How does wind affect a turn for an aircraft?

[thetexan]

TL;DR Summary

wind effects on aircraft track

First, I understand how wind effects aircraft tracking. I can do the calculations involving wind correction angles, effects on ground speed, headwind or tailwind calculations, etc. I teach it in a college piloting program.

I am wondering how the ground tract of turn is effected by the wind. I used autocad to graph what I believe would be the ground track but I don't think it is correct. For example, it is simple to plot the starting and stopping point of the circle. The end point of a 360 degree turn will be displaced by the wind direction and speed. This is because the track within the moving airmass itself will be a true circle, but that circle will be moved by the wind over the ground and therefore trace out a oblong and distorted shape. It is that shape I am currious about.

Let's say I begin a turn away from the direction of the wind with a 90 degree crosswind. As I begin the turn my headwind is not effected since is a direct crosswind. But as I get farther into the turn my ground speed increases as the tailwind component becomes greater. Then when I am 90 degrees into the turn my tailwind component will be at its greatest and the same with my groundspeed. Then as I begin the second 90 degrees my tailwind component decreases but my crosswind increases.

When I drew this in autocad I simply created an arc and moved the end point to the place where the wind would have carried it. But the arc itself as I drew it in autocad was of constant radius.

Here is my question. From the ground reference point of view, wouldn't the actual ground track be of varying radii as groundspeed changes? How would I figure out what the actual shape of the curve along the ground would look like taking into account these varying speeds? I don't think it would be a constant radius elongated arc. Am I thinking right or would the ground track simply be an elongated arc of the true circle being flown in the moving airmass?

tex

 

[jrmichler]

I pondered that exact same problem one day when the wind at 12,000 feet was enough to get my GPS groundspeed down into the single digits on the upwind portion. I tackle problems like this by first seeking to understand, then digging into the math. Here's how I work on the understand part:

Draw a circle and an arrow representing the wind. The circle represents your path relative to the air mass.
Assume that the wind velocity is equal to airspeed.
Mark a series of points on the circle. Pick a starting point.
The ground position matches the air position at the starting point.
Move to the next point. Since the wind speed is equal to the airspeed, the ground position is displaced in the direction of the wind by an amount equal to the path length along the circle.
Repeat around the circle.
Connect the dots.

Now search using search terms cycloid, curtate cycloid, and prolate cycloid.

 

[Lnewqban]

You can find some good illustrations here:
https://www.faa.gov/regulations_pol...iation/airplane_handbook/media/08_afh_ch7.pdf

The shape of the trajectory will depend on the rate of velocities of airplane and wind.
You could select several equidistant points in the circle and do a vectors summation.
You will see the unidirectional ground air speed vector and the rotating vector airplane to mass of air vector.

http://hyperphysics.phy-astr.gsu.edu/hbase/vect.html

The resultant shape will not be exactly real, as the drag coefficient has different values for headwind, crosswind and tailwind.

 

[thetexan]

What would be the name of the type of curve created? Is it a cycloid, curtate cycloid, prolate cycloid or epicycloid. I ask because there are many good youtubes on how to create these in autocad.

tex

 

[DrJohn]

I help run, score and analyse flights at gliding competitions. I can examine 30 - 60 flight recorder traces per day for up to ten days (weather dependant sport) and can show you exactly what the track over the ground looks like. Airspeed once in the thermal will be kept constant as will angle of bank, to stay in the thermal - bubble of rising air - but ground speed will vary as you got round the circle in the air, producing the sort of thing you see in the image. I simply opened the first one I found, this is perfectly normal when examining the flight recorder traces. Glider are typically circling at about 50 knots in the thermal, some models of glider will be slower, some faster. On windy days the distortion to the circle can be quite significant, looking as if the glider charging along in one direct, then almost turning on the spot and repeating this for five to ten minutes as it gains height. In this image the gliders comes in from bottom left, turns and eventually leaves at top right. Wind is blowing from bottom to top in the image. Just walk yourself along the trace to see what is happening.

