05 gradually varied flow slide

IWM 321: Hydraulic Engineering
Chapter 6: Gradually varied Flow
Ajoy Kumar Saha
Assistant Professor
Dept of Irrigation and water management
1

Outline of this course
• General feature of gradually varied flow (GVF)
• Basic Assumption for GVF
• Dynamic equation of GVF
• Classification of Water Surface Profiles
2

General assumption for GVF
• 1 . Flow is steady i.e hydraulic characteristics remains
constant for the time interval under consideration
• 2. The streamlines are practically parallel, so that
hydrostatic distribution of pressure prevails over the
channel section.
3

Basic Assumption for GVF
• 1. Head loss for GVF as similar for a uniform flow
- So Uniform flow formula may be use for energy
slope
• 2. The slope of the channel is small
- vertical depth of flow consider from bottom
- cos is equal to unity
- No air entrance to occurs
4

• 3. flow in prismatic channel having constant
alignment and shape
• 4. velocity-distribution coefficients are constant
5

• 5. Similarly for GVF, conveyance K and section
factor Z are exponential functions of the depth of
flow
• 6. roughness coefficient is independent of depth and
constant throughout the channel reach
6

Dynamic equation of Gradually varied flow
• Consider the profile of gradually
varied flow
• dx elementary length of the open
channel
• Now, total head above the
datum of the channel
............. (1)
7
dx
Fig. 01

Dynamic equation of GVF
• Taking the bottom of the channel as x-axis and differentiating
equation (1) with respect to x, we get
..............(2)
Now Sf is the slope of energy line, it is assumed to be positive
if it is descends in the direction of flow and negative if it
ascends.
Hence, from Fig. energy slope (as flow descending)
8

• And the slope of the channel bottom
Substituting these slopes in quation (1) we found,
......................(3)
9

......................(3)
This the general differential equation for gradually varied flow
also known as the dynamic equation for gradually varied flow.
Here depth d is measured from the bottom of the channel and
channel bottom is considered as x-axis.
10

......................(3)
Thus
Sf = S0 if
Sf < S0 if
Sf < S0 if
In other word water surface profile parallel to the channel bottom
when , rising when , and lowering
11

......................(3)
For small  , cos   1, and d  y and
Putting these values in equation (3) we get,
....................(4)
12

....................(4)
• The term in the varied flow equation represents the
change in velocity head and coefficient  has been assumed
to be constant from section to section of the channel reach.
Since, V= Q/A where, Q is constant
and dA/dy= T
 Velocity term,
13

 Velocity term,
................(5)
Suppose that a critical fow of discharge equal to Q occurs at the
section, then we can write,
...............................(6)
From equation (5) and (6) we get,
...............................(6)
14

So we can rewrite our equation (4) as
...............(7)
From Mannings formula we know, S
When Manning’s or Chezy’s formula express in terms of
conveyance, K the discharge can be written as
 ....................(8)
15

• Suppose that a uniform flow of a discharge equal to Q occurs
in the section. The energy slope would be equal to the bottom
slope, and equation (9) may be written as

Substituting above value in equation (8), we found that
16

17
Classification of Water Surface Profile

Classification of Water Surface Profile
• 5 rows: slope increasing from “adverse” to “steep”
• 3 columns:
Column 1: h > hn & hc (“above”)
dh/ds > 0
Column 2: hn > h > hc or hn < h < hc (“between”)
dh/ds < 0
Column 3: h < hn & hc (“below”)
dh/ds > 0
19

• Practice for Water Surface Profile
27

34
Practice more and more........................

05 gradually varied flow slide

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05 gradually varied flow slide