Hostname: page-component-745bb68f8f-kw2vx Total loading time: 0 Render date: 2025-01-09T07:30:16.568Z Has data issue: false hasContentIssue false
  • English
  • Français

Self-similar, time-dependent flows with a free surface

Published online by Cambridge University Press:  29 March 2006

Michael S. Longuet-Higgins
Michael S. Longuet-Higgins
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge and Institute of Oceanographic Sciences, Wormley, Godalming, England
Get access Rights & Permissions [Opens in a new window]

Abstract

A simple derivation is given of the parabolic flow first described by John (1953) in semi-Lagrangian form. It is shown that the scale of the flow decreases like t−3, and the free surface contracts about a point which lies one-third of the way from the vertex of the parabola to the focus.

The flow is an exact limiting form of either a Dirichlet ellipse or hyperbola, as the time t tends to infinity.

Two other self-similar flows, in three dimensions, are derived. In one, the free surface is a paraboloid of revolution, which contracts like t−2 about a point lying one-quarter the distance from the vertex to the focus. In the other, the flow is non-axisymmetric, and the free surface contracts like t−5.

The parabolic flow is shown to be one of a general class of self-similar flows in the plane, described by rational functions of degree n. The parabola corresponds to n = 2. When n = 3 there are two new flows. In one of these the scale varies as t12/7 and the free surface has the appearance of a trough filling up. In the other, the free surface resembles flow round the end of a rigid wall; the scale varies as t−4·17.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 73 , Issue 4 , 24 February 1976 , pp. 603 - 620
Copyright
© 1976 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

Dirichlet, G. L. 1860 Untersuchungen über ein Problem de Hydrodynamik Abh. Kön. Ges. Wiss. Göttingen, 8, 3–42.Google Scholar
Garabedian, P. 1953 Oblique water entry of a wedge Comm. Pure Appl. Math. 6, 157–167.Google Scholar
Gilbarg, D. 1952 Unsteady flow with free boundaries Z. angew. Math. Phys. 3, 34–42.Google Scholar
Gilbarg, D. 1960 Jets and cavities. Handbuch der Physik, vol. IX (ed. S. Flugge), pp. 311–445. Springer. (See especially pp. 15–18, 350–810.)
John, F. 1952 An example of a transient three-dimensional flow with a free boundary Rev. Gen. Hydraul. 18, 230–232.Google Scholar
John, F. 1953 Two-dimensional potential flows with a free boundary Comm. Pure Appl. Math. 6, 497–503.Google Scholar
Kármán, T. Von 1949 Accelerated flow of an incompressible fluid with wake formation. Ann. Mat. Pure Appl. 29 (4), 247–810.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.
Longuet-Higgins, M. S. 1972 A class of exact, time-dependent, free-surface flows J. Fluid Mech. 55, 529–543.Google Scholar
Longuet-Higgins, M. S. & Cokelet, E. D. 1976 The deformation of steep surface waves: I. A numerical method of computation. Proc. Roy. Soc. A (in press).Google Scholar
Taylor, G. I. 1960 Formation of thin flat sheets of water. Proc. Roy. Soc A 259, 1–17.Google Scholar
Yih, C.-S. 1960 Finite, two-dimensional cavities. Proc. Roy. Soc A 258, 90–100.Google Scholar