Chapter 13 Open-Channel Flow
Solutions Manual for
Fluid Mechanics: Fundamentals and Applications
Third Edition
Yunus A. Çengel & John M. Cimbala
McGraw-Hill, 2013
Chapter 13
OPEN-CHANNEL FLOW
PROPRIETARY AND CONFIDENTIAL
This Manual is the proprietary property of The McGraw-Hill Companies, Inc.
(“McGraw-Hill”) and protected by copyright and other state and federal laws. By
opening and using this Manual the user agrees to the following restrictions, and if the
recipient does not agree to these restrictions, the Manual should be promptly returned
unopened to McGraw-Hill: This Manual is being provided only to authorized
professors and instructors for use in preparing for the classes using the affiliated
textbook. No other use or distribution of this Manual is permitted. This Manual
may not be sold and may not be distributed to or used by any student or other
third party. No part of this Manual may be reproduced, displayed or distributed
in any form or by any means, electronic or otherwise, without the prior written
permission of McGraw-Hill.
13-1
PROPRIETARY MATERIAL. © 2014 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use.
Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or
posted on a website, in whole or part.
, Chapter 13 Open-Channel Flow
Classification, Froude Number, and Wave Speed
13-1C
Solution We are to define normal depth and how it is established.
Analysis In open channels of constant slope and constant cross-section, the fluid accelerates until the head loss due
to frictional effects equals the elevation drop. The fluid at this point reaches its terminal velocity, and uniform flow is
established. The flow remains uniform as long as the slope, cross-section, and the surface roughness of the channel remain
unchanged. The flow depth in uniform flow is called the normal depth yn, which is an important characteristic parameter
for open-channel flows.
Discussion The normal depth is a fairly strong function of surface roughness.
13-2C
Solution We are to discuss how pressure changes along the free surface in open-channel flow.
Analysis The free surface coincides with the hydraulic grade line (HGL), and the pressure is constant along the
free surface.
Discussion At a free surface of a liquid, the pressure must be equal to the pressure of the gas above it.
13-3C
Solution We are to determine if the slope of the free surface is equal to the slope of the channel bottom.
Analysis No in general. The slope of the free surface is not necessarily equal to the slope of the bottom surface
even during steady fully developed flow.
Discussion However, there are situations called uniform flow in which the conditions here are met.
13-4C
Solution We are to discuss some reasons for nonuniform flow in open channels, and the difference between rapidly
varied flow and gradually varied flow.
Analysis The presence of an obstruction in a channel such as a gate or a change in slope or cross-section causes
the flow depth to vary, and thus the flow to become varied or nonuniform. The varied flow is called rapidly varied flow
(RVF) if the flow depth changes markedly over a relatively short distance in the flow direction (such as the flow of water
past a partially open gate or shortly before a falls), and gradually varied flow (GVF) if the flow depth changes gradually
over a long distance along the channel.
Discussion The equations of GVF are simplified because of the slow changes in the flow direction.
13-2
PROPRIETARY MATERIAL. © 2014 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use.
Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or
posted on a website, in whole or part.
, Chapter 13 Open-Channel Flow
13-5C
Solution We are to discuss the driving force in open-channel flow and how flow rate is determined.
Analysis Flow in a channel is driven naturally by gravity. Water flow in a river, for example, is driven by the
elevation difference between the source and the sink. The flow rate in an open channel is established by the dynamic
balance between gravity and friction. Inertia of the flowing fluid also becomes important in unsteady flow.
Discussion In pipe flow, on the other hand, there may be an additional driving force of pressure due to pumps.
13-6C
Solution We are to discuss the difference between uniform and nonuniform flow.
Analysis The flow in a channel is said to be uniform if the flow depth (and thus the average velocity) remains
constant. Otherwise, the flow is said to be nonuniform or varied, indicating that the flow depth varies with distance in the
flow direction. Uniform flow conditions are commonly encountered in practice in long straight sections of channels with
constant slope and constant cross-section.
Discussion In uniform open-channel flow, the head loss due to frictional effects equals the elevation drop.
13-7C
Solution We are to explain how to determine if a flow is tranquil, critical, or rapid.
Analysis Knowing the average flow velocity and flow depth, the Froude number is determined from Fr V / gy .
Then the flow is classified as
Fr < 1 Subcritical or tranquil flow
Fr = 1 Critical flow
Fr > 1 Supercritical or rapid flow
Discussion The Froude number is the most important parameter in open-channel flow.
13-8C
Solution We are to discuss whether the flow upstream of a hydraulic jump must be supercritical, and whether the
flow downstream of a hydraulic jump must be subcritical.
Analysis Upstream of a hydraulic jump, the upstream flow must be supercritical. Downstream of a hydraulic
jump, the downstream flow must be subcritical.
Discussion Otherwise, the second law of thermodynamics would be violated. Note that a hydraulic jump is analogous
to a normal shock wave – in that case, the flow upstream must be supersonic and the flow downstream must be subsonic.
