The present work deals with the mathematical modelling and analytical solution of the problem of standing waves in rectangular curved channels under supercritical flow regime. To date, several numerical models and some analytical solutions have been proposed, however the restrictions and complexities of these models developed during the last decades lead to an increased need for a simple and reliable tool for practical use. Using the perturbation method in power series of the relative curvature, the curvilinear 2-D Saint-Venant’s equations have been reduced to a linear form. The analytical solution of the non-homogeneous wave equation was obtained by the D'Alembert method after adding the boundary conditions. The validation of the proposed solution was made on the case of a curved channel where it was a question of predicting the configuration of the free surface in a rectangular cross-section profile. The data used as a test come from the experiences of Ippen (1936), Poggi (1956) and Beltrami et al. (2007) and the results were very satisfactory for this purpose compared to the experimental measurements.


Curved channels, Analytical solution, Supercritical flow, Perturbative series.

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AMARA L., BERREKSI A., ACHOUR B. (2020). Approximate analytical solution for supercritical flow in rectangular curved channels. Applied Mathematical Modelling, Vol.80, pp.191-203.

BELTRAMI G. M., DEL GUZZO A. REPETTO R. A. (2007). Simple method to regularize supercritical flow profiles in bends, Journal of Hydraulic Research, Vol. 45, Issue 6, pp.773-786.

BHALLAMUDI M.S. (1989). Numerical modeling of open-channel flows with fixed and movable beds, Thèse de Doctorat, Université de Washington.

HESSAROEYEH M.G., TAHERSHAMSI A. (2009). Analytical model of supercritical flow in rectangular chute bends, Journal of Hydraulic Research, Vol. 47, Issue 5, pp.566-573

IPPEN A.T. KNAPP R.T. (1938). Experimental Investigations of Flow in Curved Channels, Report to Los Angeles County Flood Control District, reproduced by US Engineering Office, Los Angeles.

IPPEN A.T. (1936). An analytical and experimental study of high velocity flow in curved sections of open channels, Thèse de Doctorat, California Institute of Technology.

KÁRMÁN TH. V. (1938). Eine praktische Anwendung der Analogie zwischen Überschallströmung in Gasen und überkritischer Strömung in offenen Gerinnen, ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 18, Issue 1, pp.49-56.

LENAU C.W. (1979). Supercritical flow in bends of trapezoidal section, Journal of the Engineering Mechanics Division, Vol. 105, Issue 1, pp.43-54.

MYINT-U T., DEBNATH L. (2007). Linear partial differential equations for scientists and engineers, Springer Science & Business Media.

POGGI B. (1956). Correnti veloci nei canali in curva, L'energia Elettrica, Vol. 34, pp.465-480.

STEFFLER P.M., RAJARATNAM N., PETERSON A. (1985). Water surface at change of channel curvature, Journal of Hydraulic Engineering, Vol. 111, Issue 5, pp.866-870

VAN DYKE M. (1975). Perturbation methods in fluid mechanics, USA, The parabolic Press.


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