Varried flow: GVF
Gradually Varied flow (G.V.F.)
Definition: If the depth of flow in a channel changes gradually over a long length of the channel, the flow is said to be gradually varied flow and is denoted by G.V.F.
The document discusses gradually varied flow in open channels. It defines gradually varied flow as flow where the depth changes gradually along the channel. It presents the assumptions and governing equations for gradually varied flow analysis. It also describes different types of water surface profiles that can occur, such as mild slope, steep slope, critical slope, and adverse slope profiles. The key methods for analyzing water surface profiles, including direct integration, graphical integration, and numerical integration are summarized.
This document discusses gradually varied flow in open channels. It begins by defining gradually varied flow as flow where the water depth changes gradually along the length of the channel, as opposed to rapidly varied or uniform flow. It then provides classifications for open channel slopes as mild, steep, critical, horizontal, or adverse for analysis of gradually varied flow. Finally, it outlines methods for analyzing and computing gradually varied flow profiles, including the direct step method and standard step method which use finite difference approaches to solve the governing equations for gradually varied flow.
This document provides an overview of open channel hydraulics and discharge measuring structures. It discusses various open channel flow conditions including uniform flow, gradually varied flow, rapidly varied flow, subcritical flow, critical flow and supercritical flow. It introduces concepts such as specific energy, critical depth, energy equations, and hydraulic principles that govern open channel design. Formulas for discharge measurement using weirs and flumes are presented, such as the Chezy and Manning's equations. Common channel shapes and examples of flow through contractions and over humps are also summarized.
The document provides an overview of open channel hydraulics and discharge measuring structures. It discusses:
- Uniform and non-uniform open channel flow conditions, including gradually varied, rapidly varied, subcritical, critical and supercritical flows.
- Basic equations for uniform flow such as the continuity, energy and momentum equations.
- Hydraulic principles and formulas used to design channels and structures, including the Chezy and Manning's equations.
- Characteristics of gradually varied flow and methods for analyzing water surface profiles.
- Phenomena such as flow over a hump, through a contraction, and hydraulic jumps; and equations for analyzing conjugate depths.
This document discusses open channel hydraulics and specific energy. It defines key terms like head, energy, hydraulic grade line, energy line, critical depth, Froude number, specific energy, and gradually varied flow. It explains the concepts of critical depth, alternate depths, and how specific energy relates to critical depth for rectangular and non-rectangular channels. It also discusses surface profiles, backwater curves, types of bed slopes, occurrence of critical depth with changes in bed slope, and the energy equation for gradually varied flow. An example problem is included to demonstrate calculating distance between depths for gradually varied flow.
Open Channel Flow of irrigation and Drainage Department .pptiphone4s4
This document provides an overview of open channel flow, including relevant topics such as uniform flow, critical flow, gradually varied flow, and classification of flows. It also discusses important relationships for open channel flow, including the conservation of energy and momentum equations, dimensional analysis, and equations for discharge as a function of depth such as the Manning, Chezy, and Darcy-Weisbach equations. Key concepts introduced are the specific energy of a channel, critical depth, and the relationships between flow parameters at critical depth in a rectangular channel.
The document discusses gradually varied flow in open channels. It defines gradually varied flow as flow where the depth changes gradually along the channel. It presents the assumptions and governing equations for gradually varied flow analysis. It also describes different types of water surface profiles that can occur, such as mild slope, steep slope, critical slope, and adverse slope profiles. The key methods for analyzing water surface profiles, including direct integration, graphical integration, and numerical integration are summarized.
This document discusses gradually varied flow in open channels. It begins by defining gradually varied flow as flow where the water depth changes gradually along the length of the channel, as opposed to rapidly varied or uniform flow. It then provides classifications for open channel slopes as mild, steep, critical, horizontal, or adverse for analysis of gradually varied flow. Finally, it outlines methods for analyzing and computing gradually varied flow profiles, including the direct step method and standard step method which use finite difference approaches to solve the governing equations for gradually varied flow.
