Probability Density Function PIV Particle Image Velocimetry

RANS

Reynolds-Averaged Navier-Stokes

• Attached cavitation. It appears in a low-pressure separated region close to the solid surface. The typical flow scenarios are cavities forming on the suction side of a hydrofoil or an impeller blade. If the length of the attached cavity exceeds the body upon which it develops, it is called super cavitation, otherwise it is termed as partial cavitation.

• Vortex cavitation. The fluid in the cores of vortices is prone to vaporizing due to low pressures prevailing there. It is commonly observed at the tips of propeller blades or in the free shear layers where Kelvin-Helmholtz vortices can develop.

In most hydraulic applications like turbo-machinery and marine propellers, cavitation is an undesired phenomenon, since the unsteady behavior and violent collapse of cavitation can result in detrimental effects, such as noise, system vibrations, performance decrease, material erosion, etc. However, in some cases, cavitation can have positive consequences. For instance, the torpedoes are enveloped in a stable super-cavity for reducing frictions with water, so that they are able to reach very high speeds (about 400 km/h). The intensive energy released from a collapsing cavity could be utilized to destroy kidney stones for treatment or kill live bacteria for cleaning of waste water.

Although cavitation has been investigated extensively for more than a century, a full understanding of the physical processes underlying the cavitating flows is still far from being realized at the present time. This is mainly due to the lack of quantitative experimental data on two-phase structures and dynamics of cavitation. Therefore, high-fidelity and detailed measurements of cavitating flow fields, especially in the opaque diphasic mixture areas, are extremely desired for a better knowledge of the physical mechanisms governing the cavitation instabilities. This will help to deduce effective means for controlling the negative consequences of cavitation and increasing its positive influence. Furthermore, the quantitative experimental data can be used to validate and improve the numerical simulation models for cavitation. Once reliable predictions are achieved, the costs and time concerning cavitation tests will be reduced substantially.

Outline of the thesis

, partial cavitation developed in small convergent-divergent (Venturi) channels were studied experimentally using an ultra-fast synchrotron X-ray imaging technique aided with

. Introduction to cavitation phenomenon

Cavitation is a unique phase change phenomenon in liquid flows involving processes of the explosive vaporization of liquid in low pressure regions and the subsequent implosions when the pressure increases again. In contrast to boiling where the vaporization of liquid is driven by a temperature change, cavitation could be approximated as an isothermal process starting when the local liquid pressure is reduced below its saturated vapour pressure. Moreover, cavitation inception is also subjected to cavitation nuclei like non-condensable gases or other pollutants in liquid. Their existence weakens cohesion of liquid molecules and thus serves as starting locations for the liquid breakdown with decreasing the ambient pressure. The cavitation nuclei content is also a primary factor determining the difference between the vapour pressure and the actual pressure at cavitation inception [START_REF] Franc | Fundamentals of Cavitation[END_REF].

Cavitation can be classified, according to generation mechanisms, into acoustic cavitation, laser-induced cavitation and hydraulic cavitation. The first two types are produced through either ultrasonic waves or lasers depositing high amounts of energy into the liquid locally. The last one is caused by low pressures associated with flow dynamics, such as flow accelerations and strong vortical motions. In the literature and also in the present thesis, cavitation, if there is no specific note, usually refers to hydraulic cavitation since it is the most common and important case encountered in engineering applications.

Hydraulic cavitation can take different forms depending on how the low-pressure regions are generated, and they can be divided into three groups:

• Travelling bubble cavitation. Individual bubbles arise from regions of cavitation inception as a result of rapid growth of cavitation nuclei. They are first transported by the flow and then implode when they reach zones of high pressure. conventional high speed photography as well as particle image velocimetry (PIV). Depending on different operating conditions and geometry dimensions, partial cavitation generally exhibited two regimes with distinct behaviors: quasi-stable sheet cavitation and periodic cloud cavitation with large vapour shedding. The measurement results in the present study provided a detailed description of the two-phase flow structures and dynamics in sheet cavitation. Three mechanisms responsible for the transition of sheet-to-cloud cavitation were revealed and the differences between them were highlighted. This thesis is composed of the following 7 chapters. Chapter 2 is a literature review on the physical background of sheet/cloud cavitation and the existing experimental techniques applied to cavitating flows. Chapter 3 provides a brief description of the experimental setup and the fast X-ray imaging technique. The emphasis of this chapter is put on the procedures from visualizations to velocity and void fraction field measurements in cavitating flows. In Chapter 4, the complex two-phase flow structures inside the sheet cavity are revealed in detail based on the data from the X-ray measurements. Chapter 5 deals with a comparative study of sheet cavitation at three stages (the early stage, intermediate stage and developed stage). The comparison shows the effect of re-entrant jet behaviors on sheet cavity structures and dynamics.

The influence of cavitation on turbulent fluctuations is also discussed. Chapter 6 analyzes the cavitating flows in a Venturi channel with side gaps. It is found that cloud cavitation can be suppressed by altering the propagation path of the re-entrant jet. In Chapter 7, three mechanisms (i.e. re-entrant jet mechanism, condensation shock mechanism and collapseinduced pressure wave mechanism) to initiate cloud cavitation are described in detail, and the reasons causing the scale effect on Venturi cavitating flows are discussed. The thesis is ended with a general conclusion in Chapter 8 summarizing the main contributions of this work and several perspectives for future research are proposed.

Introduction au phé nomè ne de cavitation

La cavitation est un phé nomè ne de changement de phase unique dans les é coulements de liquide impliquant des processus de vaporisation explosive de liquide dans les ré gions à basse pression et les implosions ulté rieures lorsque la pression augmente à nouveau. Contrairement à l'é bullition où la vaporisation du liquide est entraî né e par un changement de tempé rature, la cavitation pourrait ê tre approché e comme un processus isotherme commenç ant lorsque la pression locale du liquide est ré duite en dessous de sa pression de vapeur saturé e. De plus, l'amorce de la cavitation est é galement soumise à des noyaux de cavitation comme des gaz non condensables ou d'autres polluants dans le liquide. Leur existence affaiblit la cohé sion des molé cules liquides et sert ainsi de points de dé part pour la dé gradation du liquide avec diminution de la pression ambiante. Le contenu des noyaux de cavitation est é galement un facteur principal dé terminant la diffé rence entre la pression de vapeur et la pression ré elle au dé but de la cavitation [START_REF] Franc | Fundamentals of Cavitation[END_REF].

La cavitation peut ê tre classé e, selon les mé canismes de gé né ration, en cavitation acoustique, cavitation induite par laser et cavitation hydraulique. Les deux premiers types sont produits soit par des ondes ultrasonores, soit par des lasers dé posant localement de grandes quantité s d'é nergie dans le liquide. Le dernier est causé par les basses pressions associé es à la dynamique de l'é coulement, telles que les accé lé rations de l'é coulement et les forts mouvements tourbillonnaires. Dans la litté rature comme dans la pré sente thè se, la cavitation, s'il n'y a pas de note spé cifique, fait gé né ralement ré fé rence à la cavitation hydraulique car c'est le cas le plus courant et le plus important rencontré dans les applications d'ingé nierie.

La cavitation hydraulique peut prendre diffé rentes formes selon la faç on dont les ré gions à basse pression sont gé né ré es, et elles peuvent ê tre divisé es en trois groupes:

• Cavitation à bulles itiné rantes. Les bulles individuelles proviennent de ré gions de dé but de cavitation suite à la croissance rapide des noyaux de cavitation. Ils sont d'abord transporté s par le flux puis implosent lorsqu'ils atteignent des zones de haute pression.

• Cavitation attaché e. Il apparaî t dans une ré gion sé paré e par basse pression proche de la surface solide. Les scé narios d'é coulement typiques sont des cavité s se formant du côté aspiration d'un hydroptè re ou d'une aube de turbine. Si la longueur de la cavité attaché e dé passe le corps sur lequel elle se dé veloppe, on parle de super cavitation, sinon on parle de cavitation partielle.

• Cavitation vortex. Le fluide dans les noyaux des tourbillons a tendance à se vaporiser en raison des basses pressions qui y rè gnent. Il est couramment observé aux extré mité s des pales d'hé lices ou dans les couches de cisaillement libre où peuvent se dé velopper des tourbillons de Kelvin-Helmholtz.

Dans la plupart des applications hydrauliques comme les turbomachines et les hé lices marines, la cavitation est un phé nomè ne indé sirable, car le comportement instable et l'effondrement violent de la cavitation peuvent entraî ner des effets né fastes, tels que le bruit, les vibrations du systè me, la diminution des performances, l'é rosion des maté riaux, etc. dans certains cas, la cavitation peut avoir des consé quences positives. Par exemple, les torpilles sont enveloppé es dans une super-cavité stable pour ré duire les frottements avec l'eau, afin qu'elles puissent atteindre des vitesses trè s é levé es (environ 400 km/h). L'é nergie intensive libé ré e par une cavité qui s'effondre pourrait ê tre utilisé e pour dé truire les calculs ré naux pour le traitement ou pour tuer les bacté ries vivantes pour le nettoyage des eaux usé es.

Bien que la cavitation ait é té largement é tudié e depuis plus d'un siè cle, une compré hension complè te des processus physiques sous-jacents aux é coulements de cavitation est encore loin d'ê tre ré alisé e à l'heure actuelle. Ceci est principalement dû au manque de donné es expé rimentales quantitatives sur les structures biphasé es et la dynamique de la cavitation. Par consé quent, des mesures haute fidé lité et dé taillé es des champs d'é coulement de cavitation, notamment dans les zones de mé lange diphasique opaque, sont extrê mement recherché es pour une meilleure connaissance des mé canismes physiques ré gissant les instabilité s de cavitation.

Cela permettra de dé duire des moyens efficaces pour contrôler les consé quences né gatives de la cavitation et augmenter son influence positive. De plus, les donné es expé rimentales quantitatives peuvent ê tre utilisé es pour valider et amé liorer les modè les de simulation numé rique pour la cavitation. Une fois que des pré visions fiables sont ré alisé es, les coûts et le temps relatifs aux essais de cavitation seront considé rablement ré duits.

Aperç u de la thè se

Suite aux travaux pré cé dents de notre groupe (Coutier-Delgosha et al.2009; Khlifa et al.2017), la cavitation partielle dé veloppé e dans de petits canaux convergents-divergents (Venturi) a é té é tudié e expé rimentalement à l'aide d'une technique d'imagerie par rayons X synchrotron ultra-rapide assisté e. avec la photographie conventionnelle à grande vitesse ainsi que la vé locimé trie d'image de particules (PIV). En fonction des conditions de fonctionnement et des dimensions gé omé triques diffé rentes, la cavitation partielle pré sentait gé né ralement deux ré gimes avec des comportements distincts: la cavitation en nappe quasi-stable et la cavitation nuageuse pé riodique avec une grande é vacuation de vapeur. Les ré sultats de la mesure dans la pré sente é tude ont fourni une description dé taillé e des structures et de la dynamique de l'é coulement diphasique dans la cavitation en feuille. Trois mé canismes responsables de la transition de la cavitation feuille à nuage ont é té mis en é vidence et les diffé rences entre eux ont é té mises en é vidence.

Cette thè se est composé e des 7 chapitres suivants. Le chapitre 2 est une revue de la litté rature sur le contexte physique de la cavitation en nappe / nuage et les techniques expé rimentales existantes appliqué es aux é coulements de cavitation. Le chapitre 3 fournit une brè ve description de la configuration expé rimentale et de la technique d'imagerie rapide par rayons X. L'accent de ce chapitre est mis sur les procé dures allant des visualisations aux mesures de vitesse et de champ de fraction de vide dans les é coulements de cavitation. Dans le chapitre 4, les structures d'é coulement diphasiques complexes à l'inté rieur de la cavité de la feuille sont ré vé lé es en dé tail sur la base des donné es des mesures aux rayons X. Le chapitre 5 traite d'une é tude comparative de la cavitation des plaques à trois stades (stade pré coce, stade intermé diaire et stade dé veloppé ). La comparaison montre l'effet des comportements des jets rentrants sur les structures et la dynamique des cavité s de la feuille. L'influence de la cavitation sur les fluctuations turbulentes est é galement discuté e. Le chapitre 6 analyse les é coulements de cavitation dans un canal Venturi avec des espaces laté raux. On constate que la cavitation des nuages peut ê tre supprimé e en modifiant le trajet de propagation du jet rentrant. Dans le chapitre 7, trois mé canismes (à savoir le mé canisme de jet rentrant, le mé canisme de choc de condensation et le mé canisme d'onde de pression induite par l'effondrement) pour initier la cavitation des nuages sont dé crits en dé tail, et les raisons provoquant l'effet d'é chelle sur les é coulements de cavitation Venturi sont discuté es. La thè se se termine par une conclusion gé né rale au chapitre 8 ré sumant les principales contributions de ce travail et plusieurs perspectives de recherches futures sont proposé es.

Chapter 2

Physical background and measurement techniques

Sheet and cloud cavitation

Generally speaking, partial cavities have two forms of appearance as well in case of internal flows (in a Venturi) as in case of external flows (on a hydrofoil). For small incident angles (hydrofoilangle of attack; Venturidivergent and convergent angles) and high freestream cavitation numbers, the attached cavity appears to be stationary at a fixed location, and the observed cavity length is rather constant. This situation is referred to as sheet cavitation. A typical sheet cavitation forming on the suction side of a hydrofoil or on the divergent wall of a Venturi channel is presented in Figure 2.1. When the cavitation number is decreased or/and the incident angle is increased to a certain extent, the stable sheet cavity cannot be sustained. A large portion of the cavity is shed periodically from the main cavity forming a cloud-like structure in the cavity wake, and as a result the cavity length undergoes significant oscillations. This phenomenon is commonly called cloud cavitation. Although cavitation is inherently unsteady, sheet cavitation is usually stated to be stable or quasi-stable since the shedding of small vapour-filled vortices is confined in the cavity closure region, whose characteristic length scale is much smaller than the whole cavity length. Sheet cavitation is sometimes described to be an open cavity due to its frothy appearance at the closure as classified by Laberteaux & Ceccio (2001a). In contrast, cloud cavitation results in large fluctuations of cavity volume and thus is stated to be unstable. The violent collapse of the shed cloud in the downstream wake region can emit pressure waves of high amplitude, which is considered as the main source of noise and erosion [START_REF] Reisman | Observations of shock waves in cloud cavitation[END_REF][START_REF] Dular | On the mechanisms of cavitation erosion-Coupling high speed videos to damage patterns[END_REF].

Therefore, cloud cavitation is much more destructive than a stable sheet cavity.

Mechanisms for cloud shedding

In a classical point of view, the different behaviors of a partial cavity depend on the existence or absence of re-entrant flow originating from a stagnation point behind the cavity closure.

As for sheet cavitation (open partial cavities), [START_REF] Gopalan | Flow structure and modeling issues in the closure region of attached cavitation[END_REF], [START_REF] Callenaere | The cavitation instability induced by the development of a re-entrant jet[END_REF] and Laberteaux & Ceccio (2001a) concluded that no clear re-entrant jet or only the weak reverse flow existed at the trailing edge of the cavity due to weak adverse pressure gradient. [START_REF] Leroux | An experimental study of unsteady partial cavitation[END_REF] did not detect a clear sign of a pressure wave traveling from the cavity closure towards the leading edge inside a stable sheet cavity through ten aligned pressure transducers flush-mounted along the suction side of a hydrofoil, and they attributed it to the absence of the re-entrant jet. [START_REF] Barre | Experiments and modeling of cavitating flows in venturi: attached sheet cavitation[END_REF] measured a clear re-entrant flow in a globallysteady sheet cavitation using a double optical probe technique. However, in their simultaneous numerical simulation, the re-entrant jet was not predicted, and eventually they did not further clarify the role played by the re-entrant jet in stable sheet cavitation. In general, the absence of re-entrant flow was regarded as the main reason for the stable flow regime of sheet cavitation.

The periodic shedding of large cloud was observed firstly by [START_REF] Knapp | Recent investigations of the mechanics of cavitation and cavitation damage[END_REF] and he proposed a re-entrant jet model to explain the transition from stable sheet cavitation to periodic After the establishment of the re-entrant jet model, many studies have been conducted in order to verify the existence, development, the correlation of re-entrant jet with cavitation dynamics. [START_REF] Furness | Experimental and theoretical studies of twodimensional fixed-type cavities[END_REF] predicted the development of re-entrant jet using a potential flow model. The injection of ink was used by [START_REF] Le | Partial cavities: global behavior and mean pressure distribution[END_REF] to visualize the reentrant flow, and the ink was observed near the leading edge, confirming an upstream flow component. [START_REF] Kawanami | Mechanism and control of cloud cavitation[END_REF] placed a small obstacle on the suction side of a hydrofoil to prevent the re-entrant jet from moving upstream, and the large cloud shedding was not observed in the experiment, demonstrating the re-entrant jet was the primary cause of cloud cavitation. [START_REF] Pham | Investigation of unsteady sheet cavitation and cloud cavitation mechanisms[END_REF] placed a series of six electrical impedance probes spaced equally on the upper flat surface of the hydrofoil to detect the liquid front corresponding to the reentrant jet. They found that the mean velocity of the re-entrant jet attained a maximum value near the cavity closure region and was of the same order of magnitude as the free stream velocity. [START_REF] Laberteaux | Partial cavity flows. Part 2. Cavities forming on test objects with spanwise variation[END_REF] observed that a geometry with spanwise variation can sustain stable cavities with re-entrant flow since the re-entrant flow was directed away from the cavity. The conditions necessary for the development of the re-entrant jet has been explored

by [START_REF] Callenaere | The cavitation instability induced by the development of a re-entrant jet[END_REF]. They confirmed the critical role of the adverse pressure gradient at the cavity closure in the onset of the re-entrant jet instability.

In the numerical simulation of cloud cavitation, the RANS models generally overestimate the turbulent viscosity in the rear part of the cavity. The re-entrant jet is consequently stopped too early and it does not result in any cavity break off. Coutier-Delgosha et al. (2003) reproduced the periodic cloud shedding in a Venturi-type section by using a correction initially (1964) based on the fact that the sound speed in a two-phase mixture is significantly lower than that in either component, i.e. water or water vapour [START_REF] Brennen | Cavitation and Bubble Dynamics[END_REF]. However, the direct experimental observation was made only recently by [START_REF] Ganesh | Bubbly shock propagation as a mechanism for sheet-to-cloud transition of partial cavities[END_REF]. As shown in Figure In agreement with the original experiment, all these works found that with a sufficient reduction of the cavitation number, condensation shocks overtaking re-entrant jets became the dominant mechanism for large-scale cloud shedding. Nevertheless, they had divergence on the cause for the onset of condensation shock. In the PhD thesis of [START_REF] Ganesh | Bubbly shock propagation as a cause of sheet to cloud transition of partial cavitation and stationary cavitation bubbles forming on a delta wing vortex[END_REF], it was described that a rapid growth in cavity length and vapour content, hence reduced speed of sound and increased compressibility, resulted in the production of shock waves at the rear of the cavity. Wu et al.

(2017), [START_REF] Jahangir | Dynamics of partial cavitation in an axisymmetric converging-diverging nozzle[END_REF] and [START_REF] Bhatt | Numerical investigation of partial cavitation regimes over a wedge using large eddy simulation[END_REF] observed that the propagation of condensation shocks was triggered by the impingement of collapse-induced pressure waves from previously shed clouds. This was also found by [START_REF] Budich | Numerical simulation and analysis of condensation shocks in cavitating flow[END_REF], but in addition they also captured the initiation of condensation shocks in the absence of cloud collapsing. It implied that both the rapid cavity growth and the collapse-induced pressure wave could contribute to the formation of overpressure behind the cavity which was sufficient to induce a condensation shock front. It should be noted that a pressure wave with high amplitude emanating from a large cloud collapse might crush the growing cavity suddenly as described by [START_REF] Leroux | A joint experimental and numerical study of mechanisms associated to instability of partial cavitation on twodimensional hydrofoil[END_REF], which was different from the propagation of condensation shock through the cavity.

Studies on cavitation-turbulence interactions

The cavitation dynamics is also strongly related to the cavitation / turbulence coupling: the effect of cavitation (including formation and collapse of vapour cavities) on turbulence has been investigated numerically and experimentally.

As for the numerical aspect, Dittakavi et Regarding the experimental aspect, the acquisition of quantitative velocity fields mainly relied on PIV measurements. [START_REF] Gopalan | Flow structure and modeling issues in the closure region of attached cavitation[END_REF] and [START_REF] Laberteaux | Partial cavity flows. Part 1. Cavities forming on models without spanwise variation[END_REF] observed the largest turbulent fluctuations in the region downstream of the cavity which were regarded as the impact of vapour collapse. [START_REF] Iyer | The influence of developed cavitation on the flow of a turbulent shear layer[END_REF] investigated the effect of developed cavitation on the flow downstream of the cavitating shear layer. They found that the collapse of vapour bubbles led the streamwise velocity fluctuations to be increased but the cross-stream fluctuations and the Reynolds shear stress to be decreased. Aeschlimann et al.

(2011b) performed velocity measurements in a 2D cavitating shear layer. They observed that a complex combination of the production of vapour bubbles coupled with their collapse added additional velocity fluctuations, mostly in the main flow direction, while the turbulent shear stresses almost remained constant. Based on the observation at different stations along the hydrofoil chord, they found that the internal structure close to the leading edge was characterized by large vapour bubbles with a similar critical size and then they were rapidly split into smaller bubbles downstream; most of the bubbles do not have a spherical shape.

A review of measurement techniques for cavitating flows

Particle Image Velocimetry (PIV)

Different from intrusive and pointwise measurements using probes, PIV enables a wholefield acquisition of instantaneous velocity vectors with little perturbation on the flow. It has thus been applied to a wide range of fluid flows, but in cavitating flows, the strong scattering and reflection from the liquid/vapour mixture will obscure the scattering light from the surrounding tracer particles. This contaminating effect on the PIV measurements can be avoided by injecting laser-induced fluorescent (LIF) particles which emit light with different wavelength from the laser. The reflected and scattered light, at the wavelength of the laser, is blocked by the optical filter mounted in front of the lens and only the light emitted by the particles is recorded by the camera. However, if there are many vapour bubbles passing between the laser sheet and the camera, the optical paths starting from the fluorescent particles will be deviated severely or blocked completely. (2005) shows that if the position of the laser sheet was close enough to the observation window (~5 mm), the detected particles would be sufficient to evaluate the velocity field inside the sheet cavity. However, the measured velocity field is not representative since it is strongly subjected to the wall effects.

X-ray densitometry based on absorption contrast

Both X-rays and visible light are, in nature, part of the electromagnetic spectrum. However, due to having a much shorter wavelength than visible light, X-rays can penetrate most optically opaque media with weak interactions. This distinct advantage makes X-ray radiography a powerful method to visualize opaque multiphase flows [START_REF] Heindel | A review of X-ray flow visualization with applications to multiphase flows[END_REF]).

For cavitating flows, X-ray radiography can be used to measure local void fraction, i.e.

density because of the absorption difference between water and vapour. [START_REF] Stutz | X-ray measurements within unsteady cavitation[END_REF] applied firstly the X-ray densitometry to cloud cavitation formed in a Venturi-type test section.

They found that the mean void fraction varies regularly from 25% at the upstream end of the mean cavity to 10% in the downstream part. ray absorption in vapour and water). In this chapter, a brief description of the fast X-ray imaging technique will be given, and the emphasis will be put on the procedures to obtain the velocity and void fraction fields, from the flow visualizations.

Hydraulic test rig

The studied sheet cavitation is generated in a small size Venturi-type test section installed During the experiments, the test loop is filled with water driven by a circulating pump. The flow rate measured by a turbine flow meter is regulated by a frequency inverter varying the pump rotation speed. A secondary recirculation loop is added to attain small flow rates while avoiding the pump operating in off-design conditions. Different extent of cavitation could be set properly by adjusting the pressure on the free surface of the tank with the aid of a vacuum pump or a pressurizer. A heater combined with a cooling system is employed enabling the flow temperature to stabilize at a given value that is measured by a thermocouple located upstream of the test section. where 𝑃 vap is the vapour pressure at the flow temperature, 𝑢 in is the average velocity at the cross section of upstream pressure sensor and 𝜌 𝑙 is the liquid density. The average velocity is computed according to the volume flow rate measured by a flow meter with a 2% reading uncertainty. The pressure sensors are calibrated in a range of 0-3 bar with a full scale uncertainty of 0.25%. The accuracy of the pressure and velocity measurements leads to an uncertainty of 3.5% in the cavitation number. The two downstream ones are used to detect possible oscillations of the sheet cavity formed at the throat.

X-ray imaging mechanisms

The X-ray imaging in this work is based on two different mechanisms: absorption contrast and phase contrast. The arrangement of in-line X-ray imaging system is depicted in Figure 3.3(a). The radiograph will be projected onto the x-y plane perpendicular to the optic axis z. R1

and R2 are the source to object distance and object to detector distance, respectively.

Considering a sample of a spherical vapour bubble in water, the working principle of absorption-based X-ray imaging is illustrated in Figure 3.3(b). In this method, the detector is placed at the object exit surface z = 0. The contrast in the resulting image comes from the difference in the attenuation of X-ray energy since vapour has a smaller absorption coefficient than water. In phase-contrast X-ray imaging via free-space propagation as shown in Figure 3.3(c), the sharp contrast at the periphery of the recorded image results from Fresnel diffraction and the interference between neighboring points of the wavefront at a certain distance from the sample. To make phase effects detectable an appropriate propagation distance R2 between the object and the detector is required and the X-ray beam must be (at least partially) spatially coherent. Note that absorption contrast still contributes to the intensity image acquired by the propagation-based phase contrast technique.

X-ray imaging technique

The experiments were performed using the third generation synchrotron radiation at the Advanced Photon Source (APS) in USA where a high-energy and spatially coherent X-ray source is available. A complete description of the experimental method can be found in Khlifa (2014). Here a brief description is provided.

As illustrated in Figure 3.4(a), the X-ray source is aligned with the test section on one side and the X-ray detector (a scintillator) on the other side converting X-ray beam into visible light which is then recorded by the high speed CCD camera. The X-ray source emits two types of In the beamline, the slow shutter, operating at a frequency of 1 Hz with an opening time of 24 ms, is equipped to protect the test section and the detector by reducing heat load on them.

The fast shutter used is a mechanical rotating chopper which operates at a frequency of 6035

Hz with an opening time of 9 μs [START_REF] Gembicky | A fast mechanical shutter for submicrosecond time-resolved synchrotron experiments[END_REF]). Figure 3.4(b) shows the synchronization scheme of the X-ray flashes, the two shutters, and the camera frames, to obtain appropriate pairs of images for PIV analysis. The camera frame transfer is triggered by the secondary pulse 3, so that each image in the same pair could obtain nearly identical illumination.

The acquisition frequency of the camera is set to 12 070 fps (twice the fast shutter operating frequency), which enables a spatial resolution of 704×688 pixels with a pixel size of 2 μm. It should be noted that only a packet of 144 image pairs is recorded during the opening period of the slow shutter (24 ms per second). Silver-coated hollow glass spheres with nominal mean diameters of 10 and 17µm were injected into the flow as tracers of the liquid phase. Because of the small cross section of the X-ray beam, successive but not simultaneous acquisitions at different positions are necessary to obtain the complete image for the flow field of interest as shown in Figure 3.5(a). To that end, the test section was moved parallel to the divergent floor of Venturi in front of the X-ray beamline by a motorized platform. At each position, 1872 pairs of images are recorded, which are actually divided equally into 13 packets (corresponding to the slow shutter opening 13 times), and thus the time between packets is not continuous.

