How billow increases the net geometric anhedral of a swept wing, and other related topics

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 is an early edition of this article; look for a subsequent edition with more photographs to follow.

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". If you are unfamiliar with hang gliders and are trying to understand the passages that relate to "billow", it will help if you know that a "VG" (variable geometry) is a device that tightens the "sail" (wing) and reduces billow.)

 

Let's begin:

*It's a misconception that anhedral or dihedral create a roll torque because the left wing becomes "more horizontal" than the right wing, or vice versa, or one wing ends up with a "more vertical lift vector" or a "greater projected area", when the aircraft banks to one side.

*In reality, in the aircraft's reference frame, banking can be thought of as a change in the direction of the weight vector, acting at the CG, while all the aerodynamic force vectors (drag, lift from left wing, lift from right wing, etc) initially remain unchanged in the aircraft's own reference frame, at least if we imagine that the aircraft has not yet began to move sideways through the airmass. This situation cannot produce a roll torque. If the aircraft does not move sideways through the air, anhedral or dihedral cannot produce a roll torque when the aircraft banks.

*However, if the aircraft begins moving sideways through the airmass, then anhedral or dihedral will create a roll torque. We'll explain the full reason for this in just a bit.

*The roll torque created by anhedral or dihedral when an aircraft is tipped into a bank is entirely dependent upon the fact that immediately after leaving the wings-level condition, the aircraft will initially begin to move sideways through the air to at least a small degree. This sideways motion is created by the fact that the flight path has started to curve, because the banked wing is generating a sideways (centripetal) force, but the aircraft's yaw stability or "weathervane effect" has not yet exerted sufficient yaw torque to overcome the aircraft's yaw rotational inertia and create the yaw rotation that is required to keep the nose of the aircraft aligned with the direction of travel at any given instant, i.e. to keep the nose of the aircraft pointing straight into the changing direction of the relative wind. So the flight path starts to curve, the heading of the aircraft initially tends to remain constant, and this creates a sideways airflow over the aircraft.

*If the aircraft is banking because the pilot is using weight-shift or ailerons to make an intentional roll control input, then the nose will tend to initially yaw in the "wrong" direction due to adverse yaw, and again this will create a sideways airflow over the aircraft.

*The main purpose of a rudder is to prevent this sideslip due to adverse yaw and yaw rotational inertia when the pilot is intentionally banking the aircraft. Most rudderless aircraft--unless they are using spoilerons for roll control--will experience some sideslip whenever the pilot is intentionally entering a turn, as well as when the aircraft is tipped into a bank by turbulence.

*More generally, the fundamental purpose of a rudder in most aircraft is simply to give the pilot a way to "help" the aircraft's inherent yaw stability or "weathervane effect" keep the nose of the aircraft directly aligned with the aircraft's actual flight path through the airmass, so that the nose points directly into the airflow (relative wind), and there is no sideways component in the airflow over the aircraft. A sideways component in the airflow (relative wind) would create drag, and would also make the wing less efficient. In most aircraft, the fundamental purpose of the rudder is not to create a turn, i.e. a curvature in the flight path. Banking the wing, not yawing the nose to the side, is the most efficient way to create a curvature in the flight path. In most aircraft, the pilot should turn the aircraft (i.e. make a change in the direction of the flight path) by banking the wing, while applying whatever rudder inputs are needed to counteract adverse yaw and other related effects and keep the nose directly aligned with the actual direction of the flight path and pointing directly into the airflow (relative wind) at any given moment. However, the rudder can also be used as a roll control in many aircraft--more on this below.

*The dynamics that we've been discussing up to this point take place mainly while the aircraft's bank angle is changing, or immediately after the aircraft's bank angle has changed. But even after quite a few seconds have passed with no further change in bank angle, and the aircraft's yaw stability mechanisms have had plenty of time to act, there will still typically be a very slight sideslip when a rudderless aircraft is banked, because a bank always creates a curvature in the flight path (i.e. the aircraft is turning), and when the flight path is curving, the outboard, faster-moving wingtip tends to experience more drag than the inboard, slower-moving wingtip. This yaws the nose toward the outside or high side of the turn, at which point the drag torques from the left and right wings, plus any "weathervane" yaw torque from the vertical tail (if present), again come into balance. In future articles we'll use the concept of "airflow curvature" to explore this balance of yaw torques in more detail, and we'll examine how the position and size of the vertical fin (if present) affects the slight amount of sideslip that we typically see in a stabilized, constant-bank turn in a rudderless aircraft. This is a rather complex subject.

