I am doing the following exercise:
Consider inviscid, incompressible, steady flow. The Kutta-Joukowski theorem $$L' = \rho_\infty V_\infty\Gamma$$ where
- $L'$ is the lift per unit span
- $\rho_\infty$ is the freestream pressure
- $V_\infty$ is the velocity
- $\Gamma$ is the circulation taken around the body
was derived exactly for the case of the lifting cylinder. Equation $L' = \rho_\infty V_\infty\Gamma$ also applies in general to a $2$-dimensional body of arbitrary shape. Although this general result can be proven mathematically, it also can be accepted by making a physical argument as well. Make this physical argument by drawing a closed curve around the body where the closed curve is very far away from the body, so far away that in perspective the body becomes a very small speck in the middle of the domain enclosed by the closed curve.
And here is the solution from the author:
The flow over the airfoil can be syntheized by a proper distribution of singularities, i.e, point sources or vortices. The strength of vortices, added together, give the total circulation, $\Gamma$, around the airfoil. This value of $\Gamma$ is the same along all the closed curves around the airfoil. In this case, the airfoil become a speck on the page, and the distributed point vortices appears as one stronger point vortex with strength $\Gamma$. This is exactly equivalent to the single point vortex for the circular-cylinder case, and the lift on the airfoil where the circulation is taken as the total $\Gamma$ is the same as for a circular cylinder, namely, equation $L' = \rho_\infty V_\infty\Gamma$.
I understand the author's solution until the part that said:
the lift on the airfoil where the circulation is taken as the total $\Gamma$ is the same as for a circular cylinder.
The equation $L' = \rho_\infty V_\infty\Gamma$ is derived from the circular-cylinder case which we have to do the integral of the pressure distribution over the cylinder surface. But I have not understood why the lift on the airfoil is the same for a circular cylinder like author said.