Stephie Edwige

Philippe Gilotte

Jean-Luc [ Aider

Submitted

Juan Eliott Varon

Denis Guzmán Iñigo

Peter J Sipp

Jean- Luc Aider Schmid

• Experimental application of the dynamic observer to capture forced transient boundary layer dynamics

Keywords:

Real-time PIV using optical flow

This chapter introduces an original velocimetry method developed in the laboratory PMMH for five years. After detailing its algorithm functioning, the hardware and the software used are presented, especially their critical role in the frequency of the real-time measurements. The influence of different experimental and computational parameters on the quality of the instantaneous velocity fields is then investigated, using a backward-facing step flow as a benchmark.

ment de ces travaux. J'ai aussi eu l'opportunité de collaborer avec des personnes venant de divers horizons -Philippe G., Yoann et Stéphie (Plastic Omnium) ; Pierric et Stéphane (PRISME) ; Denis, Peter et Juan (ONERA et Imperial College) -qui m'ont tous considérablement apporté sur les plans scientifique et technique comme humain. Je leur en suis extrêmement reconnaissant. Je n'oublie pas non plus d'y inclure Nicolas pour m'avoir lancé dans ce grand bain il y a trois ans. La version définitive du présent manuscrit n'aurait finalement pu exister sans la lecture minutieuse et critique des rapporteurs, qu'ils en soient sincèrement remerciés, ainsi que l'ensemble du jury pour tous nos échanges constructifs.

Malgré l'isolement apparent du sujet de cette thèse, au regard du reste des thématiques traitées au PMMH, l'ensemble du laboratoire a également apporté sa pierre à l'édifice. Effectivement, du fait même de cette singularité qui consiste à réunir sous un même toit des sujets aussi riches et variés, j'y ai largement trouvé de quoi satisfaire ma soif de connaissances. Notamment en cogitant sur des problèmes de physique, d'informatique ou de plans d'expériences -Laurette, Thierry, Sylvain, Xavier-Benoît, Tahar, Olivier, Alain, Amaury, Justine et Antonios. Mais c'est aussi en discutant de tous les sujets qui nous passaient logiquement par la tête -Vincent, Clothilde, Hugo, Lukasz, Yacine, Charly, Thibault, Hadrien, Marion, Adrien, Ashraf, Matthiew, Florence, Thomas, Lucie, Marine, Jean, Gaspard, Benoît, Ali, Gab, Alex, Ludovic... (dans un ordre purement arbitraire) -, que ce soit autour d'un billard, d'un tennis de table ou d'un rock band, que j'ai définitivement apprécié de me rendre tous les jours au laboratoire. Cette ambiance agréable est en grande partie due à l'organisation humaine qui y prévaut et pour laquelle je tiens à exprimer chaleureusement ma gratitude envers Claudette, Frédérique et Philippe. Je remercie enfin tou-te-s les technicienne-s, malheureusement trop anonymes, qui facilitent tant le quotidien au laboratoire.

Pour finir, si j'ai pu être tant heureux pendant ces trois années intenses, je le dois aussi à mes proches. J'exprime donc ma plus grande affection à Juliette qui aura quotidiennement et inlassablement eu foi en moi. Pour m'avoir indéfiniment supporté et offert l'opportunité de si longues études malgré ma difficulté à leur en rendre clair le contenu, mon indéfectible reconnaissance va à mes parents, ma soeur, mes grands-parents, mes oncles et tantes, ainsi que leurs enfants. Mes pensées vont de même à Hélène, Laurent et Élisabeth. Bien entendu, et pour conclure, mes amis ont aussi énormément compté dans cette aventure par leur désir d'en savoir toujours plus sur mes avancées ("Alors, ça trouve ?") et leurs nombreuses invitations à des voyages et week-ends loin du laboratoire (bien trop déclinées hélas). Merci donc à Charles, Thomas A., Émilie, Julien L., Thomas L., Marion, Sophie, Matthieu B., Quentin, Julie, Claire M., Simon, Armand, Justine, Delphine, Flo, Jérémie, Loreleï, Manon, Lucien, Raphaël, Elsa, Roman, Arthur, Adrien, Charlotte H., Greg, Pauline, Théo, Claire R., Crevette, Rémi B., Hélène, Marco, Charlotte R., Matthieu

Résumé détaillé

Le travail réalisé pendant ces trois années au laboratoire de physique et mécanique des milieux hétérogènes (PMMH) dans l'École supérieure de physique et de chimie industrielles de la ville de Paris (ESPCI) peut se décomposer en trois parties. La première consiste en l'amélioration continue d'un outil de mesure vélocimétrique développé récemment au PMMH. La seconde partie est un exemple d'application de cet outil pour un écoulement dans un canal hydraulique. Finalement des essais réalisés en collaboration avec Plastic Omnium à la soufflerie Malavard du laboratoire PRISME à Orléans constitue le dernier volet. Tout d'abord, le chapitre 1 présente le contexte dans lequel s'inscrit cette thèse. Le phénomène d'écoulements décollés y est introduit en soulignant son omniprésence dans la nature comme dans les écoulements industriels (aérodynamique externe des véhicules, des bâtiments, écoulements autour d'aubes de turbines, aérodynamique interne dans des tuyaux...). Comme de tels écoulements sont soit nuisibles (sources de vibrations et de bruit aéroacoustique, augmentation de la traînée, réduction de la portance), soit avantageux (efficacité des mélanges), les enjeux associés à leur compréhension et à leur maîtrise sont rappelés.

Le chapitre 2 explique le principe de fonctionnement d'un capteur visuel mesurant en temps réel (∼ 10 ms) des champs denses de vitesse des écoulements. Appartenant aux techniques de vélocimétrie laser par image de particules (PIV), cette méthode expérimentale est basée sur un algorithme de flot optique issu du domaine de la vision par ordinateur. Les améliorations qui y ont été successivement apportées sont exposées, avec notamment la mise en place d'un dispositif pouvant mesurer des vitesses rencontrées en soufflerie industrielle. Les paramètres de l'algorithme sont finalement testés afin d'établir un ajustement correct entre précision des résultats et rapidité d'exécution. Pour caractériser finement les écoulements, des grandeurs sont également dérivées des champs de vitesse. Leurs calculs sont détaillés dans le chapitre 3.

Une première application de la PIV par flot optique est illustrée par le chapitre 4 dans le cadre de l'identification de système. Un algorithme d'apprentissage récemment conçu pour construire un modèle réduit d'une simulation de couche limite sur plaque plane est en effet mise en oeuvre expérimentalement. Les mesures sont réalisées dans un canal hydrodynamique au PMMH et laissent une grande liberté dans le choix des données d'entrée du système. Les données de sortie sont calculées par décomposition en modes propres orhogonaux (POD) des champs de vitesses. Le modèle retenu in fine permet bien de prédire la dynamique de la transition de la couche limite laminaire vers la turbulence. Enfin, le chapitre 5 donne les caractéristiques moyennes et instationnaires d'un sillage pleinement turbulent derrière le corps de Ahmed, une géométrie modélisant une voiture très simplifiée. La pression au culot, les vitesses de l'écoulement et les forces aérodynamiques agissant sur le corps sont mesurées. Un comportement bimodal latéral apparaît clairement pour différentes grandeurs : pression à la paroi, vitesse dans la zone de recirculation, forces de dérive. Une étude du sillage avec des outils issus de la théorie des systèmes dynamiques est proposée dans le chapitre 6. Des indices de l'existence d'un attracteur étrange faiblement chaotique dans la dynamique du sillage sont exposés, tout en soulignant le caractère stochastique de cette dernière. Ce travail est conclu dans le chapitre 7 par de nombreux essais de contrôle de l'écoulement à l'aide de micro-jets discontinus installés sur le bord supérieur du culot. La symétrisation du sillage est atteinte pour un soufflage pulsant à la fréquence correspondant au lâché tourbillonaire. Une loi de contrôle réactif consistant à suivre et forcer la recirculation grâce à la PIV temps réel est exécutée avec succés.

Thesis abstract

The chapter 1 introduces the separated flows, ubiquitous processes in nature and industrial systems, such as diffusers, airfoils, air conditioning plants and moving vehicles. As the separation can strongly influence the performances of such devices, investigating their dynamics and their possible control is of great interest.

The chapter 2 presents a new velocimetry method developed in the laboratory PMMH for five years. After explaining its algorithm functioning, the hardware and the software used are detailed. The different parameters to obtain proper velocity fields are investigated for a well known flow: the backward-facing step (BFS). Quantities which can be extracted in real-time (RT) from the velocity fields are discussed in chapter 3. These calculations are fundamental to use the camera as a visual sensor in a closed-loop experiment. Some of these quantities are related to specific flow like the recirculation area, characteristic of the wakes. Other are intended to detect structures like the Q-criterion. All these tools are implemented on the GPU and are illustrated via measurements in the BFS flow.

In chapter 4, a reduced-order model method well suitable for amplifier flows, the dynamic observer, is presented. After applying to the boundary layer over a flat plate, a closed-loop control based on the flow reconstruction is investigated.

In chapter 5, we will first summarize the cutting-edge knowledge about the flow around a simple car model. After introducing the experimental setup, the natural flow is studied. In chapter 6, the 3D full turbulent wake dynamics can be characterized by the single trajectory of its projected barycenter. After recovering a classic chaotic pattern, its inner characteristics are analyzed more thoroughly, by evaluating, in particular, the family of random process to which it belongs, the largest Lyapunov exponent of the system and the correlation dimension. The chapter 7 presents a closed-loop control of the turbulent wake behind the square-back Ahmed body described in chapter 5 and in chapter 6. After introducing the micro-jets system, different control strategies are tested.

Communications

Most of the findings obtained during the thesis have been presented at various international conferences and national meetings:

• 10 th Pacific Symposium on Flow Visualization and Image Processing June 15-18 CHAPTER 1

Introduction to flow control

The global context of the thesis is briefly presented. After introducing the separated flows, different methods to control them are exposed. It shall be mentioned that most of the previous works on the subjects studied during this thesis are reviewed in the introduction of each other chapter.

The long-lived greenhouse gases in the atmosphere followed a cyclic evolution over long periods of several thousands years as displayed in Figure 1.1(a) [START_REF] Barnola | Historical CO 2 record from the Vostok ice core[END_REF][START_REF] Ruddiman | A methane-based time scale for vostok ice[END_REF][START_REF] Schilt | Glacial-interglacial and millennial-scale variations in the atmospheric nitrous oxide concentration during the last 800,000 years[END_REF]. However, since the Industrial Revolution, their concentration has exponentially increased until reaching new maxima as shown in Figure 1.1(b) [START_REF] Macfarling Meure | Law Dome CO 2 , CH 4 and N 2 O ice core records extended to 2000 years BP[END_REF]. Such a rupture in the natural cycle of greenhouse gases has led to the current rapid climate change and has anthropic origins. Indeed, with other indicators (geochemical signatures, extinction rates. . . ), it constitutes pieces of evidence of the long-time impacts of the human activities on Earth's ecosystems, geology and climate (Waters et al., 2016), to the point where the today's epoch is now called Anthropocene [START_REF] Monastersky | Anthropocene: The human age[END_REF]. This is the reason why it is interesting to analysis the main causes of greenhouse gases emissions in order to prioritize future efforts to reduce them.

The transport sector contributes currently to a quarter of the global CO 2 emissions, which amounts to 32.3 Gt, as displayed in Figure 1. 2(a). Road vehicles are the most responsive to The main other reason to reduce the fuel consumption of a vehicle is economic. Most of the clients take indeed into account the fuel cost when buying a car and they will consequently look rather for efficient vehicles. This pushed car manufacturers to define their own standards like the Corporate average fuel economy (CAFE) to face the oil crisis in the seventies (see Figure 1.3). Modern standards are finally created by states as summarized in Figure 1.4 to limit the CO 2 emissions, which are verified through official tests like the Worldwide Harmonized Vehicles Test Procedure (WLTP). To comply with these regulations the manufacturers have to improve the vehicle engine efficiency which is estimated at 40% for conventional gasoline cars [START_REF] Carpenter | Road transportation vehicles[END_REF][START_REF] Roura | How energy efficient is your car?[END_REF]. Developing kinetic energy recovery system and lightweighting the vehicle are complementary ways to increase the fuel economy.

Another solution involves improving the car aerodynamic efficiency. At a constant speed on a highway, denoted U ∞ and called the free stream velocity -around 110 km/h -a car is indeed essentially subjected to the aerodynamic drag F x drawn in Figure 1.5, the rolling resistance being most important at lower speeds. This force is characterized through the aerodynamic coefficient C x , defined by

C x = F x 1 2 ρU 2 ∞ S x , (1.1)
where S x is the frontal area. Thus, reducing the car C x leads to save fuel consumption. Two other aerodynamic forces are evaluated: the lift F z and the derive F y .

Figure 1.6 -Drag contribution on a personal car according to PSA [START_REF] Jermann | Eclatement tourbillonnaire dans le sillage turbulent d'un véhicule générique[END_REF].

The parts of a vehicle contributes differently to the total aerodynamic drag as summarized in Figure 1. [START_REF] Jermann | Eclatement tourbillonnaire dans le sillage turbulent d'un véhicule générique[END_REF]. The present thesis focuses on the rear part where separation of the flow occurs, which will be detailed further.

Research at PMMH

Consequently, transport domain is more than ever an important subject of research. The field in fluid dynamics which especially aims at reducing the energy consumption or improving the energy transformation is the flow control. This is the reason why Plastic Omnium Auto Exterior (PO), worldwide supplier of several car manufacturers, and the laboratory Physique et Mécanique des Milieux Hétérogènes (PMMH) have decided to collaborate. PO can indeed produce small devices, like micro jets embedded in personal cars, modifying the wake behind it, while PMMH develops a visual sensor, allowing the capture in real time of the flow around various geometries.

The present thesis is part of this collaboration as a continuation of the previous works of Gautier (2014) who developed a real time PIV setup for a water channel. By adapting this setup to a wind tunnel facility, the purpose is to obtain a reactive control based on PIV measurements to manipulate a three-dimensional (3D) fully turbulent wake. This work benefits from a continuous 17-years study on flow control done at the PMMH laboratory as well as from recent experiments made by PO. The successive PhD students contributors over the years to this research are: 2011-2014 Y. Eulalie, supervised by I. Mortazavi (CNAM) and P. Gilotte (PO), on the effects of micro-jets on the wake behind a simple car model (Eulalie, 2014).

Separated flows 1.2.1 Triggering a flow separation

By introducing the boundary layer concept, [START_REF] Prandtl | Fluid motion with very small friction[END_REF] enabled to better understand how a flow interacts with a solid, as well recalled by Anderson Jr. (2005). This layer makes indeed the link between the free stream, far from the solid influence, and the flow at the solid surface. Developing over the body, this region of the flow is submitted to the viscous effects due to the presence of the body, creating strong velocity gradients. Depending on the shape and the orientation of the body, the flow will accelerate and decelerate, producing pressure gradients.

If the boundary layer encounters an adverse pressure gradient (the pressure increases along the body surface), the flow momentum inside the boundary layer decreases until the velocity near the wall cancels. The flow separation occurs, producing a recirculation bubble from the location where the velocity is negative as displayed in Figure 1.7. Downstream of the recirculation bubble, the flow will reattach where the adverse pressure gradient is not strong enough to maintain the separation. The bubble is thus delimited by the separatrix surface, joining the lines of separation and of reattachment. The bubble exists to balance the kinetic energy from the free stream into the boundary layer. Even if the topology of a separated flow highly depends on the body geometry, two categories can be distinguished. The first one contains the flow along a solid, concerned in chapter 2, chapter 3 (backward-facing step) and chapter 4 (flat plate). The second kind of separation implies two interacting shear layers produced by a bluff body, which chapter 5, chapter 6 and chapter 7 deal with. For a given body shape, the separation triggering is also determined by the state of the flow via the Reynolds number

Re L = L × U ∞ ν , (1.2)
where L is a characteristic length of the {flow,body} system and ν the kinematic viscosity of the fluid. The Reynolds number represents the ratio between inertial forces and viscous forces. For instance Re L ∼ 10 6 for a car on a highway.

Dynamics generated by the separation

The lines of separation and of reattachment fluctuate around mean positions, except when the separation line is forced by geometry discontinuities (straight edge). This unsteady behavior is due to different instabilities interacting with each other, like the Kelvin-Helmholtz one and the vortex shedding one. In the last part of chapter 2 and in chapter 3, most of these instabilities are detailed and discussed through the backward-facing step flow, a benchmark for the study of separated flows [START_REF] Aider | Large-eddy simulation study of upstream boundary conditions influence upon a backward-facing step flow[END_REF].

The dynamics differ strongly from an instability to another. To make comparisons between them, each dynamic is associated to a dimensionless frequency (or range of frequencies) called the Strouhal number,

St L = L × f U ∞ , (1.3)
where f is the characteristic frequency of the unsteady behavior. It allows for instance to experimentally locate their complex spatial structure by analysing the power spectrum of the velocity measured at different positions in the flow.

Flow control

Passive Active

Open-loop Closed-loop Adaptive Model-based Machine Learning Self-adaptive Flow separation strongly affects the aerodynamic in ground vehicles applications (cars, trucks, trains), as well as wind turbines, air vehicles (airplanes, helicopters, launchers) and turbomachineries. For example, this phenomenon is responsive for most of the drag experienced by ground vehicles and for the loss of lift of an airplane (stall). Nonetheless, it may greatly improve the mixing efficiency between fluids [START_REF] Ottino | Mixing, chaotic advection, and turbulence[END_REF], like air and fuel in a combustion engine. It is thereby of interest to properly modify the flow behavior regarding the desired effect. As depicted in Figure 1.8, control methods are classically classified into two groups (Gad-el Hak, 1989; [START_REF] Choi | Control of flow over a bluff body[END_REF]: passive control and active control. A few examples will be given here as illustrations but more are available in chapter 7.

Passive control

The idea behind passive control is to optimize the shape of a body to increase aerodynamic performance. Beaudoin & Aider (2008) A passive control strategy is usually defined for a specific flow configuration and is often associated to the idea that it cannot adapt to possible changes in the flow state. Nonetheless, the selfadaptive method, based on biomimetics [START_REF] Bhushan | Biomimetics: lessons from nature-an overview[END_REF], seems to allow a kind of natural reactive control for passive approach. For instance, replacing the rigid blades of a small wind turbine by well designed elastic ones greatly improves its energy conversion [START_REF] Cognet | Bioinspired turbine blades offer new perspectives for wind energy[END_REF]. The blades will indeed deform with respect to the incoming wind.

Active control

Active control consists in adding momentum into the flow to modify its characteristics by bringing energy through a device called actuator. Cattafesta III & Sheplak (2011) reviewed the variety of these devices used in active flow control like the fluidic, the moving surface and the plasma actuators. The actuator activity consequently requires extra costs in energy which have to be lower than the energy saved thanks to the control [START_REF] Seifert | Closed-loop active flow control systems: actuators[END_REF].

The control law driving the actuator(s) is defined according to two approaches: the open-loop control and the closed-loop control. The former determines an offline control law. For example, by putting a cylinder into rotation, its mean wake is modified in comparison to the immobile configuration [START_REF] Tokumaru | Rotary oscillation control of a cylinder wake[END_REF][START_REF] Protas | Drag force in the open-loop control of the cylinder wake in the laminar regime[END_REF][START_REF] Thiria | The wake of a cylinder performing rotary oscillations[END_REF]. This can improve the aerodynamic performances, depending on the rotation frequency.

Taking into account the state of the flow by measuring some of its characteristics in real time via sensors enables to design closed-loop control laws. The actuator will consequently force the flow regarding any noticed change, ensuring better robustness and efficiency than the open-loop approach. This requires to have access to some values of the flow like the velocity or the pressure, and/or of forces acting on the body. This is the reason why, in a laboratory environment, a real time PIV setup is of great interest. The control law may be based on a model of the flow and of the forcing, called the plant and the compensator respectively [START_REF] Sipp | Closed-loop control of fluid flow: a review of linear approaches and tools for the stabilization of transitional flows[END_REF]. A direct approach consists in considering the full dynamics of the flow. As it implies a full knowledge of the flow, this restricts this method to the simplest flows. If the flow dynamics can be related to known physical systems, a physic-based model can be built.

A recent approach consists in obtaining model-free control laws through machine learning (Duriez et al., 2017). A large variety of control laws are evaluated regarding the minimization of a cost function. The best laws are used to design new ones, tested again until converging to the best one. Even if this approach allows to get rid of the nonlinearity problem inherent to the flow dynamics, it does not ensure any stability robustness due to its functioning. Gautier et al. (2015) successfully implemented a genetic algorithm which created a control law reducing the recirculation area behind a backward-facing step.

Conclusion and discussion

The As the worldwide population exponentially growths for years, doubling in 50 years from midtwentieth as shown by Figure 1.11, needs for transport also increase (as well as the expectations for better life quality). No radical change in this trend is expected for the next 80 years since the total population will still increase according to the predictions of United Nations (2015).

15 years ago, fuel economy for tractor-trailer trucks were already estimated between 6.5 and 16.5% thanks to simple passive devices (cross-flow vortex trap, vortex strake, under-body convergent duct) allocated in the regions responsible for the highest aerodynamic drag (Wood & Bauer, 2003). Despite of all these past discoveries, Figure 1.2(b) reveals that all possible drag reduction devices invented after years of research did not instantaneously change the rising CO 2 emission due to transport. Therefore it shall be kept in mind that, even if flow control is a good option to optimize aerodynamics performances in the long run, firstly the application of a working device takes years after its discovery and secondly diminishing CO 2 concentration in the atmosphere does not depend only on it but also on political and societal decisions.

The PMMH laboratory develops a real time visual sensor for laboratory use, allowing new experimental strategies for closed-loop control in automobile applications via micro jets designed by PO. The visual sensor will be used for different kinds of separated flows.

Introduction

Velocimetry techniques

Only optical techniques measuring a velocity field will be discussed in the present section. Even if local sensors like laser Doppler velocimetry or hot-wire have often better temporal resolution and enables to store less data, the different studies during this thesis require quantitative information on spatial structure only given by velocity fields. For each of the presented method in this section, the flow is seeded by small particles acting like tracers of the fluid movements.

FFT-PIV and derivates

Figure 2.1 -Classic 2D-2C PIV setup and process (Raffel et al., 2007). Two-dimensional two-components (2D-2C) velocity fields have been commonly computed via the particle image velocimetry (PIV) technique for more than 30 years [START_REF] Meynart | Digital image processing for speckle flow velocimetry[END_REF]. This technique is reviewed by [START_REF] Adrian | Twenty years of particle image velocimetry[END_REF] and completely detailed by Raffel et al. (2007). Briefly, the principle consists in illuminating a plane with a laser sheet generated by a laser beam passing through a cylindrical lens to highlight only the particles, as shown in Figure 2.1. Two successive snapshots are recorded by a camera at time t and t = t + dt. Each snapshot is then divided into sub-windows capturing an ensemble of particles. Each sub-window of the snapshot t is cross correlated with corresponding sub-windows of the snapshot t . The resulting maximum peak in the correlation should correspond to the displacement (amplitude and direction) of the followed group of particles.

Measurements accuracy requires optimized experimental conditions such as the optical devices [START_REF] Kähler | On the resolution limit of digital particle image velocimetry[END_REF], the size and the density of the particles [START_REF] Keane | Optimization of particle image velocimeters. I. Double pulsed systems[END_REF], and the interrogation window size [START_REF] Foucaut | PIV optimization for the study of turbulent flow using spectral analysis[END_REF]. The accuracy may be improved via advanced methods like the multiple pass interrogation combined with grid refining schemes [START_REF] Scarano | Iterative multigrid approach in PIV image processing with discrete window offset[END_REF], which shift the sub-windows and iteratively decrease their size. Even so, the accuracy is limited when dealing with flows including strong gradients and vorticities, especially turbulent flows. A solution consists in iteratively deforming the sub-windows using initial displacement estimations [START_REF] Scarano | Iterative image deformation methods in PIV[END_REF], improving the cross correlation results by better following the particles.

To save post-processing time, the cross correlation is computed using a fast Fourier transform (FFT). Some advanced methods previously mentioned may even save more time for time-resolved images data set [START_REF] Shi | Improvements to time-series TR-PIV algorithms using historical displacement and displacement variation information[END_REF]. Depending on the sub-window definition and the snapshot size, computing one velocity field may take indeed from several seconds to a few minutes. When there are thousands images to process, this may result in days of calculation, depending on the computing power. When dealing with a few days experimental campaign, it becomes necessary to use large computing facilities.

More components and dimensions can be reach through derived PIV techniques. For example, for stereo-PIV it is necessary to add a second camera or even more [START_REF] Fouras | Target-free stereo PIV: a novel technique with inherent error estimation and improved accuracy[END_REF] recording the same PIV plane at a different angle to compute the third component of the velocity field (2D-3C) via geometric reconstruction [START_REF] Arroyo | Stereoscopic particle image velocimetry[END_REF]. Continuous improvements are applied to this method to compensate the various distortions introduced by the extra optical devices [START_REF] Soloff | Distortion compensation for generalized stereoscopic particle image velocimetry[END_REF]Wieneke, 2005;[START_REF] Fouras | A simple calibration technique for stereoscopic particle image velocimetry[END_REF]. Illuminating and recording a volume rather than a plane enables to reach the third spatial dimension of the velocity fields by decomposing the cross correlation function at different depths [START_REF] Fouras | Volumetric correlation PIV: A new technique for 3D velocity vector field measurement[END_REF]. The volumetric correlation PIV designation has been consequently given to this technique. Finally, 3D-3C velocity fields are computed either through holographic PIV [START_REF] Barnhart | Phase-conjugate holographic system for high-resolution particle-image velocimetry[END_REF][START_REF] Hinsch | Holographic particle image velocimetry[END_REF], or through volumetric three-component velocimetry [START_REF] Pereira | Defocusing digital particle image velocimetry: a 3-component 3-dimensional DPIV measurement technique. Application to bubbly flows[END_REF][START_REF] Pereira | Two-frame 3D particle tracking[END_REF], or through volumetric reconstruction of the particles distribution in tomographic PIV (tomoPIV) [START_REF] Elsinga | Tomographic particle image velocimetry[END_REF].

PTV

Unlike the Eularian approach of the PIV, particle tracking velocimetry (PTV) consists in tracking single particles between two or more successive images as illustrated by Figure 2.2. This Lagrangian technique requires low-seeding densities to distinguish each particle trajectory. Resulting in sparse estimation of flow displacements, the tracking can be realized either in 2D or in 3D [START_REF] Dracos | Particle tracking velocimetry (PTV)[END_REF]. The latter, similar to the tomoPIV technique, is limited in resolution by the low tracers density requirement [START_REF] Maas | Imaging Measurement Methods for Flow Analysis: Results of the DFG Priority Programme 1147[END_REF] As an alternative to the FFT-PIV, a first optical flow algorithm has been specifically designed at the LIMSI 1 to process images intended for PIV [START_REF] Quénot | Particle image velocimetry with optical flow[END_REF] and even for sPIV recently [START_REF] Faure | Stereoscopic PIV using optical flow: Investigation of a recirculating cavity flow[END_REF][START_REF] Douay | Stereoscopic PIV using optical flow: application to a cavity recirculation[END_REF]. Based on dynamic programming, this method computes iteratively the flow displacement over strips of the snapshots. At each iteration, the strips become thinner and switch between vertical and horizontal alignments. Another optical flow algorithm has also been optimized for PIV at the IRISA/INRIA 2 [START_REF] Corpetti | Fluid experimental flow estimation based on an optical-flow scheme[END_REF]. This one derives from the continuity equation and a second-order regularizer [START_REF] Corpetti | Dense estimation of fluid flows[END_REF]. Even if both algorithms give dense velocity fields (one vector per pixel), it seems that their spatial resolution is not better than advanced FFT-PIV (Stanislas et al., 2008) and their computation time is at least of the same order as standard FFT-PIV. this algorithm to fast post-process their snapshots for a turbulence jet flow study and for an investigation of the interaction between a shock wave and a turbulent boundary layer respectively.

It shall be mentioned that optical flow applications in velocimetry via the local approach concerns also PTV, enabling a better accuracy in the near-wall measurements when properly tuning the algorithm settings [START_REF] Miozzi | Particle image velocimetry using feature tracking and Delaunay tessellation[END_REF][START_REF] Miozzi | Performances of feature tracking in turbulent boundary layer investigation[END_REF].

Functioning

The method used to compute the velocity fields in the following experiments is based on the Lukas-Kanade optical flow algorithm called FOLKI-GPU, implemented first at ONERA on a graphic processor unit (GPU) by Champagnat et al. (2011). The original code has been improved at the PMMH laboratory by Gautier (2014) and during the present thesis to satisfy experimental constraints due to the material (camera, laser) and the environment (water, air).

