The behaviour of free-surface gravity waves on small Froude number fluid flow past some obstacle cannot be determined using ordinary asymptotic power series methods, as the amplitude of the waves is exponentially small. An exponential asymptotic method is used by Chapman and Vanden-Broeck (2006) to consider the problem of two-dimensional, steady flow past a submerged obstacle in the small Froude number limit, finding that a steady downstream wavetrainis switched on rapidly across a curve known as a Stokes line. Here, equivalent wavetrains on three-dimensional and unsteady flow configurations are considered, and Stokes switching causedby the interaction between exponentially small free-surface components is shown to play an important role in both cases. The behaviour of free-surface gravity waves is introduced by considering the problem of steady free-surface flow due to a line source. A steady wavetrain is shown to exist in the far field, and the behaviour of these waves is compared to existing numerical results. The problem of unsteady flow over a step is subsequently investigated, with the flow behaviour formulated in terms of Lagrangian coordinates so that the position of the free surface is fixed. Initially, the problem is linearized in the step-height, and the steady wavetrain is shown to spread downstream over time. The position of the wavefront is determined by considering the full Stokes structure present in the problem. The equivalent fully-nonlinear problem is then considered, with the position of the Stokes lines, and hence the wavefront, being determined numerically. Finally, linearized three-dimensional free-surface flow past an obstacle is considered in both the steady and unsteady case. The surface is shown to contain downstream longitudinal and transverse waves. These waves are shown to propagate downstream in the unsteady case, with the position of the wavefront again determined by considering the full Stokes structure of the problem.

Ice sheets, such as the Greenland Ice Sheet or Antarctic Ice Sheet, have a fundamental impact on landscape formation, the global climate system, and on sea level rise. The slow, creeping flow of ice can be represented by a non-linear version of the Stokes equations, which treat ice as a non-Newtonian, viscous fluid. Large spatial domains combined with long time spans and complexities such as a non-linear rheology, make ice sheet simulations computationally challenging. The topic of this thesis is the efficiency and error control of large simulations, both in the sense of mathematical modelling and numerical algorithms. In the first part of the thesis, approximative models based on perturbation expansions are studied. Due to a thick boundary layer near the ice surface, some classical assumptions are inaccurate and the higher order model called the Second Order Shallow Ice Approximation (SOSIA) yields large errors. In the second part of the thesis, the Ice Sheet Coupled Approximation Level (ISCAL) method is developed and implemented into the finite element ice sheet model Elmer/Ice. The ISCAL method combines the Shallow Ice Approximation (SIA) and Shelfy Stream Approximation (SSA) with the full Stokes model, such that the Stokes equations are only solved in areas where both the SIA and SSA is inaccurate. Where and when the SIA and SSA is applicable is decided automatically and dynamically based on estimates of the modeling error. The ISCAL method provides a significant speed-up compared to the Stokes model. The third contribution of this thesis is the introduction of Radial Basis Function (RBF) methods in glaciology. Advantages of RBF methods in comparison to finite element methods or finite difference methods are demonstrated. / eSSENCE

Bed topography identification in open channel and glacier flows is of paramount importance for the study of the respective flows. In the former, the knowledge of the channel bed topography is required for modelling the hydrodynamics of open channel flows, fluvial hydraulics, flood propagation, and river flow monitoring. Indeed, flow models based on the Shallow Water Approximation require prior information on the channel bed topography to accurately capture the flow features. While in the latter, usable bedrock topographic information is very important for glacier flow modellers to accurately predict the flow characteristics. Experimental techniques to infer the bed topography are usually used but are mostly time consuming, costly, and sometimes not possible due to geographical restrictions. However, the measurement of free surface elevation is relatively easy. Alternative to experimental techniques, it is therefore important to develop fast, easy-to-implement, and cost-effective numerical methods. The inverse of the classical hydrodynamic problem corresponds to the determination of hydraulic parameters from measurable quantities. The forward problem uses model parameters to determine measurable quantities. New one-shot and direct pseudo-analytical and numerical approaches for reconstructing the channel bed topography from known free surface elevation data is developed for one-dimensional shallow water flows. It is shown in this work that instead of treating this inverse problem in the traditional partial differential equation (PDE)-constrained optimization framework, the governing equations of the direct problem can be conveniently rearranged to obtain an explicit PDE for the inverse problem. This leads to a direct solution of the inverse problem which is successfully tested on a range of benchmark problems and experimental data for noisy and noiseless free surface data. It was found that this solution approach creates very little amplification of noise. A numerical technique which uses the measured free surface velocity to infer the channel bed topography is also developed. The