PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL1(19)

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parted a forward velocity instantaneously. As explained e~rlier, the first batch of
fiuid particles goes around the wing sections smoothly, forming the front and rear
stagnation points as shown in Fig. 1.28a. Once this flow pattern is formed, pres-
sure gradients come into existence. Of particular interest is the adverse pressure
gradient on the upper surface. As a result, the flow separates on the upper surface
upstream of the trailing edge, and the fiow coming from the lower surface cannot
go around the sharp trailing edge as it did at t - O_ Thus, the curved or vortex fiow
formed at t =O (Fig. 1.28a) is no more formed for t > 0 (Fig. 1.28b), and that
formed at t = O is swept away in the downstream direction.

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28                 PERFORMANCE, STABILJTY, DYNAMICS, AND CONTROL
a) Flow patt,ern at t - O
S
  f
                                                                  Startin8 Vof.cx
                                   b) Flow pattern for t > O
Fig. 1.28    Schematic sketch of flow over a finite wing section.
    The bound vortex induces upwash in front of the wing and downwash behind.
The trailing vortices induce downwash everywhere,including the wing span. The
downwash effect caused by the trailing vortex is negligible in the vicinity of the
wing. A schematic variation of induced fiow field around the wing is shown in
Fig. 1.29. As a result of these induced upwash/downwash effects, the effective
angle of attack varies along the wing span.
      One direct consequence of the induced flow field around a finite wing is that the
lift vectoris now perpendicular to the local velocity vector and not to the freestream
velocit3r vector as shown in Fig. 1.30. By definition, the component of the force
perpendicular to the freestream is the lift and that along the freestream d:irection
-is the drag. As shown in Fig.  1.30,  we now have a component of the local lift in
the freestream direction, which is a form of drag. This component of drag, which
is caused by lift, is called the induced drag. It is important to bear in mind that
induced drag is not caused by fluid viscosity but represents a kind of penalty to be
paid for developing the lift.
   Referring to Fig. 1.30,
aL -. a - at
(1.44)
The basic contribution of the lifting line theory is the following expression for the
induced angle of attack,
             CL         (1.45)
                                 Cti = TA
This formula is applicable only to wings of elliptical planform. For all other wing
planforms, tlus formula is modified as follows:
  CL
cr/ = n-Ae
(1.46)
REVIEW OF BASIC AERODYNAMtC PRINCIPLES                  29
a) Flow field induced by bound vortex
b) Downwash due to trailing vortices
                                          c) Combined flow fceld
Fig.l 29    Schematicillustration ofinduced flow field around a fitute wing.
where e is called the planform efficiency factor. We note that, for elliptical wings,
e -  1. With tfus, the induced-drag coefficient of a lifting wing is given by
                                 CDi = CL(ti                          (1.47)
　
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