Dissertations / Theses on the topic 'Volume of Fluid (VOF) method'

A VOF based multiphase Lattice Boltzmann method that explicitly prescribes kinematic boundary conditions at the interface is developed. The advantage of the method is the direct control over the surface tension value. The details of the numerical method are presented. The Saffman instability, Taylor instability, and flow of deformable suspensions in a channel are used as example-problems to demonstrate the accuracy of the method. The method allows for relatively large viscosity and density ratios.

Die in dieser Arbeit vorgestellten Simulationen aufsteigender fluider Partikel wurden mit dem CFD-Programm FS3D durchgeführt, welches auf der Volume-of-Fluid (VoF) Methode basiert. Die Validierung des Codes erfolgt durch Vergleich der numerischen Lösungen für schleichende Strömungen mit analytischen Lösungen, wobei eine gute Übereinstimmung festgestellt wird. Im ersten Teil der Dissertation werden Simulationen für den freien Aufstieg von Öltropfen in Wasser mit experimentellen Beobachtungen hinsichtlich der Aufstiegsgeschwindigkeit, der Tropfenform und der Bewegungsbahn verglichen. Die Aufstiegsgeschwindigkeiten und Widerstandsbeiwerte sind vergleichbar, die simulierten Tropfen sind jedoch deutlich flacher. Dieser Unterschied kann durch Verunreinigungen der Grenzfläche im Experiment verursacht sein. Der Übergang von einem gradlinigen Aufstieg zu zickzack-förmigen Aufstiegsbahnen kann mit Hilfe der Simulationen auf Instabilitäten im Nachlauf der Blasen zurückgeführt werden, die zu einer periodischen Wirbelablösung führen. Im zweiten Teil der Dissertation wird der Aufstieg von Blasen in linearen Scherströmungen untersucht. Steigen die Blasen in einer vertikalen Scherströmung auf, so beobachtet man eine seitliche Migration. Diese seitliche Migration der Blasen wird durch die sogenannte Liftkraft verursacht, deren Vorzeichen und Betrag von der Blasengröße und den Stoffeigenschaften der Flüssigkeit abhängt. Die Simulationen zeigen, daß das Vorzeichen der Liftkraft für eher sphärische Blasen durch den Bernoulli-Effekt erklärt werden kann. An stark deformierten Blasen hingegen wirkt die Liftkraft in umgekehrter Richtung. Dieses Phänomen tritt auch in den Simulationen auf. Verschiedene Hypothesen für die Ursache dieses Phänomens werden überprüft. Die bekannteste experimentelle Korrelation für die Liftkraft von Tomiyama u.a. (2002) wird durch Simulation von realen Flüssigkeiten mit bekannten Stoffeigenschaften wie auch von Modellfluiden mit willkürlichen Stoffeigenschaften validiert und weitgehend bestätigt. Die Lift-Korrelation hat demnach hinsichtlich der Stoffeigenschaften der Flüssigkeit einen größeren Geltungsbereich, als bisher experimentell überprüft wurde. The simulations presented in this thesis were performed with the CFD code FS3D which is based on the Volume of Fluid method. The code is validated using analytical solutions for creeping flows and a good agreement is observed between simulation and analytical solution. In the first part of the thesis, the free rise of oil drops in water is simulated and compared with experimental observations. The results show that the rising velocities and the drag coefficients are similar in both cases, but the simulated drops are flatter (more oblate). This difference may be caused by impurities of the particle surface (surfactants) in the experiments. The simulations show that the transition from rectilinear to periodic trajectories is caused by instabilities in the wake, which lead to a periodic vortex shedding. In the second part of the thesis, the rise of bubbles in linear shear flows is investigated. If bubbles rise in a vertical shear flow, a lateral migration can be observed. This migration is caused by the so called lift force. Sign and magnitude of the lift force depend on the size of the bubble and the material properties of the liquid. The simulation results show that the sign of the lift force on spherical bubbles can be explained by the Bernoulli effect. However, the lift force on more distorted bubbles acts in the opposite direction. This phenomenon can also be observed in the simulation. In this work several hypotheses for the reason of this phenomenon are checked. Furthermore, most common correlation for the lift force (developed by Tomiyama et al. in 2002) is validated for fluids of known material and model fluids with arbitrary material data. The correlation is valid in a wider range of fluid material properties than proved experimentally up to now.

Die in dieser Arbeit vorgestellten Simulationen aufsteigender fluider Partikel wurden mit dem CFD-Programm FS3D durchgeführt, welches auf der Volume-of-Fluid (VoF) Methode basiert. Die Validierung des Codes erfolgt durch Vergleich der numerischen Lösungen für schleichende Strömungen mit analytischen Lösungen, wobei eine gute Übereinstimmung festgestellt wird. Im ersten Teil der Dissertation werden Simulationen für den freien Aufstieg von Öltropfen in Wasser mit experimentellen Beobachtungen hinsichtlich der Aufstiegsgeschwindigkeit, der Tropfenform und der Bewegungsbahn verglichen. Die Aufstiegsgeschwindigkeiten und Widerstandsbeiwerte sind vergleichbar, die simulierten Tropfen sind jedoch deutlich flacher. Dieser Unterschied kann durch Verunreinigungen der Grenzfläche im Experiment verursacht sein. Der Übergang von einem gradlinigen Aufstieg zu zickzack-förmigen Aufstiegsbahnen kann mit Hilfe der Simulationen auf Instabilitäten im Nachlauf der Blasen zurückgeführt werden, die zu einer periodischen Wirbelablösung führen. Im zweiten Teil der Dissertation wird der Aufstieg von Blasen in linearen Scherströmungen untersucht. Steigen die Blasen in einer vertikalen Scherströmung auf, so beobachtet man eine seitliche Migration. Diese seitliche Migration der Blasen wird durch die sogenannte Liftkraft verursacht, deren Vorzeichen und Betrag von der Blasengröße und den Stoffeigenschaften der Flüssigkeit abhängt. Die Simulationen zeigen, daß das Vorzeichen der Liftkraft für eher sphärische Blasen durch den Bernoulli-Effekt erklärt werden kann. An stark deformierten Blasen hingegen wirkt die Liftkraft in umgekehrter Richtung. Dieses Phänomen tritt auch in den Simulationen auf. Verschiedene Hypothesen für die Ursache dieses Phänomens werden überprüft. Die bekannteste experimentelle Korrelation für die Liftkraft von Tomiyama u.a. (2002) wird durch Simulation von realen Flüssigkeiten mit bekannten Stoffeigenschaften wie auch von Modellfluiden mit willkürlichen Stoffeigenschaften validiert und weitgehend bestätigt. Die Lift-Korrelation hat demnach hinsichtlich der Stoffeigenschaften der Flüssigkeit einen größeren Geltungsbereich, als bisher experimentell überprüft wurde. The simulations presented in this thesis were performed with the CFD code FS3D which is based on the Volume of Fluid method. The code is validated using analytical solutions for creeping flows and a good agreement is observed between simulation and analytical solution. In the first part of the thesis, the free rise of oil drops in water is simulated and compared with experimental observations. The results show that the rising velocities and the drag coefficients are similar in both cases, but the simulated drops are flatter (more oblate). This difference may be caused by impurities of the particle surface (surfactants) in the experiments. The simulations show that the transition from rectilinear to periodic trajectories is caused by instabilities in the wake, which lead to a periodic vortex shedding. In the second part of the thesis, the rise of bubbles in linear shear flows is investigated. If bubbles rise in a vertical shear flow, a lateral migration can be observed. This migration is caused by the so called lift force. Sign and magnitude of the lift force depend on the size of the bubble and the material properties of the liquid. The simulation results show that the sign of the lift force on spherical bubbles can be explained by the Bernoulli effect. However, the lift force on more distorted bubbles acts in the opposite direction. This phenomenon can also be observed in the simulation. In this work several hypotheses for the reason of this phenomenon are checked. Furthermore, most common correlation for the lift force (developed by Tomiyama et al. in 2002) is validated for fluids of known material and model fluids with arbitrary material data. The correlation is valid in a wider range of fluid material properties than proved experimentally up to now.