This document provides an overview of open channel hydraulics and discharge measuring structures. It discusses various open channel flow conditions including uniform flow, gradually varied flow, rapidly varied flow, subcritical flow, critical flow and supercritical flow. It introduces concepts such as specific energy, critical depth, energy equations, and hydraulic principles that govern open channel design. Formulas for discharge measurement using weirs and flumes are presented, such as the Chezy and Manning's equations. Common channel shapes and examples of flow through contractions and over humps are also summarized.
The document provides an overview of open channel hydraulics and discharge measuring structures. It discusses:
- Uniform and non-uniform open channel flow conditions, including gradually varied, rapidly varied, subcritical, critical and supercritical flows.
- Basic equations for uniform flow such as the continuity, energy and momentum equations.
- Hydraulic principles and formulas used to design channels and structures, including the Chezy and Manning's equations.
- Characteristics of gradually varied flow and methods for analyzing water surface profiles.
- Phenomena such as flow over a hump, through a contraction, and hydraulic jumps; and equations for analyzing conjugate depths.
This document discusses open channel hydraulics and specific energy. It defines key terms like head, energy, hydraulic grade line, energy line, critical depth, Froude number, specific energy, and gradually varied flow. It explains the concepts of critical depth, alternate depths, and how specific energy relates to critical depth for rectangular and non-rectangular channels. It also discusses surface profiles, backwater curves, types of bed slopes, occurrence of critical depth with changes in bed slope, and the energy equation for gradually varied flow. An example problem is included to demonstrate calculating distance between depths for gradually varied flow.
Open Channel Flow of irrigation and Drainage Department .pptiphone4s4
This document provides an overview of open channel flow, including relevant topics such as uniform flow, critical flow, gradually varied flow, and classification of flows. It also discusses important relationships for open channel flow, including the conservation of energy and momentum equations, dimensional analysis, and equations for discharge as a function of depth such as the Manning, Chezy, and Darcy-Weisbach equations. Key concepts introduced are the specific energy of a channel, critical depth, and the relationships between flow parameters at critical depth in a rectangular channel.
1) The document describes using the direct step method to determine water surface profiles for trapezoidal, rectangular, and triangular channels.
2) The direct step method is an iterative process that tests different water depths to classify a channel's flow type as subcritical, critical, or supercritical based on comparisons to the normal and critical depths.
3) Two examples are provided demonstrating the use of the direct step method to calculate normal depth, critical depth, and classify the water surface profile for different channel geometries.
This document discusses gradually varied flow and water surface profiles. It contains 3 types of water surface profiles: mild slope profiles, steep slope profiles, and critical slope profiles. Each type has 3 sub-categories depending on the relationship between the normal depth, critical depth, and flow depth. The document also discusses computation methods for water surface profiles, including graphical integration, direct step method, and standard step method.
1. The document is an exam for a transport phenomena course covering topics like boundary layers, pressure gradients, and flow in conical tubes.
2. The first problem involves calculating the velocity profile and pressure gradient for flow between a moving belt and plate filling a tank.
3. The second problem involves calculating momentum flux for two boundary layer profiles and determining which would separate first under the same pressure gradient.
4. The third problem involves deriving an expression for pressure drop in terms of flow rate, fluid properties, and geometry for flow in a conical tube.
The document summarizes concepts related to gradually varied flow in open channels. It discusses:
1. The assumptions and equations used in gradually varied flow analysis, including the energy equation.
2. The different types of water surface profiles that can occur depending on factors like bed slope, including mild slope, steep slope, critical slope, horizontal slope, and adverse slope profiles.
3. Methods for computing gradually varied flow profiles, including graphical integration, direct step method for prismatic channels, and standard step method for natural channels.
This document provides a 3-paragraph summary of a course on hydraulics:
The course is titled "Hydraulics II" with course number CEng2152. It is a 5 ECTS credit degree program course focusing on open channel flow. Open channel flow occurs when water flows with a free surface exposed to the atmosphere, such as in rivers, culverts and spillways. Engineering structures for open channel flow are designed and analyzed using open channel hydraulics.
The document covers different types of open channel flow including steady and unsteady, uniform and non-uniform flow. It also discusses the geometric elements of open channel cross-sections including depth, width, area and hydraulic radius. Uniform flow
Gradually varied flow is one kind of non uniform flow . Flow parameters such as depth of flow, flow velocity , discharge change with time and space gradually. Gradually varied flow is determined by the type of the channel bottom slopes. Flow profiles can be sustained in three different flow regions . This ppt covers only mild slope flow profile.