Data extraction based on image processing

Although the two-phase information is contained simultaneously in the X-ray images, the transition from visualization to quantitative measurement (e.g. velocity and void fraction fields)

is not easy. In the conventional PIV images, a high signal-to-noise ratio (hence a distinct cross correlation peak) is achieved by large contrast in gray levels between particles and liquid background as the liquid can rarely scatter laser light. However, due to the relatively bright background associated with X-ray imaging mechanisms, the signal-to-noise ratio would drop greatly if the background is not removed from the X-ray image. In order to analyze the dynamics of each phase, image processing is required to extract the tracer particles from the liquid background and vapour structures so that velocimetry algorithms can be applied to each phase separately.

Separation of the two phases

The first attempt to separate the seeding particles and the vapour bubbles could be found in [START_REF] Khlifa | Fast x-ray imaging of cavitating flows[END_REF] where the liquid background was eliminated based on the vapour volume fraction measurements and large areas of vapour structures were removed using a high-pass filter. This image processing method was greatly dependent on the accuracy of void fraction while the error could be ±15% according to their description if measurements of void fraction were performed using each individual image without pair averaging. In the present work, we employed a new wavelet-decomposition-based image processing method to separate the seeding particles from the non-uniform background of the liquid and the vapour structures. For a smooth narrative, the detailed description of procedures of this new method is provided in Section 3.6. The final result is presented in Figure 3.6. The particle image (Figure 3.6(b))

extracted from the raw X-ray image (Figure 3.6(a)) will be used for the measurements of liquid phase velocities. Figure 3.6(c) shows the remaining vapour structures which will be used for calculating the instantaneous velocity field of vapour phase. It should be pointed out that compared to the first attempt the same raw X-ray images were also used in the present study, nevertheless this new method is not dependent on the void fraction accuracy and is much less time-consuming (approximately 1/3 processing time with the same computing resources).

Void fraction measurement

The local void fraction 𝛼 is defined as the ratio of the vapour volume to the total volume along any given beam path crossing the test section. The value represents the averaged void fraction in the spanwise direction. Since the vapour has a different X-ray attenuation coefficient from the liquid, the local vapour volume fraction 𝛼 could be estimated quantitatively using Equation 3.2, derived from Lambert-Beer's law,

𝛼 = 1 - ln ( 𝐼 0 𝐼 𝛼 ) ln ( 𝐼 0 𝐼 1 )
where I0 is the local intensity measured when the test section is completely filled with vapor (for convenience, vapour is replaced here with air as they almost have an identical absorption coefficient), I1 is the local intensity measured when the test section is full of water, and 𝐼 𝛼 is the local intensity measured when cavitation occurs in the test section. For computing 𝛼, it is thus necessary to take the calibration images of air and water under the same acquisition conditions as the cavitation tests.

Particle image velocimetry

Based on the MatPIV.1.6.1 open source toolbox, cross-correlation algorithm was applied on each pair of consecutive particle images obtained by the aforementioned image processing procedures to determine the corresponding liquid phase displacements. The time delay between the two pictures was 3.68 μs. Four interrogation passes with window offset were carried out and the interrogation window size was progressively decreased in order to increase the spatial resolution.

In the first pass the interrogation window size was set to comply with the one quarter rule to limit in-plane loss of particle pairs. For instance, in the present case where the largest displacement estimated from the flow rate is 20 pixels, a window dimension of 80×70 pixels was chosen with a larger size in the streamwise direction. It should be noted that the correlation coefficient is computed directly in the spatial domain rather than using an FFT-based approach. In spite of its low efficiency, this direct computation allows to eliminate the constraint of equal window size with the power of 2 (32×32 pixels, 64×64 pixels, etc.) required by FFT implementation, and it can be useful to reduce the velocity gradient effect in the direction perpendicular to the flow. In the next pass, the refined interrogation windows were shifted locally according to the integer displacement field evaluated from the first pass such that the proportion of matched particle pairs is increased. The final interrogation window size should be made sufficiently small to reduce velocity differences within the interrogation area, meanwhile it is also limited by the particle concentration. To ensure a good signal-to-noise ratio, at least 4 matched particle pairs are required in the last interrogation window. Thanks to the method we use to extract particles from vapour structures, it is easy to know the mean number of particles (approximately 1200) in an X-ray image. Therefore, the minimum window size is estimated to be 40×40 pixels. Considering that the particle distribution is not strictly homogeneous, the last window size was enlarged slightly in the main flow direction into 50×40 pixels. This final window size determines the actual spatial resolution in the PIV data which implies that the turbulent structures with scales smaller than this size cannot be resolved.

In the studied cavitating flows, the local velocity field exhibits a strong shear even within the smallest interrogation window. It would make the signal-to-noise ratio decrease drastically and cause a large uncertainty in velocity measurements. In order to solve this problem in regions with high velocity gradient, an image deformation scheme [START_REF] Raffel | Particle image velocimetry: a practical guide[END_REF]) was utilized in the last pass. The correlation peak was estimated with a sub-pixel accuracy by using the three-point Gaussian fitting scheme in two directions. The same cross-correlation algorithms used for the liquid phase were also applied to each pair of vapour images. Since there are no seeding particles to trace the vapour phase, the crosscorrelation is based on the variations of gray levels related to the presence of vapour bubble interfaces and the void ratio difference. Thanks to the calculated vapour volume fraction field, the liquid region was masked out. In practice the measurements of vapour velocities were activated only when the spatially-averaged void fraction within the final interrogation window is greater than 7%.

Comparison between conventional laser PIV and X-ray PIV

Unlike the standard PIV measurements where only the particles in a given plane are illuminated with a thin laser sheet, all particles and bubbles along the beam path are projected into the X-ray image due to the line-of-sight nature associated with the X-ray imaging method.

Therefore, the standard PIV result represents the 2D velocity field in the illuminated plane while the X-ray PIV result is the span-averaged velocity field containing amassed flow information in the spanwise direction. [START_REF] Lee | X-ray particle image velocimetry for measuring quantitative flow information inside opaque objects[END_REF] reports the velocity profile in an opaque circular pipe measured by the X-ray PIV method, where the measured velocity at the pipe center is only two-thirds of the theoretical value. This discrepancy is attributed to the three-dimensionality of the laminar flow boundary layer given a mean velocity of 0.5 mm/s in an opaque Teflon tube with an inner diameter of 750 μm. can be seen, the velocity profile acquired by X-ray agrees well with the standard PIV result away from the bottom wall, which indicates that the flow is almost two dimensional in this region. Nevertheless, the velocity difference adjacent to the bottom wall is relatively high indicating a certain extent of three dimensionality of the flow in this region, which might be caused by corner flows and relatively thick boundary layers on the two side walls.

Procedures of wavelet-decomposition-based image processing method

A raw X-ray image of cavitation shown in Figure 3.6(a) is taken here as an example to illustrate the procedures of extracting the particles from the raw image. As mentioned before, the imaging is based on a combination of absorption contrast and phase contrast. The identification of particles also relies on the traits of these two mechanisms.

A two-dimensional signal (an image for instance) can be decomposed into approximation components and detail components at different levels by the discrete wavelet transform. As shown in Figure 3.10(a), it is accomplished by convolving the original signal s with a low-pass filter and a high-pass filter simultaneously resulting in an approximation component a 1 and a detail component d 1 respectively. Note that the two filters are quadrature mirror filters associated with the selected wavelet function. This decomposition is repeated until the desired level to further increase the resolution scale using the same scheme, replacing the original signal s by a 1 and producing a 2 and d 2 , and so on. The level N corresponds to the scale of 2 N in the wavelet function. The phase contrast enhances the visibility of the edge. Therefore the first identification of particles is based on the detection of the edges with sharp contrast (i.e. large intensity gradient).

In this case, we wish to highlight the boundaries of the particles rather than their interiors. This can be realized by choosing appropriate threshold values for each level from 1 to 4. In general, a relatively large threshold value is set for level 2 with small threshold values for the other levels since most edge information is contained in the detail component d 2 . In practice these threshold values were determined empirically in the Matlab graphical user interface tool -Wavemenu where it is convenient to adjust threshold values to obtain a subjectively optimal result. The Canny edge detector [START_REF] Canny | A computational approach to edge detection[END_REF]) based on image intensity gradients was then used, because of its ability to derive clean, thin edges that are well connected to nearby edges. In order to facilitate the following morphological operations, the enclosed areas by the detected edges are filled and presented as a binary image where the pixels with value 1 represent the locations occupied by particles. The remaining thin edges were removed via the morphological opening operation. On the other hand, we can take advantage of absorption contrast to identify particles since the hollow glass spheres reduce the X-ray attenuation. In this second identification, we wish to highlight the interiors of the particles instead of their edges. Figure 3.12(a) and (b) show the decomposition results using a relatively large threshold value at decomposition level 3 since the interiors of particles are mainly concentrated in the detail component d 3 . Figure 3.12(c) presents the background suppression with a constant threshold value given that the background is almost homogeneous. This processing increases the contrast of particle interiors. Then the same procedures as the first identification were implemented to obtain the binary image of particles identified based on local attenuation difference as shown in Figure 3.12(d) and (e).

The number of objects in the second identification is 1060. The total number of objects recognized is 1230. This number is important for determining the interrogation window size in PIV evaluation. All tracer particles are finally restored according to their detected locations. Figure 3.13(b) illustrates the final image of particles which is used for the measurements of liquid phase velocities. It should be noted that the gray levels of particles were restored from the image shown in Figure 3.12(c) instead of the raw X-ray image because the non-uniform exposure to X-ray beam would affect the PIV calculation, given that the cross-correlation peak is dominated by brighter particle images and the bright background might remain in the particle image due to the imperfect detection of particle locations. In order to obtain vapour tracers, the identified particles were removed from the raw X-ray image after intensity equalization and then replaced by the local average intensity of their neighborhood.

Improvement of void fraction measurement accuracy

Theoretically, every reference image of air or water in a sequence has the same intensity at the corresponding pixels if the synchronization scheme (Figure 3.4(b)) is realized perfectly.

Then any reference image of air or water can be used to calibrate void fraction. However, when we observe a sequence of air reference images, it is found that the global intensity and the width of the bright band in the center vary with the images. This suggests the ideal synchronization scheme is difficult to be achieved. In fact the fast shutter, the camera and the X-ray pulses are synchronized in three different ways which are responsible for the image-to-image difference. image in the same pair, and both intensity variations diverge with the image pairs in an oscillating way which seemingly poses a convergence problem if the average image of a packet is used as the calibration image for gaseous phase. However, let us observe the variation of the average intensity of the two images belonging to the same pair shown in Figure 3.15(b), it is noted that the quasi-periodic oscillation is steady in a statistical sense, relative to the average intensity of the packet. In addition, the other 12 packets of air reference images have the same tendency (not presented). The reference images of water were also examined and the results in a packet are illustrated in Figure 3.16. Although the image-to-image variation pattern is different, the variation of the pair average intensity is still consistent with the air reference images. Therefore, the calibration image of air (or water) for calculating vapour volume fraction can be obtained through averaging all the images in a packet.

Taking into account the slight packet-to-packet difference, the calibration images for gaseous phase and liquid phase were finally obtained by averaging all the images in the thirteen packets of each phase (as shown in Figure 3.7(a) and (b), respectively). Accordingly, the local void ratio measurements in the cavitating flow cases were performed using the average of the two X-ray images belonging to the same pair as well. In this way, the non-uniform illumination caused by the synchronization problems will have a better correspondence with the reference images than the individual images, and each velocity field will correspond to only one vapour volume fraction field. Images of pure air and images of pure water can be used to estimate the accuracy of the aforementioned method for calculating vapour volume fraction since the errors are represented by the discrepancies between the calculated values and the theoretical values (100% for the vapour phase, 0% for the liquid phase). Figure 3.17 indicates the mean vapour volume fraction errors estimated from 1872 (i.e. 144×13) pairs of air images. The big errors (exceeding ±20%) are found at two ends, which is attributed to the insufficient exposure. Consequently, for a reliable result only the domain ranging from the 90 th row to the 630 th row (limited by the red dashed lines) is considered in the following analysis. Thanks to averaging image pairs, the mean error within the middle band is as small as outside the band attaining ±2%. However, the relatively large error (around ±4%) is observed clearly in the vicinity of the middle band edges.

The explanation for this error is illustrated in Figure 3.18 by comparison between the air calibration image and the averaged image of a typical pair of air images. As mentioned previously, the middle band of the first image (band 1) is wider than the second one (band 2).

So in the pair averaged image there must be a narrow transition zone between the edges of these two bands with abrupt intensity change. However, due to averaging sufficient number of image pairs, the intensity variation within the transition zone of the air calibration image becomes relatively smooth, since the middle band width varies with images. The different intensity transition from the air calibration image results in the relatively large errors. The other X-ray image in the same pair was treated using the same procedure (Figure 3.20(e)).

The average image of the pair is indicated in Figure 3.20(f): this is the one used for the calculation of the vapour volume fraction, rather than the individual one.

Chapter summary

Cavitating flow is inherently opaque due to high reflections and multiple scatterings of visible light on vapour/liquid interfaces. Conventional experimental studies of cavitation are often limited to the determination of the global flow characteristics (like geometrical description of cavity shape and cloud shedding frequencies), and/or sometimes some measurements in the surrounding liquid flow area. However, an accurate description of cavitation physics strongly relies on the visualizations of two-phase morphology and the measurements of characteristic quantities (like velocity and void fraction) inside the volume of liquid/vapor mixture.

Dedicated to the measurement of the internal flow fields of cavitation, an ultra-fast X-ray imaging technique was developed in consideration of X-rays high penetrability and weak interaction with matter. In addition to the absorption contrast mechanism, the X-ray phase contrast leads to a sharp intensity variation at the phase interface, which allows a better visualization of the two-phase flow structures. Silver-coated hollow particles were injected into the cavitating flow as the liquid phase tracers, and they were visualized simultaneously with the vaporous structures based on the combination of absorption and phase contrast. Although the two-phase information of structures and dynamics is contained in the X-ray images, the extraction of desired data is not easy. In the present work, a wavelet-decomposition-based image processing method was introduced to separate the seeding particles from the vapour structures. In this way, image cross-correlations could be applied on both seeding particles and vapour bubbles to evaluate the instantaneous velocity field of the liquid and the gaseous phase, respectively. The local vapour volume fraction could also be derived simultaneously from the difference of X-ray attenuation. Thanks to the unprecedented acquisition of these data, the detailed experimental study of the cavitating flow structure and dynamics is made possible for the first time.

Chapter 4

Structure and dynamics of developed sheet cavitation

In the present experiments, the flow temperature was kept constant at 17 ± 0. Note that periodic cloud cavitation is not observed in the studied geometry due to the scale effect on Venturi cavitating flows (which will be discussed in detail in Chapter 7). In this chapter, we analyze the structure and dynamics of sheet cavitation at the developed stage. The comparison of sheet cavitation at different stages will be presented in the next chapter.

Global behavior of sheet cavitation based on high speed photography

The cavitating flow in the convergent-divergent channel was captured from side view using a high-speed Photron Fastcam SA 1.1 camera with a Tokina 100 mm lens. The flow was illuminated by a high-power LED in two ways: back-lighting and top-lighting. In the backlighting mode, the opaque two-phase structures blocked most of the light from the back source resulting in lower grayscale intensities in the recorded photograph. The frame rate in this mode was set to 8000 fps with a very short exposure time (1/1000000 s) to freeze the fast-moving vapour structures. In the top-lighting mode, the cavitation structures reflected light from the top LED into the camera sensor on the side resulting in higher intensities in the recorded photograph. The frame rate in this mode was set to 8000 fps with a 1/50000 s exposure time.

In both modes, the field of view was 1024 × 512 pixels with a resolution of 24 μm/pixel. As the cavitation number decreases, the cavitating region grows and the liquid-vapour interface becomes increasingly wavy and unstable. In our case, the maximum standard deviation is identified at the point of x = 11.5 mm, y = 1.5 mm and thus the mean cavity length is determined to be 11.5 mm. Another reliable method is to use the contour line of 𝛼 ̅ = 0.1 in the time-averaged void-fraction field to designate the mean cavity length. As shown in Figure 4.3(a), the cavity length is also around 11.5 mm. This good agreement demonstrates that it is reasonable to use the standard deviation method to determine the cavity length. Moreover, the maximum standard deviation is also an objective way to select the probing window for estimating the cavity shedding frequency. The time evolution of gray level in Figure 4.2(c) is obtained by averaging all gray level values in a 50×50 pixel probing window with its center at the position of the maximum standard deviation. From its Fast Fourier Transform (FFT) result, no characteristic frequency is detected implying that the small vapour shedding from the cavity closure occurs in an irregular way.

So far the mainstream explanation to the relatively stable sheet cavitation is that no reentrant jet exists underneath the cavity or the re-entrant jet only exists near the cavity closure region, thereby producing intermittent shedding of small vapour structures. The conjecture of lacking re-entrant jet is considered as the primary mechanism causing the main part of the sheet cavity attaching to the wall steadily. In the next sections, we will challenge the current theory of sheet cavitation by X-ray imaging measurements and illustrate how the seemingly stable sheet cavitation is linked to a clear re-entrant jet penetrating the entire sheet cavity.

Mean void fraction and velocity fields based on X-ray imaging measurements

Since the time-resolved acquisitions by X-ray imaging are not simultaneous at different positions, it is impossible to analyze the complete instantaneous flow field. Nevertheless, the time-averaged flow field at each position can be connected well with each other except some slight mismatching which is caused by the inaccurate movement of the motorized platform clamping the test section. With regard to the evolution of the standard deviation of void fraction, the maximum value is found to be in the range of x = 1-2 mm, which is far away from the location of maximum standard deviation of gray level (x = 11.5 mm). The explanation to this great disparity is the fact that the gray level of high speed images can only indicate the presence of cavitation structures, whereas it does not directly give any indication about the void fraction. For instance, the shedding cloud appears very bright in high speed photos while its vapour volume fraction is only around 20% as reported by [START_REF] Ganesh | Bubbly shock propagation as a mechanism for sheet-to-cloud transition of partial cavities[END_REF]. Surprisingly, the standard deviation of void fraction remains nearly constant at the value of 0.1 in a wide range of x = 4-12 mm before it begins to decrease. The standard deviation is observed to be greater than the mean void fraction from x = 11.5 mm to the wake, which is attributed to the rapid collapse of shedding vapour structures causing a relatively large void fraction fluctuation. Different packets are also highlighted with the vertical blue dashed lines. The percentage in each subfigure represents the time fraction of the negative velocity, i. e. the probability of reentrant jet occurrence and the red dashed line denotes the averaged velocity. At x = 1 mm where the average velocity is positive, the probability of re-entrant jet occurrence is less than 50%.

As expected, in the mean reverse flow region, the probability of re-entrant jet occurrence is more than 50%. A continuous presence of re-entrant jet is identified at x = 8 mm as the probability there is nearly 100%.

In order to highlight the near-wall upstream moving flow, Figure 4.7 provides the evolution of the probability of re-entrant flow along the line of y = 0.1 mm. Also shown is the timeaveraged longitudinal velocity. The re-entrant flow probability of 50% occurs at around x = 1.5 mm, which corresponds to the location of maximum standard deviation of void fraction. It suggests that the most frequent switch between positive and negative velocity leads to the largest void fraction fluctuation. The probability reaches nearly 100% at x = 4 mm. This is the ultimate location the vapour pocket can grow to as the re-entrant jet can make it shrink and break up. In the range of x = 4-12 mm, the velocity magnitude of the mean reverse flow remains almost constant at its maximum value of approximately 0.6 m/s (7% 𝑢 ref ). The persistence of the re-entrant flow with a constant mean velocity in this wide range of streamwise positions results, when combined with the outer main flow, in a relatively steady shear layer.

𝑥 (mm)

𝑢 (m/s) Probability 52

Based on the above analysis, we can infer that the upstream vapour pocket oscillates within a limited range of x = 0-4 mm and under the action of re-entrant jet it detaches from the wall, transforming into a low-void-fraction bubbly mixture that is afterwards transported downstream in the relatively steady shear layer. This explains why the mean void fraction is rapidly decreased from the maximum value 75% to 20% in the region of x = 0-4 mm and then reduced progressively from 20% to 0 at a much slower rate. It should be pointed out that the re-entrant jet in the studied sheet cavitation is quite weak (the maximum mean value is only 7% 𝑢 ref ), in contrast to the measured velocity of re-entrant jet in periodic cloud cavitation, which is generally of the same order of magnitude as the free-stream velocity [START_REF] Le | Partial cavities: global behavior and mean pressure distribution[END_REF][START_REF] Pham | Investigation of unsteady sheet cavitation and cloud cavitation mechanisms[END_REF][START_REF] Callenaere | The cavitation instability induced by the development of a re-entrant jet[END_REF]). This might be the primary mechanism for sheet cavitation to remain in a quasi-stable state.

Spectral analysis of void fraction variation

In this section we will determine whether the upstream vapour pocket is oscillating randomly or regularly. To that end, two probe locations close to the cavity leading edge are selected for examining the time-varying void fraction. The first probe location is indicated in presence of reverse flow. In this region, the thickness of the re-entrant jet remains quite constant, and the mean velocity is fairly uniform spanning the whole thickness. The magnitude of the mean reverse flow velocity also remains quite constant along the streamwise direction at its where the re-entrant jet originates from. The whole re-entrant jet can be considered as a liquid film moving upstream with dispersed cavitation bubbles, since the mean void fraction in this region is close to 0. The sparse bubbles entrained by the reverse flow tend to be of spherical shape as the velocity gradient is relatively small. The mean diameter of bubbles is estimated to be of the order of 100 microns.

The re-entrant jet (especially the main part (5b)) exists persistently in the studied sheet cavitation with a relatively constant thickness and velocity. The driving force acting on the reentrant jet is the adverse pressure gradient which is related to the test section geometry and the occurrence of cavitation. The primary resistance to the re-entrant jet is the shear stresses imposed by the main flow. The re-entrant flow in the sheet cavitation oscillates only in a small size region, while the main part is characterized by a statistically steady state, which suggests the competition between the driving and drag forces, so the re-entrant jet generally reaches an equilibrium.

A strong velocity shear layer [START_REF] Aeschlimann | Velocity field analysis in an experimental cavitating mixing layer[END_REF] exists steadily between the re-entrant jet ( 5) and the main flow ( 6) outside the cavity. In this region, the mean void fraction decreases along the flow direction at a very slow rate, and the standard deviation of void fraction is almost constant. No characteristic frequency is detected in the time variations of the void fraction. The low-voidfraction bubbly mixtures originated from the upstream vapour pocket are convected in the shear layer (2) without significant morphological and phase changes as their collapse might be limited by the low pressure in the cores of vortical structures. The coherent bubbles are deformed into a large variety of shapes and sizes as a consequence of high velocity gradient in the shear layer. The mean diameter of the bubbles cannot be estimated because they are highly coherent and non-spherical.

In the range of x = 0-10 mm, the quasi-steady re-entrant jet does not lead to any clear vapour shedding from the main cavity although a large portion of the cavity has already detached from the Venturi wall. This is the reason why the main part of the sheet cavity appears to be stable but with a wavy interface in the high speed photographs.

In the cavity closure region (3), small-scale vapour structures are shed continuously from the main cavity under the entrainment of the main flow and/or the action of horseshoe vortices.

The mean sheet cavity is also terminated in this region (the mean cavity length is 11.5 mm in the present flow conditions). The part (4) is the cavity wake region where the shedding vapour structures collapse into a large number of small bubbles. The mean void fraction in the wake is basically below 5%.

Turbulent velocity fluctuations inside sheet cavity

The acquisition of time-resolved velocity fields inside the sheet cavity allows us to obtain Reynold stresses as well. It should be clarified that what will be presented in the following is different from the classical Reynold stresses, since the instantaneous velocity was integrated along the flow passage thickness.

𝜌 = 𝛼𝜌 𝑣 + (1 -𝛼)𝜌 𝑙 = 𝜌 𝑣 + (1 -𝛼)(𝜌 𝑙 -𝜌 𝑣 )
where 𝛼 is the vapour volume fraction; 𝜌 𝑙 is the water density and 𝜌 𝑣 is the vapour density.

The standard k-ω SST model [START_REF] Menter | Two-equation eddy-viscosity turbulence models for engineering applications[END_REF]) is superior to the other two-equation models in

𝜏 𝑘 = 𝑓(𝜌)𝑎 1 ⁄ 𝑓(𝜌) = 𝜌 𝑣 + (1 -𝛼) 𝑛 (𝜌 𝑙 -𝜌 𝑣 ) = 𝜌 𝑣 + ( 𝜌 -𝜌 𝑣 𝜌 𝑙 -𝜌 𝑣 ) 𝑛 (𝜌 𝑙 -𝜌 𝑣 )
The correction is based on the void fraction and leads to a fast artificial decrease of the turbulent viscosity and thus the local shear stress when the vapour starts to appear.

Chapter summary

The sheet cavitation generated in a small size Venturi channel was at first studied by means of high speed photography. The main part of the sheet cavity was observed to attach to the solid surface steadily with a wavy interface. Small vapour structures were rolled up and detached continuously from the cavity closure region at multiple high frequencies. These global behaviors of sheet cavitation observed from high speed images are consistent with the previous research about this cavitation pattern. So far for the stable character of the sheet cavitation was usually related to an absence of re-entrant jet or a re-entrant jet existing only in the rear part of the cavity, leading to a relatively stable fore part with a turbulent frothy closure. However, this conjecture was supported by very limited experimental data on the two-phase structures and velocity fields inside the sheet cavity.

(kg/m 3 )

Structure and dynamics of developed sheet cavitation 63

Based on the present data from the X-ray measurements, the complex two-phase flow structures inside the sheet cavity were revealed in detail for the first time. As shown in Figure 4.11, the sheet cavity in the present flow conditions is essentially divided into 6 parts. An oscillating vapour pocket exists in the upstream part (1) of the cavity and the mean void fraction in this region decreases from the maximum value 75% to 20% rapidly. The main part of the reentrant jet (5) exists persistently near the wall with a relatively constant thickness and velocity.

When the vapour pocket interacts with the front of the re-entrant jet, it shrinks and transforms into a low-void-ratio bubbly mixture under the action of condensation and break-up. The whole re-entrant jet can be regarded as a liquid film moving upstream since the mean void fraction in this region is close to 0. The sparse bubbles entrained by the reverse flow tend to be of spherical shape and the mean diameter is estimated to be of the order of 100 microns. A strong velocity shear layer (2) exists steadily between the re-entrant jet ( 5) and the main flow (6) outside the cavity. In this region, the mean void fraction decreases along the flow direction at a very slow rate, and the low-void-fraction bubbly mixtures originated from the upstream vapour pocket are convected downstream without significant morphological and phase changes as their collapse might be limited by low pressure in the cores of coherent vortical structures. In the part (3), small-scale vapour structures are detached continuously from the main cavity under the entrainment of the main flow and/or the action of horseshoe vortices. The part (4) is the cavity wake region where the shedding vapour structures collapse into a large number of small bubbles. The mean void fraction in the wake is basically below 5%.

