General Solution of Conjugate Depth Ratio (Power-Law Channels)
Publication: Journal of Irrigation and Drainage Engineering
Volume 143, Issue 9
Abstract
Conjugate flow depth is of high practical importance and should be accurately determined in hydraulic design projects. Quantifying the hydraulic jump phenomenon is an application of specific force relation. This relation has a simple analytical solution only for the horizontal rectangular channels. The power-law section is very versatile and allows suitable modeling for both artificial and natural channels. These channels are used in irrigation and drainage projects. No explicit solution is available in the technical literature for the conjugate depth in general power-law channels for the exponent parameter . For this channel, the conjugate depth is currently obtained by trial and error procedures. Numerical solutions of the conjugate depth problems can always be established, though such numerical solutions are devoid of any physical interpretation. In contrast, an explicit solution involves all the input variables in functional form; and thus the dependence of the conjugate flow depth on each input variable can be interpreted. This study presents accurate explicit solutions that can be used to calculate the conjugate depths of open channels with power-law sections. If the initial depth of the hydraulic jump is known, the sequent depth can be determined using the proposed solution with a maximum error less than 0.35%. Inversely, if sequent depth is known, the initial depth can be determined with a maximum error less than 0.55%. The generic results presented in this research for power-law channels can be applied to its special forms such as rectangular, triangular, and parabolic channels. The proposed direct solutions are general, simple, accurate, and useful for studying the power-law channels.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The author gratefully acknowledges the support provided by the Center of Excellence for Evaluation and Rehabilitation of Irrigation and Drainage Networks, University of Tehran.
References
Abdulrahman, A. (2008). “Direct solution to problems of open-channel transitions: Rectangular channels.” J. Irrig. Drain. Eng., 533–537.
Achour, B., and Debabèche, M. (2003). “Control of hydraulic jump by sill in triangular channel.” J. Hydraul. Res., 41(3), 319–325.
Anwar, A. A., and Clarke, D. (2005). “Design of hydraulically efficient power-law channels with freeboard.” J. Irrig. Drain. Eng., 560–563.
Anwar, A. A., and de Vries, T. T. (2003). “Hydraulically efficient power-law channels.” J. Irrig. Drain. Eng., 18–26.
Argyropoulos, P. A. (1961). “The hydraulic jump and effect of turbulence on hydraulic structures: Contribution to the research of the phenomenon.” Proc., 9th Conf. of the Int. Association for Hydraulic Research, International Association for Hydraulic Research, Dubrovnik, Yugoslavia, 173–183.
Bélanger, J. B. (1828). Essai sur la Solution Numérique de quelques Problémes Relatifs au Mouvement Permanent des Eaux Courantes [Essay on the numerical solution of some problems relative to steady flow of water], Carrilian-Goeury, Paris (in French).
Bidone, G. (1820). “Expériences sur le remous et la propagation des ondes.” Memorie della Realle Accademia dele Science di Torino, 30, 195–292 (in French).
Chanson, H. (1999). The hydraulics of open channel flows: An introduction, Edward Arnold, London.
Chaudhry, M. H. (2008). Open-channel flow, Springer, New York.
Chaurasia, S. R. (2003). “Direct equations for hydraulic jump elements in rectangular horizontal channel.” J. Irrig. Drain. Eng., 291–294.
Chow, V. T. (1959). Open-channel hydraulics, McGraw-Hill, New York.
Das, A. (2007). “Solution of specific energy and specific force equations.” J. Irrig. Drain. Eng., 407–410.
Ead, S. A., and Rajaratnam, N. (2002). “Hydraulic jumps on corrugated beds.” J. Hydraul. Eng., 656–663.
Hager, W. H., and Sinniger, R. (1985). “Flow characteristics of the hydraulic jump in a stilling basin with an abrupt bottom rise.” J. Hydraul. Res., 23(2), 101–113.
Hager, W. H., and Wanoschek, R. (1987). “Hydraulic jump in triangular channel.” J. Hydraul. Res., 25(5), 549–564.
Khan, S. A. (2013). “An analytical analysis of hydraulic jump in triangular channel: A proposed model.” J. Inst. Eng. Series A, 94(2), 83–87.