FOLKI method

t t' i j I t I t' k N(m) [W t (I t (k),V(k))-W t' (I t' (k),V(k))] 2 N(m) m W t W t'
to minimize

Feature Optical flow FFT-PIV

Matching criteria minimize SSD maximize cross correlation Matching method

Gauss-Newton method direct searching Calculation mode parallel matrixwise sequential pointwise Dense fields output yes (1 vector/px) no Sub-pixel interpolation at each pass at final pass The FOLKI algorithm consists in estimating at each pixel m the intensity displacement which minimizes the sum of square differences (SSD) between the intensity over a warped interrogation window (IW) centred in m at time t and the intensity over the warped IW at time t = t + dt as shown in Figure 2.3. The iterative least-square Gauss-Newton method is particularly well suited to solve this minimization problem on condition of initializing the computation with a zero displacement. As real displacements are far from zero, the images resolution is deliberately degraded via a Gaussian pyramid reduction (see Figure 2.6 later). This creates several copies of the original images with lower space resolutions, making the intensity displacements consequently reduced, close to zero. The iterative calculation is thus applied to the most reduced images (top of the pyramid) first. The resulting displacement is then used as the initial estimate at the higher resolution in the pyramid (lower level). The process is repeated until recovering the initial image resolution (base of the pyramid), giving thus the final displacement. Table 2.1 compares the major features between this optical flow algorithm and FFT-PIV.

Like for PIV-FFT, a common pre-processing of the raw snapshots is required before applying the optical flow algorithm to eliminate the influence of illumination inhomogeneities (Gautier & Aider, 2015). The standard local equalization of the image intensity Ĩ is thereby obtained in each pixel m(i, j) from the zero normalized sum of square differences (ZNSSD)

Ĩ(m) = I(m) -Ī(m) Ī2 (m) -I 2 (m) , ( 2.1)
where Ī(m) is a local mean for a chosen radius of 5 pixels around m.

Baker & Matthews (2004) give a large summary of the different approaches used in optical flow to solve the displacement computation: the compositional and the additive algorithms. As the additive ones only are implemented in the present code, only these methods are described here.

Forward additive method

According to [START_REF] Lucas | An iterative image registration technique with an application to stereo vision[END_REF], the most direct method consists in, for each pixel m, looking for the best displacement V (m) minimizing the SSD between the intensity field around m taken at time t, denoted I t , and the second one taken at time t = t + dt, denoted I t , and shifted over V (m)

k∈N (m) (I t (k) -I t (k + V (m))) 2 , ( 2.2)
where N (m) is a square IW centred in pixel m and of side 2R IW + 1.

The Gauss-Newton iterative descent is typically designed to solve Equation 2.2, I t (k) -I t (k + V (m)) being the residual. The estimated displacement at the iteration step i, V i (m), is thus calculated such as V i (m) ∼ V (m). Le [START_REF] Besnerais | Dense optical flow by iterative local window registration[END_REF] explain that V (m) ∼ V (k) can be assumed at each pixel k in the IW to ensure fast convergence. Referred as the constant constraint, this assumption leads to k∈N (m)

(I t (k) -I t (k + V i (k) + V (m) -V i (k))) 2 , (2.3)
which enables a first-order expansion of Equation 2.

3 around k + V i (k), denoting ∆V (k) = V (m) -V i (k), k∈N (m) (I t (k) -I t (k + V i (k)) -∇I t (k + V i (k)) T ∆V (k)) 2 .
(2.4)

As minimizing Equation 2.4 amounts to its partial derivative cancellation with respect to ∆V (k), the equation to solve becomes k∈N (m)

∇I t (k + V i ) • ∇I t (k + V i ) T ∆V -∇I t (k + V i )(I t (k) -I t (k + V i )) = 0, (2.5)
where the k precision is dropped for V i (k) and ∆V (k) for convenience. The operator H =

k∈N (m) ∇I t (k + V i ) • ∇I t (k + V i )
T is defined as the Gauss-Newton approximation to the Hessian. Finally V (m) is recovered by replacing ∆V in Equation 2.5

V (m) = H -1 k∈N (m) ∇I t (k + V i )(I t (k) -I t (k + V i ) + ∇I t (k + V i ) T V i ).
(2.6)

The computed displacement V (m) will then be used as a better estimation V i+1 (m) at the next iteration until converging. The dependency of V i on k and not on m is crucial to replace the integration over the IW, k∈N (m) , by a convolution over the entire snapshot, implying that the IW is warped instead of being shifted. Each warping value changes at each iteration until converging, avoiding especially image interpolations per pixel. Concretely it means that parallel computing is well suited to solve this minimization problem since each step of the iteration is simultaneously done for each pixel m. Figure 2.4 shows how pixels of the green IW are modified in the first snapshot t to match the second snapshot t intensity at the best.

t t'

Inverse additive method

As H and ∇I t (k + V i ) depend on V i in Equation 2.6, their values have to be computed at each iteration of the Gauss-Newton descent. In order to save computation time [START_REF] Baker | Lucas-Kanade 20 years on: A unifying framework[END_REF] give another approach based on the works of [START_REF] Hager | Efficient region tracking with parametric models of geometry and illumination[END_REF] and [START_REF] Bouguet | Pyramidal implementation of the Lucas-Kanade feature tracker: description of the algorithm[END_REF], consisting in warping the first snapshot I t over ∆V and the second one over V i k∈N (m)

(I t (k -∆V ) -I t (k + V i )) 2 .
(2.7)

Following the same steps as in the forward additive method and still assuming V (m) ∼ V i (k) for k ∈ N (m), we finally obtain

V (m) = H -1 k∈N (m) ∇I t (k)(I t (k) -I t (k + V i ) + ∇I t (k) T V i ), (2.8)
where the Hessian is H = k∈N (m) ∇I t (k) • ∇I t (k) T . Now H and ∇I t (k) need to be computed only once. Compared to the previous method, the gradient of image at t , ∇I t (k + V i ), is here replaced by the gradient of image at time t, ∇I t (k). With this method, the pixels in the IW of the second snapshot are warped to match the correct intensity of the first snapshots as illustrated in Figure 2.4 through the bottom red IW.

Symmetric additive method

Zhao & Sawhney (2002) and [START_REF] Averbuch | Fast motion estimation using bidirectional gradient methods[END_REF] propose the symmetric approach consisting of moving both IW by V (m)/2. In this case the problem amounts to the minimization of k∈N (m)

(I t (k - V (m) 2 ) -I t (k + V (m) 2 )) 2 .
(2.9)

Following the same steps as in the previous methods and denoting the average of the spatial intensity gradient between both snapshots as

∇I(k) = 1 2 (∇I t (k -V 0 2 ) + ∇I t (k + V 0 2 )), V (m) = H -1 k∈N (m) ∇I(k)(I t (k - V 0 2 ) -I t (k + V 0 2 ) + ∇I(k) T V 0 ). ( 2

Gaussian pyramid

Solving Equation 2.10 implies to properly initiate the iteration (step i = 0) and ensuring small displacement increment to validate the first-order expansion giving Equation 2.5. In order to enable the convenient estimation V 0 = 0, the displacement inside the IW has to be as small as possible. Thus, before computing V , a reduction of the snapshot is performed in the algorithm through a Gaussian pyramid reduction, the details of which are given by [START_REF] Burt | The Laplacian pyramid as a compact image code[END_REF]. Introduced in the optical flow framework by [START_REF] Enkelmann | Investigations of multigrid algorithms for the estimation of optical flow fields in image sequences[END_REF] and completely implemented by [START_REF] Bouguet | Pyramidal implementation of the Lucas-Kanade feature tracker: description of the algorithm[END_REF], this technique degrades the space resolution of a picture to reduce its dimension. It consists in locally smoothing distinct squares of four pixels into one pixel, which divides the size of the entire snapshot by four. The process can be repeated to make far particles artificially close. This creates the successive levels l of a pyramid with the original snapshot as its base (l = 0) and the last transformed image as its top (l = n level ), as illustrated in Figure 2.6.

At each increase in the pyramid, the displacement at the level l becomes

V l = V /2 l , (2.11)
where V is the original displacement. At each level l, the Gauss-Newton iteration starts from the displacement resulting from the previous level V l 0 = V l+1 . It then converges to the resulting displacement V l , which is subsequently used as the first guess of the lower level l -1:

V l = V l-1 0 .
The last level l = n level has to be chosen such as max(V n level ) ∼ 1 -2 px in order to obtain the proper initialization of the whole algorithm with V n level 0 = 0. The algorithm goes on until reaching the first level j = 0.

Real-time computation

As previously mentioned, FOLKI-GPU has been improved by Gautier & Aider (2015) to obtain real-time, meaning high frequency, computation of instantaneous experimental velocity fields. The interests of online computation are multiple:

1. Fast post-processing, 2. Parametric studies achievable, 3. Possibility to avoid storing large data sets of images, 4. Camera used as a non-intrusive sensor in a close-loop application.

The present thesis focuses mainly on the last point for its application on fully turbulent flows (chapter 7). Real-time PIV has already been reported for FFT-PIV by reducing the processed images [START_REF] Siegel | Real-time particle image velocimetry for closed-loop flow control studies[END_REF]Willert et al., 2010) or using Field-Programmable Gate Arrays (FPGA) (Yu et al., 2006;[START_REF] Munoz | Real-time particle image velocimetry based on FPGA technology[END_REF], up to 15 fps at Re = 1.5 × 10 4 . Nonetheless, these methods seem unable to provide dense data fields at higher Reynolds numbers.

The global data flow of the optical flow algorithm is summarized in Figure 2.7. It starts with the images readout: the camera sends the snapshots to the frame-grabber. These images are then stored in the random-access memory (RAM) before being copied in the GPU memory. The resulting velocity fields are finally delivered into the RAM. Outside the GPU, data are handled by the central processing unit (CPU). Images can be written from the RAM on a solid-state drive. This optional step is critical when dealing with large images. The camera and the GPU used in the different setups are detailed below. The management of the devices together with the optical flow algorithm is made possible through a LabVIEW program explained hereafter.

Camera-laser association

To reach high-speed computation of the instantaneous velocity fields, the camera must stream snapshots of intensity fields at high frequency to the RAM. Unfortunately, most of the cameras made for PIV (inter-frame time, spatial resolution, sensitivity...) store the entire set of acquired images in an internal buffer before transferring them to the computer in one go. Capturing rapid flow dynamics requires indeed to quickly free the camera sensor and this is well done by adding directly rapid memory into the camera. Nevertheless, some cameras dedicated to velocimetry are able to stream and two configurations have been designed according to the facility where the flow is studied: a hydrodynamic channel or a wind tunnel.

Hydraulic configuration

Regarding the velocities encountered in a water channel (∼ 10 cm.s -1 ), the time exposure of the camera sensor is of the order of magnitude ∼ 1 ms. The plane to study is illuminated by a laser sheet created by the 8 W continuous laser Genesis MX-6185 (Coherent, USA) operating at the wavelength 532 nm. The camera acA 2000-340km (Basler, Germany) records pictures of the illuminated particles [Figure 2.8(a)]. The sampling rate is thus related to the frame rate of the camera. An optical filter selecting only the laser frequency is mounted on the objective to reduce the light noise.

Aero configuration

In wind tunnel facilities a common PIV setup is composed of a pulsed laser and a double-frame camera, both triggered by a synchronizer. Due to the higher velocities inside a wind tunnel (∼ 10 m.s -1 ), the capture duration has to be shorter (∼ 10 ns) to avoid the spread of the particles position. This requires to illuminate the particles with high energy laser pulses (∼ 100 mJ) at two consecutive close moments separated by ∼10 µs. Double-frame cameras are specifically manufactured to record two images so close in time, by working synchronously with the pulsed laser, as illustrated by Figure 2.8(b). For other cameras, the frame straddling method enables also to capture two snapshots in a similar way but this requires a very high frame rate capability. Already available, the double-frame camera PowerView Plus 4MP (TSI, USA) has The main advantage of the hydraulic setup is the possibility to process successive images one by one, leading to a computing speed equal to the snapshots acquisition frequency. In contrast, the velocity fields computation for the aero configuration is limited to half of the double-frame camera speed. The characteristics of the used cameras and the sizes of the recorded snapshots are summarized in Table 2.2. The dimensions and the bit precision determine the size of the images. In the present case the TSI camera generates heavy data comparing to the Basler one 3 . For both configurations, images are then transferred to the frame-grabber device NI PCIe-1433 (National Instruments, USA) through Camera Link (CL) cables (510 to 600 Mb.s -1 ). 3. It was impossible to modify the bit depth of the TSI camera via LabVIEW due to an unresolved miscommunication between the device and the software despite the help of Illunis and National Instruments engineers as explained hereafter.

Cameras

Graphic Processor Unit

Optical flow algorithms are particularly well suited to be implemented on a GPU due to their highly parallel computation as demonstrated by [START_REF] Marzat | Real-time dense and accurate parallel optical flow using CUDA[END_REF]. They indeed computed 640 px × 480 px intensity displacement fields at 15 Hz by coding an algorithm as the one presented hereinbefore. The emergence of accessible programming-language dedicated to the GPU, such as the Compute Unified Device Architecture (CUDA) from NVIDIA corporation (NVIDIA, 2017) and OpenCL, was a crucial step to improve and facilitate parallel and, consequently, real-time computation.

The first operating real-time PIV setup in the laboratory has been installed by Gautier (2014). Using CUDA language, the optical flow and the ZNSSD pre-processing have been implemented on the GPU of a GeForce GTX 580 (NVIDIA, USA), a simple precision graphics card with 520 processing cores clocked at 800 Mhz (NVIDIA, 2010). The graphic card has been upgraded into a GeForce GTX 1070 (1920 cores at 1683 Hz) for the most recent experiments (see the perspectives of the chapter 7). A homemade LabVIEW (National Instruments, USA) interface has been continuously developed since 2012 to properly manage the hardware, together with the optical flow algorithm, as sketched in Figure 2.9. Choosing to write a LabVIEW software comes from the facility to set the cameras via the NI frame-grabber, to implement user C++ dynamic-link libraries (dll), and to drive various actuators (chapter 4 and chapter 7) via NI modules.

LabVIEW program

RT PIV dll

In addition to a software development kit, most of the camera manufacturers, especially Basler here, provide a peculiar file for LabVIEW programming (.icd extension) including serial commands and timing control specific to their cameras. This allows to directly command the camera through LabVIEW language. For instance, it has been possible to set the frame rate by triggering the camera via the NI-9402 digital module. Unfortunately, this file did not exist for the TSI camera, which turned out to be in fact the RMV4022 model formerly manufactured by Illunis (USA). The serial commands list available in the user manual and exchanges with Illunis engineers enabled finally to write the missing .icd file. It is noteworthy that only the last version of this camera model (post-2014) can correctly interpret the serial commands.

The program has been consequently adapted to the aero configuration by activating the doubleframe mode and by sending TTL signals from the digital module to the frame-grabber device and to the double cavity laser [Figure 2.8(b)]. The frame-grabber transmits the trig signal to the camera. The only required setting regarding the camera is the exposure time of the first frame since the second frame is exposed during the entire readout step. One TTL signal corresponds thus to a images pair. Two other successive TTL signals are separately sent to each head laser to trig the pulses, such as the first pulse occurs just before the end of the first exposure and the second pulse occurs at the beginning of the second exposure. The delay between the two pulses is therefore the main setting concerning the laser, the duration of a pulse being ∼ 10 ns.

The first version of the optical flow algorithm had been briefly provided by A. Plyer 4 . Improved by N. Gautier, the entire code is included in a C++ dll file, called RT PIV in Figure 2.9, composed of CUDA v5 and C++ functions. C++ functions concern the CPU activity, while the CUDA ones concern the GPU. In the LabVIEW program the snapshots are easily managed to feed these functions by pair to obtain velocity fields as well as their derived values (chapter 3). These values are used in control laws to command actuators through the NI-9401 digital module. During the present thesis the code has been updated, taking partly advantage of the new CUDA v8 functions and libraries. The management of data transfer between the CPU memory and the GPU memory benefits particularly from this update, using the cuBLAS library, well designed for matrices manipulations, and the Thrust library for integral computation. A new instantaneous value detailed in the next chapter is even monitored thanks to the cuFFT library, dedicated to the fast Fourier transform computations. (2015) for different flow configurations, the optical flow algorithm is efficient to compute accurate velocity fields if its parameters like the IW radius R IW , the pyramid level n level and the number of iteration n iter are well tuned. The particle density has also to be properly determined as it will be explained. To determine these settings, we investigated (A. Giannopoulos' internship) a well known geometry: the backward-facing step (BFS). It is noteworthy that comparing optical flow PIV to the FFT-PIV through synthetic [START_REF] Pan | Evaluating the accuracy performance of Lucas-Kanade algorithm in the circumstance of PIV application[END_REF] and experimental (Champagnat et al., 2011) data processing has already given recipes to obtain high accuracy. The purpose here is to well calibrate the images processing for a real-time application.

Optimal parameters

BFS in the hydrodynamic channel

(b) (a) Experiments takes place at PMMH in the closed hydraulic channel detailed in Figure 2.10. The flow is driven by gravity and the water is continuously pumped from a bottom tank to a top one. A divergent and a convergent sections, separated by honeycombs, stabilize the flow in the test section in order to ensure laminar profile and avoid undesired disturbances upstream. The test section is 80 cm long and 15 cm wide. A NACA 00020 profile is used to smoothly start the boundary layer, growing along a flat-plate before reaching the BFS 33.5 cm downstream of the leading edge. The BFS is a simple 2D geometry to study the complexity of the separated flows [START_REF] Armaly | Experimental and theoretical investigation of backward-facing step flow[END_REF] since it has a fixed separation point and the generated disturbances do not perturb its wake. The main characteristic of a step is naturally its height h, defining the expansion ratio

E = H H + h , (2.12)
where H is the channel height upstream of the BFS. In the presented experiments, H = 7 cm and h = 1.5 cm, giving E = 0.82. The Reynolds number is classically based on the BFS height

Re h = U ∞ h ν , ( 2.13)
where the free stream velocity U ∞ goes from 1.4 to 22 cm.s -1 . The maximum turbulence intensity measured for the presented runs at the step before the separation is T u = 1.2%. Nondimensionalization is applied to the spatial variables such as x * = x/h and z * = z/h.

The vertical symmetry XZ-plane is investigated as illustrated in Figure 2.11. A very thin layer of the fluorescent paint FP Rhodamine 6G has been applied on the surfaces illuminated by the laser sheet to reduce the reflections and thus to avoid near-wall miscalculations. Following experimental fields obtained via optical flow will be compared to visualizations made at the PMMH laboratory in 2010, shown in Figure 2.12. Three main interdependent regions can be defined: the shear layer, the recirculation bubble and the reattachment region.

The separated shear layer and the reattachment region

Because of the geometry, the primary separation point is located at the top of the step edge, introducing a local discontinuity of the velocity. As the velocity has to be continuous downstream The roll-up process explains how vortices appear. Along the shear layer, the strong vorticity increases certain small wavy pertubations which becomes vortices. Thanks to the linearized stability theory applied to a hyperbolic-tangent velocity profile representing the shear layer, [START_REF] Michalke | On the inviscid instability of the hyperbolictangent velocity profile[END_REF] demonstrates that the disturbances having their wave length higher than the shear layer size are amplified. This unsteady phenomenon is known as the Kelvin-Helmholtz (KH) instability and it is sensitive to the upstream condition (Aider et al., 2007). According to Troutt et al. (1984), the growth rate of the shear layer depends directly on the vortex pairing process, which gives birth to larger vortices shed downstream of the region where the flow reattaches. These structures make the shear layer transit from 2D to 3D dynamics, whereas the not paired part of the small-scale structures feed the recirculation bubble [START_REF] Bradshaw | The reattachment and relaxation of a turbulent shear layer[END_REF].

For low expansion ratio and Re h < 200, an instability above the reattachment point has been numerically highlighted by Beaudoin et al. (2004) as the origin of stationary large-scale streamwise longitudinal vortices which are experimentally observed downstream of the BFS. They demonstrate that the radius of the local streamlines curvature [START_REF] Sipp | A Criterion of Centrifugal Instabilities in Rotating Systems[END_REF] can actually makes the pressure across the boundary layer varying, triggering this so-called Görtler (or centrifugal) instability. This kind of structures is typically observed for boundary layer on concave walls.

The recirculation bubble

The entire separated region forms the recirculation bubble downward of the BFS. The mean recirculation area (Gautier & Aider, 2013), as well as the mean recirculation length (Thangam & Knight, 1989), is related to the Reynolds number. They both increase until a maximum at Re h 600 and then decrease before reaching a quasi-plateau from Re h 2100.

The recirculation bubble is also subject to the flapping motion, an instability whose frequency is one order of magnitude lower than the shedding frequency. A secondary recirculation region takes place downstream of the step edge and upstream of a secondary separation point. Its dynamics are highly detailed by [START_REF] Spazzini | Unsteady behavior of back-facing step flow[END_REF], explaining that this region is the seat of quasi-periodic events of intense activity whose frequency corresponds to the flapping one. A typical cyclic activity is the growth of the secondary recirculation bubble until its size reaches the same order of magnitude of the step height and then collapses. The flapping motion together with the Kelvin-Helmholtz instability produce the oscillation of the reattachment line [START_REF] Dejoan | Large eddy simulation of periodically perturbed separated flow over a backward-facing step[END_REF]).

Influence of optical flow parameters

First, the particles density and the IW radius are investigated together since their are related to each other regarding the displacement accuracy. Then, the number of iterations and the pyramid level are chosen as a compromise between accuracy performance and computation time. For each experiment, the flow is seeded with 20 µm neutrally buoyant polyamid particles (Dantec Dynamics, Denmark). The different parameters have been set for various Reynolds numbers. When instantaneous fields are used, the presented observations are based on several of these fields, even if only one is displayed for succinctness reason.

Interrogation window radius and particles density

According to [START_REF] Pan | Evaluating the accuracy performance of Lucas-Kanade algorithm in the circumstance of PIV application[END_REF] three factors must be taken into account to properly set the IW radius R IW : the number of particles inside the IW d p , the spatial averaging effect and the constant constraint. To estimate the particles density, the MATLAB imfindcircles function based on the circular Hough transform is used to identify the particles. Six different concentrations are studied at Re h = 820. For each one, the averaged number of particles per pixel (ppp) is estimated by counting them in 600 snapshots. The calculation is done in the entire PIV window, as well as in three different regions of interest (ROIs) represented in Figure 2.13(a): the free stream, the shear layer and the recirculation area. The detection algorithm is illustrated in Figure 2.13(b-d) for increasing concentrations. Even if the algorithm seems to miss few particles due to the noise in all cases, particles in the laser sheet are globally well detected. The estimated values will consequently slightly underestimate the real number of particles.

Entire PIV window 1.1 1.4 1.7 1.9 2.1 2.2

Free stream ROI 1.0 1.3 1.6 1.7 1.9 Table 2.3 summarizes the results with a low dispersion (∼ 0.05 ×10 -2 ppp). As expected the recirculation area concentrates more particles than the shear layer, having itself more tracers than the free stream. The large difference between the concentration in the recirculation ROI and the other ones is also due to the noise, which is more important near the wall. The values for the entire PIV window will be considered for the next calculations.

The IW setting is evaluated for the fifth data set at Re h = 820 by regarding the optical density d p , the number of particles inside an IW. The IW radius is investigated for R IW ∈ {7, 11, 15, 19, 23, 27}, the other optical flow parameters being fixed at n iter = 5 and n level = 3, following previous recommendations (Champagnat et al., 2011;Gautier, 2014). These IW settings correspond to d p ∈ {4.6, 11, 20, 31, 45, 62} respectively. The resulting instantaneous and timeaveraged fields are compared to the flow observed for close Reynolds numbers in Figure 2.12). The mean profiles converge past 300 velocity fields and are computed for 600 ones. For the zone including low velocity gradient, the more the particles are inside the IW the more accurately V can be computed, avoiding bias induced by noise. Nevertheless there is a saturation effect which makes number of particles larger than a threshold no more efficient. If R IW is even too large, the spatial averaging effect misrepresents the regions of strong vorticity as seen previously. This is due to the validity of the constant constraint assumption made to get Equation 2.4, according to [START_REF] Pan | Evaluating the accuracy performance of Lucas-Kanade algorithm in the circumstance of PIV application[END_REF] who state that this assumption holds only if max(∇V ) < 1 (in px/px), limiting the size of the IW. The IW radius and the density are fixed to R IW 10 px and d p ∼ 15 in the next measurements made during the thesis. These settings are in good agreement with Champagnat et al. (2011).

The classic accuracy-speed compromise

Using the previously determined IW radius and particles density, the minimum number of iterations n iter is now investigated for the BFS flow, fixing n level = 3. Figure 2.15 shows that the instantaneous velocity fields are not converged yet for n iter < 5 even though the global flow seems to correspond to the visualization in Figure 2.12. Figure 2.15(a) shows that the displacements near the wall and the borders are more difficult to compute accurately. Only small differences appear between the computed fields for n iter ≥ 5. To avoid useless computations and loss in calculation efficiency, the number of iterations will be therefore fixed to n iter = 5 for the following As explained previously, the level of the image reduction pyramid n level is crucial for the gradient descent computation since it determines the initial guess V 0 = 0. An original displacement V is reduced to V /2 n level after n level . For n level = 1, the intensity differences between the two IWs are still too important, whereas they enable to obtain V 0 = 0 from n level = 3. Beyond this level the reduction in size of the snapshot is too important and the real displacements are completely lost. The pyramid level has thus to be chosen regarding the sensor size as well as the free stream. As 8-10 px is often the aimed maximal displacement between two successive snapshots in PIV by setting the proper time delay between the two laser pulses (aero) or the camera frequency acquisition (hydro), n level = 3 is particularly recommended (Champagnat et al., 2011;Gautier & Aider, 2015). The performances of the optical flow algorithm are evaluated for different combinations of the optical flow parameters, the IW radius being still fixed, by computing 1000 velocity fields offline from 1920 px × 512 px 8-bit PNG written on an internal solid-state drive, operating a 4 × Xeon E5-1607 3.00 GHz (Intel, USA) and the GeForce GTX 1070. The maximal displacements between the successive snapshots constituting the data set are around 8.5 px. The results are summarized in Table 2.4. Only the velocity fields obtained for [n iter = 1, n level = 2] seem incorrectly computed, giving maximal displacements lower than 5.5 px. It appears that 50 velocity fields per second (FPS) can be accurately processed offline. Similar calculations have been done for 2048 px × 2048 px 16-bit PNG, leading to 7.5 FPS. This rate is also obtained in real-time (chapter 5 and chapter 7). Relating to Table 2.2, this reveals that the optimum FPS is reached for the aero configuration. These speeds represent 0.02 µs/vector and 0.03 µs/vector respectively, which is in good agreement with previous measurements made for online computation of smaller 8-bit snapshots (Gautier & Aider, 2015). It shall be mention that images are read from the drive through SATA III technology (4800 Mb.s -1 , 8 times faster than CL) for the offline evaluation.

n

Conclusions and perspectives

A new PIV method, based on an optical flow algorithm using a local approach, has been successfully implemented on different GPUs using CUDA language. This method allows rapid calculation of dense and accurate velocity fields from 4MP images. It is embedded in a homemade LabVIEW program which handles the devices involved in the PIV and in flow control. Extra CUDA functions allow also to very fast compute values derived from the velocity as detailed in chapter 3. This program is now operational for a hydrodynamic channel as well as for a large wind tunnel.

Regarding the material requirement, most of the set-up already installed for 2D-2C FFT-PIV in wind tunnel may be directly used with the LabVIEW program, making the optical flow technique affordable for research facilities. Nevertheless, the main current constraint for the hardware is related to the camera and the frame-grabber, which have to be compliant with the LabVIEW environment.

Even if the velocity fields obtained thanks to the optical flow algorithm have their spatial resolution tied to the size of the IW such as for advanced PIV algorithms, they are dense (one vector per pixel), giving not only better results in the vicinity of edges and obstacles but also smoother fields. To achieve such results, the algorithm parameters have to be properly set regarding the optical devices and the experimental conditions. The IW radius will thereby depend both on the tracers density and on the free stream velocity which is mainly fixed via the inter-frames time the camera can reach. The Gaussian pyramid level will depend on the free stream velocity only, being cautious regarding the obstacles position with respect to the sensor size. Finally the number of iterations can be estimated via trial and error, making a compromise between accuracy and computation speed, which is limited by 1. the LabVIEW program not enough optimized, 2. the data transfer rates (low/medium CL) and the bit depth, 3. the memory management (CPU-GPU).

