Experiments of a Free Surface Flow in a Hydraulic Channel over an Uneven Bottom

Authors: M. Bouinoun, M. Bouhadef

Abstract:

The present study is concerned with the problem of determining the shape of the free surface flow in a hydraulic channel which has an uneven bottom. For the mathematical formulation of the problem, the fluid of the two-dimensional irrotational steady flow in water is assumed inviscid and incompressible. The solutions of the nonlinear problem are obtained by using the usual conformal mapping theory and Hilbert’s technique. An experimental study, for comparing the obtained results, has been conducted in a hydraulic channel (subcritical regime and supercritical regime). 

Keywords: Free-surface flow, experiments, numerical method, uneven bottom, supercritical regime, subcritical regime.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1107275

References:

[1] Abd-El-Malek M. B, 1981, Boundary integral methods and free surface problems”, Ph. D thesis, Windsor university, Ontario, Canada
[2] Abd-El-Malek M.-B. and Smith A. C., 1983, Hilbert’s method for numerical solution of flow from a uniform channel over a shelf”, J. Engg Math, 27-39
[3] Abd-El-Malek M.-B., 1988, Local behaviour of an inviscid flow from a uniform channel over a sharp-crested weir, Acta Mechanica 72, 233-24
[4] Abd-El-Malek M.-B. and Masoud S. Z, , 1994, Approximate predictions of gravity flows over irregular topography for large Froude number”, Acta Mechanica 105, 11-21
[5] Benjamin B., 1956, On the flow in channels when rigid obstacles are placed in the stream. J. Fluid Mech., 1, pp 227-248
[6] Bouhadef M., 1993, The influence of a sheared velocity profile on the wavelength of standing small amplitude surface waves. Computer Methods and Experimental Measurements VI, Vol. 1, Heat and Fluid Flow, pp. 293-301 Computational Mechanics Publications
[7] Bouhadef M., 1988, Contribution à l'étude des ondes de surface dans un canal hydraulique. Application à l'écoulement au-dessus d'un obstacle immergé.", Thèse de Doctorat-ès-Sciences, U.E.R. C. E. A. T., Poitiers
[8] Bouinoun M., Bouhadef M., Zitoun T., 2012, Free surface flow over an uneven bottom: Experiments in a hydraulic channel, International Conference on Metallurgical, Manufacturing and Mechanical Engineering, Dubai, UAE
[9] Boutros Y.Z., Abdelmalek M.B. and Massoud S.Z., 1986, Linearized solution of a flow over a non-uniform bottom , J. of Comput and Appli. Math., 16, 105-116
[10] Bouzelha-Hammoum K, Bouhadef M, Zitoun T & Guendouzen T, 2008, Contribution to the study of a free-surface supercritical flow above an obstacle: theory laboratory work, 6th IASME/WSEAS International Conference on Fluid Mechanics and Aerodynamics
[11] Bouzelha-Hammoum K, Bouhadef. M & Zitoun. T, 2006, Numerical study of 2D free-surface waveless flow over a bump, WSEAS Transactions on Fluid Mechanics, Issue 6, Volume 1
[12] Driscoll T. A. & Trefethen L. N, 2002, Schwarz–Christoffel Mapping”, Cambridge University Press
[13] Forbes L. K., Schwartz L. W., 1982, Free-surface flow over a semicircular obstruction. J.Fluid Mech., 114, 299, 314
[14] Gazdar A. S., 1973, Generation of waves of small amplitude by an obstacle placed on the bottom of a running stream. J. Phys. Soc. Japan, 34, 530
[15] Lamb H., 1945, Hydrodynamics, 6th ed. Dover, New York
[16] Lustri C.- J., Mccue S.-W. and Binder B.-J., 2012, Free surface flow past topography: beyond-all-orders approach. European Journal of Applied Mathematics, Vol. 23, Issue 04, 441-467
[17] Lonyangapuo J. K, Elliott L, Ingham D. B, Wen X, 2000, Use of an extremal functional in solving for an unknown bottom surface given a free surface profile”, Engineering Analysis with Boundary Elements, 24, 17–30
[18] Lonyangapuo J.K, Elliott L, Ingham D.B Et Wen X, 1999, Retrieval of the shape of the bottom surface of a channel when the free surface profile is given, Engineering Analysis with Boundary Elements, 23, 457–470
[19] Muskhelishvili N. I, 1946, Singular Integral Equations Boundary problems of function theory and their application to mathematical physics”, 2nd Ed, Moscow
[20] Nehari Z, 1952, Conformal Mapping, Dover Publication INC, New York
[21] Shamin R.V, 2009, Dynamics of an ideal liquid with a free surface in conformal variables, Jjournal of Mathematical Sciences, Vol. 160, No. 5, Moscow, Russia
[22] Thomson W., 1886, On stationary waves in flowing water Part. I – Part III, in Phil.Mag. (5), vol. 22, pp. 353–357, pp. 445–452, pp. 517–530.
[23] Toison F. and Hureau.J., 1997, Calcul de la surface libre d’un canal dont le radier a une forme quelconque, Journée de l’hydrodynamique, Nantes
[24] Wazwaz .A.M, 2011, Linear and Nonlinear Integral Equations Methods and Applications, Higher Education Press, Beijing and Springer