Back in 1973 I reckoned I'd have this done by my 30th birthday (1976), but it's taken a bit longer:
Why kites (don't) fly- single line kite stability.
For a kite to fly on a single line, it must, as the most basic condition, have some way to detect which way is up.
All single line kites that aren't under some sort of remote control do this by having their centre of lift position (CL, where lift forces act) above and forward of their centre of gravity (CG, where weight forces act). The pendulum effect that this creates causes such kites to point upwards, and upwards they will fly, until they get to a line angle at which wind generated lift exactly matches the kite's weight (when the kite is said to be at its apex)- disregarding dynamic effects of course.
But unfortunately, we can't disregard dynamic effects- because they very often prevent kites from flying stably at their apex.
And, at some upper wind speed, they will always prevent kites from doing so. This is because, while the lift (and drag) forces that drive dynamic instabilities increase with the square of wind speed, the weight force (from which the kite derives its upward seeking tendency) is constant. At some wind speed therefore, the pendulum effect will be overwhelmed by aerodynamic forces and the kite will crash- if it doesn't break first.
Dynamic instabilities derive from apparent wind effects; changes to the air speed experienced by a kite that are caused by its own movements. Of particular significance for dynamic instability is the relationship by which, when a kite is turning, the lift on the faster wing will increase by more than the lift on the slower wing decreases.
It's useful to consider two main failure modes for single line kites. One, overcorrection, is when a kite reacts too aggressively while re-aligning itself with the wind and triggers dynamic effects. The other, undercorrection, is when it reacts too slowly.
An example of overcorrection is when recovery from some directional displacement (a change in wind direction for example) initiates a series of increasing amplitude lateral oscillations that build until the kite starts to loop uncontrollably.
An example of undercorrection is when a kite takes so long to recover from a directional displacement that while doing so it traverses completely to one side or other of its wind window and collapses.
In addition to the relative magnitude of a kite's pendulum effect, the four main elements that influence overcorrection/undercorrection are tail drag (tails, trailing drogues etc), laterally disposed drag (drag sources to each side), lateral area (keels, flares, dihedral, anhedral), and longitudinal dihedral (often called 'reflex').
Tails are clever because they don't begin to apply any corrective force to a kite until there is substantial angular displacement (tail drag increases with the sine of the angle of displacement, so by 10degrees, say, are providing 17% of the maximum corrective effect they are capable of). The beneficial effect of this is that tail drag allows a kite to adapt quickly to minor wind direction changes (quickly enough so that the kite will not shift too much laterally while doing so) but comes in with rapidly increasing corrective force if for some reason the kite gets seriously tipped. Tails will therefore rarely if ever make a kite's response so slow as to cause undercorrection – unless their end catches in a tree or they are REALLY long. The bad bit about tails is that they cost lift to drag ratio (L/D). (L/D is a general measure of aerodynamic efficiency. For gliders it defines how many metres they fly forward for every metre of sink. For traction kiting it measures how well you can go upwind. For single line kites, it determines line angle- in fact the tangent of the angle, relative to the horizontal, of the flying line at the kite, is exactly the kite's L/D).
Laterally disposed drag- that is, having sources of drag out to each side of the kite, also has a clever effect: Because drag rises with the square of wind speed, when a single line kite with substantial outboard drag gets into a destructive turn, the drag on the faster side will increase by more than the drag on the slower side decreases- providing active damping. Such drag elements will also decrease L/D of course, except if they are an intrinsic and essential part of the kite anyway. The insight being offered here, and it's a major one, is that aspect ratio (AR, effectively width to length ratio) is the most powerful 'costless' (by L/D) dynamic instability cure available to kite designers. A way to make this understandable is to consider a square kite, 1m on each side, lifting area 1sq.m (aspect ratio 1.0). If such a kite is built and is found to be inclined to overcorrect and go into destructive looping, then if it's rebuilt to 1.25m span x O.8m long (still 1 sq.m but now AR 1.56), it will have much less tendency to overcorrect- may even be inclined to undercorrect. This is because the drag associated with the wingtips, while still having similar cost with respect to L/D, is further out from the kite's centre of lift, so will be more effective in resisting any rotations (in the plane of its lifting surfaces) that the kite becomes subject to (that is, it slows turns). Adjusting a kite's aspect ratio is therefore a way to get correction that's neither too fast (loops out of control) nor too slow (flies off to one side or the other and crashes or stalls). Wingtip drag isn't referenced in any way to up/down, all it can do is slow down turns- and of course this can be a bad thing when it slows a desirable recovery- but on balance it is hugely beneficial because it slows down all the movements which energise dynamic instability, unplugs their power source so as to speak.
