Page 246 - Aerodynamics for Engineering Students

Finite wing theory 229

5.4.1 The use of vortex sheets to model the lifting
effects of a wing
In Section 4.3, it was shown that the flow around a thin wing could be regarded as a
superimposition of a circulatory and a non-circulatory flow. In a similar fashion the
same can be established for the flow around a thin wing. For a wing to be classified as
thin the following must hold:
0 The maximum thickness-to-chord ratio, usually located at mid-span, must be
much less than unity.
0 The camber lines of all wing sections must only deviate slightly from the corres-
ponding chord-line.
0 The wing may be twisted but the angles of incidence of all wing sections must
remain small and the rate of change of twist must be gradual.
0 The rate of change of wing taper must be gradual.
These conditions would be met for most practical wings. If they are satisfied then the
velocities at any point over the wing only differ by a small amount from that of the
oncoming flow.
For the thin aerofoil the non-circulatory flow corresponds to that around
a symmetrical aerofoil at zero incidence. Similarly for the thin wing it corresponds to
that around an untwisted wing, having the same planform shape as the actual wing,
but with symmetrical sections at zero angle of incidence. Like its two-dimensional
counterpart in aerofoil theory this so-called displacement (or thickness) effect makes
no contribution to the lifting characteristics of the wing. The circulatory flow - the
so-called lifting effect - corresponds to that around an infinitely thin, cambered and
possibly twisted, plate at an angle of attack. The plate takes the same planform shape
as the mid-plane of the actual wing. This circulatory part of the flow is modelled by
a vortex sheet. The lifting characteristics of the wing are determined solely by this
component of the flow field. Consequently, the lifting effect is of much greater
practical interest than the displacement effect. Accordingly much of this chapter
will be devoted to the former. First, however, the displacement effect is briefly
considered.

Displacement effect
In Section 4.9, it was shown how the non-circulatory component of the flow around
an aerofoil could be modelled by a distribution of sources and sinks along the chord
line. Similarly, in the case of the wing, this component of the flow can be modelled by
distributing sources and sinks over the entire mid-plane of the wing (Fig. 5.20). In
much the same way .as Eqn (4.103) was derived (referring to Fig. 5.20 for the
geometric notation) it can be shown that the surface pressure coefficient at point
(XI, y1) due to the thickness effect is given by

7
4
where x~(z) denotes the leading edge of the wing.
In general, Eqn (5.23) is fairly cumbersome and nowadays modern computational
techniques like the panel method (see Section 5.8) are used. In the special case of