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A HIGH RESOLUTION FINITE VOLUME METHOD FOR SOLVING SHALLOW WATER EQUATIONS

注意:本論文已在Journal of Hydrodynamics, Ser. B., 2000(1): 35-41發表
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Wang Jia-song   
  School of Civil Engineering and Mechanics, Shanghai JiaoTong University, Shanghai 200030, P. R. China   

Ni Han-gen
School of Civil and Structure Engineering, Dalian University of Technology, Dalian 116024, P. R. China

      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.      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 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 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 .

    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.

6.  APPLICATIONS OF DAM-BREAK COMPUTATION

Through the computation of 1D dam-break waves in a horizontal and frictionless channel and the comparison with Stoker's theoretical solution, it is shown that steep and nonoscillatory numerical solutions could be obtained using the hybrid type of TVD scheme . Two typical examples of 2D dam-break problems are solved and discussed by solving the shallow water equations using above finite volume TVD scheme.

6.1   Rectangular Dam-Break

   Consider a 2D partial dam-break model with a non-symmetrical breach. It is assumed that in the center of a 200m×200m channel, a partial dam breaking takes place instantaneously. The breach is 75m in length, which has distances of 30m from the left bank and 95m from the right. The initial water height is 10m and 5m respectively. No slope and friction are considered. The results displaying the views of the water surface elevation, contour of the surface elevation and velocity field are shown in Figure3 at time t=7.2s after the dam failure. At the instant of breaking of the dam, water is released through the breach, forming a positive wave propagating downstream and a negative wave spreading upstream. These results agree quite well with the results of using finite difference hybrid type of TVD scheme and those in Ref. .

 
Fig. 3(a) Water surface elevation for a rectangular dam-break 

     Fig. 3(b) Contour of surface elevation for a rectangular dam-break

6.2  Circular Dam-Break

    Another typical example is based on the hypothetical test case studied by Alcrudo and Garcia-Navarro [7], which involves the breaking of a circular dam. It is an important test example for the analysis of the algorithm performance and solving a complex shallow water problem. The physical model is that two regions of still water are separated by a cylindrical wall of radius 11m. The water depth inside the dam is 10m, whilst outside the dam is 1m. At the instant of dam failure the circular wall is assumed to be removed completely and no slope and friction is considered, then the circular dam-break waves will spread and propagate radially and symmetrically. The results with above method at time t=0.69s are shown in Figures 4 (a), (b) and (c) which denote the water surface elevation, contour of surface elevation and velocity field respectively. It can be clearly seen that the waves spread uniformly and symmetrically. These results agree quite well with those given by Alcrudo and Garcia-Navarro , Zhao et al. , Alastansiou and Chan and they can be tested each other. It demonstrates that the present method is reliable and fine.


Fig. 3(c) Velocity field for a rectangular dam-break


Fig. 4(a) Water surface elevation for a circular dam-break circular dam-break


Fig. 4(b) Contour of surface elevation for a circular dam-break


Fig. 4(c) Velocity field for a circular dam-break

 

7.      SUMMARY AND CONCLUSIONS

   TVD scheme is playing an important role in gas dynamics because of its high accuracy, good shock-capturing ability and nonoscillatory numerical performance. But it is constructed based on finite difference method. In this paper a new geometry and topology is defined for the extension of nodes to elements. With the conservative type of the shallow water equations, a hybrid type second order TVD scheme is applied and two-step Runge –Kutta method is adopted in time, then a finite volume TVD scheme for the shallow water equations on arbitrary quadrilateral elements is developed. The numerical results of two types of dam-break problem show that the method is sufficiently robust and can handle discontinuities and complex flow problems efficiently. The results presented in this paper are in excellent agree with those reported recently and even display sharper discontinuities and the maximum values attenuate more slowly. It can be foreseen that this method has much broader application foreground. As for further studies, such as in the cases of a channel having bend, bifurcation and inner islands, will discuss in another paper.

REFERENCES

1.          A. Harten, 1983: High Resolution Schemes for Hyperbolic Conservation Laws, Journal of Computational Physics, 49, 357-393.

2.          S. Y. Hu, W. Y. Tan, 1990: Numerical Modeling of Bores due to Dam-Break, Journal of Hydrodynamics, Ser. A., 5(2), 90~98 (in Chinese).

3.          J. H. Tao, W. D. Zhang, 1993: The Simulation of One and Two Dimensional Dam-Breaking Waves by TVNI Scheme, Journal of Tian Jin University, (1), 7~15 (in Chinese).

4.          J. Y. Yang, C. A. Hsu, and S. H. Chang, 1993: Computations of Free Surface Flows, Part 1: 1-D Dam-Break Flow, Journal of Hydraulic Research, 31(1).

5.          J. S. Wang, H. G. Ni, S. Jin and J. C. Li, 1998: Simulation of 1D Dam-Break Flood Wave Routing and Reflection by Using TVD Schemes, Journal of Hydraulic Engineering, (5), 7~11 (in Chinese).

6.          J. S. Wang, H. G. Ni, and S. Jin, 1998: A High Accurate Numerical Simulation of the Propagation and Diffraction for 2D Dam-Break Bores, Journal of Hydraulic Engineering, (10), 1~6 (in Chinese).

7.          F. Alcrudo, P. Garcia-Navarro, 1993: A High Resolution Godunov-Type Scheme in Finite Volumes for the 2D Shallow Water Equation, International Journal for Numerical Method in Fluids, 16, 489-505 1993.

8.          D. H. Zhao, H. W. Shen, J. S. Lai, and G. Q. Tabios , 1996: Approximate Riemann Solvers in FVM for 2D Hydraulic Shock Wave Modeling, Journal of Hydraulic Engineering, 692-702.

9.          K. Alastansiou, C. T. Chan, 1997: Solution of the 2D Shallow Water Equations Using the Finite Volume Method on Unstructured Triangular Meshes, International Journal for Numerical Method in Fluids, 24, 1225-1245.

10.       J. S. Wang, 1998: A Study of Numerical Simulation of Dam-Break Bores by Applying TVD Schemes, Ph.D. Thesis, Dalian University of Technology (in Chinese).


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