Lifting flow over a cylinder

Lifting flow over a circular cylinder can be synthesised as superposition of a vortex flow of strength \(\tau \) with non lifting flow over a cylinder.

The stream function for the flow can be given as\[\psi  = \left( {{V_\infty }r\sin \theta } \right)\left( {1 – \frac{{{R^2}}}{{{r^2}}}} \right) + \frac{\tau }{{2\pi }}\ln \frac{r}{R}\]and velocity components in the radial and tangential direction can be obtained as \[{V_r} = \left( {1 – \frac{{{R^2}}}{{{r^2}}}} \right){V_\infty }\cos \theta \] \[{V_\theta } =  – \left( {1 + \frac{{{R^2}}}{{{r^2}}}} \right){V_\infty }\sin \theta  – \frac{\tau }{{2\pi r}}\]The velocity on the surface of the cylinder can be given as  \[V =  – 2{V_\infty }\sin \theta  – \frac{\tau }{{2\pi R}}\] and coefficient of lift as \[{c_l} = \frac{\tau }{{R{V_\infty }}}\]Lift per unit span is given as \[{L^\prime} = {\rho _\infty }{V_\infty }\tau \]