O tema deste trabalho é o estudo de métodos numéricos para a simulação de escoamentos incompressíveis com superfície livre a baixos Reynolds, por meio da metodologia GENSMAC e suas formulações explícita e implícita. Neste contexto, temos especial interesse na formulação implícita, objetivando o enfraquecimento da restrição de estabilidade parabólica no passo temporal em escoamentos com superfície livre. O trabalho pode ser dividido em duas partes: na primeira, algumas modificações são discutidas, propostas e testadas com o objetivo de tornar a formulação implícita mais eficiente e precisa; em seguida, aproveitamos os resultados obtidos e generalizamos o método numérico existente para simular escoamentos viscoelásticos modelados pela equação constitutiva SXPP. Em ambos os casos, a formulação explícita também é usada para comparação e teste. Resultados que demonstram a eficiência e robustez das técnicas desenvolvidas são apresentados por meio da simulação numérica de complexos problemas envolvendo superfície livre.O tema deste trabalho é o estudo de métodos numéricos para a simulação de escoamentos incom- pressíveis com superfície livre a baixos Reynolds, por meio da metodologia GENSMAC e suas formulações explícita e implícita. Neste contexto, temos especial interesse na formulação implícita, objetivando o en- fraquecimento da restrição de estabilidade parabólica no passo temporal em escoamentos com superfície livre. O trabalho pode ser dividido em duas partes: na primeira, algumas modificações são discutidas, propostas e testadas com o objetivo de tornar a formulação implícita mais eficiente e precisa; em seguida, aproveitamos os resultados obtidos e generalizamos o método numérico existente para simular escoamentos viscoelásticos modelados pela equação constitutiva SXPP. Em ambos os casos, a formulação explícita também é usada para comparação e teste. Resultados que demonstram a eficiência e robustez das técnicas desenvolvidas são apresentados por meio da simulação numérica de complexos problemas envolvendo superfície livre / The purpose of this work is the study of numerical methods for low Reynolds number incompressible free surface flows, including the GENSMAC methodology and its explicit and implicit formulations. In this context, we have special interest in the implicit formulation, in order to circumvent the parabolic stability restriction in the time step for free surface flows. The work can be divided into two parts: firstly, some modifications are proposed, discussed, and tested with the purpose of making the implicit formulation more e¢ cient and accurate; secondly, we take advantage of the results derived in the first part to generalize the numerical method for simulating viscoelastic free surface flows modeled by the constitutive equation SXPP. In both cases, the explicit formulation is also used for comparison and tests. Results that demonstrate the e¢ ciency and robustness of the developed techniques are presented and illustrated by the numerical simulation of complex problems involving free surface flows

Neste projeto, é apresentado um método numérico com uma abordagem do tipoMAC para a simulação de escoamentos viscoelásticos incompressíveis tridimensionais com superfície livre governados pelo modelo de fluido SXPP. A formulação apresentada nesse trabalho é uma extensão dos resultados obtidos por Oishi et al. (2011), sobre o estudo de métodos numéricos para a simulação de escoamentos incompressíveis viscoelásticos com superfície livre a baixos números de Reynolds, para o caso bidimensional. No contexto de problemas transientes, metodologias explícitas para solução numérica das equações governantes apresentam restrições de estabilidade muito severas para a definição do passo temporal, acarretando em um custo computacional relativamente alto. Sendo assim, utilizamos um método implícito para resolver a equação de conservação da quantidade de movimento, eliminando assim, a restrição de estabilidade parabólica e diminuindo significativamente o custo computacional. Mas tal estratégia acopla os campos de velocidade e pressão. Dessa forma, para desacoplar esses campos, foi utilizado uma abordagem que combina método de projeção com uma técnica implícita para o tratamento da pressão na superfície livre. A equação constitutiva foi resolvida pelo método de Runge-Kutta de segunda-ordem. A validação do método numérico foi realizada utilizando refinamento da malha no escoamento em um canal. Como aplicação, apresentamos resultados numéricos sobre o problema do jato oscilante e do inchamento do extrudado / In this work, we present a numerical method with a MAC type approach to simulate tridimensional incompressible viscoelastic free surface flows governed by a SXPP (Single eXtended Pom-Pom) model. The formulation presented in this work is an extension to the work of Oishi et al. (2011). They have studied numerical methods for solving incompressible viscoelastic free surface flows with low Reynolds number, for the bidimensional case. In the context of transient problems, explicitmethodologies for the numerical solution of the governing equations present severe stability constraints for defining the time step, what highly increases the computational cost. Due to this fact, an implicit method is used to solve the momentum equation, eliminating the parabolic stability constraint and decreasing significantly the computational cost. However, this strategy couples velocity and pressure fields. To decouples this fields, it was used an approach that combines a projection method and an implicit technique for the treatment of the pressure at the free surface. The constitutive equation is solved by a second-order Runge-Kutta method. The numerical method validation was achieved by a mesh refinement for a flow in a channel. As applications, numerical results of the die-swell problem and the jet buckling phenomenon are presented

