 Z 0 characteristic impedance kg • m -2 • s -1 z
Cartesian or axial coordinate (m) / generic pole of the ODR s

Introduction Motivation -Aircraft noise

Aircraft noise is considered as a high-profile issue and a significant cause of adverse community reaction when dealing with aircraft operations and airports expansion. It is a concern for airplane and engine manufacturers, airline operators but also general public and health services.

Despite the tremendous progress done on aircraft noise, the continuous air traffic growth will require further improvements on aircraft noise performances to maintain a constant or lower level of annoyance around airports. International entities like ICAO (International Civil Aviation Organisation) and CAEP (Committee on Aviation Environmental Protection) are in continuous movement for setting higher goals to be achieved by the manufacturers. One of these is the regulation "Annex 16", entered in action in 1971, which is constantly updated by the CAEP with more stringent new Chapters, as shown in Figure 1, that push the aeronautical industry to think at solutions to develop in the near future. In 1969, first International Standards and Recommended Practices for Aircraft Noise were introduced. These define three reference Certification Points for noise measurements: Lateral and Flyover for take-off phase and Approach for landing phase. Figure 2 shows these points location, still used nowadays for aircraft certification. In order to improve aerial transport public opinion as well as their market attractiveness, aircraft manufacturers always took under serious attention the noise reduction issue, as shown by the 50 years acoustic development carried out by Airbus in Figure 3. There are usually two solutions which have been adopted throughout xxi Figure 2: Reference measurement points for noise certification -Adapted from [START_REF] Filippone | Aircraft noise prediction[END_REF].

aircraft design history: first and obvious, the technology effect, which comprehends all those new technologies created for reducing the perceived noise; second, the capacity effect. Intuitively, an aircraft with higher capacity allows to carry the same number of passengers with a fewer number of flights, reducing the statistical data about noise pollution over number of passengers moved. This is the reason why the noise limitations given by CAEP Chapters in Figure 1 are function of the Maximum Take-Off Weight (MTOM). Regardless the higher thrust power and larger surface exposed to aerodynamic noise of a large aircraft, its higher capacity per flight reduces the noise annoyance linked to the number of flights. In order to take into account the operational constraints, flight performance also plays a crucial role, demanding an optimization of take-off and climb procedures and reducing approach speed. Noise Abatement Departure Procedure (NADP) is an automated procedure for further noise reduction in initial climb above xxiii populated areas, where the flight path and thrust are optimized to minimize the noise over the urban agglomeration. When all this is combined to the cutting-edge innovations on engines, airframe and nacelles manufacturing, a highly more efficient and quieter aircraft is obtained.

Aircraft noise sources can be mostly associated to landing gear, high-lift systems and engine. The first two, often distinguished as "airframe noise", will not be subject of this dissertation. Concerning the engine, in the recent turbofan architectures the fan is the major noise contributor. Future needs for fuel and performance optimization brought to the technical choice of developing Ultra-High Bypass Ratio (UHBR) engines. These architectures can provide higher performances on fuel consumption and thrust. The fan diameter will increase while the exhaust velocities will decrease, hence reducing jet noise but bringing the fan-induced noise to be up to 70% of the overall engine noise in take-off conditions. In addition, its shorter intake surface will limit the implementation of classical noise reduction solutions, as later discussed in Chapter 1, and a specific development is required to achieve the noise levels expected. One of the most efficient technical solution in turbofan acoustic problems is to cover specific engine nacelle parts with sound absorbing devices called liners. Since their introduction in industrial application, acoustic liners have been object of intensive studying and improvement. Following the principle of a quarter-wavelength resonator, liners dump the noise by resonance effects at determined frequencies. In most applications, the emission spectrum is concentrated near a few prominent frequencies, thus liners can be designed to resonate at these specific frequencies and obtain an optimal noise reduction. Unfortunately, manufacturing and testing of acoustic liners is usually a time consuming and expensive practice. In order to overcome this drawback and as a complementary study for maximizing the noise reduction benefit of the treatment, an accurate numerical approach is fundamental for its design.

State of the art

Sound absorption is more naturally described using a frequency formalism, as the acoustic impedance definition is expressed in the frequency-domain. However, translating this mechanism in a time-domain formalism is more suited for an implementation in classical unsteady Computational Fluid Dynamics (CFD), time-domain Computational AeroAcoustics (CAA) and coupled CFD/CAA tools. On the other hand, time-domain formulations of acoustic liners are as much appealing as complex to handle. Direct Numerical Simulation (DNS) for real liner geometries mounted in a turbofan model is far from being applicable on an industrial scale, due to the high grid discretization accuracy needed to correctly resolve the unsteady field in the geometric details. This was investigated in [START_REF] Casalino | Turbofan Broadband Noise Prediction Using the Lattice Boltzmann Method[END_REF] with use of Lattice Boltzmann method on a limited portion of absorbing material. Time-Domain Impedance Boundary Conditions (TDIBC) have been developed in the last two decades to numerically model the sound absorption mechanism of a lined wall at a lower numerical cost and to be coupled to classical time-domain solvers. First attempts emerged with the work of Tam and Auriault [START_REF] Tam | Time-domain impedance boundary conditions for computational aeroacoustics[END_REF], whose three-parameters model have been widely used for its implementation simplicity, however backed by a limitation in the range of applicability. Different models have followed, which can be divided into two categories: based on physical parameters or on numerical multipole schemes. The first ones are commonly rational multi-parameter, single polynomial or fractional models characterized by coefficients linked to the liner's physics (resistance, reactance, ...), as in [START_REF] Özyörük | A time-domain implementation of surface acoustic impedance condition with and without flow[END_REF][START_REF] Özyörük | Time-Domain Numerical Simulation of a Flow-Impedance Tube[END_REF][START_REF] Rienstra | Impedance Models in Time Domain, Including the Extended Helmholtz Resonator Model[END_REF][START_REF] Tam | Time-domain impedance boundary conditions for computational aeroacoustics[END_REF][START_REF] Van Den Nieuwenhof | Treatment of frequency-dependent admittance boundary conditions in transient acoustic finite/infinite-element models[END_REF]. The second ones are pure numerical models, sum of elementary dynamical systems of first and second order obtained through mathematical approximations, hence losing physical meaning but easy to translate in time-domain. Some examples are in [START_REF] Dragna | A generalized recursive convolution method for time-domain propagation in porous media[END_REF][START_REF] Fung | Time-domain Impedance Boundary Conditions for Computational Acoustics and Aeroacoustics[END_REF][START_REF] Fung | Broadband Time-Domain Impedance Models[END_REF][START_REF] Reymen | Efficient Implementation of Tam and Auriault's Time-Domain Impedance Boundary Condition[END_REF][START_REF] Reymen | Time-domain acoustic simulation of 3D lined ducts with flow using an unstructured Discontinuous Galerkin method[END_REF]. Their main problem is found to be the need of solving numerically expensive convolution products and storing an accumulator. Models combining physics and multipole emerged in the last decade [START_REF] Li | Improved Multipole Broadband Time-Domain Impedance Boundary Condition[END_REF][START_REF] Troian | Broadband liner impedance eduction for multimodal acoustic propagation in the presence of a mean flow[END_REF]. In particular, the recent work of Monteghetti [START_REF] Monteghetti | Analysis and Discretization of Time-Domain Impedance Boundary Conditions in Aeroacoustics[END_REF][START_REF] Monteghetti | Energy analysis and discretization of nonlinear impedance boundary conditions for the time-domain linearized Euler equations[END_REF][START_REF] Monteghetti | Design of broadband timedomain impedance boundary conditions using the oscillatory-diffusive representation of acoustical models[END_REF] provides a reliable modelling based on a mathematical technique referred to as the Oscillatory-Diffuse Representation (ODR), which adopts auxiliary differential equations to model the lined wall effect and relies directly on the liner's geometry.

Another point of debate in the literature when speaking of acoustic impedance is which formulation to adopt: impedance Z [START_REF] Dragna | A generalized recursive convolution method for time-domain propagation in porous media[END_REF][START_REF] Gabard | A full discrete dispersion analysis of time-domain simulations of acoustic liners with flow[END_REF][START_REF] Rienstra | Impedance Models in Time Domain, Including the Extended Helmholtz Resonator Model[END_REF][START_REF] Tam | Time-domain impedance boundary conditions for computational aeroacoustics[END_REF], admittance Y [START_REF] Liu | Stability analysis and design of time-domain acoustic impedance boundary conditions for lined duct with mean flow[END_REF][START_REF] Van Den Nieuwenhof | Treatment of frequency-dependent admittance boundary conditions in transient acoustic finite/infinite-element models[END_REF][START_REF] Zhong | A Controllable Canonical Form Implementation of Time Domain Impedance Boundary Conditions for Broadband Aeroacoustic Computation[END_REF], or reflection coefficient β [START_REF] Douasbin | Delayed-time domain impedance boundary conditions (D-TDIBC)[END_REF][START_REF] Fung | Time-domain Impedance Boundary Conditions for Computational Acoustics and Aeroacoustics[END_REF][START_REF] Jaensch | On the robust, flexible and consistent implementation of time domain impedance boundary conditions for compressible flow simulations[END_REF][START_REF] Monteghetti | Energy analysis and discretization of nonlinear impedance boundary conditions for the time-domain linearized Euler equations[END_REF][START_REF] Monteghetti | Design of broadband timedomain impedance boundary conditions using the oscillatory-diffusive representation of acoustical models[END_REF]. Gabard et al. [START_REF] Gabard | A full discrete dispersion analysis of time-domain simulations of acoustic liners with flow[END_REF] suggest the use of a β-formalism to avoid mode instabilities. In fact, β is a continuous function, without asymptotic-like singularities in Bode's diagrams, and bounded in amplitude in the range [0,1]. It has been demonstrated in [START_REF] Delorme | Computational aeroacoustics applications based on a discontinuous Galerkin method[END_REF] and further confirmed by Monteghetti [START_REF] Monteghetti | Energy analysis and discretization of nonlinear impedance boundary conditions for the time-domain linearized Euler equations[END_REF] how using the β-formulation in an acoustic simulation guarantees a CFL stability condition independent on the value of the reflection coefficient, which is not true for a Z-formulation, thus making it a reasonable choice for large scale numerical applications.

TDIBC have been initially developed in the framework of CAA, through the simulation of Euler Equations with a Linearized (LEE) or Non-Linearized (NLEE) form, as first studied at ONERA in [START_REF] Delattre | Time-Domain Simulation of Sound Absorption on Curved Wall[END_REF][START_REF] Escouflaire | Theoretical and Numerical Investigation of Time-Domain Impedance Models for Computational Aeroacoustics[END_REF][START_REF] Escouflaire | Further Insights on Time-Domain Impedance Boundary Condition[END_REF]. However, implementing them in a CFD solver is a challenging task. The interest in having a coupled CFD numerical tool with a TDIBC is that it allows to take into account complex viscous and nonlinear flow features while modeling the sound absorption of the liner. Furthermore, a single time-based numerical simulation including both the noise generation and its propagation would be sufficient, without need of separately analyzing the liner's behaviour with CAA tools, hence reducing and simplifying the workload. First promising results have been achieved in [START_REF] Scalo | Compressible turbulent channel flow with impedance boundary conditions[END_REF] for impedance laws with a simple dynamics, such as the damped Helmholtz oscillator, on a multipole model applied to a delayed reflection [START_REF] Douasbin | Delayed-time domain impedance boundary conditions (D-TDIBC)[END_REF]. A Finite-Volume (FV) approach has been tested in [START_REF] Burak | Validation of a Time-and Frequency-Domain Grazing Flow Acoustic Liner Model[END_REF], where a CFD/CAA Linearized Navier-Stokes code and a Large Eddy Simulations (LES) code had implemented the impedance condition of Tam [START_REF] Tam | Time-domain impedance boundary conditions for computational aeroacoustics[END_REF]. However, these demand either a cumbersome formulation relying on two solvers or heavy calculation times due to the fine grid requirements of a LES. A successful implementation of the fully broadband ODR model has been recently achieved by Fiévet et al. [START_REF] Fiévet | Numerical Analysis of Porous Coatings Stabilizing Capabilities on Hypersonic Boundary-Layer Transition[END_REF][START_REF] Fiévet | Numerical Study of Hypersonic Boundary-Layer Transition Delay through Second-Mode Absorption[END_REF][START_REF] Fiévet | Strong compact formalism for characteristic boundary conditions with discontinuous spectral methods[END_REF] with a Spectral-Difference CFD code. Its high-order spatial scheme proved to be ideal in the discretization of such time-domain model. Sebastian et al. [START_REF] Sebastian | Numerical simulation of acoustic propagation in a turbulent channel flow with an acoustic liner[END_REF][START_REF] Sebastian | Numerical simulation of a turbulent channel flow with an acoustic liner[END_REF] implemented a mass-spring-damper model in a Navier-Stokes solver for simulating a turbulent channel flow with an acoustic liner. In the recent work of Shur et al. [START_REF] Shur | Further Evaluation of Prediction Capability of the Broadband Time-Domain Impedance Model for Sound Propagation in Turbulent Grazing Flow[END_REF][START_REF] Shur | Unsteady Simulation of Sound Propagation in Turbulent Flow Inside a Lined Duct Using a Broadband Time-Domain Impedance Model[END_REF], a purely numerical multipole TDIBC model is derived as extension of Dragna's [START_REF] Dragna | A generalized recursive convolution method for time-domain propagation in porous media[END_REF], avoiding the expensive convolution process with use of auxiliary functions. In there, turbulence fluctuations and sound level dependency on a discrete dynamic impedance law are included. Further studies on TDIBC are nowadays carried out by Naïr et al. [START_REF] Naïr | Industrial-scale time domain modelling of acoustic surface treatments for aero-engines using discontinuous Galerkin method[END_REF] for industrial purposes, with implementation of a purely numerical time-domain impedance model on the time-based CAA Discontinuous-Galerkin solver Actran-DGM.

When considering the acoustic environment of an aeronautical engine nacelle, the problem of time-domain conversion is not the only one to answer. The flow field complexity increases sensitively and nonlinear acoustics theory must be considered. At take-off conditions, a spinning shock wave pattern is generated by the fan and propagates upstream, involving linear and nonlinear pressure level attenuation. This is the so-called Buzz Saw Noise (BSN). Nowadays, linear acoustics propagation and attenuation can be computed with high fidelity computational methods able to account for all 3D geometry and flow effects present in the engine inlet. However, xxv the state of the art nonlinear propagation codes used in the aircraft industry rely on analytical approaches which are limited by several significant approximations. For example, Fernando [START_REF] Fernando | Nonlinear waves and shocks in a rigid acoustical guide[END_REF] found an analytical solution for stationary (no flow) 2D/3D configurations, and later Kassem [START_REF] Kassem | A three-dimensional cylindrical model for non-linear propagation prediction in lined intake ducts with uniform flow[END_REF] added a uniform flow with weak coupling between the attenuation of the acoustic treatment and the propagation of the acoustic waves in the inlet, all within a 2D axis-symmetrical turbofan geometry. Yet, the nonlinear shock wave propagation is highly sensitive to the flow conditions and more complex computational approaches are necessary to remove those approximations and provide an accurate prediction of fan tone noise for realistic geometries and flow conditions. To the author's knowledge, there is currently no industrial numerical code capable of performing 3D nonlinear shock wave propagation in acoustically treated ducts with non-uniform flow. However, for nonlinear propagation of shock waves in a hard wall duct, Thisse [START_REF] Thisse | Prévision du bruit d'onde de choc d'un turboréacteur en régime transsonique par des méthodes analytiques et numériques[END_REF][START_REF] Thisse | Numerical Simulations of Shock-Wave Propagation in Turbofan Intakes[END_REF] has developed a method based on a time-domain finite volume method solving the Euler equation and obtained a promising numerical and experimental validation. Advanced semi-analytical work on this topic has been carried out in the last decade from the initial model of McAlpine et al. [START_REF] Mcalpine | On the prediction of "buzz-saw" noise in aero-engine inlet ducts[END_REF]: in particular, in [START_REF] Mcalpine | Buzz-saw" noise: A comparison of modal measurements with an improved prediction method[END_REF] the modal content of buzz-saw noise was investigated and in [START_REF] Adetifa | Nonlinear Propagation of Supersonic Fan Tones in Turbofan Intake Ducts[END_REF] sawtooth waves propagation was studied. But the main missing brick in these methodologies is the simultaneous modelling of shock waves and acoustic treatments. Determining the sound absorption brought by the intake liner in presence of shock waves requires to account for the multi-frequency content and the nonlinear redistribution of energy among the multiple tones, unlike the classical frequency-domain CAA approaches. While a single frequency model would be sufficient for linear propagation (no interaction between different frequencies), a broadband impedance boundary condition is required in a nonlinear propagation context. TDIBC implemented in a CFD solver allows to account simultaneously for sound absorption and nonlinear waves propagation in presence of a boundary layer. An industrial application which would profit of all these advantages, is the numerical simulation of the shock waves generated by a transonic rotor and propagated through an acoustically treated 3D inlet. Broadband TDIBC have been a challenging topic of research for the last decade [START_REF] Escouflaire | Theoretical and Numerical Investigation of Time-Domain Impedance Models for Computational Aeroacoustics[END_REF]. However, no fully reliable solution including nonlinear effects is available up to now. In aeronautical applications, liners mounted in engine nacelles are subject to a relatively fast grazing flow (up to Mach 0.5-0.6) and high sound levels (150-160 dB), in addition to the rotating shock wave pattern.

It is clear that nonlinearities govern engine nacelle environment, not only in the shock wave propagation but also in the liner's response due to the high SPL. In fact, if a sufficiently extended lined duct has an entry grazing sound wave of very high SPL, the liner's response near the fan will not be the same as upstream close to the inlet lip, where the sound level has already been attenuated. Thus, a nonlinear liner model would be needed. An approximated but efficient method to consider nonlinearities in the liner's description was given by Guess [START_REF] Guess | Calculation of perforated plate liner parameters from specified acoustic resistance and reactance[END_REF], which adds a term to the resistance formulation, leaving untouched the reactance. However, such model doesn't take into consideration a dynamic variation of the boundary condition response to a varying sound pressure level grazing the treated surface, but rather gives a fixed response for a fixed incident noise. A nonlinear asymptotic frequency-domain model has recently been proposed by Rienstra and Singh [START_REF] Rienstra | Nonlinear Asymptotic Impedance Model for a Helmholtz Resonator of Finite Depth[END_REF][START_REF] Singh | Nonlinear asymptotic impedance model for a Helmholtz resonator liner[END_REF], solving the stationary solution. It is asymptotic in the sense that, for relatively small amplitudes, the model coincides with the linear regime, but takes in considerations nonlinear effects close to the fundamental resonance frequency. Shur in [START_REF] Shur | Further Evaluation of Prediction Capability of the Broadband Time-Domain Impedance Model for Sound Propagation in Turbulent Grazing Flow[END_REF][START_REF] Shur | Unsteady Simulation of Sound Propagation in Turbulent Flow Inside a Lined Duct Using a Broadband Time-Domain Impedance Model[END_REF] provides a discrete nonlinear impedance model, where the impedance response is interpolated among three impedance values at different SPL. The ODR method [START_REF] Monteghetti | Energy analysis and discretization of nonlinear impedance boundary conditions for the time-domain linearized Euler equations[END_REF] is currently validated for linear liners, where the impedance is independent of the acoustic level, but preliminary studies have investigated how to extend the model to the nonlinear response of the material, without a numerical implementation at this stage. Nonetheless, the ODR method provides an attractive approach in this context, thanks to its adaptability to complex liner geometries and broadband frequency applications. From a design perspective of aero-engine and aircraft manufacturers, its capability to simulate the attenuation of nonlinear broadband noise phenomena makes it a preferred choice when building an industrial numerical tool.

Objectives and outline

This dissertation addresses the implementation of a Time-Domain Impedance Boundary Condition based on the Oscillo-Diffusive Representation in the industrial CFD solver elsA and its validation against academic and industrial cases. A Navier-Stokes Characteristic Boundary Condition (NSCBC) type of acoustic treatment has been developed and coupled to the global implicit time marching scheme. Different acoustics boundary conditions based on NSCBC have been developed in parallel to reproduce the different test cases. From an aeronautical business point view, its ability to simulate the attenuation of broadband noise phenomena makes it a valuable choice when building an industrial numerical method and its associated tools. From a scientific point of view, the objective of this work is twofold. Firstly, the TDIBC model from Monteghetti is further validated for configurations where the acoustic liner is applied to realistic configurations and presents complex features. Secondly, the implementation of this model in the finite volume Euler and Navier-Stokes solver elsA allows to have an industrial numerical method able to simulate the propagation and attenuation of fan tonal noise. This manuscript is the result of the CIFRE Ph.D. thesis1 funded by Airbus and ANRT 2 , and in collaboration with ONERA 3 . It is divided into the three following parts.

Part I

This part exposes the context and reasons of this work, then gathers all the theoretical background for its understanding.

Chapter 1 opens with a discussion on acoustics in general aviation, exposing the typical problematics and solutions faced by aircraft manufacturers. Acoustic liners are introduced along their technical specifications. The problem of coupling CFD and CAA simulations is presented.

Chapter 2 brings to light all the fundamental theory useful for the comprehension of this work. First, equations of acoustic wave propagation in ducts are reminded, followed by the definition of acoustic impedance. Then, acoustic liners theory is resumed and physical cases are classified from the literature. Different time-domain models are considered to fully understand this technology. The Oscillo-Diffusive Representation is reviewed and rewritten using a simplified formalism more suited to an engineering context for an easier comprehension. Nods on nonlinear impedance model and nonlinear acoustics propagation are then given. The chapter is concluded with a detailed overview on the state-of-the-art of CFD applications with acoustic liners.

Part II

This part describes all the numerical schemes, discretization and choices for the implementation of the TDIBC in the CFD solver.

Chapter 3 is consecrated on the numerical discretization of the aforementioned theory and highlights the choices made for its practical implementation in the CFD code. NSCBC theory is given and adapted to this specific application, for which several types of boundary conditions were developed. Inspired by the pioneer work of Giles [START_REF] Giles | Nonreflecting boundary conditions for Euler equation calculations[END_REF] with Euler approximation, a suitable acoustic source term is developed and coupled to non-reflective NSCBC inlet and outlet boundary conditions. This helped to reach a stable numerical implementation of the impedance condition in presence of boundary layer, which turned out to be extremely complex to achieve especially at the impedance discontinuity.

Chapter 4 addresses the further and essential step for a correct industrialization of the aforementioned developments. The numerical schemes and choices for obtaining an implicit formulation are given, with a particular attention on the time-domain impedance model. The chapter is concluded with a preliminary 1D validation for testing the correctness of the model.

Part III

This part presents all the results achieved with CFD numerical simulations on sound absorption, starting from validation case and ending with an industrial demonstrator.

Chapter 5 provides important results on the discretization of different impedance models using the ODR. These models are then used in the following CFD simulations.

Chapter 6 focuses on validating the CFD implementation against 2D and 3D CFD computations. First, a successful complete validation against the famous GIT benchmark [START_REF] Jones | Benchmark Data for Evaluation of Aeroacoustic Propagation Codes with Grazing Flow[END_REF] is shown for a rectangular 2D case. Then, the industrial benchmark CANNELLE is tackled for assessing high order acoustic modes attenuation in a 3D cylindrical geometry is considered [START_REF] Lavieille | Impedance eduction of liners in no-flow condition and based on multimodal excitation[END_REF], thus including azimuthal and radial modes propagation. This was a more realistic numerical representation of the physical environment of engine nacelles and confirmed the applicability of the TDIBC model in these applications.

Chapter 7 concludes the results of this dissertation with a first demonstrator of an axisymmetric engine nacelle with simultaneous consideration of shock waves propagation and acoustic liner.

A final Chapter resumes the work presented in this manuscript, highlighting the main results achieved and the possible advancements on the same subject. This Chapter deals with technical notions of acoustic physical phenomena occurring in modern aircraft architectures. The acoustic environment is defined and the noise sources identified. Physical characteristics of acoustic liners are given, where they are operated and which requirements they must satisfy. This Chapter ends with a discussion on the use of CFD for acoustics problems.

Part I

Context and theoretical background

Acoustic environment

Noise in modern aircraft can be thought as caused by two main contributors: airframe noise, generated by the air interaction with airframe, wings, high-lift devices and landing gear, and propulsion noise, involving all engine components and in particular the fan module and the jet. Several studies have been conducted on the airframe noise and still nowadays research is driven by improvement in noise prediction of landing gears and high-lift devices. Similarly, intensive studies were carried out for better understanding propulsion noise and find technical solutions to improve it. The present work will focus on propulsion noise, whose related sources can be identified to be the fan, compressor, combustor, turbine and jet, as represented in Figure 1.1. Acoustics phenomena of the first generation jet propulsion engines were mostly limited to the jet nozzle itself. Therefore, jet noise was the main contributor of propulsion noise, while the compressor noise was affecting a small region in front of the nacelle. The introduction of turbofan engines displaced the major noise source to the fan, whose noise propagates upward and rearward the engine. It quickly became comparable to the jet noise, if not greater when high pressure ratio engines were introduced. The tendency within engine manufacturers of increasing the bypass air ratio (and so the fan size) brought to the increase of fan noise up to 50% of the total propulsion noise, which is propagated up-and downstream the fan. With the introduction of Ultra-High Bypass Ratio (UHBR) engines in the next generation aircraft, fan noise is estimated to increase even more and be responsible for more than half of the overall engine noise in take-off conditions.

Acoustic liners specifications

The main solution adopted in modern aviation industry to reduce the upstream fan noise (but also noise from by-pass and turbine exhaust ducts) is the placement of noise absorbing devices called liners over the nacelle walls, as shown in Figure 1.2. The choice of using this kind of material in turbofan engines is also driven by its capacity of respecting structural integrity requirements when subject to extreme operating conditions. Section 2.2.1 will detail the mathematical theory governing the liners physics. Here, their physical characteristics and implementation in an aeronautical environment are presented. The typical layout found in turbofan engines is the Single Degree Of Freedom liner (SDOF, as in the left of Figure 1.2). It consists of an array of honeycomb cavities backed by a hard sheet at one end, then acting as an acoustic resonator. The addition of a porous face sheet on top of the cavity arrays enhances the sound absorption by visco-thermal losses, while cavities are designed in a way to maximize absorption near resonance frequencies. A second layer of cavities can be added, separated by a porous septum, leading to the Double Degree of Freedom liner (DDOF). This behaves as two resonators in series and allows to tackle a broader frequency range of absorption. The honeycomb is made of aluminum or glass fiber and the cells' diameter is selected small enough to only permit the propagation of acoustic plane waves in the cell for the frequency of interest. The face sheet and back-skin can be either metallic or in carbon fiber. Porous layers (as the face sheet and the septum) can be for example a perforated plate or a wiremesh. In order to have a cylindrical barrel, several liners are structurally joined together, leading to a partial loss of the acoustic area in the junction denoted as "splice". Modern state-of-the-art acoustic treatments are designed and manufactured without splices (zero-splice liners).

CFD/CAA MATCHING PROBLEM WITH LINERS

Liners' efficiency is evaluated with the EPNL (Effective Perceived Noise Level) reduction compared to the case with no treatment. Different typologies of liner have been created in the last two decades, driven by the necessity of adapting them to an extreme environment subject to very wide ranges of conditions and without interfering or degrading the nominal aircraft performances. At the same time, liners located in different parts of the engine will be subject to different conditions. Some of these are:

• a broadband frequency spectrum, that can range between 50 Hz up to 10 kHz;

• very high pressure levels (up to 170 dB) and grazing flows (up to Mach 0.6);

• pure tone noise and their harmonics generated by rotating parts;

• resistance to temperatures ranging between -70 • C and 180 • C degrees, without deteriorating their performances;

• light weight to not penalize fuel burn and limited thickness to not be too intrusive in the boundary layer flow (maximum of around 50 mm thickness);

• ability to evacuate fluids to avoid their freezing inside the porous medium, which would cause an aircraft weight increase and damage the nacelle by dilatation;

• reliability in terms of cost for manufacturing and of easy replacement.

CFD/CAA matching problem with liners

The complex physics governing acoustic liners made prominent the choice of pursuing their design with numerical tools. Although current CFD codes can nowadays compute complex flow features (such as hydrodynamic instabilities in a Helmholtz Resonator), the small geometric details of liners generally make it out of scope of direct simulations [START_REF] Casalino | Turbofan Broadband Noise Prediction Using the Lattice Boltzmann Method[END_REF]. CAA codes were developed in parallel to focus on the spectral content of numerical analyses. However, the translation of aerodynamic CFD data (in a time-domain) into acoustic CAA data (in a time-or frequency-domain) is far from trivial. Aerodynamic data are governed by nonlinear, unsteady Navier-Stokes equations and usually coupled with a turbulence model. Acoustic data usually come from a frequency-domain solver, governed by the Euler equations and the assumption that acoustic fluctuations are much smaller than their steady counterpart, allowing the use of a linearized approach. The information transfer between the two domains is referred to as matching problem [START_REF]ACTRAN 2021 User's Guide -Volume 1: Installation, Operations, Theory and Utilities[END_REF], and issues can arise in both the CFD and CAA sides. Spurious aerodynamic fluctuations can be generated by the discrepancies between the CFD and CAA models, and they are inconsistent with purely acoustic fluctuations. Moreover, CFD boundary conditions for inflow and outflow could be acoustically imperfect, and introducing spurious waves into the numerical domain that should be distinguished from their acoustic counterpart.

Even though the assumption of using linearized Euler equations is valid for many aeronautical applications, certain flight conditions require nonlinear fluid dynamics to be taken into account. For example, during high-rating phases like take-off and initial climb, fan blade tips rotate at a supersonic relative speed, generating a rotating shock-wave pattern propagating upstream the fan, creating strong tones at the Blade Passing Frequency (BPF). An additional complexity is brought by manufacturing imperfections and inherent blade-to-blade stagger angle deviations, giving rise to an energy transfer from the BPF pure tones to the other shaft frequencies responsible to the Buzz-Saw Noise (BSN), also called Multiple Pure Tones (MPT) noise. Even if an ideal turbofan geometry (with identical blades and perfect periodicity) is considered, nonlinear effects on sound propagation must be applied. This includes the shock coalescence and propagation of N-waves (or "sawtooth" waves), which are a discontinuity in the flow field appearing at a high-enough sound level, such as those achieved in take-off conditions. For future engines with Ultra High Bypass Ratios (UHBR), the bigger engine size along with a shorter inlet length will increase significantly the risk of BSN emerging in the cabin during climb and cruise phases, while BPFs may be dominant during take-off and initial climb phases. In order to reach an acoustic strategy objective of zero acoustic annoyance in cabin, a control of this noise source during next aircraft developments will be mandatory and a high fidelity simulation method shall be developed to ensure a better understanding and prediction of MPT noise and to design robust solutions for its reduction. Besides, in the frame of highly coupled airframe/engine architecture (UHBR engine, Boundary Layer Ingestion, ...), numerical simulations carried out by aircraft manufacturers shall be able to interface with Computational Fluid Dynamic (CFD) results of the rotor noise sources performed by the engine manufacturers.

Summary

1. Acoustic liners proved to be a fundamental component for reducing aircraft engine noise. Due to their physical complexity and elevated production costs, a numerical approach is required in support of their design.

2. Different coupled CFD/CAA techniques have been developed. However, the matching problem, or in other words the interface between the CFD and CAA calculations, is complex. Noise absorption given by acoustic liners typically belongs to the frequencydomain, but important time-domain flow features (such as shock waves generation and propagation) are fundamental in the estimation of the engine noise signature. Clearly, none of them can be neglected and a solution to improve CFD/CAA techniques is looked for.

Chapter 2

Duct acoustics and impedance models Some generalities on duct acoustics theory are given as introduction to the work presented herein. The exhaustive book of Rienstra & Hirschberg [START_REF] Rienstra | An introduction to acoustics[END_REF] has been used as baseline to understand and derive the concepts proposed, but many other works have been consulted and provided helpful insights in the present development ([4, 37, 44, 57] to cite some). Then, the concept and mathematical expression of the acoustic impedance is given, followed by its possible translations in time-domain found in the literature. The Oscillo-Diffusive Representation is chosen as model for this work and is summed up from its original work [START_REF] Monteghetti | Analysis and Discretization of Time-Domain Impedance Boundary Conditions in Aeroacoustics[END_REF] and a short introduction on how to extend it to the nonlinear domain is given. Nonlinear acoustic propagation is briefly introduced. A discussion on the current state-of-the-art TDIBC for CFD computations concludes the chapter.

Throughout the manuscript, the notation s = iω will be used for the complex-Laplacian domain, in agreement with the baseline work of Monteghetti [START_REF] Monteghetti | Analysis and Discretization of Time-Domain Impedance Boundary Conditions in Aeroacoustics[END_REF].

Duct acoustics theory

Acoustic equations can be considered as a derivation of fluid dynamics conservation equations, with the addition of specific hypotheses. This section will focus on the widely used linearized wave equation, the specific case of plane waves, waves convection in ducts and acoustic duct modes.

Linearized wave equation

In classical fluid dynamics, gas and liquids are considered as a continuum, so that an integral form of conservation equations can be specified. Applying these laws to an infinitesimal volume yields the equations in differential form: time derivative and divergence of the fluid properties are then given. For example, the conservation of mass in differential form is:

∂ρ ∂t + ∇ • (ρ - → v ) = 0 (2.1.1)
with ρ the fluid density and -→ v the three-components velocity, made of [u, v, w]. Similarly, the momentum conservation is:

ρ ∂ - → v ∂t + ∇(ρ - → v ⊗ - → v ) = -∇p + ∇ • τ i,j + ρg (2.1.2)
where p is the pressure and g the external (gravitational) force. The stress tensor τ i,j is given by:

τ i,j = µ∇ - → v + 2 3 µ(∇ • - → v ) (2.1.3)
with µ the dynamic viscosity. The energy equation is:

ρT ∂s ∂t + - → v • ∇s = ∇ • (K∇T ) + ∇ • (-p - → v + τ ij • - → v ) (2.1.4)
with s the entropy, K the heat conductivity and T the temperature. In acoustic phenomena, the density and velocity fluctuations are small enough to allow a linear approximation of the fluid motion equations. If we consider a quiescent fluid (u 0 = 0) and its acoustic perturbations (p ac , ρ ac , ...), conservation equations read:

                 ∂ρ ac ∂t + ρ 0 ∇ • -→ v ac = 0 ρ 0 ∂ -→ v ac ∂t + ∇p ac = 0 ∂s ac ∂t = 0 (2.1.5a) (2.1.5b) (2.1.5c)
By subtracting the time derivative of the mass conservation law (2.1.5a) from the divergence of the momentum conservation law (2.1.5b), we eliminate -→ v ac to obtain:

∂ 2 ρ ac ∂t 2 -∇ 2 p ac = 0 (2.1.6)
Using the perfect gas relation p ac = c 2 0 ρ ac , with c 0 the speed of sound at the temperature T 0 , either p ac or ρ ac can be eliminated, yielding to the wave equation:

∂ 2 p ac ∂t 2 -c 2 0 ∇ 2 p ac = 0 or ∂ 2 ρ ac ∂t 2 -c 2 0 ∇ 2 ρ ac = 0 (2.1.7)
Remembering that the acoustic field is irrotational, we can introduce the linearized Bernoulli equation of a potential disturbance function φ ac (whose gradient is the velocity, -→ v ac = ∇φ ac ) described by:

∂φ ac ∂t + p ac ρ 0 = 0 (2.1.8)
If we apply it to the previously derived equation (2.1.7), we obtain the generalized wave equation for a fluid at rest, related to the potential disturbance function:

∂ 2 φ ac ∂t 2 -c 2 0 ∇ 2 φ ac = 0 (2.1.9)
One of the simplest and used solution to the wave equation is the 1D d'Alembert solution:

p ac = f (x -c 0 t) + g(x + c 0 t) (2.1.10a) u ac = 1 ρ 0 c 0 (f (x -c 0 t) -g(x + c 0 t)) (2.1.10b)
where f and g are here the right and left running waves, respectively, in the mono-dimensional direction, determined by boundary and initial conditions. A generic time harmonic plane wave of amplitude A, frequency ω and wave number k = ω/c 0 can be written in complex form as:

p ac = Ae iωt-ikx (2.1.11)
These equations are particularly useful for describing sound waves in hard-walled ducts. For simplicity in the mathematical derivation, let us now consider the case of a plane wave in a duct. Plane waves can also be considered as an approximation of spherical waves at a sufficient large distance from their source or as waves in hard-walled ducts at frequencies below a critical value called cut-off frequency. The latter depends on the duct cross section shape and is proportional to the duct diameter d as c 0 /d. We consider a field made of a uniform state (ρ 0 , p 0 , u 0 ) plus a perturbation (ρ ac , p ac , u ac ), allowing a linearization of the type:

       ρ = ρ 0 + ρ ac p = p 0 + p ac u = u 0 + u ac (2.1.12)
where the flow is considered one-dimensional (u 0 = u 0 (x)) and homentropic (satisfying p ac = c 0 ρ ac ). Mass and momentum conversation system for one-dimensional frictionless flow is:

         ∂ρ ∂t + u ∂ρ ∂x + ρ ∂u ∂x = 0 ρ ∂u ∂t + u ∂u ∂x + ∂p ∂x = 0 (2.1.13a) (2.1.13b)
where it is supposed no mass injection nor external forces are present. Introducing the decomposition (2.1.12) we obtain the following linear system:

∂ρ ac ∂t + u 0 ∂ρ ac ∂x + ρ 0 ∂u ac ∂x = 0 (2.1.14a) ρ 0 ∂u ac ∂t + ρ 0 u 0 ∂u ac ∂x + ∂p ac ∂x = 0 (2.1.14b)
If we now subtract the convected time derivative (∂t + u 0 ∂x) of the mass conservation (2.1.5b) to the divergence of momentum conservation equation (2.1.5c) we obtain the convected wave equation in p ac :

∂ ∂t + u 0 ∂ ∂x 2 p ac -c 2 0 ∂ 2 p ac ∂x 2 = 0 (2.1.15)

Acoustic modes

When considering a duct of constant cross section, the elementary wave equation (2.1.15) can be solved by an infinite series expansion of solutions called modes. The interest in using modes is double: mathematical because they form a complete basis which can represent any solution, and physically because they are solutions themselves, reducing complex acoustic fields into an ensemble of simpler structures. The thorough work of Lympany et al. [START_REF] Lympany | Methodology for Measuring Higher-Order Acoustic Modes in Uniform Heated Flows[END_REF] in describing acoustics modes has been used as a baseline for deriving the equations herein presented. The notation (m, n) is introduced as an ordered pair of integers defining respectively the angular and radial mode order, with m ∈ [-∞, +∞] and n ∈ [1, +∞]. It can also be found in the literature (as in [START_REF] Lympany | Methodology for Measuring Higher-Order Acoustic Modes in Uniform Heated Flows[END_REF]) that the first value for n begins from zero. This is purely a convention that will only impact how the related equations are written. In the present work, the notation n ∈ [1, +∞] was preferred for homogeneity with other acoustic numerical tools available in Airbus. The sound wave pressure in a cylindrical duct can be expressed in the frequency-domain (ω) and in cylindrical coordinate as [START_REF] Morfey | Sound transmission and generation in ducts with flow[END_REF]:

p(r, θ, z, ω) = ∞ m=-∞ ∞ n=1 A + mn (ω)e -ik + z,mn z + A - mn (ω)e -ik - z,mn z Ψ mn (r, θ) (2.1.16)
where (r, θ, z) are the cylindrical coordinates for radius, azimuthal and axial directions, respectively. A ± mn indicates the complex pressure amplitude as in (2.1.11), k ± z,mn is the axial wavenumber and Ψ mn is the mode shape function, i.e. the acoustic pressure variation in the cross section for each mode (m, n). Such functions are here taken as invariant on the axial direction from the hypothesis of uniform mean flow along the duct axis. The mode shape function for a duct with circular cross section is given by [START_REF] Morse | Theoretical Acoustics. International series in pure and applied physics[END_REF]:

Ψ mn (r, θ) = C mn J m (k r,mn r)e -imθ
(2.1.17)

where C mn is a normalization factor, J m the Bessel function of the first kind of order m and k r,mn the eigenvalue of mode (m, n), or also called "transverse" wavenumber, given by the roots of:

J m (k r,mn r D ) = 0 (2.1.18)
where J m is the first derivative of the Bessel function of the first kind of order m and r D the duct radius [START_REF] Tyler | Axial Flow Compressor Noise Studies[END_REF]. Both Bessel functions and their derivatives are detailed in the Appendix C The normalization factor C mn is defined in a way that the modes are orthonormal:

1 S S Ψ mn Ψ * m n dS = 1, (m, n) = (m n ) 0, (m, n) = (m n ) (2.1.19)
with S the duct section area and * the complex conjugate. An explicit expression can be derived (cf. [START_REF] Lympany | Methodology for Measuring Higher-Order Acoustic Modes in Uniform Heated Flows[END_REF], Appendix, eq. (A7)):

C mn =     1 - m k r,mn r D 2   J 2 m (k r,mn r D )   -1 /2 (2.1.20)
Lympany [START_REF] Lympany | Methodology for Measuring Higher-Order Acoustic Modes in Uniform Heated Flows[END_REF] provided a very comprehensible representation of these equations, as sketched in Figure 2.1. The shaded colors represent the pressure contour, with red and blue opposite phases, and black lines pressure nodes, i.e. where the acoustic pressure is null. The pressure variations along the azimuthal coordinate m are governed in (2.1.17) by the term e -imθ , which gives a counterclockwise (or clockwise) mode spin when m is positive (or negative). The azimuthal order m can also be seen as the number of lobes (or node lines) passing through the duct center.

The radial mode order n can be interpreted as the number of maxima and minima of the pressure profile in the duct cross section. The pressure variations along the radial coordinate is governed in (2.1.17) by the Bessel function J m (k r,mn r). Now that eigenfunction Ψ mn has been made explicit, the exponential term e ik ± z,mn z in (2.1.16) is discussed. It represents the acoustic pressure variation along the duct axial coordinate and it is governed by the axial wavenumber k ± z,mn , which describes the spatial frequency of the acoustic wave of mode (m, n) propagating in the positive and negative axial directions. Different models to express such wavenumber are found in the literature. Lympany reminds three of them in [START_REF] Lympany | Methodology for Measuring Higher-Order Acoustic Modes in Uniform Heated Flows[END_REF]. Morfey gave a definition of the non-dissipative axial wavenumber in the up-and downstream directions as [START_REF] Morfey | Sound transmission and generation in ducts with flow[END_REF]

k ± z,mn = k α mn ∓ M 1 -M 2 (2.1.21)
with M the Mach number in the duct (positive when in accordance to the axial coordinate), k the total wavenumber,

k = ω c 0 (2.1.22)
and α mn an eigenvalue defined as:

α mn = 1 -(k r,mn c 0 /ω) 2 (1 -M 2 ) (2.1.23)
A well-known dispersion relation is obtained, giving rise to specific cut-on/cut-off conditions with respect to the angular frequency:

ω ≥ k r,mn c 0 1 -M 2 (2.1.24)
If the inequality (2.1.24) is not respected, then the axial wavenumber is imaginary and the pressure amplitude decays exponentially along the duct axial direction. The mode is then cut-off (also called evanescent) and it does not propagate acoustic energy along the duct. The frequency corresponding to the equality of left and ride side of (2.1.24) represents the minimal frequency for the mode (m, n) to be able to propagate energy in the duct, also called cut-off frequency of the mode. It is remembered this is the theoretical formulation for a pressure signal made of different overlapped modal structures. In the case a single mode shape (given (m, n)), as for the numerical simulation in this work, the reader should refer to the equations in Section 3.4.2.

