Hostname: page-component-848d4c4894-8kt4b Total loading time: 0 Render date: 2024-06-20T10:59:14.234Z Has data issue: false hasContentIssue false
  • English
  • Français

Free-surface flow over a step

Published online by Cambridge University Press:  21 April 2006

A. C. King  and
M. I. G. Bloor
A. C. King
Affiliation:
Department of Mathematics and Statistics, City of Birmingham Polytechnic, Birmingham B42 2TH, UK Present address: Department of Theoretical Mechanics, University of Nottingham, University Park, NG7 2RD, UK.
M. I. G. Bloor
Affiliation:
Department of Applied Mathematical Studies, University of Leeds, Leeds LS2 9JT, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

A transformation technique is used to solve the problem of steady free-surface flow of an ideal fluid over a semi-infinite step in the bottom. Application of the exact free-surface condition results in a nonlinear integro-differential equation for the free-surface angle and solutions of this equation are dependent on step height and Froude number. Linearized solutions, based upon small step height are presented and indicate that the nature of the free surface formed depends on whether the upstream flow is subcritical or supercritical. As the step height is increased, solutions to the exact nonlinear equations are obtained using the predictions of the linear theory, or possibly a previous nonlinear solution, as an initial estimate.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 182 , September 1987 , pp. 193 - 208
Copyright
© 1987 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Benjamin, T. B. 1956 On the flow in channels when rigid obstacles are placed in the stream. J. Fluid Mech. 1, 227.Google Scholar
Benjamin, T. B. 1970 Upstream influence. J. Fluid Mech. 40, 49.Google Scholar
Benjamin, T. B. & Lighthill, M. J. 1954 On cnoidal waves and bores. Proc. R. Soc. Lond. A 224, 448.Google Scholar
Bloor, M. I. G. 1978 Large amplitude surface waves. J. Fluid Mech. 84, 167.Google Scholar
Bloor, M. I. G. 1984 A note on the limiting form of shallow water waves. In Advances in Nonlinear Waves (ed. L. Debnath), p. 61. Pitman.
Cokelet, E. D. 1977 Steep gravity waves in water of arbitrary uniform depth. Phil. Trans. R. Soc. Lond. A 286, 183.Google Scholar
Cumberbatch, E. 1958 Two-dimensional planing at high Froude number. J. Fluid Mech. 4, 466.Google Scholar
Forbes, L. K. & Schwartz, L. W. 1982 Free surface flow over a semi-circular obstruction. J. Fluid Mech. 114, 299.Google Scholar
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.Google Scholar
Green, A. E. & Naghdi, P. M. 1976 Directed fluid sheets. Proc. R. Soc. Lond. A 347, 447.Google Scholar
Havelock, T. H. 1927 The method of images in some problems of surface waves. Proc. R. Soc. Lond. A 15, 268.Google Scholar
Haussling, H. J. & Coleman, R. M. 1977 Finite difference computations using boundary fitted coordinates for free surface potential flow generated by submerged bodies. Proc. 2nd Intl Conf. on Numerical Ship Hydrodynamics, Berkeley, p. 211.
Kelvin, W. 1886 On stationary waves in flowing water. Phil. Mag. 22 (5), 445.Google Scholar
Korteweg, D. J. & de Vries, G. 1895 On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves. Phil. Mag. 39 (5), 422.Google Scholar
Lamb, H. 1932 Hydrodynamics. 246, p. 410. Cambridge University Press.
Miles, J. W. 1986 Stationary, transcritical channel flow. J. Fluid Mech. 162, 489.Google Scholar
Moiseev, N. N. & Ter-Krikorov, A. M. 1958 On the non-uniqueness of solution of the problem of the hydrofoil. Dokl. Akad. Nauk SSSR 119, 899.Google Scholar
Naghdi, P. M. & Vonsarnpigoon, L. 1986a Steady flow past a step. Proc. 16th Symp. on Naval Hydrodynamics, Berkeley, 1986 (preprint).
Naghdi, P. M. & Vonsarnpigoon, L. 1986b The downstream flow beyond an obstacle. J. Fluid Mech. 162, 223.Google Scholar
Rabinowitz, P. 1970 Numerical Methods for Non-Linear Algebraic Equations. Gordon & Breach.
Seeger, R. J. & Temple, G. 1965 Research Frontiers in Fluid Dynamics, p. 534. Interscience.
Shanks, S. P. & Thompson, J. F. 1977 Numerical solution of the Navier-Stokes equations for 2-D hydrofoils in or below a free surface. Proc. 2nd Intl Conf. on Numerical Ship Hydrodynamics, Berkeley, p. 202.
Squire, H. B. 1957 The motion of a single wedge along the water surface. Proc. R. Soc. Lond. A 243, 48.Google Scholar
Stokes, G. G. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441.Google Scholar
Stokes, G. G. 1880 Supplement to a paper on the theory of oscillatory waves. In Mathematical and physical Papers, vol. 1. Cambridge University Press.
Von Kerazek, C. & Salvesen, N. 1977 Nonlinear free surface effects - the dependence on Froude number. Proc. 2nd Intl Conf. on Numerical Ship Hydrodynamics, Berkeley, p. 202.
Wehausen, J. V. & Laitone, E. V. 1960 Surface waves. In Handbuch der Physik, vol. 9 (ed. S. Flugge), p. 446. Springer.