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2016 | Buch

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Finite Element Methods for Incompressible Flow Problems

verfasst von: Volker John

Verlag: Springer International Publishing

Buchreihe : Springer Series in Computational Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book explores finite element methods for incompressible flow problems: Stokes equations, stationary Navier-Stokes equations and time-dependent Navier-Stokes equations. It focuses on numerical analysis, but also discusses the practical use of these methods and includes numerical illustrations. It also provides a comprehensive overview of analytical results for turbulence models. The proofs are presented step by step, allowing readers to more easily understand the analytical techniques.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

The behavior of incompressible fluids is modeled with the system of the incompressible Navier–Stokes equations. These equations describe the conservation of linear momentum and the conservation of mass. In the special case of a steady-state situation and large viscosity of the fluid, the Navier–Stokes equations can be approximated by the Stokes equations. Incompressible flow problems are not only of interest by themselves, but they are part of many complex models for describing phenomena in nature or processes in engineering and industry.

Volker John

Chapter 2. The Navier–Stokes Equations as Model for Incompressible Flows

Abstract

The basic equations of fluid dynamics are called Navier–Stokes equations. In the case of an isothermal flow, i.e., a flow at constant temperature, they represent two physical conservation laws: the conservation of mass and the conservation of linear momentum. There are various ways for deriving these equations. Here, the classical one of continuum mechanics will be outlined. This approach contains some heuristic steps.

Volker John

Chapter 3. Finite Element Spaces for Linear Saddle Point Problems

Abstract

This chapter deals with the first difficulty inherent to the incompressible Navier–Stokes equations, see Remark 2.19, namely the coupling of velocity and pressure. The characteristic feature of this coupling is the absence of a pressure contribution in the continuity equation. In fact, the continuity equation can be considered as a constraint for the velocity and the pressure in the momentum equation as a Lagrangian multiplier. This kind of coupling is called saddle point problem.

Volker John

Chapter 4. The Stokes Equations

Abstract

The Stokes equations model the simplest incompressible flow problems. These problems are steady-state and the convective term can be neglected. Hence, the arising model is linear. Thus, the only difficulty which remains from the problems mentioned in Remark 2.19 is the coupling of velocity and pressure.

Volker John

Chapter 5. The Oseen Equations

Abstract

Oseen equations, which are linear equations, show up as an auxiliary problem in many numerical approaches for solving the Navier–Stokes equations. Applying an implicit method for the temporal discretization of the Navier–Stokes equations requires the solution of a nonlinear problem in each discrete time. Likewise, the steady-state Navier–Stokes equations are nonlinear. Applying in either situation a so-called Picard method (a fixed point iteration) for solving the nonlinear problem, leads to an Oseen problem in each iteration, compare Sect. 6.3 The application of semi-implicit time discretizations to the Navier–Stokes equations leads directly to an Oseen problem in each discrete time, see Remark 7.61. Altogether, Oseen problems have to be solved in many methods for simulating the Navier–Stokes equations. In addition, some parts of the theory of the Oseen equations are used in the analysis of the Navier–Stokes equations, e.g., for the uniqueness of a weak solution of the steady-state Navier–Stokes equations in Theorem 6.20. For these reasons, the analysis and numerical analysis of Oseen problems is of fundamental interest.

Volker John

Chapter 6. The Steady-State Navier–Stokes Equations

Abstract

The steady-state or stationary Navier–Stokes equations describe steady-state flows. Such flow fields can be expected in practice if:

all data of the Navier–Stokes equations (2.25) do not depend on the time,
the viscosity ν is sufficiently large, or equivalently, the Reynolds number Re is sufficiently small,

see Remark 2.22.

Volker John

Chapter 7. The Time-Dependent Navier–Stokes Equations: Laminar Flows

Abstract

The time-dependent Navier–Stokes equations (2.25) were derived in Chapter “The Navier–Stokes Equations as Model for Incompressible Flows” as a model for describing the behavior of incompressible fluids. From the point of view of numerical simulations, one has to distinguish between laminar and turbulent flows. It does not exist an exact definition of these terms. From the point of view of simulations, a flow is considered to be laminar, if on reasonable grids all flow structures can be represented or resolved. In this case, it is possible to simulate the flow with standard discretization techniques in space, like the Galerkin finite element method.

Volker John

Chapter 8. The Time-Dependent Navier–Stokes Equations: Turbulent Flows

Abstract

Usually, the behavior of incompressible turbulent flows is modeled with the incompressible Navier–Stokes equations (2.25). There is no mathematical definition of what is ‘turbulence’. From the mathematical point of view, turbulent flows occur at high Reynolds numbers. From the physical point of view, these flows are characterized by possessing flow structures (eddies, scales) of very different sizes. Consider, e.g., a tornado. This tornado has some very large flow structures (large eddies) but also millions of very small flow structures.

Volker John

Chapter 9. Solvers for the Coupled Linear Systems of Equations

Abstract

Remark 9.1 (Motivation) Many methods for the simulation of incompressible flow problems require the simulation of coupled linear problems for velocity and pressure of the form

$$\displaystyle{ \mathcal{A}\underline{x} = \left (\begin{array}{*{10}c} A& D\\ B &-C \end{array} \right )\left (\begin{array}{*{10}c} \underline{u}\\ \underline{p} \end{array} \right ) = \left (\begin{array}{*{10}c} \underline{f}\\ \underline{f_{ p}} \end{array} \right ) =\underline{ y}, }$$

with

$$\displaystyle\begin{array}{rcl} & & A \in \mathbb{R}^{dN_{v}\times dN_{v} },\ D \in \mathbb{R}^{dN_{v}\times N_{p} },\ B \in \mathbb{R}^{N_{p}\times dN_{v} },\ C \in \mathbb{R}^{N_{p}\times N_{p} }, {}\\ & & \underline{u},\underline{f} \in \mathbb{R}^{dN_{v} },\ \underline{p},\underline{f_{p}} \in \mathbb{R}^{N_{p} }, {}\\ \end{array}$$

such that

$$\displaystyle{\mathcal{A}\in \mathbb{R}^{(dN_{v}+N_{p})\times (dN_{v}+N_{p})},\quad \underline{x},\underline{y} \in \mathbb{R}^{dN_{v}+N_{p} }.}$$

If C = 0, then (9.1) is a linear saddle point problem.

Volker John

Backmatter

Titel: Finite Element Methods for Incompressible Flow Problems
verfasst von: Volker John
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-45750-5
Print ISBN: 978-3-319-45749-9
DOI: https://doi.org/10.1007/978-3-319-45750-5