Chapter 13 Open-Channel Flow Chapter 13 OPEN-CHANNEL FLOW Notes | EduRev

: Chapter 13 Open-Channel Flow Chapter 13 OPEN-CHANNEL FLOW Notes | EduRev

 Page 1


Chapter 13 Open-Channel Flow 
Chapter 13 
OPEN-CHANNEL FLOW 
 
Classification, Froude Number, and Wave Speed 
 
13-1C Open-channel flow is the flow of liquids in channels open to the atmosphere or in partially filled 
conduits, and is characterized by the presence of a liquid-gas interface called the free surface, whereas 
internal flow is the flow of liquids or gases that completely fill a conduit.  
 
13-2C 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.  
 
13-3C The free surface coincides with the hydraulic grade line (HGL), and the pressure is constant along 
the free surface.  
 
13-4C No, the slope of the free surface is not necessarily equal to the slope of the bottom surface even 
during steady fully developed flow. 
 
13-5C 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. 
 
13-6C 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 y
n
, which is an important characteristic parameter for open-channel flows. 
 
13-7C 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. 
 
13-8C The hydraulic radius R
h
 is defined as the ratio of the cross-sectional flow area A
c
 and the wetted 
perimeter p. That is, R
h
 = A
c
/p. Knowing the hydraulic radius, the hydraulic diameter is determined from D
h
 
= 4R
h
. 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, 
you are using it without permission.   
13-1
Page 2


Chapter 13 Open-Channel Flow 
Chapter 13 
OPEN-CHANNEL FLOW 
 
Classification, Froude Number, and Wave Speed 
 
13-1C Open-channel flow is the flow of liquids in channels open to the atmosphere or in partially filled 
conduits, and is characterized by the presence of a liquid-gas interface called the free surface, whereas 
internal flow is the flow of liquids or gases that completely fill a conduit.  
 
13-2C 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.  
 
13-3C The free surface coincides with the hydraulic grade line (HGL), and the pressure is constant along 
the free surface.  
 
13-4C No, the slope of the free surface is not necessarily equal to the slope of the bottom surface even 
during steady fully developed flow. 
 
13-5C 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. 
 
13-6C 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 y
n
, which is an important characteristic parameter for open-channel flows. 
 
13-7C 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. 
 
13-8C The hydraulic radius R
h
 is defined as the ratio of the cross-sectional flow area A
c
 and the wetted 
perimeter p. That is, R
h
 = A
c
/p. Knowing the hydraulic radius, the hydraulic diameter is determined from D
h
 
= 4R
h
. 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, 
you are using it without permission.   
13-1
Chapter 13 Open-Channel Flow 
13-9C Knowing the average flow velocity and flow depth, the Froude number is determined from 
gy V / Fr = . Then the flow is classified as 
 
 Fr < 1 Subcritical or tranquil flow 
 Fr = 1 Critical flow 
 Fr > 1 Supercritical or rapid flow 
.  
 
13-10C Froude number is a dimensionless number that governs the character of flow in open channels. It is 
defined as gy V / Fr = where g is the gravitational acceleration, V  is the mean fluid velocity at a cross-
section, and L
c
 is the characteristic length which is taken to be the flow depth y for wide rectangular 
channels.  It represents the ratio of inertia forces to viscous forces in channel flow. The Froude number is 
also the ratio of the flow speed to wave speed, Fr = V /c
o
. 
 
13-11C The flow depth corresponding to a Froude number of Fr = 1 is the critical depth, and it is 
determined from 
c
gy = V or . g V y
c
/
2
=
 
13-12C  Yes to both questions.  
 
13-13 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 gh =
0
c . 
(a) m/s 0.990 = = = ) m 1 . 0 ( ) m/s 81 . 9 (
2
0
gh c 
(b) m/s 2.80 = = = m) (0.8 ) m/s (9.81
2
0
gh c 
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. 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, 
you are using it without permission.   
13-2
Page 3


Chapter 13 Open-Channel Flow 
Chapter 13 
OPEN-CHANNEL FLOW 
 
Classification, Froude Number, and Wave Speed 
 
13-1C Open-channel flow is the flow of liquids in channels open to the atmosphere or in partially filled 
conduits, and is characterized by the presence of a liquid-gas interface called the free surface, whereas 
internal flow is the flow of liquids or gases that completely fill a conduit.  
 
