Page 247 - Aerodynamics for Engineering Students

230 Aerodynamics for Engineering Students

Fig. 5.20 Modelling the displacement effect by a distribution of sources

wings having high aspect ratio, intuition would suggest that the flow over most of the
wing behaves as if it were two-dimensional. Plainly this will not be a good approxi-
mation near the wing-tips where the formation of the trailing vortices leads to highly
three-dunensional flow. However, away from the wing-tip region, Eqn (5.23) reduces
approximately to Eqn (4.103) and, to a good approximation, the C, distributions
obtained for symmetrical aerofoils can be used for the wing sections. For complete-
ness this result is demonstrated formally immediately below. However, if this is not of
interest go directly to the next section.
Change the variables in Eqn (5.23) to % = (x - xI)/c, 21 = z1/c and Z = (z - z1)/c.
Now provided that the non-dimensional shape of the wing-section does not change
along the span, or, at any rate, only changes very slowly St = d(yt/c)/dZ does not
vary with Z and the integral I1 in Eqn (5.23) becomes

"
12
To evaluate the integral 12 change variable to x = l/Z so that

1 1

1 1 1