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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
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