13-2C 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.  
 
13-3C The free surface coincides with the hydraulic grade line (HGL), and the pressure is constant along 
the free surface.  
 
13-4C No, the slope of the free surface is not necessarily equal to the slope of the bottom surface even 
during steady fully developed flow. 
 
13-5C 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. 
 
13-6C 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 y
n
, which is an important characteristic parameter for open-channel flows. 
 
13-7C 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. 
 
13-8C The hydraulic radius R
h
 is defined as the ratio of the cross-sectional flow area A
c
 and the wetted 
perimeter p. That is, R
h
 = A
c
/p. Knowing the hydraulic radius, the hydraulic diameter is determined from D
h
 
= 4R
h
. 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, 
you are using it without permission.   
13-1
Chapter 13 Open-Channel Flow 
13-9C Knowing the average flow velocity and flow depth, the Froude number is determined from 
gy V / Fr = . Then the flow is classified as 
 
 Fr < 1 Subcritical or tranquil flow 
 Fr = 1 Critical flow 
 Fr > 1 Supercritical or rapid flow 
.  
 
13-10C Froude number is a dimensionless number that governs the character of flow in open channels. It is 
defined as gy V / Fr = where g is the gravitational acceleration, V  is the mean fluid velocity at a cross-
section, and L
c
 is the characteristic length which is taken to be the flow depth y for wide rectangular 
channels.  It represents the ratio of inertia forces to viscous forces in channel flow. The Froude number is 
also the ratio of the flow speed to wave speed, Fr = V /c
o
. 
 
13-11C The flow depth corresponding to a Froude number of Fr = 1 is the critical depth, and it is 
determined from 
c
gy = V or . g V y
c
/
2
=
 
13-12C  Yes to both questions.  
 
13-13 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 gh =
0
c . 
(a) m/s 0.990 = = = ) m 1 . 0 ( ) m/s 81 . 9 (
2
0
gh c 
(b) m/s 2.80 = = = m) (0.8 ) m/s (9.81
2
0
gh c 
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. 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, 
you are using it without permission.   
13-2
Chapter 13 Open-Channel Flow 
13-14 Water flows uniformly in a wide rectangular channel. For given flow depth and velocity, it is to be 
determined whether the flow is laminar or turbulent, and whether it is subcritical or supercritical.  
Assumptions  The flow is uniform.  
Properties  The density and dynamic viscosity of water at 20ºC are ? = 998.0 kg/m
3
 and µ  = 1.002 ×10
-3
 
kg/m ·s. 
Analysis (a) The Reynolds number of the flow is  
5
3
3
10 984 . 3
s kg/m 10 002 . 1
) m 2 . 0 )( m/s 2 )( kg/m 0 . 998 (
Re × =
· ×
= =
-
µ
?Vy
 
which is greater than the critical value of 500. Therefore, the flow is turbulent. 
 
(b) The Froude number is 
1.43
m) )(0.2 m/s (9.81
m/s 2
Fr
2
= = =
gy
V
 
which is greater than 1. Therefore, the flow is supercritical. 
 
Discussion. The result in (a) is expected since almost all open channel flows are turbulent. Also, hydraulic 
radius for a wide rectangular channel approaches the water depth y as the ratio y/b approaches zero.  
 
 
 
 
 
 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, 
you are using it without permission.   
13-3
Page 4


Chapter 13 Open-Channel Flow 
Chapter 13 
OPEN-CHANNEL FLOW 
 
Classification, Froude Number, and Wave Speed 
 
13-1C Open-channel flow is the flow of liquids in channels open to the atmosphere or in partially filled 
conduits, and is characterized by the presence of a liquid-gas interface called the free surface, whereas 
internal flow is the flow of liquids or gases that completely fill a conduit.  
 
13-2C 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.  
 
13-3C The free surface coincides with the hydraulic grade line (HGL), and the pressure is constant along 
the free surface.  
 
13-4C No, the slope of the free surface is not necessarily equal to the slope of the bottom surface even 
during steady fully developed flow. 
 
