PDE Based Enhancement of Color Images in RGB Space

International audienceA novel method for color image enhancement is proposed as an extension of scalar diffusion-shock filter coupling model, where noisy and blurred images are denoised and sharpened. The proposed model is based on using single vectors of the gradient magnitude and the second derivatives as a technique to relate different color components of the image. This model can be viewed as a generalization of Bettahar-Stambouli filter to multi-valued images. The proposed algorithm is more efficient than the mentioned filter and some previous works on color image denoising and deblurring without creating false colors

Feature-preserving image restoration and its application in biological fluorescence microscopy

This thesis presents a new investigation of image restoration and its application to fluorescence cell microscopy. The first part of the work is to develop advanced image denoising algorithms to restore images from noisy observations by using a novel featurepreserving diffusion approach. I have applied these algorithms to different types of images, including biometric, biological and natural images, and demonstrated their superior performance for noise removal and feature preservation, compared to several state of the art methods. In the second part of my work, I explore a novel, simple and inexpensive super-resolution restoration method for quantitative microscopy in cell biology. In this method, a super-resolution image is restored, through an inverse process, by using multiple diffraction-limited (low) resolution observations, which are acquired from conventional microscopes whilst translating the sample parallel to the image plane, so referred to as translation microscopy (TRAM). A key to this new development is the integration of a robust feature detector, developed in the first part, to the inverse process to restore high resolution images well above the diffraction limit in the presence of strong noise. TRAM is a post-image acquisition computational method and can be implemented with any microscope. Experiments show a nearly 7-fold increase in lateral spatial resolution in noisy biological environments, delivering multi-colour image resolution of ~30 nm

Flow pattern analysis for magnetic resonance velocity imaging

Blood flow in the heart is highly complex. Although blood flow patterns have been investigated by both computational modelling and invasive/non-invasive imaging techniques, their evolution and intrinsic connection with cardiovascular disease has yet to be explored. Magnetic resonance (MR) velocity imaging provides a comprehensive distribution of multi-directional in vivo flow distribution so that detailed quantitative analysis of flow patterns is now possible. However, direct visualisation or quantification of vector fields is of little clinical use, especially for inter-subject or serial comparison of changes in flow patterns due to the progression of the disease or in response to therapeutic measures. In order to achieve a comprehensive and integrated description of flow in health and disease, it is necessary to characterise and model both normal and abnormal flows and their effects. To accommodate the diversity of flow patterns in relation to morphological and functional changes, we have described in this thesis an approach of detecting salient topological features prior to analytical assessment of dynamical indices of the flow patterns. To improve the accuracy of quantitative analysis of the evolution of topological flow features, it is essential to restore the original flow fields so that critical points associated with salient flow features can be more reliably detected. We propose a novel framework for the restoration, abstraction, extraction and tracking of flow features such that their dynamic indices can be accurately tracked and quantified. The restoration method is formulated as a constrained optimisation problem to remove the effects of noise and to improve the consistency of the MR velocity data. A computational scheme is derived from the First Order Lagrangian Method for solving the optimisation problem. After restoration, flow abstraction is applied to partition the entire flow field into clusters, each of which is represented by a local linear expansion of its velocity components. This process not only greatly reduces the amount of data required to encode the velocity distribution but also permits an analytical representation of the flow field from which critical points associated with salient flow features can be accurately extracted. After the critical points are extracted, phase portrait theory can be applied to separate them into attracting/repelling focuses, attracting/repelling nodes, planar vortex, or saddle. In this thesis, we have focused on vortical flow features formed in diastole. To track the movement of the vortices within a cardiac cycle, a tracking algorithm based on relaxation labelling is employed. The constraints and parameters used in the tracking algorithm are designed using the characteristics of the vortices. The proposed framework is validated with both simulated and in vivo data acquired from patients with sequential MR examination following myocardial infarction. The main contribution of the thesis is in the new vector field restoration and flow feature abstraction method proposed. They allow the accurate tracking and quantification of dynamic indices associated with salient features so that inter- and intra-subject comparisons can be more easily made. This provides further insight into the evolution of blood flow patterns and permits the establishment of links between blood flow patterns and localised genesis and progression of cardiovascular disease.Open acces

