Advances in Calculation Methods for Supercritical Flow in Spillway Channels
Publication: Journal of Hydraulic Engineering
Volume 125, Issue 10
Abstract
A calculation method is presented for applications to steady supercritical and transcritical flow in spillway channels. The method solves the two-dimensional nonlinear shallow water equations using a cell-centered finite-volume approach. High spatial resolution of shock waves and other steep flow features is achieved by employing MUSCL reconstruction and an approximate Riemann solver for the flux evaluations at each cell interface. The method can be implemented on boundary-conforming meshes to more accurately map the wide range of geometries that may occur in practice. Six analytical test problems are proposed for the validation of calculation methods applied to steady supercritical flow. These problems are used to validate the proposed flow solver, which is then applied to the case of steady supercritical flow in a curved channel transition, and comparisons are made with published data. Despite limitations in the shallow water model, the results show satisfactory agreement with data for the maximum rise in water level through the standing oblique shock waves.
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
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 and Fluids, 16, 489–505.
2.
Barber, R. ( 1992). “Solving the shallow water equations using a non-orthogonal curvilinear coordinate system.” Hydraulic and environmental modelling: Coastal waters, R. Falconer, ed., Ashgate, Aldershot, U.K., 469–480.
3.
Batten, P., Lambert, C., and Causon, D. (1996). “Positively conservative high resolution convection schemes for unstructured elements.” Int. J. Numer. Methods in Engrg., 39(11), 1821–1838.
4.
Bellos, C., Soulis, J., and Sakkas, J. (1991). “Computation of two dimensional dam break induced flows.” Advances in Water Resour., 14(1), 31–41.
5.
Berger, R., and Stockstill, R. (1993). “A 2d numerical model for high velocity channels.” Hydraulic Engineering 93, Proc., 1993 Nat. Hydr. Conf., H. Shen, S. Su, and F. Wen, eds., New York, 1085–1090.
6.
Borthwick, A., and Barber, R. (1992). “River and reservoir flow modelling using the transformed shallow water equations.” Int. J. Numer. Methods in Fluids, 14, 1193–1217.
7.
Ellis, J., and Pender, G. (1982). “Chute spillway design calculations.” Proc., Instn. Civ. Engrs., Part 2, ICE, London, 73, 299–312.
8.
Falconer, R. (1980). “Numerical modeling of tidal circulation in harbors.”J. Wtrwy., Port, Coast., and Oc. Div., ASCE, 106, 31–48.
9.
Fennema, R., and Chaudhry, M. (1990). “Explicit methods for 2D transient free surface flows.”J. Hydr. Engrg., ASCE, 116(11), 1013–1014.
10.
Garcia, R., and Kahawitha, R. (1986). “Numerical solution of the St. Venant equations with the Maccormack finite difference scheme.” Int. J. Numer. Methods in Fluids, 6, 507–527.
11.
Gharangik, A., and Chaudhry, M. (1991). “Numerical simulation of hydraulic jump.”J. Hydr. Engrg., ASCE, 117(9), 1195–1211.
12.
Hager, W. (1989). “Supercritical flow in channel junctions.”J. Hydr. Engrg., ASCE, 115(5), 595–616.
13.
Hager, W., Schwalt, M., Jimenez, O., and Chaudhry, M. (1994). “Supercritical flow near an abrupt wall deflection.”J. Hydr. Res., 32(1), 103–118.
14.
Harten, A., Lax, P., and van Leer, B. (1983). “On upstream differencing and Godunov-type schemes for hyperbolic conservation laws.” SIAM Rev., 25(1), 35–61.
15.
Herbich, J., and Walsh, P. (1972). “Supercritical flow in rectangular expansions.”J. Hydr. Div., ASCE, 98(9), 1691–1700.
16.
Hinds, J. (1920). “The hydraulic jump and critical depth in the design of hydraulic structures.” Engrg. News-Rec., 85(22), 1034–1040.
17.
Ippen, A. (1951). “Mechanics of supercritical flow.” Trans. ASCE, 116, 268–295.
18.
Ippen, A., and Dawson, J. (1951). “Design of channel contractions: High velocity flow in open channels (symposium).” Trans. ASCE, 116, 326–346.
19.
Ippen, A., and Harleman, D. (1956). “Verification of theory for oblique standing waves.” Trans. ASCE, 121, 678–694.
20.
Ippen, A., and Knapp, R. (1936). “A study of high velocity flow in curved channels.” Trans. Am. Geophys. Union, Part III, 17, 516–521.
21.
Mingham, C., and Causon, D. (1998a). “Calculation of unsteady bore diffraction using a high-resolution finite volume method.”J. Hydr. Res. (in press).
22.
Mingham, C., and Causon, D. (1998b). “High-resolution finite-volume method for shallow water flows.”J. Hydr. Engrg., ASCE, 124(6), 605–614.
23.
Pearson, R., Causon, D., and Mingham, C. ( 1997). “Simulation of coastal and estuarine hydrodynamics using a high resolution finite volume technique on a quadtree cartesian mesh.” Computer modelling of seas and coastal regions III, J. Acinas and C. Brebbia, eds., Computational Mechanics Publications, Southampton, U.K., 23–32.
24.
Preiswerk, E. ( 1938). “Anwendung gasdynamischer Methoden auf Wasserstroemungen auf freier Oberflaeche,” PhD thesis, ETH, Zurich, Switzerland.
25.
Reinauer, R., and Hager, W. (1998). “Supercritical flow in chute contraction.”J. Hydr. Engrg., ASCE, 124(1), 55–64.
26.
Terzidis, G., and Strelkoff, T. (1970). “Computation of open-channel surges and shocks.”J. Hydr. Div., ASCE, 96(12), 2581–2610.
27.
van Leer, B. (1984). “On the relation between the upwind-differencing schemes of Godunov, Engquist-Osher and Roe.” SIAM J. Scientific and Statistical Computing, 5(1), 1–20.
28.
Von Karman, T. (1938). “A practical application of the analogy between supersonic flow in gases and supercritical flow in open channels.” ZAMM, 18(1), 49–56.
29.
Yang, J., and Hsu, C. (1993). “Computation of free surface flows.” J. Hydr. Res., 31(3), 403–413.
30.
Zhao, D., Shen, H., Lai, J., and Tabios, G. (1996). “Approximate Riemann solvers in FVM for 2D hydraulic shock wave modeling.”J. Hydr. Engrg., ASCE, 122(12), 692–702.
31.
Zhao, D., Shen, H., Tabios, G., Lai, J., and Tan, W. (1994). “Finite volume two dimensional unsteady flow model for river basins.”J. Hydr. Engrg., ASCE, 120(7), 863–883.
Information & Authors
Information
Published In
Journal of Hydraulic Engineering
Volume 125 • Issue 10 • October 1999
Pages: 1039 - 1050
History
Received: Nov 3, 1998
Published online: Oct 1, 1999
Published in print: Oct 1999
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.