Courant-Friedrichs Lewy

Cpu

Edwards Correction

K-H Kelvin-Helmholtz Le

Zonal Eddy Detached

Simulation

+ Pressure CC Compressibility Correction CFD Computational Fluid Dynamics CFL

Keywords:

Nomenclature

Context

The Wright brothers' flight on 17 December of the 1903 is considered the first controlled and sustained flight of an aircraft heavier than air. The Flyer covered 37 meters in 12 second i.e. a velocity of 10.98 km/h. After that, the range of aircraft velocities increased together with the development of the aviation. The "Fédération aéronautique internationale" defined the rules for the air speed record and compiled a table with the highest aircraft airspeed. In the first half of the 20th century it is possible to find a linear slope of the highest aircraft airspeed velocity in time; the increase is of about 24km/h every year. Transonic velocities, reached during the World War II, lead to several aerodynamic and aeroelastic problems which limit flight envelope and in some case have negative effects on the aircraft performance (Vos & Farokhi [158]). When an aircraft is flying at transonic velocity, the flow accelerates on the suction side of the wings until sonic velocity and creates a shock wave. For certain values of Mach number and angle of attack (M -α), the shock wave can oscillate at high-amplitude synchronised with the thickening/thinning of the detached boundary layer (figure 1.1).

The detached boundary layer could even reattache when the shock moves downstream for certain M -α. The result is large pressure fluctuations between the shock foot and the TE. The entire phenomenon is at low-frequency and it is called transonic buffet. If coupled with a structural mode, integral aerodynamic forces and moments fluctuations lead to structural vibrations of the entire wing, called buffeting. These vibrations can weaken the structure of the wings and in the worst case cause failure due to fatigue.

.1: Region of shock oscillation for BGK No. 1 airfoil (Lee [START_REF] Lee | Flow separation on a supercritical airfoil[END_REF]).

Consequently, the buffet phenomenon limits the flight envelope and diminishes the handling quality of civil aircraft. The physical mechanisms behind the phenomenon

are not yet fully understood even if the first studies on transonic buffet were performed about seventy years ago. Furthermore, modern wings are mostly based on supercritical profiles (Whitcomb [159]), which have been designed to delay the appearence of a shock wave and to reduce its strength. At the same time these kinds of profiles, very efficient at design conditions, exhibit a particularly strong shock oscillation off-design.

Therefore, predicting the buffet onset and improving the understanding of the phenomenon is a main concern for aircraft manufacturers. Generally, wind tunnel tests on models at atmospheric conditions are not fully realistic because of the smaller Reynolds number, while the flight tests are complete but available too late and too expensive. This is the reason why computational fluid dynamics is gaining increasing importance.

Simulations give a better overview of the overall flow field, allowing comparison with experimental data and filling the lack of model instrumentation in all the experiments. The timescale of the periodic motion in transonic buffet is much longer in comparison with those of wall-bounded turbulence, consequently the Unsteady RANS (Reynolds Averaged Navier-Stokes) equations approach, closed with a turbulence model, is justified. Nevertheless, more expensive DNSs (Direct Numerical Simulations), LESs (Large Eddy Simulations) and DESs (Detached Eddy Simulations, Spalart [START_REF] Spalart | Comments on the feasibility of les for wings, and on a hybrid rans/les approach[END_REF]) have been 1.1 Context 3 used to compute the transonic buffet phenomenon. DNS directly solve the entire range of turbulent length scales structures in the flow. LES are numerical simulations where the length scales of the flow are solved until a certain fixed value below which length scales are ignored or modeled. DESs are a mix of RANS and LES: the attached boundary layer and regions where turbulent lengths scales are smaller than a value, fixed a priori, are solved using the RANS approach while the other regions are solved by LES.

In the last years, two journal papers have reviewed the developments and achievements in the understanding of the buffet phenomenon: Lee [START_REF] Lee | Self-sustained shock oscillations on airfoils at transonic speeds[END_REF] and Giannelis et al. [START_REF] Giannelis | A review of recent developments in the understanding of transonic shock buffet[END_REF]. Lee concluded his review with a possible explanation of the transonic buffet phenomenon, presented in the next paragraphs, while Giannelis et al. concluded that a unique mechanism explaining the phenomenon is still lacking but also that several studies contradict each other.

Shock-induced oscillations

The transonic buffet phenomenon is a shock wave boundary layer interaction (SWBLI) (Dolling [35]). Nevertheless, Sartor et al. [START_REF] Sartor | Unsteadiness in transonic shockwave/boundary-layer interactions: experimental investigation and global stability analysis[END_REF] showed that certain SWBLIs do not result in a large-scale unsteadiness and they could be well described by steady state flow. This is the reason why the flow "category" of transonic buffet can be restricted in the shock induced oscillations (SIOs), that are the unsteady physical phenomena exhibiting oscillation due to a shock wave. The first identification of transonic SIOs over aerofoil was performed by Hilton & Folwer [START_REF] Hilton | Photographs of shock wave movement[END_REF] in 1947. They determined the period and amplitude of oscillations by consecutive photographs on a 14% thick lowdrag section" (1442/1547) with Goldstein 'roof-top' distribution. A first subdivision of transonic buffet, due to a difference in the physical mechanism, was performed over two configurations of aerofoil: a symmetrical aerofoil at zero incidence and a lifting aerofoil at incidence. SIOs over symmetrical aerofoils at zero incidence have been studied from the 1950s by several authors (Mabey et al. [START_REF] Mabey | Periodic flows on a rigid 14% thick biconvex wing at transonic speeds[END_REF], McDevitt et al. [START_REF] Mcdevitt | Transonic flow about a thick circular-arc airfoil[END_REF] and Mundell & Mabey [START_REF] Mundell | Pressure fluctuations caused by transonic shock/boundary-layer interaction[END_REF]). The flow field shows in this case a shock wave both on the pressure and suction sides of the aerofoil. The two shock waves exhibit the same high-amplitude shock movement synchronised with the detached boundary layer described above. The difference here is a phase lag of 180 • between the two shock waves on the pressure and the Introduction suction side. The physical mechanism behind this phenomenon appears to be linked with the phase lag and the communication between the two sides of the aerofoil: the forward/backward movement of a shock sustains the backward/forward movement of the shock on other side. McDevitt identified the values of onset and offset of the phenomenon over a thick circular-arc aerofoil for different values of Mach number. He underlined the possibility to virtually eliminate the unsteady phenomenon through a splitter plate at the TE which interrupts the communication between pressure and suction side. Furthermore, he showed an hysteresis effect (especially at onset) when increasing/decreasing the Mach number and an attenuation of this effect when the angle of attack is not equal to zero. Erickson & Stephenson [START_REF] Erickson | A suggested method of analyzing for transonic flutter of control surfaces based on available experimental evidence[END_REF] studied the coupling of shock movement with aileron flutter at transonic velocity and proved that the unsteadiness is totally aerodynamic and not linked with the aeroelastic unsteadiness. An investigation on the pressure fluctuations of the turbulent wake of a NACA 651-213 aerofoil was performed by Sorenson et al. [START_REF] Sorenson | Preliminary investigation of the pressure fluctuations in the wakes of two-dimensional wings at low angles of attack[END_REF]. They defined the buffet boundaries and a semi-empirical equation for the frequency of pressure fluctuations. For this kind of transonic buffet a predictive model based on the Mach number impacting the shock has been proposed by Mabey et al. [START_REF] Mabey | Periodic flows on a rigid 14% thick biconvex wing at transonic speeds[END_REF]. The physical mechanism behind buffet for symmetrical aerofoils is rather well accepted by the entire research community.

The present work is concerned exclusively with the transonic buffet over lifting aerofoil at incidence. The reason is twofold and has been already outlined: the lack of a physical model unequivocally accepted and the huge utilisation of supercritical aerofoil in the modern aircraft transportation. The transonic buffet is presented over lifting aerofoil for both two-dimensional aerofoil and three-dimensional wing.

Buffet/Buffeting prediction over two dimensional aerofoil and three dimensional wing

Once it had been figured out that the buffeting in transonic region is due to the interaction between shock and the turbulent boundary layer, several authors proposed models for buffet/buffeting prediction i.e. defined the parameters (such as Mach number and angle of attack) of the buffet onset. The predictive models are presented in the state of the art before the overview of the phenomenon because historically the aeronautic community had first investigated the appearance and only later the deep understanding of 1.1 Context 5 the physical mechanics behind the phenomenon. Aircraft manufacturers are strongly interested in buffet prediction because the definition of buffet onset has an important role in the design process of an aircraft. Cruise condition should have a margin from the buffet boundary by design standard: in the lift curve, the margin is 30% of the value at buffet onset. When studying aerodynamic loads, the information of the onset were available only a posteriori after experimental tests on aerofoils (Kacprzynski [71],

Bartels [START_REF] Bartels | Computation of shock buffet onset for a conventional and supercritical airfoil[END_REF], Coe & Mellenthin [START_REF] Coe | Buffeting forces on two-dimensional airfoils as affected by thickness and thickness distribution[END_REF] and Poletz et al. [START_REF] Polentz | The unsteady normal-force characteristics of selected naca profiles at high subsonic mach numbers[END_REF]), on reduced-scale aircraft models (Coe [START_REF] Coe | The effect of model scale on rigid-body unsteady pressures associated with buffeting[END_REF], DeAngelis & Monaghan [START_REF] Deangelis | the F-8 supercritical wing airplane[END_REF] and Ray & Taylor [START_REF] Ray | Buffet and static aerodynamic characteristics of a systematic series of wings determined from a subsonic wind-tunnel study[END_REF]) or flight investigation of full-scale aircraft (Benepe et al. [START_REF] Benepe | An investigation of wing buffeting response at subsonic and transonic speeds: Phase 1 F-111A flight data analysis[END_REF], Skopinski & Huston [START_REF] Skopinski | A semiempirical procedure for estimating wing buffet loads in the transonic region[END_REF] and Friend & Monaghan [START_REF] Friend | Flight measurements of buffet characteristics of the f-111a variable-sweep airplane[END_REF]). This is the reason why predictive techniques based on theoretical models and/or tests on models as simple as possible, have been proposed. In the model of Pearcey [START_REF] Pearcey | A method for the prediction of the onset of buffeting and other separation effects from wind tunnel tests on rigid models[END_REF] for aerofoils, the buffet onset is defined by the Mach number and angle of attack of the bubble bursting. Poletz et al. [START_REF] Polentz | The unsteady normal-force characteristics of selected naca profiles at high subsonic mach numbers[END_REF], Mabey [START_REF] Mabey | Buffeting criteria for a systematic series of wings[END_REF], Mitchell [START_REF] Mitchell | Calculation of buffeting of slender wing aircraft at low speeds[END_REF] and Mullans & Lemley [START_REF] Mullans | Buffet dynamic loads during transonic maneuvers[END_REF] used unsteady forces and pressure fluctuations to define buffet onset but it is quite sophisticated in term of data handling. Aeroelastic effects on a scaled model have been taken into account by Hanson [51]; the comparison with full-scale airplane shows precise results but tests are extremely expensive. In order to limit the costs, Skopinski & Huston [START_REF] Skopinski | A semiempirical procedure for estimating wing buffet loads in the transonic region[END_REF] developed a semi-empirical procedure for estimating wing buffet loads in the design stage. A more complete method has been proposed by Jones [START_REF] Jones | Modelling of systems with a high level of internal fluctuations[END_REF]. The technique is based on unsteady measurement of acceleration or bending moment on a wing of solid construction (typical of convectional force test) and single-degree-of-freedoom mechanical system representing each mode of the model. Jones's method predicts buffet onset and the total aerodynamic damping with suitable results (Butler & Spavins [14]). Thomas [START_REF] Thomas | The determination of the buffet boundaries of aerofoils in the transonic regimes[END_REF] and Thomas & Redeker [START_REF] Thomas | A method for calculating the transonic buffet boundary including the influence of reynolds number[END_REF] suggested a model for a 2D aerofoil more linked with the flow physics. It is based on boundary layer theory and defined buffet onset when the rear separation from the TE reached the 90% of the chord. Thomas's prediction method has been extended by severals authors. Redeker [START_REF] Redeker | Calculation of buffet onset for supercritical airfoils[END_REF] performed a modification of the Thomas's prediction method for yawed wings. Proksch [START_REF] Proksch | Ermittlung der buffeting-grenzen von kampfflugzeugen[END_REF] adapted Thomas's method for finite wing and defined a buffeting coefficient directly related to the root mean square value of the wing root bending moment in order to predict the buffeting phenomenon. All the modifications of Thomas's method have been reviewed in Redeker & Proksch [START_REF] Redeker | The prediction of buffet onset and light buffet by means of computational methods[END_REF].

A will be shown in the following, numerical simulations add a significant contribution Introduction in buffet/buffeting prediction but a unique model for every kind of geometry is still lacking. Ross [START_REF] Ross | Some features of the unsteady pressure field in transonic arifoil buffetting[END_REF] showed that aerofoils with different geometries, a NACA 0012 and a supercritical Whitcomb, have substantially different unsteady pressure fields. In another work, Ross [START_REF] Roos | The buffeting pressure field of a high-aspect-ratio swept wing[END_REF], together with Benoit & Legrain [START_REF] Benoit | Buffeting prediction for transport aircraft applications based on unsteady pressure measurements[END_REF], was one of the first to show the difference in power spectral densities (PSDs) of buffet over aerofoils or finite wings. Mullans & Lemley [START_REF] Mullans | Buffet dynamic loads during transonic maneuvers[END_REF] several years before computed fluctuating pressure spectra over a F-4E Phantom but the complexity of the wing and the huge influence of the dog-tooth vortex covered the bump of buffet. The typical idea was that 2D and 3D buffet are two different 'showing up' of the same phenomenon but with rather different parameters. Here, it is worth introducing a non-dimensional frequency, called

Strouhal number, that will be used in the following to quantify the frequency of all the phenomena found in simulations and in experimental tests:

St = f L U (1.1)
where f is the physical frequency, L a characteristic length and U is a characteristic velocity. L is usually the chord for aerofoil and the mean aerodynamics chord (M AC)

for real wing while U is the upstream velocity. Transonic buffet over an aerofoil has a typical Strouhal number of 0.06. The phenomenon appears in the PSDs as a precise peak and the amplitude oscillation can reach 20-30% of the chord (the precise value depends on M -α and geometry aerofoil). Whilst on finite wings the amplitude oscillation of the shock decreases to about 2% of the chord, convection velocities appear in the span direction towards the tips, the peak in the PSDs increases its frequency value (from 4 to 7 times) and becomes more broadband. Typical values of Strouhal numbers are around 0.2 -0.6 in 3D buffet. Another difference is found at higher values of the Mach number or the angle of attack: the amplitude of the shock oscillation in 2D buffet decreases until reaching a steady state. This phenomenon is known as buffet offset (McDevitt & Okuno [START_REF] Mcdevitt | Static and dynamic pressure measurements on a NACA 0012 airfoil in the Ames high Reynolds number facility[END_REF], Ionovich & Raveh [START_REF] Iovnovich | Reynolds-averaged Navier-Stokes study of the shock-buffet instability mechanism[END_REF] and Sartor et al. [START_REF] Sartor | Stability, receptivity, and sensitivity analyses of buffeting transonic flow over a profile[END_REF]). Sugioka et al. [START_REF] Sugioka | Unsteady PSP measurement of transonic buffet on a wing[END_REF], Lawson [START_REF] Lawson | Characterisation of buffet on a civil aircraft wing[END_REF] and Koike [START_REF] Koike | Unsteady pressure measurement of transonic buffet on NASA common research model[END_REF] performed experimental tests at high M -α but the same effect was not observed for the 3D buffet. These differences are the reason why two dimensional transonic buffet over aerofoil and 3D transonic buffet over wings will be presented separately in the following. The link between them is still an open

Two dimensional aerofoil transonic buffet

Transonic buffet phenomenon on aerofoil has been deeply studied both experimentally and numerically. The physical models of buffet proposed up to now will be outlined in the following together with the developments helping the understanding of the phenomenon. The paragraph shows also the effects of several numerical parameters on simulations.

Some pioneer experimental studies over aerofoils in the transonic regime were conducted by the Air Force and NACA/NASA in the 50-60s. Humphreys & Kent [START_REF] Humphreys | The effects of camber and leading-edge-flap deflection on the pressure pulsations on thin rigid airfoils at transonic speeds[END_REF],

Humphreys [START_REF] Humphreys | Transonic aerodynamic characteristics of an NACA 64A006 airfoil section with a 15-percent-chord leading-edge flap[END_REF] and Coe & Mellenthin [START_REF] Coe | Buffeting forces on two-dimensional airfoils as affected by thickness and thickness distribution[END_REF] analysed the fluctuations of pressure and considered several geometrical parameters: camber, LE flap, position of maximal thickness and LE radius. They found that camber effect, LE deflection and smaller thickness of aerofoil reduce pressure fluctuations. Polentz et al. [START_REF] Polentz | The unsteady normal-force characteristics of selected naca profiles at high subsonic mach numbers[END_REF] was one of the first complete studies on transonic buffet. It is an experimental investigation over 27 NACA profiles. The increase of the unsteady forces with Reynolds is observed; the effects of the geometrical parameters studied in [START_REF] Humphreys | The effects of camber and leading-edge-flap deflection on the pressure pulsations on thin rigid airfoils at transonic speeds[END_REF], [START_REF] Humphreys | Transonic aerodynamic characteristics of an NACA 64A006 airfoil section with a 15-percent-chord leading-edge flap[END_REF] and [START_REF] Coe | Buffeting forces on two-dimensional airfoils as affected by thickness and thickness distribution[END_REF] are confirmed and several PSDs are shown. Polentz et al. [START_REF] Polentz | The unsteady normal-force characteristics of selected naca profiles at high subsonic mach numbers[END_REF] showed Strouhal number values of 0.05 -0.1. The first studies intended to better understand the physical mechanism of the phenomenon were focused on the bubble formation and the separation dynamics (Pearcey [START_REF] Pearcey | A method for the prediction of the onset of buffeting and other separation effects from wind tunnel tests on rigid models[END_REF],

Pearcey & Holder [START_REF] Pearcey | Simple methods for the prediction of wing buffeting resulting from bubble type separation[END_REF] and Pearcey et al. [START_REF] Pearcey | The interaction between local effects at the shock and rear separation-a source of significant scale effects in wind-tunnel tests on aerofoils and wings[END_REF]). Pearcey summarised two kinds of separation dynamics: the first linked with the expansion of the localized shock bubble and the second mostly linked with rear separation. He concluded that the buffet onset is identified by the collapsing of the bubble at shock location and TE with a consequently divergence of pressure at TE. This is the 'bubble bursting', the first model for transonic buffet over lifting aerofoil at incidence. Although it appears to well predict buffet onset, there are several works in conflict with this model. For certain geometries aerofoil the bubble collapsing does not correspond to buffet onset (Sartor et al. [START_REF] Sartor | Stability, receptivity, and sensitivity analyses of buffeting transonic flow over a profile[END_REF] and Nitzsche [START_REF] Nitzsche | A numerical study on aerodynamic resonance in transonic separated flow[END_REF]). Another milestone work on transonic buffet has been done by Tijdeman [START_REF] Tijdeman | Investigation of the transonic flow around oscillating airfoils[END_REF] for an aerofoil with oscillating flap. Three different self-sustained periodic motion of the shock are identified and shown in figure 1.2: type A is the classical sinusoidal shock motion oscillation, type B is similar to type A but shock strength Introduction strongly varies during the period and it disappear during downstream movement, in type C the shock moves upstream increasing in strength until reaching the LE when the shock detached from the aerofoil and starts to propagate forward in the incoming flow. All these kinds of shock motion are typical of oscillating aerofoil but they have been observed even for rigid aerofoil (Tijdeman [START_REF] Tijdeman | Investigation of the transonic flow around oscillating airfoils[END_REF], Tijdeman & Seebass [START_REF] Tijdeman | Transonic flow past oscillating airfoils[END_REF] and Lee [START_REF] Lee | Self-sustained shock oscillations on airfoils at transonic speeds[END_REF]). The present work concern only about type A shock motion. Lee's model. Previous investigation by Spee [START_REF] Spee | Wave propagation in transonic flow past two-dimensional aerofoils[END_REF] with a graphical method showed that the waves generated downstream can penetrate the supersonic region upstream.

Deck [START_REF] Deck | Numerical simulation of transonic buffet over a supercritical airfoil[END_REF] validated numerically the buffet period from Lee's model and find numerically that the downstream pressure waves are hydrodynamic in nature while upstream waves are acoustic. Jacquin et al. [START_REF] Jacquin | Experimental study of shock oscillation over a transonic supercritical profile[END_REF] Two-dimensional transonic buffet has been computed with both URANS and scaleresolving approaches. Deck et al. [START_REF] Deck | Numerical simulation of transonic buffet over a supercritical airfoil[END_REF] compared URANS, DES and Zonal-DES (ZDES, Deck [START_REF] Deck | Recent improvements in the zonal detached eddy simulation (zdes) formulation[END_REF]) simulations. The three kind of simulations are able to compute transonic buffet but ZDES better simulates some detached behaviour and, of course, both ZDES and DES captured secondary fluctuations of the flow properties modeled in URANS.

Introduction

Memmolo et al. [START_REF] Memmolo | Scrutinity of buffet mechanism in transonic flow[END_REF] also agree that both DES and URANS approaches result in a consistent prediction of buffet. Grossi et al. [START_REF] Grossi | Prediction of transonic buffet by delayed detached-eddy simulation[END_REF], even if they showed that DDES captured properties not computed by URANS, concluded that URANS results are more coherent with the experimental data. It is worth to conclude the section on transonic buffet over aerofoil with an highlight of the influence on the simulations of numerical parameters such as spatial and temporal discretisation, numerical schemes, turbulence models etc. Giannelis et al. [START_REF] Giannelis | A review of recent developments in the understanding of transonic shock buffet[END_REF] well summarised the latter; here only the most relevant influences are mentioned. Furthermore, different versions of the same code can give different results depending on the turbulence model used. An interesting conclusion on turbulence models, strongly shared by recent works, has been outlined by Grossi et al. [START_REF] Grossi | Prediction of transonic buffet by delayed detached-eddy simulation[END_REF]. They showed that

Edwards & Chandra [START_REF] Edwards | Comparison of eddy viscosity-transport turbulence models for three-dimensional, shock-separated flowfields[END_REF] (EC) and compressibility corrections (CC, Spalart [START_REF] Spalart | Trends in turbulence treatments[END_REF]) of Spalart-Allmaras (SA) turbulence model (Spalart & Allmaras [139]) tends to reduce the dissipation of the model and increases the agreement between simulations and experimental data.

Context

Stability analysis

The linear stability analysis is a theory commonly used in the study of dynamical systems. The main idea is to look at the dynamical system evolution of a perturbed initial flow, called baseflow. In the last 20 years its application in the field of fluid dynamics strongly increases (Theofilis [146], Sipp et al. [START_REF] Sipp | Dynamics and control of global instabilities in open-flows: a linearized approach[END_REF]). The capability of predicting and highlighting some characteristics of flow unsteadiness helps the physical interpretation of the unsteady phenomena. Crouch et al. [START_REF] Crouch | Predicting the onset of flow unsteadiness based on global instability[END_REF] was the first to perform stability analysis of a transonic flow over an aerofoil. They considered a NACA0012 at fixed Mach and Reynolds numbers (M = 0.76 and Re = 10 7 ). They explained the appearance of the buffet phenomenon with the appearance of an unstable mode. It means that for certain values of M -α the energy balance of the self-sutained mechanism of a certain stable mode becomes positive, i.e.

the mode becomes unstable and imposes its dynamics on the flow. Results show good agreement with both experiments and numerical simulations (Jacquin et al. [START_REF] Jacquin | Experimental study of shock oscillation over a transonic supercritical profile[END_REF] and [START_REF] Deck | Numerical simulation of transonic buffet over a supercritical airfoil[END_REF]). The buffet mode becomes unstable when the angle of attack is increased from 3.1 • to 3.2 • through a Hopf bifurcation (since the mode becomes unstable at non-zero frequency). Crouch et al. [START_REF] Crouch | Global structure of buffeting flow on transonic airfoils[END_REF] and [START_REF] Crouch | Origin of transonic buffet on aerofoils[END_REF] showed that the perturbations travel upward downstream of the shock (see the pressure fluctuations in figure 1.4). Furthermore

Crouch outlined the necessity to perturb all the variables of the baseflow, including the turbulent quantities to obtain the buffet unstable mode. To the authors' knowledge, the buffet instability is the only example where the frozen eddy viscosity approach (in which turbulent quantities are not perturbed) does not give the same results as the full approach (Mettot et al. [97] [98]). The global stability analysis was repeated by Sartor et al. [START_REF] Sartor | Stability, receptivity, and sensitivity analyses of buffeting transonic flow over a profile[END_REF], Guiho [START_REF] Guiho | Analyse de stabilité linéaire globale d'écoulements compressibles: application aux interactions onde de choc/couche limite[END_REF] and Iorio et al. [START_REF] Iorio | Direct and adjoint global stability analysis of turbulent transonic flows over a NACA0012 profile[END_REF]. Sartor et al. [START_REF] Sartor | Stability, receptivity, and sensitivity analyses of buffeting transonic flow over a profile[END_REF] was the first to show complete spectra of the global buffet mode from onset to exit. Iorio [START_REF] Iorio | Global Stability Analysis of Turbulent Transonic Flows on Airfoil geometries[END_REF] furthermore performed a stability analysis of a NACA0012 in buffet conditions over a computational domain of reduced size. She showed that a reduction of the domain up to a radius of two chords around the aerofoil does not have a significant impact on the unstable buffet mode. This result suggests that the buffet mode is concentrated around the aerofoil and it does not need the entire computational field to exhibit its unstable dynamics.