The study and modelling of two-phase flow, even the simplest ones such as the bubbly flow, remains a challenge that requires exploring the physical phenomena from different spatial and temporal resolution levels. CFD (Computational Fluid Dynamics) is a widespread and promising tool for modelling, but nowadays, there is no single approach or method to predict the dynamics of these systems at the different resolution levels providing enough precision of the results. The inherent difficulties of the events occurring in this flow, mainly those related with the interface between phases, makes that low or intermediate resolution level approaches as system codes (RELAP, TRACE, ...) or 3D TFM (Two-Fluid Model) have significant issues to reproduce acceptable results, unless well-known scenarios and global values are considered. Instead, methods based on high resolution level such as Interfacial Tracking Method (ITM) or Volume Of Fluid (VOF) require a high computational effort that makes unfeasible its use in complex systems. In this thesis, an open-source simulation framework has been designed and developed using the OpenFOAM library to analyze the cases from microescale to macroscale levels. The different approaches and the information that is required in each one of them have been studied for bubbly flow. In the first part, the dynamics of single bubbles at a high resolution level have been examined through VOF. This technique has allowed to obtain accurate results related to the bubble formation, terminal velocity, path, wake and instabilities produced by the wake. However, this approach has been impractical for real scenarios with more than dozens of bubbles. Alternatively, this thesis proposes a CFD Discrete Element Method (CFD-DEM) technique, where each bubble is represented discretely. A novel solver for bubbly flow has been developed in this thesis. This includes a large number of improvements necessary to reproduce the bubble-bubble and bubble-wall interactions, turbulence, velocity seen by the bubbles, momentum and mass exchange term over the cells or bubble expansion, among others. But also new implementations as an algorithm to seed the bubbles in the system have been incorporated. As a result, this new solver gives more accurate results as the provided up to date. Following the decrease on resolution level, and therefore the required computational resources, a 3D TFM have been developed with a population balance equation solved with an implementation of the Quadrature Method Of Moments (QMOM). The solver is implemented with the same closure models as the CFD-DEM to analyze the effects involved with the lost of information due to the averaging of the instantaneous Navier-Stokes equation. The analysis of the results with CFD-DEM reveals the discrepancies found by considering averaged values and homogeneous flow in the models of the classical TFM formulation. Finally, for the lowest resolution level approach, the system code RELAP5/MOD3 is used for modelling the bubbly flow regime. The code has been modified to reproduce properly the two-phase flow characteristics in vertical pipes, comparing the performance of the calculation of the drag term based on drift-velocity and drag coefficient approaches.<br>El estudio y modelado de flujos bifásicos, incluso los más simples como el bubbly flow, sigue siendo un reto que conlleva aproximarse a los fenómenos físicos que lo rigen desde diferentes niveles de resolución espacial y temporal. El uso de códigos CFD (Computational Fluid Dynamics) como herramienta de modelado está muy extendida y resulta prometedora, pero hoy por hoy, no existe una única aproximación o técnica de resolución que permita predecir la dinámica de estos sistemas en los diferentes niveles de resolución, y que ofrezca suficiente precisión en sus resultados. La dificultad intrínseca de los fenómenos que allí ocurren, sobre todo los ligados a la interfase entre ambas fases, hace que los códigos de bajo o medio nivel de resolución, como pueden ser los códigos de sistema (RELAP, TRACE, etc.) o los basados en aproximaciones 3D TFM (Two-Fluid Model) tengan serios problemas para ofrecer resultados aceptables, a no ser que se trate de escenarios muy conocidos y se busquen resultados globales. En cambio, códigos basados en alto nivel de resolución, como los que utilizan VOF (Volume Of Fluid), requirieren de un esfuerzo computacional tan elevado que no pueden ser aplicados a sistemas complejos. En esta tesis, mediante el uso de la librería OpenFOAM se ha creado un marco de simulación de código abierto para analizar los escenarios desde niveles de resolución de microescala a macroescala, analizando las diferentes aproximaciones, así como la información que es necesaria aportar en cada una de ellas, para el estudio del régimen de bubbly flow. En la primera parte se estudia la dinámica de burbujas individuales a un alto nivel de resolución mediante el uso del método VOF (Volume Of Fluid). Esta técnica ha permitido obtener resultados precisos como la formación de la burbuja, velocidad terminal, camino recorrido, estela producida por la burbuja e inestabilidades que produce en su camino. Pero esta aproximación resulta inviable para entornos reales con la participación de más de unas pocas decenas de burbujas. Como alternativa, se propone el uso de técnicas CFD-DEM (Discrete Element Methods) en la que se representa a las burbujas como partículas discretas. En esta tesis se ha desarrollado un nuevo solver para bubbly flow en el que se han añadido un gran número de nuevos modelos, como los necesarios para contemplar los choques entre burbujas o con las paredes, la turbulencia, la velocidad vista por las burbujas, la distribución del intercambio de momento y masas con el fluido en las diferentes celdas por cada una de las burbujas o la expansión de la fase gaseosa entre otros. Pero también se han tenido que incluir nuevos algoritmos como el necesario para inyectar de forma adecuada la fase gaseosa en el sistema. Este nuevo solver ofrece resultados con un nivel de resolución superior a los desarrollados hasta la fecha. Siguiendo con la reducción del nivel de resolución, y por tanto los recursos computacionales necesarios, se efectúa el desarrollo de un solver tridimensional de TFM en el que se ha implementado el método QMOM (Quadrature Method Of Moments) para resolver la ecuación de balance poblacional. El solver se desarrolla con los mismos modelos de cierre que el CFD-DEM para analizar los efectos relacionados con la pérdida de información debido al promediado de las ecuaciones instantáneas de Navier-Stokes. El análisis de resultados de CFD-DEM permite determinar las discrepancias encontradas por considerar los valores promediados y el flujo homogéneo de los modelos clásicos de TFM. Por último, como aproximación de nivel de resolución más bajo, se investiga el uso uso de códigos de sistema, utilizando el código RELAP5/MOD3 para analizar el modelado del flujo en condiciones de bubbly flow. El código es modificado para reproducir correctamente el flujo bifásico en tuberías verticales, comparando el comportamiento de aproximaciones para el cálculo del término d<br>L'estudi i modelatge de fluxos bifàsics, fins i tot els més simples com bubbly flow, segueix sent un repte que comporta aproximar-se als fenòmens físics que ho regeixen des de diferents nivells de resolució espacial i temporal. L'ús de codis CFD (Computational Fluid Dynamics) com a eina de modelatge està molt estesa i resulta prometedora, però ara per ara, no existeix una única aproximació o tècnica de resolució que permeta predir la dinàmica d'aquests sistemes en els diferents nivells de resolució, i que oferisca suficient precisió en els seus resultats. Les dificultat intrínseques dels fenòmens que allí ocorren, sobre tots els lligats a la interfase entre les dues fases, fa que els codis de baix o mig nivell de resolució, com poden ser els codis de sistema (RELAP,TRACE, etc.) o els basats en aproximacions 3D TFM (Two-Fluid Model) tinguen seriosos problemes per a oferir resultats acceptables , llevat que es tracte d'escenaris molt coneguts i se persegueixen resultats globals. En canvi, codis basats en alt nivell de resolució, com els que utilitzen VOF (Volume Of Fluid), requereixen d'un esforç computacional tan elevat que no poden ser aplicats a sistemes complexos. En aquesta tesi, mitjançant l'ús de la llibreria OpenFOAM s'ha creat un marc de simulació de codi obert per a analitzar els escenaris des de nivells de resolució de microescala a macroescala, analitzant les diferents aproximacions, així com la informació que és necessària aportar en cadascuna d'elles, per a l'estudi del règim de bubbly flow. En la primera part s'estudia la dinàmica de bambolles individuals a un alt nivell de resolució mitjançant l'ús del mètode VOF. Aquesta tècnica ha permès obtenir resultats precisos com la formació de la bambolla, velocitat terminal, camí recorregut, estela produida per la bambolla i inestabilitats que produeix en el seu camí. Però aquesta aproximació resulta inviable per a entorns reals amb la participació de més d'unes poques desenes de bambolles. Com a alternativa en aqueix cas es proposa l'ús de tècniques CFD-DEM (Discrete Element Methods) en la qual es representa a les bambolles com a partícules discretes. En aquesta tesi s'ha desenvolupat un nou solver per a bubbly flow en el qual s'han afegit un gran nombre de nous models, com els necessaris per a contemplar els xocs entre bambolles o amb les parets, la turbulència, la velocitat vista per les bambolles, la distribució de l'intercanvi de moment i masses amb el fluid en les diferents cel·les per cadascuna de les bambolles o els models d'expansió de la fase gasosa entre uns altres. Però també s'ha hagut d'incloure nous algoritmes com el necessari per a injectar de forma adequada la fase gasosa en el sistema. Aquest nou solver ofereix resultats amb un nivell de resolució superior als desenvolupat fins la data. Seguint amb la reducció del nivell de resolució, i per tant els recursos computacionals necessaris, s'efectua el desenvolupament d'un solver tridimensional de TFM en el qual s'ha implementat el mètode QMOM (Quadrature Method Of Moments) per a resoldre l'equació de balanç poblacional. El solver es desenvolupa amb els mateixos models de tancament que el CFD-DEM per a analitzar els efectes relacionats amb la pèrdua d'informació a causa del promitjat de les equacions instantànies de Navier-Stokes. L'anàlisi de resultats de CFD-DEM permet determinar les discrepàncies ocasionades per considerar els valors promitjats i el flux homogeni dels models clàssics de TFM. Finalment, com a aproximació de nivell de resolució més baix, s'analitza l'ús de codis de sistema, utilitzant el codi RELAP5/MOD3 per a analitzar el modelatge del fluxos en règim de bubbly flow. El codi és modificat per a reproduir correctament les característiques del flux bifàsic en canonades verticals, comparant el comportament d'aproximacions per al càlcul del terme de drag basades en velocitat de drift flux model i de les basades en coe<br>Peña Monferrer, C. (2017). Computational fluid dynamics multiscale modelling of bubbly flow. A critical study and new developments on volume of fluid, discrete element and two-fluid methods [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/90493<br>TESIS

