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The induced drag equation takes the same form as the lift equation and
parasite drag equation we looked at previously: 
Our task is to visualize what factors determine the value of CDi, just as we previously realized that CDp depends on the shape of the airplane and CL depend on the angle of attack, camber, etc.
On the previous page we analyzed the causes of induced drag and we saw that aspect ratio (AR) is an important factor. So, we can predict that CDI will be proportional 1/AR. I.E. as AR becomes larger induced drag becomes smaller.
The other primary factor affecting CDI is angle of attack. Why?
The reason is that induced drag is caused by upwash which in turn is
caused by the pressure difference between the top and bottom of the wing.
Clearly anything that increases lift will also increase induced drag.
If we look at the induced drag equation ()
we see that S, rho, and V, which are all factors in the lift equation,
are already accounted for. The only missing factor is CL. So
we might predict that CDI will be proportional to CL.
It actually turns out that CDI is proportional to CL2. Why?
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The picture to the left shows the upwash angle and lift being produced at two different angles of attack. The top wing is at a small angle of attack and the lower wing is at a greater angle of attack. As you can see two things happen when AOA increases:
Each of these factors increases the induced drag. Imagine the top wing in the diagram but with the induced AOA increased. Clearly induced drag would increase. Now imagine the lower wing with the same induced AOA as the top wing but the larger amount of lift, as shown. Once again we can see that induced drag would be greater than for the top wing. Therefore, it is quite clear that because both upwash and actual force increase when AOA increases we should expect CDI to be proportional to CL2. The "first guess" at the equation for CDI is:
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The equation for CDI proposed above is very close to being correct. The greek letter pi represents its usual value (3.14) and is the theoretically correct constant of proportionality if the circulation around the wing is uniform at all locations along the span. But in reality the circulation is seldom uniform. There are many things that could distort the circulation along the span. For example the fuselage and engine nacelles. But the most important factor is the taper and sweep of the wing.
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We already saw that wing sweep will affect the induced drag, so we must expect a factor in the equation to allow for that. The other factor is the taper of the wing. |
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Taper ratio is the ratio of the tip chord divided by the root chord. You can see this in the picture to the left. The top airplane has a rectangular wing, i.e. a wing with the same chord length at the root and the tip. The taper ratio is therefore 1.0 The lower airplane has a longer chord at the root than at the tip. Therefore, the taper ratio is less than 1.0, approximately 0.5 in this case. We already know that the wingtip vortex causes more upwash near the wingtip than at the root. But common sense tells us that if the wing is highly tapered the upwash will spread out over the wing for the same reason as with the swept wing, as shown in the picture above. |
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The airplane to the left has the ultimate taper ratio - zero (0.0) Because there is not wingtip there can be no wingtip vortex. The vortex will spread out along the span and the upwash, and induced drag will actually be greatest at the root of this wing. (Think about the stall characteristic of this airplane. It will stall at the wingtip first, because the upwash will delay stall at the root. As a result the ailerons will reverse just before the stall. All-in-all this is not a good design.) You should be suspecting that there will be some taper ratio at which the vortex will be evenly distributed along the wings span. Yes and no. |
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It turns out that the only shape that will exactly distribute the vortex along the wingspan is an ellipse. The picture to the left shows a famous second world war airplane, the Supermarine Spitfire. It is one a number of famous airplanes that have elliptical wings. Returning to our equation for CDi we had proposed:
Unfortunately even the spitfire does not achieve the value of CDI projected in this equation. That is due to the distortion of the vortex by the fuselage, and some distortion due to washout of the wings. So we must add a "fudge factor" to the equation to allow for all these factors we have been discussing. The final form of the CDI equation is:
In this equation the fudge factor is represented by the letter e. Its value will always be less than 1.0. The smaller the value the less efficient the design - i.e. the more induced drag the design produces. Hence it is called the Oswald efficiency factor. e will approach 1.0 for an elliptical wing, but never actually reaches 1.0 The value of e depends primarily on the taper and sweep of the wing (it also must be experimentally adjusted for distortion due to the fuselage, engine nacelles, etc.) An elliptical wing is quite difficult to construct out of typical airplane construction materials such as aluminum, although it is easy with composite materials such as carbon fibre. Perhaps we will see more elliptical wings in the future as composite construction becomes more common, but most modern airplanes employ a taper ratio of about 0.5 which is quite a good approximation of an ellipse, but tends to produce slightly more upwash near the wingtips, which is good from a stall characteristic point of view (much of aircraft design is the art of wise compromise.) |
