End Depth in Circular Channels
Publication: Journal of Hydraulic Engineering
Volume 124, Issue 8
Abstract
The flow upstream of a free overfall from smooth circular channels is theoretically analyzed to calculate the end-depth ratio (EDR), applying the momentum equation based on the Boussinesq approximation. The present approach eliminates the need for an empirical pressure coefficient. In subcritical approaching flows, the EDR is related to the critical depth, which occurs upstream of the end section, and the value of EDR is found to be around 0.75 for a critical depth-diameter ratio up to 0.82. On the other hand, in supercritical approaching flows, the end depth is expressed as a function of the streamwise slope of the channel using the Manning formula. Simple methods are presented to estimate the discharge from the end depth in subcritical and supercritical approaching flows. A relationship of discharge to the end depth and the channel characteristics parameter is also proposed. Streamline curvature at the free surface is used to compute the flow profiles upstream of a free overfall. This paper also presents a theoretical model to analyze the free overfall from horizontally laid rough circular channels with the aid of an autorecursive search scheme.
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References
1.
Ali, K. H. M., and Sykes, A.(1972). “Free-vortex theory applied to free overfall.”J. Hydr. Div., ASCE, 98(5), 973–979.
2.
Anderson, M. V. (1967). “Non-uniform flow in front of a free overfall.”Acta Polytech. Scand., Civ. Engrg. and Construction Series 42, 1–24.
3.
ASCE Task Force on Friction Factors in Open Channels.(1963). “Friction factor in open channel.”J. Hydr. Div., ASCE, 89(2), 97–143.
4.
Chow, V. T. (1959). Open channel hydraulics. McGraw-Hill Inc., New York, N.Y.
5.
Chow, W. L., and Han, T. (1979). “Inviscid solution for the problem of the free overfall.”J. Appl. Mech., 46(Mar.), 1–5.
6.
Clarke, N. S.(1965). “On two-dimensional inviscid flow in a waterfall.”J. Fluid Mech., Cambridge, U.K., 22, 359–369.
7.
Conte, S. D., and de Boor, C. (1987). Elementray numerical analysis: an algorithmic approach. McGraw-Hill Inc., New York, N.Y.
8.
Delleur, J. W., Dooge, J. C. I., and Gent, K. W.(1956). “Influence of slope and roughness on the free overfall.”J. Hydr. Div., ASCE, 82(4), 30–35.
9.
Dey, S. (1998). “Free overfall in rough rectangular channels: a computational approach.”Proc., Inst. of Civ. Engrs., Water, Maritime and Energy, ICE, London, U.K., 130(Mar.), 51–54.
10.
Diskin, M. H.(1961). “The end depth at a drop in trapezoidal channels.”J. Hydr. Div., ASCE, 87(4), 11–32.
11.
Fathy, A., and Shaarawi, M. A. (1954). “Hydraulics of free overfall.”Proc., ASCE 80, ASCE, Reston, Va., 1–12.
12.
Gupta, R. D., Jamil, M., and Mohsin, M.(1993). “Discharge prediction in smooth trapezoidal free overfall (positive, zero, and negative slopes).”J. Irrig. and Drain. Engrg., ASCE, 119(2), 215–224.
13.
Hager, W. H.(1983). “Hydraulics of the plane overfall.”J. Hydr. Engrg., ASCE, 109(12), 1683–1697.
14.
Henderson, F. M. (1966). Open channel flow. MacMillan Book Co., New York, N.Y.
15.
Jaeger, C. (1948). “Hauteur d'eau à l'extrémité d'un long déversoir.”La Houille Blanche, Paris, France, 3(6), 518–523 (in French).
16.
Jaeger, C. (1957). Engineering fluid mechanics. St. Martin's Press, New York, N.Y.
17.
Keller, R. J., and Fong, S. S.(1989). “Flow measurements with trapezoidal free overfall.”J. Irrig. and Drain. Engrg., ASCE, 115(1), 125–136.
18.
Marchi, E.(1993). “On the free overfall.”J. Hydr. Res., Delft, The Netherlands, 31(6), 777–790.
19.
Markland, E. (1965). “Calculation of flow at a free overfall by relaxation method.”Proc., Inst. Civ. Engrs., London, U.K., 31(May), 71–78.
20.
Matthew, G. H. (1991). “Higher order, one dimensional equation of potential flow in open channels.”Proc., Inst. Civ. Engrs., London, U.K., 90(June), 187–210.
21.
McCormick, J. M., and Salvadori, M. G. (1964). Numerical methods in Fortran. Prentice-Hall Inc., Englewood Cliffs, N.J.
22.
Montes, J. S. (1992). “A potential flow solution for the free overfall.”Proc., Inst. Civ. Engrs., Water, Maritime and Energy, ICE, London, U.K., 96(Dec.), 259–266.
23.
Murty Bhallamudi, S.(1994). “End depth in trapezoidal and exponential channels.”J. Hydr. Res., Delft, The Netherlands, 32(2), 219–232.
24.
Naghdi, P. M., and Rubin, M. B.(1981). “On inviscid flow in a waterfall.”J. Fluid Mech., Cambridge, U.K., 103, 375–387.
25.
Rajaratnam, N., and Muralidhar, D.(1964a). “End depth for exponential channels.”J. Irrig. and Drain. Div., ASCE, 90(1), 17–36.
26.
Rajaratnam, N., and Muralidhar, D.(1964b). “End depth for circular channels.”J. Hydr. Div., ASCE, 90(2), 99–119.
27.
Rajaratnam, N., and Muralidhar, D.(1968). “Characteristics of the rectangular free overfall.”J. Hydr. Res., Delft, The Netherlands, 6(3), 233–258.
28.
Rajaratnam, N., and Muralidhar, D.(1970). “The trapezoidal free overfall.”J. Hydr. Res., Delft, The Netherlands, 8(4), 419–447.
29.
Rajaratnam, N., Muralidhar, D., and Beltaos, S.(1976). “Roughness effects on rectangular overfall.”J. Hydr. Div., ASCE, 102(5), 599–614.
30.
Rouse, H.(1936). “Discharge characteristics of the free overfall.”Civ. Engrg., ASCE, 6(4), 257–260.
31.
Schlichting, H. (1960). Boundary layer theory. McGraw-Hill Inc., New York, N.Y.
32.
Smith, C. D.(1962). “Brink depth for a circular channel.”J. Hydr. Div., ASCE, 88(6), 125–134.
33.
Strelkoff, T., and Moayeri, M. S.(1970). “Pattern of potential flow in a free overfall.”J. Hydr. Div., ASCE, 96(4), 879–901.
Information & Authors
Information
Published In
Journal of Hydraulic Engineering
Volume 124 • Issue 8 • August 1998
Pages: 856 - 863
Copyright
Copyright © 1998 American Society of Civil Engineers.
History
Published online: Aug 1, 1998
Published in print: Aug 1998
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