Long, D., Steffler, P. M., and Rajaratnam, N. (1990). “LDA study of flow structure in submerged hydraulic jump.” J. Hydraul. Res., 28(4), 437–460.
Ma, Z., Zhang, G., Zhao, C., Xu, J., and Hu, L. (2013). “Iterative algorithm of conjugate depth for parabolic channels.” Flow Meas. Instrum., 32, 1–4.
Narayanan, R. (1975). “Wall jet analogy to hydraulic jump.” J. Hydraul. Div., 101(3), 347–359.
Peterka, A. J. (1978). Hydraulic design of stilling basins and energy dissipaters (No. 25), United States Bureau of Reclamation, Washington, DC.
Rajaratnam, N. (1965). “The hydraulic jump as a wall jet.” J. Hydraul. Div., 91(5), 107–132.
Rajaratnam, N., and Subramanya, K. (1968). “Profile of the hydraulic jump.” J. Hydraul. Div., 94(3), 663–673.
Rashwan, I. M. H. (2013). “Analytical solution to problems of hydraulic jump in horizontal triangular channels.” Ain Shams Eng. J., 4(3), 365–368.
Singh, S. K. (2013). “Generalized analytical solutions for alternate and sequent depths in rectangular channels: Nonuniform velocity.” J. Irrig. Drain. Eng., 426–431.
Singh, S. K. (2015). “Generalized analytical solutions for alternate and sequent depths in rectangular open channels: Sine form.” J. Irrig. Drain. Eng., 04014060.
Strelkoff, T. S., and Clemmens, A. J. (2000). “Approximating wetted perimeter in power-law cross section.” J. Irrig. Drain. Eng., 98–109.
Swamee, P. K., and Rathie, P. N. (2004). “Exact solutions for sequent depths problem.” J. Irrig. Drain. Eng., 520–522.
Valiani, A., and Caleffi, V. (2008). “Depth-energy and depth-force relationships in open channel flows. I: Analytical findings.” Adv. Water Resour., 31(3), 447–454.
Valiani, A., and Caleffi, V. (2009). “Depth–energy and depth–force relationships in open channel flows. II: Analytical findings for power law cross-sections.” Adv. Water Resour., 32(2), 213–224.
Vatankhah, A. R. (2009a). “Comments on depth-energy and depth-force relationships in open channel flows. II: Analytical findings for power law cross-sections.” Adv. Water Resour., 32(6), 963–964.
Vatankhah, A. R. (2009b). “Discussion of ‘Direct solution to problems of open-channel transitions: Rectangular channels’ by Abdulrahman Abdulrahman.” J. Irrig. Drain. Eng., 704.
Vatankhah, A. R. (2010). “Analytical solution of specific energy and specific force equations: Trapezoidal and triangular channels.” Adv. Water Resour., 33(2), 184–189.
Vatankhah, A. R., and Kouchakzadeh, S. (2008). “Discussion of solution of specific energy and specific force equations.” J. Irrig. Drain. Eng., 880–882.
Vatankhah, A. R., and Omid, M. H. (2010). “Direct solution to problems of hydraulic jump in horizontal triangular channels.” Appl. Math. Lett., 23(9), 1104–1108.
Vatankhah, A. R., and Valiani, A. (2011). “Analytical inversion of specific energy-depth relationship in channels with parabolic cross-sections.” Hydrol. Sci. J., 56(5), 834–840.
Wielogorski, J. W., and Wilson, E. H. (1970). “Nondimensional profile area coefficients for hydraulic jump in sloping rectangular channels.” Water Power, 22(4), 144–150.
Zhao, Y., Liu, J., and Wang, Z. (2016). “Calculation method for conjugate depths in quadratic parabolic channels.” Flow Meas. Instrum., 50, 197–200.
Information & Authors
Information
Published In
Copyright
©2017 American Society of Civil Engineers.
History
Received: Dec 1, 2016
Accepted: Apr 6, 2017
Published online: Jun 29, 2017
Published in print: Sep 1, 2017
Discussion open until: Nov 29, 2017
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.