*Therefore if a pilot wishes to keep the nose of the aircraft pointing directly into the airflow during a stabilized, constant-bank turn, he'll typically need to apply just a touch of inside rudder, especially at low airspeeds where the turn radius will be small and the difference in airspeed between the two wingtips will be the most pronounced.

*If there is no rudder, or if the pilot doesn't apply inside rudder, this continued, slight sideslip means that dihedral will continue to create a rolling-out (stabilizing) roll torque even when the bank angle is no longer increasing. Also, this continued, slight sideslip means that anhedral will continue to create a rolling-in (destabilizing) roll torque even when the bank angle is no longer increasing. But we're jumping the gun here: the reason a roll torque arises when a sideways airflow is present is given below.

*Anhedral or dihedral create a roll torque whenever an aircraft moves sideways through the airmass, regardless of whether the sideways motion is due to the fact that the aircraft has gained a velocity component toward one wingtip and the aircraft's yaw stability mechanisms have not yet yawed the nose into alignment of the new direction of travel, or the direction of the flight path has not changed but the nose of the aircraft has yawed to point to the left or the right of the direction that the aircraft is actually moving through the airmass. Both cases have the same end result--the aircraft moves sideways through the airmass, so there will be a sideways component in the airflow over the aircraft (relative wind). In both cases, anhedral or dihedral will generate a roll torque. And in both cases the aircraft's yaw stability mechanisms will be attempting to yaw the nose to point directly into the relative wind, which would end the sideslip or skid.

*The roll torque created by dihedral or anhedral during a sideslip or skid is caused by the fact that the left wing is experiencing a higher or lower angle-of-attack than the right wing, due to the sideways component in the airflow (relative wind). This is the central point of this entire article.

*To understand this, look at these photographs of wings and models of wings, taken from the side. (Photo 1: Superfloater ultralight sailplane with dihedral.) (Photo 2: model of wing with dihedral) (Photo 3: model of wing with anhedral). In many of them, you can see the underside of the left wing and the top side of the right wing, or vs. vs.. If the aircraft were moving directly toward the camera, which would involve an extreme angle of sideways motion through the airmass, i.e. an extreme sideslip angle, then clearly there would be a very large difference in angle-of-attack between the left and right wings: one wing would be generating a positive lift force and the other side of the wing would be generating a negative lift force. In real-life situations, involving much smaller angles of sideways motion (sideslip), neither wing will be flying at a negative angle-of-attack, but the difference in angle-of-attack between the left and right wings will still exist to a smaller degree. This will create a roll torque. Use your imagination and visualize what kind of roll torque will be created as a small sideways airflow component flows over these wings. You will see that the left wing will be developing more lift than the right wing, or vs. vs., due to the difference in angle-of-attack between the left and right wings.

*Of course the outboard parts of the wing, near the tips, will tend to generate larger roll torques than the inboard parts of the wings, near the roots. This is because the outboard areas are further from the CG, and so they act at a greater moment-arm.

*Use your imagination again (diagrams will be included in future editions of this article): think about which direction the wing's lift force will be tilted when wing gets banked to one side by a bit of turbulence. Let's assume that the wing hasn't started slipping sideways just yet, so anhedral or dihedral haven't yet come into play--there's no need to think about the left and right wings individually yet, so just focus on the sideways tilt of the aircraft's entire, net lift vector as the aircraft banks. Now consider that the aircraft will be "pushed" sideways through the air by this tilted lift vector, at least for a few seconds, as we've already discussed above, until the nose has a chance to start clocking smoothly around the horizon and the aircraft ends up in a full-fledged turn with minimal sideslip. Now, as the aircraft is slipping sideways through the airmass, any anhedral or dihedral that is present will create a roll torque, as we've already seen by looking at the photos of the wings with anhedral and dihedral. Now think about the direction of this roll torque that is created by the sideways airflow over an anhedral or dihedral wing, due to the aircraft's initial, sideways motion through the airmass, immediately after a one wing drops due to a bit of turbulence. With a bit of thought we can see that when a wing drops, the aircraft will move sideways through the air in such a way that dihedral will end up creating a stabilizing roll torque that tends to return the aircraft to wings-level, and anhedral will end up creating a destabilizing roll torque that tends to roll the aircraft to a steeper bank angle. Again, this roll torque is dependent upon the fact that when one wing drops, the aircraft will sideslip a bit (due to yaw rotational inertia) before settling into a full-fledged turn, and this will create a sideways component in the airflow (relative wind).

