Curvature in the relative wind about the yaw axis
September 16, 2006 edition
During a constant-banked turn, the relative wind has a curvature about the aircraft's yaw axis. This is a reflection of the fact that different parts of the aircraft are actually travelling at slightly different linear speeds and in slightly different linear directions at any given moment. This is due to the fact that the body as a whole has a linear motion plus a rotational motion. The increased airspeed experienced by the outboard wingtip in comparison to the inboard wingtip is really just another manifestation of this effect.
We might imagine that the curving relative wind will tend to strike the rear parts of the aircraft in a way that tends to yaw the nose toward the outside of the turn, creating a slip.
In the absence of any rudder input from the pilot, unless the aircraft's wingspan is short in relation to the aircraft's length, the increased drag experienced by the faster-moving outboard wingtip will likely yaw the nose far enough toward the outside of the turn that the airflow will strike the inside surface, not the outside surface, of the vertical fin (if present) or other equivalent rear surfaces of the aircraft. In other words the entire aircraft will likely experience a slipping airflow, so the vertical tail or other equivalent rear-most surfaces of the aircraft will likely contribute a yawing-in torque, not a yawing-out torque. (Or to put it another way, the tangent point between the curving relative wind and the aircraft's longitudinal axis will likely lie somewhere behind the rearmost surfaces of the aircraft.) But to allow the aircraft to "find" a yaw orientation where the pressure of the airflow against the inside surface of the vertical fin or other equivalent surfaces of the aircraft is strong enough to balance the drag of the faster-moving outboard wingtip, the nose of the aircraft will have to yaw much further toward the outside of the turn than it would if the same difference in airspeed existed between the wingtips but the relative wind were somehow parallel in direction all across the entire planform of the aircraft. So the curvature in the relative wind about the aircraft's yaw axis does end up increasing the amount of sideslip that the aircraft as a whole will experience during a constant-banked turn, when the pilot does not use a rudder to align the aircraft as a whole with the average direction of the relative wind.
At first glance it might appear that due to the curving nature of the relative wind, reducing the size of the vertical fin might tend to reduce the aircraft's tendency to yaw toward the outside of a constant-banked turn, in the absence of any rudder input from the pilot. However, we noted above that for most "conventionally" shaped aircraft, the airflow in a constant-banked turn will tend to strike the inside, not the outside, of the vertical fin. Therefore reducing the size of the vertical fin will usually increase the amount of slip that take place during a constant-banked turn in the absence of rudder inputs from the pilot. We'll revisit this point again in "A 'holistic' view of how dihedral contributes to roll stability and anhedral contributes to roll instability".
For a good illustration of the curvature of the relative wind about the aircraft's pitch axis, see the section entitled "Long-tail slip" from John S. Denker's superb "See How it Flies" website.
For a given bank angle, the slower an aircraft flies, the more pronounced will be the curvature in the relative wind about the aircraft's yaw axis. The effect will be most pronounced at moderate bank angles: a steeply-banked turn involves a great deal of rotation around an aircraft's pitch axis and only a little rotation about the aircraft's yaw axis
A flex-wing hang glider or trike or other swept-wing flying-wing aircraft will experience essentially the same dynamics as we've described here, though the interaction of between the airflow and the wingtip regions will take the place of the interaction between the airflow and the vertical fin.
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