How billow increases the net geometric anhedral of a swept wing--abbreviated version

August 31 2005 edition
Steve Seibel
www.aeroexperiments.org
steve at aeroexperiments.org

 

NOTE August 2006: all the content in this section has now received a fresher treatment in the "Semi- Unconventional Aerophysics Tutorial" pages. Many items that are not yet covered in the main text pages of the SUAT section are covered briefly on the page entitled "Pool of images for Semi-Unconventional Aerophysics Tutorial Pages." This older material is still accurate to the best of my knowledge except for one point: I now feel that the suggestion that increasing wingtip washout (as opposed to increasing sail billow) will tend to create an anhedral geometry was unwarranted.

 

This topic is addressed in much more detail in the longer article entitled "How billow and washout increase the net geometric anhedral of a swept wing, and other related topics". Here we'll try to confine ourselves to a very few of the most important points. For more, see the entire text of the longer article.

The passages marked "*" are excerpts from the longer article and the passages marked "**" are additional notes.

Note August 2005: many of the ideas in this article are now contained in the photos and text that can be accessed from the related page on this website entitled Photos of hang gliders and models to illustrate how billow contributes to the net geometric anhedral of a swept wing.

(A note for pilots of "conventional" aircraft: the ideas discussed in this article apply to all aircraft. The only concept that doesn't really apply to "conventional" aircraft with non-flexible wings is the concept of "billow". "Billow" refers to the way that the trailing edge of Rogallo or modified Rogallo (modern hang glider or trike) wing tends to rise upward in the mid-span region due to the aerodynamic loads acting on the flexible wing. If you are only interested in the parts of this article that relate to "conventional" aircraft with non-flexible wings, just focus on the passages that deal with "washout" and skim past the passages that deal with "billow". Also, if you come across the word "sail" you can treat it as a synonym for "wing". Also, you'll need to know that the "keel" is the aluminum tube on a Rogallo wing or modern hang glider or trike that runs in the fore-and-aft direction, like the fuselage on a "conventional" aircraft--when we talk about the "keel" of our little model wing that we'll use to demonstrate the geometry of anhedral and dihedral, you can substitute the word "fuselage".)

 

*The traditional definitions of "anhedral" and "dihedral" are simply based on a measure of how much higher or lower are the wingtips than the wing root. For example if the wingtips are 1 foot above the wing root, and the horizontal distance from the root to the tip is 20 feet, than the wing is said to have a dihedral angle of 3 degrees-- (a simple trig calculation).

*This definition is completely inadequate in some cases. For example if the wings form an "W" shape as seen in a head-on view--think of a highly modified version of the old F4U Corsair--then even if the tips are exactly at the same height as the root, the wing will act as if it has dihedral when a sideways airflow is present, because the outer panels (which have dihedral) will create much more roll torque than the inner panels (which have anhedral), because the outer panels are further from the CG. Likewise, a wing that is shaped like an "M" will act as if it has anhedral, even if the tips are at the same level as the wing root, because the outer panels will have anhedral.

*We'll use the terms "net geometric anhedral" and "net geometric dihedral" to get at this idea of the total roll torque created by the wing in a sideways airflow due to the differences in angle-of-attack of the various portions of the wing. A wing shaped like a "W" has net geometric dihedral because the outboard portions have dihedral, and a wing shaped like an "M" has net geometric anhedral because the outboard portions have anhedral. This is easily seen by looking at the wing from the side, and noting the differences in the apparent angle-of-attack of the tip areas, root areas, center areas, etc. of the left and right wings, bearing in mind that differences in the angle-of-attack between the outboard areas of the left and right wings will generate much more roll torque than will differences in the angle-of-attack between the inboard areas of the left and right wings.