The turbulent velocity fluctuations inside the sheet cavity were also investigated. The domains of high-velocity fluctuations in the streamwise and cross-stream direction were found to correspond to the shear area. Both of them decrease gradually along the flow direction, which is consistent with the trend of velocity gradient. The turbulence level in the reverse flow region is as low as in the main flow, demonstrating the relatively steady status of the re-entrant jet in the sheet cavitation. Unlike the streamwise and cross-stream fluctuations, the shear stress appears to be weakly correlated with the velocity gradient. The collapse of vapour phase was found to be the main cause of the larger shear stress in the cavity wake. The phase change of vaporization at the upstream interface was also found to increase the coupling between the streamwise and cross-stream velocity fluctuations. The lowest shear stress was identified in the shear layer where vapour bubbles are convected without significant phase change.

Thanks to the simultaneous acquisition of instantaneous velocity field and density field, the well-known Reboud empirical correction was examined experimentally for the first time.

Compared with the original Bradshaw hypothesis, the Reboud correction with n=10 was found to agree much better with the X-ray experimental data. It explains why the Reboud correction has provided significant improvements in the RANS simulations of cavitating flows in the past two decades. The maximum mean void fraction in the transverse direction is plotted in Figure 5.2, and it illustrates clearly the spatial evolution of the mean void fraction along the flow direction for the three cases. Within the cavity at early stage, the mean void fraction decreases regularly from 60% to 10% at the closure of the mean cavity (x =2 mm). For the developed and intermediate cavities, the mean void fraction shows a rapid decrease from their respective maximum value to 20%, after which the void fraction further reduces at a relatively slower rate until the downstream end of the mean cavity (x =11.5 mm and x =4.6 mm, respectively). Based on an overall comparison of the three cases, the void fraction inside the cavities increases with the extent of cavitation. Nevertheless, it is interesting to note that the mean vapour volume fraction of the intermediate cavity is higher than that of the developed cavity in the region close to the leading edge, which will be interpreted in Section 5. Therefore the re-entrant jet region in the studied flow conditions may be regarded as a nearly stagnant liquid layer that separates the cavity from the solid wall. In order to have a better understanding of the unsteady behavior of the re-entrant jet, the probability of negative velocity in the near-wall region is presented in Figure 5.9. Although the mean velocity at 𝜎 = 9.13 is always positive, the turbulent fluctuations are enough for the instantaneous velocity to become negative. One example of negative velocity is given in Figure 5.10. However, we can generally say that there is no re-entrant jet in the sheet cavitation at early stage. The two other cases exhibit a region where the reverse flow is present quite constantly, but the primary distinction between them consists in the size of this region. The Comparison of sheet cavity structures and dynamics at different stages 79

Experimental results of mean velocity distributions

Analysis of cavity instability

In this section, we discuss the phenomena possibly responsible for the higher frequency 𝑓 2 .

The preliminary conjecture is that 𝑓 2 corresponds to the shear-induced flow instability, based on the fact that the re-entrant jet is absent at 𝜎 = 9.13. However, it is interesting to note that the frequency 𝑓 2 seems rather fixed at about 500 Hz, and it does not shift with different packets or flow conditions. According to this observation, it is reasonable to suspect that 𝑓 2 results from an external factor rather than the flow itself. The suspicion is verified by testing the temporal evolution of effective X-ray intensity for imaging as shown in Figure 5.14 where the dominant frequency is exactly coincident with 𝑓 2 . Therefore, we can conclude that the void fraction variation due to shear-induced flow instability is random rather than controlled by a dominant frequency. It should be pointed out that the imaging brightness fluctuations are not caused by the X-ray source itself: they actually come from the errors of synchronization between the X-ray flashes, the fast shutter and the camera frames as mentioned in Chapter 3.

Effect of cavitation on turbulent velocity fluctuations

It is impossible to generally conclude that cavitation promotes or depresses turbulence since the cavitation-related phenomena may have different effects on turbulent fluctuations including velocity field modification, phase change and the presence of vapour phase. In addition, cavitation may influence the Reynolds normal stresses and the shear stresses differently. The large difference in the mean cavity length also makes the comparison of different cases quite complicated. In spite of the difficulties mentioned, some conclusions can still be drawn, as discussed in the following part. (indicated by a dashed ellipse) similar to a recirculation zone is still formed just downstream of the throat under the influence of adverse pressure gradient in the divergent section. The intense velocity fluctuations exactly occur in this recirculation-resembling zone and then decrease rapidly towards downstream.

In order to further study the influence of cavitation on the velocity field, the comparison of mean streamwise velocity profiles at different locations is performed between the noncavitating case 𝜎 = 10.72 and the cavitating case 𝜎 = 7.54 as shown in Figure 5.16. It can be seen that cavitation leads the mean velocity gradient to decrease in the near-wall region, whereas, in turn, it increases the velocity gradient in the shear layer. In the non-cavitating case, the effect of adverse pressure gradient on the velocity field appears to be limited to a relatively small size region compared to the cavitating flow because the velocity profile at x=12 mm for the non-cavitating case has already recovered to a shape close to a turbulent boundary layer type while the cavitating flow still undergoes a reverse flow at the same location.

Let us look back at Figure 5.15 and make a qualitative comparison between the cavitating case of 𝜎 = 7.54 and the non-cavitating case. The noticeable difference can be identified in the near-wall region where the longitudinal velocity fluctuations are attenuated substantially in the cavitating case. This observation is consistent with the relatively steady state of the re-entrant jet in this region. The reduction of velocity gradient due to cavitation also explains the decrease of velocity fluctuations. In contrast, the cavitating flow in the downstream region exhibits higher velocity fluctuations than those in the non-cavitating case. This can be attributed to the impact of larger velocity gradient and vapour structure collapse. But one cannot tell which mechanism is predominant in the augmentation of velocity fluctuations. It is interesting to compare the two cavitating flows at 𝜎 = 7.54 and 𝜎 = 9.13 in their wake region (x>12 mm and x>2 mm, respectively). As the cavitation number decreases, the vapour collapse in the cavity wake becomes more intense. Nevertheless, the velocity fluctuations at 𝜎 = 7.54 are not higher than at 𝜎 = 9.13 since the velocity gradient is also an important factor for the velocity fluctuations. Therefore it can be inferred that the collapse of small vapour structures in sheet cavitation do not result in evident velocity perturbations and the degree of velocity shear plays a more important role in the turbulent fluctuations. The occurrence of cavitation causes the presence of vapour phase, which may also have an influence on the turbulent fluctuations. Figure 5.17(a) depicts the profiles of longitudinal velocity fluctuations for different cavitation numbers at the station of x=2 mm where the flow is scarcely affected by cavitation collapse. The degree of flow shear is quantified by the magnitude of velocity gradient as shown in Figure 5.17(b), and the presence of vapour phase is described via the average void fraction as illustrated in Figure 5.17 evaporates to feed the sheet cavity. The development of cavitation does not cause any significant variation in the magnitude of the shear stress, but the strip area becomes longer.

Another one corresponds well to the cavity wake region where the shed vapour structure collapses, and the magnitude of the shear stress increases considerably with the extent of cavitation. In the region where the vapour bubbles are transported without significant phase change, the shear stress remains at a low level. Based on the above observations, one may infer that the vapour collapse and the vaporization are the two dominant mechanisms to increase the correlation between the streamwise and cross-stream velocity fluctuations.

Chapter summary

In this chapter, the sheet cavitation produced in a convergent-divergent channel is studied

Towards the trial of investigating cloud cavitation

Compared to the relatively stable sheet cavitation, another form of partial cavitation, i.e.

cloud cavitation, has attracted more attention since the violent cloud shedding and subsequent collapse have a high potential to damage material and produce noise and vibration in practical applications. However, in the present convergent-divergent channel with a small contraction ratio, the transition from sheet to cloud cavitation is not observed with decreasing cavitation number. This was actually unexpected before manufacturing such a test section. In order to investigate cloud cavitation by X-ray imaging as well as other measurement techniques, we decided to make another Venturi-type test section according to an existing geometry in our lab which is proven to be able to produce a well-defined periodic cloud cavitation.

Multi-functional Venturi-type test section

The side view of the new test section is shown in Figure 6.1(a). It was specially designed to be applicable in the experiments of high-speed visualization, particle image velocimetry (PIV) and in-line X-ray imaging. The part indicated by the dashed rectangle is zoomed and shown in Figure 6.1(b). The base (1) and the lid (2) were made of non-transparent hard plastic through the 3D printing technique. Both of them constitute the main body of the test section.

The bottom insert (3) acts as the convergent-divergent floor of the Venturi section. The top insert (4) forms the ceiling of the Venturi. The inserts (3) and ( 4) were both manufactured in glass rather than plexiglass since the former one has a better light transmittance and is less likely to be burned by intense laser. The Venturi profile is depicted by the dashed lines. The heights at the inlet and the outlet are both 20 mm, and the height at the throat is 10 mm. The convergent and divergent angles of the Venturi are 18° and 8°, respectively. The pressure sensor [START_REF] Barre | Experiments and modeling of cavitating flows in venturi: attached sheet cavitation[END_REF] was flush mounted at the inlet with another one (6) at the outlet. Two identical parts as shown in Figure 6.1(c) were inserted into the slots in the base [START_REF] Aeschlimann | X-ray attenuation measurements in a cavitating mixing layer for instantaneous two-dimensional void ratio determination[END_REF] from front and back forming the observation window and also serving as the side walls of the Venturi channel. It should be noted that the two side inserts were made of plexiglass as its refractive index is close to water. In the X-ray imaging experiment, the side insert was replaced by a hollow one with a thickness of only 0.5 mm at the tip, as shown in Figure 6.1(d). Figure 6.1(e) shows the top view after removing the top insert (4). The slot here is important for the PIV experiments as it allows the calibration plate to be placed into the Venturi channel. The channel width is only 5 mm in order to minimize the 3D effect of cavitation.

Measurements

The high speed visualization of cavitation was performed in back-lighting and top-lighting modes as described in Section 4.1. Apart from this, the other measurements were also carried out in the new test section. Towards the trial of investigating cloud cavitation 91

Pressure measurements

The transient pressure at the inlet and the outlet was measured using two piezoresistive pressure sensors (Keller 10L series) calibrated in a range of 0-3bar. Signals from the pressure sensors were collected synchronously through a NI USB-6216 data acquisition card and exported to a computer by the Labview software. The sampling frequency was set to be 10 kHz and the sample size was 10,000 (corresponding to a recording time of 1 s) for all flow conditions. Figure 6 where 𝑃 𝑣𝑎𝑝 is the vapour pressure at the flow temperature, 𝑢 𝑖𝑛 is the bulk velocity at the inlet computed according to the volumetric flow rate, and 𝜌 𝑙 is the liquid density. It can be seen qualitatively from Figure 6.2 that the pressure fluctuation at the outlet is much more intense than that at the inlet due to the occurrence of cavitation. 𝑢 𝑖𝑛 =10 m/s).

PIV-LIF measurements

The velocity fields in the cavitating flows were measured using the LaVision PIV system.

The light source was a double-pulsed Nd:YAG laser with the pulse duration of 10 ns. The laser beam was expanded by a cylindrical lens into a light sheet with thickness of about 1mm for illuminating the center plane of the Venturi channel. The energy density in the laser sheet could (6.1) be adjusted to sufficiently high as it goes through the replaceable glass inserts which are less likely to be damaged.

Before starting the circulating pump, a calibration plate shown in Figure 6.3 was placed in the center plane of the Venturi channel filled with water touching the divergent floor. The purpose of this step was to define the scale (mm/pixel), and eliminate perspective and refraction errors. The coordinate system used was also defined during the calibration process. The x-axis is along the Venturi divergent floor and the origin is located at the apex of the Venturi throat. The instantaneous velocity fields were evaluated using the iterative cross-correlation algorithm in the software Davis 8.1 from LaVision. The interrogation window for the first pass was 64×64 pixels followed by 2 passes with interrogation windows of 16×16 pixels and the overlap of 50% was used. The correlation peak was determined with a sub-pixel accuracy by using the three-point Gaussian fitting scheme in two directions. The particle density was taken into account in the final pass where the vectors in the interrogation windows with less than 4 particles were rejected and replaced by a bilinear interpolation from their valid neighboring vectors. This implied that the calculated velocities in the attached cavity had a higher uncertainty as the particle concentration there was not sufficient.

Effect of the side gap on cavitation regime

Neither a relatively stable sheet cavitation nor a periodic shedding of large scale vapour clouds is observed from the high speed visualizations of cavitating flows in the new test section. This is unexpected since a clear cloud cavitation has been obtained in another test section with the same geometry. This 'strange' cavitation behavior is attributed to the influence of a gap between the bottom insert and the side wall insert, which is revealed in the high speed images using back illumination as shown in Figure 6.6. An evident reverse flow can be identified in the region indicated by the yellow frame in Figure 6.6(a) through tracking individual vapour bubbles. An example of tracking a single bubble in a sequence of high speed images is also shown in Figure 6.6(a). The effect of the gap could also be checked by the flow velocity field. As shown in Figure 6.7, we compare the mean streamwise velocity profiles at two vertical stations (x=5 mm and x=10 mm) in two test sections: one is the newly-made test section with a gap; the other is the non-detached test section without a gap. The flow is free of cavitation with a nominal velocity of 15 m/s at the throat, and the velocity is measured using the PIV technique. It can be observed that the gap alters the flow field significantly: the existence of the gap increases the flow channel width, and thus makes the velocity decreased. The 3D printing technique was employed to make the main body of the new test section considering its advantages of time-saving and low cost. However, it is beyond our expectation that the large assembly error between the printed parts and the machined glass inserts leads to the presence of a gap, which actually suppresses the cloud cavitation, and hence delayed our plan to perform fast X-ray imaging measurements in cloud cavitating flows.

On the other hand, as a positive side effect, the influence of the gap on the cloud cavitation could be a passive control to this destructive form of cavitation. [START_REF] Danlos | Study of the cavitating instability on a grooved Venturi profile[END_REF] grooved the Venturi divergent wall and found this geometrical change suppressed the shedding of largescale vapour clouds as shown in Figure 6.8. In the case of smooth divergent wall, the re-entrant jet moves upstream along the wall and pinches off the cavity as it reaches the leading edge, thereby generating a detached cloud. However, in the case of grooved wall, the behavior of the

𝑢 / 𝑚 • 𝑠 -1 𝑢 / 𝑚 • 𝑠 -1 x=5 mm x=10 mm
Towards the trial of investigating cloud cavitation 97 re-entrant jet is supposed to be similar to that described in Figure 6.6, and thus leads to a similar cavitation regime as in the test section with a gap. Compared to the violent collapse of a large shed cloud in a short duration, the continuous shedding and collapse of small vapour structures emit pressure pulses with a much lower amplitude, thus reducing the risk of damage to the surrounding material surface. In addition, this effective means of passive control to cloud cavitation also demonstrates that the re-entrant jet is an important driving mechanism to initiate cloud cavitation.

Measurement results in the new test section

Although the desired cloud cavitation is not observed in the new test section, we still obtain some interesting results, which are presented in this section.

Pressure loss versus cavitation number

The extent of cavitation is described primarily by the cavitation number which can be varied through adjusting the inflow velocity and inlet pressure in the experiments. The absolute pressure loss 𝑃 𝑙 = 𝑃 𝑖𝑛 -𝑃 𝑜𝑢𝑡 , defined as the difference between the average inlet and outlet pressure, is evidently related to the degree of the cavitation. As shown in Figure 6.9(a), decreasing the cavitation number results in an increase of the cavitation volume and consequently the absolute pressure loss is increased over the convergent-divergent channel due to the flow blockage effect. In addition, the absolute pressure loss is positively correlated with the inflow velocity 𝑢 𝑖𝑛 . Figure 6.9(b) presents the relationship between the cavitation number and the pressure loss 𝑃 𝑙_𝑐 = (𝑃 𝑖𝑛 -𝑃 𝑜𝑢𝑡 )/0.5𝜌𝑢 𝑖𝑛 2 normalized by the inlet dynamic pressure.

It can be seen that all experimental points collapse well around one curve, suggesting that the normalized pressure loss is only a function of the cavitation number, independent of the inflow velocity. Another parameter of the pressure ratio 𝑃 𝑟 = 𝑃 𝑜𝑢𝑡 /𝑃 𝑖𝑛 is also plotted as a function of the cavitation number as shown in Figure 6.9(c), and it is observed to be also independent of the inflow velocity.

Cavity length versus cavitation number

In the present work, the mean cavity length was evaluated based on the distribution of the standard deviation of gray levels in a sequence of high speed images. The location of the maximum standard deviation was used to define the position of the mean cavity closure and consequently the mean cavity length was determined as the distance from the Venturi throat to the location of the maximum standard deviation. This criterion has been applied to all the measured flow conditions regardless of the cavitation regime with small oscillation or large scale shedding. For each operating point, 2000 high speed images were recorded to estimate the mean cavity length. Figure 6.10 shows an example of the influence of the number of the images used to calculate the mean value and the standard deviation of gray levels. The convergence test was performed exactly at the position of the maximum standard deviation determined from 2000 images. As can be seen that when the image number is greater than 1500, the mean value and the standard deviation of gray levels tend to stabilize in a small range of ±0.5. Therefore, 2000 images would be sufficient to evaluate the mean cavity length. For the flow conditions with the cavitation number greater than the critical value, the attached

𝐿 𝑐𝑎𝑣 *
100 cavity appears to be relatively steady whereas when the cavitation number is lower than the critical value, the slight change could lead to a strong influence on the extent of cavitation. This phenomenon was also reported by Long et al. (2017).

Velocity and pressure fluctuations

Based on 2000 instantaneous velocity fields evaluated from the PIV measurements, we could obtain the statistical turbulence properties: the streamwise velocity fluctuation 𝑢 ′ * , the cross-stream velocity fluctuation 𝑣 ′ * and the primary Reynolds shear stress 𝑢 ′ 𝑣 ′ * . Their definitions are given as follows,

𝑢 ′ * = √ 𝑢 ′ 𝑢 ′ 𝑢 ref ⁄ 𝑣 ′ * = √ 𝑣 ′ 𝑣 ′ 𝑢 ref ⁄ 𝑢 ′ 𝑣 ′ * = √|𝑢 ′ 𝑣 ′ | 𝑢 ref ⁄
where 𝑢 ref is the bulk velocity at the Venturi throat. In order to investigate the effect of cavitation on turbulent velocity fluctuations, the mean cavity length should be of slight difference for the measured cavitation numbers to ensure the similar dynamics. In the present work, four relatively high cavitation numbers (𝜎= 2.63, 2.55, 2.49 and 2.42) were selected according to the relationship between cavity length and cavitation number in Figure 6.11. These four flow conditions yield a similar cavitation regime which is characterized by a relatively stable cavity with small vapour shedding in the rear part. The vapour volume fraction in the attached cavities is relatively low such that more tracer particles can be carried and detected for the cross-correlation operation. Since the mean cavity length is around 6 mm for the studied cases, two vertical stations were identified: x=3 mm, within the attached cavity and x=12 mm, in the cavity wake where the detached vapour clouds collapse.

The profiles of 𝑢 ′ * , 𝑣 ′ * and 𝑢 ′ 𝑣 ′ * for the different cavitation numbers at x=3 mm and x=12 mm are presented in Figure 6.12(a) and (b), respectively. It is clear that the decrease of cavitation number leads to a rise in the turbulent fluctuations for all the three components in the wake region due to the impact of cavitation collapse. This is in a good agreement with the experimental results reported by [START_REF] Gopalan | Flow structure and modeling issues in the closure region of attached cavitation[END_REF] and [START_REF] Laberteaux | Partial cavity flows. Part 1. Cavities forming on models without spanwise variation[END_REF]. On the other hand, the presence of vapour phase is observed to reduce the streamwise and crossstream velocity fluctuations in the attached cavity. This finding is consistent with the (6.2)

(6.3) (6.4)
Towards the trial of investigating cloud cavitation 101 measurements by fast X-ray imaging described in Chapter 5. The effect of the presence of vapor on the Reynolds shear stress 𝑢 ′ 𝑣 ′ * is not clear since the correlation between the streamwise and cross-stream velocity fluctuations is not dependent on the variation of the cavitation number. respectively. As expected, with decreasing the cavitation number the collapse of vapour bubbles leads to an increase in pressure fluctuations at the Venturi outlet. This trend is more

𝑢 ′ * 𝑣 ′ * 𝑢 ′ 𝑣 ′ * 𝑢 ′ * 𝑣 ′ * 𝑢 ′ 𝑣 ′ *
pronounced in the flow conditions with lower cavitation numbers. On the contrary, the pressure fluctuations at the inlet present a global trend of decrease with the extent of cavitation, suggesting that the pressure fluctuations upstream of the cavity are suppressed due to the presence of the two-phase cavitation structures. This is consistent with the effect on the velocity fluctuations. However, a different trend is observed at higher cavitation numbers: an increase in the extent of cavitation results in a rise of pressure fluctuations at the inlet. One possible interpretation is that the small amount of vapour presence at higher cavitation numbers does not yield a significant damping on the pressure fluctuations while the cavitation collapse plays a major role.

Chapter summary

In this chapter we describe the attempt to produce a classic cloud cavitation in a newlymade test section which can be used in the X-ray imaging experiment. However, the large assembly error between the printed parts and the machined glass inserts leads to the presence of a gap, which actually alters the propagation path of the re-entrant jet, thereby preventing the periodic shedding of large vapour clouds. As a positive side effect, the influence of the gap on the cloud cavitation could be a passive control to this destructive form of cavitation.

Based on the experimental results in the new test section, the global parameters of the pressure loss coefficient, pressure ratio and mean cavity length were plotted with respect to the cavitation number. It was found that these parameters were only a function of the cavitation number, independent of the inflow velocity. In order to investigate the effect of cavitation on

Experimental set-up

As mentioned in the previous chapters, the cavitating flow regime in the Venturi channel with a small contraction ratio is characterized by a quasi-stable sheet cavitation. In order to

Results

The transitional cavitation

In the previous chapters, the X-ray imaging measurements reveal the continuous existence of a weak re-entrant flow underneath the quasi-stable sheet cavity. Compared to the old Venturi geometry, an increase in the contraction ratio of the new test section results in an augmentation of the adverse pressure gradient behind the throat, which favors the formation of a stronger reentrant jet. Under the action of a re-entrant jet with more momentum, the sheet cavity is expected to exhibit a larger scale instability. a larger vapour cloud numbered 4 separates from the main cavity, which is regarded as the primary shedding within a cycle, and it is responsible for the strongest frequency peak in Figure 7.3(b). It can be seen in Figure 7.4(h) that a horseshoe vortex develops at the sheet cavity closure, which is a typical structure of cavitating flow due to the three dimensionality of the flow.

Re-entrant jet induced cloud cavitation

With a reduction in cavitation number, the sheet cavity increases both in length and thickness, which favors a stronger re-entrant jet propagating upstream underneath the cavity without significant interactions with the cavity upper interface. When the re-entrant jet reaches the leading edge, it breaks completely the cavity into two parts, and the rear part is rolled up forming large-scale cloud shedding. The time evolution of such a process over multiple cycles is presented in the t-x diagram of Figure 7.5 for 𝜎 =4.95 and 𝑢 𝑡ℎ =14.6 m/s. Compared to the transitional cavity, a well-defined periodicity can be identified visually in this classic cloud cavitation.

In Figure 7.5, the yellow solid line depicts the cavity growth process; the red solid line denotes the progression of the re-entrant jet, and the blue dashed lines indicate the shed cloud.

It can be seen that the cavity grows at a nearly constant velocity until an instant when the reentrant jet is initiated, and afterwards, the cavity growth slows down. At the instant of the reentrant jet travelling to the origin point, the cavity length attains maximum (~25 mm). The characteristic velocity of the re-entrant jet is estimated to be 4.2 m/s on average according to the slope of the red solid line. It is interesting to note that the re-entrant jet is slightly accelerated Cloud cavitation shedding mechanisms and geometry scale effect on Venturi cavitating flow 109 during its upstream motion. This implies that the driving force (i.e. adverse pressure gradient) and resistance (i. e. shear stress) acting on the reentrant jet are not statistically balanced: to be more specific, the former force prevails over the latter one. Just because of this acceleration process during the cavity growth phase, the re-entrant jet gains sufficient momentum to pinch off the cavity.

The cloud shedding frequency is determined based on the time-varying gray level at the cavity closure whose location is indicated in During the growth phase of the sheet cavity, the upstream part is increasingly occupied by vapour phase, making it appear nearly transparent. Although it is difficult to recognize the exact location of the re-entrant jet, its upstream motion can be inferred from the shrinkage of the cavity transparent portion, as indicated by the yellow arrows in

Condensation shock induced cloud cavitation

As the cavitation number further decreases, high vapour production rate leads to a fast change in the cavity length and density, which favors the formation of a condensation shock at Although in the present cavitation regime, the prevalent mechanism governing the periodic cloud shedding is the upstream propagating shock wave, the re-entrant jet induced cavity instability can still appear occasionally. Figure 7.9 shows the high speed images of a re-entrant jet induced cloud shedding event and the following cycle of large cloud shedding induced by condensation shock. Figure 7.9 (a, b) presents the incomplete cavity growth under the influence of the re-entrant jet. Meanwhile the previously shed vapour cloud is rolled up and advected downstream. When the re-entrant jet reaches the vicinity of the leading edge, it pinches off the sheet cavity forming the detached cloud, as shown Figure 7.9 (c). With the cloud being convected downstream, a new sheet cavity starts to grow, as shown Figure 7.9 (d-f). Different from the shedding process described in Figure 7.7, the so-called detached cloud is still connected to the sheet during its most period of moving downstream. This is due to the fast cavity growth since the vapour production rate is relatively high in the present flow conditions. 7.9 (g) depicts the instant of the cloud collapse, which seems to have no significant impact on the cavity growth. In Figure 7.9 (h), the sheet cavity has grown completely to its maximum length (~ 42 mm) with a well-defined closure line. Shortly after, a condensation shock forms at the trailing edge of the cavity, as can be observed in Figure 7.9 (i). The yellow solid line indicates roughly the location of the shock front. Distinct from the thin re-entrant jet beneath the cavity, the condensation shock spans almost the entire cavity height. Following the simulation results of [START_REF] Budich | Numerical simulation and analysis of condensation shocks in cavitating flow[END_REF], the flow behind the shock front is directed upstream, as indicated by the blue dashed arrow. With the condensation shock propagating upstream through the cavity, the cavity retracts towards the leading edge. This process is illustrated in the leading edge, the cavity collapses completely, while the downstream cloud is further fed by the shear layer evaporation, as shown in Figure 7.9 (m). After a momentary cessation of vapour production, another new cavity begins to grow from the Venturi throat, and the shed cloudy structures merge as they move downstream in a spanwise rotating way (Figure 7.9 (n, o)).

