Long-tailed slip

Analyzing the "long-tailed slip" effect, with notes on how "airflow curvature" affects a swept- or delta-winged aircraft

September 2006 edition
Steve Seibel
steve at aeroexperiments.org
www.aeroexperiments.org

 

 

Added note August 2007: I feel these ideas now receive a much better treatment in the newer article on this website entitled "Tail moment-arm, airflow curvature, and spiral stability".

 

 

Note: this is an early version of this article, stay tuned for future updates!

 

(Update August 2006: a year after the initial writing, the only thing I have to add at this point is that this is an extremely convoluted article! For the most part it is thoroughly hypothetical and not based on in-flight observations except where otherwise noted. Still, it makes for some very interesting food for thought.)

 

Note: for the sake of simplicity, except where we state otherwise, in this analysis we'll assume that the aircraft has no rudder or that the rudder is being kept centered.  We'll assume that an aircraft is always free to find a yaw orientation where the net yaw torque is zero, even without help from a pilot's rudder inputs.  We'll be talking about aerodynamic effects that may be foreign to many pilots, and are most pronounced at low airspeeds, where for any given bank angle, the curvature in the flight path and airflow (relative wind) is greatest.

 

Note: this subject would be much better treated with diagrams than with text; readers are strongly encouraged to read this article with a pencil in hand and sketch their own diagrams for each of the scenarios that we'll explore. Here is a link to a polar coordinate grid; a printed copy of this grid will be very helpful for these sketches. The concentric lines can represent the lines of curving airflow (relative wind) experienced by a turning aircraft, as seen by an overhead observer. Sketch the aircraft at various points on the graph to illustrate various yaw orientations including large slip angles, large skid angles, or situations where the curving airflow is tangent to fuselage or keel at some particular point (more on this below). Exaggerating the size of the aircraft in relation to the radius of curvature of the turn--i.e. sketching the aircraft much closer to the center of the grid than would actually be appropriate in relation to the size of the drawing of the aircraft--will make the airflow-curvature effects that we'll be discussing in this article much more obvious. (If it's not yet clear how to use this diagram, it likely will be after the reader visits the illustration of the "Long-tailed slip" in the link given below.)

 

Note: definitions: "slipping airflow"--an airflow that includes a component acting toward the outside or high side of the turn.  "Skidding airflow"--an airflow that includes a component acting toward the inside or low side of the turn.  "Tangent point"--the point where the curving relative wind (airflow) is tangent to a line drawn through fuselage or keel of the aircraft.  "Front" or "forward" portion of the aircraft--the part of the aircraft that is ahead of the CG.  "Rear" or "aft " portion of the aircraft--the part of the aircraft that is aft of the CG.  "Outside" or "high side" of the turn--these mean the same thing, and refer to the "left" or "right" directions in the aircraft's own reference frame, except as stated otherwise near the end of the article when we'll introduce a slightly different reference frame.  "Inside" or "low side" of the turn--these mean the same thing, and refer to the "left" or "right" directions in the aircraft's own reference frame, except as stated otherwise near the end of the article when we'll introduce a slightly different reference frame.

 

I've long been slightly puzzled by the "long-tailed slip" effect, as outlined on p. 221 in Chapter 12 of Wolfgang Langewiesche's physics-for-pilots classic "Stick and Rudder", or section 8.9 of John S. Denker's excellent "See How it Flies" website.

 

These sources appear at first glance to suggest that the "long-tailed slip effect" involves an airflow striking the outside or high side of the vertical fin, which yaws the nose toward the outside of the turn.  We'll call this the "simple interpretation" of the long-tailed slip effect.  This "simple interpretation" of the long-tailed slip effect contradicts the intuitive idea that the increased airspeed and drag experienced by the outboard wingtip creates a yawing-out torque that must be counteracted by a yawing-in torque which must be generated by the vertical fin.  This "simple interpretation" of the long-tailed slip also appears to contradict the familiar observation (for example, as illustrated in this figure from "Chapter 9: Stability and Control" from NASA's Sp-367 "Introduction to the Aerodynamic of Flight") that an overly large vertical fin tends to reduce sideslip, which reduces the rolling-out torque created by dihedral, which in many situations ends up reducing an aircraft's roll stability or increasing an aircraft's spiral instability.  

