# The optimum shape of an hydrofoil with no cavitation

The optimum shape of an hydrofoil with no cavitation
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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...