In Figure 7.9 (p), the cavity grows again to its maximum length, and its internal void fraction is higher than the preceding cavity shown in Figure 7.9 (h) as evidenced by the fore part of the cavity being nearly filled with pure vapour. The lower pressure within the cavity associated with lower density is expected to initiate a stronger condensation shock. Three representative time instants are selected in Figure 7.10, illustrating a more evident propagation of condensation shock. At the time instant t1, the shock front (indicated by the yellow line) can be identified clearly, across which, the opaque two-phase mixture transforms to the transparent liquid phase by condensation, thereby generating a discontinuity in void fraction. Just behind the shock front, a thin cavitating shear layer can be observed. At the subsequent time instant t2, the condensation shock continues to travel upstream further collapsing the sheet cavity. At the final time instant t3, the sheet cavity disappears totally, and subsequently the cloudy vortical structures are rolled up into a large cloud moving downstream.

Pressure wave induced cloud shedding

In the above examined cavitation cases, the collapse of the shed clouds does not appear to noticeably disturb the growing cavity. This implies that the collapse-induced pressure wave may attenuate fast with propagation distance to a low intensity when it impinges on the trailing edge of the sheet cavity, and thus its influence on the overall cavity dynamics is negligible. However, we can conceive that with further reduction in cavitation number, the collapse of a larger amount of vapour may emit a pressure wave strong enough to stop the cavity growth or even make it disappear abruptly. This phenomenon is clearly observed in a set of X-ray images provided by our collaborative team. Interestingly, the condensation shock mechanism is also observed in the same images, i.e. the two mechanisms trigger cloud shedding alternately in the same flow conditions. Compared to the conventional high speed images, X-ray phase contrast imaging enables a more detailed visualization of two-phase structures, and thus to distinguish these two mechanisms clearly.

Cloud cavitation shedding mechanisms and geometry scale effect on Venturi cavitating flow 117 Firstly, the processes of the condensation shock induced cavity retraction are presented in Figure 7.11. The test section is also a Venturi-type channel with the same convergent and divergent angle as the above-mentioned one. The throat height is 0.25 mm and the channel width is 0.5 mm. The free-stream velocity at the throat is 𝑢 𝑡ℎ =10 m/s. The cavitating flow was illuminated by the synchrotron X-ray beam at APS and recorded at 67890 frames per second, i.e. 14.73 μs of time interval between two consecutive frames. Due to the limitation of the beam size, only the upstream portion of the whole flow filed is captured. As a supplementary material to the conventional high speed images, Figure 7.11 provides further evidence to support the condensation shock mechanism distinct from the classic reentrant jet mechanism. In this case, the void fraction in the attached sheet cavity is 100%, i.e. pure vapour cavity. It is clear that the shock front spans the entire cavity height, across which, the vapour phase is condensed completely into the liquid phase. It also confirms that the downstream cloudy vapour structures are not separated from the main cavity, instead they are newly generated in the shear layer between the main flow and the post-shock reverse flow. The next shedding cycle just after the one presented in Figure 7.11 is shown in Figure 7.12, where the dominant mechanism of cavity detachment is found to be pressure wave emanating Cloud cavitation shedding mechanisms and geometry scale effect on Venturi cavitating flow 119 from the cloud collapse. Although we cannot see the cloud collapse in the downstream wake, the position of the pressure wave can be roughly inferred via its impact on the pure vapour cavity. In Figure 7.12(b), the rear part of the cavity has been perturbed, which is believed to be caused by the collapse-induced pressure wave. In the subsequent two frames Figure 7.12(c) and (d), the disturbed area has extended to the whole cavity, which is believed to be related to the pressure wave spreading within the cavity. After the passage of the pressure wave, the pressure rise will continue to condense the disturbed vapour/liquid mixture. In Figure 7.12(g), the vapour volume attains the minimum value. The attached sheet cavity has disappeared totally from the Venturi wall, and the main flow entrains the residual vapour downstream. As the throat region recovers to the original low-pressure status, a new cavity starts to grow, as can be observed in Figure 7.12(h). Figure 7.12(i, j) illustrate the further cavity growth until a vapour sheet is formed.

Comparing the two shedding mechanisms revealed in the X-ray images, we can find that the propagation of the pressure wave front does not result in an obvious discontinuity in the flow field as is the case for the condensation shock, which allows us to distinguish these two mechanisms easily. In addition, the propagation velocity of the pressure wave within the cavity is estimated, based on the position of the perturbations, to be 23 m/s at least, which travels much faster than the condensation shock front.

As mentioned above, the flow behind the condensation shock front is oriented upstream, and hence a shear layer between the main flow and the reversed flow is developed. Due to low pressure induced by large turbulent fluctuations in the shear layer, a large amount of vapour volume is newly produced there. In contrast, the flow does not show any signature of reversal after the passage of the pressure wave, and therefore no new vapour is generated. Different amounts of vapour collapsing downstream of the cavity emit pressure waves with distinct magnitude. This might lead the two mechanisms to alternate in the same flow conditions.

Discussion

In the literature, two mechanisms are well acknowledged to initiate the periodic cloud cavitation. One is the classical re-entrant jet mechanism, and another is the condensation shock mechanism which is experimentally demonstrated by [START_REF] Ganesh | Bubbly shock propagation as a mechanism for sheet-to-cloud transition of partial cavities[END_REF] only recently. The third independent mechanism of collapse-induced pressure wave is neglected as it is generally categorized into the condensation shock mechanism. This inappropriate classification may be caused by the fact that pressure waves emitted from cloud collapse are called shock waves in many published papers, and some researchers tend to support the condensation shock is initiated by the collapse-induced pressure wave of the previously shed cloud, e.g. Stanley [START_REF] Bhatt | Numerical investigation of partial cavitation regimes over a wedge using large eddy simulation[END_REF]. In this section, we will give a detailed discussion on the differences of the three mechanisms based on the present experimental results and the data reported in the literature.

Origin

As shown in Figure 7.13, the flow which originally moves over the cavity is forced to deflect towards the Venturi wall due to the pressure difference inside and outside of the sheet cavity. Subsequently it reattaches to the wall establishing a local stagnation point just downstream of the cavity. On the upstream side of the stagnation point, conservation of mass forces the liquid to flow underneath the cavity, which is referred to as re-entrant jet. The above description indicates that the upstream movement of the re-entrant jet is a pressure driven phenomenon and the magnitude of adverse pressure gradient at the cavity closure is directly related to the re-entrant flow strength. The origin of condensation shock waves forming at the rear of the cavity is not yet well understood. The first explanation for this phenomenon is provided by [START_REF] Ganesh | Bubbly shock propagation as a cause of sheet to cloud transition of partial cavitation and stationary cavitation bubbles forming on a delta wing vortex[END_REF] where it is suggested that a rapid increase in the cavity length and vapour content associated with higher vapour production rate results in the generation of shock waves at lower cavitation numbers.

As sketched in Figure 7.14(a), an overpressure region downstream of the cavity causes an abrupt and intense condensation of the vapour phase. This process releases a large amount of latent heat initiating further condensation towards the cavity leading edge. When the characteristic speed based on the condensation rate exceeds the local speed of sound, a void fraction discontinuity is eventually formed. The quantities across this discontinuity have been phenomenon. On the other hand, the pressure wave emitted by the downstream cloud collapse can also contribute to initiating the condensation shock, as suggested by some researchers.

However, we have observed the formation of a condensation front in the absence of cloud collapse from Figure 7.9(i), implying that the collapse-induced pressure wave is not the necessary condition for initiating the condensation shock. This is consistent with the conclusion of [START_REF] Budich | Numerical simulation and analysis of condensation shocks in cavitating flow[END_REF].

As illustrated in Figure 7.14(b), a large vapour cloud collapses abruptly in the downstream high pressure region, which creates a void and surrounding liquid rushes into the void producing a water hammer effect. This causes a large amount of pressure to be concentrated locally resulting in compression waves that travel in all directions. Although the pressure waves are named as shock waves in the literature, there is no strong evidence to show they propagate supersonically, and therefore they are more accurately termed as pressure waves rather than shock waves.

Pressure rise and propagation velocity

The propagate through a pure vapour cavity as shown in Figure 7.11. This might be attributed to that the intense phase change concentrated at the shock front reduces the local sonic speed drastically. The travelling velocity of the collapse-induced pressure wave within the cavity is estimated to be 𝑢 wa > 23 m/s (2.3 𝑢 𝑡ℎ ) based on the position of perturbations in Figure 7.12. In general, the measured condensation shock speed is comparable to the re-entrant jet velocity, but it is much lower than the propagation speed of the pressure wave. The great difference between the condensation shock and pressure wave speed further supports that they are distinct mechanisms.

Cloud shedding processes

The cloud shedding processes initiated by the re-entrant jet, condensation shock and collapse-induced pressure wave are illustrated schematically in Figure 7.13, Figure 7.14(a) and In addition to traveling within the cavity, the pressure wave also spreads in the liquid main flow at a higher velocity that is expected to be the speed of sound in the liquid phase. Unlike the condensation shock front with concentrated phase change, the pressure wave does not result in a discontinuity in void fraction. After the passage of the pressure wave, the condensation continues to take place within the cavity until the remaining vapour structures are transported downstream as a cloud.

Scale effect on Venturi cavitating flow

Problem background

The issue of scale effect was raised when a small Venturi-type section was used for X-ray imaging experiments. Limited by the small size of the X-ray beam cross section (see Section 3.3), the test section had to be designed with very small dimensions. As a consequence, a test section was manufactured with a geometry scaled down 10 times from the original one used by [START_REF] Stutz | X-ray measurements within unsteady cavitation[END_REF]. Figure 7. 15(a) shows the main dimensions of the scaled test section (referred to as Venturi 1). The convergent and divergent angles of 18° and 8° stay the same as the original test section. The channel width is 4 mm, and the height at the entrance of the Venturi profile is 5 mm with the throat height of 3.34 mm.

Unexpectedly, the cavitating flow in Venturi 1 is characterized by a quasi-stable sheet cavitation although the original geometry can produce periodic shedding of large vapour clouds.

In order to understand the reason for this flow stabilization, an experimental study was performed in six geometrically similar Venturi sections where either width or height or both were scaled, as can be found in [START_REF] Dular | Scale effect on unsteady cloud cavitation[END_REF]. The results reveal that small geometry dimensions, especially small throat heights, influence the occurrence of unsteady cloud shedding. In order to obtain classic cloud cavitation, a 12 mm spacer insert was added to Venturi 1 such that a new geometry (Venturi 2) with increased heights was produced. Cloud cavitation shedding mechanisms and geometry scale effect on Venturi cavitating flow 125

Explanations to the observed scale effect

Although Venturi 1 and Venturi 2 are largely different in geometrical parameters, especially in throat height, they are both characterized with a quasi-stable sheet cavitation. For Venturi 1 with a smaller throat height, the occurrence of a reverse flow (𝑢 re ) will lead the outer main flow to move at a higher speed (𝑢 1 > 𝑢 2 ), hence a higher velocity gradient in the shear layer, as illustrated in Figure 7. 15(b). This means that the resistance, i.e. shear stresses acting on the re-entrant jet will become larger, which then suppresses the full development of the reentrant jet. Compared to Venturi 1, the throat height of Venturi 2 is increased greatly, but meanwhile another critical parameter, i.e. the contraction ratio of entrance to throat height, is reduced from 1.5 to 1.1. This results in a decrease of adverse pressure gradient at the cavity closure. Since the driving force gets smaller, the re-entrant jet in Venturi 2 can neither be fully developed as pointed out in the previous chapters. Overall, the re-entrant jets in the both Venturi channels are too weak to pinch off the sheet cavity, thereby making the cavity relatively stable. However, the weak re-entrant jet in Venturi 1 is caused by larger resistance associated with smaller throat height while the weak re-entrant jet in Venturi 2 is due to insufficient driving force associated with decreased contraction ratio.

For the test section (Venturi 3) shown in Figure 7.1, it has an appropriate throat height and contraction ratio. As a consequence, the re-entrant jet can be fully developed and the associated cloud cavitation is observed. This in turn demonstrates the throat height and contraction ratio both play a crucial role to the Venturi cavitation dynamics.

As aforementioned, Venturi 3 experiences periodic cloud shedding driven by re-entrant jet, condensation shock and collapse-induced pressure wave in order with the reduction of cavitation number. For Venturi 1 and Venturi 2, the weak re-entrant jet cannot initiate sheetto-cloud transition due to scale effect, but it does not mean that cloud cavitation cannot take place. Let us recall the cloud shedding processes occurring in a very small Venturi (the throat height is only 0.25 mm), as shown in Figure 7.11 and Figure 7.12. Due to the restriction of the very small throat height, the re-entrant jet is not observed underneath the cavity. However, the periodic cloud shedding induced by the condensation shock or pressure wave mechanism is indeed identified. This implies that as the cavitation number decreases, sheet-to-cloud transition of partial cavities can skip the re-entrant jet mechanism and be directly induced by condensation shocks. Based on the above analysis, it can be inferred that the periodic cloud cavitation will be observed in Venturi 1 and Venturi 2 if the vapour production rate (associated with cavitation number) is high enough to initiate a condensation shock at the downstream end of the cavity.

Chapter summary

To understand mechanisms governing the transition of sheet-to-cloud cavitation is of practical importance, since the quasi-stable sheet cavitation is basically harmless while large cloud shedding and subsequent collapse in the periodic cloud cavitation can cause severe issues like vibration, noise and erosion. In addition to the classically-described re-entrant jet, another mechanism to trigger shedding of large vapour clouds is identified in the literature as an upstream-propagating condensation shock within a partial cavity.

In this chapter, we have observed two types of cloud shedding processes in a Venturi channel through the conventional technique of high speed photography. At higher cavitation numbers, the sheet cavity is relatively short and the cloud shedding is a consequence of a reentrant jet breaking off the cavity from its leading edge. At lower cavitation numbers, the reentrant jet becomes less important to cavitation instabilities and the dominant mechanism for cavity detachment is found to be the condensation shock mechanism where a void fraction discontinuity forms at the trailing edge of the cavity and then propagates upstream until collapsing the entire cavity. By comparison of these two shedding mechanisms, noticeable differences can be found although they have propagation velocities of the same order of magnitude (0.3 𝑢 𝑡ℎ ). The re-entrant jet is generally regarded as a thin physical liquid layer moving upstream beneath the cavity without significant phase change whereas the condensation shock appears as an upstream-propagating discontinuity nearly spanning the complete cavity height with intense condensation at the shock front. Moreover, the detached cloud in re-entrant jet cases is originated from the previous sheet cavity while in condensation shock cases the observed cloud comes from newly-generated vapours due to shear layer cavitation behind the shock front.

In addition to the above two mechanisms, a third mechanism, i.e. the collapse-induced pressure wave mechanism is identified responsible for cavity shedding based on a set of X-ray images of cavitation. When the shed cloud is transported to the downstream high pressure region, it would collapse violently and emit pressure waves that travel in all directions. This mechanism is usually combined with the condensation shock mechanism in the literature because the collapse-induced pressure wave is considered as the trigger of condensation shock Cloud cavitation shedding mechanisms and geometry scale effect on Venturi cavitating flow 127 when it impinges on the rear of the cavity. However, in the present experiments we have indeed observed the onset of a condensation front in the absence of cloud collapse, which implies that collapse-induced pressure wave is not the necessary condition for initiating a condensation shock. Also, the X-ray images reveal the alternate existence of these two mechanisms in the same operating conditions and the pronounced distinctions between them are highlighted: i) pressure rise across the condensation front is very weak (a few kPa) while the amplitude of collapse-induced pressure wave can be as high as hundreds of kPa, ii) The propagation velocity of collapse-induced pressure wave within the cavity is an order of magnitude faster than condensation shock, and iii) the propagation of pressure wave front does not result in an obvious discontinuity in void fraction as is the case for the condensation shock.

Regarding the scale effect, the throat height and contraction ratio of entrance to throat height are found to both play a crucial role to the Venturi cavitation dynamics. A smaller throat height causes a larger resistance acting on the re-entrant jet and a smaller contraction ratio results in a weaker driving force. Both of them would suppress the full development of the reentrant jet and thus tent to stabilize the cavity. However, as the cavitation number decreases to a certain point, sheet-to-cloud transition of partial cavities can skip the re-entrant jet mechanism and be directly induced by condensation shocks. formed steadily between the re-entrant jet (5) and the main flow [START_REF] Brennen | Cavitation and Bubble Dynamics[END_REF]. In this region, the bubbly mixtures originated from the upstream vapour pocket are convected downstream without significant morphological and phase changes as their collapse might be limited by low pressure in the cores of coherent vortical structures. In the cavity closure region (3), small-scale vapour structures are shed continuously from the main cavity under the entrainment of the main flow and/or the action of horseshoe vortices. In the cavity wake region (4), the shed vapour structures collapse into a large number of small bubbles.

Differing from cloud cavitation, the presence of re-entrant jet in sheet cavitation does not cause a large cavity detachment from the leading edge although the jet can pierce the entire cavity. This might be attributed to the re-entrant jet having a propagation speed one order of magnitude smaller than that in cloud cavitation. The weak re-entrant jet does not have sufficient momentum to break off the cavity when it impinges on the cavity interface, thereby leading sheet cavitation to remain in a quasi-stable state. On the other side, the experimental results suggest that besides capturing the presence of re-entrant jet it is also essential to predict the reentrant flow velocity correctly in numerical simulations in order to accurately simulate the transition of sheet-to-cloud cavitation.

Effect of cavitation on turbulence

It is impossible to generally conclude that cavitation promotes or depresses turbulence, and therefore the discussion is focused on several more specific points.

(1) Normal Reynolds stresses (𝑢 ′ 𝑢 ′ and 𝑣 ′ 𝑣 ′ ) On one hand, the occurrence of cavitation alters velocity field significantly, which affects turbulent velocity fluctuations indirectly due to the strong positive correlation between velocity gradient and turbulence intensity. One the other hand, the presence of vapour phase due to cavitation is observed to greatly suppress turbulent fluctuations, which could be related to two mechanisms: (ⅰ) the presence of vapor phase modifies the vortex-stretching process, and (ⅱ) the cavitation compressibility damps out the turbulent fluctuations. As the shed vapour structures are transported to the high-pressure region, the collapse happening there would cause the augmentation of velocity fluctuations. However, this influence is not apparent in sheet cavitation since small vapour structures implode progressively.

(2) Reynolds shear stress (𝑢 ′ 𝑣 ′ )

Unlike the streamwise and cross-stream velocity fluctuations, the shear stress appears to Overall summary and perspectives 133 be weakly correlated with the velocity gradient. The collapse of vapour phase was found to be the main cause of the largest shear stress in the cavity wake. The phase change of vaporization at the upstream interface of the cavity was also found to increase the shear stress noticeably.

The lowest shear stress was identified in the shear layer where vapour bubbles are convected without significant phase change.

(3) The ratio of Reynolds shear stress to turbulent kinetic energy (𝜏 𝑘 ⁄ )

The Bradshaw hypothesis states that the Reynolds shear stress 𝜏 in a boundary layer is proportional to the turbulent kinetic energy k. It is well applied to single-phase flows. However, when cavitation takes place, the linear relationship between them is no longer satisfied due to the effect of two-phase mixtures compressibility, as suggested by the Reboud empirical correction. The X-ray experimental data in the present work show that the Reboud correction with n=10 describes the ratio of 𝜏 𝑘 ⁄ in cavitating areas much better than the Bradshaw hypothesis.

Cavitating flows in a Venturi-type test section with side gaps

A new test section was designed to investigate the structure and dynamics of cloud cavitation. The existence of the side gaps was not on purpose, and they resulted from large assembly errors between the 3D-printed plastic parts and the machined glass inserts. Individual bubbles tracking analysis showed that most of the re-entrant jet was directed into the gaps instead of moving towards the cavity leading edge along the wall. As a consequence, the anticipated large cloud shedding was not observed. On the negative side, this unexpected result delayed our plan to perform fast X-ray imaging measurements in cloud cavitating flows. On the positive side, the gaps changing the propagation path of the re-entrant jet could be a passive control of transition to the destructive cloud cavitation.

Although the desired cloud cavitation was not observed in the new test section, the conventional measurements were still carried out. The time-resolved PIV-LIF measurements revealed that the collapse of vapour clouds resulted in a rise of turbulence level while the presence of the attached cavity suppressed the streamwise and cross-stream velocity fluctuations, which were consistent with the measurement results in sheet cavitation by fast Xray imaging. Based on the pressure measurement results, the intensity of pressure fluctuations at the inlet was found to present a trend of decrease with the reduction of cavitation number, suggesting that the pressure fluctuations upstream of the cavity were suppressed due to the presence of the two-phase cavitation structures.

camera frame rate would reduce the spatial resolution drastically. In the future, if a camera can operate at 271 696 Hz with its full spatial resolution, we will remove the fast shutter in the beamline and thus be able to take advantage of every X-ray pulse for imaging. In this way, the much greater temporal resolution will provide significant insights into the two-phase morphology and dynamics of cavitation. In addition, the removal of the fast shutter will simplify the synchronization scheme, thereby reducing the measurement error of vapour volume fraction. L'effondrement de la phase vapeur est la principale cause de la plus grande contrainte de cisaillement dans le sillage de la cavité. Le changement de phase de vaporisation à l'interface amont de la cavité s'est également avéré augmenter sensiblement la contrainte de cisaillement.

Résumé général et perspectives

La contrainte de cisaillement la plus faible a été identifiée dans la couche de cisaillement où les bulles de vapeur sont convectées sans changement de phase significatif.

( Guangjian ZHANG

List of Figures

Etude expé rimentale de la structure et de la dynamique des é coulements cavitants

Ré sumé

La cavitation est un phé nomè ne complexe impliquant un transfert de masse entre la phase liquide et vapeur à des tempé ratures presque constantes. es processus physiques qui contrôlent les instabilité s dans les é coulements cavitants ne sont pas encore compris, principalement en raison du manque de donné es expé rimentales quantitatives sur les structures diphasiques et la dynamique à l'inté rieur des zones de cavitation opaques. Dans cette thè se, la cavitation partielle dé veloppé e dans de petits canaux convergents-divergents (Venturi) a é té é tudié e expé rimentalement en dé tail pour é lucider ces mé canismes. Ceci a é té ré alisé en combinant une technique d'imagerie par rayons X ultra-rapide, la visualisationconventionnelle à haute fré quence et la vé locimé trie par images de particules. Les principales contributions de la pré sente é tude portent sur les quatre aspects suivants:

(1) une description dé taillé e des structures d'é coulement diphasique au sein des poches de cavitationquasi stables, qui se caracté risent par un é coulement rentrant à faible vitesse existant en continu sous la cavité ; (2) analyse de l'effet complexe de la cavitation sur les fluctuations de vitesses turbulentes; (3) identification et discussion de trois mé canismes distincts responsables de la transition vers un comportement instable pé riodique ; (4) analyse de l'effet d'é chelle sur les comportements cavitants, dans le cas des profils Venturi é tudié s.

Mots clé s: poche de cavitation, instabilité s, Imagerie par rayons X, Structure diphasique

Ré sumé en anglais

Cavitation is a complex phenomenon involving mass transfer between liquid and vapour phase at nearly constant temperature. Advances in the understanding of the physical processes of cavitating flows are challenging, mainly due to the lack of quantitative experimental data on the two-phase structures and dynamics inside the opaque cavitation areas. In this thesis, partial cavitation developed in small convergent-divergent (Venturi) channels was studied experimentally in detail for a better knowledge of the physical mechanisms governing the cavitation instabilities. This was achieved by using an ultra-fast synchrotron X-ray imaging technique aided with conventional high speed photography and Particle Image Velocimetry. The main contributions of the present study can be summarized as follows: (1) detailed description of the two-phase flow structures in quasi-stable sheet cavitation, which is characterized by a low-speed re-entrant flow existing continuously underneath the cavity;

(2) analysis of the complex effect of cavitation on turbulent velocity fluctuations; (3) identification of three distinct mechanisms responsible for the transition of sheet-to-cloud cavitation, with a discussion of the differences between them; (4) analysis of the scale effect on cavitation in the studied Venturi flows.

Key words: Sheet/cloud cavitation, X-ray imaging, Two-phase structure

Figure 2 .

 2 2 shows two examples of cloud shedding.

Figure 2 . 1 .

 21 Figure 2.1. (a) Sheet cavitation on the suction side of a hydrofoil from Foeth (2008b); (b) sheet cavitation on the divergent wall of a Venturi channel from Barre et al. (2009).

Figure 2 .

 2 Figure 2.2. (a) Cloud cavitation on the suction side of a hydrofoil from Foeth (2008b); (b) cloud cavitation on the divergent wall of a Venturi channel from Stutz &Reboud (1997a).

  This re-entrant jet mechanism is presented schematically in Figure2.3. As the attached cavity grows to a certain length, a thin re-entrant jet, mainly composed of liquid, forms near the cavity closure region and moves upstream beneath the cavity. When this jet reaches the cavity leading edge, the whole cavity is pinched off forming a rolling cavitation cloud that is then convected downstream by the main flow until it collapses. Meanwhile a new cavity begins to grow again and the entire process is repeated.

Figure 2 . 3 .

 23 Figure 2.3. Typical unsteady behavior of a partial cavity with the development of a re-entrant jet and the periodic shedding of cavitation clouds from Franc & Michel (2005).

  proposed in Reboud et al. (1998) on turbulent viscosity which actually reduces the friction losses that the re-entrant jet encounters. Pelz et al. (2017) introduced a physical model of transition from sheet to cloud cavitation based on the criterion that the transition occurs when the re-entrant jet reaches the point of origin of the sheet cavity. A good agreement was found between the model-based calculations and the experimental measurements. Their numerical work could also demonstrate the importance of re-entrant jet to initiate cloud cavitation.

Figure 2 . 4 .

 24 Figure 2.4. Instantaneous void fraction fields illustrating the condensation shock mechanism to cause sheet-to-cloud cavitation from Ganesh et al. (2016).

