Computation of Supercritical Free‐Surface Flows
Publication: Journal of Hydraulic Engineering
Volume 114, Issue 4
Abstract
Computational methods for the solution of two‐dimensional shallow‐water equations in steady, supercritical flow are presented. The limitations of these equations and criteria regarding their applicability to the solution of supercritical flows are discussed. Two explicit, shockcapturing, finite‐difference schemes—Lax and MacCormack—are investigated. The boundary conditions along walls require careful attention for a successful implementation of these schemes. Comparison of the numerical and analytical solutions indicate that, with proper treatment of the boundaries, very good agreement can be obtained. Comparison of numerical and experimental results shows that the assumption of hydrostatic pressure distribution imposes restrictions on the utility of the shallow‐water equations to represent the steady, supercritical flow. However, it appears that the basic features of many practical problems may be simulated using the procedures presented herein.
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References
1.
Abbett, M. (1971). “Boundary conditions in computational procedures for inviscid, supersonic steady flow field calculations.” Aerotherm Report 71‐41. Aerotherm Corp., Mt. View, Calif.
2.
Abbott, M. B. (1979). Computational hydraulics; elements of the theory of free surface flow. Pitman Publishing, Ltd., London, England.
3.
Anderson, D., Tannehill, J., and Pletcher, R. (1984). Computational fluid mechanics and heat transfer. Hemisphere Publishing Co. New York, N.Y.
4.
Anderson, J. D. (1982). Modern compressible flow with historical perspective. MacGraw Hill Book Co., New York, N.Y.
5.
Bagge, G., and Herbich, J. B. (1967). “Transitions in supercritical openchannel flow.” J. Hydr. Div., ASCE, 93(5), 23–41.
6.
Cunge, J. (1975). “Rapidly varying flow in power and pumping canals.” Unsteady flow in open channels, K. Mahmood and V. Yevjevich, eds., Water Resources Publications, Littleton, Colo. 539–586.
7.
Dakshinamoorthy, S. (1977). “High velocity flow through expansions.” Proc., 17th Congress IAHR, Baden‐Baden, West Germany, Vol. 2, 373–381.
8.
Demuren, A. O. (1979). “Prediction of steady surface‐layer flows,” thesis presented to the University of London, London, U.K., in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
9.
Ellis, J., and Pender, G. (1982). “Chute spillway design calculations.” Proc. Inst. Civ. Eng., London, England 73(2), 299–312.
10.
Engelund, F., and Munch‐Petersen, J. (1953). “Steady flow in contracted and expanded rectangular channels.” La Houille Blanche, 8(4), 464–474.
11.
Fennema, R., and Chaudhry, M. H. (1986). “Explicit numerical schemes for unsteady free‐surface flows with shocks.” Water Resour. Res., 22(13), 1923–1930.
12.
Flokstra, C. (1977). “The closure problem for depth‐averaged two‐dimensional flow.” Proc., 17th Congress IAHR, Baden‐Baden, West Germany, Vol. 2, 247–256.
13.
Garcia, R., and Kahawita, R. A. (1986). “Numerical solution of the St. Venant equations with the MacCormack finite‐difference scheme.” Int. J. Numer. Methods Fluids, 6(4), 259–274.
14.
Hager, W., and Altinakar, M. S. (1984). “Infinitesimal cross‐wave analysis.” J. Hydr. Engrg., ASCE, 110(8), 1145–1150.
15.
Henderson, F. M. (1966). Open channel flow. Macmillan Publishing Co., Inc., New York. N.Y.
16.
Herbich, J. B., and Walsh, P. (1972). “Supercritical flow in rectangular expansions.” J. Hydr. Div., ASCE, 98(9), 1691–1700.
17.
Ippen, A. T., et al. (1951). “High‐velocity flow in open channels: A symposium,” Trans. ASCE, 116, 265–400.
18.
Ippen, A. T., and Dawson, J. H. (1951). “Design of channels contractions.” Trans. ASCE, 116, 326–346.
19.
Ippen, A. T., and Harleman, D. R. F. (1950). “Studies on the validity of the hydraulic analogy to supersonic flow: parts I and II, USAF Technical Report No. 5985, MIT/U.S. Air Force, Wright‐Patterson Air Force Base, Dayton, Ohio.
