Wing twist calculation [Archive]

Ollie

Jul 09, 2007, 08:16 PM

See:
http://www.coloradogliders.com/tools.htm
Twist Calculations For Flying Wings - Calculate Wing Loading, Lift Coefficient, Wing Area, Mean Chord, Aspect Ratio, Taper Ratio, Aerodynamic Centre, Alpha geo, and C of G for any flying wing. Downloadable - By Joa Harrison
www.glide.net.au/on-the-wing3/127_Panknin_Rainbow.pdf
Page 1 of 13
On the ÕWing... #165
Twist Distributions for Swept Wings, Part 5
The Horten twist distribution has been the focus thus far, but itÕs now time to take a look at the
twist distributions formulated by Irv Culver and Walter Panknin, make some comparisons, and
derive a few conclusions.
The Òmiddle effectÓ
First, a small digression is necessary in order to understand one remaining concept, the Òmiddle
effect.Ó The HortensÕ later designs included geometric modi?cations aimed at reducing or
eliminating the Òmiddle effect.Ó Irv CulverÕs twist distribution is speci?cally formulated to
eliminate the reduction in lift near the center of a swept back wing. Interestingly, the Hortens and
Culver are trying to counter two different phenomena.
As the wing moves through the air, the air coming off the trailing edge is deßected downward.
This is called the downwash. As the air approaches the wing, it moves up slightly to meet the
wing. This is called the upwash. WeÕve already illustrated these two properties in previous
portions of this article series, pointing out the angle of attack is directly related to the position of
the stagnation point.
If you look at an airfoil traveling through the air, youÕll see that the air moving over the upper
surface is moving faster than the wing is moving through the air. So too, the air along the lower
surface is moving slower than the wing is moving through the air. From a vector mathematics
perspective, if you subtract the velocity of the wing from the two air ßows, the air over the upper
surface is still moving from leading to trailing edge, but the air along the bottom of the wing is
moving backward toward the leading edge. From this perspective, the air ÒcirculatesÓ around the
airfoil in a clockwise direction as a wing producing lift moves right to left. The coef?cient of lift is
directly proportional to this circulation. See Figure 1.
According to PrandtlÕs lifting line theory, you can visualize a wing moving through the air as
simply a line connecting the two wing tips along the quarter chord line with horseshoe shaped
vortices coming from it and extending back to in?nity. In this model, both downwash and upwash
are accounted for: the air inside the vortices is being deßected downward, and the air outside the
vortices is being deßected upward. The actual lifting line calculations, however, are both complex
and extensive. Schrenk expanded PrandtlÕs lifting line theory to include taper, twist and control
deßections, but not sweep. Multhopp expanded this theoretical framework further, but still did not
fully account for the effects of sweep.
A swept wing can be viewed as a series of connected small wings, the leading edge of each
slightly behind the leading edge of its inboard partner and in front of the leading edge of its
outboard partner. Each small wing has an effect on the air ßow of both its inboard and outboard
partner, but the effect on the outboard partner is very much greater than the effect on the inboard
partner. The upwash is not equal along the span but rather tends to progressively increase over the
more outboard segments. (WeÕve illustrated this concept in previous portions of this article
series.)
Twist distributions for swept wings, Part 5
Page 2 of 13
SchrenkÕs approximation does not accurately portray a swept wing, and therefore does not
account for the loss of circulation and associated loss of lift at the root and the increase of
circulation and associated increase of lift at the wing tips.
MulthoppÕs method of determining the lift distribution, which involves established Òcontrol
pointsÓ based on Òcentral difference angles,Ó does not account for sweep either, but was used by
the Hortens as the best available model at the time. The H-II was the ?rst of the Horten aircraft to
use a bell-shaped, sin
x
, lift distribution, an outgrowth of the Multhopp paradigm.
The Òmiddle effectÓ which is so often talked about regarding the Horten designs is simply an
artifact of this inability to accurately predict the sweep induced changes in circulation, speci?cally
a loss of lift at the center. This middle effect is strictly an artifact of the computation methods and
is an error in analysis. The Òmiddle effectÓ is not the loss of lift in the center area of the wing, itÕs
the
unanticipated
loss of lift in the center area of the wing.
Horten
The Hortens, in an effort to coordinate stalling behavior and center of gravity with other planform
parameters, performed the necessary mathematical computations, but always found errors in their
results. The aircraft did not behave exactly as predicted because the center of pressure was not at
the location predicted. The Hortens believed the problem to be related to the intersection of the
two quarter chord lines at the centerline, and envisioned colliding vortices. They constructed Òbat
tailsÓ which substantially increased the root chord. Their intent in using the bat tail was to reorient
the quarter chord lines of the two wings and eliminate the colliding vortices. On the H IV, the
quarter chord lines meet at right angles to the centerline, while on the H VI the quarter chord lines
actually bend backward. Despite these changes to the quarter chord line, the Òmiddle effectÓ
remained. Al Bowers has suggested that the Hortens might have realized they were looking in the
wrong direction had they actually ßown their Parabola design.
