In this paper we present detailed Euler-Euler Large Eddy Simulations (LES) of dispersed bubbly flow in a rectangular bubble column. The motivation of this study is to investigate potential of this approach for the prediction of bubbly flows, in terms of mean quantities. The set of physical models describing the momentum exchange between the phases was chosen according to previous experiences of the authors. Experimental data, Euler-Lagrange LES and unsteady Euler-Euler Reynolds-Averaged Navier-Stokes model are used for comparison. It was found that the presented modelling combination provides good agreement with experimental data for the mean flow and liquid velocity fluctuations. The energy spectrum made from the resolved velocity from Euler-Euler LES is presented and discussed.

Measuring the liquid velocity and turbulence parameters in multiphase flows is a challenging task. In general, measurements based on optical methods are hindered by the presence of the gas phase. In the present work, it is shown that this leads to a sampling bias. Here, particle image velocimetry (PIV) is used to measure the liquid velocity and turbulence in a bubble column for different gas volume flow rates. As a result, passing bubbles lead to a significant sampling bias, which is evaluated by the mean liquid velocity and Reynolds stress tensor components. To overcome the sampling bias a window averaging procedure that waits a time depending on the locally distributed velocity information (hold processor) is derived. The procedure is demonstrated for an analytical test function. The PIV results obtained with the hold processor are reasonable for all values. By using the new procedure, reliable liquid velocity measurements in bubbly flows, which are vitally needed for CFD validation and modeling, are possible. In addition, the findings are general and can be applied to other flow situations and measuring techniques.

Cette thèse est consacrée aux écoulements homogènes de bulles, ainsi qu'à leur couplage avec le transport d'un scalaire et la turbulence. Elle s'intéresse plus spécifiquement aux effets de taille finie, des interactions hydrodynamiques et de la microstructure de la suspension qui sont étudiés à l'aide de simulations numériques directes à l'échelle d'une seule bulle. La dynamique d'une suspension laminaire de bulles induite par la seule gravité est d'abord revisitée. L'influence de la fraction volumique sur la vitesse de dérive des bulles est établie analytiquement et numériquement pour une suspension parfaitement ordonnée, puis des ressemblances entre suspensions ordonnées et suspensions désordonnées sont mises en évidence. Ces résultats sont ensuite mis à profit pour la modélisation du transport d'un scalaire passif au sein d'une suspension laminaire, tel que décrit par une diffusivité effective tensorielle, et des différences essentielles entre systèmes ordonnés et systèmes désordonnés concernant le transport de scalaire sont mises en exergue. Enfin, la turbulence est prise en compte dans les simulations et son interaction avec une bulle de taille finie est caractérisée. Il est montré que le comportement dynamique d'une bulle de taille comparable à la microéchelle de Taylor ressemble qualitativement à celui d'une microbulle, avec, notamment, une préférence pour certaines régions caractéristiques de l'écoulement. Une définition de l'écoulement vu par la bulle compatible avec les modèles standards de masse ajoutée et de portance est finalement proposée / This thesis is devoted to the study of homogeneous bubbly flows and their coupling with scalar transport and turbulence. It focuses on the effects of finite size, hydrodynamic interactions, and suspension microstructure, which are investigated using direct numerical simulations at the bubble scale. The dynamics of laminar buoyancy-driven bubbly suspensions is first revisited. More specifically, the effect of volume fraction on the bubble drift velocity is clarified by connecting numerical results to theory for dilute ordered systems, and similarities between perfectly ordered and free disordered suspensions are evidenced. These results are then used for the modeling of passive scalar transport in laminar suspensions as described by an effective diffusivity tensor, and crucial differences between ordered and disordered systems with respect to scalar transport are highlighted. Lastly, turbulence is included in the simulations, and its interaction with a finite-size bubble is characterized. The behavior of a bubble as large as Taylor microscale is shown to share a number of common features with that of a microbubble, most notably, the flow sampled by the bubble is biased. A definition of the liquid flow seen by the bubble, as it enters in usual models for the added mass and the lift forces, is finally proposed

