Friday, February 10, 2017

I can't get my holding pattern timing to work!

As I was preparing for a lesson with an instrument student on holding patterns the other day, I got to thinking - why does the timing seldom seem to work out with several recent students? How bad could my teaching be?

As a refresher, for most non-RNAV holding patterns, you want the inbound leg to be 1 minute long.

To make it 1 minute, you adjust the outbound leg timing to be more or less than a minute depending on the wind. You know, you see it in all the training references, and they give examples like "if your inbound time is 50 seconds, then make your outbound leg 1 minute and 10 seconds" or similar examples where the time difference is reasonably small.

And it usually works out that way.

But I distinctly remember noticing with my last instrument student that it didn't seem to work out very well, and I couldn't explain why! He'd time the inbound leg, adjust the outbound leg, and still be quite off. It wasn't his airspeed control, that was just fine.

So I started digging into it and ran a few mathematical simulations, because that's the kind of thing I do.

Turns out, the "add or subtract the difference in time" only really works in light wind. Now, none of us expect that in 50 kt gale force wind any of this would work, but what I found is that the simple method really starts to break down at some fairly routine wind speeds, at least my part of the country (Oklahoma).

Remember turns around a point from Private Pilot training, and how you keep correcting for the wind to maintain that constant radius from the point? If there was no wind, you would maintain a constant bank angle and end up right where you started. Your ground track would draw a perfect circle.

However, what would happen if you had a headwind and maintained a constant bank angle? Your ground track would look roughly like this, the only thing that would change is the amount of elongation depending on the wind speed.

Can we calculate the distance between the beginning of the turn and the end of the turn? Of course we can, and it's especially easy when we're talking about instrument flight and using a standard rate turn of 3 degrees per second, or a "2-minute turn". Since the 360 degrees of turn will take two minutes, the downwind displacement in that amount of time is simply the distance that the wind will move in that amount of time.

For example, if the wind is 10 knots, then the distance between start and end points is 0.33 nm.

At the midpoint of the turn we are half that distance "downwind" from the start point.

Of course, in Private Pilot training we adjust our bank angle to keep that ground track looking like a nice circle. But we don't have that luxury in instrument flying (there being no ground references), so the same principles greatly affect our holding patterns.

And that's what was causing my student's (and my) confusion.

Here in Oklahoma, as with much of the central U.S., most of our holding is done somewhere around 3000 MSL (as in the example above). As I write this, the winds aloft at 3000 at 230 at 45 knots! Now, it is certainly a windy day today, but it is very commonplace here for the wind at 3000 to be about 20-30 knots. And this makes getting quality practice on holding patterns very tricky.

Let's assume a not-unusual wind from the north at 30 knots and a holding speed of 90 knots, also typical for airplanes used in instrument training, like a Cessna 172 or many of the Piper PA-28s. Our mind's conception of a holding pattern looks basically like this:

Yes, this is actually TO SCALE! That might be a first for this blog.
That's great for no wind. We know any wind is going to distort it some. But how much?

With a 30-knot headwind, our outbound turn adopts a shape more like this (in red):

Keeping the nominal holding pattern on there for reference. Everything is approximately to scale.

The end of the outbound turn is clearly quite some distance down the outbound leg - since this turn took 1 minute (standard rate) and the wind speed was 30 knots, that means the rollout point is 0.5 nm past the abeam point. Also, since we start timing the outbound leg when we are abeam the holding fix, this rollout point is somewhere about 17 seconds into the nominal 1-minute outbound leg (the wind is on our tail for this portion of the maneuver, and it's almost a complete tailwind, meaning our average ground speed is close to but not quite 120 knots which would cover the 0.5 nm in 15 seconds).

So we fly outbound for 43 more seconds (at a ground speed of 120 knots), traveling another 1.4 nm downwind, and begin our turn back to the inbound course. Of course, now we're turning into a headwind, so our ground track is a mirror image of what it was before, and our ground speed is slowing. We now have to claw back about 2.4 nm at a ground speed of only 60 knots, which will take about 2 min and 24 seconds!

Huh. Well, we want to get a nice 1 minute inbound leg, so we have to take some time off the outbound leg. But, the normal guidance is you subtract the amount of time from the outbound leg that you were over on the inbound leg - which in this case, is 2:24 - 1:00 = 1:24. Subtract 1:24 from our 1:00 outbound leg? That's a problem. My stopwatch doesn't work on "negative time".

About the best we can do is never stop turning - literally, make a aerial "circle" once we cross the fix the first time. What happens when we do this?

Our ground track looks exactly like the first image in this article!

It turns out that we now are left with the "perfect" case, where we roll out on the inbound course 1 nm from the fix, which at 60 knots ground speed will take right at 1 minute.

So, somewhere between a light breeze and 30 knots of wind, the rule-of-thumb for adjusting timing fails us. That's why it's a "rule of thumb" of course - an attempt to perform easy calculations that work most of the time. Unfortunately, as I have found, it's pretty far off at just 20 knots of wind as well, which is a normal day here in the plains states.

Interestingly, I was recently up with an instrument student on a day when the wind aloft was 40 knots. Pretty smooth actually, but a lot of wind. He was in the holding portion of the syllabus but I knew we weren't going to get much useful practice in that wind. However, for an even worse example of the situation discussed above, we flew a holding pattern directly into the wind. I had him do as stated above and just keep a standard rate turn going through all 360 degrees after hitting the fix. Sure enough, we rolled out on the inbound course and it took about 1:30 to get back to the fix!

It seems that once the wind speed at your holding altitude gets to 1/3 of your holding true airspeed, you would need to do the continuous turn. Any wind speed greater than that will cause you to not be able to obtain a 1-minute inbound leg using standard holding methods.

But the headwind is actually the easier case! At least you have time to adjust and get established on the inbound course.

What about a tailwind?

About this time you probably regret starting instrument training...

This is about what it looks like if you time a 1-minute outbound leg.

15 second inbound leg, that's not much time to get established on the course! That's basically entirely within the "cone of confusion" at these speeds.

For this one, I'll skip ahead to the answer.

To make a 1-minute inbound leg using this tailwind, we will need to head outbound for 2:30 before turning in. That's a lot longer than the rule-of-thumb would give you!


Yes, I know. We fly real airplanes in real conditions and nothing works out perfectly like the math says - there are a million additional variables involved. But I did find it interesting to see how some of these scenarios would work even if everything was "ideal". And finding out that the headwind scenario doesn't work at all when the wind is greater than 1/3 of your TAS was pretty interesting!

What if the wind was a crosswind?

I think that will have to wait for another day!