13-5C 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. 
 
13-6C 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 y
n
, which is an important characteristic parameter for open-channel flows. 
 
13-7C 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. 
 
13-8C The hydraulic radius R
h
 is defined as the ratio of the cross-sectional flow area A
c
 and the wetted 
perimeter p. That is, R
h
 = A
c
/p. Knowing the hydraulic radius, the hydraulic diameter is determined from D
h
 
= 4R
h
. 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, 
you are using it without permission.   
13-1
Chapter 13 Open-Channel Flow 
13-9C Knowing the average flow velocity and flow depth, the Froude number is determined from 
gy V / Fr = . Then the flow is classified as 
 
 Fr < 1 Subcritical or tranquil flow 
 Fr = 1 Critical flow 
 Fr > 1 Supercritical or rapid flow 
.  
 
13-10C Froude number is a dimensionless number that governs the character of flow in open channels. It is 
defined as gy V / Fr = where g is the gravitational acceleration, V  is the mean fluid velocity at a cross-
section, and L
c
 is the characteristic length which is taken to be the flow depth y for wide rectangular 
channels.  It represents the ratio of inertia forces to viscous forces in channel flow. The Froude number is 
also the ratio of the flow speed to wave speed, Fr = V /c
o
. 
 
13-11C The flow depth corresponding to a Froude number of Fr = 1 is the critical depth, and it is 
determined from 
c
gy = V or . g V y
c
/
2
=
 
13-12C  Yes to both questions.  
 
13-13 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 gh =
0
c . 
(a) m/s 0.990 = = = ) m 1 . 0 ( ) m/s 81 . 9 (
2
0
gh c 
(b) m/s 2.80 = = = m) (0.8 ) m/s (9.81
2
0
gh c 
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. 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, 
you are using it without permission.   
13-2
Chapter 13 Open-Channel Flow 
13-14 Water flows uniformly in a wide rectangular channel. For given flow depth and velocity, it is to be 
determined whether the flow is laminar or turbulent, and whether it is subcritical or supercritical.  
Assumptions  The flow is uniform.  
Properties  The density and dynamic viscosity of water at 20ºC are ? = 998.0 kg/m
3
 and µ  = 1.002 ×10
-3
 
kg/m ·s. 
Analysis (a) The Reynolds number of the flow is  
5
3
3
10 984 . 3
s kg/m 10 002 . 1
) m 2 . 0 )( m/s 2 )( kg/m 0 . 998 (
Re × =
· ×
= =
-
µ
?Vy
 
which is greater than the critical value of 500. Therefore, the flow is turbulent. 
 
(b) The Froude number is 
1.43
m) )(0.2 m/s (9.81
m/s 2
Fr
2
= = =
gy
V
 
which is greater than 1. Therefore, the flow is supercritical. 
 
Discussion. The result in (a) is expected since almost all open channel flows are turbulent. Also, hydraulic 
radius for a wide rectangular channel approaches the water depth y as the ratio y/b approaches zero.  
 
 
 
 
 
 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, 
you are using it without permission.   
13-3
Chapter 13 Open-Channel Flow 
13-15 Water flow in a partially full circular channel is considered. For given water depth and average 
velocity, the hydraulic radius, Reynolds number, and the flow regime are to be determined.  
Assumptions  Flow is uniform. 
Properties  The density and dynamic viscosity of water at 20ºC are ? = 998.0 kg/m
3
 and µ  = 1.002 ×10
-3
 
kg/m ·s. 
Analysis From geometric considerations, 
 
3 360
2
60 60              5 . 0
1
5 . 0 1
cos
p p
? ? = = ° = ? =
-
=
-
=
R
a R
 
Then the hydraulic radius becomes 
m 0.293 =
-
=
-
= = m) 1 (
3 / 2
) 3 / cos( ) 3 / sin( 3 /
2
cos sin
p
p p p
?
? ? ?
R
p
A
R
c
h
 
