Manning equation calculator for open channel flow

Give a channel’s section, its roughness and its slope, then the flow or the depth. For a flow, Rischio’s engine finds the normal depth that carries it, as the studio does. For a depth, the page works out the flow with Manning’s equation and the engine adds the critical depth. It’s free and needs no account.

How it calculates

Manning’s equation gives the mean velocity of uniform flow in an open channel, in SI units:

$A$ is the flow area, $P$ the wetted perimeter, $R$ the hydraulic radius, $S$ the slope of the bed (in uniform flow the water surface runs parallel to it) and $n$ Manning’s roughness coefficient. $V$ comes out in m/s with $R$ in m.

For a trapezoid with a bottom width $b$ and side slopes of $z$ horizontal to 1 vertical, at a depth $y$:

A rectangle is the case $z = 0$ and a triangle the case $b = 0$. $T$ is the width of the water surface.

• For a flow, the engine finds the normal depth $y_n$, the depth at which the equation carries $Q$. A trapezoid has no closed form for it, so the engine solves it numerically (Brent’s method) to about $10^{-12}$ m.
• For a depth, the page computes $A$, $P$ and $R$, then $V$ and $Q$ directly. The engine then takes that $Q$ and returns the same depth with the rest of the results.

The Froude number on the hydraulic depth, $Fr = V/\sqrt{g A / T}$, gives the regime: subcritical below 1, supercritical above it. The critical depth $y_c$ is the one where $Q^2 T / (g A^3) = 1$. The engine warns when the flow is near critical ($0.8 < Fr < 1.2$), where the surface can be unstable, and when the flow is more than the channel carries full to its banks. The method, step by step, is in the docs: normal and critical flow and bank-full capacity.

How to use it

1. Choose what to solve for: the normal depth for a flow, or the flow at a depth.
2. Pick the section and give its dimensions: the bottom width, the side slope and the channel’s depth from the bed to the top of its banks.
3. Give Manning’s $n$ for the lining and the slope of the bed in m/m (0.001 is 0.1%).
4. Enter the flow in m³/s, or the depth in m.

The results show the depth or the flow with the area, wetted perimeter, hydraulic radius, top width, velocity, Froude number, specific energy and critical depth, and the section drawn with its water and the critical depth. Restore the example brings back a 3 m wide earth channel carrying 8 m³/s. To draw any section, give each part of it its own lining or get a calculation report with every step, use Rischio’s open channel tool.

Frequently asked questions

What is Manning’s equation used for?

To relate the flow, the depth and the velocity of uniform flow in open channels: canals, ditches, gutters, streams and pipes that run partly full, such as storm drains, sewers and culverts. It’s the usual way to size a channel or to find how deep a given flow will run.

What does R stand for in Manning’s equation?

The hydraulic radius: the flow area divided by the wetted perimeter, $R = A/P$. The wetted perimeter is the length of the channel’s boundary in contact with the water; the free surface doesn’t count.

What is Manning’s n?

An empirical coefficient for the roughness of the channel’s lining. Typical values are about 0.013 for trowelled concrete, 0.022 for a clean earth channel and 0.027 for earth with short grass. Rougher, weedier or more winding channels go higher. A wrong $n$ moves the result as much as any other input, so it’s worth choosing with care.

Can you use Manning’s equation for pipe flow?

For a pipe running partly full under gravity, such as a sewer or a culvert, yes: the flow has a free surface and behaves as open channel flow. For a pipe flowing full under pressure, the usual equations are Darcy-Weisbach or Hazen-Williams.

What is Manning’s equation in US units?

$V = \frac{1.49}{n} R^{2/3} S^{1/2}$, with $R$ in feet and $V$ in feet per second. The 1.49 converts the units, and $n$ keeps the same value. This calculator works in SI units: metres and cubic metres per second.