How this instrument works
Hydraulic radius is the ratio of a flowing channel's cross-sectional area to its wetted perimeter: R_h = A ⁄ P. It condenses an entire cross-section — however lopsided, trapezoidal, or part-full a pipe happens to be — into one number that says how much wetted boundary each square metre of flowing water has to drag against. A wide, shallow channel spreads the same area over more wetted boundary and returns a small R_h; a deep, narrow one concentrates it and returns a larger one, even while carrying identical flow.
The ratio falls directly out of balancing forces in steady, uniform open-channel flow. Gravity drives the water along the slope, and that driving force per unit channel length scales with cross-sectional area A; wall friction resists it, and the resisting shear force per unit length scales with wetted perimeter P. Setting the two in equilibrium — as Antoine Chézy did around 1770 and Robert Manning refined in 1889 — leaves A ⁄ P as the one geometric factor multiplying velocity in both the Chézy and Manning equations. That is why R_h, not A or P alone, is the shape term open-channel hydraulics keeps reaching for.
The name is a little misleading: R_h is not a geometric radius in the everyday sense, and the mismatch shows up cleanly for a pipe flowing completely full. Area is πr², and since there's no free surface, wetted perimeter is the whole circumference, 2πr, so R_h = πr² ⁄ 2πr = r ⁄ 2 = D ⁄ 4 — a quarter of the diameter, not half. Hydraulic diameter, D_h = 4R_h, was defined specifically to fix that mismatch and hand Reynolds-number formulas a figure that behaves like an actual diameter; feed R_h into a Reynolds number in place of D_h and the result comes out four times too small.
- Enter the Cross-sectional flow area — the area actually occupied by moving water, not the full pipe or channel section if it is running partly empty.
- Enter the Wetted perimeter — measure only the boundary touching water: the bed and the banks or walls, never the open free surface at the top.
- Read off Hydraulic radius in metres, or switch its unit menu to centimetres for a small drain or pipe.
- Carry that Hydraulic radius into Manning's equation, V = (1⁄n) R_h^(2/3) S^(1/2), to turn it into a flow velocity for a given slope and lining.
Worked example — a 2 m² channel with 4 m wetted perimeter
Picture a concrete-lined irrigation channel with a trapezoidal cross-section. At the design flow, the water occupies a Cross-sectional flow area of 2 m², and the Wetted perimeter — bed plus both sloped banks, measured along the wet concrete — comes to 4 m. Hydraulic radius is simply the ratio: R_h = A ⁄ P = 2 ⁄ 4 = 0.5 m, exactly, with nothing left to round.
Feed that 0.5 m hydraulic radius into Manning's equation for a concrete lining (n = 0.013) laid on a 1-in-1000 slope (S = 0.001): V = (1 ⁄ 0.013) × 0.5^(2⁄3) × 0.001^(1⁄2) ≈ 76.92 × 0.630 × 0.0316 ≈ 1.53 m/s. Multiplied by the 2 m² flow area, that channel carries roughly 3.06 m³/s — about 3065 litres of water every second.
Questions
Is hydraulic radius the same as a pipe's actual radius?
No. For a pipe flowing completely full, area is πr² and wetted perimeter is the full circumference 2πr, so R_h = πr² ⁄ 2πr = r ⁄ 2 = D ⁄ 4 — a quarter of the diameter, not half. The name comes from open-channel hydraulics, where R_h behaves as a shape factor rather than a literal distance from centre to wall.
What's the difference between hydraulic radius and hydraulic diameter?
Hydraulic diameter is D_h = 4A ⁄ P = 4R_h, defined so it collapses to the true diameter for a full circular pipe. Reynolds-number and friction-factor formulas are written expecting D_h, not R_h; substitute hydraulic radius where hydraulic diameter belongs and every result comes out four times too small.
Why does wetted perimeter leave out the free water surface?
Because the open top is not a solid boundary, so it contributes essentially no friction against the flow — only the bed and the banks or pipe wall do. Wetted perimeter counts just those solid, water-contacting edges of the cross-section; the air-water interface across the top is excluded even though it closes the shape geometrically.
What is hydraulic radius actually used for?
It is the geometry term in the Chézy and Manning equations for open-channel flow, letting engineers turn a channel's slope, roughness and cross-section into a predicted velocity and discharge. Anyone sizing an irrigation canal, drainage ditch, culvert, or a sewer meant to run partly full needs it to check whether a proposed shape carries the design flow.
Does a bigger hydraulic radius always mean faster flow?
For a given slope and surface roughness, yes — Manning's equation scales velocity with R_h raised to the two-thirds power, so more flow area per metre of wetted boundary means less friction dragging on each litre of water. That is why engineers favour wide, shallow-ish or semicircular channel shapes over narrow, deep ones when they want to maximise R_h for a fixed cross-sectional area.
Does hydraulic radius stay constant as depth changes in a part-full pipe?
No, and that is exactly when it matters most. As a culvert or sewer fills from empty toward full, both the flow area and the wetted perimeter grow, but not at the same rate, so R_h changes continuously with depth rather than sitting fixed the way a pipe's geometric radius does. Partial-flow design charts exist precisely to track that relationship.