Coming back to the acoustic pressure decomposition of (2.1.16), if the pressure signal is given (from microphones in experiments or output of a numerical simulation), N modal amplitudes A ± mn can be determined, N being the number of cut-on modes at a given frequency. The system of equations to be resolved is in matrix form:

   p 1 . . . p j    =     W + 1,00 W - 1,00 . . . W + 1,mn W - 1,mn . . . . . . . . . . . . W + j,00 W - j,00 . . . W + j,mn W - j,mn             A + 00 A - 00 . . . A + mn A - mn         (2.1.25)
with j the spatial measurement location (physical or numerical probe) and W j,mn the weight of each mode shape at each location, given by:

W ± j,mn = Ψ mn (r i , θ i )e ∓ik ± z,mn z i (2.1.26)
Additionally, a scattering matrix can be defined to assess the reflection and transmission coefficients through a "test section" (for example, an acoustically absorbing lined wall), or simply to assess the acoustic energy losses through a portion of hard walled duct. Such test section is represented in gray in Figure 2.2. It will not be object of analyses itself, but modal decomposition on the two portions of duct upstream and downstream the test section will determine how its introduction affects the acoustic pressure waves. Acoustic modal decomposition has been thoroughly studied by Ábom [START_REF] Åbom | Measurement of the scattering-matrix of acoustical two-ports[END_REF], who gave a first mathematical form of the scattering matrix as:

A - A A + B = R A-→A T A-→B T B-→A R B-→B A + A A - B (2.1.27)
where each term A ± X is a N × 1 amplitude vector of the mode propagating in the positive or negative axial direction on the X side of the duct. Each term R X-→X is a N × N matrix of pressure reflection coefficients between modes of opposite direction on the same X side. T X-→Y is a N × N matrix of pressure transmission coefficients between modes of same direction on side X and side Y . The left-hand (right-hand) side of (2.1.27) represents the outgoing (ingoing) acoustic waves, propagating away from (towards to) the test section. The central (scattering) Figure 2.2: Acoustic reflection and transmission modes in a duct. matrix determines the reflection and transmission of acoustic waves, hence how the incoming waves are transformed into outgoing waves. If we consider the only incoming waves to be the ones on side A (i.e. A + A ), then (2.1.27) reduces to:

A - A = R A-→A A + A A + B = T A-→B A + A (2.1.28)
where the first line refers to the pressure reflection on the same side A and the second line to the pressure transmission from side A to side B. Both lines can be expanded in terms of their individual modal amplitudes and reflection coefficients as

    A - A,00 . . . A - A,m n     =    R A,00-→A,00 . . . R A,mn-→A,00 . . . . . . R A,00-→A,m n . . . R A,mn-→A,m n        A + A,00
. . .

A + A,mn     (2.1.29)     A + B,00 . . . A + B,m n     =   
T A,00-→B,00 . . . T A,mn-→B,00 . . . . . .

T A,00-→B,m n . . . T A,mn-→B,m n        A + A,00
. . .

A + A,mn     (2.1.30)
In these matrices, elements on the main diagonal of the pressure reflection and transmission matrices, for which (m, n) = (m , n ), represent respectively the pure reflection of the incident mode into the same reflected mode and the pure transmission of the single mode. On the other side, off-diagonal elements represent scattering from one incident mode into a different reflected or transmitted mode, respectively. In the case that the only incoming mode is A mn , the system (2.1.28) further simplifies into:

             R A,mn-→A,m n = A - A,m n A + A,mn T A,mn-→B,m n = A + B,m n A + A,mn (2.1.31)
Different methods have been proposed to solve the modal amplitudes equations system [START_REF] Åbom | Measurement of the scattering-matrix of acoustical two-ports[END_REF][START_REF] Munjal | Theory of a two source-location method for direct experimental evaluation of the four-pole parameters of an aeroacoustic element[END_REF][START_REF] Sitel | Multiload procedure to measure the acoustic scattering matrix of a duct discontinuity for higher order mode propagation conditions[END_REF], but they are not here discussed in detail for the sake of brevity. In the present work, modes amplitude decomposition of unsteady CFD pressure results is achieved with the tool Actran iTM from Free Field Technologies (FFT). Section 6.3 will give a practical explanation and application of such modal decomposition applied on numerical CFD results.

Acoustic impedance

A widely used quantity in acoustics is the impedance. Rienstra & Hirschberg [START_REF] Rienstra | An introduction to acoustics[END_REF] give a wide overview of the impedance concept: it represents the amount by which an acoustic wave propagation is "impeded" by a surface, or in other words, its sound level reduced. Since friction forces are the main contributor of impediment and are proportional to the velocity normal to the surface (also called "acoustic velocity"), it has been chosen to express the impedance as the ratio between acoustic pressure p and wall-normal acoustic velocity u n in the frequency-domain:

Z( - → x , ω) = p( - → x , ω) u n ( - → x , ω) (2.2.1)
with -→ x the three-dimensional location vector. The measure unit is kg.m 2 .s -1 . The acoustic velocity will be considered in the following as the average normal component (pointing inward the surface) of the three dimensional velocity vector and simply called normal velocity u n .

In the most general case, Z is a number without great physical relevance, as it is not only a material characteristic, but it also depends on the acoustic field. However, in the specific case of the so-called "locally reacting" materials, impedance is pointwise and linearly proportional to the material's properties, so providing the same behaviour independently of the acoustic environment. For these materials, the acoustic wave is assumed to impact locally in one point, not generating any lateral reaction inside the material and then reflecting the wave normal to the surface (as in the right picture of Figure 2.3). In the case of non-locally reacting materials (left picture of Figure 2.3), the absorption mechanism is achieved by a an energy dissipation due to friction with the rigid structure of the material. In this case, the acoustic wave propagates in all directions inside the material, including transversely. In this work, only locally-reacting absorbing devices are considered. Impedance is a complex-valued property, composed by a real part (Resistance R) and imaginary part (Reactance X ):

Z = R + iX (2.2.2)
Both quantities are dependent on frequency, intensity of incident sound wave and mean flow. In a practical example of an acoustic liner, resistance depends on surface opening and perforations diameter (the perforated plate or wiremesh, discussed in Section 2.2.4). In normal incidence configuration, the higher the resistance, the wider the maximum absorption frequency band but the lower the maximum absolute level of absorption. Resistance is always a positive parameter, meaning the liner is never amplifying the incident wave. Reactance is related to the phase shift of the reflected wave and helps in defining the maximum level of absorption, which is achieved when reactance is null. The use of a complex quantity is needed in this case to account for the damping and phase shift imparted on the sound waves by the treated surface. More details on these two parameters will be given in Section 2.2.2. A reflection coefficient (noted here with β) can be defined as the ratio between the amplitudes of the reflected (A -) and incident (A + ) waves. After some rearrangement, the reflection coefficient for a one-dimensional wave at normal incidence can be written as:

β = A - A + = Z -ρ 0 c 0 Z + ρ 0 c 0 = Z d -1 Z d + 1 = 1 - 2 Z d + 1 (2.2.3)
where Z d is the dimensionless impedance, normalized for the characteristic impedance Z 0 = ρ 0 c 0 . In the following, Z d will be simply written Z and its dimensional character will be explicit from the context. It is clear from this formulation that the maximum absorption (minimum reflection) is given for Z = 1 (null reactance) and maximum reflection for Z = 0 (null resistance). It must be pointed out that this reasoning is only true for a wave at normal incidence. However, throughout this work, we will consider valid the β value defined by (2.2.3) whatever the incidence of the considered wave. It must then be considered as an "equivalent" reflection coefficient.

Basic impedance models

The most general and basic impedance models for cavities and perforations will now be described. Their understanding is fundamental as most of the modern acoustic liners are made of a combination of the two.

Cavity impedance

A cavity impedance (Z c ) can be seen as a combination of a free tube and a backed impedance. Kuttruff [START_REF] Kuttruff | Acoustics: An Introduction[END_REF] analyzes in detail this phenomenon. Taking in consideration a generic pipe section with constant cross section and length x, two plane waves will be traveling in the section in opposite directions, as in Figure 2.4. Omitting the time factor e iωt which is common to both waves, sound pressure in a one-dimensional tube can be derived from (2.1.16) as:

p(x) = Ae -ikx + Be ikx (2.2.4)
with wavenumber k and A and B arbitrary constants. The velocity, which is considered tangential to the cavity walls (u = u x ) is related to pressure by the factor ±1/Z 0 , where Z 0 is the medium characteristic impedance and the sign relates respectively to the first part (proportional to A) and second part (proportional to B).

Z 0 u(x) = Ae -ikx -Be ikx (2.2.5)
Figure 2.4: One-dimensional waves traveling in a tube -adapted from [START_REF] Kuttruff | Acoustics: An Introduction[END_REF].

Thanks to the mathematical relation e ±ikx = cos(kx) ± i sin(kx), equations (2.2.4) and (2.2.5) can be reformulated:

p(x) = (A + B) cos(kx) -i(A -B) sin(kx) (2.2.6a) Z 0 u(x) = (A -B) cos(kx) -i(A + B) sin(kx) (2.2.6b)
from which it is immediate to have p(0) = (A + B) and Z 0 u(0) = (A -B), and consequently:

p(x) = p(0) cos(kx) -iZ 0 u(0) sin(kx) (2.2.7a) Z 0 u(x) = Z 0 u(0) cos(kx) -ip(0) sin(kx) (2.2.7b)
If we now consider the cavity backed by a wall at one end (at x = l c , the cavity length) with impedance Z b , and we define the cavity impedance Z c at the other end (at x = 0) as p(0)/u(0), we have (after normalization by u(0)):

Z b = p(l c ) u(l c ) = Z 0 Z c cos(kl c ) -iZ 0 sin(kl c ) Z 0 cos(kl c ) -iZ c sin(kl c ) (2.2.8)
After a rewriting of (2.2.8), we obtain the cavity impedance:

Z c = p(0) u(0) = Z b cos(kl c ) + iZ 0 sin(kl c ) Z 0 cos(kl c ) + iZ b sin(kl c ) Z 0 (2.2.9)
Having a rigid backed cavity (Z b = ∞), the latter can be reduced to the final equation for cavity impedance:

Z c = -Z 0 i cot(kl c ) = Z 0 coth(ikl c ) (2.2.10)

Basic Perforation impedance

A widely known model for perforations impedance Z p is provided by Crandall [START_REF] Crandall | Theory of Vibrating Systems and Sound[END_REF] and it relies on three hypotheses:

• the perforation consists of an infinite axisymmetric cylinder of diameter d p ;

• the pressure gradient between tube's entry and exit is longitudinal, independent of the radial coordinate;

• the flow obeys the Stokes equation:

ρ 0 ∂ t u(r, x, t) = -∂ x p(x, t) + ρ 0 ν∆u(r, x, t) (2.2.11)
with ν the kinematic viscosity and a no-slip boundary condition at r = d p /2.

After tedious mathematical calculations (found in [START_REF] Crandall | Theory of Vibrating Systems and Sound[END_REF] -Appendix A), it can be demonstrated the perforation impedance to be:

Z p (l p , d p ) := ρ 0 l p s[1 -Φ(k ν dp /2)] -1 (2.2.12a) = +∞ 3 ρ 0 l p ν ( dp /2) 2 + 2 ρ 0 l p √ ν dp /2 √ s + ρ 0 l p s + O 1 |k ν dp /2| (2.2.12b) = 0 8 ρ 0 l p ν ( dp /2) 2 + 4 3 ρ 0 l p s + O |k ν dp /2| 4 (2.2.12c) with: Φ(s) = 2 s J 1 J 0 (s) (2.2.13)
being s = iω the complex Laplacian variable, J m the modified Bessel function of the first kind of order m and k ν = s/ν the wavenumber of viscous diffusion. The three equations (2.2.12) are respectively for: definition, high frequency limit and low frequency limit. The first hypothesis at the basis of this definition (infinite cylinder) sounds not physical and not of industrial interest. Orifice end corrections have been proposed in the literature [START_REF] Ingård | Acoustic Circulation Effects and the Nonlinear Impedance of Orifices[END_REF][START_REF] Sivian | Acoustic Impedance of Small Orifices[END_REF] to add the effect of a finite length to this simplified assumption.

Quarter-wavelength resonator

An interesting and widely discussed study case for acoustic propagation in ducts is the quarterwavelength resonator, detailed in [START_REF] Malmary | Étude théorique et expérimentale de l'impédance acoustique de matériaux en présence d'un écoulement d'air tangentiel[END_REF][START_REF] Primus | Détermination de l'impédence acoustique de matériaux absorbants en écoulemen par méthode inverse et mesures LDV[END_REF] to cite some. Let us focus our attention on Figure 2.5 for a practical explanation of this device. In here, the reactance and sound absorption (in terms of Transmission Loss, i.e. the decibel reduction brought by the acoustic device) are shown for a cavity impedance as defined by (2.2.10). The red curve represents the classic sound attenuation of a cavity (in decibel), while the blue curve shows the attenuation with an additional resistive constant term equal to 0.5. The "pure cavity" case presents a strong sound attenuation, however limited to a narrow frequency range. The addition of a constant resistance decreases the maximum absorption, but increases the frequency range of absorption of the treatment. The cavity maximum absorption is achieved for a value of kh = π/2. Remembering that the wavenumber is related to the wavelength by k = 2π/λ, we can state that the maximum absorption is achieved for h = λ/4, a quarter wavelength. The reactance is in this case null, as:

cot(kh) = cot 2π λ λ 4 = cot π 2 = 0
The respective frequency and all its multiples f = (2n+1)c 0 4h

, with (n ∈ N), correspond to the n-th quarter-wavelength mode of the cavity. Similarly, we can also state that for kh = 0 and kh = π the reactance tends to the infinite and absorption is null. This defines the anti-resonance frequencies f = (2n)c 0 4h at which the liner will behave as a hard wall. Even though these equations are also valid for real acoustic liner, it needs to be reminded that perforation/cavity interactions may displace the resonance (and anti-resonance) frequency.

Helmholtz Resonator (HR)

The precursor of modern liners can be found in the Helmholtz Resonator, here reviewed to make clear the liner's physical behaviour. When considering an acoustic wave traveling in a duct, the dissipation is the largest at and near resonance, and so narrow-band sound absorption is achieved for frequencies close to resonance [START_REF] Rienstra | Nonlinear Asymptotic Impedance Model for a Helmholtz Resonator of Finite Depth[END_REF]. In order to achieve resonance conditions in a duct, the tube length should be of the same order of magnitude of the acoustic wave length, as just shown for the quarter-wavelength cavity. This translates in cumbersome devices of scarce practical utility in industrial applications. A solution is the Helmholtz Resonator (HR), a container (usually a cavity) filled with air or a porous media with a small opening called neck. When excited with a fluctuating external pressure, the mass of the air plug inside the neck moves against the large volume of compressible air inside the cavity, while viscous forces and vortex shedding cause dissipation of energy. In its simplest form, for wavelengths much larger than the size of the cavity, this establishes a weakly nonlinear zero-dimensional mass-spring-damper system, where the mass inertia is given by the neck flow and the cavity air acts as a spring. The damping is normally relatively small in a way that a resonance frequency ω 0 can be identified by the equation:

ω 2 0 = S n c 2 0 (l + 2δ)V (2.2.14)
being l, S n and V the neck's length, section and cavity volume (as in Figure 2.6) and δ a small end correction [START_REF] Rienstra | An introduction to acoustics[END_REF]. A single resonator has a narrow sound absorption bandwidth near its resonant frequency, reason why usually a series of resonators is used instead for enlarging the bandwidth of absorption [START_REF] Yang | Optimising the acoustic damping of multiple Helmholtz resonators attached to a thin annular duct[END_REF]. Therefore, aeronautical liners consist of a honeycomb array of small cells, recreating the effect of an air filled cavity with the structural advantages of a hexagonal geometry, topped by a perforated plate which creates the necks. In order to give a physical meaning and modern use of a HR-type liner, the most common typologies are now presented.

Impedance model of physical acoustic liners

Now that the theoretical background behind acoustic liners is given, some existing examples of acoustic liners are presented along their impedance laws definition.

Ceramic Tubular liners (CT)

Thoroughly described and analyzed by Jones et al. in [START_REF] Jones | Benchmark Data for Evaluation of Aeroacoustic Propagation Codes with Grazing Flow[END_REF], they consist of an array of long and narrow cavities, as illustrated in Figure 2.7. Their impedance operator is linear for a wide range of SPL, making them a valuable validation tool for numerical and experimental acoustics studies [START_REF] Primus | Détermination de l'impédence acoustique de matériaux absorbants en écoulemen par méthode inverse et mesures LDV[END_REF]. The Ceramic Tubular liner can simply be expressed by a backed cavity impedance, as in equation (2.2.10):

Z CT = Z c σ c coth(ikl c ) (2.2.15)
with σ c , l c and k the cavity porosity, length and wavenumber, respectively. A suitable cavity wavenumber proposed by Bruneau is (cf. [START_REF] Bruneau | Fundamental of Acoustics[END_REF], §3.7):

ik(s) = s c 0 1 + (γ -1)Φ(k κ dc /2) 1 -Φ(k ν dc /2) 1 /2 (2.2.16a) = +∞ s c 0 1 + √ ν dc /2 γ -1 √ P r + 1 1 √ s + O[1] (2.2.16b) = 0 s c 0 2 √ 2 √ γν dc /2 1 √ s + O |k ν dc /2| 3 (2.2.16c)
Figure 2.7: Ceramic Tubular liner scheme -Source from [START_REF] Jones | Benchmark Data for Evaluation of Aeroacoustic Propagation Codes with Grazing Flow[END_REF].

with P r the Prandtl number.

Micro-Perforated liners (MP)

Micro-Perforated (or Multi-Perforated), Single (or Double) Degree of Freedom (SDOF or DDOF) liners are the most commonly used in jet engine nacelle. The classical SDOF layout is made of a honeycomb core structure in aluminum or glass fiber covered by a perforated (or microperforated) plate and backed by a hard wall, as shown in Figure 2.8. The DDOF layout consists of two honeycomb cores, separated by a porous septum and backed by the same perforated plate and hard wall on the two extremities. The cavities act as in a Helmholtz Resonator, setting the resonant frequency, while the perforations set the viscous dissipation and may also introduce non-linearities. A classic SDOF impedance representation is the combination of a cavity and perforation impedance, as previously described in (2.2.10) and (2.2.12b), respectively:

Z SDOF = Z p (l p , d p ) σ p + Z c σ c i coth(kl c ) (2.2.17)
where σ p , l p and d p are the perforated plate porosity, length (or thickness) and diameter, respectively. In this case, the honeycomb cavity porosity is close to unity (σ c 1) because the cavity walls are very thin.

Wiremesh liners

A modern upgrade of SDOF liners is the SDOF Wiremesh liner, which substitute or add to the perforate plate a film (or mesh) with a resistive capacity. Multi-perforated and wiremesh liners demonstrated to be the most efficient for aeronautical applications, mounted in engine inlet and bypass ducts. These devices can be set to resonate at specific frequencies helping in reducing Figure 2.8: SDOF and DDOF liner scheme -source from [START_REF] Piot | Design, manufacturing and demonstration of acoustic liners for air conditioning systems[END_REF][START_REF] Rienstra | Impedance Models in Time Domain, Including the Extended Helmholtz Resonator Model[END_REF].

the BPF tonal noise. It has been shown [START_REF] Lidoine | Numerical prediction of SDOF-Perforate Plate Acoustic Treatment Impedance. Part 1: Linear domain[END_REF] that a decrease in aircraft noise up to 5dB during take-off and 2dB in landing phases can be achieved with an optimal setup. Its impedance is described as for a SDOF liner, however with a different resistive term given by the porous sheet, which is often chosen empirically.

Z WM = Z mesh + Z cavity σ c i coth(kl c ) (2.2.18)
Figure 2.9: Wiremesh liner scheme -source from [START_REF] Piot | Design, manufacturing and demonstration of acoustic liners for air conditioning systems[END_REF].

Impedance in Time-Domain

Since impedance is a concept belonging to the frequency-domain, where a single discrete value of impedance is needed at each frequency value, its definition reads:

Z( - → x , ω) = p( - → x , ω) ûn ( - → x , ω) (2.3.1)
p and ûn being respectively the Fourier-transformed acoustic pressure and normal velocity perturbations on the surface normal -→ n , -→ x a three-dimensional point location on the surface. In a numerical acoustics simulation, the impedance value can be inserted from analytic or experimental data to obtain the damped acoustic solution. Nevertheless, a time-domain formulation of sound absorption reveals useful. In fact, it is able to handle broadband frequency problems with a lower computational cost and can be coupled with nonlinear partial differential equations. This means it can be used with most of the current CFD solvers, and hence capable of analyzing simultaneously complex flows, transient states and nonlinear phenomena. Equation (2.3.1) translated in time-domain reads (omitting the location -→ x ):

p(t) = [Z u n ](t) = 1 2π t 0 Z(t -τ )u n (τ )dτ (2.3.2)
where is a convolution product and the Laplace-transform is defined as below:

p(s) = ∞ 0 p(t)e -st dt (2.3.3a) ûn (s) = ∞ 0 u n (t)e -st dt (2.3.3b) Ẑ(s) = ∞ 0 Z(t)e -st dt (2.3.3c)
being s the Laplacian variable. Mathematical background [START_REF] Monteghetti | Analysis and Discretization of Time-Domain Impedance Boundary Conditions in Aeroacoustics[END_REF] supports the choice made in this study to assimilate the Fourier p(ω) and the Laplace p(s) by taking s = iω. In the following, the superscript notation "ˆ" for Fourier-(or Laplace-) transforms will be neglected for ease of read, and the dependency on a frequency or temporal variable will be explicit.

Admissibility

Time-domain modeling is however not of easy representation. Using an impedance in a timedomain solver requires its transformation into an impulse response, done by an inverse Laplacian (or Fourier) transform [START_REF] Sebastian | Numerical simulation of a turbulent channel flow with an acoustic liner[END_REF]. Unfortunately, an impedance law satisfying stability criteria in the frequency-domain may not satisfy certain physical properties in time-domain and generate instabilities [START_REF] Fung | Broadband Time-Domain Impedance Models[END_REF]. As suggested by Rienstra [START_REF] Rienstra | Impedance Models in Time Domain, Including the Extended Helmholtz Resonator Model[END_REF], in order to derive an admissible, physical and numerically stable impedance model in the time-domain, the impedance must be extended in the complex frequency-domain and satisfy necessary conditions of causality, reality and passivity:

1. Z(ω) is analytic and non-zero in Im(ω) < 0 (causality condition)

2. Z * (ω) = Z(-ω) for all ω ∈ R (reality condition)

3. Re(Z(ω)) ≥ 0 for all ω ∈ R (passivity condition)

Condition 1 ensures that p(0, t) does not depend on v(0, t) of the future and vice versa. In other words, a phenomenon (cause) causes another phenomenon (effect) in a temporal order. Condition 2 states that real part (resistance) and imaginary part (reactance) are respectively an even and odd functions in ω; so that if the input is a real, the output will be real. This can be translated, together with condition 3, in making sure the impedance wall is passive, absorbing energy for any frequency and never creating or amplifying energy. These conditions are necessary for impedance admissibility, but not sufficient. The recent work of Monteghetti [START_REF] Monteghetti | Energy analysis and discretization of nonlinear impedance boundary conditions for the time-domain linearized Euler equations[END_REF] demonstrated how these conditions become sufficient for admissibility property when working with "system theory", using then Laplace transform instead of Fourier. The correlation between the two is given both by the identity s = iω and the correspondence between the two complex half domains Im(ω) < 0 and Re(s) > 0 (later defined as C + 0 ). System theory shows that for an impedance function real and positive, the conditions of causality, reality and passivity are automatically verified. Such complex function is defined in C + 0 as:

• f is an analytic function in C + 0 • f (s) ∈ R for any real s > 0 • Re[f (s)] ≥ 0 for Re(s) > 0
This applies with no changes also to the admittance operator:

u n (t) = [Y p](t) (2.3.4)
When working with a reflection coefficient, we can link directly acoustic waves instead of acoustic pressure p ac and velocity u ac :

u ac - p ac ρ 0 c 0 (t) = β u ac + p ac ρ 0 c 0 (t) (2.3.5)
where the acoustic fluctuation of a variable φ ac is defined as the difference between its instantaneous value φ and its time-averaged value φ:

φ ac = φ -φ
Furthermore, a strong difference lies in the fact that β is a bounded real function, which converts the third of the conditions above in:

• 0 ≤ f (s) ≤ 1 for Re(s) > 0
Consequently, the Laplace formulation is considered throughout all this work, going to substitute in all previous equations s to iω.

Time-Domain Impedance Models

When discretizing a TDIBC, three steps can be defined. They are here proposed for a general case, and later applied in details for the chosen model:

1. First the impedance model has to be chosen, i.e. the physical relation that one wishes to apply at the boundary to link pressure and normal velocity fluctuations.

2. A numerical algorithm used to evaluate the said relation must be developed. This can translate, for example, in building a time-domain operator through convolution product or auxiliary functions as described in the following models.

3. Finally, the previous algorithm has to be coupled with the flow solver numerical scheme at treated boundaries.

Concerning the point 1, as already discussed in the Introduction, a unanimous choice has not been found in the literature, where three different ways of expressing acoustics impedance are described, depending on their formulation. These are based on impedance Z ( [START_REF] Dragna | A generalized recursive convolution method for time-domain propagation in porous media[END_REF][START_REF] Gabard | A full discrete dispersion analysis of time-domain simulations of acoustic liners with flow[END_REF][START_REF] Rienstra | Impedance Models in Time Domain, Including the Extended Helmholtz Resonator Model[END_REF][START_REF] Tam | Time-domain impedance boundary conditions for computational aeroacoustics[END_REF]) admittance Y ( [START_REF] Liu | Stability analysis and design of time-domain acoustic impedance boundary conditions for lined duct with mean flow[END_REF][START_REF] Van Den Nieuwenhof | Treatment of frequency-dependent admittance boundary conditions in transient acoustic finite/infinite-element models[END_REF][START_REF] Zhong | A Controllable Canonical Form Implementation of Time Domain Impedance Boundary Conditions for Broadband Aeroacoustic Computation[END_REF]) or reflection coefficient β ( [START_REF] Douasbin | Delayed-time domain impedance boundary conditions (D-TDIBC)[END_REF][START_REF] Fung | Time-domain Impedance Boundary Conditions for Computational Acoustics and Aeroacoustics[END_REF][START_REF] Jaensch | On the robust, flexible and consistent implementation of time domain impedance boundary conditions for compressible flow simulations[END_REF][START_REF] Monteghetti | Energy analysis and discretization of nonlinear impedance boundary conditions for the time-domain linearized Euler equations[END_REF]). Gabard [START_REF] Gabard | A full discrete dispersion analysis of time-domain simulations of acoustic liners with flow[END_REF] suggests the use of a β formalism to avoid mode instabilities. In fact, β is a continuous function without asymptotic-like singularities in Bode's diagrams, and bounded in amplitude in the range [0,1]. It has been demonstrated in [START_REF] Delorme | Computational aeroacoustics applications based on a discontinuous Galerkin method[END_REF] and further confirmed by Monteghetti [START_REF] Monteghetti | Energy analysis and discretization of nonlinear impedance boundary conditions for the time-domain linearized Euler equations[END_REF] how using the βformulation in a CAA simulation guarantees a CFL stability condition independent on the value of the reflection coefficient, which is not true for a Z-formulation, thus making it a reasonable choice for large scale numerical applications. However, Z-formulations are much easier to analyse physically than a β-one. That is why it is the mostly used in the literature. In the following, some of these widely used Z-formulations will be presented.

Rational "three-parameters" model

The first works on time-domain impedance modeling comprehended single polynomial forms or rational fractions, solvable in the time-domain with an ODE. Tam and Auriault [START_REF] Tam | Time-domain impedance boundary conditions for computational aeroacoustics[END_REF] pioneered in this domain proposing a proportional-integral-derivative model under the form:

Z(s) = a -1 s + a 0 + a 1 s (2.3.6)
in which the three a i coefficients can be thought as a Mass-Spring-Damper (MSD) system respectively. The three a i parameters are non-negative for defining a positive-real function and respecting admissibility conditions. The time translation of this impedance law, hence the link between pressure and velocity in the time-domain, is quickly given for a determined frequency f by the set of equation (cf. [START_REF] Tam | Time-domain impedance boundary conditions for computational aeroacoustics[END_REF], eqs. (3)-( 4)):

       ∂p ∂t = a 0 ∂u n ∂t -a 1 2πf u n , a 1 < 0, p = Ru n + a 1 2πf ∂u n ∂t , a 1 ≥ 0 (2.3.7)
Sebastian et al. [START_REF] Sebastian | Numerical simulation of acoustic propagation in a turbulent channel flow with an acoustic liner[END_REF][START_REF] Sebastian | Numerical simulation of a turbulent channel flow with an acoustic liner[END_REF] recently adopted a similar MSD model based on a characteristics formalism to an acoustic liner on a turbulent shear simulation. Different formulations based on the reactance value a 1 are given in order to avoid spurious unstable solutions. Its ease of implementation, simple stability analysis even with a grazing flow, and its exact Fourier transform to correlate pressure and velocity are countered by its poor capability in representing narrow bands around the frequency given by the parameters choice. The three parameters are not enough to represent modern liners behaviour, which are non-monotone functions.

Rational "multi-parameter" model

Shortly after Tam's works, Özyörük [START_REF] Özyörük | A time-domain implementation of surface acoustic impedance condition with and without flow[END_REF][START_REF] Özyörük | Time-Domain Numerical Simulation of a Flow-Impedance Tube[END_REF] proposed a more detailed model considering additional coefficients from experimental data and a fitting technique. In the first work, a multivalued rational fraction is proposed in the z-transform 1 domain of the kind (cf. [START_REF] Özyörük | Time-Domain Numerical Simulation of a Flow-Impedance Tube[END_REF], eq. ( 9)):

Z(z) = a 0 + M N t=1 a t z -t 1 -M D k=1 b k z -k (2.3.8)
where a t and b k are the constants giving the best approximation of the impedance. This was then translated in time-domain by some mathematical expressions including inverse z-transform, until reaching the form (cf. [START_REF] Özyörük | Time-Domain Numerical Simulation of a Flow-Impedance Tube[END_REF], eq. ( 10)):

1 + β 1 + σ p n+1 -p n-σ ∆t + L s p n+1-σ = -a 0 1 + β 1 + σ v n+1 n -v n-σ n ∆t + R n,n-1,... ( 2.3.9)
where R n,n-1,... is the accumulator term necessary to solve the convolution product. Successively, inspired by the advancements in electromagnetism, he proposed an extension of the three-parameters model with 4th degree rational fraction reading: [START_REF] Escouflaire | Theoretical and Numerical Investigation of Time-Domain Impedance Models for Computational Aeroacoustics[END_REF]. It is worth to remark the subsequent work of Nieuwenhof [START_REF] Van Den Nieuwenhof | Treatment of frequency-dependent admittance boundary conditions in transient acoustic finite/infinite-element models[END_REF], which extended Özyörük's model to finite/infinite element method, making it appealing for industrial purposes and their complex geometries.

Z(s) = r 1 + r 2 -

Multipole broadband models

Around the years 2000, many broadband model appeared, first proposed by Fung [START_REF] Fung | Time-domain Impedance Boundary Conditions for Computational Acoustics and Aeroacoustics[END_REF][START_REF] Fung | Broadband Time-Domain Impedance Models[END_REF], and followed few years later by Reymen [START_REF] Reymen | Efficient Implementation of Tam and Auriault's Time-Domain Impedance Boundary Condition[END_REF][START_REF] Reymen | Time-domain acoustic simulation of 3D lined ducts with flow using an unstructured Discontinuous Galerkin method[END_REF][START_REF] Reymen | Time-Domain Impedance Formulation Based on Recursive Convolution[END_REF]. These are under the form of a numerically based multipole model, composed by a sum of elementary systems of first or second order like:

Z num = N k=1 a k s -s k (2.3.11)
1 Caution: z is here the notation for a z-transform (as for a Fourier-transform) and not linked to the impedance. A time-shifting technique was used to make the conversion (cf. [START_REF] Özyörük | Time-Domain Numerical Simulation of a Flow-Impedance Tube[END_REF], eq. ( 5))

The number N of systems considered, as well as their respective weights a k and complex poles s k , are setting the degrees of freedom of the TDIBC, giving a wide range of possibilities in reproducing different liners behaviour. The admissibility condition is easily verified when having all the weights and poles on the left half-complex plane. The drawback is the necessity of solving N convolution products of the form:

Z u(t) = N k=1 a k (e s k H u)(t) = N k=1 a k t 0 e s k τ u(t -τ )dτ (2.3.12)
where H is the Heaviside (or step) function.

Extended-Multipole models

Li et al. [START_REF] Li | Improved Multipole Broadband Time-Domain Impedance Boundary Condition[END_REF] proposed an improved model combining the physical advantages of the threeparameter and the multipole models. The same model was recently used by Naïr et al. [START_REF] Naïr | Industrial-scale time domain modelling of acoustic surface treatments for aero-engines using discontinuous Galerkin method[END_REF] in a TDIBC using a Discontinuous Galerkin method for aero-engines applications, solving Linearized Euler Equations. Cross-comparisons between Naïr results [START_REF] Naïr | Industrial-scale time domain modelling of acoustic surface treatments for aero-engines using discontinuous Galerkin method[END_REF] and the present developments are given in Section 6.3. The implemented impedance formulation reads:

Z(ω) = iaω + b + ic ω + J j=1 C j ω -ξ j - C * j ω + ξ * j (2.3.13)
Through integro-differential relations, pressure and wall-normal velocity can be linked using recursive convolution techniques [START_REF] Reymen | Time-domain acoustic simulation of 3D lined ducts with flow using an unstructured Discontinuous Galerkin method[END_REF], from which a time-domain formulation can be obtained (cf. [START_REF] Naïr | Industrial-scale time domain modelling of acoustic surface treatments for aero-engines using discontinuous Galerkin method[END_REF], eq. (2.3.14)):

Z u(t) = -a du n (t) dt + bu n (t) + c t 0 u n (t)dt + 2 J j=1 Im C j t 0 e -iξ j τ u n (t -τ )dτ (2.3.14)
The advantage brought by this model is to have both the versatility of a multipole model to represent complex liner physics and the simplicity of a three-parameter model to add a constant resistance or reactance to the impedance function when needed. However, the convolution product as well as the respective requirement of an accumulator are still not negligible drawbacks in terms of memory and computational cost. At the same time, two types of algorithms spread for solving the convolution product. First, a "recursive" convolution technique was proposed, originally used in electromagnetism. Then, it was proposed to compute the convolution products using ODEs, whose superiority above the recursive method is widely accepted [START_REF] Dragna | A generalized recursive convolution method for time-domain propagation in porous media[END_REF]. This technique, based on Auxiliary Differential Equations (ADE), finds its origins in the electromagnetic propagation problems, where many real materials have frequency-dependent properties. ADE methods aim at approximating frequency-dependent parameters by rational functions in the frequency-domain. The time-domain counterpart corresponds to a sum of exponentially decaying functions which permits a simplified computation of the convolutions. This methodology will be presented in more details in Section 2.4, as it is the basis of the Oscillo-Diffusive Representation approach.

Physical models -Extended Helmholtz Resonator

All the aforementioned models are easy to implement but, except for the first three-parameters model, they lost any relation to the physics involved when introducing digital formulations. An alternative solution was given with the introduction of the Extended Helmholtz Resonator (EHR) [START_REF] Rienstra | Impedance Models in Time Domain, Including the Extended Helmholtz Resonator Model[END_REF], a physical model which does not require further time derivatives to be calculated and being stable by nature satisfying the admissibility conditions as in Section 2.3.1. Its physical origins make it easy to be associated to acoustic liners parameters, as its formulation explains:

Z EHR (s) = a 0 + a 1 s + a 2 coth(b 0 + b 1 s) (2.3.15)
In here, a 0 expresses the face-sheet resistance, a 1 the face-sheet mass reactance and b 0 , b 1 model the cavity reactance with a face-sheet corrective factor a 2 . If the physical properties are highlighted, such as frequency, acoustic velocity and Mach number, we obtain the typically used model:

Z(s) = R 0 + α(|v ac | + 0.3|M |) + iX m s -i coth(kh) (2.3.16)
with R 0 the resistance at zero velocity, α and X m empirical constant parameters for flow dependency and mass reactance, respectively, v ac the acoustic velocity, and h the cell depth of the liner. The clear advantage of such model is the ease in fitting experimental data on the physical parameters. However, its translation in time-domain is cumbersome due to the memory storage and computational time demands [START_REF] Burak | Validation of a Time-and Frequency-Domain Grazing Flow Acoustic Liner Model[END_REF][START_REF] Richter | Liner impedance modeling in the time domain with flow[END_REF].

Physical models -acoustic liners

A modification to the EHR model for modern multi-perforated liners is:

Z phys (s) = 1 σ p Z p (s) + 1 σ c Z c (s) (2.3.17)
where Z p and Z c are the perforation and cavity impedances, respectively given in (2.2.12) and (2.2.10). CT liners are obtained with Z p = 0 and MP liners with σ c = 1. The perforation model is given by the fractional polynomial form:

Z p Z 0 (s) = a 0 + a1 /2 √ s + a 1 s (2.3.18)
where a 0 models frequency-independent losses, a1 /2 frequency-dependent losses from visco-thermal dissipation, and a 1 is the mass reactance not influencing any loss but causing phase shift. A model for nonlinearities of orifices is given in [START_REF] Ingård | Acoustic Nonlinearity of an Orifice[END_REF], so that the resistance due to high sound amplitude may be taken in consideration as:

R NL = 1 -σ 2 σ |v ac | c 0 (2.3.19)
The three coefficients can then be estimated with different theoretical and empirical models.

The high frequency approximation of a model derived by Crandall reads [START_REF] Crandall | Theory of Vibrating Systems and Sound[END_REF]:

a 0 = 3 l p ν c 0 ( dp /2) 2 + 0.3 M + 1 -σ 2 σ |v ac | c 0 , a1 /2 = 2 (l p + d p ) √ ν c 0 ( dp /2) , a 1 = l p c 0 (2.3.20)
The cavity is modeled through a mono-dimensional wave equation:

Z c Z 0 (s) = coth(ik(s)l c ) (2.3.21)
with fractional wavenumber k(s):

ik(s)l c = b 0 + b 1 2 √ s + b 1 s (2.3.22)
with coefficients derived by Bruneau [START_REF] Bruneau | Fundamental of Acoustics[END_REF]:

b 0 = 0, b1 /2 = l c √ ν c 0 ( dc /2) γ -1 √ P r + 1 , b 1 = l c c 0 (2.3.23)
Conclusively, the impedance of a modified EHR can be written as:

Z(s) = 1 σ p a 0 + a 1 2 √ s + a 1 s + 1 σ c coth b 0 + b 1 2 √ s + b 1 s (2.3.24)
When considering industrial applications, it is important to remark that the coefficient choices proposed in the literature (such as the one here adopted) rely on idealized parameters [START_REF] Lavieille | Numerical simulations of perforate liners: Part I -Model description and impedance valdidation[END_REF]. In fact, the holes diameter, length and porosity are supposed to be identical throughout the whole acoustic liner, while in reality manufacturing processes generate an imprecision not taken into account by these parameters. A part for not correctly discretizing the true liner's behaviour, using idealized parameters may also be a limit in the development of innovative liner concepts. Furthermore, in the previous models, a dependency on the mean flow is exposed (such as the 0.3M coefficient in (2.3.20)). However, experimental studies (Kirby and Cummings [START_REF] Kirby | THE IMPEDANCE OF PERFORATED PLATES SUB-JECTED TO GRAZING GAS FLOW AND BACKED BY POROUS MEDIA[END_REF] among others) investigated the effect of a grazing flow on a perforated plate backed by an air cavity, finding that it induces an increase in resistance and a decrease in mass reactance. As Monteghetti highlighted in [START_REF] Monteghetti | Analysis and Discretization of Time-Domain Impedance Boundary Conditions in Aeroacoustics[END_REF], this experimental correlation does not yield a positive-real function, hence it cannot be used as is in the time-domain. However, an added parametric dependence can be empirically modeled, preserving the locally reacting nature of the impedance boundary condition. More details about this and its comparison with an Ingard-Myers boundary condition are given in his dissertation (cf. [START_REF] Monteghetti | Analysis and Discretization of Time-Domain Impedance Boundary Conditions in Aeroacoustics[END_REF], pg. [START_REF] Dragna | A generalized recursive convolution method for time-domain propagation in porous media[END_REF][START_REF] Escouflaire | Theoretical and Numerical Investigation of Time-Domain Impedance Models for Computational Aeroacoustics[END_REF]. Transforming (2.3.24) in time-domain is far from trivial. The complexity of this problem motivated the developments conducted by Monteghetti [START_REF] Monteghetti | Analysis and Discretization of Time-Domain Impedance Boundary Conditions in Aeroacoustics[END_REF][START_REF] Monteghetti | Energy analysis and discretization of nonlinear impedance boundary conditions for the time-domain linearized Euler equations[END_REF][START_REF] Monteghetti | Design of broadband timedomain impedance boundary conditions using the oscillatory-diffusive representation of acoustical models[END_REF] on Oscillo-Diffusive Representation, which will be presented in detail in the next section.

Oscillatory-Diffusive Representation

It appears clear that there is a lack in the scientific community of a simple representation of a TDIBC respecting the physical behaviour through an appropriate numerical discretization, overtaking difficulties of parameters choice and expensive convolution calculations. Taking as example the EHR formulation, its only need is the consideration of fractional terms (s) for diffusive properties of liner perforations and cavities. It is with Monteghetti's work [START_REF] Monteghetti | Analysis and Discretization of Time-Domain Impedance Boundary Conditions in Aeroacoustics[END_REF][START_REF] Monteghetti | Energy analysis and discretization of nonlinear impedance boundary conditions for the time-domain linearized Euler equations[END_REF][START_REF] Monteghetti | Design of broadband timedomain impedance boundary conditions using the oscillatory-diffusive representation of acoustical models[END_REF] that we obtain an Oscillatory-Diffusive Representation of irrational functions, following the developments in fractional convolution calculation of Hélie and Matignon [START_REF] Hélie | Diffusive representations for the analysis and simulation of flared acoustic pipes with visco-thermal losses[END_REF], that will avoid the use of convolution products for the translation in time-domain. The modified EHR model defined in Section 2.3.2 will here be derived in a time-local formulation of classes of irrational transfer (Laplace transform) functions. The model is presented in three parts for simplicity.

A first part where only the diffusive term is considered, and a second with both diffusive and oscillatory parts, using ODEs (Ordinary Differential Equations). The third part will focus on the time delay formulation expressed through mono-dimensional transport equation.

Diffusive part

First details of diffusive representations are given in [START_REF] Montseny | Diffusive representation of pseudo-differential time-operators[END_REF] and extensive derivation of the form adopted herein is found in [START_REF] Monteghetti | Analysis and Discretization of Time-Domain Impedance Boundary Conditions in Aeroacoustics[END_REF][START_REF] Monteghetti | Energy analysis and discretization of nonlinear impedance boundary conditions for the time-domain linearized Euler equations[END_REF][START_REF] Monteghetti | Design of broadband timedomain impedance boundary conditions using the oscillatory-diffusive representation of acoustical models[END_REF]. Starting from the observation of the heat (or diffusion) equation, diffusive equations are convenient for describing many input-output behaviors of pseudodifferential nature, such as modeling non-standard noises or in general when a memory dynamics effect is present. A reduced form of a diffusive representation h reads:

h(t) = ∞ 0 e -ξt H(t)µ(ξ)dξ (2.4.1)
with H(t) the Heavyside function. Considering the following notation, where a Laplace transform links the two expressions:

e x (t) := e -xt H(t) (x ∈ C, t ∈ R) (2.4.2) e x (s) = 1 s + x (s ∈ C + x , C + x := {s ∈ C|Re(s) > x}) (2.4.3)
then the diffusive representation (2.4.1) can be rewritten as:

h(t) = ∞ 0 e ξ (t)µ(ξ)dξ (2.4.4)
or in Laplace form as:

h(s) = ∞ 0 µ(ξ) s + ξ dξ (2.4.5)
with diffusive poles ξ negative real numbers and diffusive weights µ(ξ) real numbers.

Fractional integrals such as 1/ √ s have a diffusive representation, as well as the fractional derivative √ s after some mathematical steps (cf. [START_REF] Monteghetti | Analysis and Discretization of Time-Domain Impedance Boundary Conditions in Aeroacoustics[END_REF][START_REF] Monteghetti | Design of broadband timedomain impedance boundary conditions using the oscillatory-diffusive representation of acoustical models[END_REF]). To obtain a relation for an acoustic impedance which has a diffusive representation, the convolution product between the latter time-formulation of the impedance and velocity is introduced:

Z(t) u n (t) = ∞ 0 µ(ξ) e ξ (t) u n (t) dξ (2.4.6)
We can now introduce a powerful alternative to the temporal discretization of the convolution.

