calculating coefficients

A century ago, after the first flight: What had the Wrights really proved? Well, for starters, more accurate math.

A mid-December dawn near the North Carolina town of Kitty Hawk. Wilbur and Orville Wright haul an ungainly craft from a shed and ready it for another attempt at powered, manned flight.

A north wind blowing 20 to 30 mph makes the near-freezing air feel even colder. The steady wind—which had enticed the brothers here in the first place—this day blows rawer, stronger than usual.

William Tate, who has witnessed their earlier attempts, doesn't show this time. "No one but a crazy man would try to fly in such a wind," he explains.

Perfect flying weather had come earlier that week, but the brothers had promised their father, a bishop of the United Brethren Church, that they wouldn't try to fly on Sunday. They make an attempt on Monday, the 14th, and damage the flyer. Three days pass before they can try again.

Weather or not, the boys are expected home in Dayton, Ohio, for Christmas.

The Wrights attempted to fly under such conditions because they believed flight was possible. They backed up this confidence with mathematics.

Wilbur would later write, "One of the most gratifying features of the trials was the fact that all of our calculations were shown to have worked out with absolute exactness."

What the Brothers Knew

When lift overcomes an airplane's weight and thrust exceeds its drag, flight becomes possible. Lift acts perpendicularly to the flight path, counterbalancing weight. Drag acts opposite to the flight path, the result of air friction and pressure distribution.

When the brothers first became interested in flight, they studied the results of their predecessors, such as Germany's Otto Lilienthal. He had experimented with gliders and developed tables for the lift achieved by various airfoils. He used two established formulae for calculating the lift and drag of these airfoils:

L = kSV²c_L and D = kSV²c_D.

L is lift (in pounds), D drag (in pounds), and k the Smeaton coefficient of air pressure. S is the surface area of the airfoil (in square feet), V the velocity relative to the wind (in miles per hour), c_L is the coefficient of lift and c_D the coefficient of drag.

The Wrights used these two equations as well. The brothers could easily determine two of the variables, surface area and velocity. But, on their way to achieving flight, they would struggle with the other two variables in these formulas.

The coefficient of lift, c_L, gives a ratio of the pressure on an airfoil to the pressure on a flat plate oriented perpendicularly to the wind. Its value is different for each shape and angle of attack, the angle between the relative wind and the chord of the wing.

The Wright gliders of 1900 and 1901 used wings similar to Lilienthal's and the brothers relied on his calculations for determining c_L. The lift and drag generated by these gliders were disappointingly lower than the amounts predicted by the equations. After their Kitty Hawk trials, the brothers, despairing over the rift between theoretical and actual values, returned home to Dayton, ready to quit. Wilbur predicted that if flight were possible, it wouldn't be realized for a thousand years. "Not within my lifetime," he said.

In the winter of 1901-02, the Wrights began to reconsider the problem of flight. Could Lilienthal's calculations have been wrong?

The Wright brothers plotted out the relationship among lift, thrust, weight, and drag.

To the front of a horizontal wheel mounted on a bicycle they attached a curved wing with a surface area of one square foot set at an angle of attack of 5 degrees. They also attached to the wheel a flat plate with a surface area of 0.66 square feet set vertically one-quarter of the way around the wheel from the curved wing at a 90-degree angle to the airflow.

As the brothers peddled around Dayton, they reasoned that the pressure of the two airfoils would balance each other and the wheel wouldn't move, if Lilienthal were correct. But, the pressure on the flat plate consistently exceeded the lift produced by the curved wing.

What the Wrights needed was a more precise method for measuring c_L. So they built a wind tunnel. Then, they constructed a "lift balance" to measure the lift of their airfoil in terms of the pressure on a square plate of equivalent area that was held perpendicular to the wind. They tested more than 50 different airfoils at 14 different angles of attack.

According to the Wrights, "The lift at a given angle of incidence [angle of attack] is to the pressure on a square plane of equal area at 90 degrees as the sine of the angle indicated by the pointer [on the lift balance] is to one." The sine of this angle is c_L.

Airfoil No. 12—which was similar to the final construction of the Flyer—yielded, at an angle of attack of 5 degrees (also similar to the angle of attack used for the first flight), a c_L of 0.515.