Computational modeling of TGF- signaling

Tese de mestrado integrado em Engenharia Biomédica e Biofísica (Engenharia Clínica e Instrumentação Médica), Universidade de Lisboa, Faculdade de Ciências, 2020O fator de transformação do crescimento beta ou TGF-β é uma citocina que consta nos 33 membros da superfamília do TGF-β, envolvida em processos fundamentais da célula, nomeadamente na proliferação, migração, diferenciação das células e apoptose celular, assim como na homeostasia dos tecidos. Consequentemente, o mau funcionamento da via de sinalização do TGF-β está associado a diferentes patologias. Tendo em conta este facto e a grande diversidade de respostas biológicas por si originadas, esta cascata de sinalização é um grande alvo de pesquisas, da parte de diversos autores, através da implementação de diversos modelos computacionais. A morfologia da célula não é tida em conta na implementação da maioria dos modelos correntes da via de sinalização canónica do TGF-β. Porém, esta característica celular é normalmente vista como uma medida de grande relevância para ditar o modo como a célula responde a estímulos mecânicos externos. A resposta das células ao ambiente extracelular, caracterizada pela conversão de estímulos mecânicos em sinas bioquímicos, tem a designação de mecanotrasdução. As integrinas, que são uma família de receptores de sinais e de adesão células, convertem estímulos mecânicos em sinais bioquímicos, constituindo um meio de comunição entre as células e o ambiente extracelular. Deste modo, as interacções entre a via de sinalização do TGF-β e as integrinas constituem uma forma de mecanotransdução. Um entendimento mais profundo da dinâmica da via de sinalização do TGF-β através de um modo dependente da morfologia da célula e um entendimento do papel que o crosstalk entre as integrinas e a via de sinalização canónica do TGF-β tem sobre a expressão genética permitem, consequentemente, perceber melhor a conexão entre esta via e o comportamento celular perante estímulos mecânicos. Tal conhecimento detém o potencial para prever interações entre as células e biomateriais e poderá, possivelmente, ser aplicado a diferentes terapias, tais como terapias associadas ao controlo do desenvolvimento de tumores. Devido às razões previamente referidas, neste trabalho é apresentado um novo modelo computacional para a via de sinalização canónica do TGF-β, contendo um módulo de crosstalk com as integrinas. O modelo construído é espacial, sendo implementado através de equações diferenciais parciais. As equações utilizadas são do tipo reação-difusão. Os principais objectivos do presente trabalho são avaliar o impacto que a morfologia da célula detém sobre a sinalização downstream e perceber qual é a influência da crosstalk entre a via de sinalização canónica do TGF-β e as integrinas sobre a expressão genética. Deste modo, o modelo implementado neste trabalho divide-se em dois módulos – Módulo I e Módulo II. O modelo standard, que consiste no modelo da via de sinalização canónica do TGF-β, corresponde ao Módulo I. O Módulo I do modelo foi construído com base em dois artigos: Claus et al., 2013 e Nicklas & Saiz, 2013. As equações que envolvem as diferentes espécies da via de sinalização canónica do TGF-β, assim como as condições de fronteira aplicáveis às membranas citoplasmática e nuclear, foram adaptadas de Claus et al., 2013, e a secção do modelo respeitante ao trafficking de receptores foi adaptado de Nicklas e & Saiz, 2013. As equações de reação-difusão implementadas no Módulo I traduzem os passos gerais da cascata de sinalização do TGF-β referidos por diversos autores: os recetores do TGF-β são ativados pelo ligando TGF-β, recrutando e fosforilando o Smad2/3 junto à membrana citoplasmática. O Smad2/3 fosforilado, pSmad2/3, pode associar-se ao Smad mediador, o Smad4, formando um complexo, o pSmad2/3 + Smad4. Este complexo migra para o núcleo, onde se associa ao DNA, induzindo expressão genética. Quando o pSmad2/3 não se associa ao Smad4, pode migrar diretamente para o núcleo, tal como moléculas de Smad2/3 e moléculas