To find the regions in space where the instability develops is an important task in the framework of global stability analysis and several methods were developed in this direction. The first who introduced this idea were Huerre & Monkewitz [START_REF] Huerre | Local and global instabilities in spatially developing flows[END_REF]. They introduced the concept of wavemaker as the region where the instability waves are intrinsically generated for globally unstable flows. The interpretation by Koch [START_REF] Koch | Local instability characteristics and frequency determination of selfexcited wake flows[END_REF] of global instability uses a similar idea. Today, the most accepted definition of the wavemaker comes from Giannetti & Luchini [START_REF] Giannetti | Structural sensitivity of the first instability of the cylinder wake[END_REF]: it is a structural sensitivity that quantifies how an eigenvalue is affected by the introduction of localized forcing. Then the concept was extended to nonlinear global modes by way of Floquet theory (Luchini et al. [START_REF] Luchini | Structural sensitivity of linear and nonlinear global modes[END_REF]). Marquet et al. [START_REF] Marquet | Sensitivity analysis and passive control of cylinder flow[END_REF] investigated the sensitivity with respect to localised modifications of the baseflow which is more relevant for flow control. In summary, a lot of authors studied the problem of the instability localisation but always in the context of sensitivity analyses. In chapter 2, an alternative technique is introduced, which does not target the eigenvalue variation but the eigenvalue itself.

Three dimensional wing transonic buffet

The early studies conducted on 3D configurations were mainly experimental. Hwang & Pi [START_REF] Hwang | Northrop F-5A aircraft transonic buffet pressure data acquisition and response analysis[END_REF] presented an analysis of PSD distributions obtained with unsteady pressure transducers during flight tests of a Northrop F-5A aircraft. The spectra showed a

Strouhal number of about 0.23 for 3D buffet at well-established buffet conditions but globally the study was more focused on structural response. The description of the spectral content was then largely improved by Reneaux et al. [START_REF] Reneaux | A combined experimental and numerical investigation of the buffet phenomenon and its control through passive and active devices[END_REF] and Roos [START_REF] Roos | The buffeting pressure field of a high-aspect-ratio swept wing[END_REF].

Reneaux et al. [START_REF] Reneaux | A combined experimental and numerical investigation of the buffet phenomenon and its control through passive and active devices[END_REF] presented a combined experimental and numerical study on both 2D and 3D buffet while Roos [START_REF] Roos | The buffeting pressure field of a high-aspect-ratio swept wing[END_REF] computed a high-aspect-ratio swept half-wing at Mach number 0.827 and an angle of attack of 11 the structural response of the model and the shock unsteadiness across the span.

Two campaigns, analysed in chapter 4 of the present work, have already been presented in previous studies. Molton et al. [START_REF] Molton | Control of buffet phenomenon on a transonic swept wing[END_REF] showed the spectral analysis and results of buffet control by VGs for the ONERA research project BUFET'N Co. Dandois [START_REF] Dandois | Experimental study of transonic buffet phenomenon on a 3D swept wing[END_REF] performed a complete analysis of the BUFET'N Co and AVERT databases. He gave values for the buffet onset at different values of M -α and characterized the frequency spectra evolution in the chordwise and spanwise directions. By using different signal processing tools, the convection velocities of the buffet phenomenon and of the Kelvin-Helmholtz (K-H) instability were obtained. A large range investigation of M -α has been analysed in Koike et al. [START_REF] Koike | Unsteady pressure measurement of transonic buffet on NASA common research model[END_REF]. A classification of the shockwave oscillation in three regions was proposed. The first region is before buffet onset and without separation.

The second region is the classical buffet phenomenon with a bump in the spectra at a Strouhal number around 0.3. The third region is at high M -α with large shock oscillations and a broadband bump in the spectra at low frequency. The results of chapter 4 of the present work are entirely in the second region defined by Koike et al. [START_REF] Koike | Unsteady pressure measurement of transonic buffet on NASA common research model[END_REF]. It is however found a behaviour coherent with the third region: the bump in the spectra broadens and the frequency decreases when increasing α.

Context

With the increase of computational resources, more and more unsteady numerical simulations of 3D configurations are performed. Brunet & Deck [START_REF] Brunet | Zonal-detached eddy simulation of transonic buffet on a civil aircraft type configuration[END_REF], to the author's knowledge, were the first to perform a high fidelity Zonal DES (see Deck [START_REF] Deck | Recent improvements in the zonal detached eddy simulation (zdes) formulation[END_REF] for more details) of the 3D transonic buffet phenomenon. They showed a good prediction of the time-averaged field compared with experiments. The same high-fidelity simulation was repeated by Deck et al. [START_REF] Deck | High-fidelity simulations of unsteady civil aircraft aerodynamics: stakes and perspectives. application of zonal detached eddy simulation[END_REF] showing the latest developments and trends for unsteady civil aircraft applications. Raveh [START_REF] Iovnovich | Numerical study of shock buffet on threedimensional wings[END_REF] studied the phenomenon on 3D wings at different sweep angles and aspect ratio. They were the first to give an interpretation of the path from 2D to 3D buffet.

At zero or small sweep angles the results are similar to the 2D phenomenon. When the sweep is increased over 15 • , the Strouhal number also increases and reaches typical values of the 3D phenomenon with spanwise-propagating waves appearing on the wing, called "buffet cells". This convective phenomenon was then observed in experiments by Dandois [START_REF] Dandois | Experimental study of transonic buffet phenomenon on a 3D swept wing[END_REF], Koike et al. [START_REF] Koike | Unsteady pressure measurement of transonic buffet on NASA common research model[END_REF], Sugioka et al. [START_REF] Sugioka | Experimental analysis of transonic buffet on a 3D swept wing using fast-response pressure-sensitive paint[END_REF], Masini et al. [START_REF] Masini | Influence of vane vortex generators on transonic wing buffet: further analysis of the bucolic experimental dataset[END_REF]. They computed the outboard convection velocity by using a cross-spectrum analysis in the spanwise direction at buffet frequency. Dandois [START_REF] Dandois | Experimental study of transonic buffet phenomenon on a 3D swept wing[END_REF], Koike et al. [START_REF] Koike | Unsteady pressure measurement of transonic buffet on NASA common research model[END_REF] and Sugioka et al. [START_REF] Sugioka | Experimental analysis of transonic buffet on a 3D swept wing using fast-response pressure-sensitive paint[END_REF] found the same value of the convection velocity in the spanwise direction around 0.35U ∞ . Masini et al. [START_REF] Masini | Influence of vane vortex generators on transonic wing buffet: further analysis of the bucolic experimental dataset[END_REF] found a smaller value of convection velocity of 0.26U ∞ . These buffet cells are typical of the 3D transonic buffet. Sartor & Timme [START_REF] Sartor | Reynolds-Averaged Navier-Stokes simulations of shock buffet on half wing-body configuration[END_REF] also observed these complex structures in the spanwise direction typical of the 3D buffet. They studied the effects of different parameters such as Mach number, angle of attack and turbulence model on URANS simulations. Furthermore in Sartor & Timme [START_REF] Sartor | Delayed detached-eddy simulation of shock buffet on half wing-body configuration[END_REF], a comparison between delayed-DES and URANS simulations was presented and even though delayed-DES showed obviously a deeper description of the flow, a good agreement of the main features of the flow was found with URANS modelling.

Recently, Plante et al. [START_REF] Plante | Study of three-dimensional transonic buffet on swept wings[END_REF] performed URANS simulations of three-dimensional transonic buffet for different sweep angles. In the unswept case, they observed an Introduction & Theofilis [START_REF] Rodríguez | On the birth of stall cells on airfoils[END_REF] through global stability analysis. They showed for incompressible flows that these cells originate from a three-dimensional global unstable mode. The recent published Ohmichi et al. [START_REF] Ohmichi | Modal decomposition analysis of three-dimensional transonic buffet phenomenon on a swept wing[END_REF] performed a POD and DMD (dynamic mode decomposition [START_REF] Schmid | Dynamic mode decomposition of numerical and experimental data[END_REF]) analysis and outlined the dominant fluid structures and unsteady behaviour of the 3D transonic buffet. They found two dominant fluid structures at the typical frequencies and with the similar flow structures of the 2D and 3D transonic buffet, respectively. Furthermore, they suggested the existence of a three dimensional global instability since DMD is related to global stability analysis.

Context

Stability analysis

Theofilis [START_REF] Theofilis | Global linear instability[END_REF] reviewed the achievement in the linear instability analysis over 3D configurations. He presented classical theories as well as new-generation algorithms.

BiGlobal analysis is a two-dimensional global stability analysis where the shape of the perturbation in the third dimension is imposed (a so-called 2.5-dimensional analysis where spanwise perturbations have a sinusoidal form) while in TriGlobal analysis no assumptions are made about the three dimensional perturbations. Robinet [START_REF] Robinet | Bifurcations in shock-wave/laminar-boundary-layer interaction: global instability approach[END_REF] performed a BiGlobal stability analysis of an oblique shock wave/ laminar boundary layer interaction computed with a DNS. He found a three-dimensional unstable stationary global mode. The first example of three dimensional global stability analysis is the study of the flowfield around a spheroid by Tezuka and Suzuki [START_REF] Tezuka | Three-dimensional global linear stability analysis of flow around a spheroid[END_REF].

The application of these kinds of stability analysis to transonic buffet is recent because the numerical method has been developed in the last years and, at the same time, buffet is not a simple test case. There are few papers on this subject and a lot of questions are still open. Transonic buffet has been studied for the first time on a rectangular unswept finite-wing through three dimensional stability analysis by Iorio et al. [START_REF] Iorio | Direct and adjoint global stability analysis of turbulent transonic flows over a NACA0012 profile[END_REF]. They found the two dimensional unstable mode but no appearance of three dimension perturbations. Similarly Timme & Thormann [START_REF] Timme | Towards three-dimensional global stability analysis of transonic shock buffet[END_REF] performed a three-dimensional global stability analysis of the transonic buffet phenomenon on a realistic half-wing body configuration and only found a weakly damped global mode.

It is only recently that Timme [START_REF] Timme | Global instability of wing shock buffet[END_REF] found an unstable three-dimensional mode on the same configuration but still did not answer the main question addressed in the present work: what is the link between two-dimensional and three-dimensional buffet?

Crouch et al. [START_REF] Crouch | Global instability analysis of unswept-and swept-wing transonic buffet onset[END_REF], simultaneously and independently from the present work, is one of the first trying to answer this question. They presented very recently in a conference the so-called BiGlobal stability analysis over swept and unswept infinite wings. In addition to the two-dimensional buffet mode already observed, they found three-dimensional modes. Two ranges of spanwise wavelengths are unstable. They are centered around a wavelength of one and one-tenth chord length, respectively. These modes are steady for unswept wings and become unsteady with the introduction of sweep and their frequency increases with sweep. The three-dimensional mode centered around a wavelength of one chord shows a structures and a frequency coherent with the three dimensional buffet mode.

Introduction

Outline and scope

The present study aims to increase the understanding of the transonic buffet phenomenon and explain the relation of the latter between two-dimensional aerofoil and three-dimensional wing configurations.

RANS equations

The governing equations are presented in a general conservative discretised form:

dq dt = R(q), (2.1)
where q ∈ R N represents the set of state variables of the flow field in every cell of the computational domain D and R : D ∈ R N → R N is the compressible Navier-Stokes operator derivable over D which represents the discrete residuals. The boundary conditions are included in the operator R. Considering the finite volume space discretisation the size of the vector q corresponds to the size of the computational mesh multiplied by the number of state variables.

In the following the governing RANS equations are presented with a separation of the mean (superscripts mf ) and turbulent (superscripts tf ) fields:

d dt    q mf q tf    = •    F c,mf + F d,mf F c,tf + F d,tf    +    0 T    (2.
2)

The compressible Navier-Stokes operator R is also decomposed in terms of the convective fluxes F c , the diffusive fluxes F d and the turbulence source term T . The state variable linked with the mean flow dynamics are

q mf = (ρ i , ρU i , ρV i , ρW i , ρE i ) T ,
where ρ is the density, (U, V, W ) the three components of the velocity U, E the total energy of the flow per unit mass and i = 1, ..., N cells . The continuous form of the mean field fluxes is:

F c,mf = -       ρU ρU × U + p I ρE U + p U       , F d,mf =       0 τ + τ r τ U + τ r U -q h -q t      
, where:

p = ρRT, τ = - 2 3 µ( • U)I + 2µD, q h = - c p µ Pr T τ t = - 2 3 µ t ( • U)I + 2µ t D, q t = - c p µ t Pr t T
p is the pressure, R the perfect gas constant, c p the heat capacity at constant pressure, µ the dynamic viscosity, T the temperature, τ the viscous tensor, q h the heat flux, D and I the rate of strain and identity tensors respectively, µ t the eddy viscosity (computed with the chosen turbulence model), τ t the Reynolds stress tensor, q t the flux of diffusion of turbulent enthalpy, Pr and Pr t the classical and turbulent

Prandtl number considered constants and respectively equal to 0.72 and 0.9. The perfect gas relations is considered and the viscosity is computed using Sutherland's Numerical approach law: with FastS in chapter 5. Furthermore, Edwards-Chandra correction [START_REF] Edwards | Comparison of eddy viscosity-transport turbulence models for three-dimensional, shock-separated flowfields[END_REF] (EC) and the compressibility correction [START_REF] Spalart | Trends in turbulence treatments[END_REF] (CC) of the classical SA turbulence model have also been considered in elsA and FastS, respectively. This is probably the main difference between the two applications of the solvers. The reason is because CC is not available in elsA and equally EC in FastS ; use of both SA corrections is preferred. Indeed, it has been shown by Grossi et al. [START_REF] Grossi | Prediction of transonic buffet by delayed detached-eddy simulation[END_REF] that the agreement between URANS simulations of transonic buffet and the experimental data from Jacquin et al. [START_REF] Jacquin | Experimental study of shock oscillation over a transonic supercritical profile[END_REF] was improved using the SA turbulence model with EC and CC.

µ = µ s T T s 3/2 T s + C s T + C s (2.3) µ s = 1.

Spalart-Allmaras turbulence model

The SA turbulence model uses only one transport equation for the quantity ν which is a transformed eddy kinematic viscosity (q tf = ρν i with i = 1, ..., N cells ). Far from the walls, ρν tends towards the eddy viscosity µ t = ρν t . The model has been chosen because previous studies proved its ability to correctly reproduce the challenging buffet phenomenon (Brunet [12], Deck [START_REF] Deck | Numerical simulation of transonic buffet over a supercritical airfoil[END_REF], Thierry & Coustols [START_REF] Thierry | Numerical prediction of shock induced oscillations over a 2D airfoil: Influence of turbulence modelling and test section walls[END_REF]). It had also been used to reproduce buffet with a non-linear eddy viscosity model (Barakos & Drikakis [4]). The equation for ν results from a step by step construction by addition of terms intended for taking into account more and more physical phenomena. On the basis of a convection = production + dif f usion -dissipation form, the Spalart-Allmaras model adds the terms necessary to obtain a logarithmic zone in the velocity profiles, then those for wall effects. The turbulent fluxes and source terms are then given by:

F c,tf = -ρνU, F d,mf = - µ + ρν σ ν ν T = C b1 Sρν + C b2 σ ν (ρν) • ν -C w1 f w ρ ν2 η 2
The eddy viscosity is then evaluated by the following relations :

ν t = f ν1 ρν ρ , f ν1 = χ 3 χ 3 + C 3 ν1 , χ ≡ ρν µ (2.4)
and the other variables of the model are:

S = Ω + f ν2 ρν ρK 2 η 2 , f ν2 = 1 - χ 1 + χf ν1 f w = g 1 + C 6 w3 g 6 + C 6 w3 1/6 , g = r + C w2 (r 6 -r), r = ρν ρ SK 2 η 2
Constant terms: R specific gas constant, γ = C p /C v specific heat ratio, S is the magnitude of the strain tensor, Ω is the vorticity magnitude, η distance to the wall. And the others:

σ ν = 2/3, K = 0.41, C b1 = 0.1355, C b2 = 0.622, C w1 = C b1 K 2 + (1 + C b2 ) σ ν , C w2 = 0.3, C w3 = 2.0, C ν1 = 7.1.
The Edwards-Chandra modification [START_REF] Edwards | Comparison of eddy viscosity-transport turbulence models for three-dimensional, shock-separated flowfields[END_REF] has been proposed to increase numerical stability. It has been used in the simulations with elsA in chapter 3. It differs from the baseline Spalart-Allmaras model because f t2 is ignored (it has been already done in the formulation above) and two variables are redefined:

S = S 1/2 1 χ + f ν1 and r = tanh[ν/(K 2 η 2 S)] tanh(1.0) (2.5)
where S 1/2 is used instead of vorticity Ω. The effect of this correction is a reduction of the eddy viscosity µ t in the near-wall regions.

The compressibility correction of Secundov (discussed in the paper by Spalart [START_REF] Spalart | Trends in turbulence treatments[END_REF])

enhances the SA behaviour in compressible mixing layer. It has been used in the simulations with FastS in chapter 5. It increases the destruction term by tending to lower the eddy viscosity levels in turbulent regions of high deformation and velocity magnitude, such as compressible mixing layers. The only difference with the classical SA turbulence model is an additional term in the right hand side of the equation.

-C 5 ρν 2 a 2 U i,j U i,j (2.6)
where a is the speed of sound, C 5 = 3.5 and U i,j is the velocity vector in Einstein notation.

k-kl turbulence model The k-kl turbulence (Daris & Bézard [START_REF] Daris | Four-equation models for reynolds stress and turbulent heat flux predictions[END_REF]) model uses two equations to provide closure for the averaged Reynolds stresses (q tf = (ρk i , ρkl i ) with i = 1, ..., N cells ). One equation for the transport of the turbulent kinetic energy k and the other for the transport of a quantity that is the product of the turbulent kinetic energy with the length scale kl = k 5/2 / , where is the dissipation (a wall correction is also considered for the dissipation). The capability of this turbulence model to simulate the buffet phenomenon has been shown by Thierry & Coustols [START_REF] Thierry | Numerical prediction of shock induced oscillations over a 2D airfoil: Influence of turbulence modelling and test section walls[END_REF]. It is based on the classical form convection = production + dif f usion -dissipation. The turbulent fluxes and source terms are then given by:

F c,tf = -    ρkU ρklU    , F d,tf =     µ + µ t σ k k µ + µ t σ kl (kl)     , T =       P k -ρ k 5/2 kl - 2µk η 2 C kl1 kl k P k -C kl2 ρk 3/2 -C klw f w ρ 3 (kl) 5/2 µ 2 η 1/2 + C D      
,

where

C D = 4C klkl µ t (kl) 1/2 (kl) 1/2 + 2C klk µ t k 1/2 (kl) (k 1/2
) and the production term of turbulent kinetic energy

P k = min(P u k , M ρ ) with P u k = (τ t ) ij U i,j
. M is a fixed value from 10 to 20 which limits the production of k with respect to the dissipation. The damping functions are:

f w = exp(-C w1 R η ), f µ = 1 -exp(-C w2 R 3/2 η )
where

R η = ηk 1/2 ν .
The eddy viscosity is so define as:

ν t = C µ kl k 1/2 (2.7)
The constants of the model are:

C kl1 = 1, C kl2 = 0.58, C µ = 0.09, C klk = 0.96, C klkl = -1.72, σ k = 1.8, σ kl = 1.03.

Numerical schemes

The Navier-Stokes equations (2.2) are solved by the finite volume method in each cell of the computational domain. Concerning the spatial schemes, a second-order accurate AUSM+(P) upwind scheme is used for the mean convective fluxes. The versions by

Edwards & Liou [START_REF] Edwards | Low-diffusion flux-splitting methods for flows at all speeds[END_REF] and Mary & Sagaut [START_REF] Mary | Large eddy simulation of flow around an airfoil near stall[END_REF] are respectively implemented in elsA and

FastS. The Jameson center scheme [START_REF] Jameson | Numerical solution of the euler equations by finite volume methods using runge kutta time stepping schemes[END_REF] is also been used for the mean convective fluxes in the simulations computed with elsA. It is a centered scheme with a scalar artificial viscosity. The convective flux associated to the turbulence equation is discretised using the first-order Roe scheme with Harten's correction to prevent the occurrence of low eigenvalues (Harten & Hyman [START_REF] Harten | Self adjusting grid methods for one-dimensional hyperbolic conservation laws[END_REF]), whilst a central difference scheme is used for the turbulent diffusive flux. The viscous flux of the mean field is calculated at the interface by averaging cell-centred values of flux density, computed with a 5-point stencil. The source terms are discretised using estimates of gradients and variables at cell centres.

More details about the numerical method of FastS solver are available in Pechier et al. [START_REF] Pechier | Magnus effect over finned projectiles[END_REF]. Both an upwind and a centered scheme have been used during this work in order to show the robustness of the solvers when simulating the buffet phenomenon. On the other hand, this work does not aim at pointing out the sensitivity of transonic buffet to numerical parameters. This is the reason why results with Jameson scheme are not presented. A phenomenological reproduction of buffet is the aime and the singular effect of each numerical parameter is not analysed. However, in terms of sensitivity to numerical parameters it is noted something already highlighted by previous studies:

"the highest sensitivity is in the choice of the turbulence model".

For time-stepping, a first-order backward-Euler scheme is used with local timestepping to obtain the steady solution q 0 . The values of Courant-Friedrichs-Lewy (CF L) number changes in each simulation as a function of the angle of attack; values between 1 and 10 have been used. For the unsteady simulations, a dual-time stepping method DTS (Jameson [START_REF] Jameson | Time dependent calculations using multigrid, with applications to unsteady flows past airfoils and wings[END_REF]) with a non-dimensional time step ∆t

U ref c = 1.08 * 10 -3
Numerical approach is used (14285 time steps per buffet period), achieving a second-order accuracy. DTS allows the resolution of unsteady problems using sub-iterations of calculation between two physical instants. It belongs to the family of the sub-iterative methods and resorts to the introduction of a fictitious time step, the dual time. During sub-iterations cycle of each time step the norm of the residual decreases by a factor of 10. The value of the time step yields a maximum CF L of about 25 in the attached boundary layer, 50 in the wake and less than one in most of the domain (the values are mostly the same for the different computational grids). These values of CF L have been computed for the first computational grid presented in the next paragraph.

Computational grid

The computational domain is a multi-block C-type structured grid. The aerofoil geometry is ONERA's OAT15A transonic aerofoil with a chord length equal to 0.23 m and the far-field conditions are imposed at 50 chords away from the profile. The mesh contains 72000 cells with a refinement around the time-averaged shock location (constant grid refinement in the stream-wise direction has been used). The grid definition in the shock region is ∆x/c = 0.003. The first mesh point in the boundary layer is below y + = 0.9. To check the spatial convergence of the computations, other meshes are considered: a coarser grid with 36000 cells and a finer one with 100000 cells. The study on the mesh convergence are shown for the stability analysis in section 3.2.1.

The grid with 72000 cells is considered as the reference for chapter 3.

The 3D computational domain used in chapter 5 results from the extrusion in span of the above 2D grid. The 2D mesh has been reduced to 50000 cells with the same values of y + . The computation over the 3D domain has been conducted with 12 planes in span which gives a total of 600000 cells (the details of the grid convergence are shown in section 5.2.1).

Boundary and flow conditions

The conditions of the flow are the following: a far-field total pressure of 101325 P a,

a
velocity W ∞ = U n∞ tan(Λ), where U n∞ = (U 2 ∞ + V 2 ∞ ) 1/2 = U ref cos(Λ)
and U ref is the amplitude of the free-stream velocity of the numerical simulation. To maintain the same normal flow conditions with respect to the wing, the incoming flow and the angle of attack are modified when Λ = 0 according to:

M ∞ = M n∞ / cos(Λ) and α = arctan [tan(α n ) cos(Λ)].
For more details of the 3D base flow configuration see section 5.1 and figure 5.1. To summarise, for the aerofoil reference frame (x, y, z), the free-stream velocity vector is:

U ∞ = (U ∞ , V ∞ , W ∞ ) = (U ref cos(α) cos(Λ), U ref sin(α) cos(Λ), U ref sin(Λ)). (2.8)

Global stability analysis

Global stability analysis studies the evolution of a small amplitude perturbation q around a base flow q 0 which is a steady solution of equation (2.1):

q(t) = q 0 + q (t), ||q || << ||q 0 ||. (2.9)
Linearising the perturbed equations to the first order :

dq dt = Jq J kl = ∂R k ∂q l q=q 0 (2.10)
where J ∈ R N ×N is the linearised discrete Navier-Stokes operator R around the base flow, R k is the k th component of the residual and q l are the state variables in every cell of the mesh. In this work the computation of the linearised operator is based on a finite difference method. Indeed the analytic derivation of the linearised equations Numerical approach remains a difficult task for the complexity of the equations with the turbulence models, the boundary conditions and the spatial discretisation techniques. The computation of the matrix Ju, with u an arbitrary vector, results from a repeated evaluations of the residual of the governing equations. The code used to perform the CFD simulations may then be used in a black box manner (this is the reason why both elsA and FastS have been used for stability analysis). Considering a valid discrete residual R(u) generated by the code, one may obtain Ju with the following first order approximation:

Ju = R(q 0 + εu) -R(q 0 ) ε (2.11)
with ε a small constant. The computation of the Jacobian matrix is consequently linked with a long and repeated evaluation of residual after the small perturbation εu. The structure of the matrix is intrinsically dependent on the discretisation stencil, which in the used CFD code are compact, ensuring the sparsity of the matrix. Finally, the computation of the matrix can be optimised with a suitable definition of the vectors u.

By considering the shape of the stencil discretisation of the residual R, it is possible to adapt the vectors u in order to drastically reduce the residuals evaluations for the matrix J coefficients. For more details on the numerical strategy, the suitable value of ε and the shape of u see Mettot et al. [START_REF] Mettot | Computation of eigenvalue sensitivity to base flow modifications in a discrete framework: Application to open-loop control[END_REF].