*We can also see that if the aircraft has very high degree of yaw stability--i.e. a very strong "weathervane effect"--due perhaps to a very large vertical fin--then when it gets tipped into a bank by a bit of turbulence, instead of slipping sideways to any appreciable degree, it will tend to quickly "weathervane" into a full-fledged turn, with the nose clocking smoothly around the horizon and little sideways component in the airflow over the wing. So increasing the size of the vertical fin, or increasing the aircraft's "weathervane" stability torque by increasing the amount of sweep in the wing, will tend to minimize the roll torque created by any anhedral or dihedral that is present, and this will have a definite effect on the aircraft's roll stability or spiral stability. If the aircraft has dihedral, then a large vertical fin will tend to decrease the aircraft's roll stability or increase the aircraft's roll or spiral instability. If the aircraft has anhedral, then a large vertical fin will tend to increase the aircraft's roll stability or decrease the aircraft's roll or spiral instability.

*For some diagrams and additional notes on the way that the stabilizing effect of dihedral is entirely dependent upon the fact that an inadvertent bank always leads to a sideslip, see section 9.3 of John S. Denker's superb "See how it flies" website. Note however that Denker focusses on a slightly different driving mechanism for the sideslip than the two factors that we've emphasized here--he invokes the "long-tailed slip effect" while we've focussed on yaw rotational inertia and the difference in airspeed between the inboard and outboard wingtips. Denker actually views the vertical tail as driving the sideslip that arises during an unintentional bank, rather than mitigating the sideslip! Following Denker's logic, in an aircraft with a relatively small wingspan, and a relatively small amount of rotational inertia in the yaw axis, and a relatively long tail moment arm, increasing the size of the vertical tail could actually end up having the opposite effect as we've described above, in the context of unintentional banks due to turbulence. (This idea wouldn't apply to long-spanned flying-wing aircraft like hang gliders and trikes). However, with any aircraft, a large vertical tail will also minimize the adverse yaw that arises from a pilot's control inputs, which will increase the responsiveness of an aircraft with dihedral and decrease the responsiveness of an aircraft with anhedral, as we'll see below.

*Just as dihedral tends to return an aircraft to wings-level whenever a wing drops in turbulence and the aircraft starts sliding sideways, so too does dihedral tend to make an aircraft respond sluggishly to a pilot's roll inputs (unless a rudder or spoilerons are being used to overcome adverse yaw and/or yaw rotational inertia and avoid a sideslip).

*So the "coordinating" rudder input that a pilot typically makes to overcome adverse yaw and/or yaw rotational inertia and keep the aircraft pointing straight into the airflow (relative wind) as he banks the aircraft into a turn, will also improve the aircraft's roll rate, if the wing has a significant amount of dihedral or sweep.

*Just as anhedral tends to make an aircraft roll into a tighter bank whenever a wing drops in turbulence and the aircraft starts sliding sideways, so too does anhedral tend to make an aircraft respond more quickly to a pilot's roll inputs (at least in cases where a rudder or spoilerons are not being used to overcome adverse yaw and/or yaw rotational inertia and avoid a sideslip.)

*The roll torque created when a wing with anhedral or dihedral experiences a sideways airflow leads to a "coupling between yaw and roll". What does this mean? If a pilot applies heavy right rudder to yaw the nose of the aircraft toward the right, or if adverse yaw or any other phenomenon yaws the nose of the aircraft toward the right, then the aircraft will be moving sideways through the air. The nose will not be pointing the same direction that the aircraft is actually moving through the airmass. There will be a sideways component in the airflow (relative wind) over the aircraft, blowing from the left wingtip toward the right wingtip. Then dihedral or anhedral will create a roll torque:

*If the wing has dihedral, then when the pilot's rudder input (or adverse yaw or any other factor) yaws the nose of the aircraft toward the right (in relation to the actual direction of the flight path and airflow) as described above, the resulting left-to-right sideways component in the airflow will cause the left wing to experience a higher angle-of-attack than the right wing, and this will create a roll torque toward the right. So an initial yaw toward the right ends up creating a roll torque toward the right. This is a "positive coupling" between yaw and roll.

