Numerical Modeling in Open Channel Hydraulics

verfasst von: Romuald Szymkiewicz

Verlag: Springer Netherlands

Buchreihe : Water Science and Technology Library

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

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Über dieses Buch

Open channel hydraulics has always been a very interesting domain of scienti c and engineering activity because of the great importance of water for human l- ing. The free surface ow, which takes place in the oceans, seas and rivers, can be still regarded as one of the most complex physical processes in the environment. The rst source of dif culties is the proper recognition of physical ow processes and their mathematical description. The second one is related to the solution of the derived equations. The equations arising in hydrodynamics are rather comp- cated and, except some much idealized cases, their solution requires application of the numerical methods. For this reason the great progress in open channel ow modeling that took place during last 40 years paralleled the progress in computer technique, informatics and numerical methods. It is well known that even ty- cal hydraulic engineering problems need applications of computer codes. Thus, we witness a rapid development of ready-made packages, which are widely d- seminated and offered for engineers. However, it seems necessary for their users to be familiar with some fundamentals of numerical methods and computational techniques applied for solving the problems of interest. This is helpful for many r- sons. The ready-made packages can be effectively and safely applied on condition that the users know their possibilities and limitations. For instance, such knowledge is indispensable to distinguish in the obtained solutions the effects coming from the considered physical processes and those caused by numerical artifacts.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Open Channel Flow Equations

Abstract

The chapter begins with the basic definitions and nomenclature used in open channel hydraulics and applied in the book. In the next sections the governing equations for flow and transport are derived, including unsteady gradually varied flow equations, steady gradually varied flow equations, storage equation and advection-diffusion mass and heat transport equations. All equations are derived from general equations of conservation by their simplifications to the one dimensional form. In the last section overview of the types of equations occurring in open channel hydraulics is presented.

Romuald Szymkiewicz

Chapter 2. Methods for Solving Algebraic Equations and Their Systems

Abstract

This chapter presents some basic numerical techniques to solve nonlinear algebraic equations and systems of linear and nonlinear equations. For non-linear algebraic equations the bisection, false position, Newton, fixed point iteration and some hybrid methods are described. Application of these methods is shown for typical open channel problems, like computation of the normal depth, critical depths or the depth of water over sharp-crested weir. Next the standard methods of solution of the system of linear equations are presented. The last section of the chapter is devoted to the solution of systems of non-linear equations, including the Newton and Picard iterative methods.

Romuald Szymkiewicz

Chapter 3. Numerical Solution of Ordinary Differential Equations

Abstract

The initial value problem and the boundary value problem for the ordinary differential equations are discussed in this chapter. Derivation of simple numerical methods as well as a general approach for approximating formulas is described. The basic numerical methods applicable for open channel flow (i.e. for non-uniform grid) are developed. The problem of accuracy and stability including A-stability is discussed. Time integration of the systems of ordinary differential equations arising while solving the partial differential equations using the finite element method is described. A couple examples illustrating application of the presented methods for solving some typical problems of open channel hydraulics are included. At the end of chapter the shooting method and the difference method for solving the boundary value problem are presented.

Romuald Szymkiewicz

Chapter 4. Steady Gradually Varied Flow in Open Channels

Abstract

This chapter begins with derivation of the governing equations. Instead of the ordinary differential equation with regard to depth, commonly used for prismatic channels, the ordinary differential energy equation is proposed. It is showed that the standard step method used for computation of the flow profiles in natural channel is in fact the differential energy equation integrated numerically with the implicit trapezoidal rule. Consequently this approach is applicable for solving the steady varied flow in both prismatic and non-prismatic channels. Analysis of the non-linear equations obtained while solving the energy equation showed that it can have one, two or even three roots. Appropriate choice of the root allows us to obtain all types of the flow profiles occurring in open channels. The same approach based on the energy equation is developed for a single channel as well as for a channel network of both branched and looped types. Several examples illustrate the possibilities of the method.

Romuald Szymkiewicz

Chapter 5. Partial Differential Equations of Hyperbolic and Parabolic Type

Abstract

This chapter is devoted to the partial differential equations applicable in open channel hydraulics, which can be of hyperbolic or parabolic type. The role of characteristics for hyperbolic equations is underlined. The conditions of well posed solution problem for both types of equations are presented. The finite difference and the finite element methods are introduced. The chapter ends with basic information on the convergence, consistency and stability.

Romuald Szymkiewicz

Chapter 6. Numerical Solution of the Advection Equation

Abstract

This chapter presents a number of schemes for solution of 1D advection equation, which are based on the finite difference method, the finite element method and the method of characteristics. The roots of numerical errors in the form of numerical diffusion and dispersion generated in the solution of hyperbolic equations are discussed. The method of analysis and graphical presentation of the numerical properties of applied schemes is described. The chapter ends with basic information on the accuracy analysis using the modified equation approach.

Romuald Szymkiewicz

Chapter 7. Numerical Solution of the Advection-Diffusion Equation

Abstract

In this chapter the numerical consequences of hybrid character of the transport equation leading to advection or diffusion dominated problems are shown. The Peclet number is introduced to distinguish the two cases. Some algorithms for the solution of 1D advection-diffusion equation are presented. They are based on the finite difference method, the finite element method and the splitting technique. The last one allows using the best numerical solvers applied separately for the advective and diffusive parts of transport equation as well as for the part containing source term. Equivalent role of the numerical and physical diffusion in the numerical solution is emphasized.

Romuald Szymkiewicz

Chapter 8. Numerical Integration of the System of Saint Venant Equations

Abstract

This chapter begins with brief review of the numerical methods applicable for the Saint Venant equations. Detailed description of the finite difference Preissmann scheme and of the modified finite element method used for a channel with fixed bed is provided. For both methods stability analysis using the Neumann approach and accuracy analysis using the modified equations approach is carried out. The problem of the boundary condition required at the downstream end is discussed. Some practical aspects of solution the unsteady flow equations are underlined. Application of the Saint Venant equations for particular cases as flow in channel with moveable bed and propagation of the steep waves are presented as well. Numerous examples of solution show the flexibility and large area of application of the Saint Venant equations.

Romuald Szymkiewicz

Chapter 9. Simplified Equations of the Unsteady Flow in Open Channel

Abstract

The system of Saint Venant equations derived in Chapter 1 in the form of Eqs. (1.77) and (1.78) or Eqs. (1.79) and (1.80) is called the dynamic wave model or the complete dynamic model. This model of unsteady open channel flow gives reliable results if the underlying assumptions are satisfied. On the other hand, the Saint Venant model requires rather complex methods of solution and relatively large number of data characterizing both the channel and the flow conditions. For this reasons hydrologists tried to simplify the system of Saint Venant equations to obtain models, which require less input information.

Romuald Szymkiewicz

Backmatter

Titel: Numerical Modeling in Open Channel Hydraulics
verfasst von: Romuald Szymkiewicz
Verlag: Springer Netherlands
Electronic ISBN: 978-90-481-3674-2
Print ISBN: 978-90-481-3673-5
DOI: https://doi.org/10.1007/978-90-481-3674-2