Specific Energy and Specific Force | Civil Engineering Optional Notes for UPSC PDF Download

 Page 1


 
 
 
   
 
 
 
  
  
  
 
Specific energy: 
 
The concept of specific energy was first introduced by Bakhmeteff in 1912 and is defined as the 
average energy per unit weight of water at a channel section as expressed with respect to the 
channel bottom. Since the piezometric level coincides with the water surface, the piezometric 
head with respect to the channel bottom is 
 
y z
g
P
? ?
?
, the water depth       (1) 
So, the specific energy head can be expressed as:   
    
g
V
y E
2
2
? ? ?        (2) 
We find that the specific energy at a channel section equals the sum of the water depth (y) and 
the velocity head, provided of course that the streamlines are straight and parallel. Since
A
Q
V ? , 
Equation (2) may be written as 
2
2
2gA
Q
y E ? ? ?       (3) 
Where, A, the cross-sectional area of flow, can also be expressed as a function of the water 
depth, y. From this equation it can be seen that for a given channel section and a constant 
discharge (Q), the specific energy in an open channel section is a function of the water depth 
only. Plotting this water depth (y) against the specific energy (E) gives a specific energy curve as 
shown in Figure 1. 
 
Fig. 1  The Specific Energy Curve 
 
The curve shows that, for a given discharge and specific energy there are two ‘alternate depths’ 
of flow. At Point C the specific energy is a minimum for a given discharge and the two alternate 
depths coincide. This depth of flow is known as ‘critical dept’ (y
c
). 
 
When the depth of flow is greater than the critical depth, the flow is called sub-critical; if it is 
less than the critical depth, the flow is called super-critical. The curve illustrates how a given 
discharge can occur at two possible flow regimes; slow and deep on the upper limb, fast and 
shallow on the lower limb, the limbs being separated by the critical flow condition (Point C). 
 
Page 2


 
 
 
   
 
 
 
  
  
  
 
Specific energy: 
 
The concept of specific energy was first introduced by Bakhmeteff in 1912 and is defined as the 
average energy per unit weight of water at a channel section as expressed with respect to the 
channel bottom. Since the piezometric level coincides with the water surface, the piezometric 
head with respect to the channel bottom is 
 
y z
g
P
? ?
?
, the water depth       (1) 
So, the specific energy head can be expressed as:   
    
g
V
y E
2
2
? ? ?        (2) 
We find that the specific energy at a channel section equals the sum of the water depth (y) and 
the velocity head, provided of course that the streamlines are straight and parallel. Since
A
Q
V ? , 
Equation (2) may be written as 
2
2
2gA
Q
y E ? ? ?       (3) 
Where, A, the cross-sectional area of flow, can also be expressed as a function of the water 
depth, y. From this equation it can be seen that for a given channel section and a constant 
discharge (Q), the specific energy in an open channel section is a function of the water depth 
only. Plotting this water depth (y) against the specific energy (E) gives a specific energy curve as 
shown in Figure 1. 
 
Fig. 1  The Specific Energy Curve 
 
The curve shows that, for a given discharge and specific energy there are two ‘alternate depths’ 
of flow. At Point C the specific energy is a minimum for a given discharge and the two alternate 
depths coincide. This depth of flow is known as ‘critical dept’ (y
c
). 
 
When the depth of flow is greater than the critical depth, the flow is called sub-critical; if it is 
less than the critical depth, the flow is called super-critical. The curve illustrates how a given 
discharge can occur at two possible flow regimes; slow and deep on the upper limb, fast and 
shallow on the lower limb, the limbs being separated by the critical flow condition (Point C). 
 
