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Separated inviscid sheet flows

Published online by Cambridge University Press:  03 May 2011

BUM-SANG YOON  and
YURIY A. SEMENOV
BUM-SANG YOON
Affiliation:
School of Naval Architecture and Ocean Engineering University of Ulsan, 102 Daehakro, Ulsan 680-749, Republic of Korea WCU team, University of Ulsan, 102 Daehakro, Ulsan 680-749, Republic of Korea
YURIY A. SEMENOV*
Affiliation:
School of Naval Architecture and Ocean Engineering University of Ulsan, 102 Daehakro, Ulsan 680-749, Republic of Korea
*
†Email address for correspondence: [email protected]
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Abstract

A steady sheet flow of an inviscid incompressible fluid along a curvilinear surface ending with a rounded trailing edge is considered in the presence of gravity. The effect of surface tension is ignored. The formulation of the problem is applicable to the study of free-surface flows over obstacles in channels, weirs and spillways, and pouring flows. An advanced hodograph method is employed for solving the problem, which is reduced to a system of two integro-differential equations in the velocity modulus on the free surface and in the slope of the bottom surface. These equations are derived from the dynamic and kinematic boundary conditions. The Brillouin–Villat criterion is applied to determine the location of the point of flow separation from the rounded trailing edge. Results showing the effect of gravity on the flow detachment and the geometry of the free boundaries are presented over a wide range of Froude numbers including both subcritical and supercritical flows. For supercritical flows two families of solutions for an arbitrary bottom shape are reproduced. It is shown that the additional condition requiring the free surface to be flat at a finite distance from the end of the channel selects a unique solution for a given bottom height and geometry for supercritical flows. This solution is continuous in going from the subcritical to the supercritical flow regime.

JFM classification

Waves/Free-surface Flows: Channel flow Waves/Free-surface Flows: Critical layers Waves/Free-surface Flows: Solitary waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 678 , 10 July 2011 , pp. 511 - 534
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Ackers, P., White, W. R., Perkins, J. A. & Harrison, A. J. M. 1978 Weirs and Flumes for Flow Measurements. Wiley.Google Scholar
Benjamin, T. B. 1966 Internal waves of finite amplitude and permanent form. J. Fluid Mech. 25 (2), 241–270.CrossRefGoogle Scholar
Birkhoff, G. & Zarantonello, E. H. 1957 Jets, Wakes, and Cavities. Academic.Google Scholar
Brillouin, M. 1911 Les surfaces de glissement de Helmholtz et la resistance des fluids. Ann. Chim. Phys. 23, 145–230.Google Scholar
Chaplygin, S. A. 1910 About Pressure of a Flat Flow on Obstacles. On the Airplane theory. Moscow University, 49 p.Google Scholar
Dias, F., Vanden-Broeck, J.-M. & Keller, J. B. 1988 Flows over rectangular wiers. Phys. Fluids 31, 2071–2076.CrossRefGoogle Scholar
Dias, F. & Vanden-Broeck, J.-M. 1989 Open channel flows with submerged obstructions. J. Fluid Mech. 206, 155–170.CrossRefGoogle Scholar
Dias, F. & Vanden-Broeck, J.-M. 1990 Flows emerging from a nozzle and falling under gravity. J. Fluid Mech. 213, 465–477.CrossRefGoogle Scholar
Dias, F. & Tuck, E. O. 1991 Weir flows and waterfalls. J. Fluid Mech. 230, 525–539.CrossRefGoogle Scholar
Dias, F. & Vanden-Broeck, J.-M. 2004 Trapped waves between submerged obstacles. J. Fluid Mech. 509, 93–102.CrossRefGoogle Scholar
Forbes, L. K. & Schwartz, L. W. 1982 Free-surface flow over a semicircular obstruction. J. Fluid Mech. 114, 299–314.CrossRefGoogle Scholar
Gurevich, M. I. 1965 Theory of Jets in Ideal Fluids. Academic.Google Scholar
Joukovskii, N. E. 1890 Modification of Kirchoff's method for determination of a fluid motion in two directions at a fixed velocity given on the unknown streamline. Math. Coll. 15, 121–278.Google Scholar
Keller, J. B. 1957 Teapot effect. J. Appl. Phys. 28, 859–864.CrossRefGoogle Scholar
Reiner, M. 1956 The teapot effect. Phys. Today 9 (9), 16.CrossRefGoogle Scholar
Semenov, Y. A. 2003 Complex potential of unsteady free-boundary flow. Vestnik Khersonskogo Universiteta (in Russian), 2, 384–387.Google Scholar
Semenov, Y. A. & Yoon, B.-S. 2009 Onset of flow separation at oblique water impact of a wedge. Phys. Fluids 21, 112103–112111.Google Scholar
Vanden-Broeck, J.-M. 1987 Free-surface flow over an obstruction in a channel. Phys. Fluids 30, 2315–2317.Google Scholar
Vanden-Broeck, J.-M. & Keller, J. B. 1986 Pouring flows. Phys. Fluids 29 (12), 3958–3961.CrossRefGoogle Scholar
Vanden-Broeck, J.-M. & Keller, J. B. 1987 Weir flows. J. Fluid Mech. 176, 283–293.CrossRefGoogle Scholar
Vanden-Broeck, J.-M. & Keller, J. B. 1989 Pouring flows with separation. Phys. Fluids A 1, 156–158.Google Scholar
Villat, H. 1914 Sur la validite des solutions de certains problemes d'hydrodynamique. J. Math. Pure Appl. 20, 231–290.Google Scholar