Um problema auxiliar crucial à análise do problema de Stokes Compressível é estudado via a técnica de potenciais de camada dupla em regiões Lipschitz através de um método primeiro utilizado por Verchota e subseqüentemente estendido ao caso parabólico por Brown e Shen. Desse modo, mediante a utilização e cálculo da condição de salto na fronteira é possível estabelecer a existência e unicidade da solução em apropriados espaços funcionais via o estudo de potenciais de camada. / An auxiliary problem crucial to the analysis of the compressible Stokes problem is studied by means of the technique of double layer in Lipschitz regions through a method first used by Verchota and subsequently extended to the parabolic case by Brown and Shen. In this way through the use and calculation of the boundary jump condition it is possible to establish the existence and unicity of the solution in appropriate function spaces via the study of boundary layer potentials.

We consider mathematical modeling of thin film flow between two rough surfaces which are in relative motion. For example such flows take place in different kinds of bearings and gears when a lubricant is used to reduce friction and wear between the surfaces. The mathematical foundations of lubrication theory is given by the Navier--Stokes equation, which describes the motion of viscous fluids. In thin domains several approximations are possible which lead to the so called Reynolds equation. This equation is crucial to describe the pressure in the lubricant film. When the pressure is found it is possible to predict vorous important physical quantities such as friction (stresses on the bounding surfaces), load carrying capacity and velocity field. In hydrodynamic lubrication the effect of surface roughness is not negligible, because in practical situations the amplitude of the surface roughness are of the same order as the film thickness. Moreover, a perfectly smooth surface does not exist in reality due to imperfections in the manufacturing process. Therefore, any realistic lubrication model should account for the effects of surface roughness. This implies that the mathematical modeling leads to partial differential equations with coefficients that will oscillate rapidly in space and time. A direct numerical computation is therefore very difficult, since an extremely dense mesh is needed to resolve the oscillations due to the surface roughness. A natural approach is to do some type of averaging. In this PhD thesis we use and develop modern homogenization theory to be able to handle the questions above. Especially, we use, develop and apply the method based on the multiple scale expansions and two-scale convergence. The thesis is based on five papers (A-E), with an appendix to paper A, and an extensive introduction, which puts these publications in a larger context. In Paper A the connection between the Stokes equation and the Reynolds equation is investigated. More precisely, the asymptotic behavior as both the film thickness <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" /> and wavelength <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmu" /> of the roughness tend to zero is analyzed and described. Three different limit equations are derived. Time-dependent equations of Reynolds type are obtained in all three cases (Stokes roughness, Reynolds roughness and high frequency roughness regime). In paper C we extend the work done in Paper A where we compare the roughness regimes by numeric computations for the stationary case. In paper B we present a mathematical model that takes into account cavitation, surfaces roughness and compressibility of the fluid. We compute the homogenized coefficients in the case of unidirectional roughness.In the paper D we derive a mathematical model of thin film flow between two close rough surfaces, which takes into account cavitation, surface roughness and pressure dependent density. Moreover, we use two-scale convergence to homogenize the model. Finally, in paper E we prove the existence of solutions to a frequently used mathematical model of thin film flow, which takes cavitation into account.