*In the same situation, if the wing has anhedral rather than dihedral, the left-to right sideways component in the airflow will cause the right wing to experience a higher angle-of-attack than the left wing, and this will create a roll torque toward the left. So an initial yaw toward the right ends up creating a roll torque toward the left. This is a "negative coupling between yaw and roll.

*Therefore, over-enthusiastic use of a rudder or wingtip drag device to "skid" the nose in the direction of the intended turn (as opposed to using the "right" amount of rudder to overcome adverse yaw and keep the nose aligned with the flight path and airflow (relative wind), or using no rudder and allowing the nose to swing the "wrong" way due to adverse yaw) will create a helpful roll torque in the case of a wing with dihedral--this is why many aircraft such as Gentle Lady RC sailplanes, Dragonfly tugs and even Cessna 152's can be controlled with the rudder alone, without use of the ailerons. (Granted this technique is not always very efficient, and in some cases can be an invitation to a spin).

*By the same logic, over-enthusiastic use of a rudder to "skid" the nose in the direction of the intended turn (as opposed to using the "right" amount of rudder to overcome adverse yaw and keep the nose aligned with the flight path and airflow (relative wind), or using no rudder and allowing the nose to swing the "wrong" way due to adverse yaw) will create an unfavorable roll torque in the case of a wing with anhedral. In an aircraft with anhedral, it is very possible for a left rudder input to cause a right bank followed by a right turn--this vividly demonstrates that "yawing" (i.e. swinging the nose to one side in relation to the direction of the flight path and relative wind) and "turning" (i.e. creating a curvature of the flight path) are not at all the same thing! And by the same logic, on an aircraft with anhedral, adverse yaw will actually create a helpful roll torque! All this will be discussed in much more detail in the areas of this website that deal with experiments with rudders on flex-wing hang gliders with anhedral.

*Sweep is very similar to dihedral: it creates a "positive coupling between yaw and roll". When a swept-wing aircraft yaws to the left in relation to the actual direction of the flight path and airflow (relative wind), this creates a right-to-left sideways component in the airflow over the aircraft. The right wing becomes "less swept" in relation to the airflow, and the left wing becomes "more swept" in relation to the airflow. (Illustration center right of page). The right wing creates more lift and the left wing creates less lift, and this creates a roll torque toward the left, just as would a wing with dihedral in the same airflow.

*Swept-wing aircraft often have a mild amount of anhedral to reduce the excessive "positive coupling between yaw and roll" that would otherwise be created by the sweep.

*The roll torque created by a swept wing in a sideways airflow is highly dependent on the magnitude of the G-load or lift force that the wing is generating. If the wing is "unloaded" to the zero-lift angle-of-attack, then the left and right wings will create the same amount of lift (none), and there will be no roll torque. In a maneuver where the aircraft is creating multiple G's, the aircraft will create more roll torque in the presence of a sideways airflow than it would in ordinary 1-G flight.

*In ordinary 1-G flight, the roll torque created by a swept wing in a sideways airflow is still highly dependent on the wing's angle-of-attack. This illustration, (top of page) shows that at high angles-of-attack, changing the sweep angle of each wing in relation to the airflow will have a large effect on each wing's lift coefficient, while at low angles-of-attack this relationship will be less pronounced.

*The roll torque created by dihedral or anhedral is more pronounced when the wing is generating a large G-loading. But in ordinary 1-G flight, the roll torque created by dihedral or anhedral is relatively independent of the wing's angle-of-attack.

*Therefore if an aircraft has both sweep and anhedral, it's quite possible for the anhedral to dominate at high airspeeds (low angles-of-attack), and for the sweep to dominate--creating a dihedral-like effect--at low airspeeds (high angles-of-attack). The aircraft's roll stability characteristics--and its response to rudder inputs--would be very different at low airspeeds than at high airspeeds. I've seen this effect while experimenting with a rudder on a hang glider with sweep and anhedral, and also while experimenting with a rudder on a modified variable-geometry Zagi RC glider with sweep and anhedral. In these aircraft, at low airspeeds a left rudder input created a left bank and a left turn, while at high airspeeds, a left rudder input created a right bank and a right turn.

*Now let's look more closely at the detailed geometry of dihedral and anhedral.

