# The general theory for wings of high aspect ratio

A start is made by considering the influence of the end effect, or downwash, on the lifting properties of an aerofoil section at some distance z from the centre-line of the wing. Figure 5.34 shows the lift-versus-incidence curve for an aerofoil section of

 Cl = <3oo[«oo — c*o] = <з[а <*о] Taking the first equation with = a — є Cl = «ос [(a – ao) – Fig. 5.34 Lift-versus-incidence curve for an aerofoil section of a certain profile, working two-dimensionally and working in a flow regime influenced by end effects, i. e. working at some point along the span of a finite lifting wing

 a certain profile working two-dimensionally and working in a flow regime influenced by end effects, i. e. working at some point along the span of a finite lifting wing. Assuming that both curves are linear over the range considered, i. e. the working range, and that under both flow regimes the zero-lift incidence is the same, then

 (5.56) (5.57) (5.58)  and since  This is Prandtl’s integral equation for the circulation Г at any section along the span in terms of all the aerofoil parameters. These will be discussed when Eqn (5.59) is reduced to a form more amenable to numerical solution. To do this the general series expression (5.45) for Г is taken:

r = 45F]T An sin пв The previous section gives Eqn (5.48):

VEnA„ sin пв

V Y nAn sin пв

cax ‘ ‘ sin в

Cancelling V and collecting cax/Bs into the single parameter ц this equation becomes:

ЭО

ц{а – a0) = J An sinn6»(l +^j)

Я=1

The solution of this equation cannot in general be found analytically for all points along the span, but only numerically at selected spanwise stations and at each end.