Understanding air flow and resistance

[thetexan]

TL;DR

What kind of curve is shown here

In a recent thread, I thought I understood what was going on but alas, I am still confused. So I offer this diagram and ask for more help please. Shown is an aircraft holding pattern. The one on the left is flown in still air. The one on the right is flown with a wind coming from the left. In still air the aircraft flies a perfect circle within the air mass, and since the plane is not being displaced by the airmass it is flying in the ground track is also a perfect circle.

In the right example, the aircraft is flying a perfect circle in a moving air mass (left to right) and as a result the ground track is an elongated version of the perfect circular track in the moving airmass. The question is...what type of curve represents the ground track?

The very helpful folks so far has said a cycloid. A typical cycloid curve is usually shown as a curve representing the track of a point on the rim of a wheel as it unslippingly rolls along a path. But every cycloid illustration I have seen involves a full 360 rotation. As you can see from the above diagram only 180 degrees of turn is involved. AND, AND, since the aircraft begins that semi-circle wings level, the angle at which the curve meets the straight flight path of the long leg is 90 degrees to the center of the turn. If you look at a cycloid diagram the curve at the 90/270 point of rotation is more like 45 degrees and not 90 degrees.

So what type of curve begins in line with the wings level track and ends up that way on the other side and represents the elongated, wind-blown circular path of the airplane along a semi-circle?

I note that it is possible that my ignorance of the subject is the main causal factor of my frustration, so any help is appreciated.

Texx

 

[hmmm27]

You aren't confused ; just, apparently nobody bothered to think of using your example when they went about making diagrams. Every cycloid example you've seen is crap.

So, starting with this diagram...

we can see that for the common cycloid, the pen is on a point on the circumference of the rolling circle ; for the curtate cycloid, it's inside the circle ; the prolate cycloid has it outside the circle.

Translating to your problem, the length of the circle's radius represents the windspeed ; the length of the radial line to the pen represents the airspeed of the aircraft ; the line it transcribes, the ground track.

From which we can tell that for both common and curtate cycloids you need to find another airfield, somewhat downwind.

The ratio of windspeed to airspeed on the prolate cycloid figure looks to be about 1 to 1.25, whereas yours is more like 1:4.

Long story short, the diagram you won't be able to find would be the prolate one, but with *huge*, almost more circular'ish loops which overlap each other 4 times, instead of the isolated itty-bitty ones.

It would look pretty much exactly like you think it should.

Which would be easy enough to roughly diagram if the Trackpoint I use instead of a mouse produced anything other than humorous results in M$-Paint.

 

[thetexan]

I think I have discovered why some people here have been confused about the curve.

There IS slippage!

The aircraft will make a circle defined by a constant rate of 3 degrees per second at its velocity, say 120 kts.

The wind will blow at some random speed. Any wind speed produces some form of elongated or contracted circle. Which is an ellipse.

Tex

 

[hmmm27]

There's two people in this thread ; one of whom is not confused.

Repeat after me : "prolate cycloid"

Use the sliders : set r=1, d=6, hit play(bottom left corner)

Now, there's two.