William Benguigui

Keywords: Two-phase flow, flow regime, flow induced vibration, CFD, tube bundle, cross-flow

A mio papà, mia mamma e mio fratello Alla memoria delle mie nonne

Vorrei esprimere la mia più profonda gratitudine alla mia famiglia per il suo incrollabile sostegno, amore e incoraggiamento durante tutto questo percorso. La vostra comprensione, la vostra pazienza e i vostri sacrifici sono stati alla base del mio successo. Ai miei genitori, la vostra guida e la vostra fiducia in me sono state la mia forza. Ovunque questo percorso mi porterà, avere il vostro successo e carattere sarà sempre la mia massima aspirazione. A mio fratello più piccolo anagraficamente ma grande uomo che mi ha mostrato supporto e fiducia nei momenti più difficili di questo percorso. Alle mie nonne, venute a mancare durante la strada che ha portato al mio dottorato. Sarebbero state felici e orgogliose di vedermi raggiungere questo obiettivo.

Siete stati tutti i miei pilastri di forza e non avrei potuto raggiungere questo obiettivo senza di voi. Grazie per essere la mia costante fonte di ispirazione e per esserci sempre per me.

Ringrazio i miei tanti amici, tra i quali Domenico, Mara, Piervito e Vitalba per avermi sostenuto e sempre ben consigliato. Di amici ce ne sono tanti, ma voi siete la mia seconda famiglia.

Vorrei spendere qualche parola per Alice. Sono fortunato ad averti conosciuta, hai reso la fine di questo percorso più leggero. Voglio anche ringraziarti per la tua pazienza, rara gentilezza, il tuo affetto e per la tua capacità di ascolto.

I would like to express my deepest gratitude to my PhD thesis director, Prof. Catherine Colin. Your expertise, mentorship, and constructive feedback have not only enriched my research but also inspired me to strive for excellence. I am truly fortunate to have had the opportunity to work under your supervision. Thank you for believing in

List of Figures

The provided illustration depicts a cross-sectional view of a SG, highlighting the process of steam production through a series of blue (0% of void fraction) to red (100% of void fraction) arrows. The red circles indicate the locations where vibrations resulting from cross-flows are present [START_REF] Benguigui | Numerical simulation of two-phase flow induced vibration[END_REF] [START_REF] Pettigrew | Vibration of tube bundles in two-phase cross-flow: Part 1-hydrodynamic mass and damping[END_REF][START_REF] Pettigrew | Vibration of tube bundles in two-phase cross-flow: Part 2-fluid-elastic instability[END_REF][START_REF] Pettigrew | Vibration of a tube bundle in two-phase freon cross-flow[END_REF][START_REF] Pettigrew | Vibration behaviour of rotated triangular tube bundles in two-phase cross-flow[END_REF]. Mass flux needed to reach the instability as a function of the void fraction. . . . . . . . . . . . . . . . . . . . . . 1.7 Damping data from Pettigrew et al. [START_REF] Pettigrew | Vibration of a tube bundle in two-phase freon cross-flow[END_REF][START_REF] Pettigrew | Vibration behaviour of rotated triangular tube bundles in two-phase cross-flow[END_REF][START_REF] Pettigrew | Vibration of tube bundles in two-phase cross-flow: Part 1-hydrodynamic mass and damping[END_REF] in air-water and Freon, from Axisa et al. [START_REF] Axisa | Vibration of tube bundles subjected to air-water and steam-water cross flow: Preliminary results on flu1delastic instability[END_REF][START_REF] Axisa | Vibration of tube bundles subjected to steamwater cross flow: a comparative study of square and triangular pitch arrays[END_REF] in steam-water. Image from Pettigrew et al. [START_REF] Pettigrew | Vibration analysis of shell-and-tube heat exchangers: an overviewpart 1: flow, damping, fluidelastic instability[END_REF] 12 Three images to characterize the two-phase flow in a tube bundle by Ulbrich et al. [START_REF] Ulbrich | Vertical, upward gas-liquid two-phase flow across a tube bundle[END_REF]:

bubbly flow on the left, intermittent flow in the middle, and dispersed flow on the right. . 1.13 Flow regime maps comparisons in square pitch tube bundle among Noghrehkar et al. [START_REF] Noghrehkar | Investigation of two-phase flow regimes in tube bundles under cross-flow conditions[END_REF] (Figure 1.13a), Xu et al. [START_REF] Xu | Hydrodynamics of two-phase flow in vertical up and down-flow across a horizontal tube bundle[END_REF] (Figure 1.13b), Murakawa et al. [START_REF] Murakawa | Evaluation of void fraction around a tube in two-phase flow across horizontal tube bundle[END_REF] (Figure 1.13c) with Ulbrich et al. [START_REF] Ulbrich | Vertical, upward gas-liquid two-phase flow across a tube bundle[END_REF]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.14 Flow regime maps identified by Karnizawa et al. [START_REF] Kanizawa | Two-phase flow patterns across triangular tube bundles for airwater upward flow[END_REF]. The black lines on the flow regime map in Figure 1.14b refers to the transition identified in Figure 1.14b, and the black lines on the flow regime map in Figure 1.14c refers to the transition identified in Figure 1.14b. 1. [START_REF] Nakamura | Dynamics of an in-line tube array subjected to steam-water cross-flow. part 1: two-phase damping and added mass[END_REF] Comparison of flow regime maps in square pitch tube bundle among: Ulbrich et al. [START_REF] Ulbrich | Vertical, upward gas-liquid two-phase flow across a tube bundle[END_REF],

Noghrehkar et al. [START_REF] Noghrehkar | Investigation of two-phase flow regimes in tube bundles under cross-flow conditions[END_REF], Murakawa et al. [START_REF] Murakawa | Evaluation of void fraction around a tube in two-phase flow across horizontal tube bundle[END_REF], Xu et al. [START_REF] Xu | Hydrodynamics of two-phase flow in vertical up and down-flow across a horizontal tube bundle[END_REF], Piteau et al. [START_REF] Piteau | Experimental investigation of in-flow fluidelastic instability for rotated triangular tube bundles subjected to single-phase and twophase transverse flows[END_REF]. Flow regimes: B=Bubbly, C=Churn, S=Slug, I=Intermittent, A=Annular, T=T ransition . 1. [START_REF] Xu | Hydrodynamics of two-phase flow in vertical up and down-flow across a horizontal tube bundle[END_REF] Comparison of flow regime maps in staggered pitch tube bundle among: Kanizawa et al. [START_REF] Kanizawa | Two-phase flow patterns across triangular tube bundles for airwater upward flow[END_REF], Murakawa et al. [START_REF] Murakawa | Evaluation of void fraction around a tube in two-phase flow across horizontal tube bundle[END_REF], Noghrehkar et al. [START_REF] Noghrehkar | Investigation of two-phase flow regimes in tube bundles under cross-flow conditions[END_REF], Aprin et al. [START_REF] Aprin | Experimental analysis of local void fractions measurements for boiling hydrocarbons in complex geometry[END_REF], Grant et al. [START_REF] Grant | Two-phase flow on the shell-side of a segmentally baffled shellandtube heat exchanger[END_REF].

3.16

Fast camera results on bubble size measurement at bubble injection up to the tube bundle inlet. In the y-axis 0 m is the level at which the bubble are generated and 1. [START_REF] Janzen | Fluidelastic instability and workrate measurements of steam-generator u-tubes in air-water cross flow[END_REF] for water mass-flow rate of ṁwater = 2500 g s -1 at tube bundle outlet. . . . . . . . . . . . . 3.31 Fast camera results on bubble size measurement at tube bundle inlet and tube bundle outlet. 4.14 Flow regimes maps from Ohnuki et al. [START_REF] Ohnuki | An experimental study on developing air-water two-phase flow along a large vertical pipe: effect of air injection method[END_REF][START_REF] Ohnuki | Experimental study on transition of flow pattern and phase distribution in upward air-water two-phase flow along a large vertical pipe[END_REF]. Comparison with Mishima and Ishii [START_REF] Mishima | Flow regime transition criteria for upward two-phase flow in vertical tubes[END_REF] flow regime map. In the figure are reported the test conditions of the present experimental campaign. In the figures, the authors use the notation J g and J l , which corresponds to the notation of U sg and U sl in the present study. . . . . . . . . . . . . . . . . . . . . . . . . . 4.15 Flow regime map by Ali [START_REF] Ali | Two phase flow in large diameter vertical riser[END_REF]. Tee and Ann refers to the two different air injection systems.

"B" referrers to bubbly flow, "DB" to dispersed bubbly, "AB" to agitated bubbly and "C" to churn flow. In the figure, the authors use the notation J g and J l , which corresponds to the notation of U sg and U sl in the present study. . . . . . . . . . . . . . . . . . . . . . . . . . 4. [START_REF] Xu | Hydrodynamics of two-phase flow in vertical up and down-flow across a horizontal tube bundle[END_REF] The flow patterns in pipe (D h =0.195 m) taken with wire mesh tomography [START_REF] Prasser | Topflow tests on the structure of the gas-liquid interface in a large vertical pipe[END_REF] for U sl =1.0 m s -1 . In the figure, the authors use the notation J air , which corresponds to the notation of U sg in the present study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.17 Distribution parameter C 0 for large rectangular channel according to Kataoka et al. [START_REF] Kataoka | Drift flux model for large diameter pipe and new correlation for pool void fraction[END_REF]. [START_REF] Janzen | Fluidelastic instability and workrate measurements of steam-generator u-tubes in air-water cross flow[END_REF].18 Distribution parameter C 0 for large rectangular channel according to Hibiki et al. [START_REF] Hibiki | One-dimensional drift-flux model for two-phase flow in a large diameter pipe[END_REF]. In the figure J + g and J + correspond to the notation U + sg and U + s in the present study. . . . . . 4. [START_REF] Murakawa | Evaluation of void fraction around a tube in two-phase flow across horizontal tube bundle[END_REF] Comparison between drift-flux correlations with experimental results obtained with optical probe (OP) and WMS. The black-dashed lines represent the 30% bias for the experimental data. The water superficial velocity for the tests is constant (0.039 m s -1 ). . . . . . . . . . 4.20 Comparison between drift-flux correlations for gas velocity with experimental results obtained with optical probe (OP). The black-dashed lines represent the 30% bias for the experimental data. The water superficial velocity for the tests is constant (0.039 m s -1 ). . 7 Mesh sensitivity for Profile P1. M 1 is the reference mesh, while M 0.6 and M 1.3 are the coarser and finer mesh respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Mesh sensitivity for Profile P2. M 1 is the reference mesh, while M 0.6 and M 1.3 are the coarser and finer mesh respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Two-dimensional computational domain for tube bondle and free duct. . . . . . . . . . . . 5.10 Three-dimensional computational domain for tube bundle. . . . . . . . . . . . . . . . . . . . 5.11 Three-dimensional computational domain for free duct. "WMS" indicates the profile used to compare the data with the WMS experimental results. . . . . . . . . . . . . . . . . . . . 5.12 Comparison between numerical results for void fraction distribution with images from fast camera. The numerical domain is cropped in order to compare the two images. . . . . . . 5. [START_REF] Li | An fluidelasticinstability model with an extension to full flexible multispan tube arrays[END_REF] Comparison between numerical results with optical probe data: vertical profiles (points 1, 8, 2, 9, 3 and 4 in Figure 2.24). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.14 Comparison between numerical results with optical probe data focusing on the difference between gas and liquid velocity (respectively v g and v l ). v LO l is the liquid velocity from the single-phase numerical simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.15 Boundary conditions sensitivity results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.16 Numerical domain for the long inlet case. The profiles "TB inlet" and "P1", from which the results are extracted, are indicated in red. They are positioned respectively 1 pitch below the tube bundle and 30 tube diameters above the inlet boundary conditions. . . . . 5.17 Flow evolution between "P1" and "TB inlet" from numerical results numerical results (NCFD). In addition, a comparison between numerical and experimental results at tube bundle inlet is reported ("TB inlet"). "LO" indicates results from single phase (liquid only) simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.18 Flow evolution between "P1" and "TB inlet" from numerical results numerical results (NCFD). In addition, a comparison between numerical and experimental results at tube bundle inlet is reported ("TB inlet"). . image for test T5, the void fraction ranges from 0 to 0.5 for the dispersed phase, while an isosurface for void fractions greater than 0.5 is applied to identify the large gas structures. 5.21 Comparison of experimental data with numerical results for test T4. "OP" stands for experimental data obtained with optical probe, "Camera" indicates experimental results obtained with fast camera, "Neptune CFD" the numerical results. . . . . . . . . . . . . . . 5.22 Comparison of experimental data with numerical results for test T5. "OP" stands for experimental data obtained with optical probe, "WMS" indicates experimental results obtained with wire mesh sensor, "Neptune CFD" the numerical results. . . . . . . . . . . 5.23 Instantaneous void fraction distribution for tests T5. Comparison between WMS result (left figure) and numerical simulation result (right image). . . . . . . . . . . . . . . . . . . . 5.24 Void fraction time evolution recorded by the monitoring point in the core of the numerical domain, located near the outlet of the free duct ("WMS" in Figure 5.11). . . . . . . . . . . 5. [START_REF] Piteau | Experimental identification of the fluid-elastic coupling forces on a flexible tube within a rigid square bundle subjected to single-phase cross-flow[END_REF] Comparison between numerical results for Void Fraction distribution with images from fast camera from visualization window W5. The numerical domain is cropped in order to compare the two images. The numerical void fraction range between 0 and 0.5 for the dispersed phase, the large structures are in white. . . . . . . . . . . . . . . . . . . . . . . . . 5.26 Comparison between numerical results for Void Fraction distribution with images from fast camera from visualization window W3. The numerical void fraction range between 0 and 0.5 for the dispersed phase, the large structures are in brown. The numerical domain is in gray with reduced opacity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.27 Comparison between numerical results for Void Fraction distribution with images from fast camera from visualization window W3. The numerical void fraction range between 0 and 0.5 for the dispersed phase, the large structures are in brown. 3.1 Test conditions. In the table are reported: the test identification code, the air volumetric flow rate (Q air ) in L min -1 , the number and the type (described in Section 2.2) of tubes used in the gas injector to achieve the desired flow rate, the water flow rate ( ṁwater ) in g s -1 . J l and J g are the water and air gap superficial velocities according to Equation 2.1. U sl and U sg indicate the water and air superficial velocities in the homogenization section (free vein) according to Equation 2.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Test results in term of average Eötvös number ( Ēo), average bubble equivalent diameters deq , average aspect ratio ( R), average bubble velocity vB and Weber number W e. The uncertainty is reported in [mm] for the deq , and in [m s -1 ] for vB , v l and vB -v l . In brackets is reported, in percentage, the underestimation due to bubble cluster appearance. . . . . . 3.3 Tube bundle inlet conditions in term of average bubble equivalent diameters deq , average bubble size, sauter mean diameter d sauter , void fraction α, and gas velocity v g . The uncertainty is reported: in [mm] for deq , Bubble size, d sauter ; in % for α; in [m s -1 ] for v g .

Introduction

Two-phase flow-induced vibration is a phenomenon that occurs when two-phase gas-liquid flows encounter mechanical systems such as pipes, tubes, or heat exchangers. This dynamic interaction between the fluid flow and the structural components can lead to various types of vibrations, which can have detrimental effects on the integrity and performance of the system. Understanding and mitigating two-phase flowinduced vibration is crucial in ensuring the safe and reliable operation of industrial processes involving multiphase flows.

In the nuclear industry, these phenomena are of great interest as they can affect the integrity of the Steam Generators (SG). The core of nuclear pressurized water reactors (PWR) generates heat, which is then conveyed through a primary liquid water circuit. Subsequently, the power is transferred to a boiling secondary water circuit by means of SG. Prior to entering the turbine, the resulting steam is appropriately dried. These SG can accommodate from 3000 to 16,000 U-tubes, which have a diameter that can vary depending on the type of NPP but is generally approximately 20 mm. Figure 1 provides an illustration of the SG utilized in French PWR. In conventional recirculating SG for PWR, the primary coolant flows through U-tubes, a straight tube sheet in their lower portion and U-bends in the upper zone. The primary coolant is typically introduced into the steam generator at temperatures ranging from 315 to 330 ○ C on the hot leg side and exits at approximately 288 ○ C on the cold leg side. The secondary system water is introduced through a feedwater nozzle, then distributed via a feedwater distribution ring into the downcomer, where it combines with recirculating water draining from the moisture separators. This downcomer water descends to the bottom of the steam generator and subsequently ascends through the tube bundle, leading to the generation of steam. Roughly 25% in mass of the overall secondary water content is converted into steam as it ascends through the tube bundle, while the remaining portion is recirculated. The void fraction in the shell side of the SG varies from 0% at the bottom up to 90% upstream of the swirl vane separators. The water on the shell side is under a pressure of approximately 60 to 70 bar and is at the saturation temperature.

Within the context of SG in nuclear power plants, it is particularly the cross-flow induced vibration mechanisms that demand in-depth investigation, especially in the U-tube region. Two-phase flow-induced vibrations in SG can lead to structural fatigue, component wear, increased maintenance costs, reduced efficiency, and even failures. Therefore, investigating the underlying mechanisms, predicting and mitigating the vibration effects are of paramount importance for ensuring the safety and reliability of these systems [START_REF] Pettigrew | Vibrational analysis of heat exchanger and steam generator designs[END_REF].

To address the complexities of cross-flow induced vibrations, it is essential to bridge the gap between theoretical models and real-world behavior. While there have been valuable insights gained from existing experimental campaigns, the intricacies of two-phase cross-flow and tube vibration response require further exploration. Previous studies have provided foundational knowledge, but the dynamic and evolving nature of nuclear technology necessitates continuous research and refinement of our understanding. Therefore, there is a continous need for new experimental campaigns explicitly designed to dig into the complexities of two-phase cross-flow-induced vibrations. Such campaigns should focus on collecting data that can be used to validate computational fluid dynamics (CFD) models, which are indispensable tools for predicting complex flow patterns and vibration responses in these systems. By synergizing experimental data with advanced CFD simulations, researchers can develop more accurate predictive models and, in turn, enhance the safety, efficiency, and longevity of steam generators and nuclear power plants as a whole.

Figure 1: The provided illustration depicts a cross-sectional view of a SG, highlighting the process of steam production through a series of blue (0% of void fraction) to red (100% of void fraction) arrows. The red circles indicate the locations where vibrations resulting from cross-flows are present [START_REF] Benguigui | Numerical simulation of two-phase flow induced vibration[END_REF].

An overview of steam generator tube failures

Due to the substantial steam-water flow rate, the large array of tubes constituting the tube bundle, and the particular tube design characterized by long, slender tubes with a U-shaped configuration, flowinduced vibration (FIV) phenomena may manifest in nuclear SGs. Among various SG configurations (such as vertical U-tube, vertical once-through, horizontal, etc.), pressurized water reactor (PWR) Utube steam generators have historically faced more vibration-related issues. MacDonald et al. [START_REF] Macdonald | Steam generator tube failures[END_REF] provides a comprehensive overview of reported SG tube degradation and failures due to FIV until 1996, including all reported Steam Generator Tube Rupture (SGTR) events up to that year.

Turbulence-induced vibrations, generated by the high-speed vapor-water mixture, exhibit small amplitudes. However, even these minor vibrations can lead to long-term fretting wear at the tube-to-support plate and tube-to-anti-vibration bar (AVB) contact points. Excessive fretting can ultimately result in tube failure. Furthermore, fretting wear on the tube-AVB contact point can alter the AVB support condition, diminishing its effectiveness against FIV. By December 1993, 116 nuclear power plants (NPPs) with recirculating SGs had reported tube failures due to AVB wear/fretting, leading to the plugging of 4,663 tubes.

In contrast, high-cycle fatigue, characterized by high vibration amplitudes, leads to more severe and rapid tube ruptures compared to fretting. This is attributed to the fluid-elastic instability (FEI) mechanism, occurring when the energy added per vibration cycle by the fluid-elastic displacement overcome the energy dissipated through damping effects, as described by Connors [START_REF] Connors | Fluidelastic vibration of arrays excited by cross flow. Flow-induced vibration of heat-exchangers[END_REF]. In 1987, the North-Anna Nuclear Power Plant in the United States experienced an SGTR accident attributed to FEI. This accident highlighted the vulnerability of U-bend regions in steam generators to FIV-induced failures. A similar incident occurred at the Japanese Mihama Nuclear Power Plant in 1991, where an SGTR was linked to FEI. This event raised awareness of FIV as a potential threat to the integrity of steam generator tubes.

The French Cruas Nuclear Power Plant faced a series of reactor shutdowns between 2004 and 2008 due to primary-to-secondary reactor coolant leaks caused by high-cycle tube fatigue, due to modified thermal-hydraulic conditions resulting from clogging of tube support plates. This accident underscored the importance of maintaining proper thermal-hydraulic performance to mitigate FIV-related risks (Bodineau et al. [START_REF] Bodineau | Tube support plate clogging up of french pwr steam generators[END_REF]). In 2012, the San Onofre Nuclear Generating Station (SONGS) in the United States experienced extensive tube wear attributed to in-plane FEI (Hawkins [48]). This event was unexpected because the steam generators had been recently replaced. It emphasized the need for thorough analysis and understanding of FIV in different operational conditions.

In light of these nuclear steam generator accidents and issues linked to FIV, ongoing research into two-phase FIV remains of paramount importance. It not only helps prevent SGTR accidents but also contributes to the continuous improvement of SG design and performance assessment. As the nuclear industry evolves, ongoing research in this area remains essential for ensuring the safe and efficient generation of nuclear power.

Manuscript structure

The present manuscript is organized in five chapters as follows:

• The first Chapter provides an overview of fluid-induced vibration phenomena. It focuses on the current state of experimental investigations into two-phase flow-induced vibrations in cross-flow tube bundles and the experimental examination of two-phase flow regimes within tube bundles. The existing review highlights a critical gap in our understanding of vibration excitation mechanisms in tube bundles subjected to two-phase cross flow. Building upon the last sentence, details of the motivations behind this work are discussed.

• The TREFLE facility is described in Chapter 2. The design process of specific components is presented in detail. Furthermore, the different measurement systems and their characteristics are reported.

• Chapter 3 focuses on investigating the behavior of the bubbly flow regime and the tube vibration response. The study of the two-phase flow extends from the inlet conditions of the facility, to within the tube bundle. Once the two-phase flow is characterized, the vibration response of the central tube is explored.

• In Chapter 4 an experimental study of the intermittent flow regime and the tube vibration response is presented. The two-phase flow is analyzed in detail and a new flow regime map in tube bundle is proposed. Additionally, the vibration response of the tube and its link with the two-phase flow is investigated.

• In the last Chapter, numerical simulations conducted using the Neptune CFD code are presented, comparing the results with the findings from the experimental campaigns.

Chapter 1

Review of the literature

Flow-induced vibration is a complex phenomenon that arises from the interaction between fluid flow and structures, leading to dynamic oscillations or vibrations. Over the years, extensive research has been conducted to understand the causes, effects, and mitigation strategies associated with flow-induced vibrations. The primary objective of this chapter is to provide an introduction to the topic of flowinduced vibrations and an overview of the different studies, revealing a gap in our understanding of vibration excitation mechanisms in tube bundles subjected to two-phase cross flow. The chapter begins with an overview of fluid-induced vibration phenomena in Section 1.1, followed by a focus on two-phase flow-induced vibration in Section 1.2. Section 1.3 reviews the characterization of two-phase flow regimes in tube bundles, concluding with an outline of the thesis.

Flow-Induced Vibration: An overview

To fully investigate these phenomena, it is essential to review the various mechanisms contributing to flow-induced vibrations. According to Pettigrew et al. [START_REF] Pettigrew | Two-phase flow-induced vibration: An overview[END_REF], the vibration mechanisms can be subdivided in two classes: vibration mechanisms in axial flow and vibration mechanisms in cross-flow. For the purpose of the present work, only cross-flow-induced vibration will be analyzed. Vibration mechanisms caused by cross-flow is a serious concern of shell and tube heat exchangers [START_REF] Kuppan | Heat exchanger design handbook[END_REF]. According to Chen et al. [START_REF] Chen | Crossflow-induced vibrations of heat exchanger tube banks[END_REF] the excitation mechanisms can be classified as:

• Vortex shedding

• Fluid elastic instability

• Turbulent excitation

• Acoustic excitation

Vortex shedding

Vortex-induced vibrations (VIV) occur when alternating vortices shed from bluff bodies, such as cylinders or tubes, induce structural oscillations. As the fluid flows towards a cylinder it generates a systematic pattern of vortices in the wake that interacts with the cylinder, the vortex fluctuation from side to side of the cylinder engenders periodic forces acting on the structure [START_REF] Blevins | Flow-induced vibration in nuclear reactors : A review[END_REF] (as depicted in Figure 1.1). As the vortex shedding process synchronizes with the tube oscillation (lock-in condition), i.e. the shedding frequency matches the natural frequency of the structure, the oscillations amplitude starts to grow [START_REF] Chen | Crossflow-induced vibrations of heat exchanger tube banks[END_REF].

The vortex shedding frequency (f v ) is proportional to the fluid stream velocity (U ) and the cylinder diameter (D) [START_REF] Blevins | Flow-induced vibration in nuclear reactors : A review[END_REF]. The Strouhal number is the key parameter for design criterion to identify the vortex shedding excitation mechanism, and it is defined as St = f v D U [START_REF] Kuppan | Heat exchanger design handbook[END_REF]. Hence, if the St is known for a single cylinder, the flow velocity at which the resonance take place can be predicted [START_REF] Blevins | Flow-induced vibration in nuclear reactors : A review[END_REF]. However, as found experimentally by Pettigrew et al. [START_REF] Pettigrew | Two-phase flow-induced vibration: An overview[END_REF], the tube array geometry and the two-phase flow characteristics bring into play some complications in determining the actual vortex shedding mechanism.

Turbulent excitation

Turbulence-induced vibrations arise from the unsteady flow patterns generated by turbulent fluid motion (as depicted in Figure 1.2). The turbulence of the flow generates pressure fluctuations that promote the oscillations of the structure. The variable fluctuations in time and space due to turbulence, make impossible to define the turbulent excitation mechanism in a deterministic way [START_REF] Blevins | Flow-induced vibration in nuclear reactors : A review[END_REF]. It is possible to predict the phenomena by means of semi-empirical formula found through experimental campaigns, for example by Owen et al. [START_REF] Owen | Buffeting excitation of boiler tube vibration[END_REF]. According to Chen et al. [START_REF] Chen | Crossflow-induced vibrations of heat exchanger tube banks[END_REF] and Wever et al. [START_REF] Weaver | Cross-flow induced vibrations in a tube bank: turbulent buffeting and fluid elastic instability[END_REF], Owen's formula is the most reliable approach for turbulent excitation in single-phase flow. However, recent studies on two-phase turbulent flow-induced vibrations show how the flow regimes play an important role in this excitation mechanism [START_REF] Briceno | Validation of turbulence induced vibration design guidelines in a normal triangular tube bundle during two-phase crossflow[END_REF].

Acoustic excitation

Acoustic-induced vibrations occur due to the interaction of fluid-induced pressure waves with structures.

The major source of acoustic resonance in industrial application are pumps and rotating machines. Vortex shedding from slender structures is another cause of acoustic noise [START_REF] Kuppan | Heat exchanger design handbook[END_REF]. When the natural frequency of vortex shedding overlaps with the acoustic frequencies, fluid flow and acoustic field emphasize each other [START_REF] Chen | Crossflow-induced vibrations of heat exchanger tube banks[END_REF]. However, since the speed of sound in water is significantly higher than in gas phase, the acoustic natural frequency in the liquid phase are usually beyond the natural frequency of the structure. Therefore, acoustic vibrations are excitation mechanisms that rise a concern only for single phase flow, in particular for gas flow [START_REF] Blevins | Flow-induced vibration in nuclear reactors : A review[END_REF].

Fluid Elastic Instability

Fluid Elastic Instability (FEI) is a phenomenon in which hydrodynamic forces and structural dynamics interact to cause self-excited vibrations. The FEI with its large amplitudes tube oscillations (1.3), is the major concern, among the excitation mechanisms, since it can lead to short-term failure [START_REF] Lever | On the stability of heat exchanger tube bundles, part ii: numerical results and comparison with experiments[END_REF][START_REF] Weaver | A review of cross-flow induced vibrations in heat exchanger tube arrays[END_REF][START_REF] Pettigrew | Two-phase flow-induced vibration: some unanswered questions? 11th International Conference on Flow-Induced Vibration[END_REF]. One of the first authors to recognize and investigate FEI was Connors [START_REF] Connors | Fluidelastic vibration of arrays excited by cross flow. Flow-induced vibration of heat-exchangers[END_REF]. It was demonstrated experimentally by Connors [START_REF] Connors | Fluidelastic vibration of arrays excited by cross flow. Flow-induced vibration of heat-exchangers[END_REF][START_REF] Connors | Fluidelastic vibration of heat exchanger tube arrays[END_REF] that tube vibration amplitudes increase with flow velocity, until the growth becomes exponential when FEI sets in. According to Connors [START_REF] Connors | Fluidelastic vibration of heat exchanger tube arrays[END_REF] the excitation mechanism takes place when the energy extracted from the fluid by the structure overcomes the energy dissipated by the damping of the system. The FEI phenomena is strictly related to the flow conditions and the tube bundle array geometry (pitch, diameter, configuration etc.). For design purposes the stability limit was found by Connors by means of a semi-empirical formula (Equation 1.1) that takes into account the previous mentioned parameters:

U c Df o = K m 0 δ 0 ρD 2 n (1.1)
Where U c is the critical free stream velocity for departure of instability, ρ the density of the fluid, D, δ 0 and f 0 are respectively the diameter of the tube, the damping, the natural frequency. m 0 is the hydrodynamic mass plus the mass of the structure, n is a constant usually equal to 0.5 and K is the Connors constant that depends on the array configuration and flow conditions (single phase or two-phase flow). Experiments on a array of cylinders in air and water single-phase cross-flow have been performed by Paidoussis et al. [START_REF] Paidoussis | Flow-induced vibrations and instabilities in a rotated-square cylinder array in cross-flow[END_REF] in order to progress on understanding flow-induced vibration problems in heat exchangers. The authors found the development of instability for both water and air cross-flow. A formulation to predict the instability was proposed based on the potential flow solution. However, the authors concluded that despite extensive research, understanding and predicting the behavior of cylinder arrays in cross-flow remains a challenge, emphasizing the need for further investigation.

Two-phase flow-induced vibration experimental studies

In the open literature theoretical models for the prediction of instabilities are available [START_REF] Chen | Guidelines for the instability flow velocity of tube arrays in crossflow[END_REF][START_REF] Paidoussis | A review of flow-induced vibrations in reactors and reactor components[END_REF][START_REF] Weaver | A review of cross-flow induced vibrations in heat exchanger tube arrays[END_REF][START_REF] Price | A review of theoretical models for fluidelastic instability of cylinder arrays in cross-flow[END_REF]. However, the scientific community has been pushed to explore the mechanism of two-phase flow-induced vibrations through numerous experimental campaigns due to the risk associated with the phenomenon and the absence of a universally applicable stability criterion for all bundle geometries and flow characteristics. The properties of two-phase flow increase the challenges associated with flow-induced vibration phenomena. As a matter of fact, the two-phase flow introduces additional parameters that influence the vibration mechanisms, including void fraction and flow pattern [START_REF] Pettigrew | Two-phase flow-induced vibration: some unanswered questions? 11th International Conference on Flow-Induced Vibration[END_REF].

Various tube bundle configurations were tested in the upcoming experimental campaigns. To provide clarity regarding these different configurations, a schematic representation of the bundle geometries is shown in Figure 1.4 (the configuration angle is the angle between the flow direction and the geometry). In this context, we also define the drag and lift directions for the tube direction of vibration: The drag direction is parallel to the mean flow direction, while the lift direction is perpendicular to the flow. It is also important to define in-plane and out-of-plane modes of vibration for the U-bend configurations (Figure 1.5). A vibration mode is considered in-plane when its modal deformation primarily occurs within the plane designated by the tube. Conversely, when its modal deformation predominantly occurs perpendicular to the plane of the tube, it is referred to as an out-of-plane mode of vibration. Figure 1.5: Out-of-plane and in-plane modes of vibrations [START_REF] Janzen | Fluidelastic instability and workrate measurements of steam-generator u-tubes in air-water cross flow[END_REF].

With the objective of investigate SG operative conditions, Axisa et al. [START_REF] Axisa | Vibration of tube bundles subjected to air-water and steam-water cross flow: Preliminary results on flu1delastic instability[END_REF][START_REF] Axisa | Vibration of tube bundles subjected to steamwater cross flow: a comparative study of square and triangular pitch arrays[END_REF] and Antunes et al. [START_REF] Antunes | Cross-flow induced vibration of u-bend tubes of steam-generator tubes[END_REF] performed a series of experiments in the EVA facility. In the set-up section were tested different bundles of Inconel 600 tubes measuring 19.05 mm in diameter. Straigth tubes in normal square pitch configuration with P D = 1.44 were used in [START_REF] Axisa | Vibration of tube bundles subjected to air-water and steam-water cross flow: Preliminary results on flu1delastic instability[END_REF], normal square, parallel and normal triangular tube bundle with P D = 1.47 in [START_REF] Axisa | Vibration of tube bundles subjected to steamwater cross flow: a comparative study of square and triangular pitch arrays[END_REF] (6 rows by 10 columns measuring 1190 mm in length), while a U-tube square pitch configuration (80 tubes) with P D = 1.44 was used by Antunes et al. [START_REF] Antunes | Cross-flow induced vibration of u-bend tubes of steam-generator tubes[END_REF]. Experiments were performed both in air-water at ambient pressure and temperature and in steam-water two-phase flow at 25 bar and 210 ○ C. γ ray densitometric probes were used to measure chord averaged void fraction upstream and downstream of the tube bundle. Void fraction varies between 0.5 and 1 for water-steam experiments and from 0.9 to 1 for air-water tests. The flow regime was found to be relatively homogeneous based on measured void fraction. Axisa et al. [START_REF] Axisa | Vibration of tube bundles subjected to air-water and steam-water cross flow: Preliminary results on flu1delastic instability[END_REF] found that the two-phase flow damping shows no substantial difference between air-water and steam-water mixture. Concerning the values of the instability factor (K from Connors formula [START_REF] Connors | Fluidelastic vibration of arrays excited by cross flow. Flow-induced vibration of heat-exchangers[END_REF] in equation 1.1), they lay between 4 and 5 for steam-water, instead a K factor around 7 was obtained for air-water flow. While there were no significant differences in the lift and drag directions, the amplitudes may vary by up to a factor of about 2 among different tubes, and this variation did not show any clear correlation with the tube location [START_REF] Axisa | Vibration of tube bundles subjected to air-water and steam-water cross flow: Preliminary results on flu1delastic instability[END_REF][START_REF] Axisa | Vibration of tube bundles subjected to steamwater cross flow: a comparative study of square and triangular pitch arrays[END_REF]. The tests conducted by Axisa et al. [START_REF] Axisa | Vibration of tube bundles subjected to air-water and steam-water cross flow: Preliminary results on flu1delastic instability[END_REF] did not detect periodic vortex shedding or exceptionally high turbulent excitation when compared to single-phase experiments, suggesting that these two phenomena are of lesser significance in two-phase flow. However, in [START_REF] Axisa | Vibration of tube bundles subjected to steamwater cross flow: a comparative study of square and triangular pitch arrays[END_REF] regarding the normal triangular bundle, it appears to be more susceptible to some form of periodic wake shedding response compared to the other bundles, particularly at high void fractions. Antunes et al. [START_REF] Antunes | Cross-flow induced vibration of u-bend tubes of steam-generator tubes[END_REF] added also three sets of flat anti-vibration bars (AVB). Five inlet velocity profiles were investigated to understand their impact on the tube vibration response: uniform, convex, concave, asymmetrical left, and asymmetrical right. The tube response was found strongly dependent on flow velocity profiles. For certain profiles, response peaks corresponding to the first fully supported out-of-plane bend mode were observed with a K factor of about 3.4. No significant response was observed for in-plane modes. In the turbulent excitation regime, the vibration level was relatively low. The data from Axisa et al. [START_REF] Axisa | Vibration of tube bundles subjected to steam-water cross flow: a comparative study of square and triangular pitch arrays[END_REF] suggested that random forces in air-water and steam-water cross-flows are of the same order of magnitude, indicating that valuable insights can be gained from studying flow-induced vibration in air-water flow.

Experiments were conducted by Nakamura et al. [START_REF] Nakamura | Fluidelastic vibration of u-bend tube bundle in hcfc-123 two-phase flow. 9. international conference on nuclear engineering[END_REF] to simulate PWR steam generator conditions using a small-scale SG facility (1/10 of Japanese SG). A U-tube bundle of 46 × 5 tubes was employed to scale the flow rate to 1/10 of SG, and HCFC-123 was used as a two-phase mixture to replicate the density ratio of water-steam. The study examined the role of anti-vibration bars, i.e. designed to dampen tube vibrations, and found that the critical factor K in Connors formula was significantly larger than that reported for short straight tubes in previous studies. The influence of so-called intermittent two-phase flow was studied by Nakamura et al. [START_REF] Nakamura | Study on the vibrational characteristics of a tube array caused by two-phase flow. part 1: Random vibration[END_REF][START_REF] Nakamura | Study on the vibrational characteristics of a tube array caused by two-phase flow. part 2: Fluidelastic vibration[END_REF]. In the first part of the work the excitation forces acting on a tube in a tube bundle 20 × 5 in a square array (pitch 27 mm) was studied. The tubes had a diameter of 19 mm (air-water tests). Similar tests sections were used for steam-water tests (up to 58 bar and 272 ○ C), with 174 mm and 500 mm long tubes. The results indicate that the forces in steam-water flow resemble those in air-water flow, mainly caused by impacting water slugs. The excitation forces decrease with increasing pressure and temperature. The experiments show how the intermittent slug flow regime increases of about 5 times the magnitude of the excitation forces. However, the intermittent nature of the forces make the instability boundary complex to found analytically. From the experimental point of view, it is shown how the stability boundaries for intermittent flows indicate a different type of instability, resulting from the unsteady nature of the flow, distinct from the instability observed in homogeneous or single-phase flow: the "intermittent instability" [START_REF] Nakamura | Study on the vibrational characteristics of a tube array caused by two-phase flow. part 2: Fluidelastic vibration[END_REF]. The application of Connors' formula [START_REF] Connors | Fluidelastic vibration of arrays excited by cross flow. Flow-induced vibration of heat-exchangers[END_REF], originally derived for stability criterion in single-phase flow, has prompted the authors [START_REF] Nakamura | Study on the vibrational characteristics of a tube array caused by two-phase flow. part 1: Random vibration[END_REF][START_REF] Nakamura | Study on the vibrational characteristics of a tube array caused by two-phase flow. part 2: Fluidelastic vibration[END_REF] to propose a new stability criterion for two-phase flow. The new criterion is adapted for homogeneous bubbly flow and for intermittent slug flow.

Janzen et al. [START_REF] Janzen | Fluidelastic instability and workrate measurements of steam-generator u-tubes in air-water cross flow[END_REF] conducted experiments on a U-shaped tube bundle in air-water cross-flow to study FEI. The tube bundle consisted of 5×6 tubes with a diameter of 12.7 mm, arranged in a parallel triangular array with a P/D ratio of 1.5. They experimentally found the presence of both in-plane and out-of-plane FEI, as shown in Figure 1.5, by measuring tube vibrations using displacement probes. As reported in Figure 1.5, the out-of-plane mode had a natural frequency f n = 10 Hz, while in-plane modes 1 and 2 had natural frequencies of f n = 25 Hz and f n = 50 Hz, respectively. Later the results were confirmed by Mohany et al. [START_REF] Mohany | Experimental and numerical characterization of flow-induced vibration of multi-span u-tubes[END_REF], they used a U-tube bundle of 5 × 5 tubes arranged in a rotated triangular geometry with P D = 1.5 subjected to Freon-134a two-phase flow.

Chu et al. [START_REF] Chu | Flow-induced vibration of nuclear steam generator u-tubes in two-phase flow[END_REF] observed only out-of-plane FEI, not in-plane. Their experimental setup involved a U-shaped tube bundle consisting of 11 × 7 tubes with a diameter of 19.05 mm arranged in a square array, with P D = 1.633. The bundle was subjected to two-phase air-water flow. Extensive tests were conducted across a range of void fractions from 0.7 to 0.95, the K factor was found to be in the range of 6.5-10.5. The damping observed in the U-tube configuration was notably higher compared to those of cantilever straight tubes reported in the literature. However, the hydrodynamic mass parameter aligned with previous experiences involving cantilever tubes. The measured mode shapes were also presented.

In order to enhance the investigation of the phenomenology associated with instability in two-phase cross-flow, alternative experimental campaigns were conducted that deviated from the U-shaped geometry of steam generator tubes, along with their restrictive temperature and pressure conditions. Pettigrew et al. [START_REF] Pettigrew | Vibration of tube bundles in two-phase cross-flow: Part 1-hydrodynamic mass and damping[END_REF][START_REF] Pettigrew | Vibration of tube bundles in two-phase cross-flow: Part 2-fluid-elastic instability[END_REF] presents experimental investigations on two-phase cross-flow induced vibrations in straight tube bundles. The objectives of these experiments were to characterize hydrodynamic mass, damping, and the stability limit for fluid-elastic instability. The experimental setup consisted of a two-phase flow loop with water-air at ambient pressure and temperature. Test-sections with cantilever tubes of 13 mm diameter and different array geometries (triangular or square pitch, pitch-to-diameter ratio of 1.32 and 1.47) were tested. The results revealed differences in instability behavior for various flow regimes. In subsequent studies, Pettigrew et al. [START_REF] Pettigrew | Vibration of a tube bundle in two-phase freon cross-flow[END_REF][START_REF] Pettigrew | Vibration behaviour of rotated triangular tube bundles in two-phase cross-flow[END_REF] used a different two-phase mixture and tested a test section with tubes of 12.7 mm diameter and a pitch-to-diameter ratio of 1.5 under Freon two-phase flow. In Figure 1.6, we present a comparison of the experimental campaigns conducted by Pettigrew et al. [START_REF] Pettigrew | Vibration of a tube bundle in two-phase freon cross-flow[END_REF][START_REF] Pettigrew | Vibration behaviour of rotated triangular tube bundles in two-phase cross-flow[END_REF][START_REF] Pettigrew | Vibration of tube bundles in two-phase cross-flow: Part 1-hydrodynamic mass and damping[END_REF][START_REF] Pettigrew | Vibration of tube bundles in two-phase cross-flow: Part 2-fluid-elastic instability[END_REF] The two-phase flow experimental campaigns provided valuable data regarding the critical velocity and correlation of these parameters with the flow regimes. Upon comparing the experimental campaigns (Figure 1.6), it is evident that the mass flux required to induce instability decreases as the void fraction exceeds approximately 60%. This finding suggests that specific features of the flow regime can influence the stability limit. Additionally, tests conducted with Freon-22 two-phase flow show an early onset of instability, while tests with Freon-134 exhibit a delayed onset compared to air-water tests. This suggests that the properties of the two-phase mixture also influence the onset of instability, likely due to variations in the flow regime depending on the mixture composition. Pettigrew et al. [START_REF] Pettigrew | Vibration of tube bundles in two-phase cross-flow: Part 1-hydrodynamic mass and damping[END_REF][START_REF] Pettigrew | Vibration of tube bundles in two-phase cross-flow: Part 2-fluid-elastic instability[END_REF][START_REF] Pettigrew | Vibration of a tube bundle in two-phase freon cross-flow[END_REF] obtained valuable data on damping and hydrodynamic mass in two-phase flow. In fact, correlations for determining hydrodynamic mass and damping ratio in two-phase flow were developed and compared with the existing experimental data (as reported in Figure 1.7 and 1.8). [START_REF] Pettigrew | Vibration of a tube bundle in two-phase freon cross-flow[END_REF][START_REF] Pettigrew | Vibration behaviour of rotated triangular tube bundles in two-phase cross-flow[END_REF][START_REF] Pettigrew | Vibration of tube bundles in two-phase cross-flow: Part 1-hydrodynamic mass and damping[END_REF] in air-water and Freon, from Axisa et al. [START_REF] Axisa | Vibration of tube bundles subjected to air-water and steam-water cross flow: Preliminary results on flu1delastic instability[END_REF][START_REF] Axisa | Vibration of tube bundles subjected to steamwater cross flow: a comparative study of square and triangular pitch arrays[END_REF] in steam-water. Image from Pettigrew et al. [START_REF] Pettigrew | Vibration analysis of shell-and-tube heat exchangers: an overviewpart 1: flow, damping, fluidelastic instability[END_REF] Pettigrew et al. [START_REF] Pettigrew | Two-phase flow-induced vibration: An overview[END_REF] observed that void fraction has a dominant effect on damping. Damping increases with void fraction, reaching its maximum between 40% and 70%. Beyond 70% void fraction, damping gradually decreases to lower values. Tube bundle configuration does not noticeably affect damping. The properties of the two-phase mixture, especially surface tension, might be expected to play a significant role in two-phase damping. However, tests conducted by Pettigrew et al. [START_REF] Pettigrew | Vibration of a tube bundle in two-phase freon cross-flow[END_REF] in Freon and by Pettigrew et al. [START_REF] Pettigrew | Vibration of tube bundles in two-phase cross-flow: Part 1-hydrodynamic mass and damping[END_REF] in air-water revealed that measured damping values are very similar in all three cases. In Pettigrew et al. [START_REF] Pettigrew | Vibration of tube bundles in two-phase cross-flow: Part 1-hydrodynamic mass and damping[END_REF][START_REF] Pettigrew | Vibration of a tube bundle in two-phase freon cross-flow[END_REF][START_REF] Pettigrew | Vibration behaviour of rotated triangular tube bundles in two-phase cross-flow[END_REF] shown that generally, damping is lower in the lift direction than in the drag direction, which aligns with the observation that fluidelastic instability first occurs in the lift direction. Concerning the hydrodynamic mass in air-water two-phase flow (Figure 1.8a), Pettigrew et al. [START_REF] Pettigrew | Vibration of tube bundles in two-phase cross-flow: Part 1-hydrodynamic mass and damping[END_REF] findings indicate a strong alignment with theoretical expectations for liquid behavior when void fractions are below approximately 80 percent. However, a deviation from this agreement is observed when void fractions exceed 80%, and this discrepancy was attributed to intermittent flow occurrences during these experimental conditions. The study further reveals that the hydrodynamic mass remains relatively constant at mass flux levels below 90% void fraction but tends to decrease as void fraction overcomes 90%. This discrepancy is likely a consequence of the intermittent flow regime observed in these experiments. Pettigrew et al. [START_REF] Pettigrew | Vibration of a tube bundle in two-phase freon cross-flow[END_REF] suggest that the hydrodynamic mass is slightly greater in the lift direction compared to the drag direction, although the difference is not substantial. Additionally, in freon two-phase flow, the measured hydrodynamic mass is slightly higher than what was initially predicted. This variance may be attributed to the presence of some liquid hold-up between the tubes located upstream and downstream in the flow. In Freon two-phase flow (Figure 1.8b), the hydrodynamic mass did not exhibit a significant discrepancy from the theory even when void fractions exceeded 80%. This observation may also be attributed to the onset of the intermittent flow regime differing for various mixtures.