It is brought by the sum of infinite first-order systems of the kind e -ξt . Defining by τ the delay time for an acoustic wave to travel in the cavity, and introducing the auxiliary function ϕ:

ϕ(t, ξ) := e ξ (t) u n (t) = ∞ 0 e ξ (τ )u n (t -τ )dτ (2.4.7)
the convolution product between impedance and velocity reads:

Z u n (t) = ∞ 0 µ(ξ)ϕ(t, ξ)dξ (2.4.8)
and in discretized form, with N ξ diffusive poles ξ k , correlated weights

µ k = µ(ξ k ) and correlated diffusive variable ϕ k (t) = ϕ(t, ξ k ): Z u n (t) = N ξ k=1 ϕ k (t)µ k (2.4.9)
The interest of this formulation is that the convolution product has been substituted with a first-order ordinary differential equation (ODE), since ϕ is by definition solution of:

     ∂ϕ ∂t (t, ξ) = -ξϕ(t, ξ) + u n (t) ϕ(0, ξ) = 0 (2.4.10)
with t > 0. Advantages of this approach are:

• the possibility of choosing the frequency range on which having a more precise time resolution, fixing the values ξ min and ξ max (with a factor 2π);

• a numerically efficient method compared to previous models using recursive convolutions, as verified in [START_REF] Dragna | A generalized recursive convolution method for time-domain propagation in porous media[END_REF], without reducing the time discretization order used for solving the main PDEs system.

Oscillo-Diffusive part

Physical impedance models shown previously can be written in a structure like:

Z(s) = a 0 + a1 /2 √ s + a 1 s + 1 σ c coth b 0 + b1 /2 √ s + b 1 s (2.4.11)
with the diffusive term in √ s and the wave propagation inside cavities term in (coth), with (b 1/2 = 0) or without (b 1/2 = 0) visco-thermal losses. Monteghetti [START_REF] Monteghetti | Analysis and Discretization of Time-Domain Impedance Boundary Conditions in Aeroacoustics[END_REF] proposed an alternative way to express Z(s):

Z(s) = a 0 + a1 /2 h 1,Z (s) + a 1 s + e -τ s h 2,Z (s) (2.4.12)
where in this particular case h 1,Z (s) = √ s and h 2,Z (s) writes:

h 2,Z (s) = 1 σ c 2e -2(b 0 +b 1 /2 √ s) 1 -e -2(b 0 +b 1 /2 √ s+b 1 s) (2.4.13)
The previous formulation has been obtained thanks to the mathematical relation:

coth(s) = 1 + 2 e -2s 1 -e -2s
(2.4.14)

and remembering that τ = 2b 1 , hence simplifying the term in b 1 at the numerator. The advantage of using a β-formulation for an impedance boundary condition is demonstrated, rather than solving the complex valued time-domain convolution (Z u) in an impedance or admittance formulation [START_REF] Monteghetti | Design of broadband timedomain impedance boundary conditions using the oscillatory-diffusive representation of acoustical models[END_REF]. This is possible thanks to the fact that β does also accept an oscillatory-diffusive representation. After tedious mathematical identities, the reflection coefficient can be written from (2.2.3) and (2.4.12) as:

β phys (s) = 1 + h 1,β (s) + e -τ s h 2,β (s) (2.

4.15)

given:

h 1,β (s) = - 2 R(s) (2.4.16) h 2,β (s) = 2 R(s) e -2(b 0 +b 1 /2 √ s) (2.4.17)
with the denominator function:

R(s) = 1 + 1 σ c + Z p (s) σ p + -1 + 1 σ c - Z p (s) σ p e -2(b 0 +b 1/2 √ s+b 1 s) (2.4.18)
and the perforations impedance:

Z p (s) = a 0 + a1 /2 √ s + a 1 s (2.4.19)
The notation phys in (2.4.15) is a reminder that, at this stage, the physical EHR model is not approximated. In the following and for compactness, h 1,β and h 2,β will be simply referred to as h 1 and h 2 . The interest of using (2.4.15) is that both h 1 and h 2 accept an oscillo-diffusive representation under this form. If a 1/2 and a 1 are positive, as for a SDOF liner, the oscillodiffusive representation of h i reads:

h i (s) = n∈N r i,n s -s n + ∞ 0 µ i (ξ) s + ξ dξ (i ∈ 1, 2) (2.4.20)
These complex-valued poles s n , which can be calculated having the denominator of (2.4.17) R(s)=0, are called "oscillatory poles", have imaginary part not null and go by conjugate pairs. The complex-valued weights r i,n can be directly computed from residual of h i at each pole s n , or numerically optimized so that the right hand side term of (2.4.20) is as close as possible to the analytic known value of h i (s). The (i, n) subscript indicates that each weight is different for different function h i and pole s n . Similarly, µ i,k represents the "diffusive weight" of the "diffusive poles" ξ k for each function h i . They are directly defined by the jump of h i :

µ i (ξ) = 1 2πj h i (ξe -jπ ) -h i (ξe +jπ ) (2.4.21)
where the imaginary unit is temporarily defined by "j" to not mix it with the function index "i".

Function h 1 (s) takes a discretization of the kind:

h 1 (s) = C + n∈N r 1,n s -s n + ∞ 0 µ 1 (ξ) s + ξ dξ (i ∈ 1, 2) (2.4.22)
where C is a constant used to differentiate a CT liner from a SDOF liner (due to different mathematical formulations, cf. [START_REF] Monteghetti | Analysis and Discretization of Time-Domain Impedance Boundary Conditions in Aeroacoustics[END_REF], §6.1.2) and it takes the form:

C ≈ 1 1 + σ -1 c (2.4.23)
For a SDOF liner, which has positive a i coefficients hence it is a non-decaying formulation, C = 0 and it can be neglected. When discretizing (2.4.15):

β num (s) = β ∞ + Ns n=1 r 1,n s -s n + N ξ k=1 µ 1,k s + ξ k + e -sτ   Ns n=1 r 2,n s -s n + N ξ k=1 µ 2,k s + ξ k   (2.4.24)
The Oscillo-Diffusive Representation yields to a multipole model, but compared to precedent models the following advantages are found:

• oscillatory poles s n are known from geometrical characteristics of the material (hidden in a i and b i formulations), leaving a free choice only of diffusive poles ξ k and the number of poles N s and N ξ , adapted to the desired precision and frequency range;

• weights r n and µ k can be calculated as said above. In particular the optimization is simple as it is linear, contrarily to other multipole models which require nonlinear multiparameters optimizations;

• β num automatically respects admissibility conditions, avoiding all those numerical instabilities related to ill-posedness of boundary conditions.

Time delay formulation

In the following, the notation z is introduced for indicating a generic pole of the Oscillo-Diffusive Representation, z ∈ {ξ k , -s n }. We focus here on how to deal with the term e -sτ , generator of a delay τ . As for the previous diffusive part, auxiliary diffusive ODE functions ϕ t,z are used to discretize e -sτ h 2,β (s) in the form:

e -τ s h 2,β (s) = Ns n=1 r 2,n ϕ(t -τ, -s n ) + N ξ k=1 µ 2,k ϕ(t -τ, ξ k ) (2.4.25)
Auxiliary hyperbolic functions ψ are introduced to translate the "delayed effect" in a "transported effect". Such functions solve the three-valued initial boundary problem:

         ∂ψ ∂t (t, z, l) = c ∂ψ ∂l (t, z, l) ψ(t, z, 0) = ϕ(t, z) ψ(0, z, l) = 0 for t > 0, l ∈ (0, L τ ), z ∈ C (2.4.26)
where L τ represents the distance traveled in the time τ at a characteristic speed c: L τ = cτ . As a result, the transported variable ψ(t) equals the delayed one ϕ(t -τ ):

ψ(t, z, L τ ) = ϕ(t -τ, z) (2.4.27)
The time discretization of h 2,β (s) can then be summarized in:

e -τ s h 2,β (s) = Ns n=1 r 2,n ψ(t, -s n , L τ ) + N ξ k=1 µ 2,k ψ(t, ξ k , L τ ) (2.4.28)
The discrete impedance model of the convolution product of β, expressed as in (2.4.15) and reminding its relation with acoustic waves in (2.3.5), becomes:

u ac - p ac ρ 0 c 0 (t) = β u ac + p ac ρ 0 c 0 (t) = β ∞ u ac + p ac ρ 0 c 0 (t)+ + N ξ k=1 (µ 1,k ϕ(t, ξ k ) + µ 2,k ψ(t, ξ k , L τ )) + Ns n=1 r 1,n (ϕ(t, -s n ) + r 2,n ψ(t, -s n , L τ )) (2.4.29)
with β ∞ = 1 + C the bulk reflectivity (i.e. the frequency-independent reflectivity), equal to 1 for a SDOF liner. The latter must be resolved simultaneously to the auxiliary variables ϕ, whose boundary and initial conditions are given as in (2.4.10). A characteristic waves formulation of (2.4.29) will be detailed in Chapter 3.

Nonlinear impedance modeling

SDOF liners are subjected to a dependence on the incident sound pressure level. When the amplitude of the incident sound wave increases past a given threshold, the acoustic boundary layer separates within the perforation, creating a vortex shedding [START_REF] Lewy | Analytical and numerical prediction of harmonic sound power in the inlet of aero-engines with emphasis on transonic rotation speeds[END_REF][START_REF] Melling | The acoustic impendance of perforates at medium and high sound pressure levels[END_REF][START_REF] Roche | Simulation numérique de l'absorption acoustique de matériaux résonants en présence d'écoulement[END_REF]. A numerical example of this phenomena is shown in Figure 2.10. An energy redistribution is present: first, a conversion from acoustic energy to rotational energy, and second, dissipation of this rotational energy. Very little studies deal with this problematic, however it is of prominent importance for those high sound level applications, such as the take-off noise of turbofan engines, to understand this phenomena and be able to consider it in the design process. It is relevant to observe that nonlinear effects due to flow separation through the perforations are visible only when the grazing flow is slow (u ac ∼ u D , friction velocity) or the SPL very high. This is what creates a "bump" in the resistance curve of a liner (cf. Figure 5.1 in Section 5.1 for a practical example).

When the overall resistance increases due to high flow friction, the bump gets absorbed inside it and the added nonlinear acoustic resistance becomes negligible.

Ingard and Ising [START_REF] Ingård | Acoustic Nonlinearity of an Orifice[END_REF] pioneered experimental studies on orifice nonlinearities, demonstrating the decay of the linear formalism at sufficiently high sound pressure levels. Even though many years passed from their findings, not many studies followed on this topic. Temiz et al. [START_REF] Temiz | Non-linear acoustic transfer impedance of micro-perforated plates with circular orifices[END_REF] Figure 2.10: Vortices generation from the perforations -source from [START_REF] Roche | Simulation numérique de l'absorption acoustique de matériaux résonants en présence d'écoulement[END_REF].

focused on the transition from linear to nonlinear regimes in experimental measurements of a micro-perforated plate. Tudisco [START_REF] Tudisco | Application of the Time-Domain Impedance Boundary Condition to Large-Eddy Simulation of Combustion Instability in a Shear-Coaxial High Pressure Combustor[END_REF] used a nonlinear model for simulating injectors of a combustion chamber in a LES simulation. The scattering operator used herein is function of the time delay and of an algebraic function for the incoming characteristic wave (cf. [START_REF] Tudisco | Application of the Time-Domain Impedance Boundary Condition to Large-Eddy Simulation of Combustion Instability in a Shear-Coaxial High Pressure Combustor[END_REF], eq. (2.15)). Singh [START_REF] Singh | Nonlinear asymptotic impedance model for a Helmholtz resonator liner[END_REF] proposed a nonlinear asymptotic model for HR liner, solving the stationary solution. Asymptotic in the meaning that, for small amplitudes, it coincides with the linear regime, but takes in considerations nonlinear effects close to the fundamental resonance frequency. Rienstra [START_REF] Rienstra | Nonlinear Asymptotic Impedance Model for a Helmholtz Resonator of Finite Depth[END_REF] then expanded it solving the wave equation inside the cavity to achieve a relationship between pressure and velocity of the waves developing inside the cavity. The idea is to formulate a perturbation problem in terms of a small parameter (0 ≤ 1) dependent on the input excitation amplitude of a given frequency (cf. [START_REF] Singh | Nonlinear asymptotic impedance model for a Helmholtz resonator liner[END_REF], eq. ( 12)). They used a data set from Motsinger and Kraft [START_REF] Motsinger | Design and Performance of Duct Acoustic Treatment[END_REF] to verify their correct implementation. Grazing flow effects, in these works neglected, shall be considered in the case of a thin enough boundary layer and of a resonator outflow velocity of the same order of magnitude or higher than the mean flow velocity in the lined duct. Two impedance formulations are then provided, depending if far or near the resonance frequency (cf. [START_REF] Singh | Nonlinear asymptotic impedance model for a Helmholtz resonator liner[END_REF], eqs. ( 47) and ( 50), respectively).

The mathematical approach herein proposed focuses the attention on the translation of the nonlinear effects to the time-domain in a reflection coefficient formalism based on the Oscillo-Diffusive Representation. Monteghetti already gave an introduction to this concept in the frame of a DG solver [START_REF] Monteghetti | Energy analysis and discretization of nonlinear impedance boundary conditions for the time-domain linearized Euler equations[END_REF]. The nonlinear term investigated is proportional to a vena contracta coefficient [START_REF] Cummings | High amplitude acoustic transmission through duct terminations: Theory[END_REF], expressing the boundary layer separation in the perforation as a reduction of the original section σ to an effective cross section σ vc :

σ vc = C vc σ ≤ σ, C vc ∈ [0, 1] (2.5.1)
For mass conservation, the following is also true:

u ac = σu perf = σ vc u vc (2.5.2)
A recent numerical study from Zhang et al. [START_REF] Zhang | Numerical investigation of a honeycomb liner grazed by laminar and turbulent boundary layers[END_REF] focused on the calculation of the coefficient C vc . Assuming the flow steady, incompressible, inviscid and irrotational in the perforations, from Bernoulli's law, the energy at the cavity neck extremities is conserved:

p 1 + 1 2 ρ 0 u 2 1 = p 2 + 1 2 ρ 0 u 2 2 (2.5.3)
with 1 and 2 indexing the locations at the orifice entrance (index 1) and exit (index 2). Rearranging (2.5.3), the pressure drop through the orifice can be estimated:

p 1 -p 2 = 1 2 ρ 0 u 2 2 -u 2 1
(2.5.4) Substituting (2.5.2) in (2.5.4), the pressure drop will be expressed function of the acoustic velocity, the latter depending respectively on the global surface porosity σ and vena contracta effect C vc σ:

p 1 -p 2 = 1 2 ρ 0 u 2 ac C vc 2 σ 2 - u 2 ac σ 2 = ρ 0 C nl u 2 ac ≈ ρ 0 C nl |u ac |u ac (2.5.5)
with:

C nl = 1 -C 2 vc 2C 2 vc σ 2 (2.5.6)
and using the absolute value to keep in consideration the velocity direction [START_REF] Cummings | Transient and multiple frequency sound transmission through perforated plates at high amplitude[END_REF]. All the previous reasoning was made in the time-domain, which means one can write:

p 1 (t) = p 2 (t) + ρ 0 C nl |u ac (t)|u ac(t) (2.5.7)
Back to the impedance definition, and remembering the linear perforation impedance expression with h i (s) = µ k ϕ k a rational function as defined in Section 2.4:

Z lin (s) = a 0 + a1 /2 h 1 (s) + a 1 s + e -τ s h 2 (s) (2.5.8)
we can then write:

p(t) = (Z u ac ) (t) = (Z lin u ac ) (t) + ρ 0 C nl |u ac (t)|u ac (t) (2.5.9)
where it is highlighted the second nonlinear corrective term. We remember from (2.2.3) that the reflection coefficient can be defined by:

β = Z -1 Z + 1 = Z + 1 Z + 1 - 2 Z + 1 = 1 - 2 Z + 1 (2.5.10)
and that the scattering operator B links acoustic waves in a way that:

u ac - p ac ρ 0 c 0 (t) = B u ac + p ac ρ 0 c 0 (t) (2.5.11)
From (2.5.10), two formulations of scattering operator can then be given [START_REF] Monteghetti | Analysis and Discretization of Time-Domain Impedance Boundary Conditions in Aeroacoustics[END_REF]:

B( ) = (Z -I) • (Z + I) -1 ( ) (2.5.12a) B( ) = -2(Z + I) -1 ( ) (2.5.12b)
Now we look for a value w so that, if = p /ρ 0 c 0 + u is considered known:

w = (Z + I) -1 ( ) ⇐⇒ = (Z + I)(w) (2.5.13)
Solving the parenthesis and separating the linear and nonlinear impedance contributions:

= Z(w) + w = (Z lin w) + ρ 0 C nl w|w| + w (2.5.14)
with Z lin as in (2.4.9):

Z lin w = k µ k ϕ k (2.5.15)
After some rearrangements, the final equation system in the variables ϕ, w and u ac -pac /ρ 0 c 0 can be written:

                       ∂ϕ k ∂t = -ξ k ϕ k + w (from (2.4.10)) u ac - p ac ρ 0 c 0 = -2w (combining (2.5.11) with (2.5.12b)) = (Z lin w) + ρ 0 C nl w|w| + w (from (2.5.14)) Z lin w = k µ k ϕ k (from (2.5.15)) (2.5.16)
In comparison with the previous linear system (2.4.29), an additional equation has been added to consider the nonlinear term and resolve the additional variable w. However, the system (2.5.16) does not present a β-formalism as for (2.4.29). This is because, from (2.5.12a):

B( ) = (Z -I) • (Z + I) -1 ( ) = (Z -I)[(Z + I) -1 ( )]
(2.5.17) cannot be reversed in order to express (Z lin w) in a form of the kind (β lin w). Hence, the multipole Oscillo-Diffusive Representation must be computed on an impedance-formalism as in (2.4.9). For a weakly nonlinear case (C nl ≈ 0), the following equivalence will also be true:

                       ∂ϕ k ∂t = -ξ k ϕ k + w u ac - p ac ρ 0 c 0 = -2w = (Z lin w) + w Z lin w = k µ k ϕ k (2.5.18)
Combining (2.5.13) with the first equation of (2.5.18), we return to the linear expression:

u ac - p ac ρ 0 c 0 = -2(Z + I) -1 ( ) = B( ) (2.5.19)
One equation is then redundant, as only the left hand side of (2.5.19) and ϕ are unknown. In fact, the system (2.4.29) is quickly obtained back:

                 ∂ϕ k ∂t = -ξ k ϕ k + u ac - p ac ρ 0 c 0 = B( ) B = k µ k ϕ k
A numerical discretization of (2.5.18) is proposed in Section 3.3.5.

Theory of nonlinear propagation

Whenever the sound level is sufficiently low, the hypothesis of considering acoustics as a linear phenomena is widely accepted. However, as soon as the amplitude of acoustic waves increases towards higher values, nonlinear propagation phenomena cannot be neglected. First observations of such events are attributed to Lagrange back in 1761, then exponentially followed by great physicians such as Poisson and Riemann. The additional formulations given by Rankine-Hugoniot in the late 19th century on shock wave formation and energy conservation advanced the studies on the topic. Nayfeh et al. [START_REF] Nayfeh | Nonlinear acoustic propagation in two-dimensional ducts[END_REF] introduced nonlinear acoustics studies in ducts, differentiating it from the near-and far-field applications with a multiple scales method reduced to the case of a lined duct. In there, the semi-empirical impedance model of Zorumski and Parrott [START_REF] Zorumski | Nonlinear acoustic theory for rigid porous materials[END_REF] was used to define nonlinear resistance and reactance parameters. Shock waves amplitude decrease was detailed by Morfey and Fisher [START_REF] Morfey | Shock-Wave Radiation from a Supersonic Ducted Rotor[END_REF], who gave a clearer understanding of regular nonlinear waves (also called "N-waves" or sawtooth waves) generation, propagation and dissipation. To picture the physics involved in nonlinear acoustic propagation, we now consider for simplicity a sinusoidal acoustic wave. At a sufficiently strong sound level, the propagation speed (noted v) becomes dependent on the pressure level following the equation:

v = c 0 1 + γ + 1 2γ p -p 0 p 0 (2.6.1)
This means that points of high pressure will have a higher propagation speed, hence deforming the signal structure. Soon enough, the wave assumes the typical N-shape (or sawtooth shape). The wave distortion is stronger and faster for higher sound pressure levels. The threshold above which nonlinearities start to appear is function itself of the wavenumber (or the wave frequency), thus a precise value cannot be generally given [START_REF] Langenais | Adaptation des méthodes et outils aéroacoustiques pour les jets en interaction dans le cadre des lanceurs spatiaux[END_REF]. However, in duct acoustics applications found in the literature, this value is usually thought to be between 130 dB and 160 dB [START_REF] Nayfeh | Nonlinear acoustic propagation in two-dimensional ducts[END_REF]. Finally, N-waves amplitude decreases as in the shock wave dissipation theory. Different models on the pressure-jump dissipation have been developed in the literature. In particular, the demonstration from Morfey & Fisher [START_REF] Morfey | Shock-Wave Radiation from a Supersonic Ducted Rotor[END_REF] gives a time-envelop in which an initial regular N-Wave of finite amplitude will be contained during its dissipation. The pressure jump evolution is found to be:

∆p(t) = ∆p (t=0) 1 + γ + 1 2γ ∆p (t=0) p 0 c o t λ (2.6.2)
As a further consequence of N-waves formation, harmonic frequencies of the initial frequency emerge and energy is redistributed among them. In regards to the propagation of nonlinear acoustic waves, Kuznetsov equation describes a wave in a Newtonian fluid, homogeneous and thermo-viscous, supposing thermodynamic variables variations small and neglecting terms above the second order:

∂ 2 φ ∂t 2 -c 2 0 ∆φ = ∂ ∂t (∇φ) 2 + γ -1 2c 2 0 ∂φ ∂t 2 + b ρ 0 ∆φ (2.6.3)
with b the thermo-viscous dissipation parameter (or sound diffusivity):

b = κ 1 c v - 1 c p + 4 3 ν d + ν v
with κ the thermal conductivity coefficient, ν d and ν v the dynamic and volume viscosity coefficients, respectively. No exact solution or numerical scheme allows to solve this equation at the time of writing. Two famous alternatives which can be solved in an easier way are the Kuznetsov equation for perfect fluid (b = 0) and Burger's equation. In the case of perfect fluid, the simplified Kuznetsov equation has been solved analytically by Fernando et al. [START_REF] Fernando | Nonlinear waves and shocks in a rigid acoustical guide[END_REF] for a stationary flow in 2D/3D configurations, followed by Kassem work [START_REF] Kassem | A three-dimensional cylindrical model for non-linear propagation prediction in lined intake ducts with uniform flow[END_REF] adding a uniform flow and applying it to a turbofan geometry. Burger's equation is also used for shock propagation, and it reads:

∂p ∂t + v ∂p ∂x = 0 (2.6.4)
A full derivation of the latter is found in [START_REF] Thisse | Generation and Propagation of Multiple Pure Tones Inside Turbofans at Transonic Regime[END_REF] (cf. §1.1.5), where Thisse developed a time-domain finite volume method solving the Euler equations for nonlinear propagation of shock waves in a hard wall duct [START_REF] Thisse | Prévision du bruit d'onde de choc d'un turboréacteur en régime transsonique par des méthodes analytiques et numériques[END_REF][START_REF] Thisse | Numerical Simulations of Shock-Wave Propagation in Turbofan Intakes[END_REF]. The same CFD solver as in this work was used to compute threedimensional shock waves injection and propagation in a hard wall duct in the same conditions as in a nacelle inlet. In an aeronautic nacelle environment, two main processes are identified during the study of N-waves: the generation of N-waves and their propagation. However, two important differences must be highlighted:

• The case of a perfect rotor, in which all blades are identical, where the acoustic spectrum only contains harmonics of the Blade Passing Frequency (BPF).

• The case of a real rotor, where small blade geometrical variability (such as stagger angle variations) produces N-wave amplitude and inter-shock spacing variations that affect their nonlinear propagation. These small irregularities lead to spectral components at harmonics of the shaft rotation frequency. So, the acoustic spectrum contains not only BPF harmonics but also Multiple Pure Tones (MPT). As a consequence, the nonlinear duct propagation is characterized by an energy redistribution from BPF to MPT, leading to a complex prediction of the BPF intensity at the end of the duct propagation.

This energy redistribution from BPF to MPT during the propagation may result into some MPT harmonics to be louder than BPF. This is because they could possibly be less attenuated by the acoustic liners, since MPT are not at the frequency for which the liner is designed. An example is brought in [START_REF] Thisse | Prévision du bruit d'onde de choc d'un turboréacteur en régime transsonique par des méthodes analytiques et numériques[END_REF][START_REF] Thisse | Generation and Propagation of Multiple Pure Tones Inside Turbofans at Transonic Regime[END_REF], where the effect of different blade configurations was established for different geometries and engine ratings. In Figure 2.12, it is evident how, reshuffling the blades order of a 18-bladed turbofan, the MPT tones can drastically change in amplitude.

Figure 2.12: MASCOT2 Turbofan Model -Comparisons between measured (light) and ONERA approach (dark) spectra near the fan for two blade configurations -source from [START_REF] Thisse | Generation and Propagation of Multiple Pure Tones Inside Turbofans at Transonic Regime[END_REF].

TDIBC in CFD computations: state-of-the-art

Now that the theory behind the object of this work is given, this section aims at posing the time-domain impedance model with Oscillo-Diffusive Representation in the frame of currently developed and used impedance models in the CFD community with practical comparisons.

Fiévet et al. [START_REF] Fiévet | Numerical Analysis of Porous Coatings Stabilizing Capabilities on Hypersonic Boundary-Layer Transition[END_REF][START_REF] Fiévet | Numerical Study of Hypersonic Boundary-Layer Transition Delay through Second-Mode Absorption[END_REF][START_REF] Fiévet | Strong compact formalism for characteristic boundary conditions with discontinuous spectral methods[END_REF] implemented the same ODR reflection coefficient previously described in (2.4.29) to the Spectral Difference CFD code "Jaguar" with a characteristic waves formalism. The high-order spatial scheme proved successful for adopting this TDIBC, with great results on academic benchmark cases and on the attenuation of mode instabilities in hypersonic boundary layers, as shown in Figure 2.13.

The same physical problem is studied by Sousa in [START_REF] Sousa | Towards Direct Numerical Simulation of Hypersonic Transition Delay Via Distributed Wall Porosity[END_REF], following the works of Scalo [START_REF] Scalo | Compressible turbulent channel flow with impedance boundary conditions[END_REF] and Douasbin [START_REF] Douasbin | Delayed-time domain impedance boundary conditions (D-TDIBC)[END_REF] on a delayed-TDIBC formulation of the reflection coefficient. The latter also belongs to the characteristic boundary conditions type, and it describes the acoustic reflection coefficient by a set of complex poles and residues as:

β(ω) = 2 1 + Z(ω) ≈ n k=1 µ k s -p k + µ * k s -p * k (2.7.1)
An inverse Fourier transform gives the wall normal velocity through a convolution integral. Some results are illustrated in Figure 2.14.

Sebastian et al. [START_REF] Sebastian | Numerical simulation of acoustic propagation in a turbulent channel flow with an acoustic liner[END_REF][START_REF] Sebastian | Numerical simulation of a turbulent channel flow with an acoustic liner[END_REF] implemented a mass-spring-damper model in a Navier-Stokes solver for simulating a turbulent channel flow with an acoustic liner. This was written under an admittance formulation and with use of differential equations (rather than convolution products) in order to retain the same order of time integration. The choice of using the simpler MSD model was justified by the type of liner under consideration, which showed an instability around the resonance frequency. Hence, it was convenient to use a MSD model, function of the pressure derivative as: A recent model that was rapidly adopted is the purely multipole formalism given by Dragna [START_REF] Dragna | A generalized recursive convolution method for time-domain propagation in porous media[END_REF]. The impedance model relies on partial fraction modeling, involving a time integration of a first-order ODEs derived from poles and residues. The time-domain translation is done with an Auxiliary Differential Equation (ADE) method as a faster solution than convolution techniques. Its reflection coefficient formulation reads:

M d 2 v dt 2 + Kv + R dv dt = - dp dt (2.7.2)
2 β(ω) = β ∞ + N k=1 A k λ k -iω + 1 2 M k=1 B k + iC k α k + iβ k -iω + B k -iC k α k + iβ k -iω (2.7.3)
where similarities with the ODR model can be found: β ∞ is the same immediate response. The term in A k represents the herein called "diffusive term", which decays by an exponential rate in time and is given by real poles. The second term in B k , C k corresponds to the "oscillatory term" made by complex conjugate poles, whose period of oscillation is governed by the poles imaginary part and its decay in time by their real part (as in the diffusive poles). However, physics is no more present in the calculation of such poles and the cavity delay τ is not explicit. The poles values are then obtained by numerical methods such as vector fitting.

Dragna's model [START_REF] Dragna | A generalized recursive convolution method for time-domain propagation in porous media[END_REF] was chosen by the recent developments of Shur, Suzuki et al. [91-93, 98, 99]. Due to the strong similarities with the current work, a particular attention will now be posed on describing the points in common and differences with their numerical model. First, extensive studies were carried out on unsteady CFD simulations of aerodynamics and acoustics 2 the variable s in the original paper is substituted with β for alignment with the present notation of an engine fan and Outlet-Guide-Vane (OGV) system [START_REF] Shur | Unsteady Simulations of a Fan/Outlet-Guide-Vane System: Aerodynamics and Turbulence[END_REF][START_REF] Suzuki | Unsteady Simulations of a Fan/Outlet-Guide-Vane System: Broadband-Noise Computation[END_REF][START_REF] Suzuki | Unsteady Simulations of a Fan/Outlet-Guide-Vane System: Tone-Noise Computation[END_REF], providing a solid baseline for consequent acoustics analyses. Recently, an effort was made in vision of TDIBC implementation in the same CFD software [START_REF] Shur | Further Evaluation of Prediction Capability of the Broadband Time-Domain Impedance Model for Sound Propagation in Turbulent Grazing Flow[END_REF][START_REF] Shur | Unsteady Simulation of Sound Propagation in Turbulent Flow Inside a Lined Duct Using a Broadband Time-Domain Impedance Model[END_REF]. URANS with Spalart-Allmaras turbulence model and Wall-Modeled Large Eddy Simulations (WMLES) were run for predicting sound attenuation in a classical acoustic benchmark, then to a realistic fan geometry with inclusion of turbulence modelings. Figure 2.16 shows the liner's effect on a nacelle inlet configuration. Their time-domain impedance model, derived from the previously discussed scattering operator (2.7.3), is defined "dynamic" in respect to the sound pressure level. A set of reference values of SPL is taken, which covers the envisioned range of wall pressure fluctuations in the problem under consideration. At each of these reference values, the complex impedance dependence of the liner on frequency is approximated in the multipole form. For example, in [START_REF] Shur | Unsteady Simulation of Sound Propagation in Turbulent Flow Inside a Lined Duct Using a Broadband Time-Domain Impedance Model[END_REF], three different SPL values are taken and the multipole impedance model computed for each of them. Then, at a given computed SPL value in the range considered, a "dynamic" impedance of the liner is computed as a linear combination of the three impedances curves, and the weights values are obtained by interpolation among the reference SPL values. This allows for the impedance law to be better representative of the true pressure level measured at each point of the liner. In fact, if the sound level is strongly attenuated through the liner or the liner is sufficiently long, a single impedance value calculate at a given SPL would not be realistic of the true SPL on the lined wall. In the recent work of Shur [START_REF] Shur | Further Evaluation of Prediction Capability of the Broadband Time-Domain Impedance Model for Sound Propagation in Turbulent Grazing Flow[END_REF], an interesting TDIBC instability resolution method is tackled inspired by the work of Deng [START_REF] Deng | Characterization and suppression of the hydrodynamic instability in the time domain for acoustic propagation in a lined flow duct[END_REF]. The latter is designed for suppressing spatial hydrodynamic instabilities at the lined surfaces at the near-resonance frequencies. Shur's TDIBC is implemented in the CFD code "NTS": a cell-vertex finite-volume code accepting structured multi-block overset grids of Chimera type. Its compressible branch employs an implicit flux-difference splitting method of Roe type. The viscous fluxes in all the governing equations were approximated with the use of the 2nd-order centered scheme, while the approximation of the inviscid fluxes depends on the approach to turbulence treatment. In the framework of URANS and within the RANS sub-domain of the zonal RANS-IDDES, the inviscid fluxes were approximated with the use of the weighted 4th-order centered / 5th-order upwind-biased numerical scheme. For the time integration, in all the simulations the 2nd-order three-layer backward scheme with sub-iterations was applied. The high-order numerics just described ensure high accuracy of representation of the sound-wave propagation even on relatively coarse computational grids. In the following Chapter, the solver adopted in this dissertation and the numerical choices will be detailed and compared with those just presented from the literature.

Summary

• General theory on duct acoustics and acoustic impedance was given. The interest of using a time-domain impedance (or reflection coefficient) formulation was presented as a solution to the CFD/CAA coupled technique when computing sound absorption.

• Several examples of time-domain impedance models were classified from the literature. The Oscillo-Diffusive Representation was detailed as the impedance model chosen for this work. The three steps when building a time-domain impedance formulation were given:

The elsA solver

The elsA1 solver, developed at ONERA since 1997 and jointly owned by ONERA, Airbus and Safran until 2020, is a multi-application aerodynamic code based on a cell-centred finite-volume method for structured meshes. Solving the compressible, three-dimensional Reynolds-averaged Navier-Stokes (RANS) equations, elsA allows to simulate a great variety of aerospace configurations such as aircraft, space launchers, missiles, helicopters and turbomachines [START_REF] Cambier | ElsA: An efficient object-oriented solution to CFD complexity[END_REF][START_REF] Cambier | The Onera elsA CFD software: Input from research and feedback from industry[END_REF]. Therefore, a wide range of numerical tools and models are implemented. Conventional numerical tools are provided in elsA for turbomachinery applications such as a cell-centred space discretization scheme of Jameson with artificial viscosity, and a second-order accurate Roe scheme for transport equations of turbulence models. Several time-integration schemes are available to perform steady and unsteady computations. Explicit or implicit schemes, such as a pseudo-time approach (Dual Time-Stepping, DTS) or the Gear method, are available. Time integration can be solved either by an implicit residual smoothing phase with a 4-step Runge-Kutta technique or by an implicit LU scalar relaxation phase associated to a backward Euler scheme. Convergence acceleration techniques such as local time-stepping and multi-grid method are available for steady computations in order to reduce the total CPU time. Suitable boundary conditions for turbomachinery configurations have been implemented for steady and unsteady flow applications: coincident and non-coincident matching conditions for the treatment of periodic conditions on boundaries, and a steady multi-stage condition using pitch averaging for the treatment of the rotor-stator interface. Different types of inlet, outlet, and wall conditions are also available.

For acoustic applications, since the currently implemented non-reflecting boundary conditions are known to be rather limited, best efficiency is practically achieved by using a buffer zone close to the boundary, where the grid cells are progressively stretched in order to increase the numerical dissipation of the acoustic waves and eventual parasitic reflections. Large selections of turbulence models are presently available, from algebraic to non-Boussinesq models.

elsA is currently used for industrial applications in Airbus and Safran for external aerodynamics and turbomachines. Concerning acoustics applications, elsA is used for predicting the noise sources of turbofan and turboprop engines with consideration of nonlinear flow phenomena, such as the shock wave generation and propagation from the inlet fan. Many works are available in the literature where elsA was used: Lewy et al. [START_REF] Lewy | Analytical and numerical prediction of harmonic sound power in the inlet of aero-engines with emphasis on transonic rotation speeds[END_REF] for predicting the sound power at the inlet radiating into the free field, Le Garrec et al. [START_REF] Garrec | Tone Noise Predictions of a Full-Scale UHBR Engine at Approach Condition with Inflow Distortion Effects[END_REF] for assessing the source generation and sound propagation of the fan-OGV stage with inflow distortion in a full-scale UHBR engine, Daroukh et al. [START_REF] Daroukh | Shock Wave Generation and Radiation from a Turbofan Engine Under Flow Distortion[END_REF] continuing the latter with additional aeroacoustic investigations with a CFD-CAA chaining strategy, Thisse et al. [START_REF] Thisse | Prévision du bruit d'onde de choc d'un turboréacteur en régime transsonique par des méthodes analytiques et numériques[END_REF][START_REF] Thisse | Generation and Propagation of Multiple Pure Tones Inside Turbofans at Transonic Regime[END_REF] for assessing different methods on the generation of N-waves and their propagation through inlet ducts, producing blade passing harmonics and multiple pure tones for a real rotor geometry. Even though some non-reflective boundary conditions for acoustic applications are already available [START_REF] Thisse | Generation and Propagation of Multiple Pure Tones Inside Turbofans at Transonic Regime[END_REF], no impedance wall boundary condition was developed in this code until now. The next Chapter will detail how a characteristic-type of boundary condition is ideal for acoustic liner boundary conditions. The latter, altogether with characteristic outlet and inlet non-reflective boundary conditions, will be presented and derived up to the equations directly implemented in the code.

Navier-Stokes Characteristic Boundary Conditions

The definition of Navier-Stokes Characteristic Boundary Condition (NSCBC) is given from its original paper from Poinsot and Lele [START_REF] Poinsot | Boundary conditions for direct simulations of compressible viscous flows[END_REF]: "A well established method for boundary condition of hyperbolic systems is to use relations based on characteristic lines, which means analyzing the different waves crossing the boundaries". Figure 3.1 gives an idea of this concept and will be looked at in detail throughout this Chapter. The origin of characteristic formulation is however precedent, in fact it was introduced by Thompson [START_REF] Thompson | Time dependent boundary conditions for hyperbolic systems[END_REF] for general hyperbolic systems. Consequently, Rienstra [START_REF] Rienstra | An introduction to acoustics[END_REF] suggested the advantage of using characteristic forms in acoustics problems, as they allow an analytical solution to problems at high amplitudes (where nonlinearities appear) and they determine optimal discretization schemes and stability conditions.

NSCBC Theory

Generally speaking, we can classify boundary conditions (BC) in two types: physical and numerical. Physical BC is when the physical behaviour of one or more dependant variables is known and specified at the boundaries. As an example, specification of the axial inlet velocity or outlet pressure are physical boundary conditions. Theoretical analyses (references in [START_REF] Poinsot | Boundary conditions for direct simulations of compressible viscous flows[END_REF]) suggest the number of necessary and sufficient boundary conditions for the Euler (inviscid) or Navier-Stokes (viscous) equations, as resumed in Table 3.1. In order to build a complete set of Navier-Stokes boundary conditions, the Euler conditions are completed by viscous relations, for them to take into account viscous dissipation and thermal diffusion. In this way, physical BC may not be enough to numerically solve the problem. When the number of physical BC is

Boundary type

Euler Navier-Stokes Supersonic inflow 5 5 Subsonic inflow 4 5 Supersonic outflow 0 4 Subsonic outflow 1 4

Table 3.1: Number of physical BC required for well-posedness in a 3D flow -source from [START_REF] Poinsot | Boundary conditions for direct simulations of compressible viscous flows[END_REF].

less than the number of primitive variables, "soft" conditions (or "numerical" conditions, as they are required by the numerical implementation and not by the physics of the problem) must be introduced. An option is to extrapolate the missing values from interior points. Even though this is a simpler choice, its drawback is given by an over-defined system: in fact, the gradient of the extrapolated variable will be automatically set to zero by this numerical BC. A more rigorous method is to apply the conservative equations on the boundary as a complement of physical BC and therefore to implement NSCBC. Poinsot [START_REF] Poinsot | Boundary conditions for direct simulations of compressible viscous flows[END_REF] listed four difficulties related to NSCBC:

1. Near the boundaries, the spatial discretization accuracy has to be decreased. Central differences become one-sided difference because points are available only inside the boundary. However, this problem can be overcome if the one-sided difference is up to one-order lower than the overall scheme accuracy, or using ghost cells.

2. Some waves at the boundaries can propagate from outside the domain to the inside. This would require additional information about the solution outside the domain.

3. The amplitude of incoming waves must be evaluated from the original choice of physical BC and so expressed in terms of the outgoing waves amplitude.

4. Navier-Stokes equations are not hyperbolic and their viscous form change their system nature. However, their wave propagation is as Euler's wave, making Euler boundary conditions a first-order candidate to treat Navier-Stokes BC. The viscous terms will then be neglected in a first step in order to use Euler's form; consecutively, an additional term will model the viscous effect.

