My studies of aerodynamics have produced a fundamental theory of induced drag in wings. It turns out that the best planform for a wing is parabolic, this is something that I have suspected for a long time and now I have an exact mathematical proof arrived at using integral calculus and much experience..
In a parabolic wing planform the chord C(x) is given by the simple formula:
C(x) = C(0) ( 1 - ( 2x / S )^2 ) where x is the position along the wingspan, C(0) is the width of the wing at the root, and S is the wingspan. In this formula the value of 2x /S runs from -1 to +1.
At speeds around stall speed to roughly two times stall speed or more the induced drag is the predominant source of energy dissipation and this energy goes into the manufacture of a vortex ( or horizontal tornado ) behind the aircraft. Indeed in this portion of the flight regime the frictional drag energy losses are quite small compared to the induced drag energy losses for conventional wing shapes and the use of the parabolic wing planform provides a way to dramatically reduce the energy going into vortex generation.
In contrast to frictional drag ( which generates heat ) the induced drag makes mainly coherent motion in the air. For a normal wing the induced drag is inversely proportional to the Aspect Ratio of the wing. The Aspect Ratio is the ratio of the span to the chord at the wing root, ie S/C(0) approximately. In the case of the parabolic wing planform however, the induced drag is inversely proportional to the Aspect Ratio squared ! This is the root reason why the parabolic wing planform is superior to conventional wing designs. It is significant because it promises to reduce power requirements not just by a little bit but instead by a very large amount.
For almost any Aspect Ratio the use of the parabolic wing planform results in a very significant reduction in the power needed to maintain altitude along with a very large reduction in the energy going into trailing vortex motion.
At high altitudes airliners fly at speeds not much higher than stall speed so the use of the parabolic wing planform greatly reduces the fuel consumption requirements. This suggests the idea of using standard engines for take-off and climb but then using a tiny sustainer engine when at altitude just to be able to maintain altitude.
On approaches to landing at airports airliners are also flying at quite low speeds so because the parabolic wing planform greatly reduces the energy in the trailing vortex it is therefore possible to reduce the spacings between aircraft on final approach without compromising safety.
Also, the use of the parabolic wing planform is a good way to cut down the noise and pollution generated by aircraft in the vicinity of airports.
I am quite familiar with all kinds of aircraft and to the best of my knowledge there are no aircraft flying today that employ a parabolic wing planform. The wing designs in widespread use today date back to when kerosene sold for 5 cents per gallon or even earlier.
Dr. John Vincent Kane Jr. PhD Physicist
1088 Violet Avenue
Hyde Park, New York 12538
References; Who's Who in Science and Engineering (1982), American Institute of Physics Membership Directory.
As one result of this difference the usual 1910 Lanchester theory of induced drag which predicts that the induced drag should be proportional to one divided by the aspect ratio changes to a new relationship which is that for a parabolic wing planform the induced drag is proportional to one divided by the aspect ratio squared and it is this which leads to huge reducttions in the induced drag for wings having practical values of the aspect ratio.
Here is an explanation of my new theory which goes into greater depth hopefully to make the aspect ratio effect more understandable.
(1) Wings are finite and have to have a wing tip portion.
(2) It is therefore necessary to have taper built into any wing design even when the wing is simply chopped off at the tip this is a form of taper.
(3) The presence of taper generates vortex motion in the air at the trailing edge of the wing. Think about a section of tapered wing as viewed looking at the leading edge with the wider portion to the right and the narrower portion at the left. The airflow under the wing will be deflected to the left just because there is a taper present. This has nothing to do with the presence of sweep back or sweep forward, it is only because of the effect of change in the wing width. Similarly air flowing over the top of the wing is deflected in the opposite direction, ie, it is deflected to the right.
Therefore, taper in a wing causes vortex motion.
(4) The deflection angles of the airflow both above and below the wing are proportional directly to the angle of taper which is dC(x)/dx in calculus notation, or, the rate of change of the wing chord with spanwise distance.
(5) It is not important to know the exact values of these deflections. Of interest to me is that they correspond to sideways velocities in which the energy being transferred into vortex energy is proportional to the squares of the velocities and therefore proportional to the rate of taper dC(x)/dx squared.
(6) For each infinitesmal section of wing the mass of air involved is proportional to the square of the value of the wing chord C(x) so there is an energy value involved similar to 1/2 MV^2 in classical physics which is given by the expression ((dC(x)/dx)^2)((C(x))^2) dx. I am sorry but explaining mathematical ideas with the usual computer software is very awkward and requires a lot of parentheses to show correctly the order in which the operations are to be performed.
(7) The energy loss of an entire wing is proportional to the integral from zero to one of the last expression given above. Therefore it is necessary to know what to use for the function C(x). The function C(x) = C(0)( 1 - x^n) serves this purpose because it describes whole families of wing planforms.For example with n equal to one a wing with a straight taper from root to tip is covered. Dßiffernt values of n cover differing wing planform shapes.
(8) Evaluating the integration for the energy loss per unit area of a wing gives the following expression:
The energy loss or drag are proportional to n(n+1)(1/(2n-1) -2/(3n-1) +1/(4n-1) ). This function has a substantial minimum near the value n = 2.
(9) As a result it can be seen that a very nearly optimun wing planform is given by the expression:
C(x) = C(0)(1 - x^2) or C(x) = C(0)( 1-xx) where x is zero at the wing root and one at the wing tip.
(10) This is The Parabolic Wing Planform. Now to answer your question about why it works just think about what happenes when you double the wingspan leaving the chord at the root C(0) the same. Everywhere along the wing the taper dC(x)/dx is cut in half so the induced drag per unit area is down by one quarter. Therefore the induced drag is proportional to one divided by the aspect ratio squared.
(11) The above reasoning is good only for smallish taper angles and does not work well for squared off wing tip designs or nearly square designs such as elliptical or rounded or *beep* tips and winglets.
If you are still interested after all this I can send you graphical data. To be able to see the graphs you will need to have Curvus Pro on your computer. It can be purchased online for 39 dollars.
John Vincent Kane Jr
1088 Violet Avenue
Hyde Park, New York
Note: A sinusoidal or cosine planform is slightly more efficient than a parabolic planform by about 1.8591636 percent.