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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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. Zha
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