Advective and pressure contributions to circulatory lift ($\oint (\mathbf{u}\cdot \mathbf{n})\mathbf{u}dl=\frac{1}{2}\mathbf{\Gamma }\times \mathbf{U}$)

For uniform flow $\mathbf{U}$ plus a line vortex of counterclockwise circulation $\Gamma$, the outward advective momentum flux around a large circle is $\rho \mathbf{\Gamma }\times \mathbf{U}/2$. The Bernoulli equation supplies an equal pressure contribution. Their sum is the force on the fluid; the force on an obstacle per unit span is its negative, $\rho \mathbf{U}\times \mathbf{\Gamma }$. Retaining only the advective term loses half the lift and obscures its sign.