Despite their problems getting a handle on the Òmiddle effect,Ó the Horten twist distribution has
the potential to reduce induced drag and allow turns to be accomplished without adverse yaw. But
aircraft will operate as Dr. Horten envisioned only when all of the design parameters are utilized:
moderate sweep angle, large taper ratio, carefully chosen airfoils (pitching moment), strong
nonlinear twist distribution, Òbell-shapedÓ span load (lift distribution), and outboard ailerons of
de?ned size and con?guration.
The Horten twist distribution is such that the wing twist is concentrated over the outer portion of
the wing, in the area where the sweep generated upwash is greatest. Computing the twist
distribution is a rather complicated affair, and weÕve been so far unable to obtain formulae of use
to modelers. Mathematically inclined readers may be interested in Reinhold StadlerÕs paper,
ÒSolutions for the Bell-Shaped Lift Distribution.Ó
Culver
Unfortunately, Irv Culver did not write a comprehensive treatise on his twist formula. Rather, his
description of its use is sparse, and its derivation not explained in any detail. Still, it is possible to
understand the general thoughts behind CulverÕs paradigm.
Although Culver did not speci?cally mention the Òmiddle effect,Ó he did realize that lift of a swept
wing is depressed in the area of the root. To compensate, some amount of up trim is required of
Twist distributions for swept wings, Part 5
Page 3 of 13
the outboard elevons, depressing the lift generated by that area of the wing as well. Performance is
substantially reduced as a result. In CulverÕs view, the ideal is to make the center portion of the
wing produce more lift and thereby allow the wing tips to create more lift. At the design
coef?cient of lift, the lift distribution is near elliptical.
Another digression... The most simple method of creating a twisted wing is to use a single foam
core and root and tip templates. Twist is then imparted by setting the two templates at the
appropriate angles relative to each other. Cutting with a tensioned hot wire always creates a wing
with straight leading and trailing edges. This is quick and simple, but the angle of twist does not
change consistently across the semi-span. Rather, the angle changes at a more rapid rate near the
root for wings with no taper, and near the wing tip if the wing is moderately tapered. As Culver
uses wings with moderate taper in an effort to better achieve an elliptical lift distribution, it is the
latter situation which Culver wants to avoid.
In an effort to compensate for the loss of lift in the center area of a swept back wing, Culver
proposes placing most of the twist in the inboard 30% of the semi-span, say eight degrees. Three
more degrees of twist are then imparted in the outer 70% of the semi-span for a total of eleven
degrees. The increased angle of attack at the root increases the lift in that area. This allows the up
trim of the elevons to be reduced, increasing the lift in that area as well. The Culver twist therefore
requires constructing the semi-span of a foam wing in two parts rather than as a single panel.
As the sweep angle is increased, the Culver twist distribution calls for more twist. As the Culver
twist distribution is aimed at maintaining an elliptical lift distribution at the design coef?cient of
lift, this is in keeping with the increased upwash which is anticipated will occur over the outer
portion of the wing.
In ßight, specially designed elevons are used to trim for low coef?cients of lift. As the aircraft
approaches a stall attitude, the root will stall ?rst while the wing tips remain well below their stall
angle. This makes a full stall across the entire span very unlikely.
There are a few limitations to the Culver twist distribution: it is accurate only for wings of modest
sweep and taper, and the recommended design lift coef?cient is for very high compared with other
methodologies, particularly that of Dr. Walter Panknin. Since the Culver twist distribution is
based on maintaining a near elliptical lift distribution, adverse yaw may be noticeable, particularly
around the design coef?cient of lift.
There are reports stating that swept wing aircraft utilizing the Culver twist distribution are both
spin-proof and tumble-proof, and there is also at least one report stating the Culver twist
distribution was incorporated into the wings of a number of Boeing commercial aircraft. These
reports have not been corroborated by secondary sources, and it should be noted that Boeing
commercial aircraft are of conventional tailed con?guration and utilize both roll spoilers and
rudder to counter adverse yaw.
A six meter (236 inch) span swept wing model using an approximation of the Culver twist
distribution was constructed in Germany in 1987. The Stromburg Õwing utilized the Eppler 220
for the outboard portion of the wing and the Eppler 210 at the root, and had a sweep angle of 28.5
degrees. The twist angle at the root was 11.5 degrees, going to zero degrees at station .167 and
remaining at zero degrees to the wing tip. Elevons consisted of ÒJunkers ßapsÓ from station .833
outboard. This model performed extremely well, and was large enough to have a movie camera
Twist distributions for swept wings, Part 5
Page 4 of 13
mounted at the CG and directed at the center section. Films taken during ßight showed no air ßow
separation at the root during cruise, turning, high speed ßight, or landing.