de Smad4 podem deslocar-se até ao núcleo sem estarem associadas a outras espécies. O processo de formação do complexo pSmad2/3 + Smad4 também pode ocorrer no núcleo da célula. O Módulo II do modelo diz respeito à implementação de diferentes interações de crosstalk entre as integrinas e a via de sinalização do TGF-β. As interações de crosstalk incluídas neste módulo são: 1) A upregulation dos recetores do TGF-β despoletada pelas integrinas; 2) O aumento da constante catalítica da reacção de fosforilação do Smad2/3 mediada pelo complexo formado pelo TGF-β e os seus recetores, complexo C, sendo este aumento despoletado pelas integrinas; 3) A estabilização dos recetores do TGF-β despoletada pelas integrinas, resultando na diminuição da taxa de degradação do complexo C; 4) A formação de um complexo composto pelas integrinas e pelo complexo C, complexo IC, que se liga ao Smad2/3, provocando um aumento da constante catalítica da reação de fosforilação do Smad2/3. Foram realizadas diferentes experiências in silico para atingir os objetivos previamente referidos. Relativamente ao Módulo I, de modo a avaliar o impacto da morfologia celular sobre a expressão genética induzida pelo TGF-β, foram feitas simulações para diferentes geometrias da célula, nomeadamente para elipses e rectângulos com diferentes rácios de largura e altura. Foram ainda realizadas simulações para diferentes áreas da célula, num intervalo de valores que se estende desde a área estabelecida no modelo standard até 10 vezes a área do modelo standard. A avaliação do impacto da área da célula a nível downstream é realizada para as geometrias acima referidas. Adicionalmente, os parâmetros do modelo foram submetidos a uma análise de sensibilidade. No que diz respeito ao Módulo II, realizaram-se diferentes simulações para diferentes tipos de crosstalk que se estabelecem entre as integrinas e a via de sinalização do TGF-β para perceber a influência que um aumento dos parâmetros associados a cada tipo de crosstalk possui sobre a concentração do complexo pSmad2/3 + Smad4 no núcleo. Esta incrementação dos valores atribuídos a cada parâmetro é feita de modo independente ou para dois ou três parâmetros em simultâneo. A análise das diferentes simulações realizadas, com o intuito de avaliar apenas o impacto da geometria da célula, indica que a espécie participante desta via de sinalização que mais sofre discrepâncias na sua concentração nuclear por rácio largura/altura, devido a mudanças na geometria da célula, é o Smad2/3. As restantes espécies, nomeadamente o pSmad2/3, o complexo pSmad2/3 + Smad4 e o Smad4, sofrem discrepâncias idênticas por unidade de rácio largura/altura. Constatou-se que existe uma tendência de decréscimo na concentração nuclear do Smad2, pSmad2/3 e complexo pSmad2/3 + Smad4 com o aumento do rácio largura/altura da célula. Por outro lado, constatou-se um aumento da concentração nuclear do Smad4 por unidade de rácio largura/altura. As simulações efectuadas para avaliar o efeito que as dimensões da célula possuem sobre a cascata do TGF-β, a nível downstream, indicam que com o aumento do tamanho da célula, existe diminuição da concentração do complexo pSmad2/3 + Smad4 no núcleo da célula, havendo um declínio na expressão genética. Esta conclusão é aplicável a todas as geometrias da célula consideradas. Concretamente, considerando a passagem da área standard da célula para uma área 10 vezes superior, constata-se uma diminuição de cerca de 50% na concentração nuclear do complexo pSmad2/3 + Smad4 para todos os formatos da célula. Ainda relativamente a resultados do Módulo I, os diversos parâmetros do modelo foram submetidos a uma análise de sensibilidade que revelou que a concentração do complexo pSmad2/3 + Smad4 no núcleo é mais sensível às reacções do modelo do que a processos de difusão. No entanto, um decréscimo de 100% do valor dos parâmetros de difusão revela uma sensibilidade considerável a nível downstream da via de sinalização, o que sugere que ainda assim os parâmetros de difusão devem ser incluídos em modelos que visem estudar a dinâmica da