*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.

*Think about the shape of a wing will change as a VG is applied. If the VG is of the older "pulley" type (as opposed to the "cam" type), will the slight lowering of the tips, by a few inches, in relation to the keel, be enough to compensate for the lost billow and twist? Will the wing end up with less, or more, net geometric anhedral when the VG is applied?

*The in-flight experiments to look at coupling between yaw and roll that we'll describe elsewhere on this website, as well as rough measurements of geometry when the glider is on the ground, both show that a typical flex-wing hang glider ends up with much less net geometric anhedral when the VG is applied than when the VG is loose, even with a conventional "pulley" VG system, and more so with a "cam" VG system. More on this below.

*Some specific aerodynamic observations--such that the fact that a wing typically needs more high-siding when the VG is applied--initially appear to suggest that the net aerodynamic anhedral decreases as the VG is applied. However, there are other possible explanations for this. In a stabilized constant-bank turn, there is only a slight amount of sideslip anyway, so even a large amount of anhedral will only generate a small roll torque. (Remember, it's a misconception that anhedral or dihedral will generate a roll torque whenever the one wing is "more horizontal" and the other wing is "less horizontal". Anhedral or dihedral will only generate a roll torque when there is a sideways component in the airflow over the wing, i.e. a sideslip or skid). Here's one possible explanation for the increased amount of high-siding that is typically required when the VG is tight: tightening the VG removes washout and makes the tips "work" much harder than they do when the VG is loose. This is much like an increase in wingspan. This means that the rolling-in torque that is created by the fact that the outboard wingtip is moving faster than the inboard wingtip becomes much larger when the VG is tight. Many other subtle effects related to the flexible geometry of the entire wing may also come into play here.

*Of course, the increased sail tension when the VG is tight obviously means that the wing will respond much more slowly to a pilot's roll inputs. We aren't including this effect in our concept of "effective aerodynamic anhedral". We'll touch on the relationship between sail tension and aerodynamic damping in the roll axis in more detail elsewhere on this website. However, the decrease in net geometric anhedral that takes place when the VG is tight is another reason that a modern flex-wing hang glider becomes less responsive in the roll axis when the VG is tight.

*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.

*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).

*What if we have a hinge line that is swept forward instead of aft? 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 a forward-swept hinge line located at the panel's leading edge, we increase the net geometrical dihedral (or decrease the anhedral) 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 a forward-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.

*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)

*In this photograph we have rotated each wing (by lowering the trailing edge) around an aft-swept hinge line (the leading edge tube) and this has created net geometric dihedral. We can see the bottom surface of the near wing and the top surface of the far wing. (Photo 7)

*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.

*Now imagine that a swept-wing aircraft, with spoilerons, simultaneous extends both spoilerons so that they protrude at perhaps a 45-degee angle from the top surface of each wing. Assuming that the spoilerons are mounted on a swept hinge-line, this will create an anhedral effect, much like the anhedral effect that is visible in this model (Photo 12). Looking at the wing from the side, we would see the top surface of the spoileron nearest the camera, and the bottom surface of the spoileron farthest from the camera, assuming that the spoilerons were mounted on a swept hinge line.

*Consider this picture of a Swift ultralight sailplane (Photo 13). We can clearly see the bottom surface of the lowered, inboard, wing flap on the aircraft's left wing, and if we look very close we can see the top surface of the other flap. Clearly, this will tend to create a dihedral effect--if the airflow (relative wind) were to have a component that were blowing sideways from the left wingtip toward the right wingtip--i.e. if the nose of the aircraft were yawed to point toward the right of the direction that the aircraft were actually moving through the airmass--then the airflow (relative wind) would tend to strike the bottom surface of the left flap very squarely and to strike the bottom surface of the right flap in a much less direct manner. Clearly this would create a roll torque toward the right--away from the camera--at least so long as the flaps were deployed at some deflection that was significantly less than 90 degrees. The lowered trailing edges of the flaps are adding to the wing's net geometric dihedral or subtracting from the wing's net geometric anhedral. If this isn't clear, imagine what would happen if the aircraft were actually sliding directly toward the camera, so that the airflow was actually striking the top surface of the right flap, and the bottom surface of the left flap. Clearly this would generate a roll torque toward the right--away from the camera--because the average angle-of-attack of the entire inboard area of the left wing (including the flap) would be very positive and the average angle-of-attack of the entire inboard area of the right wing (including the flap) would be somewhat negative. None of this is related to the fact that the flaps are mounted on the inboard parts of the wing--we'd get the same dihedral effect if we drooped the trailing edges of flaps that were located on the outboard portions of the wings, as long as the flaps were pivoting around an aft-swept hinge line. All of this is similar to what we saw when we lowered the trailing edge of the entire wing in this model (photo 7). .