NS Characteristic representation in generalized coordinates

As for all fluid dynamics problems, we start our reasoning from Navier-Stokes (NS) equations in conservative form and Cartesian coordinates:

∂U ∂t + ∂E ∂x + ∂F ∂y + ∂G ∂z = S (3.2.1)
where S denotes the source terms and U is the conservative variables vector:

U = ρ, ρu, ρv, ρw, ρ(e + | - → v | 2 /2)
with ρ density, u, v, w the three velocity components, -→ v the three-dimensional velocity vector, e the internal energy. The conservative fluxes (E, F, G) can be considered as sum of convective

(E c , F c , G c ) and diffusive (E d , F d , G d )
fluxes of U and are given by:

E c =(ρu, ρu 2 + p, ρuv, ρuw, u(ρe + p)) F c =(ρv, ρuv, ρv 2 + p, ρvw, v(ρe + p)) G c =(ρw, ρuw, ρvw, ρw 2 + p, w(ρe + p)) E d =(0, p + τ (1,1) , τ (2,1) , τ (3,1) , uτ (1,1) + vτ (2,1) + wτ (3,1) + C T ∂ x T ) F d =(0, τ (1,1) , p + τ (2,1) , τ (3,1) , uτ (1,2) + vτ (2,2) + wτ (3,2) + C T ∂ y T ) G d =(0, τ (1,1) , τ (2,1) , p + τ (3,1) , uτ (1,3) + vτ (2,3) + wτ (3,3) + C T ∂ z T ) (3.2.2)
with τ i,j viscous stress tensor, C T thermal conductivity and T temperature. Even though the first NSCBC presentation from Poinsot was done in Cartesian coordinates, in the following and without loss of generality, the boundary is considered as a generic iso-surface -→ χ = constant located at a computational domain end. The passage from the Cartesian space -→ x = (x, y, z) to the generalized coordinates -→ χ = (χ, η, ζ) is defined by the Jacobian matrix:

- → χ x =    χ x χ y χ z η x η y η z ζ x ζ y ζ z    = J    y η z ζ -y ζ z η z η x ζ -z ζ x η x η y ζ -x ζ y η y ζ z χ -y χ z ζ z ζ x χ -z χ x ζ x ζ y χ -x χ y ζ y χ z η -y η z χ z χ x η -z η x χ x χ y η -x η y χ    (3.2.3) with: J = [x χ (y η z ζ -y ζ z η ) + x η (y ζ z χ -y χ z ζ ) + x ζ (y χ z η -y η z χ )] (3.2.4)
In this, x χ is the projection of x along the χ direction, χ x the projection of χ along the x direction and so on for all other terms. NS equations in generalized coordinates (χ, η, ζ), with a null source term S = 0 and in conservative form, read: Let us now take as example a general boundary surface, whose normal direction is χ. The convective flux related to the direction χ can be expressed in terms of eigenvector and wave strength:

∂ ∂t U J + ∂ ∂χ E χ x J + F χ y J + G χ z J + ∂ ∂η E η x J + F η y J + G η z J + + ∂ ∂ζ E ζ x J + F ζ y J + G ζ z J = 0 (3.2.
χ x ∂E c ∂χ + χ y ∂F c ∂χ + χ z ∂G c ∂χ = P U L U (3.2.7)
where:

• P U is the transformation matrix from characteristic to conservative variables, or in other words, the right eigenvector of the Jacobian:

χ x ∂E c ∂U + χ y ∂F c ∂U + χ z ∂G c ∂U
• L U is the strength (or amplitude) of the characteristic wave, function of conservative variables U, and is defined by:

L U = λP -1 U ∂U ∂χ (3.2.8)
with λ the characteristic velocities, or the eigenvalues diagonal matrix of:

χ x J ∂E c ∂U + χ y J ∂F c ∂U + χ z J ∂G c ∂U
When λ is positive, the associated waves are going outside the computational domain and can be directly computed from (3.2.8). When λ is negative, waves are entering in the domain and additional boundary conditions are required to be calculated. In the following, similarly to the right boundary of Figure 3.1:

       λ 1 = λ 2 = λ 3 = u n λ 4 = u n + c λ 5 = u n -c (3.2.9)
with c the local speed of sound and u n the face normal velocity, calculated as:

u n = - → v • - → n = u • n x + v • n y + w • n z
with n x , n y , n z the components of the normal vector in Cartesian coordinates, also obtained by:

n x = χ x J , n y = χ y J , n z = χ z J
The characteristic velocities can also be defined as: λ 1 the entropy waves velocity (or the convection velocity), λ 2 and λ 3 the rotational waves velocity (or the advection velocity in the χ direction), λ 4 and λ 5 the acoustic waves velocity propagating in the positive and negative χ directions. Given the equivalence between equations (3.2.6) and (3.2.7), the waves strength can be computed by:

L U = P -1 U χ x ∂E c ∂χ + χ y ∂F c ∂χ + χ z ∂G c ∂χ (3.2.10)
In the same fashion, diffusive flux according to χ and tangent fluxes (comprehensive of a convective and a diffusive part) related to η and ζ directions can be defined, alongside with the characteristic variables W time-variations:

∂W ∂t = P -1 U ∂U ∂t (3.2.11a) D = P -1 U χ x ∂E d ∂χ + χ y ∂F d ∂χ + χ z ∂G d ∂χ (3.2.11b) T = P -1 U η x ∂E ∂η + η y ∂F ∂η + η z ∂G ∂η + ζ x ∂E ∂ζ + ζ y ∂F ∂ζ + ζ z ∂G ∂ζ (3.2.11c)
A compact form of conservation equation (3.2.6) can now be given:

∂W ∂t + L + D + T = 0 (3.2.12)
where L has been taken independent of its formulation (we will later see that a primitive variable formulation is also useful). Tangent (or transverse) fluxes T are evaluated from knowledge of the inner computational domain, while diffusive fluxes D are usually given by a Dirichlet or Neumann condition for calculating the momentum and thermal energy gradients. Importantly, we can here recognise the amplitudes for respectively the entropic wave (∂W 1 ), rotational waves (∂W 2,3 ) and the left-and right-running acoustic waves (∂W 4,5 ). They are defined by:

                         ∂W 1 = n x ∂ρ + n z ∂v -n y ∂w -(n x /c 2 )∂p ∂W 2 = n y ∂ρ -n z ∂u + n x ∂w -(n y /c 2 )∂p ∂W 3 = n z ∂ρ + n y ∂u -n x ∂v -(n z /c 2 )∂p ∂W 4 = (n x ∂u + n y ∂v + n z ∂w) + 1 ρc ∂p ∂W 5 = -(n x ∂u + n y ∂v + n z ∂w) + 1 ρc ∂p (3.2.13)
As further analysis, we can demonstrate that the approximation of L U is of same order of the numerical fluxes. For achieving this, we introduce the notation:

Ê = E χ x J + F χ y J + G χ z J (3.2.14)
From the equivalence of the two equations (3.2.5) and (3.2.7), the wave strength can also be expressed by:

L U = P -1 U J ∂ Ê ∂χ -E ∂ ∂χ χ x J + F ∂ ∂χ χ y J + G ∂ ∂χ χ z J (3.2.15)
With this formulation, a high-order approximation of L U is directly obtained when the flux terms are computed with a high-order scheme. Again, a positive eigenvalue allows to compute directly the characteristic waves from (3.2.8) with the information at the domain interior, while a negative eigenvalue will need extra boundary conditions, as it corresponds to information going into the domain.

NSCBC Strategy

In the majority of cases, there is no simple method to specify the values of L i as incoming waves (i.e. when λ is negative) for NS equations, but it can be done for 1D Euler equations. This is done examining an associated Local One-Dimensional Inviscid (LODI) problem. As the name suggests, viscous and transverse terms are neglected when describing boundaries with conservative equations of the type (3.2.6). This approximation makes the problem non-physical, however it should be seen solely as a method for correlating physical boundary conditions with the amplitude of the waves crossing the boundary. Even though waves carry information about viscosity and transverse terms, this simplification is only applied during the estimation of incoming waves amplitude variations, which is an acceptable hypothesis. Then, complete NS equations will be solved taking into consideration all effects. The LODI system can also be cast under many different forms. The same system could be derived in terms of time derivatives of the quantities of interest, or also in terms of gradients. It is important to remark that different systems can be found in the literature, however this is only a matter of definitions chosen by the different authors expressing the same concept. The NSCBC strategy implemented in elsA differs from the original of [START_REF] Poinsot | Boundary conditions for direct simulations of compressible viscous flows[END_REF]. In fact, the original strategy is valid for a finite-volume cell-vertex solver.

Since elsA is a cell-centered finite volume solver, the following strategy is considered, which can be summarised in three steps:

1. The outgoing characteristic waves L i are fully determined and calculated from knowledge of the inner domain solution and (3.2.8).

2. An inviscid boundary condition is associated to each unknown ingoing wave, and its respective LODI relation is used to compute a first estimation of the unknown L * j (incoming wave), function of the known L i (outgoing waves) and eventual external contributions.

3. Then, with the known L i and the approximated L * j , the remaining L j can be correctly calculated solving the full equation system by use of (3.2.7).

Practically, equation (3.2.5) can be discretized in a cell-centered finite volume approach as follow:

∂ ∂t U J + Êc (i + 1 /2, j, k) -Êc (i -1 /2, j, k) + + Fc (i, j + 1 /2, k) -Fc (i, j -1 /2, k) + + Ĝc (i, j, k + 1 /2) -Ĝc (i, j, k -1 /2) = 0 (3.2.16)
The indexes i, j, k are assumed to belong to the directions χ, η, ζ respectively. If the boundary is at the interface i + 1 /2, the NSCBC will consist in introducing information related to this boundary in the flux balance term in the χ direction ( Ê(i + 1 /2, j, k) -Ê(i -1 /2, j, k)), while the fluxes in the other two directions will be unchanged.

Characteristic waves amplitude

Now that the general governing equations are given, primitive variables are introduced in order to further simplify them. Denoting with Q the primitive variables vector:

Q = (ρ, u, v, w, p)
we look for characteristic waves strength function of primitive variables, L . Correctly speaking, it would be L Q , but for brevity it will be noted just L . In the following analysis, diffusive terms D will be omitted, but the same reasoning stands with only the addition of a viscous relation to the system. With P Q the eigenmatrix of the jacobian ∂E/∂Q, the characteristic wave strengths in primitive variables L and in conservative variables L U are linked by:

D = P U L U and d = P Q L (3.2.17)
being D and d auxiliary variables (not related to diffusive properties) calculated as:

D =          d 1 ud 1 + ρd 2 vd 1 + ρd 3 wd 1 + ρd 4 1 2 (u 2 + v 2 + w 2 )d 1 + ρud 2 + ρvd 3 + ρwd 4 + 1 γ -1 d 5          (3.2.18)
and:

d =        d 1 d 2 d 3 d 4 d 5        =               n x L 1 + n y L 2 + n z L 3 + ρ c √ 2 (L 4 + L 5 ) -n z L 2 + n y L 3 + n x √ 2 (L 4 -L 5 ) n z L 1 + n x L 3 + n y √ 2 (L 4 -L 5 ) -n y L 1 + n x L 2 + n z √ 2 (L 4 -L 5 ) ρc √ 2 (L 4 + L 5 )               (3.2.

19)

The eigenvalue transformation matrix from characteristic to primitive variable is given by:

P -1 Q = ∂W ∂Q = 1 0 0 0 -1/c 2 0 0 0 1 0 0 0 -1 0 0 0 1/ √ 2 0 0 1/(ρc √ 2) 0 -1/ √ 2 0 0 1/(ρc √ 2) (3.2.20)
Substituting in (3.2.1), Navier-Stokes equations can then be expressed in primitive variables as function of the wave strengths:

                                       ∂ρ ∂t + n x L 1 + n y L 2 + n z L 3 + ρ c √ 2 (L 4 + L 5 ) + T ρ = 0 ∂u ∂t -n z L 2 + n y L 3 + n x √ 2 (L 4 -L 5 ) + T u = 0 ∂v ∂t + n z L 1 -n x L 3 + n y √ 2 (L 4 -L 5 ) + T v = 0 ∂w ∂t -n y L 1 + n x L 2 + n z √ 2 (L 4 -L 5 ) + T w = 0 ∂p ∂t + ρc √ 2 (L 4 + L 5 ) + T p = 0 (3.2.21a) (3.2.21b) (3.2.21c) (3.2.21d) (3.2.21e)
where T i represents the contribution of the tangential flux balance in this primitive variables formulation. If the example of a boundary normal to the x direction is taken, (3.2.21) gives us the relation between the time variation of conservative variables and characteristic waves:

                                     ∂ρ ∂t + L 1 + ρ c √ 2 (L 4 + L 5 ) + T ρ = 0 ∂u ∂t + 1 √ 2 (L 4 -L 5 ) + T u = 0 ∂v ∂t -L 3 + T v = 0 ∂w ∂t + L 2 + T w = 0 ∂p ∂t + ρc √ 2 (L 4 + L 5 ) + T p = 0 (3.2.22a) (3.2.22b) (3.2.22c) (3.2.22d) (3.2.22e)
As for (3.2.8), characteristic wave strengths L i can be computed function of the general spatial derivative of the primitive variables Q:

L = λP -1 Q ∂Q ∂χ =        L 1 L 2 L 3 L 4 L 5        =                 ûn n x ∂ρ ∂χ + n z ∂v ∂χ -n y ∂w ∂χ - n x c 2 ∂p ∂χ ûn n y ∂ρ ∂χ -n z ∂u ∂χ + n x ∂w ∂χ - n y c 2 ∂p ∂χ ûn n z ∂ρ ∂χ + n y ∂u ∂χ -n x ∂v ∂χ - n z c 2 ∂p ∂χ (û n + ĉ) 1 √ 2 n x ∂u ∂χ + n y ∂v ∂χ + n z ∂w ∂χ + 1 ρc ∂p ∂χ (û n -ĉ) 1 √ 2 -n x ∂u ∂χ -n y ∂v ∂χ -n z ∂w ∂χ + 1 ρc ∂p ∂χ                 (3.2.23)
where ûn is the velocity normal to the surface χ in generalized coordinates:

ûn = χ x u + χ y v + χ z w (3.2.24)
and ĉ the local speed of sound in generalized coordinates:

ĉ = c χ 2 x + χ 2 y + χ 2 z (3.2.25)
In the case of a boundary normal to the x direction in Cartesian coordinates, as done in (3.2.22), the system (3.2.23) gives us the relation between primitive variables spatial variations and characteristic waves:

L = λP -1 Q ∂Q ∂x =        L 1 L 2 L 3 L 4 L 5        =                 u ∂ρ ∂x - 1 c 2 ∂p ∂x u ∂w ∂x u - ∂v ∂x (u + c) 1 √ 2 ∂u ∂x + 1 ρc ∂p ∂x (u -c) 1 √ 2 - ∂u ∂x + 1 ρc ∂p ∂x                 (3.2.26)
With the last simplified set of equations (3.2.26), a simple physical interpretation of L i can be given. Take, as example, the linearized one-dimensional and inviscid NS equations, so that the T i contributions are neglected, and define the wave amplitude as:

A 4,5 = u ac ± p ac ρ 0 c 0 (3.2.27)
Because of the following property, given by the fact that the time derivative of the average value is null:

∂p ∂t = ¡ ¡ ¡ ∂ p ∂t + ∂p ac ∂t (3.2.28)
and remembering from the fourth and fifth line of (3.2.13) the link between characteristic variable and acoustic waves, we can also write:

∂W 4,5 ∂t = ∂A 4,5 ∂t = ∂ ∂t u ac ± p ac ρ 0 c 0 (3.2.29)
Let us also focus on one wave, say the acoustic wave L 5 . Then, substituting the fifth line of (3.2.13) and the fifth line of (3.2.26) in (3.2.12), with D = T = 0 and aside the constant factors √ 2 needed for considering the effective values, we find the usual property of waves amplitude conservation along the characteristic line given by the eigenvalue:

∂W 5 ∂t + L 5 = ∂u ∂t - 1 ρ 0 c 0 ∂p ∂t + λ 5 ∂u ∂x - 1 ρ 0 c 0 ∂p ∂x = 0 ⇐⇒ ∂A 5 ∂t + λ 5 ∂A 5 ∂x = 0 (3.2.30)
This means that -L i (with i ∈ {4, 5}), represents the time variation of the wave amplitude A i : 2.31) This relation between waves' amplitudes crossing the boundaries and L i is the great advantage of adopting a characteristic formalism for acoustic problems.

∂A 4,5 ∂t = ∂W 4,5 ∂t = -L 4,5 (3.

NSCBC applications in a lined duct configuration

In this section, different types of NSCBC are given. Comments on their implementation in the current CFD solver go along with each type. To the purpose of the CFD simulations of this study, the following boundary conditions are provided: inflow with specification of velocity and temperature, outflow with prescribed pressure, no-slip wall and acoustically treated wall, the latter being the cardinal point of this dissertation.

Subsonic inflow: Tuvw

One typical and necessary boundary condition is the subsonic inflow. In here, the classic NSCBC inflow condition specifying the temperature (T ) and three velocity components (u, v, w) is given. Four characteristic waves are entering the domain (L i , i ∈ {1, 2, 3, 4} in the left boundary of Figure 3.1) and one acoustic wave is leaving the domain (L 5 ). If a boundary normal to x is considered as in (3.2.22), we will have:

                         L 1 = - ρ c √ 2 (L 4 + L 5 ) - ∂ρ ∂t -T ρ from (3.2.22a) L 2 = - ∂w ∂t -T w from (3.2.22d) L 3 = ∂v ∂t -T v from (3.2.22c) L 4 = L 5 - √ 2 ∂u ∂t -T u from (3.2.22b) (3.3.1)
where each velocity derivative in the normal direction is given by:

∂u ∂t = ∂u ∂t T + σ u (u -u ∞ ) + Γ u (û -u ∞ ) ∂v ∂t = ∂v ∂t T + σ u (v -v ∞ ) + Γ u (v -v ∞ ) ∂w ∂t = ∂w ∂t T + σ u (w -w ∞ ) + Γ u ( ŵ -w ∞ ) (3.3.2)
In (3.3.2), the first derivative in T is simply the geometrical flux conservation term along the tangential direction. Concerning the terms in σ and Γ, the following reasoning is made. As introduced in the first NSCBC appearance [START_REF] Poinsot | Boundary conditions for direct simulations of compressible viscous flows[END_REF], perfectly non-reflecting boundary conditions are very interesting but difficult to implement in a domain characterised by different types of boundary conditions and can easily lead to ill-posed problems. NSCBC allow to have a weakly-reflecting boundary condition, leading to small levels of reflection but guaranteeing a well-posedness and numerical stability. Using a strong condition at the inlet or outlet, such as imposing the static thermodynamic variables, leads to relatively strong acoustic wave reflections, reason why a soft condition is needed. This is achieved with a relaxation factor σ, which allows wave reflections in order to stabilize the instantaneous values around the imposed static value, and have a negligible effect if the computed value is already close enough to it. With this said, σ u is the classical relaxation parameter linked to the velocity values and applied to the difference between instantaneous (u) and prescribed (u ∞ ) values. Γ u is the relaxation parameter applied to the difference between time-integrated filtered (û) and prescribed values, as later described by the Giles condition in Section 3.5. In the following of this section, for simplicity, the filtered term will be omitted, but the whole discussion is identically valid for it. Finally, the first equation of (3.3.1) should be discussed. It is function of the density derivative, however, it is more convenient to specify the ingoing flow temperature rather than its density, reason why a rearrangement is done to express it function of the temperature:

L 1 = ρ c √ 2 (γ -1)(L 4 + L 5 ) - ∂T ∂t (3.3.3)
where, similarly to the velocity, the temperature derivative is given by:

∂T ∂t = ∂T ∂t T + ρ T σ T (T -T ∞ ) + Γ T ( T -T ∞ ) (3.3.4)
One could argue the inclusion or not of the tangent fluxes T in (3.3.1). It was not found in the literature any valid reason why they should or should not be taken into account for an inlet condition. In the present development, they proved to play a constructive role in the determination of characteristic waves, hence it was decided to maintain the formulation as in (3.3.1). However, in the specific case of using non-reflecting Giles condition (presented in Section 3.5), they are neglected. Further discussion is found in the relative section. Alternatively, the inflow condition could be specified by giving information on other 4 variables: static pressure, static temperature and the flow direction (two variables). Preliminary tests on this type of inflow condition provided results in agreement with the Tuvw condition just described.

Subsonic outflow: Pressure outlet

In an identical manner as for the inlet, the outlet condition can be derived. In a subsonic outflow condition, only one acoustic wave is entering the domain (L 5 in the right boundary of Figure 3.1), so one condition needs to be specified for finding this external information, and it is usually linked to the pressure value. The characteristic condition will then be:

L 5 = - 1 ρc √ 2 T p - ∂p ∂t (3.3.5)
with:

∂p ∂t = ∂p ∂t T + 1 ρ (σ p (p -p ∞ ) + Γ p (p -p ∞ )) (3.3.6)
When σ p = 0 and Γ p = 0, the outflow condition is perfectly non-reflecting (no pressure is imposed). In addition, the ingoing wave (L 5 ) cannot depend on the outgoing one (L 4 ) or reflections would be generated. Alternatively, may one need to impose a pressure value and consider the outgoing wave, a reflective formulation can be derived from (3.2.22e) and would read:

L 5 = -L 4 - 1 ρc √ 2 T p - ∂p ∂t (3.3.7)

Application to isothermal no-slip wall

One of the most classic boundary condition is the hard (or no-slip) wall. The assumption of isothermal wall is made in the following. Let us consider an orthonormal reference system related to a boundary, whose normal is directed as one coordinate axis. We define with u n the velocity normal to this direction. Tangential vectors (of the orthonormal plane) -→ t 1 and -→ t 2 are defined as follow:

• for |n z | ≤ 0.7 - → t 1 = 1 n 2 x + n 2 y    n y -n x 0    - → t 2 = 1 n 2 x + n 2 y    -n z n x -n z n y n 2 x + n 2 y    (3.3.8) • for |n z | > 0.7 - → t 1 = 1 n 2 y + n 2 z    0 -n z n y    - → t 2 = 1 n 2 y + n 2 z    n 2 y + n 2 z -n x n y -n x n z    (3.3.9)
The number 0.7 was chosen arbitrarily to distinguish two cases and never have a null denominator. In fact, when n z ≈ 1, then n x ≈ n y ≈ 0 and the denominator of (3.3.8) would be null. If we take the three characteristic equations related to the velocity time variation in (3.2.21) and make the scalar product with the normal direction, we obtain:

∂u n ∂t + 1 √ 2 (L 4 -L 5 ) = n x T u + n y T v + n z T w (3.3.10)
In the same way, the scalar product for the tangential directions gives:

• for |n z | ≤ 0.7 ∂ - → v • - → t 1 ∂t + -n z n 2 x + n 2 y (n y L 2 + n x L 1 ) + n 2 x + n 2 y L 3 = n y T u -n x T v n 2 x + n 2 y (3.3.11a) ∂ - → v • - → t 2 ∂t + 1 n 2 x + n 2 y (n x L 2 -n y L 1 ) = -n x n z T u -n y n z T v n 2 x + n 2 y + n 2 x + n 2 y T w (3.3.11b) • for |n z | > 0.7 ∂ - → v • - → t 1 ∂t + n x n 2 y + n 2 z (n z L 3 + n y L 2 ) -n 2 y + n 2 z L 1 = n y T w -n z T v n 2 y + n 2 z (3.3.12a) ∂ - → v • - → t 2 ∂t + 1 n 2 y + n 2 z (n y L 3 -n z L 2 ) = n 2 y + n 2 z T u + -n x n y T v -n x n z T w n 2 y + n 2 z (3.3.12b)
The temperature equation is written for an isothermal wall as:

1 T ∂T ∂t = 1 p ∂p ∂t - 1 ρ ∂ρ ∂t (3.3.13)
which becomes, in terms of characteristic waves:

1 T ∂T ∂t - 1 ρ (n x L 1 + n y L 2 + n z L 3 ) + 1 c √ 2 (γ -1)(L 4 + L 5 ) = 1 p T p - 1 ρ T ρ (3.3.14)
At an isotherm no-slip wall, the condition to impose is null velocity components in all directions and constant temperature. In the case of an adiabatic wall (herein not detailed), the temperature equation would be substituted with a null heat flux equation. Therefore, there are three physical inviscid conditions on the normal and tangential velocities and one isothermal condition on the temperature to impose:

- → v • - → t 1 = 0 - → v • - → t 2 = 0 - → v • - → n = 0 T = T ∞ (3.3.15)
Substituting these conditions into (3.3.10), (3.3.11) and (3.3.14), we obtain the characteristic system for the hard wall condition (for n z ≤ 0.7):

                               1 √ 2 (L 4 -L 5 ) = n x T u + n y T v + n z T w -n z n 2 x + n 2 y (n y L 2 + n x L 1 ) + n 2 x + n 2 y L 3 = n y T u -n x T v n 2 x + n 2 y 1 n 2 x + n 2 y (n x L 2 -n y L 1 ) = -n x n z T u -n y n z T v n 2 x + n 2 y + n 2 x + n 2 y T w - 1 ρ (n x L 1 + n y L 2 + n z L 3 ) + 1 c √ 2 (γ -1)(L 4 + L 5 ) = 1 T σ T (T -T ∞ ) + 1 p T p - 1 ρ T ρ (3.3.16)

Application to acoustically lined wall (TDIBC)

The objective is now to find the link between two characteristic acoustic waves in presence of an acoustically lined wall. We start our derivation from the previous methodology of a hard wall condition. However, since the objective is to model an acoustic liner, we must take into account that the normal velocity is physically not null, as a mass flow and pressure gradient are present on the liner perforations. Such perforations are not directly modelled in the CAD (Computer Aided Design) geometry as they are computationally too expensive for industrial applications. Nevertheless, a behaviour as close as possible to reality is looked for with the TDIBC modelling. Normal velocity not-null translates in a fall of the condition -→ v • -→ n = 0 and the values of L i (with i ∈ {1, 2, 3}) will be computed with use of the other conditions on tangential velocities and temperature. The latter are numerically discretized under a relaxed-form:

∂ - → v • - → t 1 ∂t + σ v ( - → v • - → t 1 ) = 0 (3.3.17a) ∂ - → v • - → t 2 ∂t + σ v ( - → v • - → t 2 ) = 0 (3.3.17b)
which, for |n z | ≤ 0.7, expand into:

-n z n 2 x + n 2 y (n y L 2 + n x L 1 ) + n 2 x + n 2 y L 3 = σ v ( - → v • - → t 1 ) + n y T u -n x T v n 2 x + n 2 y (3.3.18a) n x L 2 -n y L 1 n 2 x + n 2 y = σ v ( - → v • - → t 2 ) + -n x n z T u -n y n z T v n 2 x + n 2 y + n 2 x + n 2 y T w (3.3.18b)
A similar derivation can be made for the case |n z | > 0.7. Similarly to the velocity, a temperature relaxation parameter σ T will be specified and used in a relaxed temperature equation for specifying the isothermal wall condition:

- 1 ρ (n x L 1 + n y L 2 + n z L 3 ) + 1 c √ 2 (γ -1)(L 4 + L 5 ) = 1 T σ T (T -T ∞ ) + 1 p T p - 1 ρ T ρ (3.3.19)
Moving on to the characteristic acoustic waves, we start by reminding from (3.2.29) and (3.2.30) the link between the time variation of waves amplitude and characteristic variables:

∂A 4,5 ∂t = ∂ ∂t p ac ρ 0 c 0 ± u ac = ∂W 4,5 ∂t
We remember also the time-domain reflection coefficient formulation (2.4.29) linking acoustic waves:

u ac - p ac ρ 0 c 0 (t) = β u ac + p ac ρ 0 c 0 (t) = β ∞ u ac + p ac ρ 0 c 0 (t)+ + N ξ k=1 (µ 1,k ϕ(t, ξ k ) + µ 2,k ψ(t, ξ k , L τ )) + Ns n=1 r 1,n (ϕ(t, -s n ) + r 2,n ψ(t, -s n , L τ )) (3.3.20)
We can re-write the latter reflection coefficient equation by its time derivative:

∂ ∂t u ac - p ac ρ 0 c 0 = β ∞ ∂ ∂t u ac + p ac ρ 0 c 0 + N ξ k=1 µ 1,k ϕ k + µ 2,k ψ k + + Ns n=1 r 1,n ϕ n + r 2,n ψ n (3.3.21)
where the notation ϕ = ∂ϕ/∂t is introduced and ψ is its delayed counterpart. As direct consequence, we can consider (3.3.21) in terms of characteristic variables:

∂W 5 ∂t = β ∂W 4 ∂t ⇐⇒ ∂W 5 ∂t = β ∞ ∂W 4 ∂t + N ξ k=1 µ 1,k ϕ k + µ 2,k ψ k + Ns n=1 r 1,n ϕ n + r 2,n ψ n (3.3.22)
or also, for coming back to the present NSCBC framework and by using (3.2.12) in an inviscid case, in terms of characteristic waves:

L 5 = β ∞ (L 4 + T 4 ) -T 5 + N ξ k=1 µ 1,k ϕ k + µ 2,k ψ k + Ns n=1 r 1,n ϕ n + r 2,n ψ n (3.3.23)
where the tangential terms are remembered to be:

T 4 = - 1 √ 2 (n x T u + n y T v + n z T w ) - 1 ρc T p T 5 = 1 √ 2 (n x T u + n y T v + n z T w ) - 1 ρc T p (3.3.24)
The new ϕ variable is solution of the same ODE problem of (2.4.10), with the difference that the time derivative of the acoustic normal velocity is substituted with the characteristic flux balance:

∂ ϕ k,n ∂t = -ξ ϕ k,n + (L 4 + T 4 ) (3.3.25)
Being L 4 known from the inner computational domain, the three L i (with i ∈ {1, 2, 3}) and the fifth incoming acoustic wave L 5 are fully determined with (3.3.18), (3.3.19) and (3.3.23):

                                   -n z n 2 x + n 2 y (n y L 2 + n x L 1 ) + n 2 x + n 2 y L 3 = σ v ( - → v • - → t 1 ) + n y T u -n x T v n 2 x + n 2 y n x L 2 -n y L 1 n 2 x + n 2 y = σ v ( - → v • - → t 2 ) + -n x n z T u -n y n z T v n 2 x + n 2 y + n 2 x + n 2 y T w - 1 ρ (n x L 1 + n y L 2 + n z L 3 ) + 1 c √ 2 (γ -1)(L 4 + L 5 ) = 1 T σ T (T -T ∞ ) + 1 p T p - 1 ρ T ρ L 5 = β ∞ (L 4 + T 4 ) -T 5 + N ξ k=1 µ 1,k ϕ k + µ 2,k ψ k + Ns n=1 r 1,n ϕ n + r 2,n ψ n (3.
3.26) Preliminary computations revealed how the inclusion of tangential terms is essential for a correct solution when the acoustic perturbation is grazing the soft wall.

It remains to describe how the delayed auxiliary variable ψ in (3.3.23) is calculated. This is done within the delay problem previously introduced in Section 2.4.3, equation (2.4.26). To not lose in generality, all the auxiliary variables in the following (ϕ and ψ) are considered for a generic diffusive or oscillatory pole and without the superscript ∼ for the sake of conciseness. However, they still must be intended as their time derivative part. As introduced in Section 2.4.3, in order to establish the effect of the back-and-forth traveling of the acoustic wave inside a cavity (as in an acoustic liner), a delayed variable ψ is introduced, function of the auxiliary variable ϕ at the cavity entrance. As proposed by [START_REF] Monteghetti | Analysis and Discretization of Time-Domain Impedance Boundary Conditions in Aeroacoustics[END_REF], this equals to solve an advection problem in a 1D space dimension (of coordinate l ∈ [0, L τ ]) propagating at the speed of sound c 0 :

ψ(t, 0) = ϕ(t) ψ(t, Lτ ) = ϕ(t -τ )
The last value of such advection will be allocated to ψ. The number of elements N ψ in the 1D array is chosen dependent on the maximum frequency f max to resolve (at least 2 elements for satisfying Nyquist criterion on the cut-off frequency) or the resolution wanted in terms of points-per-wavelength N λ at a specific frequency:

N ψ = N λ f max τ (3.3.27)
N λ and f max are inputs, so the number of computational cells N ψ forming the advection domain is known a priori. As a graphical explanation, N ψ is the number of crosses in Figure 3 The ODR satisfies this condition, thanks to the exponential function for discretizing the delay term, which demands a limited number of poles for discretizing the impedance model. Now, the numerical choices for this advection problem will be presented. A 2nd-order upwind scheme in time is used, in congruence with the global time marching scheme. A 2nd-order upwind scheme in space and a 5th-order polynomial Spectral Difference (SD) scheme in space were both tested. The upwind scheme was the first and easiest solution, but the SD scheme provided faster results with similar accuracy, as later demonstrated by their comparison in Section 4.4. Thus, some generalities on the SD scheme and its implicitation are now given. Let us consider correspond to solution points, the blue triangles to flux points. In a grid, solution points coincide to the cells center and flux points to the cells interface. The state vector U is known at the solution points, and a p-th order polynomial is built (1 st step). Then, an extrapolation is done to reconstruct the state vector at the flux points (2 nd step) and fluxes are computed at flux points (3 rd step). For a generic cell element K, the value in the flux point i is given by:

ψ f p i = Nsp sp=1 ψ sp B(sp + i(p + 1)) (3.3.28)
with B the interpolation matrix between solution points and flux points (used in the 2 nd step).

A Riemann problem needs to be solved between at each interface to provide an element-toelement communication and to ensure flux continuity across the computational domain (4 th step). Assuming a positive advection velocity equal to the local speed of sound, each first flux point of an element K will simply correspond to the value of the last flux point of the element K -1. If K = 1, the first value is given by initial conditions from (2.4.26). The value of ϕ is known since it is computed before the delay ψ integration. A p + 2 polynomial of these flux values is then build (5 th step). Finally, the flux derivative, that will resolve the spatial derivative of the advection problem, can be calculated as (6 th step):

∂ψ ∂i sp = N f p i=1 ψ f p i A d (i + (sp -1)(p + 2)) (3.3.29)
where A d is interpolation derivative matrix between solution points and flux points. Later on, Chapter 4 focuses on how the latter is implemented in the numerical scheme under an implicit formalism.

Application to nonlinear TDIBC

As a complement of the numerical choices given until now, a proposition of numerical scheme for the nonlinear impedance model presented in Section 2.5 is now given. The system (2.5.16) Figure 3.3: Spectral Difference algorithm for a 2nd order polynomial in 1D -source from [START_REF] Fiévet | Numerical Study of Hypersonic Boundary-Layer Transition Delay through Second-Mode Absorption[END_REF].

is reminded as a starting point:

                 u ac - p ac ρ 0 c 0 = u ac + p ac ρ 0 c 0 -2w u ac + p ac ρ 0 c 0 = (Z lin w) + ρ 0 C nl w|w| + w dϕ k dt = -ξ k ϕ k + w
It is convenient to take the time derivative of the whole system, as in the first equation the characteristic acoustic waves will appear (cf. (3.3.23)), yielding to:

               L 5 = L 4 -2 ∂w ∂t L 4 = ∂(Z lin w) ∂t + 2ρ 0 C nl ∂w ∂t |w| + ∂w ∂t d ϕ k dt = -ξ k ϕ k + ∂w ∂t (3.3.30)
where the unknowns are the ingoing acoustic wave L 5 , the auxiliary variables ϕ (time derivative of ϕ) and the nonlinear auxiliary variable w. Remembering that Z lin is a Linear Time Invariant (LTI) operator, system (3.3.30) can be further simplified into:

                 L 5 = L 4 -2 ∂w ∂t L 4 = Z lin ∂w ∂t + 2ρ 0 C nl ∂w ∂t |w| + ∂w ∂t d ϕ k dt = -ξ k ϕ k + ∂w ∂t (3.3.31)
Even though in this way the ODR is applied under an impedance formulation, the global scheme is still expressed in the form of a reflection coefficient, linking the two characteristic acoustic waves. For this reason, it is believed that such implementation would be efficient in a NSCBC framework. The major attention should be posed in obtaining a proper discretization (i.e. numerical implementation) of the second equation, nonlinear in w. This should be done in a way that it does not penalize the global numerical stability or requirements in terms of timestep size.

Acoustic perturbation injection

The elsA code features a novel routine structure capable of introducing an acoustic perturbation (or acoustic source) to any NSCBC under the form of an additional fluctuation function f of the pressure p ac . Fluctuations and their time derivatives are computed in terms of primitive variables (ρ, u, v, w, p) and then added in the characteristics wave equations system (3.2.12) as:

∂W ∂t + L + T + ∂p ac ∂t = 0 (3.4.1)
Throughout the dissertation, three types of injections revealed useful and will now be presented: pressure plane wave, Bessel function and theoretical N-wave.

Pressure plane wave

A constant plane wave with origin on the boundary and propagating in its normal direction can be injected by simple mean of:

p ac = A sin(ωt + φ 0 ) (3.4.2)
∂p ac ∂t = Aω cos(ωt + φ 0 ) (3.4.3)
with ω the angular frequency, A the amplitude and φ 0 a constant phase angle. The fluctuation for each primitive variable is:

           f ρ = p ac c 2 f u,v,w = p ac ρc f p = p ac (3.4.4)                  ∂f ρ ∂t = ∂p ac ∂t 1 c 2 ∂f u,v,w ∂t = ∂p ac ∂t 1 ρc ∂f p ∂t = ∂p ac ∂t (3.4.5)
If we now identify with L f the fluctuation added to the characteristic equations (3.4.1), and we take as example a NSCBC inflow condition, fluctuations on the velocities and temperature equations will be added to the system (3.3.1) as:

                             L f,1 = n x f ρ + n z f v -n y f w + n x f p c 2 L f,2 = n y f ρ -n z f u + n x f w + n y f p c 2 L f,3 = n z f ρ + n y f u -n x f v + n z f p c 2 L f,4 = ± f p ρc + √ 2(n x f u + n y f v + n z f w ) (3.4.6)
where f denotes ∂f /∂t and the sign ± depends on the wave direction. For a pressure outlet condition, it will simply be:

L f,5 = ± f p ρc (3.4.7)

Bessel function

In an annular or circular duct, the acoustic pressure can be expressed as a sum of modal components whose amplitude is herein called A mn , (m,n) being the fundamental azimuthal and radial orders, respectively. Section 2.1.2 gave a broad overview of the theory behind acoustic modes.

Once the thermodynamic environment is known and the instantaneous pressure measured with microphones or numerical outputs, matrix inversion techniques can decompose the signal and provide modes amplitudes, and so their intensities and attenuation can be estimated. The circular case can be seen as a simplified version of the annular case (with an internal radius null), so the set of equations is given fully in annular coordinates and can later be simplified for the circular case. The frequency-domain modal amplitude, function of Mach number M , constant phase angle φ 0 and thermodynamic parameters (ρ, T , ...), is reported from (2.1.16) in cylindrical coordinates:

p(r, θ, z, ω) = ∞ m=-∞ ∞ n=0 A + mn (ω)e -ik + z,mn z + A - mn (ω)e -ik - z,mn z Ψ mn (r, θ)
The time-domain equation implemented in the perturbation injection is for a normalized pressure perturbation of unitary modal amplitude A ± mn = 1 (it will be scaled later to its true value):

p ± mn (z, r, φ, t) = (B mn J mn (k r,mn r) + C mn Y mn (k r,mn r)) e iωt+imφ 0 -ik ± z,mn z (3.4.8)
with ± defining if the propagation is on the positive/negative axis, J m and Y m the Bessel functions of first and second species, J m and Y m their derivatives,

B mn = cos(Θ mn ), C mn = sin(Θ mn ), Θ mn = arctan - J m (k r,mn R ext ) Y m (k r,mn R ext ) (3.4.9)
with k r,mn radial wavenumber (real positive), solution of:

J m (k r,mn R int ) × Y m (k r,mn R ext ) -Y m (k r,mn R int ) × J m (k r,mn R ext ) = 0 (3.4.10)
with R ext (R int ) the external (internal) annular radius, k z,mn axial wavenumber, real for propagative modes, imaginary for cut-off (evanescent) modes:

k ± z,mn = k 1 -M 2   -M ± 1 -(1 -M 2 ) k r,mn k 2   (3.4.11)
with k the global wavenumber as in (2.1.22). The minimum k z,mn for propagating modes is given by the cut-off frequency defined in (2.1.24). A velocity potential can be calculated as:

Φ = -i k -M k ± z,mn p (3.4.12)
and the velocity field as:

- → v = ∇Φ (3.4.13)
from which the three cylindrical components are given by:

v r = ∂Φ ∂r = -ik r,mn ρ 0 c 0 (k -M k ± z,mn ) B mn J mn (k r,mn r) + C mn Y mn (k r,mn r) e iωt+imφ-ik ± z,mn z (3.4.14a) v φ = 1 r ∂Φ ∂φ = im r Φ = m ρ 0 c 0 r(k -M k ± z,mn ) p ± mn (3.4.14b) v z = ∂Φ ∂z = ik ± z,mn Φ = k ± z,mn ρ 0 c 0 (k -M k ± z,mn ) p ± mn (3.4.14c)
To come back into Cartesian coordinates and in real space, the following transformation is done:

v x = Re r cos(φ)v r -r sin(φ)v φ r , φ = arctan(x, y) (3.4.15a) v y = Re r sin(φ)v r + r cos(φ)v φ r (3.4.15b) v z = Re(v z ) (3.4.15c)
The time derivatives which express the unitary fluctuations are:

∂v x ∂t = Re (iωv x ) , ∂v y ∂t = Re (iωv y ) , ∂v z ∂t = Re (iωv z ) (3.4.16
)

∂p ∂t = Re (iωp) (3.4.17)
These derivatives, representing the unitary fluctuation, will finally be multiplied by the pressure amplitude A of the perturbation to inject.

N-waves

The interest of this paragraph is to develop a boundary condition capable to inject a theoretical N-wave, so a discontinuous linearly increasing (or decreasing) signal as presented in Section 2.6 and illustrated in Figure 3.4. The analytical expression of a sawtooth wave in time-domain is a discontinuous function (limited to a single full oscillation) as:

               p(t) = ∆p t T /2 , if 0 < t < T /2 p(t) = ∆p t -T T /2 , if T /2 < t < T p(t) = 0, if t = T /2 (phase inversion) (3.4.18)
where ∆p is the N-wave amplitude (similar to A, here chosen for congruence with works in the literature) and T its period. This function showed to require around 400 spatial points (per period) to achieve a grid convergence resolution, instead of the classical 30-40 points per wavelength of a plane wave [START_REF] Thisse | Prévision du bruit d'onde de choc d'un turboréacteur en régime transsonique par des méthodes analytiques et numériques[END_REF]. The difficulty lies in the resolution of the discontinuous jump, which requires a much finer grid and a Riemann solver. For alignment to the present NSCBC framework, an analytical formulation of the time derivative is required for adding the source term to the characteristic equations system. When derived in time, system (3.4.18) gives a Dirac impulse δ in correspondence of the discontinuity:

       dp dt = ± 2∆p T , if 0 < t < T and t = T /2 dp dt = ∓δ, if t = T /2 (3.4.19)
where the sign ± is related to the N-wave shape (linearly increasing or decreasing between each discontinuity jump). A continuous function approximating the system in (3.4.18) must be found to avoid the Dirac treatment, in absence of a Riemann solver for the boundary condition herein adopted. The "Ideally Bandlimited Sawtooth Oscillator" is the first approach (also called "additive synthesis"), derived from Fourier theory of signal decomposition. This approach produces only the harmonics below the Nyquist limit, the number of which is given by K = F s /(2f 0 ), with F s the discretization samples and f 0 the fundamental frequency. Its waveform is then given as a sum of sines functions as:

               p(t) sin = ∆p π Fs k=1 (-1) k k sin(2πkf 0 t + φ 0 ) dp dt (t) sin = 2∆pf 0 Fs k=1 (-1) k cos(2πkf 0 t + φ 0 ) (3.4.20a) (3.4.20b)
with φ 0 an eventual initial phase angle. In (3.4.20), the different k-terms from the Fourier series decomposition form the continuous-time sawtooth waveform. A similar alternative is to consider a serial decomposition in cosine terms:

               p(t) FFT = Fs k=2 |p(k)| cos (2π(k -1)f 0 t + arg (p(k))) dp dt (t) FFT = -2πf 0 Fs k=2 (k -1)|p(k)| sin (2π(k -1)f 0 t + arg (p(k))) (3.4.21a) (3.4.21b)
With this expression, amplitude and phase of the Fourier transform appear explicit and function of the complex pressure p(ω), the Discrete Fourier Transform (DFT) of the analytical pressure signal (3.4.18). Even though a successful implementation of these equations was achieved in elsA code, this did not find a direct application in the present dissertation. In fact, in Chapter 7, a previously computed flow field with shock waves is directly imposed at the boundary and not obtained analytically. Thus, a thorough stability analysis on the N-waves injection was not carried out. The latter would be essential, since the strong discontinuity periodically injected creates strong fluctuations in the derivative and average values throughout the calculation, which can easily yield to instabilities.

Giles non-reflecting boundary condition

When solving numerical unsteady, hyperbolic, PDE on infinite domains, it is necessary to perform the calculation on a truncated finite domain. This raises the problem of choosing appropriate boundary conditions for this far-field boundaries, in a way that they don't introduce any non-physical information into the numerical domain. In external aerodynamics studies, such as the flow around an airfoil, the problem is easily solved by applying a grid stretching, i.e. increasing the grid cells size towards the boundary so that any fluctuation or instability with a high enough frequency is cut-off by the coarse spatial resolution of the mesh. However, when dealing with "enclosed" problems such as turbomachinery and duct flows, grid stretching may not be always a suitable solution and non-reflectivity must be guaranteed by the boundary condition formulation. Particularly, non-reflecting boundary condition are of paramount importance when dealing with acoustics problems. In order to ensure acoustic waves propagation, a fine enough grid is necessary, however providing a favorable environment for instabilities and parasitic reflections to propagate and amplify in the enclosed domain. In addition, since acoustics fluctuations are usually of a small order compared to the flow state variables, any spurious fluctuation introduced by the boundary condition may significantly alter the acoustic content. This problematic was recently tackled by Zaabar in [START_REF] Zaabar | A Non-reflective Boundary Condition for Prediction of Acoustic Tones in Turbomachinery using Computational Fluid Dynamics[END_REF] for predicting acoustic tones in turbomachinery with CFD.

In this framework, a thorough study on non-reflective boundary conditions was carried out by Giles [START_REF] Giles | Nonreflecting boundary conditions for Euler equation calculations[END_REF]. His non-reflecting condition consists of a two-dimensional perturbation wave of the kind: Q(x, y, t) = qe i(kx+ly-ωt) (3.5.1) solution of the hyperbolic PDE:

∂Q ∂t + A ∂Q ∂x + B ∂Q ∂y = 0 (3.5.2)
where, if Q is the primitive variables vector, A and B are 5 × 5 matrices of constants. If (3.5.1) is substituted into (3.5.2), we obtain the nonlinear dispersion relation:

det(-ωI + kA + lB) = 0 - → k = f (l, ω) (3.5.3)
with k and l the wavenumber on the two directions of the two-dimensional wave. By division for ω, the latter can be written as:

det(-I + k ω A + l ω B) = 0 (3.5.4)
which allows to distinguish some particular cases. When ω -→ 0, the boundary condition falls in the category of exact, two-dimensional condition for steady problems (cf. [START_REF] Giles | Nonreflecting boundary conditions for Euler equation calculations[END_REF], §3.6). When ω -→ ∞, we obtain an approximate, two-dimensional condition for unsteady problem. In the NSCBC context previously detailed, an approximate unsteady Giles type of non-reflecting boundary condition was recently implemented in elsA by Hugues Deniau. In this Ph.D. work, we extensively relied on this functionality. The following sections explain the equations on which this functionality is based.