The stability analysis of the baseflow turns into the study of the eigenvalues of the Jacobian matrix. The shape of normal modes is imposed for the linearised variable as:

q (t) = qexp(λt) + c.c. (2.12)
where λ is the eigenvalue whose real part is the growth rate σ of the perturbation, the imaginary part ω its angular frequency and c.c. refers to complex conjugate quantities. Q is then a real diagonal matrix for which the terms Q i correspond to the volume of each cell. The adjoint matrix J † is defined as the matrix satisfying:

ω = 2π f , with f frequency in Hz. J q = λq = (σ + iω)q (2.
a, Jb Q = J † a, b Q . (2.14)
It is straightforward to show that:

J † = Q -1 J * Q (2.15)
It follows that the eigenvalues of J † are the complex conjugates of those of J. Given an eigenvalue/eigenvector pair (λ, q), the associated adjoint global mode is defined as:

J † q † = λ * q † = (σ -iω)q † . ( 2.16)
An important property used in the following is the bi-orthogonality of the two bases composed by the entire sets of eigenvectors of J and J † with respect to the defined inner product:

q † n , qj Q = q † * n Qq j = δ nj (2.17)
where δ nj is the Kronecker symbol. It means also that the inner product of a direct eigenvector with its adjoint is normalized at one. For simplicity, the weighted adjoint eigenvector q = q † Q will be considered in the following. q correspond to the adjoint eigenvector of the adjoint problem defined on the canonical element-wise inner product.

The direct and adjoint global modes form a bi-orthogonal basis with respect to the considered scalar-product: if the eigenvalues and direct/adjoint global modes are expressed as (λ i , qi , q † j ), then one may show that (λ * i -λ j )q † * i Qq j = 0. Hence, either λ * i = λ j and q † * i Qq j = 0 or λ * i = λ j and q † * i Qq j = 0. The Jacobian matrix is non-normal because of the convection operator in the Navier-Stokes equations (Chomaz [17]). Consequently, direct global modes propagate downstream and adjoint global modes upstream (Marquet et al. [START_REF] Marquet | Direct and adjoint global modes of a recirculation bubble: lift-up and convective nonnormalities[END_REF]). The physical Numerical approach meaning of the adjoint modes corresponds to sensitive regions of the flow to force, damp or in general control the phenomenon (Sipp [132]). They underline the region in the flow where a forcing results in the strongest effect on the unstable dynamics of the flow.

Concerning the numerical strategy, both the direct and adjoint eigenvalue problems are solved with the implicitly restarted Arnoldi algorithm with a shift-and-inverted strategy, using the open source library ARPACK [START_REF] Lehoucq | ARPACK users' guide: solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods[END_REF]. The direct LU solver MUMPS [START_REF] Amestoy | A fully asynchronous multifrontal solver using distributed dynamic scheduling[END_REF] is used for the linear solver. The computational details of the eigenvalue problem are provided in chapter 5. For more details on the global stability analysis and the procedure to solve the eigenvalue problem see Mettot et al. [START_REF] Mettot | Computation of eigenvalue sensitivity to base flow modifications in a discrete framework: Application to open-loop control[END_REF] and Beneddine et al. [START_REF] Beneddine | Conditions for validity of mean flow stability analysis[END_REF]. It is interesting to underline that all the state variables in equation (2.9) are perturbed, including the turbulent variable. Crouch et al. [START_REF] Crouch | Origin of transonic buffet on aerofoils[END_REF] already showed that without perturbations on the turbulent variable (frozen µ t model, Reynolds & Hussain [START_REF] Reynolds | The mechanics of an organized wave in turbulent shear flow. part 3. theoretical models and comparisons with experiments[END_REF]), the unstable buffet mode does not appear in the spectrum. This result underlines the key role of turbulence in transonic buffet.

Identification of local hydrodynamic feedback in the unstable eigenmode

The global stability analysis is useful to determine the temporal stability of non-parallel flows, i.e. flows exhibiting large variations in several directions of the space. However,

it

Structural sensitivity analysis in a discrete setting

To determine flow regions where local feedback mechanisms are active, Giannetti & Luchini [START_REF] Giannetti | Structural sensitivity of the first instability of the cylinder wake[END_REF] performed a structural sensitivity analysis of the eigenvalue problem written in a continuous setting. Following Schmid & Brandt [START_REF] Schmid | Analysis of fluid systems: Stability, receptivity, sensitivitylecture notes from the flow-nordita summer school on advanced instability methods for complex flows, stockholm, sweden[END_REF], their analysis is rewritten here in a discrete setting. The discrete Jacobian matrix J being perturbed by an arbitrary matrix δJ; the perturbed eigenvalue problem can be written

(λ + δλ)(q + δq) = (J + δJ)(q + δq) (2.18)
where δλ and δq denote the eigenvalue and eigenvector variations, respectively. To model a local hydrodynamic feedback, they considered a specific matrix modification modeling a local hydrodynamic feedback at a point x k . The discrete form of this matrix modification, denoted here δJ k , is

δJ k =                   . . . 0 0 0 • • • 0 J 0 0 • • • 0 0 0 . . .                   , ( 2.19)
where J 0 is an (unknown) constant coefficient matrix (of size 5 × 5) representing the local feedback at the k th cell. Inserting the above matrix variation into the perturbed eigenvalue problem (2.18) and left multiplying by the adjoint eigenmode q † yields the eigenvalue variation

δλ k = q * k J 0 qk (2.20)
where qk and qk are the direct and adjoint state vectors at the k th cell. An upper bound of the eigenvalue variation is then obtained as

|δλ k | ≤ ||J 0 || w k (2.21)
where ||J 0 || a matrix-norm and w k is the k th component of the discrete wavemaker vector w, defined as

w k = ||q k || ||q k ||. (2.22)
The wavemaker vector w (or function in a continuous setting) has been used in many studies to determine the locations where the local feedback induces largest eigenvalue variation, identifiying in this way the flow regions where the instability mechanism acts.

Instead of considering a local hydrodynamic feedback, Marquet et al. [START_REF] Marquet | Sensitivity analysis and passive control of cylinder flow[END_REF] have then proposed to determine regions where base flow modifications induce large eigenvalue variations. In the next paragraph, it will be shown that an equality between the eigenvalue and an expression similar to the wavemaker can be established by considering a decomposition (rather than a variation) of the Jacobian matrix. A first attempt to obtain such an equality was proposed in Marquet & Lesshaft [START_REF] Marquet | Identifying the active flow regions that drive linear and nonlinear instabilities[END_REF].

Column-decomposition of the Jacobian matrix and eigenvalue contribution

Rather than considering a variation, the Jacobian matrix is here decomposed as

J =   N cells k=1 C k   (2.23)
the sum of column-matrices C k , defined for each cell k of the mesh, as

C k =                             0 • • • 0 ∂R 1 ∂q k 0 • • • 0 . . . • • • • • • . . . • • • • • • . . . . . . • • • • • • . . . • • • • • • . . . 0 • • • 0 ∂R l ∂q k 0 • • • 0 . . . • • • • • • . . . • • • • • • . . . . . . • • • • • • . . . • • • • • • . . . 0 • • • 0 ∂R N cells ∂q k 0 • • • 0                             , ( 2.24)
where ∂R l ∂q k is a 5 × 5 matrix representing the Navier-Stokes written at cells l for the 5 state variables and linearised with respect to the 5 state variables at cells k. The objective of the method is to quantify the contribution of the column-matrices C k to the eigenvalue λ associated to the eigenmode q. The column-matrix decomposition is introduced into the eigenvalue problem (2.13), yielding

λq =   N cells k=1 C k   q = N cells k=1 (C k q) (2.25)
The matrix-vector product of the column-matrix C k with the eigenmode q is now expanded onto the basis of the eigenmodes as

C k q = λ k q + m≥1 β km qm = λ k q + r k (2.26)
where λ k is the coefficient of the eigenmode q and β km are the coefficients for the other eigenmodes in the basis, denoted here qm . The sum over the other eigenmodes is denoted r k , since it represents the residual vector of the eigenvalue problem for the column-matrix C k . Note that the coefficients β km and the residual vector are not equal to zero, since q is not an eigenvector of C k . Nevertheless, the only interest is to determine the projection coefficient λ k , as it gives the contribution of the matrix-vector product C k q into the direction of the eigenmode q. By introducing the decomposition (2.26) into (2.25):

λq =   N cells k=1 λ k   q (2.27)
where the sum of all the residual terms vanishes by construction ( N cells k=1 r k = 0). The above relation clearly shows that the eigenvalue λ is the sum of the amplitude coefficients λ k , i.e.

λ = N cells k=1 λ k .
(2.28)

The coefficient λ k are then easily determined by projecting the expression (2.26) with the weighted adjoint eigenmode q. Using the bi-orthogonal property of the adjoint eigenmode with respect to the other direct eigenmode of the basis ( m q * • qm = 0), one obtains the following expression of the coefficient

λ k = q * C k q. (2.29)
which is the contribution of the k th column-matrix C k to the eigenvalue λ. By taking the real and imaginary part of the above expression, one obtains the contribution C k

Numerical approach

to the growth rate σ and angular frequency ω. The summation of these coefficients over all the cells yields the growth rate and frequency of the eigenmode. To visualize the local contributions to the growth rate and frequency in a computational domain, the density quantities σ k /V k and ω k /V k are considered, where V k is the volume of the k th cell (area in a two dimension field; V k are the diagonal terms of the matrix Q):

σ = N cells k=1 σ k V k V k and ω = N cells k=1 ω k V k V k . (2.30)
The scaling is simply computed by considering the adjoint eigenvector q † in equation (2.29):

λ k V k = σ k V k + i ω k V k = q † * C k q.
(2.31)

Link with wavemaker function

To establish a connection between the local contribution to the eigenvalue λ k (2.29) and the wakemaker component w k (2.22), the column-decomposition of the Jacobian matrix (2.23) is introduced into the Q-weighted adjoint eigenvalue problem (2.16).

Thus,

λ * q =   N cells k=1 C k   * q = N cells k=1 (C * k q) (2.32)
where the matrix-vector product C * k q is equal to

C * k q =   0, • • • , N cells l=1 ∂R l ∂q k * ql , 0 • • •   T , (2.33)
after taking the transconjugate of C k defined in (2.24). This is a vector of zeros except for the k th component. Now, by identification of the left and right-hand sides in the equality (2.32), it results that

C * k q = λ * [0, • • • , qk , 0 • • • ] (2.34)
Reformulating (2.29) as λ k = (C * k q) * q and inserting the above expression yields The column-matrix decomposition (2.23) of the Jacobian matrix is not unique and therefore the eigenvalue decomposition (2.28) is also not unique. However, when considering a line-decomposition of the Jacobian matrix, i.e.

λ k = λ (q * k qk ) (2.
J =   N cells k=1 L k   (2.36)
where L k denotes the line-matrix at the k th cell, the same eigenvalue decomposition is obtained:

λ k = q * L k q = λ (q * k qk ) . (2.37)
This is shown in Appendix A.

Furthermore, Appendix B shows a comparison of the wavemaker with the presented technique on a laminar circular cylinder configuration. The results totally agree with each other.

Variable-decomposition of the Jacobian matrix

Another possible decomposition of the Jacobian matrix which implies (differently from the line one) also a different eigenvalue decomposition is the variable-decomposition.

Column and line decomposition result in the definition of the local contribution to the globals eigenvalue. Following the same procedure and idea, the variables decomposition result in the definition of the contribution of each variable to the global eigenvalue. In the following the decomposition is presented for laminar/turbulent variables. The Jacobian matrix is decomposed in two contributions:

Numerical approach

J = J l + J t (2.38)
where J l and J t are the column-matrices for the laminar q l = (ρ, ρU, ρV, ρW, ρE)

and turbulent q t (one for the SA model ρν or two for the k-kl model ρk, ρkl) variables, respectively. As for the local contribution, consider the lines of the variables in the matrices results in the same conclusion (Appendix A). The Jacobian matrix is now presented in terms of the contributions of q l and q t to the laminar and turbulent dynamics. Equation (2.10) is rewritten as:

d dt    q l q t    =    J ll J lt J tl J tt       q l q t    .
The matrices J lt and J tl correspond to the effect of turbulent and laminar variables to the laminar and turbulent dynamics, respectively, while J ll and J tt correspond to the effect of the laminar ad turbulent variables on its own dynamics. Specifically, J ll corresponds to the Jacobian matrix computed with the frozen µ t approach (Reynolds & Hussain [START_REF] Reynolds | The mechanics of an organized wave in turbulent shear flow. part 3. theoretical models and comparisons with experiments[END_REF]). It is now easier to show the relation between the Jacobian matrix and the two contributions J l and J t :

J l =    J ll J lt 0 0    and J t =    0 0 J tl J tt    .
The eigenvalue is also decomposed into two unknown components:

λ = λ l + λ t (2.39)
Following the same procedure presented in sections 2.3.2 and 2.3.3, the two contributions to the global eigenvalue can be computed:

λ l = q * J l q = λ (q * l • ql ) (2.40) λ t = q * J t q = λ (q * t • qt ) (2.41)
where ql , ql qt and qt are the laminar and turbulent variables of the adjoint and direct eigenvector, respectively.

λ l and λ t are the contributions of the laminar and turbulent dynamics to the overall unstable eigenvalue. They are composed of a real and an imaginary part which in turn are the contributions to the growth rate and the angular frequency of the laminar and turbulent dynamics:

λ l = σ l + ω l (2.42)
λ t = σ t + ω t . ( 2

Numerical simulations and comparison with experiment

The 2D reference frame (x, y) is relative to the aerofoil. The origin of the reference frame is at the aerofoil leading edge, x is parallel and y perpendicular to the aerofoil chord axis.

Results from RANS and unsteady RANS (URANS) computations are presented in this section. Local time-stepping method is used to obtain the base flows. In our computations, it was possible to obtain residuals close to zero machine precision even when the base flow was unstable. At the same time the unstable unsteady solutions are simulated by URANS computations. The buffet phenomenon appears inside the range of α between 3 • and 6.5 RANS unstable steady-state solution for α = 5 • , for which:

R(q 0 ) = 0 (3.1)
As already mentioned, the numerical results from RANS and URANS simulations are validated by comparison with an experimental database. Figure 3.2 shows the comparison of the numerical results with the experimental investigation of Jacquin et al. [START_REF] Jacquin | Experimental study of shock oscillation over a transonic supercritical profile[END_REF] for two cases: before buffet onset at α = 3 • and in a buffet case at α = 3.5 • .

Figure 3.2a shows a steady flow, while figure 3.2b shows the time-averaged pressure coefficient over several cycles of established buffet. Both cases are in good agreement on the pressure side of the aerofoil, the supersonic zone and close to the TE while a difference is found for the shock position. The numerical simulations, both RANS and URANS, predict a shock position about 5% chord downstream of the experimental one.

The results are satisfying and an improvement in the simulations in comparison with Sartor et al. [START_REF] Sartor | Stability, receptivity, and sensitivity analyses of buffeting transonic flow over a profile[END_REF] is found thanks to the Edwards-Chandra correction in the SA model.

Buffet onset and insight on buffet mechanism

In this section the evolution of buffet close to the onset and a physical analysis of the URANS simulation with a comparison with experiments from Stanewsky & Basler [START_REF] Stanewsky | Experimental investigation of buffet onset and penetration on a supercritical airfoil at transonic speeds[END_REF] are presented.

Buffet onset is found by small variation of α and it is identified at about 3.3 • for the SA+EC turbulence model for which the frequency is 73 Hz. Specifically, the vertical size is synchronized with µ tmax while the position of the shock wave along the x-axis is delayed by T/10, where T is the period of one buffet cycle.

This result from URANS simulations has been already identified in experiments by

Stanewsky & Basler [START_REF] Stanewsky | Experimental investigation of buffet onset and penetration on a supercritical airfoil at transonic speeds[END_REF]. They showed that the shock has a certain delay in a cycle of buffet in comparison with the thickness at the TE of the detached boundary layer.

Global stability analysis

Results of the global stability analysis are presented below with an emphasis on the evolution of the most unstable mode as function of the angle of attack α for a fixed value of the Mach number M = 0.73. Iorio [START_REF] Iorio | Global Stability Analysis of Turbulent Transonic Flows on Airfoil geometries[END_REF], Guiho [START_REF] Guiho | Analyse de stabilité linéaire globale d'écoulements compressibles: application aux interactions onde de choc/couche limite[END_REF], Sartor [START_REF] Sartor | Unsteadiness in transonic shock-wave/boundary-layer interactions: experimental investigation and global stability analysis[END_REF] and Sartor et al. [START_REF] Sartor | Stability, receptivity, and sensitivity analyses of buffeting transonic flow over a profile[END_REF].

Mesh convergence

This section briefly presents the mesh convergence of the buffet mode from stability analysis. Table 3.1 shows the values of the unstable global buffet eigenvalue for three different cells refinement of the same computational grid: the reference with 72000 cells, a coarser with 36000 cells and a finer with 100000 cells. The eigenvalue does not appear to be converged in mesh on the coarser grid while it is well converged on both reference and finer grids. Furthermore, it is possible to note that the frequency of the unstable mode converges faster than the growth rate increasing the grid refinement. the amplitude of the solutions in the transient). On the contrary, an higher ε reduces the linear transient which can even not appear if ε is to high.

Comparison stability analysis-URANS simulations

Local contribution of the flow

This section shows the results for the buffet mode of the local contribution technique presented in section 2.3. results suggest the existence of several zones, even close to the aerofoil (for example the pressure side), that do not impact the physical mechanism at the origin of the transonic buffet. Further investigations are presented here for the density maps of the growth rate. Indeed positive values contribute to the unstable behaviour of the mode while negative values are stabilising. The shock foot always appears with a strongly unstable behaviour while the shock exhibits always a stable behaviour. The detached boundary layer may have either a stable or an unstable behaviour depending on the location in space and the values of the angle of attack. The scenario of the physical mechanism behind transonic buffet resulting from the analysis of the density maps is the following: the shock foot is the core of the instability and the zone where the unsteadiness arises, the shock has a stabilising behaviour during the unstable phenomenon which can be interpreted as a stiffness (a section of the field that is sustained by and tends to damp the shock foot motion) and the detached boundary layer is the key of the buffet onset and exit. In order to confirm the influence of the different zones in the buffet phenomenon resulting from stability density maps, a local selective filtering method is used in the following.

Variable contributions to the eigenvalue

The decomposition of the jacobian matrix has been presented both by line (appendix A) or column-decomposition (section 2.3.2) but in each case in a structure that follows the geometrical topology of the computational domain. Specifically, because each group of 5/6 (the number of state variables depends on SA+EC or k-kl turbulence model) lines and columns corresponds to the cells of the domain. In the following the results of another kind of decomposition is presented: the variable decomposition (section 2.3.4).

The reason behind this kind of analysis comes from Crouch et al. [START_REF] Crouch | Predicting the onset of flow unsteadiness based on global instability[END_REF]. They showed that, differently from global unstable bluff bodies modes (Meliga et al. [START_REF] Meliga | Sensitivity of 2-D turbulent flow past a D-shaped cylinder using global stability[END_REF], Carini et al. [START_REF] Carini | Global stability and control of the confined turbulent flow past a thick flat plate[END_REF]), the transonic buffet instability does not appear in the spectrum if the stability analysis is performed without considering the perturbations of the turbulent variables (the frozen µ t approach by Reynolds & Hussain [START_REF] Reynolds | The mechanics of an organized wave in turbulent shear flow. part 3. theoretical models and comparisons with experiments[END_REF]). The disappearance of the global unstable mode of transonic buffet with the frozen µ t approach suggests the primary role of the turbulent dynamics in the mechanism of the self-sustained loop of transonic buffet. The variable decomposition has been computed for several values of angle of attack and the results are shown in figure 3.11 for the SA+EC turbulence model. The angle of attack analyses cover the entire range of buffet: the stable flow, the onset, well-established buffet and offset. The results found are coherent for both turbulence models and show a strong coupling between the laminar and the turbulence dynamics.

In the stable flow case, before the buffet onset, the turbulent dynamics is unstable while the laminar is stable resulting in a overall stable dynamics. The same values are found in the stable flow after the buffet offset. While in the range of angle of attacks where the flow is unstable and buffet exhibits unsteadiness, the dynamics of laminar and turbulence switch. Laminar dynamic becomes strongly unstable while the turbulence one becomes stable; the results is an unstable mode.

The conclusion of this brief analysis confirms the importance of the turbulence dynamics in the appearance of buffet instability. But at the same time the mechanism behind the coupling is still unknown and furthermore giving a physical meaning of the laminar and turbulence dynamics is not an easy task. This is the reason why this analysis has still several future perspective and exploitation.

Identification of the buffet mechanisms with a Selective Frequency Damping approach

The Selective Frequency Damping (SFD) formulated by Akervik et al. [START_REF] Åkervik | Steady solutions of the Navier-Stokes equations by selective frequency damping[END_REF] for a laminar incompressible case is a useful technique to damp fluctuations whose frequencies are below a certain cut-off frequency. It has been originally formulated with the aim to get steady solutions of the Navier-Stokes equations when the lack of convergence is due to oscillations of the residual. This technique is here used to analyse the critical regions of the buffet instability by localising the damping region.

3.3 Identification of the buffet mechanisms with a Selective Frequency Damping approach 53

SFD method with localised damping

SFD is based on the idea of filtering unstable temporal frequencies by coupling the system equations with a first-order low-pass filter, as follow:

         dq dt = R(q) -χ (q -q f ) dq f dt = q -q f ∆ (3.2)
where the second equation is the low-pass filter with a cut-off angular frequency 1/∆.

In the first equation, a damping term, proportional to the difference between the solution q and the filtered solution q f , is used by Akervik et al. [START_REF] Åkervik | Steady solutions of the Navier-Stokes equations by selective frequency damping[END_REF] to stabilise the coupled system and force the solution to converge towards a steady state. The amplitude χ of the damping term is localised in space in order to determine in which flow regions it effectively modifies the global flow fluctuation (it is switched to zero in the regions not damped). Furthermore, in order to solve the coupled system (3.2) with an industrial CFD solver as elsA, this would imply to modify the numerical scheme used to discretise the original equations. This is the reason why an "encapsulated" formulation, based on the splitting of the system (3.2), has been used (see [START_REF] Jordi | Encapsulated formulation of the selective frequency damping method[END_REF] for numerical details).

Based on the results obtained in the previous section, eight flow regions are investigated to apply the damping term. They are depicted in figure 3.12: the suction side of the aerofoil (zone 1), the shock foot excursion and beginning of the separated boundary layer (zone 2), the suction side TE area and wake (zone 3), from the superior half of the supersonic zone to the end of the domain (zone 4) above the supersonic zone (zone 4'), the aerofoil wake (zone 5), the path between the TE and the shock above the boundary layer (zone 6) and finally the pressure side of the aerofoil (zone 7).

Lift fluctuation amplitudes are used as global criteria for the persistence of the buffet instability. Steady state is defined by zero machine levels of residuals while probes are used locally to verify that unsteady signals are well damped. When lift continues to oscillate and the standard deviations of the signals in the filtered zone tend to zero it is possible to state that the related zone is not necessary for the buffet instability to develop.

To choose the SFD parameters, Akervik et al. [START_REF] Åkervik | Steady solutions of the Navier-Stokes equations by selective frequency damping[END_REF] state that large χ and ∆ would make the evolution of the system very slow but the SFD would in every case converge to 3.3 Identification of the buffet mechanisms with a Selective Frequency Damping approach 55 a steady-state. Furthermore, the SFD parameter values are linked to the flow dynamics.

The temporal cut-off frequency 1/∆ should be lower than the frequency of the unstable dynamics by at least a factor two. The control parameter χ has usually a value close or higher than the growth rate of the unstable flow dynamics but can take higher values in order to increase the convergence rate of the simulation towards the steady-state.

In most cases the unstable dynamics is unknown a priori and several studies proposed different techniques to choose suitable parameters [START_REF] Cunha | Optimization of the selective frequency damping parameters using model reduction[END_REF][START_REF] Jordi | An adaptive selective frequency damping method[END_REF]. In the present work the unstable dynamics is well-known and the analysis procedure is the following: eight URANS simulations are performed with SFD for each filtered zone in figure 3.12, the value of the cut-off frequency is fixed at 13 Hz and the control parameter is increased linearly as function of time. The smallest value of the control parameter for which the simulation converges to a steady-state is called χ min . Results at different values of 1/∆ have also been investigated but they are not presented here because small effect on the convergence rate has been observed.

Results

The results from the application of SFD in the different zones are presented in this section. The numerical parameters of the URANS simulations have been described in section 2. The configuration analysed is the same as in section 3. amplitude decreases towards zero, with a slope depending on the zone where SFD is activated. When ∆C l = 0 (corresponding to χ min ) the residuals values tend to zero machine levels while intermediate points, in the range 0 < χ < χ min , correspond to non-physical solutions of the URANS dynamical system coupled with SFD that did not reach convergence. The application of the SFD on the full domain results in a value of χ min = 55 while χ min is always higher when SFD is applied on a limited zone of the domain. Figure 3.13a shows the regions in which the application of a local SFD allows to reach a steady-state, for a certain value of χ. Table 3.2 shows the χ min values for the configurations in figure 3.12. N/A is used when a value of χ min is not found, i.e. when SFD damps the entire unsteady signals of the zone where it is activated but the lift still oscillates. A steady-state is reached by application of local SFD only for three zones: shock (zone 4), suction side TE area and wake (zone 3) and shock foot with the beginning of the boundary layer (zone 2) which is the most efficient area to damp. These zones correspond exactly to the larger values of the σ and ω density maps presented in figure 3.9. It is interesting to note the low value of χ min for zone 2 even though the application area of the SFD is very small and that χ min for zone 1 is very close to the value found when SFD is applied on the entire computational domain.

