Pressurized pipe flow uses Darcy-Weisbach or Hazen-Williams, but the moment water has a free surface — a storm sewer flowing half full, a drainage swale, a concrete-lined channel, a culvert — gravity drives the flow and roughness fights it. Manning's equation is the workhorse civil engineers reach for in this regime, and it's been the standard since 1889 because it's empirical, simple, and accurate enough.

The equation (SI units):

V = (1/n) · R^2/3 · S^1/2

Where V is mean velocity (m/s), n is Manning's roughness coefficient (dimensionless), R is the hydraulic radius (flow area ÷ wetted perimeter, in meters), and S is the slope of the energy grade line (m/m, often approximated as channel slope). Flow rate Q = V · A.

The key insight: velocity depends on the shape of the wetted cross-section, not just its area. Hydraulic radius R rewards "deep and narrow" over "wide and shallow" because less perimeter means less friction per unit of flow area. A 1 m × 1 m square channel running full has R = 1/4 = 0.25 m. The same area as a 2 m × 0.5 m channel has R = 1/6 ≈ 0.17 m — 30% lower, so it carries roughly 22% less flow at the same slope.

Typical Manning's n values worth memorizing:

• Smooth concrete pipe: 0.012
• Corrugated metal pipe: 0.024 (literally double — corrugations matter)
• PVC/HDPE: 0.010
• Earth channel, clean: 0.022
• Natural stream, weedy: 0.05–0.10

Worked example: A 600 mm concrete storm pipe (n = 0.012) at 1% slope, flowing full. R for a full circular pipe = D/4 = 0.15 m. V = (1/0.012) · (0.15)^2/3 · (0.01)^1/2 = 83.3 · 0.282 · 0.1 = 2.35 m/s. Area = π(0.3)² = 0.283 m². Q = 0.67 m³/s, or about 600 L/s.

Two practical gotchas:

• Circular pipes carry maximum flow at ~94% full, not 100%. Above that, wetted perimeter grows faster than area, so a slightly-less-than-full pipe actually moves more water than a completely full one.
• Minimum velocity matters as much as maximum. Sanitary and storm sewers are typically designed for ≥0.6 m/s (2 ft/s) at design flow to keep solids in suspension. Below that, sediment drops out and you get a maintenance nightmare.

Engineers don't usually solve Manning's by hand for non-circular partial flow — it's an iterative trig problem. Spreadsheets, nomographs, or tools like HydroCAD and StormCAD do the lookup. But understanding what drives the answer (R and n, both raised to fractional powers) tells you immediately why lining a ditch with concrete can shrink it by half.