2. 4 ,

 4 using X-ray densitometry to visualize the instantaneous distribution of vapour volume fraction in the cavitating flow over a wedge, they found that the leading edge cloud shedding at lower cavitation numbers was resulted from an upstream propagating void fraction discontinuity, i.e. a condensation shock front. In order to demonstrate this new finding different from the classical re-entrant jet mechanism, Ganesh et al. (2017) also placed a small obstacle under the sheet cavity just like what Kawanami et al. (1997) did. The results showed that cloud shedding in the case of condensation shocks cannot be prevented since the condensation front spans the entire cavity height rather than near the wall only. Motivated by the pioneering work of Ganesh et al. (2016), many interesting numerical and experimental studies were performed later towards revealing the condensation shock mechanism in different configurations (e.g. Wu et al. 2017; Jahangir et al. 2018; Budich et al. 2018; Wu et al. 2019; Trummler et al. 2020; Bhatt & Mahesh 2020).

  al. (2010) used large eddy simulation (LES) to predict cavitating flows in a Venturi nozzle. By comparison of three cases at different cavitation numbers, they concluded that the vapour formation due to cavitation suppressed turbulent velocity fluctuations and the collapse of vapour structures in the downstream region was a major source of vorticity production, resulting in a substantial increase of turbulent kinetic energy. Xing et al. (2005) observed, in their numerical simulation of vortex cavitation in a submerged jet, that cavitation suppressed jet growth and decreased velocity fluctuations within the vaporous regions of the jet. Gnanaskandan & Mahesh (2016) investigated partial cavitating flows over a wedge and found that the streamwise velocity fluctuations dominated the other two components within the cavity, while all three components of fluctuations were equally significant near the cavity closure and downstream of the cavity,

  Detailed flow measurements are essential for the understanding of cavitating flows. Due to the existence of non-transparent liquid/vapour mixtures, visual observation by a fast speed camera (high speed photography) is the most straightforward and widely-used method to capture the temporal evolution of cavitation structures, thereby providing insight into the underlying physics (Foeth et al. 2008a; Aeschlimann et al. 2012). Through post-processing of Physical background and measurement techniques 13 the high speed video images, it is possible to derive some quantitative data, such as the cloud shedding frequency and the cavity growth rate (Prothin et al. 2016; Jahangir et al. 2018). Synchronized with dynamic pressure measurements, high speed images can also reveal the pressure change associated with the cavity unsteady behaviors (Wang et al. 2017; Wu et al. 2017). On one hand, cavitation visibility helps to obtain its global flow characteristics. On the other hand, cavitation opacity hinders the measurements inside the two-phase region. In order to analyze the internal flow structures of cavitation, other techniques, able to visualize the twophase morphology as well as measure quantitative data on void fraction and velocity, are required.2.4.1. Local measurements by intrusive probesCeccio & Brennen (1991) detected individual vapour bubbles using a network of silver electrodes mounted on the surface of a hydrofoil, and thus acquired their velocities.Stutz & Reboud (1997a;[START_REF] Stutz | Two-phase flow structure of sheet cavitation[END_REF] 2000) used a double optical probe to measure the time-averaged void fraction and vapour-phase velocity inside the cavity generated in a two-dimensional Venturitype section. In this technique, the local velocity was estimated by the time interval of a bubble passing two probe tips successively and the local void fraction was defined as the ratio of the cumulated time of vapour phase at the tip of the probe to a given time of observation. In spite of a relative large measurement uncertainty of about 15%, their work gave a preliminary description of the two-phase flow structure inside a sheet cavity and confirmed the presence of a reverse flow along the solid surface. Coutier-Delgosha et al. (2006) made the first attempt to visualize the two-phase morphology inside the sheet cavity by means of a new endoscopic device.

3 Fast X-ray imaging technique and quantitative data extraction based on

 3 [START_REF] Coutier-Delgosha | Analysis of cavitating flow structure by experimental and numerical investigations[END_REF] performed void fraction measurements of cavitation on a plano-convex hydrofoil using the same X-ray attenuation device. They reported that the local mean vapour volume fraction does not exceed 35 % for small sheet cavities, and 60 % for the large ones. Another application of X-ray attenuation measurements can be found inAeschlimann et al. (2011a) where the main vortex shedding frequency in a turbulent cavitating shear layer was estimated through spectral analysis of the void fraction signal. In recent works[START_REF] Ganesh | Bubbly shock propagation as a mechanism for sheet-to-cloud transition of partial cavities[END_REF][START_REF] Wu | Multimodal partial cavity shedding on a twodimensional hydrofoil and its relation to the presence of bubbly shocks[END_REF], time-resolved X-ray densitometry was used to measure the instantaneous distribution of void fraction in the cavitating flow field. A well-defined void-fraction discontinuity spanning the thickness of the cavity was observed to propagate upstream. According to the authors, this discontinuity represented a bubbly shock front which was responsible for periodic shedding of large-scale vapour clouds.A standard 2D X-ray densitometry system can only provide a projection of the sample's density distribution in the direction of the X-ray beam. This means that the 3D flow information is collapsed onto a single 2D plane. As a remedy, the actual 3D spatial distribution can be retrieved using a tomographic reconstruction. The examples of using X-ray computed tomography (CT) to measure the radial distribution of void fraction in circular nozzle cavitating flows can be found in Bauer et al. (2018) and Jahangir et al. (2019).2.4.4. X-ray velocimetry based on phase contrastX-ray phase-contrast imaging enables clear visualization of boundaries between phases with different refractive index[START_REF] Kastengren | Synchrotron X-ray techniques for fluid dynamics[END_REF]. Aside from detailed illustration of two-phase morphology[START_REF] Karathanassis | High-speed visualization of vortical cavitation using synchrotron radiation[END_REF]), X-ray phase-contrast images can also be used to perform velocimetry by tracking either seeded particles or phase interfaces. Early applications of X-ray velocimetry are for solving the so-called optical access issue. For example,[START_REF] Lee | X-ray particle image velocimetry for measuring quantitative flow information inside opaque objects[END_REF] employed a low-energy synchrotron X-ray beam instead of a laser sheet to illuminate the seeded particles in the flow in an opaque tube. The instantaneous velocity field was extracted by cross-correction similar to the conventional PIV analysis. For opaque flows with a very low speed (a few millimeters per second), they also developed a compact X-ray-based PIV system employing a medical X-ray tube as a light source (Lee et al.2009).For high speed fluid flows, a short exposure time is needed to freeze the fluid motion. The third-generation synchrotron radiation sources such as the advanced photon source (APS) at Argonne National Lab can produce a high-energy and coherent X-ray beam which satisfies the requirements of ultra-fast X-ray phase-contrast imaging. Im et al. (2007) used the APS X-ray source to greatly improve the particle image quality making single-particle tracking velocimetry possible in an optically opaque vessel. Wang et al. (2008) revealed for the first time the internal structures of high-speed (order of 60 m/s) optically dense sprays near the nozzle exit using the ultrafast APS X-ray phase-contrast imaging technique. The velocity fields were measured by tracking the movements of the phase enhanced liquid-gas boundaries. A similar application of synchrotron X-ray phase-contrast imaging in fuel injector nozzles was presented by Moon (2016) for comparing the dynamic structure of biodiesel and conventional fuel sprays. The first attempt to measure velocity field inside cavitating regions by means of fast X-ray image was described by Coutier-Delgosha et al. (2009). Later Khlifa et al. (2017) also used the similar method to measure high speed cavitating flows in a small size Venturi-type test section.In the experiment, the flows were seeded with silver-coated hollow glass sphere particles which were visualized clearly along with the bubbly structures of cavitation by combined effects of X-ray absorption and phase contrast enhancement. Cross-correlation algorithms were applied on the particles and the bubble structures to evaluate the liquid-phase and the vapor-phase velocities respectively. Furthermore, the distribution of vapour volume fraction was determined from X-ray absorption difference in vapour and liquid. Based on the experimental results, the presence of significant slip velocities between the liquid and vapour phase was demonstrated quantitatively for the first time. Their work laid a foundation for the present study on flow structures and dynamics inside sheet cavities. Fast X-ray imaging technique and quantitative data extraction based on image postprocessing Chapter image post-processing The time-resolved, whole-field measurements inside a partial cavity have always been challenging due to the opaqueness of the liquid/vapour mixture. The lack of experimental data on the internal two-phase flow structures and dynamics has been a major issue for the understanding of the cavitation physics and the validation of the numerical models. Motivated by solving this problem, our group has developed in the last 13 years an ultra-fast X-ray imaging technique applied to situations of cavitating flows in a 2D Venturi-type section. Such a technique enables to obtain by image processing i) the liquid velocity field (tracking the motion of the seeding particles), ii) the vapour velocity field (tracking the motion of the vapour bubbles), and ⅲ) the distribution of the vapor volume fraction (based on the difference of X-

  in a closed hydraulic loop designed by Coutier-Delgosha et al. (2009) as shown in Figure 3.1.

Figure 3 . 1 .

 31 Figure 3.1. Schematic of the cavitation tunnel.

Figure 3 . 2 .

 32 Figure 3.2. (a) Profile of Venturi-type test section; the convergent-divergent channel is formed by the lower insert part (2) with the confinement of the top wall (3); flow direction is from left to right, (b) detailed dimensions of the convergent-divergent section (in mm).

(3. 1 )Figure 3 . 3 .

 133 Figure 3.3. (a) Schematic of in-line X-ray imaging system; R1 and R2 are the source to object distance and object to detector distance, respectively. (b) Simple model for absorption contrast of X-ray imaging; the resulting image recorded at the exit-surface z =0 is a two dimensional distribution of intensity. (c) Simple model for free-space propagation-based phase contrast of X-ray imaging; the resulting image recorded at a certain distance downstream z =R2 is a two dimensional distribution of intensity.

  ray imaging technique and quantitative data extraction based on image postprocessing 21 pulses with a cross section of 1.7×1.3 mm 2 : a primary pulse with a duration of 500 ns and a secondary pulse with a duration of 100 ps. The time interval between two primary or secondary X-ray pulses is 3.68 μs. The source-to-object distance is about 60 m, and the object-to-detector distance is optimized to 50 cm for phase contrast enhancement.

Figure 3 . 4 .

 34 Figure 3.4. (a) Diagram showing light path of X-ray imaging system at the APS (from Khlifa et al. 2017). (b) Synchronization of the shutters, the X-ray pulses and the camera frames to acquire image pairs for PIV analysis.

Figure 3 . 5 .

 35 Figure 3.5. (a) Schematic of X-ray beam scanning positions; position 1 corresponds to the beginning of the cavity. The test section was moved to different positions enabling X-ray beam to scan the complete flow field of interest. (b) A representative raw X-ray image of cavitation recorded at the scanning window 2.

Figure 3 .

 3 Figure 3.5(b) presents a typical raw X-ray image of cavitation recorded at the second position. The bright band at the center results from extra partial exposure to a secondary X-ray pulse as indicated in Figure 3.4(b). In addition to absorption contrast, the X-ray phase contrast leads to a fringe pattern with sharp intensity variation along the phase interface, which allows a better visualization of the two-phase flow morphology and the seeded particles. As can be found in the raw radiograph, the vapour structures in the position close to the cavity leading edge are generally coherent except a few individual vapour bubbles identified. The seeded particles both inside and outside the cavity are also visualized clearly. They are much smaller and have relatively regular round shape, compared to individual vapour bubbles.

Figure 3 . 6 .

 36 Figure 3.6. Result of image processing for separation of two phases. (a) Raw X-ray image of cavitation; (b) particle image for measurements of liquid phase velocities; (c) vapour structure image for measurements of gaseous phase velocities.

Figure 3 .

 3 7(a) and (b) show the calibration images for gaseous phase and liquid phase, respectively. They were obtained by averaging a large number of reference images corresponding to each phase.

Figure 3 .

 3 7(c) presents a processed image for the computation of void fraction. The aim of processing the raw image is to reduce the effect of imperfect synchronization and fringes due to X-ray phase contrast mechanism on void fraction measurements. The detailed processing procedures are provided in Section 3.7. The final result of instantaneous vapour volume fraction field is illustrated in Figure 3.7(d).

25 Figure 3 . 7 .

 2537 Figure 3.7. Measurements of vapour volume fractions, (a) calibration image of air; (b) calibration image of water; (c) processed image for computation of void fraction; (d) vapour volume fraction field.

Figure 3 . 8 .

 38 Figure 3.8. Instantaneous velocity vector field evaluated from an image pair, (a) liquid phase; (b) vapour phase.

  ray imaging technique and quantitative data extraction based on image postprocessing 27

Figure 3 .

 3 Figure 3.8(a) shows an instantaneous velocity vector field of the liquid phase evaluated through the above cross-correlation algorithm, in which the aberrant vectors detected by median test were rejected and then replaced by bilinear interpolation from their valid neighboring vectors. The two ends where illumination is not sufficient were excluded. Each pair of processed particle images produced 26×31 local velocity vectors spaced equally by 22 pixels (44 μm).

Figure 3 .

 3 8(b) presents the computed velocity field of the vapour phase.The uncertainty of the velocity measurements for both phases were discussed in[START_REF] Khlifa | Fast x-ray imaging of cavitating flows[END_REF]. The cross-correlation accuracy of tracer particles was estimated by generating synthetic images of a cavitating flow with imposed particle displacements. The comparison between imposed and measured values shows that the mean error of the liquid phase velocity measurements is around ± 0.24 m/s (approximately 3% of the reference velocity of the studied flow). The uncertainty in vapour phase velocity measurements was also estimated by imposing known displacements on vapour structures of a cavitating flow. The results indicated an error of around ±0.46 m/s (approximately 5% of the reference velocity) for the vapour phase.

Figure 3 . 9 .Figure 3 . 9 ,Flow

 3939 Figure 3.9. Comparison of mean streamwise velocity (𝑢) profiles at two locations measured by the standard PIV and the X-ray PIV in the non-cavitating condition, (a) x=8 mm; (b) x=14 mm.

Figure 3 .

 3 Figure 3.10(c) presents the decomposition results by performing a 4-level discrete wavelet transform with the db5 wavelet function in the Matlab software. As shown in Figure 3.10(b), the part delimited by the red frame in the raw image is zoomed up for clarity. The reason of

Figure 3 . 10 .

 310 Figure 3.10. Discrete wavelet transform. (a) A binary tree of 4-level discrete wavelet transform; (b) raw X-ray image and the enlarged part; (c) wavelet decomposition results corresponding to the binary tree of 4-level discrete wavelet transform.

Figure 3 .

 3 11(a) and (b) show the decomposition results using the threshold values determined from Wavemenu. It can be seen that the boundaries of the particles and the vapour ray imaging technique and quantitative data extraction based on image postprocessing 31 bubbles are separated out while leaving the particle interiors and the large-area vapour structures in the approximation component. In addition, the non-uniformities of image intensities caused by the beam's secondary pulse are solved, which would make the edge detection more effective.

Figure 3 . 11 .

 311 Figure 3.11. Particle detection based on the phase contrast imaging mechanism, (a) reconstructed approximation component; (b) reconstructed detail component; (c) preliminary particle detection by Canny edge detector; (d) removed objects by prescribed criteria; (e) remaining particles; (f) removed region shown in grey level mode; (g) hidden particles in the removed regions; (h) final result of the first identification of particles.

Figure 3 .

 3 11(c) shows the result of the preliminary detection, which is not satisfactory due to the presence of vapour bubble edges. Criteria based on the size of the tracer particles and their shapes were applied to remove these spurious objects caused by vapour bubble edges: (1) the largest diameter of an object must not exceed 20 pixels (40 μm); (2) the eccentricity of an object must be less than 0.4 to ensure its roundness. The removed objects and the remaining particles are shown in Figure3.11(d) and (e), respectively. In fact, there are in the removed region (Figure3.11(f)), so the Canny edge detector and the criteria with slightly more strict thresholds were reapplied on the removed region to identify the hidden particles as shown in Figure3.11(g). The final result of the first identification of particles is shown in Figure3.11(h) and the number of objects recognized is 870.

Figure 3 . 12 .

 312 Figure 3.12. Particle detection based on the absorption contrast imaging mechanism, (a) reconstructed approximation component; (b) reconstructed detail component; (c) background suppression; (d) binary image of identified particles with the presence of vapour structures; (e) final result of the second identification of particles.

33 Figure 3 .

 333 Figure 3.13. (a) Combination of particles identified by the two methods; (b) particle gray level restoration; (c) vapour structures after filtering particles out.

Figure 3 .

 3 Figure 3.13(a) displays a combination of particles identified by the above two methods.

Figure 3 .

 3 Figure 3.13(c) presents the final image of vapour structures which is used for calculating the instantaneous velocity field of vapour phase.

Figure 3 .

 3 Figure 3.14 illustrates the three synchronization schemes and their corresponding image pairs.

Figure 3 . 14 .

 314 Figure 3.14. Three actual synchronization schemes, (a) Image 1 exposed to one primary pulse and one secondary pulse, Image 2 exposed to one primary pulse and two secondary pulses; (b) both Image 1 and Image 2 exposed to one primary pulse and one secondary pulse; (c) Image 1 exposed to one primary pulse and two secondary pulses, Image 2 exposed to one primary pulse and one secondary pulse. The image row-averaged intensity is plotted for a quantitative comparison between Image 1 and Image 2 in the same pair.

Figure 3 . 35 Figure 3 . 15 .

 335315 Figure 3.15. Variation of global intensity of air reference images in a packet of 144 image pairs. (a) Intensity variations of the first image and the second image; the green solid line denotes the mean intensity of the first image of each pair; the blue solid line denotes the mean intensity of the second image of each pair; the red dashed line signifies the average intensity of all images in the packet; (b) variation of the average intensity of the two images belonging to the same pair (the blue solid line) relative to the packet average value (the red dashed line).

Figure 3 . 16 .

 316 Figure 3.16. Variation of global intensity of water reference images in a packet of 144 image pairs. (a) Intensity variations of the first image and the second image; the green solid line denotes the mean intensity of the first image of each pair; the blue solid line denotes the mean intensity of the second image of each pair; the red dashed line signifies the average intensity of all images in the packet; (b) variation of the average intensity of the two images belonging to the same pair (the blue solid line) relative to the packet average value (the red dashed line).

Figure 3 . 17 . 37 Figure 3 . 18 .

 31737318 Figure 3.17. Mean vapour volume fraction error estimated from pure air images.

Figure 3 . 19 .

 319 Figure 3.19. Smooth the band edges to reduce vapour volume fraction errors within the transition zone. (a) An original X-ray image of air with sharp intensity change at the middle band edges; (b) separated detail component of band edges; (c) smoothing the intensity change at the band edges; (d) reconstructed air image with smooth intensity change at the band edges; (e) mean vapour volume fraction error estimated from processed air images, compared to the mean error estimated from air images without band edge intensity correction.

39 Figure 3 . 20 .

 39320 Figure 3.20. Image processing for vapour volume fraction measurements, (a) unprocessed Xray image of cavitation; (b) detail component of wavelet decomposition; (c) detail component processed by averaging filter; (d) first processed image in a pair; (e) second processed image in a pair; (f) pair averaged image for estimating vapour volume fraction.

Figure 3 .

 3 20(b) presents the detail component where the two edges of the middle band are also included. A low pass averaging filter was applied on the image of detail component, after which the diffraction traits of black lines and white lines are neutralized; the high frequency particles are filtered out, and the middle band edges are smoothed. The processed detail component (Figure 3.20(c)) was subsequently added to the image of approximation component reconstructing the image of particles nor vapour bubble interfaces as shown in Figure 3.20(d).

  5 °C and the flow rate Q was set to 35.09 l/min leading to a reference velocity 𝑢 ref = 9.53 m/s at the Venturi throat. The Reynolds number based on the throat height is Re = 1.45×10 5 . Three cases with different cavitation numbers (𝜎 = 7.54, 8.0 and 9.13) were tested. They correspond to the developed stage, the intermediate stage and the early stage of sheet cavitation, respectively.

Figure 4 . 1 .

 41 Figure 4.1. Time series of high-speed photographs for the cavitating case of σ = 7.54, (a) backlighting mode; (b) top-lighting mode; (c) time-space (t-x) diagram derived from 1000 consecutive cavitation snapshots in top-lighting mode.

Figure 4 .

 4 1(a) and (b) show a sequence of side-view images for the developed cavitating case of σ = 7.54 recorded in back-lighting and top-lighting modes, respectively. From these images, an open partial cavity is observed with a turbulent frothy closure. The beginning of the cavity is located close to the throat of the Venturi = 0). The upstream part (x = 0-10 mm) with interfacial waves (indicated by the yellow dashed line) appears to attach to the solid surface steadily. Small-scale vapour structures are rolled up and detached continuously from the cavity closure region (x = 10-12 mm). As these vapour structures are entrained by the flow and convected downstream, they rapidly split and/or collapse into a large number of very small bubbles in the cavity wake region (x >12 mm). This process is also indicated by the red dashed line in Figure4.1(a). The temporal-spatial (t-x) diagram is a useful data processing method to illustrate unsteady cavitation behaviors.Gray levels in each high speed photograph is averaged along the y axis generating a column of the t-x diagram.

Figure 4 .

 4 1(c) presents the t-x diagram for σ = 7.54 which is obtained by stacking 1000 consecutive snapshots in top-lighting mode. It is clear that the upstream part (x = 0-10 mm) of the cavity is fairly stable with slight length oscillations, while vapour structures are shed after x = 10 mm in an irregular fashion.

Figure 4 . 2 .

 42 Figure 4.2. (a) Mean gray level values of 2000 high speed images; (b) standard deviations of gray levels; (c) time evolution of gray level in the probing window (indicated by the red frame in (a)) and its FFT result.

  As shown in Figure4.2(a) and (b), mean values and standard deviations of gray levels of 2000 images were calculated in order to determine the cavity length and the shedding frequency information. The method of estimating the cavity length from the mean gray level image is not accurate since it is too subjective to the selection of the threshold value.[START_REF] Dular | Relationship between cavitation structures and cavitation damage[END_REF] found that the position of the maximum standard deviation of gray level is correlated well with the position of maximal cavitation structure oscillation. Therefore, one can consider the location of the maximum standard deviation as the sheet cavity termination. In recent years, this method has been widely used as could be found inDanlos et al. (2014), Prothin et al. (2016), Long et al. (2017) and Zhang et al. (2019).

Figure 4 .Figure 4 . 3 .

 443 Figure 4.3. (a) Distribution of the mean void fraction 𝛼; (b) mean void fraction profiles overlaid on the mean cavity shape; (c) evolution of the mean void fraction 𝛼 and the standard deviation of void fraction 𝛼 ′ along the streamwise direction.

  void fraction in the transverse direction is plotted in Figure4.3(c), and it illustrates clearly the spatial evolution of the mean void fraction along the flow direction.The evolution of the standard deviation of void fraction 𝛼 ′ is also plotted in the same way. In the region of x = 0-4 mm, the mean void fraction is rapidly decreased from the maximum value 75% to 20%. Thereafter, it reduces progressively from 20% to 0 in the cavity wake at a much slower rate. The distinction suggests that different two-phase flow structures exist in these two regions. The void fraction evolution along the streamwise direction in the present sheet cavity is quite different from the one reported by Coutier-Delgosha et al.(2006) for cloud cavitation on a plano-convex hydrofoil: in their configuration, the void fraction is rather constant in the entire attached cavity.

Figure 4 .

 4 Figure 4.4(a) shows the spatial distribution of the time-averaged longitudinal velocity normalized by the reference velocity 𝑢/𝑢 ref for the case of σ = 7.54. The contour lines 0 and 1 of 𝑢/𝑢 ref are superimposed with the mean void fraction field as shown in Figure 4.4(b), in order to illustrate the relationship between them. Based on the two figures, the following resultscan be obtained in a statistical sense. In this seemingly steady sheet cavitation, an obvious reverse flow (under the contour line of 0) is observed beneath the cavity along the wall. It is commonly referred to as the re-entrant jet. A strong velocity shear layer is formed between the reverse flow and the outer main flow, and it extends to the cavity wake region. The vapour is mainly contained within the shear layer. The thickness of the re-entrant jet remains quite constant in the region of x = 4-12 mm, and the ratio of the re-entrant jet thickness to the cavity thickness is around 0.4, which is out of the range of 15% and 35% measured byCallenaere et

Figure 4 . 4 .

 44 Figure 4.4. (a) Distribution of the time-averaged longitudinal velocity normalized by the reference velocity, 𝑢/𝑢 ref . The white dashed line denotes the contour line of 𝛼 = 0.1. (b) Mean void fraction field on which the contour lines of 𝑢/𝑢 ref = 0 and 1 are overlaid to indicate the mean reverse flow and the external main flow.

Figure 4 . 5 . 4 . 3 .Figure 4 . 6 .Figure 4 . 7 .

 45434647 Figure 4.5. Comparison of the mean void fraction profiles (red dashed line) and the mean longitudinal velocity profiles (black solid line) at different distances from the throat.

Figure 4 .

 4 Figure 4.8(c) by a red frame which is actually included in the X-ray beam scanning position 1 (the numbering system for scanning positions can be found in Figure 3.5 and 4.11).Figure

Figure 4 .

 4 8(a) shows the time history of spatially averaged void fraction at the position of the first probe.Due to the reason mentioned above, only the first packet of 144 samples (corresponding to a duration of 24 ms) is presented here. Taking the packet-to-packet difference into account, all the 13 packets of spectra are averaged and the mean spectrum is shown in Figure4.8(b). As can be seen, there exists a single spectral peak at approximately 167 Hz indicating that the void fraction at the first probe location varies in a quasi-periodic pattern rather than randomly. Then, let us examine the second probe location which is indicated in Figure4.9(c) by a red frame (in the scanning position 2). Just like for the first probe, only the first packet of void fraction variation is presented in Figure4.9(a), and the average of the 13 spectra obtained from the 13 packets is shown in Figure4.9(b). A dominant frequency is also identified at 167 Hz which is consistent with the one obtained at the first location. Since the void fraction in both probe locations varies regularly at the same dominant frequency, it is reasonable to speculate that the upstream vapour pocket oscillates quasi-periodically at about 167 Hz. The corresponding Strouhal number (𝑆𝑡 = 𝑓 * 𝐿 cav /𝑢 ref ) based on the mean cavity length and the reference velocity is calculated to be 0.18.

Figure 4 . 8 .

 48 Figure 4.8. Examination of void fraction in the X-ray scanning position 1, (a) time-varying void fraction in the first probe location indicated by the red dashed frame in (c); (b) average of all the spectra of 13 packets; (c) X-ray images at five typical time instants.

Figure 4 . 9 .

 49 Figure 4.9. Examination of void fraction in the X-ray scanning position 2, (a) time-varying void fraction in the second probe location indicated by the red dashed frame in (c); (b) average of all the spectra of 13 packets; (c) X-ray images at five typical time instants.

Figure 4 . 10 .

 410 Figure 4.10. Examination of void fraction in the X-ray scanning position 107, (a) time-varying void fraction in the third probe location indicated by the red dashed frame in (c); (b) average of all the spectra of 13 packets; (c) X-ray images at five typical time instants.

Figure 4 . 4 . 5 . 56 Figure 4 .

 445564 Figure 4.11. Two-phase flow structures inside sheet cavity, (a) a reconstructed image from 22 scanning positions showing the internal structures of sheet cavity. The arrows indicate the mean flow directions in each sub-regions; the image is equally split into 3 sections which are zoomed and shown in (b-d).

  and dynamics of developed sheet cavitation 57 maximum value of approximately 0.6 m/s (7% 𝑢 ref ). The downstream part (5c) is the region

Figure 4 . 2 ⁄ 2 ⁄ 2 ⁄

 4222 12(a-c) show the distributions of streamwise velocity fluctuations 𝑢 ′ 𝑢 ′ 𝑢 ref , transversal velocity fluctuations 𝑣 ′ 𝑣 ′ 𝑢 ref 2 ⁄ and shear stress |𝑢 ′ 𝑣 ′ | 𝑢 ref , respectively, with the evolution of velocity gradient 𝜕𝑢 𝜕𝑦 ⁄ along the flow direction shown in (d). The velocity fluctuations in longitudinal and transversal directions have the similar distributions and both of their domains of high-velocity fluctuations correspond pretty well to the shear area (indicated by the two dashed lines). The turbulence level in the reverse flow region is quite low due to the relatively steady state of the re-entrant jet in sheet cavitation. With the growth of shear layer thickness, the velocity gradient decreases along the streamwise direction. The longitudinal and transversal velocity fluctuations are observed to decrease gradually along the flow direction, which is consistent with the trend of velocity gradient. In the cavity wake (x >12 mm) where the shedding vapour structures collapse, the longitudinal and transversal fluctuations do not increase compared with the upstream region. One possible explanation for this is that the decrease of fluctuations caused by the velocity gradient reduction is dominant, compared with the effect of vapour collapse, since the collapse is relatively mild in sheet cavitation. Let us recall the PIV measurements by Laberteaux & Ceccio (2001) in which they conclude that vapour collapses increase the turbulence level, leading the velocity fluctuations to be the largest in the wake of the cavity. It appears to be inconsistent with what is found in the present X-ray measurement results, while the fact is that they could not measure the internal velocity field of the sheet cavity and consequently draw that conclusion based on the comparison only outside the cavity. With regard to the distribution of shear stress |𝑢 ′ 𝑣 ′ | 𝑢 ref , it appears to be weakly correlated with the velocity gradient. The most intense area of shear stress is located in the range of x = 12-16 mm, which is exactly where the vapour collapse occurs. It implies that the collapse of vapour phase substantially increases the coupling between the streamwise and cross-stream velocity fluctuations. The second area of intense shear stress is located in a narrow Structure and dynamics of developed sheet cavitation 59 strip area (x = 0-4 mm) corresponding to the interface between the upstream vapour pocket and the liquid main flow. This interface is also the position where the sheet cavity is fed from the liquid through intense vaporization. Therefore, it is reasonable to conjecture that the phase change of vaporization also contributes to the coupling between the streamwise and crossstream velocity fluctuations. In the range of x = 4-10 mm, the vapour bubbles are convected without significant phase change and accordingly the correlation between the streamwise and cross-stream velocity fluctuations reaches the lowest level.