L’émergence de bruits auparavant inaudibles dans les réservoirs à carburants automobiles requiert des constructeurs une meilleure compréhension des phénomènes physiques intervenants au sein de leurs produits. Dans cette thèse, différents travaux ont été conduits autour de l’étude du ballottement de fluide dans une cuve rigide rectangulaire partiellement remplie de fluide et soumise à une excitation extérieure. La première partie présente un état de l’art sur le sloshing suivant trois approches complémentaires - approche analytique, approche numérique et approche expérimentale - permettant d’orienter les travaux. Dans une deuxième partie, une étude préliminaire sur le sloshing dans une cuve rectangulaire soumise à une excitation harmonique forcée est réalisée. La confrontation des résultats numériques entre une approche linéaire - basée sur la théorie d’écoulement potentiel tenant compte de la viscosité du fluide [Schotté et Ohayon, 2013] - et une approche non linéaire commerciale – basée sur la résolution des équations de Navier-Stokes - permet de définir un paramètre de linéarité. Ce dernier permet de déterminer les cas de sloshing qui nécessitent une résolution non linéaire et ceux pour lesquels la théorie linéaire suffit pour prédire le phénomène. La troisième partie de ce document présente une étude expérimentale du ballottement de fluide dans une cuve rectangulaire rigide soumise à un freinage automobile. Deux niveaux de remplissage créant deux types d’impacts contre les parois (avec et sans enfermement de poche d’air) ont été analysés. Les essais menés ont permis de mesurer les forces engendrées par le mouvement du fluide, les pressions d’impact en paroi ainsi que le champ de vitesse par méthode Particle Image Velocimetry (PIV). Ce chapitre constitue une importante base de données expérimentales ayant permis d’étudier précisément le phénomène physique. L’étude est complétée par une confrontation des résultats expérimentaux avec des résultats Computational Fluid Dynamics (CFD). Enfin, pour conclure ce mémoire, une étude du sloshing dans un réservoir en tenant compte de la Fluid-Structure Interaction (FSI) est présentée. Le choix du couplage a été porté sur un schéma partitionné itératif faible avec, dans un premier temps, une approche potentielle instationnaire, puis avec une approche Volume Of Fluid (VOF) pour la physique fluide. Les limites d’un tel couplage dans le cas d’étude d’un réservoir partiellement rempli de fluide et attaché de manière flexible en fonction du rapport de masse fluide-réservoir ont été mises en évidence. La correction du schéma de couplage par l’effet de masse ajoutée présentée dans [Song et al., 2013] permet la résolution d’un système couplé quel que soit le rapport de masse en jeu et améliore de manière significative la convergence en réduisant également fortement le temps de calcul<br>The present thesis focuses on an investigation of the sloshing phenomenon in a partially filled fuel tank submitted to a harmonic excitation motion. In the first part, the confrontation of numerical results between a linear approach - taking into account viscosity - and a nonlinear approach based on a commercial code leads to define a parameter of linearity. This parameter allows determining cases of sloshing who require non-linear resolution and those who need a linear theory to predict the phenomenon. An experimental study of fluid sloshing in a rectangular tank submitted to an automotive braking is conducted. Tests leaded allow measuring global forces engendered by the motion of the fluid, pressure of fluid impact and velocity field by PIV. This chapter provides an important data base and helps to investigate on the physical phenomenon. This study is completed by CFD results. To conclude, a numerical model for fluid-structure interactions is presented. Limits of this segregated partitioned coupling in case of sloshing in tank flexibly attached are highlighted, depending mostly on the mass ratio between fluid and tank structure. An added-mass term is integrated to the corrected staggered scheme ensuring systematically the convergence of the coupled solution and reducing significantly the iterations required

Numerical simulations utilizing the code SURFER++ with the incorporation of an insoluble surfactant in the VOF scheme were conducted to characterize the effects of surfactant on a drop in shear flow. The drop is suspended in a matrix liquid. A parameter called reduction, which specifically relates to a percentage decrease in effective surface tension, is used to measure the surfactant amount on the interface. In a model system where reduction = 0.1, viscosity ratio = 1 and density ratio = 1, it was found that stable drops tend to be more elongated and less inclined to the primary flow direction than drops unexposed to surfactant. This can be explained by the location of surfactant at the interface as the drop evolves. Breaking drops also show a flattened angle, but exhibit shorter necks and faster time to break than similar drops without surfactant. As reduction increases, various physical characteristics of the drops change across Reynolds number.<br>Ph. D.

This work presents an all-Mach method for two-phase inviscid flow in the presence of surface tension. A modified version of the Hartens, Lax, Leer and Contact (HLLC) approximate Riemann solver based on Garrick et al. [1] is developed and combined with the popular Volume of Fluid (VoF) method: Compressive Interface Capturing Scheme for Arbitrary Meshes (CICSAM). This novel combination yields a scheme with both HLLC shock capturing as well as accurate liquid-gas interface tracking characteristics. To ensure compatibility with VoF, the Monotone Upstream-centred Scheme for Conservation Laws (MUSCL) [2] is applied to non-conservative (primitive) variables, which yields both robustness and accuracy. Liquid-gas interface curvature is computed via both height functions [3, 4] and the convolution method [5]. This is in the interest of applicability to both cartesian and arbitrary meshes. The author emphasizes the use of VoF in the interest of surface tension modelling accuracy. The method is validated using a range of test-cases available in literature. The results show flow features that are in agreement with experimental and benchmark data. In particular, the use of the HLLC-VoF combination leads to a sharp volume fraction and energy field with improved accuracy (up to secondorder).

Numerical simulations of the complex flows of viscoelastic fluids are investigated. The viscoelastic fluids are modelled, primarily, via the Johnson-Segalman constitutive model. Our Numerical approach is based on finite volume method, based on the Johnson-Segalman constitutive model and implemented on the OpenFOAM® platform. The Johnson-Segalman model also easily reduces to the Oldroyd-B model under certain conditions of the material parameters. Since computations using the Oldroyd-B model have been extensively documented in the literature, we take advantage of the mathematical modelling connection between the Johnson-Segalman and Oldroyd-B models to validate the accuracy of our Johnson-Segalman solver via reduction to the Oldroyd-B model. Numerical validation of our results is conducted via the most commonly used benchmark problems. The final aim of our work is to assess the viability and efficiency of our numerical solver via an investigation into the complex fluid dynamical processes associated with shear banding.

Abstract In ladle metallurgy, gas stirring and the behaviour of the slag layer are very important for alloying and the homogenization of the steel. When gas is injected through a nozzle located at the bottom of the ladle into the metal bath, the gas jet exiting the nozzle breaks up into gas bubbles. The rising bubbles break the slag layer and create an open-eye. The size of the open-eye is very important as the efficiency of the metal-slag reactions depend on the interaction between the slag and steel created during the stirring process, and information about the position and size of the open-eye is important for effective alloying practice. Moreover, the open-eye has an effect on the energy balance since it increases heat losses. In this study, experimental measurements and numerical simulations were performed to study the effect of different operating parameters on the formation of the open-eye and mixing time in a water model and industrial ladle. Experimental measurements were performed to study the effect of the gas flow rate, slag layer thickness, slag layer densities and number of porous plugs in a 1/5 scale water model and in a 150-ton steelmaking ladle. For numerical modelling, a multi-phase volume of fluid (VOF) model was used to simulate the system including the behaviour of the slag layer. The numerical simulation of the open-eye size and mixing time was found to be in good agreement with the experimental data obtained from the water model and data obtained from the industrial measurements<br>Tiivistelmä Senkkametallurgiassa kaasuhuuhtelu ja kuonakerroksen käyttäytyminen ovat tärkeitä teräksen seostamisen ja homogenisoinnin näkökulmasta. Senkan pohjalla sijaitsevasta suuttimesta puhallettava kaasu hajoaa kupliksi, jotka rikkovat kuonakerroksen ja muodostavat avoimen silmäkkeen. Avoimen silmäkkeen koko on yhteydessä voimakkaampaan kuonan emulgoitumiseen, joka tehostaa metallisulan ja kuonan välisiä reaktioita. Tietoa avoimen silmäkkeen paikasta ja koosta tarvitaan myös tehokkaaseen seostuspraktiikkaan. Avoin silmäke vaikuttaa lisäksi prosessin energiataseeseen lisäten sen lämpöhäviöitä. Tässä tutkimuksessa tutkittiin kokeellisesti ja laskennallisesti erilaisten operointiparametrien vaikutusta avoimen silmäkkeen muodostumiseen vesimallissa ja terässenkassateollisessa senkassa. Kokeellisia mittauksia tehtiin kaasuhuuhtelun, kuonakerroksen paksuuden, ja suuttimien määrän vaikutuksen tutkimiseksi 1/5-mittakaavan vesimallissa ja 150 tonnin terässenkassa. Numeerisessa mallinnuksessa systeemin ja siihen lukeutuvan kuonakerroksen käyttäytymisen simuloimiseen käytettiin volume of fluid (VOF) –monifaasimenetelmää. Avoimen silmäkkeen kokoon ja sekoittumisaikaan liittyvien numeeristen simulointien havaittiin vastaavan hyvin vesimallista ja teollisista mittauksista saatua kokeellista aineistoa

The main goal of this work is to propose, implement, test, and refine numerical methodologies for computer simulations of two-fluid flows. These methodologies fall into the category of volume tracking methods, with piecewise-linear interface calculation (PLIC). The scope of this work is limited to the laminar, incompressible flow of immiscible, non-reacting Newtonian fluids, without phase change, in planar two-dimensional geometries.<br>The following new or enhanced procedures are proposed: a parallelogram scheme for multidimensional advection of the volume fraction field, that rigorously conserves mass; a circle fit technique for the orientation of the interface segments and the calculation of curvature; a novel contact angle treatment; and a staggered formulation for volumetric body forces that can accurately balance pressure forces in the vicinity of the interface. In addition, surface-tension-derived and hydrostatic-derived pressure corrections are introduced as a novel means of calculating accurate pressure forces in cells that contain the interface, thereby virtually eliminating parasitic currents, or the non-physical flows that afflict many available volume tracking methods.<br>A total of six test problems are presented. The first three test problems do not involve surface tension, and are used to demonstrate the ability of the proposed method to accurately simulate two-fluid flows with complex interface deformations. These test problems involve pure advection flows, a collapsing water column, and a Richtmyer-Meshkov instability. The last three test cases are employed to check the effectiveness of the surface tension modelling. Simulations of a static drop indicate that the proposed curvature calculation procedure is of reasonable, but not very high, accuracy, and it is quite successful at maintaining a smooth, high fidelity interface. Next, it is shown that the proposed method can accurately simulate an oscillating bubble. In the final test case, the formation of a meniscus between two parallel plates is simulated. The equilibrium meniscus shape is in good agreement with the analytic meniscus solution. Overall, the proposed method is shown to be capable of accurate and stable simulations of the two-fluid flows considered in this work.