In einem Verbundprojekt im Rahmen des Programms „Energie 2020+“ gefördert durch das BMBF koordiniert durch das HZDR arbeiteten 4 Universitäten, 2 Forschungszentren und ANSYS zusammen. Der vorliegende Bericht beschreibt die Arbeiten des HZDR, die im Zeitraum September 2009 bis Januar 2013 durchgeführt wurden. Das Vorhaben war auf die Entwicklung und Validierung von CFD-Modellen von unterkühltem Sieden bis zu Filmsieden gerichtet. Im Bericht werden die entwickelten und verwendeten Modelle dargestellt. Anhand der Nachanalyse von Experimenten wird auf die vorgeschlagene Kalibrierung der Modelle eingegangen. Wichtig ist hierbei eine genauere Beschreibung der Zwischenphasengrenzfläche, die durch Kopplung des Wandsiedemodells mit einem Populationsmodell erreicht werden kann. Anhand der Analyse von Bündelexperimenten konnte gezeigt werden, dass die gemessenen querschnittsgemittelten Messwerte mit einem Satz im Rahmen der Modellunsicherheiten kalibrierter Modellparameter reproduziert werden kann. Für die Berechnung der Verteilungsmuster des Dampfgehaltes im Kanalquerschnitt muss die Modellierung der Turbulenz beachtet werden. Die experimentellen Arbeiten waren auf die Untersuchung eines Brennelementbündels gerichtet. An einer Versuchsanordnung zu einem Brennelementbündel werden die turbulente einphasige Geschwindigkeit (PIV), der mittlere Gasgehalt (Gamma-Densitometrie) sowie der zeitlich und räumlich aufgelöste Gasgehalt (Hochgeschwindigkeits-Röntgentomographie) gemessen. Letztere Methode wurde in Rossendorf entwickelt.

The evolution of numerical methods and computational facilities allow re- searchers to explore complex physical phenomenons such as multiphase flows. The specific regime of incompressible, turbulent, bubbly two-phase flow (where a car- rier fluid is infused with bubbles or particles) is also receiving increased attention due to it’s appearance in major industrial processes. The main challenges arise from coupling individual aspects of the physics into a unified model and to provide a robust numerical framework. The presented work aimed at to achieve the second part by employing the most frequently used dispersed two-phase flow model and another incompressible, turbulent single phase solver as a base flow provider for coupled Lagrangian or surface tracking tools. Among the numerical techniques, the finite element method is a powerful can- didate when the need arises for multiphysics simulations (for example coupling with an electrochemical module) where the counterpart has a node based ap- proach. Stabilization schemes such as PSPG/SUPG/BULK provide remedies for the pressure decoupling and the inherent instability of the central discretization when applied for convective flow problems. As an alternative to unsteady solvers based upon an explicit or a fully im- plicit nonlinear treatment of the convective terms, a semi-implicit scheme results in a method of second order accurate in both space and time, has absolute linear stability and requires only a single or two linear system solution per time step. The application of the skew symmetric approach to the convective term further stabilizes the solution procedure and in some cases it even prevents divergence. The Eulerian-Eulerian two-phase flow model poses various issues to be over- come. The major difficulty is the density ratio between the phases; for an ordinary engineering problem it is in the order of thousands or more. The seemingly minus- cule differences in the formulation of the stabilizations can cause very different end results and require careful analysis. Volume fraction boundedness is of concern as well, but it is treatable by solving for its logarithm. Since the equations allow jumps (even separation of the phases) in the volume fraction field, discontinuity capturing techniques are also needed. Besides the standard ’spatial’ stabilization temporal smoothing is also necessary, otherwise the limitation in time step size becomes too stringent. Designing a flow solver is one side of the adventure, but verification is equally important. Comparison against analytical solution (such as the single and two- phase Taylor-Green testcase) provides insight and confirmation about the mathe- matical and physical properties. Meanwhile comparing with real life experiments prove the industrialization and usability of a code, dealing with low quality meshes and effective utilization of computer clusters. / Doctorat en Sciences de l'ingénieur et technologie / info:eu-repo/semantics/nonPublished

