Skip to main content Accesibility Help
×
×
Home

Free-surface flow over a step

  • A. C. King (a1) (a2) and M. I. G. Bloor (a3)
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.

Copyright
References
Hide All
Benjamin, T. B. 1956 On the flow in channels when rigid obstacles are placed in the stream. J. Fluid Mech. 1, 227.
Benjamin, T. B. 1970 Upstream influence. J. Fluid Mech. 40, 49.
Benjamin, T. B. & Lighthill, M. J. 1954 On cnoidal waves and bores. Proc. R. Soc. Lond. A 224, 448.
Bloor, M. I. G. 1978 Large amplitude surface waves. J. Fluid Mech. 84, 167.
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.
Cumberbatch, E. 1958 Two-dimensional planing at high Froude number. J. Fluid Mech. 4, 466.
Forbes, L. K. & Schwartz, L. W. 1982 Free surface flow over a semi-circular obstruction. J. Fluid Mech. 114, 299.
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.
Green, A. E. & Naghdi, P. M. 1976 Directed fluid sheets. Proc. R. Soc. Lond. A 347, 447.
Havelock, T. H. 1927 The method of images in some problems of surface waves. Proc. R. Soc. Lond. A 15, 268.
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.
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.
Lamb, H. 1932 Hydrodynamics. 246, p. 410. Cambridge University Press.
Miles, J. W. 1986 Stationary, transcritical channel flow. J. Fluid Mech. 162, 489.
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.
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.
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.
Stokes, G. G. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441.
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.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed