Why is the tendency to yaw in the direction of the roll more desirable (since it's the "proverse" yaw) than the lack of yaw in response to roll? Is it even possible to have no yawing tendency at all during rolls in a plane?
I would add that it depends on what you mean by "proverse yaw", and that vertical fin size is key.
Proverse implies "too much" yaw in excess of the appropriate yaw rate for the angle of bank, which would be a skidding turn. That's bad. If proverse just means an into-turn yaw rate that is appropriate for the bank, that's good.
Airplanes with spoilers-only for roll control suffer can suffer from "bad" proverse yaw in that sense, because the drag of the down-wing spoiler, with nothing sticking up on the opposite wing to offset it, pulls the nose into the turn too much, requiring out of turn rudder to keep coordinated flight (I've heard that the spoilers-only Mitsubishi MU-2 is like that). But almost all airplanes have the opposite, adverse yaw, to one degree or another.
Anyway, what we want is a natural yaw rate that is appropriate to the bank angle so the tail naturally wants to stay lined up behind the nose in the air stream during the turn. Asymmetric drag effects of flight controls aside, the yaw rate you get in concert with bank to get a turn is a function of the vertical fin (and the rudder, to the extent that it's providing a passive fin-like effect, like with a hydraulic rudder, or a mechanical rudder with centering springs).
It's basically a big weathervane. If you had no fin/rudder, no weathervaning effect at all, you'd just skew sideways when you bank (even ignoring the dynamic yawing effects of the rolling action or asymmetric aileron drag, which make the slipping worse).
You would think it's desirable to have a fin that can make the tail follow the nose immediately and precisely, when side slip occurs as a result of banking, so the bigger the vertical fin, the better. But there's a problem with that. To exploit the self-righting effect of dihedral, to give wings-level stability in roll, you need to have a little bit of side slip occur when the wings are banked, to create the self-righting lift differential.
So you need a weathervane, but not too big a weathervane. The sizing of the fin itself is a balancing act over the need for strong into-turn yaw, and for roll stability due to dihedral effect. If the fin is too big, into-turn yaw is too strong and immediate; the airplane will immediately yaw (weathervane) into the turn with little or no initial side slip, and it will want to enter a spiral dive any time it's banked. Vertical fin too small, and the weathervaning effect is too weak and the airplane hunts and slithers around in yaw too much.
There is a sweet spot for fin sizing that gives good "proverse" yaw to give a naturally coordinated turn that only needs bits of rudder to counteract adverse yaw while the ailerons are displaced, while still allowing just enough initial side slip to allow dihedral effect to apply its self-righting tendency when the airplane is banked slightly by a bump.
Turns with balanced flight at less than 90 degrees angle of bank (but more than zero) are actually rotations about two axes at once in steady state. It is a continuous combination of pitch and yaw. For it to be a level turn, the yaw has to be in the same direction as the angle of bank. With proverse yaw, on rolling into a turn the yaw is in the right direction.
For an aileron roll, you can get zero yaw on rolling with a little use of rudder, or if the aircraft uses a mix of spoilers and ailerons.
In the landing pattern, a strong proverse yaw while low and slow could contribute to an approach turn stall, same as if you tried to “rudder” the turn. Not a good outcome.
Is it even possible to have no yawing tendency at all during rolls in a plane?
Once the aircraft is in a steady roll, the roll damping will fully compensate for the differential lift caused by aileron deflection. Now the rolling motion will increase the local angle of attack on the downward moving wing the farther you are away from the center, and decrease it on the upward moving wing. Since lift is perpendicular to local airflow, the air force vector on the downward moving wing is pointing slightly forward, and that on the upward moving wing slightly backwards. Please note that the total lift produced by each wing is roughly the same! The rolling moment of both is exactly zero once the roll rate is steady. However, the aileron deflection will decrease lift on the downward moving wing such that it just compensates for the increased angle of attack due to the rolling motion, and the same holds for the upward moving wing with its trailing-edge-down aileron deflection.
What is different is not the magnitude, but the direction of lift on both wings. This pulls the downward-moving wing forward and the upward moving wing back, causing a negative yawing moment in a positive rolling motion. That is why you will always have a yawing moment in a roll.
A two part answer here.
- Yawing in the direction of the roll in a turn is known as a "coordinated turn". How to do it depends a lot on the design of your plane. In a high wing dihedral (trainer), with slight aileron deflection for a left roll, the dihedral and pendulum will neutralize the roll and the downward right aileron may actually start to skid the plane to the right! This is why you "center the ball" with left rudder and push the leading right wing into the relative wind. This overcomes the planes stability and gets it to roll and turn to the left.
Very important is that with any movement in an aerogravimetric environment (earth and sky) any acceleration reaches "steady state" when DRAG forces match accelerating force. The pilot can also "ease up" on the input to lower the rate of change.
- Yawing while rolling. Let's sort out effect of CG on aircraft during roll first. A forward set CG requires rudder and elevator inputs during turn to keep the nose level to the horizon. Notice, during the roll, they swap roles as the plane rolls through 90, 180, 270, 360 (0). Moving CG back (to a nuetral static stability) eliminates this effect.
What we are left with is the yawing tendency due to differences in AOA of upward and downward rolling wings, cancelled by aileron deflection as Peter pointed out. Depending on the design of the plane, it might not be much.