 

I've recently had some insights into this apparent contradiction.  Here's how the geometry seems to work out: 

 

First let's take a simplified case of a linear (non-curving) airflow, and let's assume that something is increasing the drag of the right wingtip.  The aircraft will yaw toward the right, until the airflow striking the left side of the fuselage and vertical fin, acting at the center-of-pressure of the fuselage plus vertical fin, which is well aft of the CG, creates enough left roll torque to bring things into balance.  When this has happened, the aircraft has found the yaw orientation where the net yaw torque is zero.

 

Now let's take another simplified case.  Now we'll introduce a curvature in the airflow--in the direction that would be associated with a left turn--but we'll assume that the aircraft has no wings.  Note that the fuselage can only be tangent to the curving airflow at only one point, at most.  In this case, all the parts of the fuselage forward of this tangent point will experience a slipping airflow, and all the parts of the fuselage aft of this tangent point will experience a skidding airflow.  Alternatively, a line drawn through the fuselage can be tangent to the curving airflow at a point forward of the nose (meaning that the entire aircraft is experiencing a skidding airflow) or at a point aft of the tail (meaning that the entire aircraft is experiencing a slipping airflow component).

 

If we start with the curving airflow tangent to the fuselage at the CG of the aircraft, we'll see that the slipping airflow over the front of the aircraft will create a yawing-out torque, and the skidding airflow over the rear of the aircraft will also create a yawing-out torque.  The nose of the aircraft will yaw toward the outside of the turn (i.e. toward the right).  Things will only come into balance when the nose yaws so far toward the outside (right side) of the turn that the rear parts of the aircraft, taken as a whole, experience a slipping airflow rather than a skidding airflow.  When this happens, the rear parts of the aircraft, taken as a whole, will generate a (left) yawing-in torque rather than a (right) yawing-out torque.  In other words, the point where the line of the fuselage is tangent to the curving airflow must lie well aft of the CG, so that most of the aft portions of the aircraft lie ahead of this tangent point, so that they experience a slipping airflow, so that they generate a yawing-in torque.  If the aircraft has a long nose with a lot of surface area, this tangent point may actually need to lie behind the aft end of the fuselage, so that the entire aft portion of the aircraft including all of the vertical fin can experience the slipping airflow and can contribute to the yawing-in torque.  Regardless of the exact location of this tangent point, the balance of forces works out as follows: the portion of the aircraft that lies forward of the CG will experience a large slip angle, but the surface area will be small and the moment-arm will be small.  This portion of the aircraft will create a yaw torque toward the outside of the turn.  The portion of the aircraft that lies aft of the CG and forward of the tangent point will be experience a small slip angle, but the surface area will be large and the moment-arm will be large.  This portion of the aircraft will create a yaw torque toward the inside of the turn.  The portion of the aircraft (if any) that lies aft of the tangent point will experience a skidding airflow.  The skid angle will be small, but the moment-arm will be large, so the surface area will also have to be fairly small (though the surface-area-per-unit-length of this portion of the aircraft will be large, because this is where the vertical fin will be located).  This portion of the aircraft will create a yaw torque toward the outside of the turn.  Clearly the part of the aircraft that lies between the CG and the tangent point is the part that must do the "heavy lifting" to counterbalance both the part of the aircraft that lies ahead of the CG, and the part (if any) that lies aft of the tangent point, and this is why the aircraft will clearly need to adopt a yaw attitude where the nose points rather far toward the outside or high side of the turn, so that the tangent point is quite far aft on the fuselage, or in some cases is even behind the aft end of the fuselage.

