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 conse­quence 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 back­wards 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.