Built on Facts

De Plane!

Here’s a rather harrowing video I recently watched. It’s a commercial airliner on approach to a runway during a severe crosswind. The plane is attempting to perform a maneuver to keep itself flying over the runway without drifting off course, but this requires flying at an angle into the wind. It’s difficult and the pilot narrowly averts disaster by aborting the landing at about the last possible second.

The difficulty is one of relative motion. If you’re in a boat crossing a river to a point directly opposite your starting point, merely plowing straight ahead will result in your landing downstream from your destination. You have to angle upstream slightly to offset the drift. In particular, your angle is going to have to be such that the upstream component of your velocity with respect to the water is equal and opposite to the velocity of the water with respect to the shore.

Which is a slightly awkward way of saying that if the boat’s speed is v and the water’s speed is w, and the angle measured with respect to the straight line between the origin and destination,

i-d4bb9fe09d6b78eb832985821d8a307f-1.png

More saliently,

i-88516ca505d84f969b7ab15a965c3fb3-2.png

Which says that the angle increases as the ratio of wind or water speed to vehicle speed ratio increases. If it goes over 1, the inverse cosine is undefined. Physically that means the wind/water is going faster than you and even a 90 degree angle can’t keep you on the straight and narrow. In practice I’m sure even considerably smaller angles are not possible to safely perform on an aircraft during landing.

On the other hand I have the luxury of sitting and thinking and typing about it. The associated situation in the plane probably became somewhat more difficult about the time the left wing sheared its tip off on the runway.

Comments

  1. #1 Ken Clark
    February 27, 2009

    Actually, most aircraft have a crosswind limit, for heavy aircraft with fast stall speeds like 737 it can be relatively high, 45 mph+(Boeing 737 FCOM). On a related note, the B-52 can rotate its landing gear up to 20 degrees to be parallel to the runway, while crabbing.

  2. #2 Paul browne
    February 27, 2009

    Yikes, as a glider pilot whose done his share of crosswind landings I have sympathy for the pilot. As you say the trick is to crab in at an angle to the runway and then kick the rudder just before toughdown to line up with the runway (without dropping a wing).

    It’s interesting to see that despite the very strong crosswind the pilot was doing well until touchdown, when the wind seemed to get under the right wing, and the undercarriage became a fulcrum around which the plane could pivot. I suspect a slight left wing drop as the aircraft yawed to the left to line up with the runway might have contributed, when a plane yaws to the left without banking (e.g. on landing) it causes the left wing to move “downstream” relative to the wind direction and the resulting lower airspeed and lift relative to the other wing (which moves upstream by a corresponding amount) can result in a small unintentional bank to the left and the left undercarriage to touch down first. We call this a secondary effect of the rudder.

    Or it might just have been a gust!

    We usually try to land into wind. Landing directly downwind isn’t good either wince you land with greater ground speed, so hit the ground harder! Even more importantly you loose control authority, if you are travelling at 20 kts with a 20 kt wind behind you you have ) kt airspead…and no control (unless your undercarriage is steerable).

  3. #3 Uncle Al
    February 27, 2009

    Mount landing gear wheels on swivels. Now the plane can land at any angle to the runway and still go “straight” down the concrete. A declared emergency is no big deal unless nuns start crying. If so, grab one as a cushion.

  4. #4 yaque
    February 27, 2009

    I used to drive a 150cc Vespa motor scooter.
    During the summer,sometimes I had to drive outside the city in rough country. Windy, oh yeah! Had to drive leaning into the wind, steering at a pronounced angle. Fun! not.

  5. #5 co
    February 27, 2009

    Note that the inverse sine (or cosine) is undefined—on the Reals—for an argument greater than unity. In the complexes, the arcsine is perfectly well defined for angles >1 or < -1; this fact is used all the time for diffraction phenomena (for an example).

  6. #6 co
    February 27, 2009

    Huh. Preview fail on that last post. Meant to say that arcsine is well-defined for |argument| > 1, and this fact is used all the time in, e.g., analysis of any oscillators (with real physical consequences!). Very neat video.

  7. #7 Eric Lund
    February 27, 2009

    Or it might just have been a gust!

    I believe it was. I’ve seen this video before. The airport in question is in northern Germany (my recollection is hazy on whether it was Hamburg or Hannover), and apparently there was a gust of wind right at the time this plane was attempting to land.

    Many airports in the United States have crossing runways, allowing air traffic control to choose an approach direction within 45-60 degrees of any wind direction. For example, O’Hare has seven runways oriented in four different directions. And of course you can choose to run the planes in one direction or the other; e.g., Runway 4L is the same as Runway 22R viewed from the other direction. (As Paul #2 said, you want to take off or land into the wind when possible, because you have a lower ground speed for a given air speed.) But most airports I have been to outside the US (and some in the US, such as Seattle–which happens to be the major US city whose climate most closely resembles that of northern Germany) have runways in only one orientation. If you get a strong wind nearly perpendicular to the runways, this scenario can happen.

  8. #8 magista
    March 1, 2009

    Fabulous timing, as my class has just finished a lesson on relative motion. I’m going to drop a copy of this onto their online notes…

  9. #9 Jim_K
    March 21, 2009

    I was on a flight landing at New York Kennedy. We were in the middle of lightning and high cross-wind. It took the pilot three tries to make the landing. Kinda scary.