Computation of Gradually Varied Flow | Civil Engineering Optional Notes for UPSC PDF Download

 Page 1


 Computation of Gradually Varied Flow
• The computation of gradually-varied flow profiles involves basically 
the solution of dynamic equation of gradually varied flow. The main 
objective of computation is to determine the shape of flow profile.
• Broadly classified, there are three methods of computation; 
namely:
1. The graphical-integration method,
2. The direct-integration method,
3. Step method.
The graphical-integration method is to integrate the . dynamic 
equation of gradually varied flow by a graphical procedure. There 
are various graphical integration methods. The best one is the 
Ezra Method.
Page 2


 Computation of Gradually Varied Flow
• The computation of gradually-varied flow profiles involves basically 
the solution of dynamic equation of gradually varied flow. The main 
objective of computation is to determine the shape of flow profile.
• Broadly classified, there are three methods of computation; 
namely:
1. The graphical-integration method,
2. The direct-integration method,
3. Step method.
The graphical-integration method is to integrate the . dynamic 
equation of gradually varied flow by a graphical procedure. There 
are various graphical integration methods. The best one is the 
Ezra Method.
The direct-integration method: Thje differentia equation of GVF 
can not be expressed explicitly in terms of y for all types of flow 
cross section; hence a direct and exact integration of the equation 
is practically impossible. In this method, the channel length under 
consideration is divided into short reaches, and the integration is 
carried out by short range steps.
• The step method: In general, for step methods, the channel is 
divided into short reaches. The computation is carried step by step 
from one end of the reach to the other.
• There is a great variety of step methods. Some methods appear 
superior to others in certain respects, but no one method has been 
found to be the best in all application. The most commonly ised 
step methods are:
1. Direct-Step Method, 
2. Standart-step Method.
Page 3


 Computation of Gradually Varied Flow
• The computation of gradually-varied flow profiles involves basically 
the solution of dynamic equation of gradually varied flow. The main 
objective of computation is to determine the shape of flow profile.
• Broadly classified, there are three methods of computation; 
namely:
1. The graphical-integration method,
2. The direct-integration method,
3. Step method.
The graphical-integration method is to integrate the . dynamic 
equation of gradually varied flow by a graphical procedure. There 
are various graphical integration methods. The best one is the 
Ezra Method.
The direct-integration method: Thje differentia equation of GVF 
can not be expressed explicitly in terms of y for all types of flow 
cross section; hence a direct and exact integration of the equation 
is practically impossible. In this method, the channel length under 
consideration is divided into short reaches, and the integration is 
carried out by short range steps.
• The step method: In general, for step methods, the channel is 
divided into short reaches. The computation is carried step by step 
from one end of the reach to the other.
• There is a great variety of step methods. Some methods appear 
superior to others in certain respects, but no one method has been 
found to be the best in all application. The most commonly ised 
step methods are:
1. Direct-Step Method, 
2. Standart-step Method.
Direct-Integration Methods
We have seen that the flow equation
is true for all forms of channel section, provided that the Froude number 
F
r
is properly defined by the equation:
and the velocity coefficient, a = 1, channel slope q is small enough so 
that cos ? =1.
We now rewrite certain other elements of this equation with the aim of 
examining the possibility of a direct integration. It is convenient to use 
here the conveyance K and the section factor Z.
2
1
r
f o
F
S S
dx
dy
?
?
?
3
2 2
2
A
T
g
Q
A
T
g
V
F
r
? ?
Page 4


