 Kinetic energy correction factor

For unidirectional flow through area $A$ with mean speed $\overline{u}$, the outgoing kinetic-energy flux is $\frac{1}{2}\rho A\alpha {\overline{u}}^3$, where $\alpha =(A{\overline{u}}^3{)}^{-1}{\int }_A{u}^{3}dA$. Cubing the mean speed generally differs from averaging the cube. A uniform profile has $\alpha =1$; circular laminar Hagen-Poiseuille flow has $\alpha =2$.