# Finite wing theory

Preamble

Whatever the operating requirements of an aeroplane may be in terms of speed; endurance, pay-load and so on, a critical stage in its eventual operation is in the low-speed flight regime, and this must be accommodated in the overall design process. The fact that low-speed flight was the classic flight regime has meant that over the years a vast array of empirical data has been accumulated from flight and other tests, and a range of theories and hypotheses set up to explain and extend these observations. Some theories have survived to provide successful working processes for wing design that are capable of further exploitation by computational methods.

In this chapter such a classic theory is developed to the stage of initiating the preliminary low-speed aerodynamic design of straight, swept and delta wings. Theoretical fluid mechanics of vortex systems are employed, to model the loading properties of lifting wings in terms of their geometric and attitudinal characteristics and of the behaviour of the associated flow processes.

The basis on which historical solutions to the finite wing problem were arrived at are explained in detail and the work refined and extended to take advantage of more modern computing techniques.

A great step forward in aeronautics came with the vortex theory of a lifting aerofoil due to Lanchester[21] and the subsequent development of this work by Prandti. t Previously, all aerofoil data had to be obtained from experimental work and fitted to other aspect ratios, planforms, etc., by empirical formulae based on past experi­ence with other aerofoils.

Among other uses the Lanchester-Prandtl theory showed how knowledge of two-dimensional aerofoil data could be used to predict the aerodynamic charac­teristics of (three-dimensional) wings. It is this derivation of the aerodynamic characteristics of wings that is the concern of this chapter. The aerofoil data can either be obtained empirically from wind-tunnel tests or by means of the theory described in Chapter 4. Provided the aspect ratio is fairly large and the assump­tions of thin-aerofoil theory are met (see Section 4.3 above), the theory can be applied to wing planforms and sections of any shape.