Commentary on the mathematics of circles in wind
September 14, 2006 edition
steve at aeroexperiments.org
The "Mathematics of
circles in wind" page was created by Dr. Dan Tyler and includes a
mathematical proof that when an aircraft flies at a constant airspeed through a
series of circles of constant radius, whose centers are stationary with respect
to the surrounding airmass, then the acceleration acting on the aircraft is
constant, regardless of whether the airmass is moving with a constant
horizontal velocity (i.e. there is a wind), or the airmass is stationary (i.e.
there is no wind). The sole source of this acceleration, naturally, is the
sideways force component created by the banked wing. This proof is easy when
one adopts a reference frame that is fixed with respect to the airmass; Dan's
proof tackles the problem from the standpoint of a ground-based reference
frame. Dan's treatment of the problem brings out an interesting point--at first
glance, the rise and fall in the aircraft's groundspeed seems to imply that the
aircraft is experiencing more drag when flying into the wind, and is being
somehow "pushed" when flying with a tailwind, but this cannot really
be the case. Instead, the rise and fall in the aircraft's groundspeed is caused
solely by the fact that the sideways force component created by the banked
wing--though always perpendicular to the aircraft's flight path--often includes
a small component which is parallel to the aircraft's ground track.
The accompanying diagram, also created by Dr. Dan Tyler, illustrates this point very clearly, and also reveals many other interesting details. For example, let's assume that the diagram illustrates an aircraft circling in the clockwise direction, with a wind blowing from right to left. In this case, when the aircraft is "above" the dotted centerline of the diagram, it is pointing upwind, and when the aircraft is "below" the dotted centerline, it is pointing downwind. As the aircraft flies through a series of 360-degree circles with respect to the airmass, it spends an equal amount of time "above" and "below" the dotted centerline, but it covers much more distance over the ground when it is "below" the dotted centerline. Note also that the dotted centerline connects all the points where the aircraft's flight path through the airmass is perpendicular to the wind. Along the dotted centerline, the aircraft's groundspeed is higher than its airspeed--to convince yourself of this, sketch the closed vector triangle depicting the airspeed, wind speed, and groundspeed vectors, with a 90-degree angle between the wind speed and airspeed vectors.
To supplement the dotted centerline, we can also draw a second, higher horizontal line connecting all the points where the aircraft's ground track is perpendicular to the wind. This second line is not shown on the diagram, but an "eyeballing" of the curving ground track suggests that this line will fall in the vicinity of the
"y=.3" line. Along this second horizontal line, the aircraft's airspeed is higher than its groundspeed--to convince yourself of this, sketch the closed vector triangle depicting the airspeed, wind speed, and groundspeed vectors, with a 90-degree angle between the wind speed and groundspeed vectors.
It follows that there must be a third horizontal line, located above the dotted centerline but below the second horizontal line described immediately above, along which the groundspeed and airspeed are equal. Above this third line, the aircraft's groundspeed is being diminished by the wind, and below this third line, the aircraft's groundspeed is being augmented by the wind. Since this third line lies above the dotted centerline, it's clear that the circling aircraft spends more time below this third line than above it. In other words, as the aircraft circles, it is more frequently true that the wind is augmenting the aircraft's groundspeed than it is true that the wind is subtracting from the aircraft's groundspeed, even though the gain in groundspeed at the moment of maximum tailwind is no larger than the loss of groundspeed at the moment of maximum headwind. Or to put it yet another way, the time-averaged groundspeed of a circling aircraft increases when there is a wind. If we were looking at the aircraft's circling flight path respect to the airmass rather than the aircraft's "curlicue" flight path with respect to the ground, we would reach the same conclusion by observing that as the aircraft flies through the circling path of a constant-rate turn, it spends more time in the (larger) portion of the circle where its groundspeed is increased by the wind, than it spends in the (smaller) portion of the circle where its groundspeed is decreased by the wind. Naturally, the line dividing these two unequally sized parts of the circle is none other than the "third line" mentioned immediately above, connecting the two points on the circle where the groundspeed is equal to the airspeed.
Of course, the "curlicue" shape of the ground track in the accompanying diagram will be very familiar to soaring pilots who fly with GPS units--this is the ground track that is laid down during a long series of thermalling circles in the presence of wind.
On a related note, soaring pilots who fly with a GPS-based groundspeed-recording barograph set to a high sampling frequency will be familiar with the zig-zag groundspeed trace that results from circling in wind. With continuous sampling, this zig-zag groundspeed trace would be centered around the aircraft's airspeed and the would extend above and below the airspeed value by an interval equal to the windspeed. With non-continuous sampling, there will be some sampling bias in favor of the higher groundspeeds. Since the lower-groundspeed data points are more likely to be omitted from the zig-zag groundspeed trace, this make the airspeed appear higher, and the total variation in the ground speed appear lower (which also means that the windspeed appears lower), than was actually the case. Here's the reason for this sampling bias: as noted immediately above in our discussion of the "third line", as the aircraft flies through series of 360-degree turns at a constant turn rate with respect to the airmass (i.e. a constant yaw rotation rate), it spends more time in the (larger) portion of the circle where its groundspeed is being increased by the wind, than it spends in the (smaller) portion of the circle where its groundspeed is being decreased by the wind.
The above paragraph assumes that the airspeed is greater than the windspeed. If the reverse is true, the zig-zag groundspeed trace laid down during circling flight (assuming continuous recording) is centered around the windspeed and extends above and below the windspeed value by an interval equal to the airspeed. In the case the sampling bias will tend to make the windspeed appear higher, and the airspeed appear lower, than was actually the case.
When the windspeed is greater than the airspeed, the "circlicues" illustrated in the accompanying diagram become stretched so much that the ground track is just a "scalloped" line, without any actual closed "loops".
Larger image for
mathematics of circles in wind graphic
For more, see these related articles on the Aeroexperiments website:
Brain teasers for those who believe that downwind turns are "different"--i.e. that an aircraft can "feel" the wind direction in flight
Downwind turns ARE "different"!
The never-ending myth of the "dangerous downwind turn"
And for more still more, see these articles from the "Ask J and D" feature of the "DJAerotech" website:
Downwind -- debunking the myth of the dangerous downwind turn
Wind_plane -- more on the above topic, with some interesting notes on wind shear
And for more still more, see these articles:
Challenging the wind by Martin Hepperle-- an interesting little article on the best strategy for flying in wind during a pylon race