Approximate Riemann Solvers in FVM for 2D Hydraulic Shock Wave Modeling
Publication: Journal of Hydraulic Engineering
Volume 122, Issue 12
Abstract
This paper presents three approximate Riemann solver schemes, namely: the flux vector splitting (FVS), the flux difference splitting (FDS), and the Osher scheme. Originally used to solve the Euler equations in aerodynamic problems, these Riemann solvers based on the characteristic theory are used in the finite volume method (FVM) for solving the two-dimensional shallow water equations. The three solvers are compared in this paper according to theoretical development, difference schemes, practical applications to shock wave problems, and sensitivity analysis on the computational stability of the methods. The effects of changes in bed elevations on the solutions are also investigated. Comparison of numerical and analytical solutions indicates that very good agreement can be obtained by all three approximate Riemann solvers. Differences in accuracy, computer time, and numerical stability among the three schemes are not significant. For practical purposes, all of them can satisfactorily simulate the hydraulic phenomena in subcritical and supercritical flows as well as in smooth and discontinuous flows, especially shock wave modeling. These solvers are useful for studying levee failure or dam break due to extreme flood events, or the sudden opening or closing of sluice gates in a channel.
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References
1.
Akanbi, A. A., and Katopodes, N. D.(1988). “Model for flood propagation on initially dry land.”J. Hydr. Engrg., ASCE, 114(7), 689–706.
2.
Alcrudo, F., and Garcia-Navarro, P.(1993). “A high-resolution Godunov-type scheme in finite volumes for the 2D shallow-water equations.”Int. J. Numer. Methods in Fluids, 16, 489–505.
3.
Bellos, C. V., Soulis, J. V., and Sakkas, J. G.(1991). “Computation of two-dimensional dam-break induced flows.”Advances in Water Resour., 14(1), 31–41.
4.
Engquist, B. E., Osher, S., and Somerville, R. C. J. (1985). “Large-scale computations in fluid mechanics, Volume 22-Part 1,”Lectures in applied mathematics, Am. Math. Soc., R.I., 58–68.
5.
Fennema, R. T., and Chaudhry, M. H.(1990). “Explicit methods for 2D transient free-surface flows.”J. Hydr. Engrg., ASCE, 116(11), 1013–1014.
6.
Garcia, R., and Kahawita, R. A.(1986). “Numerical solution of the St. Venant equations with the MacCormack finite difference scheme.”Int. J. Numer. Methods in Fluids, 6, 507–527.
7.
Glaister, P.(1988a). “Approximate Riemann solutions of the shallow water equations.”J. Hydr. Res., 26(3), 293–306.
8.
Glaister, P.(1988b). “Flux difference splitting for the Euler equations with axial symmetry.”J. Engrg. Math., 22, 107–121.
9.
Hager, W. H., Schwalt, M., Jimenez, O., and Chaudry, M. H.(1994). “Supercritical flow near an abrupt wall deflection.”J. Hydr. Res., 32(1), 103–118.
10.
Hirsch, C. (1988). Numerical computation of internal and external flows, Vol. 1 . John Wiley & Sons, Inc., New York, N.Y.
11.
Hirsch, C. (1990). Numerical computation of internal and external flows, Vol. 2 . John Wiley & Sons, Inc., New York, N.Y.
12.
Katopodes, N. D., and Strelkoff, R.(1978). “Computing two-dimensional dam-break flood waves.”J. Hydr. Div., ASCE, 104(9), 1269–1288.
13.
Osher, S., and Solomone, F.(1982). “Upwind difference schemes for hyperbolic systems of conservation laws.”Math. Comp., 38, 339–374.
14.
Roe, P. L.(1981). “Approximate Riemann solvers, parameter vectors, and difference schemes.”J. Computational Phys., 43, 357–372.
15.
Savic, L. J., and Holly, F. M. Jr.(1993). “Dambreak flood waves computed by modified Godunov method.”J. Hydr. Res., 31(2), 187–204.
16.
Spekreijse, S. P. (1988). Multigrid solution of steady Euler equations . CWI Tract 46, Amsterdam, The Netherlands.
17.
Steger, J. L., and Warming, R. F.(1981). “Flux vector splitting of the inviscid gas dynamic equations with application to finite-difference methods.”J. Computational Phys., 40, 263–293.
18.
Stoker, J. J. (1957). Water waves . Pure and Appl. Math. Ser., Vol. IV, Wiley-Interscience, New York, N.Y.
19.
Sweby, P. K.(1984). “High resolution schemes using flux limiters for hyperbolic conservation laws.”SIAM J. Num. Anal., 21(5), 995–1005.
20.
Tan, W. (1992). Shallow water hydrodynamics . Elsevier, Amsterdam, The Netherlands.
21.
Van Leer, B.(1974). “Towards the ultimate conservative difference scheme. II: Monotonicity and conservation combined in a second order scheme.”J. Computational Phys., 14, 361–370.
22.
Van Leer, B. (1982). “Flux-vector splitting for the Euler equations.”Proc., 8th Int. Conf. on Numer. Methods in Fluid Dynamics, E. Krause, ed., Springer Verlag, Berlin, Germany, 507–512.
23.
Yang, H. Q.(1992). “A comparative study of advanced shock-capturing schemes applied to Burger's equation.”J. Computational Phys., 102, 139–159.
24.
Zhao, D. H., Shen, H. W., Tabios III, G. Q., Lai, J. S., and Tan, W. Y.(1994a). “A finite volume two-dimensional unsteady flow model for river basins.”J. Hydr. Engrg., ASCE, 120(7), 863–883.
25.
Zhao, D. H., Lai, J. S., and Shen, H. W. (1994b). “2D contaminant transport model with high-resolution upwind schemes.”Proc., Nat. Conf. on Hydr. Engrg., ASCE, New York, N.Y.
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Published In
Journal of Hydraulic Engineering
Volume 122 • Issue 12 • December 1996
Pages: 692 - 702
Copyright
Copyright © 1996 American Society of Civil Engineers.
History
Published online: Dec 1, 1996
Published in print: Dec 1996
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