When there is a rapid change in depth of flow from a high to a low stage, a steep depression will 
occur in the water surface; this is called a ‘hydraulic drop’. On the other hand, when there is a 
rapid change from a low to a high stage, the water surface will rise abruptly; this phenomenon is 
called a ‘hydraulic jump’ or ‘standing wave’. The standing wave shows itself by its turbulence 
(white water), whereas the hydraulic drop is less apparent. However, if in a standing wave the 
change in depth is smal1, the water surface will not rise abruptly but will pass from a low to a 
high level through a series of undulations (undular jump), and detection becomes more difficult. 
The normal procedure to ascertain whether critical flow occurs in a channel contraction - there 
being sub-critical flow upstream and downstream of the contraction - is to look for a hydraulic 
jump immediately downstream of the contraction. 
 
From Figure 1 it is possible to see that if the state of flow is critical, i.e. if the specific energy is a 
minimum for a given discharge, there is one value for the depth of flow only. The relationship 
between this minimum specific energy and the critical depth is found by differentiating Equation 
(3) to y, while Q remains constant. 
 
dy
dA
gA
Q
dy
dE
3
2
1 ? ? ?       (4) 
Since dA = Bdy, this equation becomes 
   
gA
B V
dy
dE
2
1 ? ? ?        (5) 
 
If the specific energy is a minimum, then 0 ?
dy
dE
, we may write 
   
c
c c
B
A
g
V
2 2
2
? ?        (6) 
Equation (6) is valid only for steady flow with parallel streamlines in a channel of small slope. If 
the velocity distribution coefficient, ? is assumed to be unity, the criterion for critical flow 
becomes 
2
1
?
?
?
?
?
?
?
?
?
c
c
c
B
gA
V                  (7) 
 
Provided that the tail water level of the measuring structure is low enough to enable the depth of 
flow at the channel contraction to reach critical depth, continuity equations, energy equation and 
Equations (7) allow the development of a discharge equation for each measuring device, in 
which the upstream total energy head (H
1
) is the only independent variable. 
 
Equation (6) can be written for critical depth, y
c
 (assuming, ? is to be unity) as: 
3
1
2
2
?
?
?
?
?
?
?
?
?
c
c
gB
Q
y                 (8) 
Alternatively, critical depth, y
c
 can be written in terms of alternate depth as: 
 
?
?
?
?
?
?
?
?
?
?
2 1
2
2
2
1
2
y y
y y
y
c
       (9) 
Alternatively, critical depth, y
c
 can be written in terms of critical specific energy, E
c
 as: 
 
c c
E y
3
2
?        (10) 
 
At critical flow the average flow velocity is given by Equation (7). It can be proved that this flow 
velocity equals the velocity with which the smallest disturbance moves in an open channel, as 
measured relative to the flow. Because of this feature, a disturbance or change in a downstream 
level cannot influence an upstream water level if critical flow occurs in between the two cross-
sections considered. 
 
Page 3


 
 
 
   
 
 
 
  
  
  
 
Specific energy: 
 
The concept of specific energy was first introduced by Bakhmeteff in 1912 and is defined as the 
average energy per unit weight of water at a channel section as expressed with respect to the 
channel bottom. Since the piezometric level coincides with the water surface, the piezometric 
head with respect to the channel bottom is 
 
y z
g
P
? ?
?
, the water depth       (1) 
So, the specific energy head can be expressed as:   
    
g
V
y E
2
2
? ? ?        (2) 
We find that the specific energy at a channel section equals the sum of the water depth (y) and 
the velocity head, provided of course that the streamlines are straight and parallel. Since
A
Q
V ? , 
Equation (2) may be written as 
2
2
2gA
Q
y E ? ? ?       (3) 
Where, A, the cross-sectional area of flow, can also be expressed as a function of the water 
depth, y. From this equation it can be seen that for a given channel section and a constant 
discharge (Q), the specific energy in an open channel section is a function of the water depth 
only. Plotting this water depth (y) against the specific energy (E) gives a specific energy curve as 
shown in Figure 1. 
 
Fig. 1  The Specific Energy Curve 
 
The curve shows that, for a given discharge and specific energy there are two ‘alternate depths’ 
of flow. At Point C the specific energy is a minimum for a given discharge and the two alternate 
depths coincide. This depth of flow is known as ‘critical dept’ (y
c
). 
 