Future improvements

The different elements slowing down the data flow and the code itself should be soon improved. The latter will be modified to include an adapting IW radius to deal with the different constraints imposed by the investigated flow. Such approach has been suggested and successfully tested in the last version of FOLKI, called e-FOLKI [START_REF] Plyer | Massively parallel Lucas Kanade optical flow for real-time video processing applications[END_REF]. Before, the code will completely benefit from the new CUDA libraries since its update is still in progress. This implies an upgrade of the 32-bit LabVIEW program to a 64-bit version (especially for the cuFFT library). Actually, this has been already done for the hydro configuration but requires a change in the camera for the aero configuration, due to an apparent incompatibility of the TSI camera with 64-bit program.

Saving the images processed in real-time, instead of just getting rid of them, decreases also the performances since it globally adds a step in the images management process. Grouping several solid-state drives into redundant array of independent disks (RAID 0) may considerably reduce the induced extra time. On the other hand, if only closed-loop control is of interest, the implementation of the optical flow code directly in the camera may be considered. This is the reason why writing a version of the algorithm for a FPGA configuration will also be investigated.

The LabVIEW program allows to target several ROIs to define multi input, either for closed-loop control laws (chapter 3) or for reduced-order models (chapter 4). For the moment the ROIs are created once the RAM receives the image. The time to obtain the aimed quantities may therefore be reduced by directly selecting the ROIs in the camera sensor to avoid useless data transfer. This is what is planned with a new fast camera, which benefits also from the CoaXPress high speed interface technology (1.25 to 6.25 Gb.s -1 ), which seems to stretch the transfer rate limit between the camera and the frame-grabber.

Stanislas, M., Okamoto, K., Kähler CHAPTER 3

Velocity fields analysis for real-time PIV

This short chapter details all the quantities which can be extracted in real-time (RT) from the velocity fields. This a fundamental step to be able to use the camera as a visual sensor in a closed-loop experiment. Some of these output are related to specific flow like the recirculation area for wakes. Other are intended to detect structures like the Q-criterion. All these tools are implemented on the GPU and will be illustrated via measurements in the previously seen BFS flow for Re H = 2 × 10 3 (T u = 1.2%).

Velocity fields

The Reynolds decomposition is used to have access to the turbulent part of the flow. Its application to the 2D-2C velocity fields V = (u, w) resulting from the optical flow is

V = V + V , (3.1)
where V = (U, W ) and V = (u , w ) are the mean velocity and the fluctuations respectively. The former can be easily used to compute profiles free from noise, giving direct access to the general flow characteristics of the boundary layer (see chapter 4). The latter carry the flow dynamics.

Time-averaged velocity

By cumulating the instantaneous velocity fields V (n) in the same variable, the time-averaged velocity V is straightforwardly obtained. This online post-process can adapt to the dynamics via optional moving-average window and user-defined conditions to only keep specific velocity fields. This way, the RT computed conditional-averaged fields are easily obtained without storing large amount of data, even if the time-resolution of the streamed images is inadequate to capture the main flow dynamics (see chapter 5).

Velocity fluctuations

Vortex detection

An objective when studying unsteady flows is to follow coherent structures, especially vortices, since they are peculiar patterns influencing and moving in the fluid [START_REF] Jeong | On the identification of a vortex[END_REF]. Detecting such structures enables to divide the flow in regions of different dynamics but distinguishing a large-scale vortex from the small-scale turbulent motion embedded inside remains a difficult task. How is mathematically defined a vortex? There is no absolute answer to this question, which is the reason why there are many criteria for vortex identification.

Two historic approaches have given birth to several detection tools: the Eulerian and the Lagrangian techniques. On one hand, Lagrangian techniques delimit regions of different dynamics from a finite set of successive velocity fields. For instance, in space oceanography, the finite-time Lyapunov exponent (FTLE) [START_REF] Haller | Lagrangian coherent structures from approximate velocity data[END_REF][START_REF] Shadden | Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows[END_REF] is used to identify mixing regions and to predict oil slick extent from the geostrophic velocities [START_REF] Mezić | A new mixing diagnostic and gulf oil spill movement[END_REF]. On the other hand, Eulerian methods detect regions of vortices, defined from the instantaneous velocity field and its gradients. The presented tools come from this side.

Swirling strength criterion

Chong et al. (1990) proposed to identify the 2D vortex core as a region where the velocity gradient has complex conjugate eigenvalues. They define thus the swirling strength criterion λ Ci from the characteristic polynomial of ∇V as Gautier & Aider (2015a) implemented this criterion on the GPU to capture the shedding frequency downstream of the BFS flow at Re h ∼ 10 3 . They aimed at injecting momentum in the flow at this frequency since it is effective to reduce the recirculation area. A vertical ROI located

λ Ci = 1 2 4 det(∇V ) -tr(∇V ) 2 , if defined (3.

Q-criterion

Since the vorticity alone is not enough to distinguish vortices from strain, Hunt et al. (1988) defined the Q-criterion as

Q = 1 2 ( Ω 2 -S 2 ) > 0, (3.4)
where Ω = 1 2 (∇V -∇V T ) and S = 1 2 (∇V + ∇V T ) are the vorticity tensor and the rate of strain tensor respectively. This criterion can be related to the pressure Laplacian Q = ∆p/(2ρ) via the Poisson equation for the pressure. The main issue with the above described criteria appears for 3D fully turbulent flows, where large-scale and small-scale vortices tangle. This is due to the velocity gradient which tends to amplify the inherent noise of turbulence. In these cases, a user-defined threshold is required to detect the vortex cores. This is critical because it can lead to misinterpretation of the flow dynamics by removing apparently low Q values. It shall be stressed that velocity fields obtained via the optical flow allow to derive directly accurate gradients. For velocity fields computed from FFT-PIV this requires time consuming methods (Raffel et al., 2007).

Separated flow activity

The previous quantities are widely used to characterize and analyze the activity of separated flows. Two important properties already discussed in chapter 2 are the recirculation area A rec , a valuable target for experimental control practice (Gautier, 2014), and the Kelvin-Helmholtz (KH) instability.

Recirculation area The instantaneous recirculation area A rec (t) is defined as

A rec (t) = H(-u * (x * , z * , t) dx dz, (3.5)
where H is the Heaviside function. It allows the study of the time evolution of the recirculation bubble. In chapter 5, the tracking of its barycenter will be of particular interest to understand the dynamics of a 3D fully turbulent wake. This value will even be used as a sensor in a closed-loop control in chapter 7. The KH frequency has been estimated for 2000 snapshots, giving f * KH = 0.79 ± 0.12, as displayed in Figure 3.5. Regarding the vortex shedding frequency (see after), this result is in good agreement with previous experimental [START_REF] Norberg | Effects of Reynolds number and a low-intensity freestream turbulence on the flow around a circular cylinder[END_REF] and numerical [START_REF] Dong | A combined direct numerical simulation-particle image velocimetry study of the turbulent near wake[END_REF][START_REF] Kim | Large-eddy simulation of the turbulent near wake behind a circular cylinder: Reynolds number effect[END_REF] estimations done for the wake behind a cylinder. It even supports well the Re 0.67 h scaling proposed by [START_REF] Prasad | The instability of the separated shear layer from a bluff body[END_REF], predicting f * KH = 0.76.

The first advantage of such a measurement is the possibility to reach a characteristic frequency of the separated flow usually captured only by high sampling rate sensors (see chapter 5). The second advantage resides in the instantaneous measurement itself, giving the information in real-time. It enables indeed to compute the shedding frequency online without building time series. Moreover, the measurements are spatially averaged when extracted from the instantaneous fields. This is even better than a local sensor, especially an intrusive one like the hot-wire, which is critical for a noise amplifier flow. For example, a second, larger, ROI can be defined in the second part of the shear layer as shown by Figure 3.6(a). The results are displayed in Figure 3.6(b), where the mean estimated shedding frequency is f * shed = 0.2 ± 0.04, in good agreement with the expected value. The high standard deviation is due to the low spatial resolution for the FFT, the ROI being limited in the streamwise direction.

Conclusions and perspectives

Building linear controllers to manipulate the recirculation area in a BFS flow at Re h ∼ 10 3 , N. Gautier has implemented numerous quantities derived from the instantaneous fields in the CUDA/LabVIEW environment (Gautier & Aider, 2013, 2014a,b, 2015a). These RT processes have even enabled an experimental application of machine learning on this flow (Gautier et al., 2015b). The CUDA codes computing these variables have been improved, especially to fulfil the requirements of the measurement conditions of a bluff-body in a wind-tunnel (see chapter 7): higher Reynolds number, 3D wake and low frame rate of the streaming double-frame camera. Quantities like the Reynolds stress tensor, the Q-criterion and the boundary layer characteristics (see chapter 4) have been added. A new method to capture rapid dynamics is now available and needs to be intensively tested on different geometries (see chapter 5).

Known structures in unsteady turbulent flows have low vorticity Ω, such as the counter-rotating longitudinal vortices downstream of a slanted bluff-body (see chapter 5), making them difficult to detect using the presented criteria, highly sensitive to the small-scale turbulence. A solution, introduced by Graftieaux et al. (2001), consists in only dealing with the fields topology, excluding the velocities magnitude. This approach defines two scalar functions, detecting both the centre and the boundary of vortices. For instance, using this criterion, Venning et al. (2017) have recently highlighted the presence of a twin-horseshoe structure in the wake behind a slanted bluff-body, rather than the usually admitted toroidal vortex (see chapter 5).

A complementary Lagrangian technique could be added to the previous Eulerian tools. This would require data storage since the vortex detection is based on an ensemble of velocity fields. Furthermore, it is a time-dependent method, meaning that the frame rate and the computation speed of the optical flow have to be higher than the characteristic frequencies of the flow. Nonetheless, [START_REF] Rockwood | Correlation of the surface pressure distribution on a circular cylinder with objective identification of vortex formation and shedding[END_REF] have recently coupled the FTLE analysis to RT pressure sensor, reaching online accurate tracking of the vortices behind a cylinder. Furthermore, Guéniat (2013) proposed a faster computation of the FTLE, based on GPU interpolation and applied to an open cavity flow.

Introduction

The presented work, done in collaboration with J. Guzmán Iñigo (Imperial College London, former ONERA), D. Sipp (ONERA DAAA) and P. J. Schmid (Imperial College London), is the experimental realization of the numerical investigation into dynamic observer by Guzmán Iñigo et al. (2014); Guzmán Iñigo et al. (2016). The theory will therefore only be summarized in this section since complete details can be found in the given references.

Reduced-order model

In their classification of unsteadiness in open flows, [START_REF] Huerre | Local and global instabilities in spatially developing flows[END_REF] define two main categories exhibiting strong non-linearities. On one hand, the oscillator-type flows are characterized by a global instability resulting in self-sustained oscillatory fluid behaviour (wake, jets). On the other hand, the noise amplifiers category contains fluid systems which are globally stable even if they selectively amplify upstream perturbations by convective instabilities (boundary layer).

Directly controlling a fluid system requires to handle its numerous degrees of freedom, which is not realistic yet. The idea consists thus in simplifying the system as a reduced-order model (ROM). If the physics are deeply understood, then a physical model can be set. Unfortunately, such level of knowledge is still difficult to reach.

Galerkin projection

Projecting the Navier-Stokes (NS) equations onto a reduced basis, purposebuilt to capture the most essential dynamics, is another way called Galerkin projection. The reduced basis may be computed either through global stability analysis or through statistical decomposition like proper orthogonal decomposition [START_REF] Berkooz | Galerkin projections and the proper orthogonal decomposition for equivariant equations[END_REF]) and dynamic mode decomposition [START_REF] Tissot | Model reduction using dynamic mode decomposition[END_REF]. This technique is appropriate for oscillator-type flows (Jackson, 1987), like the cylinder wake [START_REF] Bergmann | Control of the circular cylinder wake by trust-region methods and POD reduced-order models[END_REF]) and the open cavity [START_REF] Barbagallo | Closed-loop control of an open cavity flow using reduced-order models[END_REF], but not for amplifier flows. Indeed, due to their nature, the dynamics driven by environmental noise cannot be distinguished from the intrinsic dynamics by global stability analysis [START_REF] Trefethen | Hydrodynamic stability without eigenvalues[END_REF][START_REF] Chomaz | Global instabilities in spatially developing flows: non-normality and nonlinearity[END_REF][START_REF] Brandt | Effect of base-flow variation in noise amplifiers: the flat-plate boundary layer[END_REF]. In the same manner, a statistical decomposition cannot detect the statistically steady noise, which prevents the system from creating the corresponding output. In fact, the model would need an access to the noise distribution to avoid this issue, which is experimentally realizable with data-based techniques. It shall be mentioned that the transient growth of the perturbations, characteristic of such a flow, can be captured using the resolvent operator [START_REF] Sipp | Characterization of noise amplifiers with global singular modes: The case of the leading-edge flat-plate boundary layer[END_REF].

System identification By highlighting the role of system identification as a design problem, especially for control, rather than solely a tool to match the real dynamics, [START_REF] Ljung | System identification: Theory for the user[END_REF] proposed, amongst other things, an alternative approach to Galerkin projection. The system identification algorithm aims at predicting the dynamics of the entire flow through a statistical learning process from local measurements of input and output data, as explained hereafter. For instance, Hervé et al. (2012) used the system identification method to numerically predict and control the dynamics of a back-facing step between an upstream sensor and a downstream one. For the same flow, [START_REF] Henning | Feedback control applied to the bluff body wake[END_REF] delayed the vortex shedding using an array of pressure sensors as an input of a model identification.

Dynamic observer principle

From the amplifier flow dynamics to a linear system

The complete dynamics of a forced amplifier flow are represented by the incompressible NS equations including an unknown upstream forcing term F w w(t)

∂ t V + V .∇V = -∇P + Re -1 ∆V + F w w(t), (4.1a) ∇.V = 0, (4.1b)
where the velocity V and the pressure P are dimensionless and the physical meaning of w(t) is given in the following section. Expressing the velocity and the pressure in Equation 4.1 such as V = V base + v and P = P base + p, where V base and P base are respectively the velocity and the pressure when w = 0 (base-flow), and v and p are the velocity and pressure disturbances, leads to

∂ t v + V base .∇v + v .∇V base = -∇p + Re -1 ∆v + F w w(t), (4.2a) ∇.v = 0, (4.2b)
where linear perturbation dynamics are assumed (second order perturbations are negligible).

The objective is thus to obtain the following linear time invariant (LTI) multiple-input-multipleoutput (MIMO) system

x(n + 1) = Ax(n) + Bu(n) + w(n), (4.3a) y(n) = Cx(n) + Du(n) + v(n), (4.3b)
where y(n) is the system output, x(n) is the system state, u(n) is the system input, w(n) is the state noise, v(n) is the sensor noise, and A, B, C and D are the system matrices. w(n) and v(n) being unknown, the state space model described in Equation 4.3 rather predicts an estimated output denoted y e when w = v = 0. The system matrices are consequently chosen to minimize the difference between the estimated output, and the measured output y.

POD reduced-order model

N consecutive velocity fields {V (n)} n=1...N are computed to form the so-called learning data set. The proper orthogonal decomposition (POD) of this data set enables to build a ranked and orthonormal basis containing N modes (Lumley, 1967;Sirovich, 1987) thanks to the snapshots method. The first K modes {Φ k } k=1...K with K ≤ N are chosen to compute the approximated velocity field Ṽ (n)

Ṽ (n) = K k=1 Φ k , V (n) Φ k , ( 4.4)
where the scalar product , is the energy-based inner product. By gathering all the POD coefficients a k (n), defined as a k (n) = Φ k , V (n) , the system output is obtained through the reduced state vector

Y (n) = [a 1 (n) a 2 (n) ... a K (n)] T .
The modes dynamics are contained in their coefficients.

The dynamics of the estimation Y e (n) of the reduced state vector Y (n) can therefore be captured from sensor measurements s(n) by

Y e (n + 1) = Ãs Y e (n) + Ls(n), (4.5a
)

s(n) = CY e (n), (4.5b)
where Ãs is the linear evolution matrix, L is the observer matrix, while C is the measurement matrix. These matrices are determined using system identification techniques based on the inputoutput relation only and described hereafter. Assuming D = 0, the matrices of the desired LTI MIMO system in Equation 4.3 are related to Equation 4.5 through Ãs = CAC -1 and L = CB.

Actually, the model has to consider the physical perturbations rather than the measurements in Equation 4.5a and also the noise in Equation 4.5b to capture the real dynamics

Y e (n + 1) = Ãw Y e (n) + Bw w(n) (4.6a) s(n) = CY e (n) + g(n), ( 4.6b)
where Bw w(n) is the unknown driving term and g(n) is the measurement noise. The real linear evolution matrix Ãw is directly computed from the other matrices, using Equation 4.6a, Equation 4.5a and Equation 4.6b Ãw = Ãs + L C. (4.7)

System identification

The system identification comprises three steps summarized in Figure 4.1. The POD modes are first computed from the learning data set which is made of velocity fields recorded while the flow is randomly perturbed upstream. This data set must contain enough events to build an efficient reduced-order model. The number of velocity fields to retain depends consequently on the perturbation generation settings and the sampling frequency of the PIV setup. According to Sirovich (1987), the number of POD modes to conserve is defined through the percentage of energy they contain. However the most important criteria to select the appropriate number of modes are the sensor nature and position. The position is thereby verified by computing the relative error between the measures by the sensors and the reconstructed measures from a certain amount of POD modes (observability of the system). Then a subspace identification algorithm [START_REF] Qin | An overview of subspace identification[END_REF] is applied to calculate the state matrices of the stochastic linear system in Equation 4.3. The so-called "N4SID" algorithm (Van Overschee & De [START_REF] Van Overschee | N4SID: Subspace algorithms for the identification of combined deterministic-stochastic systems[END_REF][START_REF] Van Overschee | A unifying theorem for three subspace system identification algorithms[END_REF] is well suited for this step. Finally the model is tested on a validation data set by comparing the predicted output POD coefficients to the real ones. Contrary to the learning step, a high temporal resolution is required because of the nature of the system itself.

Experimental application

As it has been specifically designed for amplifier flows, the dynamic observer is applied to the transient boundary layer. The results are evaluated as the ones numerically obtained by Guzmán Iñigo et al. (2014).

The boundary layer over a flat plate

The noise amplifier studied in the present chapter is the zero pressure gradient two-dimensional (2D) boundary layer flow over a flat plate. Even if this flow is less complex than the adverse pressure gradient case, it contains interesting dynamics to capture, which have been intensively studied and theorized [START_REF] Kachanov | Physical mechanisms of laminar-boundary-layer transition[END_REF]. Thereby, this flow may be delimited in three successive regions: the laminar one also called the Blasius flow (Schlichting & Gersten, 2017), the transient one and the turbulent one. Two convective instabilities work in the two first regions. They produce different structures described hereafter, depending on the nature of the small upstream perturbations.

Dynamics description

Due to its viscosity ν and the free-stream U ∞ , the flow near the wall undergoes shear stress τ w , as explained by Prandtl first. At low Reynolds numbers the dissipative effect of the viscosity is more important than the shearing: this is the laminar regime. Increasing the Reynolds number inverts the trend and the kinetic energy rises: this is the turbulent regime. It produces vorticity which is diffused vertically and advected downstream and the boundary layer thickness δ grows consequently. 2), going from two-dimensional stable to three-dimensional unstable structure [START_REF] Klebanoff | The three-dimensional nature of boundary-layer instability[END_REF]. The waves finally breakdown into hairpin eddies, becoming then a fully turbulent flow with small-scale structures (spots, eddies) and intense fluctuations. This instability is referred as a receptivity problem since it linearly amplifies small specific upstream perturbations [START_REF] Kachanov | Physical mechanisms of laminar-boundary-layer transition[END_REF].

Another transition process may occur if the turbulence intensity of the flow is higher than 1% [START_REF] Morkovin | Bypass transition to turbulence and research desiderata[END_REF]: the bypass transition. It is referred as the secondary instability and it generates groups of streaks, which are streamwise structures periodically distributed in the spanwise direction [START_REF] Matsubara | Disturbance growth in boundary layers subjected to free-stream turbulence[END_REF]. This instability can be well forced and observed in a laboratory environment as Duriez et al. sition looks like the 3D shape of the T-S waves and the natural transition spectrum signature is also identified in the streaks dynamics [START_REF] Hughes | Natural transition phenomena on an axial compressor blade[END_REF]. Modelling these instability mechanisms in a control purpose is therefore of interest since the growing structures advected downstream and tilted by the free-stream provoke finally the flow breakdown into turbulence [START_REF] Butler | Three-dimensional optimal perturbations in viscous shear flow[END_REF]. Delaying this transition is thus a good strategy to reduce the drag related to the skin friction. For instance, at moderate turbulence intensity, the T-S waves can be stabilized by streaks of low amplitude forced [START_REF] Cossu | Stabilization of Tollmien-Schlichting waves by finite amplitude optimal streaks in the Blasius boundary layer[END_REF][START_REF] Fransson | Experimental study of the stabilization of Tollmien-Schlichting waves by finite amplitude streaks[END_REF] or appearing naturally [START_REF] Martin | Numerical studies of non-linear intrinsic streaks in the flat plate boundary layer[END_REF]. The reconstruction and the prediction of the T-S waves only are investigated in the present experiment.

Flow characteristics

Defining some characteristics of the flow allows to know the instant when the different dynamics described above are triggered and the position where they are. The most imaginable value is the boundary layer thickness δ. It conventionally corresponds to the height where

v x (y = δ) = 0.99U ∞ . (4.8)
The displacement thickness δ * (x) is the vertical deflection imposed by the boundary layer existence to the potential flow 2 incoming over the flat plate. It is defined as

δ * (x) = ∞ 0 1 - v x (x, y) U ∞ dy. (4.9)
The Reynolds number Re δ * (x) in Equation 4.2a will be consequently based on the displacement thickness

Re δ * (x) = δ * (x)U ∞ ν . ( 4

.10)

The streamwise position allows also a common alternative definition Re x .

In a similar way as the displacement thickness, the momentum thickness θ(x) is the vertical shift required for the boundary layer flow to recover from the lost in momentum due to the shear stress τ w , in comparison of the potential flow. It is defined as

θ(x) = ∞ 0 v x (x, y) U ∞ 1 - v x (x, y) U ∞ dy, (4.11)
whose flux deficit is equal to the 2D skin friction drag ρU 2 ∞ θ = x 0 τ w dx.

The last two thickness definitions lead to the shape factor H(x)

H(x) = δ * (x) θ(x) . (4.12)
The turbulent separation is assumed to occur when 1.8 < H < 2.4 according to [START_REF] Head | Entrainment in the turbulent boundary layer[END_REF] and [START_REF] Cebeci | Calculation of separation points in incompressible turbulent flows[END_REF]. Theoretical values for the laminar and the turbulent 3 regions are summarized in Table 4.1.

Flow in absence of vorticity.

3. There is also another power-law used in some engineering books: Rex 1/5 .

Characteristics

Experimental setup

Water channel

The experiments are carried out in the closed hydrodynamic channel mentioned in the previous chapter and sketched with the investigated flat plate installed in Figure 4.5. For the present study, the free-stream velocity range is U ∞ ∈ [3; 7; 12] cm.s -1 .

Boundary layer

A homemade leading edge, designed as a NACA 00020 profile, is used to smoothly start the boundary layer. The flat plate along which the boundary layer growths entirely spans the test section width to manage 2D dynamics at best. The channel height is H = 8.5 cm. The origin is located at the beginning of the leading edge (x = 0), in the vertical symmetry plane (y = 0) and on the flat plat (z = 0).

Forcing noise

The perturbation is induced by a span-wise inclined slotted jet, whose angle to the wall is 45 • (Figure 4.3). The injection system is composed of glass beads between two pierced grids to avoid undesired structures. It is located 5 cm downstream the leading edge (Figure 4.5). The jet outlet has a rectangular cross-section which is 0.4 cm long (in the streamwise direction) and 9 cm wide (spanwise direction). Thanks to this geometry, the perturbation can be considered as homogeneous along the spanwise direction.

As described in Figure 4.4, the slotted jet is supplied by a pressurized water tank, monitored by an electro-pneumatic regulator to control the jet velocity u f orcing , which can be set regarding the frequency f f orcing , the amplitude U f orcing , and the length δT f orcing . This setup enables to induce random perturbations by changing these parameters after each pulse. The command signal between the impulses remains at the value avoiding any perturbation generation, as if the slot does not exist.

Real-time PIV as a sensor

The flow is seeded with 20 µm neutrally buoyant polyamid seeding particles. The vertical symmetry plane of the test section is illuminated by a laser sheet created by the 8 W continuous laser beam operating at a 532-nm wavelength and passing through a cylindrical lens. A very thin layer of the fluorescent paint FP Rhodamine 6G has been applied on the surface illuminated by the laser sheet to reduce the reflections. The snapshots of the illuminated particles are recorded The two dimensional two components (2D-2C) velocity fields v x (x, z) and v z (x, z) (respectively the streamwise and wall normal components) are computed at the frequency f P IV using the optical flow algorithm described in chapter 2. Finally a low-pass filter has been applied to the raw data sets with a cutoff frequency f cut at least ten times higher than the highest measured frequency of the forced boundary layer dynamics.

Instead of the wall sensor as in the numerical work of Guzmán Iñigo et al. (2014), a different sensor has been used in the present experiment. Indeed, different values such as the vorticity ω y and the λ Ci -criterion, have been monitored directly from the local fluctuations of the velocity fields in the region of the instantaneous PIV snapshot. The size and location of the sensor may play a role in the efficiency of the learning process because the former will act as a spatial filter and the later will be crucial for its ability to detect relevant flow variations [START_REF] Guzmán Iñigo | Recovery of the inherent dynamics of noise-driven amplifier flows[END_REF]. This point will be discussed later.

Flow characterization

Base flow

The unforced flow is first studied for three Reynolds numbers, Re δ * 0 = [224, 297, 337], detailed in this section. The base flow V base = (v x , vz ) is processed over t * = 50, t * = 250 and t * = 500 respectively. Like the velocity profile, Figure 4.7(d-f) highlight that the computed displacement thickness progress from the laminar model to the turbulent one as soon as the corresponding Reynolds number increases Figure 4.7(j-l). The shape factor leads consequently to expected values, decreasing from H ∼ 2 at Re δ * 0 = 224 to H ∼ 1.5 at Re δ * 0 = 337. As the flow in the observed region is desired to be transitional, the perturbations will be generated for Re δ * 0 337 but not too high to remain in the linear approximation imposed by the system identification method.

Forced flow

Experimentally the forcing design is one of the toughest aspects since the same perturbations cannot be brought out as in Guzmán Iñigo et al. (2014). In the present work, a random 2D jet is created by sending a square signal, similar to the one in Figure 4.9, to the air pressure controller with the following settings (0 being equivalent to a closed slot): the jet frequency The kinetic energy (KE) k * , previously defined in chapter 3, is analyzed to study the generated perturbations. Figure 4.10 shows the KE evolution in time at two streamwise positions, x * W and x * W + 25. The lag between the two curves gives the length of time which perturbations travel the distance between the measurements locations. The resulting velocity of the T-S waves is 0.373, in agreement with Guzmán Iñigo et al. (2014). As expected most of the perturbations are increased by the instability, even if some of them seem to decrease downstream. It may be due to a too high jet amplitude, generating a strong forcing but without the proper dynamics to be amplified [START_REF] Luchini | Reynolds-number-independent instability of the boundary layer over a flat surface: optimal perturbations[END_REF]. A surprising aspect of these results concerns the Reynolds value at which the T-S waves are obtained since it is under the theoretical one (Re crit δ * = 520). At the time of the thesis submission, this is still under investigation and it is planned to be discussed in further works.

Experimental dynamic observer

The retaining learning data set is composed of 1500 velocity fields sampled at f * P IV = 1.1 with a spatial resolution of 0.24 mm.px -1 , containing about ten events perturbing the base flow. The length of the learning data set is related to the occurrence of recorded events: it is useless to

POD modes

The POD energy and the normalized POD modes of the streamwise velocity v *

x are given in Figure 4.12. The first 32 modes accumulate more than 80% of the total energy, the first one being at 46%. Beyond the eleventh POD mode, each other mode contains less than 1% of the energy of the first one. The spatial organization of the first mode contains all the structures advected downstream. The T-S waves are globally well captured by the following modes. The most energetic structures, rather downstream, look indeed like the most developed T-S waves [Figure 4.12(c-e)], whereas the lowest energetic modes contain weaker shapes of the T-S waves localized upstream [Figure 4.12(f) and (g)].