Turbulence is ubiquitous in the flows of fluids and plasmas. This thesis is devoted to studies of the statistical properties of turbulence in the three-dimensional (3D) Hall magnetohydrodynamic (Hall-MHD) equations, the two-dimensional (2D) MHD equations, the one-dimensional (1D) hyperviscous Burgers equation, and the 3D Navier-Stokes equations. Chapter 1 contains a brief introduction to statistically homogeneous and isotropic turbulence. This is followed by an over-view of the equations we study in the subsequent chapters, the motivation for the studies and a summary of problems we investigate in chapters 2-6. In Chapter 2 we present our study of Hall-MHD turbulence [1]. We show that a shell-model version of the 3D Hall-MHD equations provides a natural theoretical model for investigating the multiscaling behaviors of velocity and magnetic structure functions. We carry out extensive numerical studies of this shell model, obtain the scaling exponents for its structure functions, in both the low-k and high-k power-law ranges of 3D Hall-MHD, and find that the extended-self-similarity procedure is helpful in extracting the multiscaling nature of structure functions in the high-k regime, which otherwise appears to display simple scaling. Our results shed light on intriguing solar-wind measurements. In Chapter 3 we present our study of the inverse-cascade regime in two-dimensional magnetohydrodynamic turbulence [2]. We present a detailed direct numerical simulation (DNS) of statistically steady, homogeneous, isotropic, two-dimensional magnetohydrodynamic (2D MHD) turbulence. Our study concentrates on the inverse cascade of the magnetic vector potential. We examine the dependence of the statistical properties of such turbulence on dissipation and friction coefficients. We extend earlier work significantly by calculating fluid and magnetic spectra, probability distribution functions (PDFs) of the velocity, magnetic, vorticity, current, stream-function, and magnetic-vector-potential fields and their increments. We quantify the deviations of these PDFs from Gaussian ones by computing their flatnesses and hyperflatnesses. We also present PDFs of the Okubo-Weiss parameter, which distinguishes between vortical and extensional flow regions, and its magnetic analog. We show that the hyperflatnesses of PDFs of the increments of the stream-function and the magnetic vector potential exhibit significant scale dependence and we examine the implication of this for the multiscaling of structure functions. We compare our results with those of earlier studies. In Chapter 4 we compare the statistical properties of 2D MHD turbulence for two different energy injection scales. We present systematic DNSs of statistically steady 2D MHD turbulence. Our two DNSs are distinguished by kinj, the wave number at which we inject energy into the system. In our first DNS (run R1), kinj = 2 and, in the second (run R2) kinj = 250. We show that various statistical properties of the turbulent states in the runs R1 and R2 are strikingly different The nature of energy spectrum, probability distribution functions, and topological structures are compared for the two runs R1 and R2 are found to be strikingly different. In Chapter 5 we study the hyperviscous Burgers equation for very high α, order of hyperviscosity [3]. We show, by using direct numerical simulations and theory, how, by increasing α in equations of hydrodynamics, there is a transition from a dissipative to a conservative system. This remarkable result, already conjectured for the asymptotic case α →∞ [U. Frisch et al., Phys. Rev. Lett. 101, 144501 (2008)], is now shown to be true for any large, but finite, value of α greater than a crossover value α crossover. We thus provide a self-consistent picture of how dissipative systems, under certain conditions, start behaving like conservative systems, and hence elucidate the subtle connection between equilibrium statistical mechanics and out-of-equilibrium turbulent flows. In Chapter 6 we show how to use asymptotic-extrapolation and Richardson extrapolation methods to extract the exponents ξ p that characterize the dependence of the order-p moments of the velocity gradients on the Reynolds number Re. To use these extrapolation methods we must have high-precision data for such moments. We obtain these high-precision data by carrying out the most extensive, quadruple precision, pseudospectral DNSs of the Navier-Stokes equation.

Fluid dynamics considers the physics of liquids and gases. This is a branch of classical physics and is totally based on Newton's laws of motion. Nevertheless, the equation of fluid motion, Navier-Stokes equation, becomes very complicated to solve even for very simple configurations. This thesis treats mainly analytical vortex solutions to Navier-Stokes equations. Vorticity is usually concentrated to smaller regions of the flow, sometimes isolated objects, called vortices. If one are able to describe vortex structures exactly, important information about the flow properties are obtained. Initially, the modeling of a conical vortex geometry is considered. The results are compared with wind-tunnel measurements, which have been analyzed in detail. The conical vortex is a very interesting phenomenaon for building engineers because it is responsible for very low pressures on buildings with flat roofs. Secondly, a suggested analytical solution to Navier-Stokes equation for internal flows is presented. This is based on physical argumentation concerning the vorticity production at solid boundaries. Also, to obtain the desired result, Navier-Stokes equation is reformulated and integrated. In addition, a model for required information of vorticity production at boundaries is proposed. The last part of the thesis concerns the examples of vortex models in 2-D and 3-D. In both cases, analysis of the Navier-Stokes equation, leads to the opportunity to construct linear solutions. The 2-D studies are, by the use of diffusive elementary vortices, describing experimentally observed vortex statistics and turbulent energy spectrums in stratified systems and in soapfilms. Finally, in the 3-D analysis, three examples of recent experimentally observed vortex objects are reproduced theoretically. First, coherent structures in a pipe flow is modeled. These vortex structures in the pipe are of interest since they appear for Re in the range where transition to turbulence is expected. The second example considers the motion in a viscous vortex ring. The model, with diffusive properties, describes the experimentally measured velocity field as well as the turbulent energy spectrum. Finally, a streched spiral vortex is analysed. A rather general vortex model that has many degrees of freedom is proposed, which also may be applied in other configurations.

<p>Fluid simulating is one of the most difficult problems in computer graphics. On the other hand, water appears in our life very frequently. This thesis focuses on water simulating. We have two main methods to do this in the thesis: the first is wave based water simulating; Sine wave summing based and Fast Fourier Transform based methods are all belong to this part. The other one is physics based water simulating. We make it based on Navier-Stokes Equation and it is the most realistic animation of water. It can deal with the boundary and spray which other method cannot express. Then we put our emphasis on implement by the physics method using Navier-Stokes Equation.</p>

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The work presented here is part of an ongoing effort to develop a highly accurate and numerically efficient turbulence simulation technique. The paper consists of four main parts, viz., the general discussion of the procedure known as Additive Turbulent Decomposition, the derivation of the "synthetic velocity" subgrid-scale model of the high wavenumber turbulent fluctuations necessary for its implementation, the numerical investigation of this model and a priori tests of said models physical validity. Through these investigations we have demonstrated that this procedure, coupled with the use of the "Poor Mans Navier-Stokes" equation subgrid-scale model, has the potential to be a faster, more accurate replacement of currently popular turbulence simulation techniques since: 1. The procedure is consistent with the direct solution of the Navier-Stokes equations if the subgrid-scale model is valid, i.e, the equations to be solved are never filtered, only solutions. 2. Model parameter values are "set" by their relationships to N.S. physics found from their derivation from the N.S. equation and can be calculated "on the fly" with the use of a local high-pass filtering of grid-scale results. 3. Preliminary studies of the PMNS equation model herein have shown it to be a computationally inexpensive and a priori valid model in its ability to reproduce high wavenumber fluctuations seen in an experimental turbulent flow.

Cette thèse porte sur la conception des méthodes robustes de réduction d’ordre de modèles numériques de type Éléments Finis (EF) avec contrôle de la précision. La réduction d’ordre est en général nécessaire pour réduire drastiquement les temps de calcul et permettre ainsi une analyse paramétrique, une étude de faisabilité ou de performance de système (avion, unité de production, procédé complexe, etc). Dans cette étude, la technique de décomposition orthogonale aux valeurs propres (POD) sera utilisée pour construire des modèles réduits locaux. Informatiquement parlant, le “modèle” sera considéré comme une base de données de résultats de calcul avec capacité d’extrapolation et d’interpolation locale. Une stratégie adaptative pour stocker et accéder à la base de données est étudiée en étendant l’algorithme In situ Adaptive Tabulation (ISAT) proposé initialement par Pope. En fonction de l’usage et des exigences en précision des résultats, la base de données est enrichie en ligne (online) par des appels au modèle fin en respectant une précision spécifiée jusqu’à couvrir le domaine paramétrique entier, après quoi l’évaluation d’une solution devient très peu couteuse. L’approche couplée POD-ISAT proposée dans cette thèse fournit une méthode de réduction de modèle EF très performante. La méthodologie est évaluée sur un cas réel de conditionnement d’air en régime stationnaire de cabine d’avion dépendant de plusieurs paramètres de conception (température et vitesse d’entrée d’air, mode de ventilation personnalisée, conductivité thermique du fuselage, etc.). Pour les problèmes d’évolution en temps, nous explorons une piste de chainage de modèles et d’utilisation d’algorithme de parallélisation en temps tel que l’algorithme pararéel initialement proposé par Lions, Maday et Turinici (2001). Nous proposons ici une variante quasi-Newton de l’algorithme pararéel que nous appelons algorithme Broyden-pararéel. Il est appliqué au calcul de la diffusion d’un gaz dans la cabine d’avion. Cette thèse s’insère dans le cadre du projet CSDL (Complex System Design Lab, Fond Unique Interministériel) visant à développer une plate-forme logicielle multidisciplinaire pour la conception de systèmes complexes. / In this thesis, an efficient Reduced Order Modeling (ROM) technique with control of accuracy for parameterized Finite Element solutions is proposed. The ROM methodology is usually necessary to drastically reduce the computational time and allow for tasks like parameter analysis, system performance assessment (aircraft, complex process, etc.). In this thesis, a ROM using Proper Orthogonal Decomposition (POD) will be used to build local models. The “model” will be considered as a database of simulation results store and retrieve the database is studied by extending the algorithm In Situ Adaptive Tabulation (ISAT) originally proposed by Pope (1997). Depending on the use and the accuracy requirements, the database is enriched in situ (i.e. online) by call of the fine (reference) model and construction of a local model with an accuracy region in the parameter space. Once the trust regions cover the whole parameter domain, the computational cost of a solution becomes inexpensive. The coupled POD-ISAT, here proposed, provides a promising effective ROM approach for parametric finite element model. POD is used for the low-order representation of the spatial fields and ISAT for the local representation of the solution in the design parameter space. This method is tested on a Engineering case of stationary air flow in an aircraft cabin. This is a coupled fluid-thermal problem depending on several design parameters (inflow temperature, inflow velocity, fuselage thermal conductivity, etc.). For evolution problems, we explore the use of time-parallel strategies, namely the parareal algorithm originally proposed by Lions, Maday and Turinici (2001). A quasi-Newton variant of the algorithm called Broyden-parareal algorithm is here proposed. It is applied to the computation of the gas diffusion in an aircraft cabin. This thesis is part of the project CSDL (Complex System Design Lab) funded by FUI (Fond Unique Interministériel) aimed at providing a software platform for multidisciplinary design of complex systems.

Submitted by (ana.araujo@ufrpe.br) on 2016-07-07T12:07:15Z No. of bitstreams: 1 Jose Rodrigo Santos Silva.pdf: 4054411 bytes, checksum: c22cd915da573fd5a5ce7b45feb85a0f (MD5) / Made available in DSpace on 2016-07-07T12:07:15Z (GMT). No. of bitstreams: 1 Jose Rodrigo Santos Silva.pdf: 4054411 bytes, checksum: c22cd915da573fd5a5ce7b45feb85a0f (MD5) Previous issue date: 2011-02-18 / The velocity-area method is a standard procedure for measurement of river discharge, with wide application in hydrometric studies, standardized at the international level by the norm ISO 748:2007 of the International Standards Organization. This method requires measurement of velocity at several verticals of the river, at different depths for each vertical. In general, a relatively high number of measurements is necessary do determine the discharge. Recently a technique was proposed which results in a robust estimate of river discharge using a reduced number of measurement points, based on elementary properties of fluid dynamics, stemming from the Navier-Stokes equations, and the use of continuous interpolation between the verticals for calculating velocity across the entire river cross section. In the present work the Monte Carlo Markov Chain (MCMC) method is used to search for the optimum positions for velocity measurement, with the objective of reducing the number of measurement points without significant loss of precision, and therefore maximizing the efficiency of the estimate. A dedicated computer algorithm was developed in C programming language and applied to measurements collected on the river Exu, state of Pernambuco, Brazil, in April 2008. It is found that the discharge estimates with three or more measurement points exhibit variations well within uncertainty limits corresponding to the full 27 point estimate using the traditional velocity-area method. Simulation results indicate that the best positions for velocity measurement are close to the surface, and that significant savings in cost and labor may be accomplished by positioning the measurements at strategic points, without precision loss. / O método área-velocidade é um procedimento utilizado para medir a descarga de rios. Esta é uma técnica bastante difundida na hidrometria, e é normatizada internacionalmente pela ISO 748:2007 da International Standard Organization. Este método requer a medição da velocidade em diversas verticais do rio, e em diferentes profundidades de cada vertical. Em geral é necessário um número relativamente elevado de medições para determinar a vazão. Recentemente foi proposta uma técnica que resulta em uma estimativa robusta da descarga fluvial com reduzido número de pontos de medida, que se baseia nas propriedades básicas da dinâmica de fluidos e nas equações de Navier- Stokes, além de utilizar uma interpolação continua para o cálculo das velocidades em toda a seção vertical. No presente trabalho, o método Monte Carlo Markov Chain (MCMC) é utilizado na busca da melhor posição das medidas de velocidade a serem realizados na seção vertical do rio, tal que seja possível reduzir o número de medições e maximizar a eficiência da estimativa. O algoritmo foi desenvolvido em linguagem C e aplicado em medidas de velocidade colhidas no riacho Exu, no estado de Pernanbuco, em abril de 2008. Estimativas de vazão realizadas a partir de 3 medidas de velocidade sobre a seção vertical mostraram-se eficientes, apresentando diferenças da estimativa obtida com 27 pontos através do método área-velocidade tradicional dentro de limites de incerteza. Os resultados de simulação indicam que os melhores locais de medição da velocidade sob a seção vertical situam-se perto da superfície do rio, e que uma economia significativa no custo e no trabalho pode ser conseguida através posicionamento dos pontos de medição em locais estratégicos, sem perda da precisão da estimativa.

<p>This thesis is a numerical investigation of two-dimensional steady flows past a circular obstacle. In the fluid dynamics research there are few computational results concerning the structure of the steady wake flows at Reynolds numbers larger than 100, and the state-of-the-art results go back to the work of Fornberg (1980) Fornberg (1985). The radial velocity component approaches its asymptotic value relatively slowly if the solution is ``physically reasonable''. This presents a difficulty when using the standard approach such as domain truncation. To get around this problem, in the present research we will develop a spectral technique for the solution of the steady Navier-Stokes system. We introduce the ``bootstrap" method which is motivated by the mathematical fact that solutions of the Oseen system have the same asymptotic structure at infinity as the solutions of the steady Navier-Stokes system with the same boundary conditions. Thus, in the ``bootstrap" method, the streamfunction is calculated as a perturbation to the solution to the Oseen system. Solutions are calculated for a range of Reynolds number and we also investigate the solutions behaviour when the Reynolds number goes to infinity. The thesis compares the numerical results obtained using the proposed spectral ``bootstrap" method and a finite--difference approach for unbounded domains against previous results. For Reynolds numbers lower than 100, the wake is slender and similar to the flow hypothesized by Kirchoff (1869) and Levi-Civita (1907). For large Reynolds numbers the wake becomes wider and appears more similar to the Prandtl-Batchelor flow, see Batchelor (1956).</p> / Doctor of Science (PhD)