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    Kramer, M.R. Maki, K.J. and Young, Y.L. 2013. Numerical prediction of the flow past a 2-D planing plate at low Froude number. Ocean Engineering, Vol. 70, p. 110.

    Maklakov, D. V. and Uglov, A. N. 1995. On the maximum drag of a curved plate in flow with a wake. European Journal of Applied Mathematics, Vol. 6, Issue. 05,

    Kacimov, A. R. 1994. Optimization of the protrusion shape for a couette-type flow. Optimal Control Applications and Methods, Vol. 15, Issue. 3, p. 193.

    Ilyinsky, N. B. and Kacimov, A. R. 1992. Problems of seepage to empty ditch and drain. Water Resources Research, Vol. 28, Issue. 3, p. 871.

    Ilyinsky, N. B. and Kacimov, A. R. 1992. The Estimation of Integral Seepage Characteristics of Hydraulic Structures in Terms of the Theory of Inverse Boundary-Value Problems. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 72, Issue. 2, p. 103.

    Ilyinsky, N. B. and Kacimov, A. R. 1992. Analytical Estimation of Ground-Water Flow Around Cutoff Walls and Into Interceptor Trenches. Ground Water, Vol. 30, Issue. 6, p. 901.

    Kacimov, A.R. and Nicolaev, A.N. 1992. Steady seepage near an impermeable obstacle. Journal of Hydrology, Vol. 138, Issue. 1-2, p. 17.

    Kacimov, Anvar 1991. Steady, two-dimensional flow of ground water to a trench. Journal of Hydrology, Vol. 127, Issue. 1-4, p. 71.

    Payne, Peter R. 1986. On the spray sheet thickness of a planning wedge hull. Ocean Engineering, Vol. 13, Issue. 1, p. 1.

    Payne, Peter R. 1982. The spray volume shed by an uncambered planing hull in steady planing. Ocean Engineering, Vol. 9, Issue. 4, p. 373.

    Payne, Peter R. 1982. Contributions to the virtual mass theory of hydrodynamic planning. Ocean Engineering, Vol. 9, Issue. 6, p. 515.

    Wu, T. Yao-Tsu and Whitney, A. K. 1973. Variational Calculus Involving Singular Integral Equations. ZAMM - Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 53, Issue. 11, p. 737.

    Whitney, Arthur K. 1972. Theory of optimum shapes in free-surface flows. Part 2. Minimum drag profiles in infinite cavity flow. Journal of Fluid Mechanics, Vol. 55, Issue. 03, p. 457.


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)
  • DOI:
  • Published online: 01 March 2006

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