*We won't use the terms "net geometric anhedral" and "net geometric dihedral" to include the dihedral-like effects created by sweep in the case of a wing with no dihedral or washout or billow. In our conception, a swept wing with no dihedral or washout or billow will have a "net geometric dihedral" of zero. This wing will definitely exhibit dihedral-like behavior (creating a roll torque in a sideways airflow), due to the sweep, but this roll torque will not be due to a difference in angle-of-attack between the left and right wings, and so it should not be considered a result of "net geometric dihedral". The ultimate behavior of a swept wing in a sideways airflow (sideslip or skid) will be governed by the combined or competing effects of sweep, and net geometric anhedral or dihedral. The relative importance of these two competing effects, at various angles-of-attack, in the case of a wing with both sweep and anhedral, will be explored in much more detail elsewhere in this website. This article will focus only on the "net geometric anhedral" or "net geometric dihedral" of the wing, and the way that this is affected by washout and billow, if the wing is swept.

*In general, with a non-swept wing, washout (twist) has a relatively small effect on the wing's net geometric anhedral or dihedral. If we look at a model of a non-swept wing, with dihedral or anhedral, from a side view, we'll see almost the same amount of top and bottom surface area of the near and far wings regardless of how much washout we give the wing by twisting the outboard portions to a lower angle-of-attack or angle-of-incidence.

*In the case of a swept wing with washout and/or billow, things get extremely interesting. For example, at first glance many hang glider wings appear to be fairly flat, i.e. they appear not to have very much anhedral, especially if we are talking about single-surface wings, and especially the old Rogallo models. But look at these photographs and notice how washout and billow are very clearly adding to the wing's net geometric anhedral. Note how we can often see the top surface of the outboard portion of the wing that is closest to the camera. And note how we can often see the undersurface of the outboard portion of the wing that is farthest from the camera. Recall how we've already seen that this will create a roll torque when there is a sideways component in the airflow. And bear in mind that the outboard areas of the wing are more important than the inboard areas of the wing, when it comes to the balance of roll torques. As we look at the wing from the side, when we can see the top surface of the outboard portion of the near wing and the bottom surface of the outboard surface of the far wing, we are seeing net geometric anhedral. (Photo 3: pilot landing at Packwood WA) (Photo 4: Dwayne Hyatt launching at Peterson Butte OR) (Photo 5: Dwayne Hyatt flying at Peterson Butte OR) (More photos illustrating how washout and billow contribute to the net geometric anhedral of a swept wing)

*It's very clear from these photos that washout and billow can greatly increase the net geometric anhedral of a swept wing.

*It's interesting to try to think more rigorously about way that washout and billow contribute to the net geometric anhedral of a swept wing. What is really going on here? What happens if we rotate the entire trailing edge of each wing upward like this (photo 6), or downward like this (photo 7)? Notice that we aren't moving the "leading edges" of our model in relation to the "keel". We are only rotating the trailing edges of the wings upward or downward, using the leading edge as a fixed hinge line. If we change our viewing angle a little bit, it's easy to see that by rotating the trailing edges upward we have created net geometric anhedral (photo 8), and by rotating the trailing edges downward we have created net geometric dihedral (photo 9).

*The traditional definition of anhedral and dihedral, based on the vertical distance between the wing tips and the wing root, still makes sense in relation to these models, but only if we specify that before measuring this vertical distance, we orient the wing in such a way that the wing chord line is horizontal. This may seem slightly arbitrary, but it is not. It is certainly much less arbitrary than assuming that the keel tube should remain horizontal. The keel tube has no aerodynamic or geometric relevance. In this picture the wing chord line is horizontal, and the fact that we've lifted the trailing edge, in relation to the leading edges and keel, has created anhedral (photo 10). We can measure this anhedral by measuring the vertical distance between the wingtips and the wing root, while holding the wing in a pitch attitude where the chord line is horizontal. In this picture the wing chord line is horizontal, and the fact that we've lowered the trailing edge, in relation to the leading edges and keel, has created dihedral (photo 11). We can measure this anhedral by measuring the vertical distance between the wingtips and the wing root, while holding the wing in a pitch attitude where the wing chord line is horizontal.

