# OPEN CHANNEL BASICS

Open channel flow is flow characterized by a free water surface. We'll restrict our study to cases where flow is in a man-made channel, as opposed to natural channels such as rivers. In our analysis, we will assume that the section of channel under study is **prismatic**, ie, the channel is unvarying in cross-section and has constant bottom slope. Also, we will focus on **steady** (), **uniform** () flow. Except for flow of very thin films of water, open channel flow is **turbulent**; we will assume this is always the case. Finally, we will assume that streamlines are always parallel to the channel bottom so that the pressure distribution in the direction normal to the streamlines is **hydrostatic**. Taken together, these conditions mean that we are studying the simplest type of open channel flow. If you are interested in learning more about open channel flow, tune in to CE 4123.

Open channel flow presents a more complicated analysis than flow in pressure conduits because the water surface can rise and fall, and because resistance ceofficients are more complicated. The Darcy-Weisbach equation can be used to calculate resistance to flow in open channels, but the **Manning equation** is more popular. In this class, we will use the latter exclusively.

*A* = cross-sectional area
*P* = wetted perimeter

*R* = hydraulic radius = *A/P*

*T* = top width

*D* = hydraulic depth = *A/T*

*Z* = section factor = *AD*^{1/2}

*S*_{o} = slope of the channel bottom

For open channel flow, gravity is the primary driving force. Consequently, the appropriate dimensionless number to study open channels (and to scale model results) is the Froude number (rhymes with "food"), which expresses the ratio of inertial forces (kinetic energy) to gravity forces. It is given by

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where *V* is the average velocity in the channel and *L* is some characteristic length of the channel, typically taken as *D*. Among other uses, the Froude number can be used to classify open channel flow into one of three types:

A related quantity is the wave celerity, *c*, which is the speed at which a small disturbance (eg, a wave caused by throwing a small stone in the water) propagates. Wave celerity can be computed from the relation

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## The Energy Equation for Open Channel Flow

In our analysis of open channel flow, we will make a few additional assumptions to simplify the equations:

Considering these assumptions and the assumption of a hydrostatic pressure distribution, we can write the energy equation between any two points in a channel as

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where *z* is the elevation from a datum to the channel bottom, *y *is the channel depth, *V* is the average velocity in the channel, and *h*_{L} is the total head loss in the channel between points 1 and 2.

Specific energy, *E*, is the energy in the channel relative to the channel bottom, ie,

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A plot of *E *versus *y* yields much information about the behavior of open channel flow.
At critical depth, the following are true. Note! Critical depth is independent of the channel slope.

*E* is a minimum for a given discharge *Q*.
*Q* is a maximum for a given specific energy.
- The velocity head is equal to one-half the hydraulic depth.
- The Froude number equals 1.
- The flow velocity equals the wave celerity.

Critical slope is defined to be the channel slope that produces a flow in the channel equal to the critical depth. We'll use the symbol *S*_{c}. If *S*_{o} > *S*_{c }, then the flow is supercritical. If *S*_{o }< *S*_{c} , then the flow is subcritical.

Critical flow is unstable because small changes in roughness or bed slope produce relatively big changes in depth. The water surface for flow at critical depth appears wavy due to these instabilities. **It is poor engineering practice to design channels so that water flows at critical depth.** The only time we want to intentionally force critical flow is either for flow measurement or flow control of some type.

The depth of water under conditions of steady, uniform flow is called the normal depth, *y*_{n}. From the energy equation, it can be shown that normal depth is achieved when *S*_{o} = *S*_{f} where *S*_{f }is the slope of the energy grade line (the "f"riction slope). This relation tells us that for uniform flow, the driving force due to gravity (*S*_{o}) is just balance by the frictional resistance of the channel walls *S*_{f} . Manning used this information coupled with numerous empirical studies to come up with the famous uniform flow formula that bears his name

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where *n* is the Manning roughness coefficient, frequently called Manning's *n*, *V *is in ft/sec, *R *is in feet, and *S*, which is technically the friction slope, is dimensionless. In SI, we have

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where *V* is in m/sec and *R* is in meters. The formula is almost universally used, is based on a lot of data, and has shown good agreement in practice.
A key to successful use of the formula is the proper choice for *n*. I recommend Chow's book as a reference. In general, *n *depends on the following factors:

kolar@ou.edu
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