The experiments performed by Feenstra et al. [START_REF] Feenstra | Fluidelastic instability in a tube array subjected to two-phase r11 cross-flow[END_REF][START_REF] Feenstra | Modelling two-phase flow-excited damping and fluidelastic instability in tube arrays[END_REF] focus on the vibration behavior of fully flexible, parallel triangular tubes array subjected to Freon R-11 two-phase flow. The test section used consisted in 10 cantilever tubes 305 mm long, 6.35 mm of diameter with a ratio P/D=1.44. The tube natural frequency in air about 38 Hz. The instability behavior of tests performed in single phase flow agrees with results obtained for water flow from other authors. However, when it comes to two-phase flow, the experiments performed with Freon exhibit a different behavior from water-steam experiments. Feenstra et al. [START_REF] Feenstra | Fluidelastic instability in a tube array subjected to two-phase r11 cross-flow[END_REF][START_REF] Feenstra | Modelling two-phase flow-excited damping and fluidelastic instability in tube arrays[END_REF] found the void fraction to be an important parameter for the stability criteria. Therefore, the authors developed a new correlation to estimate the void fraction inside the tube bundle. The model is based on the classic void fraction (α) -quality (x) relationship that leads to the equation 1.2

α = 1 + S ρ g ρ l 1 x -1 -1 (1.2)
Where ρ g and ρ l are the gas and liquid density. The key point of this methodology is to identify the slip ratio (S) based on the Capillary (Ca) and Richardson (Ri) numbers as follow:

S = 1 + 25.7 (Ri ⋅ Ca) 0.5 (P D) -1 (1.3)
The Capillary and Richardson numbers defined as follow:

Ri = ∆ρ 2 ga G 2 p Ca = µ l U g σ with U g = xG p αρ g (1.4)
with a the gap among the tubes, µ l and σ are liquid dynamic viscosity and surface tension. G p is the pitch mass flux defined as:

G p = V p ρ tp (1.5)
With V p the pitch two-phase flow velocity, and ρ tp the two-phase density. This method has found good agreement among the literature with experimental results in air-water and Freon two-phase flow. The correlation take into account the effect of the bundle geometry.

In summary, applying the void fraction model (Equation 1.2) to analyze fluidelastic instability data has led to significant insights. Feenstra et al. [START_REF] Feenstra | Fluidelastic instability in a tube array subjected to two-phase r11 cross-flow[END_REF][START_REF] Feenstra | Modelling two-phase flow-excited damping and fluidelastic instability in tube arrays[END_REF] utilized this model to calculate parameters for detecting the stability limit in Connors' formula [START_REF] Connors | Fluidelastic vibration of arrays excited by cross flow. Flow-induced vibration of heat-exchangers[END_REF] (Equation 1.1) and compared them with parameters calculated using the homogeneous equilibrium model (HEM). When using Feenstra et al.'s model, the previously observed discontinuity in the K factor as one transitions from the bubbly to the intermittent regime, as documented by other researchers, disappears. This improved model has also narrowed the range of the mass-damping parameter ( mδ ρtpD 2 ) for different modeling fluids. Furthermore, the relationship between the mass-damping parameter ( mδ ρtpD 2 ) and the reduced velocity parameter ( Vp fnD ), which remains constant in the bubbly flow regime, deviates from this linear pattern upon entering the intermittent flow regime when using the HEM model. However, when employing their void fraction model, Feenstra et al. [START_REF] Feenstra | Fluidelastic instability in a tube array subjected to two-phase r11 cross-flow[END_REF][START_REF] Feenstra | Modelling two-phase flow-excited damping and fluidelastic instability in tube arrays[END_REF] observed that this discontinuity disappears, as depicted in Figure 1.9. Feenstra et al. [START_REF] Feenstra | Fluidelastic instability in a tube array subjected to two-phase r11 cross-flow[END_REF][START_REF] Feenstra | Modelling two-phase flow-excited damping and fluidelastic instability in tube arrays[END_REF] model, employed for evaluating the mass-damping parameter and the reduced velocity parameter, has notably enhanced our understanding of the transition in stability behavior as one moves from bubbly to intermittent flow regimes. In Table 1.1 one can appreciate the effect of void fraction on the parameters to detect the stability limit in Connors formula [START_REF] Connors | Fluidelastic vibration of arrays excited by cross flow. Flow-induced vibration of heat-exchangers[END_REF] (Equation 1.1). [START_REF] Hirota | Dynamics of an in-line tube array subjected to steam-water cross-flow. part 3: fluidelastic instability tests and comparison with theory[END_REF] conducted a series of experimental campaigns using a loop capable of producing water-steam mixture up to 60 bar. The test section contained a bundle of 30×5 tubes measuring 161 mm in length, 22.23 mm in diameter, and with a pitch of 32.54 mm. The majority of the tube bundle was rigidly fixed, except for three tubes positioned in rows 2, 16, and 29. Tests were performed at 0.1 (single phase only), 5, 30, 58 bar. The authors obtained significant data on damping and hydrodynamic mass in two-phase flow which were then compared with the available experimental data and theory from Axisa et al. [START_REF] Axisa | Vibration of tube bundles subjected to air-water and steam-water cross flow: Preliminary results on flu1delastic instability[END_REF][START_REF] Axisa | Vibration of tube bundles subjected to steamwater cross flow: a comparative study of square and triangular pitch arrays[END_REF] (Figure 1.10). Significant findings were obtained regarding damping and added mass by Nakamura et al. [START_REF] Nakamura | Dynamics of an in-line tube array subjected to steam-water cross-flow. part 1: two-phase damping and added mass[END_REF]. Specifically, it was highlighted that damping in two-phase flow decreases with increasing pressure, possibly due to changes in the flow regime structure. Comparing the experimental results with the literature, a small variation in added mass was noticed between water-steam (experimental data) and air-water twophase flow (theory from Carlucci et al. [START_REF] Carlucci | Experimental studies of damping and hydrodynamic mass of a cylinder in confined two-phase flow[END_REF] Figure 1.10), which could be attributed to phase transition on the tube surface. Unsteady fluid force measurements through strain gauge on the three flexible tubes were performed by Mureithi et al. [START_REF] Mureithi | Dynamics of an in-line tube array subjected to steam-water cross-flow. part 2: unsteady fluid forces[END_REF]. Substantial difference among the measurements in single-phase flow and two-phase flow was observed. This confirms that unsteady fluid forces are complex functions of void fraction, reduced velocity and other parameters. The study of FEI was another objective of the experimental study from Hirota et al. [START_REF] Hirota | Dynamics of an in-line tube array subjected to steam-water cross-flow. part 3: fluidelastic instability tests and comparison with theory[END_REF]. The FEI phenomena was well defined in homogeneous bubbly two-phase flow not in intermittent flow condition. The studies highlighted also the appearance of intermittent instability. The onset of FEI was observed by Hirota et al. [START_REF] Hirota | Dynamics of an in-line tube array subjected to steam-water cross-flow. part 3: fluidelastic instability tests and comparison with theory[END_REF] anticipated in the lift direction compared to the drag direction.

A study was performed by Khalvatti et al. [START_REF] Khalvatti | Effect of preferential flexibility direction on fluidelastic instability of a rotated triangular tube bundle[END_REF] to examine the relationship between FEI and the angle between the flow direction and the tube flexibility direction. In order to accomplish this, the researchers set up a facility consisting of a test section with a fixed tube bundle and several flexible tubes. The findings from the experiments verified that the critical velocity decreases as the angle approaches the lift direction. Additionally, the tests revealed that an increased number of flexible tubes within a tube bundle resulted in an earlier occurrence of instability. Ricciardi et al. [START_REF] Ricciardi | Fluidelastic instability in a normal triangular tube bundle subjected to air-water cross-flow[END_REF] conducted an experimental study to explore the distinction between instability in the lift and drag directions for a normal triangular tube bundle in air-water flow P D = 1.5, with a tube diameter of 38 mm. They employed a vibrational system capable of inducing vibrations along a specific direction, allowing distinct tube natural frequencies in the lift and drag directions (around 14 Hz and 81 Hz, respectively, in air). The findings indicated that FEI manifested solely in the lift direction and not in the drag direction. The authors additionally emphasized the significance of investigating the flow regime to gain insights into FEI in two-phase flow.

A large range of void fractions and flow velocities were investigated by Delenne et al. [START_REF] Delenne | Experimental determination of motion dependent fluid forces in two-phase water freon cross-flow[END_REF] and Granger et al. [START_REF] Granger | Motiondependent excitation mechanisms in a square in-line tube-bundle subject to water cross-flow: an experimental modal analysis[END_REF] on an in-line tube bundle with P/D= 1.44, subjected to liquid water -vapor freon cross-flow. The studies investigated flow-induced vibrations on a tube array subjected to water cross-flow using a novel approach based on modal identification and indirect estimation of motion-dependent fluid forces. The results revealed distinct excitation mechanisms, such as single-mode flutter and symmetric vortexshedding resonance, and their variations with flow velocity. The experiments provided valuable data on damping variation, fluid-elastic forces, and modal parameters. The studies successfully achieved multipledegree-of-freedom modal identification for a flexible tube array, and the Global Model Approach was applied to derive fluid-elastic force data. The findings highlighted the importance of studying neighboring flexible tubes and the limitations of extrapolating results to other geometries and fluids. The analysis emphasized the importance of modal parameter variations and fluidelastic force data for understanding the physics and developing predictive models for flow-induced vibration. Deri [START_REF] Deri | Operational modal analysis of a triangular-pitch tube bundle subjected to two-phase cross-flow[END_REF] reports experimental results and physical analysis of some experiment series performed in the same experimental facility as Delenne et al. [START_REF] Delenne | Experimental determination of motion dependent fluid forces in two-phase water freon cross-flow[END_REF] and Granger et al. [START_REF] Granger | Motiondependent excitation mechanisms in a square in-line tube-bundle subject to water cross-flow: an experimental modal analysis[END_REF]. The author presented experimental results focusing on FEI and random excitation spectra. The impact of two-phase flow regime on the bundle behavior was highlighted, and a comprehensive database of coupling force coefficients was provided. Measured mode shapes obtained through operational modal analysis were presented, revealing that the most responsive mode during FEI remains the same across different void fractions, characterized by a high frequency.

Two different test sections were investigated by Zhang et al. [START_REF] Pettigrew | Detailed flow and force measurements in a rotated triangular tube bundle subjected to two-phase cross flow[END_REF][START_REF] Zhang | Vibration excitation force measurements in a rotated triangular tube bundle subjected to two-phase cross flow[END_REF][START_REF] Zhang | correlation between vibration excitation forces and the dynamic characteristics of two-phase cross flow in a rotated triangular tube bundle[END_REF] under air-water two-phase flow. The narrow test section consisted of a column of six 38 mm diameter cylinders in a rotated triangular configuration with P/D=1.5. The wider test section is similar but includes three columns of cylinders instead of one. The authors measured the forces acting on tubes of two type of tube bundles in airwater two phase flow conditions. Measurements of void fraction, bubble velocity and bubble diameters were obtained by means of an optic probe. Flow measurements reveal that the flow between cylinders forms a uniform stream resembling two-dimensional elbows, resulting in a non-uniform void fraction distribution. Force measurements indicate the presence of unexpected quasi-periodic forces in both drag and lift directions, with larger forces observed along the drag direction. These forces exhibit a relatively low frequency (3-6 Hz) independent of the flow velocity. In contrast, higher frequencies (up to 16 Hz) in the lift direction correlate with flow velocity. The observed forces correspond to periodic flow patterns such as slug and intermittent flow. Two-phase flow measurements reveal that quasi-periodic drag and lift forces arise from different mechanisms. Drag forces are linked to momentum flux fluctuations caused by void fraction variations in the main flow path, while lift forces are correlated with oscillations in the wake of the cylinders.

Experimental tests were conducted on the previous mentioned wider test section used by Zhang et al. to investigate FEI by Sawadogo et al. [START_REF] Sawadogo | Fluidelastic instability study in a rotated triangular tube array subject to two-phase cross-flow. parti: Fluid force measurements and time delay extraction[END_REF][START_REF] Sawadogo | Fluidelastic instability study in a rotated triangular tube array subject to two-phase cross-flow. parti: Experimental tests and comparison with theoretical results[END_REF]. Measurements were taken for added mass, flowindependent damping ratio, and critical velocity. A new formulation for added mass, considering the reduction of effective local void fraction, showed better agreement with experimental data. Unsteady fluid forces were used in an unsteady model to compute critical velocity, yielding good agreement. The quasi-steady model, validated against the unsteady model, incorporated experimentally determined time delay. The quasi-steady model results aligned well with experimental data, validating its use in two-phase flow. The unsteady fluid force was found to be dependent on reduced flow velocity and unaffected by void fraction within the considered range.

Among the various experiences realized with the TITAN facility, it is important to mention the experimental campaign performed by Piteau et al. [START_REF] Piteau | Experimental investigation of in-flow fluidelastic instability for rotated triangular tube bundles subjected to single-phase and twophase transverse flows[END_REF]. The authors present experimental results on a rotated triangular straight tube bundle with P/D=1.44. The test section contains fixed tube that can be substituted with several flexible tubes. The findings suggest that instability along the drag direction is a result of vibration coupling between adjacent tubes, highlighting the importance of cross-coupling force terms. Coupling forces along the drag direction were found to be smaller than those causing instabilities in the lift direction. The tests with 2 to 7 in-line flexible tubes that exhibited instability were only observed during air flow, not water flow or two-phase flow. The exact cause of this behavior remains unclear, but it can be attributed to the fact that denser fluids, especially in two-phase flows, tend to produce more damping and generate more intense random excitation. A different tube bundle configuration (P/D = 1.5) subjected to air-water two-phase flow was investigated in the AMOVI facilty by Delaune et al. [START_REF] Delaune | Experimental investigation of in-flow fluidelastic instability for square tube bundles subjected to single-phase and two-phase transverse flows[END_REF]. Experiments at large values of void fraction (80%-98%) were conducted to investigate fluidelastic instability in square bundles along the drag direction. The results showed no instability in both single and two-phase flows for all tested flexible configurations. The authors conclude that the flow patterns in square bundles are minimally affected by tube motions, leading to insufficient destabilizing coupling forces for inducing fluidelastic instability along the drag direction.

As it can be seen from the experimental studies analyzed that focuses on tube vibration investigation, we are far from understanding all the phenomena that contribute to instability. The complexity of the two-phase flow, and regime transitions play a key role in the establishment of instability. This makes it complex to adapt semi-empirical formulas, developed for single-phase flows, to two-phase cross-flow. Essential parameters such as hydrodynamic mass and two-phase damping still need empirical formulas to be determined, and these in turn need experimental validation. This has pushed the scientific community to investigate the behavior of two-phase flow inside the tube bundle configuration. The next section reviews the most relevant experimental campaigns in this direction.

Two-phase cross-flow in tube bundles: Flow regimes characterization

Shell and tube heat exchangers such as SGs of nuclear power plants experience two-phase flow on the shell side. The two-phase flow regimes in the cross-flow play a crucial role in heat transfer and, as mentioned in the previous section, in flow-induced vibrations. Therefore, accurately characterizing the flow regime is essential for designing efficient and robust heat exchangers. Before entering into the literature review of flow regime maps in tube bundles, it is important to define the various flow patterns in a tube bundle. For this purpose, we refer to the studies conducted by Mao et al. [START_REF] Mao | Flow regime transition criteria for upward two-phase cross-flow in horizontal tube bundles[END_REF], as well as Kanizawa et al. [START_REF] Kanizawa | Two-phase flow patterns across triangular tube bundles for airwater upward flow[END_REF], which provide descriptions of flow regimes in a general tube bundle configuration.

• Bubbly flow: It consists of the presence of bubbles smaller than the gap between tubes, accompanied by a continuous liquid phase (Figure 1.11a).

• Dispersed bubbles: In the dispersed bubbly flow regime, bubbles are finely scattered throughout the mixture. High liquid turbulence can lead to bubble break-up. The bubble size is relatively smaller compared to the bubbly flow regime.

• Churn: characterized by turbulent phases with intermittent upward and downward movement of small liquid portions. This flow regime is primarily influenced by gravitational effects. As the bubbles grow in size due to coalescence, the upper liquid portions lack sufficient kinetic energy to sustain upward movement and experience temporary downward movements. Subsequently, the fluids are pushed upward again by the flow dynamics (Figure 1.11b).

• Intermittent flow: Intermittent flow is characterized by alternating liquid slugs and large gas regions, both exhibiting high velocities (Figure 1.11c).

• Annular flow: Annular flow is identified by the formation of a thin liquid film along the walls of the tubes and shell, enclosing a high-velocity central gas core that carries entrained droplets (Figure 1.11d). In the open literature, numerous experimental campaigns have been conducted to investigate flow regimes inside tube bundles. Grant et al. [START_REF] Grant | Two-phase flow on the shell-side of a segmentally baffled shellandtube heat exchanger[END_REF] were precursors of these studies on two-phase flow regimes in tube bundles. The facility contained a normal triangular tube bundle 4 × 10 with P D = 1.25 and D =19 mm. Their experiments were conducted in vertical air-water cross flow, and the flow regimes were established by visual observations. One of the notable studies on air-water two-phase flow patterns, specifically focusing on square pitch tube bundle cross-flow, was carried out by Ulbrich et al. [START_REF] Ulbrich | Vertical, upward gas-liquid two-phase flow across a tube bundle[END_REF]. The facility contained 5×10 tubes with P D = 1.5 and D =20 mm. They employed measurements of two-phase mixture pressure drop to assess fluctuations in the void fraction over time. The variation of pressure drop over time exhibits differences for the different flow patterns. By comparing pressure drop measurements with visual observations using photographic techniques and video cameras (Figure 1.12), the researchers were able to construct a flow regime map, providing a detailed description of the different flow patterns and their transitional regimes. Figure 1.12: Three images to characterize the two-phase flow in a tube bundle by Ulbrich et al. [START_REF] Ulbrich | Vertical, upward gas-liquid two-phase flow across a tube bundle[END_REF]: bubbly flow on the left, intermittent flow in the middle, and dispersed flow on the right.

When examining air-water two-phase flow within a square pitch horizontal tube bundle, it is important to highlight the research conducted by Xu et al. [START_REF] Xu | Hydrodynamics of two-phase flow in vertical up and down-flow across a horizontal tube bundle[END_REF]. The test section was a 3 × 20 tube bundle with P D = 1.28 and D =9.79 mm. This study investigated both upward and downward two-phase flows. The experimental work aimed to develop correlations for the bundle's average void fraction and two-phase friction multiplier. Pressure drop measurements were utilized to determine the average void fraction, while visual observations were used to construct the flow regime map (Figure 1.13a).

Noghrehkar et al. [START_REF] Noghrehkar | Investigation of two-phase flow regimes in tube bundles under cross-flow conditions[END_REF] conducted experiments to characterize the two-phase flow inside both in-line and staggered 5 × 24 tube bundles with P D = 1.47 and D =12.7 mm (the proposed flow regime map is reported in Figure 1.13b). They utilized local void fraction measurements and the probability density function (PDF) method to identify the two-phase flow regimes. The PDF statistical method relies on the shape of the PDF, which exhibits peaks for small void fractions in the presence of bubbly flow regimes. When the intermittent flow regime occurs, the PDF spreads across various void fraction values and, with increasing gas flow rate, displays two peaks: one at small void fraction values and another at high values characteristc of slug flow. For annular flow regimes, the PDF shows a peak only at high void fraction values. The results obtained for the in-line tube bundle were compared to those reported in the open literature, particularly the findings of Ulbrich et al. [START_REF] Ulbrich | Vertical, upward gas-liquid two-phase flow across a tube bundle[END_REF]. However, discrepancies were observed due to different measurement techniques and bundle geometries, particularly in relation to the intermittent flow regime at liquid superficial velocities greater than 0.4 m s -1 . Furthermore, the establishment of the annular flow regime was anticipated compared to Ulbrich et al. [START_REF] Ulbrich | Vertical, upward gas-liquid two-phase flow across a tube bundle[END_REF].

More recent studies were conducted in 2015 by Kanizawa et al. [START_REF] Kanizawa | Two-phase flow patterns across triangular tube bundles for airwater upward flow[END_REF] and in 2016 by Murakawa et al. [START_REF] Murakawa | Evaluation of void fraction around a tube in two-phase flow across horizontal tube bundle[END_REF]. Murakawa et al. [START_REF] Murakawa | Evaluation of void fraction around a tube in two-phase flow across horizontal tube bundle[END_REF] performed experiments in both in-line and straggered 4 × 8 tube bundles with P D = 1.5 and D =15 mm. They utilized X-ray radiography and visual observations to obtain twodimensional void fraction distribution and constructed flow regime maps in a tube bundle (Figure 1.13c). The flow regime maps were found to be in good agreement with the existing literature. The experimental results from Murakawa et al. [START_REF] Murakawa | Evaluation of void fraction around a tube in two-phase flow across horizontal tube bundle[END_REF] were compared with the correlation proposed by Feenestra et al. [START_REF] Feenstra | An improved void fraction model for two-phase cross-flow in horizontal tube bundles[END_REF] for predicting the void fraction in a tube bundle. The correlation was found to be in agreement with the experimental data. Aprin et al. [START_REF] Aprin | Experimental analysis of local void fractions measurements for boiling hydrocarbons in complex geometry[END_REF] investigated two-phase flow patterns in a staggered 5×9 tube bundle with P D = 1.33 and D =19.05 mm, for three hydrocarbons at boiling conditions. The authors measured local void fraction through an optical probe. A quantitative method, based on PDF of void fraction, for analyzing flow structures was proposed. Statistical analysis identified bubbly and annular-dispersed flow regimes, with transitions at specific void fraction thresholds. A flow map based on vapor and liquid velocities was presented. The same methodology and instrumentation was used by Piteau et al. [START_REF] Piteau | Experimental investigation of in-flow fluidelastic instability for rotated triangular tube bundles subjected to single-phase and twophase transverse flows[END_REF] to identify flow regime in a square tube bundle subjected to water-air two phase flow. The test section contained 5 × 3 tubes with P D = 1.5 and D =30 mm. A summary of the geometry, working fluid and instrumentation used in the previous mentioned experimental campaigns is presented in Table 1.2. While a comparison of the previous mentioned flow regime maps is reported in Figure 1.15 for square pitch tube bundle and in Figure 1. [START_REF] Xu | Hydrodynamics of two-phase flow in vertical up and down-flow across a horizontal tube bundle[END_REF] Upon analyzing Figure 1.15, it can be concluded that most authors agree on the transition between the bubbly flow regime and the flow regime occurring at larger void fractions (churn/slug/intermittent). However, some authors, such as Xu et al. [START_REF] Xu | Hydrodynamics of two-phase flow in vertical up and down-flow across a horizontal tube bundle[END_REF] and Piteau et al. [START_REF] Piteau | Experimental investigation of in-flow fluidelastic instability for rotated triangular tube bundles subjected to single-phase and twophase transverse flows[END_REF], report the initiation of a churn/slug flow regime. This discrepancy may be attributed to the narrower test sections used in their studies, which could facilitate the occurrence of such a flow regime rather than the intermittent one. The transition to the annular flow regime varies among different authors; however, the choice of measurement systems, the inlet conditions and flow regime identification methods can impact the results.

The flow regime maps in staggered tube bundles (Figure 1.16) exhibit discrepancies among different authors. The bundle geometry, including parameters like P D, tube diameter (D), and the angle between the fluid direction and the configuration, may bring a more substantial influence on the staggered configuration compared to the square one. This suggests that the gap between the tubes can affect the transitions between flow regimes and the specific type of flow regime observed. Furthermore, the choice of measurement systems, the inlet conditions and flow regime identification methods can significantly impact the obtained results. It's essential to note that the experimental campaign conducted by Aprin et al. [START_REF] Aprin | Experimental analysis of local void fractions measurements for boiling hydrocarbons in complex geometry[END_REF] used a boiling two-phase mixture instead of air-water under adiabatic conditions. In flow boiling conditions, vapor is generated on both the tube wall and within the entire bundle. This factor, coupled with the complex geometry of the tube bundle, introduce substantial disturbances to the two-phase flow, resulting in a shift of the transition zone towards smaller void fractions, as reported by Aprin et al. [START_REF] Aprin | Experimental analysis of local void fractions measurements for boiling hydrocarbons in complex geometry[END_REF].

Thesis outline

Based on the aforementioned review, it can be deduced that a comprehensive understanding of vibration excitation mechanisms in tube bundles under two-phase cross flow has not been fully achieved. To gain deeper insights into these mechanisms, it is necessary to further investigate how void fraction, flow regime, flow pattern, and flow distribution impact flow-induced vibrations. Some work has been done in this direction (Benguigui et al. [START_REF] Benguigui | Experimental investigation on two-phase cross-flow-induced vibration of a cylinder in a confined channel[END_REF], Zhang et al. [START_REF] Pettigrew | Detailed flow and force measurements in a rotated triangular tube bundle subjected to two-phase cross flow[END_REF], Piteau et al. [START_REF] Piteau | Experimental investigation of in-flow fluidelastic instability for rotated triangular tube bundles subjected to single-phase and twophase transverse flows[END_REF] and Soussan et al. [START_REF] Soussan | Local two-phase flow measurement in an oblique tube bundle geometry[END_REF]) by instrumenting the tube bundles with instrumentation dedicated to investigate the two-phase flow.

The majority of research on tube vibrations in two-phase flow relies heavily on experimental approaches. The experimental campaigns help in the development and validation of semi-empirical models, enabling the determination of stability thresholds for design purposes (a detailed review is reported by Hassan [START_REF] Hassan | Flow-induced vibrations in nuclear steam generators[END_REF] and Blevins [START_REF] Blevins | Flow-induced vibration[END_REF]). However, determining the formulas for stability limits in two-phase flow remains a challenge, and there is considerable uncertainty regarding their extrapolation to operational conditions in steam generators. Moreover, SG prototypical steam-water mixtures pose difficulties in experimental replication due to high pressure and temperature conditions. To address this, alternative mixtures such as air-water or refrigerants have been often employed. Water-air mixtures are relatively easier to work with, while refrigerants require specialized tubing connections, heating elements, and pumping systems.

The progress on computational fluid dynamics has motivated researchers to explore the application of CFD in predicting flow-induced vibration stability thresholds. Researchers have followed two approaches: one involves CFD to acquire essential parameters for theoretical models as done by Hassan et al. [START_REF] Hassan | Damping-controlled fluidelastic instability forces in multi-span tubes with loose supports[END_REF] and Sadek et al. [START_REF] Sadek | Numerical investigation of the cross flow fluidelastic forces of two-phase flow in tube bundle[END_REF], while the other involves employing coupled fluid/structure simulations as performed by Benguigui et al. [START_REF] Benguigui | Numerical simulation of two-phase flow induced vibration[END_REF][START_REF] Benguigui | Fluid-structure interaction in two-phase flow using a discrete forcing method[END_REF]. Several approaches have been developed for multiphase CFD modeling (Merigoux et al. [START_REF] Merigoux | A generalized large interface to dispersed bubbly flow approach to model two-phase flows in nuclear power plant[END_REF] and De Santis [START_REF] Santis | A generalized multiphase modelling approach for multiscale flows[END_REF] among others). While multiphase CFD has gained significant attention as a promising tool for analyzing complex multiphase flows, it is imperative to conduct a thorough preliminary validation to assess its consistency and reliability before its widespread application (Fleau et al. [START_REF] Fleau | Validation of multifield approach for the simulations of two-phase flows[END_REF]). Validation studies play a vital role in assessing the accuracy and predictive capabilities of multiphase CFD models by comparing simulation results with experimental data. These validation efforts help in identifying limitations, improving model formulations, and enhancing the understanding of multiphase flow behavior.

Hence, the objective of the present work is to take a step forward in this direction. The primary aim of this study is to comprehend the fundamental vibration excitation mechanisms occurring in tube bundles under two-phase cross flow. Specifically, the objective is to carry out meticulous measurements of two-phase flow characteristics and tube vibration. In order to accomplish this a new test facility named TREFLE (Two-phase flow REgimes and FLuid-structure interaction Experimental facility) was installed in 2022 (Spina et al. [START_REF] Spina | Tube bundles subjected to two-phase flow: a new experimental apparatus design for flow regimes and vibration study[END_REF][START_REF] Spina | A new experimental facility for two-phase flow characterization in a tube bundle and vibration study[END_REF]). One of the purpose of TREFLE is also to gather high quality data CFD-grade for the validation of multiphase CFD simulation tools.

Organized into four parts as outlined below, the manuscript starts by describing the TREFLE facility design. It then proceeds to the experimental characterization of two-phase flow in bubbly and intermittent flow regimes across a tube bundle, along with the analysis of tube vibration response. Finally, the manuscript presents and discusses simulations conducted using the Neptune CFD code of the aforementioned experimental campaigns.

• In Chapter 2, the TREFLE facility is presented. The design process of specific components, such as the gas injector, the water inlet, and the vibrating tube system, is described in detail. Furthermore, the different measurement systems and their characteristics are reported, including the fast camera, dual-tip optical probe, wire mesh sensor (for analyzing the two-phase flow), and accelerometer (for measuring the tube vibration response).

• The bubbly flow experimental campaign is described in Chapter 3: this chapter focuses on investi-gating the behavior of the bubbly flow regime. The study of the two-phase bubbly flow begins by examining the inlet conditions of the facility. A comprehensive analysis of the bubbles generated at the gas injector is conducted using a fast camera. Additionally, the two-phase flow within the tube bundle is explored through the utilization of a fast camera and an optical dual-tip probe. The optical probe provides crucial information on the gas velocity, bubble sauter mean diameter and void fraction evolution within the bundle. Positioned upstream of the tube bundle, the probe offers insight into the inlet conditions, which are essential for establishing CFD-scale boundary conditions in the current setup. Once the two-phase flow is characterized, the central tube is replaced with a vibrating tube. The vibration response of the tube is studied using an accelerometer and the fast camera. Hence, both two-phase flow and tube vibration response are investigated.

• Chapter 4 is devoted to the intermittent flow experimental campaign: this chapter presents an experimental study of the intermittent flow regime. The two-phase flow is analyzed qualitatively using a fast camera. Furthermore, the dual-tip optical probe yields results on the gas velocity, bubble sauter mean diameter, and void fraction evolution within the bundle. The study also reports information on slug frequency and flow regime identification through probability density function analysis. The wire mesh sensor is utilized to determine the inlet conditions of the tube bundle. A new flow regime map for two-phase flow in a tube bundle is proposed. Additionally, the vibration response of the tube is investigated. Finally, the obtained results are compared to those of the experimental campaign conducted on bubbly flow. Therefore, the study includes both the behavior of two-phase flow and the tube vibration response.

• In the last chapter are presented numerical simulations conducted using the Neptune CFD code. The previous experimental campaigns provided high-quality data, enabling the establishment of accurate boundary conditions for the multiphase code. Additionally, the simulation results are compared with the findings from the experimental campaigns, revealing the capabilities of the code.

Chapter 2

TREFLE facility experimental device

This chapter presents the experimental setup and measurement techniques employed to gain deeper insights into the mechanisms of two-phase flow-induced vibrations. The objective is to describe the design of TREFLE, to go into detail about the design process of some components, and to present the measurement systems used.

TREFLE facility design

The TREFLE facility (Two-phase flow REgimes and FLuid structure-interaction Experimental facility) main goal is to investigate air-water two-phase flow induced vibration on a tube bundle. The experimental device, TREFLE, is reported in Figure 2.1, which includes a detailed view of the test section. The water is supplied from a divergent section. A honeycomb and a structure made of plates are installed inside the divergent section to ensure a flat profile of water velocity at the inlet of the facility. On the other hand, the air is delivered to the facility through a gas injector. The TREFLE facility consists of three main sections: the Homogenization section, the Test section (Tube bundle), and the Tranquilization section. In red the central tube to be substituted with the vibrating tube.

The homogenization section (approximately 1400 mm with a cross-section of 300 mm × 216 mm) aids in the homogenization of the two-phase flow prior to reaching the test section. The test section of the facility contains an in-line, square pitch 5 × 5 tube bundle of fixed (i.e. non vibrating) tubes with pitchto-diameter ratio (P D) of 1.44. The tube length (L) is 10 times the tube diameter, which measures 30 mm. The central tube is specifically designed to be substituted with a vibrating one. To prevent significant flow perturbations in the test section, a tranquillization section (about 1600 mm), provides sufficient space between the test section and the air/water outlet. The facility is equipped with several transparent panels in order to allow flow visualization. A detailed sketch of the facility, including detailed dimensions, is shown in the Figure 2.2. The water loop consists of a water storage tank, a Coriolis flow meter and a pump. Under nominal conditions, the pump can deliver a maximum flow rate of 2600 g s -1 with an uncertainty of 0.5%. On the other hand, the air loop draws air from outside the building and utilizes a compressor to achieve a mass flow rate of up to 300 g s -1 . For conducting experiments at different air flow rates, two separate air injection lines were used. The main air injection line was used for tests T5, T6 and T7 in Table 2.1. This line incorporates a control valves to regulate the air flow rate, and the uncertainty associated with the control valve varies depending on the specific flow rate, as specified in Table 2.1. The small air injection line was used for tests T1, T2, T3 and T4. This line is equipped with a Bronkhorst air mass flow meter with a range of 4 to 200 L min -1 and uncertainty specified in Table 2.1. The facility is designed to accommodate the entire range of void fractions and flow regimes. To ensure uniform air injection across the section, the number and type of tubes utilized in the gas injector vary according to the air flow rate (Further information regarding the number and type of tubes utilized can be found in Table 2 The experiments are performed with water flow rates of ṁwater = 2500 g s -1 and air flow rates in the range of 25.0 < Q air < 200.0 L min -1 using the small injection line. For the main injection line, air flow rates are in the range of 11.0 < ṁair < 29.0 g s -1 . Hence, water gap superficial velocity is J l = 0.126 m s -1 and air gap superficial velocity is in the range of 0.021 < Jg < 1.22 m s -1 . The generic superficial gap velocity (J) is calculated as reported in Equation 2.1, with a the gap between the tubes, n the number of tubes, and ṁ the generic mass flow rate.

J = ṁ ρanL (2.1)
While, considering b and c width and depth of the free duct, the generic free duct superficial velocity (U s ) is calculated as reported in Equation 2.2.

U s = ṁ ρbc (2.2)

Gas Injector

The design of this component has been developed with the objective of generating various flow regimes within the test section. The gas injector is reported in Figure 2.3, and the tubes are designed to be interchangeable, allowing for adapting the component based on the air flow rate.

Design of the gas Injectors

The air injection is made through seven pierced tubes placed perpendicularly with respect to the tube bundle. The shape and size of the two-phase structures are strongly influenced by the holes size. However, the design must consider the hydraulic constraints of the air loop, ensuring that the introduced pressure drops are within the compressor's tolerance. In addition, as the gas injector is positioned downstream of the water inlet, it generates disturbances on the water flow that should be minimized. To address these requirements, two configurations have been designed in order to generate various flow regimes and to comply with the hydraulics constraint of the air loop and water flow:

• The first configuration (Figure 2.4) was designed to produce bubbles with dimensions in the range of a few millimeters, simulating a bubbly flow regime. The outer diameter (O.D.) of the pipes is 15 mm, and each pipe contains 32 holes placed diametrically opposed on the tube sides to reduce the likelihood of bubble coalescence and enhance the bubble detachment. The hole diameter is 1 mm, enabling the compressor to deliver the required mass flow rate without exceeding the specified pressure drops. However, the spacing and diameter of the holes allow for the generation of bubble sizes in the range of a few millimeters.

Bubble diameter prediction for type 1 tubes

The prediction of bubble diameter for bubbles generated through an orifice is a widely discussed topic in the open literature. In line with Colombet's [START_REF] Colombet | Modelisation de reacteurs gaz-liquide de type colonne a bulles en conditions industrielles[END_REF] methodology for categorizing various approaches, three distinct regimes of bubble formation through an orifice can be identified: the quasi-static regime, the dynamic regime, and the jet regime.

The quasi-static regime

The prediction of bubble diameter is achieved through a force balance analysis. In the quasi-static regime, only buoyancy and surface tension forces are considered. The force balance yields Tate's law (Equation 2.3):

d b = 6d o σ ∆ρg 1 3 (2.3)
Where d b is the bubble diameter, d o the orifice diameter, σ the surface tension, ∆ρ the density difference between liquid and gas phase, g is the gravity acceleration. Because of the narrow range of validity of this law, the equation is applicable for very low gas flow rate (quasi-static injection) and still fluid. The quasi-static injection boundary is defined by a critical velocity value, which according to Oguz and Prosperetti [START_REF] Oguz | Dynamics of bubble growth and detachment from a needle[END_REF] can be defined as Equation 2.4:

u critical = 4 d 2 o 16 3g 2 1 6 σd o 2ρ L 5 6
(2.4)

The dynamic regime

Therefore, when the gas velocity overcame the critical velocity (Equation 2.4) the dynamic regime comes into play. In this regime other forces play an important role in the force balance and cannot be neglected. According to Gaddis et al. [START_REF] Gaddis | Bubble formation in quiescent liquids under constant flow conditions[END_REF], in the dynamic regime, the bubble diameter can be found through the force balance between buoyancy force, gas momentum, surface tension force, inertia and drag forces. This balance allow to find out the equation for the prediction of bubble diameter (Equation 2.5):

d b = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 6d o σ ρ L g 4 3 + 81ν L q g πg + 135q 2 g 4π 2 g 4 5 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ 1 4
(2.5)

Where ν L is the kinematic viscosity of the liquid phase, q g is the volume flow rate at the orifice. If we consider a weak value of gas flow rate, from Equation 2.5 we can obtain the Tate's law (Equation 2.3).

The equation from Gaddis et al. [START_REF] Gaddis | Bubble formation in quiescent liquids under constant flow conditions[END_REF] can be written as function of adimensional numbers such as Froude, Archimede and Eotvos at the orifice.

The jet or inertial regime

According to Gaddis et al. [START_REF] Gaddis | Bubble formation in quiescent liquids under constant flow conditions[END_REF] the transition between the dynamic and jet regime is defined when the Weber number at the orifice W e o = ρ g u 2 g d o σ lies in between 1 and 3. In the jet or inertial regime, the gas momentum prevails over the other forces, so comparing the terms in Equation 2.5 the third term prevails, leading to the Equation 2.6:

d b = 135q 2 g 4π 2 g 1 5
(2.6)

Which can be rearranged in terms of Froude number at the orifice (F r o = u 2 g gdo ), leading to the Equation 2.7:

d b d o = 1.16F r 1 5 o (2.7)
Iguchi et al. [START_REF] Iguchi | Bubble and liquid flow characteristics in a vertical bubbling jet[END_REF] determined a new correlation for the inertial regime to estimate bubble diameter starting from the bubble frequency (f b ) at the injection estimated as reported in Equation 2.8:

f b = 1.06 ρ l g 3 σ 1 4 ρ g ρ L 1 5 ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ q 2 g g 1 5 d o ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ 0.5 (2.8)
Therefore, the bubble diameter at the injection can be estimated as:

d b = 6q g πf b 1 3
(2.9)

Alternative approaches for predicting the bubble diameters

Even though the proposed methods have been developed for quiescent liquid, they show a good agreements with experiments performed with liquid co-flow by Muilwijk et al. [START_REF] Muilwijk | The limerick bubbly flow rig: Design, performance, hold-up and mixing pattern[END_REF].The correlation from Muilwijk et al. [START_REF] Muilwijk | The limerick bubbly flow rig: Design, performance, hold-up and mixing pattern[END_REF] (Equation 2.10) came from a correction of a correlation previously developed by the same author [START_REF] Muilwijk | Experimental investigation on the bubble formation from needles with and without liquid co-flow[END_REF]. The formulation of the bubble diameter prediction is based on a force balance as done by Gaddis et al. [START_REF] Gaddis | Bubble formation in quiescent liquids under constant flow conditions[END_REF].

d b d o = 0.093 u l u go + 6.18 Bo + 1.26 F r 3 5 -1 -1 3 (2.