When the depth of flow is greater than the critical depth, the flow is called sub-critical; if it is 
less than the critical depth, the flow is called super-critical. The curve illustrates how a given 
discharge can occur at two possible flow regimes; slow and deep on the upper limb, fast and 
shallow on the lower limb, the limbs being separated by the critical flow condition (Point C). 
 
When there is a rapid change in depth of flow from a high to a low stage, a steep depression will 
occur in the water surface; this is called a ‘hydraulic drop’. On the other hand, when there is a 
rapid change from a low to a high stage, the water surface will rise abruptly; this phenomenon is 
called a ‘hydraulic jump’ or ‘standing wave’. The standing wave shows itself by its turbulence 
(white water), whereas the hydraulic drop is less apparent. However, if in a standing wave the 
change in depth is smal1, the water surface will not rise abruptly but will pass from a low to a 
high level through a series of undulations (undular jump), and detection becomes more difficult. 
The normal procedure to ascertain whether critical flow occurs in a channel contraction - there 
being sub-critical flow upstream and downstream of the contraction - is to look for a hydraulic 
jump immediately downstream of the contraction. 
 
From Figure 1 it is possible to see that if the state of flow is critical, i.e. if the specific energy is a 
minimum for a given discharge, there is one value for the depth of flow only. The relationship 
between this minimum specific energy and the critical depth is found by differentiating Equation 
(3) to y, while Q remains constant. 
 
dy
dA
gA
Q
dy
dE
3
2
1 ? ? ?       (4) 
Since dA = Bdy, this equation becomes 
   
gA
B V
dy
dE
2
1 ? ? ?        (5) 
 
If the specific energy is a minimum, then 0 ?
dy
dE
, we may write 
   
c
c c
B
A
g
V
2 2
2
? ?        (6) 
Equation (6) is valid only for steady flow with parallel streamlines in a channel of small slope. If 
the velocity distribution coefficient, ? is assumed to be unity, the criterion for critical flow 
becomes 
2
1
?
?
?
?
?
?
?
?
?
c
c
c
B
gA
V                  (7) 
 
Provided that the tail water level of the measuring structure is low enough to enable the depth of 
flow at the channel contraction to reach critical depth, continuity equations, energy equation and 
Equations (7) allow the development of a discharge equation for each measuring device, in 
which the upstream total energy head (H
1
) is the only independent variable. 
 
Equation (6) can be written for critical depth, y
c
 (assuming, ? is to be unity) as: 
3
1
2
2
?
?
?
?
?
?
?
?
?
c
c
gB
Q
y                 (8) 
Alternatively, critical depth, y
c
 can be written in terms of alternate depth as: 
 
?
?
?
?
?
?
?
?
?
?
2 1
2
2
2
1
2
y y
y y
y
c
       (9) 
Alternatively, critical depth, y
c
 can be written in terms of critical specific energy, E
c
 as: 
 
c c
E y
3
2
?        (10) 
 
At critical flow the average flow velocity is given by Equation (7). It can be proved that this flow 
velocity equals the velocity with which the smallest disturbance moves in an open channel, as 
measured relative to the flow. Because of this feature, a disturbance or change in a downstream 
level cannot influence an upstream water level if critical flow occurs in between the two cross-
sections considered. 
 
The ‘control section’ of a measuring structure is located where critical flow occurs and sub-
critical, tranquil, or streaming flow passes into supercritical, rapid, or shooting flow. Thus, if 
critical flow occurs at the control section of a measuring structure, the upstream water level is 
independent of the tail water level; the flow over the structure is then called ‘modular’. 
 
If the discharge changes, the specific energy will be changed accordingly. 
 
