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2016 | Buch

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Finite Element Methods in Incompressible, Adiabatic, and Compressible Flows

From Fundamental Concepts to Applications

verfasst von: Mutsuto Kawahara

Verlag: Springer Japan

Buchreihe : Mathematics for Industry

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

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Über dieses Buch

This book focuses on the finite element method in fluid flows. It is targeted at researchers, from those just starting out up to practitioners with some experience. Part I is devoted to the beginners who are already familiar with elementary calculus. Precise concepts of the finite element method remitted in the field of analysis of fluid flow are stated, starting with spring structures, which are most suitable to show the concepts of superposition/assembling. Pipeline system and potential flow sections show the linear problem. The advection–diffusion section presents the time-dependent problem; mixed interpolation is explained using creeping flows, and elementary computer programs by FORTRAN are included. Part II provides information on recent computational methods and their applications to practical problems. Theories of Streamline-Upwind/Petrov–Galerkin (SUPG) formulation, characteristic formulation, and Arbitrary Lagrangian–Eulerian (ALE) formulation and others are presented with practical results solved by those methods.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

In general, natural phenomena can be described by analog differential equations.

Mutsuto Kawahara

Introduction to Finite Element Methods in Fluid Flows

Frontmatter

Chapter 2. Basic Concepts of Finite Element Method

Abstract

This chapter provides an overview of basic concepts and mathematical foundations of finite element method. Understanding the fundamental concepts of the finite element method is useful not only in structures but also in fluid flows. The most basic concepts of the finite element method can be best explained using a structure composed of springs, which we are going to discuss in this chapter. This spring is one dimensional and it is thus simple to understand the important concepts for the finite element method. The physical aspects of springs are also comprehensive at the intuitive level.

Mutsuto Kawahara

Chapter 3. Pipeline Systems

Abstract

Chapter 2 covered the fundamentals of the finite element method, illustrating the derivation of the global finite element method using a simple spring structure. How to drive the global finite element equation from the local finite element equation is more or less identical across all finite element analyses that apply to structures and fluids. Therefore, it is all the more critical to understand the derivation of the local finite element equation. In essence, then, the bulk of the literature on the finite element method addresses how to derive the local finite element equation. This chapter extends the previous chapter by presenting a case study using a simple pipeline system. We will go through the method to derive its local finite element equation.

Mutsuto Kawahara

Chapter 4. Potential Flow

Abstract

The previous chapters have used one-dimensional second-order differential equation to explain the basic concepts of the finite element method. As we have observed, it is possible to obtain the exact solution in the case of one-dimensional problem in general. Therefore, the advantage of deploying the finite element method may not appear to be so substantial given one-dimensional problems, although it does indeed have an advantage of allowing us to deal with the nonuniform length distributions.

Mutsuto Kawahara

Chapter 5. Advection–Diffusion

Abstract

There are a plethora of important and practical phenomena in science and engineering concerning with advection and diffusion, such as thermal convection in a furnace, chemical substance diffusion in a solution, contaminant dispersion in the environmental waters, salt water diffusion in the sea, CO\(_2\) dispersion in the atmosphere, etc. We must solve these problems not only in space but also in time, because they are time-dependent. The phenomena normally include two different and sometimes contradictory phenomena, i.e., advection and diffusion.

Mutsuto Kawahara

Chapter 6. Creeping Flow

Abstract

As the last topic of the basic concepts of the finite element method in fluid flows, creeping flow problem of the incompressible fluid is discussed in this chapter. The flow is highly viscous and moves with slow velocity, whose acceleration effect is negligible. Thus, the problem turns out to be linear. This type of flow is not practical, i.e., we can only count melting candle flow and few other examples. However, the analysis of the flow includes important technique, namely, the mixed interpolation. The interpolation for velocity is one order or more higher than that for pressure. The incompressibility is assumption of some sort of limit state. Therefore, special treatment should be required. To simplify the problem, we confine our attention to the two-dimensional analysis. However, extension to the three-dimensional analysis is straightforward.

Mutsuto Kawahara

Computational Methods and Applications of Finite Element Method in Fluid Flows

Frontmatter

Chapter 7. Continuum Mechanics of Fluid Flows

Abstract

Although numerous theories in continuum mechanics in fluid flows have been proposed in the past decades, only theories related to the later chapters will be presented in this chapter. First, definition of description and concepts of deformation, displacement, velocity, and acceleration will be discussed. Then, the main fundamental concepts of the continuum mechanics, i.e., the conservation laws of mass, momentum, and energy, are defined.

Mutsuto Kawahara

Chapter 8. Analysis of Incompressible Flows

Abstract

In the past three decades, a number of solution methods have been developed based on the finite element method of incompressible fluid flows.

Mutsuto Kawahara

Chapter 9. Analysis of Adiabatic Flows

Abstract

A large body of studies in the finite element methods have considered that the fluid is incompressible or compressible. In the case of the compressible fluid flow analyses, we must solve the equation of energy in addition to the equations of continuity and momentum. Because this substantially increases computational burdens, we may avoid it if possible.

Mutsuto Kawahara

Chapter 10. Analysis of Compressible Flows

Abstract

It is well known that the compressibility of fluids should be considered to solve high Mach number flows. Compressible flow simulation is important not only in the field of airplane engineering, in which high Mach number flows are treated, but also in the field of wind engineering, in which low Mach number flows are considered.

Mutsuto Kawahara

Chapter 11. ALE Formulation

Abstract

Finite element methods to trace the moving boundary problems may be classified into three categories: (a) Lagrangian, (b) Eulerian, and (c) arbitrary Lagrangian–Eulerian methods, which are schematically shown in Fig. 11.1. In the Lagrangian method, the mesh moves along with fluid particles. The equation and computation become simple. However, mesh is distorted, and as a consequence, sometimes, computation is unstable. In the Eulerian method, the mesh is fixed and does not move until the end of computation of free boundary. However, inventive approaches to trace the free boundary are necessary. The mesh movement does not need to coincide with the movement of fluids; we can assign the mesh independent of the fluid flows.

Mutsuto Kawahara

Backmatter

Titel: Finite Element Methods in Incompressible, Adiabatic, and Compressible Flows
verfasst von: Mutsuto Kawahara
Verlag: Springer Japan
Electronic ISBN: 978-4-431-55450-9
Print ISBN: 978-4-431-55449-3
DOI: https://doi.org/10.1007/978-4-431-55450-9

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