This Thesis describes the numerical simulation of fluid-structure interaction (FSI) problems. A finite-volume based stress analysis code was developed and coupled to an existing in-house CFD code to form a general purpose FSI solver capable of being used with the advanced turbulence and near-wall models developed within the research group. The code has been used to study a number of physiological flows in the present work, although the general nature of the solver allows it to be used for other applications also. By using the same numerical method, implemented in a consistent manner, for both fluid and solid domains, the inefficiencies associated with using separate packages for the fluid and solid were avoided. Separate packages typically store information in different data structures; some form of software interface is required to transfer information between the two packages. This additional software layer, which is called during each FSI iteration, causes a considerable overhead. By using a single numerical mesh across both domains, the inaccuracies associated with boundary interpolation were also avoided. Typically, separate packages use meshes which do not conform at their common boundary. In order to find nodal values of the fluid pressure, say, at the solid nodes, some form of interpolation is necessary. The interpolation leads to the introduction of truncation errors. These improvements allow for more accurate and efficient FSI simulations, particularly transient cases, to be performed. The solid solver was verified against analytical solutions for a number of test cases, including: planar stress distribution in a square plate with a circular hole in the centre; axisymmetric stress in a thick walled cylinder under internal pressure, and unsteady displacement of a cantilevered beam under free vibration. Before coupled analyses were performed, the flow solver was also validated through a number of rigid walled test cases, including steady flow through a stenosed tube and unsteady flow through an aneurysm. Many physiological flows are difficult to capture due to flow separation and early transition to turbulence. The use of a low-Reynolds number k-ε turbulence model was successful at capturing the flow field over a range of physiologically relevant flow rates. Once the solid body and flow solvers had been validated in isolation, they were coupled together and applied to a number of physiological flows, namely: steady flow through an initially straight tube with a compliant wall; steady flow through a compliant stenosis, and unsteady flow through a compliant aneurysm. The results from all three test cases showed good agreement with the available experimental and numerical data in terms of wall deformation. The solid body solver also proved itself to be capable of producing high quality numerical meshes for use in other simulations. The fluid mesh was considered to be a solid body with arbitrary material properties; the required deformation was specified as prescribed displacement boundary conditions. The main benefit of this method, compared to simple elliptical grid generation methods, is that near-wall grid spacing was preserved throughout the coupled simulation.

Carbonated sparkling water has been widely used from ancient age [1]. The original ideacame from natural sparkling water and people believed that taking baths at carbonatedhot springs was good for health and healed their sicknesses. This fact led people to startthinking that sparkling water could have more effective uses. Joseph Priestley success-fully produced artificial carbonated water in 1767 and sparkling water quickly becamewidely spread because it gives people refreshing feeling. The bottled and canned beverageindustry has grown from the 19th century and has become one of the biggest markets inthe world. According to Bloomberg Intelligence and Euromonitor, the global market ofthe carbonated beverages is around 350 billion dollar. One main drawback was that itwas not possible to re-cork the bottle to save the carbonation so that once it was opened,fizz was kept only for a short time. In 1813, the method to dispense a portion of carbon-ated water was invented by Charles Plinth[2]. This was the origin of the Soda Syphon.As the demand of sparkling water increased, the machine with which people could makesparkling water by themselves was introduced. Recently, it has become a very popularhome appliance, especially in Europe and North America. The most common way tocarbonate water is by injecting high-pressure CO2 into a water bottle. However, currentsystems waste a lot of CO2 during this carbonating process. In this thesis, the flow insidethe bottle during the injection of CO2 into water was studied in order to determine the pa-rameters that had most influence on the carbonation process. CFD (Computational FluidDynamics) simulations were performed in STAR-CCM+ of an axisymmetric 2D modeland a 3D model that was a 30 degree wedge of the real bottle shape. The Volume of Fluidmethod was used to solve the multiphase flow of gas and liquid. The RANS approachwas used with k 􀀀ϵ model and implicit time marching. To validate the simulations, axialpropagation of the volume fraction of CO2 was compared with the experimental visual-ization of the CO2 and H2O distribution. At the beginning of the phenomena, the gaspropagation was reasonably predicted and the results capture the features of the bubbleshape. However the results did not perfectly match with the experimental visualization.To seek the reason for the unrealistic results, the grid sensitivity study was performedand to consider the 3D effect the results with the 2D and the 3D model were compared.In addition, the bubble breakup process was deeply investigated.

The development of a fuel filler pipe is based solely on experience and physical experiment. The challenge lies in designing the pipe to fulfill the customer needs. In other words designing the pipe such as the fuel flow does not splash back on the fuel dispenser causing a premature shut off. To improve this “trial-and-error” based development a computational fluid dynamics (CFD) model of the refueling process is investigated. In this thesis a CFD model has been developed that can predict the fuel flow in the filler pipe. Worst case scenario of the refueling process is during the first second when the tank is partially filled. The most critical fluid is diesel due to the commercially high volume flow of 55 l/min. Due to limitations of computational resources the simulations are focused on the first second of the refueling process. The challenge in this project is creating a CFD model that is time efficient, thus require the least amount of computational resources necessary to provide useful information. A multiphase model is required to simulate the refueling process. In this project the implicit volume of fluid (VOF) has been used which has previously proven to be a suitable choice for refueling simulations. The project is divided into two parts. Part one starts with experiments and simulations of a simplified fuel system with water as acting liquid with a Reynolds number of 90 000. A short comparison between three different turbulence models has been investigated (LES, DES and URANS) where the most promising turbulence model is URANS, specifically the SST k-ω model. A sensitivity analysis was performed on the chosen turbulence model. Between the chosen mesh and the densest mesh the difference of streamwise velocity in the boundary layer was 2.6 %. The chosen mesh with 1.9 M cells and a time step of 1e-4 s was found to be the best correlating model with respect to the experiments. In part two a real fuel filling system was investigated both with experiments and simulations with the same computational model as the chosen one from part one. The change of fluid and geometry resulted in a lower Reynolds number of 12 000. Two different versions of the fuel system was investigated; with a bypass pipe and without a bypass pipe. Because of a larger volumetric region the resulting mesh had 3.7 M cells. The finished model takes about 230 h on a local workstation with 11 cores. On a cluster with 200 cores the same simulation takes 30 h. The resulting model suffered from interpolation errors at the inlet which resulted in a volume flow of 50 l/min as opposed to 55 l/min in the experiments. Despite the difference the model could capture the key flow characteristics. With the developed model a new filler pipe can be easily implemented and provide results in shorter time than a prototype filler pipe can be ordered. This will increase the chances of ordering one single prototype that fulfills all requirements. While the simulation model cannot completely replace verification by experiments it provides information that transforms the development of the filler pipe to knowledge based development.

An equal-order co-located Control Volume Finite Element Method (CVFEM) for the prediction of multidimensional, incompressible, viscous fluid flow problems has been formulated. CVFEMs provide the geometric flexibility traditionally associated with finite element methods. In addition, their control volume based formulation facilitates physically meaningful interpretation of the resulting discretization equations.<br>In the proposed CVFEM, the calculation domain is divided into three-node triangular and four-node tetrahedral finite elements in two- and three-dimensions, respectively. Each element is further subdivided in such a way that upon assembly of all elements, complete control volumes are formed about each node in the calculation domain. Interpolation functions for the dependent variables are prescribed in a manner that is consistent with the physical process they are intended to approximate. In this context, three different interpolation schemes of the convective flux across control volume surfaces are investigated, one of which guarantees positive contributions to the coefficients in the algebraic discretization equations. Appropriate conservation laws are imposed on the control volumes associated with the nodes. The resulting sets of integral conservation equations are then approximated by algebraic discretization equations, using the previously-mentioned interpolation functions. These nonlinear, coupled, algebraic equations are solved by a sequential solution procedure which incorporates Picard iterations.<br>The proposed method has been implemented into computer programs, and used to solve several test problems. These include convection-diffusion problems, and laminar and turbulent flow problems, in both two- and three-dimensions. The results demonstrate the ability of the proposed CVFEM to accurately solve the mathematical model used in this thesis.<br>Lastly, the CVFEM was used to predict flows similar to those found in film cooled gas turbine aerofoils. A complementary experimental program was designed and set up to investigate such flows. The numerical predictions were compared to the experimental observations of mean velocities, normal turbulent stresses and one component of the turbulent shear stress. These comparisons indicated that the high Reynolds number $k- epsilon$ turbulence model used in this thesis is unable to capture certain features of the flow. It appears as if a low-Reynolds number turbulence model, with appropriate modifications to account for streamline curvature and non-isotropy of the turbulence in the vicinity of the walls, would be better suited to the prediction of such flows. This new model can be easily incorporated into the CVFEM proposed in this thesis.

