Page 253 - Aerodynamics for Engineering Students

236 Aerodynamics for Engineering Students

(5.27)

(5.28)

(5.29)

Equations (5.27 to 5.29) are now substituted into Eqn (5.26), and terms involving
and
(6~)~ higher powers are ignored, to give

In order to obtain the velocity induced at P1 due to all the horseshoe vortex elements,
6vi is integrated over the entire wing surface projected on to the (x, z) plane. Thus
using Eqn (5.30) leads to

The induced velocity at the wing itself and in its wake is usually in a downwards
direction and accordingly, is often called the downwash, w, so that w = -Vi.
It would be a difficult and involved process to develop wing theory based on
Eqn (5.31) in its present general form. Nowadays, similar vortex-sheet models are
used by the panel methods, described in Section 5.8, to provide computationally
based models of the flow around a wing, or an entire aircraft. Accordingly, a
discussion of the theoretical difficulties involved in using vortex sheets to model wing
flows will be postponed to Section 5.8. The remainder of the present section and
Section 5.6 is devoted solely to the special case of unswept wings having high aspect
ratio. This is by no means unrealistically restrictive, since aerodynamic considera-
tions tend to dictate the use of wings with moderate to high aspect ratio for low-speed
applications such as gliders, light aeroplanes and commuter passenger aircraft. In
this special case Eqn (5.31) can be very considerably simplified.
This simplification is achieved as follows. For the purposes of determining the
aerodynamic characteristics of the wing it is only necessary to evaluate the induced
velocity at the wing itself. Accordingly the ranges for the variables of integration are
given by -s 5 z 5 s and 0 5 x 5 (c)-. For high aspect ratios S/C> 1 so that
I
Ix - XI << r over most of the range of integration. Consequently, the contributions of
terms (b) and (c) to the integral in Eqn (5.31) are very small compared to that of term
(a) and can therefore be neglected. This allows Eqn (5.31) to be simplified to

(5.32)

where, as explained in Section 5.4.1 , owing to Helmholtz's second theorem

(5.33)