# Open-Channel Flow

Looking upstream. The velocity distribution in fully developed turbulent open channel flows is given approximately by Prandtl's power law:. Chen developed a complete analysis of the velocity distribution in open channel and pipe flow with reference to flow resistance. In uniform equilibrium flows, the velocity distribution exponent is related to the flow resistance:. For a wide rectangular channel, the relationship between the mean flow velocity V and the free-surface velocity V max derives from the continuity equation:.

Further discussion is developed in Chapter In supercritical open channel flows , free-surface aeration is often observed. At uniform equilibrium, the air concentration distribution is a constant with respect to the distance x in the flow direction. The continuity equation for air in the air—water flow yields:. A first integration of the continuity equation for air in the equilibrium flow region leads to:. Assuming a homogeneous turbulence across the flow i. Advanced void fraction distribution models may be developed assuming a non-constant bubble diffusivity. Assuming that the diffusivity distribution satisfies:.

Equation 17B. In skimming flows and smooth-chute flows, the air concentration profiles have a S-shape that correspond to:. The theoretical models were compared with model and prototype experimental data on smooth and stepped chutes. In each case, the dimensionless turbulent diffusivity was deduced from the mean air content. Results are presented in Fig. Figure 17B.

Chanson a , a discussed the results in more length, including some analogies with sediment-laden open channel flows. Dimensionless air bubble diffusion coefficient in self-aerated open channel flows. Using a mixing length model, an estimate of the depth-averaged momentum exchange coefficient across the air—water flow is:. The ratio of the air bubble diffusion coefficient over the momentum transfer coefficient v T becomes:. In Fig.

Such a result suggests that scale-model studies of self-aerated flows might not describe accurately the air bubble diffusion process in self-aerated open channel flows. Prototype data include the re-analysed data of Aivazyan and Cain The above analysis is an extension of the work of Chanson a , a. Note however that there was a typographic error in this development and that the equation 17B.

## Flow in Open Channels

The simplest open channel flow configuration is one in which the bottom channel slope and the channel or river cross-section are constant along the length of the channel. Then, if the channel is long, the flow will quickly reach a uniform state where all the streamlines are parallel to the channel bottom. Since the flow is parallel, the pressure will automatically be hydrostatic and the discharge and momentum will be constant along the length of the channel. In other words, the volume, mass and momentum of water flowing into the Eulerian control volume, shown in Fig.

Figure 5. Schematic of uniform flow. The weight of the fluid above the height x 3 is given by:.

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In wide rectangular channel, the shear stress in the water column thus varies linearly with distance from the surface. We may also apply 5. If we further assume that the boundary stress is uniform over the whole wetted perimeter then:. The quantity;. In order to find the velocity of the water in such a channel, we may proceed by analogy with pipe flow where it was assumed, from dimensional reasoning,.

The hydraulic radius for a circular pipe is given by:. For the simple case of a very wide channel of depth h and discharge per unit width q , this reduced to:.

This is called the normal flow depth, the depth for the discharge q where downslope gravitational forces just balance the retarding bottom drag forces. The similarity between 5. We thus have a solid boundary at the bottom and the top and the channel is now a conduit with twice the area and twice the perimeter, so the hydraulic radius remains the same.

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There is, however, one important difference between an open channel flow and a pipe flow. In an open channel flow, the free surface allows surface waves to move along the water surface. These waves have an associated velocity field and so may induce velocity fluctuations at the channel bottom, which in turn may influence the bottom stress. The ratio of the water velocity to the wave speed is called the surface Froude Number:.

However, little information exists on the magnitude of the influence of Fr on the friction factor, but it is clear that whenever the surface undulations become large they will effect the eddy structure, and so the friction factor, in an open channel. In practical applications, the influence of Fr is, however, always neglected. The above use of the friction factor is the most rational approach for channel flow; it is dimensionally consistent and further has the advantage of unifying pipe and channel flow.

However, traditionally hydraulic engineers have used two empirical formulas and these are still used very widely so it is necessary to discuss them here. Comparison of 5. This clearly shows that the Chezy formulation is similar to the friction factor method, except that C is not dimensionless and so takes on different values for different measurement systems. Many authors see for example Chow, give values of n for both artificial channels and natural streams.

Once again we may equate 5. In unsteady open channel flows , the velocities and water depths change with time and longitudinal position. For one-dimensional applications, the continuity and momentum equations yield the Saint-Venant equations Chapter The application of the Saint-Venant equations is limited by some basic assumptions :.

With these hypotheses, the unsteady flow can be characterized at any point and any time by two variables: e. V and Y where V is the flow velocity and Y is the free-surface elevation. The unsteady flow properties are described by a system of two partial differential equations:.

Only 5 left in stock more on the way. Very good. Seattle, Washington. I was fortunate enough to find a hardcover copy of this book just a few years ago. I used it as text book in a graduate open-channel flow course and would not be without it as a practicing engineer.

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It covers all of the basics and I have yet to find a better treatment on the fundamentals of the method of characteristics. The illustrations are quite good and the text is clear and very well written. I actually came to Amazon in hopes of a more current edition for my library and ended up submitting this review. If you ever see one - buy it. Only 1 left in stock more on the way. It is an excellent introduction book for river and channel modeling. Jain Only 1 left in stock - order soon.

The first chapter hits you with complex differential equations, don't be intimidated! The author believes that it is important to derive the equations so that you will understand the conditions for which they are applied. The book is useful but hopefully, if you are using it for a class, your teacher will not use the suggested homework in it's text. Available for download now. See All Buying Options. Open Channel Flow. I liked this work as it covered the more esoteric aspects not normally covered in the standard text books.

Personally I found it excellent for what I was after. I think a prior understanding of the subject helped. Hozalski, O. Mohseni, J. Nieber, B. Wilson, P. Open channel flow transports water by gravity with a free surface exposed to the atmosphere.

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## Open Channel Flow | Flow Measurement | Ultrasonic Flow Meter UK

Any of the principal methods of discharge measurement outlined below can be used to measure open channel flow. Some methods are more accurate than others while some methods measure a large range of discharge.

Stormwater is variable and thus the method for measuring stormwater discharge must be able to measure small values of discharge accurately while also having the capacity to measure large values of discharge. For reference, the depth-discharge also called the stage-discharge relationship for six discharge measurement techniques is shown in figure 4.

## Open-channel flow

Selection of a discharge measurement method is dependent on many factors, including accuracy, cost, range of discharge, and site conditions. Bureau of Reclamation All the discharge measurement principles listed above require a measurement of water depth and a known channel or pipe, etc.

In the case of a weir, the water depth is measured behind the weir and weir equations discussed in detail below convert depth to an estimated discharge over the weir. In the case of discharge measurement probes, a water depth is needed to determine the wetted perimeter. The principal methods of depth measurement use pressure under hydrostatic conditions and density. Bubbler probes and pressure transducers, when located under the water surface, measure the pressure of water i. Ultrasonic and Doppler probes, typically positioned above the water surface, locate the water surface using the change in density from air to water because the water surface reflects the acoustic signal back to the probe.

The accuracy of any depth measurement should be verified prior to installation of equipment and re-verified each time the site is visited to ensure that the equipment is calibrated correctly and in good working condition.