fluid_mechanics
10 Open-Channel Flow CHAPTER OPENING PHOTO: Hydraulic jump: Under certain conditions, when water flows in an open channel, even if it has constant geometry, the depth of the water may increase considerably over a short distance along the channel. This phenomenon is termed a hydraulic jump (water flow from left to right). Learning Objectives After completing this chapter, you should be able to: ■ discuss the general characteristics of open-channel flow. ■ use a specific energy diagram. ■ apply appropriate equations to analyze open-channel flow with uniform depth. ■ calculate key properties of a hydraulic jump. ■ determine flowrates based on open-channel flow-measuring devices. V10.1 Off-shore oil drilling platform. Open-channel flow involves the flow of a liquid in a channel or conduit that is not completely filled. A free surface exists between the flowing fluid (usually water) and fluid above it (usually the atmosphere). The main driving force for such flows is the fluid weight—gravity forces the fluid to flow downhill. Most open-channel flow results are based on correlations obtained from model and full-scale experiments. Additional information can be gained from various analytical and numerical efforts. Open-channel flows are essential to the world as we know it. The natural drainage of water through the numerous creek and river systems is a complex example of open-channel flow. Although the flow geometry for these systems is extremely complex, the resulting flow properties are of considerable economic, ecological, and recreational importance. Other examples of open-channel flows include the flow of rainwater in the gutters of our houses; the flow in canals, drainage ditches, sewers, and gutters along roads; the flow of small rivulets and sheets of water across fields or parking lots; and the flow in the chutes of water rides in amusement parks. Open-channel flow involves the existence of a free surface which can distort into various shapes. Thus, a brief introduction into the properties and characteristics of surface waves is included. The purpose of this chapter is to investigate the concepts of open-channel flow. Because of the amount and variety of material available, only a brief introduction to the topic can be presented. Further information can be obtained from the references indicated. 534
10.1 General Characteristics of Open-Channel Flow 10.1 General Characteristics of Open-Channel Flow 535 Open-channel flow can have a variety of characteristics. Uniform flow Rapidly varying flow (photograph courtesy of Stillwater Sciences). In our study of pipe flow 1Chapter 82, we found that there are many ways to classify a flow— developing, fully developed, laminar, turbulent, and so on. For open-channel flow, the existence of a free surface allows additional types of flow. The extra freedom that allows the fluid to select its free-surface location and configuration 1because it does not completely fill a pipe or conduit2 allows important phenomena in open-channel flow that cannot occur in pipe flow. Some of the classifications of the flows are described below. The manner in which the fluid depth, y, varies with time, t, and distance along the channel, x, is used to partially classify a flow. For example, the flow is unsteady or steady depending on whether the depth at a given location does or does not change with time. Some unsteady flows can be viewed as steady flows if the reference frame of the observer is changed. For example, a tidal bore 1difference it water level2 moving up a river is unsteady to an observer standing on the bank, but steady to an observer moving along the bank with the speed of the wave front of the bore. Other flows are unsteady regardless of the reference frame used. The complex, time-dependent, wind-generated waves on a lake are in this category. In this book we will consider only steady open-channel flows. An open-channel flow is classified as uniform flow 1UF2 if the depth of flow does not vary along the channel 1dydx 02. Conversely, it is nonuniform flow or varied flow if the depth varies with distance 1dydx 02. Nonuniform flows are further classified as rapidly varying flow 1RVF2 if the flow depth changes considerably over a relatively short distance; dydx 1. Gradually varying flows 1GVF2 are those in which the flow depth changes slowly with distance along the channel; dydx 1. Examples of these types of flow are illustrated in Fig. 10.1 and the photographs in the margin. The relative importance of the various types of forces involved 1pressure, weight, shear, inertia2 is different for the different types of flows. As for any flow geometry, open-channel flow may be laminar, transitional, or turbulent, depending on various conditions involved. Which type of flow occurs depends on the Reynolds number, Re rVR hm, where V is the average velocity of the fluid and R h is the hydraulic radius of the channel 1see Section 10.42. A general rule is that open-channel flow is laminar if Re 6 500, turbulent if Re 7 12,500, and transitional otherwise. The values of these dividing Reynolds numbers are only approximate—a precise knowledge of the channel geometry is necessary to obtain specific values. Since most open-channel flows involve water 1which has a fairly small viscosity2 and have relatively large characteristic lengths, it is rare to have laminar open-channel flows. For example, flow of 50 °F water 1n 1.41 10 5 ft 2 s2 with an average velocity of V 1 fts in a river with a hydraulic radius of R has Re VR hn 7.1 10 5 h 10 ft . The flow is turbulent. However, flow of a thin sheet of water down a driveway with an average velocity of V 0.25 fts such that R h 0.02 ft 1in such cases the hydraulic radius is approximately equal to the fluid depth; see Section 10.42 has Re 355. The flow is laminar. In some cases stratified flows are important. In such situations layers of two or more fluids of different densities flow in a channel. A layer of oil on water is one example of this type of flow. All of the open-channel flows considered in this book are homogeneous flows. That is, the fluid has uniform properties throughout. Open-channel flows involve a free surface that can deform from its undisturbed relatively flat configuration to form waves. Such waves move across the surface at speeds that depend on y UF uniform flow GVF gradually varying flow RVF rapidly varying flow RVF UF RVF UF RVF GVF RVF UF F I G U R E 10.1 Classification of open-channel flow.
