Page 111 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

96 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

Hilbert type that reduces, in an acceptable approximation, to a Dirichlet
problem for the Laplace equation.

8.1 Mathematical Formulation of the Problem
Suppose that our (wing) profile is formed'' by the arcs C, and Cy of
equations

=1,2,
2
<b,j
y = 9; (2) =eh;(@),a<
where € > OQis a very small positive parameter; we admit that the
functions hy(z) and he(x) are continuous and derivable in [a,b] and
hj(a) = hj(b), 7 = 1,2. Suppose also that he(z) > hi(z),a<a<b.
This profile is placed in a uniform fluid free-stream of complex veloc-
ity Woo = Vooe™, both the magnitude of the physical (attack) velocity

at far field V,. and its angle of incidence a, sufficiently small, being inde-
pendent of time. In what follows we will look for the complex potential
of the fluid flow under the form

f(z) = Woz4+ F(z)

or, focussing on the velocity field determination, we set

df dF .
qg Wot Gaur w

with

dF ,
a UT.

The unknown function F(z), the corrective complex potential, in-
duced by the presence of the thin profile, is a holomorphic function in
the vicinity of any point at finite field, with a logarithmic singularity
at infinity. On the contrary, the derivative of this function, ae is holo-
morphic in the entire outside of the profile, vanishing at infinity, that is

(4) = 0. More, the above equality (for the velocity field) generates
the representation

u=V.cosa+U, v=Vosina+
V,
U and V playing the roles of some perturbation (corrective) velocities
due to the presence in the free-stream of the thin profile.

"Obviously it is about the cross-section of the profile in the plane xzOy.