Sensors definition

The sensors used are directly derived from the velocity fields. Various natures, positions and numbers of sensors have been tested. The streamwise v *

x and the vertical components v * z are first both monitored in the same region Ω S placed upstream at x * s = 150 as shown in Figure 4.13(a) and (b). The sensor does not contain the ten first pixels (interrogation window radius) the closest to the wall to limit noisy data. It is also not too large too avoid a smoothing effect. The nature of this sensor s = [ v *

x Ω S , v * z Ω S ] is based on the sensitivity of such values to the T-S waves. The position of Ω S has been chosen to detect the perturbations incoming in the PIV window as soon as possible. A downstream sensor is avoided in purpose in the present study since the final The observability of the sensor s regarding the POD modes {Φ i } i=1..K is given by measuring the same values in the K modes as in the velocity fields. The resulting sensor s P OD (K) is then compared to the real sensor s:

s P OD,error (K) = ||s P OD (K) -s|| ||s|| . (4.13)
Figure 4.14 shows the results for the considered sensors at different streamwise positions. Taking a low number of POD modes into account leads to slightly better results for the sensors localized in the most upstream part of the PIV window. This is due to the spatial organization of the first POD modes as previously observed. Adding more modes compensates this difference. Beyond a threshold, there is almost no more gain. For the present case, the most important gain without requiring too extra modes is obtained with the first 17 modes for x * s = 150. Nevertheless the error remains high since the noise has a huge impact for the defined sensors which fluctuate around 0 when there is no perturbation. Only considering snapshots when perturbation occurs should improve this evaluation.

Reduced-order model and validation

= [ v * x Ω S , v * z Ω S ].
The ROM is finally computed using N4SID. Different model orders N order ∈ [1 : 55] have been tested since there is no explicit rule to select it. Even if the validation data set is recorded with the same settings as the ones generating the perturbations, it is independent from the learning data set. For this state order, Figure 4.16(a) compares the KE resulting from the output POD coefficients of the model (red) and the real ones (blue). Even if the amplitudes do not match during the A passing T-S wave is depicted by three successive instantaneous fields in Figure 4.17(a,d,g). Their corresponding POD reconstructions are given in Figure 4.17(b,e,h). As expected from Figure 4.12(a), most of the large-scale structures are well contained by those 17 POD modes. The noise of the velocity fields is filtered by their projection onto the POD modes. Finally the corresponding output of the system are shown in Figure 4.17(c,f,i). The fields appear to be rather close to their real projections, confirming the results obtained with the KE.

Other tested sensors

Other sensors have been tested: the velocity components alone, the swirling strength criterion λ Ci alone (see chapter 3), then combined with the streamwise component. As shown by Figure 4.18, the streamwise and vertical velocity proved the best sensors for N order ∈ [5 : 37] and N order > 49, whereas it is the vortex identification criterion otherwise. As the vertical component lead to the worst results, the swirling strength criterion and the streamwise component have been tested together. They turn out to only improve the ROM computed for N order = 20. Even if this sensor lead to an error of the same order as the first presented one, the obtained ROM requires a higher state order.

Conclusions and perspectives

A system identification method based on the N4SID algorithm has been successfully implemented for the forced boundary layer flow developing T-S waves. The input data have been obtained from optical flow PIV measurements, retaining in a small area of the velocity fields a double visual sensor of the streamwise fluctuating velocity and either the vertical fluctuating velocity or the swirling strength criterion. The KE increase induced by the T-S waves propagating along the flat plate has been identified with acceptable agreement between the model and the real flow via a POD.

Discussion

The computation of the POD modes and coefficients requires large usage of memory regarding the dense fields obtained via optical flow. The high data density inherent to this velocimetry method may thus become a drawback in the present case, as it was foreseen by Champagnat et al. (2011). Nonetheless, spatially down-sampling the field turns out to be a good solution, giving the choice (if the hardware allows it) between high space accuracy and computation cost, the reduced-order model being still available.

A larger PIV window would be better to capture a longer T-S waves evolution, giving more distributed POD modes along the flat plate. The description of the flow dynamics would be improved but it would consequently increase the system state order. This would require to change the camera objective to keep the spatial accuracy without extending the number of vectors and therefore the size of the learning matrix.

Perspectives

The motivation of the presented work is to delay the laminar-turbulent transition by suppressing the captured T-S waves. Further works will therefore concern the closed-loop control based on the flow reconstruction (Ottonelli, 2014; Guzmán Iñigo et al., 2014) since the actuator is already installed. The region investigated in this chapter contains indeed a slotted jet designed like the one forcing the T-S waves, which was covered with a plate visible in Figure 4.5. This experimental application will first require a similar study as the one presented here, including an additional random forcing by the actuator. The command signal sent to the actuator will be associated to the sensor(s) to create the input u of the system. The reduction of the KE may be finally a good objective to fix to the built system since it can also be tracked in real-time.

Introduction

The flow over a real vehicle is very complex but most of the aerodynamics forces are related to various flow separations over the front part (A-pillar vortices, wake of the rear-view mirrors) to the massive flow separation over the rear part, which is responsible of 50% to 70% of the drag, depending on the geometry. Studying the structure and dynamics of the three-dimensional (3D) wake downstream of the vehicle is then crucial to optimize and reduce the overall drag and related fuel consumption [START_REF] Hucho | Aerodynamics of road vehicles: From fluid mechanics to vehicle engineering[END_REF]. As a consequence, one can focus on simplified geometries that could mimic the wake of a real vehicle.

The turbulent wakes downstream of 3D bluff bodies, like cylinders (circular or square) [START_REF] Williamson | Vortex dynamics in the cylinder wake[END_REF] [START_REF] Castro | The flow around a surface-mounted cube in uniform and turbulent streams[END_REF][START_REF] Klotz | Experimental investigation of flow behind a cube for moderate Reynolds numbers[END_REF], either wall-mounted or in the freestream, can also be multifaceted, exhibiting large-scale and small-scale coherent structures, either spanwise or streamwise, with strongly intermittent behaviors. Passenger cars or trucks can be considered as 3D bluff-bodies interacting with a wall (underbody flow). Free-stream direction is given by the blue arrows (no drift).

A model to

In response to the need of a simplified 3D model for automotive aerodynamics studies, the socalled Ahmed body [START_REF] Ahmed | Some salient features of the time-averaged ground vehicle wake[END_REF] has thus been designed obeying a reflectionnal symmetry.

Based on the previous works of Morel (1978), its adjustable shape has been essentially made to investigate the drag due to the rear part of ground vehicles. No wing mirrors to keep clear of structures generation. No heat exchanger to avoid a pressure drop. Wheels are replaced by cylinder supports. In this geometry, only the rear-slant angle α can be changed (0 • α 90 • ) as illustrated by Figure 5.1. This simple modification leads to drastic changes in the overall structure of the wake, together with large variations of the aerodynamic drag coefficient. This angle enables to select a simplified rear profile from the sport utility vehicle (SUV) one (slanted configuration) to the truck or bus one (square-back configuration).

Historically, the Reynolds number for this flow is based on the height of the model H,

Re H = U ∞ H ν air , (5.1)
where ν air is the kinematic viscosity of the air at ambient temperature. This definition enables to gather and compare numerous investigations, which involve reduced scales of the bluff body and different aspect ratios AR = H/W .

Until recent improvements in the experimental (TR-PIV) and the numerical instruments (LES), the first analysis of the wake rested essentially on time-and space-averaged values, without obtaining a complete description of the wake unsteady behavior.

The slanted rear configuration

The slanted configuration described in Figure 5.2 involves three mean flow arrangements, determined by two threshold angles: α m ∼ 12.5 • and α M = 30 • . Most of the studies focus on the Ahmed body with α = 25 • slant angle, since the comprehension of these wake dynamics is still challenging, as well as its control.

< α α m

The flow coming from the roof keeps being tangential to the slanted surface, before it separates at the sharp rear edge. This separation occurs also at each other edge of the base area and creates consequently a counter-rotary vortex system with a toroidal shape, defining the contour of recirculation bubbles behind the vertical base (Gilliéron, P. & Chometon, F., 1999). In this configuration, the drag is essentially due to the massive depression contained in the bubbles [START_REF] Ahmed | Some salient features of the time-averaged ground vehicle wake[END_REF]. As α increases, the flow over the slanted surface encounters a larger adverse pressure gradient, making the contribution of this area to the drag more important. Concurrently, the recirculation bubbles decrease, up to the base surface reduction, and it finally results in a total drag lowering until α ∼ α m . α m α ≤ α M A first separation of the incoming flow happens at the front edge of the inclined surface, followed by a reattachment, giving birth to a recirculation bulk over the upper part of the surface. The bulk is composed of small-scale vortices, which aggregate inside until being advected far from the surface by the shear layer between the bulk and the freestream [START_REF] Krajnović | Flow around a simplified car. Part 1: Large eddy simulation[END_REF]. The position of the reattachment lines fluctuates, such as the flow may be fully separated periodically at large values of α [START_REF] Hinterberger | Large eddy simulation of flow around the ahmed body[END_REF][START_REF] Thacker | Experimental contribution to the steady and unsteady analysis of the flow behind a bluff body[END_REF].

The separatrix adopts a D-shape, occasioned by the top corners, from where a pair of counterrotating longitudinal vortices develops downstream. Each of these vortices is supplied continuously by the flow coming from its lateral edge [START_REF] Lehugeur | Characterization and control of longitudinal vortices in automotive aerodynamics[END_REF][START_REF] Rouméas | Separated flows around the rear window of a simplified car geometry[END_REF] and intermittently by small-scale structures from the bulk [START_REF] Minguez | High-order large-eddy simulation of flow over the "ahmed body" car model[END_REF]. Then, it follows a slightly deviated line from the inclined surface [START_REF] Spohn | Flow separations generated by a simplified geometry of an automotive vehicle[END_REF][START_REF] Leclerc | Réduction de la traînée d'un véhicule automobile simplifié à l'aide du contrôle actif par jet synthétique[END_REF], going downwards into the upper wake, where it is dissipated quite far away [START_REF] Lienhart | Flow and turbulence structures in the wake of a simplified car model (ahmed modell)[END_REF]. The complex structure of such vortices above the slanted surface still interacts with the bulk by delimiting its reattachment line close to the sides [START_REF] Spohn | Flow separations generated by a simplified geometry of an automotive vehicle[END_REF][START_REF] Krajnović | Flow around a simplified car. Part 1: Large eddy simulation[END_REF][START_REF] Vino | Flow structures in the near-wake of the ahmed model[END_REF]. These vortices are well studied because their path corresponds to the lowest pressure region [START_REF] Krajnović | Large-eddy simulation of the flow around simplified car model[END_REF]Beaudoin et al., 2004), making them responsible for the so-called induced drag (like the ones produced by the end of the airplane wings).

The recirculation region behind the vertical base is composed of two bubbles: a large one above a small one. The large recirculation feed the counter-rotating longitudinal vortices [START_REF] Venning | The effect of aspect ratio on the wake of the Ahmed body[END_REF]. The interaction between the incoming underflow and the small recirculation makes the flow separate from the floor, generating a transverse bulk (Drouin V. & P., 2002; Joseph, 2012). The location of this separation seems to oscillate with the location of the recirculation reattachment. Downstream of this bulk, a horseshoe vortex arises [START_REF] Venning | The effect of aspect ratio on the wake of the Ahmed body[END_REF]. Each bubble are included in a horseshoe vortex (Venning et al., 2017). It can be noted than in some studies where the lateral cylindrical supports are missing, a longitudinal vortex develops along the lower part of each lateral surface [START_REF] Krajnović | Large-eddy simulation of the flow around simplified car model[END_REF][START_REF] Strachan | The vortex structure behind an ahmed reference model in the presence of a moving ground plane[END_REF] and merge with the pair of counter-rotating vortices and the horseshoe vortex [START_REF] Venning | The effect of aspect ratio on the wake of the Ahmed body[END_REF]. The main hypothesis to explain the absence of such structures in other studies is that the depression caused by the supports suppress them.

The unsteady flow consists thus in a competition between the large-scale streamwise longitudinal vortices, the spanwise Kelvin-Helmholtz vortices emerging from each separation zone, the recirculation bulks oscillations, and the twin horseshoe vortices over the base surface. As α rises, the the bulk enlarges in average and the two longitudinal vortices amplifies. Therefore, the global drag critically increases until α = α m . α M ≤ α The bulk and the bubbles behind the base area becomes suddenly a unique large separated region.

The square-back Ahmed body

Like the slanted configuration of the Ahmed body, the flow around the square-back version (α = 0 • ) has been considerably investigated since it corresponds to the profile of a rear truck. Yet, if the mean fields are relatively simple and well defined, the instantaneous fields remain complicated and exhibit together large-and small-scale structures, leading to a very puzzling 3D turbulent flows. Indeed, time-averaged velocity and pressure fields reveal a toroidal vortex in the near wake [START_REF] Duell | Experimental study of a ground vehicle body unsteady near wake[END_REF][START_REF] Krajnović | Numerical study of the flow around a bus-shaped body[END_REF] and spectral analyses highlight a low-frequency mechanism, the bubble pumping [START_REF] Duell | Experimental study of a ground vehicle body unsteady near wake[END_REF][START_REF] Volpe | Experimental characterization of the unsteady natural wake of the full-scale square back Ahmed body: Flow bi-stability and spectral analysis[END_REF].

Bimodality Recently, a right-left switching of the global turbulent wake has been observed experimentally (Grandemange et al., 2013a) and numerically (Östh et al., 2014), defining the so-called reflectional symmetry breaking (RSB) modes and leading to the bimodal (or bi-stable) wake. In fact, this phenomenon emerges in the laminar regime (Grandemange et al., 2012b;Evstafyeva et al., 2017) and is maintained when increasing the Reynolds number. For such dynamics, two different time scales are notable: a long one, T * RSB ∼ 10 3 H/U ∞ , is associated to the symmetry-breaking process, whereas the switching itself is characterized by a short time scale T switch which will be discussed hereafter. The bimodality refers to two metastable asymmetric states of the wake but the switching process suggests a third one, less stable. In fact it will be shown that this mode can be defined experimentally, as the one found numerically by [START_REF] Kaiser | Cluster-based reduced-order modelling of a mixing layer[END_REF] when they statistically computed a cluster-based reduced-order model of the wake. Finally the bimodality is sensitive to other geometric parameters like the aspect ratio of the bluff-body's cross-section AR and the ground clearance C, which can even trigger a vertical bimodal behavior as shown in Figure 5

Experimental setup

The dynamics of a 3D bimodal turbulent wake downstream of a square-back Ahmed body are experimentally studied in a wind tunnel through high-frequency wall-pressure probes mapping the rear of the model and a horizontal two-dimensional (2D) velocity field.

Bluff body

The blunt body is a 0. floor with a properly profiled leading edge, to ensure a fully turbulent incoming boundary layer, and an adjustable trailing edge to avoid undesired flow separations (see Figure 5.5). The ground clearance is set to C = 5 cm, higher than the boundary layer thickness (∼ 2.1 cm). In the following, the free-stream velocity is U ∞ = 30 m.s -1 , corresponding to Re H = 3.9 × 10 5 . The origin is located on the rear of model (x = 0), in the vertical symmetry plane (y = 0) and on the raised floor (z = 0). Nondimensionalization is applied to spatial variables such as x * = x/H, y * = y/H, z * = z/H, while the convective time and the Strouhal number are respectively defined as t * = tU ∞ /H and St H = f H/U ∞ . Although this kind of definition of the Strouhal number does not suit correctly to qualify all the observed dynamics in separated flows [START_REF] Roshko | On the wake and drag of bluff bodies[END_REF][START_REF] Bearman | On vortex street wakes[END_REF][START_REF] Griffin | Universal similarity in the wakes of stationary and vibrating bluff structures[END_REF][START_REF] Williamson | A series in 1/ √ Re to represent the Strouhal-Reynolds number relationship of the cylinder wake[END_REF][START_REF] Ponta | Effect of shear-layer thickness on the Strouhal-Reynolds number relationship for bluff-body wakes[END_REF], it is used here to help comparisons with other studies using it.

Measurements and the Holy Grail: No intrusion

Pressure

The wall-pressure over the rear part of the model is studied using a set of 95 taps defining an area denoted S p and covering 70% of the entire rear surface S r = H × W , as shown on Figure 5.4. Each hole is 2 cm away from each of its closest neighbors and is connected through a vinyl tube to a 32-channels microDAQ pressure scanner (Chell Instruments, UK), insuring an accuracy of ±17 Pa and located inside the body. The scanner computes the pressure difference between the rear pressures p(i, j) and the reference static pressure p ∞ , measured by a Prandtl tube located above the leading edge of the raised floor. This tube sends also the reference dynamic pressure, q ∞ = 1 2 ρU ∞ 2 , synchronously to easily calculate C p (see Equation 5.2). Therefore, a total of three scanners are linked to an external computer over Ethernet, through a cylindrical pipe of diameter 32 mm at the center of the bottom face (see the gray part in Figure 5.4). The need of such a pipe comes from the pneumatic tubes for the actuators introduced in chapter 7. Even if this pipe does not break the reflexive symmetry of the body, it has certainly an influence on the near wake, which will be discussed. The data are processed by a Chell software, then stored in the RAM during the record time and finally given in a file per scanner. The number of samples per acquisition is bounded to N = 30 000 1 , making the sampling frequency for the pressure f * P dependent on the acquisition time T * P : f * P = N/T * P . Only the 95 pressure values p(i, j) measured by the scanners are used to calculate the pressure-linked values below.

The pressure coefficient C p (t) is defined as:

C p (t, i, j) = p(t, i, j) -p ∞ (t) q ∞ (t) , ( 5.2)
where p(t, i, j) is the local pressure (i,j) measured at time t, and S r is the rear area of the bluff body mentioned above. A typical instantaneous pressure coefficient field is shown in Figure 5.7(a). It shall be mentioned that such a smooth imagery is obtained by using a cubic interpolation 2 . Denoting the space average of a quantity a over an area S as a S , we call for convenience the 1. After N recorded samples, a lag occurs, shifting the next N samples in time.

2. Thanks to the Matlab files inpaint_nans, from J. D'Errico, and interp2.

instantaneous spatially-averaged pressure coefficient as the base coefficient C b (t)

C b (t) = C p (t) Sp . (5.3)
We are interested in the large-scale dynamics of the global wake. This is the reason why we define the instantaneous wall-pressure barycenter at x * = 0, which can be seen as the footprint of the wake,

----→ OG p (t) = y * p (t) z * p (t) =    y * p(t) Sp p(t) Sp z * p(t) Sp p(t) Sp    .
(5.4)

Velocity

The air flow is seeded with oil droplets via a nozzle located just downstream of the test section.

The velocity fields are obtained using the aero configuration given in chapter 2, able to stream the 2048 px × 2048 px snapshots to the LabVIEW program. Because it is the best suited to characterize the bimodality, the main investigated PIV plane is the near wake horizontal XYplane at z * = 1, displayed in Figure 5.5, with a 0.21 mm.px -1 resolution. Other PIV planes are studied.

XZ-plane at y*=0 0 XZ-plane at y*=0

XY-plane at z*=1

x* -4.2 1.4 1.6 The 2D velocity fields are computed at the frequency f * P IV = 2.8 × 10 -2 (f P IV = 4.2 Hz) using the optical flow algorithm described in chapter 2. The interrogation window size is 16 × 16 pixels and the calculation is based on three iterations for each of the three pyramid reduction levels. An example of a 2D instantaneous velocity field is shown in Figure 5.7(b). One can see the complexity of the turbulent wake, with large and small-scale strongly fluctuating vortices.

In the same way as the pressure center we define another global indicator of the state of the wake through the computation of the instantaneous recirculation intensity barycenter (also called

* = 1, ------→ OG rec (t) = x * rec (t) y * rec (t) =   x * urec(t) Arec(t) urec(t) Arec(t) y * urec(t) Arec(t) urec(t) Arec(t)   , ( 5.5)
where u rec (t) = u rec (x * , y * , t) and A rec (t) are respectively the local streamwise component of the recirculation velocity and the recirculation area at time t. The instantaneous barycenter of the depression can thus be tracked at each time step. An example of the evolution in time is given in Figure 5.21 for y * p (a) and y * rec (b).

Aerodynamics forces

The bluff body is linked to an aerodynamic balance through a supporting turntable (see the orange part in Figure 5.5). As the influence of the free-stream orientation is not studied here, the yaw angle β is fixed to 0 • . The drag and the lift forces are respectively measured with an accuracy of 0.08 N and 0.24 N. The precision for the drift measurements is the same as the drag one.

Investigations on the sensitivity of the balance to the model vibrations, like those provoked by actuators (see chapter 7), have highlighted possible bias in measurements when forcing is active. Consequently, the mean aerodynamics indicators will be only presented for the reference case.

Time-averaged characteristics of the unforced wake

First, the incoming flow is described. The wake topology is then characterized before presenting and discussing the aerodynamics behavior together with the rear pressure maps and the velocity fields.

Incoming flow

The time-averaged velocities are first computed over 510 fields at the front of the bluff body in a XZ-plane at y * = 0. Close to the wall, it encounters an adverse pressure gradient near the end of the curvature, making it strongly slow down for a short distance. This region of the mean flow is detailed in Figure 5.9 for three different Reynolds numbers. It shall be specified that the instantaneous fields are very similar to the mean fields. The mean flow over the roof has previously been detailed by [START_REF] Lahaye | Aerodynamic characterisation of a square back bluff body flow[END_REF]. Only its behavior at the top trailing edge is thus described, moving the previous longitudinal median PIV The boundary layer is first evaluated from the dense PIV data at the top trailing edge for Re H = 3.9×10 5 . The profiles of streamwise velocity [Figure 5.10(a)] and of its standard deviation [Figure 5.10(b)] are characteristic of a turbulent boundary layer: highest fluctuations close to the wall reduce the loss of momentum with respect to the free-stream. This is confirmed by the computation of the shape factor H = 1.35, the displacement thickness and the momentum thickness being respectively δ * = 0.036 and θ * = 0.027. The nearby separation tends nonetheless to increase the loss of momentum, as attested by the change in the curvature of the velocity profile between the trailing edge and an upstream position in Figure 5.10(a).

Mean wake topology

The same average wake organization as Lahaye et al. The time-averaged drag coefficients for Re H = [2.6; 5.1] × 10 5 are displayed in Figure 5.14(a), together with data from previous studies. Each point results from two independent 2-min runs, sampled at the frequency f * aero = 6.7 (f aero = 1000 Hz). The drag coefficients are only comparable when model geometries (ground clearance, aspect ratio) are close to each other (Table 5.1), which explains some important differences between the references. It is noteworthy that the experiments involving the central pipe have the highest drag coefficients. Nevertheless, most of the studies exhibit a Reynolds effect, C x Taero depending on Re H until a critical value. In the present study, this value is Re H = 3.9 × 10 5 . The influence of the Reynolds number is also visible for the lift coefficient in Figure 5.14(b). Beyond the critical Reynolds number, this coefficient remains constant: C z Taero ∼ -4.0(±0.22) × 10 -2 . As there is no yaw angle (β = 0 • ) breaking the reflexive symmetry of the experiment, the mean drift statistically cancels: C y Taero ∼ 1.2 × 10 -4 . The base pressure is given for different Reynolds numbers and compared with previous studies in Figure 5.15(a). The depression seems curiously 3 less stronger than the ones previously measured for the same bluff body (Eulalie, 2014;[START_REF] Lahaye | Aerodynamic characterisation of a square back bluff body flow[END_REF]. Indeed Figure 5.15(b) shows a 56% mean base-pressure-to-drag ratio for the present wake, whereas it was 62%. Even so, the base suction represents the dominant part of the drag and its recover remains a good objective for drag reduction. Finally -C b Tp / C x Taero linearly increases with Re H since the viscous forces tangential to the surface decline.

Global aerodynamic indicators

Unsteady characteristics and large-scale dynamics

In this section, several representations of the measurements will be displayed to precisely describe the unsteady natural wake through the large-scale dynamics in the flow control perspective. If there is no precision, the Reynolds number is always Re H = 3.9 × 10 5 .

Unsteady characteristics

Studying successive instantaneous fields gives a more complete description of the involved in-wake phenomena. Thereby, λ Ci -criterion highlights the vortex roll-up due to the Kelvin-Helmholtz instability along the upper and lateral shear layers in Figure 5.17(d-e-f) and in Figure 5. 16(d-e-f) respectively. As well as in the backward-facing step wake (chapter 2) or in the circular cylinder wake [START_REF] Kovasznay | Hot-wire investigation of the wake behind cylinders at low Reynolds numbers[END_REF], those growing perturbations will here also cause the downstream vortex shedding. The bottom vortex of the toroidal structure can be either far from the rear surface [Figure 5.17(a)] or close to it [Figure 5.17(c)]. Even if the recirculation area globally keeps it shape and size of the same order of the bluff body height, it can be elongated [Figure 5.17(c)] or even strongly collapses [Figure 5.16(c) and Figure 5.17(b)].

In fact, the recirculation area seems to have periodic dynamics. Plotting its time evolution in Figure 5.18(a) and (c), a long-time periodic pattern, evaluated at T * ∼ 3.5 × 10 3 , partly appears for the smoothed data. This can be referred to the periodic oscillations of the recirculation length in the BFS wake as reported by Aider et al. (2007).

When analyzing the power spectra of the pressure sensors along the height of the model for various Reynolds numbers, Figure 5.19 reveals that the low frequencies dominate, especially in 3. Investigations have been a posteriori made and are still in progress to understand these values which may due to air temperature, update of the pressure scanners... Along the bottom trailing edge, there is clear peak at St H = 0.19. This peak is also present at the middle of the upper trailing edge [Figure 5.20(d) and (g)]. There is a peak at St H ∼ 0.3 along the upper edge. There is a peak at St H ∼ 0.4 along the upper edge. The centre of the rear surface seems to have higher level of energy according to Figure 5.20(b-e-h).

Bimodality and more

As we are interested in the global wake dynamics, the probability density function (PDF) is computed for both barycenters. Regarding their spanwise position, Figure 5.21 shows clearly the bimodal behavior of the turbulent wake, whether it is through the pressure barycenter PDF P y * p (b) or the recirculation barycenter PDF P y * rec (d).

Indeed each barycenter moves from one side to the other [Figure 5.21(a) and (c)], between two more probable locations, one located in the negative part of the y * -axis and the other one in the positive part. This is the kind of sidewise bimodal configurations that is expected for a square-back bluff body with such geometric parameters (aspect ratio H/W = 0.743, and ground clearance C/W = 0.248), as shown by Grandemange et al. (2013c) at Re H = 3.3 × 10 4 (Figure 5.3). The difference in the peaks value is due to the low acquisition time as observed by respectively alternate between two positions. In fact, their respective PDFs, P z * p and P z * rec , show only one mode, but smoothing the pressure data over ∆t * = 1.5 × 10 2 (and even less) exhibits a second peak. Unfortunately this second peak is difficult to obtain from the PIV data due to the sample size. Nevertheless, it suggests the existence of a weak top-down switching, hidden by the strong transverse oscillation.

The streamwise position x * rec seems to fluctuate around one mean position as its PDF is roughly Gaussian and reveals one peak.

The evolution of the wake in the XY-plane at z * = 1 and in the XZ-plane at y * = 0 during a switch from a RSB mode to another one are respectively displayed in Figure 5.24 and Figure 5.25. The two sets of successive instantaneous fields presented here have been selected regarding y p and for a transition long enough each to obtain comprehensive fields and understand the process. They are representative of most of the observed transitions in the data set but one has to keep in mind that other phenomena are captured. Figure 5. 24(a) shows that, in an asymmetric state, one of the lateral vortex defining the toroidal structure is near the rear surface, whereas the other lateral vortex is 1H distant from it. The closest vortex to the body gives the depression localization. The recirculation area in the XZ-plane spread along 0.75H in height and the bottom vortex seems 0.5H distant from the body.

Thus the same lateral symmetry-breaking mechanism is observed in the wake topology as the ones presented by Grandemange et al. (2013a); Cadot et al. (2015); [START_REF] Volpe | Experimental characterization of the unsteady natural wake of the full-scale square back Ahmed body: Flow bi-stability and spectral analysis[END_REF]. Figure 5.26 shows that other physical quantities enable to track the bimodality, such as the spanwise pressure gradient ∂C p /∂y * and the drift coefficient C y . The further analyses about the pressure center hold consequently for these quantities. The effects of the modified ground clearance on the wake reversal behind a square-back bluff body have been studied by [START_REF] Barros | Forcing symmetry exchanges and flow reversals in turbulent wakes[END_REF]. It was reported in particular that the presence of a circular cylinder, very similar to the central pipe in our experiments but closer to the rear part, changes the vertical position of the modes but does not cancel the lateral bimodality. (d-e-f) Re H = 3.9 × 10 5 , and (g-h-i) Re H = 5.1 × 10 5 .

Relation between the recirculation bubble and the pressure over the rear part

The relation between the dynamics of the wake, characterized by its instantaneous recirculation area, and the dynamics wall pressure is not straightforward because of the complex 3D flow.

