Hydraulic radius ($R=A/\ell$)

The hydraulic radius is cross-sectional flow area divided by wetted perimeter. A perimeter-averaged boundary shear stress $\tau$ exerts drag per unit fluid volume $\tau \ell /A=\tau /R$. It differs from both actual depth and the radius of a noncircular channel.

A quadratic boundary-drag model relates shear stress to mean flow velocity, with dimensionless resistance coefficient $c_d>0$. Uniform steady flow down slope $S$ balances $\rho gAS=\tau \ell$, giving $|u|=\sqrt{gRS/c_d}$. Its sign is chosen so drag opposes the motion.