Hydraulic Radius

Every friction equation in open-channel flow, Manning and Chezy included, takes hydraulic radius as an input and none of them define it, which is a strange gap in most references. It is simply the flow area divided by the wetted perimeter, and the physical reading is a ratio of what carries the water to what rubs against it. A 3 metre channel flowing 1.2 metres deep has 3.6 m² of area against 5.4 metres of wetted boundary, so $R = 0.667$ m. Note the top of the flow is not part of the perimeter, because air exerts effectively no drag.

The one result worth memorising is that a circular pipe flowing full has $R = D/4$ exactly, and the algebra is one line: $\pi D^2/4$ divided by $\pi D$. A 600 mm pipe therefore has a hydraulic radius of 150 mm whether it is carrying water or sewage. The same pipe flowing exactly half full has the same hydraulic radius, since area and perimeter both halve, which is why a half-full pipe and a full pipe carry water at the same velocity under Manning.

That last fact leads somewhere genuinely counter-intuitive. Hydraulic radius in a circular pipe peaks at about 0.81 of the diameter, not at the crown, so a sewer flowing roughly 93 percent full discharges more than the same sewer flowing brim full. The maximum velocity and the maximum discharge occur at partial depths. Designers still cap peak flow at 75 to 80 percent depth, but for ventilation and sulfide control rather than hydraulics. The other regular error is in wide channels: once the width exceeds about ten times the depth, $R$ approaches the depth itself, and using the full geometry rather than that approximation buys you nothing but arithmetic.