Page 250 - Aerodynamics for Engineering Students

Finite wing theory 233

cannot tend to equalize by spanwise components of velocity so that the streams
of air meeting at the trailing edge after sweeping under and over the wing have no
opposite spanwise motions but join up in symmetrical flow in the direction of
motion. Again no trailing vorticity is formed.
A more rigorous treatment of the vortex-sheet modelling is now considered. In
Section 4.3 it was shown that, without loss of accuracy, for thin aerofoils the vortices
could be considered as being distributed along the chord-line, i.e. the x axis, rather
than the camber line. Similarly, in the present case, the vortex sheet can be located on
the (x, z) plane, rather than occupying the cambered and possibly twisted mid-surface
of the wing. This procedure greatly simplifies the details of the theoretical modelling.
One of the infinitely many ways of constructing a suitable vortex-sheet model is
suggested by Fig. 5.21. This method is certainly suitable for wings with a simple
planform shape, e.g. a rectangular wing. Some wing shapes for which it is not at all
suitable are shown in Fig. 5.22. Thus for the general case an alternative model is
required. In general, it is preferable to assign an individual horseshoe vortex of
strength k (x, z) per unit chord to each element of wing surface (Fig. 5.23). This
method of constructing the vortex sheet leads to certain mathematical difficulties

(a 1 Delta wing ( b ) Swept - back wing

Fig. 5.22

Fig. 5.23 Modelling the lifting effect by a distribution of horseshoe vortex elements