Hostname: page-component-7c8c6479df-8mjnm Total loading time: 0 Render date: 2024-03-29T07:40:38.324Z Has data issue: false hasContentIssue false

Theory of optimum shapes in free-surface flows. Part 1. Optimum profile of sprayless planing surface

Published online by Cambridge University Press:  29 March 2006

T. Yao-Tsu Wu
Affiliation:
California Institute of Technology, Pasadena, California
Arthur K. Whitney
Affiliation:
California Institute of Technology, Pasadena, California Present address: Lockheed Palo Alto Research Laboratory, Lockheed Missiles and Space Co., Palo Alto, California.

Abstract

This paper attempts to determine the optimum profile of a two-dimensional plate that produces the maximum hydrodynamic lift while planing on a water surface, under the condition of no spray formation and no gravitational effect, the latter assumption serving as a good approximation for operations at large Froude numbers. The lift of the sprayless planing surface is maximized under the isoperimetric constraints of fixed chord length and fixed wetted arc-length of the plate. Consideration of the extremization yields, as the Euler equation, a pair of coupled nonlinear singular integral equations of the Cauchy type. These equations are subsequently linearized to facilitate further analysis. The analytical solution of the linearized problem has a branch-type singularity, in both pressure and flow angle, at the two ends of plate. In a special limit, this singularity changes its type, emerging into a logarithmic one, which is the weakest type possible. Guided by this analytic solution of the linearized problem, approximate solutions have been calculated for the nonlinear problem using the Rayleigh-Ritz method and the numerical results compared with the linearized theory.

Type
Research Article
Copyright
© 1972 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

Cumberbatch, E. 1958 Two-dimensional planing at high Froude number. J. Fluid Mech. 4, 466.Google Scholar
Miskhelishvili, N. I. 1953 Singular Integral Equations. Groningen, Holland: Noordhoff.
Rispin, P. P. A. 1967 A singular perturbation method for nonlinear water waves past an obstacle. Ph.D. thesis, California Institute of Technology.
Tricomi, F. G. 1957 Integral Equations. Interscience.
Wehausen, J. V. & Laitone, E. V. 1960 Surface Waves. Handbuch der Physik, vol. 9. Springer.
Whitney, A. K. 1969 Minimum drag profiles in infinite cavity flows. Ph.D. thesis, California Institute of Technology.
Wu, T. Y. 1967 A singular perturbation theory for nonlinear free-surface flow problems. International Shipbuilding Progress, 14, 88.Google Scholar
Wu, T. Y. & Whitney, A. K. 1971 Theory of optimum shapes in free-surface flows. Part 1. California Institute of Technology Rep. E 132 F. 1.Google Scholar