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. Then 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. INTRODUCTIONIt 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. (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 Manning’s formula (2)in which n is Manning roughness coefficient.Fig. 1. Geometric and topological relationship 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.[10] 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. However, the elements on land boundaries have only six satellite ones shown in Figure 2. 1. FINITE VOLUME TVD SCHEMEFor 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 . The vector U is assumed constant over an element. Further discretizing (3), the basic equation of the finite volume method can be obtained (4)where is the length of side k, denotes the outer normal flux vector of side k. satisfies (5) F(U) and G(U) have a rotational invariance property, so they satisfy the relation (6)or (7)where represents the angle between unit vector n and the x axis (along the counter-clockwise from the x axis), and denote transformation and inverse transformation matrices respectively (8) Eq. (4) can be rewritten as (9) Let the right terms of above equation be , then (10) Two-step Runge-Kutta method is used to discretize Eq. (10), then the second-order accuracy in time can be obtained (11)The flux at every side of any element (e.g. at the side 1 of element ) can be given through the following form (12)where is the right eigenvector component (l=1,2,3) by Roe‘s average state between the element and the satellite element 1. A hybrid type form of is used (13)where represents the characteristic speed component by Roe‘s average state between element and 1; denotes the average wave strength component; is a limiter. The MUSCL type limiter of Van Leer is used, which has moderate dissipative and compressible performance; is a dissipative function put forward by Harten. The definitions of all these variables are given in Ref.[10]. The ratio between time and space is (14)where denotes the distance of the barycenters between element and satellite element 1. Eqs. (12) and (13) concern four satellite elements around the element , but the limiter function concerns another four satellite elements, so this scheme concerns eight satellite elements in all.5. BOUNDARY CONDITIONS The boundaries of the computational domain have land boundaries (solid boundaries) and water boundaries (open boundaries) for a general shallow water problem. In the case of solid boundaries, no-slip or slip boundary conditions is considered on the basis of whether considering turbulent viscosity or not. Generally speaking, no-slip boundary conditions are given if considering turbulent viscosity, otherwise slip conditions are specified. The open boundary conditions, however, need to have a particular treatment. The local value of Froude number or whether the flow is subcritical or supercritical is the basis of determining the number of boundary conditions. For supercritical flow, three conditions at the inflow boundary and none at the outflow boundary must specified. For subcritical flow, two external conditions are specified at inflow boundary and one is required at the outflow boundary. 1 ，法语论文，法语论文 |
