Page 102 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Dynamics of Inviscid Fluids 87

of the vortex panels. References can be found, for instance, in the book

of Chow [22].

6.3 Example
Let us consider, for instance, a source panel of length 2, lying sym-

metrically on the Oy axis [22]. Assume that on it, sources of the strength
A per unit length are distributed. The velocity potential induced at every
point (z,y) by the source contained in the infinitesimal panel element
! 1
dy’ at (0,y’) is Au In [x? + (y — y’)?]? (this expression is obtained by
taking the real part of the source complex potential).
The potential induced by the entire panel is

Xd L
®(x,y) = z/ In {a? + (y — y')"] dy’
TS-L

and the velocity components can be obtained by derivation with respect
to xz, respectively y,

u(z,y) = + [arctg (#22) — arctg (44)

x? + (y+L)?
— A
v(z,y) = 7 In PE lyo Lye

Considering a point (z,y) such that z > 0 and y € (~-L,L),if c> 0

from the right of the panel we obtain the limit u(+0,y) = 3. On the
other hand, by a similar approach from the left, we obtain the limit
u(—0,y) = —3. Thus the panel generates a flow having an outward
normal velocity of magnitude A, The tangential velocity v is the same
on both sides of the panel and it is zero at the midpoint and infinite at

the edges of the panel.
If such a panel with sources of strength 4 = 2U is placed normal
to a uniform flow of speed U, the induced normal velocity cancels the
oncoming flow on the left side and thus the resultant flow is tangent to
the surface. So, the panel becomes coincident with one of the streamlines
of the flow.

If the panel makes an angle @ with the uniform stream, the generated
flow cancels the normal induced flow if its strength is A = 2U sin @.
Let now m be the number of the panels. On each panel are distributed
uniform sources of strength A1,...,Am (strength per unit length) respec-
tively. The velocity potential of the resultant flow at every point (z;, y;)
from the flow field, generated by the sources from the 7-th panel is, as
above, Au J, Inrijds; where J is the panel and A,;ds; is the strength of