La compréhension des écoulements multiphasiques en milieu poreux revêt une importance capitale dans de nombreuses applications industrielles et environnementales, à des échelles spatiales et temporelles variées. Par conséquent, la présente étude propose une modélisation des écoulements multiphasiques en milieu poreux par le biais de la méthode Volume de Fluide, et présente des simulations de digitations de Saffman-Taylor, motivées par l'analyse d'expériences de balayage dans des blocs de grès de Bentheimer quasi bidimensionnels initialement saturés en huile extra-lourde par de l'eau. Le code Gerris, permettant des calculs parallèles efficaces à l'aide d'un maillage de type octree, est utilisé. Des tests de précision et de rapidité de calcul sont réalisés à l'aide de divers niveaux de raffinement, ainsi qu'une comparaison avec des simulations de référence dans la littérature. Des simulations 3D dans des milieux réels numérisés sont réalisés avec des résultats encourageants. Même s'il n'est pas encore possible d'atteindre des nombres capillaires réalistes, des écoulements dans des domaines cubiques de 1 mm de côté sont simulés, avec un temps de calcul raisonnable. Des simulations 2D de digitations visqueuses avec injection centrale ou latérale sont également présentées, basées sur la loi de Darcy. L'aspect fractal des digitations est étudié aussi bien à l'aide de la dimension fractale que de la variation de l'aire des motifs obtenus par rapport à leur périmètre. Enfin, des balayages à l'aide de polymères suivant des balayages à l'eau dans un processus en deux temps sont simulés à partir d'une modélisation darcéenne<br>Understanding multiphase flow in porous media is of tremendous importance for many industrial and environmental applications at various spatial and temporal scales. The present study consequently focuses on modelling multiphase flows by the Volume-of-Fluid method in porous media and shows simulations of Saffman-Taylor fingering motivated by the analysis of waterflooding experiments of extra-heavy oils in quasi-2D square slab geometries of Bentheimer sandstone. The Gerris code which allows efficient parallel computations with octree mesh refinement is used. It is tested for accuracy and computational speed using several levels of refinement and comparing to reference simulations in the literature. Simulations of real rocks are realised in three dimensions with very promising results. Though it is not yet possible to attain realistic capillary numbers, it is possible to simulate flows in domains of physical size up to 1 mm3 in reasonable CPU time. 2D simulations of viscous fingering with both central and lateral injection are also presented in this study, based on Darcy's law. The fractal aspect of this fingering is studied by considering both its fractal dimension and the variation of the area of the resulting pattern with respect to its arclength. Finally, polymer flooding following waterflooding in a two-step process is simulated with Darcy modelling

This study focuses on developing an efficient, fully three dimensional Finite Volume Method for viscoelastic fluid flow problems, and on numerical investigations of some complex rheological phenomena.

Thesis (Ph. D.)--Mechanical Engineering, Georgia Institute of Technology, 2004.<br>Kok-Meng Lee, Committee Chair ; Andrei G. Fedorov, Committee Member ; William E. Singhose, Committee Member ; David G. Taylor, Committee Member ; Zhi Zhou, Committee Member. Includes bibliographical references.

This thesis presents the development of a parallel finite volume numerical method to analyse thermoelastic and hyperelastic materials and applied problems with mutual interaction between a fluid and a structure. The solid problem follows a cell-centred finite volume formulation for three-dimensional unstructured grids under the same framework that is frequently devoted to computational fluid dynamics. Second-order accurate schemes are used to discretise both in time and space. A direct implicit time integration promotes numerical stability when facing vibration and quasi-static scenarios. The geometrical non-linearities, encountered with the large displacements of both Saint Venant-Kirchhoff and neo-Hookean models, are tackled by means of an updated Lagrangian approach. Verification of the method is conducted with canonical cases which involve: static equilibrium, thermal stress, vibration, structural damping, large deformations, nearly incompressible materials and high memory usage. Significant savings in computation time are achieved owing to the acceleration strategies implemented within the system resolution, namely a segregated algorithm with Aitken relaxation and a block-coupled system arrangement. The similarities between the block-coupled method and the displacement-based finite element method, with regards to the matrix form of the resulting equations, allow for including Rayleigh viscous damping within a finite volume solver. The program for structures is to be coupled with the in-house fluid numerical models in order to produce a unified fluid-structure interaction platform, where an arbitrary Lagrangian-Eulerian approach is used to solve the flow in a conforming grid. As a first step, the method for incompressible Newtonian fluids is adapted to deal with structure-coupled problems. To do so, the Lagrangian-Eulerian version of the Navier-Stokes equations is presented, and automatic moving mesh techniques are developed. These techniques are designed to mitigate the mesh quality deterioration and to satisfy the space conservation law. Besides, a semi-implicit coupling algorithm, which only implicitly couples the fluid pressure term to the structure, is implemented. As a result, numerical stability for strongly coupled phenomena at a reduced computational cost is obtained. These new tools are tested on an applied case, consisting of the turbulent flow through self-actuated flexible valves. Finally, a pioneering coupled numerical model for the thermal and structural analysis of packed-bed thermocline storage tanks is developed. This thermal accumulation system for concentrated solar power plants has attracted the attention of the industry due to the economic advantage compared to the usual two-tank system. Dynamic coupling among the thermoelastic equations for the tank shell and the numerical models for all other relevant elements of the system is considered. After validating the model with experimental results, the commercial viability of the thermocline concept, regarding energetic effectiveness and structural reliability, is evaluated under real operating conditions of the power plants.<br>Esta tesis presenta el desarrollo de un método numérico paralelo basado en volúmenes finitos para analizar materiales termoelásticos e hiperelásticos y problemas con una interacción mutua entre un fluido y una estructura. El problema del sólido sigue una formulación de volúmenes finitos centrada en las celdas para mallas no-estructuradas tridimensionales, bajo el mismo marco que se suele emplear en la dinámica de fluidos computacional. Se utilizan esquemas de segundo orden de precisión para discretizar el tiempo y el espacio. Una integración temporal directa implícita asegura estabilidad numérica al afrontar escenarios casi-estáticos o de vibración. Las no linealidades, que aparecen con los amplios desplazamientos de los modelos de Saint Venant-Kirchhoff y de neo-Hookean, son abordadas con un enfoque Lagrangiano actualizado. La verificación del método se realiza a través de casos canónicos que involucran: equilibrio estático, tensiones térmicas, vibración, amortiguación estructural, grandes deformaciones, materiales casi incompresibles y altos requerimientos de memoria. Se registra un ahorro significativo en el tiempo de cálculo gracias a las estrategias de aceleración implementadas dentro de la resolución del sistema, principalmente un algoritmo segregado con relajación Aitken y una disposición acoplada en bloques del sistema. Las similitudes entre este método acoplado en bloques y el método de los elementos finitos basados en el desplazamiento, con respecto a la forma matricial de las ecuaciones resultantes, permiten incluir la amortiguación viscosa tipo Rayleigh dentro de un solucionador de volúmenes finitos. El programa para estructuras se acoplará con los modelos numéricos internos para fluidos con el objetivo de generar una plataforma unificada de interacción fluido-estructura, donde se usa un enfoque arbitrario Lagrangiano-Euleriano sobre una malla conforme para resolver el fluido. Como primer paso, el método para flujos incompresibles Newtonianos se adapta para lidiar con problemas acoplados a una estructura. Para ello, se presenta la versión Lagrangiana-Euleriana de las ecuaciones de Navier-Stokes y se desarrollan técnicas automáticas de movimiento de malla. El diseño de estas técnicas se centra en mitigar el deterioro de la calidad de la malla y satisfacer la ley de conservación del espacio. Además, se implementa un algoritmo de acoplamiento semi-implícito, que sólo acopla implícitamente el término fluido de presión a la estructura. Como resultado, se obtiene estabilidad numérica para fenómenos fuertemente acoplados a un coste computacional reducido. Estas nuevas herramientas se prueban en un caso aplicado, que consiste el flujo turbulento a través de válvulas flexibles autoactivadas. Finalmente, se desarrolla un modelo numérico acoplado pionero para analizar estructuralmente y térmicamente los tanques termoclina de almacenamiento térmico. Este sistema de acumulación para centrales termosolares ha atraído la atención de la industria debido al ahorro económico comparado con el sistema de doble tanque habitual. Se tiene en cuenta el acoplamiento dinámico entre las ecuaciones gobernantes de la pared del tanque y las de todos los elementos relevantes del sistema. Tras validar el modelo con datos experimentales, se evalúa la viabilidad comercial de estos tanques, en cuanto a rendimiento energético y fiabilidad estructural, bajo condiciones reales de operación de las centrales.