Panknin
Dr. Panknin derived his twist paradigm from a paper by Helmut Schenk. Using airfoil zero lift
angles and pitching moments, span and chords, sweep angle and static margin, a pitch stable
tailless aircraft can be assured. The method relies heavily on MulthoppÕs approximation of the lift
distribution, but includes a correction by D. Kuechemann so that it has good accuracy for sweep
values for zero to beyond 30 degrees. (Schenk states the Òmiddle effectÓ still exists using these
calculations.)
The Panknin methodology provides only the total twist required for longitudinal stability for a
given monolithic wing with straight leading and trailing edges and a predetermined static margin.
The computed twist values have been proven in practice to be extremely accurate for sweep
angles of up to 30 degrees, tapered or constant chord wing.
Like the Culver formulae, the Panknin method lends itself quite easily to both custom written
computer programs and commercially available spreadsheet software. In fact, a scienti?c
calculator is suf?cient when there are no time constraints. The de?ned twist angle can be used on
a moderately tapered wing using the foam core construction method described previously, with
straight leading and trailing edges from root to tip. Successful applications, however, include
planforms with constant chord in which the twist begins at station 0.5, half the semispan, placing
more of the twist over the outboard portion of the wing.
All of Dr. PankninÕs designs, and our own designs based on Dr. PankninÕs paradigm, incorporate
winglets. These vertical surfaces assist in reducing oscillations in yaw in straight and level ßight
and act to reduce adverse yaw at the expense of some increase in drag. As weÕve stated in previous
columns, thermal machines seem to climb better with winglets, racers track better with a single
vertical ?n mounted on the centerline.
Conclusions
All three twist distributions have both positive and negative aspects.
The Horten twist distribution is based on the work of Prandtl and others, and has been supported
by the more recent works of R.T. Jones and Klein and Viswanathan. The Horten paradigm has the
potential to reduce induced drag and eliminate adverse yaw, but is computationally intensive and
the twist distribution itself must be used in combination with a number of additional planform
attributes.
The Culver twist distribution is centered on the elliptical lift distribution. This is a conservative
approach which provides relatively low drag and good ef?ciency within a con?ned design point,
but may be prone to adverse yaw, particularly when operating at the design coef?cient of lift.
The Panknin twist distribution has proven itself over a nearly two decade period to be an accurate
determiner of both required wing twist and center of gravity location. It has been used with great
success by a very large number of international designers. Its major limitation is that it calculates
only the twist required for pitch stability, but it can be used as a fundamental method of
determining the approximate minimum twist required for a preliminary design.
Twist distributions for swept wings, Part 5
Page 5 of 13
Figure 2A shows the elliptical lift distribution for a conventional cross-tailed design as seen from
behind. The fuselage and vertical surface have been neglected. Figure 2B shows the downwash
pattern this lift distribution produces. Keep in mind the internal structure of the wing is required to
support both itself and a fuselage and tail structure. Additionally, the fuselage must be strong
enough to support itself and the mass and aerodynamic loads of the tail.
These factors, taken in combination, paint a picture of a relatively heavy aircraft with substantial
surface and interference drag. Additionally, there is the surface and induced drag of the separate
relatively low aspect ratio horizontal and vertical stabilizers. In ßight, large amounts of drag are
created in an effort to make coordinated turns. Given this perspective, the possibility of more
ef?cient aerodynamics, as seen in Figure 2C, is obvious.
While a specially tailored single surface wing may be necessary to achieve this goal, a well
integrated design approach for tailless aircraft is certainly very close, as demonstrated by the
recent articles by Katherine Diaz in
Pilot Journal
and Carl Hoffman in
Popular Science
. It is only
a matter of time before such design paradigms and appropriate construction technologies are
available to modelers.
When designing a tailless planform, the type of twist distribution to be used should be one of the
?rst decisions to be considered, and always relative to other aspects of the design such as
prescribed task, design lift coef?cient, and planform. There are a number of design ÒßowchartsÓ
available to assist the novice designer, and we very much encourage readers to investigate their
usefulness. The information presented in this series can be used to augment these resources and
assist in developing viable, and perhaps cutting edge, designs.
__________
Ideas for future columns are always welcome.
RCSD
readers can contact us by mail at P.O. Box
975, Olalla WA 98359-0975, or by e-mail at <bsquared@appleisp.net>.
Twist distributions for swept wings, Part 5
Page 6 of 13
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P