10)

With:

Bo = ρ L gd 2 o σ and F r = u 2 go gd 2 o (2.11)
Where

u go = 4qg πd 2 o
and u l is the liquid velocity. Akita et al. [START_REF] Akita | Bubble size, interfacial area, and liquid-phase mass transfer coefficient in bubble columns[END_REF] developed a correlation (Equation 2.12) independent from the injection diameter d o , introducing the cross-section equivalent diameter D eq . As matter of fact, their correlation is intended to estimate the bubble diameter far from the injection point.

d b = 26 ρ L D 2 eq g σ -0.5 gD 3 eq ν 2 L -0.12 ⎛ ⎝ u g gD eq ⎞ ⎠ -0.12
(2.12) Table 2.2 displays the range of bubble diameters predicted at the injector by comparing the correlations mentioned earlier. The experimental conditions refer to the ones presented in Table 2.1 (for tests T1, T2, T3, T4) and the gas injector type 1 geometry presented in Figure 1.6. In our specific application, which involves estimating the bubble diameter for gas injector type 1, the correlations previously mentioned (Tate's law in Equation 2.3, Equations 2.5 and 2.6 from Gaddis et al. [START_REF] Gaddis | Bubble formation in quiescent liquids under constant flow conditions[END_REF], and Equation 2.9 from Iguchi et al. [START_REF] Iguchi | Bubble and liquid flow characteristics in a vertical bubbling jet[END_REF]), developed for bubbles detaching from an orifice in quiescent liquid, are less appropriate. However, this remains valuable for design purposes, as it helps identify a range of expected bubble diameters. Upon analyzing Table 2.2, it becomes evident that Tate's law (Equation 2.3) is valid for u g < u cr but is less applicable to the gas velocities studied in the current conditions. Additionally, as the Weber number at the orifice exceeds the unity, the jet regime can be considered established under all conditions. Consequently, Equation 2.5 from Gaddis et al. [START_REF] Gaddis | Bubble formation in quiescent liquids under constant flow conditions[END_REF] is also less appropriate for the gas velocities under investigation. Therefore, based on the experimental conditions, the most reliable correlations are Equation 2.9 from Iguchi et al. [START_REF] Iguchi | Bubble and liquid flow characteristics in a vertical bubbling jet[END_REF], Equation 2.10 from Muilwijk et al. [START_REF] Muilwijk | The limerick bubbly flow rig: Design, performance, hold-up and mixing pattern[END_REF], and Equation 2.12 from Akita et al. [START_REF] Akita | Bubble size, interfacial area, and liquid-phase mass transfer coefficient in bubble columns[END_REF].

Test no. ṁgo u g W e o d b [m] d b [m] d b [m] d b [m] d b [m] d b [m] kg s -1 m s -1 -

Divergent and honeycomb design

In order to ensure optimal experimental conditions for obtaining a reliable database to validate multiphase CFD calculation codes, meticulous attention is devoted to the two-phase flow inlet conditions. The water inlet is specifically designed to achieve a uniform velocity profile prior to reaching the gas injectors. However, the conical shape of the divergent section can induce vortex formation and asymmetries in the water flow. To address this concern, a new system that consists of plates and honeycomb structure (shown in Figure 2.6) has been implemented in the facility with the assistance of CFD analysis. The simulation results are depicted in Figure 2.7, showing the effectiveness of the implemented system. The two distinct Figures 2.7a 2.7b serve to highlight the impact of the implemented system. As it is pointed out in Figure 2.7, the sole honeycomb is not enough to avoid the asymmetry in the flow. However, the new system effect is satisfactory. The water velocity profiles along the direction of the flow are reported in Figure 2.8. The figure shows that the water profile is homogeneous across the facility's cross-section. This indicates a uniform flow and facilitates effective bubble generation at the gas injector. To validate this effect, experiments were conducted specifically targeting bubble generation. The fast camera images captured the bubble production from the central tube of the gas injector, and the results are presented in Figure 2.9. Notably, Figure 2.9a demonstrates that the absence of the plate and honeycomb system causes the bubbles to shift towards the left side of the facility due to the effect of asymmetric water flow. Conversely, the presence of the system's effect is clearly depicted in Figure 2.9b, where the bubble path remains straight.

Vibration sensor tube design

The tubes within the bundle are attached to the facility wall on one side while remaining free on the other side (refer to Figure 2.10). Due to the hydraulic conditions of the facility, the forces exerted on the tube walls are minimal, resulting in very subtle vibrations. Consequently, the tubes are considered "fixed". However, the central nine tubes in the test section can be replaced with "vibrating" tubes, enabling the measurement of their vibration behavior. These "vibrating" tubes consist of a rigid tube attached to a flexible beam, allowing for a more flexible behavior. This characteristic of the test section facilitates the study of tube vibration as separate effect (i.e. the tube vibration can be investigated following the characterization of the flow). The "vibrating" tube(s) can freely vibrate in one direction rather than another, allowing for the examination of the influence of one or more vibrating tubes. Piteau et al. [START_REF] Piteau | Experiments and computations of a loosely supported tube in a rigid bundle subjected to single-phase flow[END_REF] described as loosely supported tube (reported in Figure 2.11), a tube used to generate nearly planar vibrations. The tube is linked to a 100 mm long bar, with rectangular cross section 25 mm × 4 mm (width x thickness). The bar allows the tube to be flexible and so to vibrate along the preferential direction due to its rectangular cross-section. The flexible tube design allows the authors to compare experimental and numerical results of the tube vibration response. The tube motions has been constrained to be along the lift direction (perpendicular to the flow direction). The loosely supported tube has been modeled numerically using two opposite springs at half tube, with stiffness adjusted in order to reproduce the tube support. Lately, the flexible tube support was modified [START_REF] Piteau | Experimental identification of the fluid-elastic coupling forces on a flexible tube within a rigid square bundle subjected to single-phase cross-flow[END_REF][START_REF] Lagrange | Fluid-elastic coefficients in single phase cross flow: dimensional analysis, direct and indirect experimental methods[END_REF]. The tube has been replaced with a rigid tube mounted using two parallel flexible blades which allow almost pure planar translations (Figure 2.12). The implemented design permits to interchange the blades to study different frequency configurations adjusting the blades thickness. The experimental campaigns performed by the authors [START_REF] Piteau | Experimental identification of the fluid-elastic coupling forces on a flexible tube within a rigid square bundle subjected to single-phase cross-flow[END_REF][START_REF] Lagrange | Fluid-elastic coefficients in single phase cross flow: dimensional analysis, direct and indirect experimental methods[END_REF] lead to the identification of fluid-elastic coupling coefficients for a tube vibrating along the lift direction in a rigid bundle.

In their research, Khalvatti et al. [START_REF] Khalvatti | Effect of preferential flexibility direction on fluidelastic instability of a rotated triangular tube bundle[END_REF] investigated how the dependence of fluid-elastic instability (FEI) Figure 2.12: Two blades support tube configuration and experimental rig design from Piteau et al. [START_REF] Piteau | Experimental identification of the fluid-elastic coupling forces on a flexible tube within a rigid square bundle subjected to single-phase cross-flow[END_REF].

on the angle between the flow direction and the tube flexibility direction. To explore this relationship, they utilized a facility equipped with a test section containing both fixed and flexible tubes. The flexible tubes are designed as rectangular cross-section supported beams, offering flexibility in a single direction. Each tube is equipped with a dowel pin at the base of the flexible support, which fits into a specifically drilled hole in the facility. This ensures precise orientation of the tube in the desired direction. Granger et al. [START_REF] Granger | Motiondependent excitation mechanisms in a square in-line tube-bundle subject to water cross-flow: an experimental modal analysis[END_REF] and Deri [START_REF] Deri | Operational modal analysis of a triangular-pitch tube bundle subjected to two-phase cross-flow[END_REF] report the design of the flexible tube as a "piano wires tube" (Figure 2.14). The tube is supported from one side and clumped by means of a special clump in the other side. The special clump permits the tube motion to be considered as rigid body motion. A similar design has been implemented by Khalifa et al. [START_REF] Khalifa | A single flexible tube in a rigid array as a model for fluidelastic instability in tube bundles[END_REF] using a piano wire tube (Figure 2.15). The tube natural frequency can be adjusted by changing the tension of the wire with the relationship reported in Equation 2.13. Where f is the frequency of the tube, m its mass, T the tube tension and L its length.

f = 1 2π 2T mL (2.13)

New flexible tube design

The design of the flexible tube has been inspired by the aforementioned designs. The system is based on the concept of a tube-flexible beam (refer to Figure 2.18). The tube is designed to exhibit rigid body behavior, while the beam allows deformation in a single direction. As a result, the system enables the tube to vibrate as a rigid body, with the vibration characteristics such as frequency, stiffness, and structural damping determined by the properties of the flexible beam. The design process incorporates the application of the Connors formula [START_REF] Connors | Fluidelastic vibration of heat exchanger tube arrays[END_REF], a stability limit equation, to ensure optimal stability:

U Df 0 = K( m 0 δ 0 ρD 2 ) n (2.14)
Where K is the Connors constant, f 0 is the natural frequency, m 0 the mass of the tube per unit of length plus the hydrodynamic mass, δ 0 the damping (δ 0 = 2πζ with ζ damping ratio), ρ the density of the tube. U is the fluid velocity, and D the tube diameter. The Connors constant, two-phase damping and hydrodynamic mass are unknown in our flow condition. Therefore, to get an estimation of the mentioned values we refer to the experimental campaigns performed by Pettigrew et al. [START_REF] Pettigrew | Vibration of tube bundles in two-phase cross-flow: Part 1-hydrodynamic mass and damping[END_REF][START_REF] Pettigrew | Vibration of tube bundles in two-phase cross-flow: Part 2-fluid-elastic instability[END_REF]. The experimental campaign has the objective to provide information for the damping and added mass in two-phase airwater flow conditions, for different tubes bundles configurations. The data collected by Pettigrew et al. [START_REF] Pettigrew | Vibration of tube bundles in two-phase cross-flow: Part 1-hydrodynamic mass and damping[END_REF][START_REF] Pettigrew | Vibration of tube bundles in two-phase cross-flow: Part 2-fluid-elastic instability[END_REF] allow to provide design guidelines for damping ratio and hydrodynamic mass in two-phase flow. Therefore, given certain flow conditions (void fraction, flow velocity) the parameters which affect the design of the vibrating system are the natural frequency and the tube mass. The velocity of a mixture is influenced by its density, with different factors coming into play at different void fractions. At small values of void fraction the water velocity is predominant, while at larger values of void fraction the gas velocity prevails. The objective is to create a design able to produce important vibration with respect to the "fixed" tubes, while ensuring the vibration amplitudes remain below the distance between the tubes of the bundle. The system properties plays an opposite role regarding those objectives. Hence, according to the considerations made, the design process is iterative. The geometry of the system is adjusted according to the constraints.

The flexible beam 30 mm long, has been chosen to be realized in stainless steel. The beam cross section is 2 mm × 20 mm. Those characteristics are able to give a small system natural frequency, allowing the vibration amplitudes to be sufficiently large. While, the rectangular cross section allow the mode decoupling. The tube material was chosen to allow the realization of a full tube (circular cross section with 30 mm diameter), so that the moment of inertia of the structure is sufficiently high and the tube mass still small enough. The design details of the flexible tube and its implementation in the test section are described in In order to verify the mode decoupling and anticipate the vibration mode, a modal analysis of the system was conducted using finite element Ansys Mechanical. Table 2.3 presents the various deformations of the tube along with their corresponding natural frequencies. Additionally the participation factor is reported, the participation factor of a mode shape in a modal analysis helps in understanding the relative importance of a particular mode shape in the overall dynamic behavior of the structure. Referring to Figure 2.20 for the reference system, the first four detected mode shapes are:

• First mode (11 Hz), deformation along the x-direction. The system exhibits a bending deformation in the x-direction (lift direction). The flexible beam undergoes bending deformation along its length in the x-direction. It forms a characteristic bending curve. The tube, being attached to the flexible beam, moves as a rigid body in response to the beam's bending deformation. It does not deform independently in this mode. Instead, it translates as a whole while maintaining its cylindrical shape. The beam and tube deform together, resulting in a combination of bending and translation. This mode, with a participation factor of 0.64, has a significant influence on the overall response of the system. When the system is subjected to dynamic loads or vibrations, this mode will be a dominant factor in defining the system's behavior.

• Second mode (80 Hz), deformation along the y-direction. The flexible beam experiences bending deformation along its length in the y-direction (drag direction). Similar to the previous mode, it forms a characteristic bending curve. The tube, being attached to the flexible beam, maintains its cylindrical shape and moves as a rigid body. It translates in response to the bending deformation of the beam. This mode, with a very low participation factor of 1 ⋅ 10 -5 , has minimal influence on the system's overall response. In practical terms, this mode is essentially negligible in its contribution to the system's behavior.

• Third mode (220 Hz), deformation along x with one node. The term "one node" refers to a point along the structure where there is little or no displacement during vibration, resulting in a sinusoidal wave-like deformation pattern. The flexible beam undergoes bending deformation along its length in the x-direction. This bending pattern takes the form of a single sinusoidal wave. The tube also exhibits a sinusoidal movement along its length, following the bending pattern of the beam. The third mode, with a participation factor of 0.31, has a moderate influence on the system's overall response. While not as dominant as the first mode, it still plays a significant role in shaping the system's deformation patterns.

• Fourth mode (278 Hz), rotation/torion around the z-axis. The flexible beam experiences twisting or torsional deformation around its longitudinal axis (z-axis). The tube, attached to the flexible beam, also participates in the rotational deformation rotating around its own axis. The tube maintains its cylindrical shape but undergoes a twisting motion. This mode, with an extremely low participation factor of 1 ⋅ 10 -6 , has an almost negligible impact on the system's overall response.

The system proprieties and geometry allow to keep the performance below the stability limit. Table 2.4 reveals that the available flow velocity (U av ) consistently remains below the critical velocity (U cr ), which is calculated using Equation 2.14 and two-phase properties obtained from Figure 2.19. However, the tube implementation inside the facility requires to solve some issues. In order to respect the same wettability in all the tubes of the bundle, the surface proprieties of the tube must be the same as the fixed tubes. To solve this issue, a special design was developed. The design allows the same surface properties to be maintained and the mass of the tube to be changed according to the needs. The flexible beam is wrapped by the tube itself, ensuring that it does not affect the two-phase flow. Once the vibrating tube is implemented in the test section, its characteristics (natural frequency f , stiffness k, mass m and damping δ) are determined:

• The tube mass m is selected according to the design needs.

• The natural frequency of the first vibration mode, denoted as f , is measured within the tube bundle under two distinct conditions: in still water and in a still air environment. These measurements are essential to determine the natural frequency when submerged in water, denoted as f w , and when exposed to air, denoted as f a . Additionally, the objective is to account for any potential influence that neighboring tubes may have on the tube natural frequency. This information is crucial as boundary condition for the CFD code to accurately simulate the tube vibrational behavior.

• The damping δ is calculated by means of the half-power bandwidth method, i.e. by measuring the width of the frequency response curve at the points where the power level drops to half of its peak value. Then, the damping ratio can be calculated using the following formula:

ζ = 1 2 (1 -(f 1 f 2 ) 2 )
where ζ is the damping ratio, f 1 is the lower frequency at which the response is reduced by √ 2 (half-power point), and f 2 is the higher frequency at which the response is reduced by √ 2.

• To experimentally determine the stiffness k of the system, a rigorous procedure is employed. Several calibrated weights are incrementally attached to the tip of the tube, creating a precisely controlled loading condition. The displacement of the tube is tracked and measured using a high-resolution camera system. This comprehensive experimental setup enables the generation of a detailed forcedisplacement (F -x) curve. The force-displacement curve serves as a fundamental tool for characterizing the system's stiffness. It represents the relationship between the applied force (F ) and the corresponding displacement (x). The stiffness k is accurately derived from this curve by calculating the slope, which directly relates to the relationship expressed as F = kx.

All measurements were repeated several times and averaged to ensure reliable values. The aforementioned quantities are reported in Table 2

[Hz] f w [Hz] m [kg] ζ a [ -] ζ w [ -] k [N m -1 ]
13.4 10.4 0.39 0.01 0.04 600 Table 2.5: Proprieties of the new vibrating system.

Instrumentation and measurements

The following section is dedicated to the description of the available instrumentation in the TREFLE facility.

Fast camera

The TREFLE facility is equipped with several transparent windows in order to allow flow visualization. In order to visualize the two-phase flow a Fast camera was used. The JAI SP-12000M-CXP4 camera provides 12-megapixel (4096 × 3072) resolution at 189 fps (frame per second). The frequency of acquisition can increase by decreasing the resolution. It is equipped with a wide angle (24 mm) lens fixed focus. The diaphragm opening ranges from 1.4 to 22. Measurements were performed in the visualization windows reported in Figure 2.21: at the air injection, at the tube bundle inlet, in the tube bundle (lateral and frontal view), at tube bundle outlet (respectively W1, W2, W3, W5 and W4 in Figure 2.21). The positions of the high-speed camera and the lighting system with respect to the test section depend on the chosen visualization windows.

Dual-tip optical probe

The fiber-optical device developed by RBI Instrumentation has been used to measure bubble size, bubble velocity and void fraction. The declared sampling frequency of 20 kHz. The optical dual tip probe is composed by two sapphire tips (30 µm of diameter) spaced 1.004 mm apart in the direction of the flow and 0.84 mm in the direction perpendicular to the flow. The probe has two 5 m long optic fiber cables. The diameter of the probe is of 6.35 mm and the total length is of 445 mm. The tips detail is reported in Figure 2.23, the distance between the two tips was measured by means of an electrical microscope. The operating principle of the optical probe is based on the law of refraction: a liquid-gas interface passing through the probe tip makes the system change from a refractive state to a reflection state [START_REF] De Moura | Local measurements in two-phase flows using a double optical probe technique[END_REF]. We utilized an optical dual probe to measure at 13 different positions within the tube bundle. The locations of these points are illustrated in Figure 2.24 (The probe positioning device provides an uncertainty of 1 mm).

The measurement points presented in Figure 2.24 enable the construction of horizontal and vertical profiles for various parameters measured by the optical probe within the tube bundle. These parameters include the bubble mean Sauter diameter, bubble velocity, bubble size distribution, and void fraction. The optical probe signal allows us to calculate the average void fraction, denoted as α = T g (T l + T g ), where T g and T l represent the total gas and liquid time recorded from one end of the probe, respectively. Additionally, the Sauter mean diameter (d sm ) is calculated as d sm = 6α A i , where A i corresponds to the interfacial area value calculated according to Kataoka et al. [START_REF] Kataoka | Local formulation and measurements of interfacial area concentration in two-phase flow[END_REF]. The average bubble velocity (v b ) is determined as v b = d T f light , taking into account the distance between the primary and secondary tips (d) and the average transit time required for bubbles to travel from the reference sensor to the secondary sensor (T f light ). The average transit time is obtained by cross-correlating the signals from the two tips. Further details regarding the measured quantities via the optical probe can be found in Garnier et al. [START_REF] Garnier | Measurement of local flow pattern in boiling r12 simulating pwr conditions with multiple optical probes[END_REF]. Signal acquisition yields the "phase-time histogram", which represents the distribution of recorded gas residence time and liquid residence time on the probe tip. Multiplying each gas-time by the velocity results in the histogram of intercepted chord lengths. By employing a mathematical model that considers the non-spherical nature of bubbles (as described by Garnier et al. [START_REF] Garnier | Measurement of local flow pattern in boiling r12 simulating pwr conditions with multiple optical probes[END_REF]), the "bubble size distribution" can be obtained.

Wire Mesh Sensor

The WMS used in this work is presented in Figure 2.25. The WMS contains 78 × 56 wires with a spatial resolution of 3.95 mm × 4.00 mm. The wire diameter is of 0.3 mm. The design is a compromise between measurement accuracy and flow disturbance. The local fluid conductivity u (i,j,k) measured at the time instant t = k∆t at the crossing point identified by the indexes j (transmitter wire) and i (receiver wire) can be converted into void fraction α (i,j,k) according to Equation 2.15

α i,j,k = u i,j,k -u G,j,k u L,j,k -u G,j,k (2.15)
Where u i,j,G and u i,j,L are the conductivity that would be measured if the area around the crossing point (i, j) would be completely filled with gas or liquid respectively. The sensor consists of two electrode grids placed at a small axial distance behind each other. Two measurement positions were performed with the WMS (reported as dashed lines in Figure 2.27): at the tube bundle inlet, i.e. 230 mm below the test section; at the tube bundle outlet, i.e. downstream the test section. To acquire precise quantitative insights into the flow, one approach involves performing both temporal and spatial averaging of void fraction data. This averaging process relies on weight coefficients that assign the contribution to each intersection point (i, j) within the sensor matrix, relative to the overall domain size where the averaging is applied. Figure 2.28 illustrates the definition of these weight coefficients (a i,j ) crucial for computing the cross-section averaged void fraction. For each discrete sampling period (k), the cross-section average can be calculated independently using Equation 2.16. A sensor is the surface of the considered cell [START_REF] Prasser | A new electrode-mesh tomograph for gas-liquid flows[END_REF].

ᾱk = i j a i,j ⋅ α i,j,k (2.16)
The data evaluation using Equation 2.16 yields a series of instantaneous average void fractions. It is also possible to perform temporal averaging. Two-dimensional averaged void fraction distributions can be computed using Equation 2.17.

ᾱi,j = 1 k max kmax k=1 α i,j,k (2.17)
With Equations 2.16 and 2.17 an average void fraction for the total measurement cross-section can be obtained as reported in Equation 2.18

ᾱ = i j a i,j ⋅ ᾱi,j (2.18)
Furthermore, it is possible to determine a radial void fraction profile by averaging the instantaneous void fractions locally over the measurement period within a series of concentric ring-shaped domains (m). This calculation can be performed using Equation 2. [START_REF] Murakawa | Evaluation of void fraction around a tube in two-phase flow across horizontal tube bundle[END_REF].

ᾱm = 1 k max k i j a i,j,m ⋅ α i,j,k (2.19)
where a i,j,m are weight coefficients indicating the contribution of each measurement point with the indexes i, j to a ring with the number m.

The process of identifying bubbles and extracting their key characteristics relies on specialized evaluation algorithms. In this context, a bubble is defined as a connected region of gas-containing elements within the void fraction data α i,j,k that is entirely enclosed by liquid-phase elements. Each element belonging to a specific bubble is assigned a unique identification number, denoted as b i,j,k . Utilizing the bubble identification matrix in conjunction with void fraction information, we can calculate the equivalent diameter for each individual bubble. To begin, we compute the area occupied by each bubble within the xy plane. This procedure is carried out independently for each sampling time, characterized by the index k.

A xy,n,k = ∆x∆y i,j α i,j,k ∀ [i, j, k] ∶ b i,j,k = n (2.20)
Afterwards, the maximum area is found and converted into the diameter of an area equivalent circle:

D xy,n = 4A xy,n,max π with A xy,n,max = max(A xy,n,k ) (2.21)

Accelerometer

In order to record the tube vibration an accelerometer is placed at the free end of the vibrating cylinder.

The HBK piezoelectric accelerometer (Figure 2.29) has a frequency range from 0.2 Hz to 12 800 Hz with a voltage sensitivity of 10.2 mV s 2 m -1 .

Figure 2.29: HBK piezoelectric accelerometer.

Conclusion

In this chapter, the design of The TREFLE facility was presented. The design process for its various components was detailed. The gas injector design involved the utilization of correlations found in the open literature to estimate the possible bubble diameter for the upcoming bubbly flow experimental campaign. CFD played a crucial role in designing a device to achieve a flat velocity profile at the facility's inlet. Together, these two systems allowed for the precise design of the inlet conditions for the TREFLE facility.

Incorporating designs from existing literature, a new vibrating tube was developed, adapted to meet the specific requirements of the facility. The mode shapes of this new design were examined using the finite element analysis tool, Ansys Mechanical solver. Additionally, its properties, including natural frequency (f ), stiffness (k), mass (m), and damping (δ), were determined through experimental methods.

Finally, a comprehensive description of the instrumentation used in the experimental campaign was provided, with a particular focus on the extracted quantities. In the next chapter, we will present the results of the bubbly flow experimental campaign.

Chapter 3

Bubbly flow regime: experimental investigation

In this chapter, the experimental analysis performed on the bubbly flow regime (tests T1 to T4 in Table 2.1 in Section 2.1) in the TREFLE facility is reported. Specifically, the chapter is organized as follows: In Section 3.1, the focus lies on investigating the dynamics of bubble generation at the gas injector. Section 3.2.1 is dedicated to the analysis of the two-phase flow at the inlet of the tube bundle. Within Section 3.2.2, the characteristics of the two-phase flow within the tube bundle itself are explored. Section 3.2.3 shifts the examination to the outlet of the tube bundle for further analysis. Throughout these sections, both qualitative and quantitative measurements, acquired through the use of a fast camera, are presented. The exploration of the two-phase flow within the tube bundle also involves the use of a dualtip optical probe for a more comprehensive understanding. Finally, the attention turns to the vibrating behavior of the tube. In Section 3.3, an accelerometer is employed in conjunction with the fast camera to provide an in-depth analysis of the dynamic response of the tube.

Inlet conditions: gas injector characterization

To characterize the inlet conditions, a study was conducted on the bubble injections downstream of the gas injector. This characterization provides crucial information about the facility inlet conditions, which are necessary for determining CFD-scale boundary conditions. The experimental procedure involved the use of a fast camera (described in Section 2.5.1) with a resolution of 800 × 800 pixels and an acquisition frequency of 500 frames per second to characterise the bubbles immediately after injection. The camera was positioned in front of the window W1 (refer to Section 2.5.1), downstream of the gas injector, and illumination was provided by a light in the same direction as the camera, as shown in Figure 3 With the aim of producing a bubbly flow inside the facility, the dedicated gas injector type 1 (described in Section 2.2.1) was used to realize this experimental campaign. For the sake of clearness, tests T1 to T4 are presented in Table 3.1, even though they have already been discussed in Table 2.1 in Section 2.1. 3.1: Test conditions. In the table are reported: the test identification code, the air volumetric flow rate (Q air ) in L min -1 , the number and the type (described in Section 2.2) of tubes used in the gas injector to achieve the desired flow rate, the water flow rate ( ṁwater ) in g s -1 . J l and J g are the water and air gap superficial velocities according to Equation 2.1. U sl and U sg indicate the water and air superficial velocities in the homogenization section (free vein) according to Equation 2.2.

Test no. Q air No. of tubes & Q o air ṁwater J l J g U sl U sg L min -1 type of tubes L min -1 g s -1 m s -1 m s -1 m s -1 m s -1 T1 25 ±0.125 2
To characterize the bubble injections for the test series T1 to T4, we let the bubble injection from only one of the tubes composing the gas injector. This approach was chosen to reduce the number of bubbles in the section compared to a full gas injection, thus making the post-processing of the obtained images more efficient. The air flow rate for each test corresponds to the total mass flow rate divided by the number of tubes, i.e.:

• Test T1 and T2 (refer to Table 3.1): one tube with an air flow rate of 12.5 L min -1 (= 25 2 for T1 and = 50 4 for T2) was used for the injector characterization.

• Test T3 (refer to Table 3.1): one tube with an air flow rate of 14.3 L min -1 (= 100 7) was used for the injector characterization.

• Test T4 (refer to Table 3.1): one tube with an air flow rate of 28.6 L min -1 (= 200 7) was used for the injector characterization.

For the image post-processing and bubble identification it is important that the contrast between the bubbles and the image background stands out. One of the most common techniques used is the ombroscopy: the test section is positioned in between the camera and the light (in the order: camerasection-light). In our configuration it is not possible to make use of this technique as the section is only provided with one transparent face in this position. Therefore, as reported in Figure 3.1, the light is positioned in front of the window and the camera behind (in the order: camera-light-section), while the viewing window is parallel to the injection tube.

Image post-processing

The image post-processing was performed using the Matlab tool. The process of image analysis is schematized in Figure 3.2 and can be described in a few steps:

• The initial image, with a resolution of 800 × 800 pixels, is cropped using a window size of 135 × 145 pixels. This allows for the selection of a smaller region with more uniform lighting and simplifies the contrast adjustment procedure. The window of 135 × 145 pixels is then shifted, and the contrast is adapted accordingly.

• Next, the image is binarized to obtain a binary representation.

• Subsequently, the binarized image is inverted to identify the bubbles as "objects", and any holes within the objects, caused by light reflection, are filled.

• The image is then filtered, eliminating objects in contact with the borders and removing background noise.

• Finally, all objects are identified. However, an additional filtration process is applied to remove bubble clusters, as detailed in subsection 3.1.2.

Bubble identification

To discriminate bubbles from bubble clusters, a more sophisticated process has been implemented based on the concept introduced by Laupsien et al. [START_REF] Laupsien | Image processing for bubble morphology characteristics in diluted bubble swarms[END_REF]. The process relies on the "solidity" parameter, which measures the proportion of pixels in the smallest possible polygon (convex hull) enclosing the object that are also part of the object of interest. The concept of solidity is illustrated in Figure 3.3: for instance, Object "1" has a smaller solidity value compared to Object "2"." Therefore, based on the property of solidity, it can be assumed that all complex objects (bubble clusters) have a solidity value smaller than the objects of interest (real bubbles). To determine the solidity thresholds, a study on the shape of the bubbles is performed using the diagram from Clift et al. [START_REF] Clift | Bubbles, drops and praticles[END_REF], as shown in Figure 3.4. The Equivalent diameter (d eq ) is defined as:

d eq = 4A π (3.1)
Here, A represents the 2D area of the object. Furthermore, the Reynolds number (Re), Eötvös number (Eo), and Morton number (M ) are defined as follows:

Re = ρ l (v B -v l )d eq µ l (3.2) Eo = g∆ρd 2 eq σ (3.3) M = g∆ρµ 4 l ρ 2 l σ 3 (3.4)
Where ρ l and µ l denote the liquid density and viscosity, v B represents the bubble velocity, v l represents the liquid velocity, g denotes the gravitational acceleration, ∆ρ represents the difference in density between liquid and gas, and σ represents the surface tension. In order to calculate the Reynolds number the bubble velocity is known from fast camera measurement (as detailed in the following explanation), while the water velocity is estimated to be v l = ṁwater Sαρ l , where S represents the flow area and α denotes the homogeneous void fraction. To allow for bubble identification independent of its shape, the solidity threshold is used in combination with the object size. If the detected object has an Eötvös number Eo < 0.5, it is classified as a spherical bubble, requiring a solidity value (S) greater than 0.99 for accurate identification. For 0.5 < Eo < 30, the bubble is classified as "wobbling" due to its irregular shape, and a threshold of S > 0.9 is determined to be optimum. For Eo > 30, the bubble enters the spherical cap regime, and the adopted threshold is S > 0.95. The threshold values have been determined through analysis of different images, ensuring that all detected objects are reliably identified as bubbles.

To calculate the bubble velocity, two consecutive images are analyzed. The center of mass (CM) of the objects detected in the first image is compared with the CM of the second image. By knowing the distance traveled by the bubble and the image acquisition frequency in frames per second, the bubble velocity can be estimated. To avoid interference from objects detected only in one of the images, a maximum distance of 5 pixels between the two objects is allowed. An example of this procedure is illustrated in Figure 3.5, where the object "1" was not detected in the first image and therefore is not considered in the calculation of bubble velocity. Only the vertical direction is taken into consideration for velocity determination.

Results

The results for the analyzed quantities can be presented in terms of Probability Density Function (PDF). For a generic quantity X and its associated probability P (X), the average value can be calculated using the following equation:

X = ∑ X ⋅ P (X) ∑ P (X) (3.5)
To determine the appropriate number of images for acquisition during the image post-processing, we compare the PDFs of the equivalent diameter for different numbers of detected bubbles. An example is illustrated in Figure 3.6, where the PDF can be considered converged with approximately 2000 detected bubbles. The results of the experiments performed with one gas injection tube are reported in Figure 3.7. The results are presented in terms of PDF for bubble equivalent diameter (Equation 3.1), Eötvös number (Equation 3.3), bubble velocity and aspect ratio. In order to determine the bubble aspect ratio, the object detected are approximated to ellipses (an example is reported in Figure 3.8). Hence, the aspect ratio is defined as: The small air flow rate difference between the test performed with an air flow rate of 12.5 L min -1 and the test performed with an air flow rate of 14.3 L min -1 leads to a negligible difference in the PDF shape. However, the test performed with an air flow rate of 28.6 L min -1 shows a PDF shape distinct from the rest of the tests. Analyzing the PDF of the bubble equivalent diameter (refer to Figure 3.7a), it is evident how the bubble size moves towards smaller values when increasing the gas flow rate. This is due to the presence of small bubbles in the images. The discrepancy between the different gas flow rates in bubble formation can be appreciated in Figure 3.9. Hence, increasing the gas injection flow rate above a certain value, the gas momentum leads to the breakup of bubbles into smaller dispersed bubbles. The results are in line with by the Eötvös PDF and the aspect ratio PDF, which are shifted towards spherical bubbles. However, the increasing air mass flow rate leads to the formation of bubble clusters, which may be due to a gas jet formation at the holes of the injection tubes. These clusters are not identified by the image post-processing, resulting in an underestimation of the number of bubbles with larger equivalent diameters. In fact, the shape of the PDF 3.7a shows how the second peak (diameter around 6 mm) appears but is significantly smaller than the first peak (diameter around 3 mm). For each detected bubble, corresponding bubble velocity and bubble equivalent diameter values can be obtained. These values can be used to determine the Reynolds number of each bubble. To calculate the Reynolds number (Equation 3.2), the bubble velocity is determined through fast camera measurements, while the water velocity is estimated as v l = ṁwater Sαρ l , where S represents the flow area and α denotes the homogeneous void fraction. The results are presented in the form of PDF for relative velocity and Reynolds number in Figure 3.10. The low value of liquid velocity lead to a small difference between the relative velocity vBv l in Figure 3.10a and the bubble velocity vB in Figure 3.7d. With low liquid velocity, bubbles have a longer residence time within the column. This prolonged interaction allows bubbles to come into close proximity with each other. When bubbles collide, they may coalesce or merge together. In a low velocity liquid flow shear forces are weak, bubbles are less likely to undergo breakup, and instead, they tend to remain intact or coalesce with nearby bubbles. This may lead to the formation of large bubbles right after the gas injector that may be unstable due to their size, leading to breakup later along the section.

When the Reynolds number is calculated as mentioned, the plots may exhibit negative values, which can be attributed to bubbles moving slower than the water flow. As matter of fact, in regions of low liquid velocity within the bubble column, stagnant zones may develop. However, assuming a constant value for v l in our calculations may result in negative relative velocities. The PDF of the Reynolds number demonstrate that the test performed with an air flow rate of 28.6 L min -1 is shifted towards smaller values compared to the other tests. This shift is due to the fact that bubbles with smaller equivalent diameters are recognized and, as a result, exhibit lower velocity values.

The results in terms of a two-dimensional PDF (bubble velocity vs. equivalent diameter) are presented in Figure 3.11. It is observed that as the probability peaks of bubble diameter shift towards smaller values, the velocity follows the same trend. Consequently, smaller bubbles exhibit lower momentum compared to larger ones. With the data collected so far, we can refer to the bubble shape regime map from Clift et al. [START_REF] Clift | Bubbles, drops and praticles[END_REF] in Figure 3.4 and observe that the majority of the bubbles exhibit a wobbling shape, while some appear spherical. These findings are consistent with the flow visualization obtained from the fast camera, as shown in the previous images (Figure 3.9). The red dot in Figure 3.4 represents the averaged values (calculated as described in Equation 3.5) reported in Table 3.2. Table 3.2 presents the various results in terms of average values, along with their relation to the tests outlined in Table 3.1. Additionally, we have included the uncertainty associated with the measurement of the equivalent diameter and velocity, accounting for an error of 1 pixel in detecting the bubble contour. The values in parentheses reflect the underestimation error resulting from bubble clustering. Furthermore, Table 3.2 provides the Weber number, calculated according to Equation 3. [START_REF] Pettigrew | Vibration of a tube bundle in two-phase freon cross-flow[END_REF].

W e = ρ l (v B -v l ) 2 d eq σ (3.7)
The Weber number is an important parameter which provides insight into the dominant forces govern- ing the bubble deformation and helps classifying the bubble shape, motion, and interaction with the fluid. It helps in understanding the transition from surface tension-dominated behavior to inertial-dominated behavior and provides valuable insights into bubble dynamics in various fluid flow situations. When the Weber number is significantly larger than 1 (as in the present case), the inertial forces dominate over surface tension forces. In this regime, the bubble behavior is influenced primarily by the fluid motion and inertia. The bubble can experience deformation, fragmentation, or breakup due to the strong influence of fluid flow, resulting in non-spherical shapes and higher velocities. Therefore, the Weber number values confirm what we see visually in terms of bubble generation at the injector.

Test no. Ēo deq R vB v l vB -v l W e F r [-] [mm] [-] [m s -1 ] [m s -1 ] [m s -1 ] [-] [-] T1
The critical Weber number for bubble breakup (W e cr ) represents the threshold value of the Weber number at which a bubble is most likely to break into smaller bubbles. Above the critical value, inertial forces become more significant, causing the bubble to break into smaller fragments. For air bubbles rising in water, the critical Weber number typically falls within the range 1 < W e cr < 12 [START_REF] Hartunian | On the instability of small gas bubbles moving uniformly in various liquids[END_REF][START_REF] Martinez-Bazan | On the breakup of an air bubble injected into a fully developed turbulent flow. part 2. size pdf of the resulting daughter bubbles[END_REF][START_REF] Jain | Flow-induced breakup of drops and bubbles[END_REF]. Hence, considering that the values of W e reported in Table 3.2 fall within the range of W e cr , it is expected that fragmentation will occur before the bubbles reach the tube bundle.

Additionally, the Froude number is reported in Table 3.2, defined according to Equation 3.8. The Froude number for bubbles is defined as the ratio of the inertial forces and gravitational forces acting on the bubbles. When the Froude number for bubbles is greater than 1, it implies that the inertial forces are more significant than gravity. High Froude number can result in non-spherical deformations of bubbles. Instead of maintaining a perfect spherical shape, bubbles may elongate or deform due to the influence of surrounding shear and inertial forces.

F r = ρ l (v B -v l ) 2 ∆ρgd eq (3.8)
Finally, the obtained bubble equivalent diameter results can be compared with the correlations for bubble diameter prediction presented in Table 2.2 in Chapter 2. The comparison, shown in Figure 3.12, reveals a lack of agreement between the experimental data and the correlations developed by Gaddis et al. [START_REF] Gaddis | Bubble formation in quiescent liquids under constant flow conditions[END_REF], Muilwijk et al. [START_REF] Muilwijk | The limerick bubbly flow rig: Design, performance, hold-up and mixing pattern[END_REF], and Iguchi et al. [START_REF] Iguchi | Bubble and liquid flow characteristics in a vertical bubbling jet[END_REF]. This discrepancy can be attributed to the fact that these authors developed their correlations based on conditions significantly different from the present study. Specifically, their investigations focused on gas injection through a single orifice, and most of them did not consider liquid flow conditions (stagnant water) except for Muilwijk et al. [START_REF] Muilwijk | The limerick bubbly flow rig: Design, performance, hold-up and mixing pattern[END_REF], who conducted experiments with co-current liquid flow. However, the correlation from Muilwijk et al. [START_REF] Muilwijk | The limerick bubbly flow rig: Design, performance, hold-up and mixing pattern[END_REF], possibly because was originally designed for bubble detachment near the injection point, does not match the experimental results.

Among the authors, the correlation proposed by Akita et al. [START_REF] Akita | Bubble size, interfacial area, and liquid-phase mass transfer coefficient in bubble columns[END_REF], which was developed for conditions more similar to the present experiments, provides the best approximations of the experimental data. In fact, the correlation introduced by Akita et al. [START_REF] Akita | Bubble size, interfacial area, and liquid-phase mass transfer coefficient in bubble columns[END_REF] is specifically designed to determine the average bubble size at locations distant from the injection points. Additionally, the experimental conditions involved test sections with equivalent diameters up to 30 mm (29 mm in the present study) and gas superficial velocities (U sg ) up to 0.042 m s -1 (U sg up to 0.051 m s -1 in the present study). However, the correlation proposed by Akita et al. [START_REF] Akita | Bubble size, interfacial area, and liquid-phase mass transfer coefficient in bubble columns[END_REF], does not always adequately match the experimental results, probably due to the fact that it was developed for a bubble swarm generated by a gas (air) sparger in stagnant water. In fact, it is the only correlation where the gas velocity is not proportional to the bubble diameter, since a larger gas velocity lead to bubbles breakup in a bubble column. However, since the correlation is semi-empirical and their experiments primarily focused on small void fractions, it did not account for scenarios with larger void fractions that could result in bubble coalescence. Tate's Law (Equation 2.3) does not provide a good approximation of our experimental data. This empirical law, which typically balances buoyancy and surface tension forces, is primarily influenced by fluid properties and orifice diameters. However, as highlighted by the values of F r and W e in Table 3.2, inertial forces largely overcome buoyancy forces (F r > 1) and surface tension forces (W e >> 1). This implies a significant contribution of inertial forces to bubble diameter, which could explain why Tate's Law underestimates the observed bubble diameters. Nonetheless, for design purposes, the examined correlations can provide a useful range of expected bubble diameters.

In conclusion, the bubble detachment mechanisms is influenced by many factors, the gas flow rate and liquid velocity relationship is important for this mechanism. As the gas flow rate increases, more gas is entrained into the liquid, leading to the formation of larger and more frequent bubbles. At higher liquid velocities, smaller bubbles are more likely to detach from the orifice. This is because the higher liquid velocity generates stronger shear forces, which can break smaller bubbles away from the orifice more easily. Conversely, at lower liquid velocities as in the present application (refer to Table 3.2 for the liquid and relative velocity), larger bubbles may form and detach due to reduced shear forces. In fact, at low liquid velocities, bubbles may detach primarily due to buoyancy, with larger bubbles being more stable and less affected by shear forces, while at higher liquid velocities, bubbles are more likely to detach due to shear-induced forces. The shear forces from the flowing liquid can overcome buoyancy, causing bubbles to detach. The liquid velocity also affects the rise behavior of bubbles. At higher liquid velocities, bubbles rise more rapidly through the liquid, potentially leading to quicker detachment. In contrast, at lower liquid velocities, bubbles may rise more slowly, and some of them might not detach immediately, leading to the formation of bubble plumes or clusters.

The comprehensive analysis of bubble generation presented in this section offers valuable insights into how our air injection system produces bubbles in the bubbly flow regime. These information can be utilized by two-phase CFD codes to establish appropriate boundary conditions for simulating the current experimental campaign. Once the gas injection was characterized and inlet conditions defined, we can move to the investigation of the two-phase flow inside the tube bundle.

Tube bundle two-phase flow characterization

This section presents the experimental analysis of the two-phase flow upstream, downstream, and inside the tube bundle. The analysis was conducted using a fast camera and a dual-tip optical probe. Table 3.1 provides information about the flow rates, number of tubes in the gas injector, type of gas injector, and superficial velocities used in the experiments. Figure 3.13 illustrates the different positions for image acquisition, while Figure 3.14 shows the various probe positions. For further details regarding the instrumentation and their placement in the facility, the reader can refer to Section 2.5.1 (for information on the fast camera) and Section 2.5.2 (for details about the dual-tip optical probe).

Tube bundle inlet conditions

Measurements of bubble equivalent diameter in the visualization window upstream of the tube bundle (W2 in Figure 3.13a) were performed using the fast camera and the methodology outlined in Section 3.1.1. However, due to the high void fraction and widespread dispersion of bubbles across the entire cross section, bubble identification using this method proved to be impossible. As a result, a Matlab routine was developed to manually select and consider specific objects as "actual bubbles" for further analysis. The results are reported in terms of PDF of bubble equivalent diameter, Eötvös number and bubble aspect ratio in Figure 3.15. This methodology do not provide velocity measurement.

The PDF of bubble diameters (Figure 3.15a) and the Eötvös number (Figure 3.15b) suggest that the bubbles still have a wobbling shape for the different tests. However the PDF of the aspect ratio (Figure 3.15c), suggests that tests T1 and T2 have bubbles with larger aspect ratio with respect to tests T3 and T4. Using the average value of the bubble equivalent diameter (calculated with Equation 3.5), these results can be compared with the measurements performed downstream of the gas injector in Section 3.1.3: we can conclude that the bubble equivalent diameter tends to decrease as the bubbles approach the tube bundle inlet as it can be seen in Figure 3.16. This result agrees with the observations made in the previous section about the Weber number: the bubble are generated with unstable shape and size, therefore, along their path in the homogenization section, the break-up mechanism prevails.

