Brief discussion about open channel flow

1.  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.
1/7/2017 1MODASSAR ANSARI

2.  BY MODASSAR ANSARI
 2nd Year
 Department of civil Engineering
 SUBJECT- HYDRAULICS & HYDRAULIC
MACHINES
 SUBJECT CODE-NCE 403
1/7/2017 2MODASSAR ANSARI

3.  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.
1/7/2017 3MODASSAR ANSARI

4.  Find Change in Depth wrt x
1/7/2017
2 2
1 2
1 2
2 2
o f
V V
y S x y S x
g g
      
energy equation for non-
uniform, steady flow
12 yydy 
2
2
f o
V
dy d S dx S dx
g
 
   
 
P
A
T
dy
y
dy
dx
S
dy
dx
S
g
V
dy
d
dy
dy
of 









2
2
 
2 2
2 1
2 1
2 2
o f
V V
S dx y y S dx
g g
 
     
 
Shrink control volume
4MODASSAR ANSARI

5.  Governing equation
 So and Sf are positive when sloping down in
direction of flow
 y is measured from channel bottom
 dy/dx =0 means water depth is _______
1/7/2017
2
1 Fr
SS
dx
dy fo



yn is when o fS S=
constant
5MODASSAR ANSARI

6.  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
1/7/2017 6MODASSAR ANSARI

7. Normal depth
Steep slope (S2)
Hydraulic Jump
Sluice gate
Steep slope
Obstruction
1/7/2017
2
1 Fr
SS
dx
dy fo


 S0 - Sf 1 - Fr2 dy/dx
+ + +
- + -
- - + 0
1
2
3
4
0 1 2 3 4
E
y
yn
yc
7MODASSAR ANSARI

8. 1/7/2017
S0 - Sf 1 - Fr2 dy/dx
1 + + +
2 + - -
3 - - +
2
1 Fr
SS
dx
dy fo



8MODASSAR ANSARI

9. 1/7/2017
xS
g
V
yxS
g
V
y fo 
22
2
2
2
2
1
1
of SS
g
V
g
V
yy
x



22
2
2
2
1
21
energy equation
solve for x
1
1
y
q
V 
2
2
y
q
V 
2
2
A
Q
V 
1
1
A
Q
V 
rectangular channel prismatic channel
9MODASSAR ANSARI

10. 1/7/2017
2 2
4/3f
h
n V
S
R
=
2 2
4/3
2.22
f
h
n V
S
R
=
2
f
8
f
h
V
S
gR
=
Manning Darcy-Weisbach
Si units
english units
10MODASSAR ANSARI

11. 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)
Method
 identify type of profile (determines whether Dy is + or -)
 choose Dy and thus yi+1
 calculate hydraulic radius and velocity at yi and yi+1
 calculate friction slope given yi and yi+1
 calculate average friction slope
 calculate Dx
1/7/2017 11MODASSAR ANSARI

12.  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 direction
 Usually solved upstream for subcritical
 Usually solved downstream for supercritical
 Find a depth that satisfies the energy equation
1/7/2017
xS
g
V
yxS
g
V
y fo 
22
2
2
2
2
1
1
12MODASSAR ANSARI

13. 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
05101520
elevation(m)
distanceupstream(m)
bottom
surface
yc
yn
1/7/2017
S1
S3
is there a curve between yc and yn that
increases in depth in the downstream
direction? 13MODASSAR ANSARI

14. 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0510152025303540
elevation(m)
distanceupstream(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?
1/7/2017
Rapidly varied flow!
When dy/dx is large
then V isn’t normal
to cs
Hydraulic jump!
Check conjugate
depths
14MODASSAR ANSARI

15.  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)
1/7/2017
free surface
location
0
1
2
3
4
0 1 2 3 4
E
y
15MODASSAR ANSARI

16.  Methods of calculating location of free surface (Gradually
varying)
◦ Direct step (prismatic channel)
◦ Standard step (iterative)
◦ Differential equation
 Rapidly varying
◦ Hydraulic jump
1/7/2017
2
1 Fr
SS
dx
dy fo



16MODASSAR ANSARI

17.  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?
1/7/2017
2 2
1 2
1 2
2 2o f
V V
y S x y S x
g g
+ + D = + + D
2
2
2
q
y
gy
= +
g
V
yE
2
2

2
2
2
Q
y
gA
= +
17MODASSAR ANSARI

18. 1/7/2017
0
1
2
3
4
0 1 2 3 4
E
y
18MODASSAR ANSARI