This document discusses open channel flow, which is the flow of liquid through a conduit with a free surface driven only by gravity. It compares open channel flow to pipe flow, describes different types of open channel flows, parameters used in analysis like hydraulic radius and Froude number, and formulas like Chezy's and Manning's equations used to analyze open channel flow characteristics. Examples are provided to demonstrate how to apply these concepts and formulas to calculate quantities like velocity, discharge, slope, and critical depth in open channel flow problems.
The document provides an introduction to open channel flow. It defines open channel flow and distinguishes it from pipe flow. Open channels are exposed to atmospheric pressure and have a cross-sectional area that varies depending on flow parameters, while pipe flow is enclosed and has a constant cross-sectional area. The document discusses different types of channel flows including steady/unsteady and uniform/non-uniform flow. It also defines geometric elements of open channel sections such as depth, width, wetted perimeter, and hydraulic radius. Critical depth is introduced as the depth where specific energy is minimum. Specific energy, defined as the total energy per unit weight of flow above the channel bottom, is also summarized.
This document discusses gradually varied flow (GVF) in open channels. It defines key terms related to GVF including normal depth, critical depth, flow zones, and profile classifications. It also covers topics like energy balance, mixed flow profiles, rapidly varied flow, hydraulic jumps, and applying GVF concepts to storm sewer analysis and hydraulic modeling with examples.
This document discusses open channel flow, including:
1) Key parameters like hydraulic radius, channel roughness, and types of flow profiles.
2) Empirical equations for open channel flow including Chezy and Manning's equations.
3) Concepts of critical flow including critical depth, specific energy, and the importance of the Froude number.
4) Measurement techniques for discharge like weirs and sluice gates.
5) Gradually and rapidly varied flow, water surface profiles, and hydraulic jumps.
Gradually varied flow and rapidly varied flowssuserd7b2f1
This document discusses gradually varied flow in open channels. It defines gradually varied flow and rapid flow, and lists some common causes of gradually varied flow including changes in channel shape, slope, obstructions, and frictional forces. It also lists the assumptions of gradually varied flow models including prismatic channels, constant Manning's n, hydrostatic pressure, and fixed velocity distribution. The basic differential equation for gradually varied flow relates the water surface slope, energy slope, channel bed slope, discharge, conveyance, and hydraulic radius. Channel bed slopes are also classified.
This document provides an overview of gradually varied flow and water surface profiles in open channels. It discusses the following key points in 3 sentences:
Uniform flow can be considered a control as it relates depth to discharge. Gradually varied flow occurs when controls pull flow away from uniform conditions, resulting in a transition between the two states. The water surface profile classification is based on the channel slope and how the flow depth compares to the normal and critical depths.
This document discusses gradually varied flow in open channels. It begins by defining gradually varied flow and providing examples. It then outlines the basic assumptions and develops the basic differential equation used to analyze water surface profiles. The document classifies channel types and divides the flow space into regions. It discusses the characteristics and asymptotic behaviors of different water surface profile types, including M1, M2, and M3 curves. An example problem is also included to demonstrate determining the profile type for a given channel and flow conditions.
Chapter 6 energy and momentum principlesBinu Karki
1) Open channel flow concepts such as specific energy, critical depth, Froude number, and their relationships are introduced. Critical depth corresponds to the minimum specific energy for a given discharge and occurs when the Froude number is 1. (2)
2) Flow over humps and through contractions is analyzed. For subcritical flow over a hump, the water surface lowers; above a critical hump height the water surface rises upstream in a "damming" effect called afflux. Through contractions, depth decreases for subcritical flow and increases for supercritical flow if losses are negligible. (3)
3) Broad crested weirs and Venturi flumes, which rely on critical flow principles, are commonly
This document discusses uniform flow in open channels. It defines an open channel as a stream that is not completely enclosed by solid boundaries and has a free surface exposed to atmospheric pressure. The document describes different types of open channels, types of flow, and geometric properties of channels. It also presents the Chezy and Manning formulas for calculating velocity and discharge under conditions of uniform flow in open channels.