The local criterion based on the standard deviation (sd) for the streamwise momentum (ρu) is presented for zone 4 in figure 3.13b. The same slope is found for all other state variables. For zones 1, 2 and 3, the standard deviation of the state variables decreases constantly while increasing χ, until χ min (figure omitted). Four different probes are used in zone 4, two above the supersonic zone and two inside the area swept by the shock. Figure 3.12d shows the location of all the probes. Figure 3.13b shows that the standard deviation of ρu drastically decreases when SFD is activated for the probes outside the supersonic zone (probes C and D) while remaining on a plateau and then strongly decreasing to zero for the probes inside the supersonic zone (probes A andB). These results suggest that in the zone 4, the field converges towards a steadystate as soon as SFD is activated, except for the shock which continues to oscillate.

Consequently the perturbations around the supersonic zone are more a consequence than a cause of the buffet. The shock is in a certain way entrained by the shock foot.

It is considered as a slave zone with a certain role in the buffet scenario because the instability can be suppressed by filtering exclusively the shock. To conclude, the shock is a slave zone but behaves as a stiffness on the instability phenomenon. If some part of the shock is prevented from moving, the rest of the shock, even if it is free to move, has a harder time doing it.

The analysis continues with zones where it is not possible to suppress buffet instability by local SFD. This is the case for zones 4 , 5, 6 and 7. For example zone 6 at χ = 4000 shows a reduction of 22% in lift coefficient amplitude with a reduction of 98.5% in the standard deviation of ρu.

Discussion on the physical mechanism behind transonic buffet

The physical mechanisms presented in the introduction are now discussed in the light of the results from the present work. The topology of the active and passive zones is discussed looking at the several models of transonic buffet.

The numerical results have shown, close to the aerofoil, that some zones are not strictly necessary for the buffet instability: the pressure side of the aerofoil (figure 3.12h), region above the supersonic zone (figure 3.12e), the aerofoil wake (figure 3.12f) and the path between the TE and the shock above the boundary layer (figure 3.12g). The unsteady signals found by Jacquin et al. [START_REF] Jacquin | Experimental study of shock oscillation over a transonic supercritical profile[END_REF] on the lower surface of the aerofoil do not appear to contribute to the transonic buffet mechanism. Some self-sustained mechanisms of transonic buffet are based on perturbations circumventing the supersonic zone, such as Crouch's interpretation of transonic buffet (figure 1.4) and the mechanism highly localised around the shock [START_REF] Memmolo | Scrutinity of buffet mechanism in transonic flow[END_REF] suggesting acoustic rays (Lee et al. [START_REF] Lee | Role of Kutta waves on oscillatory shock motion on an airfoil[END_REF] and Spee [START_REF] Spee | Wave propagation in transonic flow past two-dimensional aerofoils[END_REF]). Figure 3.13b suggests that the perturbations around the supersonic zone are more a consequence than a cause of the buffet phenomenon. Conclusions suggest that these models do not focus on the origin of the buffet. The last zone to look at is probably the most important one: the path between the TE and the shock above the boundary layer (figure 3.12g). It is indeed the key-zone involved in several models to close the self-sustained loop with backward acoustic waves impacting the shock. The

Summary and conclusion

59

present results show that the unsteady signals in this zone are not necessary for buffet instability. The unsteady signals in the Lee's model [START_REF] Lee | Oscillatory shock motion caused by transonic shock boundary-layer interaction[END_REF], in the Hartmann's model [START_REF] Hartmann | On the interaction of shock waves and sound waves in transonic buffet flow[END_REF],

and in the model based on acoustic rays originating from TE and passing through zone 6 (Lee et al. [START_REF] Lee | Role of Kutta waves on oscillatory shock motion on an airfoil[END_REF] and Spee [START_REF] Spee | Wave propagation in transonic flow past two-dimensional aerofoils[END_REF]) contribute weakly to the buffet mechanism. However, the emission of acoustic waves resulting from the diffraction at the TE can be superimposed on the buffet phenomenon. This is the reason why unsteady simulations [START_REF] Deck | Numerical simulation of transonic buffet over a supercritical airfoil[END_REF] observe such acoustic emissions.

The only self-sustained closed-loop still possible is a model similar to Lee's one but with a feedback passing through the boundary layer. Indeed it is shown that by filtering the separated boundary layer, the buffet instability may be completely damped. It is worth mentioning that in the article of Deck [START_REF] Deck | Numerical simulation of transonic buffet over a supercritical airfoil[END_REF], both numerical and experimental results showed presence of downstream and upstream traveling waves in the separated boundary layer (se figure 14 of Deck [START_REF] Deck | Numerical simulation of transonic buffet over a supercritical airfoil[END_REF]). A second possible mechanism, still open to discussion, is a more localised kind of instability. The mechanism is supposed to be strongly localised at the shock foot with the shock and the boundary layer driven by the shock foot oscillation and acting only as a stiffness on the overall mechanism.

Summary and conclusion

Transonic buffet is a phenomenon largely studied during the last 70 years but which is

still not yet fully understood. Several studies have contributed to the understanding of -zones which are not strictly necessary for the instability (not at the origin, at best a Numerical investigation on two-dimensional transonic buffet consequence): the zone on the lower side of the aerofoil, on the upper side above the shock, downstream of the boundary layer and the path between the TE and the shock outside the boundary layer.

-zones which are absolutely necessary for the instability: the shock and the separated boundary-layer. More precisely, the shock wave appears as a slave zone with a stiffness effect while the separated boundary-layer has a more active role in the buffet mechanism scenario.

-core of the instability: the shock foot. In this chapter the 3D transonic buffet is presented through the analysis and the comparison of four different experimental databases. The review of Giannelis et al. [START_REF] Giannelis | A review of recent developments in the understanding of transonic shock buffet[END_REF] pointed out one of the main objective for future research on 3D buffet: "gain a comprehensive understanding of geometric effects". Chapter 3 highlighted, among others, the physical characteristics of the 2D transonic buffet: the value of the Strouhal number, the peaked-shape of the power spectral density, the shock amplitude oscillation etc.

62

Experimental investigation on three-dimensional transonic buffet

The objective of the following chapter is to define the same characteristic parameters for the 3D buffet and underline the differences with the 2D one. Furthermore, the possibility to compare different wing geometry allows to study the sensitivity of the phenomenon with the shape of the wing. It must be pointed out that the present chapter is presented in an "aeronautical" reference frame, where the z-axis is the lift-axis.

Consequently, the z-axis and the y-axis have been switched compared with chapters 3 and 5.

The chapter is organized as follows. Section 4.1 defines the experimental set-up used in each database. Buffet onset for the four wind tunnel tests is described in section 5.2.3. Section 4.3 gives information on the evolution of the separated zone with the flow conditions. In section 4.4, a spectral analysis of the Kulite data is performed. In section 4.5, signal processing tools like cross-spectra and frequency-wavenumber spectra of the Kulite data are used to compute convection velocities of the buffet phenomenon.

Finally, the conclusion presents a synthesis of the results with physical discussions.

Experimental setup

Four different campaigns are analysed and compared. These campaigns were performed in three wind-tunnel tests with four different half wing/fuselage-body models. They correspond to the following projects:

Separated flow evolution

Oil flow visualizations are available for the BUFET'N Co and AVERT campaigns.

Therefore it is possible to describe the evolution of the separated flow on the suction side of the wing.

Power Spectral Densities

The power spectral densities (PSDs) of Kulites data are presented in this section. PSDs are computed with the same procedure as in Dandois [START_REF] Dandois | Experimental study of transonic buffet phenomenon on a 3D swept wing[END_REF]: Welch's method with Hamming window and 50% overlapping blocks. The number of overlapping blocks and the frequency resolution are given in table 4.1 for each campaign. As already said, the main difference between 2D and 3D buffet is the increase of shock frequency oscillation and the broadening of the buffet frequency range. Consequently, the identification of a precise value of buffet frequency is more complicated in 3D than in 2D, where PSD exhibits clear peaks. Nevertheless, it has been chosen to define the buffet frequency as the center of gravity of these bumps. The PSDs are analysed on the whole wing, i.e. in the chordwise (iso-x/c) and spanwise (iso-y/b) directions, in order to get the spatial variations of the buffet frequency. The PSDs, depending on the frequency sampling, can also show other physical phenomena. The Kelvin-Helmholtz (K-H) instability appears in the frequency range [1000 -4000 Hz]. The theoretical frequency is around f KH = 0.135 Ū /δ ω , where Ū is the average velocity above and below the shear layer and δ ω the vorticity thickness (see Huerre & Rossi [START_REF] Huerre | Hydrodynamic instabilities in open flows[END_REF] for more details). The K-H phenomenon exists in all the tests but spectra of AVERT, FLIRET and DTP Tremblement overlook the phenomenon due to a low-pass filter applied to each signal. Nevertheless it is still possible to observe the K-H instability in the frequency-wavenumber spectra for AVERT and FLIRET tests. This is because the frequency-wavenumber (f -k)

PSDs for the AVERT model

During the AVERT campaign, several values of α were tested at Mach numbers: 0.78, 0.8, 0.82, 0.84 and 0.86. This database has already been analysed by Dandois [START_REF] Dandois | Experimental study of transonic buffet phenomenon on a 3D swept wing[END_REF], so The figures show that the buffet Strouhal number decreases and bumps tend to become thinner when increasing α at a fixed Mach number. x/c=50% x/c=60% x/c=70% x/c=76% x/c=80% x/c=85% x/c=90%

PSDs for the DTP Tremblement model

PSDs for the FLIRET model

Cross-Spectral analysis

In this section, signal processing tools are presented, like cross-spectra and frequencywavenumber spectra. They are an efficient way to determine the convection velocities of the 3D buffet phenomenon (and of the K-H instability). U Cc and U Cs are the convection velocity components in the chordwise direction towards TE and in spanwise direction towards wing tip, respectively. ϕ is the angle between the wave propagation and the chordwise direction. U Cc and U Cs can be identified by two different analyses: cross-spectra and frequency-wavenumber spectra. Then the resulting convection velocities U C and the directions ϕ are computed These velocities are defined for a fixed value of frequency, or at least a range, in order to link the velocity to the physical phenomenon appearing at the considered frequencies.

Cross-Spectrum

Cross-spectrum is the Fourier transform of the cross-correlation of two stochastic processes: in the present case, the measured signals from the Kulites. If x 1 (t) and x 2 (t) are two continuous signals, the cross-correlation R x 1 x 2 (τ ) is the convolution of the signals.

R x 1 x 2 (τ ) = ∞ -∞ x 1 (t)x 2 (t + τ )dt (4.1)
The Fourier transform F converts the cross-correlation R x 1 x 2 (τ ) from the time to the frequency domain defining in this way the cross-spectrum Rx

1 x 2 (f ). Rx 1 x 2 (f ) = F{R x 1 x 2 (τ )} = ∞ -∞ R x 1 x 2 (τ )e -2iπf τ dτ (4.2)
The cross-spectrum Rx 1 x 2 (f ) in polar coordinates can be decomposed into an amplitude Âx 1 x 2 (f ) and a phase Φ x 1 x 2 (f ). The latter is used to compute the convection velocity while the square of the amplitude divided by the spectra of the two signals

gives the coherence γ 2 . Rx 1 x 2 (f ) = Âx 1 x 2 (f )e iΦx 1 x 2 (f ) (4.3) γ 2 x 1 x 2 (f ) = Â2 x 1 x 2 (f ) Rx 1 x 1 (f ) * Rx 2 x 2 (f ) (4.4)
The coherence allows identifying the range in the frequency domain where there are convective phenomena, and from the phase difference, it is possible to compute their velocities. Normally, there are high values of the coherence at the buffet Strouhal number. Then, there are two ways to estimate the convection velocities. The two methods are based on the same idea and give almost the same results. The first one consists in selecting the frequency for which the coherence is the highest and then look at linear variations of the phase at this frequency in space. It is very precise in terms of frequency while it is averaged in space. Convection velocity is obtained using the relation U C = 2πf∆x/∆Φ where f is the selected frequency in Hertz, ∆x the length of the line of sensors used in meters and ∆Φ the phase difference in radians. The second Experimental investigation on three-dimensional transonic buffet method is computed specifically between two sensors. The range of frequencies with high coherence shows linear variation of phase in the frequency domain. From this slope, it is possible to obtain the convection velocity: U C = 2π∆x∆f/∆Φ where the variables are the same as above except for ∆f which is here a range and not a single value. This case is more precise in space because just two sensors are analysed but averaged in the range of frequencies where the coherence is maximal. This is the reason why it is important to consider only the frequency range of the interesting phenomenon. it is possible to compute the full velocity vector, and hence its norm and direction.

The chordwise and spanwise velocities are not combined in the classical vectorial way but following Larchevêque [START_REF] Larchevêque | Simulation des grandes échelles de l'écoulement au-dessus d'une cavité[END_REF]:

       U C = U Cc cos ϕ U C = U Cs sin(ϕ+Λ x/c ) (4.5)

Frequency-wavenumber spectra

The analysis of the frequency-wavenumber spectra is another way to compute the convection velocities. Theoretically, it is based on the two dimensional Fourier transform of the spatio-temporal cross-correlation R x 1 x 2 (∆, τ ).

R x 1 x 2 (∆, τ ) = ∞ -∞ ∞ -∞ x 1 (x, t)x 2 (x + ∆, t + τ )dτ d∆ (4.6) Rx 1 x 2 (k, f ) = F{R x 1 x 2 (∆, τ )} (4.7)
Since the definition of a transform in space is not possible, because the sensors are not equidistant, an estimator ψ(f, k) is defined. It is based on the cross-spectral matrix Ψ(f ) of the sensors (see Larchevêque [START_REF] Larchevêque | Simulation des grandes échelles de l'écoulement au-dessus d'une cavité[END_REF] for more details):

ψ(k, f ) = η H (k)Ψ(f )η(k) (4.8)
where H is the Hermitian transpose, k is the wavenumber and η(k) and Ψ(f ) are defined by:

       (Ψ(f )) ij = Rx i x j (f ) (η(k)) i = e -ikx i (4.9)
Experimental investigation on three-dimensional transonic buffet

The obtained estimator is a function of frequency and wavenumber. Convective phenomena are identified by regions of constant ratio ω/k, where ω is the pulsation frequency and k the wavenumber. The slope of these lines in the f -k plane corresponds to the convection velocity. Phase velocities (U p = 2πf /k) are found with the crossspectra, while in the f -k spectra it is possible to find both phase and group velocities.

The group velocity is the propagation of the real information of the waves, the envelope of a signal. It is defined as the variation of the angular frequency δf with the wavenumber, U g = 2πδf /δk. The values of phase and group velocities found for 3D buffet are similar. A phenomenon with similar values of phase and group velocities is considered having low dispersion.

The results of f -k spectra are presented in the following paragraph. In all figures, the Strouhal number based on M AC is on the horizontal axis and the non-dimensional wavenumber is on the vertical axis (it has been divided by the chord or the span depending of the direction of the line of sensors analysed). shown and the resulting velocity is obtained thanks to equation (4.5). In this case, the Kelvin-Helmholtz wave propagates mostly in the chordwise direction (indeed it does not appear in the spanwise direction). The resulting velocity is typical of this instability 180 m.s -1 or 0.65U ∞ (see Dandois [START_REF] Dandois | Experimental study of transonic buffet phenomenon on a 3D swept wing[END_REF], Dandois et al. [START_REF] Dandois | Numerical simulation of active separation control by a synthetic jet[END_REF] and Larchevêque [START_REF] Larchevêque | Large-eddy simulation of a compressible flow in a three-dimensional open cavity at high reynolds number[END_REF]).

Concerning the buffet phenomenon, • . The range of M -α tested is smaller than for the other campaigns and it is not possible to look at the evolution of these velocities with M -α. This is not the case for FLIRET and AVERT tests where the M -α range is larger.

The f -k spectra of FLIRET differ a bit from BUFET'N Co since the buffet convection velocity is more oriented towards the chordwise direction and K-H instability is not only convected in chordwise but also in spanwise direction. In the reference test at M = 0.85 and α = 3.37 • (figures 4.17(a) and 4.17 and for K-H instability of 0.61U ∞ with ϕ = 4 • . The effects of higher M -α are a K-H velocity more oriented towards the chordwise direction, while buffet convection velocity is more oriented in spanwise direction towards the wing tip and its norm is also slightly smaller than the typical value of 0.24U ∞ .

The f -k spectra analysis for the AVERT test adds some interesting phenomena in comparison with the other databases and with the previous work of Dandois [START_REF] Dandois | Experimental study of transonic buffet phenomenon on a 3D swept wing[END_REF].

Above all, it is interesting to look at the precise evolution of the convection velocity for different angles of attack. The analysis is performed at a line position in chord and span with the highest density sensors of Kulites and at a Mach number for which data are clearer and uniformly distributed in α. Finally at α = 5 • (not shown) buffet velocity is 47 m.s -1 (0.17U ∞ ) with ϕ = 30 • and K-H instability velocity is 170 m.s -1 (0.6U ∞ ) with a ϕ = 15 • . In summary, it seems that the buffet convection velocity decreases when the angle of attack increases.

St MAC K 0 0.5 1 1.5 2 2.5 3 3.5 -1.5 -1

Convection velocities

The convection velocities are now presented at some characteristics points over the wings. The results of cross-spectra and f -k spectra are consistent with each other. All models exhibit buffet convection velocities both in the chordwise and spanwise directions. The convection in span is characteristic of the 3D transonic buffet and it is probably the main cause of the "buffet cells" convection towards the wing tip as

Results synthesis

Power spectral densities, cross-spectra and frequency-wavenumber spectra of the unsteady pressure transducers have been analysed as well as static pressure and accelerometers. The main results of the analysis of the four databases are summarised in Table 4.2

Experimental investigation on three-dimensional transonic buffet and they are essentially consistent with each other. [START_REF] Dandois | Experimental study of transonic buffet phenomenon on a 3D swept wing[END_REF]). (b) Phase of cross-spectra at x/c=85% for five selected frequencies (reference sensor: y/b=80%).

Strouhal numbers and convection velocities summary

Buffet cell wavelengths summary

The discovery of a convective phenomenon of so-called "buffet cells" in spanwise direction on a wing during buffet is very recent. As already said in the introduction, Iovnovich & Raveh [START_REF] Iovnovich | Numerical study of shock buffet on threedimensional wings[END_REF] were the first to observe this convection numerically in 2015 and introduce the name of "buffet cells". Dandois [START_REF] Dandois | Experimental study of transonic buffet phenomenon on a 3D swept wing[END_REF] computed the convection velocity on the AVERT model by using a cross-spectrum analysis. From the values of the phase difference it is possible to define the wavelength of the cells λ. lengths characterizing the buffet cells λ = U C /f = 2π∆x/∆Φ. Table 4.3 shows all the values of wavelengths computed and a comparison with Iovnovich & Raveh [START_REF] Iovnovich | Numerical study of shock buffet on threedimensional wings[END_REF].

Wavelengths are presented here as non-dimensional numbers. It is possible to state that the wavelength decreases with the angle of attack but its value depends on the wing geometry. Finally, even if the geometrical dependency is very high, it is still possible to define a range of λ/M AC between 0.6 and 1.3 for these supercritical wings.

Table 4.3: Non-dimensional wavelength for the convective buffets cells. Database:

BUFET'N Co AVERT FLIRET Ref. [START_REF] Iovnovich | Numerical study of shock buffet on threedimensional wings[END_REF] (Λ = 30 • ) Flow condition: M = 0.82

M = 0.82 M = 0.85 M cos(Λ) = 0.73 λ/M AC 0.56 (α=3.5 • ) 1.3 (α=4.25 • ) 0.62 (α=3.6 • ) 1.3 (α=4 • ) 1.6 (α=3.5 • ) 0.55 (α=3.8 • ) λ/M AC 0.56 (α=3.5 • ) 1.6 (α=3.5 • ) 0.9 (α=3.6 • ) 1.3 (α=4 • ) λ/c (y/b = 78%) 0.7 (α=3.5 • ) 1.9 (α=3.5 • ) 1.2 (α=3.5 • ) 1.3 (α=4 • )
To conclude this overview, figure 4.22 shows a sketch of the buffet cells convection for the FLIRET model. The value of wavelength and Strouhal are presented with MAC' as reference length together with the buffet convection velocity.

Conclusion

This chapter compares the buffet phenomenon characteristics (Strouhal number, convection velocity) over four different databases. The models are different in terms of Finally, a comparison between 2D and 3D transonic buffet is assessed. The increase of the frequency (Iovnovich & Raveh [64]), the change of shock amplitude oscillation and the creation of buffet cells are not still completely understood but the convection of buffet cells in the spanwise direction of the wing is the main difference between the two types of buffet. It is crucial in 3D to look for the existence or not of an unstable global mode as found by Crouch et al. [START_REF] Crouch | Predicting the onset of flow unsteadiness based on global instability[END_REF] in 2D. Indeed the spanwise convective nature of the 3D transonic buffet does not rule out the possibility of explaining it with a global mode as in 2D; the broadband nature of the spectra stems from unmodeled noise (Bagheri [3]).

The second hypothesis is a convective instability where the convection velocities wipe out the presence of an unstable global mode and establish a noise-amplifier behaviour.

A challenging 3D global stability analysis could state which hypothesis explained the nature of the flow for the 3D transonic buffet.

Chapter 5

Three-dimensional global stability analysis It is compulsory at the beginning of this chapter to explain the similarities and differences with the 2D stability analysis of chapter 3 which is a preamble of the 3D stability analysis. In order to remain consistent with the non dimensional parameters, the normal component on the wing of the free-stream velocity U n∞ is used as the reference velocity in the definition of the Strouhal number. It corresponds to the free-stream velocity U ∞ in chapter 3. Both growth rate and frequency have been nondimensionalised with the same reference length, the aerofoil chord, and velocity. The differences between the two chapters are in the CFD solver used: the simulations in chapter 3 have been conducted with elsA while here with FastS. The solver has been changed because the implementation and the computation of the 3D Jacobian has been proved to be easier in FastS solver than in elsA. The two solvers have been presented in section 2.1. The main difference between the simulations of the base flow with elsA and FastS is in the corrections of the SA turbulence model [START_REF] Spalart | A one equation turbulence model for aerodynamic flows[END_REF] used. The Edwards-Chandra (EC) [START_REF] Edwards | Comparison of eddy viscosity-transport turbulence models for three-dimensional, shock-separated flowfields[END_REF] and compressibility correction (CC) [START_REF] Spalart | Trends in turbulence treatments[END_REF] are used in elsA and FastS, respectively (because they are not both available in the solvers). SA+CC turbulence model gives lower values of µ t in the simulations resulting in a buffet onset at α = 3 • (section 5.2.3) instead of 3.3 • (section 3.1.1) which is in better agreement with the experimental data (Jacquin et al. [START_REF] Jacquin | Experimental study of shock oscillation over a transonic supercritical profile[END_REF]). The results of global stability analysis are in very good agreement between both solvers in terms of modes shape and frequencies while differences are found in the values of the growth rate which depend of course on the values of the angle of attack. In any case the differences are justified by the kind of study conducted: this is a "phenomenological" insight on transonic buffet and not a quantitative reproduction of the phenomenon.

The chapter is organized as follows. Section 5.1 shows the flow configuration of the base flow. The results of the chapter are presented in section 5.2: mesh convergence, wavenumber, sweep angle effect and onset are detailed as well as the adjoint modes, structural sensitivity and the local contribution of the instabilities. Finally, section 5.3

summarises the main conclusions.

Flow configuration and base flow computation

The aerofoil geometry is ONERA's OAT15A transonic aerofoil with a chord length equal to 0.23 m. The 2D mesh contains about 5 × 10 4 cells with a refinement around the time-averaged shock location. After extrusion, the total number of cells in the 3D mesh is obtained by multiplying the 2D mesh size by the number of planes in the spanwise direction N z . For the eigenvalue problem, the Jacobian matrix is extracted by considering Lperiodic boundary conditions in the spanwise z direction. Table 5.1 provides an overview of the computational details of the eigenvalue problems. The effect of the extruded length L, sweep angle Λ and number of planes N z on the Jacobian matrix size, number of non-zero elements and memory consumption of the eigenvalue problem is shown. The angle of attack does not have a significant effect on the computational cost. The reduction of the spanwise length L increases the memory consumption be-

Results

The global stability analysis of an infinite unswept wing reveals two types of unstable modes: an unsteady 2D mode (f = 0, βc = 0) and several non-oscillating 3D modes (f = 0, βc = 0, see figure 5.3a). Here β designates the spanwise wavenumber that can be obtained for each computed 3D-eigenvector by analyzing its 3D z-structure.

The 2D unstable mode corresponds to the buffet mode found by Crouch et al. [START_REF] Crouch | Predicting the onset of flow unsteadiness based on global instability[END_REF][START_REF] Crouch | Origin of transonic buffet on aerofoils[END_REF], Sartor et al. [START_REF] Sartor | Stability, receptivity, and sensitivity analyses of buffeting transonic flow over a profile[END_REF] and Iorio et al. [START_REF] Iorio | Direct and adjoint global stability analysis of turbulent transonic flows over a NACA0012 profile[END_REF] for a 2D aerofoil (figure 3.7). The mode depicts the flow structure of a shock oscillation synchronized with the thickening/thinning of the boundary layer. This mode does not exhibit a spanwise component ( Ŵ (x, y, z) = 0), nor a z-dependence in the other components. The frequency value f c/U n∞ = 0.07 (ωc/U n∞ = 0.46) is in agreement with numerical simulations (Deck [START_REF] Deck | Numerical simulation of transonic buffet over a supercritical airfoil[END_REF]) and experimental tests (Jacquin et al. [START_REF] Jacquin | Experimental study of shock oscillation over a transonic supercritical profile[END_REF]) of transonic buffet over 2D aerofoils.

The analysis also reveals that all the 3D modes (βc = 0) are non-oscillating (f = 0).

