Page 89 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

74 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

trailing edge of the profile of the affix zp, has, in general, an unbounded

modulus. This situation does not arise when Z = Zp is a zero velocity
(stagnation point) for the envisaged flow; really, Z = Zp being a simple
dF
zero for 7A and
dZ\ | 1

(a), ~ AQ HZ = Frys

a will be zero at z = zp if0 < 6 < 1 or, bounded, if 6 = 0 (this last case
corresponds to the presence, at the trailing edge, of a cuspidal point of
(c)).
To avoid the existence of infinite velocities in the neighborhood of
the sharp trailing edge (which does not have any physical support), one

states the following hypothesis, called also the Joukovski-Kutta hypothe-
sis (condition): “The circulation which, for a given incidence, should be
considered for the flow around a profile with sharp trailing edge, is that
which leads to a finite velocity at the trailing edge”.
To determine the effective value of this circulation it would be suffi-
cient to write that Zp = Re’ is a stagnation (zero velocity) point for

the transformed (associated) flow around the disk (C).
From the expression of the complex velocity on the circular boundary
in the fluid flow past the disk [69], that is

C = ie“ Vo [sin (9 ~ a) — siny],
we could see that this implies ~ = 6 — a and hence

T= 4rnVoRsiny = 4nVoRsin(B — a).

So that, taking into account the Joukovski hypothesis, there is only

one flow past a profile when the incidence is “a priori’ given. The angle
B defines the so-called zero lift direction because, if 6 = a, Tr = 0 and
the lift will be also zero by the above evaluation for I.

5.3 Theory of Joukovski Type Profiles

Let us consider the transformation z = i (Z + 3) whose derivative
is ee = 5 Gi — Zz). This transformation defines a conformal mapping
between the planes (z) and (Z) except the singular points Z = +1 where
the conformal character is lost.
It is shown that if Z = re’? (r #1 ), its image in the plane (z) will

be the ellipse

1 1 1 1l\ .
r=-—|r+-]coso,y==z={r--—]sing
2 r 2 r