I am trying to calculate the distribution of lift over the span of a rectangular, non-swept wing with a (constant) Joukowsky airfoil cross-section. The wing is rectangular in the sense that the chord length is constant. The hope is to combine results from Joukowsky airfoils and Lifting-Line theory. I have followed the derivations from the following sources:



For the Joukowsky airfoil, the circulation is fully determined by the airfoil geometry and angle of attack by imposing the Kutta condition (finite fluid velocity at the trailing edge):

$$ \Gamma = -4\pi UR\sin (\alpha + \beta)$$

where $\alpha$ is the angle of attack and $R$ and $\beta$ determine airfoil geometry.

Now, in Lifting-Line theory, we require knowledge of the slope of the airfoil lift coefficient $C_{\ell \alpha}$, which I assume is $\frac{dC_{\ell}}{d\alpha}$. However, we come up with a solution for both the induced angle of attack $\alpha_i$ and the circulation $\Gamma$ which have nothing to do with Joukowsky airfoils or the Kutta condition. It would appear that both circulation and the effective angle of attack (geometric angle of attack $-$ $\alpha_i$) are specified now, and the above relation should not be true in general.

My question is: how is the circulation found from Lifting-Line theory consistent with results of Joukowksy airfoils, and the requirements of the Kutta condition? Can the two methods be used in tandem?


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