To study the linear part of their relationship, the cross correlations between y * P and y * rec , and between z * P and z * rec have been fist computed for five T * = 8.9 × 10 3 runs and one T * = 1.8 × 10 4 run respectively:

Γ y * p ,y * rec (τ * ) = y * p (τ * ) ⊗ y * rec (τ * ) and Γ z * p ,z * rec (τ * ) = z * p (τ * ) ⊗ z * rec (τ * ), (5.6)
where τ * is the dimensionless evaluated lag. Even if the results are similar with the raw data, The wake states can therefore be characterized either by the pressure barycenter or by the recirculation intensity one. Due to the higher sampling rate, we essentially present results in the following regarding pressure measurements for several independent experiments of various lengths acquisition runs (T * P = {1.8; 4.4} × 10 4 ).

Large-scale activity

For different Reynolds numbers, the power spectra density (PSD) of the sidewise position y * p respectively, contain high amplitude at a very-low-frequency range (St H < 0.02). The observation of high energy level in this part of the spectrum is common in measurements close to the rear body (Grandemange et al., 2013a;[START_REF] Volpe | Experimental characterization of the unsteady natural wake of the full-scale square back Ahmed body: Flow bi-stability and spectral analysis[END_REF]. Considering the average time spent in a RSB mode (∼ 7 × 10 2 ), it may be associated to the lateral bimodal behavior. Recent frequency measurements in a simulated turbulent wake behind a circular disk confirm the presence of two close low frequencies: 0.02 and 0.03 [START_REF] Yang | Low-frequency characteristics in the wake of a circular disk[END_REF]. The lowest one is associated to the asymmetric changes in the wake orientation at such low frequency, whereas the highest one is related to the bubble pumping. Similar results have been obtained for a turbulent axisymmetric body wake by [START_REF] Gentile | Lowfrequency behavior of the turbulent axisymmetric near-wake[END_REF]. Through a POD analysis of the velocity fluctuations, they find very-low-frequency peaks, St H ∼ 10 -4 and St H = 10 -3 , in the dynamics of the modes associated respectively to the symmetry breaking process and to the bubble pumping. Barros et al., 2016), meaning that the z-position of the depression over the rear part is influenced by these far wake dynamics. This natural feedback inside the wake has been reported by Broze & Hussain (1994) for a turbulent axisymmetric jet, where the spectral signature of vortex pairings downstream of a nozzle appears in the velocity fluctuations measured at the nozzle exit. For the turbulent axisymmetric body wake, the characteristic vortex shedding frequency also appears in the spectrum of the POD mode describing the switching behavior [START_REF] Gentile | Lowfrequency behavior of the turbulent axisymmetric near-wake[END_REF]. Physically, the feedback is due to upstream propagating pressure fluctuations.

The vortex roll-up generated by the Kelvin-Helmholtz (KH) instability and visible for instantaneous velocity fields in Figure 5.16 and in Figure 5.17 cannot be detected through PSD. In fact, their high frequency can be estimated from the instantaneous velocity fields using the method based on the vortex detection described in chapter 3. As a reminder, computing the λ Ci -criterion in a region of interest which comprises the starting shear layer enables the detection of the KH vortices, as shown in Figure 5.30(a). Depending on the λ Ci values, a wavenumber Λ * KH,i and a convective velocity U * KH,i are estimated for the instantaneous field i, giving an estimate of the instantaneous KH frequency f * KH,i = U * KH,i Λ * KH,i [Figure 5.30(b)]. Applying this method to the upper and lateral shear layers returns respectively f * KH,up = 3.6 ± 0.8 and f * KH,lat = 3.6 ± 0.9. The deviation remains important since the instability is not steady. Nevertheless, these values are in good agreement with f * KH = 3.8 predicted by the scaling law Re 0.52 for such high Reynolds number from [START_REF] Thompson | The shear-layer instability of a circular cylinder wake[END_REF], rather than the model mentioned in chapter 3 for lower Reynolds numbers. (d-e-f-j-k-l) λ Ci -criterion of three successive instantaneous velocity fields, together with the separatrix (white), in the rear XZ-plane at y * = 0 during spanwise switching of the wake.

Thus, the dynamics of the depression seem to be rather dominated by the spanwise bimodal behavior of the recirculation and the top-bottom shear layers interaction. The central pipe may to the turbulent perturbations growth (Brackston et al., 2016). The positions of the pressure barycenter G P for successive stays in the RSB modes are displayed in Figure 5.31. The most probable location (red square plots) near which G p evolves during each stay alternates between two opposite spanwise positions, keeping the same vertical position [Figure 5.31(a-c-e)]. During most of the flipping processes, the pressure center goes around a central position y * p ∼ 0, as shown by Figure 5.31(b-d-f-h). In average, this step lasts for well shorter time than the RSB modes. It is noteworthy that, after being in this transient state, the wake may also go back in the previous RSB mode as illustrated by Figure 5.31(e-f-g).

Transient symmetric state

As mentioned hereinbefore, three states can be considered for the system: two corresponding to the RSB modes, and a third one being a transient mode. For a more quantitative analysis, two thresholds for y * p (t) has consequently been defined. Above a given positive threshold y * p,P , the wake is in the P mode, and lower than a given negative threshold y * p,N , it is in the N mode. Between these values, it is the transient state. ) , (

where P X is the PDF of the variable X, X is the temporal mean of X, and w th is a positive weight to avoid too large thresholds (w th = 0.25 is chosen). From now on, the state of the wake is defined as negative (N), positive (P) or transient symmetric (TS), where respectively y * p < y * p,N , y * p > y * p,P and y * p,N < y * p < y * p,P .

Even if the switching process seems to randomly occur, some characteristic time scales can be statistically estimated. Hence, the time the wake spent on each mode is evaluated from all runs, giving T * RSB ∼ 7 × 10 2 for each asymmetric mode, whereas the switch itself lasts for T * switch ∼ 4 × 10 1 . As expected, these results have a high dispersion caused by the turbulent noise, their respective standard deviations being σ T * RSB = 10 2 and σ T * switch = 10 1 . A mean switching frequency is also computed by counting the switching occurrences during runs: f * switch ∼ 3.4 × 10 -3 . In probability terms, the RSB modes exist 83.4% (±5.3) of the time (41.1% for the N state, and 42.3% for the P one), whereas the wake is in the unstable T S mode during the remaining 16.6% (±1.2). This is the reason why the present wake can also be considered thereafter exhibiting a trimodal dynamics. For each area a quasi-attractive center can be identified, such as

y * C 1 ∼ -y * C 2 and z * C 1 ∼ z * C 1 .
During the switch between these two positions, the pressure barycenter follows preferentially a trajectory along a well-defined path.

It is then interesting to visualize the corresponding time-averaged pressure and velocity fields using conditional averaging. Figure 5. 33(a-c) display the conditional time-averaged pressure fields of the three modes, while Figure 5.33(d-f) correspond to their conditional time-averaged velocity fields. The RSB modes presented here have identical pressure and velocity topologies as the ones without considering the transient state [START_REF] Volpe | Experimental characterization of the unsteady natural wake of the full-scale square back Ahmed body: Flow bi-stability and spectral analysis[END_REF]: the depression is localized on the same side as the vortex being the closest to the rear surface. In the TS state, the wake appears to be a perfect mean of the two RSB modes. These results are in good agreement with the transient state revealed in the cluster-based modelling by [START_REF] Kaiser | Cluster-based reduced-order modelling of a mixing layer[END_REF]. Once again, the phase opposition between the pressure and velocity barycenters clearly appears.

The conditional time-averaged velocity fields is also computed in the reflexive XZ-plane from a T * P IV = 18 × 10 3 run. The RSB modes have the same vertical configuration, which is given in Figure 5.34(a). In the TS state [Figure 5.34(b)], the toroidal structure seems to globally move closer to the rear body and the recirculation barycenter is slightly higher than for the asymmetric sates. It shall be mentioned that 80 velocity fields were detected as describing a TS mode to compute the corresponding mean field.

Coherent structures

As we are interested in the dynamics of the large-scale structures, we quantify the spatio-temporal organization of the wall pressure spatial distributions using POD on each entire studied data set. As a reminder, it is an efficient approach to detect coherent structures in turbulent flows (Lumley, 1967;Sirovich, 1987). Thus we apply the POD on the rear pressure coefficient fluctuations cp :

cp (t) ∼ k i=1 a i (t)Φ i , ( 5.9)
where k is the number of POD modes Φ i carrying most of the coherent structures energy and a i are the corresponding temporal coefficients. The energy distribution of the first 20 POD modes is given in Figure 5.35 showing that the first five modes contain 75% of the total energy. The most remarkable results are that the first mode Φ 1 represents 42% of the total energy (Figure 5.35) and that its spatial organization exhibits the global symmetry breaking presented in The spectral signature of the first POD coefficient a 1 is identical to the one of the pressure barycenter, displayed in Figure 5.37. This confirms that the pressure barycenter is a direct measure of the most energetic large-scale coherent structure governed by a long time scale. The POD modes pair Φ 2 and Φ 4 are multifaceted since they contain the peak at St H = 0.19, characteristic of S z * p z * p alone, and they partly share the same higher-frequency spectrum as S y * p y * p . As expected from the Φ 3 spatial organization, a 3 behaves like z * p . Finally, at least three POD modes carrying 30.1% of the total energy have the vortex shedding signature in their spectra, confirming the influence of the top shear layer on the depression dynamics.

Conclusions

The square-back configuration of the Ahmed body has been investigated. The barycenters of the pressure distribution over the rear part of this model and of the intensity recirculation are found highly correlated. Both described the most energetic large-scale structures dynamics, confirming the relation between the large-scale recirculation bubble and its wall-pressure footprint. The switching process and the vortex shedding from the top shear layer seem to dominate the depression dynamics. A third state, transient symmetric, is highlighted when analysing the bimodality.

CHAPTER 6 Turbulent wake as a dynamical system

In this chapter, the 3D full turbulent wake dynamics can be characterized by the single trajectory of its projected barycenter. After recovering a classic chaotic pattern, its inner characteristics are analyzed more thoroughly, by evaluating, in particular, the family of random process to which it belongs, the largest Lyapunov exponent of the system and the correlation dimension. This chapter is an extended version of the article published in Physical Review Fluids [START_REF] Varon | Chaotic dynamics of large-scale structures in a turbulent wake[END_REF]. The most exciting phrase to hear in science, the one that heralds new discoveries, is not 'Eureka!' (I found it!) but 'That's funny ...' ascribed to Isaac Asimov in the source code Fortune (1987)

Quasi strange attractor dynamics

The results, gathered in Figure 5.32, disclose two areas highly preferred by the barycenter, corresponding to the two RSB states and a third one less visited, which is referred as the TS state. This pattern can bring back the strange attractors dynamics to us since two y-axis symmetric points attracts successively the depression center, passing through a preferential path when switching. We are interested here in the characterization of the bimodal oscillation of the near wake in the framework of dynamical systems theory and, more precisely, as a chaotic system1 . A classic example of a chaotic system is the Lorenz attractor and corresponding Lorenz system, whose characteristic butterfly shape is famous (Lorenz, 1963). The so-called Lorenz system is a simplified weather model defined by the set of three ordinary differential equations:

       ẋ = σ(y -x) ẏ = x(R -z) -y ż = xy -βz . ( 6.1)
With the correct choice for the three parameters (σ = 16, R = 45.92, and β = 4), the trajectory plotted in Figure 6.1 exhibits a chaotic behavior, circling in an apparent random manner between two stable attractors.

Since the pioneering work of Lorenz, it has been shown that many biological, natural, or artificial systems, either at very small or very large scales, follow a chaotic dynamics. The brain wave activity (EEG) [START_REF] Bibliography Achermann | Correlation dimension of the human sleep electroencephalogram: Cyclic changes in the course of the night[END_REF] or the heart rate activity (ECG) [START_REF] Fojt | Applying nonlinear dynamics to ECG signal processing[END_REF] can exhibit chaotic behaviors. In some cases, chaotic excitations can be used to study the response of mechanical systems. A variation in correlation dimensions can be used as an indicator of a fracture in the overall mechanical structure [START_REF] Yan | A novel approach to detecting breathing-fatigue cracks based on dynamic characteristics[END_REF]. Chaotic behavior has also been found in trading market time series [START_REF] Orlando | A discrete mathematical model for chaotic dynamics in economics: Kaldor's model on business cycle[END_REF]. One can even find chaotic behavior for large-scale phenomena like earthquakes [START_REF] Matcharashvili | Detecting differences in temporal distribution of small earthquakes before and after large events[END_REF].

The link between turbulence and strange attractors, has been a challenging question since the introduction of the chaos concept in fluid mechanics by [START_REF] Ruelle | On the nature of turbulence[END_REF]. May the wake state be predicted at any time ? In the laminar regime, Depending on the nature and the strength of the turbulences, some dynamics can be categorized as low-dimensional chaotic [START_REF] Ruelle | The turbulent fluid as a dynamical system[END_REF]. For instance, [START_REF] Guckenheimer | Strange attractors in fluid dynamics[END_REF] report that weakly turbulent convective flows contain strange attractors. On the other hand, fully turbulent wakes are rather related to high-dimensional system and, because of this, are well captured by stochastic representations [START_REF] Swinney | Characterization of hydrodynamic strange attractors[END_REF]

Clues to weak chaos

Analysing the signal to know whether the dynamics of the wake are chaotic or stochastic is therefore of prime interest. Different statistical methods may enable to properly categorize the wake [START_REF] Guckenheimer | Strange attractors in fluid dynamics[END_REF][START_REF] Provenzale | Distinguishing between lowdimensional dynamics and randomness in measured time series[END_REF]. Since such computations are only efficient through the use of long time series regarding the natural time scales [START_REF] Eckmann | Liapunov exponents from time series[END_REF], the longest y * p signal is used in this section (T * = 45 × 10 3 ).

Structure function

An effective approach is to study its self-affinity by computing its first order (k = 1) [START_REF] Osborne | Finite correlation dimension for stochastic systems with power-law spectra[END_REF] or its second order (k = 2) (Provenzale et al., 1992) structure function S k , defined as

S k (n) = |y * p (i + n) -y * p (i)| k i , (6.2)
where n is the lag and . i stands for the average over N -n points. According to [START_REF] Mandelbrot | The fractal geometry of nature[END_REF], if y * p is fractal, S k follows a scaling law for small n:

S k (n) ∝ n kh , ( 6.3)
where h is the so-called scaling exponent. If the signal is chaotic then h = 1. If it is stochastic then its power spectrum follows a power-law P SD(St W ) ∝ St -α W and α = 2h + 1.

However, [START_REF] Provenzale | Distinguishing between lowdimensional dynamics and randomness in measured time series[END_REF] demonstrate that S k alone is not enough to conclude and the structure function of the first derivative of the signal y * p , denoted as S k,d , needs to be also computed. Thus, if it is stochastic then S k,d is almost constant and if the signal is chaotic then S k,d (n) follows a scaling law for small n.

Phase space reconstruction

As we only observe a part of the non-linear wake dynamics (rear body pressure and 2D-2C velocity fields of the near-wake), we cannot experimentally access to its complete phase space. But [START_REF] Eckmann | Ergodic theory of chaos and strange attractors[END_REF] and [START_REF] Sauer | Embedology[END_REF] provide the so-called embedding methods to reconstruct a pseudo phase space from scalar N -points time series, which are Y N = {y * p (i)} i=1...N in the present study. Thus, according to the pioneering Takens's time-delay embedding method [START_REF] Takens | Detecting strange attractors in turbulence[END_REF], we build the M state vectors { Ŷi } i=1...M : Ŷi (m) = y * p (i) y * p (i + J) ... y * p (i + (m -1)J) , (6.4) where m is the embedding dimension and J the reconstruction delay2 , implying M = N -(m -1)J.

J is determined first to avoid redundance -J too small, the extra components of Y i are not independent -and irrelevance -J too large, information is lost due to the possible chaotic separation (see the Lyapunov exponent section) -in phase space reconstruction [START_REF] Casdagli | State space reconstruction in the presence of noise[END_REF]. It is thus interesting to estimate the relation between the time series Y M and its shifted version Y M J = {y * p (i + J)} i=1...M for various time-delays J. To manage it, a practical quantity is the mutual information (MI) between two time series X 1 and X 2 , denoted I(X 1 , X 2 ), which gives the information (bits) on X 1 , contained in X 2 [START_REF] Fraser | Independent coordinates for strange attractors from mutual information[END_REF]. The best J to choose corresponds either to the first minimum of I(Y M , Y M J ) if it exists, or to the moment a significant low rate of reduction in MI. Two recently improved algorithms are finally used: one computes directly the MI [START_REF] Cellucci | Statistical validation of mutual information calculations: Comparison of alternative numerical algorithms[END_REF][START_REF] Jiang | Mutual information algorithms[END_REF], and the other derives a quantity from it, the averaged mutual information (AMI) Î(Y M , Y M J ), based on a kernel density estimation ), together with their respected minimum (red diamond) [START_REF] Silverman | Density estimation for statistics and data analysis[END_REF][START_REF] Thomas | An efficient algorithm for the computation of average mutual information: Validation and implementation in Matlab[END_REF]. Figure 6.3(a) and (b) show that the minima of MI and AMI are respectively obtained for J * ∼ 2.9 × 10 2 and J * ∼ 3.7 × 10 2 . The first value is chosen since it corresponds also to the end of the rapid AMI decay.

We then determine the minimum value of m from y * p , following Cao's method [START_REF] Cao | Practical method for determining the minimum embedding dimension of a scalar time series[END_REF], which emanates from the false nearest neighbor algorithm of [START_REF] Kennel | Determining embedding dimension for phase-space reconstruction using a geometrical construction[END_REF]. This method enables also to distinguish deterministic and stochastic time series. The idea is to evaluate how the mean distance between close state vectors in dimension m evolves in the dimension m+1. Over the real embedding dimension, their ratio, denoted E(m), does not change anymore. We look initially for the closest neighbor ŶN (i,m) (m) of each state vector Ŷi (m): min

N (i,m) =i Ŷi (m) -ŶN (i,m) (m) , i ∈ [[1; M ]], (6.5)
where . stands for the Euclidean distance3 . The ratio of the distances in m and m+1 dimensions for Ŷi is

a i,m = Ŷi (m + 1) -ŶN (i,m) (m + 1) Ŷi (m) -ŶN (i,m) (m) . ( 6.6)
Finally E(m) is computed:

E(m) = 1 M M i=1 a i,m . (6.7)
For convenience its evolution is evaluated through E 1 (m),

E 1 (m) = E(m + 1) E(m) , ( 6.8)
and if ∃ m \ ∀k ≥ m, E 1 (k + 1) = E 1 (k) then m is the minimum embedding dimension. In parallel the mean difference between the raw data E * (m) is evaluated with respect to m, relating to N (m) obtained in Equation 6.5:

E * (m) = 1 M M i=1
|y * p (i + mJ) -y * p (N (i, m) + mJ)|. (6.9)

In the same manner as Equation 6.8, the evolution is analyzed through E 2 (m):

E 2 (m) = E * (m + 1) E * (m) , ( 6.10)
and if ∃ k \ E 2 (k) = 1 then the signal is deterministic, otherwise it is stochastic. As shown by Figure 6.4, the raw signal appears stochastic but the LP filtered one is indeed deterministic, confirming the results obtained with the structure function. The minimum embedding dimension seems to be m = 12, which tends towards a high-dimensional system for the low-frequency large scales. It should be noted that a new reconstruction delay J * is computed for the filtered data (J * ∼ 4.4×10 2 and J * ∼ 4.9×10 2 through MI and AMI evaluations respectively).

Thus the complete large-scale bimodal dynamics of such a fully turbulent wake are well composed of a stochastic part and a deterministic part. In the following sections only the chaotic behavior of these dynamics is characterized through the analysis of the LP filtered signal directly denoted as y * p (t) for convenience.

A chaos portrait

There are many ways to characterize and quantify chaos. Among the most popular quantities, one can cite the correlation dimension [START_REF] Grassberger | Characteriztion of strange attractors[END_REF] which gives an estimate of the system complexity and characteristic exponents which give an estimate of the level of chaos in the dynamical system. In this section we focus on the Lyapunov exponent.

Largest Lyapunov exponent

The spectrum of Lyapunov exponents is well known for detecting and quantifying chaotic systems from the experimental time series [START_REF] Wolf | Determining Lyapunov exponents from a time series[END_REF]. Indeed chaos exists if a system is sensitive to its initial conditions. Thus the principle consists in following the evolution of the distance d between two initial neighbouring state vectors in the phase space. For a chaotic attractor, the distance d exponentially increases in time at an average rate equal to the largest Lyapunov exponent (LLE) λ 1 [START_REF] Eckmann | Ergodic theory of chaos and strange attractors[END_REF]:

d(t) = d(0)e λ 1 t . (6.11)
Due to its efficiency on small data sets, Rosenstein's algorithm [START_REF] Rosenstein | A practical method for calculating largest Lyapunov exponents from small data sets[END_REF][START_REF] Sato | Practical Methods of Measuring the Generalized Dimension and the Largest Lyapunov Exponent in High Dimensional Chaotic Systems[END_REF] is applied to the LP filtered N -points time series {y * p (i)} i=1...N , rather than Wolf's algorithm [START_REF] Wolf | Determining Lyapunov exponents from a time series[END_REF]. The M state vectors { Ŷi } i=1...M and their closest neighbor { ŶN (i) } i=1...M as previously to compute the distances evolution,

d i (j) = Ŷi+j -ŶN (i)+j , ( 6.12)
where j = [[f * p t * ]] verifies i + j ≤ M and N (i) + j ≤ M . The distance d(t) is approximated by averaging over i the distances d i (j):

d(j) = d i (j) i . (6.13)
By taking the logarithm of Equation 6.13 with respect to j the slope, extracted through a leastsquare fit, gives directly the LLE. The distance d(t) computed for the LP filtered data is given in Figure 6.5 and it appears to follow the Equation 6.11 for small t * , typically t ∈ [50 : 1500]. The LLE is thus computed in this range.

The estimated dimensionless positive LLE is λ * 1 = 2.8(±0.3) × 10 -4 . The relative error is due to the main difficulty to define the right linear region of the curve to fit. The low frequencies dynamics associated to large-scale structures may thus be considered as a weak chaotic strange attractor.

Correlation dimension

The fractal dimension (or Hausdorff dimension) D of a strange attractor can be rigorously approximated by its correlation dimension D 2 which is directly computed from experimental time series according to the works of [START_REF] Grassberger | Characteriztion of strange attractors[END_REF]. The previous M state vectors { Ŷ } i=1...M are also used to compute the correlation integral function C(r), defined as where Θ is the Heaviside function and r is an evaluated size. The correlation dimension D 2 can be derived from the correlation integral function C(r) which scales as a power-law for small r:

C(r) = 2 M (M -1) M -1 i=1 M j=i+1 Θ(r -Ŷi -Ŷj ), (6.14)
C(r) ∝ r D 2 , ( 6.15)
The correlation integral function of the LP filtered signal y * p is computed using the Grassberger-Procaccia method improved by [START_REF] Ning | Computing correlation integral with the euclidean distance normalized by the embedding dimension[END_REF], who normalize the Euclidean distance in Equation 6.14 by the embedding dimension m. Figure 6.6 shows the resulting correlation integral function in logarithmic scale. Regarding the whole experimental LP signals we obtain D 2 = 0.94 ± 0.01, from a fit over two decades. The result well verifies the validation condition on the length of the studied time series D 2 ≤ 2 log N [START_REF] Ruelle | The Claude Bernard lecture, 1989. deterministic chaos: The science and the fiction[END_REF]. This low value, close to 1, confirms the weak chaotic nature of the low-frequency dynamics, which are even so validated thanks to the phase randomization (PR) test provided by [START_REF] Provenzale | Distinguishing between lowdimensional dynamics and randomness in measured time series[END_REF]: for a stochastic signal the correlation dimension does not change when its Fourier phases are randomized. Figure 6.6 reveals a modified correlation dimension for the surrogate PR data obtained from the original LP time series, D 2,rand = 1.31 ± 0.01, which is indicative of chaotic dynamics.

Telegraph-like signal

Another interesting way to characterize chaotic oscillator is to analyze the properties of its autocorrelation function (ACF). Considering Equation 5.6, the ACF of a given function x(t) is

Γ x (τ * ) = Γ x,x (τ ) = x(τ * ) ⊗ x(τ * ), (6.16)
which will be normalized by Γ x (0). The ACF of a random process has different properties depending on the nature of the system. Two famous models used to describe the behavior of many applied random systems are the noisy harmonic oscillations and the telegraph signal [START_REF] Anishchenko | Correlation analysis of dynamical chaos[END_REF]. The model of telegraph signal is particularly well-suited to describe the statistics of random switching of a bimodal system in the presence of noise, which is a close description of the bimodal wake as previously mentioned. Two main kinds of telegraph signals can be distinguished, namely, the random and quasirandom telegraph signals. The random telegraph signal is characterized by a Poisson distribution of switching moments while quasirandom telegraph signal corresponds to random switching between two equi-probable states (probability of switching events equal to 1 2 ). For instance, the latter is very well suited to the Lorenz attractor [START_REF] Anishchenko | Correlation analysis of dynamical chaos[END_REF].

To characterize the time series as telegraph signals, the first step is then to define two states as the symmetric spanwise positions of the pressure centers given by Equation 5.8. It is then possible to construct a telegraph-like signal Y * p such as where n * 1 corresponds to the normalized mean switching frequency, while the ACF of a random telegraph signal can be linearly approximated on short times by the following function Ψ QR (τ * ): ) is a good approximation of the ACF of the telegraph signal obtained from the raw data. The estimated mean switching frequency n * 1 = 3.2 × 10 -3 is close to the directly measured one f * switch ∼ 3.4 × 10 -3 . The ACF of the telegraph-like signal extracted from the LP filtered time series is well approximated by the linear function Ψ QR (ξ * 0 = 4.0 × 10 2 ) in Figure 6.8(b). From the linear approximation we thus find ξ * 0 = 4.0 × 10 2 , which is not far from the filtered experimental results, T * min = 0.8 × 10 2 for the studied run. The same kind of ACF approximation is obtained for the Lorenz attractor described by Equation 6.1 [START_REF] Anishchenko | Correlation analysis of dynamical chaos[END_REF]. Furthermore the PDF of the switching times t * switch of the raw time series is given in Figure 6.9(a), revealing itself close to a Poisson distribution whose parameter is 2.9 × 10 3 . The mean switching time, which is the parameter of the Poisson distribution, is of the same order as the residential time T * RSB seen previously. Regarding the corresponding LP filtered data, the probability of switching P (T * ) occurring at times T * multiple to ξ * 0 is computed in Figure 6.9(b). The results show that the probability of switching at time T * = ξ * 0 is P (ξ * 0 ) = 0.45, agreeing with the ACF approximation of Equation 6.19. [START_REF] Anishchenko | Correlation analysis of dynamical chaos[END_REF] obtained P (ξ * 0 ) = 0.52 for the Lorenz system of Equation 6.1, which is also close to 0.5 as expected for such a strange chaotic attractor.

Y * p (t) =    y * A N if y * p (t) ≤ 0 y * A P if y * p (t) > 0 . ( 6
Ψ QR (τ * ) =    1 -|τ * | ξ * 0 if |τ * | < ξ * 0 0 if |τ * | ≥ ξ * 0 , ( 6.19)

Conclusions and discussion

The trajectory of the pressure barycenter over the rear part of the model is first characteristic of a stochastic behavior. But the low-frequency part of this signal reveals a chaotic aspect: the instantaneous pressure barycenter circles around two stable areas acting like strange attractors and randomly switches from one attractor to the other. The nonlinear dynamics of the pressure center are therefore investigated by reconstructing its phase space from the spanwise component. The largest Lyapunov exponent is first estimated, leading to a low positive value, characteristic of strange attractors and weak chaotic systems. The weak aspect is then confirmed by the correlation dimension found close to one. Afterwards, the low-frequency signal is transformed and studied as a telegraph signal, showing that its dynamics correspond to a quasirandom telegraph signal. All of these elements tend therefore to describe the dynamics of a complex 3D turbulent wake as a weak chaotic system with physical noise.

In fluid dynamics, the turbulence problem is associated either to chaotic [START_REF] Landau | On the problem of turbulence[END_REF] or to probabilistic [START_REF] Ruelle | On the nature of turbulence[END_REF] representation. Due to the high embedding dimension and the weak Lyapunov exponent evaluated, the "chaotic" term has to be consequently used with caution for the present turbulent wake. The deterministic model of the turbulent wake is indeed very difficult to completely prove [START_REF] Noack | On chaos in wakes[END_REF]) because of the turbulent noise itself. Nevertheless, a research team from CEA has recently obtained very close experimental results for the turbulent Von Kàrmàn flow [START_REF] Faranda | Stochastic chaos in a turbulent swirling flow[END_REF], applying the same tools from dynamical system theory. Qualifying the dynamics as "stochastic chaos", they suggest to use the stochastic Duffing attractor to model the flow.