Figure 4 .

 4 Figure 4.12. (a) Distribution of streamwise velocity fluctuations 𝑢 ′ 𝑢 ′ 𝑢 ref 2 ⁄ , (b) distribution of

2 ⁄Figure 4 . 13 .

 2413 Figure 4.13. Profiles of streamwise velocity fluctuations √ 𝑢 ′ 𝑢 ′ 𝑢 ref ⁄ (black line), transversal velocity fluctuations √ 𝑣 ′ 𝑣 ′ 𝑢 ref ⁄ (red line), and shear stress √|𝑢 ′ 𝑣 ′ | 𝑢 ref ⁄

Figure 4 .

 4 Figure 4.13 shows the profiles of streamwise velocity fluctuations, transversal velocity fluctuations, and shear stress compared with the profiles of mean streamwise velocity at different distances from the throat. It is observed that the peaks of velocity fluctuations in the streamwise and cross-stream directions, as expected, correspond approximately to the location of maximum velocity gradient. The turbulence level in the reverse flow region is of the same order of magnitude as in the outer main flow. The streamwise fluctuations are much larger than the cross-stream ones within the shear layer, while in the other regions they just have a slight disparity. That means the turbulence in the shear layer is anisotropic and is dominated by the streamwise velocity fluctuations. The isotropic hypothesis of turbulence modelling may thus require special corrections for the shear layer in cavitating flows.

  predicting turbulent flows with adverse pressure gradient since it accounts for the transport of the principle Reynolds shear stress 𝜏 based on the well-known Bradshaw hypothesis: the Reynolds shear stress 𝜏 in a boundary layer is proportional to the turbulent kinetic energy k , written as,𝜏 𝑘 = 𝜌𝑎 1 ⁄, with 𝑎 1 being a constant.Although the standard k-ω SST model is well applied to single-phase flows, it fails to predict cavity sizes and shedding frequencies in unsteady cavitating flows. Taking into account the effect of cavitation compressibility on turbulent fluctuations, the Reboud correction is to replace the mixture density used in Bradshaw hypothesis by a function of density,

Figure 4 . 14 .

 414 Figure 4.14. Comparison between the experimental data and the Reboud empirical modification with n=1, 3 and 10 (n=1 is the Bradshaw hypothesis).

5 . 1 .Figure 5 . 1 Figure 5 . 1 .Figure 5 . 2 .

 51515152 Figure 5.1 shows the measured distributions of time-averaged void fraction 𝛼 at different cavitation numbers. Also indicated are the mean cavity length 𝐿 𝑐𝑎𝑣 and thickness 𝑇 𝑐𝑎𝑣 determined from the contour of 𝛼 = 0.1 . Note that the number of X-ray scanning positions varies according to the extent of cavitation. In practice, only the positions where vaporous structures dominate were recorded due to the limitation of available beam time. It can be seen that the cavitating area grows significantly both in length and thickness with decreasing the cavitation number 𝜎 . The highest local mean void fraction is obtained immediately

3 .Figure 5 . 3 .

 353 Figure 5.3. Comparison of mean void fraction profiles at different streamwise locations.

Figure 5 .

 5 Figure 5.4 indicates a comparison of three cases for the time-averaged longitudinal velocity fields in which the mean velocity 𝑢 is normalized by the reference velocity 𝑢 ref . The mean cavity profile represented by the isoline of 𝛼 = 0.1 is also overlaid on the corresponding velocity field using the white dotted line. It is clear that the averaged velocity flow field is significantly altered by the development of cavitation. In the case of early cavity, no reverse flow is observed in the mean velocity field, which is consistent with the observation of the cavity attaching to the wall. With the expansion of cavitation area, the adverse pressure gradient at the cavity closure is supposed to increase. It results in higher probability for the onset of a reverse flow. It is clear that a reverse flow region (under the contour line of 0) is formed along

Figure 5 . 4 .

 54 Figure 5.4. Contours of the time-averaged longitudinal velocity normalized by the reference velocity 𝑢/𝑢 ref at different cavitation numbers. The white dotted line depicts the mean cavity profile.

Figure 5 . 5 .Figure 5 . 6 .

 5556 Figure 5.5. Evolution of the time-averaged longitudinal velocity 𝑢/𝑢 ref along the horizontal line of y=0.1 mm (adjacent to the Venturi wall).

Figure 5 .Figure 5 . 7 .Figure 5 . 8 .

 55758 Figure 5.7 shows the internal structure of the sheet cavity using the reconstructed X-ray image, on which the mean velocity vectors and the boundary of the mean reverse flow (indicated by the red dashed line) are overlaid. At 𝜎 = 7.54 and 8.0, the very upstream part of the cavity is characterized by an attached vapour pocket which is fed by intense vaporization

Figure 5 . 9 .Figure 5 . 10 .

 59510 Figure 5.9. Evolution of the occurrence probability of negative velocity along the line of y = 0.1 mm.

  ref large-scale cloud structure. At 𝜎 = 9.13, it is less possible to sustain a vapour pocket since the slow vapour production rate is not able to support the vapour filling process after it is partially transported downstream. The position of the maximum 𝛼 ′ is close to x=1 mm, where there is no re-entrant jet. This largest void fraction perturbation may be attributed to the shear-induced unsteadiness.

Figure 5 . 8 . 5 . 4 .

 5854 Figure 5.8. The 𝛼 ′ at 𝜎 = 7.54 is much larger than that at 𝜎 = 8.0 even though the mean void fraction at 𝜎 = 7.54 is lower before x=1.5 mm. This implies that the re-entrant jet plays an important role to make the void fraction vary in a wider range. A more detailed description of

Figure 5 .Figure 5 . 13 .

 5513 Figure 5.12. (a) Time-varying void fraction in a representative packet corresponding to a duration of 0.024 s; (b-d) FFT of the time-varying void fraction for three cavitation numbers.

Figure 5 . 14 .

 514 Figure 5.14. Brightness fluctuations when the test section is empty (full of air) and the corresponding spectrum.

Figure 5 . 2 ⁄ 2 ⁄Figure 5 . 2 ⁄Figure 5 . 16 .

 52252516 Figure 5.15(a) shows the distributions of non-dimensional longitudinal velocity fluctuations 𝑢 ′ 𝑢 ′ 𝑢 ref 2 ⁄

Figure 5 . 17 . 2 ⁄

 5172 Figure 5.17. Profiles of (a) longitudinal velocity fluctuations 𝑢 ′ 𝑢 ′ 𝑢 ref 2 ⁄ , (b) mean velocity

2 ⁄

 2 (c). In general, one can observe that the maximum intensity of longitudinal velocity fluctuations occurs at the location of the largest velocity gradient, which indicates the strong positive correlation between these two quantities. When decreasing the cavitation number, the peak value of velocity gradient, i.e. the shear intensity, increases. However, the longitudinal velocity fluctuations show a different trend: first increasing and then decreasing as cavitation develops. First, let us merely compare the three cavitating cases (𝜎 = 7.54, 8.0 and 9.13). An increase in the extent of cavitation is found to reduce the longitudinal velocity fluctuations in the shear layer although the shear degree is increased. This result suggests that the presence of vapour phase due to cavitation should be responsible for reversing the positive correlation between the velocity gradient and the fluctuations. In other words, more vapour tends to suppress the streamwise turbulent fluctuations. Then the comparison between the non-cavitating case (𝜎 = 10.72) and the smallcavity case of 𝜎 = 9.13 reveals that the occurrence of cavitation leads to a rise in 𝑢 ′ 𝑢 ′ 𝑢 ref , which differs from the preceding trend observed. This is actually another consequence of the competition between the damping effect imposed by the presence of vapour phase and the promoting effect caused by the flow shear. Indeed, as presented in Figure5.17(c), the small amount of vapour at 𝜎 = 9.13 is not expected to yield a prominent damping on the velocity fluctuations.The fact that vapour suppresses the turbulent fluctuations is in agreement with the numerical results of[START_REF] Dittakavi | Large eddy simulation of turbulentcavitation interactions in a Venturi nozzle[END_REF] in a similar Venturi channel. They attribute it to a modification of the vortex-stretching process, due to the presence of vapor. As a non-cavitating vortex is stretched, its diameter reduces. In order to satisfy the conservation of angular momentum, the rotation rate increases and consequently the pressure at the vortex core decreases. However, the core of a cavitating vortex has to maintain a constant pressure, which limits the vorticity production due to vortex stretching. Therefore the formation of vapour results in suppression of turbulent fluctuations. In addition, the local speed of sound in vapour/liquid mixtures could be reduced to a few meters per second, which makes the cavitation structures highly compressible. The compressibility of the two-phase mixture may also damp out the velocity fluctuations, as suggested by[START_REF] Barre | Compressibility effects on the structure of supersonic mixing layers: experimental results[END_REF].

Figure 5 . 18 . 2 ⁄ 2 ⁄ 2 ⁄

 518222 Figure 5.18. Distributions of normalized Reynolds shear stress |𝑢 ′ 𝑣 ′ | 𝑢 ref 2 ⁄ at four different cavitation numbers. The white dashed line depicts the mean cavity profile.

5 . 6 .

 56 systematically by a detailed comparison of the cavities at three representative stages (𝜎 = 7.54 corresponding to the developed stage, 𝜎 = 8.0 corresponding to the intermediate stage as well as 𝜎 = 9.13 corresponding to the early stage). The measurements based on the X-ray imaging indicate that the structure and dynamics of sheet cavitation are highly associated with the behavior of the re-entrant jet.The sheet cavity grows significantly both in length and thickness when the cavitation number is decreased. At the early stage of sheet cavitation, no reverse flow is observed and accordingly the cavity attaches to the wall without evident flow separation. When cavitation develops to an intermediate stage, a reverse flow is initiated in the near-wall region. It tends to displace the rear part of the cavity away from the wall, and hence contributes to the thickening of the sheet cavity. With a further reduction in cavitation number, the size of the reverse flow region increases; the re-entrant jet is strengthened in terms of the magnitude of negative velocity, and the presence of the re-entrant jet becomes more continuous. The enhanced reverse flow combined with the outer main flow generates a stronger shear layer between them where the local low pressure induced by larger turbulent fluctuations prevents the bubbly mixture from collapsing immediately after detaching from the wall, and thus contributes to the significant increase in the cavity length. As in periodic cloud cavitation, the re-entrant jet in the studied sheet cavitation, especially at the developed stage, can almost penetrate the entire cavity, but it never pinches off the cavity into two parts due to insufficient momentum.When cavitation develops from the intermediate stage to the developed stage, the mean void fraction measured in the very upstream part of the cavity becomes smaller. This unexpected phenomenon can be explained from a net effect of two competing mechanisms: vapour production and destruction. With the reduction of cavitation number, the vapour production rate is increased due to more intense vaporization, but in the same time the probability and intensity of the re-entrant jet reaching the cavity leading edge is also increased.Considering the strong destruction effect of the re-entrant jet on the vapour phase, it is therefore possible for the mean void fraction to decrease with the reduction of cavitation number. By comparing the void fraction variation in the three cases, the re-entrant jet is confirmed to be responsible for the quasi-periodic variation of void fraction. In addition, the shear-induced flow instability is another mechanism leading the void fraction to oscillate but in an irregular manner without a dominant frequency.With respect to the influence of cavitation on the longitudinal velocity fluctuations, it is highly correlated with the modification of velocity field. The longitudinal turbulent fluctuations are mainly concentrated in the shear layer with a high velocity gradient, and they are attenuated substantially in the near-wall low-velocity region. The collapse of small vapour structures in the studied sheet cavitating flows does not result in an evident perturbation of the velocity field, while the presence of vapour phase due to cavitation is observed to strongly suppress the streamwise turbulent fluctuations, which might be attributed to two mechanisms: (1) the presence of the vapor phase modifies the vortex-stretching process, and (2) the cavitation compressibility damps out the turbulent fluctuations. Regarding the effect of cavitation on the Reynolds shear stress, the vapour collapse and the vaporization are the two dominant mechanisms to increase the correlation between the streamwise and cross-stream velocity fluctuations. Ré sumé du chapitre Dans ce chapitre, la cavitation tôle ré alisé e dans un canal convergent-divergent est é tudié e systématiquement par une comparaison détaillée des cavités à trois étages représentatifs (σ = 7,54 correspondant à l'étage développé, σ = 8,0 correspondant à l'étage intermédiaire ainsi que σ = 9,13 correspondant au stade pré coce). Les mesures basé es sur l'imagerie aux rayons X indiquent que la structure et la dynamique de la cavitation de la feuille sont fortement associé es au comportement du jet rentrant. La cavité en feuille croî t de maniè re significative à la fois en longueur et en é paisseur lorsque le nombre de cavitation est diminué . Au stade pré coce de la cavitation de la feuille, pré sence de phase vapeur due à la cavitation supprime fortement les fluctuations turbulentes dans le sens du courant, qui pourraient ê tre attribué es à deux mé canismes : (1) la pré sence de la phase vapeur modifie le processus d'é tirement du vortex, et (2) la compressibilité de la cavitation amortit les fluctuations turbulentes. En ce qui concerne l'effet de la cavitation sur la contrainte de cisaillement de Reynolds, l'effondrement de la vapeur et la vaporisation sont les deux mé canismes dominants pour augmenter la corré lation entre les fluctuations de vitesse de flux et transversales. Towards the trial of investigating cloud cavitation 89 Chapter 6

Figure 6 . 1 .

 61 Figure 6.1. Test section -flow direction from left to right, (a) side view of the overall test section; (b) the zoomed convergent-divergent section, dimensions in mm; (c) side wall insert; (d) hollow side wall insert for X-ray imaging; (e) top view of the convergent-divergent section.

Figure 6 . 2 .

 62 Figure 6.2. An example of pressure signals recorded by the inlet and outlet sensors (𝜎=2.08,

Figure 6 . 3 .

 63 Figure 6.3. Calibration plate with regular distribution of marks. The dot diameter is 0.33 mmand the distance between dots is 1 mm. The coordinate system used is also defined.

Figure 6 . 4 .

 64 Figure 6.4. An example of evaluating the instantaneous velocity field in the cavitating flow (𝜎=2.49, 𝑢 𝑖𝑛 =8.75 m/s), flow direction from left to right. (a) Raw PIV image; (b) processed image with a high-pass Gaussian filter; (c) instantaneous velocity vector field (every second vector is displayed in the horizontal direction for clarity).

Figure 6 . 5 .

 65 Figure 6.5. A sequence of high speed images showing a typical cavitation regime obtained in the new test section (𝜎=2.34, 𝑢 𝑖𝑛 =10 m/s). The time interval between two images is 0.5 ms.

Figure 6 .

 6 5 shows the representative cavitation regime observed in the new test section. It is clear that the cavity consists of two distinct parts. The fore part (as shown by 22 mm Flow Towards the trial of investigating cloud cavitation 95 the red frame) is characterized by an attached sheet cavity with nearly constant length, and the rear part (indicated by the yellow frame) is a long region where the cavity seems to break up into numerous vapour bubble clusters.

Figure 6 .

 6 6(b) illustrates the schematic of flow paths in the side gap. As can be seen, the re-entrant jet (indicated by the red line with arrow) is directed into the gap instead of moving upstream along the wall due to smaller resistance in the gap. When it encounters the flow towards downstream (indicated by the yellow line with arrow), the re-entrant jet is forced to deflect upwards and then impinges on the cavity at several points. In this way, the rear part of the cavity is pinched off into a number of small vapour clouds, while the front portion remains relatively stable due to the weak interaction with the re-entrant jet.

Figure 6 . 6 .

 66 Figure 6.6. High speed images in back-lighting mode showing the flow paths in the gap between the bottom insert and the side wall insert (𝜎=2.08, 𝑢 𝑖𝑛 =10 m/s).

Figure 6 . 7 .

 67 Figure 6.7. Comparison of mean streamwise velocity profiles in two test sections: one with a gap and the other without a gap.

Figure 6 . 8 .

 68 Figure 6.8. Cloud cavitation regime on the smooth divergent wall (left) versus sheet cavity without large cloud shedding on the grooved wall (right) from Danlos et al. (2014).

Figure 6 . 9 .

 69 Figure 6.9. The pressure loss as a function of the cavitation number 𝜎, (a) absolute pressure loss 𝑃 𝑙 ; (b) pressure loss coefficient 𝑃 𝑙_𝑐 ; (c) outlet-to-inlet pressure ratio 𝑃 𝑟 .

Figure 6 . 10 .

 610 Figure 6.10. Convergence test of the mean value and the standard deviation of gray levels (𝜎=2.23, 𝑢 𝑖𝑛 =10 m/s).

uFigure 6 . 11 .

 611 Figure 6.11. The nondimensional mean cavity length as a function of the cavitation number.The curve fits through the measurement points.

Figure 6 .

 6 Figure 6.12. (a) Profiles of 𝑢 ′ * , 𝑣 ′ * and 𝑢 ′ 𝑣 ′ * for the different cavitation numbers at x=3 mm; (b) profiles of 𝑢 ′ * , 𝑣 ′ * and 𝑢 ′ 𝑣 ′ * for the different cavitation numbers at x=12 mm.

Figure 6 .

 6 13(a) and (b) illustrate the intensity of pressure fluctuations (𝑃 𝑖𝑛 ′ * , 𝑃 𝑜𝑢𝑡 ′ * ) normalized by the inlet dynamic pressure at the inlet and outlet,

Figure 6 . 13 .

 613 Figure 6.13. Nondimensional pressure fluctuation intensity with respect to the cavitation number, (a) 𝑃 𝑖𝑛 ′ * at the Venturi inlet; (b) 𝑃 𝑜𝑢𝑡 ′ * at the Venturi outlet.

6 . 5 . 7 Cloud

 657 , two vertical stations were selected: one within the attached cavity and another in the cavity wake. It was observed that the collapse of vapour clouds resulted in a rise of turbulence level while the presence of the attached cavity suppressed the velocity fluctuations. The effect of cavitation on the intensity of pressure fluctuations was found to be consistent with the effect on the velocity fluctuations. Ré sumé du chapitre Dans ce chapitre, nous dé crivons la tentative de produire une cavitation nuageuse classique dans une nouvelle section de test qui peut ê tre utilisé e dans l'expé rience d'imagerie aux rayons X. Cependant, la grande erreur d'assemblage entre les piè ces imprimé es et les inserts en verre usiné s conduit à la pré sence d'un interstice, qui modifie en fait le trajet de propagation du jet rentrant, é vitant ainsi le rejet pé riodique de gros nuages de vapeur. En tant qu'effet secondaire positif, l'influence de l'é cart sur la cavitation des nuages pourrait ê tre un contrôle passif de cette forme destructrice de cavitation.Sur la base des ré sultats expé rimentaux de la nouvelle section d'essai, les paramè tres globaux du coefficient de perte de charge, du rapport de pression et de la longueur moyenne de la cavité ont é té tracé s par rapport au nombre de cavitation. Il a é té constaté que ces paramè tres é taient uniquement fonction du nombre de cavitation, indé pendant de la vitesse d'entré e. Afin d'é tudier l'effet de la cavitation sur les fluctuations de vitesse turbulentes, deux stations verticales ont é té sé lectionné es: une dans la cavité attaché e et une autre dans le sillage de la cavité . On a observé que l'effondrement des nuages de vapeur entraî nait une augmentation du niveau de turbulence tandis que la pré sence de la cavité attaché e supprimait les fluctuations de vitesse. L'effet de la cavitation sur l'intensité des fluctuations de pression s'est avé ré cohé rent avec l'effet sur les fluctuations de vitesse.Chapter cavitation shedding mechanisms and geometry scale effect on Venturi cavitating flowThe transition from sheet to cloud cavitation is, in the classic view, related to the presence of a strong re-entrant jet which has sufficient momentum to break off the original sheet cavity from the leading edge, thereby producing periodic shedding of large clouds. In addition to the classic re-entrant jet mechanism, the recent experimental work of[START_REF] Ganesh | Bubbly shock propagation as a mechanism for sheet-to-cloud transition of partial cavities[END_REF] demonstrates that the condensation shock propagating upstream within the bubbly mixture is also an important driving mechanism for periodic cloud cavitation. It should be noted that the pressure wave emitted from the collapse of the downstream vapour clouds may favor the formation of the condensation shock, but it can also cause cloud shedding in a different manner, which is regarded as the third mechanism for cloud cavitation.In this chapter, the three mechanisms (i.e. re-entrant jet mechanism, condensation shock mechanism and clouds-collapse-induced pressure wave mechanism) to initiate periodic cloud cavitation are illustrated in detail through high speed visualization of cavitating flows in a convergent-divergent channel. Moreover, the distinction between the condensation shock and clouds-collapse-induced pressure wave mechanism is highlighted with the help of a set of Xray images from the team of Prof. Matevž Dular and Dr. Martin Petkovšek at University of Ljubljana. Also, we apply these three mechanisms to explain the observed geometry scale effect on Venturi cavitating flows.

  Cloud cavitation shedding mechanisms and geometry scale effect on Venturi cavitating flow 105 produce well-defined periodic cloud cavitation, we use a new Venturi test section without any gap considering the small contraction ratio might limit the cavitation dynamic range, whose schematic is shown in Figure 7.1. The convergent and divergent angles are 18° and 8°, same as the old test section, while it has a 21 mm inlet height and a 10 mm throat height producing a larger contraction ratio of 2.1. The channel width is 5 mm. The static pressure at the inlet is measured at the position of 40 mm before the Venturi throat for determining the inlet cavitation number. The entire section was made with transparent Plexiglas in order to enable clear visualization of cavitation. The high speed photography experiments were performed in the cavitation tunnel described in Section 3.1. The cavitating flow was illuminated by the highpower LED from top, and the high speed camera recorded the side view images at a frame rate of 16000 fps.

Figure 7 . 1 .

 71 Figure 7.1. Schematic of the Venturi section with dimensions in mm, flow from right to left.

Figure 7 .Figure 7

 77 Figure 7.2 shows a time-space (t-x) diagram for 𝜎 =5.76 corresponding to the intermediate regime between stable sheet cavitation and periodic cloud cavitation. In contrast to the continuous shedding of small vapour structures in sheet cavitation, the tendency of larger cloudshedding becomes more apparent and the location where the clouds separate from the main cavity tends to be more upstream. Based on preliminary visual inspection, no clear periodicity can be discerned. Therefore, it is referred to as the transitional cavitation regime.

Figure 7 . 3 .

 73 Figure 7.3. Spectral analysis of the transitional cavity shedding for 𝜎 =5.76 and 𝑢 𝑡ℎ =11.17 m/s. (a) Distribution of the standard deviation of gray level. The probe window for tracing gray level variation is located at the maximum standard deviation. (b) Time-varying spatially averaged gray level within the probe window and the corresponding FFT result.

Figure 7 .

 7 3(b) presents the time history of spatially averaged gray level within the probe area during 125 ms (2000 images) and its spectrum calculated by FFT. A maximum peak can be found at the frequency of 280 Hz, but with the presence of other higher frequency peaks with similar magnitudes. This suggests that the main cloud shedding is accompanied by several secondary sheddings that have a comparable contribution to the cavity oscillations. In the present transitional cavitation, the Strouhal number based on 𝐿 cav and 𝑢 𝑡ℎ is St = 0.23.

Figure 7 . 4 .FlowFigure 7 . 4

 7474 Figure 7.4. High speed photography images showing the main and secondary cloud sheddings of a transitional cavity for 𝜎 =5.76 and 𝑢 𝑡ℎ =11.17 m/s. The numbers 1-3 denote the secondary clouds, and 4 denotes the main cloud. The corresponding time instants are indicated in the above t-x diagram that is a small portion of the one in Figure 7.2.

Figure 7 .

 7 Figure 7.5. T-x diagram obtained by stacking 1000 successive high speed images of re-entrant jet induced cloud cavitation (𝜎 =4.95, 𝑢 𝑡ℎ =14.6 m/s). The yellow solid line depicts the cavity growth process. The red solid line denotes the progression of the re-entrant jet, and the blue dashed lines indicate the shed cloud.

Figure 7 . 6 .

 76 Figure 7.6. Spectral analysis of re-entrant jet induced cloud cavitation for 𝜎 =4.95 and 𝑢 𝑡ℎ =14.6 m/s. (a) Distribution of the standard deviation of gray level. The probe window for tracing gray level variation is located at the maximum standard deviation. (b) Time-varying spatially averaged gray level within the probe window and the corresponding FFT result.

Figure 7 .

 7 6 (a). The transient gray level signal at the probe position and the corresponding spectrum are shown in Figure7.6 (b). The spectrum at the cavity closure reveals that the periodic cloud shedding occurs at a dominant frequency of 144 Hz, which corresponds to a period of 6.95 ms. The smaller peaks at 288 Hz and 432 Hz are not caused by secondary cloud shedding and they are just the harmonics of the dominant frequency. This is in accordance with the well-defined periodicity observed in Figure7.5. In this case, the mean cavity length is 𝐿 cav =20.5 mm, also determined by the position of the maximum standard deviation of gray level. The Strouhal number based on 𝐿 cav and 𝑢 𝑡ℎ is computed to be St = 0.20.

Figure 7 .

 7 Figure 7.7 shows a sequence of high speed images for the re-entrant jet induced cloud cavitation. The corresponding time instants are indicated in the above t-x diagram with the same alphabetic markings.Figure 7.7(a, b) depict the growth of an attached sheet cavity, with

Figure 7 .

 7 7(a, b) depict the growth of an attached sheet cavity, with a vapour cloud detached from the previous cavity being transported downstream.

Figure 7 .

 7 7(cf) show a re-entrant jet forms at the cavity closure and then moves towards the leading edge.

Figure 7 .

 7 7 (d-f). This is because the pressure rise brought by the re-entrant jet can cause the transparent vapour condensing into the opaque mixture. When the re-entrant jet reaches the throat, the entire cavity is peeled off forming the detached cloud and a new cavity appearing at the leading edge begins to grow.Due to the counter-directional velocity of the main flow and the re-entrant jet, the multiple smaller clouds coalesce into a larger cavitating spanwise vortex denoted by the red curve with arrow, as shown in Figure7.7 (g-i). During the same period, the new cavity grows slower than the cloud roll-up, thereby generating a clear spatial separation between them. As the shed cloud is convected downstream in a rolling motion, it collapses in the high pressure wake region and the new sheet cavity continues to grow, as shown in Figure7.7 (j).

