Part
3. Dragonfly wings: Responding to
pressure (continued).
In the last installment we learned that the pressure of the
air that provides the aerodynamic lift on gliding wings is not distributed
uniformly from costal margin to trailing edge, but is highest near the leading
edge of the wing. The structure of
wings reflects this anterior positioning of the lift in the clustering of the
major longitudinal veins and their corrugations near the costal margin.
Figure 1 depicts the pressure distribution as it might appear on a
dragonfly wing cross section, with the pressure below atmospheric above the wing
and above atmospheric below the wing.
In
Figure 2 I have added to the picture two lines terminated by arrowheads.
The first is a nearly vertical line directed upward and labeled “Total
Aerodynamic Force”. It originates
at the “quarter-chord” point, 25% of the way from the leading edge of the
wing, which is called the “center of lift” of the wing and is the point at
which the net effect of the aerodynamic pressure distributed over the wing can
be considered to act. The arrow
itself is called a “force vector”, and is a line for which both its length
and its direction are representative. The
line length represents the total amount of aerodynamic force (in Newtons)
generated by the pressure acting on the wing.
The direction of the line indicates the direction in which the force
acts. This force is the net result
of the pressure distribution, and is essentially equivalent to the pressure
distribution in mechanical terms. The
second line is horizontal, and is labeled “Velocity”.
This is the “Velocity Vector”, and, again, the length of the line
represents the speed of the wing through the air (or, equivalently, of the air
moving past the wing), e.g. in meters per second, and the direction the
direction in which the air is moving.
Now,
just as we can effectively replace the pressure distribution on the wing by an
equivalent aerodynamic force vector acting at the center of lift, we can also
represent the total force vector by two other vector forces, one acting in a
direction parallel to the velocity vector, and one in a direction perpendicular
to the velocity vector. This is
shown in Fig. 3, where we have labeled the vector parallel to the velocity as
the “Drag” force, and the vector perpendicular to the velocity as the
“Lift” force. The two mutually perpendicular vectors that add up to the
total aerodynamic force vector are called force “components”, and resolving
the total force into components in line with and at 90 degrees to the velocity
vector and calling them drag and lift, respectively is the standard way of
depicting the net aerodynamic force on a wing.
Now let’s look at the way the air pressure supporting the
insect in flight varies along the wing length, or span, from base to tip.
One way to picture this would be to draw lots of wing cross sections
adjacent to one another along the wing from base to tip as in Fig. 4a, and to
imagine that each section is actually adjacent to the two adjoining sections so
that the pressure distribution is continuous.
An equivalent way to depict the wing would be to show the sections
adjacent to one another, but with the aerodynamic forces shown rather than the
pressure distribution, as in Fig. 4b.
If we were to choose any point along the wing span, for
example, somewhere near the middle of the wing. we would see that the inboard
and outboard sections had the same pressure distribution, so the constant
pressure distribution along the span makes sense. But as we near the end of the wing, this distribution no
longer holds. At the wing tip, the
wing surface ends and there is nothing to keep the positive pressure below the
wing from leaking around the tip to the top where the pressure is lower.
This results in a net flow of air toward the wing tip beneath the wing
and toward the wing base above the wing at the tip, and a rotational flow around
the tip as shown in Fig. 5.
This leakage of pressure would result in the pressure distribution decreasing near the wing tip, so that this region supports less of the weight of the insect. The spanwise pressure distribution would thus look more like Fig. 6 tha Fig. 4a.
The leakage of pressure from below to above the wing near the wing tip also results in a movement of air around the wing tip. Fig. 7 shows the air movement from below to above around the wing tip. Not shown in this picture is the movement of air past the wing from front to rear. If we add these two flows together, we find (Fig. 8) that the air flow beneath the wing (gray arrows) is deflected toward the wingtip while the airflow above the wing (black arrows) is deflected toward the wing base. The net flow twirls around itself, forming what is called the wing tip vortex.
The
developed tip vortex looks like a miniature "tornado" that originates
at the wing tip and extends downstream behind the wing, as shown in Figure
9. The trailing vortex extends far behind the wing, and persists for a
long time (vortices are very effective at transferring momentum in
fluids). Some photographs of airplanes in flight in conditions in which
their trailing vortices are rendered visible may be found on the efluids.com web
site:
http://www.efluids.com/efluids/gallery/trailing_vortices_2.html
http://www.efluids.com/efluids/gallery/trailing_vortices_c130.htm
http://www.efluids.com/efluids/gallery/cessnajet_1.htm
http://www.efluids.com/efluids/gallery/Trailing_vortices_1.html
Vortices
are key structures in the flight of insects, and we will look at other kinds of
vorticity and other vortex structures associated with wings in future
installments of this series. For now, let's make one further observation
about the spanwise distribution of pressure and air forces near wingtips.
The tip vortex causes the air near the tip of the wing to flow downward.
If we add this downward component to the incident airflow velocity, as shown in
Fig. 10, we see that the effective angle of incidence of the airflow at the
wingtip is less than that near the wing base. Thus the wing lift, the
component of the aerodynamic force that is parallel to the local airflow vector,
is effectively tilted backwards with respect to the lift vector for segments of
the wing near the base (Fig. 10). Thus the lift force at the wing tip
produces an effective small amount of extra drag, called the induced drag.
Early aerodynamics researchers were able to show that it is possible to minimize
the total induced drag for a wing by choosing an appropriate wing shape.
In fact, it was shown that if the variation of the wing chord with span was
elliptical, the induced drag would be minimized. An example of such a wing
is that of the World War II British fighter, the Supermarine Spitfire, shown in
Fig. 11 with the beautiful elliptical-shaped wings and tail surfaces. Also
on this picture is a dragonfly with its pretty tapered wings - which don't
differ a great deal from being elliptical out near the tip. Is this to
make the wing more efficient or just a product of the way wing cells develop
into a wing, or both?
Key facts: