## Use of the Manning Equation for Open Channel Flow in Natural Channels

written by: Harlan Bengtson โข edited by: Lamar Stonecypher โข updated: 5/19/2010

The Manning equation is widely used for open channel flow calculations. It can be used for water flow rate and flow velocity calculations in man made open channels, and for river discharge and river flow velocity in terms of the slope, size & shape and roughness characteristics of natural channels.

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### Introduction

The Manning equation can be used for a variety of open channel flow calculations for natural channels, like rivers, streams and canals. Parameters like river discharge, water flow, and flow velocity can be related to slope and size & shape and roughness characteristics of natural channels. For flow in a natural channel, however, the bottom slope and channel size, shape & roughness are less clearly defined and more variable than for water flow in a man made open channel.

The meaning of uniform open channel flow along with details of the Manning equation and the parameters in the equation are presented in the first article of this series, Open Channel Flow Basics I: The Manning Equation and Uniform Flow, so they won't be repeated here.

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### Uniform Flow in a Natural Open Channel

There must be uniform open channel flow in a natural channel (river, stream, etc) in order to use the Manning equation. That is, the bottom slope, cross-section size, shape & roughness characteristics of natural channels must be   at least approximately constant. River channel characteristics are usually not all that uniform over an extended length, but a particular section (called a reach) of a river channel may have reasonably constant slope, size, etc. If that is the case, then the water flow depth and flow velocity will also be constant for that channel reach, and the Manning equation [ Q = (1.49/n)(A)(RH2/3)(S1/2) ] can be used for water flow calculations involving river discharge, Q; cross-sectional area of flow, A; hydraulic radius, RH; channel bottom slope, S; and Manning roughness coefficient, n. The relationship, V = Q/A can be used to calculate average flow velocity.

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### Manning Roughness Coefficient Values for Natural Channels

The Manning roughness coefficient, n, is an important parameter for open channel flow calculations with the Manning Equation. The value of n has an effect on variables like river discharge (water flow rate) and flow velocity. A reasonably accurate value of n can be obtained for most man made open channels, but obtaining good values of the Manning roughness coefficient for a reach of natural channel is a bit more of a challenge, because of the greater variability in the nature of the bottom and side surfaces of a natural river channel. Tables with values of n for natural channels are available in many textbooks & handbooks and on the internet. The table in this section is an example from a state agency. It is part of a table from the Indiana Department of Transportation Design Manual. The internet reference is given in the references section at the end of this article.

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### Example Calculation

Problem Statement: A stream on a plain has a reach that is described as clean and winding with some pools and some weeds. This reach has a reasonably constant slope of 0.0002. The stream cross-section of flow can be approximated as a trapezoid with bottom width equal to 5 feet and side slopes having horiz:vert = 3:1. Use the Manning equation with estimated maximum and minimum values of n for open channel flow in this stream to find the range of river discharge and water flow velocity to be expected for a 3 ft water depth.

Solution: As given above: bottom width, b = 5 ft; channel bottom slope, S = 0.0003; side slope, z = 3; and flow depth, y = 3 ft. From the table in the previous section, for a 'stream on plain, winding with some pools or shoals and some weeds or rocks', the maximum expected value of n is 0.050 and the minimum is 0.35. The second article in this series, 'Calculation of Hydraulic Radius for Open Channel Flow' gives an equation for hydraulic radius of a trapezoidal channel as follows:

RH = (by + zy2)/[ b + 2y(1 + z2)1/2 ].

Substituting known values gives: RH = [ (5)(3) + 3(3)2]/[5 + (2)(3)(1 + 32)1/2] = 1.75 ft.

Also, the trapezoid area is A = by + zy2 = (5)(3) + 3(32) = 42 ft2.

Values can now be substituted into the Manning equation [Q = (1.49/n)A(RH2/3)S1/2] to give:

For minimum n (0.035): Qmax = (1.49/0.035)(42)(1.752/3)(0.00031/2) = 45.0 cfs

For maximum n (0.050): Qmin = (1.49/0.050)(69)(2.882/3)(0.00031/2) = 31.5 cfs

Using V = Q/A: Vmax = 45.0/42 = 1.08 ft/sec; Vmin = 31.5/42 = 0.750 ft/sec

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### Reference and Image Credit

Indiana Department of Transportation Design Manual: http://www.in.gov/dot/div/contracts/standards/dm/english/Part4/ECh30/ch30.htm

#### Uniform Open Channel Flow and the Manning Equation

The Manning equation is widely used for uniform open channel flow calculations with natural or man made channels. The Manning equation is used to relate parameters like river discharge and water flow velocity to hydraulic radius, and open channel slope, size, shape, and Manning roughness.