Page 100 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Dynamics of Inviscid Fluids 85

b
1
© = —— | Oryds.
sz | Oras
a
Analogously, the circulation around the vortex sheet from s = a to
s = b is the sum of the strength of the elemental vortices, that is
b
Tl = fds. Another property of this vortices distribution is that the

a
tangential component of the fluid velocity experiences a discontinuity
across the sheet in the sense that, for every s, y = uj —Ue, u, and ug be-
ing the tangential velocities “above” and “below” the sheet respectively.
This last relation is used to demonstrate that, for flow past a wing
profile, the value of + is zero at the trailing edge, which means yp = 0. In

fact this relation is one form of the Joukovski condition which fixes the
values of the circulation around the profile with a sharp trailing edge, the
lift force L being related to this circulation through the Kutta—Joukovski
theorem, that is L = poo¥ ql". The goal of this method is to find + (s)
such that the body (profile) surface (boundary) becomes a streamline of
the flow. At the same time we wish to calculate the amount of circulation
and, implicitly, the lift on the body.

As in the case of sources, we will approximate the vortex sheet by
a series of n panels (segments) of constant strength (per unit length)
which form a polygonal contour “inscribed” in the profile contour. Let
us denote by 1, ya, .-- , Yj, » Yn the constant vortex strength over each
panel respectively. Our aim is to determine these unknown strengths
such that both the slip-condition along the profile boundary and the

Joukovski condition are satisfied. Again the midpoints of the panels are
the control points at which the normal component of the (total) fluid
velocity is zero.
Let P(x, y) be a point located a distance rp; from any point of the j-th
panel, the radius rp; making an angle 8; to the Oz axis. The velocity
potential induced at P due to all the panels is

nr nm
@(P) = } 7; =— rk | sds
j=l j=1

where 0); = arctg 7 ,

If P is the control point of the 2-th panel, then

2d
® (24,41) = - wt | %sasi, 6:5 = arctg 222