Page 87 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

72 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

oO »
hand, for |z| large enough, using z = Z+ > a and F(Z) = Vo(Ze**+
n=0
2 pia T
am )- =~ log Z we also have

. ~1
_
ff _dFdZ fy in i 1 Reet 1 yn an

dz didz \° MZ @F Zn

_y pie Hl _y,-ia EL,
= Voe ma = Voe nn ’

the unwritten terms being infinitesimally small of second order in z7!
and Z—-!. Hence

2 . —i

z
7
dz
(a) — Vee-ria _ Woe I
such that
; iTV, —1O .
F =~ in) (-=—) = ipl Vee".
2 T

So, we can see that the general resultant is acting on a direction which
is perpendicular to the attack (far field) velocity, its algebraic magnitude
being —pI'Vo. This result is known as the Kutta—Joukovski theorem and,

according to it the resultant component on the velocity direction — the
so-called drag —, is zero, which represents D’Alembert’s paradox, while
the normal component vs. the velocity direction, the so-called lift, would
be zero if the flow is without circulation.
D’ Alembert’s paradox also holds for three-dimensional potential flows.
This “weakness” of the mathematical model could be explained not only
by accepting the inviscid character of fluid and, implicitly, the slip-

condition on rigid walls but also by assuming the potential (irrotational)
character of the entire fluid flow, behind the obstacle too. However ex-
perience shows that, behind the obstacles, there are vortices separations.
That is why we will consider, in the next sections, the case of the almost
(nearly) potential flows — that is with vortices separation — and when
D’ Alembert’s paradox does not show up.

5.2 Profiles with Sharp Trailing Edge.
Joukovski Hypothesis

Many aerodynamics profiles have “behind” an angular point, the plane
trace of the sharp edge of the wing with infinite span. Let zp be the