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# A HIGH RESOLUTION FINITE VOLUME MET

БЯвЕТлЮФ

ABSTRACT: A high-resolution finite volume numerical method for solving the shallow water equations is developed in this paper. In order to extend finite difference TVD scheme to finite volume method, a new geometry and topology of control bodies is defined considering the corresponding the relationships between nodes and elements. This solver is implemented on arbitrary quadrilateral meshes and their satellite elements, and based on a second-order hybrid type TVD scheme in space discretization and a two-step Runge-Kutta method in time discretization. n it is used to deal with two typical dam-break problems and very satisfactory results are obtained comparing with other numerical solutions. It can be considered as an efficient implement for the computation of shallow water problems, especially concerning those having discontinuities, subcritical and supercritical flows and with complex geometries.

KEY WORDS: shallow water equations, finite volume, TVD scheme, dam-break bores

1. INTRODUCTION

It is necessary to conduct fluid flow analyses in many areas, such as in environmental and hydraulic engineering. Numerical method becomes gradually the most important approach. The computation for general shallow water flow problems are successful, but the studies of complex problems, such as having discontinuities, free surface and irregular boundaries are still under development. The analysis of dam-break flows is a very important subject both in science and engineering.

For the complex boundaries, the traditional method has usually involved a kind of body-fitted coordinate transformation system, whilst this may make the original equations become more complicated and sometimes the transformation would be difficult. It is naturally desirable to handle arbitrary complex geometries on every control element without having to use coordinate transformations. For the numerical approach, the general methods can be listed as characteristics, implicit and approximate Riemann solver, etc. The TVD finite difference scheme is playing a peculiar role in such studies , but it is very little in finite volume discretization.

The traditional TVD schemes have different features in the aspects of constructive form and numerical performance. Some are more dissipative and some are more compressive. Through the numerical studies it is shown that good numerical performance and the complicated flow characteristics, such as the reflection and diffraction of dam-break waves can be demonstrated by using a hybrid type of TVD scheme with a proper limiter. In this paper, such type of scheme is extnded to the 2D shallow water equations. A finite volume method on arbitrary quadrilateral elements is presented to solve shallow water flow problems with complex boundaries and having discontinuities.

2. GOVERNING EQUATIONS

The governing equations of shallow water problems can be derived by depth averaging of the Navier-Stokes equations. The conservative form of the shallow water equations is given by

(1a)

where

(1b)

where h is water depth, are the discharges per unit width, bottom slopes and friction slopes along x- and y- directions respectively. The friction slopes and are determined by ManningЁЏs formula

(2)

in n is Manning roughness coefficient.

ЁЁ

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Fig. 1. Geometric and topologicalrelationship between elements

Fig. 2 Relationship between elements on land boundaries

3. GEOMETRICAL AND TOPOLOGICAL RELATIONSHIPS OF ELEMENTS

The second-order TVD schemes belong to five-point finite difference scheme and the unsolved variables are node-node arrangement. In order to extend them to the finite volume method, it is necessary to define the control volume. The types of traditional control volume have element itself, such as triangle, quadrilateral and other polygons or some kinds of combinations, and polygons made up of the barycenters from the adjacent elements. In this paper we consider that a node corresponds to an element and the middle states between two conjunction nodes correspond to the interface states of public side between two conjunction elements. A new geometrical and topological relationship is presented for convenience to describe and utilize the TVD scheme. An arbitrary quadrilateral element is defined as a main element and the eight elements surrounding this main element are named as satellitic elements. If the number of all the elements and nodes is known, the topological relations between the main elements and the satellite ones can be predetermined (see Ref. in detail). Then the numerical fluxes of all the sides of the main element can be determined. The relationships between the main and the satellite elements are shown in Figure 1. , the elements on land boundaries have only six satellite ones shown in Figure 2.

1. FINITE VOLUME TVD SCHEME

For the element , the integral form of equation (1a) for the inner region and the boundary can be written as

(3)

where A represents the area of the region , dl denotes the arc length of the boundary , and n is a unit outward vector normal to the boundary .



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