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Computation Of Gradually Varied Flow

In the last chapter, we discussed how to qualitatively sketch water-surface profiles in channels having gradually varied flows. For engineering applications, however, it is necessary to compute the flow conditions in these flows. These computations, generally referred to as water-surface profile calculations, determine the water-surface elevations along the channel length for a specified discharge. The water-surface elevations are required for the planning, design, and operation of open channels to assess the effects of various engineering works and channel modifications. The addition of a dam, for example, raises water levels upstream of the dam and it is necessary to know the flow depths in the upstream area to determine the extent of flooding.

In addition, steady-state flow conditions are needed to specify proper initial conditions for the computation of unsteady flows. Improper initial conditions introduce false transients into the simulation, which may lead to incorrect results. Unsteady-flow algorithms may be used directly to determine the initial conditions by continuing the computations until the flow conditions become steady. However, such a procedure is computationally inefficient and may not converge to the proper steady-state solution if the finite-difference scheme is not consistent.

In this chapter, methods to compute gradually varied flows are presented. Preference is given to the methods suitable for a computer solution. Two traditional methods – commonly referred to as the direct and standard step methods – are first presented. The computations progress step by step from one section to the next in these methods. Then, numerical methods to integrate the governing differential equation are introduced. A procedure is then presented that computes the flow conditions at all specified locations of a channel system simultaneously instead of computing them from one section to the next.