 
Fig. 2  Specific Energy Curves for Different Discharges 
 
Dividing both sides of equation (3) by y
c
 and after simplification, one gets 
 
2
2
1
?
?
?
?
?
?
?
?
? ?
y
y
y
y
y
E
c
c c
       (11) 
 
Equation (11) is the generalized form of the relationship between specific energy and depth of 
flow in which each term is dimensionless.  
 
Fig. 3  Dimensionless Specific Energy Curve 
 
 
Specific Force: 
 
 
Fig. 4 Specific-force curve supplemented with specific-energy curve. (a) Specific energy curve; 
(b) channel section; (c) specific-force curve. 
 
Page 4


 
 
 
   
 
 
 
  
  
  
 
Specific energy: 
 
The concept of specific energy was first introduced by Bakhmeteff in 1912 and is defined as the 
average energy per unit weight of water at a channel section as expressed with respect to the 
channel bottom. Since the piezometric level coincides with the water surface, the piezometric 
head with respect to the channel bottom is 
 
y z
g
P
? ?
?
, the water depth       (1) 
So, the specific energy head can be expressed as:   
    
g
V
y E
2
2
? ? ?        (2) 
We find that the specific energy at a channel section equals the sum of the water depth (y) and 
the velocity head, provided of course that the streamlines are straight and parallel. Since
A
Q
V ? , 
Equation (2) may be written as 
2
2
2gA
Q
y E ? ? ?       (3) 
Where, A, the cross-sectional area of flow, can also be expressed as a function of the water 
depth, y. From this equation it can be seen that for a given channel section and a constant 
discharge (Q), the specific energy in an open channel section is a function of the water depth 
only. Plotting this water depth (y) against the specific energy (E) gives a specific energy curve as 
shown in Figure 1. 
 
Fig. 1  The Specific Energy Curve 
 
The curve shows that, for a given discharge and specific energy there are two ‘alternate depths’ 
of flow. At Point C the specific energy is a minimum for a given discharge and the two alternate 
depths coincide. This depth of flow is known as ‘critical dept’ (y
c
). 
 
When the depth of flow is greater than the critical depth, the flow is called sub-critical; if it is 
less than the critical depth, the flow is called super-critical. The curve illustrates how a given 
discharge can occur at two possible flow regimes; slow and deep on the upper limb, fast and 
shallow on the lower limb, the limbs being separated by the critical flow condition (Point C). 
 
When there is a rapid change in depth of flow from a high to a low stage, a steep depression will 
occur in the water surface; this is called a ‘hydraulic drop’. On the other hand, when there is a 
rapid change from a low to a high stage, the water surface will rise abruptly; this phenomenon is 
called a ‘hydraulic jump’ or ‘standing wave’. The standing wave shows itself by its turbulence 
(white water), whereas the hydraulic drop is less apparent. However, if in a standing wave the 
change in depth is smal1, the water surface will not rise abruptly but will pass from a low to a 
high level through a series of undulations (undular jump), and detection becomes more difficult. 
The normal procedure to ascertain whether critical flow occurs in a channel contraction - there 
being sub-critical flow upstream and downstream of the contraction - is to look for a hydraulic 
jump immediately downstream of the contraction. 
 
From Figure 1 it is possible to see that if the state of flow is critical, i.e. if the specific energy is a 
minimum for a given discharge, there is one value for the depth of flow only. The relationship 
between this minimum specific energy and the critical depth is found by differentiating Equation 
(3) to y, while Q remains constant. 
 
dy
dA
gA
Q
dy
dE
3
2
1 ? ? ?       (4) 
Since dA = Bdy, this equation becomes 
   
gA
B V
dy
dE
2
1 ? ? ?        (5) 
 
If the specific energy is a minimum, then 0 ?
dy
dE
, we may write 
   
c
c c
B
A
g
V
2 2
2
? ?        (6) 
Equation (6) is valid only for steady flow with parallel streamlines in a channel of small slope. If 
the velocity distribution coefficient, ? is assumed to be unity, the criterion for critical flow 
becomes 
2
1
?
?
?
?
?
?
?
?
?
c
c
c
B
gA
V                  (7) 
 
Provided that the tail water level of the measuring structure is low enough to enable the depth of 
flow at the channel contraction to reach critical depth, continuity equations, energy equation and 
Equations (7) allow the development of a discharge equation for each measuring device, in 
which the upstream total energy head (H
1
) is the only independent variable. 
 
