Mannings Equation Calculator

Manning equation describes uniform, steady flow, the condition in which depth and velocity stay constant along the reach so the energy slope equals the bed slope. In US customary form V = (1.486/n) R^(2/3) S^(1/2), velocity comes out in ft/s when the hydraulic radius R = A/P is in feet and the slope S is a dimensionless rise over run. The constant 1.486 is 3.28084^(1/3), the factor that lets the same tabulated n values serve foot units, since n carries hidden dimensions of seconds per foot to the one third. Discharge follows as Q = V x A. Representative n values are 0.010 for PVC, 0.013 for finished concrete, 0.025 for gravel, 0.030 for a clean natural channel and 0.050 for heavy brush. The exponents explain where the uncertainty lives. Velocity varies with the square root of slope, so doubling the grade buys only a 41 percent increase, whereas velocity is inversely proportional to n, so a roughness estimate that is off by a factor of two is off by a factor of two in the answer. The companion result, boundary shear stress tau = gamma R S with gamma taken as 62.4 lb/ft3, is what governs lining choice: permissible velocities run 2 to 3 ft/s for fine sand, 5 to 6 ft/s for stiff clay and 4 to 8 ft/s for grass linings. Caveats: n varies with stage, the formula assumes fully rough turbulent flow, and it must not be applied through hydraulic jumps or backwater reaches. The Froude number here substitutes hydraulic radius for hydraulic depth, valid only for wide sections.

Open channel resistance was first formalized by Antoine Chezy in 1769, while investigating a canal on the Yvette intended to supply Paris with water. He proposed V = C sqrt(R S), which correctly isolated hydraulic radius and slope but left the coefficient C to be determined. The next century was spent trying to predict C. Philippe Gauckler in 1867 and Gotthilf Hagen in 1881 had separately noted that a simple power of the hydraulic radius fit the data well. Robert Manning, an Irish engineer who became chief engineer of the Office of Public Works in Ireland, presented his paper to the Institution of Civil Engineers of Ireland in 1889, with publication following in 1891. He sought the simplest expression that still matched observation, and the two-thirds power of hydraulic radius emerged from that fitting. Manning was uneasy with the result precisely because it is not dimensionally homogeneous, and the form now bearing his name was popularized by later writers. In the United States the version with the 1.486 conversion factor spread through Horton and King handbook tables, and Ven Te Chow assembled the definitive roughness catalogue in Open-Channel Hydraulics in 1959. The USGS extended it in 1967 with Water Supply Paper 1849, which paired photographs of gauged natural reaches with their back-calculated n values so engineers could estimate roughness by visual comparison.