Additionally, the optical probe placed upstream of the bundle (point 5 in Figure 3.14) was used to measure the tube bundle inlet conditions. The bubble size distribution obtained through the optical probe (Section 2.5.2) can be compared with the probability density function (PDF) of the equivalent diameter obtained using a fast camera. The comparison, illustrated in Figure 3.17, presents both distributions in a normalized form. However, it is important to note that a different scale must be employed to compare the two measurements due to the different numbers of classes used for plotting the PDF. The results show a narrow PDF for both the measurements systems for test T1 and T2. However, in test T3 and T4 the optical probe starts to record bubble size larger than 15 mm. This discrepancy can be attributed to the measurement locations of the two systems. The optical probe is placed in the center of the section (300 mm × 216 mm of cross section), while the fast camera can only detect bubbles near the wall. This difference in measurement locations can impact the comparison of results, particularly because large gas structures can appear in the middle of the two-phase flow during test T3 and T4.

This hypothesis can be confirmed by the comparison in terms of bubble average equivalent diameter obtained via fast camera and sauter mean diameter obtained via optical probe presented in Figure 3.18 for different tests. The comparison demonstrates a satisfactory agreement between the two measurement systems. However, it should be noted that test T4 (air flow rate Q air =200 L min -1 ) shows a significant bias. The image post-processing technique adopted is not able to record large gas structures that cannot be classified as dispersed bubbles. Hence, as the gas flow rate increases (from T1 to T4 Q air from 25 L min -1 to 200 L min -1 ), the formation of large structures also increases. Consequently, the difference between the two measurement systems in bubble size increases. This can also confirm the occurrence of large gas structures in the middle of the two-phase flow during test T4 that are not detected by the fast camera but captured by the probe (in the middle of the flow). However, post-processing of the optical probe signal could also be questionable (the two different methods in post-processing the bubble diameter and the sauter mean diameter are described in Section 2.5.2). The quantitative data obtained from the fast camera and optical probe can be supported by visual observations of the two-phase flow upstream of the tube bundle. Figure 3.19 illustrates the formation of significant gas structures that may not be detected through image post-processing but can be identified by the optical dual-tip probe.

A summary of the comparison of the two measurement systems is presented in data about void fraction and gas velocity collected with the dual-tip optical probe are reported. The "Probe Bubble size" is the average value of the bubble size distribution reported in Figure 3.17 calculated according to Equation 3.5. The evaluation of uncertainty for the values measured with the optical dual-tip probe are calculated according to Cartellier [START_REF] Cartellier | Optical probes for local void fraction measurements: characterization of performance[END_REF] and Bachar et al. [START_REF] Bachar | Breaking wave bubble measurements around ship model by optical probe[END_REF].

The void fraction measured by the probe at the tube bundle inlet serves as a crucial parameter for establishing CFD-grade boundary conditions. In fact, estimating the void fraction using the Homogeneous Equilibrium Model (HEM) can lead to significant errors, as illustrated in Figure 3.20. The figure presents a comparison between the measured void fraction and the one estimated using the HEM model. The discrepancy between the measured values and the calculated values may be attributed to the HEM model's assumption of no slip between the phases, where gas and liquid velocities are assumed to be equal (v g = v l ). In experiments conducted under water-air two-phase flow, this assumption is found to be overly restrictive for the present experimental conditions. both the superficial velocity and the void fraction is presented in Figure 3.21. The actual gas velocity can be estimated using the superficial velocity U sg , either with the void fraction measured by the probe (U sg α) or with the void fraction obtained from the HEM (U sg α HEM ). The values of U sg α HEM do not agree with the experimental data, which is not surprising, considering that the HEM model, as shown in Figure 3.20, is not suitable for estimating void fractions under these conditions. However, the gas velocity estimated by U sg α performs well for tests T1 and T2 (25 L min -1 and 50 L min -1 ), but it does not align with the experimental data for tests T3 and T4 (100 L min -1 and 200 L min -1 ). It's important to note that superficial velocity provides an approximation of gas behavior within a multiphase system, assuming uniform distribution. This assumption can lead to inaccuracies in scenarios where phases are not evenly distributed. Therefore, the behavior of U sg α in Figure 3.21 may be attributed to the non-uniform gas velocity and void fraction profiles resulting from the appearance of large gas structures in tests T3 and T4, while tests T1 and T2 exhibit more uniform bubbly flow across the cross-section.

A thorough analysis of the inlet conditions yields CFD-grade boundary conditions for the tube bundle. Following a detailed examination of the two-phase flow upstream of the tube bundle, we can then proceed with characterizing the two-phase flow within the bundle.

Two-phase flow characterization inside the tube bundle

In this section, we analyze the flow evolution within the tube bundle and discuss the impact of the inlet conditions examined in the previous section.

In ) is a bi-stable behavior exhibited by the two-phase flow. This phenomenon was previously observed experimentally in single phase cross-flow in the literature by Olinto et al. [START_REF] Olinto | Experimental study of the bistable flow in tube arrays[END_REF][START_REF] Olinto | Experimental study of the characteristics of the flow in the first rows of tube banks[END_REF] and Keogh et al. [START_REF] Keogh | Bi-stable flow in parallel triangular tube arrays with a pitch-todiameter ratio of 1.375[END_REF]. In our case, this phenomenon is evident when tracking the paths of the bubbles. The bubble flow tends to follow a preferential direction within the tube bundle; however, any disturbance in the flow, such as the passage of an air plug through the bundle, can alter this preferential direction. In order to realize a more quantitative analysis of the two-phase flow in the tube bundle, measurements with the double-tip optical probe were performed. The measurement points reported in Figure 3.14, allow us to plot horizontal and vertical profiles for bubble mean sauter diameter, bubble velocity and void fraction in the tube bundle. The acquisition time (1000 seconds) for the measurements was established in such a way that those values are statistically converged.

The results reported in Figure 3.25 show the horizontal profiles in the tube bundle for the 4 different flow configurations (Test T1 to T4). Bubble diameter and bubble velocity profiles keep increasing as the air flow rate increases from test T1 to T4. The same behavior is reported for the void fraction. Since the water flow rate remains constant, an increase in gas flow rate clearly leads to an increase in these parameters. However, profiles downstream and upstream of the gap between the central tube and the one to its right (green and red respectively) in Figures 3.25j and 3.25n feature in an asymmetrical behavior for the void fraction, that can be related to the bi-stable nature of the flow (as already observed in the analysis of the fast camera images in Figure 3.24). It is interesting to notice that, even if the measurements have been performed in different moments and conditions, the void fraction profiles downstream and upstream of the gap (green and red respectively) always tend to be tilted to the left (the highest value is recorded on the left). This suggests a favorite leftward equilibrium position in the flow over a 1000-second acquisition period. This results cannot be confirmed by the velocity and bubble diameter profiles, since the uncertainty in the measurements are larger than the difference between two consecutive values. To confirm the assumptions made, additional measurement points within the gap could potentially provide more evidence for the asymmetry observed in the horizontal profile of the gap void fraction. Vertical profiles for bubble sauter mean diameter, bubble average velocity and average void fraction are reported in Figure 3.26. The profiles show how the different variables tend to increase in between two tubes and to decrease once the measurement point is shifted outside the gap between two tubes. Concerning the gas velocity, for test T3 and T4 this behavior is in line with the theory, since the velocity increases when the flow area reduces. For test T1 and T2 the gas velocity is about constant (the variation is not significant compared to the uncertainty range). This may be attributed to the fact that for tests T1 and T2, the bubble population is relatively small, which may not significantly be influenced by the flow constriction effect caused by the gaps between the tubes. In fact, when we consider a single bubble within the gap, its vertical motion remains unaffected by the presence of the tubes. However, as the number of bubbles increases, the reduction in available space starts to have a more pronounced impact on their velocity field. On the other hand, the trend observed in the void fraction suggests a possible correlation with a geometrical effect. The increase in void fraction within the gap can be attributed to the reduction in the cross-sectional area of the flow passage. This difference in gas velocity and void fraction behavior can be attributed to the methods used to calculate these quantities by the probe (α = T g (T l + T g ) and v g = d T f light , as reported in Section 2.5.2). In fact, the void fraction is directly related to the number of bubbles that impact the probe. Since the sauter diameter is a quantity derived from the void fraction, the results of the void fraction profiles are reflected on the diameter profiles. Furthermore, it's observed that the void fraction, bubble Sauter mean diameter, and gas velocity at the tube bundle inlet are smaller than those inside the tube bundle. This may be attributed to the geometrical effects of the tube bundle on these quantities. As mentioned in the literature review (Section 1.2), Feenstra et al. [START_REF] Feenstra | Modelling two-phase flow-excited damping and fluidelastic instability in tube arrays[END_REF] developed a model to calculate the void fraction in tube bundles. The authors found agreement with experimental data in Freon twophase flow. Comparing experimental data from the present work with the model from Feenstra et al. [START_REF] Feenstra | Modelling two-phase flow-excited damping and fluidelastic instability in tube arrays[END_REF] (Figure 3.27), we can conclude that, for the bubbly flow regime experimental campaign, the model agrees with the experimental data.

The evolution of the bubble size distribution within the tube bundle is illustrated in Figure 3.28. In the case of tests T1 and T2 (Figures 3.28b and 3.28c), the size distributions exhibit a similar trend to the Sauter diameter in Figure 3.26c. Specifically, the measurement point at the tube bundle inlet registers smaller values compared to those inside the bundle, likely due to bubble coalescence within the tube bundle, a phenomenon influenced by its geometry. Whereas within the bundle the size increases inside the gap while decreasing outside the gap. In tests T3 and T4 (Figures 3.28e and 3.28f), this behavior is not clearly depicted by the PDFs. However, it should be noted that the uncertainty in the measurements tends to increase as the void fraction rises and the flow direction becomes more chaotic [START_REF] Kiambi | Measurements of bubble characteristics: Comparison between double optical probe and imaging[END_REF].

Tube bundle outlet conditions

To comprehensively analyze the two-phase bubbly flow, we can now proceed to examine the outlet of the tube bundle. The images presented in Figure 3.29, show how the difference between the inlet and the outlet of the tube bundle increases with the air flow rate. Specifically, Figures 3.29c Eötvös number and bubble aspect ratio in Figure 3.30. The results show no substantial difference among the different tests. However, the chaotic nature of the flow at the tube bundle outlet leads to large uncertainty in the bubble detection methodology for tests T3 and T4. Additionally, when comparing the bubble equivalent diameter at the tube bundle inlet and outlet, no significant difference is observed (Figure 3.31). It should be noted that the measurement position at the tube bundle outlet is located in the visualization window W4, located approximately 110 mm above the last tube of the bundle. Based on this observation, we can conclude that once the bubbles exit the tube bundle, they tend to regain the original shape and size they had at the inlet of the bundle.

Tube vibration

In order to investigate the tube vibration in TREFLE, the central tube of the bundle is replaced with a flexible tube (Figure 3. The results are presented in Figure 3.34. The spectra reported were calculated using the accelerometer output recorded over a duration of 1000 seconds. Additionally, tests were performed under "free vibration" conditions, i.e., in static water and air, to observe the natural vibrating modes of the tube (exciting the tube manually from outside the facility). In parallel, finite element analysis (FEA), detailed in Section 2.4.1, was conducted to understand the modes of vibration of the tube.

Analyzing the tube spectra obtained from various tests (Figure 3.34a), distinct frequency peaks can be observed. Proceeding in ascending order of frequency, we aim to assign meanings to these peaks. The first peak appears at approximately 10 Hz for all the investigated air flow rates. This peak corresponds to the first mode of vibration of the tube (i.e., the tube natural frequency), in the lift direction, along the x axis (coordinates shown in Figure 3.32). The second peak, occurring at 48 Hz in Figure 3.34a, is not present in the free vibration tests (Figure 3.34b) as it is associated with the rotational speed of the water pump (2890 rpm = 48 Hz). The second mode of vibration (109 Hz in Figure 3.34b) does not appear during the experiments in two-phase flow. This second mode is a translation along the y direction. It is found that this mode is stressed by forces acting along the y direction. Even though The accelerometer is positioned to record displacements along the x direction, a small angle in the accelerometer position allows the recording of this mode in free vibration. However, it is overshadowed by the background ). The discrepancy between the recorded frequency can be related to the hydrodynamic mass effect. This effect also influences the first mode of vibration at approximately 10 Hz. In order to confirm our theory, we can calculate the hydrodynamic mass using the formula provided by Carlucci et al. [START_REF] Carlucci | Experimental studies of damping and hydrodynamic mass of a cylinder in confined two-phase flow[END_REF] derived from the relative change in tube natural frequency:

m h = m s f a f T P 2 -1 (3.9)
Where m s is the structural mass of the vibrating cylinder per unit of length, f a and f T P are the frequency in air and two-phase mixture. The calculated hydrodynamic mass was compared with the formula proposed by Pettigrew et al. [START_REF] Pettigrew | Vibration of tube bundles in two-phase cross-flow: Part 1-hydrodynamic mass and damping[END_REF]:

m h = ρπd 2 4
(D e d) 2 + 1

(D e d) 2 -1 (3.10)
Where ρ = ρ l (1α HEM ) + ρ g α HEM is the two-phase density, α HEM the void fraction calculated using the homogeneous model (HEM) and D e d = (1.07 + 0.56p d) p d. The comparison between the calculated hydrodynamic mass (Equation 3.9) and the results from the theory (semi-empirical correlation, Equation 3.10) are reported in Figure 3.35a. The results are presented in terms of hydrodynamic mass ratio i.e. the ratio between m h and the hydrodynamic mass in liquid (m h for α HEM = 0). Being the semiempirical correlation from Equation 3.10 derived from experiments performed without any measuraments on the void fraction, the HEM model is employed by Pettigrew et al. [START_REF] Pettigrew | Vibration of tube bundles in two-phase cross-flow: Part 1-hydrodynamic mass and damping[END_REF]. The hydrodynamic mass was calculated using the frequencies of the first mode of vibration (f 1) for the different tests (around 10 Hz in Figure 3.34a) and the third mode of vibration (f 3) for the different tests (around 188 Hz in Figure 3.34a). The results agree with the semi-empirical correlation proposed by Pettigrew et al. [START_REF] Pettigrew | Vibration of tube bundles in two-phase cross-flow: Part 1-hydrodynamic mass and damping[END_REF]. However, for void fractions exceeding 0.4, discrepancies between the experimental data and Equation 3.10 begin to emerge. Pettigrew et al. [START_REF] Pettigrew | Vibration of tube bundles in two-phase cross-flow: Part 1-hydrodynamic mass and damping[END_REF] conducted a comparison between Equation 3.10 and experimental data in air-water cross flow, showing that discrepancies become more pronounced, particularly for square bundle configurations, when void fractions exceed approximately 0.45. The difference between the results with f 1 and f 3 is interesting to analyze. Equation 3.10 from Pettigrew et al. [START_REF] Pettigrew | Vibration of tube bundles in two-phase cross-flow: Part 1-hydrodynamic mass and damping[END_REF] does not take into account the mode of vibration of the tube and its natural frequency. However, different modes of vibration can have varying degrees of interaction with the fluid, resulting in different hydrodynamic mass contributions. In this case, mode corresponding to f 1 experiences a more pronounced hydrodynamic mass effect compared to mode corresponding to f 3. This can occur due to the mode alignment with the fluid flow direction, vibration direction or its sensitivity to flow-induced forces.

Additionally, in Figure 3.35b, the same comparison as in Figure 3.35a is presented, but this time using the measured values of void fraction. The void fraction values used to calculate the hydrodynamic mass in Equation 3.10 represent an average obtained from measurements within the tube bundle. The results reveal that the model fails to accurately predict the hydrodynamic mass when actual void fraction values are considered. This outcome is not unexpected, as the semi-empirical correlation in Equation 3.10 was originally developed by Pettigrew et al. [START_REF] Pettigrew | Vibration of tube bundles in two-phase cross-flow: Part 1-hydrodynamic mass and damping[END_REF] based on the HEM model. In Figure 3.34a, an additional peak emerges at around 250 Hz exclusively for tests T1 and T2. This peak is also observed in the accelerometer data obtained from the facility, indicating that it represents a displacement of the entire facility. Interestingly, the peak is only present in tests with lower void fractions, as the vibration of the structure gains intensity with an increased presence of water (resulting in a greater mass). This change in mass affects the natural frequency of the facility, potentially bringing it closer to the frequencies of the forces applied to the facility. A similar observation can be made for the last peak detected at around 359 Hz in test T1.

Tube vibrations in two-phase flow were also recorded using a fast camera. By employing image post-processing techniques, we were able to track and record the tube displacement. Subsequently, the results obtained from the fast camera recordings were compared to the measurements obtained using the accelerometer. The first mode around 10 Hz has been found with the fast camera, too. However, being limited by the image acquisition frequency because of memory issues, the remaining higher modes could not be acquired. On the other hand, being the fast camera not influenced by the facility structure or the pump rotational speed, the resulting signal is "clearer" than the one obtained by accelerometers. Nevertheless, the tube displacement recording is limited by the resolution of the image, i.e. if the displacement is less than 1 pixel it is not recorded by the camera. Moreover, the tube displacement recording must account for an uncertainty of 1 pixel in the image post-processing. In the present case, 1 pixel corresponds to 1 17 mm. The comparison in terms of RMS (root mean square) of the tube displacement between the signal from the accelerometer (integrated twice) and the signal from image post-processing is reported in Figure 3.36. Despite the fact that the camera can be immune from disturbances being outside the facility, its limitations are evident as the higher modes are not detected. The results reported in Figure 3.36 show that the RMS increases between test T3 and test T4. RMS for test T4 increases by 190% of RMS of test T3, while the maximum increase between tests T1 to T3 is of 85% between T1 and T2.

The study done in Section 3.2 can help us to better understand the link between tube vibration behavior and two-phase flow. More information can be extracted from the accelerometer signal to further investigate the tube vibration behavior. Figure 3.37 reports the PDF of the accelerometer signal. The probability density function (PDF) of a signal describes the probability of the signal values falling within different intervals. In other words, it gives a mathematical representation of the likelihood of various acceleration values occurring in the signal. The shape of the PDF suggests that test T4 is different from tests T1 to T3. It can be deduced that the peculiar behavior of test T4 reported in Section 3.2 (such as sporadically presence of large gas structures and flow bi-stability) has an influence on the tube vibration. 3.1.

Therefore, from Figure 3.38a it can be deducted that test T4 increases its vibration amplitude in non-linear way compared to test T1, T2 and T3. Hence, this could be an indication of flow regime transition and it explains the effect of air plug traversing the tube bundle. However, since the kurtosis can be positive even for symmetric signals, the skewness comes into play to analyze the asymmetries in the acceleration signals. Hence, from Figure 3.38b, it can be understood the effect noted in Section 3.2.2 where the void fraction was found to have an asymmetric profile between the tubes. This effect could play a role on the symmetry of the tube vibration and thus allowing the tube to have larger displacements in one side rather than another. It could be assumed that due to some effect related to the configuration of the facility the bi-stability of the flow tends in one direction rather than another. However, these hypotheses require more data and tests to be confirmed.

Conclusion

In this particular chapter, we detailed and reported the comprehensive results from an extensive experimental analysis that was conducted on the bubbly flow regime within the TREFLE facility.

The study of the two-phase flow starts from the inlet conditions of the facility: a comprehensive analysis of the bubbles generated at the gas injector has been performed by means of fast camera. The bubbles produced at the injector are unstable, i.e. their shape and size are such that break-up is enhanced. Hence, the bubble equivalent diameter at the tube bundle inlet is smaller than the bubble diameter measured right after the bubble generation.

The two-phase flow in the tube bundle has been investigated by means of fast camera and optic dual tip probe. From qualitative analysis of fast camera images significant findings have been reported about the bi-stable behavior of the flow. As previously described in literature in single-phase, the bi-stability is also present in two-phase, this is illustrated by qualitative measurements. The optical probe provides information about the gas velocity, bubble sauter diameter and void fraction evolution in the bundle. The role of the gap on the two-phase behavior is illustrated, and give an interesting highlight on its effect on gas structures. The probe, positioned upstream the tube bundle provides information about the inlet conditions, necessary to have CFD scale boundary conditions on the present setup.

Once the two-phase flow is characterized, the central tube is substituted with a vibrating tube. The vibration response of the tube has been investigated with an accelerometer and the fast camera. Interesting findings have been reported for the third mode of vibration that can change its frequency depending on the void fraction of the two-phase flow. This behavior is related to the hydrodynamic mass effect, its value has been calculated and compared with the theory with satisfactory agreement. An excessive increase of the tube vibration amplitude has been found for the test performed with the highest air mass flow rate. This behavior can be related to the flow regime onset of transition in the tube bundle as pointed out from the PDF of accelerometer signal. Regarding spectra, it appears that the two-phase behavior on vibration is highlighted at the natural frequency of the tube. For low void fraction, the peak is narrow and similar to a single phase flow one. By increasing the void fraction, its shape is larger, it illustrates that the two-phase flow has a wider range of gas structure size.

Additional insights into the flow regime transition and its impact on the tube's vibration behavior will be presented in the upcoming chapter. This chapter will explore scenarios involving higher air flow rates and void fractions.

Chapter 4

Intermittent flow regime: experimental investigation

In this chapter, the findings and results of the experimental campaign investigating the intermittent flow regime in the tube bundle are presented. A comparison is made between the experimental data obtained in this campaign and the data collected from the previous bubbly flow experimental campaign. Additionally, a flow regime map is proposed based on the analysis of the two-phase flow. The test matrix for the analysis of the intermittent flow regime is reported in Table 4 In order to characterize the two-phase flow, the measurements systems used are: the fast camera (Section 2.5.1), the dual tip optical probe (the measurement points were from 1 to 5, 8 and 9 according to Section 2.5.2) and the wire mesh sensor (Section 2.5.3). The instrumentation position adopted in this experimental campaign is reported in In this chapter, the two-phase flow in the free duct (i.e., the homogenization section) is analyzed to study the inlet conditions for the tube bundle in Section 4.1. The experimental results are compared to those found in the literature, focusing on drift flux correlations developed under such conditions. Section 4.2 is dedicated to the investigation of the behavior of the two-phase flow within the bundle, establishing connections between the findings and the previously explored inlet conditions. Subsequently, Section 4.3 presents an experimental campaign aimed at developing a new flow regime map for the tube bundle. This section includes a comparison of the results and methodologies employed with those available in existing literature. Finally, in Section 4.4, the study explores the tube vibration response, examining the influence of the flow regime on the tube vibration behavior.

Two-phase flow characterization: tube bundle inlet

In order to investigate the two-phase flow upstream of the tube bundle, videos were recorded using a fast camera. The two-phase flow is highly chaotic, and the continuous generation of large gas structures makes the bubble detection routine in Matlab inefficient. As a result, the method described in Section 3.1.1 cannot be utilized. Nevertheless, a qualitative analysis of the flow can still be conducted.

In Figure 4.2, photos taken upstream of the tube bundles are presented. The images predominantly show dispersed bubbles on the wall, occasionally alternating with larger gas structures. However, upon video analysis, it becomes evident that the occurrence of large gas structures is primarily concentrated in the core of the two-phase flow. Due to the considerable size of the section (300 mm × 216 mm), it is not feasible to solely rely on analyzing external images for investigating this type of regime in the twophase flow. Nevertheless, during the analysis of the videos recorded by the fast camera, an interesting phenomenon was observed: the downward motion of the bubbles closer to the wall. In fact, as the air flow rate increases the difference in velocity between the gas and liquid phases increases (as indicated in Table 4.1 where U sg is much larger than U sl ). When the air velocity is much higher than the water velocity, the bubbles experience a net upward drag force due to the air flow. This upward drag force tends to push the bubbles upwards in the central region of the duct, where the air velocity is highest. As the bubbles approach the duct wall, they enter a region known as the boundary layer. The boundary layer is a thin layer of fluid near the wall where the velocity of the fluid increases gradually from zero at the wall to the free-stream velocity. In this region, the water velocity is relatively lower compared to the central region of the duct. Because of the lower water velocity near the wall and the upward drag force from the fast-moving air, the bubbles experience a net upward motion in the central part of the duct. However, as the bubbles approach the boundary layer, they encounter a downward pressure gradient. This pressure gradient is a result of the decreasing water velocity near the wall. In the central part of the duct, the dominant force is the upward drag force from the fast air flow. Near the wall, the dominant force becomes the downward pressure gradient caused by the slowing water flow. When these forces balance, the bubbles appear to move downward or remain relatively stagnant near the wall. Additionally, interactions between bubbles and the complex turbulence patterns near the wall can further influence their motion.

A more detailed investigation can be obtained by analyzing the data obtained from the wire mesh sensor (WMS). The WMS was positioned about 260 mm below the tube bundle in order to avoid any disturbance due to the presence of the bundle. In order to select the acquisition frequency and time, several tests were conducted. By obtaining the results from the optic probe positioned upstream of the tube bundle, we can evaluate the average gas transition time (g t ) by dividing the sauter mean diameter (d sm calculated as reported in Section 2.5.2) by the average gas velocity (v g calculated as reported in Section 2.5.2) as reported in Equation 4.1.

g t = d sm v g (4.1)
By multiplying the gas transition time by the acquisition frequency (f a ), we can estimate how many times one gas structure (of average size) is recorded by the WMS along its path (Equation 4.2).

n B = g t ⋅ f a (4.2)
For test T7, with a frequency of 1000 Hz, a general gas structure is recorded 10 times on average in one second of acquisition. This frequency was determined to be a suitable compromise between result accuracy and storage memory usage. The time duration of acquisition was set to be in the same order of magnitude as the optical probe acquisitions, specifically 800 seconds.

Valuable insight into the flow structure can be obtained by analyzing the instantaneous two-dimensional (2D) void fraction distribution. In order to have an idea on how the flow structure evolves during the experiments, virtual cross-sectional side-views can be extracted. These side-views are derived by extracting the distribution of void fractions across the cross-section of the duct from the 2D distribution, and then arranging time series of these distributions as lines in a vertical sequence. In this case, the z-axis represents the time scale. However, the time axis can be converted into a virtual z * -axis by scaling it with the average velocity of the gas phase derived from the average (in space and time) void fraction ( ᾱ) as reported in Equation 4.3.

z * = t ⋅ U sg ᾱ (4.3)
In Figures 4.3, 4.4, and 4.5, snapshots of the 2D void fraction distribution obtained from the WMS and their corresponding side-views are presented. In these cases, a time scale of 1 second (equivalent to 1000 frames) was used for tests T5, T6, and T7. However, the z * -axis varies among the different tests because Equation 4.3 depends on U sg and ᾱ, which change depending on the specific test. From a qualitative analysis, the main difference among the different tests (T5 in Figure 4.3, T6 in Figure 4.4, and T7 in Figure 4.5) is the increased amount of the dispersed phase and the large gas structures within the images. Such plotting provides an estimation of the size of the large gas structures within the flow. However, based on qualitative analysis alone, we cannot definitively conclude that the number of large gas structures increases from test T5 to T7. The impression conveyed by the image might be influenced by the fact that although the time scale remains the same, the varying gas velocities result in different z * scales. Therefore, it is still plausible that the number and frequency of large gas structures are of a similar magnitude. From the images in Figures 4.3, 4.4, and 4.5, the two-phase flow morphology appears to be characterized by a churn-like motion of the flow with large gas structures or slugs within it. The flow exhibits a chaotic, churning behavior, and the large gas structures appear irregularly distributed. From the image analysis, it is clear that large gas structures frequently arise in the central part of the flow. The bubbles and large gas structures can be isolated by plotting an isosurface using a void fraction threshold of 30% (refer to Figure 4.6). This method allows for the visualization of a three-dimensional distribution of the gas phase by removing the dispersed phase detected with a void fraction less than 30%. In the z-axis is reported the number of frame, while the x and y-axis are reported as the number of wires (56×78).

Figure 4.6 shows how the void fraction increases from test T5 to T7. The amount of smaller bubbles appears to increases from test T5 to T7. The gas bubbles and larger structures are irregularly distributed along the z-axis (frame number, in this case). However, when examining the x-y cross-section, the larger gas structures appear to be concentrated in the center of the section, with smaller bubbles surrounding them. A more quantitative analysis can confirm these results. The void fraction at each point (x, y) in the WMS grid is averaged over a period of 800 seconds. Consequently, it is evident from the tests that the presence of large gas structures in the center of the flow significantly impacts the distribution of void fraction, resulting in a pronounced peak in the central region and a decrease in the near-wall zone. This behavior is clear also from the radial distribution of the void fraction (ᾱ(r) calculated according Section 2.5.3 from WMS measurements) reported in Figure 4.8. Based on the information gathered so far, it is evident that relying solely on a fast camera, on such large test section, may have limitations in extracting comprehensive information about the two-phase flow and performing a qualitative analysis of the flow regime. These limitations arise from the presence of a large channel, where a significant amount of information within the center of the two-phase flow remains concealed and inaccessible to the fast camera.

Analyzing the space-averaged void fraction provides valuable insights into the characteristics of twophase flow (ᾱ(t) calculated according Section 2.5.3 from WMS measurements). Figure 4.9 presents the evolution of the void fraction over time, only part of the signal is displayed for a more comprehensive understanding. 4.9a presents a comparison between the different tests, illustrating a signal that exhibits an average increase with higher air flow rates. Notably, the signal displays peaks and depressions, as highlighted in Figure 4.9b. The peaks correspond to the passage of large gas structures, while the depressions correspond to the dispersed phase. By following this approach, we analyze the two-phase flow and calculate the power spectral density (PSD) of the signal representing the evolution of the space-averaged void fraction, denoted as ᾱ(t) (calculated according Section 2.5.3 from WMS measurements). The PSD results are shown in Figure 4.10.

The PSD exhibits a prominent peak around 3 Hz for all tests. Specifically, 3.4 Hz for test T5, 3.5 Hz for test T6 and 3.2 Hz for test T7. This frequency can be interpreted as the characteristic frequency of large gas structure passage within the flow. The PSD was obtained by finding optimum values of Hann windows. However, due to the smoothness of the peak, it is difficult to determine if the identified frequency represents the exact value or if the mentioned values are within an order of magnitude. In fact, by performing a fast fourier transform, it becomes clear that the peak in the PSD includes a set of peaks ranging from 3 Hz to 4 Hz. Nevertheless, an intriguing observation is the presence of a periodicity in the large gas structure that appears to be relatively independent of the air flow rate. As described in Section 2.5.2, the optical probe can be positioned upstream of the tube bundle, approximately 110 mm below the last row of tubes, to investigate the inlet conditions. However, it is important to note that the output of the probe is a binary signal, where 0 represents the liquid phase and 1 represents the gas phase. To obtain a time-varying evolution of the void fraction α(t) and enable similar analysis as the space-averaged void fraction evolution ᾱ(t) obtained from the WMS, the probe signal, acquired at a frequency of 20 kHz, can be integrated over a time window ∆t. The void fraction time-evolution (α(t)) is calculated using the gas time over a small time window (T g (∆t)), hence α(t) = T g (∆t) ∆t. This integration allows for the calculation of the void fraction evolution in time. To facilitate the comparison, the integration time window was set to ∆t = 0.001, ensuring that it matches the acquisition frequency of the WMS (1000 Hz). The optical probe provides a local measurement of the void fraction in the core of the flow, while the WMS provides a surface-averaged distribution. However, in terms of comparing the two measurement systems in recording the frequency of the large gas structures, it can be considered a reliable approach. In fact, a closer look to the probe signal post-processed in Figure 4.11b, shows (red rectangles) the large structures recorded according to this type of post-processing. The PSD obtained from the probe in Figure 4.11a shows the same results as the PSD obtained with the WMS (Figure 4.10). The PSD exhibits a frequency peak around 3 Hz, indicating that the frequency of passage of large structures is confirmed by the optical probe measurements in the center of the channel.

From the WMS and optical probe we can extract also information about the bubble/large gas structure size. In Figure 4.12 is presented the PDF of bubble size distribution obtained with WMS and optical probe. The interpretation of a PDF of bubble sizes can provide valuable insights into the characteristics of the bubble population and their distribution. It should also be noted that the two measurement systems use different methods for calculating bubble sizes (as reported in Section 2.5.2 and 2.5.3), besides the fact that one is a local measurement of two-phase flow (the optical probe) and the other reports a two-dimensional measurement (the WMS). This can lead to differences in results. One of which is the minimum diameter detected, the WMS is limited by its mesh, while the probe is limited by the distance between the tips. From Figure 4.12, both measurement systems detect bubble sizes that exhibit a PDF peak consisting of small dispersed bubbles in the order of a few millimeters, accompanied by a tail representing the contribution of large gas structures in all the tests. The length of the tails and peaks increases progressively from test T5 to test T7 both for probe and WMS results. Despite this, the peaks corresponding to dispersed bubbles remain consistently around the same values. In fact, several peaks are detected between 3 mm and 5 mm for the WMS results and a peak around 1.5 mm for optical probe results. This behavior suggests an increase in the number of dispersed bubbles of the same size and an enlargement of the large gas structures. The constant dispersed bubble sizes across all tests can be attributed to reaching the critical Weber number, limiting their further growth. Notably, the main disparity between the optical probe and WMS results lies in the dispersed bubble peak, which appears shifted towards smaller values for the optical probe results. However, as noted by Kiambi et al. [START_REF] Kiambi | Measurements of bubble characteristics: Comparison between double optical probe and imaging[END_REF], measurements of bubble size by the optical probe during chaotic two-phase flows are not entirely reliable. In fact, the recirculations upstream the bundle affect the measurements, potentially causing the flow to not align with the probe tips. Moreover, in the case of the optical probe, bubble size is not detected, but it is post-processed as described in Section 2.5.2 by using the distribution of recorded gas residence time and the gas velocity in order to yeld the bubble size distribution. Hence, while the qualitative shape of the PDF remains similar between the two measurement systems, the average diameter differs significantly. To gain further insights into the characteristics and contribution of large gas structures, the WMS measurements allow the extraction of the bubble size contribution to the void fraction, represented as ∆α i ∆d Bi (void fraction of the bubbles of this class referring to bubble class). The bubble diameters were discretized considering a ∆d B = 0.25 mm, hence, the number of class of void fraction was adapted at the number of class of diameters. The corresponding results are presented in Figure 4.13. Bubble size distributions are constructed by summarizing the contributions of bubbles within specific diameter ranges to the integral volumetric void fraction. These partial void fractions, denoted as ∆α i ∆d Bi , are plotted against the equivalent bubble diameter, ∆d Bi . Integrating over this type of bubble size distribution yields the average void fraction (ᾱ = ∑ ∆α i ). Often, the resulting curves exhibit unevenness at larger diameters due to the presence of only a few large bubbles, which account for most of the void fraction. From this type of bubble size presentation, four distinct peaks can be observed in all tests. The peaks are highlighted through an envelope of the histograms. The first peak (P1) corresponds to dispersed bubbles around 4 mm. The second peak (P2) increases its prominence as the air flow rate increases from the T5 to T7 tests, reaching sizes on the order of 20 mm. The third peak (P3) shows a progressive increase in size and power from the T5 to T7 tests, ranging between 40 mm and 50 mm. Finally, the fourth peak becomes more pronounced as the air flow rate increases from test T5 to T7, indicating sizes greater than 60 mm. The last three peaks shift towards larger values as the tests progress from T5 to T7, suggesting the presence of three distinct types of large gas structures. The data collected with WMS and optical probe can be summarized in the Table 4.2.

The comparison reveals clear discrepancies between the optical probe and WMS results, attributed to differences in measured quantities. These disparities can in fact be attributed to the placement of the optical probe in the center of the two-phase flow, enabling local measurements, while the WMS provides 2D averaged quantities. Under these conditions, where the two-phase flow is not spatially homogeneous, substantial differences in the results arise. Specifically, the large gas structures tend to flow in the center of the flow, while the dispersed bubbles are distributed around them. As a result, the probe detects significantly larger diameters and void fractions compared to the measurements obtained by the WMS in the entire section ( ᾱ). In fact, the void fraction measured by the probe closely aligns with the void fraction detected in the center of the flow by the WMS ᾱc (Figure 4.7). Additionally, it should be noted that measurements of bubble size by the optical probe during chaotic two-phase flows are not entirely reliable, while gas velocity measurements are more accurate but have large uncertainties. Moreover, an oscillating phenomenon is observed in the ascending gas structures, in which large gas structures move back and forth horizontally. Consequently, the probe, with its fixed measurement point, may also overlook some of these substantial structures.

Based on the findings in this section, we classify the flow regime in the free duct for test T5, T6, and T7 as churn flow. Churn flow, according to Taitel et al. [START_REF] Taitel | Modelling flow pattern transitions for steady upward gas-liquid flow in vertical tubes[END_REF] definition, is characterized as chaotic, frothy, and disordered. In this flow regime, the large gas structures become distorted, and the continuity of the liquid between successive large gas structures is frequently disrupted due to high local gas concentrations. This disruption causes liquid slugs to fall, accumulate, form bridges, and then be lifted by the gas. A distinctive feature of churn flow is the oscillatory or alternating direction of liquid motion. Some observers describe this regime as resembling a froth flow pattern with finer dispersion.

According to the analysis conducted on the upstream two-phase flow in the tube bundle, it is both intriguing and valuable to compare our findings with the existing literature on this subject.

Two-phase flow in vertical ducts

The study of two-phase flow regimes in vertical pipes has been extensively explored in the literature. Numerous investigations have examined a diverse range of two-phase flow mixtures and pipe geometries, resulting in the availability of various two-phase flow regime maps. In the context of the current study, the analysis focuses on air-water mixture under ambient temperature and pressure conditions, specifically in pipe configurations with large hydraulic diameters (D h = 0.287 in the present work for the free duct with L/D=5). Flow regime maps from Ohnuki et al. [START_REF] Ohnuki | An experimental study on developing air-water two-phase flow along a large vertical pipe: effect of air injection method[END_REF][START_REF] Ohnuki | Experimental study on transition of flow pattern and phase distribution in upward air-water two-phase flow along a large vertical pipe[END_REF] are reported in Figure 4.14. In this case, the general superficial velocity (U s ) is calculated using the area of the free duct (A ∞ ):

U s = ṁ ρA ∞ (4.4)
Ohnuki et al. [START_REF] Ohnuki | An experimental study on developing air-water two-phase flow along a large vertical pipe: effect of air injection method[END_REF] conducted an investigation on flow regimes in a large diameter pipe with D h = 0.48 m and length L = 2 m. They employed high-speed video systems for flow pattern visualization, along with 6 differential pressure sensors and an optical void probe for local void fraction measurements. The study also utilized two air injection systems to analyze the impact of injection system configuration on the flow pattern. In a subsequent study [START_REF] Ohnuki | Experimental study on transition of flow pattern and phase distribution in upward air-water two-phase flow along a large vertical pipe[END_REF], the original test section was replaced with a smaller diameter pipe of D h = 0.2 m and length 12.3 m. To deepen their measurements, a resistivity probe was added for bubble velocity measurement, and bubble diameter was obtained through the optical probe. Notably, measurements taken at L/D=60 and L/D=10 exhibited different results in the flow regime map. The authors observed that in large diameter pipes, the slug flow regime is absent, and the transition occurs directly from bubbly to churn flow. The findings from the initial experimental campaign [START_REF] Ohnuki | An experimental study on developing air-water two-phase flow along a large vertical pipe: effect of air injection method[END_REF], as depicted in Figure 4.14a, align with the results obtained in the current experimental campaign. However, in the flow regime map developed for the smaller section pipe [START_REF] Ohnuki | Experimental study on transition of flow pattern and phase distribution in upward air-water two-phase flow along a large vertical pipe[END_REF], as shown in Figure 4.14a, the agreement is only evident for measurements taken at L/D=10. [START_REF] Ali | Two phase flow in large diameter vertical riser[END_REF]. Tee and Ann refers to the two different air injection systems. "B" referrers to bubbly flow, "DB" to dispersed bubbly, "AB" to agitated bubbly and "C" to churn flow. In the figure, the authors use the notation J g and J l , which corresponds to the notation of U sg and U sl in the present study.

Ali [START_REF] Ali | Two phase flow in large diameter vertical riser[END_REF] also observed a direct transition from bubbly (specifically, agitated bubbly) to churn flow in large diameter pipes. However, in that study, the transition occurred at gas flow superficial velocities greater than 1.5 m s -1 . In contrast, in the present study, the cases exhibiting churn flow feature gas super-ficial velocities ranging from 0.13 < U sg < 0.4 m s -1 . This discrepancy may be attributed to the different range of liquid superficial velocities investigated (U sl = 0.04 m s -1 in the present work). Furthermore, as previously noted (Figure 4.14), the discrepancy in the flow regime transition can be attributed to the larger L/D (L/D=5 in the present work, while for Ali [START_REF] Ali | Two phase flow in large diameter vertical riser[END_REF] L/D=47).

The facility TOPFLOW [START_REF] Schaffrath | Topflow -a new multipurpose thermalhydraulic test facility for the investigation of steady state and transient two-phase flow phenomena[END_REF] is used by Prasser et al. [START_REF] Prasser | Topflow tests on the structure of the gas-liquid interface in a large vertical pipe[END_REF] to study air-water two-phase flow in a vertical pipe of D h =0.195 m and L/D=40. This experiments have been performed with variable gas injection distance and the two-phase flow has been characterized by means of a Wire Mesh Sensor (64 x 64 wires, with a pitch of 3 mm). The flow regimes are reported in Figure 4.16. The significant differences in liquid superficial velocity and L/D ratio between the investigation conducted by Prasser et al. [START_REF] Prasser | Topflow tests on the structure of the gas-liquid interface in a large vertical pipe[END_REF] and the present work do not allow a realistic comparison between the two studies. However, it is worth noting that, similar to our findings, no slug flow regime was identified in their study as well. In fact, according to the authors, the flow regime taking place between U sg =0.34 m s -1 and U sg =0.53 m s -1 was classified as churn flow. In the In summary, TREFLE shares similarities with other authors in terms of flow pattern analysis, the utilization of high-speed cameras, the application of optical probes and the use of WMS. Although the present study investigates a relatively narrower range of velocities, its utilization of multiple measurement systems and a thorough approach to data post-processing enables a comprehensive analysis of the twophase flow. TREFLE features an approximate L D ratio of 5, indicating a shorter pipe length compared to some other authors, but it remains within the range typically found in the literature.

Drift flux correlation for large diameter pipe in low flow rate condition

The drift-flux correlation is a powerful method for predicting void fraction. Numerous correlations are available in the literature for different two-phase flow conditions and mixtures. However, most of these correlations are validated for small diameter pipes (tipically up to 60 mm) and flow velocities outside the range of the present study. Therefore, a careful selection of correlations compatible with the flow conditions and pipe geometry of the present study was made. The basic theory of the drift-flux model was initially presented by Zuber and Findlay [START_REF] Zuber | Average volume concentration in two phase systems[END_REF]. This model is based on the relative motion between the two phases and employs a constitutive relation. The drift-flux model, used for evaluating void fraction, can be expressed as:

⟨V g ⟩ = ⟨U sg ⟩ ⟨α⟩ = ⟨V gi ⟩ + C 0 ⟨U s ⟩ (4.5)
In Equation 4.5, the symbol ⟨⟩ represents the averaging of the variable over the cross-sectional area. U sg denotes the superficial velocity of the gas phase, which is the ratio of the gas volumetric flow rate to the entire cross-sectional area. U s corresponds to the mixture superficial velocity, obtained by summing the superficial velocities of the gas and liquid phases. α denotes the void fraction. C 0 and V gi respectively represent the distribution parameter (that takes into account the profile of the void fraction) and the void fraction-weighted and area-averaged local drift velocity. Several correlations exist in the literature for predicting C 0 and V gi .

