The Spell of Kutta-Zhukovsky’s Circulation Theory

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This post is related to Paradoxes from Misunderstanding Mahematics and Why It Is Possible to Fly. The standard explanation of why it is possible to fly on wings presented in the aerodynamics literature is captured in the following picture showing that circulation of air around a wing section generates lift according to the circulation theory by Kutta and Zhukovsky:



This is because the speed of air will by the circulation be higher above the wing than below, which will correspond to lower pressure above than below by Bernouilli’s principle , which will generate lift according to Kutta-Zhukovsky’s famous formula (with a suitable normalization):

(KZ)                                                    lift = circulation,

more precisely according to the following logic reading the formula (KZ) from right to left:
  • If there is circulation, then there is lift.
This is an implication of the form: If A then B, which means: If A is true, then also B is true.
But the statement: If A then B must not be confused with: If B then A. For example the statement: If you bang your head into the wall, then you get a headache, is different from: If you have a headache, then you have banged your head into the wall. As we know, one can get a headache also by drinking too much wine, for example.
However, state-of-the-art in fluid mechanics expresses:
  • The circulation theory of lift is still alive… still evolving today, 90 years after its introduction [2].
  • If the airfoil experiences lift, a circulation must exist [1],
which is the same as reading (KZ) from left to right. In other words, modern aerodynamics of flight is based on the principle that lift and circulation are intimately coupled by (KZ): Lift is equivalent to circulation, which means that circulation implies lift, and lift implies circulation.
But interpreting the formula (KZ) as an equivalence, forgetting that it was derived only as a statement that if there is circulation then there is lift, amounts to a misunderstanding of mathematics and logics.
More precisely, it represents a statement which is not true: In Why It Is Possible to Fly it is shown that the lift you experience when you fly, comes without circulation, as displayed in the following figure showing the lift and circulation of a Naca0012 wing as function of the angle of attack, computed by solving the Navier-Stokes equations for the flow around the wing:

We see that the lift increases linearly with the angle of attack up to 16 degrees, while the circulation stays basically zero up to 10 degrees: Lift and circulation are not equivalent as in Kutta-Zhukovsky’s formula (KZ). There is lift but no circulation! Lift does not come from circulation!  Why It Is Possible to Fly exhibits the true origin of lift.

The statement above indicates that circulation theory is still alive, which can give an incentive to ask the pilot before take-off if he believes in the spell of circulation theory, and if the pilot says yes, what do you then do?
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