Page 236 - Aerodynamics for Engineering Students

Finite wing theoly 21 9

To find the velocity at P due to the length AB the sum of induced velocities due to
all such elements is required. Before integrating, however, all the variables must be
quoted in terms of a single variable. A convenient variable is $ (see Fig. 5.10) and the
limits of the integration are
(; 1
to OB=+--@
$,4 = -(;-a)
since $ passes through zero when integrating from A to B.

sin 6’ = cos 4, r2 = h2 sec’ $
ds = d(h tan 4) = h sec’ $d $

The integration of Eqn (5.8) is thus

-
-- (cosa+cosp)
47rh (5.9)
This result is of the utmost importance in what follows and is so often required that it
is best committed to memory. All the values for induced velocity now to be used in
this chapter are derived from this Eqn (5.9), that is limited to a straight line vortex of
length AB.
The influence of a semi-infinite vortex (Fig. 5.11~) If one end of the vortex stretches
to infinity, e.g. end By then p = 0 and cos p = 1, so that Eqn (5.9) becomes
r
v = - (cos0 + 1) (5.10)
47rh
When the point P is opposite the end of the vortex (Fig. 5.11b), so that
CY = 7r/2, COSQ = 0, Eqn (5.9) becomes
r
)I=- (5.11)
47rh
The influence of an infinite vortex (Fig. 5.11~) When a = ,8 = 0, Eqn (5.9) gives
r
v=- (5.12)
27rh
and this will be recognized as the familiar expression for velocity due to the line
vortex of Section 3.3.2. Note that this is twice the velocity induced by a semi-infinite
vortex, a result that can be seen intuitively.
In nature, a vortex is a core of fluid rotating as though it were solid, and around
which air flows in concentric circles. The vorticity associated with the vortex is
confined to its core, so although an element of outside air is flowing in circles the
element itself does not rotate. This is not easy to visualize, but a good analogy is with
a car on a fairground big wheel. Although the car circulates round the axis of the
wheel, the car does not rotate about its own axis. The top of the car is always at the
top and the passengers are never upside down. The elements of air in the flow outside
a vortex core behave in a very similar way.