Cette thèse porte sur l'étude de la dynamique et de la morphologie de bulles non confinées dans des colonnes à bulles et de bulles confinées dans des microcanaux, ainsi que sur l'étude du transfert de matière entre une bulle sphérique confinée dans un microcanal et le liquide environnant. <p><p>Un dispositif expérimental d'imagerie et une méthode de posttraitement des images brutes ont été développés afin d'analyser la dynamique et la morphologie de bulles non confinées ayant une trajectoire non rectiligne de leur centre de masse en évitant les effets de perspectives et en déterminant un seuil pour la binarisation des images brutes sur base d'un critère bien défini. Ce dispositif expérimental et cette méthode de posttraitement des images brutes ont permis de générer des données relatives à la dynamique et la morphologie de bulles ellipsoïdales isolées et non confinées, pour des nombres d'Eötvös (Eo) et de Morton (Mo) de ces bulles tels que 0.8 < Eo < 8 et 10 <SUP>-11</SUP> < Mo < 10 <SUP>-7</SUP>. L'analyse de ces données a permis de cartographier la nature de la trajectoire d'une bulle et la présence d'une éventuelle oscillation de son interface en fonction de ses nombres d'Eötvös et de Morton. Les bulles ayant une trajectoire hélicoïdale sans oscillation de leur interface ont été sélectionnées afin de proposer des corrélations pour calculer l'amplitude et la fréquence de leur trajectoire en fonction de leurs nombres d'Eötvös et de Morton. Concernant les bulles ayant une trajectoire en zigzag ou hélicoïdale sans oscillation de leur interface, l'analyse des données a permis de montrer l'alignement entre le vecteur vitesse de leur centre de masse et leur petit axe. Les rayons de courbure de l'avant et l'arrière de l'interface de ces bulles ont été évalués. Pour les bulles ayant une trajectoire en zigzag, une corrélation a été établie pour calculer le rapport des rayons de courbure à l'avant et à l'arrière de leur interface en fonction de leurs nombres d'Eötvös et de Morton. Une pulsation dans la composante verticale du mouvement du centre de masse d'une bulle a été observée dans le cas d'une trajectoire en zigzag de la bulle et ce à une fréquence égale au double de celle de sa trajectoire. Une telle pulsation n'a pas pu être identifiée dans le cas d'une trajectoire hélicoïdale d'une bulle. <p><p>Concernant l'analyse de la dynamique de bulles sphériques confinées dans des microcanaux de sections carrée et circulaire, ainsi que du transfert de matière entre ces bulles et le liquide environnant, une méthode numérique a été développée dans laquelle deux conditons aux limites sont considérées sur l'interface liquide-gaz: une condition de contrainte tangentielle nulle et une condition de non glissement. Les résultats obtenus avec cette méthode ont permis de caractériser les champs de vitesse et de concentration autour des bulles considérées, et de montrer leurs interactions. Grâce à ces résultats, des corrélations ont été établies, dans ces microcanaux et pour ces deux conditions aux limites, pour calculer la vitesse des bulles et pour caractériser le transfert de matière entre ces bulles et le liquide en fonction des paramètres définissant le système. Sur base de ces corrélations et de bilans de matière et de quantité de mouvement, un modèle pour la dissolution de bulles le long de microcanaux de sections carrée et circulaire a été proposé, pour le régime bubbly flow, et comparé avec des données expérimentales disponibles dans la littérature. Ce modèle permet de prédire, pour une bulle se mouvant le long d'un microcanal de section carrée ou circulaire, les évolutions des pressions dans le liquide et le gaz, de son diamètre, de sa vitesse, de la concentration du gaz dissous dans le liquide, de la distance de séparation entre cette bulle et la bulle qui la suit et du coefficient de transfert de matière entre cette bulle et le liquide environnant.<p><p> / Doctorat en Sciences de l'ingénieur / info:eu-repo/semantics/nonPublished