one-dimensional shallow water equations along with an empirical relationship between the free surface and the depth averaged velocities are used for the inverse problem analysis. It is shown that after a series of algebraic manipulation and integration, the equation governing the inverse problem simplifies to a simple integral equation. The proposed method is tested on a range of analytical and experimental benchmark test cases and the results confirm that, it is possible to reconstruct the channel bed topography from a known free surface velocity distribution of one-dimensional open channel flows. Following the analysis of the case of one-dimensional shallow water flows, a numerical technique for reconstructing the channel bed topography from known free surface elevation data for steep open channel flows is developed using a modified set of equations for which the zero-inertia shallow water approximation holds. In this context, the shallow water equations are modified by neglecting inertia terms while retaining the effects of the bed slope and friction terms. The governing equations are recast into a single first-order partial differential equation which describes the inverse problem. Interestingly, the analysis shows that the inverse problem does not require the knowledge of the bed roughness. The forward problem is solved using MacCormack’s explicit numerical scheme by considering unsteady modified shallow water equations. However, the inverse problem is solved using the method of characteristics. The results of the inverse and the forward problem are successfully tested against each other. In the framework of full two-dimensional shallow water equations, an easy-to-implement and fast to solve direct numerical technique is developed to solve the inverse problem of shallow open channel flows. The main underlying idea is analogous to the idea implemented for the case of one-dimensional reconstruction. The technique described is a “one-shot technique” in the sense that the solution of the partial differential equation provides the solution to the inverse problem directly. The idea is tested on a set of artificial data obtained by first solving the forward problem. Glaciers are very important as an indicator of future climate change or to trace past climate. They respond quickly compared to the Antarctica and Greenland ice sheets which make them ideal to predict climate changes. Glacier bedrock topography is an important parameter in glacier flow modelling to accurately capture its flow dynamics. Thus, a mathematical technique to infer this parameter from measured free surface data is invaluable. Analogous to the approaches implemented for open channel flows, easy-to-implement direct numerical and analytical algorithms are developed to infer the bedrock topography from the knowledge of the free surface elevation in one space dimension. The numerical and analytical methods are both based on the Shallow Ice Approximation and require the time series of the ablation/accumulation rate distribution. Moreover, the analytical method requires the knowledge of a non-zero glacier thickness at an arbitrary location. Numerical benchmark test cases are used to verify the suitability and applicability of the algorithms.

The subject of this study is the determination of the resistance of an inland vessel engaged in cargo transport in the lower course of the Magdalena River, considering that the hydrodynamic effects in shallow water navigation are very different compared to the effects in deep water navigation. The hydrodynamic analysis is realized numerically using Computational Fluid Dynamics (CFD). The Reynolds-Averaging Navier-Stokes equation (RANS) solver is applied to simulate viscous and pressure effects around a tank and a hull in confined tank considering the wall bottom and side effects in shallow water navigation. For turbulence effects, realizable k-? model is used. The motion of the vessel causes elevations of the free surface, in which, is captured using the Volume of Fluid method (VOF). For discretization of flow domain, the Finite Volume Method (FVM) is applied. The motion of the fluids is updated for each time step that allows the calculation of the resistance acting on the hull. The numerical simulation results are compared with experimental data obtained by the Technological Research Institute of the State of São Paulo (IPT, acronym in Portuguese) together with the existing empirical methods for this type of cases. / É apresentado um estudo para determinar a resistência de uma barcaça empregada no transporte de carga que poderia operar no setor baixo do rio Magdalena. Os efeitos hidrodinâmicos de um navio em águas rasas são muito diferentes, comparados a esses efeitos em águas com profundidade infinita. A análise hidrodinâmica é realizada numericamente usando a Dinâmica dos Fluidos Computacional (CFD, acrônimo em inglês). A solução das equações de Navier-Stokes (NS) junto com a decomposição do Reynolds (RANS, acrônimo em inglês) é aplicada para simular os efeitos viscosos e de pressão em torno de um tanque e de uma embarcação em um tanque confinado que é caracterizado pelos efeitos do fundo e das paredes. Para efeitos de turbulência, o modelo realizado k-? é usado. O movimento da embarcação do rio provoca elevações da superfície livre que são capturadas usando o método do Volume de Fluido (VOF, acrônimo em inglês). Para a discretização do domínio de fluxo, o Método dos Volumes Finitos (FVM, acrônimo em inglês) é utilizado. O movimento dos fluidos é atualizado para cada intervalo de tempo o que permite o cálculo da resistência atuando no casco. Os resultados da simulação numérica são comparados com dados experimentais obtidos pelo Instituto de Pesquisas Tecnológicas do Estado de São Paulo (IPT), juntamente com os métodos empíricos existentes para esse tipo de casos.