Equation (6) can be written for critical depth, y
c
 (assuming, ? is to be unity) as: 
3
1
2
2
?
?
?
?
?
?
?
?
?
c
c
gB
Q
y                 (8) 
Alternatively, critical depth, y
c
 can be written in terms of alternate depth as: 
 
?
?
?
?
?
?
?
?
?
?
2 1
2
2
2
1
2
y y
y y
y
c
       (9) 
Alternatively, critical depth, y
c
 can be written in terms of critical specific energy, E
c
 as: 
 
c c
E y
3
2
?        (10) 
 
At critical flow the average flow velocity is given by Equation (7). It can be proved that this flow 
velocity equals the velocity with which the smallest disturbance moves in an open channel, as 
measured relative to the flow. Because of this feature, a disturbance or change in a downstream 
level cannot influence an upstream water level if critical flow occurs in between the two cross-
sections considered. 
 
The ‘control section’ of a measuring structure is located where critical flow occurs and sub-
critical, tranquil, or streaming flow passes into supercritical, rapid, or shooting flow. Thus, if 
critical flow occurs at the control section of a measuring structure, the upstream water level is 
independent of the tail water level; the flow over the structure is then called ‘modular’. 
 
If the discharge changes, the specific energy will be changed accordingly. 
 
 
Fig. 2  Specific Energy Curves for Different Discharges 
 
Dividing both sides of equation (3) by y
c
 and after simplification, one gets 
 
2
2
1
?
?
?
?
?
?
?
?
? ?
y
y
y
y
y
E
c
c c
       (11) 
 
Equation (11) is the generalized form of the relationship between specific energy and depth of 
flow in which each term is dimensionless.  
 
Fig. 3  Dimensionless Specific Energy Curve 
 
 
Specific Force: 
 
 
Fig. 4 Specific-force curve supplemented with specific-energy curve. (a) Specific energy curve; 
(b) channel section; (c) specific-force curve. 
 
It is defined as the force at any section which is equal to sum of the momentum of the flow 
passing through the channel section per unit time per unit weigh of water and the force per unit 
weight of water.  
Specific force for any channel section may be expressed as: 
A z
gA
Q
F ? ?
2
        (12) 
Where, z = vertical distance of the centroid of the cross section from the free surface 
A = cross sectional area of flow 
F = specific force 
 
By plotting the depth against the specific force for a given channel section and discharge, a 
specific-force curve is obtained (Fig. 4c). This curve has two limbs AC and BC. The limb AC 
approaches the horizontal axis asymptotically toward the right. The limb BC rises upward and 
extends indefinitely to the right. For a given value of the specific force, the curve has two 
possible depths y
1
 and y
2
. At point G on the curve the two depths become one, and the specific 
force is a minimum. It can be easily prove that the depth at the minimum value of the specific 
force is equal to the critical depth, y
c
. 
 
Dividing both sides of Equation (12) by b y
c
2
and simplifying one can obtains  
2
2
2
1
?
?
?
?
?
?
?
?
? ?
c
c
c
y
y
y
y
b y
F
       (13) 
Equation (13) is the generalized specific force equation and each term of this equation is 
dimensionless. 
 