cascata do TGF-β. As simulações realizadas para estudar o efeito da crosstalk entre as integrinas e a via de sinalização do TGF-β sugerem que de todos os tipos de interação implementados no modelo construído, o tipo de interação que causa maior upregulation da concentração nuclear do complexo pSmad2/3 + Smad4 é o decréscimo da taxa de degradação dos receptores devido à estabilização dos receptores do TGF-β por parte das integrinas. Por outro lado, o tipo de interacção de crosstalk que menos contribui para a expressão genética é o aumento da constante catalítica de fosforilação do Smad2/3 mediada pelo complexo IC. Em adição, a expressão genética sofreu mais upregulation, em geral, quando houve uma intensificação simultânea dos efeitos de dois tipos de crosstalk com as integrinas, comparativamente com uma forma de crosstalk ou três tipos de interação em conjunto. Os próximos passos associados ao trabalho apresentado neste documento são o estabelecimento de um conjunto de experiência in vitro para analisar o efeito da morfologia da célula e da mecanotransdu-ção, com o intuito de validar os resultados e conclusões obtidos através do conjunto de experiências in silico. Adicionalmente, outras formas de crosstalk entre as integrinas e a cascata do TGF-β deverão ser incluídas no modelo, tais como a interação da via de sinalização com proteínas reguladas pela actina-G e actina-F do citoesqueleto da célula. Estas proteínas são libertadas quando existe remodelação do cito-esqueleto e um desequilíbrio nos seus níveis resulta na expressão de receptores de ligandos da família TGF-β.The TGF-β is a cytokine involved in fundamental cell processes, such as cell migration, proliferation and apoptosis. A deeper insight into the dynamics of the TGF-β pathway through a cell morphology-dependent manner and into the role that the crosstalk with the integrins has upon downstream signaling of this pathway allows a better understanding of the link between the TGF-β pathway and mechanotransduction cues, therefore holding the potential to uncover interactions between biomaterials and cells and apply this knowledge to different therapies. In this document, a spatial computational model for the canonical TGF-β pathway is presented, including a module for the crosstalk with the integrins. The equations used to model this system are partial differential equations of the reaction-diffusion type. The model was created to evaluate the impact of cell shape and size, as well as crosstalk with the integrins, on genetic expression. The results of the experiments suggest that a bigger width/height ratio of the cell induces less con-centration of the nuclear Smad2/3, nuclear phosphorylated Smad2/3 and nuclear phosphorylated Smad2/3 + Smad4 complex, for elliptical and rectangular cell shapes. Conversely, the concentration of the nuclear Smad4 increases as the width/height ratio of the cell increases, for elliptical and rectangular cell shapes. The nuclear species of the TGF-β pathway that suffers more variation in its nuclear concentration across cell shapes is the Smad2/3. Considering a range of values that go up to 10 times the original area of the cell for different cell shapes, as the area of the cell increases, the concentration of the nuclear phosphorylated Smad2/3 + Smad4 complex decreases by approximately 50%. Sensitivity analysis of the model parameters indicate that genetic expression elicited by the TGF-β when considering diffusive processes differs considerably from results without considering this phenomenon. It is suggested that the way of crosstalk which more heavily increases genetic expression is the decrease of the degradation rate of the receptors due to stabilization from the integrins. Experimental procedures also suggest that the concentration of the nuclear phosphorylated Smad2/3 + Smad4 complex undergoes more upregulation, in general, by the joint behavior of two ways of crosstalk interaction than by the effects of one way of crosstalk independently or the combination of three ways of crosstalk out of the considered interactions

