Equation of gradually varied flow and its limitations, Flow classification and surface profiles, Integration of varied flow equation by analytical, graphical
and numerical methods, Flow in channels of non-linear alignment specifically for the
case of a bend.
UNIT 3
05/03/2023 1MODASSAR ANSARI
BY MODASSAR ANSARI 2nd Year Department of civil Engineering SUBJECT- HYDRAULICS & HYDRAULIC
MACHINES SUBJECT CODE-NCE 403
HYDRAULIC & HYDRAULIC MACHINES
05/03/2023 2MODASSAR ANSARI
The flow in an open-channel is termed as gradually varied flow (GVF) when the depth of flow varies gradually with longitudinal distance. Such flows are encountered both on upstream and downstream sides of control sections. Analysis and computation of gradually varied flow profiles in open-channels are important from the point of view of safe and optimal design and operation of any hydraulic structure.
Gradually varied flow
05/03/2023 3MODASSAR ANSARI
Find Change in Depth wrt x
Gradually Varied Flow
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2 21 2
1 22 2o fV Vy S x y S xg g
energy equation for non-uniform, steady flow
12 yydy
2
2 f oVdy d S dx S dxg
PA
T
dyy
dydxS
dydxS
gV
dyd
dydy
of
2
2
2 2
2 12 1 2 2o f
V VS dx y y S dxg g
Shrink control volume
4MODASSAR ANSARI
Governing equation
So and Sf are positive when sloping down in direction of flow
y is measured from channel bottomdy/dx =0 means water depth is _______
Gradually Varied Flow:
05/03/2023
21 Fr
SS
dx
dy fo
yn is when o fS S=constant
5MODASSAR ANSARI
Mild slope (yn>yc)◦ in a long channel subcritical flow will occur
Steep slope (yn<yc)◦ in a long channel supercritical flow will occur
Critical slope (yn=yc)◦ in a long channel unstable flow will occur
Horizontal slope (So=0)◦ yn undefined
Adverse slope (So<0)◦ yn undefined
Surface Profiles
05/03/2023 6MODASSAR ANSARI
Normal depthSteep slope (S2)Hydraulic Jump
Sluice gateSteep slope
Obstruction
Surface Profiles
05/03/2023
21 Fr
SS
dx
dy fo
S0 - Sf 1 - Fr2 dy/dx
+ + +
- + -
- - + 0
1
2
3
4
0 1 2 3 4
E
y
ynyc
7MODASSAR ANSARI
More Surface Profiles
05/03/2023
S0 - Sf 1 - Fr2 dy/dx
1 + + +
2 + - -
3 - - +
21 Fr
SS
dx
dy fo
8MODASSAR ANSARI
Direct Step Method
05/03/2023
xSg
VyxS
g
Vy fo
22
22
2
21
1
of SS
g
V
g
Vyy
x
22
22
21
21
energy equation
solve for x
1
1y
qV
2
2y
qV
2
2A
QV
1
1A
QV
rectangular channel prismatic channel
9MODASSAR ANSARI
Direct Step Method Friction Slope
05/03/2023
2 2
4 /3fh
n VSR
=2 2
4/32.22fh
n VSR
=
2
f8f
h
VSgR
=
Manning Darcy-Weisbach
Si units
english units
10MODASSAR ANSARI
Limitation: channel must be PRISMATIC (channel geometry is independent of x so that velocity is a function of depth only and not a function of x)
Methodidentify type of profile (determines whether Dy is + or
-)choose Dy and thus yi+1calculate hydraulic radius and velocity at yi and yi+1calculate friction slope given yi and yi+1 calculate average friction slope calculate Dx
Direct Step
05/03/2023 11MODASSAR ANSARI
Given a depth at one location, determine the depth at a second given location
Step size (x) must be small enough so that changes in water depth aren’t very large. Otherwise estimates of the friction slope and the velocity head are inaccurate
Can solve in upstream or downstream directionUsually solved upstream for subcriticalUsually solved downstream for supercritical
Find a depth that satisfies the energy equation
Standard Step
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xSg
VyxS
g
Vy fo
22
22
2
21
1
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0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
05101520
elev
atio
n (m
)
distance upstream (m)
bottom
surface
yc
yn
What curves are available?Steep Slope
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S1
S3
is there a curve between yc and yn that increases in depth in the downstream direction? 13MODASSAR ANSARI
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0510152025303540
elev
atio
n (m
)
distance upstream (m)
bottom
surface
yc
yn
If the slope is mild, the depth is less than the critical depth, and a hydraulic jump occurs, what happens next?
Mild Slope
05/03/2023
Rapidly varied flow!
When dy/dx is large then V isn’t normal to csHydraulic jump!
Check conjugate depths
14MODASSAR ANSARI
All the complications of pipe flow plus additional parameter... _________________
Various descriptions of energy loss◦ Chezy, Manning, Darcy-Weisbach
importance of Froude Number◦ Fr>1 decrease in e gives increase in y◦ Fr<1 decrease in e gives decrease in y◦ Fr=1 standing waves (also min e given Q)
Summary
05/03/2023
free surface location
0
1
2
3
4
0 1 2 3 4
E
y
15MODASSAR ANSARI
Methods of calculating location of free surface (Gradually varying)◦ Direct step (prismatic channel)◦ Standard step (iterative)◦ Differential equation
Rapidly varying◦ Hydraulic jump
Summary
05/03/2023
21 Fr
SS
dx
dy fo
16MODASSAR ANSARI
Specific energy
Two depths with same energy◦ How do we know which depth is the right one?◦ Is the path to the new depth possible?
Energy equation
05/03/2023
2 21 2
1 22 2o fV Vy S x y S xg g
+ + D = + + D
2
22qygy
= +g
VyE
2
2
2
22QygA
= +
17MODASSAR ANSARI
Specific Energy-Depth graph
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0
1
2
3
4
0 1 2 3 4
E
y
18MODASSAR ANSARI