Page 112 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Dynamics of Inviscid Fluids 97

Just the regularity of the function ue in the whole outside of the
considered profile leads to the idea of determining of this function instead
of the corrective potential F(z). To reach this purpose we need first
to formulate the boundary conditions of the problem in terms of the
functions U and V.
Since the unit normal vector to the contour C, of equation g; (z)—y =

0,is n 9; (2), -1], the slip-condition along the walls C; can be written

vin=ugj(z)-v=0, j=1,2.

Taking into account the above relationship between (u,v) and (U,V) we
have finally the condition

V = —Vo sina + gj (x) (Voo cos a + U) on Cj, 7 =1,2,

such that the velocity field determination comes to the solving of a
Hilbert boundary value problem associated to the Laplace equation.
It is obvious that, additionally, we should observe the Joukovski con-
dition to ensure the boundness of the velocity at sharp trailing edge
(that is, at x = b).
So far we have not formulated, in the mathematical model associated

to the problem, any simplifying hypothesis. Now we assume that |U| is
small enough to be neglected in the presence of V. cosa which agrees
with the fact that the considered profile is thin and the incidence itself
a is small. On the other hand we may assimilate the profile with the
segment AB of the real axis and designating by C” this segment, by

C4, its side corresponding to y = +0 and by C{ that corresponding to
y = —0, the above boundary (slip-) condition could be approximated by

V =—Vosina+Vo.cosa-g;(z)=lj(r) pe Cy, j=1,2.

Thus we are led, in view of the determination of the harmonic function
V(z,y), to a Dirichlet problem for the entire plane Oxy with a cut along

the segment C” of the real axis.

8.2 Solution Determination
The solving of a Dirichlet problem joined to the Laplace operator for

the whole plane with a cut along the segment AB of the real axis, to