The third main useful stabilising element, lateral area (flares, keels, dihedral, anhedral etc), is also relatively costless by L/D, and can be very effective at damping out any incipient overcorrection but has to be of appropriate magnitude and carefully positioned. If a kite with substantial lateral area (as a proportion of its lifting area) is subject to an angular disturbance (that is the longitudinal axis of the kite gets out of alignment with the wind direction), the aero forces acting on this lateral area can cause the kite to move a long way sideways across the wind window before the pendulum effect gets it back in line- that is, excessive lateral area can promote undercorrection. Clearly, the longitudinal placement of lateral area will have an effect also. If disposed mainly behind the kite's CG, it can promote rapid re-alignment but may also exacerbate dynamic effects (overcorrection). If in front of the kite's CG, it will tend to cause undercorrection and make it very difficult for the kite to fly centrally (that is, directly downwind of the line tether point). Although dihedral (upward angled wings) and anhedral (downward angled wings) have some different effects on how single line kites react, they are primarily both just ways to get lateral area. There is a mistaken belief that dihedral is 'stable' while anhedral is 'unstable' but this comes from aeroplane experience and doesn't generally apply to kites. When an aeroplane rotates around its longitudinal axis, if the downside wing loses projected area at a faster rate than the upside wing gains projected area then the rotation will become self promoting. Aeroplanes are made with dihedral so that they are auto-stable in rotation about their longitudinal axis. For kites, bridles generally prevent this sort of rotation anyway. Kites with centre line bridling (most diamond kites for example), require dihedral for the same reason that aeroplanes do, but kites with laterally disposed bridles (like sleds for example) don't.
Longitudinal dihedral, or reflex, the fourth and last major single line kite stabilising element has the obviously beneficial effect of reducing or eliminating luffing tendencies, but its underlying influence is more profound: Because aerodynamic lift forces drive instability (of both the overcorrection and undercorrection types), anything that decreases lift without changing other things too much, will generally improve a kite's stability. "More longitudinal dihedral" is just another way of saying "less camber"- and having less camber will cause less lift to be generated, (a generally applicable aerodynamic effect). Introducing longitudinal dihedral therefore deals directly to overcorrection, but it's a rather ugly solution, a last resort (usually taken when graphics considerations don't permit other more efficient form changes), because it also directly reduces L/D, and by a lot if it's to be effective. It's influence on undercorrection is equivocal: Reducing lift does reduce the driving force that makes a kite traverse off to the side before it's pendulum effect can straighten it up- less lift means that it won't get as far before correction occurs. But, longitudinal dihedral also shifts the kite's CL rearward (nearer to its CG), which reduces the effective pendulum length and therefore its corrective effect (while adding to its usefulness against overcorrection of course).
This is a brief description of a complex and indeterminate field. Like all things that are subject to turbulent flow (the weather for example), single line kites will never be fully predictable.
But, there are some things that are both true and useful that can be established- which is what I've tried to do.
I've tested the above against the kites I see flying, and don't think I've seen anything that falsifies any of it. However, there are so many overlapping effects and other influences that it's sometimes difficult to see through all this fog to the fundamental relationships. No doubt I've made errors in at least some respects.
I'll modify and correct when these come to light, and plan to add descriptions of various special conditions as time and opportunity permits (von Karman affects for example).
Peter Lynn, Ashburton, New Zealand, January 1 '09.
PS: Just had a terrible thought; what if it's all wrong- wasted life, AAGH!