I have recently read that Induced drag is related to wing area as a square root function. That is, all else being equal, that if you reduce the wing area to 50% of the original value, the induced drag will be reduced to 25% of the original value. YES?-NO?-Perhaps if--? Can you reference?

From : Don Stackhouse

Well, you were headed in sort of the right direction, but took a few wrong turns. Induced drag is inversely related to the square of the wing span. In other words, if you doubled the wing span, you would have about one fourth the induced drag. More span (not area) equals less induced drag.

There is no direct relationship between wing area and induced drag. Whether there is an indirect relationship between induced drag and wing area depends on how you go about changing the wing area, in particular what happens to the wing span.

There are frequent references to induced drag being related to aspect ratio (the span squared, divided by the wing area; aspect ratio is a measure of how skinny the wing is). There are even some perfectly valid equations that calculate the induced drag as a function of aspect ratio and lift coefficient. However, this is very misleading. It works with the assumption that wing area is fixed. If you increase the aspect ratio (make the wing skinnier) while holding the area constant, the only way this can happen is by decreasing the chord and increasing the span. It's the increase in span that is the real reason the induced drag goes down.

If you don't hold the wing area constant in the relationship, then that forces a direct relationship between wing area and lift coefficient. Either way you approach the analysis, the net result is that induced drag ultimately ends up as a function of the effective wing span squared, the air speed squared, the air density, and the amount of lift you're trying to make.

A wing makes lift by grabbing chunks of air and accelerating them downwards. The wing pushes air down, so the air reacts by pushing the wing up, in accordance with Newton's third law, the one about action and reaction. All that stuff about Bernoulli, and high pressure on the bottom and low pressure on top, merely explains just how the wing manages to grab hold of the air in order to accelerate it downwards.

You can make a given amount of lift by grabbing a little chunk of air and giving it a violent shove, or by grabbing a big chunk of air and giving it a gentle push. You get the same amount of lift, but that second method generates a lot less induced drag.

The key then is understanding how big these "chunks of air" are. You can visualize it by imagining a cylinder of air, lying on its side, so the axis of the cylinder is aligned with the plane's flight path. The diameter of the cylinder is equal to the wing span, and the length of the cylinder is equal to the distance the plane travels in one second. The volume of that cylinder is a representation of the size of the chunk of air that wing processes each second to make lift. What really matters is the mass of air per second, so we also need to multiply the volume of the cylinder by the air's mass density to get the actual mass of the air inside the cylinder.

The volume of that cylinder is the cross-sectional area (the radius squared times Pi, or the diameter squared times Pi divided by 4) times the length. If you fly faster, the length of the cylinder gets longer, so the "mass flow" to the wing increases. If you double the airspeed, the cylinder becomes twice as long, contains twice as much air, and so you have twice as much mass flow, and a reduction in induced drag. This is why induced drag is usually less of a problem in cruise than it is at lower speeds such as takeoff and climb.

However, if you plan to cruise at extremely high altitudes, such as where jet airliners typically cruise, the air is very thin. Even though the cylinder is very long because of the plane's high speed, and therefore has lots of volume, the thin air's very low density means that the mass of the air in the cylinder is relatively low. This is why devices that help reduce induced drag, such as winglets, can help an airliner's high-altitude cruise performance.

The mass of the air in the cylinder is linear with speed, and inversely linear with air density. However, it's proportional to the square of the diameter (the plane's wing span). Thus, even small changes in the wing span can have dramatic effects on the induced drag. This is why airplanes that are especially sensitive to induced drag, such as sailplanes, and airplanes like the Voyager and the Lockheed U-2 (a relatively slow airplane that cruises at extremely high altitudes, where the air is extraordinarily thin) need such long wings.

Don Stackhouse
DJ Aerotech