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10.1 General Characteristics of Open-Channel Flow<br />
10.1 General Characteristics of Open-Channel Flow 535<br />
Open-channel flow<br />
can have a variety<br />
of characteristics.<br />
Uniform flow<br />
Rapidly varying flow<br />
(photograph courtesy<br />
of Stillwater Sciences).<br />
In our study of pipe flow 1Chapter 82, we found that there are many ways to classify a flow—<br />
developing, fully developed, laminar, turbulent, and so on. For open-channel flow, the existence of<br />
a free surface allows additional types of flow. The extra freedom that allows the <strong>fluid</strong> to select its<br />
free-surface location and configuration 1because it does not completely fill a pipe or conduit2 allows<br />
important phenomena in open-channel flow that cannot occur in pipe flow. Some of the<br />
classifications of the flows are described below.<br />
The manner in which the <strong>fluid</strong> depth, y, varies with time, t, and distance along the channel, x, is<br />
used to partially classify a flow. For example, the flow is unsteady or steady depending on whether the<br />
depth at a given location does or does not change with time. Some unsteady flows can be viewed as<br />
steady flows if the reference frame of the observer is changed. For example, a tidal bore 1difference it<br />
water level2 moving up a river is unsteady to an observer standing on the bank, but steady to an observer<br />
moving along the bank with the speed of the wave front of the bore. Other flows are unsteady regardless<br />
of the reference frame used. The complex, time-dependent, wind-generated waves on a lake are in this<br />
category. In this book we will consider only steady open-channel flows.<br />
An open-channel flow is classified as uniform flow 1UF2 if the depth of flow does not vary<br />
along the channel 1dydx 02. Conversely, it is nonuniform flow or varied flow if the depth varies<br />
with distance 1dydx 02. Nonuniform flows are further classified as rapidly varying flow 1RVF2<br />
if the flow depth changes considerably over a relatively short distance; dydx 1. Gradually<br />
varying flows 1GVF2 are those in which the flow depth changes slowly with distance along the<br />
channel; dydx 1. Examples of these types of flow are illustrated in Fig. 10.1 and the photographs<br />
in the margin. The relative importance of the various types of forces involved 1pressure, weight,<br />
shear, inertia2 is different for the different types of flows.<br />
As for any flow geometry, open-channel flow may be laminar, transitional, or turbulent,<br />
depending on various conditions involved. Which type of flow occurs depends on the Reynolds<br />
number, Re rVR hm, where V is the average velocity of the <strong>fluid</strong> and R h is the hydraulic radius<br />
of the channel 1see Section 10.42. A general rule is that open-channel flow is laminar if Re 6 500,<br />
turbulent if Re 7 12,500, and transitional otherwise. The values of these dividing Reynolds<br />
numbers are only approximate—a precise knowledge of the channel geometry is necessary to<br />
obtain specific values. Since most open-channel flows involve water 1which has a fairly small<br />
viscosity2 and have relatively large characteristic lengths, it is rare to have laminar open-channel<br />
flows. For example, flow of 50 °F water 1n 1.41 10 5 ft 2 s2 with an average velocity of<br />
V 1 fts in a river with a hydraulic radius of R has Re VR hn 7.1 10 5 h 10 ft<br />
. The flow<br />
is turbulent. However, flow of a thin sheet of water down a driveway with an average velocity of<br />
V 0.25 fts such that R h 0.02 ft 1in such cases the hydraulic radius is approximately equal to<br />
the <strong>fluid</strong> depth; see Section 10.42 has Re 355. The flow is laminar.<br />
In some cases stratified flows are important. In such situations layers of two or more <strong>fluid</strong>s<br />
of different densities flow in a channel. A layer of oil on water is one example of this type of flow.<br />
All of the open-channel flows considered in this book are homogeneous flows. That is, the <strong>fluid</strong><br />
has uniform properties throughout.<br />
Open-channel flows involve a free surface that can deform from its undisturbed relatively<br />
flat configuration to form waves. Such waves move across the surface at speeds that depend on<br />
y<br />
UF uniform flow<br />
GVF gradually varying flow<br />
RVF rapidly varying flow<br />
RVF UF RVF UF RVF GVF RVF UF<br />
F I G U R E 10.1<br />
Classification of open-channel flow.