Dihedral Effect on low Aspect Ratio Wings

[Ran D. St. Clair]

I am looking for some serious understanding of "dihedral effect" as seen on low to very low aspect ratio wings.

For example, a PBF (Pizza Box Flyer) is a 2ft. Sq. piece of Choroplast, and is absolutely flat, but it fly's like it has a ton of dihedral. It's a really fun plane to fly in high alpha (harrier) flight because it strongly wants to turn its belly into the airflow. The dihedral effect (yaw roll coupling) is so pronounced that it does some fun tricks that I would not dare attempt on any other plane. For example, it will take off, hang on the prop at about 80 degrees, safely turn 180 degrees in the space of about 3 ft., and then drop to a smooth landing in the same place it just took off from.

It also has some "bad" characteristics which are only bad in the sense that it won't do certain things. You can't even begin to fly knife edge because it desperately wants to lay flat on its back or its belly. This roll stability is so extreme that even big elevons with extreme throws will not make it roll until you pick up some airspeed, at which point it rolls and fly's like any other plane. Even a roll at moderate airspeed, with the nose only 10 degrees up will tick over with hesitations when the wing is flat upright or inverted.

I have designed and built other more conventional planes with low to fairly low aspect ratios and they all have this characteristic to a lesser degree. I have a Morris "The Knife" which is a fairly conventional profile with lots of side area and a thick wing. As its name implies, it will knife edge pretty well, but it is much happier to be upright or inverted. It wants to roll out of knife edge one way or the other. That, and if you give it hard rudder in level flight it wants to roll into the turn. Likewise it wants to roll into the turn when inverted as well (which if you think about it is the opposite roll direction for the same yaw input, meaning is not a simple asymmetry effect.)

I have just one theory to explain all of this and it involves massive wing tip vortices. I am thinking that the wing tip vortices are rolling over the top of a fair portion of the wing, especially for the PBF. This spinning cyclone of negative pressure is creating a lot of lift, but also a ton of drag. It also energizes the airflow over much of the wing, allowing it to achieve a ridiculous angle of attack before the flow over the top surface fully separates. In level (high alpha) flight the vortices are equal over both wings (left and right) so there is no induced roll. If the plane should "slip" sideways for any reason, like the application of rudder, then the upwind vortex is pushed more over the wing, while the downwind vortex is pushed more off the end of the wing tip. That implies a big pressure differential, with the upwind wing making much more lift (and drag). The result is opposite roll and the plane turns its belly into the airflow.

It plays out like this"
1. High alpha straight ahead.
2. Left rudder and the plane falls off (yaws) to the left.
3. The plane rolls right turning its belly into the new airflow.
4. High alpha and straight, but on a new heading (to the left)

It's odd to think you need to roll right to turn left, but that's the way it works in very high alpha.

Keep the left rudder applied, and perhaps just a little right aileron for good measure and the plane does the toilet bowl spiral, without loosing altitude.

So, is my vortex theory even close to the mark? So why does this effect seem to go away almost completely at more normal aspect ratios? Is this one reason that many sport aerobatic designs have fairly severe wing taper, to minimize the area at the tips and reduce this effect? (I know about span loading and structural reasons as well.)

Comments?