13-3
PROPRIETARY MATERIAL. © 2014 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use.
Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or
posted on a website, in whole or part.
, Chapter 13 Open-Channel Flow
13-9C
Solution We are to define critical length, and discuss how it is determined.
Analysis The flow depth yc corresponding to a Froude number of Fr = 1 is the critical depth, and it is determined
from V gy c or y c V 2 / g .
Discussion Critical depth is a useful parameter, even if the depth does not actually equal yc anywhere in the flow.
13-10C
Solution We are to define and discuss the usefulness of the Froude number.
Analysis Froude number, defined as Fr V / gy , is a dimensionless parameter that governs the character of
flow in open channels. Here, g is the gravitational acceleration, V is the mean fluid velocity at a cross-section, and Lc is a
characteristic length (Lc = flow depth y for wide rectangular channels). Fr represents the ratio of inertia forces to viscous
forces in open-channel flow. The Froude number is also the ratio of the flow speed to wave speed, Fr = V /co.
Discussion The Froude number is the most important parameter in open-channel flow.
13-11
Solution A single wave is initiated in a sea by a strong jolt during an earthquake. The speed of the resulting wave is
to be determined.
Assumptions The depth of water is constant,
Analysis Surface wave speed is determined the wave-speed relation to be
c 0 gh (9.81 m/s 2 ) (2000 m) 140 m/s
Discussion Note that wave speed depends on the water depth, and the wave speed increases as the water depth
increases. Also, the waves eventually die out because of the viscous effects.
13-12
Solution The flow of water in a wide channel is considered. The speed of a small disturbance in flow for two
different flow depths is to be determined for both water and oil.
Assumptions The distance across the wave is short and thus friction at the bottom surface and air drag at the top are
negligible,
Analysis Surface wave speed can be determined directly from the relation c 0 gh .
(a) c 0 gh (9.81 m/s 2 ) (0.25 m) 1.57 m/s
(b) c 0 gh (9.81 m/s 2 ) (0.8 m) 2.80 m/s
Therefore, a disturbance in the flow will travel at a speed of 0.990 m/s in the first case, and 2.80 m/s in the second case.
Discussion Note that wave speed depends on the water depth, and the wave speed increases as the water depth
increases as long as the water remains shallow. Results would not change if the fluid were oil, because the wave speed
depends only on the fluid depth.
13-4
PROPRIETARY MATERIAL. © 2014 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use.
Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or
posted on a website, in whole or part.
Solutions Manual for
Fluid Mechanics: Fundamentals and Applications
Third Edition
Yunus A. Çengel & John M. Cimbala
McGraw-Hill, 2013
Chapter 13
OPEN-CHANNEL FLOW
PROPRIETARY AND CONFIDENTIAL
This Manual is the proprietary property of The McGraw-Hill Companies, Inc.
(“McGraw-Hill”) and protected by copyright and other state and federal laws. By
opening and using this Manual the user agrees to the following restrictions, and if the
recipient does not agree to these restrictions, the Manual should be promptly returned
unopened to McGraw-Hill: This Manual is being provided only to authorized
professors and instructors for use in preparing for the classes using the affiliated
textbook. No other use or distribution of this Manual is permitted. This Manual
may not be sold and may not be distributed to or used by any student or other
third party. No part of this Manual may be reproduced, displayed or distributed
in any form or by any means, electronic or otherwise, without the prior written
permission of McGraw-Hill.
13-1
PROPRIETARY MATERIAL. © 2014 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use.
Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or
posted on a website, in whole or part.
, Chapter 13 Open-Channel Flow
Classification, Froude Number, and Wave Speed
13-1C
Solution We are to define normal depth and how it is established.
Analysis In open channels of constant slope and constant cross-section, the fluid accelerates until the head loss due
to frictional effects equals the elevation drop. The fluid at this point reaches its terminal velocity, and uniform flow is
established. The flow remains uniform as long as the slope, cross-section, and the surface roughness of the channel remain
unchanged. The flow depth in uniform flow is called the normal depth yn, which is an important characteristic parameter
for open-channel flows.
Discussion The normal depth is a fairly strong function of surface roughness.
13-2C
Solution We are to discuss how pressure changes along the free surface in open-channel flow.
Analysis The free surface coincides with the hydraulic grade line (HGL), and the pressure is constant along the
free surface.
Discussion At a free surface of a liquid, the pressure must be equal to the pressure of the gas above it.
13-3C
Solution We are to determine if the slope of the free surface is equal to the slope of the channel bottom.
Analysis No in general. The slope of the free surface is not necessarily equal to the slope of the bottom surface
even during steady fully developed flow.
Discussion However, there are situations called uniform flow in which the conditions here are met.
13-4C
Solution We are to discuss some reasons for nonuniform flow in open channels, and the difference between rapidly
varied flow and gradually varied flow.