This document summarizes uniform flow in open channels. It defines open channels as streams not completely enclosed by boundaries with a free water surface. Open channels can be natural or artificial with regular shapes. Uniform flow occurs when the depth, area, velocity and discharge remain constant in a channel with a constant slope and roughness. The Chezy and Manning formulas are presented to calculate mean flow velocity from hydraulic radius, slope and conveyance factors. Examples are given to solve for velocity, flow rate, and channel slope using the formulas.
Within the framework of the theory of plane steady filtration of an incompressible fluid according to Darcy’s law, two limiting schemes modeling the filtration flows under the Joukowski tongue through a soil massive spread over an impermeable foundation or strongly permeable confined water bearing horizon are considered.
Basic equation of fluid flow mechan.pptxAjithPArun1
This document discusses the basic equations of fluid flow, including:
- The continuity equation, which states that the rate of mass entering a fluid system equals the rate leaving under steady conditions.
- The momentum equation, which relates the rate of change of momentum of a fluid to the forces acting on it.
- Bernoulli's equation, which states that the total head (pressure head, velocity head, and elevation head) remains constant in an inviscid, incompressible, steady flow.
This document discusses the design of open channel sections to convey water flow in the most economical way. It examines rectangular, trapezoidal, triangular, and circular channel cross-sections. For rectangular channels, the most economical section is when the base width is twice the flow depth. For trapezoidal channels, the most economical section is when the side slopes are at an angle of 60 degrees from horizontal and the half top width is equal to the flow depth. Empirical flow equations like Chezy's and Manning's formulas are also presented to estimate normal flow velocities based on hydraulic radius and channel slope.
1) The document describes using the direct step method to determine water surface profiles for trapezoidal, rectangular, and triangular channels.
2) The direct step method is an iterative process that tests different water depths to classify a channel's flow type as subcritical, critical, or supercritical based on comparisons to the normal and critical depths.
3) Two examples are provided demonstrating the use of the direct step method to calculate normal depth, critical depth, and classify the water surface profile for different channel geometries.
This document discusses gradually varied flow and water surface profiles. It contains 3 types of water surface profiles: mild slope profiles, steep slope profiles, and critical slope profiles. Each type has 3 sub-categories depending on the relationship between the normal depth, critical depth, and flow depth. The document also discusses computation methods for water surface profiles, including graphical integration, direct step method, and standard step method.
1. The document is an exam for a transport phenomena course covering topics like boundary layers, pressure gradients, and flow in conical tubes.
2. The first problem involves calculating the velocity profile and pressure gradient for flow between a moving belt and plate filling a tank.
3. The second problem involves calculating momentum flux for two boundary layer profiles and determining which would separate first under the same pressure gradient.
4. The third problem involves deriving an expression for pressure drop in terms of flow rate, fluid properties, and geometry for flow in a conical tube.
The document summarizes concepts related to gradually varied flow in open channels. It discusses:
1. The assumptions and equations used in gradually varied flow analysis, including the energy equation.
2. The different types of water surface profiles that can occur depending on factors like bed slope, including mild slope, steep slope, critical slope, horizontal slope, and adverse slope profiles.
3. Methods for computing gradually varied flow profiles, including graphical integration, direct step method for prismatic channels, and standard step method for natural channels.
This document provides a 3-paragraph summary of a course on hydraulics:
The course is titled "Hydraulics II" with course number CEng2152. It is a 5 ECTS credit degree program course focusing on open channel flow. Open channel flow occurs when water flows with a free surface exposed to the atmosphere, such as in rivers, culverts and spillways. Engineering structures for open channel flow are designed and analyzed using open channel hydraulics.
The document covers different types of open channel flow including steady and unsteady, uniform and non-uniform flow. It also discusses the geometric elements of open channel cross-sections including depth, width, area and hydraulic radius. Uniform flow
Gradually varied flow is one kind of non uniform flow . Flow parameters such as depth of flow, flow velocity , discharge change with time and space gradually. Gradually varied flow is determined by the type of the channel bottom slopes. Flow profiles can be sustained in three different flow regions . This ppt covers only mild slope flow profile.