To calculate the coefficient of drag, the Wrights created a "drift balance" to measure an angle they called the tangential, which "gives the inclination of the chord [of the wing] above or below the horizon."

The Wrights built this lift balance after discovering a discrepancy between actual and predicted values for lift and drag.

They added the tangential to the angle of attack to find the gliding angle, the angle between the horizontal and the relative wind. For example, using airfoil No. 12 at a 5-degree angle of attack, the tangential was measured at 1 degree, so the gliding angle was 6 degrees.

The tangent of the gliding angle measures the ratio of drag to lift, so c_D can be calculated.

tan G = D/L = c_D/c_L
c_D = c_L tan G

For airfoil No. 12 at an angle of attack of 5 degrees, c_D is computed as follows:

c_D = c_L tan G = (0.515) tan 6° = (0.515) (0.1051) = 0.0541

When the Wrights compared their results with those of Lilienthal, they found only small disagreements. Wilbur wrote, "It would appear that Lilienthal is very much nearer the truth than we have heretofore been disposed to think."

With the coefficients of lift and drag holding up to their scrutiny, the Wrights turned their attention to the only other possible source of error in the equations, the Smeaton coefficient of air pressure.

Questioning Authority

English civil engineer John Smeaton published a paper in 1759 on the efficiency of windmill blades. It won a Royal Society Gold Medal. In an appendix, Smeaton included a number that, when multiplied by the square of the wind velocity (in miles per hour), gave the pressure in pounds per square foot on any flat surface presented at right angles to the wind. Smeaton had determined the number to be a constant approximately equal to 0.00492. This value had been in use for more than 140 years when the Wrights began tinkering with it.

The Wrights would later mention the difficulty in measuring the Smeaton coefficient: "When this simplest of measurements presents so great difficulties, what shall be said of the troubles encountered by those who attempt to find the pressure at each angle as the plane is inclined more and more edgewise to the wind?"

Lilienthal referred to the Smeaton coefficient as being "generally known." Orville Wright's copy of Lilienthal's book shows that Orville had underlined those words and put a question mark in the column next to them.

The Wright brothers tested more than 50 different airfoils and various angles of attack in the tunnel using their lift and drift balances (top). Number 12 (above) would eventually emerge as the champ.

The Wrights started questioning the value of the Smeaton coefficient as early as the summer of 1901 and by the end of that flying season they were convinced that the Smeaton coefficient was a major cause for error in their calculations. If they had known at the beginning of their experiments that this coefficient was so much in error, Tom Crouch, the author of The Bishop's Boys, writes, "They would never have begun."

The Wrights' value for the Smeaton coefficient was 0.0033. The value currently accepted for this coefficient is 0.00327. How did they achieve such an accurate result?

No record of the exact calculations exists. However, by measuring the drag on a glider directly with a spring scale and restraining ropes, the Wrights were able to calculate the Smeaton coefficient from the equation for drag: D = kSV²c_D. Solving for k gives

k = D/SV²c_D.

According to Harry Combs, the author of Kill Devil Hill, this calculation was "the key to the whole show."

December 17, 1903

The surface area of the Wright Flyer is 512 square feet. Assume a wind velocity of 25 mph and a ground speed of 7 mph (the ground speed actually achieved on December 17), and the velocity relative to the wind becomes 32 mph.

Lift (0.0033 x 512 x 322 x 0.515) came to 891 pounds. The Flyer weighed 605 pounds and Orville 140 for a total weight of 745 pounds.

Drag (0.0033 x 512 x 322 x 0.0541) equaled 94 pounds, and thrust was 132 pounds.

Since lift was greater than weight and thrust was greater than drag, flight was possible. Using information similar to this, Orville and Wilbur attempted flight.

If we assume a wind velocity of 22 mph for the flight, the formula gives a lift of only 732 pounds, not enough to overcome weight. Similarly, if we assume a wind velocity of 32 mph, for a relative velocity of 39 mph, the drag generated is 139 pounds, which is more than the thrust.

The Wright Flyer was only marginally flyable. That the Wrights figured out these margins is a tribute to their genius. Perhaps all they proved in 1903 was that flight was possible on a cold and windy day in North Carolina. But, that was enough.

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