Block that Coefficient!

**I have to admit that when I encounter equations in a book, I usually just skip over them. Often some phrase like, “It may be readily seen that…” precedes them, and so I figure that since plenty of other people must be readily seeing whatever there is to see, there’s no reason that I need to as well.**

Nevertheless, I think of equations as something like fibre in food: nutritionally null, but indispensable to mental digestion. So…

It may be readily seen that *L = .00119V ^{2}C_{L}S*

That thing that just shot past was the classical equation for the lift force. What it’s saying is so beautifully simple that it’s worth a close look even from hardened equation-evaders.

L stands for Lift. V stands for speed (“velocity”) in feet per second, and S for area (“surface”) in square feet. CL is the lift coefficient – I’ll get back to that later. The curious little number .00119 stands for what air is like at sea level, basically, how heavy and thick it is. The equation says that sea level air moving one foot per second exerts a pressure of .00119 pound, or about 1/50th of an ounce, on an area of one square foot. This force is called the “dynamic pressure” of the moving air.

We can conveniently measure the dynamic pressure with a pitot tube and a manometer, a fluid-filled transparent U-shaped tube, or with an airspeed indicator, which amounts to the same thing. If we take several measurements at different speeds, we discover that the pressure is proportional to the square of the speed; double the speed and the pressure quadruples. If you go 100 times the speed – 100 feet per second or about 68 mph – the pressure is 10,000 times greater than at one foot per second: 11.9 pounds per square foot. Now we are in the range of pressures that can support an aeroplane, provided that our wing has a lift coefficient of 1.0.

In pre-Wright days, what lift coefficients could be obtained from wings was an unsettled, and unsettling, question. Wilbur Wright built a very ingenious wind tunnel in which some simple levers made it possible to compare the lift exerted by a model wing with the force exerted on a flat plate facing the wind. The beauty of this arrangement was that it found a ratio, not an absolute value, and so it was independent of the speed of the wind in the tunnel and also of the density of the air. It would have worked just as well under water. Wright could measure the actual force on the reference flat plate separately, using a spring balance, and compare it with the air pressure measured by a manometer. In this way he was able to find out how different variables, such as camber, angle of attack, and aspect ratio, affected the lift – and the drag – generated by his model wings.

What the Wrights and other early experimenters found was that only two things really mattered: aspect ratio and angle of attack. Lift increased regularly with angle of attack, but it increased less rapidly at lower aspect ratios because of spillage around the wingtips.

Later, more sophisticated measurements demonstrated that aerofoil shape, thickness and camber had marginal effects on the performance of wings, but that a wing without tip losses – simulated by running the test section from one wall of the wind tunnel to the other – gained about 0.1 unit of lift coefficient per degree in a range between minus and plus 10 degrees of angle of attack.

Every aerofoil has some angle of attack at which its lift is zero. For symmetrical aerofoils, it’s zero; for cambered ones it’s some small negative angle. The venerable NACA 23012, for example, has a zero-lift angle of about -1.5 degrees; the NACA 4415, which we would think of as an old-fashioned high-lift aerofoil, has a zero-lift angle of -4 degrees. The real angle of attack of a cambered aerofoil, therefore, is not the apparent or “geometric” angle of the chord line to the direction of flight, but rather the angle of the chord line to the zero-lift angle. The aforementioned 4415 aerofoil, when its chord line is parallel to the direction of flight, has a lift coefficient of .4. (In inverted flight, on the other hand, it needs an apparent angle of attack of eight degrees to achieve the same lift coefficient.)

There exists a great variety of aerofoil shapes, and the maximum lift coefficients they can produce vary a good deal. Generally, the thicker the section the higher its maximum lift, and, for a given thickness, the greater the camber the greater the maximum lift as well. For a typical thickness ratio of 12 percent, the maximum lift coefficient of an unflapped aerofoil might be around 1.3 to 1.5 at an angle of attack of around 15 degrees. The NACA 23012 – and this is one reason designers like it so much – gets up to almost 1.8. Specially designed aerofoils can achieve coefficients higher than two; with sufficiently effective flaps and leading-edge devices, values nearing four are possible before artificial means, such as blowing the flaps with engine air, must be resorted to for further improvement. As a practical matter, however, the lift coefficients of general aviation wings, even with slotted flaps, seldom get much higher than two.

In level, unaccelerated flight, the lift coefficient is the ratio of the wing loading to the dynamic pressure. At 150 knots (a.k.a. 252 feet per second), for instance, the dynamic pressure is .0019 x 252 x 252, or about 121 lb/sq ft. If the wing loading of the aeroplane is 20 lb/sq ft, then the lift coefficient must be 20/121, or .165. Cruising lift coefficients are typically in the range of .1 to .3 for low-altitude aeroplanes with low to moderate wing loadings; they get up to the range of .5 to .7 or more for highly loaded aeroplanes flying at high altitudes. In fact, some jets, cruising at very high altitude, are not too far from the stall.

You can also work the calculation the other way. If you suppose that an aeroplane has a maximum lift coefficient of two with flaps down, and that its wing loading is 20 lb/sq ft, then its stalling speed will be 91.7 fps or 54.6 knots (because .00119 x 2.0 x 91.7 x 91.7 = 20).

Now, let us gratefully meditate for a few moments upon the happy fact that deflecting air with a wing can yield a greater force than the air itself could exert by merely pushing, say, against the broadside of a barn, and what’s more, it can do it in exchange for much less effort.

Hence the demise of the paddle steamer. To propel a boat with a paddle, you had to put into the paddlewheel the same amount of force as it exerted upon the water. To propel one with a screw – that is, a propeller – on the other hand, you had only to overcome the drag of the screw. It happens that the lift obtainable from a wing may exceed the drag by a huge factor – more than 100. In practical aeroplanes the drag of non-wing elements, like fuselages, stabilizing surfaces, engine cooling, cabin ventilation and so on, nibbles away at that value, but there are sailplanes with lift-drag ratios of 60. Even jet airliners invest only a little more than a twentieth as much effort to overcome drag as they get back in lift.

Next month – Fermat’s last theorem!