We present a numerical scheme to model two-phase flow in porous media where capillary forces dominate over viscous effects. The volume-of-fluid method is employed to capture the fluid-fluid interface whose dynamics are described based on a finite volume discretization of the Navier--Stokes equations. Interfacial forces are calculated directly on reconstructed interface elements such that the total curvature is preserved. The computed interfacial forces are explicitly added to the Navier--Stokes equations using a sharp formulation which effectively eliminates spurious currents. The numerical model is validated in terms of physics, robustness, and mesh convergence, using an extensive hierarchy of static and dynamic test cases including wetting effects at the solid interface in two and three space dimensions. Next we provide an extensive study of viscous coupling effects in porous media flows, where the flow of one phase in the centre of a pore affects the flow of phases in layers or corners and vice versa. We perform two-phase flow simulations for different fluid configurations in non-circular capillary tubes to investigate viscous coupling effects as a function of viscosity ratio, contact angle, wetting phase saturation and wettability. We demonstrate the accuracy of our code in determining fluid velocities and capillary pressures, even for slow flows, where previous approaches fail. We specifically show the dependence of velocity profile and consequently flow conductivities on viscosity ratio and interface boundary condition, by modelling immiscible two-phase flow through an equilateral triangular capillary tube with sandwiched layers. We also demonstrate that imposing no-flow or free-slip interface boundary conditions at a clean fluid-fluid interface with zero interfacial shear viscosity, may lead to under- or over-estimation of flow conductance in layers compared to the physically correct continuity boundary condition at the interface. We use two-phase direct numerical simulation results in conjunction with basic arguments from fluid mechanics to present parametric models that estimate fluid conductivities as a function of the geometry and viscosity ratio. These scaling models, which take into account the flow coupling, can then be incorporated into pore-to-Darcy-scale flow models, for example two-phase pore-network models, to study the effects of viscous coupling on macroscopic flow properties such as relative permeabilities. These network models provide a much more computationally efficient framework for the simulation of flows to the centimetre or larger scales. Furthermore, in future work our methods can be used to assess local recovery and displacement mechanisms in multiphase flow using pore-scale images of different rocks.

This thesis is about studying the finite volume method for hyperbolic conservation laws system. Starting from the one dimensional Euler equations, we rewrite them from the form in Eulerian coordinates into the form in the Lagrangian coordinates. This technique transforms a moving grid in Eulerian co- ordinates into a fixed grid in Lagrangian coordinates, thus allowing easier imple- mentation of boundary conditions. The thesis consists of three parts: • piston problem, • multi-fluid models, • asymptotic behavior of Euler equations. In the first part, we consider the piston problem in the paper where the authors Yoshinori Inoue and Takeru Yano study the nonlinear propagation of plane waves radiated into a semi-infinite space filled with a perfect gas, by the sinusoidal motion of an infinite plate. They use the Euler equations in Eulerian coordinates describing the conservation of mass, momentum, and energy then approximate the solution. From this idea, we consider the waves propagating into a semi- infinite tube, which is filled with a perfect gas, closed by a piston on one end and extending along the x-axis at infinity. We use the mass Lagrangian coordinates to obtain the Euler equations rewritten in Lagrangian coordinates and reproduce the results following the piston problem in Yano's paper. The goal is to perform the computation in a finite computational domain, and to develop non-reflecting boundary conditions to impose on the right boundary. In order to reduce the impact of the reflected wave, we propose to combine the Burgers equation in few additional cells of the computational domain. The numerical error caused by the reflected wave is reduced by an order of magnitude by using this approach. In the second part, we consider the tube filled periodically by a large number of pairs of two immiscible fluids. We use Roe’s solver, which is described in Munz's paper, in Lagrangian coordinates, to study the motion of multi-fluid problem and then compare this detailed numerical solution with two isentropic homogeneous models. The first one is a 2 × 2 isentropic system and the second model is a 3 × 3 system which takes into account some turbulent effects. The goal is to check which homogenized model gives better prediction. We study two cases according to the ratio of densities of the two fluids: moderate ratio and large ratio. For each case, we perform the test with smooth and discontinuous initial condition in pressure and velocity. For the problem with smooth initial conditions before the shock formation, the detailed numerical solutions and the numerical results of the two isentropic homogeneous models are in very good agreement. After the shock formation, the detailed numerical solution is strongly oscillatory and we have to use the average values, namely smoothed numerical solution, for the comparison with the two models. We observe the difference between the predictions of the two models. For moderate density ratio the 2×2 model gives a better prediction of the shock position, while for large density ratio, the turbulent 3×3 model is in better agreement with a smoothed out version of the detailed numerical solution compared with the simple 2 × 2 model. In the third part, we study long time behavior of the solutions to the Euler equations by using two different numerical methods: the second order finite volume method in Lagrangian coordinates adopted in the previous chapters, and a high order finite volume method in Eulerian coordinate. In particular, the latter is based on WENO (Weighted Essentially Non-Oscillatory) reconstruction. Then we perform the comparison of the numerical solutions obtained at a final time, when pressure and velocity profiles are almost flat.

The problem of real-time simulation of ocean surface waves, ship movement and the coupling in between is tackled, and a number of different methods are covered and discussed. Among these methods, the finite volume method has been implemented in an attempt to solve the problem, along with the compressible Euler equations, an octree based staggered grid which allows for easy adaptive mesh refinement, the volume of fluid method and a variant of the Hyper-C advection scheme for compressible flows for advection of the phase fraction field. The process of implementing the methods that were chosen proved to be tricky in many ways, as they involve a large number of advanced topics, and the implementation that was implemented in this thesis work suffered from numerous issues. There were for example problems with keeping the interface intact, as well as a harsh restriction on the time step size due to the CFL condition. Improvements required to make the method sustainable for real-time applications are discussed, and a few suggestions on alternative approaches that are already in use for similar purposes are also given and discussed. Furthermore, a method for compensating for gain/loss of mass when solving the incompressible flow equations with an inaccurately solved pressure Poisson equation is presented and discussed. A momentum conservative method for transporting the velocity field on staggered grids without introducing unnecessary smearing is also presented and implemented. A simple, physically based illumination model for sea surfaces is derived, discussed and compared to the Blinn–Phong shading model, although it is never implemented. Finally, a two-dimensional partial differential equation in the spatial domain for simulating water surface waves for mildly varying bottom topography is derived and discussed, although it is deemed to be too slow for real-time purposes and is therefore never implemented.<br><p>This publication differs from the printed version of the report in the sense that links are blue in this version and black in the printed version.</p>

In this work, a code has been developed that solves the Navier-Stokes equations using the finite volume method with unstructured triangular grids. A cell-centred, finite volume method is used and the pressure-velocity coupling is treated using both the SMTLE and the MAC algorithms. The major advantage of using triangular grids is their applicability to complex geometry. A special treatment is developed to ensure good quality triangular elements around the boundaries. The numerical simulation of incompressible flow at low Reynolds number is studied in this thesis. A code for generating triangular grids using the tri-tree algorithm has been written and an adaptive finite volume method developed for calculating laminar fluid flow. The grid is locally adapted at each time step, with grid refinement and derefinement dependent on the vorticity magnitude. The resulting grids have fine local resolution and are economical in reducing the numerical simulation time. The discretised equations are solved by using an iterative point by point Gauss-Seidel solver. For calculating the values of velocity and pressure at vertices of triangular grids, special interpolation schemes (averaged linear-interpolation and scattered interpolation) are used to increase the accuracy. To avoid the well known checkerboard error problems, i. e., the oscillations occurring in the pressure field, third derivative terms in pressure, first introduced by Rhie-chow (1983), are added to the mass flux velocity. Convective terms are approximated using a QUICK (Quadratic Upstream Interpolation for Convective Kinematics) differencing scheme which has been developed here in for unstructured grids. Three cases of two-dimensional viscous incompressible fluid flow have been investigated: the first is channel flow, in which the numerical results are compared with the analytical solution; the second case is the backward-facing step flow; and the third case is flow past circular cylinders at low Reynolds number (Re). The numerical results obtained for the last two cases are compared with published data. The evolution of vortex shedding is presented for the case of unidirectional flow past a circular cylinder at Re=200. In addition, drag and lift force coefficients are calculated and compared for single and multiple cylinders in unidirectional flow.

Computational fluid dynamics (CFD) simulations can provide tremendous insight into complex physical processes and are often faster and more cost-effective to execute than experiments. However, each CFD result inherently contains numerical errors that can significantly degrade the accuracy of a simulation. Discretization error is typically the largest contributor to the overall numerical error in a given simulation. Discretization error can be very difficult to estimate since the generation, transport, and diffusion of these errors is a highly nonlinear function of the computational grid and discretization scheme. As CFD is increasingly used in engineering design and analysis, it is imperative that CFD practitioners be able to accurately quantify discretization errors to minimize risk and improve the performance of engineering systems. In this work, improvements are made to the accuracy and efficiency of existing error estimation techniques. Discretization error is estimated by deriving and solving an error transport equation (ETE) for the local discretization error everywhere in the computational domain. Truncation error is shown to act as the local source for discretization error in numerical solutions. An equivalence between adjoint methods and ETE methods for functional error estimation is presented. This adjoint/ETE equivalence is exploited to efficiently obtain error estimates for multiple output functionals and to extend the higher-order properties of adjoint methods to ETE methods. Higher-order discretization error estimates are obtained when truncation error estimates are sufficiently accurate. Truncation error estimates are demonstrated to deteriorate on grids with a non-smooth variation in grid metrics (e.g., unstructured grids) regardless of how smooth the underlying exact solution may be. The loss of accuracy is shown to stem from noise in the discrete solution on the order of discretization error. When using conventional least-squares reconstruction techniques, this noise is exactly captured and introduces a lower-order error into the truncation error estimate. A novel reconstruction method based on polyharmonic smoothing splines is developed to smoothly reconstruct the discrete solution and improve the accuracy of error estimates. Furthermore, a method for iteratively improving discretization error estimates is devised. Efficiency and robustness considerations are discussed. Results are presented for several inviscid and viscous flow problems. To facilitate the study of discretization error estimation, a new, higher-order finite-volume solver is developed. A detailed description of the code base is provided along with a discussion of best practices for CFD code design.<br>Ph. D.<br>Computational fluid dynamics (CFD) is a branch of computational physics at the intersection of fluid mechanics and scientific computing in which the governing equations of fluid motion, such as the Euler and Navier-Stokes equations, are solved numerically on a computer. CFD is utilized in numerous fields including biomedical engineering, meteorology, oceanography, and aerospace engineering. CFD simulations can provide tremendous insight into physical processes and are often preferred over experiments because they can be performed more quickly, are typically more cost-effective, and can provide data in regions where it may be difficult to measure. While CFD can be an extremely powerful tool, CFD simulations are inherently subject to numerical errors. These errors, which are generated when the governing equations of fluid motion are solved on a computer, can have a significant impact on the accuracy of a CFD solution. If numerical errors are not accurately quantified, ill-informed decision-making can lead to poor system performance, increased risk of injury, or even system failure. In this work, research efforts are focused on numerical error estimation for the finite -volume method, arguably the most widely used numerical algorithm for solving CFD problems. The error estimation techniques provided herein target discretization error, the largest contributor to the overall numerical error in a given simulation. Discretization error can be very difficult to estimate since these errors are generated, convected, and diffused by the same physical processes embedded in the governing equations. In this work, improvements are made to the accuracy and efficiency of existing discretization error estimation techniques. Results are presented for several inviscid and viscous flow problems. To facilitate the study of these error estimators, a new, higher-order finite -volume solver is developed. A detailed description of the code base is provided along with a discussion of best practices for CFD code design.

The world population is expected to increase to approximately 11 billion by 2100. The ageing population (aged 60 and over) is projected to exceed the number of children in 2047. This will be a situation without precedent. The number of citizens with disorders of old age like Dementia will rise to 115 million worldwide by 2050. The estimated cost of Dementia will also increase, from $604 billion in 2010, to $1,117 billion by 2030. At the same time, medical expertise, evidence-driven policymaking and commissioning of services are increasingly evolving the definitive architecture of comprehensive long-term care to account for these changes. Technological advances, such as those provided by computational science and biomedical engineering, will allow for an expansion in our ability to model and simulate an almost limitless variety of complex problems that have long defied traditional methods of medical practice. Numerical methods and simulation offer the prospect of improved clinically relevant predictive information, and of course optimisation, enabling more efficient use of resources for designing treatment protocols, risk assessment and urgently needed management of a long term care system for a wide spectrum of brain disorders. Within this paradigm, the importance of the relationship of senescence of cerebrospinal fluid transport to dementia in the elderly make the cerebral environment notably worthy of investigation through numerical and computational modelling. Hydrocephalus can be succinctly described as the abnormal accumulation (imbalance between production and circulation) of cerebrospinal fluid (CSF) within the brain. Using hydrocephalus as a test bed, one is able to account for the necessary mechanisms involved in the interaction between cerebral fluid production, transport and drainage. The current state of knowledge about hydrocephalus, and more broadly integrative cerebral dynamics and its associated constitutive requirements, advocates that poroelastic theory provides a suitable framework to better understand the disease. In this work, Multiple-network poroelastic Theory (MPET) is used to develop a novel spatio-temporal model of fluid regulation and tissue displacement in various scales within the cerebral environment. The model is discretised in a variety of formats, through the established finite difference method, finite difference – finite volume coupling and also the finite element method. Both chronic and acute hydrocephalus was investigated in a variety of settings, and accompanied by emerging surgical techniques where appropriate. In the coupled finite difference – finite volume model, a key novelty was the amalgamation of anatomically accurate choroid plexuses with their feeding arteries and a simple relationship relaxing the constraint of a unique permeability for the CSF compartment. This was done in order to account for Aquaporin-4 sensitisation. This model is used to demonstrate the impact of aqueductal stenosis and fourth ventricle outlet obstruction. The implications of treating such a clinical condition with the aid of endoscopic third (ETV) and endoscopic fourth ventriculostomy (EFV) are considered. It was observed that CSF velocity in the aqueduct, along with ventricular displacement, CSF pressure, wall shear stress and pressure difference between lateral and fourth ventricles increased with applied stenosis. The application of ETV reduced the aqueductal velocity, ventricular displacement, CSF pressure, wall shear stress and pressure difference within nominal levels. The greatest reversal of the effects of atresia come by opting for ETV rather than the more complicated procedure of EFV. For the finite difference model incorporating nonlinear permeability, qualitatively similar results were obtained in comparison to the pertinent literature, however, there was an overall amplification of ventriculomegaly and transparenchymal pressure difference using this model. A quantitative and qualitative assessment is made of hydrocephalus cases involving aqueductal stenosis, along with the effects to CSF reabsorption in the parenchyma and subarachnoid space. The finite element discretisation template produced for the n^th- dimensional transient MPET system allowed for novel insight into hydrocephalus. In the 1D formulation, imposing the breakdown of the blood-CSF barrier responsible for clearance resulted in an increase in ventricular displacement, transparenchymal venous pressure gradient and transparenchymal CSF pressure gradient, whilst altering the compliance proved to markedly alter the rate of change of displacement and CSF pressure gradient. The influence of Poisson's ratio was investigated through the use of the dual-grid solver in order to distinguish between possible over or under prediction of the ventricular displacement. In the 2D model based on linear triangles, the importance of the MPET boundary conditions is acknowledged, along with the quality of the underlying mesh. Interesting results include that the fluid content is highest in the periventricular region and the skull, whilst after longer time scales, the peak CSF content becomes limited to the periventricular region. Venous fluid content is heavily influenced by the Biot-Willis constant, whilst both the venous and CSF/ISF compartments show to be strongly influenced by breakdown in the blood-CSF barrier. Increasing the venous compliance effects the arterial, capillary and venous compartments. Decreasing the venous compliance shows an accumulation of fluid, possibly helping to explain why the ventricles can be induced to compress rather than expand under decreased compliance. Finally, a successful application of the 3D-MPET template is shown for simple geometries. It is envisaged that future observations into the biology of cerebral fluid flow (such as perivascular CSF-ISF fluid exchange) and its interaction with the surrounding parenchyma, will demand the evolution of the MPET model to reach a level of complexity that could allow for an experimentally guided exploration of areas that would otherwise prove too intricate and intertwined under conventional settings.

An unstructured-grid, h-adaptive control-volume finite element method (CVFEM) was formulated, implemented, and tested for the simulation of steady, viscous, incompressible fluid flow and heat transfer in arbitrarily shaped two-dimensional domains. The h-adaptive procedure is based on a posteriori error estimation using a superconvergent recovery or smoothing technique. The CVFEM is based on a co-located, equal-order formulation that deals directly with the primitive variables. A set of sufficient conditions is presented for guaranteeing that the algebraic approximations to the diffusion terms in the integral conservation equations contribute positively to the coefficients in the discretized equations. An additive-correction multigrid method, based on an adaptive volume-agglomeration technique, was implemented for the purposes of accelerating iterative convergence in the solution of the discretized equations.<br>An extended Richardson extrapolation technique for unstructured grids is proposed: A discrete solution is obtained from an h-adaptive analysis in which the estimated error is reduced to a predetermined level and uniformly distributed over each element. The resulting mesh is then used as a base grid in constructing a hierarchy of grids via uniform element subdivision.<br>A closed-form, analytical solution to the steady, two-dimensional, incompressible Navier-Stokes equations was obtained by prescribing a divergence-free velocity field and a consistent pressure field, substituting them into the momentum equations, and deriving the implied volumetric source terms. This analytical solution was used to test several numerical algorithms implemented in this work. Lid-driven, viscous flow in a triangular cavity was also used as a test problem.