 

The shape of the fuselage will definitely come into play here.  If the aircraft has a protruding nose and a long slender tail boom, then the vertical fin as a whole must experience a slipping airflow (i.e. the tangent point must fall near or aft of the aft edge of the vertical fin), because that is the only way that the yawing-out torque created by the slipping airflow over the nose can be counteracted.  If the aircraft has a stubby nose and a wide, slab-sided aft fuselage, the tangent point will fall further forward, and a vertical fin located at the extreme rear of the fuselage will experience a skidding airflow and create a yawing-out torque.  The (hypothetical) extreme case of this would be an aircraft with finless, rectangular, slab-sided fuselage and a CG located at the extreme forward end of the fuselage.  In this case, since no part of the aircraft is forward of the CG, for the net yaw torque to be zero, the tangent point would have to be somewhere aft of the center of pressure of the fuselage (because the more rearward parts of the rear fuselage act at a greater moment-arm from the CG than do the more forward parts of the rear fuselage), but still well forward of the aft end of the fuselage.

 

So the conclusion we're reaching is that the parts of the aircraft that are aft of the CG, taken as a whole, definitely experience a slipping airflow, but the very rearmost parts of the aircraft may experience a skidding airflow.  Therefore the vertical fin itself, taken as a whole, may experience an overall airflow that is either slipping or skidding.

 

It's interesting to note that the fact that a pilot invariably has to apply a touch of inside rudder (except in the case of prop-driven plane experiencing strong torque and P-factor effects, or the case of a small aircraft flying at high speeds where "airflow curvature" effects are negligible) to center the yaw string (which measures the direction of the airflow a the nose) or to center the slip-skid ball (which measures the aerodynamic sideforces arising from the overall sideways airflow component affecting the aircraft as a whole) cannot be taken as evidence that, in the absence of the rudder input, the airflow would strike the high side or outside of the vertical fin.

 

Let's "zoom out" a bit and take a look at the larger picture.  Let's ignore the fact that the very rearmost parts of the aircraft may experience a skidding airflow, and let's focus on the fact that the parts of the aircraft the aircraft that are aft of the CG, taken as a whole, experience a slipping airflow. Very loosely we speaking, we could say that the aircraft's inherent yaw stability or "weathervane effect" is trying to yaw the aircraft into an attitude where the rear parts of the aircraft are streamlined with the airflow.  So it's definitely true that the airflow curvature effect causes the nose to yaw toward the outside of the turn. But if we look closer, we see that this yawing tendency is not actually being driven by an airflow component impacting against the outside or high side of the rear parts of the aircraft, taken as a whole.  The reverse is actually true--in order for the net yaw torque to be zero, the airflow must be striking the inside or low side of the rear parts of the aircraft, taken as a whole.  But as we've noted, the vertical fin itself sits so far aft that it may in theory be experiencing either a net slipping airflow or a net skidding airflow, depending on the shape of the rest of the aircraft.

 

Let's return to our example of a left turn, and let's add the wings back into the picture.  Now we have an additional outside (right) yaw torque, created by the increased drag experienced by the faster-moving outside (right) wingtip.  This means that the nose tends to yaw even further toward the outside of the turn.  When the yaw (slip) angle is sufficiently large, the force of the slipping airflow against the parts of the fuselage and tail that lie between the CG and the tangent point will create enough inside (left) yaw torque to counterbalance the outside (right) yaw torque that is generated by the high drag of the outside (right) wingtip, plus the outside (right) yaw torque that is generated by the slipping airflow around the forward fuselage, plus the outside (right) yaw torque that is generated by skidding airflow around the part of the fuselage and tail (if any) that is aft of the tangent point.  This seems to be a rather tall order--unless the aircraft has a very slab-sided fuselage and a very small vertical tail, it seems very likely that the slip angle will need to be large enough that the tangent point will fall near or behind the aft end of the fuselage, so that the vertical fin as a whole will experience a slipping airflow and contribute to the inside (left) yaw torque.

