Learn: Manning discharge
Manning’s equation estimates the discharge carried by an open channel under uniform-flow conditions. Roughness slows flow, while hydraulic radius and energy slope increase its carrying ability.
What the formula is saying
Calculate R^(2/3) and √S, multiply them, and divide by Manning roughness n. Finally multiply the mean velocity by flow area A to obtain discharge.
Read the symbols in plain language
- n
- Manning roughness
Empirical resistance coefficient for the stated channel surface. In this SI form its unit is s/m^(1/3); it is not a percentage.
s/m^(1/3)Use s/m^(1/3) as the base unit shown here. Use the SI form: R in m, S as a dimensionless energy slope and n in s/m^(1/3). A is m² and discharge is m³/s. Hydraulic radius R = area/wetted perimeter is not generally water depth.
- A
- Flow area
Flow area. Multiply mean velocity by the wetted cross-sectional area.
m²Square metres measure area; square the length conversion factor.
- R
- Hydraulic radius
Flow cross-sectional area divided by wetted perimeter. Do not include the open free surface in that perimeter.
mMetres measure length; 1 m = 1000 mm.
- S
- Energy slope
Energy slope. Energy slope enters through its square root, using a decimal slope rather than whole percent.
ratio / no unitA dimensionless ratio has no physical unit; 0.01 as a ratio is 1% when the percent option is selected.
- Q
- Result to find
Manning discharge. Multiply mean velocity by the wetted cross-sectional area.
m³/s
Sort out the units first
Use the SI form: R in m, S as a dimensionless energy slope and n in s/m^(1/3). A is m² and discharge is m³/s. Hydraulic radius R = area/wetted perimeter is not generally water depth.
Assumptions before calculating
Assume a prismatic reach with steady uniform free-surface flow and an appropriate roughness value. Energy slope equals bed slope only for the uniform-flow approximation; use S ≥ 0 as a magnitude.
Let’s solve one together
Read the given values, follow each operation, then check what the result means.
Read the supplied values as one complete study case. Find Q and explain the result in the stated output unit.
- n · Manning roughness
- 0.013 s/m^(1/3)
- A · Flow area
- 2 m²
- R · Hydraulic radius
- 0.5 m
- S · Energy slope
- 0.001
Calculate the hydraulic-radius contribution
The two-thirds power represents the geometric contribution in the Manning resistance model.
(0.5)^(2 ÷ 3) ≈ 0.6299605249 m^(2/3)Take the square root of energy slope
Energy slope enters through its square root, using a decimal slope rather than whole percent.
√((0.001)) ≈ 0.0316227766Allow for channel roughness
Dividing by the SI roughness coefficient gives the mean velocity for this reach.
(0.6299605249) × (0.0316227766) ÷ (0.013) ≈ 1.532392381 m/sConvert velocity to discharge
Multiply mean velocity by the wetted cross-sectional area.
(2) × (1.532392381) ≈ 3.064784761 m³/s
Does this worked answer make sense?
Doubling n halves the result at unchanged geometry and slope. Multiplying S by four doubles it; multiplying S by two does not double it because the equation uses √S.
A second worked example — different values
A second case uses different data. Predict which way the answer will change, then calculate it without reusing the first answer.
- n · Manning roughness
- 0.02 s/m^(1/3)
- A · Flow area
- 3 m²
- R · Hydraulic radius
- 0.7 m
- S · Energy slope
- 0.002
Calculate the hydraulic-radius contribution
The two-thirds power represents the geometric contribution in the Manning resistance model.
(0.7)^(2 ÷ 3) ≈ 0.7883735163 m^(2/3)Take the square root of energy slope
Energy slope enters through its square root, using a decimal slope rather than whole percent.
√((0.002)) = 0.04472135955Allow for channel roughness
Dividing by the SI roughness coefficient gives the mean velocity for this reach.
(0.7883735163) × (0.04472135955) ÷ (0.02) ≈ 1.762856774 m/sConvert velocity to discharge
Multiply mean velocity by the wetted cross-sectional area.
(3) × (1.762856774) ≈ 5.288570322 m³/s
Now try your own values
Change a value or its unit. The same method will show your calculation, step by step.
Results update only when you calculate. The lesson example above stays unchanged.
Your turn — check your understanding
Solve this separate case yourself. Use only the values below; the two worked examples use different data. Give the requested result in the selected unit.
- n · Manning roughness
- 0.015 s/m^(1/3)
- A · Flow area
- 1.5 m²
- R · Hydraulic radius
- 0.4 m
- S · Energy slope
- 0.0015
Find: Learn: Manning discharge
A hint, not the answer
Calculate R^(2/3) and √S, multiply them, and divide by Manning roughness n. Finally multiply the mean velocity by flow area A to obtain discharge.
Use the SI form: R in m, S as a dimensionless energy slope and n in s/m^(1/3). A is m² and discharge is m³/s. Hydraulic radius R = area/wetted perimeter is not generally water depth.
Show the full practice solution
Compare the steps with your work; revealing a solution does not mark the lesson complete.
Calculate the hydraulic-radius contribution
The two-thirds power represents the geometric contribution in the Manning resistance model.
(0.4)^(2 ÷ 3) ≈ 0.5428835233 m^(2/3)Take the square root of energy slope
Energy slope enters through its square root, using a decimal slope rather than whole percent.
√((0.0015)) ≈ 0.03872983346Allow for channel roughness
Dividing by the SI roughness coefficient gives the mean velocity for this reach.
(0.5428835233) × (0.03872983346) ÷ (0.015) ≈ 1.40171923 m/sConvert velocity to discharge
Multiply mean velocity by the wetted cross-sectional area.
(1.5) × (1.40171923) ≈ 2.102578845 m³/s
Avoid the common trap
Do not use ordinary pipe radius instead of hydraulic radius, put slope percent directly into S, or treat Manning n as a dimensionless percentage. A slope of 0.1% is 0.001.
When this method applies — and when it does not
This is an empirical resistance equation, not a complete gradually varied or rapidly varied flow analysis. Roughness depends on the channel, vegetation and flow stage. Do not import the US customary coefficient into this SI equation.
For study and understanding, not approval of a real structure, site operation or design. Apply the correct standard, National Annex and professional review to actual engineering work.
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Sources & further reading
References open in a new tab and explain the underlying principles. The teaching text and examples here are SimpleFlick’s own; the source organisations have not endorsed this calculator.
Reading focus: Manning discharge. Water-measurement principles; for discharge devices, read the orifice/weir chapters and the installation and head-measurement conditions, not only the coefficient formula.
- U.S. Bureau of Reclamation — Water Measurement Manual, 3rd edition (1997; revised reprint 2001)
- Dawei Han, University of Bristol — Concise Hydraulics (2008, Ventus Publishing)
Lesson updated: · Both examples and the separate practice case are checked against an independent high-precision numerical implementation. This verifies arithmetic for the stated model, not engineering certification.