In the context of our experimental conditions, Shoukri et al. [START_REF] Shoukri | Two-phase bubbly flow structure in large-diameter vertical pipes[END_REF] conducted a series of experiments in a large vertical pipe (D h = 0.2 m) using an air-water two-phase mixture at ambient pressure and temperature. The experiments covered a range of superficial velocities, specifically 0 < U sl < 0.4 m s -1 for the liquid phase and 0.02 < U sg < 0.1 m s -1 for the gas phase. Measurements of bubble velocity, size, and void fraction were obtained using a dual-sensor optical probe. Shoukri et al. [START_REF] Shoukri | Two-phase bubbly flow structure in large-diameter vertical pipes[END_REF] compared the drift-flux correlations available in the literature with their experimental results and found that the correlation developed by Kataoka et al. [START_REF] Kataoka | Drift flux model for large diameter pipe and new correlation for pool void fraction[END_REF] provided the best fit for their data:

V + gj = ⟨V gi ⟩ σg∆ρ ρ 2 l 1 4
(4.6) with

V + gj = 0.03 ρ g ρ l -0.157 N -0.562 µl f or D * h > 30 (4.7) N µl = µ l ρ l σ σ g∆ρ 1 2 (4.8) D *
h being the dimensionless hydraulic diameter

D * h = D h σ g∆ρ (4.9)
With σ the surface tension, g the gravitational acceleration, ∆ρ the density difference of liquid and gas phase (ρ lρ g ). While Kataoka et al. [START_REF] Kataoka | Drift flux model for large diameter pipe and new correlation for pool void fraction[END_REF] provided a method to calculate the distribution parameter C 0 based on the channel geometry and on the density ratio of the two phase mixture (Figure 4.17), according to Shoukri et al. [START_REF] Shoukri | Two-phase bubbly flow structure in large-diameter vertical pipes[END_REF] the most appropriate value for C 0 in their configuration is 0.8 (bubbly flow condition). Therefore, combining equations 4.5, 4.6, 4.7, 4.8, 4.9 and using C 0 = 0.8 the authors [START_REF] Shoukri | Two-phase bubbly flow structure in large-diameter vertical pipes[END_REF] obtained averaged values of void fraction in agreement with the experimental results. Hibiki et al. [START_REF] Hibiki | One-dimensional drift-flux model for two-phase flow in a large diameter pipe[END_REF] developed a drift-flux correlation specifically designed for two-phase flow in large diameter pipes at low flow rates. The correlation was validated using a comprehensive database of experimental campaigns, encompassing a wide range of gas superficial velocities from 0.01 to 1.16 m s -1 and liquid superficial velocities from 0 to 0.387 m s -1 . The majority of the pipes considered in the experiments had hydraulic diameters ranging from 0.102 to 0.48 m. The experimental data primarily pertained to air-water or nitrogen-water two-phase mixtures at ambient pressure and temperature. To establish a dimensionless drift-flux correlation, Hibiki et al. [START_REF] Hibiki | One-dimensional drift-flux model for two-phase flow in a large diameter pipe[END_REF] introduced dimensionless gas and liquid superficial velocities for dimensionless drift-flux correlation. Being U s * the generic superficial velocity and U + s * the generic dimensionless superficial velocity:

⟨U + s * ⟩ = ⟨U s * ⟩ σg∆ρ ρ 2 l 1 4
(4.10)

Therefore, equation 4.5 can be expressed in dimensionless form:

⟨V + g ⟩ = ⟨U + sg ⟩ ⟨α⟩ = ⟨V + gi ⟩ + C 0 ⟨U + s ⟩ (4.11)
According to the authors V + gi (equation 4.12) can be written as a combination between V + gi,B and V + gi,P , being respectively the drift velocity for bubbly flow and pool boiling systems.

V + gi = V + gi,B exp A ⟨U + sg ⟩ + V + gi,P 1 -exp A ⟨U + sg ⟩ (4.12)
With the coefficient A estimated to be -1.39. V + gi,P is given by equations 4.6-4.9 (indicated as V + gj ), and V + gi,B is: The work conducted by Bhagwat et al. [START_REF] Bhagwat | A flow pattern independent drift flux model based void fraction correlation for a wide range of gas-liquid two phase flow[END_REF] is worth mentioning as they have adapted the driftflux correlation to explore a broad spectrum of two-phase mixtures, pipe geometries, flow rates, and pipe orientations. Consequently, the correlation has been extensively validated against a wide range of experimental data, specifically for air-water mass flux in the range of 10 < G < 8450 (with G the ratio between the mixture mass flow rate and the pipe cross-section). In the context of the present work, a specific model for the distribution parameter has been obtained for large vertical pipes with rectangular cross-sections:

V + gi,B = √ 2 (1 -α) 1.75
C 0 = 2 - ρg ρ l 2 1 + Retp 1000 2 + C 0,1 1 + Retp 1000 2 (4.14) with C 0,1 = 0.4 -0.4 ρ g ρ l (2.6 -β) 0.15 -f tp (1 -x) 1.5 (4.15)
Where x is the two phase flow quality and β = U sg U s the volumetric injection ratio. The two phase mixture Reynolds number is defined as Re tp = U s ρ l D h µ l . f tp is the two phase friction factor defined as function of wall surface roughness ( ), hydraulic pipe diameter (D h ) and Re tp :

1

f tp = -4.0log 10 ⎛ ⎝ D h 3.7 + 1.256 Re tp f tp ⎞ ⎠ (4.16)
The drift velocity, proposed by Bhagwat et al. [START_REF] Bhagwat | A flow pattern independent drift flux model based void fraction correlation for a wide range of gas-liquid two phase flow[END_REF] is defined as a function of pipe orientation, pipe diameter, fluid properties and void fraction. For the present case of study it reduces to:

V gi = 0.35 gD h (ρ l -ρ g ) ρ l (1 -α) 0.5 (4.17)
Therefore, combining equations 4.14 to 4.17 and equation 4.5, the average void fraction can be estimated according to Bhagwat et al. [START_REF] Bhagwat | A flow pattern independent drift flux model based void fraction correlation for a wide range of gas-liquid two phase flow[END_REF]. The drift-flux correlations analyzed and their validation ranges are compared in Table 4.4. In Figure 4.19, the correlation proposed by Kataoka et al. [START_REF] Kataoka | Drift flux model for large diameter pipe and new correlation for pool void fraction[END_REF] has been modified by Shoukri et al. [START_REF] Shoukri | Two-phase bubbly flow structure in large-diameter vertical pipes[END_REF], who introduced a correction factor of C 0 = 0.8 for the bubbly flow regime (up to U sg = 0.05 m s -1 ). For the churn flow regime (with U sg ranging from 0.13 m s -1 to 0.37 m s -1 ), the distribution parameter was estimated using the data presented in Figure 4.17. The correlation proposed by Shoukri et al. [START_REF] Shoukri | Two-phase bubbly flow structure in large-diameter vertical pipes[END_REF] performs less accurately than the one proposed by Hibiki et al. [START_REF] Hibiki | One-dimensional drift-flux model for two-phase flow in a large diameter pipe[END_REF] for U sg = 0.05 m s -1 and below, but it exhibits better performance for U sg ranging from 0.13 m s -1 to 0.37 m s -1 . Consequently, despite offering a more accurate approximation than the distribution parameter proposed by Kataoka et al. [START_REF] Kataoka | Drift flux model for large diameter pipe and new correlation for pool void fraction[END_REF] for the bubbly flow regime, the correction proposed by Shoukri et al. [START_REF] Shoukri | Two-phase bubbly flow structure in large-diameter vertical pipes[END_REF] is still found to be inadequate. This discrepancy may be attributed to factors such as the channel geometry, which Shoukri et al. [START_REF] Shoukri | Two-phase bubbly flow structure in large-diameter vertical pipes[END_REF] did not consider when evaluating C 0 , and the flow regime effects, which were not accounted for by Kataoka et al. [START_REF] Kataoka | Drift flux model for large diameter pipe and new correlation for pool void fraction[END_REF].

On the other hand, the results from the correlation by Bhagwat et al. [START_REF] Bhagwat | A flow pattern independent drift flux model based void fraction correlation for a wide range of gas-liquid two phase flow[END_REF] do not take into account the flow regime effect. Their correlation does not perform well for the bubbly flow campaign but is in agreement with the churn flow regime. In fact, their drift velocity V gi is specifically developed for slug flow regimes. However, due to the broad range of experimental data for validation, experimental campaigns in large rectangular pipes are limited to steam-water conditions.

The gas velocity ⟨V g ⟩ obtained from the drift flux correlation from the authors was compared to the experimental data obtained with the optical probe in the present experimental campaign in Figure 4.20. The results for the churn flow regime (U sg ranging from 0.13 m s -1 to 0.37 m s -1 ) are clarly in disagreement with the drift flux correlations. These results was expected because the probe provides local measurements within the core of the flow, which do not represent the entire mixture. This is particularly evident due to the pronounced core-peaked shape of the void fraction profile (refer to Figure 4.8). As a result, for the churn flow regime, the experimental data from the optical probe cannot be considered for direct comparison with drift flux correlations, which are intended to provide cross-section-averaged values of gas velocity. In fact, the core-peaked profile of the void fraction results in a corresponding core-peaked profile of gas velocity, influenced by the buoyancy of larger structures. Consequently, this core-peaked gas velocity profile leads to an underestimation of the gas velocity from the drift flux correlations when compared with optical probe measurements for this specific flow regime.

On the other hand, for bubbly flow regime, the measurements obtained through the optical probe within the core of the flow can be considered as representative of the entire mixture. In fact, the gas velocities calculated using the aforementioned correlations for U sg < 0.05 m s -1 align with the general behavior of the void fraction. However, while the correlation proposed by Hibiki et al. [START_REF] Hibiki | One-dimensional drift-flux model for two-phase flow in a large diameter pipe[END_REF] in Figure 4.19 provides the best approximation for the entire dataset, it becomes less accurate for gas velocities in the range of U sg from 0.025 m s -1 to 0.05 m s -1 . This discrepancy may be attributed to the adoption of a rectangular cross-section in the present study. In fact, the corners and edges of a rectangular duct can introduce boundary effects that significantly impact the velocity profile. Rectangular ducts generally have a higher perimeter-to-area ratio compared to circular ducts, meaning that for a given cross-sectional area, a rectangular duct has longer wall segments (perimeter). This characteristic results in higher frictional losses along the duct walls, which, in turn, may lead to higher pressure drops for the same flow rate as compared to circular ducts. Consequently, the uneven pressure distribution within a rectangular duct can influence the velocity profile.

Hence, it can be concluded that pipe geometry, flow regime, and two-phase mixture play an important role in the void fraction and gas velocity evaluation that should be taken into consideration for the driftflux correlation.

Two-phase flow characterization within the tube bundle

Once the two-phase flow has been characterized upstream the tube bundle, we can proceed with the analysis of the two-phase flow within the tube bundle. The investigation of the two-phase flow upstream of the tube bundle can give a valuable insight on how the flow condition in the free section affect the two-phase flow inside the tube bundle. The objective of this section is to employ the optical probe (Section 2.5.2) and fast camera (Section 2.5.1) to experimentally investigate the two-phase flow behaviour in the tube bundle. Optic probes provide valuable real-time measurements of local parameters such as void fraction, interfacial area (in the bubbly flow regime), and gas velocity. Fast cameras, on the other hand, enable the visualization and analysis of rapid fluid motion with high temporal resolution. Figure 4.21 shows a visual observation of the intermittent flow regime within the tube bundle during tests T5, T6, and T7.

The images reveal an interesting distribution pattern within the two-phase flow. The majority of small dispersed bubbles tend to accumulate near the wall, while larger gas structures can be observed behind them. This distribution pattern is a direct outcome of the inlet condition. The churn-like flow observed upstream of the tube bundle, characterized by prominent gas structures in the core of the flow, persists within the bundle. However, as the two-phase flow encounters the obstacle posed by the tube bundle, the large gas structures are disrupted, as evidenced by the lateral views of the tube bundle depicted in Figure 4.22 in comparison with the images obtained with the WMS (Figure 4.3).

The images clearly exhibit large gas structures that have undergone shape distortion upon entering the tube bundle (refer to Figure 4.3 4.4 4.5 for images upstream the tube bundle). As discussed in the literature review (Section 1.3), different definitions of this flow regime exist within flow regime maps. Some authors, such as Piteau et al. [START_REF] Piteau | Experimental investigation of in-flow fluidelastic instability for rotated triangular tube bundles subjected to single-phase and twophase transverse flows[END_REF], Kanizawa et al. [START_REF] Kanizawa | Two-phase flow patterns across triangular tube bundles for airwater upward flow[END_REF], Mao et al. [START_REF] Mao | Flow regime transition criteria for upward two-phase cross-flow in horizontal tube bundles[END_REF], and Chou et al. [START_REF] Chu | Flow-induced vibration of nuclear steam generator u-tubes in two-phase flow[END_REF], define a transition between the bubbly and churn/slug flow regimes. Conversely, Grant et al. [START_REF] Grant | Two-phase flow on the shell-side of a segmentally baffled shellandtube heat exchanger[END_REF], Ulbrich et al. [START_REF] Ulbrich | Vertical, upward gas-liquid two-phase flow across a tube bundle[END_REF], Noghrehkar et al. [START_REF] Noghrehkar | Investigation of two-phase flow regimes in tube bundles under cross-flow conditions[END_REF], and Murakawa et al. [START_REF] Murakawa | Evaluation of void fraction around a tube in two-phase flow across horizontal tube bundle[END_REF] establish a transition between the bubbly and intermittent flow regimes. In the present study, the flow regime, based on qualitative analysis using the fast camera, can be described as a churn-like flow regime, characterized by the presence of both large gas structures and dispersed bubbles within a continuous liquid phase. However, owing to the periodic intermittence observed during the passage of large gas structures upstream of the tube bundle, the video footage also reveals an intermittent-like behavior of the flow, with distinct large gas structures alternating with the liquid phase, even within the bundle itself. To further investigate this behavior, the utilization of optical probes for void fraction measurements comes into play. As demonstrated in Section 4.1, the signal obtained from the optical probe can be discretized over a time interval ∆t (= 0.001 second also in this case). In this particular case, we consider measurement point 2 (Figure 2.24), which lies within the tube bundle, specifically in the gap between the central tube and the adjacent one. The PSD of the void fraction fluctuation is reported in Figure 4.23.

The PSD analysis, for tests T5, T6 and T7, reveals prominent peaks occurring at frequencies approximately around 3 Hz. These frequencies can be attributed to the passage frequency of the large gas structures within the tube bundle. A comparison of this PSD analysis with the PSD obtained upstream of the tube bundle (Figures 4.10 and 4.11) demonstrates that the signal obtained is similar in terms of the frequency of void fraction oscillations. This signal reflects the bundle inlet conditions, as the frequency of large gas structures remains similar inside the tube bundle. This behavior leads to an intermittent nature of the flow. From this analysis, a significant conclusion can be drawn: the interaction between the gas and liquid phases, the influence of inlet conditions, and the obstructive nature of the tube bundle all contribute to the potential overlap of churn flow and intermittent flow regimes. The presence of obstacles created by the tube bundle disrupts the flow dynamics and can result in the breakup of large gas struc- tures characteristic of churn flow. This disruption leads to the formation of intermittent flow patterns within the tube bundle, where distinct slugs or bubbles of gas and liquid coexist. Vertical profiles of void fraction, gas velocity and sauter mean diameter can be obtained from the probe measurement points 1, 8, 2, 3, 4, 5 in Figure 2.24. The results are reported in Figure 4.24.

In the vertical profile, it is evident that the void fraction increases when transitioning from upstream of the tube bundle to within the bundle. The larger void fraction within the tube bundle is coherent with the measurement of a smaller gas velocity, compared to the measurements upstream. However, the void fraction inside the tube bundle is only significantly affected by the bundle geometry in test T5, where it increases as the flow area decreases and vice versa. On the other hand, for tests T6 and T7, the void fraction is less influenced by the bundle geometry and tends to remain constant. This phenomenon could be attributed to the increase in the presence of large gas structures and the instrumentation employed. In fact, the optical probe calculates the void fraction as ᾱ = T g (T l + T g ), where T g and T l represent the total gas and liquid time recorded by the probe. Specifically, when a large gas structure moves from a region with a certain flow area to a zone with a smaller flow area, it has a relatively minor impact on the local void fraction. Conversely, when a bubble swarm passes through a smaller flow area, the bubbles are more likely to be detected by the probe, resulting in an increase in the local void fraction. The evolution of gas velocity within the bundle, as depicted in Figure 4.24b, supports the notion that a smaller flow area leads to a larger flow velocity. However, upstream of the tube bundle, the gas velocity is larger than within the bundle. This behavior is attributed to the presence of large gas structures in this regime. Due to the substantial size of these gas structures and the relatively low liquid velocity, the two-phase flow upstream is primarily buoyancy-driven. As the gas structures move into the tube bundle, they undergo fragmentation or breakup. The smaller size of gas bubbles inside the bundle reduces their buoyancy effect. Smaller bubbles have less buoyancy force acting on them compared to larger ones. Consequently, within the tube bundle, the buoyancy-driven component of the flow becomes weaker and the gas velocity decreases.

The experimental campaign conducted to explore the intermittent flow regime has unveiled intriguing findings, particularly in relation to the void fraction parameter. While many experiments documented in the literature have focused on studying tube vibration response and Fluid-Elastic Instability, these studies often rely on two-phase flow information using the HEM model [START_REF] Pettigrew | Vibration of tube bundles in two-phase cross-flow: Part 1-hydrodynamic mass and damping[END_REF][START_REF] Pettigrew | Vibration of tube bundles in two-phase cross-flow: Part 2-fluid-elastic instability[END_REF][START_REF] Axisa | Vibration of tube bundles subjected to air-water and steam-water cross flow: Preliminary results on flu1delastic instability[END_REF][START_REF] Axisa | Vibration of tube bundles subjected to steamwater cross flow: a comparative study of square and triangular pitch arrays[END_REF][START_REF] Delaune | Experimental investigation of in-flow fluidelastic instability for square tube bundles subjected to single-phase and two-phase transverse flows[END_REF]. However, this model, which assumes equal gas and liquid velocities, proves inadequate as the phase slip intensifies. In the context of water-air two-phase flow under ambient temperature and pressure conditions, such as in our present experimental campaign, the HEM model cannot accurately estimate the void fraction as reported in Figure 4.25.

The comparison depicted in Figure 4.25 reveals the inadequacy of the HEM model in accurately predicting the void fraction within the tube bundle. On the other hand, the model proposed by Feenstra et al. [START_REF] Feenstra | An improved void fraction model for two-phase cross-flow in horizontal tube bundles[END_REF] serves as a significantly improved approximation; however, its precision diminishes when applied to the intermittent flow regime tests (T5, T6, and T7). This discrepancy may be attributed to variations in the experimental conditions under which the correlation was validated. In their study, Feenstra et al. [START_REF] Feenstra | An improved void fraction model for two-phase cross-flow in horizontal tube bundles[END_REF] employed gamma ray attenuation to obtain an average void fraction measurement within the gap, while in our experiment, we consider the central value of the void fraction within the gap. As a result, the correlation underestimates the void fraction in our setup. Additionally, our experimental conditions are unique due to the large slip ratio (see Figure 4.26). Consequently, as the air flow rate in our experiments increases while the water flow rate remains constant, the disparity between Feenstra et al. model and the experimental data becomes more pronounced. This may be because the elevated slip ratio falls outside the validation range for their model. In fact, as reported by Feenstra et al. [START_REF] Feenstra | An improved void fraction model for two-phase cross-flow in horizontal tube bundles[END_REF], their correlation does not consistently perform well for data obtained in an air-water mixture under atmospheric conditions.

The slip ratio (S) is calculated starting from Equation 4.18. Where ρ g and ρ l are gas and liquid . The results are reported within the tube bundle and in the free duct.

density and x is the dynamic quality.

α = 1 1 + S ρg ρ l 1 x -1 (4.18)
Wich rearranged yelds Equation 4. [START_REF] Murakawa | Evaluation of void fraction around a tube in two-phase flow across horizontal tube bundle[END_REF].

S = 1 α -1 ρg ρ l 1 x -1 (4.19)
The void fraction, denoted as α, is measured using different methods for tube bundle and free duct results. In the tube bundle, it is obtained by the optical probe at point 2 (in the gap of the 3rd row). For tests T1 to T4 in the free duct, it is determined at the tube bundle inlet using the probe (point 5), while for tests T5 to T7, it is the space-time-averaged void fraction obtained using the WMS. As observed in Figure 4.26, the slip ratio increases with larger air flow rates, reaching values as large as 20 in test T7 in the free duct.

The experimental data collected in this Section, in conjunction with the findings presented in Section 3.2.2, provide us with the necessary resources to further explore the two-phase flow regimes and their transition boundaries within the TREFLE facility. This, in turn, enables to propose a new flow regime map.

Two-phase flow regime map

Accurate characterization and prediction of two-phase flow regimes in tube bundles play a crucial role in various engineering applications, such as nuclear reactors, heat exchangers, and chemical reactors. This knowledge ensures safety, enhances efficiency, and improves heat transfer, contributing to optimized processes, resource conservation, and reduced environmental impact. In recent years, significant research efforts have been dedicated to developing flow regime maps for tube bundles, aiming to provide a comprehensive understanding of the flow patterns and their transitions under different operating conditions. Traditionally, flow regime maps have been constructed based on subjective analysis, relying on visual observations and expert judgment. However, with the advancements in data acquisition and analysis techniques, objective analysis methods have emerged as a promising approach to complement subjective assessments. This section presents a proposition for constructing two-phase flow regime maps in tube bundles by incorporating both subjective and objective analyses, leveraging the strengths of each method to enhance the accuracy and reliability of the predictions. The data acquired from Chapter 3 and the present Chapter can be utilized to propose a methodology for identifying the flow regime and developing a flow regime map specific to the current bundle geometry. Additionally, the analysis of the inlet conditions provides valuable insights into how these conditions influence the flow pattern within the bundle.

Subjective identification of flow regime through visual observations is the most commonly employed method, but it necessitates a predefined definition of the flow regime. Although various definitions have been proposed in the literature, it is crucial to classify the flow regime based on the distinct regimes identified in the present experimental campaign, considering the specific geometry and flow conditions:

• Bubbly flow regime: In the bubbly flow regime, the two-phase flow consists of discrete gas bubbles dispersed within a continuous liquid phase. The gas bubbles are relatively small and uniformly distributed throughout the flow domain. The bubble size is smaller than the gap between the tubes, no gas structures larger than the gap are present.

• Transition regime: The transition regime represents a transitional state between different flow regimes. It occurs when the flow conditions change, causing a shift in the dominant flow behavior.

In the context of a tube bundle, the transition regime may occur when there is a change in flow velocity, void fraction, or gas structure sizes. During the transition, the flow characteristics may exhibit a combination of features from multiple flow regimes, making it challenging to categorize the flow into a specific regime.

• Intermittent/Churn flow regime: This regime is characterized by the periodic coexistence of large gas structures and liquid. In this regime, the flow pattern exhibits intermittent variations, with gas or liquid slugs intermittently occupying most of the tube cross-section. The large gas structures can have different lengths and velocities, and their presence causes periodic fluctuations in flow parameters. The gas structures are characterized by a churn motion. The churn motion causes the liquid film to experience intense agitation and disruption. Gas bubbles within the liquid film coalesce and break up, continuously changing the bubble size distribution. This regime is typically unsteady and exhibits complex flow behavior due to the continuous interaction between the liquid and the gas phase. In this scenarios, intermittent and churn flow regimes mix and coexist within the tube bundle.

In order to characterize the flow regime using a subjective methodology, fast camera videos are analyzed, as reported in Section Chapter 3 and the present Chapter. Additionally, the availability of optical probe measurements allows for a more quantitative approach to identify the flow regime correcting the subjective approach.

A statistical analysis using PDF of void fraction fluctuation was performed to characterize the flow regimes. Statistical analysis employing Probability Density Function (PDF) is widely utilized in the literature. Jones and Zuber [START_REF] Jones | The interrelation between void fraction fluctuations and flow patterns in two-phase flow[END_REF] were pioneers in employing a non-visual method based on X-ray measurements of void fraction. They successfully employed PDF analysis to identify various flow regimes such as bubbly, slug, and annular flow in an upward air-water flow within a narrow rectangular channel. The PDF distributions displayed distinctive patterns, with bubbly flow exhibiting a single peak at low void fraction, annular flow showing a single peak at high void fraction, and slug or intermittent flow displaying two peaks representing characteristics of both bubbly and annular flow. Ali et al. [START_REF] Ali | Two phase flow in large diameter vertical riser[END_REF] employed PDF analysis of void fraction measured by electrical resistivity probes as an objective approach to identify two-phase flow regimes in circular and narrow channels. Similarly, Noghrehkar et al. [START_REF] Noghrehkar | Investigation of two-phase flow regimes in tube bundles under cross-flow conditions[END_REF] utilized this technique to identify flow regimes in a tube bundle subjected to air-water two-phase flow, successfully identifying bubbly, intermittent, and annular flow regimes. In another study, Aprin et al. [START_REF] Aprin | Experimental analysis of local void fractions measurements for boiling hydrocarbons in complex geometry[END_REF] employed optical probe measurements to identify flow regimes across a horizontal tube bundle for three different hydrocarbons under saturated conditions, specifically identifying bubbly and annular dispersed flow regimes.

Before discussing the obtained data, it is valuable to clarify the physical interpretation of the PDF. If we denote the probability that the void fraction α is less than some specific value as P (α), then dP (α) dα = p(α) would be the probability per unit void fraction that the void fraction ranges between α and α + dα. Considering a void-time trace record where the void scale is discretized into equal increments of ∆α i , and the time scale is partitioned into equal intervals of ∆t j . If, the void fraction is observed to fall within ∆α i a total of n i times in the total time T , and N is the total number of void fraction data registered during T , then:

n i N ∆α i = 1 ∆α i ni j=1 ∆t j T (4.20)
In the case of bubbly flow, a large count ratio n i N would be observed at low void fraction values, while smaller ratios would exist in ∆α i intervals at higher α values. Conversely, in annular flow, the opposite pattern emerges. Since the ratio ∑ t j T represents the probability that the void fraction falls within the given interval ∆α i , we can see that:

lim ∆αi→0 1 T ∆α i ni j=1 ∆t j (4.21)
represents the PDF of the specific record under examination. The PDF for this particular record, as given by Equation 4.21, will exhibit a distinctive configuration dependent on the nature of the flow passing through the optical probe during the sampling period. Consequently, the outcome of Equation 4.21 is expected to produce a low-void peak in the PDF for bubbly flow and a high-void peak for annularlike flow. Conversely, in a churn-like flow, the optical probe may perceive all water with possibly some entrained small bubbles at one instant, yielding a PDF characteristic of bubbly flow. In the next instant, the large gas structure could pass the optical probe, creating a wall film and a center core of gas, resulting in a PDF resembling annular flow. The average is expected to generate a twin-peaked PDF.

In summary, Jones and Zuber [START_REF] Jones | The interrelation between void fraction fluctuations and flow patterns in two-phase flow[END_REF] propose that the PDF will exhibit the following behavior:

• **Bubbly-like flows:** A single-peaked PDF at low void fraction (Figure 4.27a).

• **Annular-like flows:** A single-peaked PDF at high void fraction.

• **Churn/Intermittent-like flows:** A twin-peaked PDF, with one peak characteristic of bubbly flow and the other characteristic of annular flow (Figure 4.27b). In the present work, the optical probe signal from the measurement point 2 (Figure 2.24) within the tube bundle was used to perform a PDF analysis of the void fraction time-evolution. As explained before, the void fraction time-evolution (α(t)) is calculated using the gas time over a small time window (T g (∆t)); hence α(t) = T g (∆t) ∆t. However, as also reported by Aprin et al. [START_REF] Aprin | Experimental analysis of local void fractions measurements for boiling hydrocarbons in complex geometry[END_REF], the PDF shapes are closely related to the windows ∆t. To choose the value of ∆t that provides the most accurate information about the flow regime, it is necessary to discriminate between large gas structures and small dispersed bubbles.

Within the tube bundle, a "large gas structure" can be defined as a gas structure that overcomes the size of the gap between two tubes (a). Therefore, by obtaining the average gas velocity v B through the probe itself, the optimal time window for analysis can be determined as ∆t = a v B . This approach allows the values of void fraction associated with the large gas structure to be very close to 1, while the gas structures smaller than the gap size are related to the void fraction values from 0.1 to 0.99. The void fraction scale is divided into 20 intervals (∆α i = 0.05 as done by Aprin et al. [START_REF] Aprin | Experimental analysis of local void fractions measurements for boiling hydrocarbons in complex geometry[END_REF]). The PSD serves as an indication of the frequency at which large gas structures pass through the tube bundle. Notably, tests T1 to T3 exhibit no low-frequency peaks, suggesting the absence of significant gas structures within the bundle, thus indicating a bubbly flow regime. In contrast, test T4 shows an increase in low-frequency power, implying the presence of some large gas structures traversing the bundle. Tests T5 to T7 clearly demonstrate the excitation of low frequencies, indicating the regular presence of large gas structures traversing the bundle.

It is important to highlight that the PSD presented in Figure 4.29 differs from that reported in Figure 4.23 due to the selection of different time windows to extract the void fraction time-evolution from the probe signal. Specifically, in Figure 4.29, the time windows are chosen to record each gas structure larger than the gap, while in Figure 4.23, a smaller time window (∆t = 0.001 s) allows for the recording of large gas structures smaller than the gap size. To enhance the comprehension of regime boundaries, several additional experiments were conducted alongside tests T1 to T7 reported in Table 4.5 listed as N1 to N19.

Test no. Air flow rate Water flow rate By combining the previously mentioned methods for flow regime classification (visual observation, statistical analysis using PDF, and PSD analysis), a flow regime map for the current bundle geometry and flow mixture is proposed in Figure 4.30. A sensibility on the ∆t used for the discretization of the probe signal (T g (∆t)) shows no difference for the flow regime identification. The flow regimes boundaries identified in this study differ from those proposed by most other authors. The geometry of the bundle plays a crucial role in determining the flow regime, and the gap between the tubes is a key factor influencing this effect. Performing a sensitivity analysis on the gas injection system for test T4, no substantial difference was reported regarding the flow regime identification. Optical probe measurements within the bundle (Figure 3.26 and Figure 4.24) demonstrate that void fraction and gas velocity. Consequently, different gap sizes could lead to varying evolution of these parameters, which are essential for establishing the flow regime.

J l J g g s -1 m s -1 m s -1 N1 5 g s -
The present configuration has a gap size of 13.2 mm, while Noghrehkar et al. [START_REF] Noghrehkar | Investigation of two-phase flow regimes in tube bundles under cross-flow conditions[END_REF] used a gap of 5.97 mm. Therefore, even if a similar methodology based on PDF statistical analysis is used, the flow regime boundaries may differ. Ulbrich et al. [START_REF] Ulbrich | Vertical, upward gas-liquid two-phase flow across a tube bundle[END_REF], however, used a gap of 10.0 mm, which is closer to the present configuration. However, their identification method relied mainly on visual observation, which could be misleading as crucial information might not be accessible using fast cameras.

Murakawa et al. [START_REF] Murakawa | Evaluation of void fraction around a tube in two-phase flow across horizontal tube bundle[END_REF] reported the presence of a transition zone similar to the present study, but the regime boundary may differ due to their gap size (7.0 mm). Xu et al. [START_REF] Xu | Hydrodynamics of two-phase flow in vertical up and down-flow across a horizontal tube bundle[END_REF], on the other hand, did not report a transition from the bubbly to churn flow regime for J l < 0.2m s -1 , possibly due to the gap between the tubes being 2.74 mm, and their experimental campaign performed for J g > 0.4m s -1 .

In comparison, there was agreement with the flow regime map from Piteau et al. [START_REF] Piteau | Experimental investigation of in-flow fluidelastic instability for rotated triangular tube bundles subjected to single-phase and twophase transverse flows[END_REF], as they used a similar method for flow regime identification and had a similar bundle geometry with a gap of 15.0 mm and tube diameter of 30.0 mm. However, Piteau et al. [START_REF] Piteau | Experimental investigation of in-flow fluidelastic instability for rotated triangular tube bundles subjected to single-phase and twophase transverse flows[END_REF] found a slug flow regime established in the bundle, possibly due to the larger water superficial velocities used in their tests. However, in the test section adopted by Piteau et al. [START_REF] Piteau | Experimental investigation of in-flow fluidelastic instability for rotated triangular tube bundles subjected to single-phase and twophase transverse flows[END_REF] for flow regime identification, there were only 3 columns of tubes (compared to 5 columns in our present work), resulting in a narrower test section and free duct. It's important to note that the inlet conditions may play a role here, as demonstrated in Section 4.1.1, where the slug flow regime is observed only in small diameter pipes.

In conclusion, several factors can affect the flow regime: the bundle geometry, the identification mechanism, the two-phase mixture, and the inlet conditions such as the gas injector and homogenization length. Therefore, the flow regime boundaries should be considered as indications, and each flow regime map is case-dependent.

After conducting a thorough analysis of the two-phase flow, identifying various flow regimes, it is interesting to explore the impact of flow characteristics on tube vibration.

Tube vibration

Understanding the relationship between flow dynamics and tube vibration is of significant interest and importance. By replacing the central tube in the facility by a vibrating tube equipped with an accelerometer, as described in Section 3.3, it becomes feasible to examine the vibration behavior of the tube subjected to the previously studied two-phase flow. In this section, to complete the analysis, we will reference the tests conducted during the bubbly flow and intermittent flow experimental campaigns, specifically, tests T1, T2, T3, T4, T5, T6, and T7, as listed in Table 2.1. In Figure 4.32 are reported the tube acceleration spectra of the different tests. The spectra for T1 to T4 were analyzed in Section 3.3. The same considerations for the peaks identification can be applied also for tests T5, T6, and T7. The vibration modes of the tube correspond to the first peak around 10 Hz (first mode) and the peak around 180 Hz (third mode). By analyzing the first mode of vibration, it can be noted how the peak gains power and becomes broader as the air flow rate increases. The peak power is related to the excitation forces and the amplitude of vibration, both of which increase with the air flow rate. The broadening of the peak can be attributed to the flow regime and the void fraction. In fact, as the void fraction increases and the flow regime acquires an intermittent behavior, the hydrodynamic mass effect is subjected to variations. The continuous changes in the effective mass due to variations in void fraction and intermittent flow behavior cause shifts in natural frequencies and alterations in vibration amplitudes, resulting in broader peaks in the spectrum (as explained also by Moran et al. [START_REF] Moran | On the damping in tube arrays subjected to two-phase cross-flow[END_REF]). Hence, the peak broadening in the tube spectrum can be related to a flow regime transition.

A quantitative approach to estimate the peak broadening is the half power bandwidth (HPB) of the peak corresponding to the first mode of vibration. The bandwidth at half power is the width of the frequency range over which the PSD is greater than or equal to its maximum value divided by the square root of 2. The results are reported in Figure 4.33. From the results shown in Figure 4.33, the impact of the flow regime on the tube vibration response can be appreciated. In fact, the tests in bubbly flow (T1, T2, T3), the test corresponding to a flow regime transition (T4), and the tests performed in the intermittent flow regime (T5, T6, T7) can be distinguished, as they are positioned at different levels of bandwidth. More insight on how the flow regime impacts the tube vibration response can be obtained by the PDF of the acceleration data and their full width at half maximum (FWHM), reported in Figure 4.34. In this case the FWHM is the width of the acceleration range calculated at half of the maximum PDF value.

The flow regime effect is reflected also on the PDF of the acceleration data. The peak becomes broader as the air flow rate increase. An abrupt increase of FWHM is observed as the flow regime changes. In fact, by fitting the data in Figure 4.34b with different mathematical functions, it is possible to determinate the best curve that approximate the data. A key factor is the R-squared value (R 2 in Equation 4.22) that provides a way to quantify the goodness of fit of a regression model to the observed data points.

R 2 = 1 - ∑(y -ŷ) 2 ∑(y -ȳ) 2 (4.22)
In this equation, y represents the actual data values, ŷ represents the predicted values from the regression model, and ȳ represents the mean of the data values. The closer the R-squared value is to one, the better the mathematical function approximates the data. Hence, it was found that in Figure 4.34b, the FWHM increases linearly between T1 and T3. However, between T1 and T4, and between T1 and Similar behavior is reflected on the root mean square (RMS) plot of the vibration amplitude (Figure 4.36). The vibration amplitude can be found by integrating twice the accelerometer signal. Double integration of an accelerometer signal to estimate displacement can introduce noise and errors into the displacement signal. This noise and error can have an impact on the evaluation of the RMS of the displacement, affecting the accuracy. To increase the accuracy of the results, filtering techniques are applied, and the displacement is compared to the one recorded by the fast camera (as reported in Section 3.3) to ensure the quality of the results.

Again, in this case, the impact of the flow regime transition on the vibration amplitude can be observed. Specifically, from test T4 (transition flow regime), the RMS of vibration amplitude start to increase exponentially. The intermittent flow regime induces a significant amplitude of vibration, which could result in the unstable behavior of the tube vibration response. Therefore, based on the analysis conducted so far, it is evident that the transition in flow regime significantly affects the vibration behavior of the tube.

The damping is influenced by the dynamic interactions between the gas and liquid phases, which can vary depending on the predominant flow regime. Damping in two-phase flows plays a crucial role in the stability of tube arrays subjected to instabilities. The interactions between gas and liquid phases lead to energy dissipation, affecting the system ability to sustain vibrations. Accurate determination of damping is challenging but essential for predicting the behavior of tube arrays under various flow conditions. Consequently, design guidelines and predictive models have been developed [START_REF] Pettigrew | Two-phase flow-induced vibration: An overview[END_REF]. However, measuring damping in two-phase flow poses difficulties, leading to measured values characterized by high uncertainties [START_REF] Moran | On modeling fluidelastic instability in tube arrays subjected to two-phase cross-flow[END_REF]. In the open literature the most common and accepted method [START_REF] Axisa | Vibration of tube bundles subjected to air-water and steam-water cross flow: Preliminary results on flu1delastic instability[END_REF][START_REF] Nakamura | Dynamics of an in-line tube array subjected to steam-water cross-flow. part 1: two-phase damping and added mass[END_REF][START_REF] Pettigrew | Vibration of tube bundles in two-phase cross-flow: Part 1-hydrodynamic mass and damping[END_REF] to measure the damping in two-phase flow (ζ T P ) is the one reported by Carlucci et al. [START_REF] Carlucci | Experimental studies of damping and hydrodynamic mass of a cylinder in confined two-phase flow[END_REF]:

ζ T P = ζ -(ζ ν + ζ f + ζ s ) (4.23)
Here ζ is the measured value of the damping (using the half power method in this case), ζ s is the structural damping, and ζ f the flow damping, due to the normal component of the fluid drag force on the tube, which is very small in this situation and can be considered negligible in most of the cases [START_REF] Pettigrew | Two-phase flow-induced vibration: An overview[END_REF], ζ ν is the viscous damping calculated according Equation 4. [START_REF] Piteau | Experiments and computations of a loosely supported tube in a rigid bundle subjected to single-phase flow[END_REF].

ζ ν = π √ 8
ρ tp D 2 m + m h 2ν tp πf D 2 ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 1 + (D D e ) 3 1 -(D D e ) 2 2 ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ (4.24)
With m the tube mass, m h the hydrodynamic mass, ν tp the two-phase kinematic viscosity (defined as Equation 4.25 with ν l and ν g liquid and gas kinematic viscosity), ρ tp the two-phase density (defined as Equation 4.25 with ρ l and ρ g liquid and gas density), f the natural frequency, D the tube diameter and the ratio D D e is defined by Pettigrew et al. [START_REF] Pettigrew | Two-phase flow-induced vibration: An overview[END_REF] as D D e = [(1.07 + 0.56P D)P D] -1 for a square pitch tube bundle.

ν T P = ν l (1 -α) + ν g α and ρ T P = ρ l (1 -α) + ρ g α (4.25)
Following this formulation for the two-phase damping calculation, Pettigrew et al. [START_REF] Pettigrew | Vibration analysis of shell-and-tube heat exchangers: an overviewpart 1: flow, damping, fluidelastic instability[END_REF] developed a semi-empirical formulation of damping in two-phase flow (Equation 4.26). The formulation is based on the available database in the literature, with most of the experiments performed in air-water conditions and pitch-to-diameter ratios ranging from 1.22 to 1.47. The expression, take into account the effect of flow regime effect with a function of the void fraction f (α).

ζ T P = A ρ l D 2 m + m h f (α) ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 1 + (D D e ) 3 1 -(D D e ) 2 2 ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ (4.26)
In this formulation A is a coefficient determined experimentally (A = 4 according to Pettigrew et al. [START_REF] Pettigrew | Vibration analysis of shell-and-tube heat exchangers: an overviewpart 1: flow, damping, fluidelastic instability[END_REF]). The value of f (α) take into account the flow regime effect, is estimeted to be = α 0.4 for α < 0.4, = 1 for 0.4 < α < 0.7 and = 1 -(α -0.7) 0.3 for α > 0.7 according to Pettigrew et al. [START_REF] Pettigrew | Vibration analysis of shell-and-tube heat exchangers: an overviewpart 1: flow, damping, fluidelastic instability[END_REF]. Therefore, the measured values of damping in two-phase flow (Equation 4.24) and the semi-empirical formulation (Equation 4.26) can be compared in Figure 4.37. To facilitate a comparison between the formula and the experimental values, the HEM void fraction is employed. This is because Equation 4.26 is designed to estimate damping without requiring precise knowledge of the exact void fraction values. Howerver, in Figure 4.37 the damping values calculated with the measured values of void fraction (α m ) are also reported. The semi-empirical formulation successfully predicts the general trend of two-phase damping. In fact, between tests T1 and T3 (with HEM void fractions of 14% and 40%, respectively), the damping increases. Meanwhile, a change in flow regime occurs at test T4 (57% of HEM void fraction), which corresponds to a decrease in damping as the void fraction increases. However, since the equation serves as a design guideline for steam generators, it is intended to represent a lower bound for damping in two-phase flow. Nonetheless, the damping values obtained for the tests with lower void fractions (T1, T2, and T3) are overestimated by the formula.

Concerning the damping values evaluated with the measured values of void fraction (α m in Figure 4.37), the trend up to test performed with α m = 39% is well reproduced (in a conservative way) by the formula provided by Pettigrew et al [START_REF] Pettigrew | Vibration analysis of shell-and-tube heat exchangers: an overviewpart 1: flow, damping, fluidelastic instability[END_REF]. However, the measured values of the damping for tests with α m > 39% anticipate the decreasing effect due to the intermittent flow regime. This can be attributed to the values of f (α) that were estimated by Pettigrew et al. [START_REF] Pettigrew | Vibration analysis of shell-and-tube heat exchangers: an overviewpart 1: flow, damping, fluidelastic instability[END_REF] for the HEM model. Moreover, the experimental damping values (calculated according to Equation 4.24) are slightly smaller when calculated with the measured void fraction compared to the HEM void fraction. This difference becomes more evident as the void fraction increases, indicating that calculating the damping with the HEM model is not a conservative approach.

Conclusion

A comprehensive analysis of the two-phase flow at the tube bundle inlet and within the tube bundle reveals interesting findings for the intermittent flow regime: large gas structures concentrate in the core of the flow, while small dispersed bubbles are closer to the wall. This distribution makes identifying the flow regime with a fast camera less efficient. The intermittence of large gas structures remains unaffected by the presence of the tube bundle, as their frequencies remain consistent both at the tube bundle inlet and within the bundle. This suggests that the intermittence of this flow regime is likely not caused by the pressure drop introduced by the tube bundle and may be scalable to steam generator conditions. However, further tests should be conducted in this direction by adding more measurement points.

The extensive use of measurement systems provides a database for comparing our data with correlations in the literature to predict void fractions. The HEM model for evaluating the void fraction is ineffective in water-air conditions due to the significant phase slip. Therefore, the development of new models based on the drift-flux approach is encouraged, especially in large pipe configurations. Moreover, the exploration of new models for predicting void fraction in bundles, such as the one developed by Feenstra et al. [START_REF] Feenstra | Modelling two-phase flow-excited damping and fluidelastic instability in tube arrays[END_REF], holds promise and enhances the need for experimental campaign aimed at measuring void fraction in a tube bundle.

The change in flow regime has a noticeable impact on the tube vibration response and damping characteristics. However, predicting damping in two-phase flow remains challenging, and the measurement of damping in such conditions presents some difficulties, despite its critical importance in determining stability limits.

