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Do split flaps produce lift? I don't see how, because there is no change in camber. It seems like an upside down speed break, producing only drag

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Yes, deploying split flaps increases lift, because they do increase camber. It's as if the airfoil became solid between its (now two) trailing edges. Between those two trailing edges there's a lot of turbulence, of course, but that doesn't negate the fact that camber increased.

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Do split flaps produce lift? I don't see how

Well, after all lift is created by deflecting air downward, which is exactly what a split flap does - although in a very inefficient way i.e. with a lot of drag.

This NACA TN shows how $C_l$ and $C_d$ increase with the deflection $\delta_f$ of the split flap (left plot):

enter image description here

At stall (at around 10° AoA) $C_l$ goes from 1.3 to 2.2 (plus 70%) while $C_d$ goes from 0.1 to more than 0.4 (plus 300%!).

For comparison, the following plots show the behaviour for a double slotted flap, like seen on almost every jetliner (source this NASA TN):

enter image description here enter image description here

With this more efficient flap configuration, to get the same increase of 70% $C_l$ at stall, the flap has to be deflected of 35° ($C_l$ increases from 1.6 to 2.9); anyway $C_d$ remains almost the same at 0.026!

It seems like an upside down speed break, producing only drag

A speed brake does not increase drag only but decreases lift as well.

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    $\begingroup$ These hard numbers are more convincing than my handwaving :-) $\endgroup$ Commented Jun 18 at 15:55
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    $\begingroup$ @CamilleGoudeseune Nevertheless, your intuitive picture you drew was really helpful to me. I like reading answers that come from different directions. $\endgroup$ Commented Jun 19 at 20:30
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A split flap is a very good choice for the landing configuration. Simple and effective.

For the approach you want that extra drag. It helps to slow down and allows to keep the engine running at a higher thrust level, so spoolup will be faster once you decide to go around (and raise the flaps, hopefully).

Split flaps do increase lift and are even more effective for the same chord and deflection angle than a simple camber flap!

In the words of Sighard Hoerner (see section 5.7 in his book Fluid Dynamic Lift):

Figure 7 reveals, however, that split flaps are somewhat superior to plain flaps. The straight upper side of the airfoil section evidently leads the flow right to the trailing edge, while past a plain flap, the flow is exposed to interference (mixing) with the highly turbulent "dead" space. Another advantage of the split type might also be a more stable wake pattern; the separation points are clearly fixed.

Figure 6 in section 5 of Fluid Dynamic Lift

Figure 7 in section 5 of Fluid Dynamic Lift

And these are Figures 6 and 7.

If you want even more lift for a bit of added complexity, make the hinge point move back when the flap is lowered. This is the Zap flap, named after Eduard F. Zápařka, who invented it in 1931. Moving the flap back in effect increases wing chord. It is not only the Fowler flap (which was invented by Gustav Lachmann, btw.) which performs that trick.

Figure 8, from the same book and chapter.

Figure 8, from the same book and chapter.

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