 

[thetexan]

I see two curves here. One, the result of turning and flying with the wind (the yellow) and the other when turning and flying into the wind (green). Here are a couple of questions...

1. Are these two of the same kind of curve (for example, cycloid, if that is indeed what it is, or whatever) with different parameters?
2. Are these actually elipses (compressed circles) each with different major and minor axes? (I'm sorta leaning in that direction, by the way). And if and elipse, how would you go about determining the minor axis? One axis would have one end at the start of the turn and the other at the end of the turn. The other axis would depend on the relation of the wind direction to the turn, I guess.

The reason I want to know all of this is because my plan was to settle some arguments about holding patterns and wind corrections needed to fly the perfect holding pattern in wind. I was using autocad to draw it out when I realized this curve was an issue.

tex

 

[A.T.]

Is it a cycloid, curtate cycloid, prolate cycloid or epicycloid.

Depends on the windspeed. But usually the windspeed is smaller than the airspeed of the plane, so it's a prolate cycloid:

View attachment 325713

I see two curves here

You can divide it in as many parts you want. But under ideal conditions (perfect circles relative to a uniformly moving air-mass) it's all just one prolate cycloid.

 

[thetexan]

Here is the curve in question. The center of the "wheel" travels along a straight path. The height of the curve is equal to the radius of the circle. The length of the curve is determined by the distance between the starting point of the curve (let's call it the 270 degree point on the circle) and the ending point of the curve (let's call it the 90 degree point on the circle).

Is this not the very definition of an ellipse? I may be referring to it incorrectly by name. But autocad has a ellipse function so it is easy to draw. Other types of cycloids are more difficult.

Isn't this an ellipse with foci located based on the distance between the starting and ending points. In other words, can I use the ellipse function to acurrately draw this curve?

te

 

[Lnewqban]

You could build a series of points and join all with a single spline.
The actual trajectory is more like an elipse rolling on a flat surface, due to the differences in drag as the airplane rotates respect to the direction of the wind, as previously explained.

Please, see:
http://edpstuff.blogspot.com/2010/07/cycloids-and-their-construction.html

 

[jbriggs444]

Is this not the very definition of an ellipse? I may be referring to it incorrectly by name. But autocad has a ellipse function so it is easy to draw. Other types of cycloids are more difficult.

Isn't this an ellipse with foci located based on the distance between the starting and ending points. In other words, can I use the ellipse function to acurrately draw this curve?

No. This is not an ellipse. No. This is not any definition of an ellipse. Cycloids and ellipses are not the same thing.

An ellipse may be defined as the set of points where the sum of the distances from the two foci to each point is the same as the sum to any other point. Mechanically, It is the figure that is traced out if you put a closed loop of string around two pegs, one at each focus and run a pencil around while keeping the loop taut.

Note the tips of the "ellipse" in the figure that you have provided. Those tips are at a 45 degree angle to the horizontal. In a real ellipse, the tips would be at a 90 degree angle to the horizontal.

An ellipse may also be defined as a "conic section" -- the intersection of a plane with a circular cone.

It turns out that an ellipse is also the intersection of a plane with a right circular cylinder.

The wiki page on ellipse gives a bunch of other constructions. Pay special attention to the one using the Tusi Couple.

Edit: I'd wondered for many years whether an ellipse is the same thing as a circle that is linearly stretched in one dimension. Today I've confirmed that this is indeed the case. Here is one page that I Googled up. I swear, you learn more things from answering questions than from asking them.An airplane following an ideal circular path relative to a uniformly moving air mass while maintaining constant air speed will trace out a cycloidal ground track, not an elliptical ground track.

 

[A.T.]

Is this not the very definition of an ellipse?

No, but if it looks close enough to you, just use ellipse segments. Or does it have to be prefectly accurate?

