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At the same time, the air on the upper surface has a tendency to flow in toward the fuselage and off the trailing edge. This air current forms a similar vortex at the inboard portion of the trailing edge of the airfoil, but because the fuselage limits the inward flow, the vortex is insignificant.

_(FAA Pilot’s Handbook of Aeronautical Knowledge, chapter:5 ,Pg 8)

This got me thinking if there is no fuselage, as in the case of high wing aircraft, the inward flow is not restricted anymore. So what will happen to them, do they form new significant vortex? Or there will be another pressure drag?

P.S. I could not find any image of a high wing aircraft with vortices in middle. So if you get one (if it happens so) please link it in the answer. I am too eager to see.

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Short answer

If there are two wingtips, there are two vortices.


Vortices - summary

There are only two vortices, produced due to the very existence of ends to the wing - the wingtips. Pressure distribution that exists over the wing ceases to exist beyond the wingtips (since there's no wing beyond the wingtips). To understand this in detail, first consider 2D flow.

In 2D, the low pressure region on the upper surface of the wing sucks the air from all four directions: It pulls the air from ahead, causing it to accelerate and upwash. It pulls the air from behind, causing it to decelerate (this is the pressure recovery). It pulls the air from up above, causing it to downwash. And in an attempt to pull the air from underneath, it pulls the wing up, thus producing lift.

In 3D, this low pressure also tries to pull the air from sideways. On an infinite wing, the pressure distribution is homogeneous spanwise, and so there is no spanwise flow. On a finite wing, there are two wingtips, beyond both of which there isn't a similar pressure distribution. This causes the low pressure region above the wing to pull in the air from beyond the wingtips. The high pressure region underneath the wing similarly expells air outboard, beyond the wingtips. This is what causes wingtip vortices.


High Wing Configuration

On a high wing aircraft, on the part of wing above the fuselage, there's still a region of low pressure (and even if that portion on its own didn't have a low pressure region, the pressure from the consecutive parts of the wing will be carried over there). More importantly, this part of the wing is not like a wingtip, and shouldn't be treated like one; there is only one wing, and two wingtips (the "left wing" doesn't terminate at the fuselage, it extends from there all the way to the right wingtip). Since there are no wingtips in the middle, there are no vortices.

You can think of it this way: both the "left wing" and the "right wing" try to form a vortex in the middle; the two POTENTIAL vortices are equal in magnitude but opposite in direction. The two potential vortices completely annihilate each other, and so there's no vortex in the middle. (you can apply this idea at any point on the span, except that the two opposing vortices won't always be equal in magnitude; they are only equal at the middle of the span. If you do this for every point, the end result will be the two wake vortices as we know them)

As far as your handbook's claim goes that the fuselage acts as an end plate, restricting the vortex formation - yes, that's technically true: the fuselage will act as an end plate for the left wing¹, largely inhibiting it's vortex. But the right wing will do a better job at this, eliminating the left wing's vortex altogether, as described earlier. If anything, fuselage is actually making the situation worse by not letting the right wing properly do it's job (that's why the two small vortices form in first place).

¹same is applicable for right wing

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  • $\begingroup$ Thank you for such an explanation. I did understand the concept of vortices. But the point where two equal or almost equal in magnitude inward flow carrying some momentum meets and cancel each other without any energy transformation like drag or something is a little hard to imagine, may you please enlighten about this too. Thank you. $\endgroup$ Apr 2, 2023 at 3:13
  • $\begingroup$ @NoorulQuamar If a wing was to be cut in half, two entirely seperate wings will be obtained - both of which will independently produce two tip vortices (four in total). As we attempt to join the wings back together, we will observe that the two vortices at the centre will cancel out (because they are counter-rotating). $\endgroup$ Apr 2, 2023 at 5:05
  • $\begingroup$ Actually since the half-wings are joined, the pressure discontinuity which existed at the edges of both wings no longer exists. Earlier the low pressure above the wing ended at the edge, and was able to pull air from beyond the edge; but now, the low pressure extends beyond the edge, since there's the other wing beyond the edge. This means that there is no pressure difference across the edge and so the vortex cannot form $\endgroup$ Apr 2, 2023 at 5:17
  • $\begingroup$ There is no energy transformation (drag) because there is no pressure difference. If there was a pressure difference, like there was when the wings were not connected, then the vortices would form and then there will be drag. Now there is pressure equilibrium at the centre, therefore that particular drag is avoided. $\endgroup$ Apr 2, 2023 at 5:28
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FAA handbooks are known to make scientifically inaccurate statements on occasion. This may be one of those times. What they probably intended is two separate wings magically flying with a gap where the fuselage would normally be, rather than a continuous span like a B2 flying wing. This 'two wings with a gap' situation would indeed have inboard vortices, because it would have inboard wingtips. However, including such a contrivance in a practical manual may be more ridiculous than if they had made a simple scientific error.

Anyway, flying wings ( Even two in close formation ;p ) produce vortices at the wingtips as the air is accelerated by the difference in pressure between the lower and upper surfaces. Air has mass [inertial resistance] and it is very compressible and extensible; so it creates a pressure gradient that has effects far inboard from the wingtips. The amount of effect (primarily spanwise flow) is a continuous [non-linear] gradient from maximum near the tips to zero at the center.

There are cases where a sudden sectional change can create smaller additional vortices, such as the outboard end of a large extended flap. These are caused by the same type of pressure effects as those at the wingtips.

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Where does air flow?

Air flows from high to low pressure

Underneath the wing, air pressure tends to be higher than ambient. The fuselage acts as an endplate, and pressure tends to leak out the wingtip and aft. The front is a ram.

So, this "engine on a half-shell", or sail, has its highest pressure next to the fuselage. It gets better with flaps.

"because the fuselage limits the inward flow, the vortex is insignificant"

Essentially, this is saying the fuselage acts as a "fence", or endplate. This idea has been carried forward with little fuselages on wingtip, know as "tip tanks".

The thought seems to be an extension of the logic that each half wing ends in 2 places. In reality, the fuselage has lifting properties as well.

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