Non-reflecting characteristic conditions

We already saw that for NSCBC certain primitive variables (i.e. pressure for an outlet condition) are imposed with a relaxation factor for calculating ingoing waves, while outgoing waves can be directly computed with the information from the inner domain. Giles non-reflecting conditions will be expressed here function of the characteristic variables W and in a relaxed form. In order to simplify the formulation, we consider a boundary in a Cartesian reference system normal to the axis x, while y and z will correspond to the boundary tangent directions. In this reference system, characteristic equations are written as:

                                     ∂W 1 ∂t + u ∂W 1 ∂x + v ∂W 1 ∂y + w ∂W 1 ∂z = 0 ∂W 2 ∂t + u ∂W 2 ∂x + v ∂W 2 ∂y + w ∂W 2 ∂z = - 1 ρ ∂p ∂z ∂W 3 ∂t + u ∂W 3 ∂x + v ∂W 3 ∂y + w ∂W 3 ∂z = 1 ρ ∂p ∂y ∂W 4 ∂t + (u + c) ∂W 4 ∂x + v ∂W 4 ∂y + w ∂W 4 ∂z = - c √ 2 ∂W 2 ∂z - ∂W 3 ∂y ∂W 5 ∂t + (u -c) ∂W 5 ∂x + v ∂W 5 ∂y + w ∂W 5 ∂z = - c √ 2 ∂W 2 ∂z - ∂W 3 ∂y (3.5.5)
These equations are derived directly from Euler equations, and the characteristic variables W i are calculated from the primitive variables Q = (ρ, u, v, w, p) T with the relation:

∂W = P -1 Q ∂Q (3.5.6)
with P -1 Q as in (3.2.20). Characteristic waves amplitudes are reminded to be noted with L i and given by:

L i = u ∂W i ∂x for i ∈ {1, 2, 3} L 4 = (u + c) ∂W 4 ∂x L 5 = (u -c) ∂W 5 ∂x (3.5.7)
The objective of this section is not to expose in detail the Giles formulation, for which the reader is pointed to the original work [START_REF] Giles | Nonreflecting boundary conditions for Euler equation calculations[END_REF], but to present the results recently obtained by Medida [START_REF] Medida | Curvilinear Extension to the Giles Non-reflecting Boundary Conditions for Wall-bounded Flows[END_REF] and the advantages in choosing this modified formulation. Giles boundary condition approximate non-reflection conditions for Linearized Euler Equations (LEE). Medida, as previously Giles, introduces perturbations in relation to the mean flow state, which by definition is never clearly defined for an unsteady condition. If LEE are solved, this perturbation can be supposed to be the mean flow. These fluctuations are noted Q = (ρ , u , v , w , p ) (note that the previous p ac and p stands for the same physical property, the latter is here chosen for a lighter notation). He also defines some characteristic variables C i as (cf. [START_REF] Medida | Curvilinear Extension to the Giles Non-reflecting Boundary Conditions for Wall-bounded Flows[END_REF], eq. (5.78)):

       C 1 C 2 C 3 C 4 C 5        =        -c 2 0 0 0 1 0 0 ρc 0 0 0 0 0 ρc 0 0 ρc 0 0 1 0 -ρc 0 0 1               ρ u v w p        (3.5.8)
Properties noted with an overbar φ represents the variable φ in the mean flow. Then, Medida presents non-reflecting conditions for an inlet and outlet frontiers, with attention to the corner points treatment (i.e. those elements sharing an inlet/outlet and hard wall conditions). As given in Section 3.3.1, in an inlet condition, 4 relations must be specified, which take the form of:

∂ ∂t      C 1 C 2 C 3 C 4      =      0 0 0 0 0 0 v 0 1 2 (c + u n ) 1 2 (c -u n ) 0 0 v 0 0 0 -u n 0 v 0      ∂ ∂y        C 1 C 2 C 3 C 4 C 5        + +      0 0 0 0 0 0 w 0 0 0 0 0 w 1 2 (c + u n ) 1 2 (c -u n ) 0 0 -u n w 0      ∂ ∂z        C 1 C 2 C 3 C 4 C 5        (3.5.9)
Medida studied the well-posedness of these boundary conditions and he showed that, for certain frequencies, system (3.5.9) yields to an ill-posed problem. He then proposed a correction to Giles's formulation, adding some terms to the generalized curvilinear formulation, and trans-formed (3.5.9) in:

∂ ∂t      C 1 C 2 C 3 C 4      =      0 0 0 0 0 0 v 0 1 2 (c + u n ) 1 2 (c -u n ) 0 0 v 0 0 0 1 2 (c -u n ) 0 v 0      ∂ ∂y        C 1 C 2 C 3 C 4 C 5        + +      0 0 0 0 0 0 w 0 0 0 0 0 w 1 2 (c + u n ) 1 2 (c -u n ) 0 0 1 2 (c -u n ) w 0      ∂ ∂z        C 1 C 2 C 3 C 4 C 5        (3.5.10)
For the outlet condition, only one expression must be specified, and it reads:

∂C 5 ∂t = 0 u n 0 0 v ∂ ∂y        C 1 C 2 C 3 C 4 C 5        + 0 0 u n 0 w ∂ ∂z        C 1 C 2 C 3 C 4 C 5        (3.5.11)
When studying the theoretical reflection coefficient at the boundary, he realized it could be reduced by modifying the previous condition into:

∂C 5 ∂t = 0 1 2 (u n + c) 0 0 v ∂ ∂y        C 1 C 2 C 3 C 4 C 5        + 0 0 1 2 (u n + c) 0 w ∂ ∂z        C 1 C 2 C 3 C 4 C 5        (3.5.12)
The next step consists of establishing a relation between W i and the C i as given by Medida. It is easily deducted that:

                   C 1 = -c 2 ∂W 1 C 2 = -ρc∂W 3 C 3 = ρc∂W 2 C 4 = ρc √ 2∂W 4 C 5 = ρc √ 2∂W 5
(3.5.13)

Derivation of Giles/Medida conditions

Taking in consideration the previous relations, Giles conditions can be written under the following 2nd-order formulation, compatible with the characteristic formalism:

                                     ∂W 1 ∂t = 0 ∂W 2 ∂t + v ∂W 2 ∂y + w ∂W 2 ∂z + 1 √ 2 (c + u n ) ∂W 4 ∂z + 1 √ 2 (c -u n ) ∂W 5 ∂z = 0 ∂W 3 ∂t + v ∂W 3 ∂y + w ∂W 3 ∂z - 1 √ 2 (c + u n ) ∂W 4 ∂y - 1 √ 2 (c -u n ) ∂W 5 ∂y = 0 ∂W 4 ∂t + v ∂W 4 ∂y + w ∂W 4 ∂z + 1 2 √ 2 (c -u n ) ∂W 2 ∂z - 1 2 √ 2 (c -u n ) ∂W 3 ∂y = 0 ∂W 5 ∂t + v ∂W 5 ∂y + w ∂W 5 ∂z + 1 2 √ 2 (c + u n ) ∂W 2 ∂z - 1 2 √ 2 (c + u n ) ∂W 3 ∂y = 0 (3.5.14)
The comparison between exact characteristic (3.5.5) and non-reflective (3.5.14) equations would allow the computation of ingoing waves intensities. However, characteristics spatial derivatives ∂W i are now dependent on mean properties φ in the non-reflecting conditions (3.5.14) and on instantaneous properties in the exact relations (3.5.5). Linearizing equations (3.5.5) yields to replacing spatial derivative coefficients with their mean value, allowing the identification of two equations. For an outlet condition, for which only the value of L 5 is given by the boundary condition, we obtain:

L 5 = - 1 2 √ 2 (u n -c) ∂W 3 ∂y + 1 2 √ 2 (u n -c) ∂W 2 ∂z (3.5.15)
For an inlet condition, 4 values of L i must be given by the boundary condition:

                           L 1 = -v ∂W 1 ∂y -w ∂W 1 ∂z L 2 = - 1 ρ ∂p ∂z + 1 √ 2 (c -u n ) ∂W 5 ∂z + 1 √ 2 (c + u n ) ∂W 4 ∂z L 3 = 1 ρ ∂p ∂y - 1 √ 2 (c -u n ) ∂W 5 ∂y - 1 √ 2 (c + u n ) ∂W 4 ∂y L 4 = 1 √ 2 (c + u n ) ∂W 3 ∂y - 1 √ 2 (c + u n ) ∂W 2 ∂z
(3.5.16)

Hixon decomposition for Giles conditions

As highlighted in the previous discussion, Giles conditions do not allow to maintain a prescribed value, what could yield to a divergence, but, at the same time, prescribing a value would bring to a reflective condition. This problem was tackled by Hixon in [START_REF] Hixon | Deterministic period mean flow boundary condition for unsteady flow predictions[END_REF][START_REF] Hixon | Mean flow boundary conditions for computational aeroacoustics[END_REF], where he proposed a solution to this issue. The principle lies in decomposing the different phenomena in function of their frequency. He considers that a relaxation towards a prescribed value will not be reflective, if it concerns only those low frequencies (low in relation to the acoustic waves frequencies) that will be cut-off by the numerical resolution. A decomposition of the time derivative is hence done at the Boundary Condition (BC) under the form:

∂Q ∂t BC = ∂Q ∂t Nref + ∂Q ∂t fix + ∂Q ∂t mean (3.5.17)
where ∂Q ∂t Nref represents the contribution of a Giles-type non-reflecting condition, the term ∂Q ∂t mean is the contribution of prescribing a value and ∂Q ∂t fix represents the perturbation injection throughout this frontier. He also suggested a formulation for velocity perturbation injection without generating parasitic acoustic waves named VGBC (Vortical Gust Boundary Condition). This comes useful when the injection does not present an analytic form. In the case of a duct mode injection, as seen in Section 3.4.2, all primitive variables are given analytically and the VGBC method is not needed. For this reason, it is not detailed in this section but reported in Appendix D.1 for completeness.

Determination of ∂Q ∂t mean

The first step consists of defining a time-averaged value of a quantity φ. For this purpose, we apply a causal filter to φ, so to obtain the filtered value:

φ = 1 ∆ t -∞ exp τ -t ∆ φ(τ )dτ (3.5.18)
where ∆ is the filter characteristic time and is given in input. Nonetheless, computing φ from (3.5.18) is not straightforward, reason why its derivative is considered so that φ is given by an ODE:

∂ φ ∂t + 1 ∆ φ = 1 ∆ φ(t) (3.5.19)
In a recent article, Hixon [START_REF] Hixon | Deterministic period mean flow boundary condition for unsteady flow predictions[END_REF] advances two definitions of the mean operator:

• φ = 1 t t 0
φdτ , which corresponds to a long-time average,

• φ = 1 ∆ T +∆
T φdτ , which the author calls a short-time average.

He also suggests to take ∆ of the same order of the acoustic wave period. T is the starting time for computing the short-time average. The causal filter defined by (3.5.18) belongs to the second category, because the exponential reduces the integration range which is of the order of ∆. Hixon also precised that using a short-time average allows a faster stabilisation of the mean value on the prescription. A simple test using (3.5. [START_REF] Dragna | A generalized recursive convolution method for time-domain propagation in porous media[END_REF]) with an acoustic wave in a 2D duct confirmed these developments. The outlet condition of Section 3.3.2 was first set to a classical relaxation function towards a prescription, and then to a Hixon-type function with time-filtered value:

1. σ(p -p cons ), classical relaxation, 2. Γ(p -p cons ), Hixon filtered relaxation.

In the first case, strong reflections were observed in the outlet, while the second formulation did not show any reflection. It was also verified that reflected waves intensity becomes greater if ∆ becomes smaller that the wave's period. This quick test confirmed that the filter operator φ is a good candidate for calculating {∂Q/∂t} mean . Finally, Hixon proposed a slight modification in [START_REF] Hixon | Deterministic period mean flow boundary condition for unsteady flow predictions[END_REF][START_REF] Hixon | Mean flow boundary conditions for computational aeroacoustics[END_REF]. The relaxation is now applied to the filtered value only if the differences (p -p fix ) and (p -p fix ) between filtered and instantaneous pressure, respectively, have the same sign, for avoiding too strong fluctuations when around the prescribed value. With this latter modification, the relaxation on the time-averaged value writes:

Γ p (p -p fix ) max [0, sign ((p -p fix )(p -p fix ))] (3.5.20)
This filtering technique was also verified for an inlet condition, where velocity and temperature are the prescribed values. Practically, in the case of a simulation restart, this boundary condition will require a read/write step for keeping track of the filtered values throughout the different simulations.

For resuming, in the present work, the term {∂Q/∂t} mean is computed with NSCBC on the mean flow defined by the time-filtered values (as in (3.3.2), (3.3.4), ...).

Determination of ∂Q ∂t Nref

We look for an expression of the time variation of the primitive state vector Q for ensuring the non-reflecting character of the boundary condition. This contribution derives from the previous Giles conditions, but this time the coefficients will be calculated with the filtered variables just described. We then substitute φ = φ in equations (3.5.15) and (3.5.16). The essential difficulty consists in identifying Giles relations (3.5.14) and characteristic relations (3.5.5), and then the characteristic variables determination. In Giles conditions, characteristic variables are defined by (3.5.8) and the fluctuation state vector is defined by Q = Q -Q. With knowledge of the expression of Q, it is possible to compute the different terms of Giles equations (i.e. the spatial derivatives). If we want to linearize (3.5.5), a fluctuation of characteristic variables W i must be introduced, for example posing W i = W i -W i . Let us take as practical example the linearization of the equations for W 2 and W 4 in (3.5.5):

∂ W 2 ∂t + u ∂ W 2 ∂x + v ∂ W 2 ∂y + w ∂ W 2 ∂z + 1 ρ ∂ p ∂z + + ∂W 2 ∂t + u ∂W 2 ∂x + v ∂W 2 ∂y + w ∂W 2 ∂z + 1 ρ ∂p ∂z + + u ∂ W 2 ∂x + v ∂ W 2 ∂y + w ∂ W 2 ∂z + 1 ρ ∂ p ∂z = 0 (3.5.21) ∂ W 4 ∂t +( u n + c) ∂ W 4 ∂x + v ∂ W 4 ∂y + w ∂ W 4 ∂z + c √ 2 ∂ W 2 ∂z - ∂ W 3 ∂y + + ∂W 4 ∂t +( u n + c) ∂W 4 ∂x + v ∂W 4 ∂y + w ∂W 4 ∂z + c √ 2 ∂W 2 ∂z - ∂W 3 ∂y + +(u n + c ) ∂ W 4 ∂x + v ∂ W 4 ∂y + w ∂ W 4 ∂z + c √ 2 ∂ W 2 ∂z - ∂ W 3 ∂y = 0 (3.5.22)
In equations (3.5.21) and (3.5.22), only the second rows are identifiable with Giles relations.

Even if all the terms can be computed, no simplifications can be made a priori, because nothing proves that the mean state Q satisfies the Euler equations, so that the first rows of (3.5.21) and (3.5.22) cannot be eliminated. However, the time derivative ∂ W 4 /∂t corresponds to the prescribed relaxation and hence it is known. The contribution of the relaxation of the ingoing wave's intensity L 4 , which is noted L 4 , is then chosen so that:

∂ W 4 ∂t mean + L 4 + v ∂ W 4 ∂y + w ∂ W 4 ∂z + c √ 2 ∂ W 2 ∂z - ∂ W 3 ∂y = 0 (3.5.23)
In the case of LODI equation, the last expression reduces to:

∂ W 4 ∂t mean + L 4 = 0 (3.5.24)
The additional terms in (3.5.23), which are absent in (3.5.24), correspond to tangential terms in the NSCBC description (noted T ). It is remembered that such terms are neglected in the LODI calculation. One of these two relations determines the amplitude of the "low-frequency-part" of the ingoing wave. The unsteady part of L 4 , noted L 4 , is then determined directly with Giles condition, yielding to:

L 4 = 1 √ 2 ( c + u n ) ∂W 3 ∂y - 1 √ 2 ( c + u n ) ∂W 2 ∂z (3.5.25)
The main difference between equation (3.5.25) for L 4 and (3.5.16) for L 4 lies in the presence of W i instead of W i , which complicates Giles conditions implementation.

For resuming, for an inflow boundary, Giles conditions coupled with a prescription are formulated as follows. First, ingoing waves' intensities are divided in two parts, a time-filtered average and a fluctuation:

L i = L i + L i (3.5.26)
Then, the contribution resulting from the values prescription is calculated by a classical NSCBC:

                                   ∂ W 1 ∂t mean + L 1 + α 1 v ∂ W 1 ∂y + w ∂ W 1 ∂z = 0 ∂ W 2 ∂t mean + L 2 + α 1 v ∂ W 2 ∂y + w ∂ W 2 ∂z + 1 ρ ∂ p ∂z = 0 ∂ W 3 ∂t mean + L 3 + α 1 v ∂ W 3 ∂y + w ∂ W 3 ∂z - 1 ρ ∂ p ∂y = 0 ∂ W 4 ∂t mean + L 4 + α 1 v ∂ W 4 ∂y + w ∂ W 4 ∂z + c √ 2 ∂ W 2 ∂z - ∂ W 3 ∂y = 0 (3.5.27)
where α 1 is a boolean parameter, defining respectively:

• α 1 = 0: LODI conditions

• α 1 = 1: account of tangential terms
Consideration of tangential terms in an inlet condition with a Giles/Hixon formulation is not popular in the literature. Even though this option was not tested throughout the present works, it lies the bases for interesting future studies on the subject. However, it must be reminded that for a classic NSCBC without Giles treatment, the consideration of tangential terms proved well suited for acoustics applications (cf. discussion in Section 3.3.1).

Concerning the unsteady part, we have the following system: Time-dependent ordinary and partial differential equations can be numerically approximated with explicit or implicit approaches. Explicit methods calculate the solution at a later time than the system state at the current time, while implicit methods calculates the solution involving simultaneously the current and future states. In mathematical terms, if Y (t) is the current state function and Y (t + ∆t) the later one, we will have:

                                                                 L 1 = -v ∂W 1 ∂y -w ∂W 1 ∂z -α 2 v ∂ W 1 ∂y + w ∂ W 1 ∂z L 2 = - 1 ρ ∂p ∂z + 1 √ 2 ( c -u n ) ∂W 5 ∂z + 1 √ 2 ( c + u n ) ∂W 4 ∂z + -α 2 u ∂ W 2 ∂x + v ∂ W 2 ∂y + w ∂ W 2 ∂z + 1 ρ ∂ p ∂z L 3 = 1 ρ ∂p ∂y - 1 √ 2 ( c -u n ) ∂W 5 ∂y - 1 √ 2 ( c + u n ) ∂W 4 ∂y + -α 2 u ∂ W 3 ∂x + v ∂ W 3 ∂y + w ∂ W 3 ∂z + 1 ρ ∂ p ∂y L 4 = 1 √ 2 ( c + u n ) ∂W 3 ∂y - 1 √ 2 ( c + u n ) ∂W 2 ∂z + -α 2 (u n + c ) ∂ W 4 ∂x + v ∂ W 4 ∂y + w ∂ W 4 ∂z + c √ 2 ∂ W 2 ∂z - ∂ W 3 ∂y (3.
Y (t + ∆t) = F (Y (t)) explicit form G(Y (t), Y (t + ∆t)) = 0 implicit form
Notably, implicit methods are much more complex to solve than explicit ones, since extra computation and specific coupling need to be included. In fact, a nonlinear equation system, which has no exact solution, must be resolved. Two types of algorithm are widely used to this scope: the Newton's method and the pseudo-steady method. The second one will be considered in the following. At each iteration (from one timestep to another), a series of inner iterations (also called sub-iterations) with a pseudo-timestep are run as a pseudo-steady calculation. In this way, the problem convergence is given by these sub-iterations, whose CFL number is given in input, while the global timestep will determine the temporal resolution of the physics, as for example the timesteps per acoustic wave period. The reason why we consider such complex algorithms instead of the simpler explicit methods is that many real problems are stiff by nature. In these, an explicit formalism would require an impractically small timestep ∆t to achieve numerical stability in the smallest cells, typically near the walls (where CFL < 1). Implicit methods pay back their complexity of implementation with a much larger timestep size accepted, and so a faster converging simulation. Even if more calculations are done per iteration, the overall computing time advantage can be notable. This is the case for turbofan engine simulations. The complex and unsteady physics involved applied on massive grid sizes would make difficult if not impossible to build an affordable industrial-scale model with an explicit method. The implicit solution method of elsA is then used for these applications. In vision of including the TDIBC in turbofan simulations, an appropriate implicitation of such boundary condition is optimal to ensure stability and fast convergence. A huge effort has been made to translate the TDIBC into a fully implicit form, aligned with the global time marching scheme of 2nd-order. A great merit in this development goes to Hugues Deniau, without whose expertise this would have not been achievable in such a short time.

Implicitation scheme for NSCBC

Introduction

As a starting point, for simplifying the reasoning, let us consider a structured grid surface at constant index: i = -1 /2 as in Figure 4.1. The boundary condition will modify the flux balance in the normal i-direction for all boundary cells (0, j). The normal i-flux balance in these cells will be:

(F ( 1 /2,j) -F (-1 /2,j) ) = D (4.1.1)
with D as in (3.2.18) and F the global flux (convective, tangential and diffusive). If the notation of (3.2.1) is taken and the E-flux is supposed aligned with the i-direction, then F = E. In fact, it is reminded that the boundary condition does not modify the flux value F ( 1 /2,j) nor the fluxes in the j direction F (i,j± 1 /2) . Since elsA is a finite-volume solver based on ADI (Alternate Directions Implicit), for a point (0, j) we need to compute the influence of the points (0, j -1), (0, j), (0, j + 1) and (1, j). For certain conditions requiring ghost cells, the point (-1, j) will be also included. Be V (read "V") the notation for these points ensemble. The calculation of a point's influence on another is done computing the Jacobian matrices of a quantity in relation to the state vector at the neighboring points. This allows to calculate the influence of a state vector variation at the points V on the flux balance at the point (0, j). The calculation of the point (0, j) solving NS equations is given by:

V (0,j) ∂U ∂t + F (0,j+ 1 /2) -F (0,j-1 /2) + D = 0 (4.1.2)
where V (0,j) stands for the volume of the cell (0, j). We now define with U n the conservative state vector at the known timestep and with U n+1 at the consequent timestep that we want to calculate. If a second-order in time scheme is considered, the latter translates into the following implicit form:

V (0,j) 3U n+1 -4U n + U n-1 2∆t + F n+1 (0,j+ 1 /2) -F n+1 (0,j-1 /2) + D n+1 = 0 (4.1.3)
As said in this Chapter's introduction, sub-iterations are introduced for resolving this nonlinear system with a method pseudo-unsteady. The following equality can be written from the first term of (4.1.3):

3U n+1,n +1 -4U n + U n-1 2∆t = 3(U n+1,n +1 -U n+1,n ) 2∆t + 3U n+1,n -4U n + U n-1 2∆t (4.1.4)
In the last equation:

• n represents the index of time iteration of global timestep ∆t,

• n represents the index of dual iterations of pseudo-timestep ∆τ (to not confuse with τ delay time of an acoustic liner),

• the state vectors U n-1 , U n and U n+1,n are known,

• the state vector U n+1,n +1 is the unknown.

Substituting in (4.1.3), after few passages introducing the pseudo-time derivative:

V (0,j) 1 ∆τ + 3 2∆t (U n+1,n +1 -U n+1,n ) - 3U n+1,n -4U n + U n-1 2∆t + + F n+1,n +1 (0,j+ 1 /2) -F n+1,n +1 (0,j-1 /2) + D n+1,n +1 = 0 (4.1.5)

Linearization of tangential fluxes

The implicit scheme here presented is added to the DDADI (diagonally dominant ADI) algorithm already available in elsA. The latter's principle is to represent the numerical schemes with 3points stencils, in accordance with the tridiagonal matrix system to resolve. Thus, for solving (4.1.5), the fluxes F n+1,n +1 (0,j+ 1 /2) are linearized into a 3-points stencil scheme:

F n+1,n +1 (0,j+ 1 /2) =F n+1,n (0,j+ 1 /2) + ∂F (0,j+ 1 /2) ∂U (0,j) - ∂F (0,j-1 /2) ∂U (0,j) δU (0,j) + + ∂F (0,j+ 1 /2) ∂U (0,j+1) δU (0,j+1) - ∂F (0,j-1 /2) ∂U (0,j-1) δU (0,j-1) (4.1.6)
in which we see no influence on the points (1, j), as expected by the tangential fluxes definition, and where the following is valid for every j:

δU (0,j) = U n+1,n +1 (0,j) -U n+1,n (0,j) (4.1.7)
These variations of state vector between sub-iterations n and n + 1 are the new unknowns. If the Roe scheme is used, then fluxes in (0, j + 1 /2) are expressed by:

F (0,j+ 1 /2) = 1 2 F (0,j+1) + F (0,j) - 1 2 B roe (0,j+ 1 /2) U (0,j+1) -U (0,j) (4.1.8)
where B roe is the typical Roe-flux matrix. It follows that their derivative in terms of the conservative variables takes the form:

∂F (0,j+ 1 /2) ∂U (0,j) = ∂F (0,j) ∂U (0,j) + 1 2 B roe (0,j+ 1 /2) ∂F (0,j+ 1 /2) ∂U (0,j+1) = ∂F (0,j+1) ∂U (0,j+1) - 1 2 B roe (0,j+ 1 /2) (4.1.9)
In this way, the flux balance contribution in the point (0, j) is divided in three parts:

∂F (0,j+ 1 /2) ∂U (0,j+1) = ∂F (0,j+1) ∂U (0,j+1) - 1 2 B roe (0,j+ 1 /2) ∂F (0,j+ 1 /2) ∂U (0,j) - ∂F (0,j-1 /2) ∂U (0,j) = 1 2 B roe (0,j+ 1 /2) + 1 2 B roe (0,j-1 /2) ∂F (0,j-1 /2) ∂U (0,j-1) = - ∂F (0,j-1) ∂U (0,j-1) - 1 2 B roe (0,j-1 /2) (4.1.10)

Linearization of normal fluxes

Similarly to (4.1.6), we can linearize the term D n+1,n +1 , where all four neighbour points must be considered:

D n+1,n +1 =D n+1,n + 1 /2 + ∂D ∂U (0,j)
δU (0,j) + ∂D ∂U (1,j) δU (1,j)

+ ∂D ∂U (0,j-1)
δU (0,j-1) + ∂D ∂U (0,j+1) δU (0,j+1) + (4. 1.11) In there, and n + 1 /2 represent minor details of the code implementation and their full mathematical discussion is given in Appendix D.2. Since they are terms introduced by the implicitation of the acoustic liner, later discussed in Section 4.3, they are here supposed known and their understanding will be clear at the end of this Chapter.

If we define with "Rhs" the Right-Hand Side term, which does not depend on values at the iteration to compute (n + 1, n + 1), but only on what is currently available:

Rhs = V 0,j 3U n+1,n -4U n + U n-1 2∆t -F n+1,n (0,j+ 1 /2) -F n+1,n (0,j-1 /2) -D n+1,n + 1 /2 (4.1.12)
the implicit system can be written under the form of a linear system for the increments δU (i,j) :

Rhs = C 0 δU (0,j) + C 1 δU (1,j) + C 2 δU (0,j-1) + C 3 δU (0,j+1) (4.1.13)
with:

                                   C 0 = 3 2 V 0,j • I + 1 2 B roe (0,j+ 1 /2) + 1 2 B roe (0,j-1 /2) + ∂D ∂U (0,j) C 1 = ∂D ∂U (1,j) C 2 = - ∂F (0,j-1) ∂U (0,j-1) - 1 2 B roe (0,j-1 /2) + ∂D ∂U (0,j-1) C 3 = ∂F (0,j+1) ∂U (0,j+1) - 1 2 B roe (0,j+ 1 /2) + ∂D ∂U (0,j+1) (4.1.14)
We then have to determine, for each conservative variable in U:

∂D ∂U (0,j-1)
, ∂D ∂U (0,j)

, ∂D ∂U (0,j+1)

, ∂D ∂U (1,j) It is reminded from the previous Chapter that D can be expressed as:

D = P U L U
Hence, we are looking for a general solution of:

∂D ∂U = ∂ ∂U ∂Q ∂U P -1 Q L (4.1.15)
with ∂Q/∂U the transformation matrix from primitive Q to conservative U variables, P -1 Q transformation matrix from characteristic W to primitive variables and L the characteristic wave amplitude. The next Section will deal with the calculation of ∂L /∂U of different boundary conditions.

Implicitation of characteristic waves

This Section focuses on the computation of ∂L /∂U as needed in (4.1.15). The following analysis will look at the different characteristic waves singularly: first outgoing acoustic waves, then entropic and rotational waves. A dedicated Section will follow for the ingoing waves and the time-domain impedance condition.

Outgoing waves

On a general NSCBC boundary, two waves types are considered: the waves entering the domain, function of the NSCBC formulation, and the waves exiting the domain, simply extrapolated from the inner solution. For the known outgoing waves we have:

L = ΛP Q ∆Q, ∆Q = ρ 1 -ρ 0 u 1 -u 0 v 1 -v 0 p 1 -p 0 (4.2.1)
with 0 and 1 the indexes related to the points with i = 0 and i = 1 as in Figure 4.1. This notation will be used throughout this Section. Λ is the vector of characteristic velocities λ and P Q the transformation matrix from primitive to characteristic variables. If we identify with q one of the primitive variables {ρ, u, v, w, p} and with u n the wall normal velocity u n = n x u 0 + n y v 0 + n z w 0 , we now have:

L 1 = u n n x ∆ρ + n z ∆v -n y ∆w - n x c 2 ∆p L 1 (4.2.2)
as in (3.2.23), and so the derivative on primitive variables:

∂L 1 ∂q = u n n x ∂∆ρ ∂q + n z ∂∆v ∂q -n y ∂∆w ∂q - n x c 2 ∂∆p ∂q -u n n x ∆p ∂(1/c 2 ) ∂q + L 1 ∂u n ∂q (4.2.3)
Similarly for L 2 :

L 2 = u n n y ∆ρ -n z ∆u + n x ∆w - n y c 2 ∆p L 2 (4.2.4) ∂L 2 ∂q = u n n y ∂∆ρ ∂q -n z ∂∆u ∂q + n x ∂∆w ∂q - n y c 2 ∂∆p ∂q -u n n y ∆p ∂(1/c 2 ) ∂q + L 2 ∂u n ∂q (4.2.5)
and for L 3 :

L 3 = u n n z ∆ρ + n y ∆u -n x ∆v - n z c 2 ∆p L 3 (4.2.6) ∂L 3 ∂q = u n n z ∂∆ρ ∂q + n z ∂∆u ∂q -n x ∂∆v ∂q - n z c 2 ∂∆p ∂q -u n n z ∆p ∂(1/c 2 ) ∂q + L 3 ∂u n ∂q (4.2.7)
In the same fashion, if we consider L 4 to be the outgoing acoustic wave:

L 4 = (u n -c) 1 √ 2 -n x ∆u -n y ∆v -n z ∆w + 1 ρc ∆p L 4 (4.2.8) ∂L 4 ∂q = (u n -c) 1 √ 2 -n x ∂∆u ∂q -n y ∂∆v ∂q -n z ∂∆w ∂q + 1 ρc ∂∆p ∂q + +(u n -c)∆p ∂ ∂q 1 ρc √ 2 + L 4 ∂u n ∂q - ∂c ∂q (4.2.9)
In Appendix D.3, the complete expression of the derivatives for each primitive variable q is given. It is important to remark that the L i just extrapolated are independent on the points (i, j -1) and (i, j + 1), so their Jacobians are null.

Entropic and rotational waves

After numerical verification, it was found that entropic and rotational waves should be considered as "ingoing" waves, hence not fully determined prior the time advancement solution. Their expression is given by resolving the following system:

                           σ u (n y u 0 -n x v 0 ) = -n z (n y L 2 + n x L 1 ) + (n 2 x + n 2 y )L 3 -n y T u -n x T v T 1 σ u (-n z n x u 0 -n z n y v 0 + (n 2 x + n 2 y )w 0 ) = n x L 2 -n y L 1 + n z n x T u + n z n y T v + (n 2 x + n 2 y )T w T 2 σ T (T -T ∞ ) = p ρ 2 (n x L 1 + n y L 2 + n z L 3 ) + c γ √ 2 (γ -1)(L 4 + L 5 ) - 1 ρ T p + p ρ 2 T ρ T 3
(4.2.10) with σ relaxation parameters defined as in Chapter 3. For computing the Jacobians of L i , we are going to solve 5 linear systems for each primitive variable (ρ, u, v, w, p). For computing ∂/∂ρ, for example, we need to solve:

                           0 = -n z n y ∂L 2 ∂ρ -n z n x ∂L 1 ∂ρ + (n 2 x + n 2 y ) ∂L 3 ∂ρ + ∂T 1 ∂ρ 0 = n x ∂L 2 ∂ρ -n y ∂L 1 ∂ρ + ∂T 2 ∂ρ -σ T p Rρ 2 = p ρ 2 n x ∂L 1 ∂ρ + n y ∂L 2 ∂ρ + n z ∂L 3 ∂ρ + c γ √ 2 (γ -1) ∂L 5 ∂ρ + ∂L 4 ∂ρ + - ∂T 3 ∂ρ -2 p ρ 3 (n x L 1 + n y L 2 + n z L 3 ) + γ -1 γ √ 2 ∂c ∂ρ (L 4 + L 5 ) (4.2.11)
It is important to remark that the system (4.2.11) (and the other 4 systems for the remaining 4 primitive variables) is quadruple, since it is necessary to compute the derivatives on ρ (0,j) , ρ (1,j) , ρ (0,j-1) and ρ (0,j+1) at each line. In particular, the dependence on the two terms ρ (0,j-1) and ρ (0,j+1) is linked by:

• the presence of tangential fluxes in the expression of T i ,

• the presence of L 4 and L 5 which are function themselves of the tangential fluxes. In addition, L 5 includes the impedance condition, making its linearization very complex.

Unfortunately, the previous derivation is cumbersome and can easily become source of error in a numerical scheme, reason why a simplification is now introduced. Analyses on the robustness of such simplification were not made at this stage. One could say, as first hypothesis, to neglect the influence of the terms in T i , L 4 and L 5 . With this condition, only the point (0, j) will have an influence on the increment of L i (i ∈ {1, 2, 3}), which is not acceptable. In order to avoid the complex equations linked to the liner condition, it is proposed to neglect the influence of the variation of T i and L 5 but to conserve the one of L 4 , supposing that the incident (outgoing) wave is the cause for the other quantities variation. With this hypothesis and for the influence given by point (0, j), the previous system (4.2.11) reduces to:

                           0 = -n z n y ∂L 2 ∂ρ 0 -n z n x ∂L 1 ∂ρ 0 + (n 2 x + n 2 y ) ∂L 3 ∂ρ 0 0 = n x ∂L 2 ∂ρ 0 -n y ∂L 1 ∂ρ 0 -σ T p 0 rρ 2 0 = p 0 ρ 2 0 n x ∂L 1 ∂ρ 0 + n y ∂L 2 ∂ρ 0 + n z ∂L 3 ∂ρ 0 + c 0 γ √ 2 (γ -1) ∂L 4 ∂ρ 0 + -2 p 0 ρ 3 0 (n x L 1 + n y L 2 + n z L 3 ) + γ -1 γ √ 2 ∂c 0 ∂ρ 0 (L 4 + L 5 ) (4.2.12)
where, in this specific case, the subscript 0 stands for the point location and not static thermodynamic state. Similarly, the influence given by point (1, j) would be:

                   0 = -n z n y ∂L 2 ∂ρ 1 -n z n x ∂L 1 ∂ρ 1 + (n 2 x + n 2 y ) ∂L 3 ∂ρ 1 0 = n x ∂L 2 ∂ρ 1 -n y ∂L 1 ∂ρ 1 0 = p ρ 2 n x ∂L 1 ∂ρ 1 + n y ∂L 2 ∂ρ 1 + n z ∂L 3 ∂ρ 1 + c γ √ 2 (γ -1) ∂L 4 ∂ρ 1 (4.2.13)
With the same hypotheses, the 8 linear three-equations systems (4 for ρ 0 and 4 for ρ 1 , as for the quadruple system (4.2.11)) can be established for all the primitive variables and are given in Appendix D.4. It is reminded that in all these systems the derivatives of L 4 are known since they are extrapolated.

Implicitation of delayed-impedance condition 4.3.1 Ingoing waves

The objective is now to express the incoming waves amplitude in an implicit form. In other words, for a TDIBC condition, we want to implicit the acoustic liner function β as described in (2.4.29). In the same frame of a finite-volume solver, fluxes will be divided as in (3.2.12) into convective L and tangential T (and diffusive D, if any).

L 5 + T 5 + D 5 = β (L 4 + T 4 + D 4 ) (4.3.1)
where the subscripts 4 and 5 are respectively for the outgoing and ingoing acoustic waves, as the right side of Figure 3.1. When passing to an ODR discretization defined as in (2.4.29), the relation between the aforementioned waves becomes:

F 5 = β ∞ F 4 + N ξ k=1 µ 1,k ϕ ξ,k + Ns n=1 r 1,n ϕ s,n + N ξ k=1 µ 2,k ψ ξ,k + Ns n=1 r 2,n ψ s,n (4.3.2)
where with F is intended the sum of all fluxes (convective, tangential and diffusive) and β ∞ is the bulk reflection coefficient. The variables ϕ and ψ are themselves function of F 4 and will be singularly discussed in the next sections. It is reminded that diffusive terms (ξ k , µ k , ϕ k , ψ k ) are real values, while oscillatory terms (s n , r n , ϕ n , ψ n ) are complex values. The final discretization of 4.3.2, upon a simplification of complex calculus1 given by handling both complex conjugates of each variable, yields the real value:

L 5 = β ∞ F 4 -(T 5 + D 5 ) + N ξ k=1 µ 1,k ϕ ξ,k + µ 2,k ψ ξ,k + 2Re   Ns /2 n=1 r 1,n ϕ s,n + r 2,n ψ s,n   (4.3.3)
The missing step, of a non-negligible complexity, is the implicitation of the calculation of auxiliary variables ϕ and ψ.

Implicitation of auxiliary variable ϕ

This section focuses on how the auxiliary variables ϕ of the time-domain impedance model are implicited in the code. After a first introduction, the algorithm is given step by step in order to help the understanding.

We suppose to be calculating the iteration n + 1 (i.e. the unknown in an implicit scheme) and to have just computed the sub-iteration n . The previous two iterations n and n -1 are also known and stored in memory. We are now looking to advance to the sub-iteration n + 1. Before descending in the algorithm details, it is useful to remind the expression of ϕ from (3.3.25):

∂ϕ ∂t = -zϕ(t, z) -F 4 (t)
where z is a generic pole of the ODR and F 4 is the known outgoing flux term. The updated ϕ n+1,n +1 would be written as:

ϕ n+1,n +1 = ϕ n+1,n + δ (t) ϕ (n -→n +1) + δ (F ) ϕ (n -→n +1) (4.3.4)
where:

• ϕ n+1,n is the known value at the previous sub-iteration;

• δ (t) ϕ (n -→n +1) is its time variation between the sub-iterations n and n + 1, discretized with a 2nd-order upwind scheme as: • δ (F ) ϕ (n -→n +1) is the variation of ϕ brought by the simultaneous variation of δF between the sub-iterations n and n + 1. This is possible thanks to the linear character of F in relation to ϕ. The effect brought by the full implicit consideration of F 4 can then be written as:

δ (t) ϕ (n -→n +1) = ∆τ 2∆t -3ϕ n+1,n + 4ϕ n -ϕ n-1 + zϕ n+1,n -F n+1,n
δ (F ) ϕ (n -→n +1) = δF 4 ∆τ 1 -z∆τ + 3 2 ∆τ ∆t = δF 4 β (4.3.6)
where β regroups all the constants.

The only unavailable information in the previous equations is the flux variation δF [START_REF] Bruneau | Fundamental of Acoustics[END_REF] , not yet computed at this point, since it intrinsically depends on the new flux value. For this reason and for the sake of the following derivation, we define an intermediate ϕ n+1,n + 1 /2 as in (4.3.4) but without consideration of the flux variation term:

ϕ n+1,n + 1 /2 = ϕ n+1,n + δ (t) ϕ (n -→n +1) (4.3.7)
With these properties defined, we can now detail the algorithm structure that allows us to update the auxiliary variables.

Step 1: The numerical fluxes F n+1,n i are calculated at the sub-iteration n , which is fully determined at this stage.

Step 2: Compute the auxiliary variable ϕ n+1,n solving the equation (4.3.4) for the previous sub-iteration, where all variables (included δ (F ) ) are known:

ϕ n+1,n = ϕ n+1,n -1 + δ (t) ϕ (n -1-→n ) + δ (F ) ϕ (n -1-→n ) (4.3.8)
Step 3: We then compute the intermediate auxiliary variable ϕ n+1,n + 1 /2 with use of (4.3.7), since we still do not know the value of δ (F ) ϕ (n -→n +1) .

Step 4: We now want to determine the derivative of L 5 as for the other characteristic waves of Section 4.2. From (4.3.3) we can write:

δL 5 = β ∞ δF 4 -δT 5 + A δ (F ) ϕ (4.3.9)
with A a constant. Theoretically, all the effects brought by the auxiliary variables ϕ should be included in the implicit representation of (4.3.9), including the term δ (F ) ϕ, and not appear explicitly. However, for the choice exposed in (4.3.7), the flux variation term was still not taken into account when updating in Step 3. The advantage of this formulation is that, thanks to (4.3.6), the variation given by the flux can be incorporated in the whole flux resolution as:

δL 5 = (β ∞ + A β )δF 4 -δT 5 (4.3.10)
In this way, instead of coupling an additional equation to the implicit system (i.e. solving (4.3.4) with the updated flux contribution), we take into account its effect in the global flux formulation.

Step 5: Fluxes F n+1,n +1 i are now fully determined, convergence criteria can be assessed and a new iteration or sub-iteration can be started.

From the previous equations, it remains to define the value of δF 4 and δT 5 . The first is reminded to be F = L + T and so:

δF 4 = ∂L 4 ∂Q δQ + ∂T 4 ∂Q δQ (4.3.11)
with δQ the primitive variables variation between the sub-iterations n and n + 1.

