Compute discharge and velocity in a rectangular or trapezoidal open channel using Manning's equation, V = (1/n)·R^(2/3)·S^(1/2). Enter the geometry, roughness, and bed slope; the calculator derives the flow area, wetted perimeter, and hydraulic radius for you, in SI units.
Manning's equation is the workhorse formula of open-channel hydraulics — used to size storm sewers, drainage ditches, culverts, irrigation canals, and natural streams flowing partly full under gravity. It relates the average flow velocity to the channel's roughness, shape, and slope, and when multiplied by the flow area it gives the discharge. This calculator handles rectangular and trapezoidal cross-sections and reports every intermediate quantity so you can check your work.
V = (1/n) · R^(2/3) · S^(1/2), and the discharge Q = V · A.
Here V is the mean velocity (m/s), n is Manning's roughness coefficient (dimensionless), R is the hydraulic radius (m), and S is the channel (energy) slope (m/m). In SI units the leading constant is exactly 1.0. In US customary units the same equation carries a conversion factor of 1.486 (so V_fps = (1.486/n)·R^(2/3)·S^(1/2) with R in feet) — the roughness n itself is the same number in both systems, which is why the unit factor is needed.
For a rectangular channel of bottom width b and flow depth y: area A = b·y and wetted perimeter P = b + 2y. For a trapezoidal channel with side slope z (horizontal:vertical): A = (b + z·y)·y and P = b + 2y·√(1 + z²). The hydraulic radius is R = A / P — the ratio of the flow area to the length of boundary in contact with the water. R, not depth, is the geometric variable that governs friction, which is why a wide shallow channel and a deep narrow one of the same R convey the same unit velocity.
The roughness coefficient n captures all the boundary resistance. Typical values: smooth finished concrete 0.012, ordinary concrete 0.013–0.015, brick 0.015, corrugated metal 0.022–0.027, earth (clean, straight) 0.022, earth with weeds/stones 0.030–0.035, natural streams (clean) 0.030, and major streams with vegetation and irregularity 0.050 or more. Because discharge is inversely proportional to n, a 20% error in n produces a 20% error in Q — so selecting n is often the largest source of uncertainty in an open-channel calculation.
Manning's equation describes uniform (normal) flow — the steady condition where the water-surface slope equals the bed slope and the depth (the normal depth, y_n) stays constant along a prismatic channel. To find normal depth for a target discharge you fix Q, n, S, and the geometry and solve for the y that makes the equation balance, usually by iteration since A and R both depend on y. Real channels also experience gradually-varied flow near transitions, weirs, and changes in slope, where backwater profiles must be computed separately.
Yes — Manning's n is treated as a dimensionless property of the boundary and uses the same numerical value in both systems. The unit difference is absorbed entirely by the leading constant: 1.0 in SI (metres, m/s) versus 1.486 in US customary (feet, ft/s). Forgetting that factor is a common error when converting calculations between systems.
The hydraulic radius R = A/P is the flow area divided by the wetted perimeter. It represents the effective flow depth from the standpoint of boundary friction. Using R lets a single equation handle any cross-sectional shape — narrow, wide, rectangular, or trapezoidal — because it normalises area against the amount of channel wall doing the retarding.
Lined channels in concrete or masonry are often rectangular because vertical walls are structurally easy and save right-of-way. Unlined earthen channels are almost always trapezoidal because soil cannot stand vertically — the side slope z is set by the soil's angle of repose (commonly 1.5:1 to 3:1). Trapezoidal sections also convey more flow for a given lining and resist erosion better.
Manning's equation is for open-channel (free-surface) flow, including circular pipes flowing partly full, which is the normal design case for gravity sewers. For a pipe flowing completely full under pressure you should switch to a pressure-flow method such as Darcy-Weisbach or Hazen-Williams, since there is no free surface and the driving slope is the hydraulic grade line, not the pipe invert.
Check the slope units (S must be m/m, not percent — a 1% grade is 0.01, not 1) and the roughness (using n = 0.13 instead of 0.013 will drop velocity ten-fold, and the reverse inflates it). Extremely steep slopes also push flow into the supercritical regime where Manning's uniform-flow assumption may not hold and a hydraulic jump can form downstream.