 

[hmmm27]

The actual trajectory is more like an elipse rolling on a flat surface, due to the differences in drag as the airplane rotates respect to the direction of the wind, as previously explained.

It's not. In the aircraft reference-frame nothing changes : drag is a constant. If you wanted to "reverse" the cycloid to trace a circle in the ground reference-frame, yes, drag would vary.

---

Both ellipse and cycloid can be thought of as projections of a circle function.

Circle : $(x,y) = r( \sin\theta, \cos\theta )$

Ellipse : $(x,y) = r( a \cdot \sin\theta , b \cdot \cos\theta )$

Cycloid: $(x,y) = ( a\cdot(\theta - \sin\theta) , b(1 - \cos \theta) )$

I'm a bit shaky on that last one, but it should be close : $a$ would be windspeed, $b$ airspeed.

 

[DrJohn]

You could build a series of points and join all with a single spline.
The actual trajectory is more like an elipse rolling on a flat surface, due to the differences in drag as the airplane rotates respect to the direction of the wind, as previously explained.

Please, see:
http://edpstuff.blogspot.com/2010/07/cycloids-and-their-construction.html

No, the drag on the aircraft is not varying due to changes in drag from the wind as its direction is changing.
The aircraft's frame of reference is the portion of the atmosphere it is moving within. The fact that the atmosphere is moving across the ground is only relevant to the path a ground based observer would see. The aircraft is simply moving in a circle within that chunk of air. It is moving within a box that happens to be moving.

Think of a person walking along a train corridor while the train moves at 100mph. An outsider can see the person moving along at 104 mph if walking in the direction the train is moving, or walking backwards at 96mph if walking in the opposite direction to the trains motion. The walker sees that they are walking at 4mph, in whatever direction they are going, as do all the onboard passengers.

This is a common old problem when explaining how an aircraft moves through the air and turns. We have to explain it to non-scientists, and expect scientists to understand what we say very quickly. Most scientists I teach to fly do understand it very quickly.

 

[berkeman]

Most scientists I teach to fly do understand it very quickly.

You're a flight instructor? Very cool.

 

[tech99]

No, the drag on the aircraft is not varying due to changes in drag from the wind as its direction is changing.
The aircraft's frame of reference is the portion of the atmosphere it is moving within. The fact that the atmosphere is moving across the ground is only relevant to the path a ground based observer would see. The aircraft is simply moving in a circle within that chunk of air. It is moving within a box that happens to be moving.

Think of a person walking along a train corridor while the train moves at 100mph. An outsider can see the person moving along at 104 mph if walking in the direction the train is moving, or walking backwards at 96mph if walking in the opposite direction to the trains motion. The walker sees that they are walking at 4mph, in whatever direction they are going, as do all the onboard passengers.

This is a common old problem when explaining how an aircraft moves through the air and turns. We have to explain it to non-scientists, and expect scientists to understand what we say very quickly. Most scientists I teach to fly do understand it very quickly.

Agree. Once clear of the ground, an aeroplane does not know if the wind is blowing or not. Similarly, a boat does not know if there is a current.

 

[A.T.]

I'd wondered for many years whether an ellipse is the same thing as a circle that is linearly stretched in one dimension.

Weirdly that's how I always thought of ellipses. Maybe because in most graphics software the same tool is used for circles and ellipses.

The ellipse construction via a 3D intersection with a plane seems far more complex. And it's not intuitive, that it works with both: cylinders and cones.

 

[A.T.]

And it's not intuitive, that it works with both: cylinders and cones.

My hand-wavy explanation why a cone-plane-intersection also produces a symmetrical shape across the tilting axis goes like that:

Where the cone is wider one might expect less curvature of the intersection curve, than where the cone is more narrow. But at the wider end the angle of the tilted plane to the cone surface is also smaller, than at the narrow end. Miraculously the two effects exactly cancel each other, so the curvature of the intersection curve is the same at the opposite ends.