This document discusses open channel flow, which is the flow of liquid through a conduit with a free surface driven only by gravity. It compares open channel flow to pipe flow, describes different types of open channel flows, parameters used in analysis like hydraulic radius and Froude number, and formulas like Chezy's and Manning's equations used to analyze open channel flow characteristics. Examples are provided to demonstrate how to apply these concepts and formulas to calculate quantities like velocity, discharge, slope, and critical depth in open channel flow problems.
The document provides an introduction to open channel flow. It defines open channel flow and distinguishes it from pipe flow. Open channels are exposed to atmospheric pressure and have a cross-sectional area that varies depending on flow parameters, while pipe flow is enclosed and has a constant cross-sectional area. The document discusses different types of channel flows including steady/unsteady and uniform/non-uniform flow. It also defines geometric elements of open channel sections such as depth, width, wetted perimeter, and hydraulic radius. Critical depth is introduced as the depth where specific energy is minimum. Specific energy, defined as the total energy per unit weight of flow above the channel bottom, is also summarized.
This document discusses gradually varied flow (GVF) in open channels. It defines key terms related to GVF including normal depth, critical depth, flow zones, and profile classifications. It also covers topics like energy balance, mixed flow profiles, rapidly varied flow, hydraulic jumps, and applying GVF concepts to storm sewer analysis and hydraulic modeling with examples.
This document discusses open channel flow, including:
1) Key parameters like hydraulic radius, channel roughness, and types of flow profiles.
2) Empirical equations for open channel flow including Chezy and Manning's equations.
3) Concepts of critical flow including critical depth, specific energy, and the importance of the Froude number.
4) Measurement techniques for discharge like weirs and sluice gates.
5) Gradually and rapidly varied flow, water surface profiles, and hydraulic jumps.
Gradually varied flow and rapidly varied flowssuserd7b2f1
This document discusses gradually varied flow in open channels. It defines gradually varied flow and rapid flow, and lists some common causes of gradually varied flow including changes in channel shape, slope, obstructions, and frictional forces. It also lists the assumptions of gradually varied flow models including prismatic channels, constant Manning's n, hydrostatic pressure, and fixed velocity distribution. The basic differential equation for gradually varied flow relates the water surface slope, energy slope, channel bed slope, discharge, conveyance, and hydraulic radius. Channel bed slopes are also classified.
This document provides an overview of gradually varied flow and water surface profiles in open channels. It discusses the following key points in 3 sentences:
Uniform flow can be considered a control as it relates depth to discharge. Gradually varied flow occurs when controls pull flow away from uniform conditions, resulting in a transition between the two states. The water surface profile classification is based on the channel slope and how the flow depth compares to the normal and critical depths.
This document discusses gradually varied flow in open channels. It begins by defining gradually varied flow and providing examples. It then outlines the basic assumptions and develops the basic differential equation used to analyze water surface profiles. The document classifies channel types and divides the flow space into regions. It discusses the characteristics and asymptotic behaviors of different water surface profile types, including M1, M2, and M3 curves. An example problem is also included to demonstrate determining the profile type for a given channel and flow conditions.
Chapter 6 energy and momentum principlesBinu Karki
1) Open channel flow concepts such as specific energy, critical depth, Froude number, and their relationships are introduced. Critical depth corresponds to the minimum specific energy for a given discharge and occurs when the Froude number is 1. (2)
2) Flow over humps and through contractions is analyzed. For subcritical flow over a hump, the water surface lowers; above a critical hump height the water surface rises upstream in a "damming" effect called afflux. Through contractions, depth decreases for subcritical flow and increases for supercritical flow if losses are negligible. (3)
3) Broad crested weirs and Venturi flumes, which rely on critical flow principles, are commonly
This document discusses uniform flow in open channels. It defines an open channel as a stream that is not completely enclosed by solid boundaries and has a free surface exposed to atmospheric pressure. The document describes different types of open channels, types of flow, and geometric properties of channels. It also presents the Chezy and Manning formulas for calculating velocity and discharge under conditions of uniform flow in open channels.