Figure 7 . 7 .

 77 Figure 7.7. High speed photography images showing re-entrant jet induced cloud shedding for 𝜎 =4.95 and 𝑢 𝑡ℎ =14.6 m/s. The corresponding time instants are indicated in the above t-x diagram that is a portion of the one in Figure 7.5.

  t=26.125 ms (j) t=27.375 ms the cavity closure. In cavitating cases with lower cavitation numbers, the large-scale cloud shedding is mainly associated with the propagation of the condensation shock rather than the classically-described re-entrant jet.

Figure 7 .Figure 7 . 8 .

 778 Figure 7.8. (a) Temporal evolution of the cloud shedding processes mainly induced by condensation shocks for 𝜎 =4.31 and 𝑢 𝑡ℎ =20.83 m/s. (b) Time-varying gray level signal and the corresponding FFT result probed at the cavity closure.

Figure 7 . 9 .

 79 Figure 7.9. High speed photography images showing processes of a small cloud shedding induced by re-entrant jet and a large cloud shedding induced by condensation shock for 𝜎 =4.31 and 𝑢 𝑡ℎ =20.83 m/s. The corresponding time instants are indicated in the above t-x diagram that is a portion of the one in Figure 7.8(a).

Figure

  Figure 7.9 (g) depicts the instant of the cloud collapse, which seems to have no significant

Figure 7 .

 7 Figure 7.9 (j-l). During the same period, a shear layer develops between the outer main flow (indicated by the white dashed arrow) and the reversed flow across the shock, exhibiting classical Kelvin-Helmholtz instabilities. The coherent vortical motions within the shear layer

  of new cloudy structures behind the shock. Once the shock wave reaches

Figure 7 .

 7 Figure 7.11. A sequence of X-ray images showing the upstream progression of the condensation shock through the attached sheet cavity.

  propagation speed of the shock wave is estimated to be 3.2 m/s through tracking the shock front in the frame series.

Figure 7 .

 7 Figure 7.12. A sequence of X-ray images showing a cavity shedding cycle induced by pressure wave emitted from cloud collapse.

  et al. (2014), Wu et al. (2017), Jahangir et al. (2018),

Figure 7 . 13 .Figure 7 . 14 .

 713714 Figure 7.13. Schematic illustration of cloud shedding processes initiated by re-entrant jet.

  the Rankine-Hugoniot jump conditions numerically (Budich et al. 2018) and experimentally (Ganesh et al. 2016), and thus it represents a compressible shock wave

Figure 7 . 8

 78 Figure 7.8 and tracking the shock front in Figure7.11, respectively. Since the shock speed is supersonic, we can infer that the local speed of sound drops to the order of a few meters per second, which is possible in bubbly mixtures. However, the condensation shock can also

Figure 7 .

 7 Figure 7.14(b) respectively. For the re-entrant jet mechanism, the cloud shedding is caused by the physical re-entrant jet pinching off the sheet cavity. This process does not involve significant phase change. For the condensation shock mechanism, the original sheet cavity collapses completely due to the upstream-propagating condensation shock while the vapour content in the shed cloud is originated from the post-shock cavitating shear layer. For the collapse-induced pressure wave mechanism, it represents an external forcing of the cavity. In

Figure 7 .

 7 15(a) shows also the main dimensions of Venturi 2. The entrance and throat heights were increased to 17 mm and 15.34 mm respectively. Other geometrical parameters were not changed. Nevertheless, as described in Chapter 4 and 5, we find that partial cavities developed in Venturi 2 exhibit the similar behaviors as in Venturi 1, i. e. remaining in a relatively steady regime without noticeable shedding in a wide range of cavitating conditions.

Figure 7 .

 7 Figure 7.15. (a) Main dimensions (in mm) of two Venturi-type test section. (b) Schematic showing the effect of throat height on shear velocity gradient.

8 . 1 . 8 . 2 .

 8182 This thesis focuses on the investigation of the internal structures and the dynamics of partial cavitation developed in two-dimensional convergent-divergent (Venturi) channels, through several experimental techniques including the state-of-the-art synchrotron X-ray imaging. In general, partial cavitation takes two forms of appearance: sheet cavitation and cloud cavitation. The former is often stated as stable in the literature due to slight variation of the observed cavity length and location, while the latter undergoes strong instabilities associated with periodic shedding of large vapour clouds when the cavity length exceeds a certain extent. The mechanisms governing transition from stable to shedding cavities are also discussed in the present work. The application of X-ray imaging technique to cavitating flows An accurate description of cavitation physics strongly relies on the visualization of twophase morphology and the measurements of characteristic quantities (e.g. velocity and void fraction) inside cavitating areas. The use of conventional optical techniques has been unsuccessful to provide this information, due to the opacity of the liquid/vapour mixture.Motivated by solving this problem, an ultra-fast synchrotron X-ray imaging technique was developed in consideration of X-rays high penetrability and weak interaction with matter.Small silver-coated hollow particles (~10 μm) were injected into cavitating flows as the liquid phase tracers for PIV. Thanks to the high-flux coherent X-ray beam provided by the APS synchrotron facility and the purposed-made test section with extremely thin Plexiglas side walls, we obtained high-quality images containing diphasic structures and tracer particles with an unprecedented spatial resolution, under the combined effects of X-ray phase contrast and absorption contrast. A wavelet-decomposition-based image processing method was developed Overall summary and perspectives 131 originally and applied to separate the seeding particles from the vapour structures, which enabled the simultaneous acquisition of time-resolved velocity and void fraction fields in the cavitating flow. Internal two-phase flow structures and dynamics of quasi-stable sheet cavitation Sheet cavitation was generated in the throat region of a convergent-divergent channel with a small contraction ratio, where the quasi-stable cavity can be sustained in a wide range of cavitation numbers. The global behavior of sheet cavitation was studied through the conventional high speed photography. It was observed that the main part of the sheet cavity attached to the solid surface steadily while small vapour structures were rolled up and shed continuously from the closure region in an irregular manner. So far, the mainstream explanation to quasi-stable sheet cavitation is that no re-entrant jet exists underneath the cavity or the reentrant jet only exists near the cavity closure region, thereby producing a relatively stable fore part with a turbulent frothy closure. However, this conjecture was challenged by the detailed X-ray imaging measurements.According to the X-ray measurement results, we classify quasi-stable sheet cavitation into three representative stages based on the status of re-entrant flow: (ⅰ) at the early stage (𝜎 = 9.13), the incipient cavity is short and no reverse flow is observed; (ⅱ) at the intermediate stage (𝜎 = 8.0), an intermittent re-entrant jet is identified underneath the sheet cavity; (ⅲ) at the developed stage (𝜎 = 7.54), the re-entrant jet becomes continuous and it can almost penetrate the entire cavity. It is clear that the sheet cavity beyond inception level can still remain quasistable under the influence of a re-entrant jet.For the developed sheet cavitation, 6 characteristic parts are identified: (1) very upstream part of the cavity, (2) shear layer, (3) cavity closure, (4) cavity wake, (5) reverse flow region, and (6) main flow region outside the cavity. An attached vapour pocket is sustained in the upstream part (1) due to intense vaporization, but it oscillates quasi-periodically under the influence of re-entrant jet. The main part of the re-entrant jet (5) exists persistently along the wall with a relatively constant thickness and velocity. When the attached vapour pocket interacts with the front of the re-entrant jet, it detaches from the wall transforming into a lowvoid-ratio bubbly mixture. The whole re-entrant jet can be regarded as a thin liquid film moving upstream underneath the cavity. The dispersed bubbles entrained by the reverse flow tend to be of spherical shape due to small velocity gradient. The strong velocity shear layer (2) is

Figure 2 . 1 . 8 Figure 2 . 2 . 8 Figure 2 . 3 . 9 Figure 2 . 4 . 10 Figure 3 . 1 . 18 Figure 3 . 2 . 18 Figure 3 . 3 . 20 Figure 3 . 4 . 21 Figure 3 . 5 . 22 Figure 3 . 6 . 24 Figure 3 . 7 . 25 Figure 3 . 8 . 26 Figure 3 . 9 . 28 Figure 3 . 10 . 30 Figure 3 . 11 . 31 Figure 3 . 12 . 32 Figure 3 . 13 . 33 Figure 3 . 14 . 34 Figure 3 . 15 .

 21822823924103118321833203421352236243725382639283103031131312323133331434315 Figure 2.1. (a) Sheet cavitation on the suction side of a hydrofoil from Foeth (2008b); (b) sheet cavitation on the divergent wall of a Venturi channel from Barre et al. (2009). ....................... 8 Figure 2.2. (a) Cloud cavitation on the suction side of a hydrofoil from Foeth (2008b); (b) cloud cavitation on the divergent wall of a Venturi channel from Stutz &Reboud (1997a). .... 8 Figure 2.3. Typical unsteady behavior of a partial cavity with the development of a re-entrant jet and the periodic shedding of cavitation clouds from Franc & Michel (2005). ..................... 9 Figure 2.4. Instantaneous void fraction fields illustrating the condensation shock mechanism to cause sheet-to-cloud cavitation from Ganesh et al. (2016). ................................................. 10 Figure 3.1. Schematic of the cavitation tunnel. ....................................................................... 18 Figure 3.2. (a) Profile of Venturi-type test section; the convergent-divergent channel is formed by the lower insert part (2) with the confinement of the top wall (3); flow direction is from left to right, (b) detailed dimensions of the convergent-divergent section (in mm). ...................... 18 Figure 3.3. (a) Schematic of in-line X-ray imaging system; R1 and R2 are the source to object distance and object to detector distance, respectively. (b) Simple model for absorption contrast of X-ray imaging; the resulting image recorded at the exit-surface z =0 is a two dimensional distribution of intensity. (c) Simple model for free-space propagation-based phase contrast of X-ray imaging; the resulting image recorded at a certain distance downstream z =R2 is a two dimensional distribution of intensity. ...................................................................................... 20 Figure 3.4. (a) Diagram showing light path of X-ray imaging system at the APS (from Khlifa et al. 2017). (b) Synchronization of the shutters, the X-ray pulses and the camera frames to acquire image pairs for PIV analysis. ...................................................................................... 21 Figure 3.5. (a) Schematic of X-ray beam scanning positions; position 1 corresponds to the beginning of the cavity. The test section was moved to different positions enabling X-ray beam to scan the complete flow field of interest. (b) A representative raw X-ray image of cavitation recorded at the scanning window 2. ......................................................................................... 22 Figure 3.6. Result of image processing for separation of two phases. (a) Raw X-ray image of cavitation; (b) particle image for measurements of liquid phase velocities; (c) vapour structure image for measurements of gaseous phase velocities. ............................................................. 24

Figure 3 . 16 . 35 Figure 3 . 17 . 36 Figure 3 . 18 . 37 Figure 3 . 19 . 38 Figure 3 . 20 .

 31635317363183731938320 Figure 3.16. Variation of global intensity of water reference images in a packet of 144 image pairs. (a) Intensity variations of the first image and the second image; the green solid line denotes the mean intensity of the first image of each pair; the blue solid line denotes the mean intensity of the second image of each pair; the red dashed line signifies the average intensity of

Figure 4 . 1 .

 41 Figure 4.1. Time series of high-speed photographs for the cavitating case of σ = 7.54, (a) backlighting mode; (b) top-lighting mode; (c) time-space (t-x) diagram derived from 1000 consecutive cavitation snapshots in top-lighting mode. .......................................................... 43

Figure 4 . 2 . 44 Figure 4 . 3 . 46 Figure 4 . 4 .

 4244434644 Figure 4.2. (a) Mean gray level values of 2000 high speed images; (b) standard deviations of gray levels; (c) time evolution of gray level in the probing window (indicated by the red frame in (a)) and its FFT result. ......................................................................................................... 44Figure 4.3. (a) Distribution of the mean void fraction 𝛼; (b) mean void fraction profiles overlaid on the mean cavity shape; (c) evolution of the mean void fraction 𝛼 and the standard deviation of void fraction 𝛼′ along the streamwise direction. .................................................. 46 Figure 4.4. (a) Distribution of the time-averaged longitudinal velocity normalized by the reference velocity, 𝑢/𝑢 𝑟𝑒𝑓 . The white dashed line denotes the contour line of 𝛼 = 0.1. (b) Mean void fraction field on which the contour lines of 𝑢/𝑢 𝑟𝑒𝑓 = 0 and 1 are overlaid to indicate the mean reverse flow and the external main flow. ....................................................................... 48

Figure 4 . 5 .

 45 Figure 4.5. Comparison of the mean void fraction profiles (red dashed line) and the mean longitudinal velocity profiles (black solid line) at different distances from the throat. ........... 49

Figure 4 . 6 . 50 Figure 4 . 7 .

 465047 Figure 4.6. Time-varying streamwise velocities at four positions along the line of y = 0.1 mm. The percentage represents the probability of re-entrant jet occurrence. The red dashed line denotes the averaged velocity. The vertical blue dashed lines separates different packets. .... 50 Figure 4.7. Evolution of the probability of re-entrant jet occurrence (red line with squares) and the time-averaged longitudinal velocity (black line) along the horizontal line of y = 0.1 mm. .................................................................................................................................................. 51

Figure 4 . 8 . 53 Figure 4 . 9 .

 485349 Figure 4.8. Examination of void fraction in the X-ray scanning position 1, (a) time-varying void fraction in the first probe location indicated by the red dashed frame in (c); (b) average of all the spectra of 13 packets; (c) X-ray images at five typical time instants. .......................... 53 Figure 4.9. Examination of void fraction in the X-ray scanning position 2, (a) time-varying void fraction in the second probe location indicated by the red dashed frame in (c); (b) average of all the spectra of 13 packets; (c) X-ray images at five typical time instants. ...................... 53

Figure 4 . 10 . 54 Figure 4 . 11 . 56 Figure 4 . 12 .

 4105441156412 Figure 4.10. Examination of void fraction in the X-ray scanning position 107, (a) time-varying void fraction in the third probe location indicated by the red dashed frame in (c); (b) average of all the spectra of 13 packets; (c) X-ray images at five typical time instants. ...................... 54Figure 4.11. Two-phase flow structures inside sheet cavity, (a) a reconstructed image from 22 scanning positions showing the internal structures of sheet cavity. The arrows indicate the mean flow directions in each sub-regions; the image is equally split into 3 sections which are zoomed and shown in (b-d). .................................................................................................................. 56Figure 4.12. (a) Distribution of streamwise velocity fluctuations 𝑢 ′ 𝑢 ′ 𝑢 𝑟𝑒𝑓 2 ⁄ , (b) distribution of

Figure 4 . 13 . 60 Figure 4 . 14 . 62 Figure 5 . 1 . 67 Figure 5 . 2 . 68 Figure 5 . 3 . 69 Figure 5 . 4 . 70 Figure 5 . 5 . 71 Figure 5 . 6 . 72 Figure 5 . 7 . 73 Figure 5 . 8 . 74 Figure 5 . 9 . 74 Figure 5 . 10 . 74 Figure 5 . 11 . 76 Figure 5 . 12 .Figure 5 . 13 .Figure 5 . 14 .Figure 5 . 15 . 2 ⁄Figure 5 . 16 .Figure 5 . 17 . 2 ⁄Figure 5 . 18 . 2 ⁄Figure 6 . 1 .Figure 6 . 2 .Figure 6 . 3 .Figure 6 . 4 .Figure 6 . 5 .Figure 6 . 6 .Figure 6 . 7 .Figure 6 . 8 .Figure 6 . 9 .Figure 6 . 10 .Figure 6 . 11 .Figure 6 . 12 .Figure 6 . 13 .Figure 7 . 1 .Figure 7 . 2 .Figure 7 . 3 .Figure 7 . 4 .Figure 7 . 5 .Figure 7 . 6 .Figure 7 . 7 .Figure 7 . 8 .Figure 7 . 9 .Figure 7 . 10 .Figure 7 . 11 .Figure 7 . 12 .Figure 7 . 13 .Figure 7 . 14 .Figure 7 . 15 .

 41360414625167526853695470557156725773587459745107451176512513514515251651725182616263646566676869610611612613717273747576777879710711712713714715 Figure 4.13. Profiles of streamwise velocity fluctuations √ 𝑢 ′ 𝑢 ′ 𝑢 𝑟𝑒𝑓 ⁄ (black line), transversal velocity fluctuations √ 𝑣 ′ 𝑣 ′ 𝑢 𝑟𝑒𝑓 ⁄ (red line), and shear stress √|𝑢 ′ 𝑣 ′ | 𝑢 𝑟𝑒𝑓 ⁄

  As a consequence, most PIV-LIF measurements have focused on the liquid flow regions outside the cavity (Laberteaux & Ceccio 2001; Foeth et al. 2006; Kravtsova et al. 2014) or turbulent cavitating regions with low void fraction (Iyer & Ceccio 2002; Aeschlimann et al. 2011b). Interestingly, the work of Dular et al.

. Validation of the Reboud empirical correction

  The simultaneous acquisition of instantaneous velocity field and density field enables us to validate the Reboud correction experimentally. In this section, we will firstly make a brief introduction of the Reboud empirical correction[START_REF] Reboud | Two-phase flow structure of cavitation: experiment and modeling of unsteady effects[END_REF][START_REF] Coutier-Delgosha | Evaluation of the turbulence model influence on the numerical simulations of unsteady cavitation[END_REF]) and then demonstrate its rationality based on the present experimental data.The local speed of sound in the liquid/vapour mixture is strongly related to the void fraction[START_REF] Shamsborhan | Experimental determination of the speed of sound in cavitating flows[END_REF]) and in some flow conditions it may decrease drastically to a few meters per second. This implies that the cavitating flow is highly compressible in the two-phase region where the Mach number is expected to exceed 1. The mixture density in cavitating flows is defined as

	4.7
	√ 𝑢 ′ 𝑢 ′ 𝑢 ref ⁄	𝑢/𝑢 ref	𝑢/𝑢 ref	𝑢/𝑢 ref
	√ 𝑣 ′ 𝑣 ′ 𝑢 ref ⁄	x=4 mm	x=6 mm
	√|𝑢 ′ 𝑣 ′ | 𝑢 ref ⁄
	x=2 mm
	𝑢/𝑢 ref	𝑢/𝑢 ref	𝑢/𝑢 ref
	x=8 mm	x=10 mm		x=14 mm

4.8. Ré sumé du chapitre

  La cavitation de la feuille générée dans un canal Venturi de petite taille a d'abord été étudiée au moyen d'une photographie à grande vitesse. On a observé que la partie principale de la cavité

de la feuille se fixait à la surface solide de manière constante avec une interface ondulée. De petites structures de vapeur ont été enroulées et détachées en continu de la région de fermeture de la cavité à de multiples hautes fréquences. Ces comportements globaux de cavitation de feuille observés à partir d'images à grande vitesse sont cohérents avec les recherches précédentes sur ce modèle de cavitation. Jusqu'à présent pour le caractère stable de la feuille, la cavitation était généralement liée à une absence de jet rentrant ou à un jet rentrant existant uniquement dans la partie arrière de la cavité, conduisant à une partie antérieure relativement stable avec une fermeture mousseuse turbulente. Cependant, cette conjecture était soutenue par des données expérimentales très limitées sur les structures à deux phases et les champs de vitesse à l'intérieur de la cavité de la feuille.

Sur la base des données actuelles des mesures aux rayons X, les structures d'écoulement diphasiques complexes à l'intérieur de la cavité de la feuille ont été révélées en détail pour la première fois. Comme le montre la figure 4.11, la cavité de feuille dans les présentes conditions d'écoulement est essentiellement divisée en 6 parties. Une poche de vapeur oscillante existe dans la partie amont (1) de la cavité et la fraction de vide moyenne dans cette zone décroît rapidement de la valeur maximale 75% à 20%. La partie principale du jet rentrant (5) existe en permanence à proximité de la paroi avec une épaisseur et une vitesse relativement constantes.

  .2 shows an example of pressure signals recorded by the inlet and outlet sensors. The averaged inlet pressure 𝑃 𝑖𝑛 was used to determine the cavitation number 𝜎:

	𝜎 =	𝑃 𝑖𝑛 -𝑃 𝑣𝑎𝑝 1 2 𝜌 𝑙 𝑢 𝑖𝑛 2

7.5. Ré sumé du chapitre

  Comprendre les mé canismes ré gissant la transition de la cavitation feuille à nuage est d'une importance pratique, car la cavitation de feuille quasi-stable est fondamentalement inoffensive tandis que l'effondrement important des nuages et l'effondrement ulté rieur dans la cavitation pé riodique des nuages peuvent causer de graves problè mes tels que les vibrations, le bruit et l'é rosion. . En plus du jet rentrant dé crit de maniè re classique, un autre mé canisme pour dé clencher la dissipation de gros nuages de vapeur est identifié dans la litté rature comme un choc de condensation se propageant en amont dans une cavité partielle.Dans ce chapitre, nous avons observé deux types de processus d'effacement des nuages dans un canal Venturi grâ ce à la technique conventionnelle de photographie à grande vitesse.À des nombres de cavitation plus é levé s, la cavité de la feuille est relativement courte et la dispersion des nuages est une consé quence d'un jet rentrant rompant la cavité de son bord d'attaque. À des nombres de cavitation infé rieurs, le jet rentrant devient moins important pour les instabilité s de cavitation et le mé canisme dominant pour le dé tachement de la cavité s'avè re ê tre le mé canisme de choc de condensation où une discontinuité de la fraction de vide se forme au bord de fuite de la cavité puis se propage en amont jusqu'à l'effondrement. toute la cavité .En comparant ces deux mé canismes de dé lestage, des diffé rences notables peuvent ê tre trouvé es bien qu'ils aient des vitesses de propagation du mê me ordre de grandeur (0,3 𝑢 𝑡ℎ ). Le jet rentrant est gé né ralement considé ré comme une fine couche de liquide physique se dé plaç ant vers l'amont sous la cavité sans changement de phase significatif alors que le choc de condensation apparaî t comme une discontinuité se propageant en amont couvrant presque toute la hauteur de la cavité avec une condensation intense au front de choc. De plus, le nuage dé taché dans les cas de jets rentrants provient de la cavité de feuille pré cé dente tandis que dans les cas de choc de condensation, le nuage observé provient de vapeurs nouvellement gé né ré es en raison de la cavitation de la couche de cisaillement derriè re le front de choc.En plus des deux mé canismes ci-dessus, un troisiè me mé canisme, c'est-à -dire le mé canisme d'onde de pression induite par l'effondrement, est identifié comme responsable de l'effondrement de la cavité sur la base d'un ensemble d'images radiographiques de cavitation.Lorsque le nuage de jet est transporté vers la ré gion de haute pression en aval, il s'effondrerait violemment et é mettrait des ondes de pression qui se dé placent dans toutes les directions. Ce mé canisme est gé né ralement combiné avec le mé canisme de choc de condensation dans la En ce qui concerne l'effet d'é chelle, la hauteur de la gorge et le rapport de contraction de l'entré e à la hauteur de la gorge jouent tous deux un rôle crucial dans la dynamique de cavitation Cloud cavitation shedding mechanisms and geometry scale effect on Venturi cavitating flow 129 du Venturi. Une hauteur de col plus petite entraî ne une plus grande ré sistance agissant sur le jet rentrant et un rapport de contraction plus petit entraî ne une force d'entraî nement plus faible.Les deux supprimeraient le dé veloppement complet du jet rentrant et tentaient ainsi de stabiliser la cavité . Cependant, à mesure que le nombre de cavitation diminue jusqu'à un certain point, la transition feuille à nuage des cavité s partielles peut sauter le mé canisme de jet rentrant et ê tre directement induite par des chocs de condensation.

	litté rature car l'onde de pression induite par l'effondrement est considé ré e comme le
	dé clencheur du choc de condensation lorsqu'elle frappe l'arriè re de la cavité . Cependant, dans
	les pré sentes expé riences, nous avons bien observé l'apparition d'un front de condensation en
	l'absence d'effondrement des nuages, ce qui implique qu'une onde de pression induite par
	l'effondrement n'est pas la condition né cessaire pour dé clencher un choc de condensation.
	Aussi, les images radiographiques ré vè lent l'existence alterné e de ces deux mé canismes dans
	les mê mes conditions de fonctionnement et les distinctions prononcé es entre eux sont mises en
	é vidence: i) la monté e en pression à travers le front de condensation est trè s faible (quelques
	kPa) tandis que l'amplitude de l'effondrement -l'onde de pression induite peut atteindre des
	centaines de kPa, ii) la vitesse de propagation de l'onde de pression induite par l'effondrement
	dans la cavité est d'un ordre de grandeur plus rapide que le choc de condensation, et iii) la
	propagation du front d'onde de pression n'entraî ne pas une discontinuité é vidente dans la
	fraction de vide comme c'est le cas pour le choc de condensation.

8.1. L'application de la technique d'imagerie par rayons X aux é coulements de cavitation

  Une description pré cise de la physique de la cavitation repose fortement sur la visualisation de la morphologie à deux phases et les mesures de grandeurs caracté ristiques (par exemple la vitesse et la fraction de vide) à l'inté rieur des zones de cavitation. L'utilisation de techniques optiques conventionnelles n'a pas ré ussi à fournir ces informations, en raison de l'opacité du mé lange liquide/vapeur. Motivé e par la ré solution de ce problè me, une technique d'imagerie par rayons X synchrotron ultra-rapide a é té dé veloppé e en tenant compte de la haute pé né trabilité des rayons X et de la faible interaction avec la matiè re. De petites particules le synchrotron APS et à la section de test spé cialement conç ue avec des parois laté rales en plexiglas extrê mement minces, nous avons obtenu des images de haute qualité contenant des structures diphasiques et des particules traceurs avec une ré solution spatiale sans pré cé dent, sous les effets combiné s du contraste de phase des rayons X et du contraste d'absorption. Une mé thode de traitement d'image basé e sur la dé composition en ondelettes a é té dé veloppé e à l'origine et appliqué e pour sé parer les particules d'ensemencement des structures de vapeur, ce qui a permis l'acquisition simultané e de champs de vitesse et de fraction de vide ré solus dans le temps dans l'é coulement de cavitation.

