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Finite wing theory 243
since
4s2 span2 - aspect ratio(AR)
-
S area
Equation (5.43) establishes quantitatively how CDv falls with a rise in (AR) and
confirms the previous conjecture given above, Eqn (5.36), that at zero lift in sym-
metric flight CD, is zero and the other condition that as (AR) increases (to infinity for
two-dimensional flow) CD, decreases (to zero).
5.5.4 The general (series) distribution of lift
In the previous section attention was directed to distributions of circulation (or lift) along
the span in which the load is assumed to fall symmetrically about the centre-line according
to a particular family of load distributions. For steady symmetric manoeuvres this is quite
satisfactory and the previous distribution formula may be arranged to suit certain cases.
Its use, however, is strictly limited and it is necessary to seek further for an expression that
will satisfy every possible combination of wing design parameter and flight manoeuvre.
For example, it has so far been assumed that the wing was an isolated lifting surface that
in straight steady flight had a load distribution rising steadily from zero at the tips to a
maximum at mid-span (Fig. 5.31a). The general wing, however, will have a fuselage
located in the centre sections that will modify the loading in that region (Fig. 5.31b), and
engine nacelles or other excrescences may deform the remainder of the curve locally.
The load distributions on both the isolated wing and the general aeroplane wing will
be considerably changed in anti-symmetric flight. In rolling, for instance, the upgoing
wing suffers a large decrease in lift, which may become negative at some incidences
(Fig. 5.3 IC). With ailerons in operation the curve of spanwise loading for a wing is no
longer smooth and symmetrical but can be rugged and distorted in shape (Fig. 5.31d).
It is clearly necessary to find an expression that will accommodate all these various
possibilities. From previous work the formula 1 = p VI' for any section of span is familiar.
Writing I in the form of the non-dimensional lift coefficient and equating to pVT:
CL
r=-vc (5.44)
2
is easily obtained. This shows that for a given steady flight state the circulation at any
section can be represented by the product of the forward velocity and the local chord.
Isolated wing in
( a m steady symmetric
flight
I
I I
I I
I
(b) I Lift distribution
modified by
fuselage effects ( d m
I Antisymmetric flight
I with ailerons
in operation
Fig. 5.31 Typical spanwise distributions of lift