Fig. 5  Dimensionless Specific Force Curve 
 
 
 
 
    
  
    
 
 
 
 
 
  
 
  
 
 
  
  
b y
F
c
2
 
Page 5


 
 
 
   
 
 
 
  
  
  
 
Specific energy: 
 
The concept of specific energy was first introduced by Bakhmeteff in 1912 and is defined as the 
average energy per unit weight of water at a channel section as expressed with respect to the 
channel bottom. Since the piezometric level coincides with the water surface, the piezometric 
head with respect to the channel bottom is 
 
y z
g
P
? ?
?
, the water depth       (1) 
So, the specific energy head can be expressed as:   
    
g
V
y E
2
2
? ? ?        (2) 
We find that the specific energy at a channel section equals the sum of the water depth (y) and 
the velocity head, provided of course that the streamlines are straight and parallel. Since
A
Q
V ? , 
Equation (2) may be written as 
2
2
2gA
Q
y E ? ? ?       (3) 
Where, A, the cross-sectional area of flow, can also be expressed as a function of the water 
depth, y. From this equation it can be seen that for a given channel section and a constant 
discharge (Q), the specific energy in an open channel section is a function of the water depth 
only. Plotting this water depth (y) against the specific energy (E) gives a specific energy curve as 
shown in Figure 1. 
 
Fig. 1  The Specific Energy Curve 
 
The curve shows that, for a given discharge and specific energy there are two ‘alternate depths’ 
of flow. At Point C the specific energy is a minimum for a given discharge and the two alternate 
depths coincide. This depth of flow is known as ‘critical dept’ (y
c
). 
 
When the depth of flow is greater than the critical depth, the flow is called sub-critical; if it is 
less than the critical depth, the flow is called super-critical. The curve illustrates how a given 
discharge can occur at two possible flow regimes; slow and deep on the upper limb, fast and 
shallow on the lower limb, the limbs being separated by the critical flow condition (Point C). 
 
When there is a rapid change in depth of flow from a high to a low stage, a steep depression will 
occur in the water surface; this is called a ‘hydraulic drop’. On the other hand, when there is a 
rapid change from a low to a high stage, the water surface will rise abruptly; this phenomenon is 
called a ‘hydraulic jump’ or ‘standing wave’. The standing wave shows itself by its turbulence 
(white water), whereas the hydraulic drop is less apparent. However, if in a standing wave the 
change in depth is smal1, the water surface will not rise abruptly but will pass from a low to a 
high level through a series of undulations (undular jump), and detection becomes more difficult. 
The normal procedure to ascertain whether critical flow occurs in a channel contraction - there 
being sub-critical flow upstream and downstream of the contraction - is to look for a hydraulic 
jump immediately downstream of the contraction. 
 
From Figure 1 it is possible to see that if the state of flow is critical, i.e. if the specific energy is a 
minimum for a given discharge, there is one value for the depth of flow only. The relationship 
between this minimum specific energy and the critical depth is found by differentiating Equation 
(3) to y, while Q remains constant. 
 
dy
dA
gA
Q
dy
dE
3
2
1 ? ? ?       (4) 
Since dA = Bdy, this equation becomes 
   
gA
B V
dy
dE
2
1 ? ? ?        (5) 
 
If the specific energy is a minimum, then 0 ?
dy
dE
, we may write 
   
c
c c
B
A
g
V
2 2
2
? ?        (6) 
Equation (6) is valid only for steady flow with parallel streamlines in a channel of small slope. If 
the velocity distribution coefficient, ? is assumed to be unity, the criterion for critical flow 
becomes 
2
1
?
?
?
?
?
?
?
?
?
c
c
c
B
gA
V                  (7) 
 
Provided that the tail water level of the measuring structure is low enough to enable the depth of 
flow at the channel contraction to reach critical depth, continuity equations, energy equation and 
Equations (7) allow the development of a discharge equation for each measuring device, in 
which the upstream total energy head (H
1
) is the only independent variable. 
 