Cette thèse porte sur le développement d’un modèle numérique de l’affouillement causée par des obstacles montes sur le lit, combinant les processus hydrodynamiques et morphologiques. Le modèle numérique est basé sur le solveur de champ d’écoulement polyphasique de l’outil CFD open-source OpenFOAMR qui est distribue par OpenCFD Ltd. Le module hydrodynamique du modèle résout les équations de Navier-Stokes avec moyennes de Reynolds (RANS) et les modèles des turbulences k-ε ou k-ω. Il existe deux interfaces dans le domaine de simulation: la surface libre entre l’eau et l’air, qui est suivi par la méthode de Volume de Fluide (VOF); et l’interface entre l’eau et le lit du sédiment, qui est représentée par un maillage de surface finie déformable construit à partir de la limite en bas du maillage de volume fini. En outre, un module morphologique qui a été développé dans le cadre du projet se compose de trois composantes: un modèle de transport de sédiments comprenant la charge suspendue et le charriage; l’équation d’Exner pour mesurer la déformation du lit; et un mécanisme de glissement du sable pour limiter la pente du lit à être plus petite que l’angle de repos du sédiment. Le changement morphologique est incorporé dans le modèle hydrodynamique par la déformation du maillage. Des conditions limites spéciales et des corrections nécessaires pour le calcul en parallèle sont également ajoutées au modèle. Chaque partie du modèle est validée séparément avec les tests préliminaires correspondants, y compris les fonctions de paroi rugueuse, les performances de la méthode VOF, le modèle de transport de charge suspendu et le mécanisme de glissement de sable. Le modèle numérique est ensuite appliqué pour étudier un affouillent bidimensionnelle cause par un jet immerge provenant d’une ouverture sous écluse. Comparaison des résultats de la simulation avec des données expérimentales prouve la capacité du modèle. Et les limites du modèle sont également discutées. Enfin, le modèle est appliqué à l’étude du champ d’écoulement tridimensionnel et de la formation d’affouillement autour d’un obstacle dans l’écoulement. Tout d’abord, la déformation du lit n’est pas activée. Le tourbillon en fer à cheval devant un obstacle et le champ d’écoulement turbulent autour d’un cylindre sur un lit lisse ou rugueux sont simulés. Deux types de simulation pour le module hydrodynamique sont effectués: une simulation qui utilise une surface fixe et rigide pour représenter l’interface air-eau, et une simulation incluant à la fois les domaines de l’eau et de l’air avec la surface libre suivie par la méthode VOF. Les influences de la surface libre sur le champ d’écoulement sont identifiées et discutées. La comparaison avec les données expérimentales confirme l’importance de la déformation de la surface libre sur le champ d’écoulement. Ensuite, le lit est autorisé à se déformer et le développement temporel de l’affouillement tridimensionnelle autour d’un cylindre sur le lit est simule. Le développement temporel d’affouillement et la profondeur maximale du trou calcule devant et derrière le cylindre conviennent assez bien avec les mesures expérimentales. Les influences de l’affouillement sur le champ d’écoulement sont aussi étudiées et la performance du modèle numérique développé est discutée. / This thesis describes the development of a numerical model for local scour caused by bed-mounted obstacles, combining the hydrodynamic and morphological processes. The basis of the numerical model is the multiphase flow field solver in the open-source CFD toolbox OpenFOAMR which is released by OpenCFD Ltd. The hydrodynamic module of the model solves the Reynolds Averaged Navier-Stokes (RANS) equations with either a k-ε or a k-ω model. There are two interfaces in the simulation domain: the free surface between water and air, which is tracked using the Volume of Fluid (VOF) method, and the interface between the water and the sediment, which is represented by a finite area mesh constructed from the bottom boundary of the finite volume mesh. A morphological module which has been developed as part of the project consists of three components: a sediment transport model which includes suspended load and bed load transport; the Exner equation to compute the bed deformation, and a sand-sliding mechanism to restrict the bed slope angle to be smaller than the angle of repose. The morphological changes are incorporated into the hydrodynamic field through deformation of the computational mesh. Additional boundary conditions and parallel computing corrections are also added into the model. Each individual part of the model has been validated separately with corresponding preliminary test cases including the rough wall functions, the performance of the VOF method, the suspended load transport model and the sand-sliding mechanism. The numerical model is then applied to study two-dimensional scour caused by a submerged jet issuing from an opening under sluice