Apart from its fundamental interest, this result is also of great practical interest. Indeed, if the wake dynamics can be modelled as a chaotic attractor, it opens the path to different closed-loop control strategies, similar to the one used to control chaotic systems. by selecting either the vortex shedding frequency for an asymmetric forcing configuration, or its subharmonic for a symmetric forcing configuration (Barros et al., 2016a). Inversely, selecting higher actuating frequencies damps the dynamics (Barros et al., 2016b). Feedback controls achieve the suppression of the RSB modes, based either on a stochastic model of the complete wake dynamics using flaps (Brackston et al., 2016), or on an opposition strategy using lateral slit jets [START_REF] Li | Feedback control of bimodal wake dynamics[END_REF]. Drag modifications have also been investigated on modified square-back bluff-bodies, closer to real ground vehicles. For example, [START_REF] Grandemange | A study of wake effects on the drag of ahmedś squareback model at the industrial scale[END_REF] chamfered rear edges and replaced the pods by wheels. They reached a 10% drag reduction via a cavity effect coupled to boat tail effect. McNally et al. (2015) achieved 2.6% by symmetrizing the wake behind a rounded Ahmed body via micro-jets distributed along the lateral and upper faces just upstream of the separation. It is interesting to note how prolific have been the experimental research teams on this subject during the last three years (Table 7.1).

Re In the previous chapter the wake dynamics have been characterized as weakly chaotic with two well defined attractors. The objective is thus to control these chaotic dynamics. Because of the sensitive dependence on initial conditions, chaos was believed for a long time to be neither predictable nor controllable. The OGY control method proposed by [START_REF] Ott | Controlling chaos[END_REF] changed completely this point of view. They showed that it is possible to use small time-dependent perturbations to force a chaotic attractors system in a time-periodic motion, defined from one of its unstable periodic orbits (UPOs) [START_REF] Grebogi | Unstable periodic orbits and the dimensions of multifractal chaotic attractors[END_REF]. Since this pioneering work, control of chaotic systems has become a very active research field and many new control methods have been proposed such as linear-feedback [START_REF] Yassen | Controlling chaos and synchronization for new chaotic system using linear feedback control[END_REF], adaptive control [START_REF] Tian | Adaptive control of chaotic dynamical systems using invariant manifold approach[END_REF], impulsive or intermittent control [START_REF] Li | Exponential stabilization of chaotic systems with delay by periodically intermittent control[END_REF].

Controlling chaotic systems is a challenge encountered in various domains such as laser, plasma, electronics, communications, brain activity or fluid dynamics. For instance, the dynamics of a convective fluid confined in a vertical toroidal channel are well described by a Lorenz system, whose one of the two strange attractors can be removed through a feedback control [START_REF] Singer | Controlling a chaotic system[END_REF]. In simulated fluid flows, the Lorenz model (Lorenz, 1963) is often used as a benchmark case for closed-loop control algorithms. Thereby, a control law is established from limited information on the full Lorenz system and can stabilize it in one of the foci [START_REF] Guéniat | A statistical learning strategy for closed-loop control of fluid flows[END_REF]. Machine learning control is also applied to such system as a training exercise (Duriez et al., 2017). Finally, a first control of a chaotic open flow has been recently experimentally achieved. After slightly forcing the dynamics of an axisymmetric jet into a low-dimensional chaotic attractor (Broze & Hussain, 1994), a closed-loop control is designed to sustain a selected UPO [START_REF] Narayanan | A dynamical systems approach to the control of chaotic dynamics in a spatiotemporal jet flow[END_REF]. The chaotic dynamics correspond to the vortex ring formation downstream of the jet nozzle exit, which includes successive vortex pairings. The control strategy enables to stabilize the ring formation.

The present study aims at controlling the dynamics of the bimodal chaotic oscillations of a 3D turbulent wake by applying a reactive control based on measurements obtained from the real-time PIV. Unlike the previous examples on the control of the Lorenz system, this implies here to steady an unstable symmetric state. In the present study, we define an opposition strategy similar to [START_REF] Li | Feedback control of bimodal wake dynamics[END_REF], but the sensors being the velocity fields and the actuators being discontinuous pulsed jets only localized at the top trailing edge. A closed-loop control using scattered velocity fields from real-time PIV has previously been successfully applied to minimize a pole-cart system plunged into water (Roberts, 2012).

Micro-jets system

To realize the present control, the rear part of the Ahmed body used by Eulalie (2014) has been re-designed by Plastic Omnium R&D engineers, P. Gilotte and Y. Eulalie, to have several independent groups of actuators. The wake of the bluff-body is indeed forced using microjets which are distributed along the rear body width, 10 mm under the upper trailing edge (z * = 1.25). The jets are either continuous or pulsed at the jet frequency f j using three solenoid valves (Matrix R ) integrated inside the model just upstream the blowing jets. Each solenoid valve controls a group of jets, which corresponds to the three regions of actuation defined in Figure 7.1. The pressurized air is supplied through pneumatic tubes passing inside a cylindrical pipe of diameter 32 mm at the center of the bottom face as the one in previous chapters (see the gray part in Figure 7.1). Even if this pipe does not break the reflexive symmetry of the body, it has certainly an influence on the near wake, which will be discussed.

The outlets of the jets have a rectangular cross-section l j × h j = 5.5 × 0.5 mm 2 . The distance between each jet is l = 12.5 mm. The jets angle θ relative to the horizontal axis can be varied but in the following it will be fixed to θ = 0 o (blowing parallel to the freestream velocity). The jet velocity u j can reach up to 75 m.s -1 , i.e. a velocity ratio u * jet = u jet /U ∞ = 2.5. Based on the Strouhal number, the normalized jet frequency is defined as

f * j = f j H U ∞ , ( 7.1)
instead of f j h j u j . The momentum coefficient, which is the ratio between the momentum injected by the actuator and the one due to the bluff body,

c µ = 2 l j h j u 2 j S r U 2 ∞ , ( 7.2)
is evaluated at c µ = 0.053% (continuous) and c µ = 0.013% (pulsed) for each jet when U ∞ = 30 m.s -1 .

The three regions of actuations are called "N", "0" and "P" (Figure 7.1). "N" contains the eight jets located in the Negative part of the y-axis, whereas "P" contains the eight jets located in the Positive part and "0" contains the four central jets (for a total of 20 jets). Due to the temporal constraints imposed for experiments carried in wind tunnel, all recorded control runs lasted only T * = 8.9 × 10 3 . Regarding the aerodynamics, only the changes in the pressure coefficient will be considered to analyze influences of the control on the wake to avoid the structural response visible in the aerodynamics measurements.

Open-loop forcing of the wake

Three different open-loop (OL) control strategies are first investigated to evaluate the forcing on the state of the wake. The three open-loop forcings are defined as"OL-P" when only the "P" solenoids group is activated, "OL-0" when only the "0" solenoids group is activated and "OL-P0" when both the "P" and "0" solenoids groups are activated (Figure 7.1). Due to the reflection symmetry the results obtained with the "P" group are expected to be the same as for the "N" configuration. First we used continuous jets, which means only the forcing region is changed.

Then pulsed jets are investigated, especially at the normalized frequency f * j = 0.2. The influence of the location of the continuous blowing on the wake dynamics is studied by analyzing directly the times series of the pressure center obtained for the three open-loop cases. On one hand, as displayed in Figure 7.2(a) and (c), blowing from one side drives the depression to remain localized in this side. On the other hand, Figure 7.2(b) reveals that the symmetric forcing seems not to affect the spanwise bimodal behavior. For all the forcing cases, the vertical position of the depression becomes closer to the upper trailing edge. The averaged z * p indeed rises 2.1% for OL-P [Figure 7.2(d)], 1.2% for OL-0 [Figure 7.2(e)] and 2.8% for OL-P0 [Figure 7.2(f)].

Continuous blowing in different regions

The previous observations are also confirmed by the 2D PDFs. Figure 7.3(a) shows that forcing with continuous jets the upper shear layer on one side of the model (OL-P) strongly stabilizes the wake oscillations. The pressure barycenter is now kept on the side of the blowing (y * p > 0) and oscillates around a single attractor. If the forcing is extended to the central region (OL-P0), the pressure barycenter is still confined on one side of the model but explores a larger region around Concerning the wake dynamics for OL-0, random switches occurs less often but the weak chaotic behavior in the low-frequency range remains (St H < 2 × 10 -3 ). Indeed, Figure 7.4(a) displays the second order-structure functions of y * p and its derivative previously seen in chapter 5. The n 2 slope of these curves is clearly conserved for higher lags than for the unforced wake. Besides, the functions E 1 and E 2 , computed from Cao's method and plotted in Figure 7.4(b), give respectively a minimal embedding dimension m = 15 and clue to deterministic dynamics, confirming the weak chaotic character. Instead of looking at the trajectories of the pressure barycenter over the rear of the model in the physical space, it is interesting to look at the trajectory in the phase space. Indeed the state of the wake can be characterized not only by the right-left oscillations (bimodality), through the state parameter y * p , but also by the normalized fluctuations of the instantaneous base pressure

coefficient ∆C b (t) ∆C b (t) = C b (t) -C b,ref C b,ref , (7.3

Pulsed blowing in different regions

Finally, the pulsed jets are used with the normalized frequency f * j = 0.2 for the three previous configurations, since this corresponds to the vortex shedding signature found in the up-down oscillations of the depression position for the reference case. At this actuating frequency, the depression behaves almost as previously for the OL-P and OL-P0 cases, but with more regular oscillations in the z-direction and a reduction of the spanwise fluctuations, amplifying the decrease in the base pressure. On the contrary, the OL-0 case is strongly modified: the average spanwise position of the depression is now centered [Figure 7.6(b)], the amplitude of the vertical fluctuations doubled while the mean vertical position increases much more than for the continuous blowing [Figure 7.6(e)]. For these reasons, the symmetric pulsed forcing will be more detailed in the following.

In average the highest pressure values are localized at the centre of the rear surface, circled by the lowest values, as shown in Figure 7.7(a). The fluctuations have a similar spatial organization Actually, the control with pulsed jets at f * j = 0.2 leads to a symmetric PDF with a single central attractor for the trajectory of the pressure barycenter, as illustrated by Figure 7.8(a), exhibiting only one central peak. The probability of being in the TS mode is multiplied by two. This has to be compared to the two attractors trajectories obtained with a constant blowing in the same region [Figure 7.3(b)]. Figure 7.8(b) shows that the pressure coefficient is globally unmodified. Figure 7.9 stresses the peculiarity of the actuation frequency at f * j = 0.2, since no other frequency enables such a symmetrization of the wake. This dynamic response of the wake to a specific perturbation reminds the one obtained for a forced axisymmetric jet resulting to UPOs (Broze & Hussain, 1994). As the pressure center of the turbulent wake acts like a strange attractor in The four first POD modes are given once again for the natural flow in Figure 7. 10(a,d,g,j). These modes are also respectively given for OL-0 in Figure 7.10(b,e,h,k) with continuous blowing and in Figure 7.10(c,f,i,l) with pulsed jets at f * j = 0.2. The spatial organization of Φ 1 is the same for each run and it is related to the bimodal behavior. Its energy increases for the continuous forcing (+14%), while it decreases with the pulsed jets at f * j = 0.2 (-50%). It appears that the total energy of POD modes having a spatial organization with a reflexive symmetry with respect to the y-axis [Figure 7.10(f,i,l)] rises for the pulsed forcing.

The sensitivity analysis made by Grandemange et al. (2014a) shows that targeting the center of and(c,f,i,l) OL-0 with pulsed jets at f * j = 0.2. For the continuous blowing and the pulsed jets, the associated POD energies are respectively 50% and 22% (Φ 1 ), 12% and 14% (Φ 2 ), 6.5% and 12% (Φ 3 ), 4.0% and 6.8% (Φ 4 ). the top or of the bottom shear layer with a control cylinder stabilizes the wake in a symmetric mode. In our case, this wake topology is obtained by forcing the top shear layer at the vortex shedding frequency. As highlighted recently, using this frequency for the actuation frequency leads to a strong increase of the pressure drag (over +25%) by enhancing the entrainment rate of the wake (Barros et al., 2016a,b). Thereby, forcing the wake in its center at one of its main resonant frequency (St H = 0.2) induces two opposite effects on the base pressure, resulting finally to its very slight increase (+0.75% in average), as shown by Figure 7.8(b). The impact of a low pulse frequency previously observed may be attenuated in our case as the exit slit of the jets is discontinuous and only localized in the middle of the upper rear part, unlike the forcing by Barros et al. (2016a,b). It is noteworthy that base pressure recovery was also obtained at similar and lower normalized jet frequencies for other bluff body wakes [START_REF] Brunn | Active control of turbulent separated flows over slanted surfaces[END_REF]Joseph, 2012), axisymmetric wakes [START_REF] Sigurdson | The structure and control of a turbulent reattaching flow[END_REF], and 2D wakes [START_REF] Pastoor | Feedback shear layer control for bluff body drag reduction[END_REF]. The principle of the closed-loop experiments is based on the computation in real-time of the recirculation barycenter in the horizontal XY-plane at z * = 1, defining y * rec as the control parameter. This tracking is then limited by the low acquisition frequency of the PIV setup (f * P IV = 2.8 × 10 -2 ). It means that we will act on the low-frequency large-scale dynamics of the wake. We then choose an opposition algorithm for the closed-loop experiments. It consists in blowing in the region where the recirculation barycenter is detected, as summarized in Figure 7.11. This is consistent with the three modes defining the state of the wake: the two RSB modes and the TS mode. To these three modes correspond three blowing regions that are activated when the recirculation barycenter is detected in the related region. For each solenoids group we define a range for the spanwise location of the recirculation intensity barycenter y * rec for which the group is activated. The instantaneous velocity fields and y * rec are computed in real-time. The control variable is updated with a period T * act , the minimum being 1/f * P IV .

To make things more simple, we focus on the detection -blowing algorithm with all other actuation parameters fixed. The parameters for a closed-loop experiment are chosen based on the best or more interesting parameters found in the open-loop experiments. The actuation frequency f * j and the mean jet velocity u jet are then chosen and set to fixed values before turning the control on.

In the following, two types of actuation are used in the closed-loop experiments: a continuous blowing, denoted "CL-CONT" ("CL" stands for closed-loop.), or a pulsed blowing with different actuation frequencies f * j . We will focus on two actuation frequencies: f * j = 0.2 and f * j = 0.8, respectively denoted "CL-LF" (for low frequency) and "CL-HF" (for high frequency). The objective of these experiments is to control the wake dynamics which are characterized by the fluctuations of the spanwise location of the recirculation barycenter y * rec (t). Typical time-series of the pressure and recirculation barycenters fluctuations for the reference case are displayed respectively in Figure 7.12(a) and (c) as a reminder. When the closed-loop actuation CL-CONT is triggered, the dynamics of the barycenters [Figure 7.12(b) and (d)] are completely changed. Instead of random large and small fluctuations, the oscillations of the barycenters become very regular and periodic. The right-left switching is now imposed by the characteristic time scale of the closed-loop control law T * act . This observation holds also for the pulsed actuations.

Stabilization of the wake

The trajectories of the pressure barycenter for CL-CONT during four successive control cycles are illustrated in Figure 7.13(a-d). It reveals that the flipping process occurs at the defined period T * act , only enforcing the presence of the two asymmetric modes, whereas the transient state becomes very short. The bimodal behavior is thus now completely driven by the opposition Introducing again pulsed jets instead of continuous blowing modifies the dynamics, depending on the jet frequency. On one hand, the dynamics obtained for CL-HF [Figure 7.14(c) and (f)] are very close to the ones for CL-CONT [Figure 7.14(a) and (d)]. On the other hand, Figure 7.14(b) shows for CL-LF slightly lower spanwise fluctuations, of the same order as the reference, whereas Figure 7.14(e) reveals a higher mean vertical position of the depression. This is probably due to the central jets which tend to symmetrize the wake and to attract the depression, as seen for the OL-0 case at this jet frequency. Instead of following the physical trajectories of the pressure barycenter over the rear part of the model, it is now possible to follow the trajectory of the wake in the state parameter (∆C b , ∆A rec ). Unfortunately, the acquisition frequency of the PIV fields is much smaller than the ones of the pressure fields, so that it is not possible to correlate all mean pressure coefficients to a corresponding recirculation area. Nevertheless, it is possible to correlate all PIV fields to a corresponding mean pressure coefficient and then plot the two state parameters.

As shown in Figure 7.15, all closed-loop experiments lead to clear reduction of the recirculation area, i.e. of the size of the wake. The type of actuation has also a strong influence: pulsed jets are more efficient (-40% and -32% in average when f * j = 0.2 and f * j = 0.8 respectively) than continuous blowing (-18%). On the contrary, the depression over the rear end increases in all cases, with the most important increase for f * j = 0.2 (+19%). The portion of the phase space explored by the wake is much larger for the CL-LF case [Figure 7.15(b)] than for the two other configurations but there is no clear relation between the changes in ∆A rec and the ones in ∆C b .

Regarding the RSB modes in particular, the more the CL controls reduce the recirculation area, the more asymmetric the wake is, as highlighted by Figure 7.16(a). Even if the TS state existence is reduced, the separatrices computed for this state [Figure 7.16(b)] have the shape similar to the mean profiles [Figure 7.16(c)]. This is due to the enhanced equiprobability of the BSR states. It is noteworthy that the recirculation barycenter comes also closer to the bluff-body for all types of actuations.

The conditional averages of the pressure and the velocity are computed and displayed in Figure 7.17 for CL-LF. The mean rear pressure of the TS state is higher than the BSR one (-0.227 and -0.234).

Creation of a multimodal state

Another way to characterize the dynamics of the wake is to look again at the full trajectories of the pressure barycenter over the rear part of the model. For the three actuation types, the corresponding 2D PDF are shown in Figure 7.18, where the previously highlighted strange attractors are displayed in blue.

When the recirculation barycenter is tracked and forced by the closed-loop algorithm with a continuous blowing, the PDF is also completely changed and exhibits two strong foci, as shown in Figure 7.18(a). The transient mode is nearly suppressed and the wake switches quickly from one attractor to the other. This modification is also observed with CL-HF [Figure 7.18(c)]. In both cases, the dynamics are marked by a decrease of 50% in the TS mode existence. Interestingly, the PDF is also strongly modified by the lower pulse frequency (CL-LF). In this case, one can see in Figure 7.18(b) a butterfly shape with four attractors instead of two for all other cases. These results suggest the occurrence of multi-modal state with a right-left oscillation and a topbottom oscillation, which are undoubtedly enlarged due to the actuating frequency corresponding to the resonant frequency found in the spectral analysis of the z * p (t) fluctuations [Figure 5. 29(b)]. This harmonic resonance is also highlighted by Barros et al. (2016a), through an antisymmetric open-loop forcing, close to the one resulting from our closed-loop control law. Figure 7.19 displays successive displacements of the pressure center, which oscillates at the constant period T * j = 1/f * j = 5 with a doubled amplitude compared to the natural dynamics.

The multi-modality of the controlled wake dynamics is also analyzed using the POD of the wallpressure spatial distribution. For the three controlled cases, the spatial organization of Φ 1 remains strictly the same as the natural flow as displayed in Figure 7.20(a-c). Nevertheless, the energy levels are different regarding the control settings: for CL-CONT and CL-HF they reach 57.9% 7.20(e) and (h) show that Φ 2 (18.8%) and Φ 3 (12.3%) have both the identical top-bottom reflexive symmetry. Φ 4 (5.3%) is also a symmetric configuration, but centred in the region of G p , as displayed by Figure 7.20(k).

Comparing the PSD of the POD modes with the ones of y * p and z * p also explains how these modes are related to the dynamics of the wake. As expected, Figure 7.21 shows that Φ 1 is clearly linked to the y * p activity only, especially for the low frequencies. The St H = 0.19-peak exclusively present in S z * p z * p appears in the spectra of the three following POD modes for each control case. Indeed, for CL-CONT and CL-HF, Φ 2 shares some peaks with S y * p y * p in the high-frequency range, but it contains particularly a similar low frequencies spectrum as S z * p z * p and a low St H = 0.19-peak. For CL-LF, this peak is clearly stronger in the Φ 2 and Φ 4 spectra. S a 4 a 4 and S z * p z * p are quasi-identical, meaning Φ 4 is rather only linked to z * p dynamics for this case.

The spatial organization and the dynamics of the POD mode enable thus to evaluate the impact of the closed-loop controls on the energy part associated to the large-scale fluctuations tracked by the barycenters. Among the four most energetic POD modes, Φ 1 solely describes a left-right behavior. The other POD modes are rather related to the TS state and slightly to the vortex shedding. The top-bottom variations of the wake fall by 30.2% and 37.9% for CL-CONT and CL-HF with 17.9% (-37.9%). So, these controls enforce vigorously the lateral motion of the wake. Concerning the CL-LF case, the top-bottom characterization is carried by Φ 2 and Φ 3 , which contain 31.2% (+8.3%) of the total energy. This control tends thus to create a multistable state, making the top-bottom dynamics visible as the right-left one.

Table 7.2 summarizes the previous interpretations. For instance, the energy associated to the RSB rises by 37.9% and 42.1% for CL-CONT and CL-HF respectively, whereas it decreases by

Conclusions and issues raised

After testing different open-loop controls with micro jets on the turbulent wake behind a squareback body, a simple opposition control has been based on the real-time computations of the location of the recirculation area barycenter. To our knowledge, this is the first implementation of a closed-loop flow control experiment based on the dense velocity fields computations at such Reynolds numbers (up to 5.1 × 10 5 ). The chaotic spanwise oscillations of the location of the recirculation (and pressure) barycenter is then forced into a well-controlled periodic oscillation

Phenomena

Ref. at the time scale imposed by the PIV setup and the control algorithm.

CL-CONT CL-LF CL-HF

RSB

Moreover, a much larger reduction of the recirculation area is obtained when a pulsed actuation is used in the closed-loop control experiments, instead of a continuous blowing. When the frequency of the pulsed actuation used in the closed-loop experiments corresponds to the natural shedding frequency the fluctuations in the phase space are strongly increased. Moreover, if the PDF of trajectories exhibits well-defined stable foci points for most of the configurations, the lowfrequency actuation case leads to a new four attractors configuration, suggesting multi-modality behavior with a right-left oscillation together with a top-bottom oscillation of the wake. These results are also confirmed by a POD analysis.

The square-back configuration of the Ahmed body is investigated in chapter 5. The barycenters of the pressure distribution over the rear part of this model and of the intensity recirculation are found highly correlated. Both described the most energetic large-scale structures dynamics, confirming the relation between the large-scale recirculation bubble and its wall-pressure footprint. The switching process and the vortex shedding from the top shear layer seem to dominate the depression dynamics. A third state, transient symmetric, is highlighted when analysing the bimodality. The influence of the central pod seems to stabilize the vertical bimodality.

In chapter 6, a stochastic behavior is first observed when analysing the trajectory of the pressure barycenter through some tools of the dynamic systems framework. But applying the same process to the low-frequency signal reveals a chaotic aspect: the instantaneous pressure barycenter circles around two stable areas acting like strange attractors and randomly switches from one attractor to the other. The nonlinear dynamics of the pressure center are thus investigated by reconstructing its phase space from the spanwise component. The largest Lyapunov exponent is first estimated, leading to a low positive value, characteristic of strange attractors and weak chaotic systems. The weak aspect is then confirmed by the correlation dimension found close to one. Afterwards, the low-frequency signal is transformed and studied as a telegraph signal, showing that its dynamics correspond to a quasirandom telegraph signal. All of these elements tend therefore to describe the dynamics of a complex 3D turbulent wake as a weak chaotic system, which can be qualified as "stochastic chaos", due to the physical noise.

Different open-loop controls with micro jets have been applied to the turbulent wake behind the square-back body in chapter 7. A simple opposition control has been then based on the real-time computations of the location of the recirculation area barycenter. To our knowledge, this is the first implementation of a closed-loop flow control experiment based on the dense velocity fields computations at such Reynolds numbers (up to 5.1 × 10 5 ). The chaotic spanwise oscillations of the location of the recirculation (and pressure) barycenter is then forced into a well-controlled periodic oscillation at the time scale imposed by the PIV setup and the control algorithm.

The chaotic spanwise oscillations of the location of the recirculation (and pressure) barycenter is then forced into a well-controlled periodic oscillation at the time scale imposed by the PIV setup and the control algorithm. Moreover, a much larger reduction of the recirculation area is obtained when a pulsed actuation is used in the closed-loop control experiments, instead of a continuous blowing. When the frequency of the pulsed actuation used in the closed-loop experiments corresponds to the natural shedding frequency the fluctuations in the phase space are strongly increased. Moreover, if the PDF of trajectories exhibits well-defined stable foci points for most of the configurations, the low-frequency actuation case leads to a new four attractors configuration, suggesting multi-modality behavior with a right-left oscillation together with a top-bottom oscillation of the wake. These results are also confirmed by a POD analysis.

Perspectives

Here come the unrealized experiments and the unanswered questions. We planned to control every flows encountered during these three years and the transient boundary layer flow requires still some adjustments due to the difficulty to properly trigger and follow T-S waves. The previously summarized results pave the way for further work from which, I hope, the society at large will benefit.

Real-time PIV anywhere

The most interesting change into the optical flow algorithm is the adapting IW radius to deal with the different constraints imposed by the investigated flow. The upgrade of the 32-bit LabVIEW program to a 64-bit version (especially for the cuFFT library) will be completed with the new camera. The protocol to transfer the images (CoaXPress) is rather new and requires some stratagems to enable fully operating communication. It shall be mentioned that the first framegrabber device supporting LabVIEW has been just announced in August 2017.

The new camera will be FPGA compliant, making the implementation of the optical flow code directly in the camera conceivable. Even if it may save time when only closed-loop applications are considered, writing a version of the algorithm for a FPGA configuration will be a weighty task due to the hardware compilation. As the current CUDA code can still be improved thanks to the new library, it could be interesting to evaluate a possible combination of both devices.

The transition from laminar to turbulent has been delayed

The region investigated in chapter 4 contains a slotted jet designed like the one forcing the T-S waves. A second external tank has been built and a second air controller has been calibrated. The closed-loop control of the boundary layer flow via the dynamic observer will therefore be soon tested at the PMMH laboratory. Control attempts during the first steps of this work showed promising results1 . This may require a larger PIV window to capture a longer T-S waves evolution, giving more distributed POD modes along the flat plate. The reduction of the TKE may be finally a good objective to fix to the built system since it can also be tracked in real-time.

The stochastic chaos

The results obtained in the chapter 6 are the most surprising. Even more when another research team finds similar conclusions for a different turbulent flow. Is the applied low filter too strong ? Is there another mathematical equation than the Langevin equation which could properly model such dynamics ? Those results still need further confirmations, knowing the two opposing representations of the turbulence recalled above. Recent improvements in the dynamic system framework via the recurrent neural network approach for short times series and the evaluation of multi-variables should help.

Control of a personal car wake

The real-time PIV setup will be used to investigate the flow over a reduced modular SUV designed by Plastic Omnium. This model has been conceived by designing various rear and front profiles, as well as different configurations for the nature of the embedded actuators (discontinuous jets, Coanda effect). Rear pressure and aerodynamic forces are planned to be synchronously measured with the velocity fields.

In fact, these experiments have already started through the characterization of the natural turbulent wake for some configurations of the SUV model. A few open-loop controls have also been applied. Instantaneous measurements of the Q-criterion in a XY-plane revealed high swirling activities in the shear layers. Even if the third velocity component is missing and the wake is 3D, simulations done by S. Edwige (PO-CNAM) showed that 2D (in the concerned plane) and 3D computations of the Q-criterion are very close. A machine learning control has even been rapidly tested to reduce the swirling activity. This finally requires last adjustments of the actuators management via the LabVIEW program. The possibility to select ROIs directly on the sensor of the new camera is well suited for this kind of application.

Figure 1 . 1 -

 11 Figure 1.1 -Concentration of carbon dioxide (CO 2 ), methane (CH 4 ) and nitrous oxide (N 2 O) in the atmosphere (a) from the year -400,000 to 0 (ice cores) and (b) from the year 1500 to 2017 (ice cores and direct air measurements). ppmv and ppbv stand for parts per million by volume and parts per billion by volume respectively.

Figure 1 . 2 -

 12 Figure 1.2 -World CO 2 emissions (a) by sector and (b) from transport in 2014, estimated by International Energy Agency (2016). The total emissions are 32.3 GtCO 2 .

Figure 1 . 3 -

 13 Figure 1.3 -Mean real gasoline value in France based on year 2007 (according to Annuaire statistique de la France).

Figure 1 . 4 -

 14 Figure 1.4 -Light-duty vehicle efficiency standards by The International Council on Clean Transportation (2015). Vertical grey line is year 2017.

