# The optimum shape of an hydrofoil with no cavitation

We consider a two-dimensional hydrofoil at rest in the xy-plane embedded in a steam with a uniform flow at infinity and we pose the problem of finding the optimum shape of the hydrofoil of a given length and prescribed mean curvature for which the lift is a maximum. Using the lifting line theory and standard variational calculus techniques we show that the slope of the mean chord of the hydrofoil has to satisfy a differential equation of the second order. The Rayleigh-Ritz method is used to solve the second order differential equation which gives the optimal values.

Tipo de documento: Artículo - Article

Palabras clave: Hydrofoil, plane, uniform steam flow, infinite line theory, standard techniques, variational calculus, differential equation, Rayleigh-Ritz method, optimum values

Source: http://www.bdigital.unal.edu.co

## Teaser

Revista Colombiana de Matematicas Vol.
XXV (1991) pgs.
103 - 122 THE OPTIMUM SHAPE OF AN HYDROFOIL WITH NO CAVITATION by A.
Y.
AL-HAWAJ and A.H.
ESSAWY ABSTRACT. We consider a two-dimensional hydrofoil at rest in the (xy)-plane embedded in a steam with a uniform flow at infinity and we pose the problem of finding the optimum shape of the hydrofoil of a given length and prescribed mean curvature for which the lift is a maximum.
Using the lifting line theory and standard variational calculus techniques we show that the slope of the mean chord of the hydrofoil has to satisfy a differential equation of the second order. The Rayleigh-Ritz method is used to solve the second order differential equation which gives the optimal values. I.
Introduction. optimum shape and The in prescribed hydrofoil a uniform ing an sional A used of The purpose of this paper is to evaluate the a two-dimensional hydrofoil of given length mean curvature which produces maximum lift. as in the accompanying diagram (Fig.
I) is placed flow infinite of space. irrotational, steady. two-dimensional to hydrofoil. simulate This an incompressible The liquid and vortex the method is taken a linearized theory distribution over two-dimensional leads non-viscous flow zero to a system liquid fill- to be two-dimenis assumed. the cavity of integral hydrofoil flow past equations is the and AL-HA W AJ and ESSA WY these nique. are solved This method exactly is similar using Carleman-Muskhelishvili techto that used by T.V.
Davies, [1], [2]. We use variational calculus techniques to obtain the optimum shape of the hydrofoil in order to maximize the lift coefficient subject problem is that extremizing a functional depending on y (the two strength), and z are related by vortex, functions solution for the unknown shape z and distributions has branch type singularities analytical singularity ends (the hydrofoil slope) when a singular integral equation. of th...