Concerning ∂L 4 , it was already determined in (4.2.8), while T 4 is:

T 4 = 1 √ 2 -n x (F (0,j+ 1 /2) -F (0,j-1 /2) ) ρu -n y (F (0,j+ 1 /2) -F (0,j-1 /2) ) ρv + -n z (F (0,j+ 1 /2) -F (0,j-1 /2) ) ρw + 1 ρc √ 2 (F (0,j+ 1 /2) -F (0,j-1 /2) ) ρe (4.3.12)
where (F (0,j+ 1 /2) -F (0,j-1 /2) ) U represents the conservative variable component of the flux balance in the boundary tangential direction. It is reminded that T 4 depends only on the points (0, j -1), (0, j) and (0, j + 1), as for all tangent fluxes terms. Let's introduce the notation J j (x) for the flux balance Jacobian in the j-direction in relation to the fluid state at the point x. J j (x) is a 5 × 5 matrix whose k-th line is noted J j k (x). We then obtain:

∂T 4 ∂Q (0,j) = 1 √ 2 -n x J j 2 (j) -n y J j 3 (j) -n z J j 4 (j) + 1 ρc √ 2 J j 5 (j)+ + (F (0,j+ 1 /2) -F (0,j-1 /2) ) ρe 1 √ 2 ∂ ∂Q (0,j) 1 ρc (4.3.13a) ∂T 4 ∂Q (0,j+1) = 1 √ 2 -n x J j 2 (j + 1) -n y J j 3 (j + 1) -n z J j 4 (j + 1) + 1 ρc √ 2 J j 5 (j + 1) (4.3.13b) ∂T 4 ∂Q (0,j-1) = 1 √ 2 -n x J j 2 (j -1) -n y J j 3 (j -1) -n z J j 4 (j -1) + 1 ρc √ 2 J j 5 (j -1) (4.3.13c)
The matrices ∂T 4 /∂Q k are lines of 5 elements. In practice, the matrices J j (x) are calculated with help of the numerical flux Jacobians in (j + 1 /2) and (j -1 /2):

J j (j) = ∂F (0,j+ 1 /2) ∂Q (0,j) - ∂F (0,j-1 /2) ∂Q (0,j) J j (j + 1) = ∂F (0,j+ 1 /2) ∂Q (0,j+1) J j (j -1) = - ∂F (0,j-1 /2) ∂Q (0,j-1) (4.3.14)
In the same fashion, the last unknown T 5 can be expressed by:

∂T 5 ∂Q (0,j) = 1 √ 2 n x J j 2 (j) + n y J j 3 (j) + n z J j 4 (j) + 1 ρc √ 2 J j 5 (j) +(F (0,j+ 1 /2) -F (0,j-1 /2) ) ρe 1 √ 2 ∂ ∂Q (0,j) 1 ρc (4.3.15a) ∂T 5 ∂Q (0,j+1) = 1 √ 2 n x J j 2 (j + 1) + n y J j 3 (j + 1) + n z J j 4 (j + 1) + 1 ρc √ 2 J j 5 (j + 1) (4.3.15b) ∂T 5 ∂Q (0,j-1) = 1 √ 2 n x J j 2 (j -1) + n y J j 3 (j -1) + n z J j 4 (j -1) + 1 ρc √ 2 J j 5 (j -1) (4.3.15c)
In the previous paragraph, we also highlighted that normal outgoing waves are affected only by the points (0, j) and (1, j):

δL 4 = ∂L 4 ∂Q (0,j) δQ (0,j) + ∂L 4 ∂Q (1,j) δQ (1,j) (4.3.16)
With the simpler notation of:

Γ 1 = β ∞ + A β -1 Γ 2 = β ∞ + A β + 1 (4.3.17)
We can finally write the variation of L 5 for all neighbour points:

δL 5 = ∂L 5 ∂Q (0,j) δQ (0,j) + ∂L 5 ∂Q (0,j+1) δQ (0,j+1) + ∂L 5 ∂Q (0,j-1) δQ (0,j-1) + ∂L 5 ∂Q (1,j) δQ (1,j) + A δ (F ) ϕ (4.3.18) with: ∂L 5 ∂Q (0,j) = - Γ 2 √ 2 n x J i 2 (i) + n y J i 3 (i) + n z J i 4 (i) + Γ 1 ρc √ 2 J i 5 (i)+ + (F (0,j+ 1 /2) -F (0,j-1 /2) ) ρe Γ 1 √ 2 ∂ ∂Q (0,j) 1 ρc + (β ∞ + A β ) ∂L 4 ∂Q (0,j) ∂L 5 ∂Q (1,j) =(β ∞ + A β ) ∂L 4 ∂Q (1,j) ∂L 5 ∂Q (0,j+1) = - Γ 2 √ 2 n x J i 2 (i + 1) + n y J i 3 (i + 1) + n z J i 4 (i + 1) + Γ 1 ρc √ 2 J i 5 (i + 1) ∂L 5 ∂Q (0,j-1) = - Γ 2 √ 2 n x J i 2 (i -1) + n y J i 3 (i -1) + n z J i 4 (i -1) + Γ 1 ρc √ 2 J i 5 (i -1) (4.3.19)

Implicitation of delayed auxiliary variable ψ

We now want to derive an implicit formulation of the Spectral Difference calculation performed in Section 3.3.4 for computing the delayed advection problem. This is performed immediately after the calculation of ϕ n+1,n +1 previously detailed, which is then at this stage known. It is reminded that, in the SD framework, we have N ψ elements in the 1D domain, p the polynomial order, sp solution points (sp

∈ [1, N sp = p + 1]) and f p flux points (f p ∈ [1, N f p = p + 2]
). The notation ψ f p will be used to determine values at the flux points, while the simple ψ will refer to values at the solution points. Furthermore, in the following, it must be taken into account that ψ is a column vector containing all the delay values at the solution points for each pole of the ODR.

In an implicit method with dual time-stepping, the Riemann problem introduced in Step 4 of Figure 3.3 for a generic element K of the delay advection becomes:

   ψ f p 1 (K) = ψ f p N f p (K -1) ψ f p 1 (K = 1) = ϕ n+1,n +1 (4.3.20)
expressing the link between extrema of each neighboring element (i.e. the last flux point of the element K -1 and the first flux point of the element K). When computing the derivative at the solution points (Step 6 in Figure 3.3), the effect given by the first interface can be isolated:

∂ψ ∂x sp = ψ 1 A d (1 + (sp -1)(p + 2)) + N f p i=2 ψ f p i A d (i + (sp -1)(p + 2)) (4.3.21)
Using the relation (3.3.28) for expressing the value at flux points, we obtain:

∂ψ ∂x sp = ψ 1 A d (1 + (sp -1)(p + 2)) + N f p i=2   A d (i + (sp -1)(p + 2
∂ψ ∂x sp = ψ 1 A d (1 + (sp -1)(p + 2)) + Nsp j=1 ψ j C sp,j (4.3.23)
with:

C sp,j = N f p i=2 A d (i + (sp -1)(p + 2))B(j + i(p + 1)) (4.3.24)
The advantage of writing this formulation is that these coefficients can be calculated at the first iteration and be reused throughout the whole calculation, as they are given by time-independent parameters. This brings a strong improvement in terms of calculation speed of the implicit scheme. Formally, the integration of ψ is written as:

1 ∆τ + 3 2∆t δψ + 1 2∆t 3ψ n+1,n -4ψ n + ψ n-1 + V ∂ψ n+1,n ∂x + V ∂δψ ∂x (4.3.25)
The

   δψ f p 1 (K) = δψ f p N f p (K -1) δψ f p 1 (K = 1) = δϕ (4.3.27)
For the first element K = 1, we know δψ 1 as we imposed the value ϕ n+1,n +1 (second line of (4.3.27)). For an element K ≥ 2, δψ 1 is equal to the variation of ψ at the last flux point of the element K -1 (first line of (4.3.20)). After having determined the δψ 1 for all K elements (remembered to be equal to N ψ ), the linear system (4.3.25) is resolved, whose i-th equation writes (with i ∈ [2, N f p ]) :

1 ∆τ + 3 2∆t (δψ) i + Nsp j=1 C i,j (δψ) j = R i (4.3.28)
with:

R i = - 1 2∆t 3(ψ n+1,n ) i -4(ψ n ) i + (ψ n-1 ) i -V ∂ψ n+1,n ∂x i -A d (1 + (i -1)(p + 2))δψ 1 (4.3.
29) where V is the element's volume. In order to create a coupled implicit scheme between the Navier-Stokes flow field and the auxiliary variables, the latter algorithm must be modified for taking into account respectively:

• the variation of L i in the computation of δψ,

• the influence of δψ in computing the variations of L i . This was realized in (4.3.4), here reminded:

ϕ n+1,n +1 = ϕ n+1,n + δ (t) ϕ (n -→n +1) + δ (F ) ϕ (n -→n +1)
where the term in δ (t) is the variation of ϕ linked to the integration of its update equation, and the term in δ (F ) is the variation of ϕ linked to the variation of F 4 . Similarly to (4.3.6), we want to determine a constant β so that:

δ (F ) ψ (n -→n +1) = β δF 4 (with ψ = ψ f p N f p (N ψ )) (4.3.30)
in order to be able of rewriting (4.3.10) as:

δL 5 = (β ∞ + A β + A β )δF 4 -δT 5 (4.3.31)
It must be highlighted that ψ in (4.3.30) is taken at the last flux point of the last element of the 1D advection discretization (i.e. K = N ψ , i = N f p ). The linearity of (4.3.23) comes again in support for determining the influence of a variation δψ 1 applied at the first flux point of the first element (K = 1, i = 1) on the variation of ψ at the last flux point of the last element (K = N ψ , i = N f p ). More precisely, the contribution of ψ on the ingoing characteristic wave is of the kind A ψ(N f p ), where ψ(N f p ) is the value of the element K at the last flux point (i = N f p ). If we note with (δψ 1 ) 1 the variation at the first flux point of the first element, the full resolution of system (4.3.25) allows to calculate ψ K in the case of (δψ 1 ) 1 = 0. We solve this same system for a value (δψ 1 ) 1 = 0 to obtain a new value ψ . The linearity of these equations allows us to write:

β = ψ -ψ (δψ 1 ) 1 β (4.3.32)

Numerical scheme validation -1D tube

A one-dimensional wave reflection in a tube has been taken as validation test of the aforementioned numerical methodology. The analytical solution of this simple case, comparable to a Tube of Kundt in experimental practices, is available and proved to be a suitable candidate for validating the numerical outcome of the elsA CFD software with TDIBC, solving in this occasion the Euler equations.

Numerical setup and results

The analytical solution has been computed through a MATLAB script developed by Monteghetti in [START_REF] Monteghetti | Analysis and Discretization of Time-Domain Impedance Boundary Conditions in Aeroacoustics[END_REF], where tube's length and its discretization, wave's frequency and amplitude are specified in input. The CFD setup is shown in Figure 4.2. It consists in a sinusoidal pulse of 1 Pa amplitude initialized in the middle of a 1D channel, with characteristic rightwards velocity u ac = p ac /(ρ 0 c 0 ). The tube's length is each time adapted to be equal to three times the pulse wavelength λ = c 0 /f , with f pulse frequency, and it is discretized with a uniform grid. Top and bottom boundaries are symmetry (slip) walls, hence not introducing any transverse effect into the computational domain. The right boundary is an acoustic absorbing wall modeled with a generic TDIBC as in (4.3.3) and the left boundary is a non-reflective outlet condition as in (3.3.5), allowing the pressure wave to leave the domain with no spurious reflections. The physics wants that, given the intrinsic rightwards velocity, the pulse will travel towards the right side of the tube, impacting the treated wall. Its amplitude will then be dampened by the TDIBC and its velocity reversed in sign, so that it will reach the outlet condition on the left side. A probe to capture the pressure evolution in time is placed right after the initialized pulse, at the green dot location of Figure 4.2. The impedance law or its discretization in multipole form are not herein given precisely, since this validation can be made with any TDIBC once it is applied equally to the analytical and numerical solvers. showed to be sufficient for resolving the delay 1D advection problem. These decisions come from a detailed convergence analysis on grid, time and delay resolution explained in the following.

Convergence study

The delay resolution deriving from the 1D advection problem of Section 4.3.3 showed to play a relevant role in the overall amplitude of the reflected signal. At the same time, the spatial discretization could introduce numerical dissipation, either due to a too refined grid, hence introducing a largely propagating error from the numerical scheme, or due to a too coarse grid, losing the acoustic properties during the wave advection. For this reason, a double convergence study was carried out, on the global channel grid size and on the 1D advection problem. A 2000 Hz pulse is considered for this exercise and with a different impedance law. Three grids were chosen (120, 240 and 480 cells, equal to respectively 40, 80 and 160 points per wavelength) and multiple delay resolutions from 16 up to 128 points per wavelength. Initially, the upwind scheme in space was used for solving the 1D problem instead of Spectral Difference for its easier implementation, however requiring a higher resolution. During the grid convergence study, shown in Figure 4.4, the delay resolution was set to a very high value (N λ = 128, or PPW, Points Per Wavelength) to not introduce any error due to such effect. The coarser grid introduces dissipation errors which are visible in presence of strong discontinuities (as in t=2.5 and t=3.5 periods). In this case, neither a very refined delay discretization can provide excellent results, since the problem comes from the wave's resolution. Figure 4.5 shows the counterpart study: the grid size is fixed to a very fine value (480 cells) and the delay discretization convergence is analysed. As expected, discrepancies from the analytical results are notable in the coarse cases, but only in the "delayed" reflected signal (i.e. for t>3 periods). Neither a very refined grid can ensure an optimal result if the delay resolution does not have enough elements in its discretization. It is then clear that grid and delay resolutions are playing a common role for achieving a correct solution. In order to overcome the necessity of a very high delay resolution, a Spectral Difference (SD) discretization was then implemented, in replacement of the 2nd-order upwind scheme for the spatial discretization of the delay advection (the ∂ψ/∂l τ in (2.4.26)). The more complicated algorithm structure allows a much coarser resolution in terms of points per wavelength (N λ ) in the delay discretization. Such advantage is brought by the 5th-order polynomial inside the scheme, as demonstrated in [START_REF] Fiévet | Numerical Analysis of Porous Coatings Stabilizing Capabilities on Hypersonic Boundary-Layer Transition[END_REF][START_REF] Fiévet | Numerical Study of Hypersonic Boundary-Layer Transition Delay through Second-Mode Absorption[END_REF][START_REF] Fiévet | Strong compact formalism for characteristic boundary conditions with discontinuous spectral methods[END_REF]. In comparison with upwind scheme, the same accuracy as in the N λ = 128 case is achieved with only N λ = 8 with the Spectral Difference scheme, with a notable advantage in computational time too. This is shown in

Summary

• The numerical algorithms and formulations for obtaining an implicit formulation of NSCBC and TDIBC boundary conditions were given in detail.

• A preliminary 1D simulation proved the correct implementation of the time-domain impedance model in the CFD solver. The minimal criteria in terms of space and time resolution for waves propagation were established. The efficiency of the Spectral-Difference discretization for the delay advection problem was also demonstrated.

Part III

Results discussion

Chapter 5 Since the objective of this work is the implementation of the TDIBC previously developed by Monteghetti [START_REF] Monteghetti | Energy analysis and discretization of nonlinear impedance boundary conditions for the time-domain linearized Euler equations[END_REF] in the elsA software, this Chapter has the purpose of extending the validation of the Oscillo-Diffusive Representation method with impedance laws of realistic industrial liners. In fact, the first step when using an ODR is to define a reliable fitting of the given impedance model over a determined frequency range. A MATLAB code, developed by Monteghetti [START_REF] Monteghetti | Analysis and Discretization of Time-Domain Impedance Boundary Conditions in Aeroacoustics[END_REF] and adapted during this work to be applicable to a wider range of acoustic treatments, is used for computing poles and weights as defined in equation (2.4.24). In the original work of Monteghetti [START_REF] Monteghetti | Analysis and Discretization of Time-Domain Impedance Boundary Conditions in Aeroacoustics[END_REF], the ODR method was validated for the Grazing Incidence Tube (GIT) CT-57 liner of [START_REF] Jones | Benchmark Data for Evaluation of Aeroacoustic Propagation Codes with Grazing Flow[END_REF] and the Grazing Flow Impedance Tube (GFIT) SDOF liner of [START_REF] Primus | ONERA-NASA cooperative effort on liner impedance eduction[END_REF]. In both cases, a very reasonable number of poles allowed the discretized model to match the analytical EHR model, which was taken as reference. The impedance law of the CT-57 liner will be here re-proposed for clarity, alongside its ODR multipole discretization as given in [START_REF] Monteghetti | Analysis and Discretization of Time-Domain Impedance Boundary Conditions in Aeroacoustics[END_REF]. Then, the ODR technique is used for discretizing two industrial-type liners (SDOF liners with either a perforated or wiremesh resistive sheet), for an additional and more realistic 3D CFD validation: the CAN-NELLE benchmark [START_REF] Lavieille | Impedance eduction of liners in no-flow condition and based on multimodal excitation[END_REF] discussed in Section 6.3. Finally, a prototype of UHBR nacelle inlet liner is proposed for the final demonstrator case of this work, the ASPIRE study case, presented in Chapter 7. Other aeronautical-type liners provided by Airbus have also been tested with the ODR and all provided satisfactory results with a limited number of poles. They are herein not detailed for confidentiality and brevity reasons, and the focus will be on those previously said. Appendix A provides all the poles and weights sets and Appendix B gives the geometrical details of the liners used in the present work.

ODR of industrial acoustical liner

The general formulation of impedance and reflection coefficient in frequency-domain writes:

Z(ω) = p(ω) u n (ω) = R + iX β(ω) = Z(ω) -ρ 0 c 0 Z(ω) + ρ 0 c 0 93
In the following, ODRs will be presented under the form of four graphics for each liner, representing respectively:

• Resistance R, the real part of Z(ω);

• Reactance X , the imaginary part of Z(ω);

• Reflection coefficient module, the absolute value of the complex variable β(ω);

• Reflection coefficient phase, the argument of β(ω).

In particular, the last two represent the continuous function that is introduced in the CFD code as representative of the liner's physics.

GIT Ceramic Tubular CT-57 liner

This liner was used to obtain the experimental data reported by Jones in [START_REF] Jones | Benchmark Data for Evaluation of Aeroacoustic Propagation Codes with Grazing Flow[END_REF]. It will also be used herein in Chapter 6.2 for the 2D validation of the TDIBC implementation in the CFD solver. It is composed of a ceramic tubular core and a rigid back plate. This liner has been object of several acoustic studies in the literature because of its similar response and frequency range of aeronautical liners. Figure 5.1 shows its Oscillo-Diffusive Representation against the analytical EHR model and the relative experimental data, at a sound level of 130 dB and for Mach numbers of 0 and 0.255. The multipole representation for this liner is taken identical as in [START_REF] Monteghetti | Analysis and Discretization of Time-Domain Impedance Boundary Conditions in Aeroacoustics[END_REF], i.e. the same poles and weights are considered. Their values are given in Appendix A in Table A.1. In this way, any difference between the CAA [START_REF] Monteghetti | Analysis and Discretization of Time-Domain Impedance Boundary Conditions in Aeroacoustics[END_REF] and CFD computations will be only due to the numerical implementation and not to the impedance discretization. While the case at Mach number equal to 0 does not present any strong discrepancy, some differences between the experimental values and the ODR are found in the case at Mach number 0.255. As explained in [START_REF] Monteghetti | Energy analysis and discretization of nonlinear impedance boundary conditions for the time-domain linearized Euler equations[END_REF], these impedance values present a very low resistance in the range [0.7 -1.1] kHz and the resistance curve suggests the presence of noise in the data (top left picture). This is a difficulty for the first step of an ODR, i.e. the impedance model definition, whose output is sensitive to the experimental points given as first guess. In fact, the EHR model cannot fit both low and high resistance regions, namely [0.7 -1.1] kHz and [2.3 -3] kHz. In addition, the 1 kHz value leaded to an unstable multipole representation in [START_REF] Monteghetti | Energy analysis and discretization of nonlinear impedance boundary conditions for the time-domain linearized Euler equations[END_REF]. Since the first step of an ODR is not sensitive itself to the initial guess, provided that the chosen data points cover the anti-resonance, it was chosen to exclude the data points in/at [0.7 -0.8], [1 -1.4], 1.8 and [2.1 -2.3] kHz. This yielded to the β E model [START_REF] Monteghetti | Analysis and Discretization of Time-Domain Impedance Boundary Conditions in Aeroacoustics[END_REF] with only 2 oscillatory poles. The increased resistance at 1 kHz compared with the identified value reduces the instability, although not enough to fit the experiment. If the attention is posed on the reflection coefficient module (bottom-left picture), we can affirm that in the range [0.5 -1.5] kHz the ODR impedance is overestimated, i.e. with a lower reflection coefficient module. This will hypothetically bring to a stronger sound reduction than in the experiment at these frequencies, what will be shown confirmed later by the CFD computations in Section 6.2. In Figure 5.2, a comparison is given on the number of poles required to obtain a well discretized ODR for the CT-57 liner at Mach 0. Contrarily to the previous ODR of Figure 5.1, where 4 poles were sufficient to discretize the model, here it is highlighted how 3 and 7 poles are not sufficient in this case, but 13 provide a perfect match. This is due to two facts. First, all the experiment points are considered in the model definition, demonstrating the previously discussed need of filtering out specific unstable or noisy points. Second, the range of interest is now extended to 10 kHz (against the 3 kHz previously considered), hence resolving until the fourth anti-resonance. Even though this will not be of practical interest in this work, it shows that ODR can not only provide a correct and light discretization for typical frequency ranges, but it also proved to be a good model for larger ranges on specific applications.

CANNELLE industrial liners

Two aeronautical-type liners are considered for a new benchmark study. These liners were used during the Airbus experimental campaign CANNELLE [START_REF] Lavieille | Impedance eduction of liners in no-flow condition and based on multimodal excitation[END_REF], detailed in Section 6.3. They are an Aircelle bypass SDOF liner (ACL), kindly provided by the acoustics department of Safran Nacelles, and a nacelle inlet wiremesh SDOF liner designed by Airbus (AIB). In Figures 5.3 and 5.4, a comparison is shown between the ODR multipole model, the analytical EHR model and the experimental measurements, for respectively the AIB and ACL liners, at Mach numbers 0 and 0.3 and Sound Pressure Level of 120 dB. Differences in impedance laws due to SPL variations (up to 140 dB) have shown to be negligible, which means that these liners are linear with respect to the SPL. Thus, only the impedance law at 120 dB is shown. As before, poles and weights are computed with a MATLAB code and their values are given in Appendix A in Tables A.2 and A.3. For both liners, 6 poles (2 real and 4 complex conjugated) are sufficient for a correct discretization, independently on the SPL and Mach number. To ensure numerical stability in the event of spurious oscillations appearing during the calculation, the impedance discretizations have been computed so to satisfy Rienstra's passivity, reality and causality conditions up to 10 kHz [START_REF] Rienstra | An introduction to acoustics[END_REF]. Three vertical dashed lines correspond to the three frequencies that are investigated in the CFD simulations (1600, 2600 and 3500 Hz), in order to verify that the reflection coefficient module is correctly discretized with the ODR at these specific values.

In AIB liner, the high resistance level derives from the wiremesh sheet, which is introducing a global higher resistive effect to overcome grazing flow sensibility. Indeed, if the Mach 0.3 flow is considered, the reflection coefficient module is practically unchanged. No major discrepancies were found, regardless the noise in the resistance of the experimental data at Mach number 0.3.

In ACL liner, the Mach 0 case shows a very low resistance and an acoustic nonlinear effect at 3 kHz (black diamond symbols in Figure 5.4) due to resonance effect, as confirmed by a null reactance at the same frequency. This physics can not be captured by the current linear impedance formulation. For simplicity, such nonlinear data have been approximated averaging the resistance level to nearby values (white diamond symbols). This approximation, applied on the range [2800-3500] Hz, did not show any evident drawback on the final results. In addition, the no-flow case shows a reflection coefficient close to 1, which makes the liner's physics hard to be exactly reproduced, being a slight derivation from a hard wall but with a not-null normal velocity component. The grazing flow case presents a stronger reflection coefficient (lower in module), masking the nonlinear resonance effect identified in the Mach 0 case. Even if some discrepancies in the resistance level are present, the reflection coefficient module and phase are correctly discretized in the frequency range of interest, hence ensuring an appropriate modelling in the CFD calculation. 1000 1500 2000 2500 3000 3500 4000 4500 5000

Frequency [Hz]

ASPIRE UHBR liner

An industrial SDOF liner that could hypothetically be mounted in future UHBR inlet nacelle architectures is here presented. Its geometrical details are given in Appendix B and its impedance law is illustrated in Figure 5.5. This liner is adopted in a Mach 0.27 flight condition, however this will not be the exact grazing flow on the lined surface. The ODR showed to be a perfect candidate to discretize its impedance law with as little as 6 complex oscillatory poles (i.e. 3 conjugated pairs). No discrepancies were found at any frequency up to 5 kHz with this discretization.

Summary

• The impedance laws and multipole discretizations were given for the acoustic liners that will be used in the following CFD simulations.

• The advantage of using an ODR was demonstrated by the low number of poles needed for different liners. However, its capacity of resolving a broad frequency range was also demonstrated by increasing the number of poles. In this Chapter, the CFD results achieved in this dissertation for validating the Time-Domain Impedance Boundary Condition are presented. First, general information on the solver parameters and useful equations are given. Second, a 2D validation against a classical acoustic benchmark provides the confirmation of the correct implementation of the time-domain condition. Third, a 3D industrial benchmark developed by Airbus brought an additional validation never tackled before in the scientific domain.

This Chapter includes the results presented in the Forum Acousticum 2020 [START_REF] Casadei | Time-Domain Impedance Boundary Condition Implementation in a CFD solver and validation against experimental data of acoustical liners[END_REF] and AIAA-Aviation 2021 [START_REF] Casadei | Towards sound absorption in a cylindrical lined duct using CFD with time-domain impedance boundary condition[END_REF] conferences.

General settings and formulas

Simulations have been run on both Onera and Airbus HPC (High-Performance Computing) clusters with the elsA software (v4.2), in-house modified to include the TDIBC modeling described in the previous chapters. The numerical settings kept for all simulations, confirmed by the preliminary convergence studies detailed in Section 4.4, are the following:

• 2D and 3D modules are used, solving the Euler equations in cases with Mach = 0 and Navier-Stokes equations when a mean flow is present, including a viscous laminar boundary layer. No turbulence model is considered at this stage. Very few studies in the literature assessed the influence of turbulence on the acoustic impedance [START_REF] Sebastian | Numerical simulation of acoustic propagation in a turbulent channel flow with an acoustic liner[END_REF][START_REF] Sebastian | Numerical simulation of a turbulent channel flow with an acoustic liner[END_REF][START_REF] Shur | Further Evaluation of Prediction Capability of the Broadband Time-Domain Impedance Model for Sound Propagation in Turbulent Grazing Flow[END_REF][START_REF] Shur | Unsteady Simulation of Sound Propagation in Turbulent Flow Inside a Lined Duct Using a Broadband Time-Domain Impedance Model[END_REF]. However, it is certain that a significant difference is found when including any form of velocity gradient near the boundary layer instead of a uniform Euler flow. This was also proven in CAA studies [START_REF] Naïr | Industrial-scale time domain modelling of acoustic surface treatments for aero-engines using discontinuous Galerkin method[END_REF]. Hence, the inclusion of a laminar boundary layer is a first major step in considering the velocity gradient into the impedance determination, that will be extended in the future with inclusion of turbulence modeling.

• The time advancing scheme is the implicit dual-time step marching scheme with LussorMat algorithm solving Backward Euler ODE, with control on the number of dual-iterations (maximum 30), dual-CFL (100) and dual-convergence criteria (10 -4 ). In particular, the latter proved to be a minimal requirement for a correct resolution of the acoustic wave attenuation.

• The spatial scheme is a second-order Roe flux reconstruction with third-order limiter and a 5 points stencil.

• Temporal discretization of 100 timesteps per acoustic period, as shown well suited in Section 4.4 and similar to literature studies [START_REF] Shur | Unsteady Simulation of Sound Propagation in Turbulent Flow Inside a Lined Duct Using a Broadband Time-Domain Impedance Model[END_REF].

• Spatial discretization of approximately 40 cells per acoustic wavelength, also confirmed in the parametric study in Section 4.4 and in other studies [START_REF] Thisse | Prévision du bruit d'onde de choc d'un turboréacteur en régime transsonique par des méthodes analytiques et numériques[END_REF].

• Conservative variables are stored at the end of each simulation for a possible restart.

Whenever a mean flow is present, a hyperbolic profile as proposed in [START_REF] Rienstra | Spatial Instability of Boundary Layer Along Impedance Wall[END_REF] is initialised in the numerical domain and imposed in the inflow boundary, given by the following law:

M hy (h) = M c tanh 1 -|h| δ + M c 1 -tanh 1 δ   1 + tanh 1 δ δ + 1 + |h|   (1 -|h|) (6.1.1)
with M c the centerline Mach, given by:

M c = M avg δ log cosh 1 δ + 1 -tanh 1 δ 1+tanh( 1 δ ) 6δ + 2 3 (6.1.2)
In there, M avg is the average Mach number, h is the coordinate on the normal distance from the wall (boundary layer height), δ ∈ [0, 1] is the non-dimensional boundary layer thickness.

It is also convenient to introduce the notion of Sound Pressure Level (SPL), defined by: SPL = 20 log 10 p ac p ref (

with p ac the acoustic pressure wave amplitude and p ref the reference pressure equal to 20 µPa. Throughout this work, SPL and its phase are obtained through use of either the MATLAB Fast Fourier Transform (FFT) algorithm or the Antares Discrete Fourier Transform (DFT) algorithm. Inversely, if the SPL is given in input (as for the initial conditions of the following CFD problems), the identity (6.1.

3) provides us with the value of the acoustic fluctuation in terms of pressure amplitude that will be used in the perturbation injection routine described in Section 3.4:

p ac = p ref 10 ( SPL 20 ) (6.1.4)
Another interesting acoustic property is the instantaneous acoustic intensity I(t), defined as the energy flux, or the power per unit area needed to move an acoustic particle with a (3-dimensional) velocity v(t) in a pressure field p(t):

I(t) = p(t)v(t) (6.1.5)
The intensity spectrum is given by the convolution product of pressure and conjugated velocity spectrum:

I(ω) = p(ω) • v * (ω) (6.1.6)
The latter is true in a quiescent case. In presence of a mean flow velocity v 0 , associated mean Mach number M 0 and under the assumption of linear acoustics, Morfey's formulation [START_REF] Morfey | Acoustic energy in non-uniform flows[END_REF] is used (where the dependency on ω is implied):

I = 1 2 Re (1 + M 0 2 )p • v * + ρ 0 v 0 |v| 2 + M 0 ρ 0 c 0 |p| 2 (6.1.7)
The acoustic power P is quickly determined as the surface integral of the intensity:

P = S < - → I • - → n >dS (6.1.8)
with S a surface of normal -→ n and < > the time average. Similarly to the SPL, the Sound Power Level (PWL) is given by: PWL = 10 log 10 P P ref (6.1.9)

where P ref is the reference power equal to 10 -12 W.

2D -Grazing acoustic wave

For this 2D validation, experimental data from the previous work of Jones [START_REF] Jones | Benchmark Data for Evaluation of Aeroacoustic Propagation Codes with Grazing Flow[END_REF] are taken as example. This experiment was motivated by the need of a complete and trustful set of Grazing Incidence Tube (GIT) data which could be used for validation of aeroacoustic duct propagation codes, allowing improved acoustic liner design. The wide data set of different Mach numbers and frequencies was used, for example, in a recent impedance eduction code validation from ONERA [START_REF] Primus | ONERA-NASA cooperative effort on liner impedance eduction[END_REF]. A preliminary study of this configuration was published in [START_REF] Casadei | Time-Domain Impedance Boundary Condition Implementation in a CFD solver and validation against experimental data of acoustical liners[END_REF] during this Ph.D. work. Herein we show consolidated results, specially for the Navier-Stokes computations. The experimental setup used by Jones is pictured in Figure 6.1: a rectangular channel with an acoustic liner mounted on top and microphones on the lower wall to capture the instantaneous pressure. This validation was crucial during the present developments, since similar results from the Discontinuous-Galerkin aeroacoustic code (DG-LEE) used by Monteghetti were available for comparison. The advantage of using these data is that the impedance model was exactly the same adopted herein: the same ODR coefficients (poles and weights) of Monteghetti have been used (namely βD and βE of Table 2 in ref. [START_REF] Monteghetti | Energy analysis and discretization of nonlinear impedance boundary conditions for the time-domain linearized Euler equations[END_REF], also reported herein in Table A.1). This permits to determine possible differences only due to the numerical implementation and not to the impedance modelling. In the experiment, the duct is three-dimensional, but here it is modelled by a 2D rectangular duct. Geometrical features are as in the original work and summarised in Table 6.1.

Duct length

L x (mm) Duct height L z (mm) Inter. length L 1 (mm) Inter. length L 2 (mm)
Cut-off frequency (Hz) GIT 812.8 51 203 609 3380 Table 6.1: GIT duct geometrical details.

The numerical layout and its boundary conditions are presented in Figure 6.2. A continuous plane wave of 130 dB (fluctuation of approximately ± 63 Pa) is injected from the left side with an inlet characteristic boundary condition given by (3.3.1), with a perturbation term as discussed in Section 3.4.1 and a Giles relaxation form as described in 3.5. Meanwhile the Euler (no flow) case did not pose any particular issue, Navier-Stokes simulations required a particular attention on non-reflecting boundary conditions in presence of a sheared boundary layer. Thus, a Giles treatment proved to be essential for ensuring a true non-reflecting boundary condition while managing the incoming waves due to physical reflections at the impedance discontinuity. The plane wave is exiting from the duct right side through the non-reflective outlet condition of 3.3.5. As in the experiment, the sound level injected was adjusted in a way that, considering the waves reflections generated in the duct (i.e. at the impedance discontinuity at the upstream edge of the liner), the measured level at the entrance was exactly 130 dB. The TDIBC covers three fifth ( 3 /5) of the top boundary and is modeled with the multipole discretization of Section 5.1. All others boundaries are rigid wall, with slip or no-slip condition depending if solving Euler or Navier-Stokes equations, respectively. Two structured rectangular grid are generated: one for the no-flow (Euler) case and one for the flow (Navier-Stokes) case, the latter requiring a boundary layer refinement. The same grid is used for all frequencies, so that the low frequencies are theoretically over-meshed. However, convergence studies showed that this grid was suited for all frequencies considered. At 3 kHz, sound waves are captured using 40 points-per-wavelength, thanks to a preliminary grid of 300 elements in the axial direction. After a first analysis in the Navier-Stokes case, it was found that the inlet/outlet conditions were located too close to the acoustic liner and their interaction had a negative effect on the solution convergence speed. Thus, the total duct length was doubled, resulting in 600 cells for the Navier-Stokes case. In the normal direction, 20 elements are taken for the Euler uniform grid to maintain an almost unitary aspect ratio, and 60 elements for the Navier-Stokes grid with boundary layer refinement. These grids are shown in Figure 6.3. In regards to this strong attenuation, a test has been made for assessing the level of accuracy that could be achieved. The no-flow (Euler) case at 600 Hz is selected for this exercise. The injected amplitude was reduced from 130 dB down to 15 dB, which after attenuation (≈ 11 dB) reached an amplitude of 4 dB, or around 5 • 10 -5 Pascal. This is shown in Figure 6.7, where the difference in SPL in regards to the sound level injected is shown for 130 and 15 dB cases. In Figure 6.8, the pressure signal is shown before and after the liner for this sound level case. A perfectly converged solution of the acoustic signal is obtained in the simulation at atmospheric pressure (101325 Pa) with fluctuation of the order of 10 -5 Pa after the acoustic treatment. Hence, the CFD code shows to achieve a precision to the fifth decimals when solving Euler equations. Furthermore, this result confirms the correct implementation of non-reflecting boundary condition for inlet and outlet. In fact, no artificial dissipation (such as with grid stretching) was used in this analysis and no spurious reflections were found. This result, however, should be taken with care. The case under study is a very simplified benchmark case with a 2D uniform geometry and solving Euler equations, far from a realistic configuration with an unsteady flow perturbation. Furthermore, the signal phase is presented in Figure 6.9 (only for the 3.0 kHz case) and showed a quasi-perfect agreement both with experiment and previous DG-CAA calculation of [START_REF] Monteghetti | Energy analysis and discretization of nonlinear impedance boundary conditions for the time-domain linearized Euler equations[END_REF].

3D -CANNELLE

The CANNELLE test bench has been developed by Airbus to experimentally study the sound propagation and attenuation of acoustic modes in a lined cylindrical duct, in view of aircraft nacelle design. Measurements from this bench have been used to present an impedance eduction technique [START_REF] Lavieille | Impedance eduction of liners in no-flow condition and based on multimodal excitation[END_REF]. This methodology aimed at being used in multi-modal excitation of the liner, in order to take into account multiple tones and propagating modes. Preliminary results on acoustic attenuation in this benchmark were published during this Ph.D. work in [START_REF] Casadei | Towards sound absorption in a cylindrical lined duct using CFD with time-domain impedance boundary condition[END_REF] using elsA CFD solver and in [START_REF] Naïr | Industrial-scale time domain modelling of acoustic surface treatments for aero-engines using discontinuous Galerkin method[END_REF] using a CAA solver and a different time-domain impedance model. In here, the analysis is further extended to the mode decomposition study and the addition of a laminar Navier-Stokes flow. The experimental layout is shown in Figure 6.10. The cylindrical bench is made of different modular parts, of which the central "Test barrel" can be acoustically treated.

An acoustic source controlled by 50 loudspeakers is mounted upstream or downstream the test section, in order to simulate a bypass duct or engine inlet environment, where the acoustic perturbation is propagating along or against the flow, respectively. amplitude and then their attenuation through the lined section. Different liners were tested in the experimental campaign. Two of them are herein considered for validating the CFD: a bypass SDOF liner by Aircelle (ACL) and a nacelle inlet SDOF liner designed by Airbus (AIB). They are presented in detail in Section 5.2, altogether with their ODR discretized model.

Numerical layout

Geometrical features retained for the numerical study are as in Table 6.2. A 3D cylindrical geometry, with a central acoustic treatment and two hard walls up-and downstream, was meshed. Since the experimental rig was modular, two acoustic liners of different lengths could be mounted. Consequently, the ratio of liner length over cylinder diameter is 0.5 and 1.0 and the overall duct length is 825 mm and 1000 mm, for AIB and ACL configurations respectively. This geometry, which represents faithfully the true dimensions of the modular parts, is the same used in generating the numerical design. The choice of using such a relatively short duct was made possible thanks to the Giles' non-reflecting boundary conditions of Section 3.5 for inlet and outlet. In this way, the mesh elements were kept at a minimum, without having to include extra stretched zones (or sponge zones), usually required for ensuring non-reflectivity. These conditions, which will later prove to perform greatly, are an essential tool for reproducing enclosed configurations at a reasonable computational cost. A structured multi-block grid is created with ANSA pre-processor. The chosen block strategy is shown with red lines in Figure 6.12: a O-H grid splitted in three sections corresponding to the hard walls and liner. Within ANSA, a smoothing step of 0.5 was run for 500 iterations on the O-H grid in order to smooth the corners of the central block and reduce the skewness in these joints. This can be seen in detail in Figure 6.13. Two groups of grids are generated: a uniform grid for the no-flow (Euler) case, and a grid uniform in azimuthal and axial directions, but with refinement in radial direction for boundary layer treatment in the flow (Navier-Stokes) case. Following the grid convergence results on acoustic wave propagation shown in Section 4.4, between 30 and 40 points per wavelength are taken in axial, azimuthal and radial directions. In the full duct length (1 meter), approximately 4.5 wavelengths are included. If 40 points per wavelength are considered, a total of 180 elements is found. The three equal segments (two hard walls and liner) have then 60 elements each. If the same reasoning is done on the azimuthal direction, considering the outer perimeter as the total length to discretize, we have:

p = 2πr ≈ 1 m (6.3.2)
hence, again, 180 elements on the total azimuthal direction are taken, subdivided into the four blocks of 45 elements each. For the radial resolution, the block structure must be looked at in detail. The whole radius is made of half the distance of the central block plus the radial direction of the outer block. With the same reasoning, we can find the total number of elements in the radial direction:

N R = 40 • r λ ≈ 32 (6.3.3)
But since 22 elements are taken from the half azimuthal direction already defined, only 10 points would be needed in the radial direction of the outer block. Anyway, 15 points have been taken as conservative approximation. It is obvious that three grids are needed for the three different frequencies, all calculations being function of the wavelength. Concerning the boundary layer, the Blasius solution for laminar flows is used for a crude determination of the boundary layer thickness δ 99 (where the velocity is 99% of the free stream velocity u 0 ):

δ 99 ≈ 5 νx u 0 ≈ 2 mm (6.3.4)
Thus, the first boundary cell size in the radial direction ∆r min is taken 0.125 mm high and a growth factor of 1.15 is used for the grid normal extrusion. These choices yield to approximately 25 points in the boundary layer, which is reasonable when solving uRANS equations in

Post-processing

As introduced in Section 1.3, the matching problem between CFD and CAA domains is complex. For the reasons exposed, CFD aerodynamic data cannot be directly fed to an acoustic simulation. Therefore, the Single Plane Pressure-Velocity matching method (SPPV-method), derived from [START_REF] Ovenden | Mode-Matching Strategies in Slowly Varying Engine Ducts[END_REF], is used in ACTRAN iTM software to convert unsteady, compressible CFD results in acoustic modes. This method relies on the knowledge of acoustic pressure and velocity on a single plane-section of a duct to determine its modal content. It samples the CFD data on the plane, Fourier-transform and project them onto a modal basis made of duct acoustic eigenfunctions, separating the amplitude values for incident and reflected modes. It adopts the non-dissipative wavenumber formulation as in (2.1.21) and takes into account the mean flow when computing modes intensities as in (6.1.7). Although this method is mathematically robust, it depends on parameters that can not be set a priori, as for example the plane location, and that can affect the result. If the plane is too close to the acoustic source, this will generate an overestimation of the modal amplitude, as nonlinearities are predominant in this region and the linear acoustic model is no longer valid. On the other hand, a too distant plane from the source would bring underestimations given by CFD numerical scheme dissipation of the acoustic information, vanishing as it travels from the acoustic source. Two modal detection rings are replicated from the experiment with two numerical surface outputs at the middle of each hard wall section, represented in Figure 6.11 by green lines, where instantaneous primitive variables are extracted at each iteration. In this way, the SPPV planes are equally distant from the acoustic source and the acoustic treatment.

The unsteady data is converted into complex amplitude and intensity of incident and reflected modes on each surface (i.e. the matching plane). This was done with the acoustics toolbox Actran iTM. It must not mislead the meaning of "incident" here. Taking as example the representation in Figure 6.15 (same as Figure 2.2), with the hypothesis that the acoustic source is on the left boundary, incident and reflected modes on a duct section correspond respectively to A + A and A - A for Side A (left), A + B and A - B for Side B (right). When talking of incident and CANNELLE experimental results are given in terms of Transmission Loss (TL), the difference in decibels between incident and transmitted acoustic intensities in the treated case. During the experiment, hard-walled configurations (without liner) were run for each mode to verify that the TL was null, hence confirming no acoustic losses were found. In the present numerical study, only few configurations were tested in a hard-wall layout for the same verification purpose, which proved confirmed. Before giving the formulation of TL (in (6.3.10)), an important observation must be made. As already introduced, loudspeakers can control the azimuthal order injected in the experiment (through a phase controlling process), but not the radial order. For this reason, each experimental run was done at a given azimuthal order only and showed the energy on all propagative radial orders. Modal detection techniques, however, can provide the distribution of a given injected signal on all propagative modes (azimuthal and radial). The acoustic perturbation injected in the CFD is expressed for a single azimuthal and radial order as in (3.4.8). To allow a comparison between CFD and experiment, a recombination of CFD results for the different radial modes is done throughout a weighting-like post-processing to fit the modal content of the experiment on the source generation side. Practically, CFD simulations are run by imposing a constant pressure amplitude |A mn | of 120 dB for a specific azimuthal and radial mode (m, n). This means having a different injected intensity level for each mode, since the latter does not scale only with A mn , but it is also function of the modal shape Ψ mn :

< I mn >= |A mn | 2 π Sρ 0 c 0 N mn Ψ ± mn |k -M k ± z,mn | 2 (6.3.5)
where S is the reference surface (the circular section), "< >" is here the spatially averaged value, N mn and Ψ ± mn are parameters linked to the modal structure calculated as (referring to the nomenclature used in Section 3.4.2):

N mn = Rext R int r [B mn J mn (k r,mn r) + C mn Y mn (k r,mn r)] 2 dr = = 1 2 r 2 - m 2 k 2 r,mn [B mn J mn (k r,mn r) + C mn Y mn (k r,mn r)] 2 r=Rext r=R int (6.3.6) Ψ ± mn = Re M 0 k 2 + k(k ± * z,mn -M 0 2 k ± z,mn ) (6.3.7)
Under the assumption of linear acoustics, the intensity levels can then be scaled to those detected in the experiment. In view of verifying this recombination process for each separate mode, it is not suitable to run CFD simulations including all radial orders of a single azimuthal order as in the experiment, so that each mode has to be injected singularly (i.e. one simulation run per mode). In the following, intensity variables I are considered as spatially averaged intensities < I > as given by (6.3.5) and taken in their dimensional form [W/m 2 ] for avoiding logarithmic expressions. A scaling factor I SF (m,n) is defined for each injected mode (m, n) as the ratio between the intensity of incident modes in the experiment I exp,inc (m,n) and the one computed in the CFD I CFD,inc (m,n) :

I SF mn = I exp,inc mn I CFD,inc mn (6.3.8)
Thereby, the transmitted intensity level after the treatment is scaled with this factor to be comparable to the experimental acoustic level. For each simulation run at mode (m, n), all the contributions of propagative modes (m, n ) are taken into account. For instance, when injecting the mode (0, 2), the transmitted propagative modes (0, 1), (0, 2), (0, n ) will be considered in the scaled transmitted intensity I sc,trn (m,n) : where the numerator could be also considered as a recombination of the CFD using (6.3.8). For further clarity, the previous steps can be summarised into:

I
1. Compute a Scaling Factor I SF mn between incident intensity in the experiment and CFD.

2. Include in the CFD transmitted intensity all propagative (mn ) radial modes for a determined (mn) simulation.

3. Scale the latter CFD transmitted intensity with the previous Scaling Factor. [START_REF] Bruneau | Fundamental of Acoustics[END_REF]. Calculate the Transmission Loss as (logarithmic) difference between the total intensity at a given azimuthal order, including all radial orders, between experimental incident intensity and scaled CFD transmitted intensity.

Each simulation is run with a temporal resolution of 100 timesteps per acoustic period, and the last period is extracted and post-processed for modal decomposition. A sensitivity study was conducted on the number of periods needed to reach a converged TL, and one period showed to be sufficient for most cases. This is true if the instantaneous pressure signal is sufficiently converged, i.e. the mean pressure value is constant.

CFD results without mean flow

In this section, we present results without mean flow obtained solving Euler equations in the CFD solver. The calculation is initialized with a defined static thermodynamic condition and run for 40 acoustic periods (4000 iterations) to achieve convergence. The acoustic perturbation source (with Giles treatment) and acoustic liner condition are active from the first iteration, what will be shown impossible for Navier-Stokes equations in the next section. First, a qualitative analysis in Figure 6.16 shows converged solutions of modes propagation and attenuation in the cylindrical duct case ACL (cf. Figure 6.11 for the configuration definition) at 3500 Hz.

The spinning acoustic pattern injected on the left side propagates rightwards, it is attenuated in amplitude through the central lined section and leaves the right outlet boundary without numerical reflections. At a first glance, one can immediately spot few interesting acoustic properties. In the high azimuthal modes cases (m ≥ 4), periodic "circles" appear on the lateral surface near the injection. They are thought to be caused by interference between the incident mode and the modes reflected at the impedance discontinuity at the liner's edge. This generates oscillations in the sound level (SPL) preceding the liner, as also proven by 2D results in Figure 6.6 (cf. range x ∈ [0, 200] mm).

Another noticeable point is that, for some modes, the sound level seems increasing after the acoustic treatment. This is clear for modes like m0n2 or m2n1, among others. An explanation was found and is shown in Figures 6.17 and 6.18, where axial sections of the two cylindrical ducts are given. In this particular case, the mode m0n2 at 3500 Hz is taken under consideration, thus there is no azimuthal variation and any axial section is equivalent. The contour is colored with acoustic pressure fluctuations, while black solid lines represent the modes nodal lines, i.e. where the acoustic fluctuation is null. Vertical dashed lines define the acoustic liner boundaries. In the first third of the duct, the acoustic modal structure is mostly preserved, with almost vertical and horizontal node lines, as expected theoretically. Some reflections due to the impedance discontinuity are seen in the ACL case. In the second duct portion, the liner Figure 6.17: Axial section of AIB duct case -3500 Hz, mode (m0n2), Mach 0. brings in the acoustic attenuation and signal deformation. However, at this stage, its effect is mostly evident only near the boundaries and not in the duct centerline, where the acoustic level is comparable to the one at the source. In the last hard-walled section, the acoustic structure is deformed from its original state, both in amplitude and modal shape. An energy redistribution happened between the inner and outer duct, hence showing an apparent sound level increase on the boundaries, as given by the previous 3D visualization. In order to establish this phenomena, the modal structure was analyzed in detail between the two rigid sections. A graphical comparison is brought in Figures 6. [START_REF] Dragna | A generalized recursive convolution method for time-domain propagation in porous media[END_REF] Next, for validating the time-domain impedance condition correctness, the Transmission Losses (TL) are evaluated. The results herein obtained with elsA CFD software are compared with the experiment [START_REF] Lavieille | Impedance eduction of liners in no-flow condition and based on multimodal excitation[END_REF] and the recent numerical results of Naïr in [START_REF] Naïr | Industrial-scale time domain modelling of acoustic surface treatments for aero-engines using discontinuous Galerkin method[END_REF]. In [START_REF] Lavieille | Impedance eduction of liners in no-flow condition and based on multimodal excitation[END_REF], the experimental bench was first presented and an inverse method described for educing the acoustic liners impedance under a multi-modal excitation. The grazing flow was not considered in the eduction method, and it was proposed as further improvement of these measurements. In [START_REF] Naïr | Industrial-scale time domain modelling of acoustic surface treatments for aero-engines using discontinuous Galerkin method[END_REF], the timebased CAA commercial software Actran-DGM is used. It solves Linearized Euler Equations in a Discontinuous-Galerkin formalism, and it considers a prescribed boundary layer as measured in the experiment. Then, a Triple-Plane Pressure method is used to perform the modal decomposition of instantaneous values, instead of the SPPV method previously detailed for this work. Both numerical results are obtained adopting the same acoustic treatments showed in Figures 5.3 and 5.4, but different time-domain models (cf. (2.3.13) for the impedance model in [START_REF] Naïr | Industrial-scale time domain modelling of acoustic surface treatments for aero-engines using discontinuous Galerkin method[END_REF]). This fruitful comparison allowed a higher fidelity validation of both numerical implementations, rather than solely basing it on the experimental measurements. Although no comprehensive uncertainty study was performed on this very complex bench, the experimental repeatability has been assessed around 1 dB, apart for unexplained instabilities on some modes and at high Mach number (Mach 0.6, not discussed in this case). The overall accuracy can be thus estimated around 1 dB on the selected measurement points used in this work. In all the following analyses, the CAA-LEE numerical results will be taken as the reference for two reasons. First, because this code showed to be accurate on several academic and industrial validations. Second and as a reminder of the present development's objective, to show the capabilities of a CFD software to reproduce what is usually achieved with a CFD/CAA coupled technique. Figure 6.21 shows the Transmission Losses for the AIB inlet liner at Mach 0, for frequencies of 1600, 2600 and 3500 Hz. A great match is found between the two codes for most of the modes at all frequencies, with only few cases at high azimuthal order where a slight higher attenuation from the CAA is found (maximum deviation of 2 dB). These cases are also presenting the strongest differences with the experiment (lower than 4 dB), but given the difficulty in reproducing such complex acoustics environment, both experimental and numerical results can be considered a close representation of the reality. The Transmission Losses of the ACL liner case at Mach 0 are shown in Figure 6.22. Stronger differences are found for high order modes, where elsA presents stronger attenuation levels. However, an overall satisfying accuracy was found in lower order modes between the two codes. One sensitive difference between the two numerical approaches is the previously detailed assumption of averaging the impedance resistance curve, since it was impossible to capture this phenomena at this stage of the TDIBC development (cf. Section 5.2). In fact, in this case without flow, the ACL liner presents nonlinearities which are not modelled in neither numerical solvers, hence providing a further justification of the differences between experimental and numerical data. Finally, it must be reminded that this SDOF liner is conceived to operate in a grazing flow condition, making experimental and numerical results without flow not comparable to a realistic application.

CFD results with mean flow

In this section, we present the results obtained solving laminar Navier-Stokes equations with the previously determined boundary layer (cf. Figure 6.14). It follows the preliminary results presented in [START_REF] Casadei | Towards sound absorption in a cylindrical lined duct using CFD with time-domain impedance boundary condition[END_REF], where an uniform flow and only the exhaust flow case were considered. As demonstrated by Naïr [START_REF] Naïr | Industrial-scale time domain modelling of acoustic surface treatments for aero-engines using discontinuous Galerkin method[END_REF], a sheared boundary layer, even if imposed numerically while solving LEE, has a non negligible effect on the sound absorption. In the time available, simulations were run only at 1600 Hz and 2600 Hz for both liners. It is remembered from Figure 6.11 that the flow velocity has the same direction as the acoustic propagation in the ACL exhaust case and the opposite direction for the AIB inlet case. As a consequence, the acoustic wavelength will be modified by the mean flow u 0 as:

λ = (c 0 ± u 0 ) f
resulting in a larger (ACL) or shorter (AIB) wavelength than in the stationary case. For simplicity, this was not considered in the grid generation, in which it was kept the axial resolution independent on the mean flow. This will translate to a grid over-refined for ACL (≈ 50 points per wavelength) and under-defined for AIB (≈ 30 points per wavelength) in the axial direction. This resolution is still acceptable in industrial applications and proved to not have a visible impact on the final result. When solving Navier-Stokes equations compared to Euler, the computational time and case complexity sensitively increased. This was an ideal environment where testing the efficiency of the implicitation method detailed in Chapter 4. First, it was verified that the implicit scheme for NSCBC ensured a robust stability of inlet and outlet non-reflective boundary conditions. In fact, when including a laminar mean flow, a proper converged and non-reflective solution of the acoustic properties showed to be hard to achieve, if not impossible without the implicit formulation. For obtaining a converged solution of the results shown in this section, the following procedure was performed:

1. A preliminary unsteady simulation was run for 50000 iterations without dual-time stepping method, without acoustic perturbation and without liner. This was governed by a CFL number of 100, which equals to a timestep of 3.4 • 10 -5 seconds. In this way, a faster convergence towards a stabilized duct flow was obtainable. Since no acoustic perturbation was included at this stage, Giles treatment was not needed and only the typical NSCBC relaxation parameter (named σ in (3.3.2)) was used.

2. The time algorithm was then switched to the dual-time stepping method and run for 10000 iterations, reducing the timestep to the acoustic timestep equal to 6.25 • 10 -6 seconds.

3. The acoustic perturbation was then activated, and with it the Giles non-reflective treatment and NSCBC implicitation scheme. In order to achieve a stabilized regime with the Giles time-filtered operator, around 150 acoustic periods were needed (15000 iterations).

4. Finally, the implicit liner condition was activated and required an additional 150 acoustic periods (15000 iterations) to ensure convergence for each acoustic mode.

These steps are illustrated in Figure 6.23 and compared to an Euler calculation of the previous section, where only 40 acoustic periods (4000 iterations) could ensure a solid convergence.

In order to establish the computational time difference brought by the new implicit scheme, Step 3: Acoustic Pert.

Step 4: Acoustic Liner two simulations were run for the 1600 Hz ACL flow case with and without implicitation of the NSCBC and TDIBC boundary conditions, but always maintaining a global implicit dualtime stepping scheme for the solution update (i.e. the Step 4). Simulations were run with 112 processors, which means approximately 17.000 elements per processor. In order to achieve a convergence of 10 -3 in the dual time-stepping method, the non-implicit case needed no less than 26 sub-iterations (per iteration), while the implicit case immediately dropped to 6 sub-iterations: an acceleration of 4 times in the total computational time. Thanks to this remarkable result, steps 3 and 4 combined required approximately 1 day to obtain the final result with acoustic treatment, whereas without implicitation they would have taken 4 days, provided that the global convergence would have been achieved in the same number of iterations. Steps 1 and 2 took approximately 1 day, making a total simulation time (from Step 1 to 4) of 2 days for each mode.

In fact, it is remembered that the solution of Step 2 was used as starting point for each acoustic mode injection (m0n1, m0n2, ...) in Step 3.

With the aforementioned numerical method, Transmission Losses are calculated in the CAN-NELLE duct for both liners and flow cases and shown in Figures 6.24 and 6.25, for respectively the AIB and ACL case. A good agreement is found for both cases at all frequencies and modes. Some remarks on the AIB case are the following:

• The azimuthal mode m0 at 1600 Hz in the experiment gave a negative TL, thus it is not considerable for comparison. Nonetheless, the present CFD and CAA from [START_REF] Naïr | Industrial-scale time domain modelling of acoustic surface treatments for aero-engines using discontinuous Galerkin method[END_REF] are in good agreement.

• The azimuthal mode m1 for 2600 Hz was unexpectedly low for the CFD and an explanation was not found in the time available.

• CFD and CAA are in good agreement for the highest azimuthal orders where the sound attenuation is the strongest, confirming a good modelling of the TDIBC with a grazing flow in the opposite direction than the acoustic propagation.

Similarly, the following comments are made for the ACL case:

• A better agreement is found for the flow case than the previous no-flow case, reinforcing the idea that calculating sound absorption with this liner without grazing flow is not suitable.

• Great accordance (lower than 1 dB) was found between the CFD and CAA for high azimuthal modes (m4 and m5), with slightly higher differences for the mode m6.

• Concerning the high azimuthal modes and their strong sound reduction, a proper numerical convergence was hard to achieve. The latter showed to play a significant effect on the calculation of the Transmission Loss, hence confirming that an appropriate numerical method without spurious reflections is essential for accurate modes attenuation analyses.

Finally, for assessing the impact of the TDIBC on the total computational time, two simulations were run, one with a hard wall and the other with a TDIBC. The simulations were run for 7000 iterations for the ACL flow case at 2600 Hz, thus using a grid of 9.8 million elements (from Table 6.3), with 224 processors. The wall clock times were 12.5 hours and 13 hours for respectively the hard and lined wall cases, with a difference between the two of approximately half hour. This is considered acceptable in industrial applications for such a complex simulation of a realistic acoustic environment. In this Chapter, a continuation of the recent work of Daroukh et al. [START_REF] Daroukh | Shock Wave Generation and Radiation from a Turbofan Engine Under Flow Distortion[END_REF] is carried out. There, the same solver elsA is used for simulating shock-wave generation and propagation in a UHBR aircraft engine. A perfect rotor is considered, hence with no stagger angle variations. However, both a baseline flow and a distorted flow (angle of attack of 0 and 15 degrees, respectively) are evaluated and their influence on the shock generation and radiation is highlighted. These simulations are carried out with a coupled CFD/CAA methodology, inspired by the work of Thisse [START_REF] Thisse | Prévision du bruit d'onde de choc d'un turboréacteur en régime transsonique par des méthodes analytiques et numériques[END_REF][START_REF] Thisse | Generation and Propagation of Multiple Pure Tones Inside Turbofans at Transonic Regime[END_REF]. Throughout this Chapter and for distinction from aeroacoustic codes, the notation CAA must be intended as a CFD simulation solving Euler equation. When dealing with shockwaves generation and propagation, the consideration of a rotating fan in an acoustic domain (i.e. with a spatial resolution suitable for acoustic propagation) is costly for industrial applications. This is especially true if the radiation of acoustic waves has to be correctly captured over a long distance outside the nacelle. Furthermore, viscous terms, turbulence modeling and boundary layer points would be uselessly resolved in the outer propagation domain. For this reason, using a coupled CFD/CAA methodology is common practice for reducing the computational cost. This chained strategy also allows to separately investigate shock waves generation and propagation mechanisms by defining the CFD/CAA chaining interface very close to the fan, where the propagation of shocks is highly non-linear. In this way, it allows to efficiently perform the optimization of an acoustic liner without having to recompute the noise sources (i.e. the rotor shocks generation). A further advantage of this strategy is the possibility for a collaboration between Engine and Aircraft manufacturers, where the rotor shock generation would be computed by the first and their propagation through and outside the intake by the second, without exchanging the sensitive rotor geometry but only the rotor shock flow field. Estimating the effect of an acoustic liner was suggested in the paper [START_REF] Daroukh | Shock Wave Generation and Radiation from a Turbofan Engine Under Flow Distortion[END_REF] as future study, where the shock generation could be retrieved and directly applied on a new CAA simulation. This is what is tackled in this Chapter. The ODR-based TDIBC is applied on the same CAA simulation of the baseline case 125 (no flow distortion) to provide information on the sound absorption of such complex but realistic geometry. It is worth to remind that an advantage of having a liner condition in a CFD solver is that the acoustic liner could be included starting from the CFD simulation with rotating fan geometry. In this way, the shock generation could be investigated with inclusion of the sound reduction effect. However, in the time available, only the CAA simulation is replicated with inclusion of acoustic liner, and the shock generation with TDIBC will be part of future studies.

Numerical setup

The ASPIRE 1 turbofan model is taken into consideration. It consists of an asymmetric intake with a small drop, 16 rotor blades, and 36 OGVs (Outlet Guide Vein). For numerical simplicity, the drop is neglected and the simulation is performed with an axisymmetric intake, shown in Figure 7.1. This assumption proved to have negligible effects on the shock generation and propagation [START_REF] Daroukh | Shock Wave Generation and Radiation from a Turbofan Engine Under Flow Distortion[END_REF]. The target bypass ratio is equal to 16 and the in-flight minimum thrust was chosen to be representative of a modern mid-sized passenger aircraft, with a nacelle radius of around 1 meter. The fan rotation speed is taken at 108% of nominal regime, at a flight Mach number of 0.27, with a maximum relative Mach number of 1.14 on the blade tip. The same numerical scheme as in Chapter 6 is used, with the only difference of using a Van-Albada limiter to correctly resolve the shock-wave discontinuity. In both CFD and CAA simulations, the computational domains are restricted to the fan only, and the nacelle is approximated by a duct with suited inlet and outlet boundary conditions to ensure the target mass flow values.

Preliminary CFD simulation -shock generation

The preliminary CFD simulation of the rotor, carried out by Daroukh in [START_REF] Daroukh | Shock Wave Generation and Radiation from a Turbofan Engine Under Flow Distortion[END_REF], is presented in order to give the context on the coupled CFD/CAA strategy. A single channel grid was computed (equal to one 16th of the whole engine), periodic conditions were applied on the azimuthal boundaries and a stretching zone was added in the flow inlet to limit numerical reflections. An artificial long cylindrical axis was considered to avoid degenerate elements in 1 Aerodynamic and acouStic for high-by-Pass ratIo tuRbofan intEgration the meshing. This computational domain is shown in Figure 7.2, where the blade geometry is included in the volume with a no-slip wall condition. The Reynolds-Averaged Navier-Stokes (RANS) equations with k-turbulence model of Launder and Sharma [START_REF] Launder | Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disc[END_REF] were solved in a rotating frame, hence shocks are stationary in this simulation frame. After convergence was reached, a section is extracted for the successive CAA computation. Such section is located at approximately half a chord from the rotor leading edge at 95% of the blade height. The chaining strategy is detailed in the next section.

CAA simulation layout -shock propagation

The same elsA CFD solver is used again, solving now the nonlinearized Euler equations, since viscous effects are negligible in shock propagation. The CAA domain for a single channel is the same as in [START_REF] Daroukh | Shock Wave Generation and Radiation from a Turbofan Engine Under Flow Distortion[END_REF] and it is shown in Figure 7.3. The grid, kindly provided by Daroukh, is designed to correctly propagate upstream-travelling waves up to the third BPF (BPF3) in the nacelle inlet region. For this purpose, a minimum of 30 points per wavelength are taken in the axial and azimuthal directions. Stretching zones are added at the domain's end to avoid reflections. With these characteristics, the single channel grid counts 23 million points. Periodic conditions are applied on azimuthal boundaries. A far-field condition is imposed in the outer boundaries with the conservative variables given in Table 7.1. A slip-wall condition is applied on the nacelle, spinner and lower span of the central axis (i.e. the "cylindrical" boundary adjacent to the spinner and extending until the far-field). As already said in the previous section, the latter is a non-physical condition necessary for creating a structured grid without degenerated elements. A specific boundary condition is developed in this work for the coupling interface near the fan. A non-reflecting type of boundary condition is used, solving 1D characteristic equations from Thompson [START_REF] Thompson | Time dependent boundary conditions for hyperbolic systems[END_REF] with a prescribed conservative variable field. Thisse in [START_REF] Thisse | Generation and Propagation of Multiple Pure Tones Inside Turbofans at Transonic Regime[END_REF] used a similar approach, where a pressure field was transformed into a conservative variable field with use of conservation equations and an isentropic hypothesis, then injected with use of a non-reflecting characteristics boundary condition [START_REF] Thisse | Numerical Simulations of Shock-Wave Propagation in Turbofan Intakes[END_REF]. In fact, if a classical NSCBC outlet condition as in 3.3.2 was used, the pressure (or its time derivative) would be the only variable to give in input. Even though the pressure is obtainable from conservative variables through the assumption of ρ ρu ρv ρw ρE 1.16 108.12 0 0 252011 Table 7.1: Far-field conservative variables. ideal gas, this kind of boundary condition does not give us control on the three-dimensional velocity, which is extrapolated from the known domain. This means that the resulting velocity profile on the boundary would be a combination of the mean flow field (constant Mach 0.27) and the pressure profile, whereas there is an important contribution produced by the rearward fan (suction effect, varying in spanwise direction) which cannot be taken into account. A solution for using NSCBC would be to fully specify the ingoing characteristic wave (i.e. solving directly (3.2.13) with knowledge of the flow field), however this was not tested at this stage for time constraints. Nonetheless, in a classic CFD simulation with rotating fan blade (as the one presented in the previous section), a typical pressure outlet condition could be used downstream the fan to ensure the desired mass flow.

In the CAA domain, conservative variables extrapolated from the preliminary CFD simulation are imposed on the outlet surface, with a shifting technique for traducing the rotating frame into a stationary one. This means that, at each iteration, the imposed conservative variables are "rotated" by an angle φ, equal to the blade rotation in a timestep ∆t. This rotation is artificially done a-priori by interpolating the points for having a complete data set of a full blade passage. This amounts to 225 files describing a blade passage, differently applied at each timestep. A comparison on the density field between the extracted preliminary CFD data and the one obtained on the coupled CAA interface is presented in Figure 7.4. Almost perfect agreement is reached, demonstrating the correct implementation of the Thompson boundary condition. Practically, this condition can be compared to a "dynamic far-field" condition, where the flow state changes at each iteration. Daroukh in [START_REF] Daroukh | Shock Wave Generation and Radiation from a Turbofan Engine Under Flow Distortion[END_REF] used a similar strategy, but solving slightly different equations in a boundary condition called "Nref" (for non-reflecting). The location of such chaining interface in the 3D domain, placed shortly upstream of the fan, is shown in the left side of Figure 7.6. Because the preliminary CFD simulation included a turbulent boundary layer treatment, a pre-processing interpolation was carried out in [START_REF] Daroukh | Shock Wave Generation and Radiation from a Turbofan Engine Under Flow Distortion[END_REF] to move to an inviscid-type of calculation, i.e. for matching the two different CFD and CAA grids. A specific discussion about the acoustic liner topology must be made. In the ASPIRE intake, the liner is split in two parts to assess the respective contribution of the base liner and the lip liner. These would be located as in Figure 7.5, the red curve being the lip liner and the light blue curve the base liner. 2 The vertical black line represents the coupling interface location, also pictured in Figure 7.6 with the density contour previously discussed. As immediate conclusion, the portion of base liner at the right of the interface cannot be included in the current CAA simulation since it is outside the numerical domain. In addition, as safety precautions during this first numerical demonstrator in terms of numerical stability, some hard wall distance was considered between the acoustic injection and the impedance boundary condition. For these reasons, the entire base liner (blue curve) is not considered at this stage, and only the lip liner (red curve) will be present. This will translate to an underestimated sound reduction at the nacelle exit and could be part of further analyses in the future, with consideration of the full nacelle duct and hence the whole base liner length. It must be noted that, in the impedance law shown in Figure 5.5 representative of this liner, no Ingard-Myers correction was used. In this way, refraction effects brought by the sheared boundary layer will be missed. However, since here we are interested only in qualitative results, it is an acceptable hypothesis.

Two simulations are run: a hard-wall case (similar to the no-distortion case of [START_REF] Daroukh | Shock Wave Generation and Radiation from a Turbofan Engine Under Flow Distortion[END_REF]) and a lined-wall case. In both cases, around one and a half engine revolution (1 revolution = 16 blade passages) proved to be sufficient to reach convergence on conservative and primitive variables. A blade passage rotation being equivalent to 225 iterations, with a ∆t = 6.44•10 -6 s and an engine rotation frequency of 690 Hz, this corresponded to approximately 5400 iterations per simulation. Simulations were run on Sator, the ONERA HPC cluster, with 55 processors. These proved to be highly under-dimensioned for the grid under consideration (23 million points), but it helped a quicker development of the chaining methodology without having to deal with splitted surfaces and the relative complications of reading and allocating a splitted file. Indeed, 55 processors was the higher limit allowed by the automatic splitter tool to maintain the chaining interface as a single block. This was purely a choice of convenience for a faster development in the time available, and immediate further improvements in this chaining methodology with elsA would be on the splitting capacities. In both simulations, the computational time was strictly influenced by the number of sub-iterations of the DTS time marching scheme. With between 8-9 sub-iterations, both simulations took approximately 2.4 hours for computing one blade passage. For the total simulation (25 blade passages), a wall clock time of approximately 60 hours was achieved. The TDIBC proved to have a negligible impact on computational resources on such a large industrial application and over-dimensioned elements per processor ratio. Indeed, the TDIBC boundary points are only a small fraction of the whole computational domain.

Results

Radial surface

A surface on the axial/azimuthal plane is extracted inside the nacelle. The surface is taken at constant j-index (radial direction of the structured block), at a radial distance of 95% of the blade height, starting at the injection surface. This surface is shown in Figure 7.7, obtained by axially duplicating the single-channel domain into the whole nacelle. In the next figures, the azimuthal angle is brought in the abscissa for half revolution ([0-180] degrees) and the axial direction is in the ordinates, with the bottom of each figure (x = -0.15m) corresponding to the fan coupling interface. In Figure 7.8, the instantaneous density gradient on the axial direction is shown, identifying clearly the shock waves angle and their propagation and dissipation. It follows in Figure 7.9 a zoomed section of the same radial surface colored with a Mach number contour, where some regions at Mach 0.95 are visible. Considering that this extraction surface is placed slightly upstream the fan, it is a confirmation that on the blade tips there is a relative supersonic condition. Furthermore, it confirms the difficulty in applying a NSCBC to this transient regime. In fact, if the Table 3.1 is recalled, a supersonic outlet would be fully determined by inner points, hence impossible to specify at the boundary information entering the domain such as a rotating shock wave pattern. In Figures 7.10 to 7.12, the real part of the complex acoustic pressure is illustrated up to BPF3 for both rigid and lined cases. The sound reduction induced by the liner is evident for BPF1 and visually perceivable for BPF2 and BPF3 too (the legend colors are saturated for showing the sound reduction achieved in the lined case). In order to provide a quantitative analysis, the SPL is calculated on an azimuthal section at the same radius r 95 equal to 95% of the maximum radius, and along the whole nacelle length. It is remembered that SPL is function of the complex pressure module, a Fourier-transformed property independent of the azimuthal coordinate thanks to the periodicity of the system (cf. equation (6.1.3)), thus any other azimuthal section would be identical. Vertical dot-dashed lines show where the acoustic liner is located. The acoustic propagation is towards the negative x-axis (leftward). At the intake exit (left side of the chart), a massive sound reduction of approximately 35 dB is achieved for the first 3 BPFs. At BPF2 and BPF3, one can observe several extinctions which are known to be due to the interference of different radial modes. Those interferences are amplified for the case with liner, which modifies the modal structure as already demonstrated in Section 6.3.

Axial section

An axial section at constant coordinate z = 0 m is extracted for analysing the noise level in the nacelle and its immediate vicinity. This section is shown in Figure 7.14. It must be reminded that the grid is fine enough for propagating the third harmonic (BPF3) in the nacelle only. For such reason, the following analysis are made not further than 1.5 m in the other two directions.

A more accurate analysis on acoustic propagation in the far-field will be proposed as future continuation of this study. Figures 7.15 to 7.17 bring in light the real part of the acoustic pressure between rigid and acoustically treated cases for the three BPFs, as done previously in the radial illustrations of Figures 7.10 to 7.12. Thanks to this view, the sound reduction near the liner and outside the nacelle is clear, especially for BPF1. To better quantify such reduction, the difference between the two cases is done for each BPF and shown in Figure 7.18 in terms of SPL (dB). The blue areas present regions where the sound level is reduced, with peaks of almost 40 dB reduction from the hard wall case. Concerning the red areas of increased noise appearing in front of the engine spinner, a detailed analysis was not performed due to lack of time but an explanation is suggested. Because this simulation is axisymmetric and in order to avoid a degenerated axis or a complex topology, a small cylinder was created on the engine rotation axis with a slip-wall boundary condition. The liner is known to deform the acoustic signal structure and its directivity. May some acoustic wave travel until the cylinder, the slip-wall condition would reflect it, justifying why a stronger level is recorded in this region. With a full 360 • topology (or one without the artificial cylinder), acoustic waves from opposite sides of the nacelle might interact with each other generating constructive/destructive phenomena. This was not object of the present study and will also be proposed as future developments.

Nonetheless, in order to confirm that the liner is passive and producing a negative energy balance, the acoustic power level was calculated on 34 surfaces at axial coordinates x ∈ [-0.17, -0.5] m, from the fan injection to the nacelle lip, with the formula (derived from (6. where I -is the real part of the acoustic intensity (calculated with (6.1.7)) propagating in the Figure 7.14: Axial-meridian plane location at z = 0.

negative x axis, P ref is the reference acoustic power 10 -12 W and the factor 16 takes into account the whole engine surface (i.e. the 16 blades). This is pictured in Figure 7.19, where the lined and hard wall cases are compared to the hard wall case of [START_REF] Daroukh | Shock Wave Generation and Radiation from a Turbofan Engine Under Flow Distortion[END_REF]. The great accordance between the two hard wall cases confirms that the simulation is correctly set up. Only small differences are observed, which are most probably due to the different chaining interface boundary conditions used for the computations. Very small differences at the injection may be amplified along the propagation when the noise level is reduced. Another difference lies in the post-processing strategies used for computing the acoustic power level, which took into account different time frames at different convergence levels for calculating the DFT. When looking at the lined wall case, the important decrease in power level confirms the passivity of the TDIBC. Overall, a decrease of around 35 dB is found for the three BPFs at the nacelle exit, as calculated in the previous SPL calculation in Figure 7.13. It can also be observed that, for the hard-wall case, the power level remains almost constant from the liner downstream edge to the inlet lip. This shows that the acoustic waves entered into a more linear behaviour with reduced nonlinear dissipation.

In conclusion, this numerical demonstrator allows to affirm that the TDIBC implemented in the CFD solver throughout this work is capable of providing accurate sound absorption of complex industrial cases. It also proved its robustness when dealing with shock-wave propagation in an aircraft nacelle environment with multi-modal excitation. Its efficiency in terms of computational time makes it ready to be applied on more complex cases with a more thorough quantitative validation. in its derivation, the present implicit boundary conditions proved correctly implemented and provided fast and numerically stable solutions.

A preliminary 1D validation case, comparable to an acoustic impedance tube, was run to verify the correctness of the TDIBC against analytical results. Different analyses on the acoustic properties convergence with the selected CFD solver were made. As general practice, 40 points per wavelength and 100 timesteps per period were found to be acceptable requirements to have a well resolved acoustic waves propagation. The choice of using a Spectral Difference method in the delay resolution was strongly confirmed.

Then, the 2D GIT benchmark [START_REF] Jones | Benchmark Data for Evaluation of Aeroacoustic Propagation Codes with Grazing Flow[END_REF] was run as further validation against experimental and numerical (CAA) data. This required the inclusion of two-dimensional effects (tangent fluxes), which proved essential in the resolution of the TDIBC. A great accordance was found between the CFD results and the references for a no-flow and sheared flow case. The latter, solved in a Navier-Stokes laminar regime, turned out to be particularly complex to implement. In particular, the greatest difficulty was the coupling between the boundary layer (no-slip walls), non-reflecting conditions and the impedance boundary conditions.

An additional level of complexity was faced with a 3D validation against the Airbus industrial benchmark CANNELLE [START_REF] Lavieille | Impedance eduction of liners in no-flow condition and based on multimodal excitation[END_REF]. In there, the sensitivity of non-reflecting and impedance conditions on the convergence level was established. Sound reduction and modal decomposition was assessed with CAA tools for two different liners and two flow cases, all providing completely satisfactory results. The consideration of the implicit formalism proved essential in the total computational time of this more complex case, dividing by a factor 4 the calculation time per iteration. All Navier-Stokes computations were run in a laminar regime, as a first simplification, in order to take into account the refractive effect of the sheared boundary layer while getting rid of potential numerical stiffness caused by the RANS turbulence modelling.

Finally, a first demonstration of the TDIBC applicability on real nacelle geometry was shown in the context of supersonic rotor noise (so-called Buzz-Saw Noise), which constituted the final objective of this thesis. The ASPIRE configuration, involving prototypes of UHBR fan and nacelle, was taken as example as continuation of Daroukh's study [START_REF] Daroukh | Shock Wave Generation and Radiation from a Turbofan Engine Under Flow Distortion[END_REF]. The same numerical setup was taken: a CFD Euler simulation with a mean flow derived from a preliminary computation, with the addition of the novel TDIBC in correspondence of what would be the "lip liner" of this configuration. Different qualitative and quantitative information were produced on the liner impact on the acoustic content inside and near the engine intake. Both Sound Pressure Level and integrated acoustic power proved strongly reduced by the liner (≈ 35 dB). Thanks to the efficient implicit scheme of the liner condition, the computational time showed to be unaffected by the presence of the acoustic liner model. Finally, this prototype computation demonstrated new possibilities of collaboration between Engine and Aircraft manufacturers, where the fan noise sources could be computed by the first and coupled to the propagation and attenuation in the intake computed by the second.

Future advancements

Different axes of future developments are identified. Some are related to the extension of the Time-Domain Impedance Boundary Condition, others to improve the robustness and applicability of such TDIBC model, and further applications of the ASPIRE case are proposed.

DDOF liner model

The physics of SDOF and wiremesh liners proved to be correctly implemented in the CFD code. However, Double Degree Of Freedom (DDOF) liners cannot be discretized with the current model. Nonetheless, they are already used in specific parts of the engine nacelle and a numerical tool supporting their design would be valuable. Monteghetti already provided a mathematical representation of these more complex acoustic devices (cf. [START_REF] Monteghetti | Analysis and Discretization of Time-Domain Impedance Boundary Conditions in Aeroacoustics[END_REF], Appendix B), in which additional auxiliary functions would model the two different cavities and their combined effect. The multipole discretization validation (as done in Chapter 5) and its adaptation to the current CFD methodology would be of great industrial interest for the development of these materials.

Nonlinear liner model

The problem of nonlinear liner was introduced in Section 3.3.5. A full nonlinear liner model could take into account the real-time impedance response, i.e. the impedance law variation due to sound level variations during the simulation. A preliminary mathematical formulation was proposed in (2.5.16). The next steps would involve the determination of the impedance law of a nonlinear liner and the modification of the CFD methodology to include the additional equation for the nonlinear response. Concerning the flow effect on the impedance determination, a mathematical formulation to include impedance variations due to velocity fluctuations in a ODR reflection coefficient formulation was not yet discussed. However, for the applications mentioned in this dissertation, the wall friction velocity does not change sensitively spatially. Since the latter is supposed to be the major player in flow effects on impedance determination, we can assume it would have a negligible impact on these results. As proposition, may one be interested to include this effect, a tabulated interpolation among discrete flow velocities could be a first approach.

Numerical robustness and stability of the TDIBC in turbulent regime

Promising results were achieved with the TDIBC in a Navier-Stokes laminar regime. However, typical unsteady simulations of aircraft nacelle includes turbulence equations, such as the k -ω model, for resolving the complex turbofan physics. A rotating frame is also a typical practice in turbomachinery. The interest of applying a TDIBC in an unsteady simulation for evaluating the noise sources (generated by the fan or the Outlet Guide Vanes) would require these additional complexities. Theoretically, nothing prohibits to use the present TDIBC in these environments. A particular attention will need to be taken when pairing the NSCBC and TDIBC with wallmodeled functions. Giles non-reflecting inflow and outflow conditions, as used in this work without grid stretching nor sponge zones, will need further testing with inclusion of turbulent fluctuations. This also opens the door to a more comprehensive study on the different parameters of these boundary conditions: relaxation parameters (classic σ and Giles Γ), tangential and diffusive fluxes contribution, dual iterations convergence, among others. In order to provide an even more versatile model for industrial applications, the already mentioned adiabatic lined wall condition would also be a valuable add to this numerical tool.

Appendix C

Bessel functions

The purpose of this Appendix is to give general definitions and properties of the Bessel functions used in Chapter 2.1.2 and 3.4.2. Bessel's differential equations are often encountered when solving boundary value problems and especially when working in cylindrical coordinates. Bessel functions are canonical solutions of the Bessel equation: For non-integer orders, J m and J -m are linearly independent and composing already a set of solutions to the Bessel equation, so any additional solution would be redundant. When the order m is an integer value (hence the Gamma function has simple poles), the following is applicable:

J -m (x) = (-1) m J m (x) (C.6)
153 which means the two solutions are linearly dependent and insufficient to solve the Bessel's equation. The Bessel function of second kind Y m (x) is then looked for. This has a singularity at the origin (x = 0), is multivalued, and related to J m (x) by: Y m (x) = J m (x) cos(mπ) -J -m (x) sin(mπ) (C.7)

In the case of an integer order m z , the function is defined as its limit with the non-integer m r tending to m z : Y mr (x) = lim mr→mz Y mr (x) (C.8)

In this assumption, the same relation as in (C.6) is also valid for the second kind functions:

Y -m (x) = (-1) m Y m (x) (C.9)
Equations (C.10a) to (C.10d) resume some useful properties of Bessel functions, such their derivatives and the link between higher and lower orders. These properties, here exposed for the function of the first kind, are valid for both first and second kind. • L 1 will not be modified because of the isentropic hypothesis • L 2 will become L 2 + ∂w ∂t fix

J m+1 (x) = 2m x J m (x) -J m-1 (x) (C.10a) J m+1 (x) = 1 2 (J m-1 (x) -J m+1 (x)) (C.10b) J m (x) = - m x J m (x) + J m-1 (x) (C.
• L 3 will become L 3 -∂v ∂t fix

• L 4 will become L 4 + ∂u ∂t fix An additional hypothesis is made on the perturbations, which are supposed entering the domain following the flow streamlines, so that we can consider a local reference system on the streamlines. A unitary vector -→ e s is defined:

- → e s = ū V - → n + v V - → t 1 + w V - → t 2 (D.1)
with V = √ ū2 + v2 + w2 , and the the new streamlined-reference system with coordinates ( -→ e s , -→ e r , -→ e q ). Perturbation equations allow to compute the three velocity components v s , v r , v q and the derivatives ∂ t v r and ∂ t v q . The remaining ∂ t v s could be also analytically given, but Hixon preferred using the latter for imposing a null pressure fluctuation (p = 0 and ∂ t p = 0), so for avoiding any spurious pressure wave. This translates into: remembering that v r = v q = 0 when -→ e s is aligned with the flow. Since the flow field is known only in the inlet plane, it is not possible to compute directly ∂ s v s , and (D.2) must be used. However, in there, derivatives of the type ∂ q φ must be computed. Knowing that:

γP
∂φ ∂q = ( - → n • - → e q ) ∂φ ∂n + ( - → t 1 • - → e q ) ∂φ ∂t 1 + ( - → t 2 • - → e q ) ∂φ ∂t 2 (D.4)
the calculation of derivatives on (s,r,q) is quickly obtained. This is true for all mean properties derivatives. For fluctuations derivatives, it is possible to directly compute ∂ t 1 φ and ∂ t 2 φ, but not ∂ n φ. This because the scheme decentering in the -→ n -direction is in an upwind sense, while for the perturbation injection a downwind sense is needed. Hence, the calculation of the derivatives along -→ n , ∂ n v r and ∂ n v q is done using the momentum equations for these two components in ∂v q ∂t + u ∂v q ∂n + w ∂v q ∂t 1 + w ∂v q ∂t 2 + u ∂v q ∂n + u ∂v q ∂t 1 + w ∂v q ∂t 2 = 0

The fluctuation components (u, v, w) in the reference system ( -→ n , -→ t 1 , -→ t 2 ) are calculated directly from the known (v s , v r , v q ). Equations (D.5) allow to compute ∂ n v r and ∂ n v q . Using (D.4), one can determine ∂ r v r and ∂ q v q , while using (D.2) one can calculate ∂ s v s , which finally allows to compute ∂ t v s .

D.2 Linearization of D

In Section 4.1.3, the term D n+1,n +1 , was linearized as: D n+1,n +1 =D n+1,n + 1 /2 + ∂D ∂U (0,j) δU (0,j) + ∂D ∂U (1,j) δU (1,j)

+ ∂D ∂U (0,j-1) δU (0,j-1) + ∂D ∂U (0,j+1) δU (0,j+1) + From (4.3.7), the notation n + 1 /2 should now be familiar as an intermediate solution update where the contribute of the flux variation has not been taken into account yet. In the same context, is defined as the contribute given by a variation of ϕ on the variation δD. Now, the variation of D can be written under the form:

δD = V M D V • M d V • M L V (δQ) V + (D.1)
where V represents the 4 points (0, j), (0, j -1), (0, j + 1) et (1, j) and the notation (•) the scalar product. M i links the variation of D, d and L in respect to the primitive variable vector Q variation. The index V is dropped in the following for a better readability, but we must not forget that each variable is quadruple for the four points in V.

