Zur Approximation der Gleichungen von Navier-Stokes

In der vorliegenden Arbeit betrachten wir die Strömung einer zähen, inkompressiblen, instationären Flüssigkeit in einem dreidimensionalen beschränkten Gebiet, deren Verhalten wird mit den instationären Gleichungen von Navier-Stokes beschrieben. Diese Gleichungen gelten für viele wichtige Strömungsprobleme, beispielsweise für Luftströmungen weit unterhalb der Schallgeschwindigkeit, für Wasserströmungen, sowie für flüssige Metalle. Im zweidimensionalen Fall konnten für die Navier-Stokes-Gleichungen bereits weitreichende Existenz-, Eindeutigkeits- und Regularitätsaussagen bewiesen werden. Im allgemeinen dreidimensionalen Fall, falls also die Daten keinen Kleinheitsannahmen unterliegen, hat man bisher lediglich Existenz und Eindeutigkeit zeitlich lokaler starker Lösungen nachgewiesen. Außerdem existieren zeitlich global so genannte schwache Lösungen, deren Regularität für den Nachweis der Eindeutigkeit im dreidimensionalen Fall allerdings nicht ausreicht. Somit bleibt die Lücke zwischen der zeitlich lokalen, eindeutigen starken Lösung und den zeitlich globalen, nicht eindeutigen schwachen Lösungen der Navier-Stokes-Gleichungen im dreidimensionalen Fall weiterhin offen. Das renommierte Clay Mathematics Institute hat dieses Problem zu einem von sieben Millenniumsproblemen erklärt und für seine Lösung eine Million US-Dollar ausgelobt. In der vorliegenden Arbeit wird ein neues Approximationsverfahren für die Navier-Stokes-Gleichungen entwickelt, das auf einer Kopplung der Eulerschen und Lagrangeschen Beschreibung zäher Strömungen beruht.

A combined BIE-FE method for the Stokes equations

A numerical algorithm for the biharmonic equation in domains with piecewise smooth boundaries is presented. It is intended for problems describing the Stokes flow in the situations where one has corners or cusps formed by parts of the domain boundary and, due to the nature of the boundary conditions on these parts of the boundary, these regions have a global effect on the shape of the whole domain and hence have to be resolved with sufficient accuracy. The algorithm combines the boundary integral equation method for the main part of the flow domain and the finite-element method which is used to resolve the corner/cusp regions. Two parts of the solution are matched along a numerical ‘internal interface’ or, as a variant, two interfaces, and they are determined simultaneously by inverting a combined matrix in the course of iterations. The algorithm is illustrated by considering the flow configuration of ‘curtain coating’, a flow where a sheet of liquid impinges onto a moving solid substrate, which is particularly sensitive to what happens in the corner region formed, physically, by the free surface and the solid boundary. The ‘moving contact line problem’ is addressed in the framework of an earlier developed interface formation model which treats the dynamic contact angle as part of the solution, as opposed to it being a prescribed function of the contact line speed, as in the so-called ‘slip models’. Keywords: Dynamic contact angle; finite elements; free surface flows; hybrid numerical technique; Stokes equations.

Evaluation of a bounded high order upwind scheme for 3D incompressible free surface flow computations

fit the context of normalized variable formulation (NVF) of Leonard and total variation diminishing (TVD) constraints of Harten. this paper presents an extension of it previous work by the authors for solving unsteady incompressible flow problems. The main contributions of the paper are threefold. First, it presents the results of the development and implementation of a bounded high order upwind adaptative QUICKEST scheme in the 3D robust code (Freeflow), for the numerical solution of the full incompressible Navier-Stokes equations. Second, it reports numerical simulation results for 1D hock tube problem, 2D impinging jet and 2D/3D broken clam flows. Furthermore, these results are compared with existing analytical and experimental data. And third, it presents the application of the numerical method for solving 3D free surface flow problems. (C) 2007 IMACS. Published by Elsevier B.V. All rights reserved,

Efficient solutions to robust, semi-implicit discretizations of the immersed boundary method

The immersed boundary method is a versatile tool for the investigation of flow-structure interaction. In a large number of applications, the immersed boundaries or structures are very stiff and strong tangential forces on these interfaces induce a well-known, severe time-step restriction for explicit discretizations. This excessive stability constraint can be removed with fully implicit or suitable semi-implicit schemes but at a seemingly prohibitive computational cost. While economical alternatives have been proposed recently for some special cases, there is a practical need for a computationally efficient approach that can be applied more broadly. In this context, we revisit a robust semi-implicit discretization introduced by Peskin in the late 1970s which has received renewed attention recently. This discretization, in which the spreading and interpolation operators are lagged. leads to a linear system of equations for the inter-face configuration at the future time, when the interfacial force is linear. However, this linear system is large and dense and thus it is challenging to streamline its solution. Moreover, while the same linear system or one of similar structure could potentially be used in Newton-type iterations, nonlinear and highly stiff immersed structures pose additional challenges to iterative methods. In this work, we address these problems and propose cost-effective computational strategies for solving Peskin`s lagged-operators type of discretization. We do this by first constructing a sufficiently accurate approximation to the system`s matrix and we obtain a rigorous estimate for this approximation. This matrix is expeditiously computed by using a combination of pre-calculated values and interpolation. The availability of a matrix allows for more efficient matrix-vector products and facilitates the design of effective iterative schemes. We propose efficient iterative approaches to deal with both linear and nonlinear interfacial forces and simple or complex immersed structures with tethered or untethered points. One of these iterative approaches employs a splitting in which we first solve a linear problem for the interfacial force and then we use a nonlinear iteration to find the interface configuration corresponding to this force. We demonstrate that the proposed approach is several orders of magnitude more efficient than the standard explicit method. In addition to considering the standard elliptical drop test case, we show both the robustness and efficacy of the proposed methodology with a 2D model of a heart valve. (C) 2009 Elsevier Inc. All rights reserved.

Estimativas do erro nas aproximações de Galerkin para as equações de Navier-Stokes

Neste trabalho são provadas algumas estimativas de erro em espaços para as aproximações de Galerkin para a solução do sistema de equações de Navier-Stokes. Mostra-se que o erro decresce em proporção inversa aos autovalores do operador de Stokes.

Alguns aspectos numéricos e teóricos das equações de Navier-Stokes na modelagem do escoamento em torno de um vórtice

Neste trabalho desenvolvemos uma metodologia numérica para a solução do escoamento em torno de um vórtice. Como a análise completa deste tipo de fluxo não é uma tarefa fácil, simplificações quanto ao escoamento e ao método numérico são necessárias. Também investigamos o comportamento das soluções das equações governantes (Navier-Stokes) quando o tempo tende ao infinito. Nesse sentido, dividimos este trabalho em duas partes: uma numérica e outra analítica. Com o intuito de resolver numericamente o problema, adotamos o método de diferenças finitas baseado na formulação incompressível das equações governantes. O método numérico para integrar essas equações é baseado no esquema de Runge- Kutta com três estágios. Os resultados numéricos são obtidos para cinco planos bidimensionais de um vórtice com números de Reynolds variando entre 1000 e 10000. Na parte analítica estudamos taxas de decaimento das soluções das equações de Navier-Stokes quando os dados iniciais são conhecidos. Também estimamos as taxas de decaimento para algumas derivadas das soluções na norma L2 e comparamos com as taxas correspondentes da solução da equação do calor.

A bounded upwinding scheme for computing convection-dominated transport problems

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Evaluation of a bounded high order upwind scheme for 3D incompressible free surface flow computations

fit the context of normalized variable formulation (NVF) of Leonard and total variation diminishing (TVD) constraints of Harten. this paper presents an extension of it previous work by the authors for solving unsteady incompressible flow problems. The main contributions of the paper are threefold. First, it presents the results of the development and implementation of a bounded high order upwind adaptative QUICKEST scheme in the 3D robust code (Freeflow), for the numerical solution of the full incompressible Navier-Stokes equations. Second, it reports numerical simulation results for 1D hock tube problem, 2D impinging jet and 2D/3D broken clam flows. Furthermore, these results are compared with existing analytical and experimental data. and third, it presents the application of the numerical method for solving 3D free surface flow problems. (C) 2007 IMACS. Published by Elsevier B.V. All rights reserved,

A FEM approach to the equations of magneto-aerodynamics

In this work, a Finite Element Method treatment is outlined for the equations of Magnetoaerodynamics. In order to provide a good basis for numerical treatment of Magneto-aerodynamics, a full version of the complete equations is presented and FEM contribution matrices are deduced, as well as further terms of stabilization for the compressible flow case.

Geração algébrica de malhas bidimensionais

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Spurious transients of projection methods in microflow simulations

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Aplicación del método de los volúmenes finitos al cálculo del parámetro de masa añadida de un cilindro vibrante en el seno de un fluido

Máster Universitario en Sistemas Inteligentes y Aplicaciones Numéricas en Ingeniería (SIANI)

Meccanismi di formazione dei tubi di lava

Two analytical models are proposed to describe two different mechanisms of lava tubes formation. A first model is introduced to describe the development of a solid crust in the central region of the channel, and the formation of a tube when crust widens until it reaches the leve\'es. The Newtonian assumption is considered and the steady state Navier- Stokes equation in a rectangular conduit is solved. A constant heat flux density assigned at the upper flow surface resumes the combined effects of two thermal processes: radiation and convection into the atmosphere. Advective terms are also included, by the introduction of velocity into the expression of temperature. Velocity is calculated as an average value over the channel width, so that lateral variations of temperature are neglected. As long as the upper flow surface cools, a solid layer develops, described as a plastic body, having a resistance to shear deformation. If the applied shear stress exceeds this resistance, crust breaks, otherwise, solid fragments present at the flow surface can weld together forming a continuous roof, as it happens in the sidewall flow regions. Variations of channel width, ground slope and effusion rate are analyzed, as parameters that strongly affect the shear stress values. Crust growing is favored when the channel widens, and tube formation is possible when the ground slope or the effusion rate reduce. A comparison of results is successfully made with data obtained from the analysis of pictures of actual flows. The second model describes the formation of a stable, well defined crust along both channel sides, their growing towards the center and their welding to form the tube roof. The fluid motion is described as in the model above. Thermal budget takes into account conduction into the atmosphere, and advection is included considering the velocity depending both on depth and channel width. The solidified crust has a non uniform thickness along the channel width. Stresses acting on the crust are calculated using the equations of the elastic thin plate, pinned at its ends. The model allows to calculate the distance where crust thickness is able to resist the drag of the underlying fluid and to sustain its weight by itself, and the level of the fluid can lower below the tube roof. Viscosity and thermal conductivity have been experimentally investigated through the use of a rotational viscosimeter. Analyzing samples coming from Mount Etna (2002) the following results have been obtained: the fluid is Newtonian and the thermal conductivity is constant in a range of temperature above the liquidus. For lower temperature, the fluid becomes non homogeneous, and the used experimental techniques are not able to detect any properties, because measurements are not reproducible.

Maximum principle preserving high order schemes for convection-dominated diffusion equations

The maximum principle is an important property of solutions to PDE. Correspondingly, it's of great interest for people to design a high order numerical scheme solving PDE with this property maintained. In this thesis, our particular interest is solving convection-dominated diffusion equation. We first review a nonconventional maximum principle preserving(MPP) high order finite volume(FV) WENO scheme, and then propose a new parametrized MPP high order finite difference(FD) WENO framework, which is generalized from the one solving hyperbolic conservation laws. A formal analysis is presented to show that a third order finite difference scheme with this parametrized MPP flux limiters maintains the third order accuracy without extra CFL constraint when the low order monotone flux is chosen appropriately. Numerical tests in both one and two dimensional cases are performed on the simulation of the incompressible Navier-Stokes equations in vorticity stream-function formulation and several other problems to show the effectiveness of the proposed method.