The Idea - a paper by Dr. Reimar Horten
Today, one rarely sees a true elliptical wing, as other factors dictate its ideal shape. The tapered wing, for instance, is lighter and easier to build, factors which outweigh the advantages of the elliptical wing.
Since lift and weight are equal in straight and level flight, one needs to find out how the weight of the wing on a cantilevered sailplane change with its shape and taper when the span is constant.
Let's look at flight characteristics. For good roll control, the airfoil should be thin in the aileron area, while a thick airfoil is needed at the root to obtain an acceptable weight/strength ratio. Since the sailplane will be thermalling near the wings maximum lift capability, its stall characteristics must be closely studied, both during turns and level flight.
If air separation first occurs near one tip, which is likely due to the thin airfoil used there, the roll will quickly stall additional portions of the wing due its downward movement, and the asymmetric lift can not be overcome by the ailerons.
In a swept back flying wing, the conditions are somewhat different. Here the flow separation occurs initially at a point about of the half span, right where the center of pressure and the aircraft's center of gravity is located, thus no upsetting moment is created. Asymmetric lift can be controlled by the ailerons, since these still work in undisturbed airflow. If the separation should occur on one side only, a momentum is created around the yaw axis, because a stalled wing has a large increase in parasite drag, which slows the wing despite the disappearance of induced drag. When ailerons are used to control the asymmetry, normal, not adverse yaw should be generated to cancel out the momentum, and directional control could be maintained even with rudders of low efficiency.
Correct moment around the pitch axis requires that the center of pressure and center of gravity lie on a line at 25% of the wing chord. The conventional elliptical shaped wing without washout has an elliptical shaped lift distribution curve at all angles of attack, and the center of pressure in Y-direction on a half-wing can be expressed as Yell. = 0.42 (b/2). This lift distribution is not desirable, since the point along the wing where the airflow first separates, can not be determined.
The desired bell shaped lift distribution curve can be obtained on any wing by the appropriate amount of twist or washout. The center of pressure will then be located near the half span, and moves along the wing with changes in angle of attack.
On a flying wing aircraft with built in washout, one can obtain the desired lift distribution simply by moving the wing tip elevators, thus obtaining a CL corresponding to the best L/D ratio. This distribution at other CL should be at or near the desired form, thereby giving us the same center of pressure in Y-direction.
The moment around the pitch axis depends therefore on a lift distribution, which at it's center of pressure (Y =1/3 x b/2 ) also has the largest loading, and depending on the taper ratio, will determine where airflow separation first occurs in the case of an excessive angle of attack. Thus, one can easily determine the fixed washout, the washout that is variable through elevon deflection, and the needed wing taper ratio.
Once the airflow separation point along the span is determined, the balance problem around the roll axis is also solved, since the ailerons remain effective, and will overcome any asymmetrical loads. Remaining is the most important problem in a flying wing; how to retain aileron effectiveness at all angles of attack, and to minimize or eliminate adverse yaw.
A swept back wing has a large skid-roll moment, and therefore it is necessary to prevent any skidding caused by aileron yaw, since the skid will cancel the desired roll moment, and aileron response will be zero! To put it simply; one must make it easy for the pilot to fly the wing coordinated. This puts one additional requirement on the lift-distribution. While the elliptical lift curve was quite suitable for a conventional aircraft, the swept-back flying wing was found to require the bell shaped curve, in order to give a slight negative angle of attack in the aileron area near the tips. This reverses the forces normally associated with wingtip vortices, and actually generates some forward thrust! Adverse yaw is also minimized when the ailerons are deflected, and therefore controllability around all three axis are assured to a degree not possible with conventional aircraft.