fluid_mechanics

9.4 Lift 519
α = 0
= 0
(a)
α > 0
= 0
(b)
α > 0
> 0
(c)
+ =
"(a) + circulation = (c)"
(d)
F I G U R E 9.36 Inviscid
flow past an airfoil: (a) symmetrical flow
past the symmetrical airfoil at a zero
angle of attack; (b) same airfoil at a
nonzero angle of attack—no lift, flow
near trailing edge not realistic; (c) same
conditions as for (b) except circulation
has been added to the flow—nonzero
lift, realistic flow; (d) superposition of
flows to produce the final flow past the
airfoil.
V9.19 Wing tip
vortices
(Photograph courtesy of
NASA.)
the flow pattern of Fig. 9.36b is changed to that of Fig. 9.36c2, and 122 the average velocity on the
upper surface of the airfoil is increased while that on the lower surface is decreased. From the Bernoulli
equation concepts 1i.e., pg V 2 2g z constant2, the average pressure on the upper surface is
decreased and that on the lower surface is increased. The net effect is to change the original zero lift
condition to that of a lift-producing airfoil.
The addition of the clockwise swirl is termed the addition of circulation. The amount of
swirl 1circulation2 needed to have the flow leave the trailing edge smoothly is a function of the
airfoil size and shape and can be calculated from potential flow 1inviscid2 theory 1see Section 6.6.3
and Ref. 292. Although the addition of circulation to make the flow field physically realistic may
seem artificial, it has well-founded mathematical and physical grounds. For example, consider the
flow past a finite length airfoil, as is indicated in Fig. 9.37. For lift-generating conditions the average
pressure on the lower surface is greater than that on the upper surface. Near the tips of the
wing this pressure difference will cause some of the fluid to attempt to migrate from the lower
to the upper surface, as is indicated in Fig. 9.37b. At the same time, this fluid is swept downstream,
forming a trailing vortex 1swirl2 from each wing tip 1see Fig. 4.32. It is speculated that the
reason some birds migrate in vee-formation is to take advantage of the updraft produced by the
trailing vortex of the preceding bird. [It is calculated that for a given expenditure of energy, a flock
of 25 birds flying in vee-formation could travel 70% farther than if each bird were to fly separately
1Ref. 152.]
The trailing vortices from the right and left wing tips are connected by the bound vortex
along the length of the wing. It is this vortex that generates the circulation that produces the
lift. The combined vortex system 1the bound vortex and the trailing vortices2 is termed a horseshoe
vortex. The strength of the trailing vortices 1which is equal to the strength of the bound
vortex2 is proportional to the lift generated. Large aircraft 1for example, a Boeing 7472 can generate
very strong trailing vortices that persist for a long time before viscous effects and instability
mechanisms finally cause them to die out. Such vortices are strong enough to flip smaller
aircraft out of control if they follow too closely behind the large aircraft. The figure in the margin
clearly shows a trailing vortex produced during a wake vortex study in which an airplane
flew through a column of smoke.