This document summarizes uniform flow in open channels. It defines open channels as streams not completely enclosed by boundaries with a free water surface. Open channels can be natural or artificial with regular shapes. Uniform flow occurs when the depth, area, velocity and discharge remain constant in a channel with a constant slope and roughness. The Chezy and Manning formulas are presented to calculate mean flow velocity from hydraulic radius, slope and conveyance factors. Examples are given to solve for velocity, flow rate, and channel slope using the formulas.
Within the framework of the theory of plane steady filtration of an incompressible fluid according to Darcy’s law, two limiting schemes modeling the filtration flows under the Joukowski tongue through a soil massive spread over an impermeable foundation or strongly permeable confined water bearing horizon are considered.
Basic equation of fluid flow mechan.pptxAjithPArun1
This document discusses the basic equations of fluid flow, including:
- The continuity equation, which states that the rate of mass entering a fluid system equals the rate leaving under steady conditions.
- The momentum equation, which relates the rate of change of momentum of a fluid to the forces acting on it.
- Bernoulli's equation, which states that the total head (pressure head, velocity head, and elevation head) remains constant in an inviscid, incompressible, steady flow.
This document discusses the design of open channel sections to convey water flow in the most economical way. It examines rectangular, trapezoidal, triangular, and circular channel cross-sections. For rectangular channels, the most economical section is when the base width is twice the flow depth. For trapezoidal channels, the most economical section is when the side slopes are at an angle of 60 degrees from horizontal and the half top width is equal to the flow depth. Empirical flow equations like Chezy's and Manning's formulas are also presented to estimate normal flow velocities based on hydraulic radius and channel slope.
Similar to Module-III Varried Flow.pptx GVF Definition, Water Surface Profile Dynamic Equation of GVF, Direct Step Method (20)
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#Abstract:
- Learn more about the real-world methods for auditing AWS IAM (Identity and Access Management) as a pentester. So let us proceed with a brief discussion of IAM as well as some typical misconfigurations and their potential exploits in order to reinforce the understanding of IAM security best practices.
- Gain actionable insights into AWS IAM policies and roles, using hands on approach.
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2. SYLLABUS
Module-III: Varied Flow
Gradually (G.V.F.):Definition, classification of channel Slopes,
dynamic equation of G.V.F. (Assumption and derivation), classification
of G.V.F. profiles-examples, direct step method of computation of
G.V.F. profiles
Rapidly varied flow (R.V.F.) Definition, examples, hydraulic jump-
phenomenon, relation of conjugate depths, parameters, uses, types of
hydraulic jump
3. Course Outcomes
CO1: Design open channel sections in a most
economical way.
CO2: Know about the non-uniform flows in open
channel and the characteristics of hydraulic jump.
CO3: Understand application of momentum principle of
impact of jets on plane.
4. Reference Books
1) Fluid Mechanics and Hydraulic Machinery By Dr.
R.K. Bansal
2) Fluid Mechanics and Hydraulic Machinery By
Modi and Seth
3) Fluid Mechanics By Kumar K.L.
4) Flow In Open Channels By K Subramanya
6. Gradually Varied flow (G.V.F.)
• Definition: If the depth of flow in a channel changes gradually over a
long length of the channel, the flow is said to be gradually varied flow
and is denoted by G.V.F.
7. FLOW CLASSIFICATION
• Uniform (normal) flow: Depth is constant at every section
along length of channel
• Non-uniform (varied) flow: Depth changes along channel
• Rapidly-varied flow: Depth changes suddenly
• Gradually-varied flow: Depth changes gradually
10. ASSUMPTIONS FOR GRADUALLY-VARIED FLOW
1. The channel is prismatic and the flow is steady.
The cross-sectional shape, size and bed slope are constant.
2. The bed slope, So, is relatively small.
3. The velocity distribution in the vertical section is uniform
and the kinetic energy correction factor is close to unity.
4. Streamlines are parallel and the pressure distribution is
hydrostatic.
5. The channel roughness is constant along its length and
does not depend on the depth of flow.
