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
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