Twist Distributions for Swept Wings
Part 4: The effect of lift distribution and aileron configurations on adverse and proverse yaw, and using bell-shaped lift distribution to reduce induced drag.
Readers who have not already done so may want to read Parts 1, 2 and 3 of this five part series before proceeding with the following. — Ed.
Defining Lift Distributions
Before describing research results related to lift distributions, another look at the elliptical lift distribution is in order.
In Part 1, the elliptical lift distribution was defined by means of a geometric construction. Figure 1 illustrates this methodology. Simply stated, vertical lines are dropped from a semicircle to the baseline. The center of these verticals are then determined and a curve drawn which connects the determined points. The curve thus defined is an ellipse. This shape is then used as a basis for the lift distribution across the span. The result of such a lift distribution is a constant downwash across the entire span and a minimization of induced drag.
As an extension of the description in Part 1, there is another method of defining the elliptical lift distribution which involves trigonometric functions. In this construction, the point P is defined by its X and Y coordinates as determined by the following formulae:
For the construction of the semicircle, K = b / 2, the semi-span. For the construction of an ellipse, K can be any value less than one. In the illustrated case, Figure 2, K = 1 / 2 in keeping with the geometric construction explained previously.
It should be noted at this point that each point P’ defines the lift generated by that wing section, the coefficient of lift times the local chord. One way of visualizing this is to consider an elliptical lift distribution and an elliptical wing operating at a coefficient of lift of one. Remember, the lift coefficient is constant across the span; that is, the local coefficient of lift for each wing segment will be one. In this case, the wing chord is directly proportional to the height of the lift distribution curve at that point along the Y-axis.
Taking this trigonometric methodology one step further, we can modify the trigonometric function by adding an exponent n. For example, rather than using sin ξ, we use sinⁿ ξ. See the included Table for an idea as to how various exponents affect the resulting points P’.
Figure 3 shows the elliptical lift distribution, sin ξ, and three other distributions, sin².⁵ ξ, sin³ ξ, and sin⁴ ξ. Because the aircraft weight is held constant, the area under each curve is identical. The latter lift distributions which utilize the n exponent are termed bell-shaped for obvious reasons.
When the bell-shaped distribution is applied to moderately swept back wings, the following generalizations apply: when the exponent n is two, the lift distribution is bell-shaped but there is no induced thrust at the wing tips. When n = 2.5, the adverse yaw disappears and proverse yaw begins to appear. As n approaches three, the induced drag begins to increase rapidly. The designer should therefore use the lowest value of n in keeping with his/her objectives. The Hortens used n = 3 for most of their designs, but n = 2.5 may be sufficient for use on models where both adverse and proverse yaw are undesirable and induced drag should be as low as possible.
Yaw Moment, Lift Distribution, and Aileron Configuration
Dr. Edward Udens analyzed the yawing moment of two swept wing planforms with differing lift distributions and control surface configurations. Figure 4 shows the various configurations, notes their lift distributions, and presents the yaw moment for each. The elliptical and sin³ x bell-shaped lift distributions were evaluated. Negative yaw moment values indicate adverse yaw, positive yaw moment values indicate proverse yaw. Both of the wings with elliptical lift distributions demonstrate adverse yaw regardless of control surface placement. Proverse yaw can be generated by using the bell-shaped lift distribution and by keeping the elevon control surface well outboard.
Dr. Udens’ results demonstrate an increasing adverse yaw moment as the elevon control surface is moved inboard. This is an important consideration. The roll control surfaces must be placed in the area of the wing which has a concave lift distribution curve; that is, outboard in the case of the bell-shaped lift distribution. Although the Hortens used the sin³ lift distribution, they included inboard elevons which may have significantly reduced the proverse yaw moment and in fact created an adverse yaw moment.
A Relevant Example
There are a number of readers who at this point desire some sort of practical example of the bell-shaped lift distribution generating proverse yaw as elevon control surfaces induce a roll moment. Ideally, we would look for a swept wing tailless model without winglets which exhibits very strong adverse yaw as an example. Those who have built and flown a Klingberg wing know well this model meets the ideal. Don Stackhouse of DJ Aerotech had the following to say about his Klingberg wing:
“My stock Klingberg, with its horrible adverse yaw and a yaw-roll coupling that essentially negates the roll response to any but the smallest elevon deflections, is essentially unsafe to fly in any place with maneuvering space restrictions or in any kind of turbulence.”
Don goes on to say that the addition of any aileron differential severely affects the aircraft in pitch. The application of down elevator to inhibit the nose up pitching reduces the differential, so it’s a Catch 22 situation. Don has not flown his Klingberg wing in several years, and in fact only takes it out of storage to serve as an exhibit model.
Michael Allen, a student at Embry-Riddle Aeronautical University and an intern at NASA Dryden Flight Research Center under Al Bowers, decided to build a Klingberg wing using a bell-shaped lift distribution. See Photo 1.
The taper ratio and other planform parameters of the two meter Klingberg wing closely match those of the Horten Xc, an advanced ultra-light glider designed by Reimar Horten while he was living in Argentina. Al was able to get the twist values for the modified Klingberg wing from Reinhold Stadler, and Michael built the model using the defined twist distribution. Additionally, Michael used an elevon planform, illustrated in Figure 5, in keeping with the results of Dr. Udens (Figure 4 No.6). This elevon planform is calculated to give a small amount of proverse yaw, Cₙ∂a = 0.001942.
Al says the wing looks very ‘organic’ in the air, and while flying directly overhead and giving full right or left stick, there is not even a hint of adverse yaw in evidence.