20.
Ippen, A. T., and Harleman, D. R. F. (1956). “Verification of theory for oblique standing waves.” Trans. ASCE, 121, 678–694.
21.
Ippen, A. T., and Knapp, R. T. (1938). “Curvilinear flow of liquids with free surfaces at velocities above that of wave propagation.” Proceedings, 5th International Congress of Applied Mechanics, Cambridge, Mass., 531.
22.
Jimenez, O. (1987). “Computation of supercritical flow in open channels,” thesis presented to Washington State University, at Pullman, Wash., in partial fulfillment of the requirements for the degree of Master of Science.
23.
Knapp, R. T. (1951). “Design of channel curves for supercritical flow.” Trans. ASCE, 116, 296–325.
24.
Kutler, P. (1975). “Computation of three‐dimensional, inviscid supersonic flows.” Progress in numerical fluid dynamics, Lecture Notes in Physics No. 41, Springer‐Verlag, New York, N.Y. 287–374.
25.
Kutler, P., and Lomax, H. (1971). “Shock‐capturing, finite‐difference approach to supersonic flows.” J. Spacecr. Rockets, 8(12), 1175–1182.
26.
Lai, C. (1977). “Computer simulation of two‐dimensional unsteady flows in estuaries and embayments by the method of characteristics.” WRI 77‐85, U.S. Geological Survey, Reston, Va.
27.
Lax, P. (1954). “Weak solutions of nonlinear hyperbolic equations and their numerical computation.” Commun. Pure Appl. Math., VII(1), 159–193.
28.
Liggett, J. A. (1975). “Basic equations of unsteady flow.” Unsteady flow in open channels, K. Mahmood and V. Yevjevich, eds., Water Resources Publications, Littleton, Colo. 29–62.
29.
Liggett, J. A., and Vasudev, S. U. (1965). “Slope and friction effects in two dimensional, high speed flow.” Proc., 11th International Congress IAHR, Leningrad, U.S.S.R., Vol. 1, paper 1.25. 1–12.
30.
Moretti, G. (1969). “Importance of boundary conditions in the numerical treatment of hyperbolic equations.” The physics of fluids, Vol. 12, Supplement II, Amer. Inst. of Physics. II‐13–II–20.
31.
Pandolfi, M. (1975). “Numerical experiments on free surface water motion with bores.” Proc. 4th International Conference on Numerical Methods in Fluid Dynamics, Lecture Notes in Physics No. 35, Springer‐Verlag, New York, N.Y.; 304–312.
32.
Preiswerk, E. (1940). “Application of the methods of gas dynamics to water flows with free surface.” Technical Memoranda Nos. 934‐935, National Advisory Committee for Aeronautics Washington, D.C.
33.
Rajar, R., and Cetina, M. (1983). “Two‐dimensional dam‐break flow in steep curved channels.” Proc., XX Congress IAHR, Moscow, U.S.S.R., Vol. II, 571–579.
34.
Roache, P. J. (1972). Computational fluid dynamics. Hermosa Publishers, Albuquerque, N.M.
35.
Rouse, H. (1938). Fluid mechanics for hydraulic engineers. MacGraw‐Hill Book Co., New York, N.Y.
36.
Rouse, H., Bhoota, B. V., and Hsu, E. V. (1951). “Design of channels expansions.” Trans. ASCE, 116, 347–363.
37.
Villegas, F. (1976). “Design of the Punchiná Spillway.” Water Power Dam Constr., 28(11), Nov., 32–34.
38.
Vreugdenhil, C. B. (1982). “Numerical effects in models for river morphology.”in Engineering applications of computational hydraulics, and J. A. Cunge, M. B. Abbott, M. B. eds., Vol. 1, Pitman Publishing Ltd., London, England, 91–110.
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Published In
Journal of Hydraulic Engineering
Volume 114 • Issue 4 • April 1988
Pages: 377 - 395
Copyright
Copyright © 1988 ASCE.
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Published online: Apr 1, 1988
Published in print: Apr 1988
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