Oxide inclusions are among the most commonly reported defects in ferrous and non-ferrous castings. They affect the surface quality, machinability, and mechanical performance of a cast part. Air entrainment during mold filling is the main source of the oxygen that is consumed in inclusion formation. A quantitative understanding of the formation mechanisms or the prediction of final amounts and locations of oxide inclusions in metal castings is not available. Ductile iron experiments are conducted to study the formation of oxide inclusions during pouring. Oxide inclusions are measured by serial sectioning of the solidified castings. The effect of different gating systems, section thicknesses, and surface orientations on the inclusion formation and final distribution is studied. In addition, a computational model is developed for predicting the formation, motion and final location of oxide inclusions during pouring of metal castings, with the focus on the important mechanism of generation of oxide inclusions due to air entrainment during mold filling. The developed model calculates the local air entrainment rate as a function of the turbulent kinetic energy and the magnitude of the normal velocity gradient of the liquid metal at the liquid-air interface. The turbulent kinetic energy is estimated from the sum of the squares of the fluctuating velocity components relative to a spatially averaged mean velocity. The air entrainment model is implemented in a casting simulation software and validated by comparing its predictions to experimental air entrainment measurements for a circular water jet plunging into a quiescent pool. The liquid velocity, diameter and the turbulence intensity dependence is determined by a single entrainment coefficient. Oxide inclusions are then generated at the liquid-air interface, transported with the melt flow under the combined influences of drag and buoyancy, and captured by the solidifying casting surface. The developed model provides a powerful technique for predicting the oxide inclusion formation and final location.

La simulation numérique des écoulements à surface libre pour les applications du génie maritime est un problème qui présente de grands défis dans le domaine de la dynamique des fluides numérique (CFD). On propose dans cette thèse une solution, qui consiste à utiliser la méthode de Boltzmann sur réseau régularisée (RLBM) avec un modèle de surface libre basé sur le volume-de-fluide (VOF), et on étudie sa faisabilité et sa fiabilité. Les connaissances théoriques de la méthode de Boltzmann sur réseau (LBM) sont présentées dans un premier temps, sur la base d'un développement polynomial d'Hermite et d'une analyse de Chapman-Enskog. De cette perspective, l’idée de la RLBM se résume comme étant la régularisation d'Hermite des fonctions de distribution. Dans les cas tests suivants du vortex de Taylor-Green et de la cavité entraînée, il est vérifié que la RLBM posse possède une précision de second ordre et une stabilité améliorée. On a alors ensuite implémenté le modèle de surface libre dans la RLBM. Sur la simulation d'une onde de gravité visqueuse stationnaire et d'un écoulement de dambreak, il est montré que la régularisation stabilise fortement le calcul en réduisant les oscillations de pression, ce qui est très bénéfique pour obtenir des écoulements à surface libre précis, et que la RLBM n'introduit pas non plus de dissipation numérique supplémentaire. De plus, une nouvelle méthode de reconstruction des fonctions de distribution à la surface libre est proposée. Le modèle proposé est ainsi plus consistent avec la RLBM, ce qui offre un moyen efficace pour simuler des écoulements à surface libre à un grand nombre de Reynolds en génie maritime. / The numerical simulation of the freesurface flows for marine engineering applications is a very challenging issue in the field of computational fluid dynamics (CFD). In this thesis, we propose a solution, which is to use the regularized lattice Boltzmann method (RLBM) with a volume-of-fluid (VOF) based single-phase free-surface lattice Boltzmann (LB) model, and we investigate its feasibility and its reliability. The theoretical insights of the lattice Boltzmann method (LBM) are given at first, through the Hermite expansion and the Chapman-Enskog analysis. From this perspective, the idea of the RLBM is summarized as the Hermite regularization of the distribution functions. On the test-cases of the Taylor-Green vortex and the lid-driven cavity flow, the RLBM is verified to have a 2nd-order accuracy and an improved stability. The adopted free-surface model is then implemented into the RLBM and validated through simulating a viscous standing wave and a dambreak flow problems. It is shown that the regularization not only strongly stabilizes the calculation by reducing spurious pressure oscillations, which is very beneficial for obtaining accurate free-surface motions, but also does not introduce any extra numerical dissipation. Furthermore, a new reconstruction method for the distribution functions at the free-surface is proposed. The present model is more consistent with the RLBM, which provides an effective way for simulating high-Reynoldsnumber free-surface flows in marine engineering.