Consequently, no spanwise propagating phenomenon appears on an unswept wing. Due to the imposed periodic conditions on the spanwise boundaries, the wavelength of these modes λ n depends on the extruded length of the base flow, such that the wavenumber is with n max < N z /3 from the Nyquist-Shannon sampling theorem. These 3D modes exhibit coherent structures in all the directions of the computational domain. Their structure in the x -y plane is similar to the 2D unstable mode but with a lower amplitude. High values of the mode amplitude are also found on the suction side of the wing in the detached boundary layer. The modes oscillate in the spanwise direction as the buffet cells observed in URANS and DES simulations. The 3D modes exhibit a flow structure similar to the stall cell phenomenon in the detached boundary layer (Manni et al. [START_REF] Manni | Numerical study of airfoil stall cells using a very wide computational domain[END_REF]). The structure of the flow is similar but the reasons behind the detachment of the boundary layer are strongly different: at low speed the boundary layer detached because of the stall phenomenon, while in transonic regime the detachment results from the interaction of the boundary layer with the shock wave in the suction side of the wing. The stall cell phenomenon has been also identified as a global unstable mode by Rodriguez & Theofilis [START_REF] Rodríguez | On the birth of stall cells on airfoils[END_REF] for an incompressible flow.

β n = n2π/L.
In the following, if not specified, only the fundamental 3D mode n = 1 is considered, which exhibits a wavelength λ 1 = L. For the harmonics n ≥ 2, better-converged results may be obtained by considering the fundamental mode on a smaller domain (a fraction of the extruded length L/2, L/3, etc, but with the same number of N z planes). Therefore, to explore the spectrum in the β direction, the extrusion length L has been varied to obtain the most converged results (see section 5.2.4).

Mesh convergence

First, it is important to check the convergence of the results with the number of cells in the spanwise direction. The effect of the number of cells in the x-y plane has also been investigated and the results in terms of growth rate and frequency are converged on the present mesh (figure not shown). The effect of the mesh refinement in the spanwise direction may have a different impact on the unstable modes. This refinement does not impact the 2D unstable mode which remains constant while it has an important effect on the three dimensional spatial modes. Figure 5.4 shows the growth rate and frequency variations of the fundamental buffet cells mode with respect to the number of spanwise planes N z . Only the fundamental mode λ 1 is considered here because it is the best discretised in span. With 16 planes in span, the variation of the growth rate for the unswept wing is smaller than 2% and the mode is considered as converged. Figure 5.4 shows also the 3D mode for a swept wing at Λ = 30 • which is considered converged for N z = 12 (3D modes at Λ = 0 will be presented in section 5.2.2). The second effect of the mesh refinement in span is the possibility to capture more harmonics λ n of the 3D mode, as explained in the previous section. But the n-th harmonic λ n will not be as well discretised in span, up to the n max -th harmonic which has only 3 points per wavelength. In the following, if not specified, the chosen mesh refinement in span is fixed at N z = 12.

Three-dimensional global stability analysis

Sweep effect

In this section the evolution of the two unstable modes (2D and 3D) for the fixed extruded length L = c, when the sweep angle Λ is increased from 0 • to 30 • , is presented. , the convection velocity is equal to 0.43U n∞ which is exactly the same value found experimentally by Dandois [START_REF] Dandois | Experimental study of transonic buffet phenomenon on a 3D swept wing[END_REF] and Koike et al. [START_REF] Koike | Unsteady pressure measurement of transonic buffet on NASA common research model[END_REF]. The flow structure of the mode is shown in figure 5.5b and corresponds to the unsteady [109] and experimentally by Dandois [START_REF] Dandois | Experimental study of transonic buffet phenomenon on a 3D swept wing[END_REF] and Sugioka et al. [START_REF] Sugioka | Experimental analysis of transonic buffet on a 3D swept wing using fast-response pressure-sensitive paint[END_REF] for the 3D buffet. The evolution of the spectrum with the sweep angle suggests that the phenomenon known as 3D transonic buffet is not linked with the 2D one: there are two distinct unstable modes in the spectrum. The 3D buffet phenomenon corresponds to the unstable and steady global mode of stall cells or buffet cells for an unswept wing which becomes unsteady with the addition of sweep. It will be justified in section 5.2.4 why βc = 2π is the right wavenumber to pick for this comparison.

Onset

The onset of the two unstable global modes is studied by the analysis of several values of the angle of attack. Figure 5.6 shows the spectra for an unswept and a swept wing at Λ = 30 • for three regimes of the flow: stable flow (α = 2.5 • ), around the onset (2.9 • < α < 3.2 • ) and unstable flow (α = 3.5 • ). Figure 5.6a shows that the 2D buffet mode becomes unstable when increasing α from 2.9 • to 3.0 • through a supercritical Hopf bifurcation [START_REF] Crouch | Predicting the onset of flow unsteadiness based on global instability[END_REF] while the non-oscillating stall cells mode becomes unstable increasing α from 3.0 • to α = 3.1 • . For the swept wing at Λ = 30 • (figure 5.6b), the scenario is slightly different: the 3D mode now becomes also unstable through a Hopf bifurcation because it is a non-zero frequency mode which becomes unstable. Furthermore there is a shift of -0.1 • in the values of onset for both modes because the effect of sweep tends to stabilize the flow. To conclude, the onset of the two unstable modes is very close to each other and even if the two dimensional mode appears first, there is not a separated onset dynamics. Another interesting point found for both unswept and swept wings is that the 2D buffet mode exists and it is clearly visible in the spectrum even in a stable case such as α = 2.5 • . On the contrary, the three dimensional mode appears only when the boundary layer is separated and in a small range of α its growth rate value quickly increases.

Wavenumber effect

In the present section, the effect of the non-dimensional spanwise wavenumber βc on the different unstable modes is analysed. 2D mode with the wavenumber, respectively. The strongest growth rate is found at βc = 0 which confirms the 2D nature of the mode. The persistence of the unstable behaviour of the mode is found only in a small range of wavenumbers: βc < 0.75 for the unswept wing and βc < 0.6 for the swept wing at Λ = 30 • . The frequency of the mode is only weakly affected by the wavenumber. Figure 5.8a shows the evolution of the 2D buffet eigenvalue for an unswept Λ = 0 • aerofoil with extruded lengths L = 8c, 9c and 12c. Note that the 3D mode is strongly stable and not visible in this plot. The mode for βc = 0 is no more purely 2D and exhibits a slight variation in span (see figure 5.8b). Figure 5.9 shows the variation of the growth rate and frequency of the 3D mode with the wavenumber. The 3D nature of the mode is confirmed because the maximum value of the growth rate is found at βc = 0. The range of wavenumber analysed has been set around the experimental and numerical values of the wavenumber for the 3D buffet mode found by Dandois [START_REF] Dandois | Experimental study of transonic buffet phenomenon on a 3D swept wing[END_REF], Sugioka et al. [START_REF] Sugioka | Experimental analysis of transonic buffet on a 3D swept wing using fast-response pressure-sensitive paint[END_REF], Iovnovich & Raveh [START_REF] Iovnovich | Numerical study of shock buffet on threedimensional wings[END_REF] Plante et al. [START_REF] Plante | Study of three-dimensional transonic buffet on swept wings[END_REF]. The most unstable 3D buffet cell mode of an unswept wing has a wavenumber of βc ≈ 2π which corresponds to a wavelength equal to one chord length.

The same wavenumber value is found for the swept wing at Λ = 30 • . This is the reason why all the spectra in sections 5.2.1, 5.2.2 and 5.2.3 have been presented for L = c. The variation of the frequency in the swept wing case is shown to be linear with the wavenumber. As mentioned in section 5.2.2, the wavenumber value of the most unstable three dimensional mode is consistent with the values found by Dandois [START_REF] Dandois | Experimental study of transonic buffet phenomenon on a 3D swept wing[END_REF],

Iovnovich & Raveh [START_REF] Iovnovich | Numerical study of shock buffet on threedimensional wings[END_REF] and Plante et al. [START_REF] Plante | Study of three-dimensional transonic buffet on swept wings[END_REF].

Adjoint problem and structural sensitivity

The results of the adjoint problem (2.16) are presented in this section together with the structural sensitivity. Figure 5.10 shows the 2D adjoint buffet mode for the unswept wing at M = 0.73 and α = 3.5 • . As already explained, the 2D mode is presented for βc = 0. The results are consistent with the ones from Sartor et al. [START_REF] Sartor | Stability, receptivity, and sensitivity analyses of buffeting transonic flow over a profile[END_REF]. The region where the values of the mode are the highest are localized on the suction side of the wing, in the attached boundary layer, the recirculation bubble, and along the characteristic line inside the supersonic zone impacting the shock foot. The 2D adjoint mode of the swept wing is not presented here because, as already mentioned for the direct mode, no relevant differences exist with respect to the adjoint mode at zero sweep (only the appearance of non-zero and constant values of ρW † in span). Figure 5.11 shows the adjoint modes for the newly found unstable global modes: the 3D mode at Λ = 0 • and Λ = 30 • . Both adjoint modes show in the (x -y) plane a shape similar to the 2D mode (the characteristic line inside the supersonic zone is even more visible) and a 3D pattern in span. The highest values are localized at the shock foot and they spread more upstream than downstream, in accordance with the adjoint nature of this mode. Looking at the y = 0 plane upstream of the aerofoil, it is noticed that the iso-lines of the adjoint mode are aligned with the direction of the freestream velocity.

To conclude, the control strategy for 2D and 3D transonic buffet modes are not so different since the highest sensitivity region of the adjoint modes are in both cases the shock foot, the attached boundary layer and the characteristic line in the supersonic region.

The computation of both direct and adjoint modes allows to perform a further analysis of the instability. The direct mode underlines the coherent oscillation of the instability while the adjoint mode highlights its most sensitive region. The region of the flow where the two modes overlap define the core of the instability, called the wavemaker (Giannetti & Luchini [46]). In this region, a localized feedback in the equations governing a global mode leads to the largest shift of its eigenvalue. The wavemaker has been presented in section 2.3.1 for a discrete setting. , respectively. The real and imaginary parts of both the direct and the adjoint 3D modes exhibit a harmonic shape in z and are shifted by 90 • . This is the reason why the norm of the modes is constant in the spanwise direction, and therefore also the wavemaker which written in a continuos setting results

Three-dimensional global stability analysis

w(x, y, z) = w(x, y).

The separation line, which also corresponds to the shock foot location, is the region with the highest values of structural sensitivity but the separated region also exhibits high values. The core of this new instability is in fact localized closer to the wing by comparison with the 2D instability: the wavemaker extends further in the separated region (see wing-body contours) and is limited to the very near-wall wall region of the shock (see z plane values). The analysis of the wavemaker therefore indicates that the instability is really linked to the separation region and not so much to the shock. This is also in agreement with the fact that the buffet cell mode is linked to the stall cell mode that appear on separated aerofoils in subsonic flows, in which case no shock is present. The presence of a non-zero sweep angle does not remarkably influence the shape of the structural sensitivity.

In order to establish a link with section 3.2.3, the local contribution technique, presented in section 2.3, is here applied. The application is unfortunately only an overview due to time constraints. A deeply analysis, as for the 2D case, is a perspective. highlights further information. The shock foot that is an important region for this instability appears to contribute to the unstable behaviour in the global mechanism of the mode, as for the 2D buffet mode. While the detached boundary layer exhibits an unstable behaviour that is a strong difference with the 2D buffet mode where the detached boundary layer has mainly a stable behaviour. This confirms the localisation in the shock foot and the detached boundary layer of the unstable core of the 3D buffet mode and it is in strong contrast with the 2D mode where the unstable core is exclusively the shock foot.

Chapter 6

Conclusions and perspectives

Conclusions

The transonic buffet is a low-frequency unsteadiness of the flow which appears on aerofoils and wings in the transonic regime for certain values of Mach number and angle of attack. It consists of a shock oscillation that implies pressure and notably lift fluctuations, thus limiting the flight envelope of civil aircraft. The phenomenon has been largely studied until nowadays but the physical mechanism behind the main unsteadiness of the flow was not fully understood. It has been shown by previous studies ( [START_REF] Roos | The buffeting pressure field of a high-aspect-ratio swept wing[END_REF], [START_REF] Benoit | Buffeting prediction for transport aircraft applications based on unsteady pressure measurements[END_REF]) that transonic buffet strongly changes, in terms of physical parameters like shock frequency and amplitude, when passing from a 2D aerofoils to a 3D swept wing.

The 2D transonic buffet over aerofoil is characterized by a peak in the spectrum at a Strouhal number of about 0.06-0.08 and an amplitude oscillation of the shock of about 10% -20% x/c. Furthermore, it has been shown by Crouch et al. ([23], [START_REF] Crouch | Origin of transonic buffet on aerofoils[END_REF]) that 2D buffet corresponds to an unstable global mode of the flow. 3D buffet exhibits different characteristic parameters in comparison with the 2D one: higher Strouhal number, a bump instead of a peak in the power spectral densities, much lower shock amplitude oscillation and 3D patterns in the detached boundary layer which are convected outboard. These differences suggest a strong evolution of the same phenomenon when passing from aerofoils to swept wings or two different physical mechanisms resulting in two different phenomena. The scope of this work was to enhance the present understanding of the physical mechanism behind transonic buffet and, above all, to explain the differences between the appearance of buffet over 2D aerofoils and 3D swept wing.

This study started with the analysis of the 2D transonic buffet. A second technique has been used to confirm the results of the spatial localisation of the instability; a selective frequency damping method has been locally applied in some regions of the flow field. This method consists in coupling an unsteady simulation with a low-pass filter on selected zones of the computational domain in order to damp the fluctuations. When the fluctuations of the filtered zone are totally damped, the flow field is analysed: if the transonic buffet unsteadiness of the flow still appears; it means that the zone is not necessary for the instability to go living on while it is necessary if the flow tends to a steady state. This technique, similarly to the previous one, allows identifying which zones are necessary for the persistence of the instability. The two different approaches give the same results: the shock foot is identified as the core of the instability, the shock and the detached boundary layer are also necessary zones while damping the fluctuations on the pressure side of the aerofoil, outside the boundary layer between the shock and the trailing edge or above the supersonic zone does not suppress the shock oscillation. A discussion on the several physical models, proposed until now for the 2D transonic buffet phenomenon, is also offered in the light of these results. Some models of the explication of the self-sustained closed-loop behind 2D transonic buffet are here shown not to be focused on the origin of buffet and there are two possible mechanisms: a self-sustained closed-loop with a feedback passing through After the in-depth analysis on 2D transonic buffet, the focus of the work moves to the 3D transonic buffet in order to first understand the influence of the angle of attack, the Mach number and the wing's geometry on the phenomenon and then establish a link with the 2D buffet. The 3D transonic buffet is studied firstly through the analysis and comparison of four different experimental databases and then numerically by global stability analysis. The experimental analysis has identified characteristic values of the buffet phenomenon such as Strouhal numbers, convection velocities, buffet onset etc. The models were different in terms of chord length, taper ratio, twist and geometry but they are all based on supercritical aerofoils. It has been shown that some non-dimensional numbers are kept constant between the different databases and consequently can be considered as characteristics of the 3D buffet, whereas others change.

The key for the understanding the transonic buffet phenomenon lies in explaining the common features but also the variability of transonic buffet parameters in different configurations. The convection velocities of the phenomena propagating on the suction side of four different wings have been computed. This study has shown that buffet is a strongly convective phenomenon in 3D and the values found are consistent among the different models. A characteristic convection velocity range of (0.245±0.015)U ∞ has been found for a sweep angle of 30 • . It has been shown that these velocities are the main cause of the buffet cell convecting in the spanwise direction. Another main result is the definition of a frequency range of the buffet phenomenon: a range of Strouhal number based on the local chord of 0.2 -0.3 has been found. The differences between 2D and 3D buffet in the Strouhal number, shock amplitude oscillation and convective nature of the flow is again highlighted. Another strong difference in terms of nature of the phenomenon between 2D and 3D buffet is in the shape of the power spectral density from Kulites. 3D buffet exhibits a broadband bump in the spectrum while 2D buffet exhibits a peak. Oscillatory type instability normally exhibits a peaked frequency spectrum which is linked with an absolute unstable flow while a noise amplifier normally exhibits more broadband bump spectra which are linked with a convectively unstable flow. The last part of the work has been conducted in order to establish the link between 2D and 3D buffet and to define the unstable nature of the flow for the latter.

Conclusions and perspectives

In the last chapter the 3D transonic buffet is analysed numerically by a 3D global stability analysis. In the case of an infinite unswept wing, the present study shows that two unstable modes actually exist: the 2D transonic buffet mode already identified by [START_REF] Crouch | Predicting the onset of flow unsteadiness based on global instability[END_REF] and a strongly amplified 3D zero-frequency mode. The latter exhibits regular patterns in the separated boundary layer, which relates to the so-called buffet cells also known as stall cells in subsonic regime. The non-zero sweep angle generates a spanwise velocity component on the wing which convects the cells outboard. This impacts both modes identified in the unswept case: the 2D mode is weakly damped by the sweep while the 3D buffet cells mode, even if weakly damped, remains strongly unstable and now exhibits a non-zero frequency which increases with the sweep angle. The Strouhal number (f c/U n∞ = 0.43 or f c/U ∞ = 0.37 ), the wavenumber (βc ≈ 2π or a wavelength equal to one chord length) and the convection velocity (0.43 U n∞ or 0.37 U ∞ ) of the most unstable 3D mode for a sweep angle of 30 • agree well with numerical and experimental values of the 3D transonic buffet on 30 • swept wings. The onset of the instabilities has also been studied for both unswept and swept wing. Two different angles of attack have been found for the onset of the 2D and 3D modes but the values are very close and a coupled onset dynamics is expected when considering a more complex study. Furthermore, the analysis of the local contribution of the 3D modes has also been computed. It indicates that the core of the instability is nearly solely located in the separated region, with a maximum along the separation line. This is in contrast with the 2D buffet mode, for which the local contribution also exhibits strong values all along the shock-wave.

In summary, the conclusion of the present study has been first to enhance the understanding of the physical mechanism behind the 2D transonic buffet by identifying the region of the flow necessary for the instability to go living on. Then, to identify the characteristic values of the 3D buffet phenomenon and define the common features but also the variability of the parameters in different wing geometry configurations.

Finally, it has been shown that the phenomenon called in the literature 3D transonic buffet corresponds to a "detached boundary" cells convection phenomenon with a zerofrequency unstable global mode which becomes unsteady with sweep. The mode is distinct from the dynamics of the 2D transonic buffet mode.

Perspectives

Appendix A

Line-decomposition of the Jacobian matrix

The eigenvalue problem is presented in a formulation where the Jacobian matrix is decomposed as function of line matrices as done in paragraph 2.3.2 for column matrices.

First the adjoint problem is analysed:

λ * q =   N cells k=1 L k   * q = N cells k=1 (L * k q) (A.1)
where L k denotes the line matrix of the Jacobian at the k th cell:

L k =                            0 • • • • • • 0 • • • • • • 0 . . . • • • • • • . . . • • • • • • . . . 0 • • • • • • 0 • • • • • • 0 ∂R 1 ∂q k • • • • • • ∂R l ∂q k • • • • • • ∂R N cells ∂q k 0 • • • • • • 0 • • • • • • 0 . . . • • • • • • . . . • • • • • • . . . 0 • • • • • • 0 • • • • • • 0                            (A.
2)

The matrix-vector product of the line-matrix l k with the adjoint eigenmode q is now expanded onto the basis of the adjoint eigenmodes as

L * k q = ϕ * k q + rk (A.3) 120
Linearised compressible Navier-Stokes equations q(x, y, z, t) = [ρ(x, y, z, t), ρu(x, y, z, t), ρv(x, y, z, t), ρw(x, y, z, t), ρE(x, y, z, t), ρν(x, y, z, t)] T .

(C.5)

The extruded base flow used in chapter 5 is considered: q 0 (x, y) = [ρ 0 (x, y), ρ 0 u 0 (x, y), ρ 0 v 0 (x, y), ρ 0 w 0 (x, y), ρ 0 E 0 (x, y), ρ 0 ν0 (x, y)] T . (C.6) Furthermore, the shape of the 3D modes found in chapter 5 is used for the linearisation in order to underline the effect of each term of the equations on the appearance of the unstable modes.

q(x, y, z, t) = q 0 (x, y) + q (x, y, z, t) For the compressibility correction there is an additional term:

||q
-C 5 ρν 2 γRT U i,j U i,j (C. [START_REF] Crouch | Origin of transonic buffet on aerofoils[END_REF] where: The constant terms are: R specific gas constant, γ = C p /C v specific heat ratio, S is the magnitude of the strain tensor, Ω is the vorticity magnitude, η distance to the wall.

U i,
f ν2 = - χ 1 + χ 0 f ν10 + χ 0 (χ 0 f ν1 + χ f ν10 ) (1 + χ 0 f ν10 ) 2 (C.32
)

f w = 1 + C 6
f ν1 - χ χ 2 0 + S 2S 0 1 χ 0 + f ν10 (C.38) r = 1 cosh( ν0 K 2 η 2 S0
) 2 tanh(1.0) Mots clés : tremblement transsonique, analyse de stabilité globale, écoulements compressibles, interaction onde de choc couche limite turbulent.

1 ρ 0 K 2 η 2 S0 ρ ν - ρ 0 ν0 ρ ρ 0 - ρ 0 ν0

INSIGHT ON TRANSONIC BUFFET INSTABILITY. EVOLUTION FROM TWO-DIMENSIONAL AEROFOILS TO THREE-DIMENSIONAL SWEPT WINGS ABSTRACT:

The transonic buffet is a complex aerodynamic instability which appears on wings and aerofoils in high subsonic regime for certain values of Mach number and angle of attack. The aim of the present study is to improve the understanding of the flow physics behind the instability and, in particular, to explain the evolution of the transonic buffet phenomenon from two-dimensional aerofoils to threedimensional swept wings. First the work by [START_REF] Crouch | Predicting the onset of flow unsteadiness based on global instability[END_REF]Crouch et al. ( , 2009) ) has been reproduced. Then, the two-dimensional transonic buffet is analysed to highlight the zones of the flow necessary for the persistence of the instability. The study has been conducted numerically through the coupling between two techniques based on different approaches: linear stability analysis and numerical simulations. The two different approaches give the same results: the shock foot is identified as the core of the instability, the shock and the detached boundary layer are also necessary zones. In order to outline the differences between two-dimensional and three-dimensional buffet, the analysis and comparison of four different experimental databases of three-dimensional wings is performed. The experimental analysis identifies the characteristic values of the buffet phenomenon such as Strouhal numbers, convection velocities, buffet onset etc. Three-dimensional buffet exhibits different characteristics in comparison with the twodimensional one: higher Strouhal number, bump instead of peak in the power spectral density, lower shock amplitude oscillation and three-dimensional patterns in the detached boundary layer which are convected outboard. These differences suggest different physical mechanisms. Finally a three-dimensional global stability analysis of a wing is performed. The two and three-dimensional buffet phenomena appear as two different unstable modes in the spectrum. The phenomenon called in the literature three-dimensional transonic buffet corresponds to a "detached boundary" cells convection phenomenon with a zerofrequency unstable global mode, which becomes unsteady with the addition of sweep.

Keywords: transonic buffet, global stability analysis, compressible flow, shock wave turbulent boundary layer interaction.

  sound, m.s -1 b = Model span, m c = Local chord length, m C p = Pressure coefficient E = Total energy per unit mass, m 2 .s -2 f = Frequency, s -1 J = Jacobian matrix k = Wavenumber unsteady signals, m -1 L = Characteristic length, m M = Mach number N z = Number spanwise planes 3D grid N cells = Number cells grid M AC = Mean aerodynamic chord, m p = Pressure, kg.m -1 .s -2 Pr = Prandtl number Pr t = Turbulent Prandtl number q = State variables q = Weighted adjoint eigenvector R = Residulas Re c = Reynolds number based on local chord Re M AC = Reynolds number based on mean aerodynamic chord sd = Standard deviation signal St = Strouhal number T = Temperature, K T = Period buffet cycle, s t = Time, s U = Velocity x-axis, m.s -1 v U c = Convection velocity, m.s -1 U g = Group velocity, m.s -1 U p = Phase velocity, m.s -1 U ref = Reference velocity, m.s -1 V = Velocity y-axis, m.s -1 W = Velocity z-axis, m.s -1 w = Wavemaker vector x, y, z = Spatial coordinates, m y+ = Non-dimensional wall distance α = Angle of attack, deg β = Spanwise wavenumber 3D buffet mode, m -1 β = Non-dimensional spanwise wavenumber 3D buffet mode γ 2 = Experimental signals coherence ∆ -1 = Cut-off frequency in selective frequency damping, s -1 ε = Small perturbation Λ = Wing sweep angle, deg Λ x/c = Angle sensor line with respect to y-axis, deg λ = Eigenvalue, s -1 λ = Wavelength in chapter 4, m λ k = Local contribution of the flow to global eigenvalue µ = Dynamic viscosity, kg.m -1 .s -1 µ t = Eddy dynamic viscosity, kg.m -1 .s -1 ν = Kinematic viscosity, m 2 .s -1 ν t = Eddy kinematic viscosity, m 2 .s -1 νt = Transformed eddy kinematic viscosity, m 2 .s -1 ρ = Density, kg.m -3 σ = Growth rate, s -1 Φ = Experimental signals phase ϕ = Convection velocity angle, deg χ = Control parameter in selective frequency damping, s -1 ω = Angular frequency, s -1 subscript n : Relative to aerofoil section frame in 3D grid subscript 0 : Relative to base flow subscript ∞ : Relative to free-stream superscript : Relative to perturbations superscript † : Relative to adjoint mode overscript ˆ: Relative to direct mode

1. 1

 1 Context 7 research topic and it is the main question of the present work.

Figure 1 . 2 :

 12 Figure 1.2: Time histories of the periodical shock-wave motions [152].

  experimentally found the presence of disturbances travelling upstream on the lower side of the aerofoil and linked these signals with a possible alternative path of a self-sustained closed-loop. More recently, in a numerical study, Garnier & Deck[START_REF] Garnier | Large-eddy simulation of transonic buffet over a supercritical airfoil[END_REF] pointed out that the buffet period computed using Lee's model was not in very good agreement with their simulation. More recently, Hartmann et al.(Hartmann et al. [53]; Feldhusen et al.[START_REF] Feldhusen | High-speed tomographic piv measurements of buffet flow over a supercritical airfoil[END_REF]) have proposed another model which can be considered as an extension of Lee's one. It suggests that the shock movement is totally driven by the change of the sound pressure level of the waves at the TE, which are in turn linked to the strength of the vortical structures convected from the shock foot to the TE. Memmolo et al.[START_REF] Memmolo | Scrutinity of buffet mechanism in transonic flow[END_REF] studied the link between the propagation of acoustic waves (both on the pressure and the suction sides) and the low-frequency dynamics. They concluded that the buffet mechanism is strongly localized around the shock or linked to the separation bubble dynamics.