12. THE EQUATIONS FOR GRADUALLY VARIED FLOW
z
+
h
+
g
2
V
=
H
2
dx
dz
+
dx
dh
+
g
2
V
dx
d
=
dx
dH 2
S
-
=
dx
/
dz
,
S
-
=
dx
/
dH 0 S=Sf: slope of EGL
h1
h2
13. • It should be noted that the slope is defined as the sine of
the slope angle and that is assumed positive if it descends
in the direction of flow and negative if it ascends. Hence,
• It should be noted that the friction loss dh is always a
negative quantity in the direction of flow (unless outside
energy is added to the course of the flow) and that the
change in the bottom elevation dz is a negative quantity
when the slope descends.
• In the other words, they are negative because H and z
decrease in the flow direction
S
=
dx
/
dz
,
S
=
dx
/
dH 0
15. DERIVATION OF GVF EQUATION
Fr
-
1
S
-
S
=
dx
dh
2
0
For any cross-section
3
c
3
o
o
y
y
1
y
y
1
S
=
dx
dh Wide rectangular section
(Using Chezy equation
for Sf) f
RS
V
C
3
c
3
/
10
o
o
y
y
1
y
y
1
S
=
dx
dh
Wide rectangular section
(Using Manning’s formula for
Sf)
16. WATER SURFACE PROFILES
• For a given channel with a known Q = Discharge, n = Manning
coefficient, and So = channel bed slope, yc = critical water depth and
yo = uniform flow depth can be computed.
• There are three possible relations between yo and yc as
1) yo > yc ,
2) yo < yc ,
3) yo = yc .
17. WATER SURFACE PROFILES CLASSIFICATION
• For each of the five categories of channels (in previous slide), lines
representing the critical depth (yc ) and normal depth (yo ) (if it exists)
can be drawn in the longitudinal section.
• These would divide the whole flow space into three regions as: (y:
non-uniform depth)
• Zone 1: Space above the topmost line,
y> yo> yc , y > yc> yo
• Zone 2: Space between top line and the
next lower line,
yo > y> yc , yc > y> yo
• Zone 3: Space between the second line
and the bed.
yo >yc>y , yc>yo>y
19. WATER SURFACE PROFILES CLASSIFICATION
For the horizontal (So = 0) and adverse slope ( So < 0) channels,
Horizontal channel: So = 0 → Q = 0
Adverse channel: So < 0 Q cannot be computed,
2
/
1
o
3
/
2
S
AR
n
1
Q
For the horizontal and adverse slope channels, the
uniform flow depth yo does not exist.
20. WATER SURFACE PROFILES CLASSIFICATION
• For a given Q, n, and So at a channel,
• yo = Uniform flow depth, yc = Critical flow depth, y = Non-uniform flow
depth.
• The depth y is measured vertically from the channel bottom, the slope
of the water surface dy / dx is relative to this channel bottom.
the prediction of surface
profiles from the analysis of
Fr
-
1
S
-
S
=
dx
dh
2
0
21. Classification of Profiles According to dy/dl
1) dy/dx>0; the depth of flow is increasing with the distance.
(A rising Curve)
2) dy/dx<0; the depth of flow is decreasing with the distance.
(A falling Curve)
3) dy/dx=0. The flow is uniform Sf=So
4) dy/dx = -∞. The water surface forms a right angle with the
channel bed.
5) dy/dx=∞/∞. The depth of flow approaches a zero.
6) dy/dx= So The water surface profile forms a horizontal line.
This is special case of the rising water profile
dl=dx
22. WATER SURFACE PROFILES CLASSIFICATION
Classification of profiles according to dy / dl or (dh/dx).
Fr
-
1
S
-
S
=
dx
dh
2
0
23. GRAPHICAL REPRESENTATION OF THE GVF
Zone 1: y > yo> yc
Zone 2: yo > y > yc
Zone 3: yo > yc > y
3
c
3
o
o
y
y
1
y
y
1
S
=
dx
dh
24. I. Example
Draw water surface profile for two reaches of the open channel given
in Figure below. A gate is located between the two reaches and the
second reach ends with a sudden fall.
(a) The open channel
and gate location.
(b) Critical and normal
depths.
(c) Water surface profile.
