Downwash distribution on basis of Lift distribution

Question

I am new here so I might make some elementary mistakes. Correct me if I am wrong. I just stumbled upon the downwash distribution for a given lift per unit span distribution along the wingspan. I have attached a photo for your reference.

I was wondering that if we look upon a $dy$, an infinitely small spanwise slice of the wing, then the lift it produces should be proportional to the downwash it created as more downwash equals more distortion which equals more lift. But we can clearly see that at the end where the lift is minimum the downwash is maximum and at the center where the lift is maximum downwash is minimum. Could anyone please explain it?

Answer 1

You are confusing downwash ($w$) with circulation ($\Gamma$). As per Kutta-Joukowski theorem, lift per unit span (in the case of a single lifting line) depends on the distribution of circulation along the wing span:

$$L'(y)=\rho_\infty V_\infty\Gamma(y)$$

If you have an elliptical lift/circulation distribution, the circulation decreases towards zero at the wingtip, therefore lift at the tip is zero. Induced downwash, on the other hand, is constant across the span.

Source: Anderson, Fundamentals of Aerodynamics

You may believe that downwash is the largest at the tip due to the presence of wingtip vortices. However, the vortices are better understood as downwash rollover at the wingtips. The vortices themselves do not contribute to lift generation (but unavoidable for finite span).