
In response to questions regarding the exponential effect of gusts on
the force generated by a kite, and also the resulting acceleration of a
kiter that happens to be attached to it:
Before getting into
this topic, it's important to clarify some of the relevant terms. Here
is a piece that I wrote about kite aerodynamics, including relevant
references to AOA: http://www.kiteboardbc.com/index.php?pr=Aerodynamic_Efficiency And here is one that defines force and power: http://www.kiteboardbc.com/index.php?pr=Force_Energy
As
I and others have stated on several occasions, the forces defined as
lift and drag are approximately exponential functions of airspeed over
the surface of a kite, all other things being equal. That means, if the
airspeed over a kite doubles, but all other parameters remain
unchanged, then the lift and drag forces will roughly quadruple.
Now,
to take a practical look at the effects of this, I'll use some
different examples to illustrate forces and acceleration. The first
example is a kite tethered to the ground, the second is a kiteboarder
riding on the water with the kite flying low, and the third is a
kiteboarder with the kite flying overhead. In each case, I'll explain
what happens if the windspeed doubles from 15 mph to 30 mph. In all of
these examples, the same moderatesize kite is used, and the trim of
the kite is kept constant (well powered):
1. KITE TETHERED TO GROUND
FORCE:
If the kite exerts a force (mainly lift, but also a small drag
component) through the lines of about 50 lbs in a 15 mph wind, then the
force will increase to about 200 lbs when the wind speed doubles to 30
mph. (This is only an approximation that doesn't take the weight of the
kite into account.)
ACCELERATION: There is no acceleration involved in this example because the kite is tethered.
2. KITEBOARDER ON A BEAM REACH
In
this rather typical example, assume that the kiteboarder is riding on a
26 mph beam reach (90 degrees to the true wind direction) in a 15 mph
(true) wind, which is about right on an efficient board (high L/D
ratio) such as a Spleene Session, with a well powered kite. A bit of
Pythagoras and trig shows that the apparent wind is passing over the
rider and kite at 30 mph, at an angle of 30 degrees from the rider's
direction of travel.
FORCE: In this example, we know that the
forces exerted by the kite are about 200 lbs, because the airflow over
the kite is 30 mph, which is the same as the example of the tethered
kite in a gust.
Now, assume that the true wind speed doubles to
30 mph. More Pythagoras and trig shows that the apparent wind will
increase to almost 40 mph (at 49 degrees), which would result in a
force of about 355 lbs. Also, the apparent wind direction will back*
toward the true wind direction, causing the AOA (angle of attack) to
increase slightly.
*The term "back" is used in relation to the
rider's direction; not prevailing hemispherical weather systems
(somebody would undoubtedly correct me otherwise).
ACCELERATION:
At this point, a rider would normally sheet out the bar a little to
reduce the force generated by the kite, but since it has been
established that the kite trim is to stay constant in these examples,
he will have to rely on his edge, the old fashioned way, to resist the
force of the kite. If he can't resist, then his body will accelerate
toward the kite, which will cause him to "bear off" (change course in a
downwind direction) somewhat. His acceleration at this point depends on
his weight and how hard he can edge. If he weighs 200 lbs, and he can
only resist 275 lbs of force by edging, then he will theoretically
accelerate downwind at (355 lbs  275 lbs) / (200 lbs/g) = 0.4 g, but
in reality he probably won't. Here's why:
The instant that he
begins to accelerate downwind, the kite's AOA will be reduced, which
will reduce the force generated by the kite, which will in turn reduce
the downwind acceleration of the rider. And in the real world, that
"instant" of theoretical 0.4 g acceleration would be so short
(dependent on linestretch) that the kite would probably take longer to
pass completely from the 15 mph wind into the 30 mph wind.
So,
if the rider maintains his kite trim, he'll gradually accelerate onto
one of those outofcontrol high speed downwind reaches, but his actual
acceleration will be very gradual compared to gravity, due to the
mitigating effect of the acceleration on the kite's force. Calculation
of the acceleration in such scenarios would be a timeintensive
geometric exercise with many variables, but suffice it to say that any
such acceleration wouldn't be dangerous. Only the resulting speed might
be (especially if there's something hard downwind).
3. KITEBOARDER WITH KITE OVERHEAD
FORCE:
As with the "tethered" example, the force during a gust from 15 mph to
30 mph would result in an increase in force from 50 lbs to 200 lbs.
ACCELERATION: As long as the rider weighs 200 lbs, there'd be no acceleration.
4. EXTRA DANGEROUS EXAMPLE
Now
let's look at a severe example. Suppose that the same kiteboarder is
flying the same kite overhead in 15 mph wind when an insane 60 mph gust
hits. Now we're talkin'.
FORCE: The theoretical kite forces
would increase 16fold from 50 lbs to 800 lbs, but that wouldn't
actually happen unless the rider's feet were anchored to something.
ACCELERATION:
It depends on how abruptly the gust srikes. There is no such thing as a
truly instantaneous gust, but if there was, and if the 200 lb rider was
dumb enough to keep the bar sheeted in, then he could theoretically get
hoisted at about 3 g (plus his own weight of 1 g). But he won't. As in
example (2.), the instant that he begins to accelerate skyward, the AOA
will be reduced, which will reduce the force, and therefore the
acceleration. We're talking a fraction of a second. That theoretical
spike of acceleration would be over within the time it takes to
(statically) unstretch the lines, which wouldn't give that spike time
to build to begin with.
The kiter's vertical acceleration in this example is a function of many variables, including the following:
a)
How quickly the gust hits. Keep in mind that slow moving air has to get
out of the way before it's replaced by fast moving air, and there are
boundary layers between different pockets of air.
b) The drag
characteristics of the kite. As the kiter begins to rise, and the AOA
reduces, the kite's effective L/D ratio will change, causing the kite's
airspeed to reduce, causing the kite to "back up" relative to the
ground, ultimately causing the rider to arc upward. How quickly this
happens depends on the kite's drag characteristics at various speeds
and AOA.
c) The length of the flying lines. When the kiter arcs
upwards as described in (b), his centripetal acceleration will be a
function of his line lengths. The shorter the lines, the higher the
acceleration. Theoretically, if the kite, with 30 m lines, immediately
slowed to an airspeed of 40.5 mph, resulting in a 60 mph "pendulum"
motion by the rider, with the kite (6040.5)/60 of the way to the
center of a 92.3 m arc (stationary relative to the true wind), then his
instantaneous centripetal acceleration would be about 0.79 g. That
means, as he begins to arc upward, his kite would have to exert a force
of 1.79 times his weight of 200 lbs, or 358 lbs, which correlates
almost exactly to what that kite would exert at 40.5 mph airspeed,
knowing that it exerts 200 lbs at 30 mph airspeed.
And that is
about as close an approximation as you're likely to get for this
example: 0.79 g acceleration (plus his weight) for a 200 lb rider,
getting lofted in a 60 mph gust by a fully powered kite that exerts 50
lbs (including its own weight) in 15 mph wind.
Of course, as the
rider climbs, his vertical acceleration will decrease throughout his
lofting experience, becoming negative somewhere before the apex.
(Previously, I've represented the maximum possible theoretical height
of jumps and loftings in "Thread 7: Biggest Jumps".)


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