Page 99 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

84 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

Certainly this approximation could be made more accurate by in-
creasing the number of panels and, if necessary, by considering panels of

different length (for instance, in the case of a profile shape, one gets a
good accuracy by considering 50 to 100 panels which are either smaller
in the leading edge region of a rapid surface curvature or longer over the
quasi-flat portions of the profile).
Obviously, following the same way, we can also obtain the tangential
components of the velocity at the same point (z,, y;), precisely

v7
. 4s fd
Vi = Voo,t + Vt = Voo sin B; + 2. On / ds (In rij) ds;.

~ j

Hence, the pressure at the same control point is calculated by the
2
Bernoulli theorem while the pressure coefficients are Cp; = 1 — (24) ,
Voo
Before ending this section it is important to give a procedure for
testing the accuracy of the above method. If S; is the length of the j-th
panel of source strength A; (per unit length), then the strength of the
entire panel will be, obviously, S;A;. But the mass conservation, in the
Tm
hypothesis of a closed contour, allows us to write $7 SjA; = 0 which
j=l
provides an independent criterion to test the obtained results.

6.2 The Vortex Panel Method for Lifting Flows
Over Arbitrary Two-Dimensional Bodies

Consider now a continuous distribution of vortices (vortex sheet) over

the surface (contour) of a body (profile) in an incompressible flow with
free-stream velocity Vy. Let y = y(s) be the strength (circulation)
of the vortex sheet, per unit length along s. Thus the strength of an
infinitesimal portion ds of the boundary (vortex sheet) is yds and this
small section could be treated as a distinct vortex of strength yds. Intro-
ducing again the point P(z, y) in the flow, located at distance r from ds,
the infinitesimal portion ds of the boundary (vortex sheet) of strength
yds induces an infinitesimal velocity potential at P, namely

and, correspondingly, the entire distribution of vortices from s = a and
s = b will generate a velocity potential