Module-III Varried Flow.pptx GVF Definition, Water Surface Profile Dynamic Equation of GVF, Direct Step Method

HYDRAULICS-II
(BTCVC401)
Module-III Varied Flow
By
Nitin U. Sonavane
Assistant Professor
Dept. of Civil Engineering
VDF School of Engineering & Technology,
Latur-413531 [MS]

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

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.

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

 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.

 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

 FLOW CLASSIFICATION
• RVF: Rapidly-varied flow
• GVF: Gradually-varied flow

Figure 1: Examples for gradually varied flow in open channels.

 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.

 ANALYSIS OF GRADUALLY-VARIED FLOW

 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

• 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



3
2
3
2
2
2
2
2
2
A
g
B
Q
dx
dh
-
=
dh
dA
A
g
Q
dx
dh
-
=
A
g
2
Q
dh
d
dx
dh
=
dh
dh
A
g
2
Q
dx
d
=
g
2
V
dx
d
























S
-
dx
dh
A
g
B
Q
-
1
=
S
-
dx
dh
+
A
g
B
Q
dx
dh
-
=
S
- 0
3
2
0
3
2








A
g
/
B
Q
-
1
S
-
S
=
dx
dh
3
2
0
Fr
-
1
S
-
S
=
dx
dh
2
0 General governing
Equation for GVF
If dh/dx is positive the depth is increasing otherwise
decreasing

 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)

 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 .

 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

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.

• 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

 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

Classification of profiles according to dy / dl or (dh/dx).
Fr
-
1
S
-
S
=
dx
dh
2
0

 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

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.

Computation of Water
Surface Profiles

 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)

 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

 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

 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

 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

 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

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.

   
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)

Module-III Varried Flow.pptx GVF Definition, Water Surface Profile Dynamic Equation of GVF, Direct Step Method

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Module-III Varried Flow.pptx GVF Definition, Water Surface Profile Dynamic Equation of GVF, Direct Step Method