26. METHODS OF SOLUTIONS OF THE GRADUALLY
VARIED FLOW
1. Direct Integration
2. Graphical Integration
3. Numerical Integration
i- The direct step method (distance from depth for regular channels)
ii- The standard step method, regular channels (distance from depth
for regular channels)
iii- The standard step method, natural channels (distance from depth
for regular channels)
27. GRADUALLY VARIED FLOW
Important Formulas
g
2
V
y
z
H
2
b
g
2
V
y
E
2
E
z
H b
dx
dE
dx
dz
=
dx
H
d
f
o S
S
=
dx
dE
Fr
-
1
S
-
S
=
dx
d
2
0 f
y
28. GRADUALLY VARIED FLOW COMPUTATIONS
• Analytical solutions to the equations above not available for the most
typically encountered open channel flow situations.
• A finite difference approach is applied to the GVF problems.
• Channel is divided into short reaches and computations are carried
out from one end of the reach to the other.
_
=
dx
f
o S
S
dE
Fr
-
1
S
-
S
=
dx
d
2
0 f
y
E: specific energy
29. DIRECT STEP METHOD
A non-uniform water surface profile
f
_
o
U
D
S
S
=
x
E
E
Sf : average friction slope in
the reach
)
S
S
(
2
1
S fD
fu
f
_
3
/
4
u
2
u
2
fu
R
V
n
S 3
/
4
D
2
D
2
fD
R
V
n
S
Manning Formula is sufficient to accurately evaluate
the slope of total energy line, S
30. DIRECT STEP METHOD
S
S
g
2
/
V
y
g
2
/
V
y
S
S
E
E
X
f
_
o
2
U
U
2
D
D
f
_
o
U
D
Subcritical Flow
The condition at the downstream is
known
yD, VD and SfD are known
Chose an appropriate value for
yu
Calculate the corresponding Vu
, Sfu and Sf
Then Calculate X
Supercritical Flow
The condition at the upstream is
known
yu, Vu and Sfu are known
Chose an appropriate value for yD
Calculate the corresponding SfD,
VD and Sf
Then Calculate X
31. Example: A trapezoidal concrete-lined channel has a constant bed
slope of 0.0015, a bed width of 3 m and side slopes 1:1. A control
gate increased the depth immediately upstream to 4.0m when the
discharge is 19 m3/s. Compute WSP to a depth 5% greater than the
uniform flow depth (n=0.017).
Two possibilities exist:
OR
32. Solution:
The first task is to calculate the critical and normal depths.
Using Manning formula, the depth of uniform flow:
2
/
1
3
/
2
S
AR
n
1
Q yo = 1.75 m
Using the critical flow condition, the critical depth:
3
2
2
gA
T
Q
Fr yc = 1.36 m
It can be realized that the profile should be M1 since yo > yc
That is to say, the possibility is valid in our problem.
33.
S
S
g
2
/
V
y
g
2
/
V
y
S
S
E
E
X
f
_
o
2
U
U
2
D
D
f
_
o
U
D
y A R E ΔE Sf Sf So-Sf Δx x
4.000 28.00 1.956 4.023 0.000054 0
3.900 26.91 1.918 3.925 0.098 0.000060 0.000057 0.001443 67.98 67.98
3.800 25.84 1.880 3.828 0.098 0.000067 0.000064 0.001436 68.14 136.11
3.700 24.79 1.841 3.730 0.098 0.000075 0.000071 0.001429 68.32 204.44
3.600 23.76 1.802 3.633 0.097 0.000084 0.000080 0.001420 68.54 272.98
3.500 22.75 1.764 3.536 0.097 0.000095 0.000089 0.001411 68.80 341.78
3.400 21.76 1.725 3.439 0.097 0.000107 0.000101 0.001399 69.09 410.87
3.300 20.79 1.686 3.343 0.096 0.000120 0.000113 0.001387 69.44 480.31
3.200 19.84 1.646 3.247 0.096 0.000136 0.000128 0.001372 69.86 550.17
3.100 18.91 1.607 3.151 0.095 0.000155 0.000146 0.001354 70.36 620.53
3.000 18.00 1.567 3.057 0.095 0.000177 0.000166 0.001334 70.96 691.49
1.800 8.64 1.068 2.047 0.066 0.001280 0.001163 0.000337 195.10 1840.24
yo + (0.05 yo)