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I'm wondering what is the formula for induced drag in a stalled regime, i.e. in a regime where the $C_L$ (coef. of. lift) has started to decrease but is still nonzero.

I've a feeling that the conventional formula for induced drag $$D_i = \frac{L^2}{\frac{1}{2}\rho V^2 \pi b^2 \epsilon}$$ (taken e.g. from this answer) which in essence depends on lift $L$, does not explain induced drag in stalling regime (if it would, it would erroneously imply that the induced drag is equal for a pair of AoAs $\alpha_{\text{before-stall}} < \alpha_{\text{after-stall}}$ which both correspond to the the same $C_L$).

I've come across the evidence that induced drag continues to grow as square of AoA even in stalled regime (e.g. schematic Fig. 4.14 from av8n.com)

Fig. 4.14 from av8n.com

but can't find more.

Thanks.

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4 Answers 4

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That formula describes the "lift-induced drag" for a planar wing. Lift-induced drag

As the definition implies, this drag is induced by the generated lift, the effect of stall (flow separation) is not contributing to this drag component.

Therefore, there is no other formula for the stalled regime.

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    $\begingroup$ Doesn't the epsilon in the formula above account for non-elliptical lift distributions? For all wings except those that create an elliptical lift distribution this value is going to be less than one, thus creating more induced drag. $\endgroup$
    – Jan
    Commented May 21, 2019 at 17:54
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    $\begingroup$ Yes indeed, that's where it differs from the wikipedia version. I overlooked that part, I'll edit my answer. This doesn't change much about the content of the answer though. $\endgroup$
    – hrobeers
    Commented May 21, 2019 at 21:48
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Beyond the stall, the wing profile is governed by flat plate aerodynamics. The airfoil does still have lift and drag, and one could express one as a function of the other, but I have not seen a useful application of it.

enter image description here

Flat plate drag is a function of Reynolds number. The picture is from this document, and shows $C_D$ as function of $Re$ and AoA.

More info, an equation and 360 deg plots of a NACA 0012 profile in the answers to this question. Looks a lot like sine & cosine graphs, doesn't it.

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If we interpret induced drag as the backward component of the lift vector (explained here and here), post stall is not much different from pre-stall.

It is simply the vector decomposition of the lift, so if we rotate the wing more, the induced angle of attack increases and induced drag increases. However, with stall, when we rotate the wing more, we actually reduce the lift and thereby the induced drag.

The final experienced induced drag is influenced by both (i.e. how big is the lift force, and how much is it tilted backward), so how the induced drag actually develops with AoA depends on the relative change in both and thus is highly dependent on the airfoil.

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From my understanding induced drag is the penalty for accelerating air (to create lift) and the creation of the wing tip vortices. If you deflect the air a lot it creates more induced drag, stronger vortices but also more lift and if you don't deflect it as much it creates less lift, less induced drag. Notice I haven't even mentioned the stall because to me that is a general cause and affect relationship. As you angle the airfoil up to create more lift you deflect the air down but this deflection of air also affects the incoming airflow (as long as we're subsonic) and because of that the net force vector starts pointing aft a little bit. It is this additional deflection of the force vector that causes a drag force, the induced drag.

Note: The lift after the stall doesn't drop to zero! There will always be quite a bit of lift throughout the full angle of attack range as can be seen in wind tunnel results (already linked by other users here but I'll repeat it: What is the performance of a flat plate wing?). Only at around 180° angle of attack and around 0° will the lift coefficient ever be close to zero and thus create no more induced drag. At 45° angle of attack the wind tunnel results show a lift coefficient of roughly 1.1, which can be more lift than if the airfoil is in normal operating range.

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