gate. Comparison of the simulation results with the experimental measurements proves the ability of the model for conducting two-dimensional simulations and the limitations of the model are also discussed. Finally, the model is applied to study the three-dimensional flow field and scour formation around an obstacle in flow. Initially, the bed deformation is not activated in the model. The horseshoe vortex formed in front of an obstacle in water and the turbulent flow field around a cylinder on smooth and rough beds are simulated. Two types of simulations for the hydrodynamic module are used: a rigid lid simulation with a slip boundary condition to represent the air-water interface, and a free surface simulation including both the water and air domains with the free surface tracked by the VOF method. The influences of the variation of the water depth on the flow field are identified and discussed. Comparison with the experimental data also confirms the importance of the water surface variation on the flow field. Next, the bed is allowed to deform in the model. The temporal development of three-dimensional scour around a cylinder on live-bed in a steady current is simulated. The development of the scour with time and the computed maximum scour depths in front of and behind the cylinder agree quite well with the experimental measurements. The influences of the scour process on the flow field are also studied and the performance of the numerical model is discussed.

Este trabalho apresenta um método numérico para simular escoamentos viscoelásticos bidimensionais governados pela equação constitutiva Giesekus [Schleiniger e Weinacht 1991]. As equações governantes são resolvidas pelo método de diferenças finitas numa malha deslocada. A superfície livre do fluido é modelada por partículas marcadoras possibilitando assim a sua visualização e localização. O cálculo da velocidade é efetuado por um método implícito enquanto a pressão é calculada por um método explícito. A equação constitutiva de Giesekus é resolvida pelo método de Euler modificado explícito. O método numérico desenvolvido nesse trabalho é verificado comparando-se a solução numérica com a solução analítica para o escoamento de um fluido Giesekus em um canal. Resultados de convergência são obtidos pelo uso de refinamento de malha. Os resultados alcançados incluem um estudo da aplicação do modelo Giesekus para simular o escoamento numa contração planar 4:1 e o problema de um jato incidindo sobre uma placa rígida, em que o fenômeno jet buckling é simulado. / This work presents a numerical method to simulate two-dimensional viscoelastic flows governed by the Giesekus constitutive equation [Schleiniger e Weinacht 1991]. The governing equations are solved by the finite difference method on a staggered grid. The free surface of the fluid is modeled by tracer particles thus enabling its visualization and location. The calculation of the velocity is performed by an implicit method while pressure is calculated by an explicit method. The Giesekus constitutive equation is resolved by the explicit modified Euler method. The numerical method developed in this work is verified by comparing the numerical solution with the analytical solution for the flow of a Giesekus fluid in a channel. Convergence results are obtained by the use of mesh refinement. Results obtained include a study of the application of the Giesekus model to simulate the flow through a 4:1 contraction and the problem of a jet flowing onto a rigid plate where the phenomenon of jet buckling is simulated.

Este trabalho apresenta um ambiente de simulação de escoamentos com simetria radial e superfícies livres, baseado no sistema Freeflow. O sistema é formado por três módulos: um modelador de moldes, um simulador e um visualizador de escoamentos. O simulador implementa o método GENSMAC para a solução das equações de Navier-Stokes em coordenadas cilíndricas, utilizando diferenças finitas em uma malha diferenciada. São introduzidos os efeitos da tensão superficial e do ângulo de contato nas simulações com simetria radial. Alguns resultados de simulações utilizando este sistema e uma validação do código são apresentados, comparando simulações com soluções analíticas e experimentais, e estudando a convergência do método. / This work presents an environment for the simulation of axisymmetric free surface flows, based on the Freeflow system. The system contains three modules: a geometric model modeller, a simulator and a visualizator. The simulator implements the GENSMAC method for the solution of the Navier-Stokes equations in cylindrical coordinates, using finite differences in a staggered grid. The effects of surface tension and contact angle are introduced in the axisymmetric simulations. Some results from simulations using this system and a validation of the code are presented, comparing the simulations with analytical and experimental solutions, and studying the convergence of the method.

