Fluid Mechanics Unit 4 Notes Flow in Open channel

Flow in Open channel

1. DEPTH ENERGY RELATIONSHIP
Dr A D Katdare

2. ■ To solve the problems in open channel flow, Bakhmete ff (1912) introduced the concept of
specific energy.
■ The specific energy of flow at any section, is defi ned as, energy per unit weight of water
measured with respect to channel bottom as datum.
■ It is denoted by E.
Specific energy
2 2
2
2 2
V Q
E y y
g gA
   
2 2
1 1 2 2
1 2
Bernoulli's equation is given as,
2 2
f
P V P V
z z h
g g g g
 
     
Since, top surface is under atmospheric pressure,
1
0
P
g


And Z is replaced by channel depth.

3. Specific energy
■ Fig. shows flow in an open
channel. The bed slope is
exaggerated for clarity of
diagram.
■ Total energy per unit weight is
given by Bernoulli’s equation.
2
cos
2v
H z d
g

  
Specific energy in a channel s defined as the energ y per unit weight of liquid at
any section of a channel measured with respect to c hannel bottom as datum.
Fig. Specific energy diagram

4. Relationship between specific energy and depth ■ The specific energy is given by,
2
2v
E y
g
 
■ But, V = (Q/A)
2
2
2
Q
E y
gA
 
Substituting in eq for E
•For a given channel, and known
discharge Q, specific energy is a
function of y.
•Thus for a given discharge, and
given channel, specific energy (E) on
X axis and depth (y) on Y axis can be
plotted.
•This curve is known as ‘specific
energy curve’.

5. Relationship between specific energy and depth
■ The specific energy is increasing as
depth is increasing.
■ It is asymptotic to X axis and line at
45
o
with horizontal.
■ The depth at which energy is
minimum is knows as ‘critical depth’
(y
c
).
■ For any other value of specific energy,
there are two values of depth y
1
and
y
2
.
■ y
1
and y
2
are known as ‘alternate
depths’.
Fig. Specific energy curve

6. ■ Flow at minimum energy is called as
‘critical flow’ (Flow corresponding to critical
depth, y
c
)
■ The flow above minimum energy is called
as ‘sub-critical flow (Depth of flow is more
than critical depth)
■ The flow below minimum energy is called
as ‘super-critical flow’ (Depth of flow is less
than critical depth)
■ For any other value of E, there are two
values of depth of flow, y
1
and y
2
.
Relationship between specific energy and depth

7. Criteria for critical flow We have,
2 2
2
2 2V Q
E y y
g gA
   
For critical flow, E should be minimum,
0
dE
dy

Differentiating w.r.t. ‘y’ and keeping ‘Q’ constant,
2
3
1 0
dE Q dA
dy gA dy
  
But, dA = T dy
Therefore, 2 3
Q A
g T

2 3
Q A
g T

This is condition for critical flow.

8. Criteria for critical flow
2 3 2
2
1
=1
Q A Q T
g T A g A
 
But,
,velocity and Hydraulic depth,
QA
V D
AT
 
2
1
V
gD

2
2 2V D
g

Thus, at critical flow, Velocity head is equal to
half the hydraulic depth.

9. Criteria for critical flow
2
1 OR 1
V V
gDgD
 

VgD
Is known as Froud Number.
Therefore,
F
R= 1, is called as
critical flow
.
F
RV 1, is called as
sub
-
critical flow
.
F
R> 1, is called as
super
-
critical flow
.
In
sub
-
critical flow
, velocity of flow is
less than critical velocity
.
In
super critical flow
, velocity of flow is
more than critical velocity
.
2
3
c
q
y
g

Also,

10. ■ With increase in discharge,
there is increase in minimum
specific energy of the flow.
Effect of discharge on specific energy curve
■ For Given Q, Specific energy
curve isconstant.

11. Specific force •The concept of specific force is application of impulse mome ntum
equationtoopenchannelflow.
•Accordingtomomentumequation,therateofchangeofmomentum
in any direction is equal to the sum of all external forces acting on
theliquidmassinthatdirection.

12. Specific force
By using Impulse momentum principle,
 
2 1 1 2
sin
F
Q
V V P P W F
g


    
Since the slope is very small.
 
2 1 1 2
Q
V V P P
g

  
But, we know that,
1 2
1 1 2 2
and
P A z P A z
 
 
D
̅P
TH N
D
̅R
are depths of CG below FLS.
But, Q= A.V = A
1
V
1
= A
2
V
2
1 2
1 2 2 2
2 1
Q Q Q
A z A z
g A A

 
 
    
 
2 2
1 2
1 2
1 2
Q Q
A z A z
gA gA
  
The two sides of
equation may be
generalized as,
2
Q
Az F
gA
 
This eq is known as specific force.

13. Specific force diagram
Similar to Specific energy curve, specific
force curve has 2 limbs AC and CB.
At point C, specific force is minimum.
For any other value of F, there are two
values of depths, which are know as
conjugate depths or sequent depths.

14. Condition for minimum specific energy ■ For minimum specific energy,
2
0 0
dF d Q
Az
dy dy gA
 
  
 
 
 
2
2
0 (1)
Q dA d
Az
gA dy dy
  
But, is moment of arrea about FLS. ther
efore, d( ) in momentum of area
about FLS corrosponding to change 'dy' in depth y.
AzAz
Now, change in momentm = New moment of a
rea abt FLS with depth (y+dy)
- O ld moment of area with depth y
OR
 
( )
2
dy
d Az a z dy Tdy Az
  
   
   
   

15. ■ Neglecting higher power of dy,
NGYDEL A YNI A NIGYDEL A Y
Substituting in (1),
2
2
0
Q
T A
gA
  
2 3 2
Q A A T

This is condition for critical flow (i.e.
when specific force is minimum)