This thesis presents the investigation of an Euler-Lagrange framework for modeling bubble coalescence in dispersed bubbly flows. The interaction between bubbles may be caused by several mechanisms. Among them, the random motion due to turbulent fluctuations is normally of major significance. One focus of this work is to apply a bubble dispersion model for modeling turbulence-induced coalescence, occurring in a certain percentage of collision events. Large bubbles appear due to coalescence, and their disturbance to the liquid phase is not negligible in most circumstances. However, the point-mass Euler-Lagrange method requires the bubble or particle size to be much smaller than the cell size when the interphase coupling is considered. Otherwise, numerical instabilities may arise. Therefore, interpolation methods between the Euler and the Lagrange phase for finite-size bubbles that are bigger than or of the same size as numerical cells are studied. The Euler-Lagrange method describes the continuous phase on the Euler grid, and the dispersed phase is treated as Lagrange points in the simulation. Bubble motion is governed by an ordinary differential equation for the linear momentum considering different forces. The turbulent dispersion of the dispersed phase is reconstructed with the continuous random walk (CRW) model. Bubble-bubble collisions and coalescence are accounted for deterministically. The time-consuming search for potential collision partners in dense bubbly flows is accelerated by the sweep and prune algorithm, which can be utilized in arbitrary mesh types and sizes. If the interphase coupling is considered in the simulations, the spatially distributed coupling method is used for the Lagrange-to-Euler coupling. For the Euler-to-Lagrange coupling, a new approach is proposed. To evaluate the dispersion and coalescence models, one-way coupled simulations of bubbly pipe flows at low Eötvös numbers are conducted. Validation against the experiments demonstrates that the one-way coupled EL-CRW dispersion model can well reproduce the bubble distribution in a typical dense bubbly pipe flow. Good agreement of the bubble size distribution at the pipe outlet between the simulation and the experiment is obtained. Two-way coupled simulations are performed to validate the interpolation methods. A combination of coupling approaches is employed in a square bubble column reactor to examine the general validity for a large-scale bubbly flow. Combining the proposed interpolation scheme with the dispersion and bubble interaction models, the coalescence and breakage in bubbly flows are studied in a turbulent pipe flow. The predicted bubble size distribution shows a good match to the measurement. The results are independent of the mesh resolution in the studied range from point-mass simulations to finite-size situations.:Nomenclature 1 Introduction 1.1 Motivation and background for the thesis 1.2 Outline 2 Equations for modeling bubbly flows 2.1 Governing equations of the continuous phase 2.2 Governing equations of the dispersed phase 2.3 Modifications to the bubble force equations 2.3.1 One-way coupled simulations with RANS modeling 2.3.2 Two-way coupled simulations 2.4 Generation of fluctuations 2.4.1 Different approaches to dispersion modeling 2.4.2 Normalized continuous random walk model 2.4.3 Employing the mean velocity field to determine forces 3 Bubble collision, coalescence and breakup 3.1 Previous studies and requirement of the interaction modeling 3.2 Detection of collisions with the sweep and prune algorithm 3.3 Coalescence modeling 3.3.1 Condition of bubble coalescence 3.3.2 Model of Kamp et al. [2001] 3.3.3 Model of Hoppe and Breuer [2018] 3.3.4 Model of Schwarz et al. [2013] 3.3.5 Comparison of coalescence models 3.4 Breakup modeling 3.4.1 Turbulence induced breakups 3.4.2 Post-breakup treatment 4 Interpolation techniques for two-way coupled simulations 4.1 Lagrange-to-Euler coupling 4.1.1 Introduction to the spatially distributed coupling 4.1.2 Intersection plane method 4.1.3 Subcell method 4.1.4 Random points method 4.2 Euler-to-Lagrange coupling 4.2.1 Approaches for computing the undisturbed velocity 4.2.2 Coarser grid method 4.2.3 Averaging the fluid velocity in front of the bubble 4.2.4 Velocity from upstream disk 4.2.5 Gradient of the undisturbed liquid velocity 5 One-way coupled simulation of bubble dispersion and resulting interaction 5.1 Implementation and verification of the continuous random walk model 5.2 Bubble dispersion in turbulent channel flows 5.3 Bubble dispersion and interaction in turbulent pipe flows 5.3.1 Overview of studied cases 5.3.2 Results of the bubble dispersion 5.3.3 Results of the bubble coalescence 6 Two-way coupled simulation of finite-size bubbles 6.1 Flow solver and algorithm 6.2 