Finally, a flow regime map was proposed. Although it did not fully align with all the findings of previous authors, it is worth noting that variations in bundle geometry can significantly influence the results. Various methods for identifying flow regimes were employed, and a new method based on the frequency of large gas structures was proposed, offering a novel approach to identifying the flow regime and studying the intermittence of the so-called "intermittent" flow regime.

Chapter 5

Numerical simulations of TREFLE

The present chapter is dedicated to the numerical simulations of the experimental campaigns detailed in Sections 3 and 4. In this chapter, a brief literature review of the hybrid approaches to numerically model two-phase flow is presented, with a focus on the approach used in neptune cfd. Subsequently, the numerical setup is detailed, and the simulation results are presented. These results offer valuable insights into the behavior of the two-phase flow under investigation. Key findings, trends, and noteworthy observations are discussed, providing a comprehensive understanding of the outcomes of the numerical simulations.

Two-phase flows modeling

Two-phase flow simulation presents a significant challenge in the field of fluid mechanics today. Numerous approaches have been developed for modeling two-phase flow, with two main models being the homogeneous model and the two-fluid model. The homogeneous model assumes that the two-phase flow can be treated as a single gas-liquid mixture, allowing the Navier-Stokes equations to be written in the conventional form using a mixture density and viscosity. This model assumes that the two phases share the same pressure and velocities. On the other hand, the two-fluid model departs from the assumption of a two-phase mixture and treats the two-phase flow as distinct phases, each with its own set of conservative equations weighted through the void fraction parameter. The interaction between the two phases is represented through interfacial transfer terms.

Two-fluid model

The Euler-Euler two-fluid model, introduced by Ishii [START_REF] Ishii | Two-fluid model and hydrodynamic constitutive relations[END_REF], formulates the approach by considering conservation equations for each phase, weighted with the k-phase volume fraction (α k ), within an Eulerian reference frame. However, since the two phases are not independent, interfacial terms are introduced to account for mass, momentum, and energy coupling. The Navier-Stokes equations for the two-fluid model are presented below, with the energy equation excluded in the scope of this work as it focuses on adiabatic flow.

∂ ∂t (α k ρ k ) + ∇ ⋅ (α k ρ k u k ) = p≠k Γ p→k (5.1) ∂ ∂t (α k ρ k u k ) + ∇ ⋅ (α k ρ k u k ⊗ u k ) = -α k ∇P + α k ρ k g+ ∇ ⋅ [α k µ k ∇u k + (∇u k ) T ] + p≠k I p→k (5.2)
Where P is the pressure, ρ k , u k and µ k are the density, velocity and dynamic viscosity of the k-field. Γ p→k indicates the mass exchanges at the interface between the field k an p (equal to zero in the present case of study). I p→k represents the momentum exchange rate between phase k and p.

The interaction at the interface plays a crucial role in this formulation. As a result, the interaction terms appearing in the equations need to be modeled. These transfer terms must adhere to the balance law, ensuring that the momentum transfer from phase k is equal to the momentum received by phase p, and vice versa. The rate of momentum exchange is significantly influenced by the interfacial area between the phases. Specifically, the momentum transfer at the interfaces should accurately define the momentum exchange between the phases at several scales.

Handling interfaces

Two-phase flows present modeling complexities mainly due to the importance of taking interfacial transfer phenomena into account and the discontinuity of the interface between phases. It is therefore practical to categorize multiphase flows according to the characteristic scale of their interfaces (Figure 5.1) for modeling purposes. This classification results in a diversification between dispersed flows with small interfacial scales (bubbly flow, droplet flow in Figure 5.2a) and segregated flows with large interfacial scales (film flow, stratified flow in Figure 5.2b). It should also be pointed out that in the context of numerical modeling, the terms "large" and "small" interfaces are relative to the size of the local mesh. This categorization lead to two different approaches for modeling multiphase systems: interface averaging and interface resolving. Nevertheless, in many practical multiphase flows, there exists a wide spectrum of interfacial scales (Figure 5.2c), which can result in the degradation of the assumptions made in both interface-averaging and interface-resolving modeling approaches. Over the recent years, this resulted in the introduction of several "hybrid" models designed to address the multiscale characteristics of multiphase flows. Typically, this is accomplished by combining elements from both interface-averaging and interface-resolving approaches.

Dispersed flows: interface averaging approach

In the context of the two-field model developed by Ishii [START_REF] Ishii | Two-fluid model and hydrodynamic constitutive relations[END_REF], the majority of research focuses on studying the bubble regime [START_REF] Brennen | Fundamentals of multiphase flows[END_REF]. While the liquid phase is treated as a continuous phase, the gas phase is considered as a dispersed phase. In this approach, the explicit resolution of the interface is not performed. As a result, the interface structure is lost during the volume-averaging process applied to the governing equations. Consequently, the coupling between the phases is achieved by introducing closures laws for the interfacial terms appearing in the momentum equations. The information regarding the interface is incorporated into these interfacial terms and relies on statistical models.

Segregated flows: interface resolving approach

In gas-liquid two-phase flow, the generation of large structures can occur, and the transition between the dispersed phase and these structures is often not well defined. Numerically, this transition can be influenced by the mesh employed, making interface tracking essential for modeling large structures. According to Denèfle [START_REF] Denefle | Modelisation locale diphasique eau-vapeur des ecoulements dans les generateurs de vapeur[END_REF], three methods can be identified for large interface treatment. The first methodology [START_REF] Unverdi | A front-tracking method for viscous, incompressible, multi-fluid flows[END_REF] involves discretizing the flow field on a fixed grid, while a separate, unstructured grid describes the interface, moving through the fixed grid. However, as the interface deforms frequently, its grid needs to be reconstructed during the calculations.

The second approach [START_REF] Hyman | Numerical methods for tracking interfaces[END_REF] is known as the Lagrangian grid method, which utilizes moving meshes that deform and adapt to the shape of the interface.

The third methodology is a region-following scheme, where the method identify the volume fraction of cells to determine the location of cells relative to the interface. These methods for interface tracking fall under the category of Volume of Fluid (VOF) methods [START_REF] Hirt | Volume of fluid (vof) method for the dynamics of free boundaries[END_REF], which employ the void fraction as the key quantity for interface transport. Consequently, the interface is reconstructed based on the local void fraction and its gradient.

However, the main constraint of the interface-resolving approaches in the simulation of complex multiscale flows is the spatial grid resolution, and the associated computational costs, required to resolve the full range of interfacial scales.

Hybrid models

In several industrial applications, two-phase flows involve a wide range of interfacial scales. To accurately model the various flow regimes encountered, it is crucial to consider both the dispersed phase and large interfaces. Consequently, significant efforts have been made to develop hybrid models capable of effectively capturing the complex topology of multiphase flow within the same time free surface, bubbles or large gas structures.

One of the early explorations in this direction is the work done by Cerne et al. [START_REF] Cerne | Coupling of the interface tracking and the two-fluid models for the simulation of incompressible two-phase flow[END_REF]. The model involved the integration of two different approaches by utilizing a switching function based on a dispersion function. In this work, the authors implemented a dispersion function that utilized the average gradient of a volume fraction across a 3 × 3 cell stencil for two-dimensional problems. The VOF formulation was applied in cells where the dispersion function value was below a user-specified critical threshold, while a standard two-fluid formulation was used in the remaining cells. Determining an appropriate value for this user-defined dispersion threshold was considered a fundamental aspect of the model. Additionally, a one-fluid approach was enforced in the VOF regions, resulting in numerical complexities when transitioning between the one-fluid and two-fluid approaches. The integration of a two-fluid model with an interface-resolving technique for gas-liquid systems was accomplished by Yan et al. [START_REF] Yan | A coupled model for simulation of the gas-liquid two-phase flow with complex flow patterns[END_REF]. Within this methodology, the gas phase was divided into distinct components: one representing dispersed bubbles on a smaller scale, and the other representing larger gas regions. A VOF approach for capturing interfaces was employed specifically for the larger gas regions, leading to the development of a three-field/two-fluid model. Subsequently, other researchers, including Tomiyama et al. [START_REF] Tomiyama | Modeling and hybrid simulation of bubbly flow[END_REF] and Alajbegovic et al. [START_REF] Alajbegovic | Simulation of multiphase flows in complex geometry using a hybrid method combining the multi-fluid and the volume-of-fluid (vof) approaches[END_REF], have explored the coupling of interface-resolving features and multifluid formulations. This method simplifies the model formulation by eliminating the need for complex transitions from one-fluid to multifluid approach within the same computational domain. Consequently, the recent trend has been to adopt this latter approach. In order to simulate free surface flows characterized by large interfacial scales, Strubelj et al. [START_REF] Štrubelj | Two-fluid model with interface sharpening[END_REF] integrated interface sharpening into a two-fluid model. The large interface modeling accounted for drag forces and surface tension forces assessed using the local volume fraction as the weighting factor.

Hansch et al. [START_REF] Hansch | A multi-field two-fluid concept for transitions between different scales of interfacial structures[END_REF] introduced a new concept called GENTOP (GENeralized TwO-Phase flow). In this model, a three-field/two-fluid approach is utilized, dividing the gas phase into two separate phases: a dispersed gas phase and a continuous gas phase. The GENTOP model extends the existing Multiple Size Group approach (developed by Krepper et al. [START_REF] Krepper | The inhomogeneous musig model for the simulation of polydispersed flows[END_REF]) for modeling different flow structures, adding a continuous gas phase to model large gas structures. A free surface indicator field is used to identify these large gas structures, which is defined based on the gradient of the volume fraction.

A hybrid multiphase CFD approach was developed by Wardle et al. [START_REF] Wardle | Hybrid multiphase cfd solver for coupled dispersed/segregated flows in liquid-liquid extraction[END_REF]. The model combines the Euler-Euler multifluid approach with a sharp interface capturing method VOF on specific large interfaces. The activation of the VOF-like methodology can be determined based on either the local gradient of the void fraction, the resolution of the interfacial scale, or the resolution of the dispersed phase length scale.

Mathur et al. [START_REF] Mathur | A hybrid dispersed-large interface solver for multi-scale two-phase flow modelling[END_REF] introduced a hybrid solver based on the Eulerian-Eulerian framework, which combines dispersed and segregated two-phase flow regimes: Hybrid Dispersed-Large Interface Solver (HD-LIS). The identification of large interface regions is achieved by utilizing flow regime maps that rely on the local void fraction value. The solver employs weight functions that rely on volume fractions to incorporate suitable numerical and physical models for each flow regime. By combining interface sharpening and turbulence damping in the segregated regime, and selecting drag and non-drag momentum exchange forces in the dispersed regime, the solver becomes capable of effectively modeling hybrid flows.

A novel approach called GEeneralized Multifluid Modelling Approach (GEMMA) for simulating multiphase flows with a wide range of interfacial scales was introduced by De Santis et al. [START_REF] Santis | A generalized multiphase modelling approach for multiscale flows[END_REF]. Unlike other models, it doesn't rely on predefined thresholds but dynamically determines the local interface resolution required to activate the large-interface model. A blending method is used to control interfacial transfer terms, allowing for both dispersed and large-interface formulations based on the switch status. The model detects localized interfaces based on the resolution of both interfacial morphology and dispersed phase scales and applies a physically accurate large-interface formulation. In the dispersed regime, a standard multifluid formulation is used.

In this work the Generalized Large Interface Model (GLIM) proposed by Merigoux et al. and Davy et al. [START_REF] Merigoux | A generalized large interface to dispersed bubbly flow approach to model two-phase flows in nuclear power plant[END_REF][START_REF] Davy | Euler-euler simulations of condensing two-phase flows in mini-channel: Combination of a sub-grid approach and an interface capturing approach[END_REF][START_REF] Merigoux | Multiphase eulerian-eulerian cfd supporting the nuclear safety demonstration[END_REF] implemented in neptune cfd has been used. The authors aim to model the transitions between dispersed bubbles and large interfaces in neptune cfd. They achieved this by blending the momentum interfacial closure terms between two fields based on the local void fraction and its corresponding gradient. The model is briefly described in the next section.

Multi-regime approach in neptune cfd

The neptune cfd code solves the three-dimensional governing equations based on the two-fluid model developed by Ishii et al. [START_REF] Ishii | Two-fluid model and hydrodynamic constitutive relations[END_REF]. These equations have been described in Section 5.1.1 (Equations 5.1 and 5.2). In this model, the momentum exchange rate between field k and p (I p→k ) can be expressed as the sum of two contributions:

I p→k = I ′ p→k + Γ p→k u I pk (5.3)
Here, I ′ p→k indicates the momentum exchange related to the hydrodynamic forces, while Γ p→k u I pk accounts for the momentum exchange due to the mass exchange between phase k and p (equal to zero in the present case of study). u I pk denotes the interfacial velocity between the two phases. One of the main challenges when dealing with different flow regimes is modeling the interfacial momentum exchange term. In neptune cfd, this challenge has been tackled by combining two models: the dispersed approach [START_REF] Mimouni | Combined evaluation of second order turbulence model and polydispersion model for two-phase boiling flow and application to fuel assembly analysis[END_REF] and the Large Interface Model (LIM) [START_REF] Coste | A large interface model for two-phase cfd[END_REF].

Dispersed approach in neptune cfd

The dispersed approach falls in the category of interface averaging approach described in Section 5.2.1. In this model, the interfacial area is determined by the polydispersed model developed by Ruyer et al. [142]. This model follows the moment-density approach, which consists in assuming a second order polynomial for the bubble diameter distribution function. The transport equation for the interfacial area (a i ) is defined as:

∂a i ∂t + ∇ ⋅ (a i ⃗ u g ) = 2 3 a i αρ g -α dρ g dt + Φ CO + Φ BK (5.4)
where Φ CO and Φ BK are the source terms that account for bubble coalescence and breakup respectively. Consequently, once the interfacial area is estimated, the forces exerted on the dispersed phase can be evaluated. The primary forces acting on the particles are as follows:

• The Drag force, i.e. the force generated by the continuous phase, acting as resistance to the particle motion. For bubbles Ishii et al. [START_REF] Ishii | Drag coefficient and relative velocity in bubbly, droplet or particulate flows[END_REF] presented the following expression of the drag force:

F Drag = 1 8 a i ρ l C D ⃗ u g -⃗ u l (⃗ u g -⃗ u l ) (5.5)
Where ⃗ u g and ⃗ u l are gas and liquid velocity, ρ l is the liquid density. C D is the drag coefficient determined according to Ishii et al. [START_REF] Ishii | Drag coefficient and relative velocity in bubbly, droplet or particulate flows[END_REF]:

C D = 2 3 d g ρ g -ρ l σ ⎛ ⎝ 1 + 17.67 (f (α)) 6 7
18.67f (α)

⎞ ⎠ with f (α) = (1 -α) 1.5
f or distorted bubbles (5.6)

C D = 1 8 (1 -α) 2
f or churn turbulent regime (5.7)

• The Lift force is the force acting on the perpendicular direction to the particle motion. Hence, considering C L the lift coefficient, according to Tomiyama et al. [START_REF] Tomiyama | Transverse migration of single bubbles in simple shear flows[END_REF], the force can be expressed as:

F Lif t = C L αρ l (⃗ u g -⃗ u l ) ∧ ⃗ ∇ ∧ ⃗ u l (5.8)
Where C L is a function of Eötvös (Eo) and Reynolds (Re) number:

For Eo < 4:

C L = min 0.288tanh (0.121Re) , 0.00105Eo 3 -0.0159Eo 2 -0.0204Eo + 0.474 (5.9)
For 4 ≤ Eo ≤ 10:

C L = 0.00105Eo 3 -0.0159Eo 2 -0.0204Eo + 0.474 (5.10)

For Eo > 10:

C L = -0.27 (5.11)
• The added-mass or Virtual-Mass force, is the inertia force due to the volume of liquid that must be displaced as a particle moves through it. Considering C A the added mass coefficient, the force, according to Zuber [START_REF] Zuber | On the dispersed two-phase flow in the laminar flow regime[END_REF], can be formulated as follow:

F V irtualM ass = C A 1 + 2α 1 -α αρ l ∂ ⃗ u g ∂t + ⃗ u g ∇⃗ u g - ∂ ⃗ u l ∂t + ⃗ u l ∇⃗ u l (5.12)
where C A is equal to 1/2 for a spherical bubble and the factor (1 + 2α) (1α) takes into account the effect of the bubbles concentration.

• The turbulent dispersion force take into account for turbulent fluctuations in the flow field acting on the dispersed phase. The turbulence in the continuous phase usually tend to scatter the dispersed phase particles. In the momentum equation, the turbulent dispersion force is generally introduced by a term proportional to the gradient of the dispersed phase volume fraction. Therefore, considering C T D the dispersion coefficient, the turbulent dispersion force can be expressed as reported by Lavieville et al. [START_REF] Lavieville | A generalized turbulent dispersion model for bubbly flow numerical simulation in neptune cfd[END_REF]:

F T D = - 2 3 C T D α l α g ∂α g ∂x i (5.13)
With C T D determined according to Lance et al. [START_REF] Lance | Phase distribution phenomena and wall effects in bubbly two-phase flows[END_REF].

Considering the gas and liquid phase, the interfacial momentum exchange can be written as I l→g and it can be expressed as:

I D l→g = F Drag + F Lif t + F V irtualM ass + F T D (5.14)

Large Interface Model in neptune cfd

The LIM approach [START_REF] Coste | A large interface model for two-phase cfd[END_REF] is applicable in cases where the interfaces between liquid and gas are much larger than the size of computational cells, commonly known as large interfaces. The LIM model consists of two main steps:

• Identification of the large interfaces.

• Application of specific models to these interfaces.

The method used to recognize the large interfaces is based on the refined volume fraction gradient. When the calculated void fraction gradient between two neighboring cells exceeds a certain threshold value, the interface is identified. To handle the residual phase in the cells surrounding the interface cell, the three-cell stencil method is employed (see Figure 5.3). This involves selecting three cells: one on the liquid side (STL), one crossed by a large interface (ST), and one on the gas side (STG). After locating the large interface, the drag force (Equation 5.5) is applied only in the normal direction of the interface. However, to account for friction along the interface, the friction model is applied in the tangent direction. Considering the friction velocity of the gas phase, u * g ,the frictional force density is expressed as:

F I l→g = ρ g u * 2 g a i (u l -u g ) u l -u g (5.15)
Therefore, the interfacial momentum exchange at the large interface is defined as:

I LIM l→g = F I l→g + F I P (5.16)
Here, F I P is a penalized term. The role of this term is to ensure the equality of the normal components of the two velocities u l and u g for numerical purposes. Further details on the models applied at the interface can be found in the literature [START_REF] Coste | A large interface model for two-phase cfd[END_REF].

The Generalized Large Interface Model

The GLIM model [START_REF] Merigoux | A generalized large interface to dispersed bubbly flow approach to model two-phase flows in nuclear power plant[END_REF][START_REF] Davy | Euler-euler simulations of condensing two-phase flows in mini-channel: Combination of a sub-grid approach and an interface capturing approach[END_REF][START_REF] Merigoux | Multiphase eulerian-eulerian cfd supporting the nuclear safety demonstration[END_REF] combines the LIM and Dispersed approach. The method begins by identifying the large interface using the previously mentioned technique. The momentum exchange, I LIM l→g , is then applied to the large interface. In the remaining part of the domain, where α ≠ 1 and α ≠ 0, the interfacial forces are computed according with Section 5.2.1. Identifying the cells in which the large interface is located is a critical step because they may also contain the dispersed phase simultaneously. Therefore, in those cells, the momentum exchange computed using the LIM approach (I LIM l→g in Equation 5. [START_REF] Xu | Hydrodynamics of two-phase flow in vertical up and down-flow across a horizontal tube bundle[END_REF]) and the Dispersed approach (I D l→g in Equation 5.14) are weighted using a weighting factor γ (Equation 5.17).

I p→k = γI D l→g + (1 -γ)I LIM l→g
(5.17) The part labeled as "Tube bundle" in green, as depicted in Figure 5.4, is intended to study the twophase flow in the tube bundle. To study the evolution of two-phase flow in the homogenization section, we extrude the numerical domain in the negative y-direction and crop it, resulting in a new numerical domain labeled "Free duct" in red, also depicted in Figure 5.4. Combining these two domains yields what we later refer to as "long inlet". This combination allows us to assess the code's capability to track the evolution of two-phase flow along the homogenization section and the tube bundle, providing boundary conditions downstream of the gas injector. Further extruding these domains in the z-direction to a total depth of 2 ⋅ D results in the creation of different three-dimensional domains.

Tube bundle: bubbly flow

A two-dimensional computational domain (Figure 5.5) was used to perform the numerical simulations of the bubbly flow experimental campaign. The distance between the inlet and outlet conditions from the tube bundle was chosen to be of 10 times the tube diameter. The choice is the result of a compromise: do not to allow the result in the tube bundle to be influenced by the boundary conditions, and to reduce the size of the domain to reduce computational time. The air and water mass flow rates are set as uniform inlet boundary conditions, i.e., with flat velocity profiles. The void fraction and bubble Sauter mean diameter boundary conditions are measured quantities with the optical dual-tip probe for the bubbly flow experimental campaign (Chapter 3), the turbulent intensity is estimated to be I = 2% (based on the Reynolds number I = 0.16Re -0.125 ). The numerical simulations presented in this chapter were conducted using the following strategy: first, a single-phase simulation with zero void fraction was performed to establish "time-converged" conditions for the water phase (30 seconds for this domain), given the slower velocity compared to the gas phase. Subsequently, the gas phase was introduced by adjusting the void fraction at the inlet boundary condition to its desired value. Sufficient time (30 seconds for this domain) was allowed for the system to reach "time-converged" conditions, after which various quantities were processed and extracted. The main difference among the three different defined meshes can be appreciated in the bubble diameter, as shown in Figure 5.8d. It is worth noting that, despite the very small difference between mesh M 1 and M 1.3, this parameter is highly sensitive to mesh size.

In Figure 5.7, an asymmetry in the velocity and void fraction profiles can be observed. Although these profiles have reached a time-converged state, the asymmetry within the tube bundle still persists. This asymmetry was also reported experimentally in Section 3.2.2.

Nevertheless, the sensitivity study shows satisfactory agreement among the different types of meshes. Therefore, mesh type M 1 was chosen for the numerical simulations, assuming that the results are not affected by the mesh size.

Tube bundle and free duct: bubbly flow

For simulating the bubbly flow experimental campaign, an additional computational domain was created: the inlet of the two-dimensional domain (reported in Figure 5.5) was extended to match the entire facility's length up to the gas injector. This yields what we later refer to as "long inlet" (Figure 5.9), i.e. a twodimensional domain of m = 508, 900 fully hexahedral cells. Consequently, the bubble inlet diameter was determined based on measurements obtained downstream of the gas injector using a fast camera. The void fraction, calculated using Equation 5.21. The calculation involved using gas and liquid velocity (detailed in Table 3.2) to determine the slip ratio. The air and water mass flow rates are defined as uniform inlet boundary conditions, meaning they have flat velocity profiles.

α = 1 1 + vg v l ρg ρ l 1 x -1 (5.21)
Where α is the local value of the void fraction and v g the gas velocity obtained from optical probe measurements, x is the flow quality (x = ṁa ( ṁa + ṁw )), ρ g and ρ l the gas and liquid density. The turbulent intensity is estimated to be I = 2% (based on the Reynolds number I = 0.16Re -0.125 ). Figure 5.9: Two-dimensional computational domain for tube bondle and free duct.

The numerical simulations presented later were conducted using the following strategy: first, a singlephase simulation with zero void fraction was performed to achieve stable conditions for the water phase (60 seconds for this domain), given the smaller liquid velocity compared to the gas phase. Subsequently, the gas phase was introduced by adjusting the void fraction at the inlet boundary condition to its desired value. Sufficient time (50 seconds for this domain) was allowed for the system to reach "time-converged" conditions, after which various quantities were processed and extracted.

Tube bundle: churn/intermittent flow

In order to simulate experiments conducted at higher void fractions, a three-dimensional domain was created by extruding the tube bundle domain in the z direction to a depth of 2 ⋅ D, with the application of periodic boundary conditions (Figure 5.10). This choice was a compromise between computational efficiency, considering the number of cells, and the code's capability to account for the three-dimensional phenomena observed in these experiments. In fact, at elevated void fractions, the gas phase tends to form large structures or bubbles, which can significantly impact flow dynamics. These large gas structures not only displace liquid but also influence the local flow patterns and turbulence characteristics in three dimensions. As a result, simulating such scenarios necessitates a three-dimensional approach to accurately capture the interactions between the phases and their effect on flow behavior.

This results in m = 6, 792, 000 fully hexahedral cells. The air and water mass flow rates are set as uniform inlet boundary conditions, i.e., with flat velocity profiles. For the intermittent flow experimental campaign the inlet boundary conditions data, such as void fraction and dispersed bubble size, are measured via WMS (Chapter 4).The turbulent intensity is estimated to be I = 5% (based on the Reynolds number I = 0.16Re -0.125 ).

Figure 5.10: Three-dimensional computational domain for tube bundle.

The numerical simulations described later followed this approach: initially, a single-phase simulation with zero void fraction was executed to establish stable conditions for the water phase, requiring 30 seconds for convergence due to the smaller liquid velocity compared to the gas phase. Following this, the gas phase was introduced by adjusting the void fraction at the inlet boundary condition to the desired level. Adequate time (5 seconds for this domain) was allocated to ensure the system reached steady-state conditions, following which various quantities were computed and extracted.

Free duct: churn/intermittent flow

To simulate the intermittent experimental campaign, extending the long inlet case in the z direction to a total depth of 2 ⋅ D requires a computational effort that is impractical with such fine mesh. Consequently, the three-dimensional numerical domain was divided into two parts: the first part simulates the homogenization section (free duct depicted in Figure 5.4, computational domain in Figure 5.11) from the gas injector up to the measurements performed with the WMS, while the second part simulates the tube bundle (presented in Section 5.4). The inlet boundary conditions were established as described in Section 5.4. The free duct numerical domain has a total length of 1.4 m, resulting in m = 4, 030, 560 fully hexahedral cells. The numerical simulations presented later were conducted using the following strategy: initially, a single-phase simulation with zero void fraction was executed to establish stable conditions for the water phase. This required 30 seconds for convergence, mainly due to the slower velocity of the water phase compared to the gas phase. Following this, the gas phase was introduced by adjusting the void fraction at the inlet boundary condition to its desired value. A sufficient period (45 seconds for this domain) was allowed for the system to reach "time-converged" conditions, after which various quantities were processed and extracted.

Having established the numerical setup, we now turn our attention to the presentation of the numerical results. In the following sections, we will analyze the outcomes of our simulation experiments.

Bubbly flow numerical results

The experimental data used as inlet boundary conditions to run two-dimensional (2D) numerical simulations for tests T1 and T2 are detailed in Chapter 3. The numerical domain correspond to "Tube bundle" in green in Figure 5.4, as described in Section 5.4. The inlet boundary conditions are reported in Table 5.1.

As described in Chapter 2, measurements taken via an optical probe at the tube bundle inlet are performed in the center of the section, approximately 110 mm below the first row of tubes. However, numerically, the inlet boundary conditions are imposed 300 mm below the first row of tubes. This difference has no effect on the precision of the boundary conditions because, due to the long homogenization section (approximately 1400 mm), a discrepancy of 190 mm in the measurement position at the top of the homogenization section is negligible when compared to the precision of the instrumentation for the characteristics of the two-phase flow. The water flow rate is measured by means of a Coriolis flow meter in g s -1 , while the air flow rate by means of a Bronkhorst air flow meter in L min -1 converted in g s -1 by assuming an air density of 1.2 kg m -3 . Qualitative comparison between the numerical and experimental results are reported in Figure 5.12. The void fraction distribution in the numerical domain is compared with the images obtained via fast camera. The void fraction distribution in the numerical domain follows the bubble path recorded in the images. In particular, for test T1 (Figures 5.12a and 5.12c), the bubbles follow the gap between the tubes, and some bubbles become trapped at the stagnation point under the cylinder. With further increases in air flow rate (test T2 in Figures 5.12b and 5.12d), the numerical void fraction within the bundle starts to increase in line with the experimental observations (the number of bubbles increases). Some of these bubbles begin to accumulate below the cylinders, which is numerically replicated by an increase in void fraction in this zone. The mesh is so thin at the wall that it is possible to resolve the air filaments and the bubble on the first row of cylinders. In conclusions, the numerical results show satisfactory qualitative agreement with the experimental data. The numerical results for void fraction, gas velocity and bubble diameter are averaged over a period of 20 seconds of simulation after reaching a time-converged condition. From the optical probe measurements a quantitative comparison with numerical results can be performed. Figure 5.13 shows the vertical profile in the tube bundles for the analyzed quantities. The tubes are shown in the graphics to provide a visual reference for the locations of the measured quantities within the bundle geometry, with the central tube highlighted in red. From a general analysis, it can be noticed how the numerical results from both tests T1 and T2 predict the vertical void fraction profile trend.

As the two-phase mixture encounters the tubes, the available space for the flow is reduced, causing changes in the flow dynamics. This flow constriction can lead to increased velocity, void fraction and Sauter mean diameter. As matter of fact both experimentally and numerically, these quantities increase as the flow progresses from the bundle inlet to inside the bundle. The flow constriction has a direct impact on the distribution of the gas and liquid phases within the tube bundle. Specifically, it is observed both experimentally and numerically, that it tends to increase the void fraction in regions where the flow is constrained, such as the gaps between the tubes. When the void fraction increases, it means that a larger proportion of the cross-sectional area is occupied by the gas phase (bubbles) compared to the liquid phase. This can lead to the coalescence of bubbles and the formation of larger bubbles within these constrained regions. The influence of flow constriction on void fraction is crucial because it directly affects the behavior of the Sauter mean diameter (d sm = 6α A i ). However, the numerical void fraction and bubble diameter results are in agreement with the experimental data, consistently falling within the measurement uncertainty range. Some discrepancy can be observed in the bubble velocity profiles. In the experiments, these profiles remain relatively constant across the tube bundle. In contrast, simulation results show that velocity profiles follow the trend of void fraction profiles. While both void fraction and bubble diameter from numerical simualtions influence gas velocity, the post-processing of gas velocity through optical probes is not affected by the void fraction. Therefore, it's possible that the amount of bubbles in the experiments is too small to reflect the geometric effects of the bundle on bubble velocity, as explained in Section 3.2.2. In contrast, in the numerical results, these effects are linked to the void fraction parameter. In fact, the numerical gas velocity profile shape is only hinted by the experimental profiles. Furthermore, the small value of the liquid velocity used in the experimental campaign minimizes its contribution concerning this geometrical effect, especially when compared to the gas velocity.

To gain a deeper understanding, it is interesting to plot the liquid velocity profiles. Figure 5.14 presents a comparison between numerical and experimental gas and liquid velocity profiles, along with the liquid velocity profile obtained from single-phase numerical simulations. This simulation is conducted with an inlet void fraction of 0 and a water mass flow rate of 2500 g s -1 to examine the mutual influence of the gas and liquid phases. The experimental water velocity profile is calculated using the following Equation 5.21. Hence, rearranging the terms, the liquid phase velocity v l can be estimated as follows:

v l = v g ρg ρ l 1 x -1 1 α -1 (5.22)
By analyzing the results from Figure 5.14, it can be observed that the liquid velocity from the singlephase simulation (Neptune v LO l in Figure 5.14) increases from the tube bundle inlet to within the tube bundle and then remains almost constant. The liquid velocity from the two-phase simulation (Neptune v l in Figure 5.14) follows the gas velocity trend. This confirms the previous hypothesis: the evolution of the water velocity within the bundle is not affected by the constriction effect of the gaps; rather, it is the void fraction (profile reported in Figure 5.13) that affects the liquid velocity profile inside the tube bundle. In fact, the calculated v l (Exp v l Figure 5.14) is influenced by the void fraction because it is determined according to Equation 5.22. Consequently, numerically, void fraction and gas/liquid velocity are strongly linked, while experimentally, they are weakly linked. This leads to a bias between the numerical and experimental gas-liquid velocity difference (v gv l ) due to the fact that the gas velocity trend is not fully predicted by the numerical simulations. This disparity can impact the calculation of momentum exchange at the interfaces between the two phases in the numerical simulations. However, the numerical and experimental slip ratio (v g v l ) profiles are in agreement, emphasizing good agreement for the void fraction parameter. Even though some experimental points are not well predicted by the numerical simulations, the error between them do not overcome the 35 % of the measured value. However, these results still provide an encouraging outlook. In order to simulate numerically tests T3 and T4 (in Table 5.1) a three-dimensional domain will be necessary. Since the flow is mainly bubbly but 3D effects and larger two-phase structure start to take place.

Boundary conditions sensitivity

In order to understand the effect that different boundary conditions may have on the simulation results, four cases were set up. Two of these cases tested different turbulent intensities as the inlet boundary condition, one examined the effect of air compressibility, and the last one involved extruding the inlet for approximately 1000 mm to impose the boundary conditions measured with the fast camera in the bubbly flow experimental campaign during the gas injector study (Section 3.1). Numerical simulations were conducted only for test T1, with the results of the reference case presented in Figure 5.13.

• "I 5%" case: in this scenario, the numerical domain and boundary conditions were identical to the reference case, with the only difference being an increase in the inlet turbulent intensity from 2% to 5%.

• "I 10%" case: similar to the "I 5%" case, the numerical domain and boundary conditions matched those of the reference case, but the inlet turbulent intensity was increased from 2% to 10%.

• Compressible case: in this instance, the numerical domain and boundary conditions were the same as the reference case, except that it was no longer incompressible. Air density varied with pressure P (while temperature T remained constant) following the equation ρ = P ⋅ M (R ⋅ T ), where M represented the air molar mass, and R was the universal (or perfect) gas constant.

• Long inlet case: for this case, the inlet of the two-dimensional domain was extended to match the entire facility's length up to the gas injector (details about the numerical domain and simulation strategy are reported in Section 5.4). Consequently, the boundary conditions were determined based on measurements obtained downstream of the gas injector using a fast camera. This included setting the bubble inlet diameter to 5 mm and calculating the void fraction using Equation 5.21. The calculation involved using gas and liquid velocity (detailed in Table 3.2) to determine the slip ratio. The air and water flow rates remained consistent with the reference case.

The results, presented in Figure 5.15, are compared to both the numerical results obtained from the reference case and the experimental results in terms of vertical profiles for gas velocity, void fraction, and bubble diameter. The vertical profiles obtained from "I 5%", "I 10%", and compressible cases show no substantial difference from the reference case. However, the long inlet case deserves further investigation. The velocity profile trend obtained is closer to the experimental results than the reference case. However, this behavior is a consequence of the void fraction profile, which in this case does not match the experimental results properly and is not subjected to the geometrical effect of the tube bundle. The mismatch suggests that the simulation or model might not accurately predict the distribution of gas bubbles. In fact, the bubble diameter arrives at the tube bundle inlet with a value larger than the measured value and remains almost constant within the bundle due to a significant coalescence effect. As the two-phase flow move through the bundle, the bubble diameter mostly remains the same but tends to decrease as they reach the outlet. It appears that, in this case, the code identifies bubble diameters large enough for breakup to occur after interacting with the tube bundle. It is important to remind the reader, that 2 cylinder are present to inject the air, introducing a modification of the flow and consequently an effect on the bubble diameter which is not reproduced on the present simulation (flat inlet) profile.

Focus on the long inlet case

Further investigation of the long inlet case can be conducted by comparing the tube bundle inlet conditions (TB inlet in Figure 5.16) with the experimental data (optical probe measurements at the tube bundle inlet, point 5 in Figure 2.24) and analyzing the flow evolution from few centimeters after the inlet boundary conditions (P1 in Figure 5.16). The results of such comparison are reported in Figures 5.17 From the gas velocity profiles (Figure 5.17a), it can be observed how the numerical gas velocity decreases as the flow evolves from the "P1" profile to the "TB inlet" profile. This observation aligns with experimental findings where the gas velocity is higher at the injection point, likely due to the smaller void fraction values, and lower at the inlet of the tube bundle. However, there is an approximately 60% error between the experimental and numerical results for the gas velocity at the tube bundle inlet.

On the other hand, the liquid velocity profile remains relatively constant on average, even though its shape differs. The velocity profile, peaking closer to the wall, may be attributed to the gas phase behavior. Both the gas and liquid velocity profiles exhibit this wall-peak effect, as they mutually influence each other through the drag force. The impact of gas velocity on liquid velocity profiles is evident when examining the profiles generated through single-phase (liquid-only) numerical simulations in Figure 5.17b. Figure 5.17: Flow evolution between "P1" and "TB inlet" from numerical results numerical results (NCFD). In addition, a comparison between numerical and experimental results at tube bundle inlet is reported ("TB inlet"). "LO" indicates results from single phase (liquid only) simulations.

The void fraction profile (Figure 5.18a) increases as we move from "P1" to "TB inlet". This increase was also observed experimentally by calculating the void fraction at the gas injector based on the slip velocity, using the gas and liquid velocities detailed in Table 3.2, and comparing it with the void fraction measured by the optical probe. The wall-peaked void fraction profile is a common phenomenon in the literature, as reported by Serizawa et al. [START_REF] Serizawa | Phase distribution in two-phase flow. Transient Phenomena in Multiphase Flow[END_REF], Liu et al. [START_REF] Liu | Structure of air-water bubbly flow in a vertical pipe-i. liquid mean velocity and turbulence measurements[END_REF], and Ohnuki et al. [START_REF] Ohnuki | An experimental study on developing air-water two-phase flow along a large vertical pipe: effect of air injection method[END_REF], which is consistent with the gas velocity profile.

However, the bubble diameter profile (Figure 5.18b) exhibits a different behavior compared to the experimental data, where smaller bubble diameters are observed at the tube bundle inlet compared to the gas injection point. This particular experimental behavior involves the formation of bubbles with a critical Weber number, and along their path, their breakup is promoted. This behavior is not replicated by the numerical simulations, and instead, the bubble diameters increase as the flow progresses through the homogenization section, resulting in a core-peaked profile. .18: Flow evolution between "P1" and "TB inlet" from numerical results numerical results (NCFD). In addition, a comparison between numerical and experimental results at tube bundle inlet is reported ("TB inlet").

Additionally, a three-dimensional domain is employed by extruding the meshes in the z direction to a total depth of 2⋅D and applying periodic boundary conditions. This is done to investigate potential threedimensional effects that may influence these bubbly flow simulations. However, as shown in Figure 5.19, the results closely are very similar to those of the reference case, indicating that three-dimensional effects are negligible in this scenario. In terms of sensitivity to boundary conditions, the main outcome is that simulations conducted using the conditions described for the reference case provide the best approximation of the two-phase flow in the tube bundle. Nevertheless, sensitivity analysis could be extended by exploring different gas velocities, void fractions, and bubble diameters within the range of uncertainty. Furthermore, simulations carried out with the long inlet configuration reveal certain shortcomings in the models used to predict bubble diameters, which can in turn affect gas velocity and void fraction. Despite these limitations, the results are promising. Consequently, the use of CFD, once validated, is of paramount importance in complementing experimental data and advancing our understanding of this topic.

Free duct: churn/intermittent simulations

Extending the long inlet case presented in Section 5.5.1 in the z direction to a total depth of 2 ⋅ D requires a computational effort that is impractical with such fine mesh. Therefore, the numerical domain was divided into two parts: the first part simulates the homogenization section (free duct depicted in Figure 5.4) from the gas injector up to the measurements performed with the WMS, while the second part simulates the tube bundle (presented in Figure 5.4) as described in Section 5.4. This section is dedicated to the numerical simulation of the free duct for test T4 and T5.

For test T4, the boundary conditions regarding void fraction and bubble diameters were characterized by means of a fast camera in Section 3.1. Therefore, they are imposed as specified in the long inlet case of the previous section (5.5.1), with the bubble inlet diameter set to 5 mm and the void fraction calculated using Equation 5.21. The calculation involved gas and liquid velocities (as detailed in Table 3.2) to determine the slip ratio.

In the case of test T5, the inlet conditions for the free vein were not analyzed due to the inability to use the fast camera to characterize injector type 2. Therefore, certain assumptions were made. The inlet bubble diameter was set to 5 mm, as determined in test T4, while the void fraction was calculated using Equation 5.21. In this case, the slip ratio was calculated by using the gas velocity obtained with the optical probe at the tube bundle inlet and the water velocity calculated as the ratio of the water superficial velocity to the liquid fraction (v l = U sl (1α)).

The results show that the code is capable to replicate the transition between the bubbly/transition flow regime for test T4 and the churn flow regime for test T5 (Figure 5.20). Bubbly/transition flow is characterized by the presence of small gas bubbles dispersed within the liquid phase with larger bubbles identified in the core of the duct, while churn flow represents a more complex regime with larger gas structures and increased interaction between phases. for test T5, the void fraction ranges from 0 to 0.5 for the dispersed phase, while an isosurface for void fractions greater than 0.5 is applied to identify the large gas structures.

The measurements obtained with the probe and fast camera at the tube bundle inlet for test T4 can be compared with profiles extracted near the outlet of the free duct numerical domain, as shown in Figure 5.21. The profiles are averaged over 10 physical seconds of simulation. In terms of void fraction (Figure 5.21a), there is approximately a 30% discrepancy between the experimental and numerical values, as reflected in the velocity profile (Figure 5.21b). It was observed that the wall-peaked void fraction and gas velocity profiles observed in the bubbly flow (Figure 5.18a) are transformed into core-peaked profiles in this case. This behavior is attributed to the transition flow regime, characterized by the presence of larger bubbles concentrated in the flow core. As the gas volume fraction increases and the bubble size distribution changes, the flow dynamics can shift, leading to core-peaked profiles.

The velocity profile follows the void fraction profile, but it is noteworthy that negative velocities are recorded near the wall. This observation aligns with experimental findings of stagnant or downwardflowing bubbles along the wall, as captured by the fast camera. This behavior can be explained by the fact that the gas velocity in the core of the flow drags the liquid, which is less affected by the lower velocities near the wall, potentially leading to the formation of recirculation zones closer to the wall. Hence, the drag forces exerted by the gas phase on the liquid can create regions of reverse flow or recirculation near the wall. This phenomenon is physically consistent with the large slip ratio (about 5) between the phases due to low liquid velocities.

Concerning the bubble diameter behavior, the simulation aligns well with the experimental data. Figure 5.18a presents measurements obtained through the fast camera and optical probe, alongside with the numerical simulation result. The difference between the optical probe measurement (d sm ) and the average value of the "bubble size distribution" ( d eq ) is discussed in Section 2.5.2 ( d eq is the average value of the "bubble size distribution"). The equivalent diameter measured with the fast camera concerns bubbles closer to the walls, while the optical probe is positioned within the flow core. In the case of test T4, both experimental and numerical results exhibit a core-peaked distribution of bubble diameters, indicating the presence of a transition flow regime with larger bubbles concentrated in the core. This consistency between experimental and numerical findings reinforces the reliability of the simulations.

The measurements acquired using the probe and WMS at the inlet of the tube bundle in test T5 are suitable for comparison with profiles extracted near the outlet of the numerical domain corresponding to the free duct, as depicted in Figure 5.22. These profiles represent averages over a 10 seconds simulation duration.

(a) (b) Also in test T5 the core-peaked void fraction profile (Figure 5.22a) is consistent with the churn flow regime experimentally detected by the WMS. Hence, the gas velocity profile follow the void fraction profile trend. However, in this case (profiles in Figure 5.22) the error committed by the code is slightly larger with respect the previously analyzed test T4. This may be a limitation of the fictitious three-dimensional approach. In fact, the large gas structures recorded by the WMS have sizes up to 90 mm, while the depth of the numerical domain is of 60 mm with periodicity boundary conditions. Hence, due to the large structures the buoyancy force prevails over the inertia forces. however, the code faces limitations in accurately modeling the sizes of these structures, resulting in an underestimation of gas velocity and an overestimation of the void fraction.