 

Let's take a little detour to look at a rather subtle issue. In addition to the difference in wingtip airspeeds, the curvature in the relative wind (airflow) causes the direction of the relative wind as experienced by various points across the span of the wing to be different, unless the aircraft has adopted a yaw orientation where the fuselage is tangent to the curving airflow at a point that is right in the middle of the wing chord. For example, let's take the case of an aircraft that has a "conventional" shape, like a sailplane. If the fuselage is tangent to the curving airflow at a point that lies well aft, near or behind the tail, then the outside wing will experience a relative wind that flows slightly toward the outside of the turn, but the inside wing will experience a relative wind that flows more strongly (at a larger angle from the "head-on" direction) toward the outside of the turn. (Readers who are having trouble visualizing this are strongly encouraged to sketch out the situation for themselves, using a printed copy of the polar coordinate grid provided in the link at the beginning of this article.) Although a discussion of the full balance of torques in roll axes is a really beyond the scope of this article, it's interesting to give the subject a bit of thought: the fact that the inside wing experiencing a stronger sideways airflow component will tend to make it generate less lift than the outside wing, unless the aircraft has sweep or dihedral, in which case the sideways airflow over both wings will act to increase the lift of the inside wing and decrease the lift of the outside wing, which would offset some (or possibly all) of the rolling-in torque created by the difference in airspeed between the two wingtips. (Aircraft with anhedral and no sweep will experience the opposite effect: a rolling-out torque.) To the extent that dihedral or sweep act to increase the inside wing's lift, this will also tend to increase the inside wing's drag, creating a yawing-in torque component. However we need to be careful with this line of analysis, because at the end of the day, if the bank angle is constant, both wings must in fact be generating the same amount of lift, when all aerodynamic effects (including pilot roll inputs) are included. If both wings are generating the same amount of lift, and the outboard wing is operating in a faster airflow (and at a lower, less efficient angle-of-attack with a lower L/D ratio) than the inboard wing, it appears that the outboard wing will normally end up creating more drag than the inboard wing. Of course, in the case of a flying-wing aircraft this cannot be true, because here the wing is the only source of yaw torques, and the net yaw torque must be zero in a constant-banked turn. We'll consider some particular types of flying-wing aircraft later in this article.

 

Let's "zoom out" once again and take a look at the larger picture.  Again, we'll assume for now that the aircraft has a "conventional" shape, like a sailplane. Again, let's ignore the fact that the very rearmost parts of the aircraft may experience a skidding airflow, and let's focus on the fact that the parts of the aircraft that are aft of the CG, taken as a whole, experience a slipping airflow. Very loosely we speaking, we could again say that the aircraft's inherent yaw stability or "weathervane effect" is trying to yaw the aircraft into an attitude where the rear parts of the aircraft are streamlined with the airflow.  So it's definitely true that the airflow curvature effect causes the aircraft to settle into an attitude that is yawed toward the outside of the turn. But if we look closer, we again see that this yawing tendency is not actually being driven by an airflow component impacting against the outside or high side of the rear parts of the aircraft, taken as a whole.  The reverse is actually true--in order for the net yaw torque to be zero, the airflow must be striking the inside or low side of the rear parts of the aircraft, taken as a whole, to create a yawing-in torque.  This yawing-in torque must counterbalance the increased drag of the faster-moving outboard wing, plus the yawing-in torque create by the sideways (slipping) airflow over the forward parts of this aircraft. Therefore this slipping airflow over the rear parts of the aircraft taken as a whole, and the resulting yawing-in torque, must actually be greater when the airflow is curving than it would need to be in an imaginary situation where the airflow over the fuselage were somehow linear and the increased drag of the outboard wingtip were the only yaw torque that needed to be counterbalanced.  But once again, as we've noted, in theory the vertical fin itself may be experiencing either a net slipping airflow or a net skidding airflow, though the former seems far more likely if the wingspan is large in comparison to the fuselage length and the difference in airspeed and drag between the two wingtips is creating a strong yaw torque. 

