Description of Open-Channel Flow http://edugen.wiley.com/edugen/courses/crs2436/crowe9771/crowe9771...
15.1 Description of Open-Channel Flow
Flow in an open channel is described as uniform or nonuniform, as distinguished in Fig. 15.1. As defined in
Chapter 4, uniform flow means that the velocity is constant along a streamline, which in open-channel flow
means that depth and cross section are constant along the length of a channel. The depth for uniform-flow
conditions is called normal depth and is designated by yn. For nonuniform flow, the velocity changes from
section to section along the channel, thus one observes changes in depth. The velocity change may be due to a
change in channel configuration, such as a bend, change in cross-sectional shape, or change in channel slope.
For example, Fig. 15.1 shows steady flow over a spillway of constant width, where the water must flow
progressively faster as it goes over the brink of the spillway (from A to B), caused by the suddenly steeper slope.
The faster velocity requires a smaller depth, in accordance with conservation of mass (continuity). From reach B
to C, the flow is uniform because the velocity, and thus depth, are constant. After reach C the abrupt flattening
of channel slope requires the velocity to suddenly, and turbulently, slow down. Thus there is a deeper depth
downstream of C than in reach B to C.
Figure 15.1 Distingishing uniform and nonuniform flow. This example shows steady flow over a
spillway, such as the emergency overflow channel of a dam.
The most complicated open-channel flow is unsteady nonuniform flow. An example of this is a breaking wave
on a sloping beach. Theory and analysis of unsteady nonuniform flow are reserved for more advanced courses.
Dimensional Analysis in Open-Channel Flow
Open-channel flow results from gravity moving water from higher to lower elevations, and is impeded by
friction forces caused by the roughness of the channel. Thus the functional equation Q = f(µ, Á, g, V, L) and
dimensional anaysis as presented in Chapter 8 lead to two important independent p-groups to characterize
open-channel flow: the Froude number and the Reynolds number. The Froude number is the ratio of inertial
force to gravity force:
(15.1)
(15.2)
The Froude number is important if the gravitational force influences the direction of flow, such as in flow over a
spillway, or the formation of surface waves. However, it is unimportant when gravity causes only a hydrostatic
pressure distribution, such as in a closed conduit.
The use of Reynolds number for determining whether the flow in open channels will be laminar or turbulent
1 of 4 1/15/2009 1:30 AM
Description of Open-Channel Flow http://edugen.wiley.com/edugen/courses/crs2436/crowe9771/crowe9771...
depends upon the hydraulic radius, given by
(15.3)
where A is the cross-sectional area of flow and P is the wetted perimeter. The characteristic length Rh is
analagous to diameter D in pipe flow. Recall that for pipe flow (Chapter 10), if the Reynolds number
(VDÁ/µ = VD/½) is less than 2000, the flow will be laminar, and if it is greater than about 3000, one can expect
the flow to be turbulent. The Reynolds number criterion for open-channel flow would be 2000 if one replaced D
in the Reynolds number by 4Rh, where Rh is the hydraulic radius. For this definition of Reyholds number,
laminar flow would occur in open channels V(4Rh)/½ < 2000.
However, the standard convention in open-channel flow analysis is to define the Reynolds number as
(15.4)
Therefore, in open channels, if the Reynolds number is less than 500, the flow is laminar, and if Re is greater
than about 750, one can expect to have turbulent flow. A brief analysis of this turbulent criterion (see Example
15.1) will show that water flow in channels will usually be turbulent unless the velocity and/or the depth is very
small.
It should be noted that for rectangular channels (see Fig. 15.2), the hydraulic radius is
(15.5)
Example 15.1 shows that for very wide, shallow channels the hydraulic radius approaches the depth y.
Figure 15.2 Open-channel relations.
Most open-channel flow problems involve turbulent flow. If one calculates the conditions needed to maintain
laminar flow, as in Example 15.1, one sees that laminar flow is uncommon.
2 of 4 1/15/2009 1:30 AM
Description of Open-Channel Flow http://edugen.wiley.com/edugen/courses/crs2436/crowe9771/crowe9771...
EXAMPLE 15.1 CO DITIO S FOR LAMI AR
OPE -CHA EL FLOW
Water (60°F) flows in a 10 ft wide rectangular channel at a depth of 6 ft. What is the Reynolds
number if the mean velocity is 0.1 ft/s? With this velocity, at what maximum depth can one be
assured of having laminar flow?
Problem Definition
Situation: Constant velocity in rectangular channel, so uniform flow.
Find:
1. Reynolds number for given mean velocity.
2. 2 Maximum depth for which flow is laminar.
Properties: ½ = 1.22 × 10-5 ft2, from Table A.5.
Plan
1. Calculate Reynolds number using Eq. (15.4)
2. Find the depth for which Re = 500 using Eq. (15.5).
Solution
1. Reynolds number
where
since Re > 500, flow is turbulent.
2. Depth for which Re = 500.
For a rectangular channel,
Review
3 of 4 1/15/2009 1:30 AM
Description of Open-Channel Flow http://edugen.wiley.com/edugen/courses/crs2436/crowe9771/crowe9771...
1. Note: Velocity or depth must be very small to yield laminar flow of water in an open channel.
2. Note: Depth and hydraulic radius are virtually the same when depth is very small relative to
width.
Copyright © 2009 John Wiley & Sons, Inc. All rights reserved.
4 of 4 1/15/2009 1:30 AM