Equation (6) can be written for critical depth, y
c
 (assuming, ? is to be unity) as: 
3
1
2
2
?
?
?
?
?
?
?
?
?
c
c
gB
Q
y                 (8) 
Alternatively, critical depth, y
c
 can be written in terms of alternate depth as: 
 
?
?
?
?
?
?
?
?
?
?
2 1
2
2
2
1
2
y y
y y
y
c
       (9) 
Alternatively, critical depth, y
c
 can be written in terms of critical specific energy, E
c
 as: 
 
c c
E y
3
2
?        (10) 
 
At critical flow the average flow velocity is given by Equation (7). It can be proved that this flow 
velocity equals the velocity with which the smallest disturbance moves in an open channel, as 
measured relative to the flow. Because of this feature, a disturbance or change in a downstream 
level cannot influence an upstream water level if critical flow occurs in between the two cross-
sections considered. 
 
The ‘control section’ of a measuring structure is located where critical flow occurs and sub-
critical, tranquil, or streaming flow passes into supercritical, rapid, or shooting flow. Thus, if 
critical flow occurs at the control section of a measuring structure, the upstream water level is 
independent of the tail water level; the flow over the structure is then called ‘modular’. 
 
If the discharge changes, the specific energy will be changed accordingly. 
 
 
Fig. 2  Specific Energy Curves for Different Discharges 
 
Dividing both sides of equation (3) by y
c
 and after simplification, one gets 
 
2
2
1
?
?
?
?
?
?
?
?
? ?
y
y
y
y
y
E
c
c c
       (11) 
 
Equation (11) is the generalized form of the relationship between specific energy and depth of 
flow in which each term is dimensionless.  
 
Fig. 3  Dimensionless Specific Energy Curve 
 
 
Specific Force: 
 
 
Fig. 4 Specific-force curve supplemented with specific-energy curve. (a) Specific energy curve; 
(b) channel section; (c) specific-force curve. 
 
It is defined as the force at any section which is equal to sum of the momentum of the flow 
passing through the channel section per unit time per unit weigh of water and the force per unit 
weight of water.  
Specific force for any channel section may be expressed as: 
A z
gA
Q
F ? ?
2
        (12) 
Where, z = vertical distance of the centroid of the cross section from the free surface 
A = cross sectional area of flow 
F = specific force 
 
By plotting the depth against the specific force for a given channel section and discharge, a 
specific-force curve is obtained (Fig. 4c). This curve has two limbs AC and BC. The limb AC 
approaches the horizontal axis asymptotically toward the right. The limb BC rises upward and 
extends indefinitely to the right. For a given value of the specific force, the curve has two 
possible depths y
1
 and y
2
. At point G on the curve the two depths become one, and the specific 
force is a minimum. It can be easily prove that the depth at the minimum value of the specific 
force is equal to the critical depth, y
c
. 
 
Dividing both sides of Equation (12) by b y
c
2
and simplifying one can obtains  
2
2
2
1
?
?
?
?
?
?
?
?
? ?
c
c
c
y
y
y
y
b y
F
       (13) 
Equation (13) is the generalized specific force equation and each term of this equation is 
dimensionless. 
 
Fig. 5  Dimensionless Specific Force Curve 
 
 
 
 
    
  
    
 
 
 
 
 
  
 
  
 
 
  
  
b y
F
c
2
 
3. Compute E/y
c
 and F/(y
c
2
b) for each of the sections using equations (11) and (13) 
respectively. 
4. Plot y/y
c
 vs. E/y
c
 and y/y
c
 vs. F/(y
c
2
b) on plain graph papers to get the generalized specific 
energy and specific force curves. 
 
 
 
Fig. 6  Sub-critical and Supercritical Flow Regions in a Channel 
 
  
Assignments 
 
? Derive the equation of critical depth 
c c
E y
3
2
? and 
?
?
?
?
?
?
?
?
?
?
2 1
2
2
2
1
2
y y
y y
y
c
 
? What are the general features of specific energy curve of an open-channel flow? 
? How can you calculate the critical depth, y
c 
of an open-channel using Froude No.?  
      
 
 
 
Discussion 
 
Comment on the results obtained, practical significance of the results, interpretation of flow 
patterns, sources of error, etc.