Figure 1 . 5 -

 15 Figure 1.5 -Sketch of the drag force F x , inspired by Eulalie (2014).

  2001-2004 J. F. Beaudouin, supervised by J.-L. Aider (PSA Peugeot-Citroën) and J. E. Wesfreid (PMMH), 2002-2005 B. Thiria, supervised by J. E. Wesfreid (PMMH), on the dynamic and stability properties of forced open flows (Thiria, 2005); 2006-2009 T. Duriez, supervised by J. E. Wesfreid (PMMH) and J.-L. Aider (PMMH), on the control of flow over a ramp via vortex generator devices (Duriez, 2009); 2009-2012 P. Joseph, supervised by J.-L. Aider (PMMH) and X. Amandolese (IAT), within the CARAVAJE project, on control of the external aerodynamic of automotive vehicles by pulsed jets (Joseph, 2012); 2008-2012 T. Cambonie, supervised by J.-L. Aider (PMMH), on the interactions between one or many jets with a boundary layer (Jets in Cross Flow) (Cambonie, 2012); 2011-2014 N. Gautier, supervised by J.-L. Aider (PMMH), on closed-loop controls based on real-time PIV (Gautier, 2014);

Figure 1 . 7 -

 17 Figure 1.7 -Sketch of a time-averaged flow separation for a rounded backward-facing ramp. Dashed lines are the limits of the fluctuations of the separation (red) and the reattachment (green) lines.

Figure 1 . 8 -

 18 Figure 1.8 -Summary of the flow control categories.

Figure 1 . 9 -

 19 Figure 1.9 -(a) Side view of the simplified car model. (b) Model equipped with the different flaps on all the rear edges. Courtesy of J.L. Aider.

Figure 1 . 10 -

 110 Figure 1.10 -Active vortex generators (a) on the simplified car model, (c) on a Citroën C4 (2004) and (d, e) on the concept-car Citroën C-Airlounge (2003). (b) Sketch of the active vortex generators system. Courtesy of J.L. Aider.

  context of the present work has been globally introduced. The freedom of movement is clearly internationally recognized by most of the countries since the article 13 of the Universal Declaration of Human Rights in 1948, the article 12 of the International Covenant on Civil and Political Rights in 1966 and the article 45 of the Charter of Fundamental Rights of the European Union proclaimed in 2000. Ensuring this right by developing travel means -bicycle, car, bus, metro, train, boat, plane. . . -contributes actually to improve the society by giving us the possibility of new encounters, cultural exchanges, world exploration and collaborative works. What the people produce is also part of the transport challenge through the design of specific vehicles for the transport of goods trade such as: trucks, container ships, freight trains and cargo planes.

Figure 1 . 11 -

 111 Figure 1.11 -History and projection of the total population by major area. (United Nations, 2015).

  but recent improvements have enhanced the 3D reconstructing particles Cornic et al. (2015).

Figure 2 . 2 -

 22 Figure 2.2 -Difference in particle density between (a) PIV and (b) PTV, from Gautier (2014).

2.1. 2

 2 Optical flow for PIV: the origin of good Optical flow algorithms have been first introduced by Horn & Schunck (1981) in the computer science community for the challenging machine vision problem (Fennema & Thompson, 1979). The objective is to measure the visual displacement field by analysing any video sequence, which has been succesfully achieved first by Horn & Schunck (1981) for synthetic images, and by Fitzpatrick (1988) for experimental snapshots. Jara-Wilde et al. (2015) recently reviewed the three existing optical flow approaches: the global one assuming that images have time-invariant total intensity (Horn & Schunck, 1981), the local one assuming a constant flow within a user-defined neighbourhood (Lucas & Kanade, 1981), and the last one combining the two previous methods (Bruhn et al., 2002).

Following improvements in the

  displacement estimations of the local approach (Le Besnerais & Champagnat, 2005), Champagnat et al. (2011) proposed a fast optical flow algorithm dedicated to PIV application and detailed below. Davoust et al. (2012) and Sartor et al. (2012) used

Figure 2 . 3 -

 23 Figure 2.3 -Principle of the optical flow algorithm based on the local approach.

Figure 2 . 4 -

 24 Figure 2.4 -Forward (green, solid, top) and inverse (red, dashed, bottom) additive warping illustrations for R IW = 1.

Figure 2 . 5 -

 25 Figure 2.5 -Additive methods comparison from Pan et al. (2015).

Figure 2 . 6 -

 26 Figure 2.6 -Gaussian pyramid applied to a 1920 × 512 snapshot with n level = 3 (left) and iterative computation of the displacement field for the two highest pyramid levels (right).

Figure 2 . 7 -

 27 Figure 2.7 -Sketch of the data flow: I and V stand for Images and Velocity fields respectively. Dashed lines are optional data transfer.

Figure 2 . 8 -

 28 Figure 2.8 -Standard sequence diagrams for (a) the hydro configuration and (b) the aero configuration. Blue arrows shows the triggered processes.

Figure 2 . 9 -

 29 Figure 2.9 -Sketch of the LabVIEW program organization regarding the different devices related to in the aero configuration. Blue arrows are trig (TTL) signals, dashed black arrows are command signals, solid black arrows are images data, light red arrows are velocity fields and dark red arrow is the set of values derived from the velocity (introduced in the next chapter, a Q-criterion is given as an example).

  As rigorously demonstrated by Champagnat et al. (2011); Gautier & Aider (2015) and Pan et al.

Figure 2 .

 2 Figure 2.10 -(a) CAD sketch and (b) diagram of the hydrodynamic channel at the PMMH laboratory, from Cambonie (2012) and Gautier (2014) respectively.

Figure 2 . 11 -

 211 Figure 2.11 -BFS flow sketch and PIV plane, inspired from Spazzini et al. (2001).

Figure 2 . 12 -

 212 Figure 2.12 -BFS flow visualized with fluorescent dye at different Reynolds numbers: 230 ≤ Re h ≤ 1200.

Figure 2 .

 2 Figure 2.13 -(a) Positions of the different ROIs in the PIV window for the particles detection: the free stream (red rectangle), the shear layer (green rectangle) and the recirculation area (blue rectangle). Detection of the particles in an enlarged region inside the shear layer for different particles concentrations (×10 -2 ppp): (b) 1.1, (c) 1.7 and (d) 2.2.

Figure 2 .

 2 Figure 2.14 display one instantaneous and the time-averaged velocity fields. It is stressed that the instantaneous fields are computed from the same snapshots pairs. The recirculation length and vortex shedding are expected to be around L * rec 7 for this Reynolds number. The former is only obtained for d p = {11, 20} [Figure 2.14(c-f)], while the latter is visible for d p = {4.6, 11, 20, 31} [Figure 2.14(a,c,e,g)].It clearly appears that increasing the IW radius smooth the fields. This effect is amplified in the recirculation area since the particles concentration is here the most important. It may lead to make the constraint displacement assumption wrong near the reattachment region [Figure2.14(g-l)]. Due to the simultaneous computation of the displacement in each pixel, such errors are propagated far from their origin. In the shear layer, this even suppresses a real vortex [Figure2.14(i) and (k)]. For too low radius, the SSD minimization score seems to be insufficient to compute the free stream velocities [Figure2.14(a) and (b)].

Figure 2 .

 2 Figure 2.14 -Instantaneous (left) and mean (right) streamwise component of the velocity at Re h = 820, together with the separatrix (white), computed with (a,b) R IW = 7 px, (c,d) R IW = 11 px, (e-f) R IW = 15 px, (a,b) R IW = 19 px, (c,d) R IW = 23 px, and (e-f) R IW = 27 px.

Figure 2 .

 2 Figure 2.15 -Instantaneous (left) and mean (right) streamwise component of the velocity at Re h = 360, together with the separatrix (white), computed with (a,b) n iter = 3, (c,d) n iter = 5, and (e-f) n iter = 7.

Figure 2 .

 2 Figure 2.16 -(a)Instantaneous snapshot at Re h = 981, together with an IW (red square) and its corresponding contents at time (b) t and t + dt. Contents for the same IW after (c) n level = 1, (d) n level = 3 and (e) n level = 5.

Figure 2 .

 2 16(c-e) display the resulting IWs of two successive snapshots located in Figure 2.16(a) for different levels compared to the original ones [Figure 2.16(b)].

3. 1

 1 Velocity fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.1.1 Time-averaged velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.1.2 Velocity fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.2 Vortex detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.2.1 Swirling strength criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.2.2 Q-criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.3 Separated flow activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.4 Conclusions and perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Figure 3 . 1 -

 31 Figure 3.1 -Time-averaged (a) streamwise and (b) normal to the wall components of the velocity over 2000 fields, together with the mean separatrix (white line). Re H = 2 × 10 3 .

Figure 3 . 2 -

 32 Figure 3.2 -Instantaneous fluctuations of (a) the streamwise and (b) the normal to the wall components of the velocity, together with the instantaneous separatrix (white line). Re H = 2 × 10 3 .

  3)and λ Ci = 0 otherwise. It means that an element of fluid at the position (x,z) will rotate around the nearest vortex core in 2π/λ Ci (x, y).[START_REF] Cambonie | Experimental study of counter-rotating vortex pair trajectories induced by a round jet in cross-flow at low velocity ratios[END_REF] demonstrated the effectiveness of the 3D version of this criterion[START_REF] Zhou | Mechanisms for generating coherent packets of hairpin vortices in channel flow[END_REF] to detect vortices despite the presence of shear. Figure3.3(a) illustrates the criterion for the BFS flow. The activity in the shear layer and the vortex shedding starting at x * ∼ 4.9 are both well detected.

Figure 3

 3 Figure 3.3 -(a) λ Ci and (b) Q-criterion derived from the same instantaneous velocity field. Re H = 2 × 10 3 .

  Figure 3.3(b) gives the Q-criterion derived from the same previous velocity field. Both criteria are in good agreement.

Figure 3 . 4 -

 34 Figure 3.4 -(a) Instantaneous velocity field, together with contour of λ Ci (gray) and the rectangular ROI (white). (b) Integrated λ Ci along the z-axis. (c) FFT of the integrated λ Ci . (d) Previous resulting FFT multiplied by the mean velocity of the vortices. Re H = 2 × 10 3 .

Figure 3 . 5 -

 35 Figure 3.5 -(a) Instantaneous velocity field, together with contour of λ Ci (gray) and the rectangular ROI (white). (b) Estimated KH frequency for each velocity field f * KH (t * ) (highlighted in blue).

Figure 3 . 6 -

 36 Figure 3.6 -(a) Instantaneous velocity field, together with contour of λ Ci (gray) and the rectangular ROI (white). (b) Estimated shedding frequency for each velocity field f * shed (t * ) (highlighted in blue). Re H = 2 × 10 3 .

Figure 4 . 1 -

 41 Figure 4.1 -Sketch of a system identification method using POD coefficients a i as output.

Figure 4 . 2 -

 42 Figure 4.2 -Sketch of a transient boundary layer undergoing a T-S instability.

  (2009) did to exhibit the self-sustaining process between the steaks and the streamwise vorticity. The streaks finally collapse into turbulent spots[START_REF] Schlatter | On streak breakdown in bypass transition[END_REF]) like the T-S waves.Both transitions are not completely uncorrelated since the streaky structures of the bypass tran-1. History of the works leading to their study is well detailed by[START_REF] Eckert | A kind of boundary-layer 'flutter': The turbulent history of a fluid mechanical instability[END_REF].

Figure 4 .

 4 Figure 4.3 -(a) Computer-aided design of the modified flat plate used in the water channel. (b) Inclined slot sketch. (c) 3D view and (d) side view of the jet injection system.

Figure 4 . 4 -

 44 Figure 4.4 -Jet supply circuit.

Figure 4 . 5 -

 45 Figure 4.5 -Experimental setup with the PIV plane (red rectangle) and forcing jet (pink arrow) positions. Sketch of the water channel is displayed on the bottom right.

Figure 4 . 6 -

 46 Figure 4.6 -Experimental (solid blue), theoretical laminar (solid black) and theoretical turbulent (solid red) time-averaged velocity profiles, together with the fluctuations (dashed blue) for (a) Re δ * 0 = 224, (b) Re δ * 0 = 297 and (c) Re δ * 0 = 337. Profiles measured at x * W .

Figure 4 .

 4 6(a-b) show a laminar behavior for Re δ * 0 = 224 and Re δ * 0 = 297 since the maximum of their fluctuations is located in the free-stream. On the other hand the fluctuations are slightly stronger near the wall for Re δ * 0 = 337 [Figure 4.6(c)]. According to Morkovin (1985), this enables to obtain the natural transition to turbulence with T-S waves as desired. This indeed keeps only 2D dynamics by avoiding the bypass transition characterized by streaks. The streamwise velocity profiles are also compared to the theoretical ones in Figure 4.6. At Re δ * 0 = 224, the experimental velocity is rather well approximated by the Blasius profile. It begins to go slightly closer to the turbulent approximation when increasing the Reynolds number. At Re δ * 0 = 337 it is clearly in the transitional dynamics [Figure 4.6(c)]. These observations are confirmed by computing the boundary layer characteristics.

Figure 4 . 7 -

 47 Figure 4.7 -Flow characteristics for the different Reynolds numbers, Re δ * 0 = 224 (left), Re δ * 0 = 297 (middle) and Re δ * 0 = 337 (right): (a-c) the streamwise velocity,(d-f) the displacement thickness, (g-i) the shape factor and (j-l) the Reynolds number along the streamwise direction. Theoretical values are also indicated for the displacement thickness: laminar (black) and turbulent (red).

Figure 4 . 8 -

 48 Figure 4.8 -Successive instantaneous streamwise (left) and vertical (right) components of the velocity fields while a T-S wave propagates.

Figure 4 . 9 -

 49 Figure 4.9 -Example of a random signal created to perturb the flow upstream.

Figure 4 .

 4 Figure 4.10 -Mean kinetic energy k * (t * ) in the vertical pixels at x * W (blue) and at x * W + 25 (red).

Figure 4 . 11 -

 411 Figure 4.11 -Sketch of the learning matrix (red) construction from the data set (gray), explaining how the data are stored.

Figure 4 .

 4 Figure 4.12 -(a) POD energy (red) and cumulated energy (blue). The 85%-threshold is displayed (dashed black). Normalized POD modes of the streamwise velocity v * x : (b) Φ 1 , (c) Φ 2 , (d) Φ 3 , (e) Φ 4 , (f) Φ 5 and (g) Φ 6 .

Figure 4 . 13 -

 413 Figure 4.13 -Instantaneous (a) streamwise and (b) vertical components of the fluctuating velocities at t * = 310, together with the sensors area Ω S (rectangles). Time series of (c) the streamwise and (d) the vertical components of the fluctuating velocity, both space-averaged over Ω S .

Figure 4 . 14 -

 414 Figure 4.14 -Relative error between the real and the POD sensor at different streamwise positions.

Figure 4 .

 4 Figure 4.15 -Difference between the real POD coefficients and the ones estimated by ROMs computed with various state orders N order for the sensor s = [ v *x Ω S , v * z Ω S ].

Figure 4 .

 4 Figure 4.16 -(a) KE from the POD coefficients k * P OD for the real validation data (blue) and the output of the dynamic observer (red). Time series of (b) the streamwise and (c) the normal components averaged over Ω S .

Figure 4 .

 4 Figure 4.17 -Real (left), reconstructed by POD (middle) and estimated (right) streamwise component of successive instantaneous velocity fields (from top to bottom) separated of ∆t * = 18.

Figure 4 . 18 -

 418 Figure 4.18 -Difference between the real POD coefficients and the ones estimated by ROMs computed with various state orders N order for different natures of sensor.

Figure 5 . 1 -

 51 Figure 5.1 -Ahmed body sketched with (a) a slanted and (b) a square-back rear surfaces. Free-stream direction is given by the blue arrows (no drift).

Figure 5 . 2 -

 52 Figure 5.2 -(a) Sketch of the different large coherent structures generated by the Ahmed body with a slanted rear surface with 12.5 • < α < 30 • . (b) Resulting time-averaged drag coefficient with respect to the slant angle, together with some sketches of streamlines over the slant surface. Figure extracted from the works of Choi et al. (2014).

  Wake behaviors similar to the bimodality are reported for other 3D bluff bodies. The 3D turbulent wakes behind a sphere(Grandemange et al., 2014a) or a bullet-like body (Grandemange et al., 2012a; Rigas et al., 2014) adopt actually instantaneous asymmetric patterns. Due to the axisymmetric shape of those obstacles, there are an infinite number of RSB modes. As well as the square-back body, such dynamics arise in the laminar regime (Bury & Jardin, 2012), occasioned by a supercritical pitchfork bifurcation (Fabre et al., 2008; Meliga et al., 2009). Through long time observation, it is possible to statistically recover a symmetric wake since each state is equiprobable. These observations lead Rigas et al. (2015) to use a generic Langevin equation to capture the large-scale dynamics of the axisymmetric wake downstream of the bullet-like body. This equation is based on a deterministic model for the persistent laminar instabilities coupled with a stochastic representation of the turbulent fluctuations. The same approach has subsequently been applied to the turbulent wake behind the square-back Ahmed body by Brackston et al. (2016), who stress the key role of the turbulence intensity in the switching trigger. Indeed, they highlighted that the switching time T switch is larger than the vortex shedding period (T * shedding ∼ 5H/U ∞ ) and depends on the increase in the turbulent fluctuations, which thus drive the large-scale instability. The general theory of the noise-induced breaking symmetry has been demonstrated by Van den Broeck et al. (1994).

Figure 5 . 3 -

 53 Figure 5.3 -Present study configuration (black star) in the instabilities map experimentally made by Grandemange et al. (2013c). D y and D z are the domains of spanwise and vertical bimodality respectively. H * = H/W and C * = C/W .

Figure 5 . 4 -

 54 Figure 5.4 -Upper view (top figure) and side view (lower left figure) and view from behind (lower right figure) of the Ahmed body. The PIV measurement plane is shown on top (green rectangle) and lower left (green line) figure. The rear part of the model is mapped with 95 pressure sensors (red circles on lower right figure). Cables pass through a supplementary pipe (grey part on lower figures).

Figure 5 . 5 -

 55 Figure 5.5 -Sketch and picture of the 0.7 scaled square-back Ahmed body mounted inside the Malavard wind tunnel facility (Orléans, France).

Figure 5 . 6 -

 56 Figure 5.6 -PIV planes investigated in this chapter.

Figure 5 . 7 -

 57 Figure 5.7 -Typical (a) instantaneous pressure field over the rear part of the model with the pressure center G p indicated as a white circle, and (b) instantaneous normalized longitudinal velocity field with streamlines and the recirculation barycenter G rec shown as a white diamond in the horizontal XY-plane at z * = 1.

  Figure 5.8(a) shows a slowing down of the flow until reaching the bluff body. Then it starts to accelerate along the curved edge, where a depression takes place [Figure 5.8(b)].

Figure 5 . 8 -

 58 Figure 5.8 -(a) Streamwise and (b) normal components of the mean velocity field, together with streamlines, in the front XZ-plane at y * = 0. Re H = 3.9 × 10 5 .

Figure 5 .

 5 9(b) and (c) show that the upstream area where the speed decreases is reduced when increasing the Reynolds number, as reported numerically by Krajnović S. (2005) and experimentally by Spohn & Gilliéron (2002); Grandemange et al. (2013a).

Figure 5 . 9 -

 59 Figure 5.9 -Streamwise component of the mean velocity field and streamlines in the front XZ-plane at y * = 0 for (a) Re H = 2.6 × 10 5 , (b) Re H = 3.9 × 10 5 , and (c) Re H = 5.1 × 10 5 .

Figure 5 .

 5 Figure 5.10 -(a) Mean streamwise velocity profile and (b) its standard deviation at the top trailing edge (x * , y * ) = (0, 0) (blue solid lines), and at an upstream position (x * , y * ) = (-0.15, 0) (red dashed lines).

  (2014) is unsurprisingly found. Figure 5.11(a) and (b) show indeed two vortices on both sides of the recirculation, the bottom

Figure 5 .

 5 Figure 5.11 -(a) Streamwise and (b) normal components of the mean velocity field, together with streamlines (black) and the separatrix (white), in the rear XZ-plane at y * = 0. Re H = 3.9 × 10 5 .

Figure 5 .

 5 Figure 5.12 -(a) Streamwise and (b) spanwise components of the mean velocity field, together with streamlines (black) and the separatrix (white), in the XY-plane at z * = 1. Re H = 3.9 × 10 5 .

Figure 5 . 13 -

 513 Figure 5.13 -Spatial distribution of (a) the time-averaged pressure coefficient < C p > Tp , and of (b) the pressure fluctuations C p ST D over the rear of the Ahmed body.

Figure 5 .

 5 Figure 5.14 -(a) C x Taero and (b) C z Taero with errorbars (blue circles) for different Re H . Extra data come from Grandemange et al. (2013c) ( ), Eulalie (2014) ( ), Lahaye et al. (2014) (♦), Volpe et al. (2015) ( ), and Barros (2015) ( ).

Figure 5 .

 5 Figure 5.15 -(a) C b Tp with errorbars (blue circles) and (b) base-pressure-to-drag ratio -C b Tp / C x Taero for different Re H . Extra data come from Grandemange et al. (2013c) ( ), Eulalie (2014) ( ), Lahaye et al. (2014) (♦), Volpe et al. (2015) ( ), and Barros (2015) ( ).

Figure 5 .

 5 Figure 5.16 -(a-c) Streamwise velocity component with superposed streamlines and (d-f) λ Cicriterion of three successive instantaneous velocity fields, together with the separatrix (white), in the rear XY-plane at z * = 1.

Figure 5 .

 5 Figure 5.17 -(a-c) Streamwise velocity component with superposed streamlines and (d-f) λ Cicriterion of three successive instantaneous velocity fields, together with the separatrix (white), in the rear XZ-plane at y * = 0.

Figure 5 . 18 -

 518 Figure 5.18 -Evolution of the recirculation area and the separatrix computed in (a-b) the XZ-plane at y * = 0 and (c-d) the XY-plane at z * = 1. Red curves are smoothed data over ∆t * = 300.

Figure 5 .

 5 Figure 5.19 -Spatial PSD of the pressure fluctuations for (a-b) Re H = 2.6 × 10 5 , (c-d) Re H = 3.9 × 10 5 , and (e-f) Re H = 5.1 × 10 5 .

Figure 5 .

 5 Figure 5.20 -Spatial PSD of the pressure fluctuations for (a-b-c) Re H = 2.6 × 10 5 ,(d-e-f) Re H = 3.9 × 10 5 , and (g-h-i) Re H = 5.1 × 10 5 .

Figure 5 .

 5 27 is obtained from time-series smoothed over ∆t * = 150 for visibility. The results show a strong correlation and an almost non-existent delay between the two measurements as illustrated by Figure 5.27. The values of the strongest peak are indeed Γ y * p ,y * rec (τ * ∼ 60) = -0.84 ± 0.05 and Γ z * p ,z * rec (τ * ∼ 77) = -0.53, meaning that both barycenters are in phase opposition. When the depression is located in the negative part of the rear surface (y * p < 0), the recirculation bubble is located in the positive one (y * rec > 0) and vice versa.

Figure 5 . 21 -

 521 Figure 5.21 -Pressure barycenter spanwise position y * P : (a) evolution in time and (b) normalized PDF, P * yp . Recirculation intensity barycenter spanwise position y * rec : (c) evolution in time and (d) normalized PDF P y * rec . Red curves show the data smoothed over ∆t * = 150.

p[Figure 5 .

 5 29(a)] and the vertical position z * p , noted S y * p y * p and S z * p z *

Figure 5 . 22 -

 522 Figure 5.22 -Pressure barycenter vertical position z * P : (a) evolution in time and (b) normalized PDF, P * zp . Recirculation intensity barycenter vertical position z * rec : (c) evolution in time and (d) normalized PDF P z * rec . Red curves show the data smoothed over ∆t * = 150.

  The PSD of the vertical position z * p , S z * p z * p , alone reveals a peak at St H = 0.19, independent from the Reynolds number, as displayed in Figure5.29(b). This Strouhal number is characteristic of the signature of vortex shedding from the top-bottom shear layers interaction(Parezanović & Cadot, 2012;[START_REF] Lahaye | Aerodynamic characterisation of a square back bluff body flow[END_REF][START_REF] Volpe | Experimental characterization of the unsteady natural wake of the full-scale square back Ahmed body: Flow bi-stability and spectral analysis[END_REF]

Figure 5 . 23 -

 523 Figure 5.23 -Recirculation intensity barycenter spanwise position x * rec and its normalized PDF P x * rec in (a-b) the XZ-plane at y * = 0 and (c-d) the XY-plane at z * = 1. Red curve shows the data smoothed over ∆t * = 150. Unlike the previous figures, these measurements come from different runs but are done at the same Reynolds number.

Figure 5 .

 5 Figure 5.24 -(a-b-c-g-h-i) Streamwise velocity component with superposed streamlines and(d-e-f-j-k-l) λ Ci -criterion of three successive instantaneous velocity fields, together with the separatrix (white), in the rear XZ-plane at y * = 0 during spanwise switching of the wake.

Figure 5 .Figure 5 . 26 -

 5526 Figure 5.25 -(a-f) Streamlines and streamwise component of six successive instantaneous velocity fields in the rear XZ-plane at y * = 0 during spanwise switching of the wake.

Figure 5 .

 5 Figure 5.27 -Cross-correlation of Γ y * p ,y * rec (τ * ) (blue solid line) and Γ z * p ,z * rec (τ * ) (black dashed line).

Figure 5 .

 5 Figure 5.28 -(a) Normalized time series of |y * p | (blue solid line) and z * P (black dashed line) and (b) their cross-correlation Γ |y * p |,z * p (τ * ). Plotted data are smoothed over ∆t * = 150 for clarity.

Figure 5 . 29 -

 529 Figure 5.29 -PSD of the fluctuations of (a) y * p and (b) z * p for five Reynolds number: 2.6 × 10 5 (circle), 3.2 × 10 5 (diamond), 3.9 × 10 5 (square), 4.5 × 10 5 (star) and 5.1 × 10 5 (cross). Curves are shifted for clarity. The frequency resolution is ∆St H = 2 × 10 -3 .

Figure 5 .

 5 Figure 5.30 -(a) Instantaneous velocity field, together with contour of λ Ci (gray) and the rectangular ROI (white). (b) Estimated KH frequency for each velocity field f * shed (t * ) (highlighted in blue).

Figure 5 . 31 -

 531 Figure 5.31 -Successive trajectories of G p (y * p , z * p ) during periods of (a) ∆t * = 367, (b) ∆t * = 105, (c) ∆t * = 126, (d) ∆t * = 145, (e) ∆t * = 194, (f) ∆t * = 111, (g) ∆t * = 414, and (h) ∆t * = 96. The solid blue line shows the barycenter trajectory and the grey dashed line displays its previous one. The green circle and the red square are respectively the last position and the most probable location in the corresponding sub-figure of G p .

Figure 5 .

 5 Figure 5.32 -Normalized 2D PDF of G p (y * p , z * p ) for a 2-min run, together with the conditionalmean positions of G p (red squares) for each identified mode.

Figure 5 . 33 -

 533 Figure 5.33 -Conditional averages of the pressure coefficient at the rear body and the velocity field at z * = 1 for the modes (a-d) N, (b-e) TS and (c-f) P. The average position of the pressure center G p (white circle) and of the recirculation barycenter G rec (white diamond) are displayed. y * = 0 is also displayed (dash-dotted grey lines). Some streamlines are plotted over the streamwise component of the velocity to localize the coherent large-scale structures.

Figure 5 .

 5 Figure 5.34 -Conditional averages of the velocity field at y * = 0 for the modes (a) N and P, and (b) TS. The average position of the recirculation barycenter G rec (white diamond) is also displayed. Some streamlines are plotted over the streamwise component of the velocity to localize the large-scale structures.

Figure 5 . 35 -

 535 Figure 5.35 -Energy distribution (black) and cumulative energy (blue) of the first 20 POD modes of the rear pressure fluctuations.

Figure 5 . 36 -

 536 Figure 5.36 -Four first POD modes of the rear pressure fluctuations: (a) Φ 1 (42%), (b) Φ 2 (17%), (c) Φ 3 (8.8%) and (d) Φ 4 (4.3%).

Figure 5 .

 5 36(a). Perry et al. (2016) found a similar result by analyzing the PDF of its temporal coefficient a 1 , exhibiting two peaks like in Figure 5.21(b) and Figure 5.21(d). Φ 2 (17%) and Φ 4 (4.3%) can be paired, since they display the same central symmetry and their amplitude are in opposition. Both have a spatial singularity centred in the same position as the G p resident zone. With 8.8% of the total energy, Φ 3 has an uniform distribution with a strong singularity localized in the middle of the top edge.

Figure 5 . 37 -

 537 Figure 5.37 -PSD of the four first POD coefficients of the rear pressure fluctuations, together with S y * p y * p (solid blue line) and S z * p z * p (dashed black line).