*We can attempt to extend this definition of anhedral and dihedral to a more complex 3-dimensional situation, involving billow and/or washout as well as sweep, by specifying that when we say the "wing chord line", we really mean the "mean" or "average" chord line, or better yet, an "average" chord line that is calculated in a way that is weighted according to the (chord length * moment-arm) of each little slice of the wing from the root to the tip. For example, the wing area immediately adjacent to the keel would be given minimal weight when we calculate this "average" chord line because this part of the wing has little moment-arm and cannot generate very much roll torque, and the extreme tip of a sharply pointed wing would be given little weight when we calculate this "average" chord line because the chord (width) at this point on the wing is so small. Or we can use a cruder, simpler definition of anhedral: for example we can specify that the wing should be held in a pitch attitude where the wing chord line at mid-span is horizontal--we'll return to this idea later. One way or another, we need to specify what part or parts of the wing should form our reference line when we put the wing in a "level" pitch attitude to measure the vertical distance between the wing tips and the wing root in an effort to quantify the wing's anhedral or dihedral. If the wing is swept, our choice of a reference line for "leveling" the aircraft will make a very large difference in the amount of anhedral or dihedral that we come up with when we measure the vertical distance from the wing tips to the wing root.

Note August 2005: for more photos relating to the idea of placing a glider in a pitch attitude where the chord line at the mid-span point of each wing is horizontal, see this page.

*However all this talk about placing the wing in a specific pitch attitude and making measurements of the vertical distance between the tips and the root is really missing a deeper point. What is really going on here? Why do things seem to get so complex when we start talking about a swept wing with washout and billow?

*Here's the answer in a nutshell: any time we rotate any portion or panel of a wing in a way that raises the panel's trailing edge, and pivots the panel around an aft-swept hinge line located at the panel's leading edge, we increase the net geometrical anhedral (or decrease the dihedral) of that part of the wing. Any time we rotate any portion or panel of a wing in a way that lowers the panel's trailing edge, and pivots the panel around an aft-swept hinge line located at the panel's leading edge, we increase the net geometrical dihedral (or decrease the net geometric anhedral) of that part of the wing.

* (This idea may break down if the wing portion or panel is rotated so far that it starts to approach a vertical orientation--we're assuming that the chord lines of all portions or panels of the wing will remain oriented within 45 degrees or so of the average chord line of the wing as a whole).

*If we rotate a portion or panel of a wing around a hinge line that has no sweep, this will have no effect on the wings net geometric anhedral or dihedral. (Twisting an unswept wing to create washout can be thought of as a rotation of the wing around a hinge line that has little or no sweep, and this has little effect on the wing's net geometric anhedral or dihedral).

*All this may seem like quite a mouthful but it is very easy to see in photographs.

*For example, in this photograph we have rotated each wing (by raising the trailing edge) around an aft-swept hinge line (the leading edge tube) and this has created net geometric anhedral. We can see the top surface of the near wing and the undersurface of the far wing. (Photo 8)

*A very important point is that it doesn't matter whether the panels we are moving are located on the inboard or outboard areas of the wing. We can increase the wing's net geometric anhedral even if we only raise the trailing edges of the outboard parts of the wing, while leaving the inboard portions alone, so long as we are rotating these outboard parts of the wing around aft-swept hinge line. And we can increase the wing's net geometric dihedral even if we only lower the trailing edges of the outboard parts of the wing, so long as we are rotating these parts of the wing around aft-swept hinge lines. With a swept wing, it's definitely a misconception to imagine that any time we raise any part of the outboard areas of the wing, we create a dihedral effect, or that any time we lower any part of the outboard portions of the wing, we create an anhedral effect.

