I read that the circulation at the wingtip for a finite wing is $0$. I am a little bit confused about this statement. First, it is said that drag due to lift (aka induced drag) generates vortices at the wingtip. But on the other hand in Prandtl's lifting line theory the circulation at the wingtip dies off. I thought about some explanations but I would like to know which one is really true. I also welcome any additional explanations.

  1. Pressure difference at wingtip is $0 \implies $ no lift $\implies \Gamma = 0$.
  2. Indirect: If the circulation at the wingtip is not equal to $0$, then the downwash due to wingtip vortex is infinitely large, which is not possible. This gives $\Gamma =0$ by contradiction.

Prandtl's lifting line assumes that the form of the solution for its equation is similar to an elliptical solution with small variations.

$$ \Gamma(\Theta)=4sV_{\infty} \sum_{n=1}^{\infty}A_nsin(n\Theta) $$

Which, essentially, assumes that the circulation at the tips is 0, so the tip is not producing lift at all in any assumemd solution. The theory assumes that the tip pressure difference is 0 and actually no speed normal to the surface.

This theory explains that the drag due to lift is NOT a drag produced at the tip, is drag of the complete wing as a single element and the tip vortex is not produced at the vortex, is produce down the wing due to the circulation variation along the wing.

Which drives the typical picture (via helicel)

enter image description here


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