Manning Equation for Open Channel Flow

In SI units the Manning equation is V = (1/n) · R^(2/3) · S^(1/2), and the discharge is Q = V · A. V is the mean velocity in m/s, n is Manning's roughness coefficient, R is the hydraulic radius in metres, S is the bed slope as a ratio (0.001 for 0.1 %) and A is the flow area in m².

The hydraulic radius is the flow area divided by the wetted perimeter, R = A / P. It is the term that rewards a compact section: for the same area, the less lining the water touches, the faster it runs.

In US customary units the same formula is written V = (1.486/n) · R^(2/3) · S^(1/2), with feet and ft³/s. The 1.486 is not a different formula. It is the cube root of 3.2808, the number of feet in a metre, and it is there so the same n works in both systems.

Take a concrete-lined trapezoidal channel with a 2 m bottom width, side slopes of 1.5 horizontal to 1 vertical, a bed slope of 0.1 % and water running 1 m deep. With n = 0.014 the flow area is 3.5 m², the wetted perimeter 5.61 m and the hydraulic radius 0.624 m.

The Manning equation gives a velocity of 1.65 m/s and a discharge of 5.78 m³/s. These are the numbers the calculator above opens with, so you can change one input at a time and watch what moves.

Change only the lining to shotcrete (n = 0.017) and the same section at the same depth carries 4.76 m³/s, about 18 % less. Discharge is inversely proportional to n: nothing else in the equation is chosen with so little measurement behind it, and nothing else moves the answer so directly.

Manning's n is not computed. It is chosen from the surface the water runs against, and published tables give a range for each surface rather than a single value. The figures below are typical design values of the kind found in those tables; they are the same figures STREAM's channel module attaches to each lining.

Three things push n above the table value in a real channel: vegetation that was not there when the channel was built, irregular banks and bed, and shallow flow over large stone. A channel sized on the low end of the range has no margin for any of them.

The honest way to handle the uncertainty is to run the design twice, once with a low n to check the velocity against what the lining can take, and once with a high n to check that the section still has the capacity and the freeboard.

Design usually starts from the other end. The discharge is known, from a design storm or a pumping rate, and the question is how deep the water will run. That depth is the normal depth: the depth of uniform flow for a given discharge, section, slope and lining.

For a rectangular or trapezoidal section the Manning equation cannot be rearranged to give depth directly, because depth sits inside both the area and the hydraulic radius. It is solved by iteration. Discharge rises steadily with depth, so a simple bisection always converges on the one depth that carries the target flow.

On the example channel, 5 m³/s runs 0.93 m deep in concrete. Line the same channel with 150 mm riprap (n = 0.035) and the same 5 m³/s needs 1.47 m of water at about half the velocity, 0.81 m/s instead of 1.59 m/s. The lining decides the depth of the excavation as much as the discharge does.

The Froude number compares the velocity with the speed of a small wave on the surface: Fr = V / √(g · A / T), where T is the top width. Below 1 the flow is subcritical, deep and slow; above 1 it is supercritical, shallow and fast, and it erodes.

Critical depth is the depth at which Fr = 1 for a given discharge. It depends only on the discharge and the shape of the section, not on slope or roughness. In the example, 5.78 m³/s has a critical depth of 0.775 m against a normal depth of 1 m, so the flow is subcritical with Fr = 0.63.

Critical slope is the bed slope at which normal depth equals critical depth. For the example it is 0.27 %. Lay the same channel at 0.5 % and 1 m of water moves at 3.69 m/s with Fr = 1.41: supercritical, with a hydraulic jump wherever the flow is forced back to subcritical.

Between roughly Fr 0.9 and 1.1 the water surface is unstable and small disturbances grow into standing waves. A design is kept clear of that band on one side or the other.

The Manning equation describes uniform flow: one section, one slope and one lining, long enough for the water surface to run parallel to the bed. A culvert, a weir, a change of grade or a tight bend breaks that assumption locally, and the depth there is not the normal depth.

It does not check the lining. A velocity the section can carry may still be more than grass or a small riprap can resist, and a velocity that is too low lets sediment settle. Both are separate checks against the lining's limits.

It does not include freeboard. The computed depth is the water surface; the channel is built deeper than that.

And it knows nothing about the ground. A single calculation uses a single slope, while a real channel follows terrain that changes at every station.

STREAM's channel module applies the same equation along a whole alignment. You draw the channel on the surveyed terrain and set the bed width, the depth, the left and right side slopes and the lining; the roughness follows from the lining. The bed elevations come from the terrain, so every station has its own slope.

At each station it solves the normal depth and velocity with Manning and reports the critical depth, the Froude number and the critical slope alongside. The result is then checked against freeboard, flow regime, siltation and the velocity the lining permits, and each finding names the value, the limit and the remedy.

The same design gives the excavation, fill and lining quantities from the intersection with the ground, and a setting-out table. Everything runs on the machine; no survey is uploaded.