 

Now let's get to the heart of the matter and see how this analysis affects our understanding of stability of control.  Again, we'll assume for now that the aircraft has a "conventional" shape, like a sailplane. What happens if we reduce the size of the vertical fin?  In cases where the difference in airspeed between the wingtips creates a strong yaw torque, it seems likely that the "tangent point" between the fuselage and the curving airflow is aft of the center of the vertical fin, especially if the aircraft has a slender tail boom rather than a slab-sided fuselage.  If the "tangent point" is aft of the center of the vertical fin, reducing the size of the fin will increase the slip angle that we see in a constant-banked turn.  On the other hand, if the wingspan is small and the difference in airspeed between the two wingtips does not create such a large yaw torque, and if the fuselage has a slab-sided shape, then the "tangent point" may be forward of the vertical fin.  If this is the case, reducing the size of the fin will decrease the slip angle that we see in a constant-banked turn.

 

In general, if we decrease the amount of slip that is present during a turn, we also decrease the rolling-out torque created by dihedral (if present), so the aircraft becomes less stable in roll, or becomes more spirally unstable. With most aircraft--and especially with slow-flying, long-spanned aircraft that have small turn radiuses and strongly experience the "airflow curvature effect"--if we increase the size or moment-arm of the vertical tail to an excessive point, the aircraft becomes less stable in roll, or becomes more spirally unstable. Free-flight model airplane enthusiasts in particular are very familiar with this idea. This suggests that the vertical tails of these aircraft do in fact experience a slipping airflow, not a skidding airflow, during turning flight.

 

We've concluded that there are some cases (when the wingspan is small and the fuselage is boxy) where the vertical tail actually experiences a skidding rather than a slipping airflow in a steady, constant-banked turn, and reducing the size or moment-arm of the vertical tail will actually reduce the amount of sideslip that occurs in this situation.  In theory, this should be destabilizing--again, the dihedral (if present) will create less rolling-out torque.  However, in a dynamic situation immediately after an aircraft has tipped into a bank, when the flight path has just started to curve, the aircraft's yaw rotational inertia may be playing a significant roll in creating a sideslip, by keeping the nose on its original heading.  (For example in the situation depicted in this illustration from "Chapter 9: Stability and Control" from NASA's Sp-367 "Introduction to the Aerodynamic of Flight", it appears that yaw rotational inertia may be one important factor preventing the nose of the aircraft from immediately yawing into alignment with the sideways component in the airflow.) Therefore it seems that even if an aircraft is configured in such a way that the vertical fin actually does experience a skidding airflow in a constant-banked turn, in a more dynamic situation an excessively large vertical fin may once again be destabilizing, because the aircraft may tend to "weathervane" to face the new direction of flight before the dihedral has a chance to act.

 

* Added note September 2006: the above comments seem unnecessarily complicated, unless we are dealing with aircraft with very long fuselages and very short wings. In actual practice I suspect that the increased drag experienced by the outboard, faster-moving wingtip nearly always tends to yaw the nose of the aircraft far enough toward the outside or high side of the turn that the airflow strikes the inside of surface of the vertical tail. Note that in cases where the turn radius is large, only a very slight amount of yaw will be required to accomplish this, but when the turn radius is small, much more yaw can take place before the airflow strikes the inside surface of the vertical tail. When we look at things this way, we tend to see the "long-tailed slip effect" not as the root cause of a slip, but rather as something that reduces the "effectiveness" of the tail and allows the increased drag experienced by the outboard, faster-moving wingtip to cause a slip. The sideways airflow experienced by the parts of the aircraft that lie in front of the CG also plays a small roll in causing the slip. Since the airflow is still striking the inside surface of the vertical tail, the vertical tail is still creating a yaw torque that acts to swing the nose toward the inside of the turn, so reducing the size of the vertical tail will still increase the amount of sideslip that we see during a constant-banked turn, and increasing the size of the vertical tail will still decrease the amount of sideslip that we see during a constant-banked turn. This is exactly what is seen in actual practice in the design of "conventionally" shaped model and full-scale aircraft. This is also what allows an aircraft with enough dihedral, and with a small enough vertical fin, to experience a strong roll torque toward wings-level when the aircraft is banked (turning) and the pilot is not making an "inside" rudder input to keep the aircraft (or specifically the wings of the aircraft) aligned with (or "square" to) the actual direction of the airflow at any given moment.