Together with the world economic growth is the increasing of energy generation demand. However, the upgrade of world power production capability could affect the environment negatively. Even the clean and renewable sources, such as hydroelectricity and wind powers have socio-economic and environmental disadvantages. For example, the required flooded area for a hydro power plant construction could devastate entire forests, and the installation of a wind farm power plant could affect migratory rotes of birds and generate high levels of noise. Hence, for the balancing of advantages and disadvantages of each power generation source, it is necessary to diversify, which requires investments in new power sources. In this context, the energy generation in the ocean is highlighted. The first point concerning the ocean energy is that there is no need of population removal from the installation area, such as the onshore based methods and the second point is that most of the population is concentrated in coastal areas. Therefore the production occurs near to the demand, decreasing the costs with energy distribution. The two main methodologies for harassing energy from oceans are based on gravity waves and in tides. And since the tidal cycles are governed mainly by the gravitational interaction between oceans, Moon and Sun, they are easily predictable, which increases the reliability of such systems. These works explores methodologies to analyse the power generation from a single axial tidal current turbine through a Steady State RANS methodology. Are discussed the effects of flow directionality, inlet velocity profile and turbulence levels and the results are compared with an experimental scheme. It is proposed an alternative methodology for free surface modelling in the CFD analysis. The usual methodology, VOF, it is based on a homogeneous, biphasic approach which requires an additional mesh refinement and is computationally expensive. This new methodology introduces an elastic wall approach in the free surface region in which the stiffness is calculated to provide the same restoring effect as gravity. In general, the results for open domain matched with the experimental results, validating the numerical model and the confined domain has shown a higher power and thrust coefficients if compared with the open domain, which is in accordance with the actuator disk theory approach. The elastic free surface presented convergence problems related to high Froude numbers and therefore to high deformations. However, a simulation with 10% of the original inlet velocity was performed, achieving reasonable results for both power and thrust coefficients evaluation. / O crescimento econômico mundial e o aumento na demanda pela geração de energia andam juntos. No entanto, uma maior capacidade de produção de energia poderia afetar negativamente o meio ambiente. Mesmo as fontes limpas e renováveis, como a hidrelétrica e a eólica acarretam em impactos socioeconômicos e ambientais. Por exemplo, a construção de uma usina hidrelétrica demanda uma imensa área alagada que pode devastar florestas inteiras e a instalação de uma usina eólica pode afetar a migração de certas espécies de pássaros e produzir altos níveis de barulho. Portanto, para equilibrar as vantagens e desvantagens devidas a cada meio de produção de energia, é necessária a diversificação, que demanda de investimentos em novas fontes. Neste contexto, a geração de energia nos oceanos é destacada. O primeiro ponto a respeito desta fonte é de que não há a necessidade de remoção da população na área de instalação, tal como os métodos de geração dentro do continente. O segundo principal ponto é a respeito da distribuição de energia. A maior parte da população mundial vive em regiões costeiras, diminuindo, portanto, a distância entre a produção e demanda, reduzindo assim, seus custos. As duas principais metodologias para se explorar a energia proveniente dos oceanos são: Energia de Ondas e Energia de Marés. E considerando que os ciclos de mare são governados principalmente pela interação gravitacional entre os oceanos, lua e sol, eles são facilmente previsíveis, o que aumenta a confiabilidade dos sistemas de geração de energia baseados em marés. Este trabalho explora as metodologias para analisar a geração de energia a partir de uma única turbina axial de corrente de maré através de uma metodologia baseada nas equações de Navier-Stokes com a média de Reynolds, analisadas em regime permanente. São discutidos efeitos da direção do escoamento, perfil de velocidades na entrada e nos níveis de turbulência. Os resultados são comparados com experimentos. É proposta uma metodologia alternativa para a modelagem da superfície livre com CFD uma vez que a metodologia atual é baseada em um escoamento bifásico que demanda de um refinamento adicional da malha e é computacionalmente caro. A nova metodologia usa uma parede elástica na região da superfície livre com a rigidez ajustada para se obter o mesmo efeito de restauração que a gravidade. De maneira geral, os resultados para o domínio aberto se aproximaram dos resultados experimentais, validando o modelo numérico e além disso, o modelo considerando confinamento da turbine mostrou maiores valores para os coeficientes de potência e empuxo, estando portanto, de acordo com a teoria do disco atuador. O modelo com a superfície livre elástica apresentou problemas de convergência, relacionados com números de Froude elevados, uma vez que isto se relaciona com maiores deformações na região da superfície livre. Uma simulação com 10% da velocidade original foi realizada, obtendo-se resultados coerentes para ambos coeficientes de potência e empuxo.

Este trabalho tem como objetivo o desenvolvimento de um método numérico para simular escoamentos viscoelásticos tridimensionais com superfícies livres governados pelo modelo constitutivo Giesekus. As equações governantes são resolvidas pelo método de diferenças finitas numa malha deslocada. A superfície livre do fluido é modelada por partículas marcadoras, possibilitando assim a visualização e localização da superfície livre do fluido. A equação constitutiva de Giesekus é resolvida utilizando as seguintes formulações: método de Runge-Kutta de segunda ordem (também conhecido como método de Euler modificado) e transformação logarítmica do tensor conformação. O método numérico apresentado é verificado comparando-se os resultados obtidos por meio de refinamento de malha para os escoamentos em um tubo e de um jato incidindo em uma placa plana. Resultados de convergência foram obtidos por meio de refinamento de malha do escoamento totalmente desenvolvido em um tubo. Os resultados numéricos obtidos incluem a simulação de um jato incidindo em uma caixa vazia e a simulação do inchamento do extrudado (dieswell) para vários números de Weissenberg utilizando diferentes valores do fator de mobilidade do fluido. Resultados adicionais incluem simulações do fenômeno delayed dieswell para altos números de Weissenberg e altos valores do número de Reynolds. Uma comparação qualitativa com resultados experimentais é apresentada. / In this work, a numerical method for simulating viscoelastic free surface flows governed by the Giesekus constitutive equation is developed. The governing equations are solved by the finite difference method on a staggered grid. The fluid free surface is approximated by marker particles which enables the visualization and location of the free surface fluid. The Giesekus constitutive equation is solved by the following techniques: second-order Runge-Kutta, conformation tensor and logarithmic transformation of the conformation tensor. The numerical method is verified by comparing the numerical solutions obtained on a series of embedding meshes of the flow in a tube and by the flow produced by a jet flowing onto a planar surface. Additional verification and convergence results are obtained by solving tube flow employing several meshes. Results obtained include the simulation of a jet flowing into a three dimensional container and the simulation of extrudate swell using several values of the Reynolds and Weissenberg numbers and different values of the mobility parameter a. Furthermore, we present results from the simulation of the phenomenon know as delayed dieswell using highWeissenberg and Reynolds numbers. Comparisons with experimental results are given.

Em inúmeras aplicações práticas da engenharia a camada interfacial entre dois fluidos é de extrema importância para a correta caracterização do escoamento. Especificamente na Engenharia Ambiental e Hidráulica na interface entre o ar e a água existem propriedades capazes de relacionar a aeração e oxigenação. Focado na dinâmica dos processos turbulentos, o estudo proposto se aplica a problemas de sub-pressão em vertedores e visa analisar a troca de gases da interface ar-água em situações de fases dispersas e avaliar a qualidade da água a jusante dos mesmos. O objetivo maior é descrever a relação entre o comportamento da superfície da água, o desenvolvimento da camada limite e os regimes de escoamento de acordo com a vazão volumétrica. Para representar a distribuição da fração de vazios da fase dispersa, foram usados os princípios da formulação dos escoamentos em vertedores escalonados e da formulação para enlaçamento de ar. As argumentações e justificativas das formulações propostas para os perfis de concentrações foram baseadas na distribuição da fração de vazios prevista numericamente. A abordagem numérica do problema fluidodinâmico diferencia da maioria dos estudos concentrados na literatura, por estar baseada na quebra da superfície livre. / In many pratical aplications of engineering the interfacial layer between two fluids has a strong relevance in the process of right measurements of the flow. Espeficically in the enviromental and hydraulics engineering the interface between air and water is the property related with oxigen and areation of the flow. With focus on the dynamics of the turbulent processes, the proposed study is applied to sub-pressure problems in spillways. It investigates the gases exchange in the air- water interface in the dispersed phase and evaluate the water quality downstream. The major goal is to describe a relation between the boundary layer development and the flow regimes with discharge. To represent the distribution of void fractions of the dispersed phase, the physical formulation concepts of stepped spillways and entrapped air were used. The arguments and reasons to justify the proposed concentration profile formulation are based on the numerical provided void fraction distribution. The results as well as the physical model are in good agrement with described literature data. The Numerical approach of the fluid dynamics problem differs from major of described literature studies because it is based on the surface breakup.