Computation of Rapidly Varied Flow

Typical examples of natural and man-made open channels having rapidly varied flows are mountainous streams, rivers during periods of high floods, spillway chutes, conveyance channels, sewer systems, and outlet works. Unlike the case of gradually varied flows, a number of difficulties, such as the formation of roll waves, air entrainment, and cavitation, are encountered in the analysis of these flows. In addition, instabilities may develop if the Froude number exceeds a critical value, giving rise to roll waves or slug flow. Standing wave and large surface disturbances, commonly referred to as shocks or standing waves, are important aspects of rapidly varied flows and need to be considered in the analysis and design.

In this chapter, we present finite-difference methods for the computation of rapidly varied flows. These are shock-capturing methods and do not require any special treatment if a shock develops in the solution. Three different formulations are discussed. The St. Venant equation, also referred to as the shallow-water equations, are assumed to describe these flows in the first two formulations and Boussinesq terms are included in the third to account for nonhydrostatic pressure distribution. The validity of these computational procedures is verified by comparing the computed results with the analytical solutions and with the experimental measurements.


Froude Number Boussinesq Equation Hydraulic Jump Bottom Slope Courant Number 
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  1. Abbett, M., 1971, “Boundary Conditions in Computational Procedures for Invis-cid, Supersonic Steady Flow Field Calculations,” Aerotherm Report 71-41.Google Scholar
  2. Abbott, M. B., 1979, Computational Hydraulics; Elements of the Theory of Free Surface Flow, Pitman Publishing Limited, London.Google Scholar
  3. Abbott, M. B., Marshall, G., and Rodenhuis, G. S., 1969, “Amplitude-Dissipative and Phase-Dissipative Scheme for Hydraulic Jump Simulation,” Proc. 13th Congress, Inter. Assoc. Hyd. Research, Tokyo, vol. 1, Aug., pp. 313-329.Google Scholar
  4. Anderson, D. A., Tannehill, J. D., and Pletcher, R. H., 1984, Computational Fluid Mechanics and Heat Transfer, McGraw-Hill, New York, NY.MATHGoogle Scholar
  5. Bagge, G. and Herbich, J. B., 1967, “Transitions in Supercritical Open-Channel Flow,” Jour. Hydr. Div., Amer. Soc. Civ. Engrs., vol. 93, no. 5, pp. 23-41.Google Scholar
  6. Basco, D. R., 1983, “Introduction to Rapidly-Varied Unsteady, Free-Surface Flow Computation,” Water Resources Investion Report, U.S. Geological Survey, Report No. 83-4284.Google Scholar
  7. Bhallamudi, S. M., and Chaudhry, M. H., 1992, “Computation of Flows in Open-Channel Transitions,” Jour. Hydraulic Research, Inter. Assoc. Hyd. Research, vol. 30, no. 1, pp. 77-93.Google Scholar
  8. Chaudhry, M. H., 1987, Applied Hydraulic Transients, 2nd ed., Van Nostrand Reinhold, New York, NY.Google Scholar
  9. Chow, V. T., 1959, Open Channel Hydraulics, McGraw-Hill Book Co., New York, NY.Google Scholar
  10. Cunge, J., 1975, “Rapidly Varying Flow in Power and Pumping Canals,” in Unsteady Flow in Open Channels, (Eds. Mahmood, K. and Yevjevich, V.), Water Resources Publications, pp. 539-586.Google Scholar
  11. Dakshinamoorthy, S., 1977, “High Velocity Flow through Expansions,” Proc. 17th Congress, Inter. Assoc. Hyd. Research, Baden-Baden, vol.2, pp. 373-381.Google Scholar
  12. Demuren, A. O., 1979, “Prediction of Steady Surface-Layer Flows,” Ph.D. dis-sertation, University of London.Google Scholar
  13. Ellis, J. and Pender G., 1982, “Chute Spillway Design Calculations,” Proc. Inst. Civ. Engrs., Part 2, vol. 73, June, pp. 299-312.Google Scholar
  14. Engelund, F. and Munch-Petersen, J., 1953, “Steady Flow in Contracted and Expanded Rectangular Channels,” La Houille Blanche, vol. 8, no. 4, Aug-Sept, pp. 464-474.Google Scholar
  15. Fennema, R. J. and Chaudhry, M. H., 1986, “Explicit Numerical Schemes for Unsteady Free-Surface Flows with Shocks,” Water Resources Research, vol. 22, no. 13, pp. 1923-1930.CrossRefGoogle Scholar