 

Now let's revisit some of these issues, taking the case of a swept-wing or delta-winged aircraft with a moderately high or high aspect ratio. We'll assume that readers are familiar with the way that a swept wing has an inherent "weathervane" effect that is stabilizing in yaw, and with the way that a sideways airflow component over the aircraft as a whole will interact with sweep, dihedral, or anhedral to create a roll torque. Our focus will be on how the local changes in airflow (relative wind) due to the "airflow curvature" inherent in a turn will modify these effects. We'll assume that the aircraft has the general planform of a modern flex-wing hang glider, but we'll simplify things by assuming that the trailing edge of the wing is unswept. We'll assume that the aircraft may or may not have vertical fins at the wingtip or a short-coupled vertical fin on the centerline, not very far aft of the trailing edge of the wing.

 

If the aircraft has a fuselage, we'll see some of the same effects as outlined above: due to airflow curvature, the forward part of the fuselage will experience a slipping airflow, which will create a yawing-out torque. The difference in airspeed between the two wingtips will also create a yawing-out torque. In very general terms, we can think of the rear of the aircraft as acting like a "tail". Following our analysis above of a more "conventionally" shaped aircraft, this might lead us to think that to bring the net yaw torque to zero, the aircraft must adopt a yaw orientation where the tangent point between the curving airflow and the line of the fuselage or keel is located quite far aft, so that most or all of the aircraft experiences a slipping airflow, which would interact with the aft parts of the aircraft to generate a yawing-in torque. In the absence of fuselage surface area or vertical fins located well behind the trailing edge of the wing, we might even reach the conclusion that the tangent point would need to be well aft of the trailing edge of the wing, so that the entire aircraft would experience a rather strong slipping airflow component, so that the wingtip areas could generate the required yawing-in torque. But we need to consider more carefully the fact that the stabilizing yaw torque is being generated not only by the rearmost part each wing, but rather by the entire swept leading edge of each wing.

 

We'll now change our reference frame--now when we speak of an airflow component coming from or flowing toward the "inside" or the "outside" of the turn, we'll mean in relation to the direction that is perpendicular or "square" to the leading edge of the wing, not in relation to line of the fuselage or keel. (This new reference frame is only valid for examining the yaw and roll torques created by sweep. It not valid for examining the yaw and roll torques created by anhedral or dihedral.) For example, in wings-level non-turning flight, the swept leading edge of the left wing experiences a relative wind that flows toward the left in relation to a line that is "square" or perpendicular to swept leading edge of the wing, and the swept leading edge of the right wing experiences a relative wind that flows toward the right in relation to a line that is "square" or perpendicular to swept leading edge of the wing. Similarly, in a turn with a very large radius where airflow curvature is not a significant factor, and assuming that the aircraft as a whole is not yawed toward either the outside of the turn (in a slip) or toward the inside of the turn (in a skid), the swept leading edge of the outside wing experiences a relative wind that flows toward the outside of the turn in relation to a line that is "square" or perpendicular to swept leading edge of the wing, and the swept leading edge of the inside wing experiences a relative wind that flows toward the inside of the turn in relation to a line that is "square" or perpendicular to swept leading edge of the wing.