L'écoulement autour d'un objet partiellement immergé comme une carène ou une pile de pont est une configuration fondamentale au regard de la dynamique de la surface libre. La caractérisation de cet écoulement est essentielle dans des applications environnementales, ou pour des systèmes d'énergies marines renouvelables. De telles structures sont soumises à des efforts de traînée et de portance provenant de l'interaction avec le courant, la houle, et la surface libre. Ce travail s'inscrit dans la compréhension des efforts s'exerçant sur des objets partiellement immergés. Notre problématique est simplifiée en considérant une géométrie cylindrique. Nous présentons des résultats expérimentaux obtenus en canal, où le cylindre est tracté ainsi que des résultats numériques obtenus à l'aide du code YALES2, basé sur la méthode des volumes finis. Le cylindre vertical est partiellement immergé et le sillage généré derrière le cylindre est caractérisé par une déformation de la surface libre. L'écoulement derrière le cylindre est gouverné par les nombres de Reynolds et de Froude, caractérisant l'importance des effets inertiels et gravitaires. Ces nombres sans dimension sont définis à l'aide du diamètre du cylindre. La gamme de vitesse balayée et les diamètres utilisés permettent d'atteindre des nombres de Reynolds jusqu'à 240 000, et des nombres de Froude jusqu'à 2.4 correspondant à un sillage turbulent. L'attention est portée sur les fortes déformations de surface libre allant jusqu'à sa rupture et l'entraînement d'air. En particulier, deux modes d'entrainement d'air ont été observés : (i) dans la cavité le long du cylindre et (ii) dans le sillage du cylindre. La vitesse critique à l'entraînement d'air dans la cavité a été mesurée et son évolution est comparée avec une loi d'échelle proposée par Benusiglio. Nous avons observé l'influence de cet entraînement d'air sur les efforts de traînée grâce à des mesures à l'aide de capteurs piézoélectriques. Nos résultats sont comparés avec des résultats expérimentaux à des Reynolds et Froude inférieurs et des résultats sans surface libre. Une comparaison avec des résultats numériques est également réalisée. La présence de la surface libre et de l'entraînement d'air dans la cavité entraîne une diminution des efforts de traînée par rapport au cas monophasique. Numériquement, le code utilise une méthode level-set pour la description de la surface libre et permet de reproduire les phénomènes d'entraînement d'air, la déformation de la surface libre et la dynamique de l'écoulement autour du cylindre. Ce travail étend la gamme de paramètres adimensionnels parcourus expérimentalement et numériquement, et met en évidence l'effet de l'entrainement d'air sur les efforts de traînée. / The flow past ships or an emerged body such as bridge pilar, is a fundamental, familiar and fascinating sight.Measurements and modelling of this simple flow can have relevance to offshore structures and renewable energy systems. The interaction of such structures with marine environment lead to drag, lift forces and free-surface effects.Our current problem is simplified by considering cylindrical geometry. This work presents experimental results, in which vertical cylinders are translated at constant speed through water initially at rest, and numerical results using YALES2 computing code based on finite volume method. The cylinders are partially immersed, then the motion induces turbulent wake and free-surface deformation. The flow is governed by the Reynolds and Froude numbers defined with cylinder diameter. The explored range of parameters are in the regime of turbulent wake with experiments carried out for Reynolds number up to 240 000, and Froude number up to 2.4. The focus here is on drag force measurements and strong free-surface deformation up to rupture and air-entrainment. Two modes of air-entraiment have been observed: (i) in the wake of the cylinder and (ii) in a cavity along the cylinder wall. Results are as follows. First, a scaling for the critical velocity for air-entrainment in the cavity proportional to D1/5 proposed by Benusiglio is recovered. Secondly, drag coefficients measured by piezoelectric sensors are smaller in two phase flow compared to monophasic case, and air-entrainment in the cavity enhances this decrease. Numerically, YALES2 uses level-set method for the descirption of the free-surface, and is able to reproduce air-entrainment phenomenon, free-surface deformations and flow dynamics around the cylinder. The present work expands the range of dimensionless parameters and highlights free-surface effects on drag forces.