*In this photograph we have only raised the outboard trailing edges of each wing, by rotating the outer panels around the aft-swept hinge line formed by the leading-edge tubes. This has increased the wing's net geometric anhedral (Photo 12). A good side view of this model will be included in a future edition of this article--from the right angle, we can see the top surface of outboard portion of the near wing and the bottom surface of the outboard portion of the far wing, making it obvious that the wing will generate a strong roll torque toward the camera if the wing is sliding sideways toward the camera, i.e. if the airflow (relative wind) is blowing from the camera toward the wing. This particular model is really the key to understanding how billow and washout contribute to the net geometric anhedral of a swept wing. At first glance, an observer might be tempted to think that by raising a portion of the outboard areas of each wing, we must be creating dihedral, not anhedral. This would be a misconception--the three-dimensional geometry of this swept wing is such that we are clearly increasing the wing's net geometric anhedral.

*So far we've been dealing with discrete, flat wing panels. The full 3-dimensional effects of billow and washout on the geometry of a swept wing are complex. Let's do a little thought experiment. (Diagrams will be included in future editions of this article).

**(See the longer article for the entire "thought experiment")

(Note August 2005: the basic idea we were trying to convey in the "thought experiment" was that the wing would need to bend along the blue lines on this model.)

*This is a very crude mental picture but it does give us a handle on why billow tends to create a dihedral-like effect in the inboard areas of each wing, where the "hinge line" is swept forward, and an anhedral-like effect in the outboard areas of each wing, where the "hinge line" is swept aft. The net effect is almost as if the leading edge tube itself were being bent into a "M" shape as seen from a front view of the whole, entire aircraft, though of course in reality only the trailing edge, not the leading edge, of the wing is bending into the "M" shape as seen from a front view of the whole, entire aircraft.

**Recall again that the outboard parts of the wing are the most important in determining whether the wing will have "net geometric anhedral" or "net geometric dihedral", because they act such a large moment-arm from the C.G..

*These pictures give a very clear illustration of the way that billow tends to create geometric dihedral in the inboard parts of each wing and tends to create geometric anhedral in the outboard parts of each wing (Photo 14). We can see the undersurface of the inboard portion of the near wing and we can see the upper surface of the inboard portion of the far wing. We can see the top surface of the outboard portion of the near wing and we can see the under surface of the outboard portion of the far wing. Altogether, the wing will have net geometric anhedral, because the outer portions create more roll torque than the inboard portions when a sideways airflow is present.

*Added notes April 2005: this material will undergo some revision in the near future. One item that is currently lacking is a photograph of a 3-dimensional, continuous model--not made up of discrete surfaces--of a "conventional", rigid, swept wing with an exaggerated amount of washout. Photographs of such a model would convincingly demonstrate that in any case where the washout can be modeled as a rotation or twisting of the wing surface about an aft-swept axis, the washout will increase the wing's "net geometric anhedral". On the other hand, if the wing has such a very modest amount of sweep that the washout can be modelled as a rotation or twisting of the wing surface about a non-swept or forward-swept axis, then the washout will have no effect on the wing's "net geometric anhedral", or in the latter case, will actually decrease the wing's "net geometric anhedral". Again, 3-dimensional models are needed to convincingly illustrate these ideas.

*Added notes April 2005 continued: in the case of a flexible hang glider wing with both sail billow and washout, it appears that the geometry is such that increasing the washout at the wingtips can actually end up diminishing the very pronounced net geometric anhedral that would otherwise be created by the sail billow in the mid-span area of the trailing edge of each wing. This relationship is apparent in some of the photos used in this article. So in the case of a flexible hang glider wing, the increased sail billow, not the increased wingtip washout, is the key factor that increases the wing's "net geometric anhedral" when the VG is loose, and decreases the wing's "net geometric anhedral" when the VG is tight. These relationships will be explored more rigorously in a future edition of this article.

 

 

For more photos illustrating that billow contributes to anhedral, and illustrating that a hang glider has more "net geometric anhedral" when the VG is loose than when the VG is tight, see the related page on this website entitled "Photos of hang gliders and models to illustrate how billow contributes to the net geometric anhedral of a swept wing".