The Reynolds number of the flow is  
5
10 5.84 × =
· ×
= =
-
s kg/m 10 002 . 1
) m 293 . 0 )( m/s 2 )( kg/m 0 . 998 (
Re
3
3
µ
?
h
VR
 
which is greater than the critical value of 500. Therefore, the flow is turbulent.  
 When calculating the Froude number, the hydraulic depth should be used rather than the 
maximum depth or the hydraulic radius. For a non-rectangular channel, hydraulic depth is defined as the 
ratio of the flow area to top width, 
? 
R = 1 m
a=0.5 m
2 2 2
m 6142 . 0 )] 3 / cos( ) 3 / sin( 3 / [ m) 1 ( ) cos sin ( = - = - = p p p ? ? ? R A
c
 
m 3546 . 0
m)sin60 1 ( 2
m 6142 . 0
sin 2 width Top
2
=
°
= = =
? R
A A
y
c c
h
 
1.07
m) )(0.3546 m/s (9.81
m/s 2
Fr
2
= = =
gy
V
 
which is greater than 1. Therefore, the flow is supercritical (although, very close to critical). 
 
Discussion Note that if the maximum flow depth were used instead of the hydraulic depth, the result would 
be subcritical flow, which is not true. 
 
 
 
 
 
 
 
 
 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, 
you are using it without permission.   
13-4
Page 5


Chapter 13 Open-Channel Flow 
Chapter 13 
OPEN-CHANNEL FLOW 
 
Classification, Froude Number, and Wave Speed 
 
13-1C Open-channel flow is the flow of liquids in channels open to the atmosphere or in partially filled 
conduits, and is characterized by the presence of a liquid-gas interface called the free surface, whereas 
internal flow is the flow of liquids or gases that completely fill a conduit.  
 
13-2C 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.  
 
13-3C The free surface coincides with the hydraulic grade line (HGL), and the pressure is constant along 
the free surface.  
 
13-4C No, the slope of the free surface is not necessarily equal to the slope of the bottom surface even 
during steady fully developed flow. 
 
13-5C 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. 
 
13-6C 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 y
n
, which is an important characteristic parameter for open-channel flows. 
 
13-7C 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. 
 
13-8C The hydraulic radius R
h
 is defined as the ratio of the cross-sectional flow area A
c
 and the wetted 
perimeter p. That is, R
h
 = A
c
/p. Knowing the hydraulic radius, the hydraulic diameter is determined from D
h
 
= 4R
h
. 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, 
you are using it without permission.   
13-1
Chapter 13 Open-Channel Flow 
13-9C Knowing the average flow velocity and flow depth, the Froude number is determined from 
gy V / Fr = . Then the flow is classified as 
 
 Fr < 1 Subcritical or tranquil flow 
 Fr = 1 Critical flow 
 Fr > 1 Supercritical or rapid flow 
.  
 
13-10C Froude number is a dimensionless number that governs the character of flow in open channels. It is 
defined as gy V / Fr = where g is the gravitational acceleration, V  is the mean fluid velocity at a cross-
section, and L
c
 is the characteristic length which is taken to be the flow depth y for wide rectangular 
channels.  It represents the ratio of inertia forces to viscous forces in channel flow. The Froude number is 
also the ratio of the flow speed to wave speed, Fr = V /c
o
. 
 
13-11C The flow depth corresponding to a Froude number of Fr = 1 is the critical depth, and it is 
determined from 
c
gy = V or . g V y
c
/
2
=
 
13-12C  Yes to both questions.  
 
13-13 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 gh =
0
c . 
(a) m/s 0.990 = = = ) m 1 . 0 ( ) m/s 81 . 9 (
2
0
gh c 
(b) m/s 2.80 = = = m) (0.8 ) m/s (9.81
2
0
gh c 
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. 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, 
you are using it without permission.   
13-2
Chapter 13 Open-Channel Flow 
13-14 Water flows uniformly in a wide rectangular channel. For given flow depth and velocity, it is to be 
determined whether the flow is laminar or turbulent, and whether it is subcritical or supercritical.  
Assumptions  The flow is uniform.  
Properties  The density and dynamic viscosity of water at 20ºC are ? = 998.0 kg/m
3
 and µ  = 1.002 ×10
-3
 
kg/m ·s. 
Analysis (a) The Reynolds number of the flow is  
5
3
3
10 984 . 3
s kg/m 10 002 . 1
) m 2 . 0 )( m/s 2 )( kg/m 0 . 998 (
Re × =
· ×
= =
-
µ
?Vy
 
which is greater than the critical value of 500. Therefore, the flow is turbulent. 
 
