What is Manning's equation?

Manning's equation gives the discharge of steady uniform flow in an open channel: Q = (1/n) A R^(2/3) S^(1/2) in SI units, where A is flow area, R is hydraulic radius (area divided by wetted perimeter), S is bed slope and n is the roughness coefficient. In US customary units the constant 1 becomes 1.49 (more precisely 1.486), so Q = (1.49/n) A R^(2/3) S^(1/2) with Q in cubic feet per second.

How do you calculate normal depth?

Normal depth is the depth at which Manning's equation, at the given slope and roughness, carries exactly the design flow. Because area and hydraulic radius both depend on depth, the equation cannot be rearranged for depth except in special cases, so it is solved by iteration. The calculator brackets the answer and bisects until the computed flow matches the target to within 0.000001 m³/s.

What is the difference between normal depth and critical depth?

Normal depth depends on slope and roughness: it is the uniform-flow depth. Critical depth depends only on flow and section shape: it is the depth of minimum specific energy, where the Froude number equals 1. If normal depth is above critical depth the slope is mild and uniform flow is subcritical. If it is below, the slope is steep and uniform flow is supercritical.

Why does a pipe carry more water just before it runs full?

Near the crown the wetted perimeter grows faster than the flow area, so the hydraulic radius falls and Manning's equation gives less flow. A circular pipe reaches its largest open-channel capacity at about 94% of its diameter, about 8% more than the full-bore value. The calculator searches below that depth so it returns the stable normal depth, and it tells you when a flow exceeds the maximum.

Can I use Manning's equation for a pipe flowing full under pressure?

Manning's equation describes free-surface flow, so this calculator stops at the pipe crown. A pipe under pressure needs a pressure-flow treatment such as Darcy-Weisbach or Hazen-Williams with entrance and exit losses. For a culvert, where the barrel may run full under a headwater, use the culvert calculator, which applies the FHWA HDS-5 energy balance.

Manning's Equation Calculator: Open Channel and Pipe Flow

Section properties at normal depth
Flow regime	Subcritical
Critical depth, yc	0.7143m
Critical slope, Sc	0.00311
Flow area, A	2.574m²
Wetted perimeter, P	4.896m
Hydraulic radius, R	0.5258m
Top width, T	4.41m

Energy and shear
Specific energy, E	0.9955m
Velocity head, V²/2g	0.1923m
Mean boundary shear, τ = γRS	10.32Pa
Maximum bed shear, τ = γyS	15.76Pa

Using the calculator above

The calculator opens on the trapezoidal example worked below.

Section. Trapezoid, rectangle, triangle or pipe. Side slope z is horizontal distance per 1 vertical, so 1.5 means 1.5H:1V, applied to both banks.
Solve for. Normal depth (you give the flow) or flow (you give the depth).
Lining. Picking a lining fills its typical Manning’s n; the table further down this page lists most of the options. You can type any n.
Bed slope. A ratio, not a percentage: 0.002 is 0.2%, or 2 m of fall per kilometre.
Results. Normal depth or flow, velocity and Froude number first, then the section properties at that depth, critical depth and critical slope, specific energy, and two shear stresses: the mean over the wetted perimeter and the maximum on the bed. The rating curve plots depth against flow for the section, and the calculator warns when the Froude number falls between 0.85 and 1.15, where the water surface is unstable.

Manning’s equation in SI and US customary units

Manning’s equation relates the discharge of steady uniform flow to section geometry, roughness and slope:

Q = \frac{1}{n} A R^{2/3} S^{1/2} \quad \text{(SI)} \qquad\qquad Q = \frac{1.49}{n} A R^{2/3} S^{1/2} \quad \text{(US customary)}

Symbol	Meaning	SI	US customary
$Q$	Discharge	m³/s	ft³/s
$n$	Manning’s roughness coefficient	–	–
$A$	Flow area	m²	ft²
$P$	Wetted perimeter	m	ft
$R = A/P$	Hydraulic radius	m	ft
$S$	Bed slope, equal to the friction slope in uniform flow	m/m	ft/ft
$V = Q/A$	Mean velocity	m/s	ft/s

The 1.49 (more precisely 1.486) is $3.281^{1/3}$ , the cube root of the number of feet in a metre. It appears because $n$ keeps the same value in both systems. The calculator works in SI internally and converts US customary inputs on entry.

For a trapezoid with bottom width $b$ and side slope $z$ , $A = by + zy^2$ and $P = b + 2y\sqrt{1 + z^2}$ . For a pipe of diameter $D$ flowing at depth $y$ , the water surface subtends an angle $\theta = 2\arccos(1 - 2y/D)$ , and $A = \tfrac{D^2}{8}(\theta - \sin\theta)$ , $P = D\theta/2$ .

Normal depth, critical depth and the Froude number

Normal depth $y_n$ is the depth at which Manning’s equation carries the design flow. Area and hydraulic radius both change with depth, so the calculator finds $y_n$ by bisection. Critical depth $y_c$ depends only on flow and shape, and satisfies

\frac{Q^2 T}{g A^3} = 1 \qquad\qquad Fr = \frac{V}{\sqrt{g A / T}}

where $T$ is top width and $A/T$ is the hydraulic depth. If $y_n > y_c$ the slope is mild and uniform flow is subcritical ( $Fr < 1$ ). If $y_n < y_c$ it is steep and supercritical. The critical slope $S_c$ is the bed slope at which normal depth would equal critical depth.

Boundary shear follows from the same force balance: the mean value is $\tau = \gamma R S$ and the maximum on the bed is $\tau = \gamma y S$ , with $\gamma$ = 9810 N/m³.

