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Finite-Difference TVD Scheme for Computation of Dam-Break Problems

注意:本論文已在Journal of Hydraulic Engineering, ASCE, 2000, 126(4), 253-262發表
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By J. S. Wang 1 , H. G. Ni 2 and Y. S. He 3

1 Post-Doctoral Fellow, School of Civ. Engrg. and Mech., Shanghai Jiao Tong Univ., Shanghai 200030, P.R.China.
2 Prof., Dept. of Civ. Engrg., Dalian Univ. of Tech., Dalian 116024, P.R.China.
3 Prof., School of Civ. Engrg. and Mech., Shanghai Jiao Tong Univ., Shanghai 200030, P.R.China.

ABSTRACT: A second-order hybrid type of total variation diminishing (TVD) finite-difference scheme is investigated for solving dam-break problems. The scheme is based upon the first-order upwind scheme and the second-order Lax-Wendroff scheme, together with the one-parameter limiter or two-parameter limiter. A comparative study of the scheme with different limiters applied to the Saint Venant equations for 1D dam-break waves in wet bed and dry bed cases shows some differences in numerical performance. An optimum-selected limiter is obtained. The present scheme is extended to 2D shallow water equations by using an operator-splitting technique, which is validated by comparing the present results with the published results, and good agreement is achieved in the case of a partial dam-break simulation. Predictions of complex dam-break bores, including the reflection and interactions for 1D problems and the diffraction with a rectangular cylinder barrier for a 2D problem, are further implemented. The effects of bed slope, bottom friction, and depth ratio of tailwater/reservoir are discussed simultaneously.
Key words: Saint Venant equations, shallow water equations, TVD, finite-difference, dam-break waves, reflection, diffraction, one-parameter limiter, two-parameter limiter

INTRODUCTION

Floods caused by dam failures always lead to a great amount of property damage and loss of human life. Therefore,considerable efforts have been made in the past years to obtain satisfactory solutions for this problem.Mathematically, the dam-break problem is commonly described by the shallow water equations (Also named the SaintVenant equations for the 1D case). One feature of hyperbolic equations of this type is the formation of bores (i.e., therapidly varying discontinuous flow). It is an important basis for validating the numerical method whether the scheme can capture the dam-break bore waves accurately or not. This gives rise to an increasing interest in solving such a problem.

Afrom 1980 to 1990, several finite-difference schemes that handle discontinuities effectively were used to compute open-channel flows, such as the approximate Riemann solver (Glaister 1988) the modified Lax-Friedrich scheme (Rao and Latha 1992, Nujic 1995), the Godunove method (Savic and Holly, 1993). Recently, a space-time conservation

method of Chang (1995) was applied successfully to solve the Saint Venant equations by Molls and Molls (1998). In recent studies, some satisfying results that were applied on a natural channel can be found by using the finite-volume method based on a high-resolution scheme, such as the Godunov method, approximate Riemann solver, etc. (Alcrudo and Garcia-Navarro 1993; Zhao et al. 1996; Anastasiou and Chan 1997; Hu et al. 1998; Mingham and Causon 1998). During the last decade another shock-capturing scheme, the so-called total variation diminishing (TVD) scheme, which was put forward by Harten (1983) and developed by Sweby (1984), Yee (1987) and others, was applied widely in gas dynamics. The main property of this kind of scheme is that it has second-order accuracy, is oscillation-free across discontinuities, and does not require additional artificial viscosity. It began to be applied in hydrodynamics for free- surface flow, in particular, for recent complex dam-break flow. Garcia-Navarro et al. (1992) used TVD-MacCormack scheme to compute open-channel flows, particularly those involving hydraulic jumps and bores. Yang et al. (1993a,b) solved numerically 1D and 2D free-surface flows by using second-order TVD and essentially nonoscillatory schemes. Delis and Skeels (1998) made a comparison with several different TVD schemes (i.e., symmetric, upwind, TVD-MacCormack and MUSCL scheme) to predict 1D dam-break flows.(以下省略,需要全文,請下載)

 

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