Wednesday, March 25, 2009

TAS - HW != GS

A couple of days ago I posed this puzzle:

Here is a wind speed puzzle. Your course is due north, 000 degrees. That is you will proceed in a straight line over the ground to a point that is due north of where you are now. The wind is blowing from 100 degrees, that is from mostly off your right wing, but slightly behind you. With no wind at all, your groundspeed would be 100 knots. Will your groundspeed in the described conditions be greater than, less than or equal to 100 kts?

Some who posted was smart enough to know that the answer is, "it depends on the windspeed." That may seem strange when all I asked for was the sign of the change in speed, but it's true.

As a general rule of thumb, if the wind is blowing from ahead of you, it slows you down, and if it is blowing from behind you it speeds you up, just as you'd probably expect. In rough flight planning I estimate that an n knot wind from m degrees behind me resolves into nSIN(m) worth of tailwind and then add the tailwind to the true airspeed to get groundspeed. (Don't think I'm some kind of walking trig table: when I say I estimate the sine of an angle we're talking "negligible," "a bit," "a lot," "half," "most," and "all.") But when the crosswind component represents a significant proportion of the speed of the airplane, things become more complex.

When there is a significant crosswind, the airplane heading is noticeably not the same as the course. The airplane travels diagonally over the ground with respect to the direction it is pointing. This means that some of the thrust of the airplane is being used to keep the airplane from being blown off course. Logically, then, that reduces the thrust available to maintain groundspeed along the track. So the component of the wind along the track is increasing the groundspeed of the airplane, but the crab is simultaneously decreasing the groundspeed.

If you haven't got your own flight computer handy, try this online one. Scroll down to the third section: Heading, Ground Speed, And Wind Correction Angle. Put in a wind direction of 100, a true airspeed of 100 and a course of 0. Now enter a wind speed of 20 and press Calculate. The calculated groundspeed is 102, so that's two knots of tailwind. Try it again with 37 knots of wind, for a calculated groundspeed of 100. Try it again with sixty fricking knots of wind, like I had the other night, and you calculate a groundspeed of 91 kts. Yeah, the wind is behind you, and you're still down nine knots.

For a given wind speed and direction the amount of head or tailwind is fixed, but the crab angle and thus the proportional reduction in speed due to the crosswind is related to the ratio between the true airspeed of the airplane and the strength of the wind. So the same wind could provide an increase in speed for a fast airplane and a decrease in speed for a slow airplane going the same direction at the same altitude. Bizarre, eh?

My favourite answer was provided by Paul, who amusingly pointed out that all your calculations will be in vain because ATC will vector you into the wind no matter what.

It's disorienting enough flying in the dark in turbulence with a whopping big wind correction angle, but when I found that I needed to fly at a high power to maintain a groundspeed of 140 whether I was going north or south it just felt unfair. You get cheated in wind anyway, because the time lost being slowed down by a headwind is not regained flying the same distance with the same wind on the tail. But when you're slowed down by wind going both ways. Well I want to complain.

The controller working Houston Center was being funny, telling every pilot who checked in that he had "reports of intermittent chop at all flight levels, so just tell me what altitude you want to be bumpy at." Around ten in the evening airline movements diminish and the frequency quiets. I called up Houston and told him I'd like to complain about the sixty knot winds. "Okay," he said, "But I'm not going to do anything about it." As I told him: yeah, I knew. It just makes me feel better to complain.

Does the fact that I'm amused that the sign of the change depends on the sine of the wind direction a sign that I'm a word geek or a math geek? Signed, Aviatrix.

P.S. The bruise is lots of pretty colours, but it's just a bruise, nothing to worry about.

13 comments:

Anonymous said...

Whatever you are, you are certainly colorful.

J said...

So why is "the time lost being slowed down by a headwind...not regained flying the same distance with the same wind on the tail"?

Thanks for the info...

Aviatrix said...

So why is "the time lost being slowed down by a headwind...not regained flying the same distance with the same wind on the tail"?

Classic range and endurance question. It's because it takes less time to fly distance X with a headwind than with a tailwind, so on an out and back course, you spend a longer time with the wind in your face than at your back.

Imagine a 100 kt airplane on a 100 nm out and back course in calm winds: 2 hours. The same airplane in a 50 kt headwind takes two hours just to get out. No way of making up lost time.

Aviatrix said...

The first "less" in the above answer should be a "more".

Matthew Flaschen said...

Well, I'm glad it wasn't a simple yes/no answer. I actually spent substantial time trying to figure it out. However, I couldn't get past the fact that you seemingly specified the wind velocity with respect to the moving airplane ("mostly off your right wing, but slightly behind you"), rather than with respect to the ground.

Aviatrix said...

Ahh, you're right, Matthew, I did say that. I guess I had in mind "before the crab" or some kind of virtual airplane that has its nose in line with its track. I'm sorry to have muddied the problem for you.

Sarah said...

Does the fact that I'm amused that the sign of the change depends on the sine of the wind direction a sign that I'm a word geek or a math geek?

Very punny. Sounds like 'word geek' to me. Now, if you were musing about the origin and deep meaning of negativity itself ( or worse yet, its square root, 'i' ) you'd be a math geek. Too.

Phil said...

Wait'll you start flying around IN TIME.

Sarah said...

When the hell is Phil?

Anonymous said...

That's really interesting.

Have you noticed that the same's true on your bicycle? You're slower in a sidewind, and your round trip takes longer in any significant wind. This has been proved experimentally.

Anonymous said...

Smiled wide on this puzzle!

This is so much fun to watch. (Cursing stupid pipes and whatnot in the man cave).

Excellent job! Ever see my answer to the posters of Dave's post? I was proud of that....

chris said...

These are math geeks.

chephy said...

Even though a crosswind slows you down going there AND back, it still works out to be better overall than having a headwind and then a tailwind of the same speed. Simple geometry and algebra will show that.