(b) The Froude number is 
1.43
m) )(0.2 m/s (9.81
m/s 2
Fr
2
= = =
gy
V
 
which is greater than 1. Therefore, the flow is supercritical. 
 
Discussion. The result in (a) is expected since almost all open channel flows are turbulent. Also, hydraulic 
radius for a wide rectangular channel approaches the water depth y as the ratio y/b approaches zero.  
 
 
 
 
 
 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, 
you are using it without permission.   
13-3
Chapter 13 Open-Channel Flow 
13-15 Water flow in a partially full circular channel is considered. For given water depth and average 
velocity, the hydraulic radius, Reynolds number, and the flow regime are to be determined.  
Assumptions  Flow is uniform. 
Properties  The density and dynamic viscosity of water at 20ºC are ? = 998.0 kg/m
3
 and µ  = 1.002 ×10
-3
 
kg/m ·s. 
Analysis From geometric considerations, 
 
3 360
2
60 60              5 . 0
1
5 . 0 1
cos
p p
? ? = = ° = ? =
-
=
-
=
R
a R
 
Then the hydraulic radius becomes 
m 0.293 =
-
=
-
= = m) 1 (
3 / 2
) 3 / cos( ) 3 / sin( 3 /
2
cos sin
p
p p p
?
? ? ?
R
p
A
R
c
h
 
The Reynolds number of the flow is  
5
10 5.84 × =
· ×
= =
-
s kg/m 10 002 . 1
) m 293 . 0 )( m/s 2 )( kg/m 0 . 998 (
Re
3
3
µ
?
h
VR
 
which is greater than the critical value of 500. Therefore, the flow is turbulent.  
 When calculating the Froude number, the hydraulic depth should be used rather than the 
maximum depth or the hydraulic radius. For a non-rectangular channel, hydraulic depth is defined as the 
ratio of the flow area to top width, 
? 
R = 1 m
a=0.5 m
2 2 2
m 6142 . 0 )] 3 / cos( ) 3 / sin( 3 / [ m) 1 ( ) cos sin ( = - = - = p p p ? ? ? R A
c
 
m 3546 . 0
m)sin60 1 ( 2
m 6142 . 0
sin 2 width Top
2
=
°
= = =
? R
A A
y
c c
h
 
1.07
m) )(0.3546 m/s (9.81
m/s 2
Fr
2
= = =
gy
V
 
which is greater than 1. Therefore, the flow is supercritical (although, very close to critical). 
 
Discussion Note that if the maximum flow depth were used instead of the hydraulic depth, the result would 
be subcritical flow, which is not true. 
 
 
 
 
 
 
 
 
 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, 
you are using it without permission.   
13-4
Chapter 13 Open-Channel Flow 
13-16 Water flows uniformly in a wide rectangular channel. For given values of flow depth and velocity, it 
is to be determined whether the flow is subcritical or supercritical.  
Assumptions 1 The flow is uniform. 2 The channel is wide and thus the side wall effects are negligible. 
Analysis  The Froude number is 
51 . 4
m) )(0.08 m/s (9.81
m/s 4
Fr
2
= = =
gy
V
 
which is greater than 1. Therefore, the flow is supercritical. 
 
Discussion Note that the Froude Number is not function of any temperature-dependent properties, and thus 
temperature. 
 
 
 
 
 
 
13-17 Rain water flows on a concrete surface. For given values of flow depth and velocity, it is to be 
determined whether the flow is subcritical or supercritical.  
Assumptions 1 The flow is uniform. 2 The thickness of water layer is constant.   
Analysis  The Froude number is 
93 . 2
m) )(0.02 m/s (9.81
m/s 1.3
Fr
2
= = =
gy
V
 
which is greater than 1. Therefore, the flow is supercritical. 
 
Discussion Note that this water layer will undergo a hydraulic jump when the ground slope decreases or 
becomes adverse.   
 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, 
you are using it without permission.   
13-5
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