Thermal convection with evolving surface in a rotating icy satellite Master's Thesis author: Miroslav Kuchta∗ supervisor: Doc. RNDr. Ondřej Čadek, CSc.† Keywords: Stokes-Fourier system, Free surface, Finite-differences Abstract This thesis is concerned with modeling the surface deformations and thermal convection in a rotating icy satellite. The system of gov- erning equations, that we derive from general balance laws, is solved numerically using the finite-difference method on a staggered grid. Free surface is understood as implicitly described interface between the satellite and an almost massless medium with viscosity orders of magnitude smaller than ice. We design a numerical method capable of tracking the deforming surface. The numerical method is applied to models with temperature-dependent viscosity. ∗ Mathematical Institute of Charles University, Faculty of Mathematics and Physics, Charles University in Prague, Czech Republic. miroslav.kuchta@gmail.com † Department of Geophysics, Faculty of Mathematics and Physics, Charles University in Prague, Czech Republic. oc@karel.troja.mff.cuni.cz 1

Aeration issued through a ventilation shaft is an important measure to prevent cavitation and large gauge pressure in flood discharge tunnels. In order to dimension the ventilation shaft appropriately, itis necessary to have a good understanding of the air-water flow in the tunnel. In this study, the multiphase flow through a discharge tunnel was simulated in the computational fluid dynamics (CFD) software ANSYS Fluent. Since the flow was separated, the simulation setup used the volume of fluid (VOF) multiphase model, that could track the water surface. Furthermore, the so called RNG k-epsilon turbulence model was used. The CFD model was validated with measured data provided from two open channel experiments carried out on a scaled model at Sichuan University. To ensure mesh independence, grid convergence index (GCI)studies were performed for the two validating cases. After the validation, a top wall and a ventilation shaft was added to the CFD model. The flow was then simulated for four different shaft designs and four different water inlet velocities. The air demand and air supply for the various scenarios could thereby be calculated. The results of this study were also compared with previous research on multiphase flow through tunnels with similar design. It was concluded that the air flow downstream in the tunnel converged for the two larger designs. It was also concluded that the air demand in the tunnel was satisfied for the larger ventilation shafts. A smaller study on cavitation was made and the risk was considered non-existent for all the simulated cases.