Assessing the Lagrange-to-Euler coupling methods 6.2.1 Previous studies 6.2.2 Simulation setups for a single bubble in quiescent liquid 6.2.3 Results and discussion 6.3 Assessing the Euler-to-Lagrange coupling methods 6.3.1 Simulation of two bubbles rising inline 6.3.2 Simulation of a bubble rising in linear shear flows 6.4 Large-eddy simulation for a square bubble column 6.5 Bubble coalescence in a turbulent pipe flow 7 Conclusions and outlook Appendices A.1 Equations of turbulence models A.2 Numerical implementation of the full CRW drift term A.3 Results of bubble coalescence modeling for case B to case E A.4 Search algorithm of the upstream disk method Bibliography / Diese Arbeit stellt die Untersuchung eines Euler-Lagrange-Rahmens zur Modellierung der Blasenkoaleszenz in dispergierten Blasenströmungen vor. Die Interaktion zwischen Blasen kann durch mehrere Mechanismen verursacht werden. Unter ihnen sind die zufälligen Bewegungen aufgrund von turbulenten Fluktuationen von großer Bedeutung. Ein Schwerpunkt dieser Arbeit ist die Anwendung eines Blasendispersionsmodells zur Modellierung der turbulenzinduzierten Koaleszenz, die in einem bestimmten Prozentsatz der Kollisionsereignisse auftritt. Große Blasen entstehen durch Koaleszenz und ihre Störung der flüssigen Phase ist in den meisten Fällen nicht zu vernachlässigen. Die Punkt-Masse-Euler-Lagrange-Methode erfordert jedoch, dass die Blasengröße viel kleiner als die Zellgröße ist, wenn die Interphasenkopplung berücksichtigt wird. Andernfalls kann es zu numerischen Instabilitäten kommen. Daher werden Interpolationsmethoden zwischen den zwei Phasen untersucht. Die kontinuierliche Phase wird auf dem Euler-Gitter beschrieben und die dispergierte Phase wird als Punkte behandelt. Die Blasenbewegung wird durch eine gewöhnliche Differentialgleichung unter Berücksichtigung verschiedener Kräfte bestimmt. Die turbulente Dispersion der Blasen wird mit dem CRW-Modell (continuous random walk) rekonstruiert. Blasen-Blasen-Kollisionen werden deterministisch berücksichtigt. Die Suche nach potentiellen Kollisionspartnern wird durch den Sweep- und Prune-Algorithmus beschleunigt, der in beliebigen Gittertypen und -größen eingesetzt werden kann. Wird die Interphasenkopplung berücksichtigt, so wird für die Lagrange-zu-Euler-Kopplung die spatially distributed coupling verwendet. Für die Euler-zu-Lagrange-Kopplung wird ein neuer Ansatz vorgeschlagen. Um die Dispersions- und Koaleszenzmodelle zu bewerten, werden Einweg-gekoppelte Simulationen von blasenbeladenen Rohrströmungen bei niedriger Eötvös-Zahl durchgeführt. Die Validierung zeigt, dass das einseitig gekoppelte EL-CRW-Dispersionsmodell die Blasenverteilung in einer typischen dichten, blasenbeladenen Rohrströmung gut reproduzieren kann. Es wird eine gute Übereinstimmung der Blasengrößenverteilung am Rohrauslass zwischen der Simulation und dem Experiment erzielt. Zur Validierung der Interpolationsmethoden werden Zweiweg-gekoppelte Simulationen durchgeführt. Eine Kombination von Kopplungsansätzen wird in einem Blasensäulenreaktor eingesetzt, um die allgemeine Gültigkeit zu untersuchen. Durch Kombination des vorgeschlagenen Interpolationsschemas mit den Dispersions- und Blasenwechselwirkungsmodellen werden die Koaleszenz und der Zerfall in einer turbulenten blasenbeladenen Rohrströmung untersucht. Die berechnete Blasengrößenverteilung zeigt eine gute Übereinstimmung mit der Messung und erweist sich als unabhängig von der Netzauflösung im untersuchten Bereich von PunktMasse-Simulationen bis zu Situationen mit Blasen endlicher Größe.:Nomenclature 1 Introduction 1.1 Motivation and background for the thesis 1.2 Outline 2 Equations for modeling bubbly flows 2.1 Governing equations of the continuous phase 2.2 Governing equations of the dispersed phase 2.3 Modifications to the bubble force equations 2.3.1 One-way coupled simulations with RANS modeling 2.3.2 Two-way coupled simulations 2.4 Generation of fluctuations 2.4.1 Different approaches to dispersion modeling 2.4.2 Normalized continuous random walk model 2.4.3 Employing the mean velocity field to determine forces 3 Bubble collision, coalescence and breakup 3.1 Previous studies and requirement of the interaction modeling 3.2 Detection of collisions with the sweep and prune algorithm 3.3 Coalescence modeling 3.3.1 Condition of bubble coalescence 3.3.2 Model of Kamp et al. [2001] 3.3.3 Model of Hoppe and Breuer [2018] 3.3.4 Model of Schwarz et al. [2013] 3.3.5 Comparison of coalescence models 3.4 Breakup modeling 3.4.1 Turbulence induced breakups 3.4.2 Post-breakup treatment 4 Interpolation techniques for two-way coupled simulations 4.1 Lagrange-to-Euler coupling 4.1.1 Introduction to the spatially distributed coupling 4.1.2 Intersection plane method 4.1.3 Subcell method 4.1.4 Random points method 4.2 Euler-to-Lagrange coupling 4.2.1 Approaches for computing the undisturbed velocity 4.2.2 Coarser grid method 4.2.3 Averaging the fluid velocity in front of the bubble 4.2.4 Velocity from upstream disk 4.2.5 Gradient of the undisturbed liquid velocity 5 One-way coupled simulation of bubble dispersion and resulting interaction 5.1 Implementation and verification of the continuous random walk model 5.2 Bubble dispersion in turbulent channel flows 5.3 Bubble dispersion and interaction in turbulent pipe flows 5.3.1 Overview of studied cases 5.3.2 Results of the bubble dispersion 5.3.3 Results of the bubble coalescence 6 Two-way coupled simulation of finite-size bubbles 6.1 Flow solver and algorithm 6.2 Assessing the Lagrange-to-Euler coupling methods 6.2.1 Previous studies 6.2.2 Simulation setups for a single bubble in quiescent liquid 6.2.3 Results and discussion 6.3 Assessing the Euler-to-Lagrange coupling methods 6.3.1 Simulation of two bubbles rising inline 6.3.2 Simulation of a bubble rising in linear shear flows 6.4 Large-eddy simulation for a square bubble column 6.5 Bubble coalescence in a turbulent pipe flow 7 Conclusions and outlook Appendices A.1 Equations of turbulence models A.2 Numerical implementation of the full CRW drift term A.3 Results of bubble coalescence modeling for case B to case E A.4 Search algorithm of the upstream disk method Bibliography

The thesis deals with experimental research of mechanism of liquid breakup at twin-fluid atomizers. Four different atomizers were examined at the beginning of the research. Two of them were of standard design (Y-jet and effervescent nozzles), and the rest two atomizers were developed as a part of the thesis (so called CFT and inversed effervescent atomizers). Results show that only the inversed effervescent atomizer was capable of generating stable spray under examined conditions due to the specific breakup mechanism. This mechanism is similar to what was observed in effervescent atomizers. However, the mixing process inside the inversed effervescent atomizer is very different. The specific breakup mechanism was then defined as the main scope of the thesis. It was investigated by the high-speed imaging. The images were then processed by proper orthogonal decomposition and by fast Fourier transformation. Spray spatial development was examined using phase Doppler anemometer. The data was analyzed to describe the dynamics of the spray. A detailed description of the breakup mechanism is made from the combination of the experimental and post-processing techniques. The thesis brings new insight into the understanding of the liquid breakup mechanism and shows a potential for a further development of the inversed effervescent atomizer.

In einem Verbundprojekt im Rahmen des Programms „Energie 2020+“ gefördert durch das BMBF koordiniert durch das HZDR arbeiteten 4 Universitäten, 2 Forschungszentren und ANSYS zusammen. Der vorliegende Bericht beschreibt die Arbeiten des HZDR, die im Zeitraum September 2009 bis Januar 2013 durchgeführt wurden. Das Vorhaben war auf die Entwicklung und Validierung von CFD-Modellen von unterkühltem Sieden bis zu Filmsieden gerichtet. Im Bericht werden die entwickelten und verwendeten Modelle dargestellt. Anhand der Nachanalyse von Experimenten wird auf die vorgeschlagene Kalibrierung der Modelle eingegangen. Wichtig ist hierbei eine genauere Beschreibung der Zwischenphasengrenzfläche, die durch Kopplung des Wandsiedemodells mit einem Populationsmodell erreicht werden kann. Anhand der Analyse von Bündelexperimenten konnte gezeigt werden, dass die gemessenen querschnittsgemittelten Messwerte mit einem Satz im Rahmen der Modellunsicherheiten kalibrierter Modellparameter reproduziert werden kann. Für die Berechnung der Verteilungsmuster des Dampfgehaltes im Kanalquerschnitt muss die Modellierung der Turbulenz beachtet werden. Die experimentellen Arbeiten waren auf die Untersuchung eines Brennelementbündels gerichtet. An einer Versuchsanordnung zu einem Brennelementbündel werden die turbulente einphasige Geschwindigkeit (PIV), der mittlere Gasgehalt (Gamma-Densitometrie) sowie der zeitlich und räumlich aufgelöste Gasgehalt (Hochgeschwindigkeits-Röntgentomographie) gemessen. Letztere Methode wurde in Rossendorf entwickelt.

<div>The current study is aimed at developing a well-posed and objective, i.e., frame invariant, Eulerian one-dimensional (1D) Two-Fluid Model (TFM) to predict flow regime transition from dispersed to clustered bubbly and slug flow for vertical adiabatic two-phase flows. Two-phase flows in general are characterized by local material wave or void fraction wave instabilities and flow regime transitions are one of the important consequences of these instabilities. The physical mechanism of wake entrainment for clustering of dispersed bubbles is proposed, leading to formation of bubble clusters and Taylor bubbles. The focus of the work is on simulation of the local interfacial structures for bubble clusters and Taylor bubbles, using a well-posed, unstable and non-linearly bounded 1D Shallow Water TFM.</div><div><br></div><div>The first part of the current study investigates the dynamic behavior of the well posed 1D mechanistic TFM obtained from the averaging approach of Ishii [1], due to wake entrainment instability. For this, a 1D Shallow Water TFM derived from the 1D mechanistic TFM is used, which retains the same dynamic behavior as that of the latter at short wavelengths and the required wake entrainment force is derived mechanistically. Three stability approaches are followed to study the dynamic behavior of the 1D Shallow Water TFM: characteristics, dispersion analysis, and nonlinear numerical simulations. An in-house code is used for the 1D numerical simulations of the growth of void fraction waves due to wake entrainment. The simulation results are validated with the experimental data of Cheng and Azzopardi [2] and Song et al. [3] To conclude the first part, the 1D results of the two-equation Shallow Water TFM are carried over to the complete four-equation TFM for quasi 1D simulations using the commercial CFD code of ANSYS Fluent.</div><div><br></div>As an alternative to the mechanistic approach, which is based on Newtonian mathematics, a variational approach based on Lagrangian and Hamiltonian mathematics is used in the second part of the thesis. While the mechanistic approach operates in terms of forces acting on the two-phase mixture, the variational approach operates in terms of energies of the two-phase system. To derive the equations of motion using the variational approach, the extended Hamilton principle of least action is applied to the Lagrangian density of the two-phase mixture. One of the appealing features<br>17<br>of this procedure is that the derived equations of motion are objective (Geurst [4]), in particular the added mass terms.<br>Thus, the second part of the current study focuses on deriving an objective, well-posed and unstable 1D TFM as well as developing a constitutive model for the wake entrainment effect using the variational method. Additional momentum transfer terms present in both the liquid phase and gas phase momentum equations, which render the variational TFM objective, are discussed. The variational method is then used to derive the 1D Shallow Water TFM using the fixed flux assumption. The conservative interfacial momentum transfer terms require formulation of the inertial coupling between the phases. Potential flow theory is first used to derive the inertial coupling coefficient for a single bubble and then for a pair of bubbles to consider interaction between the two bubbles. Then, a lumped parameter model is used to derive the inertial coupling coefficient for the wake entrainment effect. A local drag coefficient is obtained for the non-conservative interfacial drag force from the experimental data using kinematic approximation, i.e., force balance between drag and gravity. The linear and non-linear stability analyses are used to address the stability of the 1D variational Shallow Water TFM. The presence of appropriate short-wave physics makes the 1D Shallow Water TFM hyperbolic well-posed and kinematically unstable. Finally, numerical simulations are performed to demonstrate the development of void fraction waves due wake entrainment. The growth of void fraction waves is non-linearly bounded, i.e., Lyapunov stable. The simulation results are compared with the experimental data to validate the propagation properties of void fraction waves for bubble clusters and Taylor bubbles. This work illustrates the short-wave two-phase flow simulation capability of the TFM for the bubbly to slug flow regime transition.