+Finbar Sheehy Having had a chance to think about this, I think weāre looking at an interesting coincidence, here. There are three distinct parts to this, which Iāll call A, B and C.
Part A: The usual question about āminimum dragā says, āgiven a maximum wingspan, what is the lift distribution that gives the highest L/D ratio?ā The answer to this is, āelliptical lift distributionā, and thatās not wrong.
Prandtl, though, changed the question, and asked, given a maximum lift/bending moment, what is the lift distribution that gives the highest L/D ratio?"
In this second question, wingspan is not constrained. So, hereās a thought experiment: take a wing - any wing - and then add two sections to each wingtip, one lifting up and the other, outboard of it, lifting down by exactly the same amount. The net lift hasnāt changed, but the bending moment at the wing root is lower, because the additional downward force is outboard of the additional upward force. And what about drag? Well, on the one hand, thereās more wetted surface, but on the other hand thereās more span, so itās not intuitively obvious what has happened to L/D.
What Prandtl found was that the L/D will have gone up, and he went on to find the distribution that gives optimum L/D if the span is not constrained but the maximum bending moment is. Thatās all - notice that it has nothing to do with swept wings, and isnāt about preventing tip stalling (the purpose of washout).
Part B: This is the coincidence part. What happens when you put ailerons on an un-swept Prandlt wing? If the aileron is all on the outboard section, where the lift force is downward, then when you deflect the aileron downward you reduce the local (downward) lift force, and reduce the local drag. And vice versa on the other side. This is the opposite of the normal effect of adverse yaw on ailerons: itās proverse yaw. But, it only works if the ailerons are out at the tip, in the downforce region. If you have large-span ailerons, it wonāt work. Also, as the wing actually starts to roll, youāll still see a reduction in the āproverse yawā and it may even reverse. Why? Suppose you roll right. You deflect the right aileron upward on a tip that is generating downforce. This increases the drag at the tip, and tends to yaw the aircraft to the right. But, as the roll starts, the (negative) angle of attack on the outboard tip will be reduced by the rolling motion, reducing the āproverse yawā effect. Itās conceivable to me - I havenāt tried to prove it - that you could choose the aileron length so that you could get proverse yaw at the start of the roll (creating a skid), and some adverse yaw as you stop the rolling motion, so that the nose would stay close to - but not exactly - where it would point if there were a good glider pilot at the controls. Close enough to not really require a rudder for normal turns. But, because a straight wing is not yaw stable, you still need a vertical stabilizer for yaw stability in straight flight.
Part C: A swept wing is yaw-stable in straight flight, without a vertical stabilizer. Normally, swept wings - like hang gliders - are yaw stable but have problems with adverse yaw during roll inputs unless you fit them with a vertical tail and rudder. With a swept Prandtl wing you can get yaw stability during straight flight and approximately zero yaw error during roll inputs. As long as the residual yaw deviations are acceptable, you can dispense with the vertical stabilizer and rudder. Of course, the properties of swept wings mean that, to achieve the Prandtl lift distribution, youāll need more geometric washout than you would have needed with straight wings, and this will narrow the speed range (or angle of attack range) where the arrangement will be optimal.
Now for an observation. Weāre talking about a wing with quite a lot of twist (you can do some of it with airfoil selection, but the local zero-lift-coefficient incidence angle will have a lot of twist in it), and as you move away from the design airspeed (change the AoA) the lift distribution will move away from optimal - all the more so if the wing is swept. As you speed up there will be too much downforce at the tips, too much upforce in the center, and too much drag from both. As you slow down, you will lose the proverse yaw benefits. So thereās a trade-off here: you do get to remove the vertical stabilizer, with its associated drag, but you narrow the efficient speed range of the aircraft. Sailplanes actually need wide speed ranges, so despite eliminating the drag of the vertical stabilizer, I doubt that this is going to be a winner in a sailplane application, even in the Open class (unlimited span).