M D =                    ∂D 1 ∂Q ∂D 2 ∂Q ∂D 3 ∂Q ∂D 4 ∂Q ∂D 5 ∂Q                    M d =                    ∂d 1 ∂Q ∂d 2 ∂Q ∂d 3 ∂Q ∂d 4 ∂Q ∂d 5 ∂Q                    M L =                    ∂L 1 ∂Q ∂L 2 ∂Q ∂L 3 ∂Q ∂L 4 ∂Q ∂L 5 ∂Q                    (D.2)
Additionally, the variation of L 4 also depends on the variation of the auxiliary variable ϕ for the impedance condition, as explained in Section 4.3. This ϕ variation yields to a variation of d as (with A a constant):

(δd) ϕ =                  ρ c √ 2 n x √ 2 n y √ 2 n z √ 2 ρc √ 2                  Aδϕ (D.3)
Finally, we define the contribution given by the variation of ϕ on the variation of D:

=          1 0 0 0 0 u 0 ρ 0 0 0 0 v 0 0 ρ 0 0 0 w 0 0 0 ρ 0 0 1 2 (u 2 0 + v 2 0 + w 2 0 ) ρ 0 u 0 ρ 0 v 0 ρ 0 w 0 1 γ -1          (δd) ϕ (D.4)
We now want to express the Jacobians of d as function of the Jacobians of L (respectively the second and third columns of (D.2)). We remind from (3.2.17) that d = P Q L . For the first component we have:

d 1 = n x L 1 + n y L 2 + n z L 3 + ρc √ 2 (L 4 + L 5 ) (D.5)
Introducing the following notation:

Λ α = 3ρ 2 √ 2γp 0 0 0 ρ 3 2 2γp 3 Λ k = ∂L k ∂ρ (0,j)
∂L k ∂u (0,j)

∂L k ∂v (0,j)

∂L k ∂w (0,j)

∂L k ∂p (0,j) ∂L k ∂u (0,j-1) ∂L k ∂v (0,j-1) ∂L k ∂w (0,j-1) ∂L k ∂p (0,j-1)

Λ k = ∂L k ∂ρ (1,
Λ 5 = ∂L k ∂ρ (0,j+1) ∂L k ∂u (0,j+1) ∂L k ∂v (0,j+1) ∂L k ∂w (0,j+1) ∂L k ∂p (0,j+1)
it is possible to express directly the Jacobians of d i (i ∈ {1, 2, 3, 4}) in relation to the state vector Q at the points (0, j), (1, j), (0, j -1) and (0, j + 1):

                               ∂d 1 ∂Q (0,j) = n x Λ 1 + n y Λ 2 + n z Λ 3 + ρ c √ 2 (Λ 4 + Λ 5 ) + (L 4 + L 5 )Λ α ∂d 1 ∂Q (1,j) = n x Λ 1 + n y Λ 2 + n z Λ 3 + ρ c √ 2 (Λ 4 + Λ 5 ) ∂d 1 ∂Q (0,j-1) = ρ c √ 2 Λ 5 ∂d 1 ∂Q (0,j+1) = ρ c √ 2 Λ 5 (D.7)                                ∂d 2 ∂Q (0,j) = -n z Λ 2 + n y Λ 3 + n x √ 2 (Λ 4 -Λ 5 ) ∂d 2 ∂Q (1,j) = -n z Λ 2 + n y Λ 3 + n x √ 2 (Λ 4 -Λ 5 ) ∂d 2 ∂Q (0,j-1) = - n x √ 2 Λ 5 ∂d 2 ∂Q (0,j+1) = - n x √ 2 Λ 5 (D.8)                                ∂d 3 ∂Q (0,j) = n z Λ 1 -n x Λ 3 + n y √ 2 (Λ 4 -Λ 5 ) ∂d 3 ∂Q (1,j) = n z Λ 1 -n x Λ 3 + n y √ 2 (Λ 4 -Λ 5 )
∂d 3 ∂Q (0,j- we can conclusively obtain d 5 : Now that all the Jacobians of d are known, we move back to the relation between D and d (3.2.18) to find the Jacobians of D. We note with K one of the three point locations (1, j), (0, j + 1) or (0, j + 1). Since D 1 = d 1 , the first Jacobian is already known. For the other components we have:

                              
        
∂D 2 ∂Q (0,j) = u 0 ∂d 1 ∂Q (0,j) + ρ 0 ∂d 2 ∂Q (0,j) + 1 0 0 0 0 d 2 + 0 1 0 0 0 d

∂D 2 ∂Q K = u 0 ∂d 1 ∂Q K + ρ 0 ∂d 2 ∂Q K (D.14)         
∂D 3 ∂Q (0,j) = v 0 ∂d 1 ∂Q (0,j) + ρ 0 ∂d 3 ∂Q (0,j) + 1 0 0 0 0 d 3 + 0 0 1 0 0 d

∂D 3 ∂Q K = v 0 ∂d 1 ∂Q K + ρ 0 ∂d 3 ∂Q K (D.15)         
∂D 4 ∂Q (0,j) = w 0 ∂d 1 ∂Q (0,j) + ρ 0 ∂d 4 ∂Q (0,j) + 1 0 0 0 0 d 4 + 0 0 0 1 0 d

∂D 4 ∂Q K = w 0 ∂d 1 ∂Q K + ρ 0 ∂d 4 ∂Q K (D.16)                                            ∂D 5 ∂Q (0,j) = 1 2 (u 2 0 + v 2 0 + w 2 0 )
∂d 1 ∂Q (0,j) + +ρ 0 u 0 ∂d 2 ∂Q (0,j) + ρ 0 v 0 ∂d 3 ∂Q (0,j) + ρ 0 w 0 ∂d 4 ∂Q (0,j) + 1 γ -1 ∂d 5 ∂Q (0,j) + + 0 u 0 v 0 w 0 0 d 1 + u 0 ρ 0 0 0 0 d 2 + + v 0 0 ρ 0 0 0 d 3 + w 0 0 0 ρ 0 0 d 4

∂D 5 ∂Q K = 1 2 (u 2 0 + v 2 0 + w 2 0 ) ∂d 1 ∂Q K + +ρ 0 u 0 ∂d 2 ∂Q K + ρ 0 v 0 ∂d 3 ∂Q K + ρ 0 w 0 ∂d 4 ∂Q K + 1 γ -1 ∂d 5 ∂Q K (D.

D.3 NSCBC Jacobians expressions

In Section 4.2, the full implicit formulation of characteristic waves was given. In there, Jacobians of normal characteristic waves L i were found for the primitive variable. In this Appendix, these Jacobians are solved for each variable. It is remembered that, since the speed of sound is function of pressure and density as c 2 = γp ρ , we have:

         ∂c ∂ρ = - c 2ρ ∂( 1 /c 2 ) ∂ρ = 1 γp          ∂c ∂p = γ 2ρc ∂( 1 /c 2 ) ∂p = - 1 pc 2 (D.
                                     ∂L 1 ∂ρ 0 = -u n n x -u n n x ∆p 1 γp ∂L 1 ∂u 0 = n x L 1 ∂L 1 ∂v 0 = -n z u n + n y L 1 ∂L 1 ∂w 0 = n y u n + n z L 1 ∂L 1 ∂p 0 = n x c 2 u n + u n n x ∆p 1 pc 2                                      ∂L 1 ∂ρ 1 = u n n x ∂L 1 ∂u 1 = 0 ∂L 1 ∂v 1 = n z u n ∂L 1 ∂w 1 = -n y u n ∂L 1 ∂p 1 = - n x c 2 u n (D.
                                     ∂L 3 ∂ρ 0 = -u n n z -u n n z ∆p 1 γp ∂L 3 ∂u 0 = -n y u n + n x L 3 ∂L 3 ∂v 0 = n x u n + n y L 3 ∂L 3 ∂w 0 = n z L 3 ∂L 3 ∂p 0 = n z c 2 u n + u n n z ∆p 1 pc 2                                      ∂L 3 ∂ρ 1 = u n n z ∂L 3 ∂u 1 = n y u n ∂L 3 ∂v 1 = -n x u n ∂L 3 ∂w 1 = 0 ∂L 3 ∂p 1 = - n z c 2 u n (D.
∂Q                                        ∂L 4 ∂ρ 0 = (u n -c)∆p -ρ -3/2 2 √ 2γp + L 5 c 2ρ ∂L 4 ∂u 0 = (u n -c) n x √ 2 + L 5 n x ∂L 4 ∂v 0 = (u n -c) n y √ 2 + L 5 n y ∂L 4 ∂w 0 = (u n -c) n z √ 2 + L 5 n z ∂L 4 ∂p 0 = -(u n -c) 1 ρc √ 2 + (u n -c)∆p -p -3/2 2 √ 2γρ -L 5 γ 2ρc                                      ∂L 4 ∂ρ 1 = 0 ∂L 4 ∂u 1 = -(u n -c) n x √ 2 ∂L 4 ∂v 1 = -(u n -c) n y √ 2 ∂L 4 ∂w 1 = -(u n -c) n z √ 2 ∂L 4 ∂p 1 = (u n -c) 1 ρc √ 2 (D.5)

D.4 Entropic and rotational waves -linear systems for primitive variables derivation

Following the derivation done in Section 4.2.2 for the systems (4.2.11), this section aims at completing the discussion with the remaining 8 linear 3-equations systems. It is remembered that each system is quadruple itself, as it must be resolved at the index locations (0,j), (1,j), (0,j-1) and (0,j+1). For u:

                 σ u n y = -

  Mots clés: Conditions limites d'impédance en domain temporel, Représentation Oscillo-Diffusive, Bruit d'avion, Traitement acoustique, Acoustique des conduits, Propagation non-linéaire, Conditions limites acoustique, Buzz Saw Noise, Dynamique des fluides computationnelle (CFD), Conditions limites caractéristiques de Navier-Stokes (NSCBC). Introduction xxi from [103]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13 Instantaneous contours of static pressure for the (top) solid and (bottom) porous wall configurations. The sonic isoline is shown in dashed black and the impedance wall upper location is shown in dashed green -source from [23]. . . . . . . . . . . 2.14 Comparison between flow quantities in the transitional region, from x = 0.48 m to x = 0.52 m, between the solid wall case, shown in the left, and the porous surface, shown in the right. From top to bottom it is shown: Q-criterion isosurfaces colored by the Mach number in the azimuthal direction, the instantaneous heat flux to the wall, the pressure oscillations and the acoustic flux -source from [97]. . . . . 2.15 Slices of instantaneous fluctuations of u+ (left) at wall distance y=0.03, and of v+ (center) and p/Ru τ (right) at the impedance wall. Blue (red) color corresponds to negative (positive) values, and the considered range is -6≤ u+ ≤6, -1≤ v+ ≤1 and -6≤ p/Ru τ ≤6 for all cases -source from [90]. . . . . . . . . . . . . . . . . . 2.16 Effect of inlet liner on sound-wave pattern in a meridian plane. Left: hardwall nacelle; right: with inlet liner -source from [93]. . . . . . . . . . . . . . . . . . . 3.1 NSCBC scheme -adapted from [76]. . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Delay 1D advection discretization. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.

  definition ∇ nabla symbol for differential operator, ∇f := [∂ i f ] i∈ 1,d ∈ element of ( • ) scalar product tensor product of vectors, satisfies (a b) • c = (b • c)a with a, b, c ∈ R d convolution product | |norm, absolute value Laplace/Fourier transform / Giles time-filtered value vector (or matrix) transposed

Figure 1 :

 1 Figure 1: Progression of ICAO Noise Standards -source from ICAO Environmental Report.

Figure 3 :

 3 Figure 3: Airbus fleet noise evolution in relation to CAEP chapters -source from Airbus.

1 1 . 2

 112 Acoustic environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Acoustic liners specifications . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 CFD/CAA matching problem with liners . . . . . . . . . . . . . . . . . . 5

Figure 1 . 1 :

 11 Figure 1.1: Noise sources in a turbofan engine -Adapted from [3].

Figure 1 . 2 :

 12 Figure 1.2: Acoustic liner location in a turbofan engine -sources from [84, 109].

Figure 2 . 1 :

 21 Figure 2.1: Different mode shapes for circular ducts -source from [53].

Figure 2 . 3 :

 23 Figure 2.3: Example of locally and non-locally reacting materials -source from [20].

Figure 2 . 5 :

 25 Figure 2.5: Sound attenuation and reactance in a cavity of height h=0.05 m and resistance of 0 and 0.5 -adapted from [77]j is here the imaginary unit.

Figure 2 . 6 :

 26 Figure 2.6: Helmholtz Resonator conceptual diagram.

Figure 2 .

 2 11 represents a scheme of such evolution.

Figure 2 .

 2 Figure 2.11: N-wave generation -source from [102].

Figure 2 .

 2 Figure 2.15 presents some results on the different velocity fluctuations caused by different acoustic liners.

Figure 2 . 13 :

 213 Figure 2.13: Instantaneous contours of static pressure for the (top) solid and (bottom) porous wall configurations. The sonic isoline is shown in dashed black and the impedance wall upper location is shown in dashed green -source from [23].

Figure 2 . 14 :

 214 Figure 2.14: Comparison between flow quantities in the transitional region, from x = 0.48 m to x = 0.52 m, between the solid wall case, shown in the left, and the porous surface, shown in the right. From top to bottom it is shown: Q-criterion isosurfaces colored by the Mach number in the azimuthal direction, the instantaneous heat flux to the wall, the pressure oscillations and the acoustic flux -source from [97].

Figure 2 .

 2 Figure 2.15: Slices of instantaneous fluctuations of u+ (left) at wall distance y=0.03, and of v+ (center) and p/Ru τ (right) at the impedance wall. Blue (red) color corresponds to negative (positive) values, and the considered range is -6≤ u+ ≤6, -1≤ v+ ≤1 and -6≤ p/Ru τ ≤6 for all cases -source from [90].

Figure 2 .

 2 Figure 2.16: Effect of inlet liner on sound-wave pattern in a meridian plane. Left: hardwall nacelle; right: with inlet liner -source from [93].

Figure 3 . 1 :

 31 Figure 3.1: NSCBC scheme -adapted from [76].

Figure 3 . 3 .

 33 Figure 3.2: Delay 1D advection discretization.

Figure 3 . 4 :

 34 Figure 3.4: Sawtooth wave form of unitary amplitude and unitary period.

Figure 4 . 1 :

 41 Figure 4.1: Finite-volume scheme of implicit flux calculation.

  global timestep and ∆τ the local timestep;

  Rearranging the sum operator, it yields to:

Figure 4 . 2 :

 42 Figure 4.2: 1D case -CFD setup of the sinusoidal pulse of 1 Pa amplitudet = 0 s.

Figure 4 .

 4 Figure 4.3 shows the pressure signal captured by the aforementioned probe for frequencies of 1000 and 2000 Hz and compared to the exact analytical solution.A detailed analysis of the reflected signal shows that the first dampened wave comes from the liner's direct sound absorption (between 2 and 2.5 periods of time for 1000 Hz, between 2 and 3 periods for 2000 Hz), while the remaining reflected wave includes the delay effect from the cavity resonance, modifying the signal structure. Computational time is barely affected by the TDIBC in this 1D case, since the few additional calculations are done only in the single right boundary cell. Figure4.3 is obtained with an ultra-refined 160 cells per wavelength (480 grid points for 1000 Hz) for enhancing precision in critical discontinuous points, however a sufficient resolution was found to be 40 cells per wavelength, as it will be later demonstrated. Time resolution was set at 100 timesteps per period (∆t = 1 • 10 -5 s for 1000 Hz), which showed to be enough to suppress any spurious numerical dissipation and is coherent with typical industrial applications. Concerning the delay treatment, 8 points per wavelength (N λ ) with the Spectral Difference scheme were used, which

Figure 4 .Figure 4 . 3 :

 443 Figure 4.3 shows the pressure signal captured by the aforementioned probe for frequencies of 1000 and 2000 Hz and compared to the exact analytical solution.A detailed analysis of the reflected signal shows that the first dampened wave comes from the liner's direct sound absorption (between 2 and 2.5 periods of time for 1000 Hz, between 2 and 3 periods for 2000 Hz), while the remaining reflected wave includes the delay effect from the cavity resonance, modifying the signal structure. Computational time is barely affected by the TDIBC in this 1D case, since the few additional calculations are done only in the single right boundary cell. Figure4.3 is obtained with an ultra-refined 160 cells per wavelength (480 grid points for 1000 Hz) for enhancing precision in critical discontinuous points, however a sufficient resolution was found to be 40 cells per wavelength, as it will be later demonstrated. Time resolution was set at 100 timesteps per period (∆t = 1 • 10 -5 s for 1000 Hz), which showed to be enough to suppress any spurious numerical dissipation and is coherent with typical industrial applications. Concerning the delay treatment, 8 points per wavelength (N λ ) with the Spectral Difference scheme were used, which

Figure 4 . 4 :

 44 Figure 4.4: 1D case -Tube grid convergence study. N λ = 128 in the 2nd-order upwind scheme discretization for the delay advection.

  grid points, 16 PPW ElsA -481 grid points, 32 PPW ElsA -481 grid points, 64 PPW ElsA -481 grid points, 128 PPW (a) Full signal structure.

  grid points, 16 PPW ElsA -481 grid points, 32 PPW ElsA -481 grid points, 64 PPW ElsA -481 grid points, 128 PPW (b) Zoom on reflected signal.

Figure 4 . 5 :

 45 Figure 4.5: 1D case -Delay discretization convergence study at 480 tube grid points. 2nd-order upwind scheme for the delay advection.

Figure 4 Figure 4 . 6 :

 446 Figure 4.6: 1D case -Comparison between 2nd-order upwind and Spectral Difference schemes for the delay advection discretization.

Contents 5 . 1 GIT

 51 Ceramic Tubular CT-57 liner . . . . . . . . . . . . . . . . . . . . . . 94 5.2 CANNELLE industrial liners . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.3 ASPIRE UHBR liner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Figure 5 . 1 :Figure 5 . 2 :

 5152 Figure 5.1: ODR of GIT CT-57 liner, 130dB, Mach 0.0 and Mach 0.255.

Figure 5 . 3 :Figure 5 . 4 :

 5354 Figure 5.3: Oscillo-Diffusive Representation of AIB liner, 120dB, Mach 0.0 and Mach 0.3.

Figure 5 . 5 :

 55 Figure 5.5: Oscillo-Diffusive Representation of ASPIRE liner, Mach 0.27.

Chapter 6 CFD results on benchmark cases Contents 6 . 1

 661 General settings and formulas . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.2 2D -Grazing acoustic wave . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.3 3D -CANNELLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6.3.1 Numerical layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.3.2 Post-processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 6.3.3 CFD results without mean flow . . . . . . . . . . . . . . . . . . . . . . . . . 114 6.3.4 CFD results with mean flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Figure 6 . 1 :

 61 Figure 6.1: Experimental rig for GIT acoustic measurements -source from [41].

Figure 6 . 2 :

 62 Figure 6.2: Numerical layout and boundary conditions of the 2D case.

Figure 6 . 3 :

 63 Figure 6.3: GIT grids: no-flow Euler case (top) and flow Navier-Stokes case (bottom).

Figure 6 .

 6 Figure 6.4 presents successive time frames of the sound signal attenuation caused by the TDIBC on the top wall. Immediately after the TDIBC activation, the pressure profile starts to be deformed following the impedance law. After around 10 periods, the pressure in the duct reaches a stabilized converged regime with instantaneous fields not far from the ones after 4 periods. However, 10 additional periods have been calculated for stability bulletproofing, for a total of 20 periods time simulation (or 2000 iterations, having 100 timesteps per acoustic period). Six frequencies are tested (0.6, 1.0, 1.5, 2.0, 2.5, 3.0 kHz) at two different average Mach numbers (0 and 0.255). The flow boundary layer profile imposed is the hyperbolic profile of Equation (6.1.1), also represented in Figure6.5. Even though the hyperbolic profile doesn't fit perfectly on the experimental values, where a fully developed turbulent flow was achieved, the average Mach numbers integrated on the vertical section are found to be close. Since the calculation is done with laminar Navier-Stokes equations, the expected boundary layer profile will be closer to an hyperbolic law than a fully turbulent profile, as later demonstrated by the CFD. Pressure signals are extracted at 100 equidistant locations on the lower hard wall, as done in the experiment with microphones. Comparisons in terms of SPL reduction between the experimental values, DG-CAA results and the current CFD implementation are presented in Figure6.6, respectively for the six frequencies (from top to bottom) and two Mach numbers (from left to right). In the abscissa, the longitudinal coordinate is given (i.e. the sound propagation direction), while sound pressure level is in the ordinates. Vertical dashed lines indicate the liner's limits, i.e. the central treated region. A good accordance is found between the CFD and the references CAA

Figure 6 . 5 :

 65 Figure 6.5: GIT boundary layer profile.

Figure 6 . 6 :

 66 Figure 6.6: SPL comparison of GIT benchmark: experiment, CAA-DG [61] and CFD (elsA)average Mach of 0 (left) and 0.255 (right).

Figure 6 . 7 :

 67 Figure 6.7: Comparison of sound reduction -input SPL of 130 dB and 15 dB.

Figure 6 . 8 :Figure 6 . 9 :

 6869 Figure 6.8: Acoustic pressure signal at 15 dB before and after the liner.

Figure 6 .

 6 Figure 6.10: CANNELLE experimental layout -adapted from [48].

( a )Figure 6 . 11 :

 a611 Figure 6.11: CANNELLE test configurations. In black arrows the flow direction, in blue the acoustic waves propagation, in red the acoustic treatment, in green the numerical outputs.

Figure 6 .

 6 Figure 6.12: CANNELLE structured mesh block strategy (AIB case, where the central liner is half the hard walls).

Figure 6 . 15 :

 615 Figure 6.15: Modes decomposition in a cylindrical duct with a lined section.

  Transmission Loss can be computed as the difference (in logarithm) of all propagative modes of a specific azimuthal order: TL (m) = 10 log 10

Figure 6 . 16 :

 616 Figure 6.16: CANNELLE pressure fields for different modes. ACL duct, 3500 Hz, Mach 0 (I).

Figure 6 . 16 :

 616 Figure 6.16: CANNELLE pressure fields for different modes. ACL duct, 3500 Hz, Mach 0 (II).

Figure 6 . 18 :

 618 Figure 6.18: Axial section of ACL duct case -3500 Hz, mode (m0n2), Mach 0.

  and 6.20 for modes m0n2 and m2n1, respectively, at 3500 Hz in the ACL duct. With refer to Figure6.15, incident and transmitted modes are those propagating in the fundamental direction (left to right), with incident before the liner and transmitted after the liner. Reflected modes are those propagating in the opposite direction. Looking at the absolute values (left images), we have confirmation that the injected mode (blue bar) is predominant, with almost no distribution onto other modes. However, the transmitted modes (in orange) show a distribution on other radial modes of almost the same amplitude as the fundamental mode, justifying the mode shape deformation shown in the previous figures. Concerning the reflected modes, they are of negligible amplitude in most cases, specially those created at the exit section. This confirms the good quality of the non-reflective boundary condition.

Figure 6 . 19 :

 619 Figure 6.19: Incident mode m0n2 decomposition. Absolute (left) and logarithmic (right) scales.

Figure 6 . 20 :

 620 Figure 6.20: Incident mode m2n1 decomposition. Absolute (left) and logarithmic (right) scales.

Figure 6 . 21 :

 621 Figure 6.21: Transmission Losses of AIB liner -Mach 0.

Figure 6 .

 6 Figure 6.22: Transmission Losses of ACL liner -Mach 0.

Figure 6 . 23 :

 623 Figure 6.23: Comparison on numerical convergence of the instantaneous pressure -ACL, 1600 Hz, mode m0n1 (plane wave).

Figure 6 . 24 :

 624 Figure 6.24: Transmission Losses of AIB liner -Mach 0.3.

Figure 6 . 25 :

 625 Figure 6.25: Transmission Losses of ACL liner -Mach 0.3.

Figure 7 . 1 :

 71 Figure 7.1: ASPIRE turbofan simplified geometry -source from [14].

Figure 7 . 2 :

 72 Figure 7.2: CFD single channel domain -source from [14].

Figure 7 . 3 :

 73 Figure 7.3: CAA single channel domain -source from [14].

Figure 7 . 4 :

 74 Figure 7.4: Density profile comparison for coupling surface.

Figure 7 . 5 :

 75 Figure 7.5: Acoustic liner location.Figure 7.6: Interface location.

Figure 7 . 6 :

 76 Figure 7.5: Acoustic liner location.Figure 7.6: Interface location.

Figure 7 . 7 :

 77 Figure 7.7: Radial surface location in the nacelle.

Figure 7 . 8 :

 78 Figure 7.8: Axial density gradient at r 95 .

Figure 7 . 9 :

 79 Figure 7.9: Mach contour at r 95 .

Figure 7 . 10 :

 710 Figure 7.10: Real pressure contour at r 95 -BPF1 -Hard wall (top) and lined wall (bottom).

Figure 7 . 11 :

 711 Figure 7.11: Real pressure contour at r 95 -BPF2 -Hard wall (top) and lined wall (bottom).

Figure 7 . 12 :Figure 7 . 13 :

 712713 Figure 7.12: Real pressure contour at r 95 -BPF3 -Hard wall (top) and lined wall (bottom).

  1.9)): PWL = 10 log 10 16 S I -dS P ref (7.2.1)

Figure 7 . 15 :

 715 Figure 7.15: Real part of pressure at axial section z = 0 m -BPF1.

Figure 7 . 16 :

 716 Figure 7.16: Real part of pressure at axial section z = 0 m -BPF2.

Figure 7 . 17 :

 717 Figure 7.17: Real part of pressure at axial section z = 0 m -BPF3.

Figure 7 .Figure 7 . 19 :

 7719 Figure 7.18: ∆SPL (lined vs rigid) contour at axial section z = 0 m -BPF1 (top left), BPF2 (top right), BPF3 (bottom).

x 2 d 2 y dx 2

 2 + x dy dx + (x 2 -m 2 )y = 0 (C.1)where m is here a complex number called order of the Bessel function, not to confuse with the integer number azimuthal order of the acoustic mode. In fact, when m is an integer, Bessel functions are also called cylinder functions (or cylinder harmonics), as they are solution of the Laplace's equation in cylindrical coordinates. Since (C.1) is a second-order linear differential equation, two linearly independent solutions must exist. Such solutions are generally called Bessel functions of the first kind (J m ) and of the second kind (Y m , or also called Weber function):y = AJ m (x) + BY m (x) (C.2)with A and B arbitrary constants. Bessel functions of the first kind J m are finite at the origin (x = 0) and are defined by the infinite power series expansion:J m (x) = ∞ k=0 (-1) k (x/2) 2k+m k!Γ(k + m + 1) (C.3)with Γ(z) the Gamma function, a shifted generalization of the factorial function to non-integer values. An integral representation of the Bessel function of first kind with trigonometric functions is: x sin τ -mτ ) dτ (C.5)

  Figures C.1 and C.2 represent Bessel functions of first and second kind for integer order m ∈ (0, ..., 4), and Figures C.3 and C.4 their derivatives. They both can be considered as decaying oscillating trigonometric function.

Figure C. 1 :

 1 Figure C.1: Bessel function of the first kind.

Figure C. 2 :

 2 Figure C.2: Bessel function of the second kind.

Figure C. 3 :

 3 Figure C.3: Derivative of the Bessel function of the first kind.

Figure C. 4 :

 4 Figure C.4: Derivative of the Bessel function of the second kind.

4 + 2 (Λ 4 +

 424 Λ 5 ) + (L 4 + L 5 )Λ c ∂d 5 ∂Q (1,j) = ρc √ Λ 5 ) + (L 4 + L 5 )Λ c

17

 17

  )

.2 Acoustic impedance

  Linearized wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.2 Acoustic modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2

Contents 2.1 Duct acoustics theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2.1 Basic impedance models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2.2 Quarter-wavelength resonator . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.3 Helmholtz Resonator (HR) . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.4 Impedance model of physical acoustic liners . . . . . . . . . . . . . . . . . . 18 2.3 Impedance in Time-Domain . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.1 Admissibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.2 Time-Domain Impedance Models . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4 Oscillatory-Diffusive Representation . . . . . . . . . . . . . . . . . . . . . 26 2.4.1 Diffusive part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4.2 Oscillo-Diffusive part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.4.3 Time delay formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29 2.5 Nonlinear impedance modeling . . . . . . . . . . . . . . . . . . . . . . . . 30 2.6 Theory of nonlinear propagation . . . . . . . . . . . . . . . . . . . . . . . 33 2.7 TDIBC in CFD computations: state-of-the-art . . . . . . . . . . . . . . 35

63 3.5 Giles non-reflecting boundary condition . . . . . . . . . . . . . . . . . . . 64 3

	Chapter 3
	Numerical developments for acoustic
	propagation in lined ducts
	Part II
	TDIBC numerical implementation in
	a CFD solver
	the modified
	Extended Helmholtz Resonator in (2.3.24));
	2. the discretization into a multipole model (as in (2.4.24)) and its conversion in
	time-domain (as in (2.4.29));
	3. the translation and implementation in a CFD or CAA solver (as done later in
	(3.3.23)).
	• Nods on nonlinear impedance modelling were given for an eventual future development
	and generalities on nonlinear acoustics (N-waves and shock waves) generation and
	propagation were given.

1. the choice of a physical impedance model in the Laplace domain (i.e.

• The chapter was concluded with recent examples from the literature on time-domain impedance models with different applications in CFD. Contents 3.1 The elsA solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.2 Navier-Stokes Characteristic Boundary Conditions . . . . . . . . . . . . 45 3.2.1 NSCBC Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2.2 NS Characteristic representation in generalized coordinates . . . . . . . . . 46 3.2.3 NSCBC Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2.4 Characteristic waves amplitude . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.3 NSCBC applications in a lined duct configuration . . . . . . . . . . . . . 52 3.3.1 Subsonic inflow: Tuvw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.3.2 Subsonic outflow: Pressure outlet . . . . . . . . . . . . . . . . . . . . . . . . 54 3.3.3 Application to isothermal no-slip wall . . . . . . . . . . . . . . . . . . . . . 54 3.3.4 Application to acoustically lined wall (TDIBC) . . . . . . . . . . . . . . . . 56 3.3.5 Application to nonlinear TDIBC . . . . . . . . . . . . . . . . . . . . . . . . 59 3.4 Acoustic perturbation injection . . . . . . . . . . . . . . . . . . . . . . . . 60 3.4.1 Pressure plane wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.4.2 Bessel function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.4.3 N-waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.1 Non-reflecting characteristic conditions . . . . . . . . . . . . . . . . . . . . . 65 3.5.2 Derivation of Giles/Medida conditions . . . . . . . . . . . . . . . . . . . . . 67 3.5.3 Hixon decomposition for Giles conditions . . . . . . . . . . . . . . . . . . . 68

  .2. The auxiliary variable ϕ (yellow cross) is computed at the boundary, and the bottom green cross corresponds to the delayed variable ψ. The domain size is twice the cavity length to keep into account the back-and-forth distance, since the delay τ is defined as the back-and-forth time. It is imperative that the frequency f max chosen to size the 1D domain is also correctly resolved by the relative impedance model for ensuring numerical stability. In other words, in the frequency range chosen, the reflection coefficient of the impedance model must be a passive function, i.e. bounded within [0,1]. In Chapter 5, practical examples are given on this matter. Finally, on the memory allocation, this problem translates in storing a number of additional variables equal to N ψ + 1 (where 1 refers to the ϕ variable) for local memory and 6(N ψ + 1) additional variables for global memory (3(N ψ + 1) for each ϕ and ψ). The latter are required for the previous two time instants, in alignment with the 2nd-order temporal scheme). These numbers are then multiplied by the number of poles in the ODR, since each auxiliary variable is function of the pole value. It becomes then evident the interest in having a model requiring the least number of poles possible.

  resolution of system (4.3.25) is done progressively starting from the first element K = 1 until K = N

	and, from (4.3.20):
	Nsp
	δψ j C sp,j	(4.3.26)
	j=1

ψ . The relation

(4.3.23)

is linear, so that one can write:

∂δψ ∂x sp = δψ 1 A d (1 + (sp -1)(p + 2)) +

Table 6 .

 6 2: CANNELLE geometry details.

			Diameter	Length of the Hard Wall sections	Length of the Acoustic Liner	Length of the Duct (total)
	Geometry (mm)	AIB ACL	350	325	175 350	825 1000

Table 6 .

 6 mode, hence without turbulence modelling. In the latter case, a much finer resolution near the wall would be required. The normal extrusion is made with a different number of elements, depending on what frequency needs to be resolved. It is reminded that the boundary layer thickness is independent on the acoustic properties, but dependent only on the mean flow parameters. Contrarily, a too fine grid, as it could be in the boundary layer, could yield to the propagation of parasitic frequencies that would normally be cut-off (i.e. much higher than the fundamental frequency). This is a further reason for requiring appropriate non-reflecting boundary conditions in inlet and outlet regions in presence of a boundary layer refinement. The boundary layer profile, available from the experimental measurements, is discretized with the hyperbolic law (6.1.1) with δ = 0.0045, and imposed in the inlet boundary condition (3.3.1), symmetric in respect to the azimuthal direction. The flow profile obtained in the CFD calculation with the aforementioned grid hypotheses is shown to be correct, as illustrated in Figure6.14. A total of 12 grids are generated: for the 2 flow cases, 2 liners and 3 frequencies. The grid details are given below in Table6.3. 3: Number of elements in the CANNELLE meshes in the three cylindrical directionswithout (top) and with (bottom) boundary layer.

	Z		Z
	X	Y	X	Y

laminar

Chapter 7 CFD results on ASPIRE turbofan model Contents 7.1 Numerical setup

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 7.1.1 Preliminary CFD simulation -shock generation . . . . . . . . . . . . . . . . 126 7.1.2 CAA simulation layout -shock propagation . . . . . . . . . . . . . . . . . . 127 7.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 7.2.1 Radial surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 7.2.2 Axial section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

  ∂Q (0,j) = -n y Λ 1 + n x Λ 2 + n z √ 2 (Λ 4 -Λ 5 ) ∂d 4 ∂Q (1,j) = -n y Λ 1 + n x Λ 2 +

	                              	∂d 4 ∂d 4 ∂Q (0,j+1) ∂d 4 ∂Q (0,j-1)	= -= -	√ n z 2 √ 2 n z	Λ 5 Λ 5	n z √ 2	(Λ 4 -Λ 5 )	(D.10)
	Finally, knowing that:
				d 5 =	ρc √ 2	(L 4 + L 5 )	(D.11)
	and defining with Λ c :
		Λ c =	1 2	γp 2ρ		0 0 0	ρ 2	γ 2p	(D.12)
		1)	= -	√ n y 2	Λ 5			(D.9)
	∂d 3 ∂Q (0,j+1)	= -	n y √ 2	Λ 5

  = -u n n y -u n n y ∆p 1 γp ∂L 2 ∂u 0 = n z u n + n x L 2

			                                    	∂L 2 ∂ρ 0 ∂L 2 ∂v 0 ∂L 2 ∂w 0 ∂L 2 ∂p 0	= n y L 2 = -n x u n + n z L 2 = n y c 2 u n + u n n y ∆p	1 pc 2	                                    	∂L 2 ∂ρ 1 ∂L 2 ∂u 1 ∂L 2 ∂p 1 ∂w 1 ∂L 2 ∂L 2 ∂v 1	= u n n y = -n z u n = -c 2 u n n y = n x u n = 0	(D.3)
	For L 3 from (4.2.7):
		∂L 3 ∂Q	= u n n z	∂∆ρ ∂Q	+ n z	∂∆u ∂Q	-n x	∂∆v ∂Q	-	n z c 2	∂∆p ∂Q	-u n n z ∆p	∂(1/c 2 ) ∂Q	+ L 3	∂u n ∂Q
	∂L 2 ∂Q	= u n n y	∂∆ρ ∂Q	-n z	∂∆u ∂Q	+ n x	∂∆w ∂Q	-	n y c 2	∂∆p ∂Q	-u n n y ∆p	∂(1/c 2 ) ∂Q	+ L 2	∂u n ∂Q

2)

For L 2 from (4.2.5):

Conventions Industrielles de Formation par la REcherche, contract n •

2018/0772 2 Association Nationale de la Recherche et de la Technologie 3 Office National d'Etudes et de Recherches Aérospatiales

Ensemble Logiciel de Simulation Aérodynamique, Software Package for Aerodynamics Simulation

Sum of complex and complex conjugate products: a • b + a * • b * =

2Re(a • b), a, b ∈ C

Airbus internal document reference: ME1914331

Acknowledgments

Foremost, I would like to express my gratitude to Airbus and ANRT for financing this intriguing project that I had the luck to undertake, and Onera for welcoming me in their laboratory.

Summary

• The context of the CFD solver adopted and the reasons of choosing a NSCBC formalism were given. NSCBC theory was recovered and adapted for duct acoustics applications. The time-domain impedance model based on ODR is coupled to the characteristic waves in (3.3.23), which is the third and last point when building a time-domain impedance condition.

• The different boundary conditions useful for this work were presented in their numerical aspect: inlet, outlet, hard and lined wall.

• Three possible acoustic perturbations were given in their analytical form: plane wave, Bessel mode and N-wave.

• A non-reflective type of acoustic boundary conditions was derived from Giles' formulation with Medida and Hixon corrections:

• The time-averaged term {∂Q/∂t} mean is computed with NSCBC on the mean flow defined by the time-filtered values.

• Using the previous term, the fluctuation term {∂Q/∂t} Nref is computed by classical NSCBC formulations.

Summary

• The general numerical settings common to all CFD simulations were provided.

• The 2D GIT benchmark gave a first confirmation of the TDIBC implementation correctness in the CFD solver. Both Euler and laminar Navier-Stokes results are in good agreement with the experiment and CAA results for all the frequencies considered.

The TDIBC proved to be precise in the Euler case down to an acoustic signal of 10 -4 Pascal. The dual-iterations convergence threshold demonstrated to play a significant role in the acoustic pressure value.

• The 3D CANNELLE benchmark allowed to extend the TDIBC validation to a more complex and realistic acoustic duct flow case. Again, the CFD solver proved to be able to compute Transmission Losses in the duct in accordance with experimental and CAA results for different frequencies and flow speeds, with few discrepancies at high-order acoustic modes where the attenuation is the strongest. The redistribution of acoustic modes due to the acoustic liner presence was evaluated in the no-flow case.

The advantage brought by the implicit formulation was shown in terms of stability and computational time when solving laminar Navier-Stokes equations.

Summary

• The sound absorption given by a lip liner in a UHBR nacelle intake in presence of shock-waves propagation and uniform flow was calculated with CFD.

• The coupling technique to move from the CFD to CAA (CFD solving Euler equations) simulation was detailed.

• Different outputs at radial and axial coordinates provided different but coherent information on the noise reduction in the nacelle and in its external proximity. The computation of axial acoustic power level ensured that the simulation was set up as the benchmark taken for example and that the TDIBC is acting passively, with a noise reduction of almost 40 dB at the nacelle exit.

• The minimum impact on added computational time is a further proof of the implicit scheme efficiency on large scale industrial application.

Conclusion and perspectives

141

Summary

In this thesis dissertation, a deep research was carried out on Time-Domain Impedance Model (TDIM) under a Navier-Stokes Characteristics Boundary Conditions (NSCBC) formalism for unsteady CFD applications. The interest of using a Time-Domain Impedance Boundary Condition (TDIBC) was exposed in comparison to the classical CFD/CAA coupled methodologies. This finds a relevant application in an industrial aeronautical environment, where acoustic liner design plays a major role in the overall research and development of aircraft noise reduction techniques. After an introduction on duct acoustics theory and TDIM available in the literature, the Oscillo-Diffusive Representation (ODR) [START_REF] Monteghetti | Analysis and Discretization of Time-Domain Impedance Boundary Conditions in Aeroacoustics[END_REF][START_REF] Monteghetti | Energy analysis and discretization of nonlinear impedance boundary conditions for the time-domain linearized Euler equations[END_REF] was chosen as the TDIM to implement in the CFD solver under a reflection coefficient formulation. The choice of using this multipole model was justified by numerous advantages:

• the possibility of modelling different type of liners (Single Degree Of Freedom with perforated plate or wiremesh) given their geometry or an experimental data set;

• the intrinsic stability property brought by its reflection coefficient formulation;

• the relatively low number of poles needed, hence its efficiency in a numerical code.

Consequently, the three steps for building a TDIBC using an ODR were discussed:

1. Choosing a discrete impedance model, usually based on the liner's geometry and flow properties (sound level and average Mach number). The modified Extended Helmholtz Resonator (EHR) model described in (2.3.24) was chosen for its ease of adaptability to liners typically found in aircraft nacelles.

2. Solving an algorithm to obtain a discretization with a multipole formulation, facilitating its time-domain transformation. The EHR model is discretized into the multipole model (2.4.24), where the delayed and non-delayed effects are explicit. Such model is made of a set of ODEs and a transport equation for solving the delay problem.

3. Translating and implementing the latter multipole model in a finite-volume CFD solver, here done in a characteristic formalism in (3.3.23).

Meanwhile steps 1 and 2 were the main results of Monteghetti [START_REF] Monteghetti | Analysis and Discretization of Time-Domain Impedance Boundary Conditions in Aeroacoustics[END_REF], step 3 represents the novelty brought by this thesis. It was chosen to express the TDIBC in a NSCBC formalism, hence linking the ingoing and outgoing characteristic acoustic waves, notably (u + c) and (u -c) waves. In other words, the TDIBC will simply express the relation between incident and reflected acoustic waves on a boundary. Its complete formulation was given, with a thorough discussion on certain peculiar properties such as tangential fluxes influence and the wall-normal velocity determination. The numerical algorithm used for solving the delay advection problem was detailed.

Different types of NSCBC were developed for the specific duct acoustics applications of this thesis. Specifically, an inflow, outflow and hard wall conditions were tackled, along the TDIBC previously said. A particular attention was posed on how to introduce an acoustic perturbation from inflow/outflow frontiers, which can now be modelled by an analytical function (plane wave, Bessel mode or N-wave) or by a pressure or flow field given in input. Another topic of paramount importance in flow duct acoustics is the non-reflectivity of such boundary conditions. Thanks to a modified Giles-type boundary condition, this was successfully achieved without use of any numerical artifice (such as sponge zones or grid stretching) in the same NSCBC framework. Finally, all these numerical developments were transformed in a fully implicit formalism. This is well-known to be a preferred choice when solving complex industrial cases. Although complex

Practical applications

Chapter 6 confirmed the correct implementation of the liner condition in the CFD solver and Chapter 7 brought to light first qualitative results on noise reduction of the ASPIRE numerical benchmark in view of future UHBR architectures. Unfortunately, in the time available, no further studies could be carried out and are here proposed as continuation of this thesis. Some further developments in this context could be the following:

• In simple benchmarks such as the rectangular duct GIT or circular duct CANNELLE, a fine investigation could be carried out on the impact of the boundary layer and turbulence in the acoustic propagation in lined ducts. This would also help to better exploits the experimental data. For example, the GIT benchmark provided acoustic attenuation in presence a fully developed turbulent flow profile, while in the CANNELLE benchmark the modal decomposition was performed assuming a uniform flow.

• Extend the present frequency-domain ASPIRE results to the far-field with aid of an aeroacoustic propagation code (i.e. a FFowcs-Williams Hawking solver), in order to correctly resolve those areas outside the nacelle where the present numerical grid was too coarse to propagate the sound signature.

• Always in the ASPIRE configuration, run the preliminary CFD simulation of [START_REF] Daroukh | Shock Wave Generation and Radiation from a Turbofan Engine Under Flow Distortion[END_REF] with inclusion of the TDIBC. As first step, the axisymmetric case could be considered, where prominent difficulties would already be found in including turbulent equations and the rotating frame. This simulation would allow to cross-compare the noise source generation (i.e. the conservative flow field discussed in Section 7.1.2) between the hard and lined wall cases. Furthermore, the entire liner (composed of base and lip liners, as in Figure 7.5) could be taken into account.

• In the long term, after having validated the liner model with other turbomachinery data, this numerical tool would allow to perform liner design optimization directly on the CFD simulations of turbofan. In fact, by simple modification of geometrical parameters and consequently of the poles and weights defining the discrete impedance model, the impact of new acoustic liners could directly assessed in a nacelle CFD computation.

Part IV

Appendices

Appendix A

Poles and weight

A.1 GIT CT-57

Table A.1: Poles of the Oscillo-Diffusive Representation of the GIT-CT57 liner -cf. β E in Table 2 of [START_REF] Monteghetti | Energy analysis and discretization of nonlinear impedance boundary conditions for the time-domain linearized Euler equations[END_REF].

A.2 CANNELLE -AIB

A.3 CANNELLE -ACL

-2.71049e+02 2.23646e+02 -6.28318e+04 5.74535e+03 -5.46570e+03 -9.49996e+03 ± 1.88108 e+04i -6.42290e+03 ± 6.02607e+03i 6.30385e+03 ± -6.04095e+03i -2.01231e+03 ± 8.93580e+04i -4.74709e+03 ± 6.67809e+03i 4.46491e+03 ± -6.55873e+03i

Liner ACL -Mach = 0.3 β ∞ = 1 τ = 6.7826-05 [s] -7.53982e+03 -1.17115e+03 7.07204e+02 -6.28318e+04 6.44417e+03 -5.63542e+03 -2.24031e+04 ± 1.15974e+04i -7.24172e+03 ± 9.51452e+03i 6.52749e+03 ± -9.80006e+03i -5.37529e+03 ± 9.82738e+04i -4.36261e+03 ± 5.11727e+03i 3.54252e+03 ± -5.03670e+03i

A.4 ASPIRE

7435e-04 [s] -4.63944e+03 ± 6.60575e+01i -1.80740e+03 ± 7.56952e+02i 1.78213e+03 ± 1.17923e+03i -2.89972e+03 ± 3.77994e+04i -2.28077e+03 ± 1.35338e+03i 2.15460e+03 ± 1.38612e+03i -1.75101e+03 ± 7.36788e+04i -3.38095e+03 ± 5.23262e+03i 3.00121e+03 ± 4.87497e+03i The objective of this section is to complement the discussion of Hixon decomposition of Giles conditions in Section 3.5.3. In here, it is detailed the VGBC (Vortical Gust Boundary Condition) developed by Hixon [START_REF] Hixon | Deterministic period mean flow boundary condition for unsteady flow predictions[END_REF][START_REF] Hixon | Mean flow boundary conditions for computational aeroacoustics[END_REF] for injecting velocity fluctuation without parasitic waves. It is remembered this term is not needed when the acoustic perturbation is known analytically. The VGBC method allows to determine the time derivatives of the three velocity components (u, v, w) in the orthonormal local reference system at the boundary ( -→ n , -→ t 1 , -→ t 2 ). These derivatives will then be added to the ingoing waves amplitudes calculated with Giles conditions. The hypothesis of isentropic waves is added to complete the fourth relation needed to specify the four ingoing waves. Resuming, we have: For p: