Page 269 - Aerodynamics for Engineering Students

252 Aerodynamics for Engineering Students

equation 0 = 0 and are, therefore, useless for the present purpose. Generally four
coefficients are sufficient in the symmetrical case to produce a spanwise distribution
that is insignificantly altered by the addition of further terms. In the case of sym-
metric flight the coefficients would be AI, A3, As, A7, since the even harmonics do
not appear. Also the arithmetic need only be concerned with values of 0 between
0 and 42 since the curve is symmetrical about the mid-span section.
If the spanwise distribution is irregular, more harmonics are necessary in the series
to describe it adequately, and more Coefficients must be found from the integral
equation. This becomes quite a tedious and lengthy operation by ‘hand’, but being
a simple mathematical procedure the simultaneous equations can be easily pro-
grammed for a computer.
The aerofoil parameters are contained in the expression
chord x two-dimensional lift slope
P=
8 x semi-span
and the absolute incidence (a - ao). p clearly allows for any spanwise variation in the
chord, i.e. change in plan shape, or in the two-dimensional slope of the aerofoil
profile, i.e. change in aerofoil section. a is the local geometric incidence and will vary
if there is any geometric twist present on the wing. ao, the zero-lift incidence, may
vary if there is any aerodynamic twist present, i.e. if the aerofoil section is changing
along the span.

Example 5.3 Consider a tapered aerofoil. For completeness in the example every parameter is
allowed to vary in a linear fashion from mid-span to the wing-tips.
Mid-span data Wing-tip data
3.048 Chord m 1.524

5.5 per radian 5.8
5.5 absolute incidence a’ 3.5
Total span of wing is 12.192m
Obtain the aerofoil characteristics of the wing, the spanwise distribution of circulation,
comparing it with the equivalent elliptic distribution for the wing flying straight and level at
89.4 m s-l at low altitude.
From the data:
3.048 + 1.524
Wing area S = x 12.192 = 27.85m2
2

span’ 12.192’ - 5.333
Aspect ratio (AR) = - -
area 27.85
(;)I
At any section z from the centre-line [B from the wing-tip]
[
chord c = 3.048 1 - 3.048 - 1.524 = 3.048[1 + OSCOSB]
3.048
(2)m=a=5.5[1+- 5’55~~’8 (31 = 5.5[1 - 0.054 55 cos B]

ao=5.5 [ 1-- 5*55T:’5 (31 = 5.5[1 + 0.363 64 cos e]