In the first installment of this series, we covered wing loading – the weight of the dragonfly divided by its total wing area – which is a measure of the average pressure that the air must exert on the wings to support the weight of the insect. You may remember that we are assuming the dragonfly is gliding and not flapping his wings, in order to simplify things here at the beginning of our queries.
But all three of these representations are quite ficticious – the aerodynamic pressure acting on a wing is not constant at all. In general, we can separate the problem of how the pressure on the wing varies from the leading edge to the trailing edge from the problem of how the pressure varies from the base of the wing to the tip.
Let’s begin by looking at how the pressure is distributed along the chord of the wing – the leading to trailing edge (Figure 2). We will be looking at the wing end-on, from the wing tip, at a representative location about midway out on the wing. We will represent the wing cross section as a straight line to begin with, and will use more representative cross sections later.
We will show the air movement past the wing by using what are called streamlines. Streamlines are lines which are always parallel to the air velocity. This means that there is no air flowing across the streamlines. We could demonstrate streamlines in a wind tunnel by injecting smoke into the airstream through evenly spaced tubes so that the smoke enters parallel to the flowing air. In the steady flow of the tunnel, the lines formed by the smoke would be equivalent to streamlines.
Figure 3 shows the streamlines for air flowing past a circular cylinder. The streamlines are parallel and horizontal far to the left of the cylinder, representing a horizontal uniform flow of air in the tunnel. The streamlines are once again parallel and horizontal far to the right. However, they curve around the cylinder as they approach it, and the streamlines get farther apart just before they reach the cylinder and as they begin to go around it. They get closer together as they pass by the top and bottom of the cylinder. Then they spread apart again , finally ending up parallel once the air is far enough beyond the cylinder.
The air flowing along or near the line pasing through the center of the cylinder slows down because the cylinder blocks the flow. As the air flows up or down and around the cylinder, it gets squeezed into a smaller area and accelerates. The flow speed is depicted in Fig. 3 by the arrows. Far to the left they are all the same length. At the top and bottom of the cylinder, the arrows are longer. Far to the right they are the same length again.
Since no air flows across the streamlines, the mass of air per second passing through any point between two streamlines must be the same. Since the air at these low speeds has constant density, that means that when streamlines get closer together, the air passing between them must be flowing faster than where they are farther apart.
You may recall Bernoulli’s Equation from your high school physics class: p + ˝ρV2 = constant. It basically states that the energy of the flow between any two streamlines is conserved. It also says that, wherever two streamlines get further apart, the velocity goes down and thefore the pressure goes up and wherever two streamlines get closer together, the velocity goes up and the pressure goes down.
The bottom half of Figure 3 shows the pressure distribution over the upper and lower surfaces of the cylinder resulting from the flow pattern shown in the top half of the figure. In this case the pressure is given in psi and a horizontal line at 14.7 psi marks atmospheric pressure. Since the flow is symmetrical, the pressure is the same on the upper and lower halves of the cylinder, and the curves overlap. The curve shows that the pressure near the front and back sides of the cylinder is greater than atmospheric, and that over the rest of the surface is less than atmospheric. Since the pressure is symmetrical, there is no net force acting on the cylinder.
Figure 4 shows the flow past a horizontal flat plate. When the plate is horizontal, there is no differrential pressure generated and the pressure acting on the plate is simply the atmospheric pressure (in this and the rest of the figures, the pressure is in metric units – atmospheric pressure is 101.26 kilo-Pascals). In order to generate aerodynamic lift, we need to either alter the shape of our wing or incline it at an angle to the flow.
Figure 5 shows the flow past our plate when it is inclined
at a positive angle of attack of 15 degrees.
Note how the streamlines beneath the wing widen (flow decelerates and
pressure increases) and those above the wing get closer together (flow
acelerates and pressure decreases). The
narrowest region is that of the streamline that curves sharply around the
leading edge of the wing. Look at
the pressure plot. The solid curve
is the pressure on the upper surface of the wing, the broken curve the pressure
on the lower side. Note that the
pressure on the upper side is less than atmospheric pressure, and it becomes
very much less near the leading edge. 
The
pressure on the lower surface is greater than atmospheric for most of the wing
chord, becoming less than atmospheric pressure in the small area just below the
leading edge. The combination of
higher pressure below the wing and lower above the wing yields a net supporting
pressure that equals the lift of the wing.
That very large lower than atmospheric pressure dip on theupper wing
surface at near the leading edge is called “leading edge suction”, and
occurs on all wings. It is the need
for the flow to accelerate around the leading edge that eventually causes the
wing to “stall” and lose its lift at very high angles of attack (more on
this in a later installment).
We can make a wing in a shape that will generate lift without inclining it at an angle to the flow. Figure 6 shows a flat plate that has been curved to have a convex upper surface (called positive “camber”). Again we see wide spacing of lower surface streamlines and narrow spacing of upper surface ones. This yields a greater than atmospheric pressure on the lower surface and a lower pressure on the upper surface and therefore a net lift force.
In Figure 7 this cambered wing is inclined at 15 degrees. Compare this flow field and pressure distribution to that of Fig.. 5 and you will see that the cambered wing generates higher lift than the flat wing at the same angle of attack. We see that it would be beneficial for insects to have wings that curve into a cambered shape under load rather than to stay flat.
It is usually customary to subtract the pressure on the upper wing surface from the pressure on the lower wing surface and to show the net pressure distribution on the wing as we have done in Figure 8 for a flat plate wing at 15 degrees inclination. We still see that much of the lift on the wing is generated by large pressure at the leading edge – certainly not a uniform or constant pressure – but note that the average pressure still needs to add up to the wing loading – 5 N/sq m for our Anax junius.
The final picture, Figure 9, shows the pressure distribution over a wing having the cross sectional shape of an Anax junius hind wing. You can see the corrugations at the leading edge (left end). Those corrugations, of course, stiffen and strengthen the wing so that it resists bending under pressure. And, amazingly enough, they are concentrated in the part of the wing where they need to be – near the front or leading or costal edge, where the pressure is the greatest. A very nice demonstration of functional morphology – adaptation of shape to function. The gray arrow locates the center of lift – the point where all the lift could be concentrated at a point and have the same net effect as the net pressure distribution.
The figures in this installment were generated using a marvelous bit of software provided on the internet by NASA for anyone to use or to download. It is an airfoil (wing section) simulator that can analyze cylinders, flat wings, cambered wings, or airplane-like wings, and plot the flow field and pressure distribution for you. You can access or download the software package on the web at:
<http://www.grc.nasa.gov/WWW/K-12/airplane/foil2.html>.
Key facts:
Next time: The pressure distribution along the length of the wing from base to tip.