The Sixth Copper Mountain Conference on Multigrid Methods, part 2

The Sixth Copper Mountain Conference on Multigrid Methods was held on April 4-9, 1993, at Copper Mountain, Colorado. This book is a collection of many of the papers presented at the conference and so represents the conference proceedings. NASA Langley graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The multigrid discipline continues to expand and mature, as is evident from these proceedings. The vibrancy in this field is amply expressed in these important papers, and the collection clearly shows its rapid trend to further diversity and depth

Multiphase flow modelling for enhanced oil and gas drilling and production

From the exploration to the abandonment of an oil and gas discovery, operators and engineers are constantly faced with the challenge of achieving the best commercial potential of oil fields. Although the petroleum engineering community has significantly contributed towards maximising the potential of discovered prospects, the approach adopted so far has been compartmentalised with little (heuristics-based) or no quality integration. The highly interconnected nature of the decision factors affecting the management of any field requires increased implementation of Computer-Aided Process Engineering (CAPE) methods, thus presenting a task for which chemical engineers have the background to make useful contributions. Drilling and production are the two primary challenging operations of oilfield activities, which span through different time horizons with both fast and slow-paced dynamics. These attributes of these systems make the application of modelling, simulation, and optimisation tasks difficult. This PhD project aims to improve field planning and development decisions from a Process Systems Engineering (PSE) perspective via numerical (fluid dynamics) simulations and modelbased deterministic optimisation of drilling and production operations, respectively. Also demonstrated in this work is the importance of deterministic optimisation as a reliable alternative to classical heuristic methods. From a drilling operation perspective, this project focuses on the application of Computational Fluid Dynamics (CFD) as a tool to understand the intricacies of cuttings transport (during wellbore cleaning) with drilling fluids of non-Newtonian rheology. Simulations of two-phase solid-liquid flows in an annular domain are carried out, with a detailed analysis on the impact of several drilling parameters (drill pipe eccentricity, inclination angle, drill pipe rotation, bit penetration rate, fluid rheology, and particle properties) on the cuttings concentration, pressure drop profiles, axial fluid, and solid velocities. The influence of the flow regime (laminar and turbulent) on cuttings transport efficiency is also examined using the Eulerian-Eulerian and Lagrangian-Eulerian modelling methods. With experimentally validated simulations, this aspect of the PhD project provides new understanding on the interdependence of these parameters; thus facilitating industrial wellbore cleaning operations. The second part of this project applies mathematical optimisation techniques via reduced-order modelling strategies for the enhancement of petroleum recovery under complex constraints that characterise production operations. The motivation for this aspect of the project stems from the observation that previous PSE-based contributions aimed at enhancing field profitability, often apply over-simplifications of the actual process or neglect some key performance indices due to problem complexity. However, this project focuses on a more detailed computational integration and optimisation of the models describing the whole field development process from the reservoir to the surface facilities to ensure optimal field operations. Nonlinear Programs (NLPs), Mixed-Integer Linear Programs (MILPs), and Mixed-Integer Nonlinear Programs (MINLPs) are formulated for this purpose and solved using high-fidelity simulators and algorithms in open-source and commercial solvers. Compared to previous studies, more flow physics are incorporated and rapid computations obtained, thus enabling real-time decision support for enhanced production in the oil and gas industry

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Computational fluid dynamics and experimental study of the hydrodynamics of a bubble column and an air-water jet-stirred cell

A large number of flows encountered in nature and in many industrial processes areintrinsically multiphase flows. The efficiency and the effectiveness of multiphase flow processes strongly depend on the ability to model the fluid flow behaviour. Thus, a robust and accurate description of multiphase flow can lead to an increase in performance, a reduction in cost, and an improvement in safety for engineering systems. In recent years, Computational Fluid Dynamics (CFD) has become an indispensable predictive tool for gathering information to be used for design and optimization for fluid systems. In this thesis the hydrodynamics of two bubbly flow systems, a bubble column and a waterjet-agitated flotation cell (Hydrojet cell), were studied by means of numerical simulations. In order to validate the bubble column CFD simulations Particle Image Velocimetry (PIV) was used. An experimental investigation about bubble size distribution (BSD) along a water jet was carried out by means of image analysis. Because of high gas fraction and high velocity of the air/water streams used to agitate the Hydrojet cell, with the available equipment, no experimental measurements could be done to evaluate the velocity field of the cell. The thesis consists of three parts: theoretical part, bubble column study and Hydrojet cell study. In the theoretical part, first, a summary of fluid dynamics principles and an overview of the principal issues related to multiphase flow modelling were presented. Then a brief introduction to PIV and its application to two phase bubbly flow were given. Finally a review of the principle of the flotation process and its modelling were done in order to highlight the reasons for the low recovery of fine particles. Then the potentialities offered by the use of waterjets to fine particles flotation were presented. In the second part experimental and numerical studies of a bubble column were presented. PIV technique was used to determine the velocity field of a laboratory bubble column. A separation method for multiphase PIV was developed and tested. By means of the proposed method, the acquired mixed-fluid images were processed to obtain two sets of single phase images before PIV analysis. The velocity field was determined using a multi-pass crosscorrelation. Following three-dimensional time-dependent CFD simulations of a lab-scale bubble column were presented. The simulations were carried out using the Euler - Euler approach. Two different multiphase turbulence models, Shear Stress Transport (SST) and Large Eddy Simulation (LES), were tested, and different interfacial closure models reported in the literature were examined. When LES were used to model the turbulence instead of the SST model, much better agreement with the experimental data was found, provided that the drag, lift and virtual mass forces were taken into account. In the third part a preliminary experimental study, carried out in a rectangular flat cell, was presented. It was carried out to investigate the size distribution of bubbles generated by a moderate pressure water jet, by means of image analysis. This study showed the ability of water jets at moderate pressure to break an air stream into small bubbles. Increasing the pressure of the pump, smaller and more uniform bubbles were obtained. Then three-dimensional CFD simulations of the Hydrojet cell are presented. The Hydrojet cell, due to the exceeding computational burden, was simulated as a two-phase (gas-liquid) system, although actually it is a three-phase (gas-liquid-solid) system. Also in this case simulations were carried out using the Euler - Euler approach. The turbulence of the liquid phase was modelled with the SST model. The single reference frame technique was used to describe the movement of the waterjet lance. To achieve a homogeneous aeration in the region near the inlets different inlet velocity and rotational speed were tested. The results gave useful indications about the role of the four principal operating parameters: nozzles diameter, velocity of rotation of the lance, speed of the water jets and then pressure of the pump and inlet air flow rate. What emerges is the need of high rotational speed of the waterjet lance in order to ensure an uniform gas distribution within the mixing zone. This is not possible with the current apparatus. Thus in order to make the system suitable to produce an appropriate environment for the full development of the flotation process it is necessary to modify the system