Figure 1 . 3 :

 13 Figure 1.3: (a) Lee's model of self-sustained shock oscillation [78]. (b) Schematic of wavefronts and rays emanating from source disturbances at TE of aerofoil [80].

  Stability analysis was first used in fluid dynamics under the local approximation (Batchelor & Gill[START_REF] Batchelor | Analysis of the stability of axisymmetric jets[END_REF] and Crighton & Gaster[START_REF] Crighton | Stability of slowly diverging jet flow[END_REF]). It concerns open shear flows, whose properties only depend on the cross-stream direction, such as mixing layers, jets, wakes, boundary layers, plane Poiseuille flow, but even flows in boxes such as Rayleigh-Bernard convection in finite-size cells, Taylor-Couette flow between concentric rotating cylinders, etc. Under this approximation a given crosswise profile of the flow is considered to model the entire configuration of the flow. Studying the evolution of localised disturbances Huerre & Monkewitz[START_REF] Huerre | Local and global instabilities in spatially developing flows[END_REF] outlined two kinds of instabilities: a locally absolute unstable flow when the disturbances spread upstream and downstream, contaminating the entire flow, and a locally convective unstable flow when the disturbance is swept away from the source. Huerre & Monkewitz[START_REF] Huerre | Local and global instabilities in spatially developing flows[END_REF] also studied the way to overcome the local approximation and link local stability analysis with a global stability analysis, in which no particular assumption are made on the flow. They state that local absolute instability is a necessary condition for global instability. At the same time the increase of computational capabilities allow to perform global stability analysis over more and more complex geometries. In the context of linear stability analysis, Huerre & Rossi[START_REF] Huerre | Hydrodynamic instabilities in open flows[END_REF] distinguished two different behaviours of a flow unsteadiness: an instability that does not need an external force to oscillate and impose its own dynamics on the flow field, and another acting as a filter or amplifier of every kind of external disturbance acting on the flow. The first one is called an oscillator while the second one is called a noise-amplifier. The main difference between the two behaviours is the existence of an unstable global mode. Drazin[START_REF] Drazin | On a model of instability of a slowly-varying flow[END_REF] defines a global mode as a state variable in which the system executes coherent oscillations. In chapter 2 the mathematical meaning of global mode is explained. Oscillators are driven by an unstable global mode and more precisely by the self-sustained mechanism behind the mode. Noise-amplifiers are the result of stable global modes and the Navier-Stokes operator has to be analysed Introduction through other technique showing optimal growth phenomena (Cossu et al.[START_REF] Cossu | Optimal transient growth and very largescale structures in turbulent boundary layers[END_REF]). In this way it is possible to highlight the main characteristic of flow dynamics.

1. 1 13 Figure 1 . 4 :

 11314 Figure 1.4: Contours of the pressure fluctuation at eight steps during the oscillation cycle for the conditions M = 0.76, α = 3.2 • , Re = 10 7 [25].

Figure 1 .

 1 5 shows the ZDES simulation of Deck et al. [34]. Lutz et al. [83] found also a good agreement between a new kind of Zonal DES and experiments in the European Transonic Windtunnel (ETW) over the NASA CRM. Results showed a precise description of the unsteady development of the massively separated wing flow. More recently Iovnovich & Raveh [64] and Sartor & Timme [128] [127] performed URANS simulations of the transonic buffet. Iovnovich &

Figure 1 . 5 :

 15 Figure 1.5: ZDES of transonic buffet over a civil aircraft configuration (Re M AC = 2.8 10 6 , M = 0.82, 1.9 10 8 points). Isosurface of Q-criterion featuring the coherent structures and vorticity and Mach contours by Deck et al. [34].

71 10 - 5 2 . 1 . 1 . 1

 52111 Pa s, C s = 110.4 K and T s = 273 K. The variables U, E are Favre averages whereas the other ones correspond to the classical Reynolds average. Turbulence models Two different kinds of turbulence models have been used to provide closure for the averaged Reynolds stresses with elsA in chapter 3: Spalart-Allmaras [139] (SA) and k-kl (Daris & Bézard [30]) turbulence model. Only the SA turbulence model has been used

  total temperature of 300 K and a Mach number of M ∞ = 0.73. The boundary conditions are imposed directly in the fluxe evaluation at the boundary. An adiabatic walls is considered for the aerofoil and a non-reflective boundary condition for the far-field 50 chord length away from the aerofoil. No transition term is imposed on 2.2 Global stability analysis 29 the profile where the boundary layer is fully turbulent. The Reynolds number based on the chord length is equal to Re c = 3.2 × 10 6 . Several 2D simulations have been performed with increasing values of the angle of attack from 3 • to 7.5 • . α is increased on the same mesh configuration by changing the velocity vector components in the far-field conditions: tan(α) = V ∞ /U ∞ where U ∞ and V ∞ are the velocity components in the aerofoil reference frame (x, y) of the free-stream velocity vector U ∞ . The 3D simulations have been computed with the same flow conditions on the aerofoil section frame (x, y): (M ∞ ) 2D = (M n∞ ) 3D , (α) 2D = (α n ) 3D . The angle of attack α n = 3.5 • (except when studying onset) and (M n∞ , Re c = U n∞ c/ν) = (0.73, 3.2 × 10 6 ). The sweep angle Λ is simulated in the base flow by the addition of a constant spanwise

13 )

 13 In appendix C it is possible to find the complete set of the linearized three-dimensional compressible Navier-Stokes equations closed with the classical SA turbulence model.It is so possible to highlight the effect of each terms of the Navier-Stokes equations in the linearized formulation.The adjoint eigenvalue problem associated with equation (2.13) is introduced in the following. Let us consider an inner product based on a real symmetric positive definite matrix Q such that a, b Q = a * Qb, where a and b are arbitrary vectors and the superscript * refers to the transconjugate. In the following the matrix Q is chosen so that a, a Q represents the square of the function L 2 norm. With a finite volume approach,

  does not provide further information to understand the feedback mechanisms at the origin of self-sustained (global) instability. Two feedback mechanisms are often invoked to explain global instabilities arising in hydrodynamic flows: a local hydrodynamicfeedback and non-local pressure-feedback[START_REF] Chomaz | Bifurcations to local and global modes in spatially developing flows[END_REF]. The local hydrodynamic feedback is responsible for the global instability in wake flows behind bluff bodies, for instance the circular cylinder flow, while the non-local pressure feedback is responsible for the global instability in cavity flows. The existence of local hydrodynamics feedback in cylinder flow has first been identified with local stability analysis and was connected to the appearance of an absolute instability in the wake flow. Giannetti & Luchini[START_REF] Giannetti | Structural sensitivity of the first instability of the cylinder wake[END_REF] first proposed to identify local feedback mechanisms in global eigenmode using a structural sensitivity analysis of the eigenvalue problem. After recalling the structural sensitivity in section 2.3.1, a different approach is exposed in section 2.3.2. The new method also aims to determine the flow regions where the instability mechanisms acts, 2.3 Identification of local hydrodynamic feedback in the unstable eigenmode 33 but it is based on a decomposition of the Jacobian matrix.

2. 3

 3 Identification of local hydrodynamic feedback in the unstable eigenmode 35

  • with a frequency which increases with α from 75 to 80 Hz for the SA+EC turbulence model. This compared with α between 3.5 • and 7.5 • with a frequency which increases with α from 73 to 85 Hz for the k-kl turbulence model. The largest lift amplitude is found in the middle of the unstable range at α = 5 • for the SA+EC model, for which the buffet frequency is equal to 79 Hz and α = 6 • for the k-kl model, for which the buffet frequency is equal to 80 Hz.

Figure 3 .

 3 1b and c show two Mach number fields of a URANS simulation, more precisely the fields with the most downstream (figure 3.1b) and the most upstream position (figure 3.1c) of the shock at α = 5 • for the SA+EC model. For this angle of attack, buffet is well established and the shock oscillation amplitude is about 35% of the chord. Figure 3.1a shows the 3.1 Numerical simulations and comparison with experiment

Figure 3 . 1 :

 31 Figure 3.1: Mach number field at α = 5 • . (a) RANS steady-state solution. Most downstream (b) and upstream (c) shock position during one buffet period of the URANS solution.

Figure 3 . 2 :

 32 Figure 3.2: Comparison of CFD results (lines) with experimental (points) investigation of Jacquin et al. [65]. (a) Pressure coefficient for the steady state at α = 3 • . (b) Timeaveraged pressure coefficient for the unsteady state at α = 3.5 • .

Figure 3 . 3 :

 33 Figure 3.3: Evolution of the lift coefficient at buffet onset α = 3.3 • for the SA+EC turbulence model.

Figure 3 .

 3 3 shows the evolution of the lift coefficient at buffet onset for an URANS simulation which starts from a RANS converged base flow. The evolution of the instability is very long because the simulation is very close to the onset and because the initial perturbation is small. Indeed, in theory the base flow solution should not evolve, if not perturbed, during an URANS simulation because it is solution of the system. It evolves because there are perturbations coming from the numerics (DTS, grid, etc.) which are outside the basin of attraction of the base flow solution. It will be shown in section 3.2.2 that the size of the transient towards the unsteady buffet cycle depends on the amplitude of the initial perturbation. An interesting result in the analysis of the URANS simulations quantifying how certain physical and geometrical variables change during a cycle of buffet has been obtained. The variables considered are the position of the shock wave along the x-axis, the size of the detached boundary layer along the y-axis and the values of the maximal dynamic viscosity µ t . The evolution of these variables over a cycle of transonic buffet is presented in figure 3.4. It is possible to identify a certain delay between the variables.

Figure 3 . 4 :

 34 Figure 3.4: Relation between shock movement and vertical size boundary layer, maximal value of dynamic viscosity µ t (a), trailing edge boundary layer thickness (Stanesky & Basler [141]) (b).

Figure 3 .

 3 Figure 3.4(b) shows the position of the shock along the x-axis in comparison with δ T E , defined in the same figure.The delay of the shock movement in comparison with the boundary layer suggests that the mechanisms of the transonic buffet unsteadiness are driven by the viscosity component of the field. The viscosity mechanisms that sustain transonic buffet appears to be generated in the detached boundary layer. The main conclusion of this paragraph is to highlight that the unsteady RANS approach succeeds in the reproduction of this mechanism.

Figure 3 .Figure 3 . 5 :

 335 Figure 3.5: Evolution of angular frequency (a) and temporal growth rate (b) with the angle of attack for the SA+EC turbulence model at M = 0.73.

Figure 3 . 6 :

 36 Figure 3.6: Evolution of angular frequency (a) and temporal growth rate (b) with the angle of attack for the k-kl turbulence model at M = 0.73.

3. 2 47 Figure 3 . 7 :

 24737 Figure 3.7: ρE component of the buffet mode at α = 3.5 • and M = 0.73. Real (a) and imaginary (b) part of the direct buffet mode. Real (c) and imaginary (d) part of the adjoint buffet mode. Solid and dashed lines are respectively the supersonic bubble and the boundary layer of the steady state solution.

  figure 3.8. Specifically, a smaller ε results in a smaller value of (C L -C L0 ) t=0 and consequently a longer linear transient (non-linear effects appear at the same values of

Figure 3 . 8 :

 38 Figure 3.8: Log-scale evolution of the lift coefficient C L for an URANS simulations with SA+EC at α = 3.5 • . In red the linear slope resulting from global stability analysis σ = 7.5 s -1 . C L0 is the lift coefficient of the base flow, used as initial condition.

Figures 3 .Figure 3 . 9 :

 339 Figure 3.9: (a) Growth rate and (b) angular frequency density maps for the buffet mode at M = 0.73 and α = 5 • for the SA+EC turbulence model, with a zoom on the shock foot. Solid and dashed lines respectively depict the supersonic bubble and the boundary layer of the steady state solution.

Figures 3 .

 3 Figures 3.10a, 3.10b and 3.10c show the growth rate density maps before buffet onset at α = 3 • , in a well-established buffet regime at α = 4.5 • and after buffet exit at α = 7 • . Figures 3.10a and 3.10c both show a detached boundary layer with negative values of the density growth rate while figure 3.10b, in a condition of well-established buffet, shows an area with positive values completely inside the detached boundary layer. The analysis of the density maps of the growth rate at different angles of attack suggest that the detached boundary layer has an important role on the buffet instability scenario. At buffet onset the behaviour, in terms of contribution to the growth rate, of the detached boundary layer changes with the appearance of destabilising zones (which disappear at buffet exit) and it can be explained as the active key of buffet instability.

Figure 3 . 10 :

 310 Figure 3.10: Growth rate density maps for the buffet mode at M = 0.73 with the SA+EC turbulence model, α = 3 • (a), 4.5 • (b) and 7 • (c). Solid and dashed lines are respectively the supersonic bubble and the boundary layer of the steady state solution.

Figure 3 . 11 :

 311 Figure 3.11: Frequencies (a) and growth rate (b) contributions of the conservative and turbulent variables to the global unstable buffet eigenvalue. SA+EC turbulence model is used. The grey vertical lines defined the region of α where the buffet mode is unstable.

Figure 3 . 12 :

 312 Figure 3.12: Zones of the computational domain where SFD is locally activated. Solid and dashed lines are respectively the supersonic zone and the separated boundary layer of the steady state solution. Black points are the probe positions. (a) zone 1. (b) zone 2. (c) zone 3. (d) zone 4. (e) zone 4 . (f) zone 5. (g) zone 6. (h) zone 7.

2 . 3 : 2 :

 232 (Re, M, α) = (3.2 10 6 , 0.73, 5 • ). As already said, the effect of a particular zone on the global instability is assessed through two criteria: a global one based on the lift oscillation amplitude and a local one based on the standard deviation of signals from probes which quantifies how much unsteady signals are damped in the zones where SFD is locally activated.Once the local SFD is activated, there are two types of results. In the first case a steady-state is reached, residuals decrease towards zero machine values. In the second case a steady-state is not reached. It is then possible to force the convergence towards the steady-state solution by increasing the control parameter χ, but the achievement of the converged solution depends on the zone where the SFD is activated. Figure3.13a shows the lift oscillation amplitude for increasing values of the control parameter χ, for the application of SFD in the entire computational domain and three cases of local application (zones 2, 3 and 4). When χ = 0, the lift oscillation amplitude corresponds to the URANS one without SFD. For increasing values of χ, the lift oscillation Numerical investigation on two-dimensional transonic buffet Minimal value of the control parameter χ to reach a steady-state for the different zones where SFD is activated.

3. 3 57 Figure 3 .

 3573 Figure 3.13: (a) Amplitude of lift coefficient oscillation as function of the control parameter χ for SFD activated on the entire domain and three local applications. (b) The standard deviation of the streamwise momentum ρu as function of the control parameter χ for the SFD activated on zone 4; for probes position see figure 3.12d.

Figure 3 .

 3 14a shows the global criterion based on the lift oscillation amplitude as a function of the control parameter χ for zone 7. For χ = 6400 the shock still oscillates even if the lift coefficient amplitude is reduced of about 25%. At the same time, figure 3.14b shows the local criterion based on the standard deviation for the streamwise momentum (ρu) at the point depicted in figure 3.12h; it is decreased by 98% for the same value of χ. This indicates that the SFD technique effectively suppresses fluctuations in the region where the filter is applied but the buffet instability is still present. The same conclusion was found in Memmolo et al. [95] by freezing the URANS solution on the same zone. To freeze a solution is equivalent to use SFD with a control parameter χ = ∞. Results for the other zones are not presented because they are very similar to the ones of zone 7.

Figure 3 . 14 :

 314 Figure 3.14: Amplitude of the lift coefficient oscillation (a) and standard deviation of the state variable ρu (b) as function of the control parameter χ for the SFD activated on zone 7. For probes location see figure 3.12h.

  this complex phenomenon. Experimental data, powerful modern CFD tools and technique based on stability analysis have produced a lot of information on the physical mechanism behind transonic buffet. Today there are many different hypothese and physical models to explain buffet. The purpose of the present work was to improve the understanding of the phenomenon by the definition of the regions in the flow field necessary for the persistence of the buffet instability. For this, a new technique which aims at quantifying the local contributions in space to the stability quantities has been presented. The results have been compared with URANS simulations locally filtered with an SFD technique, showing a good consistency of results between them. Conclusions have been compared with literature trying to discuss and update the physical mechanisms proposed until now. It is possible to summarise the conclusions as follows:

Finally in the light 4 . 1 4 . 2 4 . 3 4 . 4 4 . 4 . 1 4 . 5 4 . 6 4 . 6 . 1 4 . 7

 41424344441454646147 of these results, two possible explanations of transonic buffet have been presented. A modification of the self-sustained closed loop model from Lee with a feedback path inside the separated boundary layer or a more localised instability driven by the shock foot oscillation. The present work gives a contribution to the understanding of the mechanism behind the transonic buffet and open at the same time the path for several analysis. First of all, further investigations are necessary to confirm the proposed mechanism. Then, it remains several way to analyse the results of the local contribution to global instability: by looking separately to the growth rate and the angular frequency density maps or by investigating how the density maps evolves with the Mach number. Experimental setup . . . . . . . . . . . . . . . . . . . . . . . Buffet onset . . . . . . . . . . . . . . . . . . . . . . . . . . . . Separated flow evolution . . . . . . . . . . . . . . . . . . . . Power Spectral Densities . . . . . . . . . . . . . . . . . . . . PSDs for the BUFET'N Co model . . . . . . . . . . . . . . . 4.4.2 PSDs for the AVERT model . . . . . . . . . . . . . . . . . . . 4.4.3 PSDs for the DTP Tremblement model . . . . . . . . . . . . 4.4.4 PSDs for the FLIRET model . . . . . . . . . . . . . . . . . . Cross-Spectral analysis . . . . . . . . . . . . . . . . . . . . . 4.5.1 Cross-Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Frequency-wavenumber spectra . . . . . . . . . . . . . . . . . 4.5.3 Convection velocities . . . . . . . . . . . . . . . . . . . . . . . Results synthesis . . . . . . . . . . . . . . . . . . . . . . . . . Strouhal numbers and convection velocities summary . . . . . 4.6.2 Buffet cell wavelengths summary . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 .

 1 an ONERA research project called "BUFET'N Co" launched in 2007 in the ONERA S3Ch wind tunnel over a half wing-body configuration based on the OAT15A aerofoil (figure 4.1(a)). 2. an European project called "AVERT" launched in 2007 in the ONERA S2MA wind tunnel over a half wing-body configuration based also on the OAT15A aerofoil (figure 4.1(b)).

3. a

 a French project called "DTP Tremblement" launched in 2004 in the ONERA S2MA wind tunnel over a half wing-body Dassault Aviation model (figure 4.1(c)). 4. an European project called "FLIRET" launched in 2005 in the European Transonic Windtunnel (ETW) over a half wing-body Airbus model (figure 4.1(d)).

Figure 4 .

 4 Figure 4.1 shows pictures of the four models inside their corresponding wind-tunnels and figure 4.2 shows the locations of the equipment on the suction side of the wing and

Figure 4 . 1 :

 41 Figure 4.1: Overview of the four models inside their respective wind tunnels. (a) BUFET'N Co model in S3Ch wind tunnel. (b) AVERT model in S2MA wind tunnel. (c) DTP Tremblement model in S2MA wind tunnel. (d) FLIRET model in ETW.

  (a) and 4.3(b) show the analysis of the lift curve and the root mean square (RMS) values of the accelerometer at the wing tip, respectively. Concerning the lift curve, the buffet onset is defined by the intersection between the lift curve and a straight line parallel to the linear part of the lift curve shifted by +0.1 • . Concerning the analysis of the accelerometer, the buffet onset is defined in the present study when the RMS value exceeds 1.4 times (defined empirically) the rest value. The two criteria agree well and give a buffet onset value of α ∼ = 3 • .The local buffet criteria are based on the analysis of the mean pressure value at the trailing edge (Pearcey et al.[START_REF] Pearcey | The interaction between local effects at the shock and rear separation-a source of significant scale effects in wind-tunnel tests on aerofoils and wings[END_REF]) or the RMS of the unsteady pressure transducers(Mundell & Mabey [102]). These criteria have to be applied on each section of the wing in order to find in which section buffet appears first. The onset is defined when the static value of the pressure coefficient (Cp) at the trailing edge diverges more than 0.05 (Pearcey et al.[START_REF] Pearcey | The interaction between local effects at the shock and rear separation-a source of significant scale effects in wind-tunnel tests on aerofoils and wings[END_REF]) or when the RMS value exceeds 2.5 times (defined empirically) the initial plateau. It is possible to define an initial plateau because far before buffet onset the RMS of Kulites are constant as clearly shown in figure4.3(c).

Figures 4 .Figure 4 . 2 :

 442 Figures 4.3(d) and 4.3(e) do not show clearly the initial plateau because the α range is not large enough. For these cases, the value of the initial plateau is taken respectively

Figure 4 . 3 :

 43 Figure 4.3: Buffet onset criteria. (a) Lift curve for FLIRET project at M =0.85. (b) RMS of accelerometer close to the wing tip for FLIRET project at M =0.85. (c) RMS of Kulites for FLIRET close to TE at M =0.85. (d) RMS of Kulites for BUFET'N Co close to TE at M =0.82. (e) RMS of Kulites for DTP Tremblement close to TE at M =0.8.

Figure 4 . 4 :

 44 Figure 4.4: Buffet onset limit in the M -α plane. (a) FLIRET tests. (b) AVERT tests from Dandois [28].

Figure 4 .

 4 [START_REF] Bartels | Computation of shock buffet onset for a conventional and supercritical airfoil[END_REF] shows five oil flow visualizations for the BUFET'N Co case at M = 0.82. The blue oil is coming from the pressure side, consequently the size of the blue area reveals the extent of the separated zone.

  Figures 4.5(a) and 4.5(b) at α = 2.5 • and 2.8 • show a flow fully attached on the suction side of the wing with the exception of a zone at the wing tip caused by the vortex tip. In figure 4.5(b) the shock moves upstream and skin-friction lines tend to be parallel to the TE in the area of y/b = 55%.

4. 3 Figure 4 . 5 :

 345 Figure 4.5: Oil visualizations for BUFET'N Co test at M=0.82 for increasing values of α. (a) α = 2.5 • . (b) α = 2.8 • . (c) α = 3.0 • . (d) α = 3.0 • . (e) α = 3.5 • .

Figure 4 .

 4 Figure 4.6(b) shows the oil flow visualization at α = 3.5 • . The flow is separated between y/b = 42.5% and y/b = 82.5%. Both visualizations show the detachment due to the vortex tip. The separated flow condition at a higher Mach number of 0.86 is similar (figures omitted). At α = 0 • the flow is fully attached. When approaching buffet onset, a separated zone appears at about y/b = 75%; i.e. more outboard than for M = 0.82.

Figure 4 . 6 :

 46 Figure 4.6: Oil flow visualization for AVERT test at M=0.82. Thick dashed line shows the shock location. (a) α = 3 • . (b) α = 3.5 • .

  spectrum is based on signal coherence, so even if signals are filtered, large coherent zones remain. The non-dimensional Strouhal number St = (fL)/U ∞ has been defined to compare the results over different models. Three different lengths are considered here for L: the local chord, M AC the M AC . Consequently, three Strouhal numbers can be defined. The reason lies in the different point of views on the phenomenon: 4.4 Power Spectral Densities 71 local or global in space.Local Strouhal number means an analysis only at a given section, so with the value of the chord at this section. The Strouhal number based on M AC tries to define a global value for the entire wing, as if there was an unstable global mode which synchronizes all sections. Furthermore, for wings with a high value of the taper ratio, it is compulsory to consider the local Strouhal number in order to perform comparisons with small taper ratios (like FLIRET and BUFET'N Co tests).

4. 4 . 1

 41 PSDs for the BUFET'N Co model BUFET'N Co tests are performed at a Mach number of 0.82 and α ranges from 2.5 • to 3.5 • . Figures 4.7(a) and 4.7(b) show the PSDs of Kulites data in the chordwise direction at y/b = 60% for the test cases at α = 3.2 • and 3.5 • , respectively. The section at y/b = 60% has been chosen because it is where a separation at the TE first appears and because it is the best equipped with sensors. The Kelvin-Helmholtz instability clearly appears in both spectra at its typical Strouhal number (1 -4) when approaching the TE, because the flow is more separated (figure 4.5). The intensity of the KH instability increases approaching the wing tip, except for the final flow reattachment due to the wing tip vortex (figures 4.5(c) and 4.5(e)). Close to the onset at α = 3.2 • , the bump in the spectra is very large and centered around St M AC = 0.34. Here the variations of the buffet Strouhal number in span and chord are very weak (it is even difficult to visualize buffet peak when approaching the TE). At α = 3.5 • , corresponding to well-established transonic buffet conditions, the situation is relatively different. It is easier to identify a bump in the spectra and the variations in the chordwise and spanwise directions are clearer. In the chordwise direction the buffet Strouhal number decreases from around St M AC = 0.34 at x/c = 60% to St M AC = 0.2 at the TE. A spanwise variation of the buffet Strouhal number is observed with an oscillation between the critical section and the wing tip: it decreases up to y/b = 60% (where the K-H instability is the strongest), then increases before a final decrease at the wing tip. The map in figure 4.8 shows the overall variations of buffet Strouhal number on the wing.