Keywords

Euler Method Channel Network Head Loss Channel Cross Section Bottom Slope 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Bakhmeteff, B. A., 1932, Hydraulics of Open Channels, McGraw-Hill Book Co., New York, NY.Google Scholar
  2. Berztiss, A.T., 1971, Data Structures-Theory and Practice, Academic Press, Inc., New York, NY.MATHGoogle Scholar
  3. Boudine, E.J., 1861, De l’axe hydraulique des cours d’eau contenus dans un lit prismatique et des dispositifs realisant, en pratique, ses formes diverses (The flow profiles of water in a prismatic channel and actual dispositions in various forms), Annales des travaux publiques de Belgique, Brussels, vol. 20, 397-555.Google Scholar
  4. Bresse, J.A.C., 1860, Cours de Mecanique Appliquee, Hydraulique, 2e. partie, Mallet-Bachelier, Paris, France.Google Scholar
  5. Chapra, S.C., and Canale, R.P., 1988, Numerical Methods for Engineers, second edition, McGraw-Hill Book Co., New York, NY.Google Scholar
  6. Chaudhry, M. H., and Schulte, A., 1986, “Computation of Steady-State, Gradu-ally Varied Flows in Parallel Channels,” Canadian Jour. of Civil Engineering, vol. 13, no. 1, pp. 39-45.CrossRefGoogle Scholar
  7. Choi, G. W., and Molinas, A. (1993). “Simultaneous Solution Algorithm for Channel Network Modeling. Water Resources Research, vol. 29, no. 2, pp. 321-328.CrossRefGoogle Scholar
  8. Chow, V. T., 1959, Open-Channel Hydraulics, McGraw-Hill Book Co., New York, NY.Google Scholar
  9. Davis, D.W., and Burnham, M.W., 1987, “Accuracy of Computed Water Sur-face Profiles,” Proc., National Hydraulic Engineering Conference, Amer. Soc. Civil Engrs., pp. 818-823.Google Scholar
  10. Deo, N., 1974, Graph Theory with Applications to Engineering and Computer Science, Prentice-Hall, Inc., Englewood Cliffs, NJ.MATHGoogle Scholar
  11. Eichert, Bill S., 1970, “Survey of Programs for Water-Surface Profiles,” Jour. Hyd. Div., Amer. Soc. Civ. Engrs., no. 2, pp. 547-563.Google Scholar
  12. Epp, R., and Fowler, A.G., 1970, “Efficient Code for Steady- State Flows in Networks,” Jour. Hyd. Div., Amer. Soc. Civ. Engrs., vol. 96, no. HY1, Jan., pp. 43-56.Google Scholar
  13. Federal Emergency Management Agency, 1995, “Guidelines and Specifications for Study Contractors,” January.Google Scholar
  14. French, R. H., 1985, Open-Channel Hydraulics, McGraw-Hill Book Co., New York, NY.Google Scholar
  15. Hauck, G.F., and Novak, R.A., 1987, “Interaction of Flow and Incrustation in the Roman Aqueduct of Nimes,” Jour. Hydraulic Engineering, Amer. Soc. Civil Engrs., vol 113, no 2, pp. 141-157.CrossRefGoogle Scholar
  16. Henderson, F. M., 1966, Open-Channel Flow, Macmillan Publishing Co., New York, NY.Google Scholar
  17. Humpidge, H. B., and Moss, W. D., 1971, “The Development of a Comprehensive Computer Program for the Calculation of Flow Profiles in Open Channels,” Proc. Inst. Civ. Engrs., vol. 50, Sept., pp. 49-65.Google Scholar
  18. Kumar, A., 1978, “Integral Solutions of the Gradually Varied Equations for Rectangular and Triangular Channels,” Proc. Inst. Civ. Engrs., vol. 65, pt. 2, Sept., pp. 509-515.Google Scholar
  19. Kumar, A., 1979, “Gradually Varied Surface Profiles in Horizontal and Adversely Sloping Channels,” Proc. Inst. Civ. Engrs., vol. 67, pt. 2, Jun., pp. 435-452.Google Scholar
  20. Kutija, V. (1995). “A Generalized Method for the Solution of Flows in Networks, Jou. Hyd. Research, vol. 33, no. 4, pp. 535-555.Google Scholar
  21. Laurenson, E.M., 1986, “Friction Slope Averaging in Backwater Calculations,” Jour. Hydraulic Engineering, Amer. Soc. Civil Engrs., vol. 112, no. 12, pp. 1151CrossRefGoogle Scholar
  22. McBeans, E., and Perkins, F., 1975, “Numerical Errors in Water Profile Com-putation,” Jour. Hyd. Div., Amer. Soc. Civ. Engrs., vol. 101, no. 11, pp. 1389-1403.Google Scholar
  23. McCracken, D. D. and Dorn, W. S., 1964, Numerical Methods and FORTRAN Programming, John Wiley and Sons, Inc., New york, NY.MATHGoogle Scholar
  24. Molinas, A., and Yang, C.T., 1985, “Generalized Water Surface Profile Compu-tations,” Jour. Hydraulic Engineering, Amer. Soc. Civil Engrs., vol. 111, no. 3, pp. 381-397.CrossRefGoogle Scholar
  25. Naidu, B. J.., Bhallamudi, S. M., and Narasimhan, S. (1997). “GVF Computation in Tree-Type Channel Networks,” Jour. Hyd. Engineering, Amer. Soc. Civ. Engrs., vol. 123, no. 8, pp. 700-708.CrossRefGoogle Scholar
  26. Nguyen, Q. K., and Kawano, H., 1995, “Simultaneous Solution for Flood Rout-ing in Channel Networks,” Jour. Hyd. Engineering, Amer. Soc. Civ. Engrs., vol. 121, no. 10, pp. 744-750.CrossRefGoogle Scholar
  27. Paine, J. N., 1992, “Open-Channel Flow Algorithm in Newton-Raphson Form,” Jour. Irrigation and Drainage Engineering, Amer. Soc. Civ. Engrs., vol. 118, no. 2, pp. 306-319.CrossRefGoogle Scholar
  28. Prasad, R., 1970, “Numerical Method of Computing Flow Profiles,” Jour. Hyd. Div., Amer. Soc. Civ. Engrs., vol. 96, no. 1, pp. 75-86.Google Scholar
  29. Reddy, H. P., and Bhallamudi, S. M., 2004, “Gradually Varied Flow Compu-tation in Cyclic Looped Channel Networks, Jour. Irrigation and Drainage Engineering, Amer. Soc. Civ. Engrs., vol.130, no. 5, pp. 424-431.CrossRefGoogle Scholar
  30. Rhodes, D. G., 1993, Discussion of “Open Channel Flow Algorithm in Newton-Raphson Form, by John N. Paine, Jour. Irrigation and Drainage Engineer-ing, Amer. Soc. Civ. Engrs., vol. 119, no. 5, pp. 914-922.Google Scholar
  31. Rhodes, D. G., 1995, “Newton-Raphson Solution for Gradually Varied Flow, Jour. Hyd. Research, International Assoc. for Hydraulic Research, vol. 33, no. 2, pp. 213-218.Google Scholar
  32. Rouse, H., (ed.) 1950, Engineering Hydraulics, John Wiley & Sons, New York, NY.Google Scholar
  33. Schulte, A. M., and Chaudhry, M.H., 1987, “Gradually Varied Flows in Open Channel Networks,” Jour. of Hydraulic Research , Inter. Assoc. for Hydraulic Research, vol. 25, no. 3, pp. 357-371.Google Scholar
  34. Sen, D. J., and Garg, N.K., 1998, Efficient Solution Technique for Dendritic Channel Networks using FEM,” Jour. Hyd. Engineering, Amer. Soc. Civ. Engrs., vol. 124, no. 8, pp. 831-839.CrossRefGoogle Scholar
  35. Sen, D. J., and Garg, N.K., 2002, “Efficient Algorithm for Gradually Varied Flows in Channel Networks,” Jour. Irrigation and Drainage Engineering, Amer. Soc. Civ. Engrs., vol. 128, no. 6, pp. 351-357.CrossRefGoogle Scholar
  36. Soil Conservation Service, 1976, WSP-2 Computer Program , Technical Release No. 61.Google Scholar
  37. Subramanya, K., 1986, Flow in Open Channels, Tata McGraw-Hill, New Delhi, India, pp. 149-155.Google Scholar
  38. United States Army Corps of Engineers, 1982, HEC-2, Water Surface Profiles, User’s Manual, Hydrologic Engineering Center, Davis, CA. Google Scholar
  39. Unites States Geological Survey,1976,“Computer Applications for Step-Backwater and Floodway Analysis,” Open file Report 76-499.Google Scholar
  40. Wylie, E.B., 1972, “Water Surface Profiles in Divided Channels,” Jour. of Hyd. Research, Inter. Assoc. for Hydraulic Research, vol. 10, no. 3, pp. 325-341.MathSciNetCrossRefGoogle Scholar

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