The void fraction profile slightly skewed to the positive x-coordinates is an interesting finding. In fact even numerically the profile does not appear exactly symmetrical. It could be related to the bubbles swarm behavior that follows a path that is S-shaped. In fact it can be appreciated by the WMS experimental results and from the numerical simulation. This comparison is reported in Figure 5.23. The WMS image, as detailed in Section 4.1, is generated by extracting void fraction distributions across the cross-section of the free duct from the 2D distribution and arranging these distributions as horizontal lines in a time sequence. To convert the time-axis into meters and match the domain length numerically, the formula y = t ⋅ U sg ᾱ is applied, where y represents the distance in the domain, t is the time sequence, U sg is the gas superficial velocity, and ᾱ is the average void fraction. With y determined, along with U sg and ᾱ, we can calculate the number of frames required to scale the image to match the size of the numerical domain.

The comparison presented in Figure 5.23 shows qualitative agreement between the numerical and experimental results, with both replicating the size of the large gas structures and the S-shaped distribution. For a more quantitative analysis, we examine the void fraction signal at the monitoring point in the core of the numerical domain, located near the outlet of the free duct, as shown in Figure 5.24. The time evolution of void fraction in Figure 5.24a displays the characteristic behavior of a churn flow regime, with peaks representing the passage of large gas structures and depressions meaning the passage of the dispersed phase through the monitoring point (highlighted in Figure 5.24b). Performing a Power Spectral Density (PSD) analysis on the signal (Figure 5.24c) reveals a frequency peak at approximately 8 Hz. Apparently, the code successfully captures a periodicity in the frequency of passage of large gas structures. However, this frequency differs from the experimental finding in Section 4.1, which was approximately 3 Hz. This discrepancy arises due to the limitations of the domain size compared to the gas structure size in the z-direction, leading to errors in predicting void fraction and gas velocity by the code. Additionally, it's important to note that the analyzed signal starts at the 35th second of the simulation, providing only 10 seconds for PSD analysis. In contrast, the experimental data had 1000 seconds available for analysis, which may influence the results by allowing the detection of frequencies that require more time to reach significant power levels.

In conclusion, the numerical results demonstrate a qualitative agreement with the experimental data. The code successfully reproduces the flow regime in the large duct for both test T4 and T5. Nevertheless, a closer examination reveals discrepancies between the numerical results and the experimental data. In fact, the error in predicting the void fraction and gas velocity increases as the air flow rate rises (from test T4 to test T5). However, it's worth noting that this error can be attributed to limitations in the numerical domain size (details about the numerical domain and simulation strategy are reported in Section 5.4). Now that we have thoroughly discussed and analyzed the numerical results from the free duct simulations, our focus can shift to the numerical simulations conducted in the tube bundle.

Tube bundle: churn/intermittent simulations

The experimental data are used to run numerical simulations for tests T4 and T5 (detailed in Chapters 3 and 4). The inlet boundary conditions imposed at the tube bundle inlet are reported in In the case of test T5, the boundary conditions are determined using different measurement systems. The water flow rate is quantified using a Coriolis flow meter, expressed in g s -1 , while the air flow rate is measured with an air flow meter, also expressed in g s -1 . The void fraction is assessed using the WMS, which provides an average over both time and space. For dispersed bubble diameters, Figure 4.13 is utilized to establish a threshold distinguishing between the dispersed phase and large gas structures. This figure illustrates the contribution of various bubble diameters to the void fraction (∆α ∆d B ). The first peak, occurring around 4 mm, is chosen as the average diameter of the dispersed phase.

Regarding test T4, the water flow rate is measured using a Coriolis flow meter, denominated in g s -1 . In contrast, the air flow rate is quantified with a Bronkhorst air flow meter, initially recorded in L min -1 , and then converted to g s -1 by assuming an air density of 1.2 kg m -3 . The void fraction, on the other hand, is determined using an optical probe located approximately 110 mm below the first row of tubes.

The void fraction distribution inside the tube bundle, as shown in Figures 5.25, 5.26 and 5.27, is compared with video frames obtained using a fast camera. The numerical simulation results are in agreement with the experimental data. To highlight large gas structures, an isosurface for void fraction values exceeding 0.5 is traced.

The numerical instantaneous void fraction distribution for test T4 is depicted in Figure 5.25a, while the corresponding experimental image is presented in Figure 5.25c. Both the experiment and the simulation reveal that the dispersed phase is predominant within the bundle. In contrast with the previous examined bubbly flow regime, the regular distribution of bubbles is disrupted and regions of higher gas concentration are generated. In fact, the simulation results exhibit larger structures approximately the size of the gap, consistent with sporadic observations of larger gas structures (on the order of the gap size) traversing the tube bundle in the experiment.

This behavior is also observable in Figure 5.26, where the numerical domain fully displays the tube bundle, providing a lateral view. Although the numerical results do not initially depict the large gas structures at the tube bundle inlet, they form within the bundle due to the interaction between the two-phase flow and the tubes. This phenomenon can be attributed to the finer mesh inside the bundle, ensuring an adequate number of cells for the LIM model to be effective. In fact, within the bundle, even small bubbles are represented as accumulations of high void fraction zones (depicted in brown).

Transitioning to the numerical results of test T5 (Figure 5.25b) and their comparison with the experimental observations (Figure 5.25d), a substantial increase in the presence of large gas structures is evident. Both the numerical results and experimental observations indicate a change in the flow regime compared to the previously examined conditions. The flow pattern is characterized by the formation of large, irregular gas structures that can vary in size and shape, exhibiting a chaotic behavior.

The comparison presented in Figure 5.27 shows agreement between experimental and numerical results. The resemblance between the lateral view of the tube bundle captured with a high-speed camera and the numerical simulation is remarkable. The numerical results show that in this case as well, the emergence of large structures is observed only within the bundle. This phenomenon can be attributed to the relatively small distance between the tube bundle and the inlet boundary condition, which prevents the development of large structures before reaching the bundle. As we observed in the previous section, simulations conducted in the free duct exhibit substantial gas structures. Furthermore, the code successfully preserves these large gas structures at the bundle outlet. To conduct a more comprehensive analysis, we compare the numerical results with experimental data obtained using an optical probe within the bundle. For consistency in our comparisons, the code's results are averaged over a few seconds.

The results for test T4 are presented in Figure 5.28, depicting gas velocity and void fraction profiles within the bundle. The tubes are shown in the graphics to provide a visual reference for the locations of the measured quantities within the bundle geometry, with the central tube highlighted in red. In this particular case, the CFD results exhibit overall consistency with the experimental data. Nevertheless, there is a small difference in the trend of the gas velocity profile compared to the experimental result.

As observed in Figure 5.26, the limited size of the computational domain, characterized by a small length in the z-direction and insufficient periodicity, results in inadequate resolution of the structures. Consequently, this leads to larger gas structures and higher velocities within the gap. Instead of breaking within the gap, the gas structures maintain their size, enabling them to accelerate between two rows. This discrepancy may be also attributed to the inherent complexity of the transition flow regime. In two-phase flows, especially within this regime, the interactions between gas and liquid phases can give rise to intricate turbulence patterns that are not accurately captured by conventional CFD RANS models. As a result, the void fraction profile slightly overestimates the experimental values, but the evolution through the tube bundle is well reproduced. These limitations become more evident when transitioning to test T5 (Figure 5.29). In this case, the periodicity condition and the small size of the domain, does not allow the quantity involved to benefit from the entire size of the large gas structures. In fact, in this case, the void fraction does not predict the experimental trend well.

As the intermittent flow regime comes into play, the complexities of the two-phase pattern increase. Even though some limitations rise in test T5, it's important to highlight the potential of CFD modeling. In this challenging regime, where the flow undergoes sporadic transitions between gas and liquid phases, CFD remains a valuable tool for gaining insights into the dynamics of two-phase flows. While conventional RANS models may face difficulties capturing every aspect of the highly dynamic and intricate flow behavior, they still provide a foundational understanding of the system. Furthermore, it's crucial to emphasize the role of boundary conditions. Notably, the gas velocities at the inlet of the tube bundle differ from experimental results. Numerically, the inlet gas velocity measures approximately 0.15 m s -1 for both tests, while experimentally, it reaches 0.51 m s -1 for test T4 and 1.17 m s -1 for test T5. This disparity may arise from the limited length of free duct simulated in these cases, which is only 10 times the tube diameter. In contrast, when the entire free duct is considered in the previous section (as indicated in the outlet velocity profile figures, Figures 5.21b and 5.22b), the difference in inlet velocity is much smaller. The close proximity between the inlet boundary condition and the tube bundle restricts the formation of large gas structures, which are formed by the code only when the two-phase flow interacts with the bundle. The numerical simulations required substantial computational effort and time that did not allow for the use of a larger domain in the context of this work. However, an improvement in the accuracy of the numerical results in predicting the experimental values may be achieved by using actual velocity profiles as inlet boundary conditions.

Conclusion

In this chapter simulations performed with the neptune cfd code are compared with the experimental results. The literature review of the various multiregimes models available in the open literature is presented with a particular focus on the neptune cfd models.

Few cases from the experimental campaigns are selected and numerically simulated with the CFD code: two cases bubbly flow, one case in the transition regime and one in the intermittent flow regime. Different numerical domain were setup in order to simulate the different cases and to provide an overview of the two-phase flow in the homogenization section and on the tube bundle.

For the bubbly flow cases the analysis reveals a satisfactory qualitative agreement between the numerical and experimental results. However, some differences exist, especially in the gas velocity profiles, where numerical predictions do not fully capture the experimental trends. These discrepancies are attributed to the intricate nature of the flow, the influence of void fraction, and the limited geometric effects considered in the simulations. Nevertheless, the overall agreement for the void fraction parameter is promising. Further insights into the relationship between void fraction and gas/liquid velocity, along with a discussion of the slip ratio, underline the complexities of this study.

An investigation of the boundary conditions sensitivity reveals that several cases, including "I 5%", "I 10%" and compressible conditions, do not substantially differ from the reference case. However, the long inlet case exhibits closer alignment with experimental velocity profiles, mainly due to variations in the void fraction profile. This mismatch suggests that the simulation or model might not accurately predict the distribution and breakup of gas bubbles.

Concerning tests at higher air flow rates the chapter provides simulation results for the free duct and the tube bundle section. Concerning the free duct, comparisons between experimental and numerical data for test T4 reveal small discrepancies. Gas velocities exhibit negative values near the wall, consistent with stagnant or downward-flowing bubbles. In test T5, the core-peaked void fraction profile matches the churn flow regime observed experimentally. Gas velocity discrepancies are slightly larger than in test T4, potentially due to limitations in modeling large gas structures. A skewed void fraction profile suggests S-shaped bubble swarm behavior. Qualitative agreement with experimental data is demonstrated, with both capturing large gas structure sizes and distribution. A Power Spectral Density analysis of void fraction signals reveals a periodicity discrepancy between numerical and experimental findings, attributed to domain size limitations.

In the tube bundle simulations for test T4, both numerical and experimental data show a transition in flow regime within the bundle. In the case of test T5, both numerical and experimental results indicate a churn flow regime characterized by chaotic, irregular gas structures of varying sizes and shapes. The study also discusses the comparison of gas velocity and void fraction profiles within the bundle. While there is overall consistency between the CFD results and experimental data, a slight difference is noted in the profile trends. This variation is primarily attributed to the size of the domain and, consequently, the small periodicity that does not allow the resolution of the full gas structures. This, in turn, leads to differences in the numerical results when compared to the experimental data. However, the numerical simulations required substantial computational effort and time that did not allow for the use of a larger domain in the context of this work.

Finally, the role of boundary conditions is emphasized, with particular attention to the differences in gas velocities at the inlet of the tube bundle between numerical and experimental results. The proximity of the inlet boundary condition to the tube bundle restricts the formation of large gas structures in the numerical simulations, contributing to the disparity in inlet velocities.

In conclusion, the study highlights the strengths and limitations of CFD modeling in capturing complex two-phase flow behaviors within a tube bundle and underlines the importance of boundary conditions in obtaining accurate simulations. Despite challenges, CFD remains a valuable tool for understanding the dynamics of such intricate two-phase flow patterns.

Chapter 6

General conclusions

The current state of understanding regarding vibration excitation mechanisms in tube bundles under two-phase cross flow was discussed. It was conclusively demonstrated that there is still much to learn in this area, particularly regarding the impact of factors like void fraction, flow regime, flow pattern, and flow distribution on flow-induced vibrations.

The main objective of the presented work was to gain a deeper understanding. This was achieved through accurate experimental measurements of two-phase flow characteristics and tube vibration using a new test facility called TREFLE, which was installed in 2022. TREFLE also aims to provide high-quality data for validating multiphase CFD simulation tools.

TREFLE facility design summary

The TREFLE facility is well-equipped to conduct experiments on two-phase flow-induced vibration, and its key components were carefully designed and validated to ensure reliable and meaningful results.

In particular, a new gas injector was designed. Additionally, various methods for predicting bubble diameter were discussed, with an emphasis on their applicability to the experimental conditions.

A special divergent with honeycomb and plates for achieving uniform velocity profiles in the water inlet was designed. CFD analysis was used to optimize the design and suppress flow asymmetries. Experimental results confirmed the effectiveness of the implemented system in guaranteeing a nearly flat water velocity profile downstream of the honeycomb.

A newly designed flexible vibrating tube system was conceived, describing its mode shapes and natural frequencies. The design process incorporated iterative adjustments to ensure optimal performance. The vibrating tube's properties, such as mass, natural frequency, and damping, were determined experimentally.

Bubbly flow experimental campaign: key findings

The results obtained from the extensive analysis of bubble generation and behavior provide valuable insights into the complex dynamics of the air injection system and its impact on bubbly flow within the TREFLE facility. The analysis of bubble generation considered various parameters, including gas flow rates, liquid velocities, bubble equivalent diameters, aspect ratios, Eötvös, Weber, Froude and Reynolds numbers. The results revealed how these factors influence the formation, size, and behavior of bubbles in the bubbly flow regime.

The Weber number and Froude number were used to assess the dominant forces governing bubble behavior. The large Weber values observed indicate the dominance of inertial forces over surface tension forces, leading to bubble deformation and fragmentation. The large Froude values observed suggest that inertial forces are more significant than gravity, potentially causing bubbles to elongate or deform.

The experimental data for bubble equivalent diameter were compared with existing correlations. While some correlations provided better approximations than others, none of them fully matched the experimental results. The correlations by Akita et al. [START_REF] Akita | Bubble size, interfacial area, and liquid-phase mass transfer coefficient in bubble columns[END_REF] appeared to be the most suitable for the present experiments, but discrepancies still existed, likely due to variations in experimental conditions.

A comprehensive analysis of the tube bundle inlet conditions was conducted, focusing on the behavior of bubbles in the two-phase flow as it approaches the tube bundle.

To assess the characteristics of bubbles entering the tube bundle, measurements of bubble equivalent diameter, Eötvös number, and bubble aspect ratio were performed. The probability density function of bubble diameters and the Eötvös number indicated that bubbles maintained a wobbling shape as they approached the tube bundle. An analysis of bubble size showed a decreasing trend in bubble equivalent diameter as bubbles approached the tube bundle, aligning with observations of unstable bubble shapes and breakup mechanisms along their path.

Measurements from the optical dual-tip probe placed upstream of the bundle were compared with those obtained from the fast camera. The comparison indicated that while the probe provided data for larger bubble sizes, the fast camera primarily detected bubbles near the wall. This discrepancy was particularly noticeable in tests with higher gas flow rates, where larger bubbles appeared in the core of the two-phase flow and were captured by the probe.

Valuable measurements of void fraction and gas velocity at the tube bundle inlet were also provided. These measurements are crucial for establishing CFD-grade boundary conditions. Importantly, the study revealed that using the Homogeneous Equilibrium Model (HEM) to estimate void fraction can lead to significant errors, emphasizing the limitations of this model. Within the tube bundle, a detailed analysis of the two-phase flow behavior provides interesting findings. The images captured within the tube bundle for different test configurations reveal varying bubble paths and behaviors as the air flow rate increases. A notable observation is the sporadic presence of air plugs crossing the bundle in Test T4 (gas superficial gap velocity J g = 0.168 m s -1 ), indicating a complex and dynamic flow.

Tests T4, in particular, exhibits a bi-stable behavior in the two-phase flow, where the flow direction can switch between two equilibrium positions. This phenomenon can be influenced by disturbances in the flow, such as the passage of air plugs.

Vertical profiles inside the tube bundle indicate that bubble diameter and bubble velocity tend to increase in the gaps between tubes and decrease outside these gaps, suggesting a possible correlation with geometrical effects. This behavior is more pronounced in tests at higher gas superficial velocity, correlating with increased bubble population.

Concerning the investigation of tube vibration response within the TREFLE experimental setup, frequency peaks associated with tube vibration are identified and analyzed. A comparison is made between the hydrodynamic mass calculated using experimental data and the semi-empirical correlation proposed by Pettigrew et al. [START_REF] Pettigrew | Vibration of tube bundles in two-phase cross-flow: Part 1-hydrodynamic mass and damping[END_REF]. While the semi-empirical formula successfully captured the general trend, discrepancies are observed for void fractions exceeding 0.4 when using actual void fraction values. Different modes of vibration of the tube can interact with the fluid differently, affecting the hydrodynamic mass contributions.

Intermittent flow experimental campaign: key findings

The exploration of two-phase flow characteristics upstream of the tube bundle revealed a complex and dynamic fluid behavior. The highly chaotic nature of the flow, marked by the continuous generation of large gas structures, posed significant challenges for conventional bubble detection methods with fast camera. Nevertheless, qualitative insights into the flow regime were obtained, shedding light on its intriguing features.

An interesting phenomenon was observed during the video analysis: the downward motion of bubbles closer to the wall. This motion was explained by the interplay between the upward drag force from fastmoving air in the central region and the downward pressure gradient near the wall, within the boundary layer.

To gain a deeper understanding of the flow structure, a wire mesh sensor (WMS) was employed. The large gas structures exhibited a churn-like motion within the flow, and their irregular distribution was a prominent feature. Power spectral density analysis identified a characteristic frequency associated with the passage of large gas structures within the flow, which remained relatively constant across different air flow rates. This suggested a periodicity in the behavior of large gas structures, relatively independent of air flow rates. Comparison with an optical probe positioned upstream of the tube bundle provided corroborative results.

The analysis of bubble size distributions revealed peaks corresponding to dispersed bubbles and various sizes of large gas structures.

Different flow regime maps available in the literature were investigated derived from experiments conducted in large-diameter pipes. They revealed that in such pipes, the slug flow regime was absent, and the transition occurred directly from bubbly to churn flow.

The study of two-phase flow in vertical ducts was a subject of extensive research, leading to the development of various flow regime maps and drift-flux correlations. Several correlations were considered and validated against our experimental data. The results indicated that the Hibiki et al. [START_REF] Hibiki | One-dimensional drift-flux model for two-phase flow in a large diameter pipe[END_REF] correlation was the best approximation for the experimental data in the present study. However, it was noted that the rectangular geometry of the pipe used in this research may have influenced the results, as rectangular ducts can introduce boundary effects and impact the velocity profile. Hence, The performance of these correlations can be influenced by factors such as pipe geometry, flow regime, and the specific characteristics of the two-phase mixture.

The investigation of two-phase flow behavior within the tube bundle has provided valuable insights into the complex dynamics of this flow regime. The analysis revealed that the flow conditions upstream of the tube bundle influence the two-phase flow behavior inside it. The interaction between the gas and liquid phases, along with the obstructive nature of the tube bundle, leads to a unique flow pattern characterized by a blend of churn flow and intermittent flow regimes.

The periodic intermittence observed upstream of the bundle persists within it, and the frequency of void fraction oscillations reflects the inlet conditions. This overlap between churn flow and intermittent flow regimes highlights the importance of considering the influence of both the tube bundle geometry and the inlet conditions when characterizing two-phase flow in such configurations.

Vertical profiles of void fraction and gas velocity further emphasized the impact of the bundle geometry on the flow behavior. While void fraction increased within the tube bundle, gas velocity, on the other hand, decreased due to the fragmentation of large gas structures and the weakening of buoyancy-driven flow effect.

The analysis also demonstrated the limitations of the Homogeneous Equilibrium Model (HEM) in accurately predicting void fraction within the tube bundle mainly due to the large slip ratio. While the model proposed by Feenstra et al. [START_REF] Feenstra | Modelling two-phase flow-excited damping and fluidelastic instability in tube arrays[END_REF] provided improved approximations, it also exhibited limitations when applied to intermittent flow conditions.

Moreover a new flow regime map in tube bundle was proposed. A combination of methods was employed, including visual observations using fast camera and statistical analyses using probability density function (PDF) and power spectral density (PSD) analyses of void fraction time-evolution obtained from the optical probe. The PDF analysis allowed us to distinguish between different flow regimes based on the shape and characteristics of the PDF, while the PSD analysis provided insights into the frequency at which large gas structures passed through the tube bundle. It is important to note that the specific boundaries of flow regimes can vary based on factors such as tube bundle geometry, two-phase mixture properties, and inlet conditions.

Finally, the investigation of tube vibration in the context of two-phase flow has provided valuable insights into the complex interplay between fluid dynamics and structural response. The analysis of tube acceleration spectra from the tests showed that the first mode peak exhibited changes in power and width as the air flow rate increased. The increased power and broadening of the peak were associated with the intermittent flow regime and variations in void fraction. This phenomenon can be attributed to alterations in the hydrodynamic mass effect caused by the continuous changes in effective mass during intermittent flow.

Quantitatively, the half-power bandwidth of the first mode peak was employed to assess the impact of flow regimes on tube vibration. Further examination of tube vibration behavior involved analyzing PDF of the accelerometer data and their full width at half maximum (FWHM). These analyses reaffirmed the influence of flow regimes on tube vibration. The PDFs became broader with increasing air flow rate, and abrupt increases in FWHM were observed during flow regime transitions. Root mean square (RMS) plots of vibration amplitude also highlighted the impact of flow regimes on tube behavior.

To further explore the effects of flow regime on tube damping, a semi-empirical formulation proposed by Pettigrew et al. [START_REF] Pettigrew | Vibration analysis of shell-and-tube heat exchangers: an overviewpart 1: flow, damping, fluidelastic instability[END_REF] was employed. The formulation considered the impact of void fraction on damping and provided predictions for two-phase flow damping. While the semi-empirical formula successfully captured the general trend of two-phase damping, there were some discrepancies with experimental data, particularly for tests with lower void fractions.

Numerical simulation: main results

The Generalized Large Interface Model (GLIM) was used as a hybrid approach used in the Neptune CFD code in order to perform numerical simulations of the experimental campaigns. The GLIM combines the strengths of both dispersed phase and large interface models, allowing for a smooth transition between these models based on local flow conditions. This approach offers flexibility in handling various flow regimes, from fully segregated to dispersed structures.

Concerning the bubbly flow regime, the numerical domain, corresponding to the tube bundle section, was established, and inlet boundary conditions were specified based on experimental measurements.

Qualitative comparisons between numerical results and experimental data were presented, showing agreement between the void fraction distribution within the numerical domain and images obtained via a fast camera.

Quantitative analyses were conducted, focusing on void fraction, gas velocity, bubble diameter, and liquid velocity profiles. Generally, the numerical results aligned with experimental trends. The simulations effectively predicted the increase in void fraction, gas velocity, and Sauter mean diameter as the two-phase flow encountered the tube bundle's constriction effect. However, some discrepancies were noted in the bubble velocity profiles. While experimental profiles remained relatively constant across the tube bundle with a peak between the tubes, the numerical profiles exhibited differences. These disparities were attributed to the influence of void fraction on gas velocity in the simulations.

A deeper analysis of the liquid velocity profiles revealed that the constriction effect of the gaps in the tube bundle did not significantly impact the water velocity within the bundle. Instead, the void fraction profile was found to be the primary driver of liquid velocity behavior inside the bundle. This difference in the relationship between void fraction and gas/liquid velocity in numerical and experimental data was acknowledged and discussed.

A sensitivity analysis showed that the boundary conditions specified for the reference case provided the most accurate approximation of two-phase flow in the tube bundle, maintaining good overall agreement with experimental data. However, it is important to recognize that the simulation models may have limitations in predicting certain aspects of the flow, especially concerning bubble dynamics.

The transition and intermittent flow numerical simulations provide valuable insights into the complex behavior of two-phase flow in the TREFLE facility. To capture the intricate interactions between the phases and the formation of large gas structures, a three-dimensional domain was employed.

Concerning the free duct section, the numerical simulations successfully replicated the transition from bubbly/transition flow (test T4) to the churn flow regime (test T5) in the free duct. These regimes are characterized by distinct void fraction profiles and gas velocity patterns, with larger gas structures dominating the flow in churn flow. While the simulations provided qualitative agreement with the experiments, some quantitative discrepancies were observed.

The simulations accurately predicted the core-peaked void fraction profiles characteristic of transition and churn flow regimes, aligning well with experimental observations. However, there were discrepancies in the magnitude of void fraction, with numerical values tending to overestimate the void fraction compared to experimental measurements. The gas velocity profiles in the numerical simulations closely followed the trends of the void fraction profiles, with some regions near the wall exhibiting negative velocities. This behavior was consistent with the experimental findings of stagnant or downward-flowing bubbles along the wall. Power spectral density analysis of the numerical void fraction signal revealed a frequency peak associated with the passage of large gas structures. However, the frequency detected in the simulations differed from the experimental frequency, likely due to limitations in the numerical domain size and analysis duration.

Regarding the tube bundle numerical domain, the simulations successfully captured the distribution of void fraction within the tube bundle and showed agreement with experimental observations. They revealed that dispersed bubbles were predominant within the bundle, but as the flow regime transitioned to intermittent flow (test T5), there was a notable increase in the presence of large, irregular gas structures.

In test T4, while the simulations exhibited overall consistency with the experimental data, there were minor discrepancies in the trend of the gas velocity profile, attributed to the complexity of the transition flow regime. In test T5, as the intermittent flow regime became dominant, the challenges of capturing highly dynamic flow behavior were evident.

The difference in inlet gas velocities between numerical simulations and experimental results was noted. This discrepancy was primarily attributed to the limited depth of the simulated free duct and its proximity to the tube bundle. The boundary conditions significantly influenced the formation of large gas structures, which were only captured by the code when the two-phase flow interacted with the bundle.

While challenges exist, especially in capturing intricate flow patterns within transition and intermittent flow regimes, CFD provides an understanding of the system behavior.

Perspectives

The extensive analysis of bubble generation and behavior provides valuable insights into how various parameters affect bubble dynamics. Future research can focus on developing more accurate predictive models for bubble size, shape, and behavior under different conditions. This could lead to better design and control of two-phase flow systems. While some existing correlations showed promise, none fully matched the experimental data. Future work can involve refining and developing correlations that better account for the specific conditions and complex dynamics observed in the TREFLE experiments.

The observed bi-stable behavior in test T4 indicate the need for a deeper understanding of such phenomena. Future research can explore the causes and implications of this phenomena. Experimental work could focus on studying how the inlet conditions impact this behavior, investigating the gas velocity and void fraction horizontal profiles within the bundle can provide valuable information about the frequency of equilibrium position switching correlated to the inlet conditions. Subsequent research can aim to determine whether this effect is linked to the facility configuration or if it can be scaled to steam generator conditions.

The study revealed that existing drift-flux correlations perform differently in the context of large free duct in two-phase flow. Researchers can work on refining and validating correlations that specifically account for this flow conditions. In particular, they can explore how pipe geometry, flow regime, and the two-phase mixture play important roles in the evaluation of void fraction and gas velocity, and how these parameters could be considered when developing the drift-flux correlation.

The limitations of existing void fraction models in a tube bundle, especially in intermittent flow regimes, call for the development of more accurate predictive models. Due to the importance of this parameter in the prediction of fluid-elastic instability, research efforts can focus on creating models that consider the unique characteristics of flow within tube bundles, leading to improved void fraction predictions. The experimental results reported in this manuscript can serve as valuable information regarding the evolution of two-phase flow in a tube bundle. In particular, the evolution of fundamental parameters such as void fraction and gas velocity.

The methodology employed to study two-phase flow and its relationship with tube vibration response is highly valuable. Expanding the scope of the experimental campaigns by encompassing tube bundle geometry variations, a wider range of gas and liquid velocities, two-phase mixture density ratios, and surface tension would be of significant interest for scaling the phenomena to steam generator conditions. Additionally, by adding more vibrating tubes, investigating the effects of nearby tubes on each other's vibration responses can provide valuable insights.

The interplay between fluid dynamics and structural vibrations within tube bundles is a complex area that warrants further investigation. Parameters like damping and hydrodynamic mass in two-phase flow are critical factors in the prediction of stability and performance of vibrating systems. The semi-empirical formulations proposed by Pettigrew et al. [START_REF] Pettigrew | Vibration analysis of shell-and-tube heat exchangers: an overviewpart 1: flow, damping, fluidelastic instability[END_REF][START_REF] Pettigrew | Vibration of tube bundles in two-phase cross-flow: Part 1-hydrodynamic mass and damping[END_REF] (for damping and hydrodynamic mass) provide valuable insights into two-phase damping and hydrodynamic mass but may need further refining to better match experimental data. These models should consider the specific flow characteristics associated with different regimes and the mode shape effect.

The sensitivity of simulations to boundary conditions underscores the importance of accurately defining these conditions, especially int the context of churn/intermittent flow regime. Further investigations

  . . . . . . . . . . . . . . . . . . . . . 1.8 Comparing theory and experiments: hydrodynamic mass in two-phase cross flow for different two-phase mixtures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Critical flow velocities for fluidelastic instability for the data of Pettigrew et al. [6] in air-water cross-flow. ◽, ◾ normal square array P D = 1.47, ▾, ▿ normal triangular array P D = 1.32, ▴, ▵ normal triangular array P D = 1.47. The black symbols indicate the intermittent flow regime test. The black line refers to Connors theory [12], the dashed line to Li theory [13]. Images takenf from Feenstra et al. [14] . . . . . . . . . . . . . . . . . . . . 1.10 Comparing theory and experiments: hydrodynamic mass and damping in two-phase cross flow [15]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11 Flow regimes representation by Xu et al. [16]. . . . . . . . . . . . . . . . . . . . . . . . . . . 1.

  Flow regimes: B=Bubbly, DB=DispersedBubbly, C=Churn, S=Slug, I=Intermittent, A=Annular, T=T ransition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Layout of the TREFLE facility (Figure 2.1a) with details of the test section (Figure 2.1b). In red the central tube to be substituted with the vibrating tube. . . . . . . . . . . . . . . . 2.2 Detailed sketch of the TREFLE facility with dimensions reported in mm. . . . . . . . . . 2.3 Sketch of the gas injector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Sketch of the type 1 of tubes used for the gas injection. . . . . . . . . . . . . . . . . . . . . .

3 . 32

 332 Position of vibrating tube in the tube bundle with reference system. . . . . . . . . . . . . . 3.33 Temporal signal and PSD of the accelerometer in the tube and on the facility wall for test T1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.34 Tube spectra for the various tests performed according to Table 3.1 and in free vibration. 3.35 Comparison between calculated hydrodynamic mass (Equation 3.9) and the results from the theory (Equation 3.10). f 1 and f 3 refers to the hydrodynamic mass calculated using the first and second frequency peak (around 10 Hz and 188 Hz in Figure 3.34a). . . . . . . 3.36 Comparison of the RMS displacement obtained by camera and accelerometer signals. . . . 3.37 PDF of the accelerometer signals for tests T1 to T4. . . . . . . . . . . . . . . . . . . . . . . 3.38 Kurtosis and skewness of accelerometer signal. Air flow rate of tests T1 to T4 are reported in Table 3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Instrumentation position with respect the TREFLE facility for intermittent flow investigation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Photos upstream the tube bundle, form visualization windows W2 in Figure 2.21. . . . . . 4.3 Void fraction distribution upstream the tube bundle for test T5. . . . . . . . . . . . . . . . 4.4 Void fraction distribution upstream the tube bundle for test T6. . . . . . . . . . . . . . . . 4.5 Void fraction distribution upstream the tube bundle for test T7. . . . . . . . . . . . . . . . 4.6 Isosurface for void fraction threshold of 30%. In the z-axis is reported the number of frame, while the x and y-axis are reported as the number of wires (56×78). . . . . . . . . . . . . . 4.7 Time average of void fraction 2D distribution ᾱ(x, y) from WMS. . . . . . . . . . . . . . . 4.8 Time-average of the radial distribution of the void fraction ᾱ(r) from WMS. . . . . . . . . 4.9 Space-averaged void fraction evolution ᾱ(t) from WMS. . . . . . . . . . . . . . . . . . . . . 4.10 Power spectral density (PSD) of ᾱ(t) for tests T5, T6, T7 from WMS measurements. . . 4.11 PSD of α(t) obtained with the optical probe for ∆t = 0.001 s over 1000 seconds. . . . . . . 4.12 Comparison of PDF of bubble size distribution obtained with optical probe and WMS. The average diameter dB is obtained according to Equation 3.5: dB = ∑ d B ⋅ P (d B ) ∑ P (d B ). 4.13 Plot of the contribution of bubble size to the void fraction, ∆α i ∆d Bi , as a function of bubble size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143 5. 5

 1435 4.21 Images within the tube bundle, form visualization windows W5 in Figure 2.21. . . . . . . 4.22 Images within the tube bundle, form visualization windows W3 in Figure 2.21. . . . . . . 4.23 PSD of void fraction evolution within the tube bundle obtained by the OP signal. . . . . . 4.24 Vertical profiles of void fraction and gas velocity obtained with optical probe within the tube bundle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.25 Void fraction comparison among the model from Feenstra et al. [40], the HEM model and experimental results obtained by optical probe at point 2. . . . . . . . . . . . . . . . . . . . 4.26 Slip ratio of the present experimental conditions as function of the air flow rate, the water flow rate is constant 2500 g s -1 . The results are reported within the tube bundle and in the free duct. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.27 PDF from Jones and Zuber [41] for flow regime identification compared with visual observation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.28 PDF of void fraction time-evolution to identify the flow regime. . . . . . . . . . . . . . . . . 4.29 PSD of void fraction evolution within the tube bundle for flow regime identification. . . . 4.30 Proposed flow regime map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.31 Proposed flow regime map, comparison with the literature. The nomenclature according to the authors: B-bubbly, T-transition, I-intermittent, S-slug, C-churn, A-annular, D-dispersed.129 4.32 Tube vibration response spectra (power spectral density) for the different tests T1, T2, T3, T4, T5, T6, T7 according to Table 2.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.33 Half power bandwidth for the different tests T1 to T7. In the y-axis is reported the bandwidth where the PSD magnitude correspond to the maximum value divided by the square root of 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.34 PDF of the accelerometer signals for tests T1 to T7 in Figure 4.34a and its FWHM in Figure 4.34b. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.35 Exponential and linear fit of the data point reported in Figure 4.34b with the R 2 values reported. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.36 RMS of vibration amplitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.37 Normalized two-phase damping comparison between experimental values (calculated according Equation 4.24) and theory according to Pettigrew et al. [11] (Equation 4.26). Both, experimental and calculated values are evaluated using the HEM void fraction model and the measured values of void fraction (α m ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Example of two-phase flow regime for dispersed phase (bubbly flow), large structures (slug flow), hybrid (churn flow) [42]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Schematic representation of different interfacial scales compared to the numerical grid. . . 5.3 The three-cell stencil method: schematic representation [43]. . . . . . . . . . . . . . . . . . . 5.4 Schematic representation of the various numerical domains overlaid on the TREFLE facility sketch, highlighting the instrumentation positions for extracting experimental measurements.Computational domain. P1 and P2 are the extracted profiles for mesh sensitivity study. . 5.6 Mesh detail. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.19 Comparison of vertical profile results between simulations employing a three-dimensional domain and a two-dimensional domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.20 Instantaneous void fraction distribution for tests T4 and T5. Concerning the most left

  ) and critical velocity (U cr ) for different void fraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Proprieties of the new vibrating system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4. 1

 1 Test configurations for the investigation of the intermittent flow regime. . . . . . . . . . . . 4.2 Summary of average data obtained from WMS and optical probe. ᾱ is the average void fraction. ᾱc indicates the void fraction detected in the center of the flow by the WMS (Figure 4.7). dB is the average diameter, d sm is the sauter mean diameter and v B is the gas phase velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Comparisons among the experimental campaigns investigated. . . . . . . . . . . . . . . . . . 4.4 Comparisons among drift-flux correlations analyzed. The validity range reported refers to air-water two-phase mixture in vertical upward flow. . . . . . . . . . . . . . . . . . . . . . . 4.5 Test configurations for the investigation of the intermittent flow regime. . . . . . . . . . . .

Figure 1 . 1 :

 11 Figure 1.1: Representation of vortex shedding phenomena.

Figure 1 . 2 :

 12 Figure 1.2: Representation of turbulent excitation mechanism [2].

Figure 1 . 3 :

 13 Figure 1.3: Structure vibration response with increasing flow velocity to the various vibrations mechanisms in single-phase axial or cross-flow. The FEI has clearly a major impact on the vibration amplitude [3].

Figure 1 . 4 :

 14 Figure 1.4: Scheme of the different tube bundles configurations. P indicates the pitch, D the tube diameter. Drag and lift directions of tube vibration are also reported.

Figure 1 . 6 :

 16 Figure 1.6: Experimental results comparisons from Pettigrew et al. [5, 6, 7, 8]. Mass flux needed to reach the instability as a function of the void fraction.

Figure 1 . 7 :

 17 Figure 1.7: Damping data from Pettigrew et al.[START_REF] Pettigrew | Vibration of a tube bundle in two-phase freon cross-flow[END_REF][START_REF] Pettigrew | Vibration behaviour of rotated triangular tube bundles in two-phase cross-flow[END_REF][START_REF] Pettigrew | Vibration of tube bundles in two-phase cross-flow: Part 1-hydrodynamic mass and damping[END_REF] in air-water and Freon, from Axisa et al.[START_REF] Axisa | Vibration of tube bundles subjected to air-water and steam-water cross flow: Preliminary results on flu1delastic instability[END_REF][START_REF] Axisa | Vibration of tube bundles subjected to steamwater cross flow: a comparative study of square and triangular pitch arrays[END_REF] in steam-water. Image from Pettigrew et al.[START_REF] Pettigrew | Vibration analysis of shell-and-tube heat exchangers: an overviewpart 1: flow, damping, fluidelastic instability[END_REF]

  (a) Hydrodynamic mass in air-water two-phase cross flow from Pettigrew et al. [5]. (b) Hydrodynamic mass in freon two-phase cross flow from Pettigrew et al. [7].

Figure 1 . 8 :

 18 Figure 1.8: Comparing theory and experiments: hydrodynamic mass in two-phase cross flow for different two-phase mixtures.

  (a) Parameters calculated with void fraction model from Feenstra et al. [44, 14]. (b) Parameters calculated with HEM void fraction model.

Figure 1 . 9 :

 19 Figure 1.9: Critical flow velocities for fluidelastic instability for the data of Pettigrew et al. [6] in air-water cross-flow. ◽, ◾ normal square array P D = 1.47, ▾, ▿ normal triangular array P D = 1.32, ▴, ▵ normal triangular array P D = 1.47. The black symbols indicate the intermittent flow regime test. The black line refers to Connors theory[START_REF] Connors | Fluidelastic vibration of arrays excited by cross flow. Flow-induced vibration of heat-exchangers[END_REF], the dashed line to Li theory[START_REF] Li | An fluidelasticinstability model with an extension to full flexible multispan tube arrays[END_REF]. Images takenf from Feenstra et al.[START_REF] Feenstra | Modelling two-phase flow-excited damping and fluidelastic instability in tube arrays[END_REF]

Figure 1 . 10 :

 110 Figure 1.10: Comparing theory and experiments: hydrodynamic mass and damping in two-phase cross flow [15].

Figure 1 . 11 :

 111 Figure 1.11: Flow regimes representation by Xu et al. [16].

Figure 1 . 13 :

 113 Figure 1.13: Flow regime maps comparisons in square pitch tube bundle among Noghrehkar et al. [18] (Figure 1.13a), Xu et al. [16] (Figure 1.13b), Murakawa et al. [19] (Figure 1.13c) with Ulbrich et al. [17].

  (a) Flow regime map identified subjectively. (b) Flow regime map identified objectively based on pressure drop signal. (c) Flow regime map identified objectively based on pressure drop and capacitive sensoring signals.

Figure 1 . 14 :

 114 Figure 1.14: Flow regime maps identified by Karnizawa et al. [20]. The black lines on the flow regime map in Figure 1.14b refers to the transition identified in Figure 1.14b, and the black lines on the flow regime map in Figure 1.14c refers to the transition identified in Figure 1.14b.

Figure 1 . 15 :

 115 Figure 1.15: Comparison of flow regime maps in square pitch tube bundle among: Ulbrich et al. [17], Noghrehkar et al. [18], Murakawa et al. [19], Xu et al. [16], Piteau et al. [21]. Flow regimes: B=Bubbly, C=Churn, S=Slug, I=Intermittent, A=Annular, T=T ransition

Figure 1 . 16 :

 116 Figure 1.16: Comparison of flow regime maps in staggered pitch tube bundle among: Kanizawa et al. [20], Murakawa et al. [19], Noghrehkar et al. [18], Aprin et al. [22], Grant et al. [23]. Flow regimes: B=Bubbly, DB=DispersedBubbly, C=Churn, S=Slug, I=Intermittent, A=Annular, T=T ransition

Figure 2 . 1 :

 21 Figure 2.1: Layout of the TREFLE facility (Figure 2.1a) with details of the test section (Figure 2.1b).In red the central tube to be substituted with the vibrating tube.

Figure 2 . 2 :

 22 Figure 2.2: Detailed sketch of the TREFLE facility with dimensions reported in mm.

Figure 2 . 3 :

 23 Figure 2.3: Sketch of the gas injector.

Figure 2 . 4 :

 24 Figure 2.4: Sketch of the type 1 of tubes used for the gas injection.

Figure 2 . 5 :

 25 Figure 2.5: Sketch of the type 2 of tubes used for the gas injection.

  (a) Divergent system with plates and honeycomb (b) Cross-section A-A (c) Top view

Figure 2 . 6 :

 26 Figure 2.6: Plate and honeycomb system installed in the divergent section of the facility.

Figure 2 . 7 :

 27 Figure 2.7: Results of the CFD simulations that show the velocity field along the direction of the flow.

Figure 2 . 8 :

 28 Figure 2.8: Velocity profiles (z component, along the direction of the flow).From bottom to top: profile downstream the honeycomb system (in green), profile downstream the injector tubes (in blue) and profile at 0.5 m downstream the injector tubes (in red).

Figure 2 . 9 :

 29 Figure 2.9: Effect of the system with plates and honeycomb on the bubble generation with the central injection tube.

Figure 2 . 10 :

 210 Figure 2.10: Test section side view.

Figure 2 . 11 :

 211 Figure 2.11: Flexible cantilever tube vibrations amplitude in the lift direction and experimental rig design from Piteau et al. [24].

Figure 2 . 13 :

 213 Figure 2.13: Flexible tube design by Khalvatti et al. [26].

Figure 2 . 14 :

 214 Figure 2.14: Piano wire tube design from Deri [27].

Figure 2 . 15 :

 215 Figure 2.15: Piano wire tube design from Khalifa et al. [28].Hirota et al.[START_REF] Hirota | Investigation on in-flow fluidelastic instability of an array of tubes[END_REF] have investigated flow induced vibration for five different bundle configuration. The bundle geometry consists in 30 × 5 stainless steel tubes of 19 mm diameter, with pitch over diameter ratio of 1.34. The 10 central tubes of the bundle have been designed to be replaced with flexible tubes. The flexible tubes designs are based on the tube-beam principle, i.e. the tube are connected to a flexible beam clumped to the facility wall to allow the rigid body movement of the tubes. The clearance between the tube and the wall is considered negligible. The tube vibration is measured through strain gauges. Three different design of flexible tubes have been used in order to investigate several configuration (Figure2.16). The first design consist in a rectangular cross section beam that allows the authors to study the tubes vibrations in the drag and lift direction separately (Figure2.16a). The second configuration reported in Figure2.16b rely on a square cross section beam, to allow the movement of the tubes in the lift and drag direction. The third design (Figure2.16c) add the anti vibration bars to the previous described configuration in order to study the effect of the anti vibration bars on the tube vibration response.

