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I was just reading up about different methods of reducing drag through the different wingtip designs when I come across Hoerner Wingtips.

So after some research, I have two questions.

  1. According to this article on pg 1, it states that "The convex underside accelerates the speed of the air passing under the tip to a velocity more equal to that of the air flowing over the top of the tip, thereby creating streamline flow." How exactly does acceleration of air occur just by having a convex underside?
  2. As we can see, the effective wingspan using Hoerner Wingtips can be increased without increasing geometric wingspan.

    enter image description here

    How does a Hoerner Wingtip enable vortices to be generated further away, therefore increasing effective wingspan?

Thank you! :)

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How exactly does acceleration of air occur just by having a convex underside

Just like the curved forward surface of a wing does it. Imagine for a moment that the air flows along a straight path: Now it would "see" that the surface below it curves away from its path of travel, and if that path would remain unchanged, a vacuum between the wing and the air would form. Reluctantly (because it has mass and, therefore, inertia), the air will change course and follow the wing's contour. This requires lower pressure, to make the molecules overcome their inertia and change direction.

Near the tip the flow has a sideways component, flowing away from the center on the lower side. This sideways flow, coupled with the sideways curvature on the lower wingtip, creates suction. This area of lowered pressure will suck in more air from inside. In essence, the curved lower side of the Hoerner wingtip extends the suction which is found on the upper surface of the wing to the side of the wingtip.

As we can see, the effective wingspan using Hoerner Wingtips can be increased without increasing geometric wingspan.

Well, that is at least what Hoerner claims. Maybe it does move the peak suction of the wingtip vortex a little outside, but this has no effect on the flow field behind the wing and the eventual location of the trailing vortex. Below is a graphical overview of different wingtip shapes and their sideways location of the core of the tip vortex.

Wing-tip shape and tip-vortex location for a family of wings
Wing-tip shape and tip-vortex location for a family of wings (picture source)

However, this is not relevant for induced drag. In the end, the strength and width of the rolled-up vortex sheet behind the wing indicates the amount of induced drag. Induced drag itself is the horizontal component of the resulting pressure forces on the wing (see here or here for more on that). The tip vortex is sucked into that vortex system regardless of its initial location. See below for an illustration:

B-747 with contrails
B-747 with contrails (picture source)

You can see that the outer contrails of this Boeing 747's engines wrap around the contrails of the inner engines. This shows how the air is pushed down in the wake of the wing and that the centers of the vortices are slightly inboard of the contrails of the outer engines.

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    $\begingroup$ That photo is the best illustration of vortex behavior ever. You can really see the sheet rolling up from the sides behind the plane $\endgroup$ – TomMcW Jan 29 '17 at 1:27
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    $\begingroup$ @TomMcW: Yes, and there is so much more to see: The initial widening of the distance between the inner contrails shows how much they are already pushed sideways (instead of straight down) by the downward-moving air in the center. The very high relative humidity that day produced both strong engine contrails and a visible aerodynamic contrail. The mist in the middle is from condensation in the low-pressure center glinting in the sunshine. $\endgroup$ – Peter Kämpf Jan 29 '17 at 9:55
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Wings create a miniature weather system around themselves. The Angle of Attack pressurizes air under the wing and depressurizes it over the wing. Air seeks to equalize this pressure all around the wing at the speed of sound -- faster than the wing travels, slowed by the momentum shift (acceleration) of new air entering the system. So in front of the wing air below is pressurized below the altitude of the wing and air above the altitude of the wing is depressurized. This pressure difference creates an updraft ahead of the wing. The wing constantly flies into an updraft of its own creation. You can see this in the rise of streamlines ahead of the wing in wind tunnel tests. The slipstream dividing air that flows over vs. under the wing strikes the underside of the leading edge of the wing, well below where one would expect, if you ignore the weather-like effects of the wing on air around it.

Outboard of the wingtip, this same pressure difference releases pressure outward below the wing and pulls air inward above the wing, creating the vortex. The convex ramp of the Hoerner wingtip changes the angle of momentum of this outward flow of air, and makes the outward flow below and inward flow above asymmetrical. The center of the vortex is higher and farther outboard of the wingtip.

Lift is amplified beyond the cord of the wing by the thin, turbulent boundary layer behind the wing, ended by the wingtip vortex. Air pressure differential maintained by this boundary layer is a major contributor to lift. This is why Ground Effect happens. If the boundary layer strikes the ground before it disintegrates tens of yards behind the wing, the wing generates more lift and less drag. Move the vortexes out with Hoerner wingtips, and that boundary layer effectively acts like an extension of the wing lengthening the wingspan. Air mass can't flow through that invisible boundary layer any more than it could flow through the wing, and that pressure difference is where lift comes from. The boundary layer maintains that pressure difference, just like the wing does.

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