  16. Fennema, R. J. and Chaudhry, M. H., 1990, “Numerical Solution of Two-Dimensional Transient Free-Surface Flows,” Jour. of Hydr. Eng., Amer. Soc. Civ. Engr., Vol. 116, Aug., pp. 1013-1034.CrossRefGoogle Scholar
  17. Garcia, R. and Kahawita, R. A., 986, “Numerical Solution of the St. Venant Equations with the MacCormacK Finite-Difference Scheme,” Int. Jour. Numer. Meth. in Fluids, vol. 6, pp. 259-274.Google Scholar
  18. Gharangik, A. and Chaudhry, M. H., 1991, “Numerical Simulation of Hydraulic Jump,” Jour. Hydraulic Engineering, Amer. Soc. Civ. Engrs., vol 117, no. 9, pp. 1195-1211.CrossRefGoogle Scholar
  19. Gottlieb, D. and Turkel, E., 1976, “Dissipative Two-Four Methods for Time-Dependent Problems,” Mathematics of Computation, Vol. 30, No. 136, Oct., pp. 703-723.MATHCrossRefMathSciNetGoogle Scholar
  20. Henderson, F. M., 1966, Open Channel Flow, MacMillan, New York, NY.Google Scholar
  21. Herbich, J. B. and Walsh, P., 1972, “Supercritical Flow in Rectangular Expan-sions,” Jour. Hydr. Div., Amer. Soc. Civ. Engrs., vol. 98, no. 9, Sept., pp. 1691-1700.Google Scholar
  22. Ippen, A. T., 1951, et al., Proceedings of a Symposium on High-Velocity Flow in Open Channels, Trans. Amer. Soc. Civ. Engrs., vol. 116, pp. 265-400.Google Scholar
  23. Jameson, A., Schmidt, W., and Turkel, E., 1981, “Numerical Solutions of the Eu-ler equations by Finite Volume Methods Using Runge-Kutta Time-Stepping Schemes,” AIAA 14th Fluid And Plasma Dynamics Conference, Palo Alto, California, AIAA-81-1259.Google Scholar
  24. Jimenez, O. F. and Chaudhry, M. H., 1988, L“Computation of Supercritical Free-Surface Flows,” Jour. of Hydr. Eng., Amer. Soc. Civ. Engr., vol. 114, no. 4, Apr., pp. 377-395.CrossRefGoogle Scholar
  25. Katopodes, N. D., 1984, “A Dissipative Galerkin Scheme for Open-Channel Flow,” Jour. Hyd. Engineering, Amer. Soc. Civ. Engrs., vol. 110, no. 4, April, pp. 450-466.CrossRefGoogle Scholar
  26. Knapp, R. T., 1951, “Design of Channel Curves for Supercritical Flow,” Sym-posium on High-Velocity Flow in Open Channels, Trans. Amer. Soc. Civ. Engrs., vol. 116, pp. 296-325.Google Scholar
  27. Kutler, P.,1975,“Computation of Three-Dimensional, Inviscid Supersonic Flows,” in Progress in Numerical Fluid Dynamics, Lecture Notes in Physics No. 41, Springer-Verlag, pp. 287-374.Google Scholar
  28. Liggett, J. A. and Vasudev, S. U., 1965, L“Slope and Friction Effects in Two Dimensional, High Speed Flow,” Proc. 11th Int. Congress, Inter. Assoc. Hyd. Research, Leningrad, vol. 1, paper 1.25.Google Scholar
  29. MacCormac, R. W., 1969, “The Effect of Viscosity in Hypervelocity Impact Cratering,” Amer. Inst. Aero. Astro., Paper 69-354, Cincinnati, Ohio.Google Scholar
  30. McCorquodale, J. A. and Khalifa, A., 1983, L“Internal Flow in Hydraulic Jumps,” Jour. Hyd. Engineering, Amer. Soc. Civ. Engrs., vol. 109, no. 5, May, pp. 684-701.CrossRefGoogle Scholar
  31. McCowan, A. D., 1987, “The Range of Application of Boussinesq Type Numer-ical Short Wave Models,” Proc. 22nd Congress, Inter. Assoc. Hyd. Research, pp. 378-384.Google Scholar
  32. Pandolfi, M., 1975, “Numerical Experiments on Free Surface Water Motion with Bores,” Proc. 4th Int. Conf. on Numerical Methods in Fluid Dynamics, Lec-ture Notes in Physics No. 35, Springer-Verlag, pp. 304-312.Google Scholar
  33. Roache, P. J., 1972, Computational Fluid Dynamics, Hermosa Publishers, Albuquerque, NM.MATHGoogle Scholar
  34. Stoker, J. J., 1957, Water Waves, Interscience Publishers, New York, NY.MATHGoogle Scholar
  35. Tseng, M. H., Hsu, C. A, and Chu, C. R., 2001, L“Channel Routing in Open-Channel Flows with Surges, Jour. Hyd. Engineering, Amer. Soc. Civ. Engrs., vol. 127, no. 2, pp. 115-122.CrossRefGoogle Scholar
  36. Villegas, F., 1976, “Design of the Punchiná Spillway,” Water Power & Dam Construction, Nov. 1976, pp. 32-34.Google Scholar
  37. Verboom, G. K., Stelling, G. S. and Officier, M. J., 1982, “Boundary Conditions for the shallow water Equations,” Engineering Applications of Computational Hydraulics, Vol. 1, (Abbott, M. B. and Cunge, J. A., eds.), Pitman, Boston.Google Scholar

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