Page 230 - Aerodynamics for Engineering Students

Finite wing theory 21 3

each of these two vortices will equal the streagth of the vortex replacing the wing
itself.
The existence of the trailing and starting vortices may easily be verified visually.
When a fast aeroplane pulls out of a dive in humid air the reduction of pressure and
temperature at the centres of the trailing vortices is often sufficient to cause some of
the water vapour to condense into droplets, which are seen as a thin streamer for
a short distance behind each wing-tip (see frontispiece).
To see the starting vortex all that is needed is a tub of water and a small piece of
board, or even a hand. If the board is placed upright into the water cutting the
surface and then suddenly moved through the water at a moderate incidence, an eddy
will be seen to leave the rear, and move forwards and away from the ‘wing’. This is
the starting vortex, and its movement is induced by the circulation round the plate.

5.1.3 The bound vortex system
Both the starting vortex and the trailing system of vortices are physical entities that can
be explored and seen if conditions are right. The bound vortex system, on the other
hand, is a hypothetical arrangement of vortices that replace the real physical wing in
every way except that of thickness, in the theoretical treatments given in this chapter.
This is the essence of finite wing theory. It is largely concerned with developing the
equivalent bound vortex system that simulates accurately, at least a little distance away,
all the properties, effects, disturbances, force systems, etc., due to the real wing.
Consider a wing in steady flight. What effect has it on the surrounding air, and
how will changes in basic wing parameters such as span, planform, aerodynamic or
geometric twist, etc., alter these disturbances? The replacement bound vortex system
must create the same disturbances, and this mathematical model must be sufficiently
flexible to allow for the effects of the changed parameters. A real wing produces
a trailing vortex system. The hypothetical bound vortex must do the same. A conse-
quence of the tendency to equalize the pressures acting on the top and bottom
surfaces of an aerofoil is for the lift force per unit span to fall off towards the tips.
The bound vortex system must produce the same grading of lift along the span.
For complete equivalence, the bound vortex system should consist of a large
number of spanwise vortex elements of differing spanwise lengths all turned back-
wards at each end to form a pair of the vortex elements in the trailing system. The
varying spanwise lengths accommodate the grading of the lift towards the wing-tips,
the ends turned back produce the trailing system and the two physical attributes of
a real wing are thus simulated.
For partial equivalence the wing can be considered to be replaced by a single
bound vortex of strength equal to the mid-span circulation. This, bent back at each
end, forms the trailing vortex pair. This concept is adequate for providing good
estimations of wing effects at distances greater than about two chord lengths from
the centre of pressure.

5.1.4 The horseshoe vortex
The total vortex system associated with a wing, plus its replacement bound vortex
system, forms a complete vortex ring that satisfies all physical laws (see Section
5.2.1). The starting vortex, however, is soon left behind and the trailing pair stretches
effectively to infinity as steady flight proceeds. For practical purposes the system
consists of the bound vortices and the trailing vortex on either side close to the wing.
This three-sided vortex has been called the horseshoe vortex (Fig. 5.3).