*For a bit of mind-teaser, imagine what the photo of the Swift would look like if the flaps were somehow mounted on hinge lines that were swept forward instead of aft. The effect would be reversed. If the airflow had a component that were blowing sideways from the left wingtip toward the right wingtip, the flaps would contribute a roll torque toward the left, toward from the viewer.

*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).

*We'll imagine that we start with a flat, taut, untwisted, swept wing--for example let's assume that we have a hang glider whose keel and leading edge tubes all lie in the same plane. Let's imagine that our glider has defined wingtip struts, and that they lay in the same plane as the keel and the leading edges (i.e. there is no washout).

*For the time being we'll leave the defined tip struts alone, lying in the same plane as the keel--we won't give our wing any washout. We will give it some billow. To give the wing billow, imagine that we draw a little "X" at the mid-span point of each leading edge tube. Now imagine that we take the left wing and start at the place where the trailing edge meets the keel. Now imagine that we draw a straight line forward and outboard to the "X" at the mid-span point of the leading edge of the left wing. Now imagine that we draw a second straight line from this "X" outboard and rearward till it meets the corner where the trailing edge of the left wing meets the defined wingtip strut. We've drawn a forward-pointing "V" shape across the top surface of the left wing, as seen from above.

*Now we'll repeat this with the right wing. We're treating each wing separately and assuming that the sail is attached tightly to the keel--we don't have a keel pocket or floating crossbar in this thought experiment.

*Now to give each wing some billow we'll take hold of the trailing edge at the mid-span point of each wing and pull upward. The sail will need to stretch in a complex way. For a very crude mental picture, we can imagine that it is essentially bending along the solid "V" line that we've drawn on each wing. The inboard part of the sail is bending along a forward-swept hinge line and the outboard part of the sail is bending along an aft-swept hinge line. Meanwhile the sail is stretching all over, but especially on a line running straight aft from each of the "X's" that we've drawn at the mid-span of each leading edge tube, because the hinge lines run in such a way that the sail must stretch in the mid-span region of the trailing edge of each wing.

*Again, diagrams will be included in later editions of this article! (Note August 2005: the basic idea we were trying to convey in the above description was that the wing would need to bend along the blue lines on this model, especially as drawn on the wing on the viewer's left.)

*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.

*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.

*Now let's return to our imaginary wing, which we left without any washout--the defined tip struts were laying in the same plane as the keel and leading edge tubes. Now let's twist the defined tip struts upwards. This will essentially bend the outboard portions of the sail upward along a hinge line that is very roughly parallel to the leading edge tube. Since this hinge line is swept, this will further add to the wing's net geometric anhedral, as suggested by this model (photo 12) and this photo (photo 15). (Note August 2005: the basic idea we were trying to convey in the above description was that the wing would need to bend along the blue lines on this model, especially as drawn on the wing on the viewer's right. However if the sail has a large amount of billow, it's no longer clear to me that adding washout to the tip will add to the wing's net geometric anhedral.)

*The reader need not worry too much about the details of "hinge lines" etc., as long as he can look at the photos and see that we're generally on the right track here. Clearly billow and washout are adding to the glider's net geometric anhedral.