Studying the interactions of solid particles and deformable gas<br />bubbles in viscous liquids is very important in many applications,<br />especially in mining and chemical industries. These interactions<br />involve liquid-solid-air multiphase flows and an<br />arbitrary-Lagrangian-Eulerican (ALE) approach is used for the direct<br />numerical simulations. In the system of rigid particles and<br />deformable gas bubbles suspended in viscous liquids, the<br />Navier-Stokes equations coupled with the equations of motion of the<br />particles and deformable bubbles are solved in a finite-element<br />framework. A moving, unstructured, triangular mesh tracks the<br />deformation of the bubble and free surface with adaptive refinement.<br />In this dissertation, we study four problems. In the first three<br />problems the flow is assumed to be axisymmetric and two dimensional<br />(2D) in the fourth problem.<br /><br />Firstly, we study the interaction between a rising deformable bubble<br />and a solid wall in highly viscous liquids. The mechanism of the<br />bubble deformation as it interacts with the wall is described in<br />terms of two nondimensional groups, namely the Morton number (Mo)<br />and Bond number (Bo). The film drainage process is also<br />considered. It is found that three modes of bubble-rigid wall<br />interaction exist as Bo changes at a moderate Mo.<br />The first mode prevails at small Bo where the bubble deformation<br />is small. For this mode, the bubble is<br /> hard to break up and will bounce back and eventually attach<br />to the rigid wall. In the second mode, the bubble may break up after<br />it collides with the rigid wall, which is determined by the film<br />drainage. In the third mode, which prevails at high Bo, the bubble<br />breaks up due to the bottom surface catches up the top surface<br />during the interaction.<br /><br />Secondly, we simulate the interaction between a rigid particle and a<br />free surface. In order to isolate the effects of viscous drag and<br />particle inertia, the gravitational force is neglected and the<br />particle gains its impact velocity by an external accelerating<br />force. The process of a rigid particle impacting a free surface and<br />then rebounding is simulated. Simplified theoretical models are<br />provided to illustrate the relationship between the particle<br />velocity and the time variation of film thickness between the<br />particle and free surface. Two film thicknesses are defined. The<br />first is the thickness achieved when the particle reaches its<br />highest position. The second is the thickness when the particle<br />falls to its lowest position. The smaller of these two thicknesses<br />is termed the minimum film thickness and its variation with the<br />impact velocity has been determined. We find that the interactions<br />between the free surface and rigid particle can be divided into<br />three regimes according to the trend of the first film thickness.<br />The three regimes are viscous regime, inertial regime and jetting<br />regime. In viscous regime, the first film thickness decreases as the<br />impact velocity increases. Then it rises slightly in the inertial<br />regime because the effect of liquid inertia becomes larger as the<br />impact velocity increases. Finally, the film thickness decreases<br />again due to Plateau-Rayleigh instability in the jetting regime.<br />We also find that the minimum film thickness corresponds to an<br />impact velocity on the demarcation point between the viscous and<br />inertial regimes. This fact is caused by the balance of viscous<br />drag, surface deformation and liquid inertia.<br /><br />Thirdly, we consider the interaction between a rigid particle and a<br />deformable bubble. Two typical cases are simulated: (1) Collision of<br />a rigid particle with a gas bubble in water in the absence of<br />gravity, and (2) Collision of a buoyancy-driven rising bubble with a<br />falling particle in highly viscous liquids. We also compare our<br />simulation results with available experimental data. Good agreement<br />is obtained for the force on the particle and the shape of the<br />bubble.<br /><br />Finally, we investigated the collisions of groups of bubbles and<br />particles in two dimensions. A preliminary example of the oblique<br />collision between a single particle and a single bubble is conducted<br />by giving the particle a constant acceleration. Then, to investigate<br />the possibility of particles attaching to bubbles, the interactions<br />between a group of 22 particles and rising bubbles are studied. Due<br />to the fluid motion, the particles involved in central collisions<br />with bubbles have higher possibilities to attach to the bubble. / Ph. D.

When a sphere or a stream of water hits the surface of a pool of water and enters a crater or air cavity often forms. This topic has been studied, both formally and informally, for a long time. This dissertation investigates four areas of water impact that are still poorly understood using high-speed photography. First, it examines a stream of droplets impacting on a pool of water, similar to a faucet drizzling into a full bucket. For these types of impacts we predict the depth, diameter, velocity, and shape of the cavities that the droplet stream forms. Second, it examines what occurs when a sphere impacts a pool of soapy water, such as a bubble bath or kitchen sink. The minimum velocity for a cavity to form decreases when soap is present. If the water has bubbles on the surface, the sphere will always form a cavity. Third, it examines how different coatings on a sphere (car wax, etc.) affect whether the sphere forms a cavity, and it shows how the coating affect the shape of that cavity. Fourth, when objects impact a water surface they experience a large force, which many people have noticed when participating in cliff jumping, high diving, and belly flop competitions. We show that the force of impact can be reduced by 75% simply by allowing a mass of water to impact in front of the object.

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