 Computation of Gradually Varied Flow
• The computation of gradually-varied flow profiles involves basically 
the solution of dynamic equation of gradually varied flow. The main 
objective of computation is to determine the shape of flow profile.
• Broadly classified, there are three methods of computation; 
namely:
1. The graphical-integration method,
2. The direct-integration method,
3. Step method.
The graphical-integration method is to integrate the . dynamic 
equation of gradually varied flow by a graphical procedure. There 
are various graphical integration methods. The best one is the 
Ezra Method.
The direct-integration method: Thje differentia equation of GVF 
can not be expressed explicitly in terms of y for all types of flow 
cross section; hence a direct and exact integration of the equation 
is practically impossible. In this method, the channel length under 
consideration is divided into short reaches, and the integration is 
carried out by short range steps.
• The step method: In general, for step methods, the channel is 
divided into short reaches. The computation is carried step by step 
from one end of the reach to the other.
• There is a great variety of step methods. Some methods appear 
superior to others in certain respects, but no one method has been 
found to be the best in all application. The most commonly ised 
step methods are:
1. Direct-Step Method, 
2. Standart-step Method.
Direct-Integration Methods
We have seen that the flow equation
is true for all forms of channel section, provided that the Froude number 
F
r
is properly defined by the equation:
and the velocity coefficient, a = 1, channel slope q is small enough so 
that cos ? =1.
We now rewrite certain other elements of this equation with the aim of 
examining the possibility of a direct integration. It is convenient to use 
here the conveyance K and the section factor Z.
2
1
r
f o
F
S S
dx
dy
?
?
?
3
2 2
2
A
T
g
Q
A
T
g
V
F
r
? ?
The Conveyance of a channel section, K:
If a large number of calculations are to be made, it is convenient to 
introduce the concept of “conveyance” of a channel in order to calculate 
the discharge. The “conveyance” of a channel indicated by the symbol K 
and defined by the equation
This equation can be used to compute the conveyance when the 
discharge and slope of the channel are given.
When the Chézy formula is used:
where c is the Chézy’s resistance factor. Similarly when the Manning 
formula is used
2 1/
KS Q ?
S
Q
K ?
or
3 / 2
AR
n
1
K ?
2 / 1
CAR K ?
Page 5


 Computation of Gradually Varied Flow
• The computation of gradually-varied flow profiles involves basically 
the solution of dynamic equation of gradually varied flow. The main 
objective of computation is to determine the shape of flow profile.
• Broadly classified, there are three methods of computation; 
namely:
1. The graphical-integration method,
2. The direct-integration method,
3. Step method.
The graphical-integration method is to integrate the . dynamic 
equation of gradually varied flow by a graphical procedure. There 
are various graphical integration methods. The best one is the 
Ezra Method.
The direct-integration method: Thje differentia equation of GVF 
can not be expressed explicitly in terms of y for all types of flow 
cross section; hence a direct and exact integration of the equation 
is practically impossible. In this method, the channel length under 
consideration is divided into short reaches, and the integration is 
carried out by short range steps.
• The step method: In general, for step methods, the channel is 
divided into short reaches. The computation is carried step by step 
from one end of the reach to the other.
• There is a great variety of step methods. Some methods appear 
superior to others in certain respects, but no one method has been 
found to be the best in all application. The most commonly ised 
step methods are:
1. Direct-Step Method, 
2. Standart-step Method.
Direct-Integration Methods
We have seen that the flow equation
is true for all forms of channel section, provided that the Froude number 
F
r
is properly defined by the equation:
and the velocity coefficient, a = 1, channel slope q is small enough so 
that cos ? =1.
We now rewrite certain other elements of this equation with the aim of 
examining the possibility of a direct integration. It is convenient to use 
here the conveyance K and the section factor Z.
2
1
r
f o
F
S S
dx
dy
?
?
?
3
2 2
2
A
T
g
Q
A
T
g
V
F
r
? ?
The Conveyance of a channel section, K:
If a large number of calculations are to be made, it is convenient to 
introduce the concept of “conveyance” of a channel in order to calculate 
the discharge. The “conveyance” of a channel indicated by the symbol K 
and defined by the equation
This equation can be used to compute the conveyance when the 
discharge and slope of the channel are given.
When the Chézy formula is used:
where c is the Chézy’s resistance factor. Similarly when the Manning 
formula is used
2 1/
KS Q ?
S
Q
K ?
or
3 / 2
AR
n
1
K ?
2 / 1
CAR K ?
• When the geometry of the water area and resistance factor or 
roughness coefficient are given,
One of the above formula can be used to calculate K. Since the 
Manning formula is used extensively in most of the problems, in 
following discussion the second expression will be used. Either K 
alone or the product Kn can be tabulated or plotted as a function of 
depth for any given channel section: the resulting tables or curves 
can then be used as a permanent reference, which will immediately 
yields values of depth for a given Q, S and n. This conveyance factor 
concept is widely used for uniform flow computation.
Since the conveyance K is a function of the depth of flow y, it may be 
assumed that:
where 
C
1
= coefficient, and
N = a parameter called hydraulic exponent
N
y C K
1
2
?