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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)
= C2
7tAe (1.48)
'd?S2j
~
";:-.
git
-.s
j
PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
CL
w
A
Fig.130 Schematicillustration ofinduced drag.
Fig.131 Variation oflift coeffiaent with angle of attack for finite w:ings.
REVIEW OF BASlC AERODYNAMIC PRINCIPLES
31
of attack for a given lift coefficient is higher than that for a two-dimensional wing
(A = oo). In other words, the lift-curve slope effectively reduces with a decrease
in aspect ratio. Based on the lifting line theory, we can obtain an expression for
the lift-curve slope of the fi:tute wings as follows.
For a given lift coefficient,
CL - aoao
= a(LYo + at)
= aa'o (1+ -..)
(1.49)
(1.50)
(1.52)
where a(, is the sectional or two-dimensionallift-curve slope and a is the lift-curve
slope of a finite wing of aspect ratio A. In the above formula, the values of a and
ao are per rad.
1.7 Methods of Reducing Induced Drag
An obvious method of reducing induced drag is to increase the aspect ratios of a
lifting surface or install end plates (at the wing tips) to pre'vent the crossflow around
wing tips. However, either of these two approaches is not always the best option
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