Computational fluid dynamics and experimental study of the hydrodynamics of a bubble column and an air-water jet-stirred cell

A large number of flows encountered in nature and in many industrial processes areintrinsically multiphase flows. The efficiency and the effectiveness of multiphase flow processes strongly depend on the ability to model the fluid flow behaviour. Thus, a robust and accurate description of multiphase flow can lead to an increase in performance, a reduction in cost, and an improvement in safety for engineering systems. In recent years, Computational Fluid Dynamics (CFD) has become an indispensable predictive tool for gathering information to be used for design and optimization for fluid systems. In this thesis the hydrodynamics of two bubbly flow systems, a bubble column and a waterjet-agitated flotation cell (Hydrojet cell), were studied by means of numerical simulations. In order to validate the bubble column CFD simulations Particle Image Velocimetry (PIV) was used. An experimental investigation about bubble size distribution (BSD) along a water jet was carried out by means of image analysis. Because of high gas fraction and high velocity of the air/water streams used to agitate the Hydrojet cell, with the available equipment, no experimental measurements could be done to evaluate the velocity field of the cell. The thesis consists of three parts: theoretical part, bubble column study and Hydrojet cell study. In the theoretical part, first, a summary of fluid dynamics principles and an overview of the principal issues related to multiphase flow modelling were presented. Then a brief introduction to PIV and its application to two phase bubbly flow were given. Finally a review of the principle of the flotation process and its modelling were done in order to highlight the reasons for the low recovery of fine particles. Then the potentialities offered by the use of waterjets to fine particles flotation were presented. In the second part experimental and numerical studies of a bubble column were presented. PIV technique was used to determine the velocity field of a laboratory bubble column. A separation method for multiphase PIV was developed and tested. By means of the proposed method, the acquired mixed-fluid images were processed to obtain two sets of single phase images before PIV analysis. The velocity field was determined using a multi-pass crosscorrelation. Following three-dimensional time-dependent CFD simulations of a lab-scale bubble column were presented. The simulations were carried out using the Euler - Euler approach. Two different multiphase turbulence models, Shear Stress Transport (SST) and Large Eddy Simulation (LES), were tested, and different interfacial closure models reported in the literature were examined. When LES were used to model the turbulence instead of the SST model, much better agreement with the experimental data was found, provided that the drag, lift and virtual mass forces were taken into account. In the third part a preliminary experimental study, carried out in a rectangular flat cell, was presented. It was carried out to investigate the size distribution of bubbles generated by a moderate pressure water jet, by means of image analysis. This study showed the ability of water jets at moderate pressure to break an air stream into small bubbles. Increasing the pressure of the pump, smaller and more uniform bubbles were obtained. Then three-dimensional CFD simulations of the Hydrojet cell are presented. The Hydrojet cell, due to the exceeding computational burden, was simulated as a two-phase (gas-liquid) system, although actually it is a three-phase (gas-liquid-solid) system. Also in this case simulations were carried out using the Euler - Euler approach. The turbulence of the liquid phase was modelled with the SST model. The single reference frame technique was used to describe the movement of the waterjet lance. To achieve a homogeneous aeration in the region near the inlets different inlet velocity and rotational speed were tested. The results gave useful indications about the role of the four principal operating parameters: nozzles diameter, velocity of rotation of the lance, speed of the water jets and then pressure of the pump and inlet air flow rate. What emerges is the need of high rotational speed of the waterjet lance in order to ensure an uniform gas distribution within the mixing zone. This is not possible with the current apparatus. Thus in order to make the system suitable to produce an appropriate environment for the full development of the flotation process it is necessary to modify the system