Finite-Difference TVD Scheme for Computation of Dam-Break Problems
of Hydraulic Engineering, ASCE, 2000, 126(4), 253-262發表
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
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. 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)
Abott, M. D. (1979). Computational hydraulics. Ashgate publishing Co., Brookfield, Vt.
Alcrudo, F., and Garcia-Navarro, P. (1993). “A high resolution Godunov-type scheme in finite volumes for the 2d shallow water equation.” Int. J. Numer. Methods in Fluids, 16, 489-505.
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, 1225-1245.
Chang, S. C. (1995). “The method of space-time conservation element and solution element: a new approach for solving the Navier-Stokes and Euler equations.” J. Comp. Phys.,119, 295-324.
Chaudhry, M. F. (1993). Open-channel flow. Prentice-Hall, Inc., Engle-wood Cliffs, N.J.
Delis, A. I., and Skeels, C. P. (1998). “TVD schemes for open channel flow.” Int. J. Numer.Methods in Fluids, 26, 791-809.
Fennema, R. J., and Chaudhry, M. H. (1987). “Simulation of 1-d dam-break flows.” J. Hydr Res.,25(1), 41-51.
Fennema, R. J., and Chaudhry, M. H. (1989). “Implicit methods for two-dimensional unsteady free-surface flows.” J. Hydr. Res., 27(3), 321-332.
Garcia, R., and Kahawita, R. A. (1986). “Numerical solution of the St. Venant equations with MacCormack finite-difference scheme.” Int. J. Numer. Methods in Fluids, 6, 259-274.
Garcia-Navarro P., Alcrudo F., and Saviron, J. M. (1992). “1-D open-channel flow simulation using TVD-McCormack scheme.” J. Hydr. Engrg., ASCE, 118(10), 1359-1372.
Glaister, P. (1988). “Approximate Rieman solutions of shallow water equations.” J. Hydr. Res.,26(3), 293-306.
Glaister, P. (1993). “Flux difference splitting for open channel flows.” Int. J. Numer. Methods inInt. J. Numer. Methods in Fluids, 16, 629-654.
Harten, A. (1983). “High resolution schemes for hyperbolic conservation laws.” J. Comp. Phys., 49, 357-393.
Hu, K., Mingham, C. G., and Causon, D. M. (1998). “A bore-capture finite volume method for open-channel flows.” Int. J. Numer. Methods in Fluids, 28, 1241-1261.
Jeng, Y. N., and Payne, U. J. (1995). “An adaptive TVD limiter.” J. Comp. Phys., 118, 229-241.J. Hydr. Engrg
Mahmood, 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). “High-resolution finite-volume method for shallow water flows.” J. Hydr. Engrg., ASCE, 124(6), 605~614.
Molls, T., and Molls, F. (1998). “Space-time conservation method applied to Saint Venant equations.” J. Hydr. Engrg., ASCE, 124(5), 501~508.
Nujic, M. (1995). “Efficient implementation of non-oscillatory schemes for the computation of free-surface 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, 189-196.
Roe, P.L. (1986). “Characteristic-based schemes for the Euler equations.” Anal. Rev. Fluid Mech., 337-365.
Savic, L. J., and Holly Jr., F. M. (1993). “Dambreak flood waves computed by modified Gonunov method.” J. Hydr. Res., 31(2), 187-204.
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, 506-517.
Sweby, P. K (1984). “High resolution schemes using flux limiters for hyperbolic conservation laws.” SIAM J. Numer. Anal., 21, 995-1011.
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 shock-capturing schemes applied to Berger’s equation.” J. Comp. Phys., 102,139-159.
Yang, J.Y., Liu, Y., and Lomax, H. (1987). “Computation of Shock wave reflection by circular cylinder.” AIAA J., 25(5), 636-689.
Yang, J. Y., Hsu, C. A. and Chang, S. H. (1993a). “Computations of free surface flows, Part 1: 1D dam-break flow.” J. Hydr. Res., 31(1), 19-34.J. Hydr. Re
Yee, H. C. (1987). “Construction of explicit and implicit symmetric TVD schemes and their applications.” J. Comp. Phys., 68, 151-179.
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, 692-702.
返回首頁 | CIMS論文 | 并行工程 | 虛擬制造 | 敏捷制造 | 其他論文 | 項目開發 | 學術資源 | 站內全文搜索 | 免費論文網站大全 |