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FiniteDifference TVD Scheme for Computation of DamBreak Problems ע⣺ՓJournal
of Hydraulic Engineering, ASCE, 2000, 126(4), 253262l By J. S. Wang 1 , H. G. Ni 2 and Y. S. He 3 1 PostDoctoral Fellow,
School of Civ. Engrg. and Mech., Shanghai Jiao Tong Univ., Shanghai
200030, P.R.China. ABSTRACT:
A secondorder hybrid type of total variation diminishing (TVD)
finitedifference scheme is investigated for solving dambreak problems.
The scheme is based upon the firstorder upwind scheme and the secondorder
LaxWendroff scheme, together with the oneparameter limiter or
twoparameter limiter. A comparative study of the scheme with different
limiters applied to the Saint Venant equations for 1D dambreak
waves in wet bed and dry bed cases shows some differences in numerical
performance. An optimumselected limiter is obtained. The present
scheme is extended to 2D shallow water equations by using an operatorsplitting
technique, which is validated by comparing the present results with
the published results, and good agreement is achieved in the case
of a partial dambreak simulation. Predictions of complex dambreak
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. 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 dambreak 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 dambreak bore waves accurately or not. This gives rise to an increasing interest in solving such a problem. Afrom 1980 to 1990, several finitedifference schemes that handle discontinuities effectively were used to compute openchannel flows, such as the approximate Riemann solver (Glaister 1988) the modified LaxFriedrich scheme (Rao and Latha 1992, Nujic 1995), the Godunove method (Savic and Holly, 1993). Recently, a spacetime 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 finitevolume method based on a highresolution scheme, such as the Godunov method, approximate Riemann solver, etc. (Alcrudo and GarciaNavarro 1993; Zhao et al. 1996; Anastasiou and Chan 1997; Hu et al. 1998; Mingham and Causon 1998). During the last decade another shockcapturing scheme, the socalled 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 secondorder accuracy, is oscillationfree 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 dambreak flow. GarciaNavarro et al. (1992) used TVDMacCormack scheme to compute openchannel flows, particularly those involving hydraulic jumps and bores. Yang et al. (1993a,b) solved numerically 1D and 2D freesurface flows by using secondorder TVD and essentially nonoscillatory schemes. Delis and Skeels (1998) made a comparison with several different TVD schemes (i.e., symmetric, upwind, TVDMacCormack and MUSCL scheme) to predict 1D dambreak flows.(ʡҪȫՈd)
REFERENCES Abott, M. D. (1979). Computational hydraulics. Ashgate publishing Co., Brookfield, Vt. Alcrudo, F., and GarciaNavarro, P. (1993). A high resolution Godunovtype scheme in finite volumes for the 2d shallow water equation. Int. J. Numer. Methods in Fluids, 16, 489505. Anastansiou, K., and Chan, C.T. (1997). Solution of the 2d shallow water equations using the finite volume method on unstructured triangular meshes. Inter. J. Numer. Meth. in Fluids,24, 12251245. Chang, S. C. (1995). The method of spacetime conservation element and solution element: a new approach for solving the NavierStokes and Euler equations. J. Comp. Phys.,119, 295324. Chaudhry, M. F. (1993). Openchannel flow. PrenticeHall, Inc., Englewood Cliffs, N.J. Delis, A. I., and Skeels, C. P. (1998). TVD schemes for open channel flow. Int. J. Numer.Methods in Fluids, 26, 791809. Fennema, R. J., and Chaudhry, M. H. (1987). Simulation of 1d dambreak flows. J. Hydr Res.,25(1), 4151. Fennema, R. J., and Chaudhry, M. H. (1989). Implicit methods for twodimensional unsteady freesurface flows. J. Hydr. Res., 27(3), 321332. Garcia, R., and Kahawita, R. A. (1986). Numerical solution of the St. Venant equations with MacCormack finitedifference scheme. Int. J. Numer. Methods in Fluids, 6, 259274. GarciaNavarro P., Alcrudo F., and Saviron, J. M. (1992). 1D openchannel flow simulation using TVDMcCormack scheme. J. Hydr. Engrg., ASCE, 118(10), 13591372. Glaister, P. (1988). Approximate Rieman solutions of shallow water equations. J. Hydr. Res.,26(3), 293306. Glaister, P. (1993). Flux difference splitting for open channel flows. Int. J. Numer. Methods inInt. J. Numer. Methods in Fluids, 16, 629654. Harten, A. (1983). High resolution schemes for hyperbolic conservation laws. J. Comp. Phys., 49, 357393. Hu, K., Mingham, C. G., and Causon, D. M. (1998). A borecapture finite volume method for openchannel flows. Int. J. Numer. Methods in Fluids, 28, 12411261. Jeng, Y. N., and Payne, U. J. (1995). An adaptive TVD limiter. J. Comp. Phys., 118, 229241. J. Hydr. EngrgMahmood, K., and Yevjevich, V., eds. (1979). Unsteady flow in open channels. Water Resources Publications, Fort Collins, CO. Mingham, C. G., and Causon, D.M. (1998). Highresolution finitevolume method for shallow water flows. J. Hydr. Engrg., ASCE, 124(6), 605~614. Molls, T., and Molls, F. (1998). Spacetime conservation method applied to Saint Venant equations. J. Hydr. Engrg., ASCE, 124(5), 501~508. Nujic, M. (1995). Efficient implementation of nonoscillatory schemes for the computation of freesurface flows. J. Hydr. Res., 33(1), 101~111. Rao, V.S., and Latha, G. (1992). A slope modification method for shallow water equations. Int. J. Numer. Methods in Fluids, 14, 189196. Roe, P.L. (1986). Characteristicbased schemes for the Euler equations. Anal. Rev. Fluid Mech., 337365. Savic, L. J., and Holly Jr., F. M. (1993). Dambreak flood waves computed by modified Gonunov method. J. Hydr. Res., 31(2), 187204. Stoker, J. J. (1957). Wat e r Wav e s , Interscience Publications, Wiley, New York. Strong, G. (1968). On the construction and comparison of Difference Scheme. SIAM J. Numer. Anal.5, 506517. Sweby, P. K (1984). High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal., 21, 9951011. Tan, W. Y. (1992). Shallow water hydraulics. Elevier Publishing Co., New York, N.Y. Yang, H. Q., and Przekwas, A. J. (1992). A Comparative study of advanced shockcapturing schemes applied to Bergers equation. J. Comp. Phys., 102,139159. Yang, J.Y., Liu, Y., and Lomax, H. (1987). Computation of Shock wave reflection by circular cylinder. AIAA J., 25(5), 636689. Yang, J. Y., Hsu, C. A. and Chang, S. H. (1993a). Computations of free surface flows, Part 1: 1D dambreak flow. J. Hydr. Res., 31(1), 1934. J. Hydr. ReYee, H. C. (1987). Construction of explicit and implicit symmetric TVD schemes and their applications. J. Comp. Phys., 68, 151179. Zhao, D. H., Shen, H. W., Lai, J. S., and Tabios, G. Q. (1996). Approximate Riemann solvers in FVM for 2D hydraulic shock wave modeling. J. Hydr. Engrg., ASCE, 122, 692702.
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