## Thin Airfoil Theory: History in a Nutshell

Okay, some thin airfoil theory: All airfoils work the same. It took about a hundred years to think that up, so give it its due regard. It started with D'Alembert's paradox over Newton's formulation of fluid dynamics, and it ended with Lanchester and Glauert and Prandtl basically kicking Newton's stuff to the curb. What they came up with is really quite elegant: If you take the top view of a trout, you have a symmetrical airfoil. If you watch him (from above) bend his tail left and right, you have increases and decreases of "camber" which is a fancy word for curvature or bent-ness. When he bends his tail to the left, he "goes left" and when he bends his tail to the right, he goes right.

D'oh! It took mathematicians and experimenters hundreds of years to figure this out? Yes! That's why God invented audacity, arrogance, and the "can do" spirit. The Wright brothers were already flying while these guys were still scratching their heads over trout and seagulls. But, they were brilliant!

What the Wright brothers contributed to all this is that they built a wind tunnel and proved to themselves that the location of the lift of the wing wanders all over the place along the chord (fore and aft). A hundred years before, Sir George Cayley had noticed this problem and been stumped by it, because without a good fix on the location where the lift came from he couldn't figure out where to locate the CG of his aerial steamship. He wanted to locate the CG below the location or "center" of the lift so that the airplane would "hang" under the "center of lift." The Wright's contemporary, Samuel Pierpont Langley, had the same problem and tried to fix it by distributing the lift among two wings mounted fore and aft on the reasoning that the weight suspended between would be somehow magically distributed appropriately.

Cayley was stumped and Langley was wrong (the proof is in the "flight" photos), but it wasn't their faults. The airfoils studied by Cayley and Langley were either flat or arcs of circles and their centers of lift truly did wander all over the place; but the Wright's wind tunnel told them that there were some airfoil shapes that were mostly curved in the front part and basically flat in the aft part (like birds wings -imagine that!) that had special characteristics. Specifically, the Wrights noted that the center of lift of these airfoils never went forward of a certain point.

The Wrights also observed that the location of the center of lift on these airfoils was in a predictable accordance wit the angle of attack. Thus, they were able to devise to put the airplane's CG forward of the forward-most point that the center of lift would ever go to, and then to build an auxiliary wing ("tail") that would provide the small balancing force needed to keep the airplane level. Cayley had been the father of this concept; the Wrights had found the key to unlock its sublime simplicity. They chose to put their "tail" out front to help lift the weight of the machine, but in later years other considerations caused it to be moved to the back of the airplane for that classic "conventional" look.

Anyway, what the Wrights achieved was to figure out how to make airplanes stable. Then they went ahead and made their airplane unstable on purpose! Returning to the battle of the pitching moments, the Wrights reasoned that having the CG near and just in front of the forward-most location of the center of lift would ensure that their forward elevator would always be required to provide upward lift. Further, as the airplane's angle of attack decreased with increasing airspeed, they knew that the center of lift would move aft. The greater distance from the center of lift to the CG would result in a greater nose-down pitching moment to be overcome by the elevator, but the elevator would already be making more lift due to the greater airspeed.

They saw the balancing act, the battle of the pitching moments going on, and chose to solve it by letting the elevator be too large and too far forward; they opted for an unstable airplane, one in which the change of pitching moment due to the elevator was always greater than the change in pitching moment due to the wing center-of-lift/CG relationship. So, if a small disturbance pushed the nose up, the elevator's new angle of attack would ensure that it "won" the battle of the pitching moments and would cause the nose to continue to go up further.

Today, our design philosophy is the opposite. We want the aft wing (of any airplane) to always win the battle of the pitching moments. If an updraft increases the lift of the main wing, we want the increased lift of the tail as it encounters the updraft to pitch the main wing down and cancel the increased wing lift from the gust. This is called "positive static longitudinal stability."

In a roundabout way this also gives us speed stability. If the front wing gets to making too much lift because the airplane is going too fast, recall that the nose-down pitching moment made by that wing is now greater than it had been at the slower speed. We want the tail to translate the excess airspeed into a climb by overcoming the nose-down pitching moment of the wing - again, we want the tail to win the battle of the pitching moments, bringing the nose up to slow the airplane back down to its original speed.

