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

  • T. Yao-Tsu Wu (a1) and Arthur K. Whitney (a1) (a2)

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.

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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.
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.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
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