Figure 4 . 8 :

 48 Figure 4.8: Buffet Strouhal number map for BUFET'N Co test at α=3.5 • and M =0.82.

Figure 4 .

 4 Figure 4.9(c) shows two PSDs at the shock foot for α = 1.99 • and α = 3.47 • . One can

Figure 4 . 9 :

 49 Figure 4.9: PSDs for AVERT tests. (a) Buffet Strouhal number map at α=3.47 • and M =0.82. (b) Buffet Strouhal number map at α=4.99 • and M =0.86. (c) Semi-log graph of power spectral densities at shock foot and y/b=75% for M =0.86 with values of buffet Strouhal numbers in the legend.

For

  this database, only root mean square values of the sensors signal, cross-spectra (coherence and phase) and PSDs are available. The resolution of these treatments in the frequency domain is 25 Hz (St M AC = 0.023 at M = 0.8) and the frequency range of analysis is 0 -750 Hz (St M AC = 0 -0.69 at M = 0.8), too low for the visualization of the K-H instability. Tested Mach numbers are 0.75, 0.8, 0.825 and 0.85 for various α. In particular, a series of tests at M = 0.8 with nacelle, stabilizers and boundary layer transition fixed at 10% are analysed. Here a large range of buffet Strouhal number is found: from 0.28 to 0.36. The reasons of this large range are due to the large range of M -α analysed and to the double sweep geometry of the wing. Figures 4.10(a) and 4.10(b) highlight the difference in PSDs between well-established (α = 3.6 • ) and deep Experimental investigation on three-dimensional transonic buffet buffet (α = 4.3 • ) conditions at y/b = 70%. In the first case, the buffet bumps are wide and slightly vary on the wing. A value of St M AC = 0.28 (St = 0.19 based on local chord and 0.22 based on M AC ) is found. At α = 4.3 • , it is easier to identify a bump at a St M AC = 0.21 (St = 0.14 based on local chord and 0.16 based on M AC ).

Figure 4 .

 4 11(b) shows also a peak due to a structural mode (St M AC = 0.12). Similar behaviours are found in an St MAC PSD (Pa²/Hz)

Figure 4 . 10 :

 410 Figure 4.10: Semi-log graph of power spectral densities for DTP Tremblement at M=0.8 in the chordwise direction at y/b=70%. (a) α = 3.6 • . (b) α=4.3 • .

Figure 4 . 11 :Figure 4 . 12 :

 411412 Figure 4.11: Semi-log graph of power spectral densities for DTP Tremblement at M =0.8 in the chordwise direction at y/b=58%. (a) α=3.6 • . (b) α=4.3 • .

Figure 4 . 8 (

 48 Figure 4.13: Log-log graph of power spectral densities for FLIRET at α=3.37 • and M=0.85 in the chordwise direction at y/b=79%.

Figure 4 .Figure 4 . 14 :

 4414 Figure 4.14 shows an example of the analysis of a cross-spectrum for two sensors of the BUFET'N Co test at α = 3.5 • and M = 0.82. It is possible to see that the coherence is high only in the buffet Strouhal number range, where a linear slope is found in the phase plot. Once the velocity both in the spanwise and chordwise directions is defined,

4. 5

 5 Cross-Spectral analysis 79 with Λ x/c is the angle between the line of sensors at constant x/c and y-axis. Figure 4.15 shows a sketch with the variables here defined.

Figure 4 . 15 :

 415 Figure 4.15: Sketch of the measured velocity on the lines of sensors (dashed lines) in the chordwise (U Cc ) and in the spanwise (U Cs ) directions.

Figure 4 .

 4 [START_REF] Carini | Global stability and control of the confined turbulent flow past a thick flat plate[END_REF] shows the results for one selected case of BUFET'N Co test and it is in complete agreement with Dandois[START_REF] Dandois | Experimental study of transonic buffet phenomenon on a 3D swept wing[END_REF]. It is possible to identify the Kelvin-Helmholtz instability in the Strouhal number range 1 -4 as well as the buffet phenomenon in the Strouhal number range 0.15 -0.4. The magnitude of the velocity comes from the slope in the f -k spectrum U C = 2π ∆f/∆k. Both f -k spectra in chordwise and in spanwise directions are

figure 4 .

 4 figure 4.16(b) gives a velocity of 70 m.s -1 . The resulting buffet velocity has a norm of 66 m.s -1 (0.24U ∞ ) with ϕ = 43 • . The f -k spectra of BUFET'N Co show no convection velocities before buffet onset and the same results as in figure 4.16 at α = 3.3 • and 3.4 • . The range of M -α tested is smaller than for the other campaigns and it is

Figure 4 . 16 :

 416 Figure 4.16: Frequency-wavenumber spectra for BUFET'N Co test at α=3.5 • and M =0.82. (a) Spanwise direction at x/c=80%. (b) Chordwise direction at y/b=60%.

  (b)) both buffet and K-H instability velocities appear at y/b = 79% and in chord at x/c = 85%. Combining the velocities in chord and span, the results are non-dimensional convection velocities of 0.24U ∞ with ϕ = 2 • for the buffet and 0.63U ∞ with ϕ = 52 • for the K-H instability (the wind tunnel is cryogenic, so the velocities are lower and just the non-dimensional velocities are given here). Figures 4.17

  (c) and 4.17(d) show other two f -k spectra at α = 4.84 • and M = 0.87 for the same position. Combining the velocities in chord and span the results are a non-dimensional convection velocity for buffet of 0.2U ∞ with ϕ = 19 •

Figure 4 .

 4 [START_REF] Chomaz | Bifurcations to local and global modes in spatially developing flows[END_REF] shows the f -k spectra in chord and span (at x/c and y/b both 75%) for all the tests at a Mach number of Experimental investigation on three-dimensional transonic buffet

Figure 4 . 17 :

 417 Figure 4.17: Frequency-wavenumber spectra for FLIRET test. α=3.37 • and M =0.85: (a) Spanwise direction at x/c=85%. (b) Chordwise direction at y/b=79%. α=4.84 • and M =0.87: (c) Spanwise direction at x/c=85%. (d) Chordwise direction at y/b=79%.

For

  each wind tunnel test, the cases considered are the most similar ones in terms of M -α: both BUFFET'N Co and AVERT at M = 0.82 and α = 3.5 • , FLIRET at M = 0.85 and α = 3.37 • , DTP Tremblement at M = 0.8 and α = 4.3 • . The velocities found in chordwise and spanwise directions are combined following equation (4.5) and presented in figure 4.19 in dimensional units.

Figure 4 .Figure 4 . 18 :

 4418 Figure 4.18: Frequency-wavenumber spectra for AVERT test at M =0.86. Spanwise direction at x/c=75%: (a) α=0 • . (c) α=1.99 • . (e) α=3.47 • . Chordwise direction at y/b=75%: (b) α=0 • . (d) α=1.99 • . (f) α=3.47 • .

Figure 4 . 19 :

 419 Figure 4.19: Buffet convection velocities for the four databases analysed. (a) BUFET'N Co model (α = 3.5, M = 0.82). (b) AVERT model (α = 3.5, M = 0.82). (c) DTP Tremblement model (α = 4.3, M = 0.8). (d) FLIRET model (α = 3.37, M = 0.85).

at α = 3 •

 3 and M = 0.82 (figure 4.20(a)).

Figure 4 .Figure 4 . 20 :

 4420 figure 4.11, Sartor & Timme[START_REF] Sartor | Reynolds-Averaged Navier-Stokes simulations of shock buffet on half wing-body configuration[END_REF] and Sugioka et al.[START_REF] Sugioka | Unsteady PSP measurement of transonic buffet on a wing[END_REF]) while the tendency with the Mach number is less clear and needs further studies. Furthermore a convective

Figure 4 . 21 :

 421 Figure 4.21: Phase of cross-spectra along the span. (a) Values at x/c = 80%, St M AC = 0.26, M = 0.82 and α = 3.5 • for BUFET'N Co model (reference sensor at y/b = 85%). (b) Values at x/c = 85%, St M AC = 0.33, M = 0.85 and α=3.6 • for FLIRET model(reference sensor at y/b = 76.5%).

4. 7 89 Figure 4 . 22 :

 789422 Figure 4.22: Buffet cells convection on FLIRET model at α=3.6 • and M =0.85.

Contents 5 . 1 93 5. 2 95 5. 2 . 1 97 5. 2 . 2 99 5. 2 . 4 100 5. 2 . 5 102 5. 3

 51932952197229924100251023 Flow configuration and base flow computation . . . . . . . Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mesh convergence . . . . . . . . . . . . . . . . . . . . . . . . Sweep effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.2.3 Onset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wavenumber effect . . . . . . . . . . . . . . . . . . . . . . . . Adjoint problem and structural sensitivity . . . . . . . . . . . Summary and conclusion . . . . . . . . . . . . . . . . . . . . 106This chapter presents the main objective and the most challenging part of the study: the 3D global stability analysis. It has been shown that 2D and 3D transonic buffet appear on aerofoils and wings with different values of the characteristic parameters.From 2D aerofoils to 3D swept wing at Λ = 30 • , the Strouhal number increases by a factor of 4 to 7, the shock amplitude oscillation decreases by a factor 10 and 3D patterns in the detached boundary layer appear on the wing and are convected outboard. The differences can derive from an evolution of the same physical phenomenon when the geometry changes or from two distinct physical phenomena. Furthermore, the shape of the power spectral densities exhibits a peak for 2D while a broadband bump for 3D transonic buffet. The peak in the spectra for 2D buffet has been already explained with an unstable global mode, linked with an oscillatory instability of the flow, while broadband bump in the spectra are usually explained by a noise-amplifier behaviour of the flow. The 3D global stability analysis, presented in this chapter, gives an answer 92 Three-dimensional global stability analysis to the main question of the work. "what is the link between the 2D transonic buffet global mode and the so-called 3D transonic buffet?". The chapter highlights also the role of several parameters, as the angles of attack and the sweep angle, and shows the local contribution of the instabilities.

5. 1 Flow configuration and base flow computation 93 Figure 5 . 1 :

 19351 Figure 5.1: Sketch of the infinite wing with geometrical variables and two reference frames. (a) top view for α = 0 • , (b) side view for Λ = 0 • .

Figure 5 .

 5 1 shows a sketch of the flow configuration with the geometrical variables from two different view points: from above the wing (a) and a side view (b). The geometrical variables are the angle of attack α, the sweep angle Λ, the extruded length L and the chord c. Two reference frames are shown in the figure: relative to the upstream velocity (χ, η, ζ) and relative to the wing (x, y, z). The coordinate systems are the following: χ is oriented along the freestream direction, ζ is in the spanwise direction and η is vertical, x is parallel to the aerofoil chord, z is parallel to the LE and y is perpendicular to the wing. The origin of the reference frame (x, y, z) is located at the aerofoil leading edge. In the following, the results are presented in the reference frame relative to the wing. The 3D steady base flow is computed by the extrusion in the spanwise direction of the 2D solution over a length L. The obtained base flow does not exhibit spatial variations in the spanwise direction: q 0 (x, y) (see figure5.2). The 2D solution is converged to machine-epsilon values. Boundary conditions are adiabatic walls for the aerofoil, a non-reflective boundary condition for the farfield 50 chord length away from the aerofoil. The 2D computational domain is discretised using a C-type structured grid.

Figure 5 . 2 :

 52 Figure 5.2: Velocity fields in the (x, y) plane of the wing reference frame and wall pressure coefficient C p for the extruded base flow. (a) U/U n∞ and C p for the unswept wing (Λ = 0 • ), (b) U/U n∞ and C p for the swept wing at Λ = 30 • , (c) spanwise component of the velocity W/U n∞ and C p for the swept wing at Λ = 30 • with a zoom at the shock foot.

Figure 5 .

 5 Figure 5.2c shows the boundary layer on the wing generated by W ∞ . To maintain the same normal flow conditions with respect to the wing, the incoming flow and the angle of attack are modified when Λ = 0 according to: M ∞ = 0.73/ cos(Λ) and α = arctan [tan(3.5 • ) cos(Λ)].

Figure 5 . 3 :

 53 Figure 5.3: (a) Spectrum of the unswept wing with extruded length L = c and spanwise mesh refinement N z = 12. stable modes in the grey half-plane, 2D buffet mode, fundamental and the two harmonics of the buffet cells modes. (b) and (c) real part of the ρE component of the 2D buffet mode at βc = β = 0 and of the 3D buffet cells mode at βc = β = 2π, respectively.

5. 2 97 Figure 5 . 4 :

 29754 Figure 5.4: Effect of the mesh refinement in span on the growth rate and frequency of the 3D mode at βc = 2π for the unswept Λ = 0 • and swept Λ = 30 • wings.

Figure 5 .

 5 Figure 5.5a shows the spectra of an infinite wing for different values of the sweep angle Λ. The spectrum for the unswept Λ = 0 • wing has already been shown in figure 5.3 but here only the most unstable 3D mode is considered (βc = 2π in the spectrum of figure 5.3a) because it is the most discretised in span.The 2D mode for the unswept wing undergoes a slight stabilization with Λ while remaining almost at the same frequency. The growth rate σc/U n∞ decreases from 0.105 with no sweep to 0.08 at Λ = 30 • . The structure of the mode does not change in the x -y plane. For Λ > 0, a spanwise component of the velocity ρW (x, y, z) appears that is constant in span, so that the mode remains strictly 2D. In the case of a base flow which does not vary in span, the ρU and ρV dynamics are independent of that of ρW . The variation of the eigenvalue (growth rate decrease) is due to a weak coupling which is present only in compressible flows and which stems from the diffusive term in the energy conservation equation (ρE). More precisely, the coupling is due to the term div(ΣU) in the energy equation, where div is the divergence in the 3D Euclidean space and Σ is the strain tensor. The origin of the coupling has been understood by looking at the equations in appendix C and considering a two-dimensional perturbation with βc = 0.Contrary to the unswept case, the 3D stall cells mode in the swept case exhibits a non-zero frequency, which gradually increases with the sweep up to f c/U n∞ = 0.43 at Λ = 30 • (six times the 2D mode frequency). At the same time, the growth rate σc/U n∞ decreases from 0.43 to 0.36 at Λ = 30 • . The stall cells mode (called also buffet cells by Iovnovich & Raveh[START_REF] Iovnovich | Numerical study of shock buffet on threedimensional wings[END_REF]) are therefore convected outboard with a convection velocity U C proportional to the sweep angle and the amplitude of ρW component of the mode increases. The outboard convection velocity of the 3D phenomenon can be computed as U C = ω/β = 2πf /β. When a sweep angle Λ = 7.5 • is simulated on the wing, a convection velocity of 0.1U n∞ appears on the wing. It linearly increases with the sweep angle: 0.21U n∞ at Λ = 15 • and 0.32U n∞ at Λ = 22.5 • . Finally for the swept wing at Λ = 30 • , the convection velocity is equal to 0.43U n∞ which is exactly

Figure 5 . 5 :

 55 Figure 5.5: (a) Evolution of the spectrum for an aerofoil with an extruded length of L = c and a sweep angle Λ increasing from 0 • to 30 • . The harmonics of the three dimensional mode have been removed and only the fundamental mode at βc = 2π is shown in the figure. (b) Real part of ρE component of the 3D buffet mode at Λ = 30 • .

Figure 5 . 6 :

 56 Figure 5.6: Spectrum evolution of a wing (L = c) with the angle of attack α increasing from 2.5 • to 3.5 • . (a) Unswept wing Λ = 0 • and (b) swept wing Λ = 30 • . 2D buffet mode at βc = 0 and the 3D buffet cell mode at βc = 2π are outlined by dashed lines.

Figures 5. 7

 7 Figures 5.7(a) and (b) show the variation of the growth rate and frequency of the

  Figures 5.7(a) and (b) show the variation of the growth rate and frequency of the

  and

5. 2 101 Figure 5 . 7 :

 210157 Figure 5.7: Growth rate and frequency of the 2D buffet mode as a function of the wavenumber βc for the unswept Λ = 0 • and swept Λ = 30 • wing.

Figure 5 . 8 :

 58 Figure 5.8: (a) Evolution of the spectrum for an unswept wing at Λ = 0 • for three values of extruded lengths L = 8c, 9c and 12c. (b) Real part of the ρE component of the 2D buffet mode for the unswept wing Λ = 0 • and wavenumber βc = β = 2π/12 (the mode is no more 2D because βc = β = 0).

Figure 5 . 9 :

 59 Figure 5.9: Growth rate and frequency of the 3D mode as a function of the wavenumber for the unswept Λ = 0 • and the swept Λ = 30 • wing.

5. 2 103 Figure 5 . 10 :

 2103510 Figure 5.10: Real part of the ρE component of the 2D buffet adjoint mode at βc = 0 for the unswept wing at M = 0.73 and α = 3.5 • . Note the exponential colorbar.

Figure 5 . 11 :

 511 Figure 5.11: Real part of the ρE component of the 3D buffet adjoint mode at βc = 2π, M n = 0.73 and α = 3.5 • for the unswept wing Λ = 0 • (a) and the swept wing at Λ = 30 • (b). Note exponential colorbar and the three surfaces where the modes are shown: wing body, iso-z and iso-y upstream and downstream the aerofoil.

Figure 5 . 12 :

 512 Figure 5.12: Structural sensitivity of the 2D mode at βc = 0 for the unswept wing at M = 0.73 and α = 3.5 • . The colorbar is exponential.

Figure 5 . 13 :

 513 Figure 5.13: Structural sensitivity of the 3D mode at βc = 2π, M n = 0.73 and α = 3.5 • for (a) unswept wing Λ = 0 • and (b) the swept wing at Λ = 30 • . The colorbar is exponential.

Figure 5 .

 5 Figure 5.12 shows the structural sensitivity of the 2D buffet mode. The results are consistent with the ones presented by Iorio et al.[START_REF] Iorio | Direct and adjoint global stability analysis of turbulent transonic flows over a NACA0012 profile[END_REF]. The core of the 2D instability is the shock foot with lower values in the shock and in the detached boundary layer.

Figures 5. 13

 13 Figures 5.13(a) and (b) show the structural sensitivity of the 3D mode for the unswept

  Figures 5.13(a) and (b) show the structural sensitivity of the 3D mode for the unswept wing and swept wing at Λ = 30 • , respectively. The real and imaginary parts of both

5. 2 105 Figure 5 .

 21055 Figure 5.14: (a) Growth rate and (b) angular frequency growth rate density maps for the 3D buffet mode at M n = 0.73 and α = 3.5 • for the swept wing at Λ = 30 • .

Figures 5 .

 5 Figures 5.14(a) and 5.14(b) show the local contribution of growth rate and angular frequency of the 3D buffet mode for the swept wing at Λ = 30 • . The same shape is found for different values of Λ (figures not shown). Also for this instability, the results are consistent with the structural sensitivity but the growth rate in figure 5.14(a)

First a global 109 110

 109 Conclusions and perspectivesstability analysis, based on the linearised Reynolds-averaged Navier-Stokes ([START_REF] Crouch | Predicting the onset of flow unsteadiness based on global instability[END_REF],[START_REF] Crouch | Origin of transonic buffet on aerofoils[END_REF]) has been reproduced looking at the effect of some interesting numerical parameters.Crouch et al. explained the 2D transonic buffet phenomenon by the appearance of a real positive complex eigenvalue of the linearised Jacobian matrix. Then, the phenomenon has been studied numerically through the coupling between two techniques based on different approaches: linear stability analysis and numerical simulations. A recently developed technique, based on the direct and adjoint unstable global modes, has been used to compute the local contribution of the flow to the growth rate and angular frequency of the unstable global mode. It has been possible to compute two density maps respectively of the growth rate and the angular frequency in order to have information on the localisation on the computational domain of these two parameters. The results have allowed identifying which zones are directly responsible for the existence of the instability. The same technique has been used to identify the contribution of each state variable to the growth rate and the angular frequency of the global unstable mode. The key role of the turbulent variables dynamic has been highlighted.

6. 1

 1 Conclusions 111 the boundary layer or an instability more localised around the shock foot.

ρ ρ 0 w 0 ρ 0 ν0 ρ 2 0 =+-ρ C w1 f w0 - C b1 K 2 f t20 ρ 0 ν0 ρ 0 η 2 ++ ρ 0 C w1 f w - C b1 K 2 f t2 ρ 0 ν0 ρ 0 η 2 + 14 ) 3 / 2 TfC 6 w3g 6 + C 6 w3 1 / 6 ,

 022143266616 || << ||q 0 ||. (C.7)q (x, y, z, t) = q(x, y)exp(iβz+ λt) 0 + ρ 0 ν t0 ) ∂ρ -µ 0 + ρ 0 ν t0 ) iβ ρuρ -1 0 + iβ ρ ρ 0 u 0 ρ 0 + (µ 0 + ρν t0 ) µ 0 + ρ 0 ν t0 ) iβ ρvρ -1 0 + iβ ρ ρ 0 v 0 ρ 0 + (µ 0 + ρν t0 ) µ 0 + ρ 0 ν t0 ) + ρν t ) ∂ ∂x ρ 0 w 0 ρ 0 + (µ 0 + ρ 0 ν t0 ) iβ ρuρ -1 0 + iβ ρ ρ 0 u 0 ρ 0 τzy = (µ 0 + ρ 0 ν t0 ) C b1 -S0 ρ 0 ν0 ft2 + Ŝρ 0 ν0 + S0 ρν -S0 ρνf t20 + µ 0 + ρ 0 ν0 -β 2 ρν ρ 0 + β 2 ρ ρ 0 ν0 ρ ρ 0 C w1 f w0 -Linearised relation for p, µ, T, ρν t , f t2 , S, f w Equation for p(ρ, ρu, ρ E):p = (γ -1) ρEs + C s T + C s (C.17) µ = µ s T 0 T s 3/2 T s + C s T 0 + C s t2 = C t3 exp -C t4 χ 2 (C.22) S = Ω + f ν2 ρν ρK 2 η 2 , f ν2 = 1 -χ 1 + χf ν1 (C.23)f w = g 1 + g = r + C w2 (r 6 -r), r = ρν ρ SK 2 η 2 (C.24)

σ ν = 2 / 3 , 2 +( 1 +fS = Ω + f ν20 ρν ρ 0 K 2 η 2 + f ν2 ρν 0 ρ 0 K 2 η 2 - f ν20 ρν 0 ρ ρ 2 0 K 2 η 2 (

 23212222 K = 0.41, C b1 = 0.1355, C b2 = 0.622, C w1 = C b1 K C b2 ) σ ν , C w2 = 0.3, C w3 = 2.0, C ν1 = 7.1, C t3 = 1.2, C t4 = 0.5, C 5 = 3.5.The linearised turbulent variables are:ρν t = f ν1 ρν 0 + f ν10 ρν (C.27) t2 = -C t3 C t4 (2χ 0 χ ) (C.30)

g

  = r + C w2 (r 5 0 r -r ) (C.34) r = 1 ρ 0 S0 K 2 η 2 ρνj U 0i,j + ρ 0 ν2 0 2U 0i,j U i,j (C.36)For the Edwards-Chandra correction two variable are redefined:S = S 1/2 1 χ + f ν1 and r = tanh[ν/(K 2 η 2 S)]

  Rouzaud et al.[START_REF] Rouzaud | Numerical simulation of buffeting over airfoil using dual time stepping method[END_REF] used a implicit Dual Time Stepping (DTS) scheme and underlined the importance of a second order time scheme to compute buffet simulation. Rumsey et al.[START_REF] Rumsey | Efficiency and accuracy of time-accurate turbulent Navier-Stokes computations[END_REF] agreed using DTS scheme and together with

Iovnovich & Raveh

[START_REF] Iovnovich | Reynolds-averaged Navier-Stokes study of the shock-buffet instability mechanism[END_REF]

pointed out the importance of a spatial refinement around shock and detached boundary layer. Several schemes have been used to numerically compute the convective fluxes: Jameson ([67]),

Roe ([119]

), Roe with MUSCL extrapolation (

[START_REF] Van Leer | Towards the ultimate conservative difference scheme. v. a secondorder sequel to Godunov's method[END_REF]

), AUSM+P (

[START_REF] Edwards | Low-diffusion flux-splitting methods for flows at all speeds[END_REF]

,

[START_REF] Mary | Large eddy simulation of flow around an airfoil near stall[END_REF]

) and others. It is difficult to generally state that a scheme works better than others because there is a strong sensitivity with the CFD code used for the simulation. Analysing the works of Goncalves & Houdeville

[START_REF] Goncalves | Turbulence model and numerical scheme assessment for buffet computations[END_REF]

, Soda & Verdon

[START_REF] Soda | Investigation of influence of different modelling parameters on calculation of transonic buffet phenomena[END_REF]

and Sartor et al.

[START_REF] Sartor | Stability, receptivity, and sensitivity analyses of buffeting transonic flow over a profile[END_REF]

, it appears that Jameson scheme better predicts the buffet onset, Roe+MUSCL gives good experimental agreement and AUMSP is preferred for a global stability analysis. The only statement totally shared by authors which studied numerically transonic buffet, is that the biggest sensitivity among the numerical parameters is the choice of turbulence models of the URANS simulations.

Even if a lot of turbulence models have been studied it is still not possible to identify the best turbulence model for buffet because different CFD codes have been used and they do not have coherent results: Goncalves et al.

[START_REF] Goncalves | Numerical simulation of transonic buffet over an airfoil[END_REF]

and Goncalves & Houdeville

[START_REF] Goncalves | Turbulence model and numerical scheme assessment for buffet computations[END_REF]

used CANARI, Illi et al.

[START_REF] Illi | On the capability of unsteady RANS to predict transonic buffet[END_REF]

used DLR-TAU (

[START_REF] Schwamborn | The DLR TAU-code: recent applications in research and industry[END_REF]

), Thiery & Coustols

[START_REF] Thiery | Urans computations of shock-induced oscillations over 2D rigid airfoils: influence of test section geometry[END_REF]

and Sartor et al.

[START_REF] Sartor | Stability, receptivity, and sensitivity analyses of buffeting transonic flow over a profile[END_REF]

used elsA (

[START_REF] Cambier | The onera elsa cfd software: input from research and feedback from industry[END_REF]

), Deck

[START_REF] Deck | Numerical simulation of transonic buffet over a supercritical airfoil[END_REF]

used FLU3M, Grossi et al.

[START_REF] Grossi | Prediction of transonic buffet by delayed detached-eddy simulation[END_REF]

used NSMB (

[START_REF] Vos | Recent advances in aerodynamics inside the NSMB (Navier Stokes multi block) consortium[END_REF]

).

  Introduction unsteady pressure signals for different Mach numbers but without the analysis of the spectral content. More recently, unsteady Pressure-Sensitive-Paint (PSP) has been successfully used by Steimle et al. [142], Merienne et al. [96], Sugioka et al. [144] and Lawson et al. [76] to analyse the unsteady flowfield for transonic buffet over a transport-type swept-wing. The model studied by Steimle et al. [142] was flexible and results exhibited important aeroelastic effects. The PSP measurements of Merienne et al. [96] over a rigid model showed a fairly good agreement with Kulite transducers. A new fast-responding PSP has been tested by Sugioka et al. [144] over an 80%scale model of the NASA Common Research Model (CRM). The results showed a typical Strouhal number in the range 0.19 -0.25 for M = 0.85 and an angle of attack between 4.2 • and 6.8

• . The heavy buffet regime presented a large bump in the pressure spectrum in the Strouhal number range of 0.2 -0.6. Eckstrom et al.

[START_REF] Eckstrom | Unsteady pressure and structural response measurements on an elasticsupercritical wing[END_REF]

[38] presented a complete study of mean pressure coefficients and • . Lawson et al.

[START_REF] Lawson | Characterisation of buffet on a civil aircraft wing[END_REF]

presented in detail, together with unsteady PSP measurements, several buffet onset criteria. They defined two ranges of Strouhal numbers at a Mach number of 0.8: 0.08 -0.16 for α = 2.8 • and 0.22 -0.43 for α = 4 • .

The same campaign has then been analysed by Masini et al.

[START_REF] Masini | Influence of vane vortex generators on transonic wing buffet: further analysis of the bucolic experimental dataset[END_REF]

by using proper orthogonal decomposition (POD) of the PSP measurements as well as flow control by vortex generators (VGs). PODs modes showed the dominant structures of the flow:

Chapter 2 Numerical approach Contents 2.1 Numerical simulation . . . . . . . . . . . . . . . . . . . . . . 22 2

  .1.1 RANS equations . . . . . . . . . . . . . . . . . . . . . . . . .

	1.1 Context
	dimension has been extended in three dimension. In this way the link between two
	and three-dimensional transonic buffet is outlined. Finally chapter 6 summarises the
	overall study and highlights the main conclusion and perspective of each chapter.
	The study is conducted both numerically and
	experimentally through numerical simulations, global stability analysis and comparison
	of experimental databases. The purpose of the present work is to answer several main
	questions:
	How does the spectrum of two-dimensional transonic buffet over aerofoil evolve by
	changing some numerical and geometrical parameters? What is the real self-sustained
	mechanism behind two-dimensional transonic buffet over aerofoil? What are the char-
	acteristic parameters of three-dimensional transonic buffet and what is their variability
	over different kinds of swept wing? Does the stall cell type of unstable global mode
	observed in the incompressible regime also exist at transonic speeds? What is the effect
	of the sweep angle on the instabilities over an infinite wing? And, most important,
	what is the link between the two-dimensional transonic buffet global mode and the
	so-called three-dimensional transonic buffet?
	The work is organized as follows. The numerical methods used for the present work
	are presented in chapter 2: details of the numerical simulations, global stability analysis
	and local contribution technique to global instability. Chapter 3 shows the numerical
	simulations and the results of global stability analysis for two-dimensional configura-
	tions. Global stability analysis has been repeated, looking at the effects of different
	parameters (numerical scheme, turbulence model, mesh refinement etc.). Then the
	results of the local contribution technique to global instability are presented and vali-
	dated by URANS simulations with selective frequency damping. The physical models,
	presented in the state of the art, are discussed in light of these results. The compari-
	son of four experimental databases is presented in chapter 4. The analysis consists in
	an overall comparison of the characteristic parameters as buffet onset, detached zone
	size, Strouhal number and convection velocities. The aim of the chapter is to find the
	common features but also the variability of transonic buffet parameters in different
	configurations. Chapter 5 shows the results of a global stability analysis on a infinite
	wing and is the challenging part of the work. First, the shape of the transonic buffet
	in three dimensions is defined from the experimental tests analysed and the numeri-
	cal simulations from literature. Then, the global stability analysis performed in two

22 2.2 Global stability analysis . . . . . . . . . . . . . . . . . . . . 29 2.3 Identification of local hydrodynamic feedback in the un- stable eigenmode . . . . . . . . . . . . . . . . . . . . . . . . . 32

	22	Numerical approach
	operator, the Jacobian matrix. Finally, in section 2.3, a recently developed technique
	on the local contributions to global instability is presented. The technique aims to
	compute the local contributions in space of the flow to the growth rate and angular
	frequency of the unstable global mode.
	2.1 Numerical simulation
	Transonic buffet is a complex turbulent shock-wave/boundary layer interaction at high
	values of Reynolds number. The computational cost to directly solve the Navier-Stokes
	equations increases with the Reynolds number. This is the reason why RANS equa-
	tions are considered. The entire work is based on RANS simulations, consequently the
	effects of the turbulent small scales flow dynamics are modeled by an eddy viscosity
	µ t .
	2.3.1 Structural sensitivity analysis in a discrete setting . . . . . . 33 Numerical simulations are performed using two compressible CFD finite-volume
	2.3.2 Column-decomposition of the Jacobian matrix and eigen-solvers: "Ensemble Logiciel de Simulation en Aérodynamique" elsA (Cambier et al.
	value contribution . . . . . . . . . . . . . . . . . . . . . . . . 34 [15]) and "Fast Aerodynamic Solver Technology Structured grid Navier-Stokes solver"
	2.3.3 Link with wavemaker function . . . . . . . . . . . . . . . . . 36 FastS. They both solve, among others, RANS equations with various turbulence mod-
	2.3.4 Variable-decomposition of the Jacobian matrix . . . . . . . . 37 els on multiblock structured grids. elsA is a complex multi-purpose industrial solver
	owned by ONERA, Safran and Airbus while FastS is a solver focused on fast High
	Performance Computing owned by ONERA. The difference between the two solvers is
	This chapter describes the numerical methods used during this study. It has been in the implementation of the numerical schemes, turbulence models, boundary condi-
	largely shown by previous studies (Deck [32], Iovnovich & Raveh [64], Sartor & Timme tions, etc. The numerical simulations presented in chapter 3 has been conducted with
	[128] [127], Memmolo et al. [95]) that buffet instability can be simulated with a RANS elsA while the ones presented in chapter 5 has been conducted with FastS. The com-
	approach. Transonic buffet is a low-frequency instability for which the small-scale mo-putations are performed in parallel over up to 56 cores on ONERA's supercomputer
	tions, representative of turbulence, can be considered through the effect of a turbulence Sator, using 2.4 GHz Intel Xeon E5-2680 v4 processors.
	model and the mean flow. Low and high-frequencies unsteadiness are obviously linked
	with the turbulence, but in this case the target unsteady dynamics can be simulated
	without a complete computation of all the small-scale structures present in the flow. In
	this work Boussinesq's hypothesis [11] is considered to provide closure for the averaged
	Reynolds stresses and the effect of the turbulence results in an additional viscosity,
	called the eddy viscosity. RANS equations are presented in section 2.1.1 with the de-
	tails of temporal, spatial schemes and turbulence models used. Section 2.2 presents
	the global stability analysis which consists in the study of the linearised Navier-Stokes

  [START_REF] Dolling | Fifty years of shock-wave/boundary-layer interaction research: what next?[END_REF] which gives a new expression for the local contribution to the eigenvalue, similar to the definition of the wavemaker component w k , given in(2.22). It involves the local product between the direct and adjoint eigenmodes, as w k , but it is different because it establishes an equality between the local contribution and the components of the eigenvalue. Compared to(2.29), this new expression is easier to compute because it does not require to extract column-matrices.

transonic buffet Contents 3.1 Numerical simulations and comparison with experiment . 42

  .43)

	Chapter 3
	Numerical investigation on
	two-dimensional

3.1.1 Buffet onset and insight on buffet mechanism . . . . . . . . . 3.2 Global stability analysis . . . . . . . . . . . . . . . . . . . . 45 3.2.1 Mesh convergence . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Comparison stability analysis-URANS simulations . . . . . . 3.2.3 Local contribution of the flow . . . . . . . . . . . . . . . . . . 3.2.4 Variable contributions to the eigenvalue . . . . . . . . . . . . 3.3 Identification of the buffet mechanisms with a Selective Frequency Damping approach . . . . . . . . . . . . . . . . . 52 3.3.1 SFD method with localised damping . . . . . . . . . . . . . . 3.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.

4 Discussion on the physical mechanism behind transonic buffet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.5 Summary and conclusion . . . . . . . . . . . . . . . . . . . . 59

  The chapter is organized as follows. Section 3.1 shows the numerical results of the simulations and the comparison with the experiments. Then, the results of stability analysis are presented in section 3.2. The complete spectra evolution of the most unstable mode is shown for both SA+EC and k-kl turbulence models as well as the direct and adjoint mode for one selected angle of attack. In section 3.2.3, the local contribution technique explained in section 2.3 is applied on the most unstable mode found, the buffet mode. In section 3.3, the results from local contribution to stability are validated by URANS simulations with selective frequency damping. In section 3.4, all the physical models presented in the state of the art are discussed in light of the results from sections 3.2.3 & 3.3. Section 3.5 concludes the chapter with a summary of the main findings.

	The chapter presents the results of the numerical simulations computed on tran-
	sonic buffet. Steady and Unsteady RANS simulations are discussed and experimentally
	validated. The state of the art of the stability analysis on transonic buffet is reproduced
	by considering two different turbulence models. Further investigations are conducted
	on 2D buffet to enhance the understanding of the physical mechanism behind the phe-
	nomenon. Two techniques are used to localise the zone of the computational domain

strictly necessary to the instability to go living on. The two techniques are basically different: one is based on stability analysis and the other on unsteady RANS simulations.

42

Numerical investigation on two-dimensional transonic buffet

Table 3 .

 3 1: Effect of grid refinement on the values of the unstable buffet eigenvalue.

	Grid cells	Growth rate σ (s -1 ) Frequency f (Hz)
	36000	8	72
	72000	39	79
	100000	41	79

Table 4 .

 4 1: Summary of the experimental conditions for the four databases.

	Databases:	BUFET'N Co AVERT	DTP Trem. FLIRET
	Re M AC (10 6 )	2.5	2.83 -8.49 2.02 -6.25 23.5 -70.5
	M	0.82	0.78 -0.86 0.75 -0.85 0.85 -93
	α ( • )	2 -4	0 -6.5	2.5 -4.5	-0.5 -6.5
	Boundary layer tripping (x/c) 7%	7%	10%	No trip
	M AC (m)	0.22	0.3375	0.251	0.384
	M AC (m)			0.193	0.264
	b (m)	0.704	1.225	0.943	1.3167
	Sampling (Kul., acc.) (Hz)	20480	2048	2048	4096
	Anti-aliasing f ilter (Hz)	9216	921.6	750	819
	N b overlapping blocks	65	127		62
	F r. resolution (Hz)	8	2	25	2
	F r. resolution (St M AC )	0.0063	0.0024	0.023	0.0038

[START_REF] Iovnovich | Reynolds-averaged Navier-Stokes study of the shock-buffet instability mechanism[END_REF]

and Sartor et al.

[START_REF] Sartor | Stability, receptivity, and sensitivity analyses of buffeting transonic flow over a profile[END_REF]

), has not been observed. The four campaigns have not been designed to investigate the buffet offset; furthermore the tests at high M -α do not reveal the presence of buffet offset.

Different buffet onset criteria are presented. The main difference is between local and global criteria. Global criteria are common methods used in industry. They are based on the structural response of the wing or on the integral variables.

Figures 4.3

  Figure 4.7: Log-log graph of power spectral densities for BUFET'N Co test at M=0.82 in the chordwise direction at y/b=60%. (a) α=3.2 • . (b) α=3.5 • .

	(a)			(b)
	K-H					K-H
	x/c=65% x/c=70% Buffet		PSD (Pa²/Hz)	10 1 10 2	x/c=65% x/c=45% x/c=57.5% x/c=60% Buffet
	x/c=80%				x/c=75%
	x/c=85%				x/c=85%
	x/c=95%				x/c=92.5%
					x/c=98%
				10 -1		10 0
						St MAC
	-50
					St MAC
					0.39
					0.368
	0				0.346
					0.324
					0.302
					0.28
					0.258
	50 100 x/MAC(%)				0.236 0.214 0.192 0.17
	150
	200
	0	20	40	60	80	100
			y/b(%)

Table 4 . 2 :

 42 Summary of the main results for the four databases analysed.

	Database:	BUFET'N Co	AVERT	DTP Trembl.	FLIRET
	α & M :	3.5 • -0.82	3.5 • -0.82	4.3 • -0.8	3.37 • -0.85
	x/c & y/b :	80%-70%	87.5%-75%	80%-70%	85%-79%
	Onset	≈3 •	≈3 •	≈3.1 •	≈3 •
	St local c	0.24	0.22	0.26	0.27
	St M AC	0.27	0.26	0.3	0.48
	St M AC			0.23	0.33
	U C /U ∞	0.26	0.24	0.23	0.26
	It has been shown that 3D buffet appears with high frequency values in comparison

to 2D, especially for the well-established or deep buffet regime, while at the onset peaks of PSDs are not so clear. Low frequency peaks are found in some cases as in AVERT

Table 5 .

 5 1: Computational details. The size of the Jacobian matrix J is given by the product of the mesh size and the number of state variables: Mesh size ×6.cause the Jacobian matrix is less sparse. The Jacobian size increases linearly with N z and consequently the memory consumption too. Finally the presence of a sweep angle strongly increases the number of non-zero elements and the memory consumption. The eigenvalue problem is solved using up to 4000 cores on ONERA's supercomputer. The CPU time is in the range of 1800-4500 seconds per core depending on the Jacobian matrix size, as shown in table 5.1.

	5.2 Results

  The number of 3D modes n max appearing in the spectrum depends on the number of spanwise mesh planes N z . The variable n ranges within: n = 1, 2, ..., n max

  j U i,j = ∂u ∂x

	2	+	∂u ∂y	2	+	∂u ∂z	2	+	∂v ∂x	2	+		∂v ∂y	2	+		∂v ∂z	2	+
									+	∂w ∂x	2	+	∂w ∂y	2	+	∂w ∂z	2	(C.26)

  Le tremblement transsonique est une instabilité aérodynamique complexe qui apparaît sur les ailes et les profils en régime haut-subsonique pour certaines valeurs du nombre de Mach et de l'angle d'attaque. L'objectif de cette thèse est d'améliorer la compréhension de la physique de ce phénomène et, en particulier, d'expliquer l'évolution de ces caractéristiques (fréquence, amplitude d'oscillation) des profils bidimensionnels aux ailes tridimensionnelles en flèche. L'étude commence par la reproduction du travail de[START_REF] Crouch | Predicting the onset of flow unsteadiness based on global instability[END_REF]Crouch et al. ( , 2009)). Ensuite, le tremblement transsonique bidimensionnel est analysé pour mettre en évidence les zones de l'écoulement nécessaires à l'existence de l'instabilité. Cette étude a été menée numériquement en couplant deux techniques basées sur des approches différentes : l'analyse de stabilité linéaire et la simulation numérique. Ces deux approches aboutissent aux mêmes résultats : le pied du choc est identifié comme le coeur de l'instabilité, le choc et la couche limite détachée sont également des zones nécessaires à l'existence de l'instabilité. Afin de souligner les différences entre le tremblement bidimensionnel et tridimensionnel, une analyse de quatre bases des données expérimentales d'ailes tridimensionnelles est accomplie. L'analyse expérimentale identifie les valeurs caractéristiques du phénomène, telles que les nombres de Strouhal, les vitesses de convection, le seuil d'apparition du tremblement etc. Le tremblement tridimensionnel possède des caractéristiques différentes du tremblement bidimensionnel : nombre de Strouhal supérieur, bosses au lieu de pics sur la densité de puissance spectrale, amplitude d'oscillation du choc plus faible et motifs tridimensionnels dans la couche limite détachée qui sont convectés en envergure. Ces différences suggèrent des mécanismes physiques distincts. Enfin, une analyse de la stabilité globale en trois dimensions d'une aile est effectuée. Le tremblement bi et tridimensionnel apparai comme deux modes instables différents dans le spectre. Le phénomène appelé dans la littérature tremblement transsonique tridimensionnel correspond en fait à un phénomène de convection de cellules de "couche limite détachée" avec un mode global instable à fréquence nulle devenant instationnaire avec l'augmentation de l'angle de flèche.

	ETUDE DU TREMBLEMENT TRANSSONIQUE. EVOLUTION DES PROFILS
	BIDIMENSIONELS AUX VOILURES TRIDIMENSIONELLES
	RESUME :
	S
	S0	(C.39)

Three-dimensional global stability analysis

Linearised compressible Navier-Stokes equations

Bibliography

Monotonic Upwind Scheme for Conservation Laws NACA National Advisory Committee for Aeronautics NASA National Aeronautics and Space Administration NSMB Navier-Stokes MultiBlock ONERA Office National d'Études et de Recherches Aérospatiales POD

Experimental investigation on three-dimensional transonic buffet

The BUFET'N Co model (figure 4.1(a)) is composed of a swept wing attached on a half-fuselage, the sweep angle at the leading edge (LE) is 30 • and the wing is based on the supercritical OAT15A aerofoil. From root to tip, the chord varies between 0.24 m and 0.

Summary and conclusion

Evidence that the transonic buffet phenomenon on 2D aerofoils is due to a global unstable mode was first shown by Crouch et al. [START_REF] Crouch | Predicting the onset of flow unsteadiness based on global instability[END_REF]. It has been largely confirmed in the last years (Sartor et al. [START_REF] Sartor | Stability, receptivity, and sensitivity analyses of buffeting transonic flow over a profile[END_REF] and Iorio et al. [START_REF] Iorio | Direct and adjoint global stability analysis of turbulent transonic flows over a NACA0012 profile[END_REF]). In the present study, a 3D global stability analysis has been carried out on an infinite wing in transonic flow conditions. The objective was to find a link between the 2D transonic buffet global mode and the 3D transonic buffet for which frequencies are 4 to 7 times higher (Dandois [START_REF] Dandois | Experimental study of transonic buffet phenomenon on a 3D swept wing[END_REF]). The present results have shown that when perturbations are allowed to be 3D, there are actually two unstable modes: a 2D one identical to the one discovered by Crouch et al. and a 3D mode. The latter exhibits zero-frequency for an unswept wing and becomes non-zero frequency when the sweep angle increases. Its growth rate is higher than the 2D mode and its frequency increases linearly with the sweep angle up to six times the 2D one at 30 • , which explains that the buffet frequency is much higher in 3D than in 2D. This 3D mode corresponds to the buffet cell phenomenon observed by Iovnovich & Raveh [START_REF] Iovnovich | Numerical study of shock buffet on threedimensional wings[END_REF]. It is better known at low speed as the stall cell phenomenon, for which a zero-frequency unstable global mode was already found by Theofilis [START_REF] Theofilis | Advances in global linear instability analysis of nonparallel and three-dimensional flows[END_REF]. The wavenumber (βc ≈ 2π or a wavelength equal to one chord length) and the convection velocity (0.43U n∞ or 0.37U ∞ ) of this 3D mode are in very good agreement with what has been observed experimentally (Dandois [START_REF] Dandois | Experimental study of transonic buffet phenomenon on a 3D swept wing[END_REF], Sugioka et al. [START_REF] Sugioka | Experimental analysis of transonic buffet on a 3D swept wing using fast-response pressure-sensitive paint[END_REF]) and numerically (Plante et al. [START_REF] Plante | Study of three-dimensional transonic buffet on swept wings[END_REF]). The analysis of the adjoint modes shows similar regions of optimal forcing for both 2D and 3D modes. Finally, the structural sensitivities highlight the core of the instabilities which is for both modes localized near the shock foot with higher values of sensitivity in the detached boundary layer for the 3D mode.

The main conclusion of the present study is that the phenomenon called in the literature 3D transonic buffet corresponds to a stall cell convection phenomenon with a zero-frequency unstable global mode which becomes unsteady with sweep. The mode also appears to be distinct from the dynamics of the 2D mode. Hence, the analysis of the wavemaker of the 3D mode reveals that the core of the instability is nearly solely in the separated region (and only very weakly along the shock), with a maximum along the separation line. This is in contrast with the 2D mode for which the shock plays a greater role, since the wavemaker for this latter mode also exhibits strong values along the shock-wave. A future perspective of the present study is to improve the numerical 5.3 Summary and conclusion 107 method in order to analyse more complex 3D wings. It will be interesting to analyse the effect of the boundary conditions in span as well as the addition of a taper ratio and twist on the buffet frequency.

Perspectives

The present study has been able to answer some important questions about the transonic buffet phenomenon but at the same time some future works can be planned. A complete and totally accepted self-sustained model explained the 2D transonic buffet still does not exist. Good bases for the definition of the model have been here assessed.

The density maps of growth rate and angular frequency can be further analysed by precisely looking at the evolution with the Mach number and the angle of attack. Also the role of the turbulent dynamics in the onset of the 2D buffet should be deeply investigated because it appears to have a key role. The experimental comparison can be increased by the addition of more complex geometries and above all by highlighting the effect of the sweep angle. An experimental test of a 3D wing with a variable sweep angle is suggested. Then, a future perspective of the 3D stability analysis is to improve the numerical method in order to analyse more complex 3D wings. The base flow should be computed with more realistic boundary conditions and a more complex wing geometry. No-slip and subsonic/supersonic-exit boundary conditions can be considered in span. A taper ratio, twist, double sweep and different thickness aerofoils could also be considered for the geometry to evolve towards a more complex 3D wing. It would be interesting to see the effect of these parameters on the buffet frequency. More fundamentally, the present 3D stability analysis does not completely highlight whether the 3D mode is absolutely or just convectively unstable in the chord-wise direction.

If the flow is absolutely unstable, then it would be of oscillatory type with peaked frequency spectra; if not, then the flow would be more an amplifier and the frequency spectra broadband in connection with the noise environment. Further studies would help understand if on a complete and geometry complex swept wing the global mode would actually be unstable or not. Experimental results actually indicate that the frequency spectra are more broadband, which rather suggests a convectively unstable flow. Another possibility to explain the shape of the 3D transonic buffet spectra is in the variability of the buffet frequency with the chord size and sweep angle. A real complex 3D wing exhibits different values of local chord in span and of local sweep in streamwise, consequently the values of the frequency can evolve and result in more than one unstable mode of the flow. Each mode corresponds to a peak in the spectrum which in presence of strong environmental noise widens (Bagheri [START_REF] Bagheri | Effects of weak noise on oscillating flows: Linking quality factor, floquet modes, and koopman spectrum[END_REF]) resulting in a broadband bump.

Line-decomposition of the Jacobian matrix

where ϕ k is the coefficent of the adjoint eigenmode q and r † k the residual vector. By introducing (A.3) into (A.1):

The residual term vanishes by construction. Now the equation (A.3) is transconjugated and projected on the direct eigenmode q:

and the residual term is zero because of the bi-orthogonal property of the adjoint eigenmode with respect to the other direct eigenmode of the basis. The coefficient ϕ k is the contribution of the k th line-matrix L k to the eigenvalue λ. Now the main question is to establish the link between ϕ k and λ k .

The direct eigenmode problem is considered:

where the matrix-vector product L k q is equal to

Now, by identification of the left and right-hand sides in the equality (A.6), it results that

inserting in equation (A.5) it results as for equation (2.35):

Finally, line and column-decomposition result in the same values of local coefficients of the eigenvalue:

Appendix B

Comparison with cylinder

The application of the local contribution technique to a global instability and the local SFD is shown for a cylinder at Re = 60. The cylinder is a good validation case because it has largely been studied in the literature [START_REF] Giannetti | Structural sensitivity of the first instability of the cylinder wake[END_REF][START_REF] Luchini | Structural sensitivity of linear and nonlinear global modes[END_REF][START_REF] Marquet | Identifying the active flow regions that drive linear and nonlinear instabilities[END_REF][START_REF] Marquet | Sensitivity analysis and passive control of cylinder flow[END_REF][START_REF] Sipp | Dynamics and control of global instabilities in open-flows: a linearized approach[END_REF]. The convected vortices are in a slaved zone that does not have any impact on the existence of the instability. Finally if SFD is activated locally inside the core defined above, even if the filtered area is small in space, the instability is damped (figure B.3c).

Appendix C

Linearised compressible Navier-Stokes equations

The

where ρ is the density, t the time, U the velocity vector (u, v, w), p the pressure, τ the total stress tensor, g the gravitational acceleration vector, E the total energy per unit mass, q H the total heat flux vector, ν the transformed eddy viscosity. The state variables are: