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2012 | Buch

Computational Fluid Dynamics Based on the Unified Coordinates

verfasst von: Wai-How Hui, Kun Xu

Verlag: Springer Berlin Heidelberg

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

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Über dieses Buch

"Computational Fluid Dynamics Based on the Unified Coordinates" reviews the relative advantages and drawbacks of Eulerian and Lagrangian coordinates as well as the Arbitrary Lagrangian-Eulerian (ALE) and various moving mesh methods in Computational Fluid Dynamics (CFD) for one- and multi-dimensional flows. It then systematically introduces the unified coordinate approach to CFD, illustrated with numerous examples and comparisons to clarify its relation with existing approaches. The book is intended for researchers, graduate students and practitioners in the field of Computational Fluid Dynamics.

Emeritus Professor Wai-Hou Hui and Professor Kun Xu both work at the Department of Mathematics of the Hong Kong University of Science & Technology, Hong Kong, China.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

The great majority of research work in CFD, especially those in the first several decades, treats it as numerical solution to nonlinear hyperbolic partial differential equations (PDEs). For a good summary, see Hirsch^[1]. Most part of this monograph also treats CFD as numerical solution to nonlinear hyperbolic PDEs. But it is concerned mainly about the role of coordinates in CFD and, in particular, will base all CFD study on the newly discovered unified coordinates. To put it in perspective we shall first give an overview of the major developments of CFD as numerical solution to the initial value problem of nonlinear hyperbolic PDEs as follows.

Wai-How Hui, Kun Xu

Chapter 2. Derivation of Conservation Law Equations

Abstract

All fluid motions, no matter how complex, are governed by the conservation laws of physics, namely, conservation of mass, momentum and energy. This chapter is devoted to the derivation of these conservation law equations and discussing their mathematical properties.

Wai-How Hui, Kun Xu

Chapter 3. Review of Eulerian Computation for 1-D Inviscid Flow

Abstract

Let σ be a stationary surface of discontinuity and n be a unit normal of σ (Figure 3.1). We take a rectangular volume Ω for which σ cuts across Ω as shown in the figure. Let S ₊ denote the surface of Ω which lies in the positive side of σ, S ₋ that lies in the negative side, and S _l denote the lateral surfaces of Ω.

Wai-How Hui, Kun Xu

Chapter 4. 1-D Flow Computation Using the Unified Coordinates

Abstract

The gas dynamics equations in Eulerian coordinates (t, x) are written in conservation PDE form as

$$ \frac{\partial } {{\partial t}}\left( \begin{gathered} \rho \hfill \\ \rho u \hfill \\ \rho e \hfill \\ \end{gathered} \right) + \frac{\partial } {{\partial x}}\left( \begin{gathered} \rho u \hfill \\ \rho u^2 + p \hfill \\ u\left( {\rho e + p} \right) \hfill \\ \end{gathered} \right) = 0, $$

(4.1)

where

$$ e = \frac{1} {2}u^2 + \frac{1} {{\gamma - 1\rho }}\frac{p} {\rho }. $$

Wai-How Hui, Kun Xu

Chapter 5. Comments on Current Methods for Multi-Dimensional Flow Computation

Abstract

The chief advantage of Eulerian computation is that the control volumes, i.e., the cells, are fixed in space. This facilitates writing computer codes and, if we discretize the integral equation (2.17), we automatically obtain a conservative numerical scheme which ensures correct capturing of shocks.

Wai-How Hui, Kun Xu

Chapter 6. The Unified Coordinates Formulation of CFD

Abstract

We introduce arbitrary coordinates (λ, ξ, η, ζ) via a transformation from Cartesian (t, x, y, z) as follows:

$$ \left\{ \begin{gathered} dt = d\lambda , \hfill \\ dx = Ud\lambda + Ad\xi + Ld\eta + Pd\varsigma , \hfill \\ dy = Vd\lambda + Bd\xi + Md\eta + Qd\varsigma , \hfill \\ dz = Wd\lambda + Cd\xi + Nd\eta + Rd\zeta . \hfill \\ \end{gathered} \right. $$

(6.10

Wai-How Hui, Kun Xu

Chapter 7. Properties of the Unified Coordinates

Abstract

We shall call the system of coordinates (λ, ξ, η, ς) defined in (6.1) unified in the sense that it unifies the Eulerian system when Q = 0 with the Lagrangian when Q = q, and also in the sense that the system of governing equations (6.19) unites the geometrical conservation laws with the physical ones to form a closed system of PDE in conservation form.

Wai-How Hui, Kun Xu

Chapter 8. Lagrangian Gas Dynamics

Abstract

For simplicity we consider 2-D flow, when (6.1) simplifies to

$$ \left\{ \begin{gathered} dt = d\lambda , \hfill \\ dx = Ud\lambda + Ad\xi + Ld\eta , \hfill \\ dy = Vd\lambda + Bd\xi + Md\eta , \hfill \\ \end{gathered} \right. $$

(8.1)

the governing equations (6.19) also reduce to (6.21).

Wai-How Hui, Kun Xu

Chapter 9. Steady 2-D and 3-D Supersonic Flow

Abstract

As shown in Section 7.6, in the case of steady flow, for two of the unified coordinates (λ, ξ, η) to be material coordinates, the mesh velocity must be parallel to the fluid velocity, i.e., (U, V, W) = h(u, v, w).

Wai-How Hui, Kun Xu

Chapter 10. Unsteady 2-D and 3-D Flow Computation

Abstract

To illustrate the idea, we consider 2-D flow. Let us first summarize the results in Chapters 6 and 7.

Wai-How Hui, Kun Xu

Chapter 11. Viscous Flow Computation Using Navier-Stokes Equations

Abstract

In the precedent chapters, we have concentrated on inviscid flow. We now extend the unified coordinates method to viscous flow via the Navier-Stokes equations in this chapter, and via the BGK modeled Boltzmann equation in the next chapter.

Wai-How Hui, Kun Xu

Chapter 12. Applications of the Unified Coordinates to Kinetic Theory

Abstract

Besides the macroscopic governing equations—the Navier-Stokes equations, a fundamentally different approach to describing viscous flow is based on the microscopic particle (molecule) motion—the so-called Boltzmann equation.

Wai-How Hui, Kun Xu

Chapter 13. Summary

Abstract

A system of unified coordinates (UC) has been introduced via transformation (6.1). It has three degrees of freedom — the mesh velocity — and unifies the traditional Eulerian and Lagrangian systems while including them as special cases. Based on (6.1), contributions are made to CFD as follows.

Wai-How Hui, Kun Xu

Backmatter

Titel: Computational Fluid Dynamics Based on the Unified Coordinates
verfasst von: Wai-How Hui
Kun Xu
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-25896-1
Print ISBN: 978-3-642-25895-4
DOI: https://doi.org/10.1007/978-3-642-25896-1