Analysis The presence of an obstruction in a channel such as a gate or a change in slope or cross-section causes
the flow depth to vary, and thus the flow to become varied or nonuniform. The varied flow is called rapidly varied flow
(RVF) if the flow depth changes markedly over a relatively short distance in the flow direction (such as the flow of water
past a partially open gate or shortly before a falls), and gradually varied flow (GVF) if the flow depth changes gradually
over a long distance along the channel.
Discussion The equations of GVF are simplified because of the slow changes in the flow direction.
13-2
PROPRIETARY MATERIAL. © 2014 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use.
Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or
posted on a website, in whole or part.
, Chapter 13 Open-Channel Flow
13-5C
Solution We are to discuss the driving force in open-channel flow and how flow rate is determined.
Analysis Flow in a channel is driven naturally by gravity. Water flow in a river, for example, is driven by the
elevation difference between the source and the sink. The flow rate in an open channel is established by the dynamic
balance between gravity and friction. Inertia of the flowing fluid also becomes important in unsteady flow.
Discussion In pipe flow, on the other hand, there may be an additional driving force of pressure due to pumps.
13-6C
Solution We are to discuss the difference between uniform and nonuniform flow.
Analysis The flow in a channel is said to be uniform if the flow depth (and thus the average velocity) remains
constant. Otherwise, the flow is said to be nonuniform or varied, indicating that the flow depth varies with distance in the
flow direction. Uniform flow conditions are commonly encountered in practice in long straight sections of channels with
constant slope and constant cross-section.
Discussion In uniform open-channel flow, the head loss due to frictional effects equals the elevation drop.
13-7C
Solution We are to explain how to determine if a flow is tranquil, critical, or rapid.
Analysis Knowing the average flow velocity and flow depth, the Froude number is determined from Fr V / gy .
Then the flow is classified as
Fr < 1 Subcritical or tranquil flow
Fr = 1 Critical flow
Fr > 1 Supercritical or rapid flow
Discussion The Froude number is the most important parameter in open-channel flow.
13-8C
Solution We are to discuss whether the flow upstream of a hydraulic jump must be supercritical, and whether the
flow downstream of a hydraulic jump must be subcritical.
Analysis Upstream of a hydraulic jump, the upstream flow must be supercritical. Downstream of a hydraulic
jump, the downstream flow must be subcritical.
Discussion Otherwise, the second law of thermodynamics would be violated. Note that a hydraulic jump is analogous
to a normal shock wave – in that case, the flow upstream must be supersonic and the flow downstream must be subsonic.
13-3
PROPRIETARY MATERIAL. © 2014 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use.
Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or
posted on a website, in whole or part.
, Chapter 13 Open-Channel Flow
13-9C
Solution We are to define critical length, and discuss how it is determined.
Analysis The flow depth yc corresponding to a Froude number of Fr = 1 is the critical depth, and it is determined
from V gy c or y c V 2 / g .
Discussion Critical depth is a useful parameter, even if the depth does not actually equal yc anywhere in the flow.
13-10C
Solution We are to define and discuss the usefulness of the Froude number.
Analysis Froude number, defined as Fr V / gy , is a dimensionless parameter that governs the character of
flow in open channels. Here, g is the gravitational acceleration, V is the mean fluid velocity at a cross-section, and Lc is a
characteristic length (Lc = flow depth y for wide rectangular channels). Fr represents the ratio of inertia forces to viscous
forces in open-channel flow. The Froude number is also the ratio of the flow speed to wave speed, Fr = V /co.
Discussion The Froude number is the most important parameter in open-channel flow.
13-11
Solution A single wave is initiated in a sea by a strong jolt during an earthquake. The speed of the resulting wave is
to be determined.
Assumptions The depth of water is constant,
Analysis Surface wave speed is determined the wave-speed relation to be
c 0 gh (9.81 m/s 2 ) (2000 m) 140 m/s
Discussion Note that wave speed depends on the water depth, and the wave speed increases as the water depth
increases. Also, the waves eventually die out because of the viscous effects.
13-12
Solution The flow of water in a wide channel is considered. The speed of a small disturbance in flow for two
different flow depths is to be determined for both water and oil.
Assumptions The distance across the wave is short and thus friction at the bottom surface and air drag at the top are
negligible,
Analysis Surface wave speed can be determined directly from the relation c 0 gh .
(a) c 0 gh (9.81 m/s 2 ) (0.25 m) 1.57 m/s
(b) c 0 gh (9.81 m/s 2 ) (0.8 m) 2.80 m/s
Therefore, a disturbance in the flow will travel at a speed of 0.990 m/s in the first case, and 2.80 m/s in the second case.
Discussion Note that wave speed depends on the water depth, and the wave speed increases as the water depth
increases as long as the water remains shallow. Results would not change if the fluid were oil, because the wave speed
depends only on the fluid depth.
13-4
PROPRIETARY MATERIAL. © 2014 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use.
Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or
posted on a website, in whole or part.