	8.2. Structures d'é

Cette thè se se concentre sur l'investigation des structures internes et la dynamique de la cavitation partielle dé veloppé es dans des canaux bidimensionnels convergents-divergents (Venturi), à travers plusieurs techniques expé rimentales dont l'imagerie par rayons X synchrotron de pointe. En gé né ral, la cavitation partielle prend deux formes d'apparence: la cavitation en feuille et la cavitation nuageuse. Le premier est souvent dé claré stable dans la litté rature en raison de la lé gè re variation de la longueur et de l'emplacement de la cavité observé e, tandis que le second subit de fortes instabilité s associé es à la perte pé riodique de gros nuages de vapeur lorsque la longueur de la cavité dé passe une certaine é tendue. Les mé canismes qui ré gissent la transition des cavité s stables aux cavité s dé lestantes sont é galement discuté s dans le pré sent travail. creuses recouvertes d'argent (~ 10 μm) ont été injectées dans des écoulements de cavitation comme traceurs de phase liquide pour PIV. Grâ ce au faisceau de rayons X cohé rent à haut flux Overall summary and perspectives 137 fourni par

coulement à deux phases internes et dynamique de la cavitation de feuille quasi stable

  Une poche de vapeur attaché e est entretenue dans la partie amont (1) du fait d'une vaporisation intense, mais elle oscille quasi-pé riodiquement sous l'influence du jet rentrant. La partie principale du jet rentrant (5) existe en permanence le long de la paroi avec une é paisseur et une vitesse relativement constantes. Lorsque la poche de vapeur attaché e interagit avec l'avant du jet rentrant, elle se dé tache de la paroi et se transforme en un mé lange bouillonnant à faible taux de vide. L'ensemble du jet rentrant peut ê tre considé ré comme un mince film liquide se dé plaç ant en amont sous la cavité . Les bulles dispersé es entraî né es par l'é coulement inverse ont tendance à ê tre de forme sphé rique en raison du faible gradient de vitesse. La couche de cisaillement à forte vitesse (2) est formé e de maniè re ré guliè re entre le jet rentrant (5) et le flux principal[START_REF] Brennen | Cavitation and Bubble Dynamics[END_REF]. Dans cette ré gion, les mé langes bouillonnants issus de la poche de vapeur amont sont convecté s en aval sans changements morphologiques et de phase significatifs car leur effondrement pourrait ê tre limité par une basse pression dans les noyaux de structures vorticales cohé rentes. Dans la ré gion de fermeture de cavité (3), des structures de

	Selon les ré sultats de la mesure aux rayons X, nous classons la cavitation de feuille quasi-
	stable en trois é tapes repré sentatives en fonction de l'état du flux rentrant: (ⅰ) au stade précoce il est é galement essentiel de pré dire correctement la vitesse d'é coulement rentrant dans des
	(σ = 9,13), la cavité naissante est courte et non un flux inversé est observé; (Ⅱ) au stade simulations numé riques afin de simuler avec pré cision la transition de la cavitation feuille à
	intermédiaire (σ = 8,0), un jet rentrant intermittent est identifié sous la cavité de la feuille; (Ⅲ) nuage.
	au stade développé (σ = 7,54), le jet rentrant devient continu et il peut presque pénétrer dans
	toute la cavité . Il est clair que la cavité de la feuille au-delà du niveau de dé part peut encore 8.3.
	rester quasi-stable sous l'influence d'un jet rentrant.
	Pour la cavitation de feuille dé veloppé e, 6 parties caracté ristiques sont identifié es: (1)

La cavitation de la feuille a é té gé né ré e dans la ré gion de la gorge d'un canal convergentdivergent avec un faible rapport de contraction, où la cavité quasi-stable peut ê tre maintenue dans une large gamme de nombres de cavitation. Le comportement global de la cavitation des feuilles a é té é tudié grâ ce à la photographie conventionnelle à grande vitesse. On a observé que la partie principale de la cavité de la feuille é tait attaché e à la surface solide de maniè re constante tandis que de petites structures de vapeur é taient enroulé es et s'é chappaient en continu de la ré gion de fermeture de maniè re irré guliè re. Jusqu'à pré sent, l'explication principale de la cavitation de feuille quasi-stable est qu'aucun jet rentrant n'existe sous la cavité ou que le jet rentrant n'existe que prè s de la ré gion de fermeture de la cavité , produisant ainsi une partie anté rieure relativement stable avec une fermeture é cumeuse turbulente. Cependant, cette conjecture a é té contesté e par les mesures dé taillé es d'imagerie aux rayons X. partie trè s en amont de la cavité , (2) couche de cisaillement, (3) fermeture de cavité , (4) sillage de cavité , (5) ré gion d'é coulement inverse, et (6) ré gion d'é coulement principale à l'exté rieur de la cavité . vapeur à petite é chelle sont é vacué es en continu de la cavité principale sous l'entraî nement du flux principal et/ou l'action de tourbillons en fer à cheval. Dans la ré gion de sillage de cavité (4), les structures de vapeur é vacué es s'effondrent en un grand nombre de petites bulles.

A la diffé rence de la cavitation nuageuse, la pré sence de jet rentrant dans la cavitation de la feuille n'entraî ne pas un dé collement important de la cavité du bord d'attaque bien que le jet puisse percer toute la cavité . Cela pourrait ê tre attribué au jet rentrant ayant une vitesse de propagation d'un ordre de grandeur infé rieure à celle de la cavitation des nuages. Le jet rentrant faible n'a pas une impulsion suffisante pour rompre la cavité lorsqu'il heurte l'interface de la cavité , conduisant ainsi la cavitation de la feuille à rester dans un é tat quasi-stable. D'un autre côté , les ré sultats expé rimentaux suggè rent qu'en plus de capturer la pré sence d'un jet rentrant,

Effet de la cavitation sur la turbulence

  Il est impossible de conclure de maniè re gé né rale que la cavitation favorise ou diminue la turbulence, et par consé quent la discussion se concentre sur plusieurs points plus spé cifiques. part, l'apparition de cavitation modifie considé rablement le champ de vitesse, ce qui affecte indirectement les fluctuations de vitesse turbulentes en raison de la forte corré lation positive entre le gradient de vitesse et l'intensité de la turbulence. D'un autre côté , on observe que la pré sence de phase vapeur due à la cavitation supprime fortement les fluctuations turbulentes, qui pourraient être liées à deux mécanismes: (ⅰ) la présence de phase vapeur modifie le processus d'étirement du vortex, et (ⅱ) la cavitation la compressibilité amortit les fluctuations turbulentes. Lorsque les structures de vapeur de rejet sont transporté es vers la ré gion à haute pression, l'effondrement qui s'y produit provoquerait une augmentation des fluctuations de vitesse. Cependant, cette influence n'est pas apparente dans la cavitation des plaques car de petites structures de vapeur implosent progressivement.(2) Contrainte de cisaillement de Reynolds (𝑢 ′ 𝑣 ′ ) Contrairement aux fluctuations de vitesse dans le sens des courants et transversaux, la contrainte de cisaillement semble être faiblement corrélée avec le gradient de vitesse.

	Overall summary and perspectives
	D'une
	139

(1) Contraintes de Reynolds normales (𝑢 ′ 𝑢 ′ et 𝑣 ′ 𝑣 ′ )

Circuits de cavitation dans une section d'essai de type Venturi avec fentes laté rales

  Il est bien appliqué aux écoulements monophasés. Cependant, lorsque la cavitation a lieu, la relation linéaire entre eux n'est plus satisfaite en raison de l'effet de compressibilité des mélanges biphasés, comme suggéré par la correction empirique de Reboud. Les données expérimentales aux rayons X dans le présent travail montrent que la correction Reboud avec n = 10 décrit le rapport de 𝜏 𝑘 ⁄ dans les zones de cavitation bien mieux que l'hypothèse de Bradshaw. Une nouvelle section d'essai a é té conç ue pour é tudier la structure et la dynamique de la cavitation des nuages. L'existence des espaces laté raux n'é tait pas exprè s, et ils ré sultaient d'erreurs d'assemblage importantes entre les piè ces en plastique imprimées en 3D et les inserts en verre usiné s. L'analyse de suivi des bulles individuelles a montré que la majeure partie du jet rentrant é tait dirigé e dans les espaces au lieu de se dé placer vers le bord avant de la cavité le long de la paroi. En consé quence, le grand dé gagement de nuages pré vu n'a pas é té observé . Sur le plan né gatif, ce ré sultat inattendu a retardé notre projet de ré aliser des mesures d'imagerie par rayons X rapides dans des flux de cavitation dans les nuages. Du côté positif, les intervalles modifiant le chemin de propagation du jet rentrant pourraient ê tre un contrôle passif de la transition vers la cavitation destructrice des nuages. Bien que la cavitation des nuages souhaité e n'ait pas é té observé e dans la nouvelle section d'essai, les mesures conventionnelles ont toujours é té effectué es. Les mesures PIV-LIF ré solues en temps ont ré vé lé que l'effondrement des nuages de vapeur entraî nait une augmentation du niveau de turbulence tandis que la pré sence de la cavité attaché e supprimait les fluctuations de vitesse dans le sens du courant et transversal, qui é taient cohé rentes avec les ré sultats de mesure dans la cavitation de la feuille par imagerie par rayons X rapide. Sur la base des ré sultats de la

	8.4.

) Le rapport de la contrainte de cisaillement de Reynolds à l'énergie cinétique turbulente

(𝜏 𝑘 ⁄ )

L'hypothèse de Bradshaw stipule que la contrainte de cisaillement de Reynolds 𝜏 dans une couche limite est proportionnelle à l'énergie cinétique turbulente k. mesure de la pression, l'intensité des fluctuations de pression à l'entré e s'est avé ré e pré senter une tendance à la diminution avec la ré duction du nombre de cavitation, ce qui suggè re que les fluctuations de pression en amont de la cavité ont é té supprimé es en raison de la pré sence du biphasé . structures de cavitation.

8.5.

Trois mé canismes pour initier la cavitation des nuages

  Les images conventionnelles à haute vitesse de cavitation combiné es à un ensemble d'images à contraste de phase aux rayons X de notre é quipe collaborative de l'Université de Ljubljana ré vè lent la pré sence de trois types diffé rents de mé canismes responsables de la dispersion des grands nuages: mé canisme de jet rentrant, onde de choc de condensation mé canisme et mé canisme d'onde de pression induit par l'effondrement. A des nombres de cavitation plus é levé s, la cavité de la feuille est relativement courte et le dé tachement de la cavité est une consé quence d'un jet rentrant pincant la cavité de son bord d'attaque. À des nombres de cavitation infé rieurs, le jet rentrant joue un rôle plus petit dans les instabilité s de cavitation et la principale raison de l'effacement pé riodique des nuages est le mé canisme de choc de condensation où une discontinuité de la fraction de vide se forme au bord de fuite de la cavité puis se propage. en amont jusqu'à effondrement de toute la cavité . De plus, si la quantité de nuage de vapeur ré pandu atteint un certain degré , l'effondrement é mettra une onde de pression suffisamment forte pour perturber la cavité en croissance, et par la suite la dé tacher Overall summary and perspectives 141 de la paroi. C'est le troisiè me mé canisme observé dans les expé riences. Dans le pré sent travail, nous soulignons la classification inapproprié e de la combinaison des mé canismes de choc de condensation et d'onde de pression induite par l'effondrement, car nous identifions des

	diffé rences prononcé es entre eux: (i) l'augmentation de pression à travers le front de
	condensation est trè s faible (quelques kPa) tandis que l'amplitude de l'onde de pression induite
	par l'effondrement peut atteindre des centaines de kPa, (ii) la vitesse de dé placement de l'onde
	de pression induite par l'effondrement dans la cavité est d'un ordre de grandeur plus rapide que
	le choc de condensation, et (iii) l'effondrement-l'onde de pression induite n'entraî ne pas une
	discontinuité é vidente de la fraction de vide lorsqu'elle se propage à travers la cavité ,
	contrairement au cas du choc de condensation.
	8.6

. Effet d'é chelle de gé omé trie sur l'é coulement de cavitation du Venturi La

  se amé liore le taux de dé tection des particules par rapport aux algorithmes pré cé dents, la perte d'informations sur les particules due à des particules superposé es en grappes et à des bruits de fond variables doit encore ê tre prise en compte. Ces derniè res

	anné es, l'apprentissage automatique est devenu une technique dominante pour la
	reconnaissance de formes. Les travaux de Li et al. (2020) montre é galement les avantages
	d'une mé thode basé e sur l'apprentissage automatique pour analyser les images d'ombre de
	particules, en particulier pour reconnaî tre des particules incomplè tes ou fortement
	superposé es. La technique d'apprentissage automatique en plein essor nous fournit un outil
	prometteur pour une extraction plus pré cise des particules à partir d'images
	radiographiques et ainsi amé liorer les mesures de champ de vitesse à l'inté rieur des ré gions
	de cavitation.
	• La source de rayons X APS é met successivement des impulsions de rayons X à une
	fré quence de 271 696 Hz. Cependant, dans les pré sentes expé riences d'imagerie aux rayons
	8.7. Points de vue X, la fré quence d'enregistrement de la camé ra a é té fixé e à 12 070 Hz, ce qui a é té obtenu
	Afin d'optimiser l'application de l'imagerie par rayons X synchrotron aux é coulements de grâ ce à l'obturateur hacheur rapide bloquant la plupart des impulsions de rayons X. Le but
	cavitation et d'amé liorer la compré hension physique de la cavitation nappe / nuage, quelques de ceci est d'augmenter le champ de vision car l'augmentation de la fré quence d'images de
	plans supplé mentaires peuvent ê tre proposé s: la camé ra ré duirait considé rablement la ré solution spatiale. À l'avenir, si une camé ra peut
	• L'acquisition des champs de vitesse à partir d'images radiographiques repose sur fonctionner à 271 696 Hz avec sa pleine ré solution spatiale, nous supprimerons l'obturateur

cavitation nuageuse induite par le jet rentrant est fortement affecté e par les paramè tres gé omé triques du Venturi, en particulier la hauteur de la gorge et le rapport de contraction de l'entré e à la gorge. Une hauteur de col plus petite entraî nerait une force de traî né e plus grande agissant sur le jet rentrant, et un rapport de contraction plus petit entraî nerait une force d'entraî nement plus faible, c'est-à -dire un gradient de pression dé favorable. Les deux supprimeraient le dé veloppement complet du jet rentrant et conduiraient donc la cavité de la feuille à rester relativement stable. Tout comme les interstices pré cité s modifiant le trajet de propagation du jet rentrant, le passage de la cavitation nappe à nuage pourrait é galement ê tre reporté en ajustant les paramè tres gé omé triques du Venturi. Cela peut guider la conception de machines hydrauliques qui devraient fonctionner sans effusion instable de gros nuages. A noter que la cavitation des nuages peut encore avoir lieu mê me en l'absence de jet rentrant car elle peut ê tre directement initié e par les deux autres mé canismes. l'extraction de particules traceurs ensemencé es dans le flux de cavitation. Bien que la mé thode de traitement d'image basé e sur la transformé e en ondelettes dé veloppé e dans cette thè • La technique d'imagerie par rayons X synchrotron sera utilisé e pour é tudier les structures et la dynamique de la cavitation des nuages une fois qu'une nouvelle section d'essai é liminant les espaces laté raux sera fabriqué e. Les nouvelles donné es fourniront l'occasion de comparer les deux cas de dé lestage/non-effusion et fourniront des informations sur la transition d'une cavitation de feuille stable à une cavitation nuageuse instable avec effusion à grande é chelle.

• Une nouvelle section de test é quipé e d'un certain nombre de capteurs de pression aligné s le long de la paroi divergente du Venturi sera conç ue, dans laquelle on s'attend à observer l'effondrement des nuages induit par les mé canismes de jet rentrant, de choc de condensation et d'onde de pression induite par l'effondrement dans l'ordre avec la ré duction du nombre de cavitation. La monté e en pression associé e à ces trois mé canismes sera enregistré e et analysé e. rapide de la ligne de lumiè re et pourrons ainsi profiter de chaque impulsion de rayons X pour l'imagerie. De cette faç on, la ré solution temporelle beaucoup plus grande fournira des

3.8. Ré sumé du chapitre L'é coulement de cavitation est intrinsè quement opaque en raison de ré flexions é levé es et de multiples diffusions de lumiè re visible sur les interfaces vapeur/liquide. Les é tudes expé rimentales conventionnelles de la cavitation se limitent souvent à la dé termination des caracté ristiques d'é coulement global (comme la description gé omé trique de la forme de la cavité et des fré quences de dispersion des nuages) e/ou parfois à certaines mesures dans la zone d'é coulement du liquide environnant. Cependant, une description pré cise de la physique de la cavitation repose fortement sur les visualisations de la morphologie à deux phases et les mesures de grandeurs caracté ristiques (comme la vitesse et la fraction de vide) à l'inté rieur du volume de mé lange liquide / vapeur. Dé dié e à la mesure des champs de flux internes de cavitation, une technique d'imagerie ultra-rapide aux rayons X a é té dé veloppé e en tenant compte de la haute pé né trabilité des rayons X et de la faible interaction avec la matiè re. En plus du mé canisme de contraste d'absorption, le contraste de phase des rayons X conduit à une forte variation d'intensité à l'interface de phase, ce qui permet une meilleure visualisation des structures d'é coulement diphasique. Des particules creuses recouvertes d'argent ont é té injecté es dans le flux de cavitation en tant que traceurs de phase liquide, et elles ont é té visualisé es simultané ment avec les structures vaporeuses basé es sur la combinaison de l'absorption et du contraste de phase. Bien que les informations en deux phases des structures et de la dynamique soient contenues dans les images radiographiques, l'extraction des donné es souhaité es n'est pas facile. Dans le pré sent travail, une mé thode de traitement d'image basé e sur la dé composition en ondelettes a é té introduite pour sé parer les particules d'ensemencement des structures de vapeur. De cette maniè re, des corré lations croisé es d'image pourraient ê tre appliqué es à la fois aux particules d'ensemencement et aux bulles de vapeur pour é valuer le champ de vitesse instantané e du liquide et de la phase gazeuse, respectivement. La fraction volumique locale de vapeur pourrait é galement ê tre dé rivé e simultané ment de la diffé rence d'atté nuation des rayons X. Grâ ce à l'acquisition sans pré cé dent de ces donné es, l'é tude expé rimentale dé taillé e de la structure et de la dynamique des é coulements de cavitation est rendue possible pour la premiè re fois.

aucun é coulement inverse n'est observé et, par consé quent, la cavité se fixe à la paroi sans sé paration é vidente de l'é coulement. Lorsque la cavitation se dé veloppe à un stade intermé diaire, un é coulement inverse est initié dans la ré gion proche de la paroi. Il a tendance à é carter la partie arriè re de la cavité de la paroi, et contribue ainsi à l'é paississement de la cavité en feuille. Avec une ré duction supplé mentaire du nombre de cavitation, la taille de la ré gion d'é coulement inverse augmente; le jet rentrant est renforcé en termes de grandeur de vitesse né gative, et la pré sence du jet rentrant devient plus continue. L'é coulement inversé amé lioré combiné à l'é coulement principal externe gé nè re une couche de cisaillement plus forte entre eux où la basse pression locale induite par des fluctuations turbulentes plus importantes empê che le mé lange bouillonnant de s'effondrer immé diatement aprè s le dé tachement de la paroi, et contribue ainsi à l'augmentation significative de la cavité . longueur. Comme dans la cavitation nuageuse pé riodique, le jet rentrant dans la cavitation de la feuille é tudié e, en particulier au stade dé veloppé , peut presque pé né trer dans toute la cavité , mais il ne pince jamais la cavité en deux parties en raison d'une impulsion insuffisante.Lorsque la cavitation se dé veloppe de l'é tage intermé diaire à l'é tage dé veloppé , la fraction de vide moyenne mesuré e dans la partie trè s amont de la cavité devient plus petite. Ce phé nomè ne inattendu peut s'expliquer par un effet net de deux mé canismes concurrents: la production de vapeur et la destruction. Avec la ré duction du nombre de cavitation, la vitesse de production de vapeur est augmenté e en raison d'une vaporisation plus intense, mais dans le mê me temps, la probabilité et l'intensité du jet rentrant atteignant le bord d'attaque de la cavité sont é galement augmenté es. Compte tenu du fort effet de destruction du jet rentrant sur la phase vapeur, il est donc possible que la fraction de vide moyenne diminue avec la ré duction du nombre de cavitation. En comparant la variation de la fraction de vide dans les trois cas, le jet rentrant est confirmé comme é tant responsable de la variation quasi-pé riodique de la fraction de vide. De plus, l'instabilité de l'é coulement induite par le cisaillement est un autre mé canisme conduisant la fraction de vide à osciller mais de maniè re irré guliè re sans fré quence dominante.En ce qui concerne l'influence de la cavitation sur les fluctuations de vitesse longitudinales, elle est fortement corré lé e à la modification du champ de vitesse. Les fluctuations turbulentes longitudinales sont principalement concentré es dans la couche de cisaillement avec un gradient de vitesse é levé , et elles sont atté nué es sensiblement dans la ré gion de basse vitesse proche de la paroi. L'effondrement de petites structures de vapeur dans les é coulements de cavitation en feuille é tudié s n'entraî ne pas une perturbation é vidente du champ de vitesse, tandis que la

Acknowledgements

First of all, I would like to express my deepest gratitude to my supervisor Prof. Olivier Coutier-Delgosha for providing me the opportunity to pursue my PhD in his group. His continuous guidance and critical comments have improved this thesis work substantially. His work enthusiasm inspired me to never give up even though with huge frustrations. I would also like to thank my committee members including Prof. Todd Lowe (Virginia Tech), Prof. Antoine Ducoin (Ecole Centrale de Nantes), Prof. Lionel Thomas (Université de

void fraction variation within a chosen probe window is examined for the three cases in order to unveil the underlying mechanisms of the cavity unsteadiness. Figure 5.11(a) illustrates the location and size of this probe window for three cavitation numbers, and the corresponding probability density function (PDF) of void fraction is shown in Figure 5.11(b). The PDF of void fraction at 𝜎 = 7.54 has a wider distribution with finite probability for a large range of void fraction. Larger values of void fraction are more probable when the probe location is free of re-entrant jet and smaller values when the re-entrant is present. Since the probability of reentrant jet occurring at the probe position is 46% (close to 50%), the PDF at 𝜎 = 7.54 shows a quasi-symmetric distribution with respect to the most probable value. Conversely, the PDF at 𝜎 = 8.0 presents an asymmetric distribution where the larger void fraction values are much more probable than the smaller ones. This is in accordance with the small probability (17%) of re-entrant jet occurring at the probe position. Furthermore, the lower limit of void fraction (~ 0.3) for 𝜎 = 8.0 is higher than that (~ 0.1) for 𝜎 = 7.54. This indicates that the re-entrant jet for 𝜎 = 8.0 is weaker and thus leads to less destruction of vapour at the probe location. The PDF at 𝜎 = 9.13 has a Gaussian type distribution where the mean void fraction is close to the most probable value. It may result from the shear induced instability given that the flow does not see any re-entrant jet.

Due to the use of slow shutter in the experiments, only a packet of 144 pairs of images was recorded per opening of the shutter. Here we only present the time-varying void fraction within the probe window in a representative packet (out of 13 in total) as shown in Figure 5.12(a).

The corresponding spectra by means of FFT are shown in Figure 5.12(b-d). Moreover, taking into account the remaining 12 packets, all spectra of the 13 packets are averaged in one figure to give a more general description of the cavity instability as shown in Figure 5.13. By comparison of all spectra, we can find that only a single dominant spectral peak is obtained for 𝜎 = 7.54 at a lower frequency denoted as 𝑓 1 , suggesting that the void fraction at the probe position varies in a quite regular mode. The time history of void fraction in Figure 5.12(a) also evidently presents an observable periodicity. For 𝜎 = 8.0, two peaks of similar amplitude can be detected at a lower frequency 𝑓 1 and a higher one 𝑓 2 . For 𝜎 = 9.13, only a higher frequency 𝑓 2 is dominant on the spectrum. With the reduction of probability of the re-entrant jet reaching the probe position (i.e. increasing cavitation number), the frequency 𝑓 1 shifts towards small values and its amplitude becomes lower. For 𝜎 = 9.13 without re-entrant jet, 𝑓 1 cannot even be identified. These results confirm the strong correction between the re-entrant jet and the lower

Three mechanisms to initiate cloud cavitation

The conventional high speed images of cavitation combined with a set of X-ray phase contrast images from our collaborative team at University of Ljubljana reveal the presence of three different types of mechanisms responsible for large cloud shedding: re-entrant jet mechanism, condensation shock wave mechanism, and collapse-induced pressure wave mechanism. At higher cavitation numbers, the sheet cavity is relatively short and the cavity detachment is a consequence of a re-entrant jet pinching off the cavity from its leading edge.

At lower cavitation numbers, the re-entrant jet plays a smaller role in the cavitation instabilities and the primary reason for periodic cloud shedding is found to be the condensation shock mechanism where a void fraction discontinuity forms at the trailing edge of the cavity and then propagates upstream until collapsing the entire cavity. In addition, if the amount of shed vapour cloud reaches a certain extent, the collapse will emit a pressure wave strong enough to disturb the growing cavity, and subsequently make it detached from the wall. This is the third mechanism observed in the experiments. In the present work, we point out the inappropriate classification of combining condensation shock and collapse-induced pressure wave mechanisms, since we identify pronounced differences between them: (i) the pressure increase across the condensation front is very weak (a few kPa) while the amplitude of collapse-induced pressure wave can be as high as hundreds of kPa, (ii) the travelling velocity of collapse-induced pressure wave within the cavity is an order of magnitude faster than the condensation shock, and (iii) the collapse-induced pressure wave does not result in an obvious discontinuity in void fraction when it propagates through the cavity, in contrary of the case of the condensation shock.

Geometry scale effect on the Venturi cavitating flow

The re-entrant jet induced cloud cavitation is highly affected by Venturi geometrical parameters, especially the throat height and the contraction ratio from the entrance to the throat.

A smaller throat height would cause a larger drag force acting on the re-entrant jet, and a smaller contraction ratio would result in a weaker driving force, i.e. adverse pressure gradient.

Both of them would suppress the full development of the re-entrant jet and hence lead the sheet cavity to remain relatively stable. Just like the aforementioned gaps altering the propagation path of the re-entrant jet, the transition from sheet-to-cloud cavitation could also be postponed by adjusting Venturi geometrical parameters. This may guide the design of hydraulic machines that are expected to operate without unsteady shedding of large clouds. Note that cloud Overall summary and perspectives 135 cavitation can still take place even in the absence of re-entrant jet as it can be directly initiated by the other two mechanisms.

Perspectives

In order to optimize the application of synchrotron X-ray imaging to cavitating flows and improve the physical understanding of sheet/cloud cavitation, a few further plans can be proposed:

• The acquisition of velocity fields from X-ray images relies on the extraction of tracer particles seeded into the cavitating flow. Although the wavelet-transform-based image processing method developed in this thesis improves the particle detection rate compared to the previous algorithms, the loss of particle information due to overlapped particles in clusters and varying background noises still needs to be considered. In recent years, machine learning has become a prevailing technique for pattern recognition. The work of Li et al. ( 2020) also shows the advantages of a machine-learning-based method for analyzing particle shadow images, especially for recognizing incomplete or highly superimposed particles. The burgeoning machine learning technique provides us a promising tool for more accurate extraction of particles from X-ray images and thereby improving velocity field measurements inside cavitating regions.

• The synchrotron X-ray imaging technique will be employed to investigate structures and dynamics of cloud cavitation once a new test section eliminating the side gaps is manufactured. The new data will provide an opportunity to compare the two cases of shedding/no-shedding and provide insights about the transition from stable sheet cavitation to unsteady cloud cavitation with large-scale shedding.

• A new test section equipped with a certain number of aligned pressure sensors along the Venturi divergent wall will be designed, in which we expect to observe cloud shedding induced by re-entrant jet, condensation shock and collapse-induced pressure wave mechanisms in order with the reduction of cavitation number. The pressure rise associated with these three mechanisms will be recorded and analyzed.

• The APS X-ray source emits X-ray pulses successively at a frequency of 271 696 Hz. However, in the present X-ray imaging experiments, the camera recording rate was set at 12 070 Hz which was achieved through the fast chopper shutter blocking most of the X-ray pulses. The purpose of this is to increase the field of view since the increase of