Page 97 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

82 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

uous distribution of sources be along the contour (surface) of the body

and let A(s) be the source strength, per unit length, of this distribu-
tion where s is the natural parameter (the distance measured along this
contour in the edge view). Obviously an infinitesimal portion ds of the
boundary (source sheet) can be treated as a distinct source of strength
Ads. The effect induced by such a source at a point P(z,y), located a
distance r from ds, is a fluid flow with an infinitesimally small velocity

potential dé given by

Ads
dd d = —— lnr .
5

The total velocity potential at the point P, induced by all the sources
from a to b, is obtaining by summing up the above infinitesimal poten-

tials, which means

b

®(z,y) = 5 ine.

a

Obviously, the fluid velocity induced by the source distribution (sheet)
will be superposed, at any point P, on the free-stream (attack) velocity.
The problem we intend to solve (numerically) is that of the determi-
nation of such a source distribution A(s) which “observes” the surface
(boundary) of the body (profile), i.e., the combined action of the uni-
form flow and the source sheet makes the profile boundary a streamline
of the flow.

To reach this target, let us approximate the profile boundary by a
set of straight panels (segments), the source strength A per unit length
being constant over a panel but possibly varying from one to another
panel.
Thus, if there is a total of n panels and Aj, AQ, ... ,Aj, .-- » An are the

constant source strengths over each panel respectively, these “a priori”
unknown Aj; will be determined by imposing the slip-condition on the
profile boundary. This boundary condition is imposed numerically by
defining the midpoint of each panel to be the control point where the
normal component of the fluid velocity should be zero.
In what follows, for sake of simplicity, we will choose the control points
to be the midpoints of each panel (segment).
Let us denote by rp; the distance from any point (#;,y;) on the j-th

panel to the arbitrary point P(z,y). The velocity potential induced at
P due to the j-th panel of constant source strength dj; is