## Thin Airfoil Theory II

Once it was recognized that the Wrights were on to something with their curved-in-front, flat-in-back airfoil, the mathematicians took almost no time coming up with an explanation to fit the facts. By the end of WW1 the foundations of thin airfoil theory included the notion that the thin airfoil need not be thin! The concept became to reduce the thickness of the Wrights' curved-in-front, flat-in-back airfoil until it became a single curved line, then to "wrap" the nice smooth contour of the top view of a trout around that line. The line became known as the "camber" line and the trout-shape was called the thickness distribution.

Put a nice, smooth thickness distribution onto a camber line you liked and there was almost no performance difference between the camber line alone acting as an airfoil and your new, thick airfoil. Thus, wings could be made thicker and the structure could all go on the inside. Empirical airfoils and their derivatives were created and investigated by starting with a straight camber line plus a thickness distribution, then varying the camber and/or thickness in increments. (There really was a thickness distribution based on a trout, too!) Systematic investigations of camber line shapes and thickness distributions evolved under the guidance of the mathematicians and the auspices of the N.A.C.A until a large body of research was at hand. Then came a revelation.

What the Wrights had sought was to know the location of the center of pressure, and this they achieved. They mapped the location of the center of pressure and presented it in a chart versus angle of attack. That system of reporting airfoil characteristics persisted into the late 1920s. Today, we recognize that the behavior of the center of pressure is even more organized than that, lending itself to a different formulation. Indeed, as the angle of attack is increased, the lift force is increased while the center of lift moves forward; as the angle of attack is decreased, the lift force decreases while the center of lift moves aft.

If we multiply the lift force by the location of the center of lift, we get a moment. If we take a running calculation by multiplying the lift force by the location of the center of lift as we vary the angle of attack, we get a plot of moment versus angle of attack. But that presupposes that we are measuring the moment from some reference point. Historically, the reference point had been the leading edge. But nature has a surprise!

For almost all thin airfoils (up to about 20% thickness) it turns out that there is one point on that airfoil where the moment is (relatively) constant for all angles of attack. Just as Euler had discovered that the math gets easier if we use the CG as our reference point, so this new point makes an excellent reference point, too. Find this point and you discover that no matter where the center of lift runs to on the wing's chord, the moment about this point due to the lift force is "the same." Well, it gets better: It turns out that this point is in the same place on almost all thin airfoils, too. You don't need to know the particular airfoil section in question to know that the "aerodynamic center" is at the quarter-chord point, or 25% of the way back from the leading edge. (This location varies from 23% to 27%, with the vast majority of airfoils right at 25% and only some special cases and oddballs out near 23% or 27%.) A thin airfoil is a thin airfoil is a thin airfoil; they all behave essentially the same.

Okay, we have to be careful about the aerodynamic center. The last paragraph seemed to imply that the pitching moment from a particular airfoil is a constant, but that is not the case in airplane use. In the wind tunnel it is true, but not when applied to airplanes.

In wind tunnels, the airspeed is held constant. As the angle of attack is changed, the product of the lift force multiplied by the distance from the aerodynamic center to the center of lift does not change; however, the lift force changes along with the location of the center of lift. As lift increases at greater angles of attack, the center of lift moves forward, reducing the moment arm. Larger lift multiplied by smaller arm yields a constant moment; the moment turns out to be constant for all useful angles of attack. However, the changing lift force is not allowed to happen to airplanes in steady flight.

For airplanes, as the angle of attack is increased, the airspeed must be decreased in order to maintain the balance of forces (Lift + Weight = 0). Thus, for airplanes, the dynamic pressure (q) changes whereas in the wind tunnel it does not. Although the coefficient of pitching moment stays constant about the aerodynamic center both in the wind tunnel and on the airplane, the actual pitching moment does not remain constant on the airplane. In fact, on the airplane the pitching moment is at a maximum at high speed and at a minimum at slower speed. Go back and re-read the section titled "Coefficients vs. Actual Forces and Moments" if this is confusing to you. Continued in Q-Talk issue 102.

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