 

When we decrease the turn radius so that the "airflow curvature" effect becomes important, things get quite interesting. (Once again, readers are encouraged to sketch out these scenarios for themselves, using a printed copy of polar coordinate grid provided in the link at the beginning of this article.) Let's start by assuming that the curving airflow is exactly parallel to the fuselage or keel at the trailing edge of the wing. The geometry works out like this: the leading edge of the inside wing experiences a relative wind that flows slightly toward the inside of the turn (in relation to a line that is "square" to the leading edge), and the leading edge of the outside wing experiences a relative wind that flows much more strongly toward the outside of the turn (in relation to a line that is "square" to the leading edge). This effect will tend to increase the lift and drag of the inboard wing, tending to create a yawing-in torque and a rolling-out torque. We need to be careful with this type of analysis because if the bank angle is in fact constant, each wing must in fact be generating the same amount of lift (we won't introduce the complexities of weight-shift into this discussion), which as we noted above, tends to suggest that the outboard faster-moving wing will create more drag in the final analysis. But as one element in the overall mix of competing yaw and roll torques, the effect we're describing here will counteract the increase in lift and drag experienced by the outboard, faster-moving wing.

 

As we move the "tangent point" between the curving airflow and the line of the fuselage and keel forward toward the midpoint of the wing root chord line, we get weaker versions of the same airflow components and the same yaw and roll torques. If we move the tangent point forward of the midpoint of the wing root chord line, we get the opposite effects. For example, if the curving airflow is tangent to the line of the fuselage or keel at the very forward tip of the wing root (e.g. the "apex" of the aircraft in the case of hang glider or trike), so that the entire wing experiences a skidding airflow component, then in our new reference frame we can describe the geometry as follows: the outside wing experiences a relative wind that flows slightly toward the outside of the turn (in relation to a line that is "square" to the leading edge), and the inside wing experiences a relative wind that flows much more strongly toward the inside of the turn (in relation to a line that is "square" to the leading edge). This will tend to increase the lift and drag of the outboard wing, creating a yawing-out torque and a rolling-in torque.

 

It appears that as a swept-wing or delta-wing flying-wing aircraft turns, the yawing-out torque created by the difference in airspeeds between the two wingtips can be cancelled out by the yawing-in torque created by the swept leading edges even if the "tangent point" between the curving airflow and the line of the fuselage or keel is not located very far aft of the trailing edge of the wingtips, and even if the aircraft has a rather high aspect ratio. In some cases it even appears that the yaw torques might balance when the tangent point is located quite a ways forward of a line drawn between the trailing edge of the wingtips, i.e. significantly closer to (but not in front of) the center of the mean chord line of the wing as a whole. The greater the curvature in the airflow, the more the wing as whole will experience a significant slipping airflow, even if the rearmost portions of the wing do not. Bearing in mind that in a turn, the curving airflow ends up magnifying the slip angle experienced by the forward parts of an aircraft, here is one of the crucial differences between a swept-wing or delta-wing flying-wing aircraft, and one with a more "conventional" shape: on the swept-wing or delta-wing flying-wing aircraft, even large portions of the swept leading edges that are located well forward of the CG can interact with a slipping airflow to create a yaw torque toward the inside of the turn. On an aircraft with a more "conventional" shape, any part of the fuselage that is located ahead of the CG will interact with a slipping airflow to create a yaw torque toward the outside of the turn.

 

I've made some in-flight observations of the angle of the airflow during a constant-banked turn at various locations on flex-wing hang gliders. On all the flex-wing hang gliders that I've examined, a yaw string located several feet in front of the base bar always deflected at least a few degrees to the outside of a steady, constant-banked turn. My goal was to come to some conclusions about where the curving relative wind might be tangent to the line of the keel. I haven't yet finished analyzing all of this data, and the observations involve a significant amount of error, but in very general terms, it appears that with the gliders I examined, in 30- to 40-degree banked turns at angles-of-attack near min. sink the curving relative wind was tangent to the line of the keel at a point that fell somewhere between the trailing edge of the wing root, and the extreme rear of the keel (which extends aft roughly as far as the trailing edge of the wingtips). If this is accurate, then by sketching a hang-glider planform onto the polar coordinate system provided in the link at the beginning of this article, using realistic relationships between the size of the wing and the radius of the turn, readers can reach some conclusions about the direction of the airflow experienced by various parts of the wing. (According to these observations, most of the wing area will be experiencing at least a slight amount of sideslip.) Stay tuned for more details on these in-flight observations, and please contact me if you would like to share similar observations of your own.

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