Page 257 - Aerodynamics for Engineering Students

240 Aerodynamics for Engineering Students
the end-effects are more dominant. It seems therefore that a wing that is large in the
spanwise dimension, i.e. large aspect ratio, is a better wing - nearer the ideal - than
a short span wing of the same section. It would thus appear that a wing of large
aspect ratio will have better aerodynamic characteristics than one of the same section
with a lower aspect ratio. For this reason, aircraft for which aerodynamic efficiency is
paramount have wings of high aspect ratio. A good example is the glider. Both the
man-made aircraft and those found in nature, such as the albatross, have wings with
exceptionally high aspect ratios.
In general, the induced velocity also varies in the chordwise direction, as is evident
from Eqn (5.31). In effect, the assumption of high aspect ratio, leading to Eqn (5.32),
permits the chordwise variation to be neglected. Accordingly, the lifting character-
istics of a section from a wing of high aspect ratio at a local angle of incidence a(z)
are identical to those for a two-dimensional wing at an effective angle of incidence
a(z) - e. Thus Prandtl's theory shows how the two-dimensional aerofoil character-
istics can be used to determine the lifting characteristics of wings of finite span. The
calculation of the induced angle of incidence E now becomes the central problem. This
poses certain difficulties because E depends on the circulation, which in turn is closely
related to the lift per unit span. The problem therefore, is to some degree circular in
nature which makes a simple direct approach to its solution impossible. The required
solution procedure is described in Section 5.6.
Before passing to the general theory in Section 5.6, whereby the spanwise circula-
tion distribution must be determined as part of the overall process, the much simpler
inverse problem of a specified spanwise circulation distribution is considered in some
detail in the next subsection. Although this is a special case it nevertheless leads to
many results of practical interest. In particular, a simple quantitative result emerges
that reinforces the qualitative arguments given above concerning the greater aero-
dynamic efficiency of wings with high aspect ratio.

5.5.3 The characteristics of a simple symmetric
loading - elliptic distribution

In order to demonstrate the general method of obtaining the aerodynamic charac-
teristics of a wing from its loading distribution the simplest load expression for
symmetric flight is taken, that is a semi-ellipse. In addition, it will be found to be a
good approximation to many (mathematically) more complicated distributions and
is thus suitable for use as first predictions in performance estimates.
The spanwise variation in circulation is taken to be represented by a semi-ellipse
having the span (2s) as major axis and the circulation at mid-span (ro) as the semi-
minor axis (Fig. 5.30). From the general expression for an ellipse

or

(5.37)

This expression can now be substituted in Eqns (5.32), (5.34) and (5.36) to find the
lift, downwash and vortex drag on the wing.