Page 114 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Dynamics of Inviscid Fluids 99

Then, to ensure that at far distances (|z| —> oo) the solution of our

problem tends to zero, it is sufficient to choose the real constants » and
k so that , (e)
~1 PeO@=-49 ye sage
\= 5, | ep ande=o.

Finally we have for the complex velocity the representation

dF 1 f?h(é)-h(g) | In(é)+h(g) /E-a

dz wf, 2-€ dé — 5 28 fee = be

a formula given by L. I. Sedov, but obtained via other technique [134].
On the other hand, as a complex potential f(z), at far field, has an
expansion under the form

.
f(2) = Wooz + 5— logz + 49+ + F +:

and implicitly the complex velocity is

df | ri ay 1

dz "©" Orig pital)

we get for the circulation I’, necessarily'*, the value

b f—a
©)
= [ h@-b [See

This value corresponds to that obtained by the Joukovski condition
(rule), the fluid velocity being, obviously, bounded at the sharp trailing

edge. Supported by it we could also calculate the general resultant of
the fluid pressures on the thin profile, namely we have! is

. b
E—a
Ry — iBy = ipVoe* / [lo (€) — hh (€)] dé.
b-—€

Details on the theory of a thin (wing) profile and even some extensions
such as the case of the thin airfoil with jet, can be found in the book
of C. Iacob [69]. The thin profile with jet in the presence of the ground
has been studied in [113].

'4Tn virtue of the uniqueness of such a series development.
'SBy applying directly the Blasius-Chaplygin formulas,