Worked example: a concrete-lined trapezoidal channel

A float-finished concrete channel (n = 0.015) with a 2.0 m bed and 1.5H:1V side slopes runs at a slope of 0.002 and must carry 5.0 m³/s. These are the calculator’s opening values.

Result	Value
Normal depth $y_n$	0.803 m
Flow area $A$	2.574 m²
Wetted perimeter $P$	4.896 m
Hydraulic radius $R$	0.526 m
Top width $T$	4.41 m
Velocity $V$	1.94 m/s
Froude number	0.81, subcritical
Critical depth $y_c$	0.714 m
Critical slope $S_c$	0.00311
Mean / maximum bed shear	10.3 / 15.8 N/m²

Check: $(1/0.015) \times 2.574 \times 0.526^{2/3} \times 0.002^{1/2}$ = 66.67 × 2.574 × 0.651 × 0.0447 = 5.0 m³/s.

The same channel in US customary units is a 6.56 ft bed carrying 176.6 ft³/s. Normal depth is 2.64 ft, area 27.7 ft² and hydraulic radius 1.725 ft, and $(1.486/0.015) \times 27.7 \times 1.725^{2/3} \times 0.002^{1/2}$ returns the same 176.6 ft³/s.

The bed slope of 0.002 is below the critical slope of 0.00311, so the channel is mild and uniform flow is subcritical, though a Froude number of 0.81 sits close to the 0.85 warning band. Roughness matters: with n = 0.013 normal depth drops to 0.745 m, and with n = 0.017 it rises to 0.858 m.

A part-full pipe

A 600 mm concrete pipe (n = 0.013) at a slope of 0.002 carrying 0.20 m³/s runs at a normal depth of 0.380 m, which is 63% full, with a velocity of 1.06 m/s and a Froude number of 0.59. Critical depth is 0.289 m. The same pipe carries 0.275 m³/s flowing just full and 0.295 m³/s at its open-channel maximum near 94% depth.

Manning’s n table

Typical values with the range for each surface, taken from the calculator’s lining list. Use the low end for new, well-maintained surfaces and the high end for aged, weedy or debris-prone ones.

Surface	n min	n typical	n max
Concrete, trowel finish	0.011	0.013	0.015
Concrete, float finish	0.013	0.015	0.016
Concrete, unfinished	0.014	0.017	0.020
Gunite (shotcrete), good section	0.016	0.019	0.023
Asphalt, smooth	0.013	0.013	0.015
Brick in cement mortar	0.012	0.015	0.018
Cemented rubble masonry	0.017	0.025	0.030
Dry rubble masonry	0.023	0.032	0.035
Riprap, D50 = 150 mm	0.030	0.035	0.040
Riprap, D50 = 300 mm	0.035	0.040	0.050
Gabion	0.025	0.028	0.032
Earth, clean, recently completed	0.016	0.018	0.020
Earth, clean, after weathering	0.018	0.022	0.025
Earth, short grass, few weeds	0.022	0.027	0.033
Earth, dense weeds, deep channel	0.025	0.032	0.040
Rock cut, smooth and uniform	0.025	0.035	0.040
Rock cut, jagged and irregular	0.035	0.040	0.050
Natural stream, clean and straight	0.025	0.030	0.033
Natural stream, winding, pools and shoals	0.033	0.040	0.045
Sluggish reach, weedy, deep pools	0.050	0.070	0.080
Mountain stream, cobbles and boulders	0.040	0.050	0.070
Floodplain, short grass	0.025	0.030	0.035
Floodplain, scattered brush	0.035	0.050	0.070
Floodplain, dense brush	0.070	0.100	0.160
Floodplain, dense trees	0.080	0.120	0.200
PVC pipe	0.009	0.010	0.011
HDPE pipe, smooth interior	0.009	0.010	0.012
HDPE pipe, corrugated interior	0.018	0.022	0.025
Concrete pipe	0.011	0.013	0.017
Corrugated metal pipe (CMP)	0.021	0.024	0.030
Vitrified clay pipe	0.011	0.014	0.017
Steel pipe, welded, new	0.010	0.012	0.014

Flow is inversely proportional to n, so a 20% error in n is a 20% error in capacity. For natural channels, build n up from its components with the USGS procedure of Arcement and Schneider (1989) rather than picking one row.

Limitations

Uniform flow only. Near a weir, culvert, gate, change of slope or free overfall the water surface is not parallel to the bed; compute the profile with the gradually varied flow calculator.
One n for the whole wetted perimeter. A main channel with vegetated overbanks needs a composite n or separate sub-sections.
Prismatic sections. The calculator’s shapes are constant along the reach; surveyed natural sections need the Studio tool.
Free-surface flow only. Pipe results stop at the crown, and a pipe or culvert under pressure needs the culvert calculator or a pressure-flow method.
Shear stress is reported, not checked. Compare it with a lining’s permissible value in the HEC-15 channel lining calculator.
The design flow comes from elsewhere, for example the rational method calculator.

What a free Studio account adds

A free Delta Hydro Studio account includes the Channel Analysis tool, which runs the same engine with more of it exposed:

Save the calculation to a project and export a Word, Excel or PDF report.
Custom cross-sections from station–elevation pairs pasted from a survey, a HEC-RAS export or a GIS profile, and trapezoids and triangles with unequal side slopes.
Four more solve modes: the bottom width or the bed slope that carries a flow at a given depth, and the depth and flow that keep velocity or bed shear under a limit.
A recommended freeboard and configurable hydraulic curves of any two variables.
Send the result on: the velocity to riprap sizing, and the section and hydraulics into Channel Lining and GVF Profiles.

Manning's equation calculator