Figure 2 . 16 :

 216 Figure 2.16: Three different designs of flexible tubes and its use in the test section from Hirota et al. [29].

Figure 2 . 17 :

 217 Figure 2.17: Flexible tube from Rottmann et al. [30].

Figure 2 .

 2 Figure 2.18: Tube-flexible beam concept.

Figure 2 . 19 :

 219 Figure 2.19: Design guidelines for hydrodynamic mass ratio (left figure) and damping ratio (right figure) in two-phase flow [5, 6].

  Figure 2.20.

Figure 2 . 20 :

 220 Figure 2.20: Design of tube-flexible beam.

Figure 2 .

 2 [START_REF] Aprin | Experimental analysis of local void fractions measurements for boiling hydrocarbons in complex geometry[END_REF] illustrates the two configurations used. Configuration 1 (depicted in Figure2.22a) is adopted for visualization windows W2 to W5, whereas configuration 2 (shown in Figure2.22b) is used for visualization window W1.

Figure 2 . 21 :

 221 Figure 2.21: Different visualization windows used in the TREFLE facilty.

  (a) Configuration 1 (b) Configuration 2

Figure 2 . 22 :

 222 Figure 2.22: Positions of fast camera and light with respect the section.

Figure 2 . 23 :

 223 Figure 2.23: Detail of the optical probe tips.

Figure 2 .

 2 Figure 2.24: Different measurement points with the dual tip optical probe.

Figure 2 . 25 :

 225 Figure 2.25: Detailed scheme of the WMS. Measures reported in mm.

Figure 2 . 26 :

 226 Figure 2.26: Working principle of HZDR WMS [31].

Figure 2 .

 2 Figure 2.27: WMS measurement positions.

Figure 2 .

 2 Figure 2.28: Weight coefficients for the cross-section averaging of local void fractions measured by the WMS. A sensor is the surface of the considered cell [31].

  .1.

Figure 3 . 1 :

 31 Figure 3.1: Camera-light configuration with respect to the TREFLE facility gas injector.

Figure 3 . 2 :

 32 Figure 3.2: Image post-processing procedure for bubble identification.

Figure 3 . 3 :

 33 Figure 3.3: Solidity parameter: convex hull reported in green, objects in white.

Figure 3 . 4 :

 34 Figure 3.4: Bubble shape evolution as function of Eötvös, Morton and Reynolds numbers [32]. The red dotted line highlights the present range of data. The red dot is the average value of Reynolds and Eötvös numbers detected.

Figure 3 . 5 :

 35 Figure 3.5: Post-processing for bubble velocity calculation: two consecutive images overlapped, the red objects are the bubbles detected in the second image. The white points are the CM of the objects in the first image, the red dots the CM of the objects detected in the second image.

  Figure 3.6: Comparison of results for different numbers of bubbles.

  Figure 3.7: PDF of bubble equivalent diameter, bubble velocity, bubble Eötvös number, and bubble aspect ratio (R) for water mass-flow rate of ṁwater = 2500 g s -1 .

Figure 3 . 8 :

 38 Figure 3.8: Example of ellipses approximation of the detected bubbles.

1 Figure 3 . 9 :

 139 Figure 3.9: Images of bubble generation at gas injection with 1 injection tube. Test performed with water mass-flow rate of ṁwater = 2500 g s -1 .

Figure 3 .

 3 Figure 3.10: PDF of relative velocity and Reynolds number for water mass-flow rate of ṁwater = 2500 g s -1 .

1 Figure 3 .

 13 Figure 3.11: PDF of bubble velocity as function of bubble size for water mass-flow rate of ṁwater = 2500 g s -1 .

Figure 3 . 12 :

 312 Figure 3.12: Comparison of experimental data with the correlations in Section 2.2.2.

Figure 3 . 13 :

 313 Figure 3.13: Visualization windows for tube bundle analysis. Windows downstream (W4), upstream of the tube bundle (W2) and lateral windows of the tube bundle (W3) in Figure 3.13a. Window on the axis of the tube bundle (W5) in Figure 3.13b (the facility is rotated of 90 o ).

Figure 3 . 14 :

 314 Figure 3.14: Measurement points with the dual tip optical probe.

Figure 3 . 15 :

 315 Figure 3.15: PDF of bubble equivalent diameter, bubble Eötvös number, and bubble aspect ratio (R) for water mass-flow rate of ṁwater = 2500 g s -1 at tube bundle inlet.

Figure 3 . 16 :

 316 Figure 3.16: Fast camera results on bubble size measurement at bubble injection up to the tube bundle inlet. In the y-axis 0 m is the level at which the bubble are generated and 1.4 m the tube bundle inlet.

Figure 3 . 17 :

 317 Figure 3.17: Comparison of bubble size distribution obtained by optical probe and PDF of equivalent diameter obtained via fast camera.

Figure 3 . 18 :

 318 Figure 3.18: Comparison between sauter mean diameter measurements obtained from the optical probe and equivalent diameter measurements from the fast camera.

  (a) Test T4 tube bundle inlet (b) Zoom of the large gas structures

Figure 3 . 19 :

 319 Figure 3.19: Images obtained via fast camera in the visualization window W2 (refer to Figure 3.13a) for test T4. The large gas structures are highlighted.

Figure 3 . 20 :

 320 Figure 3.20: Comparison between measured void fraction by means of optical probe and HEM void fraction model.

Figure 3 . 21 :

 321 Figure 3.21: Comparison between measured gas velocity (v g ) with estimated gas velocities by means of measured void fraction (U sg α) and HEM void fraction (U sg α HEM ).

Figure 3 .

 3 [START_REF] Aprin | Experimental analysis of local void fractions measurements for boiling hydrocarbons in complex geometry[END_REF] are shown images taken in the tube bundle for the tests reported in Table3.1. It can be observed how the bubble paths are different as the air flow rate increases. In particular, for test T1 (Figure3.22a) the bubbles follow the gap in between the tube and some bubbles are stuck in the stagnation point under the cylinder. With further increases in air flow rate (Figure3.22b), the bubbles accumulate below the cylinders and a portion of them start following the horizontal gap between the cylinders. Moving to tests T3 and T4 (Figures 3.22c and 3.22d), the bubble motion becomes more chaotic, filling the entire test section except for a thin liquid layer in the region downstream of the cylinders. Although the prevailing regime is bubbly, in agreement with the flow regime maps from the literature[START_REF] Ulbrich | Vertical, upward gas-liquid two-phase flow across a tube bundle[END_REF][START_REF] Noghrehkar | Investigation of two-phase flow regimes in tube bundles under cross-flow conditions[END_REF][START_REF] Kanizawa | Two-phase flow patterns across triangular tube bundles for airwater upward flow[END_REF][START_REF] Murakawa | Evaluation of void fraction around a tube in two-phase flow across horizontal tube bundle[END_REF], in the video for test T4 plugs of air can be observed that sporadically traverse the bundle (as shown in Figure3.23).

Figure 3 . 22 :

 322 Figure 3.22: Images obtained via fast camera in the visualization window W5 (refer to Figure 3.13a).An interesting phenomenon observed in tests T3 and T4 (Figures 3.24a and 3.24b) is a bi-stable behavior exhibited by the two-phase flow. This phenomenon was previously observed experimentally in single phase cross-flow in the literature by Olinto et al.[START_REF] Olinto | Experimental study of the bistable flow in tube arrays[END_REF][START_REF] Olinto | Experimental study of the characteristics of the flow in the first rows of tube banks[END_REF] and Keogh et al.[START_REF] Keogh | Bi-stable flow in parallel triangular tube arrays with a pitch-todiameter ratio of 1.375[END_REF]. In our case, this phenomenon is evident when tracking the paths of the bubbles. The bubble flow tends to follow a preferential direction within the tube bundle; however, any disturbance in the flow, such as the passage of an air plug through the bundle, can alter this preferential direction. Figure 3.24 illustrates an example

Figure 3 .

 3 24 illustrates an example

Figure 3 . 23 :

 323 Figure 3.23: Highlight of a large gas structure in the tube bundle for test T4.

Figure 3 . 24 :

 324 Figure 3.24: Images of the test section for tests T4. In Figures 3.24a and 3.24b the bubble path is highlighted to show the bi-stable behavior of the two-phase flow.

  void fraction (g) T2 gas velocity (h) T2 sauter mean diameter (i) (j) T3 void fraction (k) T3 gas velocity (l) T3 sauter mean diameter (m) (n) T4 void fraction (o) T4 gas velocity (p) T4 sauter mean diameter

Figure 3 . 25 :

 325 Figure 3.25: Optical probe measurements: horizontal profiles for test T1 to T4.

Figure 3 . 26 :

 326 Figure 3.26: Vertical profiles of void fraction, gas velocity and sauter mean diameter within the tube bundle.

Figure 3 . 27 :

 327 Figure 3.27: Comparison of experimental data obtained with the optical probe from point 2 with the model from Feenstra et al. [14].

Figure 3 . 28 :

 328 Figure 3.28: Bubble size distribution evolution inside the tube bundle.

  and 3.29a (respectively inlet and outlet of test T1) exhibit no significant difference. However, Figures 3.29d

Figure 3 . 29 :

 329 Figure 3.29: Images taken upstream and downstream the tube bundle.

Figure 3 . 30 :

 330 Figure 3.30: PDF of bubble equivalent diameter, bubble Eötvös number, and bubble aspect ratio (R) for water mass-flow rate of ṁwater = 2500 g s -1 at tube bundle outlet.

Figure 3 . 31 :

 331 Figure 3.31: Fast camera results on bubble size measurement at tube bundle inlet and tube bundle outlet.

Figure 3 . 32 :

 332 Figure 3.32: Position of vibrating tube in the tube bundle with reference system.

Figure 3 . 33 :

 333 Figure 3.33: Temporal signal and PSD of the accelerometer in the tube and on the facility wall for test T1.

Figure 3 . 34 :

 334 Figure 3.34: Tube spectra for the various tests performed according to Table3.1 and in free vibration.

Figure 3 . 35 :

 335 Figure 3.35: Comparison between calculated hydrodynamic mass (Equation 3.9) and the results from the theory (Equation 3.10). f 1 and f 3 refers to the hydrodynamic mass calculated using the first and second frequency peak (around 10 Hz and 188 Hz in Figure 3.34a).

Figure 3 . 36 :

 336 Figure 3.36: Comparison of the RMS displacement obtained by camera and accelerometer signals.

Figure 3 . 37 :

 337 Figure 3.37: PDF of the accelerometer signals for tests T1 to T4.

Figure 3 . 38 :

 338 Figure 3.38: Kurtosis and skewness of accelerometer signal. Air flow rate of tests T1 to T4 are reported in Table3.1.

Figure 4

 4

Figure 4 . 1 :

 41 Figure 4.1: Instrumentation position with respect the TREFLE facility for intermittent flow investigation.

Figure 4 . 2 :

 42 Figure 4.2: Photos upstream the tube bundle, form visualization windows W2 in Figure 2.21.

  (a) Cross-sectional distribution of the void fraction (b) x-side-view of the void fraction evolution (c) y-side-view of the void fraction evolution

Figure 4 . 3 :

 43 Figure 4.3: Void fraction distribution upstream the tube bundle for test T5.

  (a) Cross-sectional distribution of the void fraction (b) x-side-view of the void fraction evolution (c) y-side-view of the void fraction evolution

Figure 4 . 4 :

 44 Figure 4.4: Void fraction distribution upstream the tube bundle for test T6.

  (a) Cross-sectional distribution of the void fraction (b) x-side-view of the void fraction evolution (c) y-side-view of the void fraction evolution

Figure 4 . 5 :

 45 Figure 4.5: Void fraction distribution upstream the tube bundle for test T7.

Figure 4 . 6 :

 46 Figure 4.6: Isosurface for void fraction threshold of 30%.In the z-axis is reported the number of frame, while the x and y-axis are reported as the number of wires (56×78).

Figure 4 .

 4 Figure 4.6 shows how the void fraction increases from test T5 to T7. The amount of smaller bubbles appears to increases from test T5 to T7. The gas bubbles and larger structures are irregularly distributed along the z-axis (frame number, in this case). However, when examining the x-y cross-section, the larger gas structures appear to be concentrated in the center of the section, with smaller bubbles surrounding them. A more quantitative analysis can confirm these results. Figure4.7 presents the 2D distributions of the time-averaged void fraction (ᾱ(x, y) calculated according Section 2.5.3 from WMS measurements) for the different tests T5, T6, and T7.

  (a) Test T5 (b) Test T6 (c) Test T7

Figure 4 . 7 :

 47 Figure 4.7: Time average of void fraction 2D distribution ᾱ(x, y) from WMS.

Figure 4 . 8 :

 48 Figure 4.8: Time-average of the radial distribution of the void fraction ᾱ(r) from WMS.

  (a) Comparison between the tests during 5 seconds. (b) 1-second zoom for test T6.

Figure 4 . 9 :

 49 Figure 4.9: Space-averaged void fraction evolution ᾱ(t) from WMS.

Figure

  Figure 4.9a presents a comparison between the different tests, illustrating a signal that exhibits an average increase with higher air flow rates. Notably, the signal displays peaks and depressions, as highlighted in Figure4.9b. The peaks correspond to the passage of large gas structures, while the depressions correspond to the dispersed phase. By following this approach, we analyze the two-phase flow and calculate the power spectral density (PSD) of the signal representing the evolution of the space-averaged void fraction, denoted as ᾱ(t) (calculated according Section 2.5.3 from WMS measurements). The PSD results are shown in Figure4.10.The PSD exhibits a prominent peak around 3 Hz for all tests. Specifically, 3.4 Hz for test T5, 3.5 Hz for test T6 and 3.2 Hz for test T7. This frequency can be interpreted as the characteristic frequency of large gas structure passage within the flow. The PSD was obtained by finding optimum values of Hann windows. However, due to the smoothness of the peak, it is difficult to determine if the identified frequency represents the exact value or if the mentioned values are within an order of magnitude. In fact, by performing a fast fourier transform, it becomes clear that the peak in the PSD includes a set of peaks ranging from 3 Hz to 4 Hz. Nevertheless, an intriguing observation is the presence of a periodicity in the large gas structure that appears to be relatively independent of the air flow rate.

Figure 4 . 10 :

 410 Figure 4.10: Power spectral density (PSD) of ᾱ(t) for tests T5, T6, T7 from WMS measurements.

Figure 4 .

 4 [START_REF] Pettigrew | Vibration analysis of shell-and-tube heat exchangers: an overviewpart 1: flow, damping, fluidelastic instability[END_REF] presents the PSD of the void fraction timeevolution for tests T5, T6 and T7, with a sample of the time-evolution of the void fraction highlighting the presence of large gas structures.

  α(t) 0.5-second zoom for test T7.

Figure 4 . 11 :

 411 Figure 4.11: PSD of α(t) obtained with the optical probe for ∆t = 0.001 s over 1000 seconds.

Figure 4 . 12 :

 412 Figure 4.12: Comparison of PDF of bubble size distribution obtained with optical probe and WMS. The average diameter dB is obtained according to Equation 3.5: dB = ∑ d B ⋅ P (d B ) ∑ P (d B ).

Figure 4 . 13 :

 413 Figure 4.13: Plot of the contribution of bubble size to the void fraction, ∆α i ∆d Bi , as a function of bubble size.

  (a) Test section D h =0.48 m (b) Test section D h =0.2 m

Figure 4 . 14 :

 414 Figure 4.14: Flow regimes maps from Ohnuki et al.[START_REF] Ohnuki | An experimental study on developing air-water two-phase flow along a large vertical pipe: effect of air injection method[END_REF][START_REF] Ohnuki | Experimental study on transition of flow pattern and phase distribution in upward air-water two-phase flow along a large vertical pipe[END_REF]. Comparison with Mishima and Ishii[START_REF] Mishima | Flow regime transition criteria for upward two-phase flow in vertical tubes[END_REF] flow regime map. In the figure are reported the test conditions of the present experimental campaign. In the figures, the authors use the notation J g and J l , which corresponds to the notation of U sg and U sl in the present study.

Figure 4 . 15 :

 415 Figure 4.15: Flow regime map by Ali[START_REF] Ali | Two phase flow in large diameter vertical riser[END_REF]. Tee and Ann refers to the two different air injection systems. "B" referrers to bubbly flow, "DB" to dispersed bubbly, "AB" to agitated bubbly and "C" to churn flow. In the figure, the authors use the notation J g and J l , which corresponds to the notation of U sg and U sl in the present study.

Figure 4 . 16 :

 416 Figure 4.16: The flow patterns in pipe (D h =0.195 m) taken with wire mesh tomography [37] for U sl =1.0 m s -1 . In the figure, the authors use the notation J air , which corresponds to the notation of U sg in the present study.

Figure 4 . 17 :

 417 Figure 4.17: Distribution parameter C 0 for large rectangular channel according to Kataoka et al. [38].

(4. 13 )

 13 Hence, considering the distribution parameter C 0 according to Figure4.18 based on the ratio U + sg U + s , from equation 4.11 and equation 4.12 can be estimated the average value of the void fraction.

Figure 4 . 18 :

 418 Figure 4.18: Distribution parameter C 0 for large rectangular channel according to Hibiki et al. [39]. In the figure J + g and J + correspond to the notation U + sg and U + s in the present study.

Figure 4 . 19 :

 419 Figure 4.19: Comparison between drift-flux correlations with experimental results obtained with optical probe (OP) and WMS. The black-dashed lines represent the 30% bias for the experimental data. The water superficial velocity for the tests is constant (0.039 m s -1 ).

Figure 4 . 20 :

 420 Figure 4.20: Comparison between drift-flux correlations for gas velocity with experimental results obtained with optical probe (OP). The black-dashed lines represent the 30% bias for the experimental data. The water superficial velocity for the tests is constant (0.039 m s -1 ).

Figure 4 . 21 :

 421 Figure 4.21: Images within the tube bundle, form visualization windows W5 in Figure 2.21.

Figure 4 . 22 :

 422 Figure 4.22: Images within the tube bundle, form visualization windows W3 in Figure 2.21.

Figure 4 . 23 :

 423 Figure 4.23: PSD of void fraction evolution within the tube bundle obtained by the OP signal.

  (a) Void fraction profiles.(b) Gas velocity profiles.

Figure 4 . 24 :

 424 Figure 4.24: Vertical profiles of void fraction and gas velocity obtained with optical probe within the tube bundle.

Figure 4 . 25 :

 425 Figure 4.25: Void fraction comparison among the model from Feenstra et al. [40], the HEM model and experimental results obtained by optical probe at point 2.

Figure 4 . 26 :

 426 Figure 4.26: Slip ratio of the present experimental conditions as function of the air flow rate, the water flow rate is constant 2500 g s -1 . The results are reported within the tube bundle and in the free duct.

( a )

 a Bubbly flow regime. (b) Churn-like flow regime.

Figure 4 . 27 :

 427 Figure 4.27: PDF from Jones and Zuber [41] for flow regime identification compared with visual observation.

  Figure 4.28 shows examples of PDF for bubbly, transition and intermittent flow regimes.

  (a) Test T1, bubbly flow. (b) Test T4, transition.(c) Test T7, intermittent/churn flow.

Figure 4 . 28 :

 428 Figure 4.28: PDF of void fraction time-evolution to identify the flow regime.

  (a) PSD. (b) α(t) 1-second zoom for test T7.

Figure 4 . 29 :

 429 Figure 4.29: PSD of void fraction evolution within the tube bundle for flow regime identification.

Figure 4 . 30 :

 430 Figure 4.30: Proposed flow regime map.

Figure 4 . 31 :

 431 Figure 4.31: Proposed flow regime map, comparison with the literature. The nomenclature according to the authors: B-bubbly, T-transition, I-intermittent, S-slug, C-churn, A-annular, D-dispersed.

Figure 4 . 32 :

 432 Figure 4.32: Tube vibration response spectra (power spectral density) for the different tests T1, T2, T3, T4, T5, T6, T7 according to Table 2.1.

Figure 4 . 33 :

 433 Figure 4.33: Half power bandwidth for the different tests T1 to T7. In the y-axis is reported the bandwidth where the PSD magnitude correspond to the maximum value divided by the square root of 2.

Figure 4 . 34 :

 434 Figure 4.34: PDF of the accelerometer signals for tests T1 to T7 in Figure 4.34a and its FWHM in Figure 4.34b.

Figure 4 . 35 :

 435 Figure 4.35: Exponential and linear fit of the data point reported in Figure 4.34b with the R 2 values reported.

Figure 4 .

 4 Figure 4.36: RMS of vibration amplitude.

Figure 4 . 37 :

 437 Figure 4.37: Normalized two-phase damping comparison between experimental values (calculated according Equation 4.24) and theory according to Pettigrew et al. [11] (Equation 4.26). Both, experimental and calculated values are evaluated using the HEM void fraction model and the measured values of void fraction (α m ).

Figure 5 . 1 :

 51 Figure 5.1: Example of two-phase flow regime for dispersed phase (bubbly flow), large structures (slug flow), hybrid (churn flow) [42].

Figure 5 . 2 :

 52 Figure 5.2: Schematic representation of different interfacial scales compared to the numerical grid.

Figure 5 . 3 :

 53 Figure 5.3: The three-cell stencil method: schematic representation [43].

Figure 5 . 4 :

 54 Figure 5.4: Schematic representation of the various numerical domains overlaid on the TREFLE facility sketch, highlighting the instrumentation positions for extracting experimental measurements.

Figure 5 . 5 :

 55 Figure 5.5: Computational domain. P1 and P2 are the extracted profiles for mesh sensitivity study.

Figure 5 . 6 :

 56 Figure 5.6: Mesh detail.

  (a) Gas velocity sensitivity (b) Liquid velocity sensitivity (c) Void fraction sensitivity (d) Bubble diameter sensitivity

Figure 5 . 7 :

 57 Figure 5.7: Mesh sensitivity for Profile P1. M 1 is the reference mesh, while M 0.6 and M 1.3 are the coarser and finer mesh respectively.

( a )

 a Gas velocity sensitivity (b) Liquid velocity sensitivity (c) Void fraction sensitivity (d) Bubble diameter sensitivity

Figure 5 . 8 :

 58 Figure 5.8: Mesh sensitivity for Profile P2. M 1 is the reference mesh, while M 0.6 and M 1.3 are the coarser and finer mesh respectively.

Figure 5 .

 5 Figure 5.11: Three-dimensional computational domain for free duct. "WMS" indicates the profile used to compare the data with the WMS experimental results.

Figure 5 . 12 :

 512 Figure 5.12: Comparison between numerical results for void fraction distribution with images from fast camera. The numerical domain is cropped in order to compare the two images.

Figure 5 . 13 :

 513 Figure 5.13: Comparison between numerical results with optical probe data: vertical profiles (points 1, 8, 2, 9, 3 and 4 in Figure 2.24).

Figure 5 . 14 :

 514 Figure 5.14: Comparison between numerical results with optical probe data focusing on the difference between gas and liquid velocity (respectively v g and v l ). v LO l is the liquid velocity from the single-phase numerical simulations.

Figure 5 . 15 :

 515 Figure 5.15: Boundary conditions sensitivity results.

  and 5.18.

Figure 5 . 16 :

 516 Figure 5.16: Numerical domain for the long inlet case. The profiles "TB inlet" and "P1", from which the results are extracted, are indicated in red. They are positioned respectively 1 pitch below the tube bundle and 30 tube diameters above the inlet boundary conditions.

  (a) Gas velocity profile. (b) Liquid velocity profile.

  (a) Void fraction profile. (b) Bubble diameters profile.

Figure 5

 5 Figure 5.18: Flow evolution between "P1" and "TB inlet" from numerical results numerical results (NCFD). In addition, a comparison between numerical and experimental results at tube bundle inlet is reported ("TB inlet").

Figure 5 . 19 :

 519 Figure 5.19: Comparison of vertical profile results between simulations employing a three-dimensional domain and a two-dimensional domain.

  (a) Test T4. (b) Test T5.

Figure 5 . 20 :

 520 Figure 5.20: Instantaneous void fraction distribution for tests T4 and T5. Concerning the most left imagefor test T5, the void fraction ranges from 0 to 0.5 for the dispersed phase, while an isosurface for void fractions greater than 0.5 is applied to identify the large gas structures.

Figure 5 . 21 :

 521 Figure 5.21: Comparison of experimental data with numerical results for test T4. "OP" stands for experimental data obtained with optical probe, "Camera" indicates experimental results obtained with fast camera, "Neptune CFD" the numerical results.

Figure 5 . 22 :

 522 Figure 5.22: Comparison of experimental data with numerical results for test T5. "OP" stands for experimental data obtained with optical probe, "WMS" indicates experimental results obtained with wire mesh sensor, "Neptune CFD" the numerical results.

Figure 5 . 23 :

 523 Figure 5.23: Instantaneous void fraction distribution for tests T5. Comparison between WMS result (left figure) and numerical simulation result (right image).

  (a) Void fraction time evolution. (b) Void fraction time evolution: zoom in 1 second. (c) Power spectral density.

Figure 5 . 24 :

 524 Figure 5.24: Void fraction time evolution recorded by the monitoring point in the core of the numerical domain, located near the outlet of the free duct ("WMS" in Figure 5.11).

Figure 5 . 25 :

 525 Figure 5.25: Comparison between numerical results for Void Fraction distribution with images from fast camera from visualization window W5. The numerical domain is cropped in order to compare the two images. The numerical void fraction range between 0 and 0.5 for the dispersed phase, the large structures are in white.

Figure 5 . 26 :

 526 Figure 5.26: Comparison between numerical results for Void Fraction distribution with images from fast camera from visualization window W3. The numerical void fraction range between 0 and 0.5 for the dispersed phase, the large structures are in brown. The numerical domain is in gray with reduced opacity.

Figure 5 . 27 :

 527 Figure 5.27: Comparison between numerical results for Void Fraction distribution with images from fast camera from visualization window W3. The numerical void fraction range between 0 and 0.5 for the dispersed phase, the large structures are in brown. The numerical domain is in gray with reduced opacity.

Figure 5 .

 5 Figure 5.28: Test T4. Comparison between numerical results with optical probe data: vertical profiles (points 1, 8, 2, 9, 3 and 4 in Figure 2.24).

Figure 5 .

 5 Figure 5.29: Test T5. Comparison between numerical results with optical probe data: vertical profiles (points 1, 8, 2, 9, 3 and 4 in Figure 2.24).

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Representation of vortex shedding phenomena. . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Representation of turbulent excitation mechanism [2]. . . . . . . . . . . . . . . . . . . . . . . 1.3 Structure vibration response with increasing flow velocity to the various vibrations mechanisms in single-phase axial or cross-flow. The FEI has clearly a major impact on the vibration amplitude [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Scheme of the different tube bundles configurations. P indicates the pitch, D the tube diameter. Drag and lift directions of tube vibration are also reported. . . . . . . . . . . . . 1.5 Out-of-plane and in-plane modes of vibrations [4]. . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Experimental results comparisons from Pettigrew et al.

  m the tube bundle inlet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.17 Comparison of bubble size distribution obtained by optical probe and PDF of equivalent diameter obtained via fast camera. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.18 Comparison between sauter mean diameter measurements obtained from the optical probe and equivalent diameter measurements from the fast camera. . . . . . . . . . . . . . . . . . 3.19 Images obtained via fast camera in the visualization window W2 (refer to Figure 3.13a) for test T4. The large gas structures are highlighted. . . . . . . . . . . . . . . . . . . . . . . 3.20 Comparison between measured void fraction by means of optical probe and HEM void fraction model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.21 Comparison between measured gas velocity (v g ) with estimated gas velocities by means of measured void fraction (U sg α) and HEM void fraction (U sg α HEM ). . . . . . . . . . . . . 3.22 Images obtained via fast camera in the visualization window W5 (refer to Figure 3.13a). . 3.23 Highlight of a large gas structure in the tube bundle for test T4. . . . . . . . . . . . . . . . 3.24 Images of the test section for tests T4. In Figures 3.24a and 3.24b the bubble path is highlighted to show the bi-stable behavior of the two-phase flow. . . . . . . . . . . . . . . . 3.25 Optical probe measurements: horizontal profiles for test T1 to T4. . . . . . . . . . . . . . . 3.26 Vertical profiles of void fraction, gas velocity and sauter mean diameter within the tube bundle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.27 Comparison of experimental data obtained with the optical probe from point 2 with the model from Feenstra et al. [14]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.28 Bubble size distribution evolution inside the tube bundle. . . . . . . . . . . . . . . . . . . . 3.29 Images taken upstream and downstream the tube bundle. . . . . . . . . . . . . . . . . . . . 3.30 PDF of bubble equivalent diameter, bubble Eötvös number, and bubble aspect ratio (R)

  .29 Test T5. Comparison between numerical results with optical probe data: vertical profiles (points 1, 8, 2, 9, 3 and 4 in Figure 2.24). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 Summary of results obtained by Feenstra et al. [44]. Comparison among the void fraction model developed by Feenstra et al. [14] (cyan columns) and homogeneous equilibrium model (gray columns) for determining the Connors parameters for the stability criteria. α -HEM is the void fraction calculated by means of the homogeneous equilibrium model (S = 1), α -VF model is the void fraction estimated by means of Equation 1.2, G the critical mass flux to reach instability. The parameter for Connors formula are reported under the title "HEM" or "VF model" to specify if they are calculated using the HEM model or Equation 1.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Comparison among the work done by the authors. D is the tube diameter and P D the pitch to diameter ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modal analysis of the new tube model across various vibration frequencies. . . . . . . . . . 2.4 Comparison between flow velocity available (U av

	List of Tables
	1.1

The numerical domain is in gray with reduced opacity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.28 Test T4. Comparison between numerical results with optical probe data: vertical profiles (points 1, 8, 2, 9, 3 and 4 in Figure 2.24). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 Test configurations. J l and J g are the water and air gap superficial velocities according to Equation 2.1. U sl and U sg indicate the water and air superficial velocities in the homogenization section (free duct) according to Equation 2.2. . . . . . . . . . . . . . . . . . 2.2 Comparisons between the correlations to predict bubbles diameters at the injection. ṁgo is the gas mass flow rate per hole. The water velocity in between the injector tubes is 0.075 m s -1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3

Table 1 .

 1 Images takenf from Feenstra et al.

	[14]

1: Summary of results obtained by Feenstra et al.

[START_REF] Feenstra | Fluidelastic instability in a tube array subjected to two-phase r11 cross-flow[END_REF]

. Comparison among the void fraction model developed by Feenstra et al.

[START_REF] Feenstra | Modelling two-phase flow-excited damping and fluidelastic instability in tube arrays[END_REF]

(cyan columns) and homogeneous equilibrium model (gray columns) for determining the Connors parameters for the stability criteria. α -HEM is the void fraction calculated by means of the homogeneous equilibrium model (S = 1), α -VF model is the void fraction estimated by means of Equation 1.2, G the critical mass flux to reach instability. The parameter for Connors formula are reported under the title "HEM" or "VF model" to specify if they are calculated using the HEM model or Equation 1.2. Nakamura et al. [15], Mureithi et al. [71], Hirota et al.

  for staggered pitch tube bundle.

	Authors	two-phase	bundle	D [m] -P D instrumentation &
		mixture	configuration		measurements
	Grant et al. [23]	water-air	Staggered	0.019 -1.25	fast camera
	Ulbrich et al. [17]	water-air	Square	0.02 -1.5	fast camera
					pressure drops
	Xu et al. [16]	water-air	Square upward & 0.00979 -1.28	Fast Camera
			Square downward		pressure drops
	Noghrehkar et al. [18]	water-air	Square &	0.0127 -1.47	resistivity probe
			Staggered
	Murakawa et al. [19]	water-air	Square &	0.015 -1.5	Fast Camera
			Staggered		X-ray tomography
	Kanizawa et al. [20]	water-air	Triangular	0.019 -1.263	fast camera
					pressure drops
					capacitive sensor
	Aprin et al. [22]	hydrocarbons	Staggered	0.01905 -1.33	optical Probe
	Piteau et al. [21]	Air-water	Square	0.03 -1.5	optical Probe
	Table 1.2: Comparison among the work done by the authors. D is the tube diameter and P D the pitch
	to diameter ratio.

Table 2 .

 2 .1, and specific details about the gas injector are provided in Section 2.2). 1: Test configurations. J l and J g are the water and air gap superficial velocities according to Equation 2.1. U sl and U sg indicate the water and air superficial velocities in the homogenization section (free duct) according to Equation 2.2.

	Test no.	Air flow rate	No. of tubes & Water flow rate	J l	J g	U sl	U sg
			type of tubes	g s -1	m s -1 m s -1 m s -1 m s -1
	T1	25 L min -1 ±0.125	2 type 1	2500 ±10	0.126 0.021 0.039 0.007
	T2	50 L min -1 ±0.25	4 type 1	2500 ±10	0.126 0.042 0.039 0.013
	T3	100 L min -1 ±0.5	7 type 1	2500 ±10	0.126 0.084 0.039 0.026
	T4	200 L min -1 ±1	7 type 1	2500 ±10	0.126 0.168 0.039 0.051
	T5	10 g s -1 ±0.35	7 type 2	2500 ±10	0.126 0.420 0.039 0.135
	T6	16 g s -1 ±0.85	7 type 2	2500 ±10	0.126 0.673 0.039 0.212
	T7	29 g s -1 ±0.65	7 type 2	2500 ±10	0.126 1.220 0.039 0.373

Table 2 .

 2 Tate Gaddis et al. Gaddis et al. Muilwijk et al. Akita et al. Iguchi et al. 2: Comparisons between the correlations to predict bubbles diameters at the injection. ṁgo is the gas mass flow rate per hole. The water velocity in between the injector tubes is 0.075 m s -1 .

						[98]	[98]	[100]	[102]	[99]
			[u cr = 1.6]		Eq. 2.3	Eq. 2.5	Eq. 2.6	Eq. 2.10	Eq. 2.12	Eq. 2.9
	T1	7.8 ⋅ 10 -6	8.3	1.13	0.0035	0.0069	0.0070	0.0064	0.0058	0.0068
	T2	7.8 ⋅ 10 -6	8.3	1.13	0.0035	0.0069	0.0070	0.0064	0.0053	0.0068
	T3	8.9 ⋅ 10 -6	9.4	1.47	0.0035	0.0073	0.0074	0.0067	0.0049	0.0071
	T4	1.7 ⋅ 10 -5	18.9	5.92	0.0035	0.0095	0.0098	0.0086	0.0045	0.0085

Table 2 .

 2

		void fraction void fraction void fraction void fraction
		0.3	0.5	0.7	0.9
	U av	0.17 m/s	0.24 m/s	0.41 m/s	1.2 m/s
	U cr	0.7 m/s	0.98 m/s	1.19 m/s	1.5 m/s

3: Modal analysis of the new tube model across various vibration frequencies.

Table 2 .

 2 4: Comparison between flow velocity available (U av ) and critical velocity (U cr ) for different void fraction.

Table

			type 1	0.4	2500 ±10 0.126 0.021 0.039 0.007
	T2	50 ±0.25	4 type 1	0.4	2500 ±10 0.126 0.042 0.039 0.013
	T3	100 ±0.5	7 type 1	0.45	2500 ±10 0.126 0.084 0.039 0.026
	T4	200 ±1	7 type 1	0.9	2500 ±10 0.126 0.168 0.039 0.051

Table 3 .

 3 2: Test results in term of average Eötvös number ( Ēo), average bubble equivalent diameters deq , average aspect ratio ( R), average bubble velocity vB and Weber number W e. The uncertainty is reported in [mm] for the deq , and in [m s-1 ] for vB , v l and vBv l . In brackets is reported, in percentage, the underestimation due to bubble cluster appearance.

		3.65	5.03 ±0.38	1.68	0.47 ±0.19	0.045	0.425	12.8 3.6
	T2	3.65	5.03 ±0.38	1.68	0.47 ±0.19	0.051	0.419	12.8 3.6
	T3	3.84	5.17 ±0.38	1.68	0.45 ±0.19	0.064	0.386	11.9 3.4
	T4	3.45 4.70 (+12%) ±0.38 1.53 0.45 (+8%) ±0.19	0.089	0.361	10.7 3.0

Table 3

 3

.3. Additionally,

Table 3 . 3 :

 33 Tube bundle inlet conditions in term of average bubble equivalent diameters deq , average bubble size, sauter mean diameter d sauter , void fraction α, and gas velocity v g . The uncertainty is reported: in [mm] for deq , Bubble size, d sauter ; in % for α; in [m s-1 ] for v g .

Table 4 .

 4 .1. 1: Test configurations for the investigation of the intermittent flow regime.

	Test no. Air flow rate No. of tubes & Water flow rate	J l	J g	U sl	U sg
		g s -1	type of tubes	g s -1	m s -1 m s -1 m s -1 m s -1
	T5	10 ±0.35	7 type 2	2500 ±10	0.126 0.420 0.039 0.135
	T6	16 ±0.85	7 type 2	2500 ±10	0.126 0.673 0.039 0.212
	T7	29 ±0.65	7 type 2	2500 ±10	0.126 1.220 0.039 0.373

Table 4 .

 4 2: Summary of average data obtained from WMS and optical probe. ᾱ is the average void fraction. ᾱc indicates the void fraction detected in the center of the flow by the WMS (Figure4.7). dB is the average diameter, d sm is the sauter mean diameter and v B is the gas phase velocity.

	Test no. WMS WMS center Probe WMS Probe Probe ᾱ ᾱc ᾱ dB d sm v B
		[-]	[-]	[-]	[mm]	[mm] [m s -1 ]
	T5	0.20	0.3	0.28	5.4	9.0	1.17
	T6	0.24	0.38	0.35	5.8	9.5	1.31
	T7	0.33	0.52	0.51	6.3	10.4	1.51

Table 4 .

 4 3 a summary of the mentioned operating conditions is given.

	Author	U sg [m s -1 ]	Instrumentation	analyzed	Pipe	Air Injection	L/D
	name	U sl [m s -1 ]		quantities	D h [m]	system
	Ohnuki et al.	0.02-0.87	high speed camera	flow pattern	0.48	porous tube	4.2
	[33]	0.01-0.2	optical void probe	void fraction		nozzle
			high speed video	flow pattern	0.2	porous tube	60
	Ohnuki et al.	0.03-4.68	optical probe	void fraction		nozzle
	[34]	0.06-1.06		bubble sizes
			resistivity probe	bubble velocities			10
	Ali et al.	0.06-2.26	high speed camera	flow pattern	0.25	annular sleeve	48
	[36]	0.19-1.1	differential pressure	void fraction		Tee inj.
				fluctuation
	Prasser et al.	0.04-8.0		flow pattern	0.195
	[37, 119]	0.04-1.6	WMS	void fraction		tube with
				bubble sizes		perforated head	40
	TREFLE	0.007-0.373	high speed camera	flow pattern	0.287
		0.04	WMS	void fraction		perforated
			optical dual-tip probe	bubble sizes		tubes	5
				bubble velocities

Table 4 . 3 :

 43 Comparisons among the experimental campaigns investigated.

Table 4 .

 4 5: Test configurations for the investigation of the intermittent flow regime.

	1	800	0.04	0.210

Table 5 .

 5 1: Inlet boundary conditions used for tests T1 and T2. OP indicates data from measurements performed by optical probe at tube bundle inlet.

	Test no.	ṁair	ṁwater OP α OP d sm
		g s -1	g s -1	-	mm
	T1	0.5	2500	0.0224	3.71
	T2	1.0	2500	0.0348	3.81

Table 5 .

 5 2. Details about the numerical domain and simulation strategy are reported in Section 5.4.

	Test no.	ṁair	ṁwater	α	d sm
		g s -1	g s -1	-	mm
	T4	0.004	2.5	OP 0.144 4.0
	T5	0.01	2.5	WMS 0.2 4.0

Table 5 .

 5 2: Inlet boundary conditions used for tests T4 and T5. OP and WMS denote if the data are from optical probe or wire mesh sensor measurements.

(a) System with plates and honeycomb not installed (b) System with plates and honeycomb installed

Acknowledgements

Author

U sg [m s -1 ] U sl [m s -1 The aforementioned drift-flux correlations are compared to the experimental results of the present experimental campaign, as well as with the homogeneous equilibrium model (HEM), which assumes that the phases have the same velocity and that the two-phase flow is relatively homogeneous (Figure 4. [START_REF] Murakawa | Evaluation of void fraction around a tube in two-phase flow across horizontal tube bundle[END_REF]. The data from optical probe measurements (obtained during the bubbly flow experimental campaign) and WMS time-and space-averaged void fraction (collected from the intermittent flow experimental campaign) are both used in this comparison. This approach is adopted because, as observed earlier, the local measurement of void fraction becomes less representative of the entire two-phase flow as large gas structures emerge.

The most relevant outcome from this comparison is the large difference between the experimental data and the HEM model. The significant discrepancy suggests that the assumption of a slip ratio equal to 1 and that the phases are uniformly mixed in the HEM model is definitely not consistent for this configuration. The authors declare their correlations to be accurate with a ±30% uncertainty; however, this is not always the case for the present conditions.

The correlations proposed by Hibiki et al. [START_REF] Hibiki | One-dimensional drift-flux model for two-phase flow in a large diameter pipe[END_REF] is the best approximation for the present experimental data. Although it does not take into account the rectangular geometry of the channel, since it takes into account the effect of the flow regime via the distribution parameter, it succeeds in predicting both the bubbly flow regime and the churn flow regime of the experimental campaign of the present study.

Where:

The smoothing factor β is defined based on the liquid volume fraction α l as follows:

Thus, if the location involve only the large interface, β = 0 and γ = 0 resulting in I D l→g in Equation 5.17 being zero. Therefore, the momentum exchange at the interface is associated to I LIM l→g . In locations dominated by the presence of dispersed bubbles (α l > 0.7) β = 1. Otherwise, β is defined based on the local value of volume fraction. However, in the cells where α l > 0.4, it is desirable to reduce the influence of the dispersed phase and increase that of the large interface based on the volume fraction gradient. To achieve this, the correction function f c is introduced. The correction function f c lowers γ if the rate of interfacial area associated with the large interface present in the cell (approximately ∇α l ) is of the same order of magnitude as the surface-to-volume ratio of the cell. Thus, defining S Ω the surface of the cell, V Ω the volume of the cell and n Ω the normal to the cell surface, the correction function is defined as follows:

Therefore, in this model, the dispersed and continuous phases refer to the same field. The weighting factor γ determines whether the local morphology of the flow is closer to a separated or dispersed structure.

In this work, the neptune cfd code with its hybrid model, GLIM, was used to perform numerical simulations of the previously mentioned experimental campaigns (Chapters 3 and 4). In the following, the different numerical setups are presented.

Numerical setup

In this section the numerical setup utilized for the simulations is detailed. This includes a thorough description of the computational methods and boundary conditions employed in the simulations. The GLIM model has been used even if the experiments show the dispersed nature of the two-phase flow. The choice was made to allow the code to deal with the multiscale nature of the two-phase flow in the tube bundle. The numerical simulations were performed assuming incompressible flow for both the phases. The adaptative time step allows to reach a courant number less or equal than 1 in every part of the domain and at each time step. The second-order URANS (Reynolds-Averaged Navier-Stokes) EBRSM (Elliptic Blending Reynolds Stress Model) turbulent model (detailed in Mimouni et al. [START_REF] Mimouni | A second order turbulence model based on a reynolds stress approach for two-phase boiling flow and application to fuel assembly analysis[END_REF]) was used in order to have a low Reynolds model (no wall laws) at the walls for the liquid phase. While for the gas phase no turbulence model was used considering there is no turbulent flow or developped turbulence (referring to a model such as R ij EBRSM [START_REF] Mimouni | A second order turbulence model based on a reynolds stress approach for two-phase boiling flow and application to fuel assembly analysis[END_REF]) in a closed gas structure.

Different numerical domains were used depending on the simulation's focus within the TREFLE facility. These domains are described in the following subsections, and a schematic representation of the TREFLE facility with the different parts simulated is provided in Figure 5.4.