Reducing Induced Drag
“The elliptical lift distribution is the most efficient.” We have heard this statement often over the years. Recently we’ve come to discover it is not entirely true simply because it is incomplete. More accurately, “the elliptical lift distribution is the most efficient for a wing of given lift and span.” The qualifications may not seem to be of much importance at first. But consider a wing of a given span with an elliptical lift distribution. Is there a way to reduce the induced drag of this wing, making it more efficient, while keeping the root bending moment the same?
If you simply add span and maintain an elliptical lift distribution, the wing will be more efficient because you’ve increased the aspect ratio. But the spar will need to be strengthened because the bending moment at the root will have been increased with the larger span. So the question becomes a matter of finding a means to increase the span without increasing the load at the wing root. Enter the bell-shaped lift distribution.
Ludwig Prandtl came up with the elliptical span load around 1908, but did not formally publish his work until 1918. In 1933, Prandtl published his paper On the Minimum Induced Drag of Wings in which he presented the bell-shaped lift distribution. Prandtl’s solution provided an 11% reduction in induced drag with a 22% increase in span and no increase in the root bending moment. In 1950, Robert T. Jones looked at the same problem and, unaware of Prandtl’s work, came up with a similar solution by a different means.
Jones’ computations show a 15% decrease in induced drag with a 15% increase in span when using a bell-shaped span load. Figure 6 illustrates Jones’ planform, a comparison to the standard elliptical lift distribution, and the trapezoidal shape of the produced downwash.
Also included in that illustration is a diagram showing the lift distribution for a wing with a span ratio of 1.30 and a root bending moment identical to the span ratio 1.0 elliptical wing. Jones states that while the span can be increased further, the near maximum benefit comes with a 15% increase in span.
Other investigators, notably Klein and Viswanathan, have looked at the same constant root bending moment problem but also included other constraints, such as shear. The results point to a bell-shaped lift distribution and similar reductions in induced drag.
Back to Winglets
In Part 3, we described how winglets can be a source of induced thrust. We also drew a parallel between the action of winglets and the effects of generated upwash on the outer portion of a swept wing. Consider a wing with a bell-shaped lift distribution which is producing induced thrust at the wing tips to be equivalent to a wing with winglets which is operating at its design speed.
While researching this series of articles, we ran into a document produced by Boeing as part of their publication Aero dealing with blended winglet design for various passenger and cargo aircraft. Briefly, the addition of properly designed winglets which extend the wing between ten and 16 percent can substantially increase payload and range and decrease takeoff runs, particularly near maximum gross weight. This is parallel to the effects predicted for the span extension proposed by Jones. According to the article, maximum payload increases, takeoff runs are shortened, cruise drag is decreased by four to more than five percent, and range is increased by approximately four percent.
This is evidence that, when properly designed, winglets can improve performance over a wide speed range. Additionally, blended winglets improve directional and pitch stability and longitudinal and lateral trim stability. There is no change in stall speed or Dutch roll damping. One of the interesting points covered in the article involved the toe angle of the winglet. Initially, the toe out angle was set for zero degrees. While this minimized induced drag, it imposed very high loads on the wing. A toe out angle of two degrees reduced the bending loads on the wing but did not adversely affect the drag reduction except in the flaps down position. Boeing determined this was an acceptable trade-off for reducing required structural modifications.
It’s important to realize that commercial aircraft have span limitations based on constraints imposed by airport architecture, so vertical winglets are a much more attractive option than increasing the wing span. Boeing’s blended winglets aerodynamically increase the wing span without imposing a greater root bending moment and without increasing the actual wing span.
Discussion
The following discussion recently took place on the Nurflügel Mailing List the link for which can be found in Resources, below. We think the exchange may be enlightening, particularly for those readers with some doubts as to the efficacy of the bell-shaped lift distribution as applied to reducing induced drag. Al Bowers is Chief of Aerodynamics at NASA Dryden Flight Research Center.
The Bell-Shaped Lift Distribution and Tailless RC Sailplanes
For AMA RC models outside of the Unlimited class (RC-HLG, 2m, Standard), span is limited. Designing a tailless model with a bell-shaped lift distribution in an attempt to improve performance beyond that of a conventional tailed aircraft of the same span is therefore problematic, as Al Bowers explains.
Still, for the Unlimited class, where the only limitations are wing area (2325in²), mass (5kg, 11.02lb), and wing loading (3.95–24.57oz/ft²), a competitive swept wing tailless model is certainly in the realm of possibility, and in fact, may be the best choice.
A tailless model utilizing the bell-shaped lift distribution is a particularly enticing proposition when such considerations as ground handling and construction costs are removed and modern low Reynolds airfoils, vortex-lattice computer codes, and high-tech materials and fabrication methods can be so easily added to the design and construction processes.
Our sincere appreciation goes to Al Bowers for providing substantial guidance and positive reinforcement, as well as a number of printed references, for this installment. Thanks are also due to the members of the nurflugel e-mail list for their informed questions regarding the elliptical and bell-shaped lift distributions.
What’s Next?
The next and final installment in this series will provide a summation of the Horten, Culver, and Panknin twist distribution methodologies.
©2002, 2023 Bill Kuhlman
References
Resources
- NASA Armstrong Fact Sheet: Prandtl-D Aircraft published on March 30, 2016. — “NASA’s Armstrong Flight Research Center engineers in Edwards, California, are working on an increasingly complex aircraft called the Preliminary Research Aerodynamic Design to Lower Drag, or Prandtl-D.…” This article is the source of the key photo above.
- Nurflügel Mailing List — “This mailing list serves as a discussion forum for fans of flying wings. Pretty much anything that pertains to this subject is welcome. The title Nurflügel is the German word for what in English is called a flying wing…”
All images, figures and tables by the author unless otherwise noted. Read the next article in this issue, return to the previous article in this issue or go to the table of contents. A PDF version of this article, or the entire issue, is available upon request.