The solution to partial differential equations generally requires approximations that result in numerical error in the final solution. Of the different types of numerical error in a solution, discretization error is the largest and most difficult error to estimate. In addition, the accuracy of the discretization error estimates relies on the solution (or multiple solutions used in the estimate) being in the asymptotic range. The asymptotic range is used to describe the convergence of a solution, where an asymptotic solution approaches the exact solution at a rate proportional to the change in mesh spacing to an exponent equal to the formal order of accuracy. A non-asymptotic solution can result in unpredictable convergence rates introducing uncertainty in discretization error estimates. To account for the additional uncertainty, various discretization uncertainty estimators have been developed. The goal of this work is to evaluation discretization error and discretization uncertainty estimators based on Richardson extrapolation for computational fluid dynamics problems. In order to evaluate the estimators, the exact solution should be known. A select set of solutions to the 2D Euler equations with known exact solutions are used to evaluate the estimators. Since exact solutions are only available for trivial cases, two applications are also used to evaluate the estimators which are solutions to the Navier-Stokes equations: a laminar flat plate and a turbulent flat plate using the k-Ï SST turbulence model. Since the exact solutions to the Navier-Stokes equations for these cases are unknown, numerical benchmarks are created which are solutions on significantly finer meshes than the solutions used to estimate the discretization error and uncertainty. Metrics are developed to evaluate the accuracy of the error and uncertainty estimates and to study the behavior of each estimator when the solutions are in, near, and far from the asymptotic range. Based on the results, general recommendations are made for the implementation of the error and uncertainty estimators. In addition, a new uncertainty estimator is proposed with the goal of combining the favorable attributes of the discretization error and uncertainty estimators evaluated. The new estimator is evaluated using numerical solutions which were not used for development and shows improved accuracy over the evaluated estimators.<br>Master of Science

Water management is a serious issue that affects the performance and durability of PEM fuel cells. It is known, from previous experimental investigations, that surface wettability has influence on water behaviour and fuel cell performance. This finding has lead researchers to develop numerical tools for further investigation of the liquid water behaviour in gas channels. The Volume-of-Fluid (VOF) method has been used in a wide range of studies for its advantage of showing the multi-phase interface in a Computational Fluid Dynamics (CFD) simulation to understand liquid water behaviour in gas channels. In this thesis, numerical study has been carried out to examine the behaviour of liquid water in gas channels. The dynamic movement of the liquid water in the channel and the associated pressure drop, water saturation and water coverage of the GDL have been investigated. Firstly, flow diffusion into the GDL was examined to determine its effect on liquid droplet behaviour in a small section of a gas channel. Furthermore, the effects of the percentage of flow diffusion, GDL wettability, pore size, and water inlet velocity were investigated. Fluid diffusion into GDL found to have insignificant impact on liquid water behaviour so further investigations has been carried with a solid GDL surface. Secondly, gas channel geometry effect on liquid water behaviour was studied. Square, semicircle, triangle, trapezoid with a long base and trapezoid with a short base were compared to find suitable cross section geometry to carry wall wettability investigations. Among the examined geometries, the square cross section showed reasonable results for both scenarios of geometry design, fixed Reynolds number and fixed GDL interface. The effect of wall wettability was assessed by comparing nine different wall/GDL wettability combinations for straight and bend channels. Wall wettability found to have an impact on liquid water behaviour but not as much as GDL wettability. It affects liquid water saturation in the channel by a great deal by accumulating water in the channel edges affecting water behaviour. This was also proven in the last test case of a long channel where water accumulation was investigated by running the calculation until the percentage of water saturation is stabilized. It is also concluded that changing wall wettability from hydrophobic to hydrophilic doubles the percentage of channel occupied by liquid water and increases the time to reach steady state.

In 1999, California Air Resources Board (CARB) implemented a regulation that required all gasoline cars sold in California be fitted with an Onboard Refueling Vapor Recovery System (ORVR). The ORVR system is designed to prevent Volatile Organic Compounds (VOCs) from escaping into the atmosphere during refuelling by storing the gas vapours in a carbon canister. Due to the complex nature of the fuel system, making design changes could have large implications on the ORVR performance of the vehicle. It is therefore desirable to develop a CFD model that can predict the effects of design changes, thereby reducing the need to perform physical tests on each design iteration. This master thesis project was performed at the Fuel Systems department at Volvo Cars in order to help reduce project lead times and product development costs by incorporating CFD as a part of the fuel system development cycle. The CFD results obtained were validated through experimental tests that were also performed as part of this project. In this master thesis project, a CFD model was developed to simulate the refuelling of gasoline for a California specification Volvo XC90 with an OPW-11B pump pistol. The model was set up in STAR-CCM+ using the Eulerian Volume of Fluid model for multiphase flow, the RANS realizable k − ε turbulence model and the two layer all y + wall treatment. The effects of the carbon canister were modelled as a porous baffle interface in the simulations where viscous and inertial resistances of the porous media were adjusted to obtain a desired pressure drop across the canister. This method proved to be a suitable simplification for this study. The effects of evaporation as well as a chemical adsorption model for the carbon canister have been excluded from the project due to time limitations. It was found that the CFD simulations were in good agreement with the experimental results, especially with respect to capturing the overall behaviour of the fuel system during refuelling. It was found that resolving the flow spatially (and temporally) in the filler pipe was a crucial part in ensuring solver stability. A pressure difference between experiment and simulation was also observed as a consequence of excluding evaporation from the CFD model. After the CFD model had been verified and validated, changes to different parts of the fuel system were investigated to observe their effects on ORVR performance. These included changing the recirculation line diameter, changing the carbon canister properties and changing the angle of how the pump pistol was inserted into the capless unit. It was found that the recirculation line diameter is a very sensitive design parameter and increasing the diameter would result in fuel vapour leaking back out into the atmosphere. Similarly, increasing the back pressure by swapping to a different carbon canister would result in the leakage of fuel vapour. On the other hand, insignificant changes in system behaviour were observed when the fuel pistol angle was changed.In 1999, California Air Resources Board (CARB) implemented a regulation that required all gasoline cars sold in California be fitted with an Onboard Refueling Vapor Recovery System (ORVR). The ORVR system is designed to prevent Volatile Organic Compounds (VOCs) from escaping into the atmosphere during refuelling by storing the gas vapours in a carbon canister. Due to the complex nature of the fuel system, making design changes could have large implications on the ORVR performance of the vehicle. It is therefore desirable to develop a CFD model that can predict the effects of design changes, thereby reducing the need to perform physical tests on each design iteration. This master thesis project was performed at the Fuel Systems department at Volvo Cars in order to help reduce project lead times and product development costs by incorporating CFD as a part of the fuel system development cycle. The CFD results obtained were validated through experimental tests that were also performed as part of this project. In this master thesis project, a CFD model was developed to simulate the refuelling of gasoline for a California specification Volvo XC90 with an OPW-11B pump pistol. The model was set up in STAR-CCM+ using the Eulerian Volume of Fluid model for multiphase flow, the RANS realizable k − ε turbulence model and the two layer all y + wall treatment. The effects of the carbon canister were modelled as a porous baffle interface in the simulations where viscous and inertial resistances of the porous media were adjusted to obtain a desired pressure drop across the canister. This method proved to be a suitable simplification for this study. The effects of evaporation as well as a chemical adsorption model for the carbon canister have been excluded from the project due to time limitations. It was found that the CFD simulations were in good agreement with the experimental results, especially with respect to capturing the overall behaviour of the fuel system during refuelling. It was found that resolving the flow spatially (and temporally) in the filler pipe was a crucial part in ensuring solver stability. A pressure difference between experiment and simulation was also observed as a consequence of excluding evaporation from the CFD model. After the CFD model had been verified and validated, changes to different parts of the fuel system were investigated to observe their effects on ORVR performance. These included changing the recirculation line diameter, changing the carbon canister properties and changing the angle of how the pump pistol was inserted into the capless unit. It was found that the recirculation line diameter is a very sensitive design parameter and increasing the diameter would result in fuel vapour leaking back out into the atmosphere. Similarly, increasing the back pressure by swapping to a different carbon canister would result in the leakage of fuel vapour. On the other hand, insignificant changes in system behaviour were observed when the fuel pistol angle was changed.

Unsteady heat transfer by natural convection in a closed square cavity is investigated numerically. A new finite-volume approach is developed and applied to the two-dimensional continuity, vorticity, and energy equations. The variation of the field variables is approximated by bi-quadratic interpolation formulas over the space occupied by the finite volume and the region surrounding it. These are used in the integral conservation laws for energy, vorticity and mass. The convective transport is modelled using a new upstream-weighting approach which uses volume averages for the vorticity and the energy transported across the boundaries of the finite volume. The weighting is dependent on the skewness of the velocity field to the surfaces of the finite volume as well as its strength. It is adaptive to local flow conditions. The velocities are obtained from the application of the velocity induction law. Use is made of an image system for the free vorticity of fluid. In this way, the no-penetration condition is enforced at the cavity boundaries, but at the same time it may allow a slip condition to exist. This is not permitted in a viscous flow analysis, and the slip velocity is reduced to zero by the production of free vorticity at the boundaries. Two test cases are treated which have exact solutions. The first is not new and involves a rotating shaft. The errors are less than.06% for this case. The second case is new and involves convection past a source and sink. The maximum error is 2.3%. For both test cases, the maximum error occurs at moderate values of the cell Peclet number and diminishes at the extreme low and high values. The time-development of the profiles of the vorticity, horizontal velocity, and temperature is examined at different locations within the cavity for Rayleigh numbers equal to 10³, 10⁴, and 10⁵. For these calculations, a 21 x 21 grid was used. The flow is found to approach a steady-state condition. The steady-state results are compared with a benchmark solution. In general, the agreement is excellent. The discrepancy is found to be less than 2% for the vast majority of the results for this relatively coarse grid.