Future

  investigations on the square-back version should involve only the four classic pods to avoid the clear fixing of the vertical bimodal behavior (Barros et al., 2017). They could eventually be designed with NACAA profile McArthur et al. (2016). Also, the sharp edges of the after-body are far from real modern cars design. Recently, Thacker et al. (2012) rounded the roof/backlight intersection and Rossitto et al. (2016) completed with curved sides/backlight edges for the slanted configuration of the Ahmed body. Such modifications of the square-back model should be considered in further works.

6. 1

 1 Quasi strange attractor dynamics . . . . . . . . . . . . . . . . . . . . . . . . 127 6.1.1 Clues to weak chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.1.1.1 Structure function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.1.1.2 Phase space reconstruction . . . . . . . . . . . . . . . . . . . . . . . 130 6.1.2 A chaos portrait . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 6.1.2.1 Largest Lyapunov exponent . . . . . . . . . . . . . . . . . . . . . . . 133 6.1.2.2 Correlation dimension . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.1.2.3 Telegraph-like signal . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 6.2 Conclusions and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

Figure 6 . 1 -

 61 Figure 6.1 -Lorenz attractor with the parameters (σ = 16, R = 45.92, and β = 4).

Figure 6 .

 6 2 displays S 2 and S 2,d for the raw signal y * p (a) and for the signal on which we apply a low-pass (LP) filter (b) to only select very low frequencies modes which have the highest magnitudes: St H < 10 -3 [Figure 5.29(a)]. The raw signal behaves clearly like a fractal noise with h = 1/2 implying α = 2, which is in agreement with Grandemange et al. (2013); Brackston et al. (2016). Conversely the LP filtered data appear to be chaotic since h = 1 and S 2,d is not constant. It is noteworthy that the same observations hold for k = 1.

Figure 6 . 2 -

 62 Figure 6.2 -(a) Second order structure functions of y * p (dashed blue line) and of its first derivative (solid black line). (b) Second order structure functions of the LP filtered y * p (dashed blue line) and of its first derivative (solid black line).

Figure 6 .

 6 Figure 6.3 -(a) MI I(Y M , Y M J ) and (b) AMI Î(Y M , Y M J ), together with their respected minimum (red diamond)

Figure 6 . 4 -

 64 Figure 6.4 -E 1 (m) (black circle) and E 2 (m) (blue square) computed from raw (dashed line) and LP filtered (solid line) signal y * p . The large-scale dynamics associated with the LP filtered signal show a chaotic signature.

Figure 6 . 5 -

 65 Figure 6.5ln(d(t)) i=1..M and exponential approximation e 1.8×10 -4 t * (black dashed line).

Figure 6 . 6 -

 66 Figure 6.6 -Correlation integral function C(r) of LP filtered y *p (blues lines) and their corresponding surrogate PR data (red lines). Power law approximations (dashed lines) are also given in logarithmic scale.

Figure 6 . 7

 67 Figure 6.7 shows a part of the LP filtered signal y * together with the resulting telegraph signal Y * p . The normalized ACF Γ y * p (τ * ) and Γ Y * p (τ * ) are computed for the raw signal and the LP filtered one. According to Anishchenko et al. (2002), the ACF approximation of a random telegraph signal, Ψ R (τ * ), is given by by the following function: Ψ R (τ * ) = e -2n * 1 |τ * | , (6.18)

Figure 6 . 7 -

 67 Figure 6.7 -Part of the telegraph signal Y * p (blue) obtained for the pressure barycenter spanwise position y * p (black). For clarity the telegraph signal is plotted with a 0.25 factor.

Figure 6 . 8 -

 68 Figure 6.8 -Normalized autocorrelation functions of the pressure barycenter spanwise position Γ y * p (black solid line) and of the resulting telegraph-like signal Γ Y * p (blue dotted line) for (a) the raw signal and (b) the LP filtered one, together with their respective ACF approximations: Ψ R (n * 1 = 3.2 × 10 -3 ) and Ψ QR (ξ * 0 = 4.0 × 10 2 ) (red dashed line).

Figure 6 .

 6 Figure 6.8(a) shows that Ψ R (n * 1 = 3.2×10 -3 ) is a good approximation of the ACF of the telegraph signal obtained from the raw data. The estimated mean switching frequency n * 1 = 3.2 × 10 -3 is close to the directly measured one f * switch ∼ 3.4 × 10 -3 . The ACF of the telegraph-like signal extracted from the LP filtered time series is well approximated by the linear function Ψ QR (ξ * 0 = 4.0 × 10 2 ) in Figure6.8(b). From the linear approximation we thus find ξ * 0 = 4.0 × 10 2 , which is not far from the filtered experimental results, T * min = 0.8 × 10 2 for the studied run. The same kind of ACF approximation is obtained for the Lorenz attractor described by Equation6.1

Figure 6 . 9 -

 69 Figure 6.9 -(a) PDF of the switching time t * switch for the raw data (black solid line) together with a Poisson distribution (parameter set to 2.9 × 10 3 , blue dashed line) and (b) probabilities of switching at times T * multiple to ξ * 0 = 4.0 × 10 2 for the LP filtered signal.

Figure 7 . 1 -

 71 Figure 7.1 -Upper view (top left figure), side view (lower left figure) and view from behind (lower right figure) of the Ahmed body. The jets outputs are gathered in three groups (pink): "N", "0" and "P", corresponding respectively to actuations in the left, central or right upper regions. The jets are supplied through pneumatic tubes passing inside a central pipe (grey part on lower figures).

Figure 7 . 2 -

 72 Figure 7.2 -Excerpts of y * p and z * p time series for the continuous forcing (a-d) OL-P, (b-e) OL-0, and (c-f) OL-P0 (solid red), all compared to a reference time series (dashed blue). Data are smoothed over ∆t * = 15 for clarity. The horizontal lines are the mean values.

Figure 7 .

 7 Figure 7.3 -2D PDF contour of G p (y * p , z * p ) for the open-loop controls with continuous jets compared to the reference (dashed blue contour): (a) OL-P, (b) OL-0, and (c) OL-P0 (solid red contour). The colormap is the same as Figure 5.32

Figure 7 . 4 -

 74 Figure 7.4 -(a) Second-order structure functions of LP y * p (dashed blue line) and of its first derivative (solid black line), and (b) E 1 (m) (dashed blue line) and E 2 (m) (solid black line) computed from LP y * p for OL-0.

Figure 7 . 5 -

 75 Figure 7.5 -2D PDF contour of (y * p , ∆C b ) for the open-loop controls (a) OL-P, (b) OL-0, and (c) OL-P0 with continuous blowing.

  ) where C b,ref is the time-averaged value of C b for the reference flow. In Figure 7.5(a) and (c), forcing the wake into one of the RSB modes increases roughly by 15% the depression. These results are well coherent with the sensibility analysis done by Grandemange et al. (2014a). On the contrary, acting on the symmetric mode of the wake leads to similar value for C b than the reference case [Figure 7.5(b)].

Figure 7 . 6 -

 76 Figure 7.6 -Excerpts of z * p and z * p time series for the pulsed jets at f * j = 0.2 in configurations (a-d) OL-P, (b-e) OL-0, and (c-f) OL-P0 (solid red), all compared to a reference time series (dashed blue). Data are smoothed over ∆t * = 15 for clarity. The horizontal lines are the mean values.

Figure 7 . 7 -

 77 Figure 7.7 -Spatial distribution of (a) the time-averaged pressure coefficient < C p > Tp , and of (b) the pressure fluctuations C p ST D over the rear of the Ahmed body for OL-0 with f * j = 0.2.

Figure 7 . 8 -

 78 Figure 7.8 -2D PDF contour of (a) G p (y * p , z * p ) and (b) (y * p , ∆C b ) with pulsed jets at f * j = 0.2. The first 2D PDF is compared to the reference (dashed blue contour). Colormaps are respectively the same as Figure 5.32 and Figure 7.5.

Figure 7 . 9 -

 79 Figure 7.9 -Normalized PDF for the reference flow (Ref), the continuous blow (f * j = 0) and different actuating frequencies f * j > 0 for OL-0. Curves are shifted according to the respective cases.

Figure 7 .

 7 Figure 7.10 -First, second, third, and fourth POD modes (each row of the figure) computed for (a,d,g,j) the reference, (b,e,h,k) OL-0 with continuous jets,and (c,f,i,l) OL-0 with pulsed jets at f * j = 0.2. For the continuous blowing and the pulsed jets, the associated POD energies are respectively 50% and 22% (Φ 1 ), 12% and 14% (Φ 2 ), 6.5% and 12% (Φ 3 ), 4.0% and 6.8% (Φ 4 ).

Figure 7 .

 7 Figure 7.11 -(a) Sketch of the opposition closed-loop control law, based on the detection of the spanwise position of the recirculation barycenter (white diamond), computed from the instantaneous velocity fields. (b) Illustration of the opposition control with the blowing regions chosen as a function of the position of the recirculation barycenter computed in real-time.

Figure 7 . 12 -

 712 Figure 7.12 -Time series of the y * -position of the recirculation barycenter and the pressure center for the reference case (a-c), compared to the closed-loop controlled case with continuous blowing jets (b-d). Thresholds are displayed in dash-dotted red.

Figure 7 . 13 -

 713 Figure 7.13 -Successive displacements of the pressure center G p for CL-CONT for (a) [t * 0: t * 0 + T * act ], (b) [t * 0 + T * act : t * 0 + 2T * act ], (c) [t * 0 + 2T * act : t * 0 + 3T * act ], and (d) [t * 0 + 3T * act : t * 0 + 4T * act ]. The red circle is the last position in the corresponding sub-figure, the solid blue line shows the barycenter trajectory whereas the grey dashed line displays its previous one.

Figure 7 . 14 -

 714 Figure 7.14 -Excerpts of y * p and z * p time series for (a-d) CL-CONT, (b-e) CL-LF, and (c-f) CL-HF (solid red), all compared to a reference time series (dashed blue). Data are smoothed over ∆t * = 15 for clarity. The horizontal lines are the mean values.

Figure 7 .

 7 Figure 7.15 -∆C b with respect to ∆A rec for (a) CL-CONT, (b) CL-LF, and (c) CL-HF.

Figure 7 .

 7 Figure 7.16 -Contour of the recirculation area for the reference (blue solid), CL-CONT (black dotted), CL-LF (red dash-dotted) and CL-HF (green dashed) for (a) the N state, (b) the TS state and (c) the mean velocity field.

Figure 7 . 17 -

 717 Figure 7.17 -Conditional averages of the pressure coefficient at the rear body and the velocity field at z * = 1 for the modes (a-d) N, (b-e) TS and (c-f) P in the CL-LF case. The average position of the pressure center G p (white circle) and of the recirculation barycenter G rec (white diamond) are displayed. y * = 0 is also displayed (dash-dotted grey lines). Some streamlines are plotted over the streamwise component of the velocity to localize the coherent large-scale structures.

Figure 7 .

 7 Figure 7.18 -2D PDF contour of (y * p ,z * p ) for the reference flow (blue) and various closed-loop controls (red): (a) CL-CONT, (b) CL-LF, and (c) CL-HF.

Figure 7 . 19 -

 719 Figure 7.19 -Successive displacements of the pressure center G p for CL-LF for (a) [t * 0: t * 0 +T * j ], (b) [t * 0 + T * j : t * 0 + 2T * j ], (c) [t * 0 + 2T * j : t * 0 + 3T * j ], and (d) [t * 0 + 3T * j : t * 0 + 4T * j ]. The red circle is the last position in the corresponding sub-figure, the solid blue line shows the barycenter trajectory whereas the grey dashed line displays its previous one.

Figure

Figure 7 .

 7 Figure 7.20 -First, second, third, and fourth POD modes (each row of the figure) computed for (a,d,g,j) CL-CONT, (b,e,h,k) CL-LF, and (c,f,i,l) CL-HF.

Figure 7 .

 7 Figure 7.21 -Normalized PSD of the first (diamond), the second (star), the third (square) and the fourth (circle) POD coefficients, together with the PSD of y * p (blue solid line) and z * p (black dashed line) for the (a) CL-CONT, (b) CL-LF, and (c) CL-HF. Curves are shifted for clarity.

Table 2 .

 2

1 -Comparisons of the major features between optical flow and FFT-PIV, from Pan et al. (2015).

Resolution (W (px)×H(px)) Frame rate (Hz)

				Size
				(Mb)
	acA2000-340km (Basler)	1920 × 512	340	7.7 (8-bit PNG)
	PowerPlus 4MP (TSI)	2048 × 2048	15	65.7 (16-bit PNG)

Table 2 .2 -Cameras characteristics.

 2

Table 2 .

 2

	2.1

3 -Concentration of particles (×10 -2 ppp) in different ROIs of the PIV window.

Table 2 .

 2

	iter	1	2	3	4	5	6	7	8
	n level
	2	32.3 50.0 45.5 45.5 43.5 37.0 34.5 25.0
	3	31.3 31.3 30.3 41.7 38.5 34.5 32.3 23.4
	4	52.6 50.0 43.5 40.0 37.0 34.5 32.3 29.4

4 -Averaged fields per second (FPS) reached to compute 1920 × 512 velocity fields offline.

  , C.J., Westerweel, J. & Scarano, F. (2008) Main results of the third international PIV Challenge. Experiments in Fluids 45 (1), pp. 27-71. doi:10.1007/s00348-008-0462-z. 24 Troutt, T.R., Scheelke, B. & Norman, T.R. (1984) Organized structures in a reattaching separated flow field. Journal of Fluid Mechanics 143, pp. 413-427. doi:10.1017/S0022112084001415. 37 Wieneke, B. (2005) Stereo-PIV using self-calibration on particle images. Experiments in Fluids 39 (2), pp. 267-280. doi:10.1007/s00348-005-0962-z. 23

	Thangam, S. & Knight, D.D. (1989) Effect of stepheight on the separated flow past
	a backward facing step. Physics of Fluids A: Fluid Dynamics	1 (3), pp. 604-606.
	doi:10.1063/1.857430. 38

Willert, C., Munson, M. & Gharib, M. (2010) Real-time particle image velocimetry for closed-loop flow control applications. In 15th Internation Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal. 30 Winant, C.D. & Browand, F.K. (1974) Vortex pairing: The mechanism of turbulent mixinglayer growth at moderate Reynolds number. Journal of Fluid Mechanics 63 (02), pp. 237-255. doi:10.1017/S0022112074001121. 37 Yu, H., Leeser, M., Tadmor, G. & Siegel, S. (2006) Real-time particle image velocimetry for feedback loops using FPGA implementation. Journal of Aerospace Computing, Information, and Communication 3 (2), pp. 55-62. doi:10.2514/1.18062. 30 Zhao, W. & Sawhney, H. (2002) Is super-resolution with optical flow feasible? In ECCV '02 Proceedings of the 7th European Conference on Computer Vision -Part I , pp. 599-613. Springer-Verlag London. doi:10.1007/3-540-47969-4_40. 28

Table 4 .

 4 1 -Theoretical characteristic values for the boundary layer flow.

		Laminar profile	Turbulent profile
		Blasius	Prandtl
	δ(x)/x δ * (x)/x θ(x)/x	4.91/ √ 1.73/ √ 0.664/ √ Re x Re x Re x	1/7 1/7 0.020/Re x 0.16/Re x 0.016/Re x 1/7
	H	2.59	1.29

  To evaluate the global accuracy of the prediction for the validation data set, the KE is computed from the POD coefficients k * P OD = K i=1 a 2 i (Guzmán Iñigo et al., 2014). The difference between the real POD coefficients and the ones estimated by the ROM is then estimated via k error , defined as Figure4.15 gives the resulting k error with respect to N order for the sensor s = [ v *x Ω S , v * z Ω S ]. It reveals that the best recover, k error = 38.1% is obtained for N order = 15.

	k error = 100	||k * P OD,real (t * ) -k * P OD,model (t * )|| ||k * P OD,real (t * )||	.	(4.14)

  ; Bailey et al., 2002), spheres (Gumowski et al., 2008; Yun et al., 2006; Grandemange et al., 2014b) or cubes

  The following brief presentation of the flow over different Ahmed body configurations is based on the abundant literature on the subject, hopefully recently reviewed(Joseph, 2012;[START_REF] Choi | Aerodynamics of heavy vehicles[END_REF].

	The

front surface The forebody is strongly rounded to limit structures generation, and separated zones over the roof and the lateral sides. Even if they have been experimentally observed for 1.8 × 10 3 < Re H < 9 × 10 4 (Spohn & Gilliéron, 2002;

[START_REF] Courtine | Dynamics of separation bubbles formed on rounded edges[END_REF]

and numerically obtained at Re H = 1.3 × 10 3

[START_REF] Lamballais | Direct numerical simulation of a separation bubble on a rounded finite-width leading edge[END_REF]

, their average size decreases at higher Reynolds number. Indeed, laser-Doppler anemometer measurements done by

[START_REF] Lienhart | Flow and turbulence structures in the wake of a simplified car model (ahmed modell)[END_REF]

shows that the flow remains attached to the roof at Re H = 7.3 × 10 5 .

Table 5 .

 5 .3 (Grande-mange et al., 2013c), and to experimental conditions like the yaw angle β (Cadot et al., 2015; Volpe et al., 2015).

	References	Re H	AR	C * = C/W	Remarks
	Krajnović & Davidson (2003)	2.1 × 10 5	1.0	0.08	No supports
	Rouméas et al. (2009)	7.7 × 10 5	0.74	0.13	No supports
	Grandemange et al. (2012b) Evstafyeva et al. (2017)	[3.1; 4.2] × 10 2		0.44
	Grandemange et al. (2013c) Volpe et al. (2015)	9.2 × 10 4 [5.1; 7.7] × 10 5	0.74	0.13
	Brackston et al. (2016)	[2.3; 4.4] × 10 5
	Cadot et al. (2015)	[1.8; 16] × 10 4		[0.007; 0.208]
	Barros et al. (2014) Östh et al. (2014)	3.0 × 10 5	0.85	0.14
	Barros et al. (2016)	[2.0; 4.0] × 10 5
	Lahaye et al. (2014) Eulalie (2014)	5.3 × 10 5 3.9 × 10 5	0.74	0.18	Supplementary pod across the underbody

1 -Square-back Ahmed geometries reported in the literature.

  7 scale of the original Ahmed body (L=0.731 m long, H=0.202 m high, and W =0.272 m wide), previously used by Lahaye et al. (2014) and Eulalie (2014). The rear part of the model is a square-back geometry with sharp edges (Figure 5.4), which corresponds to trucks, buses or mini-van geometries.

	5.2.2 Wind tunnel

Like some of the previous studies on a square-back bluff body

[START_REF] Lahaye | Aerodynamic characterisation of a square back bluff body flow[END_REF] Eulalie, 2014;[START_REF] Volpe | Experimental characterization of the unsteady natural wake of the full-scale square back Ahmed body: Flow bi-stability and spectral analysis[END_REF]

, our experiments are carried out in the PRISME laboratory wind tunnel (Orléans, France). The test section of this closed wind tunnel is a 2 m × 2 m square, resulting in a ratio blockage lower than 3%, and extends over 5 m. The model is mounted on a raised

Table 5

 5

	.2 indicates the different dimensionless parameters

* (×10 3 )

Runs nb. PIV planes Nb. of fields

	8.9	12	XY at z * = 1	256
	18	2	XZ at y * = 0 and XY at z * = 1	510
	45	1	None	-

Table 5 . 2 -

 52 Parameters of the different runs.

  The thresholds are computed as

	y * p,N = w th × y * p max(P	y * p |y * p <y * p	) and y * p,P = w th × y * p max(P	y * p |y * p >y * p

  . Indeed, Östh et al. (2014) need 100 POD modes to recover the bimodal behavior for a reducedorder model of the turbulent wake behind a square-back Ahmed body, while Grandemange et al. (2013) use a Markovian process and Brackston et al. (2016) rely on a Langevin equation to statistically match it.

Table 7 . 1 -

 71 Square-back Ahmed geometries for control studies reported in the literature.

		H	AR C/W Controls
	Bruneau et al. (2010)	8.3 × 10 3	0.74 0.44 Porous surface and jets (no
					supports)
	Grandemange et al. (2013b) Grandemange et al. (2015)	7.4 × 10 5 2.5 × 10 6	0.74	0.09 Horizontal rear flaps 0.12 Real wheels and chamfering
	Evrard et al. (2016)	4.0 × 10 5	0.85 0.09 Base cavity
	Evstafyeva et al. (2017) Brackston et al. (2016)	[3.1; 4.2] × 10 2 [2.3; 4.4] × 10 5	0.74	0.44 Synthetic jets 0.13 Vertical lateral flaps
	Barros et al. (2014)	3.0 × 10 5			Pulsed jets
	Barros et al. (2016a) Barros et al. (2016b)	[2.0; 4.0] × 10 5 3.0 × 10 5	0.85 0.14	Pulsed jets Pulsed jets and Coanda ef-
					fect
	Li et al. (2016)	6.0 × 10 5			Micro-jets
	Eulalie (2014)	3.9 × 10 5	0.74 0.18 Micro-jets
	McNally et al. (2015)	3.6 × 10 5	1	0.23 Micro-jets

Table 7 . 2 -

 72 , Φ 3 and Φ 4 Φ 2 , Φ 3 and Φ 4 -Φ 2 , Φ 3 and Φ 4 Effects of the different closed-loop controls on the wake through the POD modes analysis.

		Φ 1	Φ 1	Φ 1	Φ 1
			+37.9%	-20.2%	+42.1%
	TS	Φ 2 and Φ 3	Φ 2 and Φ 3 -36.8%	-?	Φ 2 and Φ 4 -47.7%
	Top-bottom interaction	Φ 4	Φ 4 -11.6%	Φ 2 , Φ 3 and Φ 4 +749%	Φ 3 +0%
	Vortex shedding	Φ 2 -33.2%	?	-40.9%

These experiments remain unpublished so far.

Former PhD student (2008-2012) and research engineer since 2012 at ONERA.

The term chaos is used to qualify a deterministic system, highly depending on the initial conditions

J is the notation index and J * is the corresponding dimensionless time.

The infinite norm has also been tested, giving same results not presented here.

Before some parts of the water channel decided to give away.

Remerciements

Je réalise à quel point je fus chanceux, pour ne pas dire privilégié, d'avoir eu un tel environnement comme cadre pour ma thèse. Je tiens en effet à saluer Jean-Luc pour ses précieux conseils, sa confiance répétée et son utile insistance à m'envoyer confronter nos idées dans de nombreuses conférences et réunions. Merci pour avoir grandement contribué à la naissance et à l'aboutisse-

CHAPTER 4

The dynamic observer This chapter presents the dynamic observer, a reduced-order model well suited for amplifier flows. After explaining its functioning in the system identification framework, this method is experimentally applied to the transient boundary layer over a flat plate.

CHAPTER 5 Turbulent square-back bluff body wake

This chapter summarizes first the cutting-edge knowledge about the flow around a simple car model: the square-back Ahmed body. After presenting the experimental setup, the natural flow is studied by investigating time-averaged and instantaneous measurements of the flow velocity in the wake, the rear pressure and the aerodynamic forces.

CHAPTER 7

Control of the chaotic dynamics of a turbulent wake

This chapter presents a closed-loop control of the turbulent wake behind the square-back Ahmed body described in chapter 5 and in chapter 6. After introducing the micro-jets system used to force the flow, different control strategies are tested. An opposition strategy is finally applied, based on the real-time PIV.

Perspectives

The analysis and the understanding of the open-loop results came after the closed-loop experiments. An interesting strategy to test would involve pulsed jets at f j = 0.2 only at the center, whereas the opposition approach would be defined on the side actuators groups. The real-time PIV setup will be used to investigate the flow over a reduced model having the complex shape of a SUV designed by Plastic Omnium. In fact, these experiments have already started through the first characterizations of the natural turbulent wake illustrated in Figure 7.22. This model has been conceived by defining various rear and front profiles, as well as different configurations for the nature of the embedded actuators (discontinuous jets, Coanda effect). A promising machine learning control, recently successfully applied to the square-back Ahmed body [START_REF] Li | Drag reduction of a car model by linear genetic programming control[END_REF], has even been rapidly tested on the SUV model. This finally requires an improved management of the actuators through the LabVIEW program.

CHAPTER 8

Conclusions and perspectives

Concluding remarks

In chapter 2, a new velocimetry method based on an optical flow algorithm has been successfully implemented on different GPUs using CUDA language. This method allows rapid calculation of dense and accurate velocity fields from 4MP images. It is embedded in a homemade LabVIEW program which handles the devices involved in the PIV as well as in different flow control experiments. Other fields characteristic to the flow are also very fast derived from the velocity as detailed in chapter 3. A new method to capture rapid dynamics like the Kelvin-Helmholtz instability is now available and gives expected results for various geometries. The program is completely operational for a hydrodynamic channel as well as for a large wind tunnel. Regarding the material requirement, most of the set up already installed for 2D-2C FFT-PIV in wind tunnel may be directly used with the program if the camera and the frame-grabber device are compliant with the LabVIEW environment.

The spatial resolution of the resulting velocity fields is tied to the size of the interrogation window (IW) such as for advanced PIV algorithms. Nevertheless, the output fields are dense (one vector per pixel), giving not only better results in the vicinity of edges and obstacles but also smoother fields and better calculations for values based on velocity gradients. To achieve such results, the algorithm parameters have to be properly set regarding the optical devices and the experimental conditions. The IW radius depends thereby both on the tracers density and on the free stream velocity which is mainly fixed via the inter-frames time the camera can reach. To enable correct initialization of the algorithm, snapshots are reduced via a Gaussian pyramid. The free stream velocity determines the pyramid level, being cautious regarding the obstacles position with respect to the sensor size. Finally the number of iterations can be estimated via trial and error, making a compromise between accuracy and computation speed.

In chapter 4, a system identification method based on the N4SID algorithm has been successfully implemented for the forced boundary layer flow developing Tollmien-Schlichting (T-S) waves. The input data have been obtained from the optical flow PIV measurements, retaining a double visual sensor of the streamwise and the vertical fluctuating velocity in a small area of the velocity fields. The output data have been defined from a proper orthogonal decomposition of the data set composed of 1500 recorded velocity fields at 60 Hz. The TKE increase induced by the T-S waves propagating along the flat plate has been identified with acceptable agreement between the estimated and the real decomposed flow. Choosing the good state order remains an open question.

Résumé

Les écoulements décollés sont omniprésents dans la nature comme dans les écoulements industriels (aérodynamique externe des véhicules, des bâtiments, écoulements autour d'aubes de turbines, aérodynamique interne dans des tuyaux...) où ils sont en général sources de nuisances (vibrations, bruit aéroacoustique, forces de traînée ou de portance). Les enjeux associés à la compréhension et à la maîtrise de tels écoulements, caractérisés par une bulle de recirculation, sont donc considérables. Un capteur visuel non invasif développé au laboratoire PMMH est d'abord amélioré afin d'accéder en temps réel aux champs de vitesses -et à leurs grandeurs dérivées -des écoulements rencontrés en soufflerie industrielle. Basée sur un algorithme de flot optique issu de la vision par ordinateur, cette approche expérimentale novatrice permet de faciliter les études paramétriques et peut être implémentée dans des boucles de contrôle réactif. Ensuite, les mesures obtenues pour un écoulement sur une plaque plane sont analysées dans le cadre de l'identification de système. Un modèle d'ordre réduit est alors construit par apprentissage, permettant de prédire la dynamique de la transition de la couche limite laminaire vers la turbulence. Enfin, le sillage pleinement turbulent derrière une géométrie modélisant une voiture simplifiée est caractérisé, de façon classique et en tant que système dynamique. Différentes modifications de l'écoulement à l'aide de micro-jets sont testées. Une loi de contrôle réactif consistant à suivre et forcer la recirculation est mise en oeuvre avec succès.

Mots Clés

Contrôle d'écoulements -Flot optique -Sillage -Système dynamique.

Abstract

Separated flows are ubiquitous in nature and industrial systems, such as diffusers, airfoils, air conditioning plants, moving vehicles... As the separation can strongly influence the performances of such devices, investigating their dynamics and their control is of great interest. A visual sensor developed at PMMH laboratory is first improved to measure in real time the velocity fields and its derived values for flows available in wind tunnels. Based on an optical flow algorithm from the computer vision domain, this new experimental approach makes easier parametric studies and may be used in closed-loop controls. The dynamics of the flow over a flat plate are then investigated. A system identification method -the dynamic observer -is successfully implemented to build a reduced-order model of the transient flow, which captures and predicts well the instabilities generated. Finally, the fully turbulent wake of the square-back Ahmed body is described. Dynamical system tools are applied to characterize it. Using continuous and pulsed micro-jets, different forcing strategies are analyzed. An opposition closed-loop control is implemented, tracking and driving the recirculation.