Manning's Equation
Manning's Equation is an empirical formula used to calculate the average velocity and discharge of water flowing in an open channel under uniform flow conditions. Developed by Irish engineer Robert Manning in 1889, it relates flow velocity to the channel's hydraulic radius, bed slope, and a roughness coefficient (Manning's n) that accounts for the resistance caused by the channel boundary. It is the most widely used formula in open-channel hydraulics for the design of drainage channels, rivers, culverts, and sewers.
Key Formula
V = (1/n) × R^(2/3) × S^(1/2)
LaTeX: V = \frac{1}{n} R^{2/3} S^{1/2}
| Symbol | Meaning | Unit |
|---|---|---|
| V | Mean flow velocity | m/s |
| n | Manning's roughness coefficient | dimensionless (s/m^(1/3)) |
| R | Hydraulic radius = A / P (flow area / wetted perimeter) | m |
| S | Longitudinal slope of energy grade line (or bed slope for uniform flow) | m/m (dimensionless) |
Worked Example
Problem
A trapezoidal irrigation channel has a base width b = 2 m, side slopes 1.5H:1V, flow depth y = 1.2 m, bed slope S = 0.0005, and Manning's n = 0.025 (earth channel). Find flow velocity V and discharge Q.
Solution
Flow area: A = (b + z·y)·y = (2 + 1.5 × 1.2) × 1.2 = (2 + 1.8) × 1.2 = 3.8 × 1.2 = 4.56 m². Wetted perimeter: P = b + 2·y·√(1 + z²) = 2 + 2 × 1.2 × √(1 + 2.25) = 2 + 2.4 × √3.25 = 2 + 2.4 × 1.803 = 2 + 4.327 = 6.327 m. Hydraulic radius: R = A/P = 4.56 / 6.327 = 0.721 m. Velocity: V = (1/n)·R^(2/3)·S^(1/2) = (1/0.025) × (0.721)^(2/3) × (0.0005)^(1/2) = 40 × 0.800 × 0.02236 = 0.715 m/s. Discharge: Q = V × A = 0.715 × 4.56 = 3.26 m³/s.
Answer
V ≈ 0.715 m/s; Q ≈ 3.26 m³/s
Typical Manning's Roughness Coefficients (n) for Various Channel Types
| Channel Type | Surface Condition | n (min) | n (max) |
|---|---|---|---|
| Concrete channel | Float finish | 0.012 | 0.014 |
| Brick-lined channel | Cement mortar | 0.014 | 0.017 |
| Earth channel | Clean, straight | 0.018 | 0.025 |
| Natural river | Clean, winding | 0.025 | 0.040 |
| Vegetated channel | Dense weeds | 0.050 | 0.120 |
| Circular pipe | Concrete | 0.011 | 0.015 |
Interactive Tools
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Related Terms
Hydraulic Jump
A hydraulic jump is a phenomenon in open-channel flow where rapidly flowing supercritical water (Froude number > 1) abruptly transitions to slower subcritical flow (Froude number < 1), accompanied by a significant rise in water depth and intense turbulence. The jump dissipates a large amount of kinetic energy as heat and sound, making it useful as an energy dissipator downstream of spillways, weirs, and sluice gates to protect channel beds from erosion. The sequent (conjugate) depths before and after the jump are related by the Belanger equation.
Road Design
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Engineering Surveying
Engineering surveying is the science of measuring and mapping the Earth's surface — its angles, distances, elevations, and positions — for the planning, design, and construction of civil engineering projects such as roads, railways, dams, buildings, and utilities. It encompasses horizontal control (establishing co-ordinates of points using triangulation or traversing), vertical control (establishing elevations by levelling), and setting out (transferring design positions from drawings to the ground). Modern engineering surveying uses total stations, GPS/GNSS receivers, digital levels, and drone-based photogrammetry to achieve centimetre or millimetre accuracy.
Named after Irish engineer Robert Manning (1816–1897), who presented his formula to the Institution of Civil Engineers of Ireland in 1889. Manning originally published a slightly different form; the formula was later popularised in its current form by Gauckler (1867) and others, leading to it also being called the Gauckler–Manning formula in European literature. 'Hydraulic' derives from Greek 'hydraulikos' (of a water organ), from 'hydor' (water) + 'aulos' (pipe).