*Why is this glider on its top (photo 16)? We're returning to our earlier idea of positioning the glider in a pitch attitude where the chord line at mid-span is horizontal. Note how the undersurface becomes invisible exactly at the mid-span point in the head-on view (photo 17)--the glider has been positioned so that an imaginary fore-and-aft chord line connecting the trailing edge to the leading edge runs horizontally (parallel to the ground) exactly at the mid-span point. The glider is inverted so that the flying wires are tight and the top wires are loose, and also so that gravity will pull the sail down into a shape that very, very loosely approximates the in-flight shape. With the glider in this orientation, we can measure the vertical distance between the quarter-chord point at the wing root and the quarter-chord point at the wing tips. (We've specified the quarter-chord point because for any given airfoil-shaped slice of the wing, the aerodynamic center is generally near the quarter-chord point, and also simply because we need to pick a fixed reference point one way or another). We can take this vertical distance figure to be a quantitative measure that is very loosely related to the wing's net geometric anhedral. Then we can tighten the VG. This removes some of the billow in the sail, lowering (in the glider's own reference frame) the trailing edge at mid-span. In the head-on view, the point where the trailing edge becomes invisible will migrate outboard on each wing, so that the wing as a whole will appear to be "flying" at a higher angle-of-attack in the glider's own reference frame. In order to return the mid-span chord line to an alignment that is parallel to the horizon, so that the trailing edge again disappears exactly at the mid-span point, we'll need to rotate the nose down (in the glider's own reference frame). This will raise the tips upward (in the glider's own reference frame), due to the sweep in the wings. On the other hand, the tips will have a tendency to lower a few inches (in the glider's own reference frame) as we tighten the VG, because this glider has a conventional "pulley" VG system, so the flying wires will tend to tighten. After we've tightened the VG, and lowered the nose of the glider (in the glider's reference frame) to return the mid-span chord line to a horizontal attitude, we can re-measure the vertical distance between the tips and the root as described above. When we actually make the measurement we find that the total vertical distance between the wingtips and wing root is now less, so our quantitative estimate of the wing's net geometric anhedral is now smaller. The upward displacement of the tips as we rotated the entire glider in a nose-down manner (in the glider's own reference frame) to keep the center-section chord line horizontal has been more significant than the downward movement of the tips (in the glider's own reference frame) due to the tightening of the side wires due to the conventional "pulley" VG system.

*By the way this "downward rotation of the nose" that we keep mentioning in the above paragraph is obvious in actual flight when we pull the VG on--the bar moves aft.

*The reader may object that the glider's pitch attitude in space, or the glider's angle-of-attack, shouldn't affect our estimate of the wing's net geometrical anhedral. This is true. And we're really not thinking about the pitch attitude or angle-of-attack per se--for example we haven't paid any attention to the fact that the wing as a whole needs to fly at a higher angle-of-attack when the VG is loose, because the highly twisted, billowed wing is less efficient aerodynamically. We're just thinking about the actual, physical chord line of various parts of the wing, and how they change when the VG is loosened or tightened. The orientation of the wing's mean chord line does have an effect on the wing's net geometrical anhedral, as we've already seen with the models. Certainly the mean chord line is a much more valid reference line than the keel tube--the keel tube has no aerodynamic or geometric significance at all. There's no reason whatsoever to assume that anhedral should be measured with the keel in a horizontal position, when washout and billow mean that most of the wing is oriented at an angle of incidence that is completely different than the line defined by the keel. The common idea that "applying the VG creates more anhedral" commits exactly this error--it takes the keel as a fixed reference line and ignores the radical changes in the 3-dimensional shape of the wing that occur as the VG is applied.

*The reader may object that by focusing on the mid-span area, we are selecting the part of the wing that undergoes the greatest changes in incidence when the VG is loosened or tightened. This may or may not be true, depending on the relative importance of washout vs. billow in the geometry of the particular wing involved. But we get the same general results if we average the anhedral measurements that we get when we level the wing at the quarter-span point, and at the mid-span point, and at the three-quarter span point. With this more complex method we again come up with a larger anhedral measurement when the VG is off than when the VG is on.

*This method of inverting the glider and orienting the glider so that the mid-span chord line is parallel to the horizon, is clearly a rather crude attempt to add a quantitative element to our theme that billow and washout contribute to a the net geometric anhedral of a swept wing. More detailed photos showing the front views and side views of the Airborne Blade in the VG-off and VG-on positions, with the glider inverted and positioned so that the mid-span chord line is horizontal, will be added to later editions of this article. So will photos comparing the appearance of a Wills Wing Skyhawk, a Wills Wing Spectrum, and the Airborne Blade when positioned in this manner. The head-on views are especially enlightening and dramatically illustrate how very much the billow and washout are contributing to the wing's net aerodynamic anhedral.

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.

*Now let's end our attempt to put the wing in a specific, defined pitch attitude, and put away our measuring tape, and simply eyeball the wing with the VG loose and with the VG tight, while the glider is inverted in order to load up the flying wires and allow the sail to billow in the normal direction. If the VG is loose, when we look at the wing in a side view, we can see a great deal of the top surface area of the outer wing panel on the wing nearest the observer, and we can see a great deal of the undersurface of the outer wing panel on the wing furthest from the observer. This is due to washout and billow. When we tighten the VG, we see much less of the top surface of the wing nearest the observer, and much less of the bottom surface of the wing farthest from the observer, especially if we concentrate our attention on the outboard portions of the wing, which are farthest from the CG and therefore most important as far as the total balance of roll torques is concerned. As we eyeball the wing from a side view, it becomes very clear that the wing has much less net geometric anhedral with the VG tight than with the VG loose. (More photos of all this will be added in the future). Again, it's clear that when we apply the VG, the decrease in washout and billow is affecting the overall three-dimensional geometry of the wing in a way that is much more important than the slight lowering of the wingtips due to the way that the conventional "pulley" VG system tends to tighten the flying wires. Also, eyeballing different gliders in this manner readily reveals that the Wills Wing Spectrum, which has very little droop in the leading edge tubes as compared to the line of the keel tube, and a very small keel pocket, and at first glance appears to have very little anhedral, actually has enough billow and washout that it ends up with more net aerodynamic anhedral than does the Airborne Blade with the VG on. The three-dimensional geometry of a swept, billowed, washed out, flexible wing is complex enough that we can really learn more from an educated "eyeballing" of the wing from a side view, than we can learn by trying to measure the vertical distance between the wing root and the wingtips while holding the wing in some specified pitch attitude.

*Of course, a wing's appearance in flight will be significantly difference than its appearance at rest, even when we invert it to put an "upward" gravitational load on the sail in the reference frame of the glider. Detailed conclusions about a flex-wing aircraft's net geometrical anhedral in actual flight are obviously best reached through in-flight experiments, such as the explorations of coupling between yaw and roll that we'll discuss in more detail elsewhere on this website, and also through careful in-flight photographs of the wing. The fundamental idea we're trying to illustrate right now with these simple visual observations is this: after a bit of thought, any person can clearly see with their own eyes that washout and billow can make a very large contribution to the net geometric anhedral of a swept-wing aircraft.

*Let's think a bit about the effects of a "cam" VG system versus a "pulley" VG system. With either system the wing ends up with less net geometric anhedral (as we've defined it above) as the VG is tightened, but this effect is more pronounced with a "cam" VG system, because the geometry is such that the wing tips do not get pulled downward in relation to the keel as the VG is applied. At first glance this doesn't seem like such a great idea--after all, anhedral helps provide better roll response on a rudderless aircraft, and roll response becomes slower as sail tension increases, so why should we want to take away even more of the net aerodynamic anhedral when the VG is tight by using a "cam" VG system? Of course, it is nice to have the side wires tight on launch when the VG is loose. And if we take the viewpoint that the glider's characteristics with the VG full on should include the maximum possible aerodynamic efficiency, regardless of the cost in terms of the roll response rate, then the choice of a "cam" VG system makes perfect sense. Also, the more we can reduce the net geometric anhedral, the less the glider will tend to enter into yaw-roll oscillations during flight at high airspeeds (low angles-of-attack).

*This entire line of investigation came as a slow "eureka" to me over many months as I attempted to understand how a glider like the Wills Wing Spectrum, which at first glance appeared to have an extremely modest amount of anhedral, could demonstrate a "negative coupling between yaw and roll" at most angles-of-attack during my experiments with the rudder and wingtip drogue chutes, while many business and military jets that at first glance appeared to have more anhedral than the Spectrum, and had about the same amount of sweep, were well known to exhibit a "positive coupling between yaw and roll" at most angles-of-attack. Unlike my thoughts on the (lack of) relationship between pitch inputs and sideslip, which essentially just represent the application of standard sailplane and airplane theory to the world of hang gliding (after careful experimental tests shifted my paradigm away from an initial inclination to accept what was reported in the main body of hang gliding training literature), the ideas presented here that deal specifically with the way that washout and billow contribute to the net geometric anhedral of a swept wing were arrived at independently of outside sources.

*(P.S.--When the glider is standing on its kingpost, the real source of the load on the sail is the upward push of the ground against the kingpost, not the downward pull of gravity!)

*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, and illustrating some of the other geometry we've discussed on this article (e.g. lowering the flaps on a swept-wing aircraft creates a dihedral effect), 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".