The Bernoulli Brigade

This Bernoulli fellow, who lived several hundred years ago, was stating a simple fact of the physics of fluids (actually, fluids flowing through pipes) that he considered more or less self-evident. He would be no better known to pilots today than d'Alembert or Torricelli had his name not come to be associated with an appealingly simple -- but unfortunately flawed -- explanation of lift. Because in fact air particles marching past a wing are not like an ordered mass of soldiers -- call them the Bernoulli Brigade -- who part at the leading edge and rejoin their buddies at the other end. There is nothing to cause the upper-surface flow to arrive at the trailing edge at the same time as the lower-surface flow, and it doesn't. Actually, it gets there sooner.

But it is not even necessary that the distance air travels along the upper surface of a wing be greater than the distance it travels along the lower, as the folded paper airplane, the butterfly and the simple balsa glider show. If the angle of attack is small enough, all that is necessary is that it be positive; that is, it is the fact that the wing is at a nose-high angle to the passing air that is fundamentally responsible for the generation of lift. Curving the flat surface, adding thickness and shaping it like an airfoil are techniques for reducing its drag and allowing it to produce more lift and achieve a greater angle of attack without stalling; but the airfoil shape is not indispensable -- it is a refinement.

What is remarkable about airfoils, however, is how well they work. They allow wings to multiply air pressure. The lift generated by an ordinary wing when it is just about to stall is about 50 percent greater than the pressure exerted by air striking a wall at the same speed. Think about that -- it's really pretty remarkable. The force exerted by wind blowing "past" an object can be greater than the force exerted by that wind blowing "against" it, provided that the object has a certain shape. It's almost as if you could change your weight by making faces while standing on the scales.

The pressure of air blowing directly against a flat surface is called the dynamic pressure. It is about 25 pounds per square foot (psf) at 100 mph (I am using mph here, rather than knots, to preserve the easily remembered 100:25 relationship). The maximum lift of an ordinary wing, no flaps, is between 30 and 50 percent more than that. The lifting force is a function of the square of speed; at 50 mph it is a quarter of 25 psf, at 200 mph four times 25. The ratio between the dynamic pressure and the lift force just before the stall is called the maximum lift coefficient, and it is around 1.5 for plain airfoils. Good slotted flaps can push it above 3.0.

For single-engine airplanes weighing less than 6,000 pounds, federal regulations require a stalling speed no higher than 61 knots. (This rule has to do with the chances of surviving a forced landing, which is considered more probable in a single-engine plane than in a multiengine one; the value of 61 knots -- 70 mph -- is arbitrary, a compromise between low landing speed and a reasonably small wing area.) The dynamic pressure at 61 knots is approximately 12 psf. An airplane without flaps can therefore weigh -- theoretically at least -- no more than about 18 pounds per square foot of wing area.

This is where flaps come in. A simple plain flap -- something similar to an aileron that moves only downward -- can add another 50 percent to the maximum force-multiplication of the wing, bringing the permissible wing loading up to 27 psf. A slotted flap can raise it to 35 psf, a multiple-slotted flap to nearly 40.

These figures are ideal ones. They suggest that a Cirrus or a Columbia with a gross weight of 3,400 pounds could make do with a wing hardly larger than the front door of your house. In fact, however, their real-life wing loadings do not exceed 25 psf. Where did the rest of the lift go?

The answer begins with the fact that wings have tips. The pressure difference between upper and lower surfaces causes spillage at the tips -- this is the reason for the tip vortex -- and robs the wing of 5 to 10 percent of its theoretical lift. Another loss occurs at the center of the wing, where the fuselage interrupts the airflow. The imaginary portion of the wing that lies within the fuselage -- reported wing area includes this hidden part -- produces, in reality, no lift. But changes in pressure are gradual, not instantaneous, and so the effect of the fuselage is to produce a dip rather than a sharp-edged gap in the spanwise distribution of lift. Depending on the fraction of the wing area that lies within the fuselage, another 10 or 15 percent of the potential lift may be lost here.

Next, flaps seldom extend over the full trailing edge of the wing. Usually, the outboard 40 or 50 percent of the span is reserved for ailerons. This outboard portion of the wing would produce only about a third of the lift, flaps up, because of the tip losses I mentioned before. But flapped wings produce their lift at a lower angle of attack than unflapped ones do -- that's why the nose comes down when you put the flaps down -- and so the unflapped outer portion of the wing never gets to its own stalling angle of attack, and it consequently yields less lift than it is really capable of.

Finally, the flap has its own tip losses. The accompanying computer-generated illustration shows a wing with a single-slotted Fowler flap -- this one moves all the way back to the trailing edge before deflecting 30 degrees. It is close to its stalling angle of attack. The colors on the fuselage and wing encode pressures; colors tending toward the red, for instance on the upper surface of the wing, indicate low pressure; you can see the lifting force on the wing that is holding the airplane up. Green is neutral, purple is high pressure. The strings are streamlines -- the paths followed by air particles flowing past the wing -- and their colors, too, indicate air pressure. You can see that low pressure is not confined to the wing surfaces, but forms a kind of cloud -- graphically, a pink glow -- above the wing.

Pressure and velocity are inversely related; this is the physical fact that we associate with Bernoulli. Thus, the reddish portions of streamlines indicate accelerated flow, and the blue-green portions retarded flow.

Think of the strings as columns of soldiers in the Bernoulli Brigade. They seem to have had a jolly time last night. (Parenthetically, "Taps," the mournful bugle tune played to summon soldiers to their barracks for the night, gets its name from the taps of beer barrels in the local pubs.) Columns of molecular soldiers are thrown into violent swirls by the ends of the flaps, each of which generates its own private tip vortex (whose core is visible, if you carefully pick your seat in a landing jet on a moist day, as a trembling gray rope). The flow over the middle of the flap veers inward above the wing and outward below it, producing a scissors-like shearing at the trailing edge. The low pressure generated by the flap distorts the ranks of streamlines on the outer panel, pulling them inward, while the vortex at the tip of the wing is quite weak, indicating that not much lift is being generated out there.

It should be apparent from this picture why the traditional account of wing lift is wrong. Air molecules do not file past a wing in neat rows and rejoin their mates at the trailing edge. Not only do the upper-surface particles outrun the lower-surface ones, they go every which way while doing so. Streamlines that curve obliquely across the wing don't even see the airfoil as it was designed, but instead a whimsically distorted version of it. And this, in plainly visible form, is why a wing whose flap can theoretically carry 36 psf at 61 knots will, in reality, support only 25.