Fluid Mechanics Unit 4 Notes Flow in Open channel

DEPTH ENERGY RELATIONSHIP
Dr A D Katdare

■ To solve the problems in open channel flow, Bakhmeteff (1912) introduced the concept of
specific energy.
■ The specific energy of flow at any section, is defined 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.

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
2
v
H z d
g

  
Specific energy in a channel s defined as the energy per unit weight of liquid at
any section of a channel measured with respect to channel bottom as datum.
Fig. Specific energy diagram

Relationship between specific energy and depth
■ The specific energy is given by,
2
2
v
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’.

■ The specific energy is increasing as
depth is increasing.
■ It is asymptotic to X axis and line at
45o with horizontal.
■ The depth at which energy is
minimum is knows as ‘critical depth’
(yc).
■ For any other value of specific energy,
there are two values of depth y1 and
y2.
■ y1 and y2 are known as ‘alternate
depths’.
Fig. Specific energy curve

■ Flow at minimum energy is called as
‘critical flow’ (Flow corresponding to critical
depth, yc)
■ 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, y1 and y2.

Criteria for critical flow
We have,
2 2
2
2 2
V 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.

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

2
1 OR 1
V V
gD gD
 
V
gD
Is known as Froud Number.
Therefore,
FR = 1, is called as critical flow.
FR V 1, is called as sub-critical flow.
FR > 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,

■ 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 is constant.

Specific force
• The concept of specific force is application of impulse momentum
equation to open channel flow.
• According to momentum equation, the rate of change of momentum
in any direction is equal to the sum of all external forces acting on
the liquid mass in that direction.

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
 
 
̅ ̅ are depths of CG below FLS.
But, Q= A.V = A1V1 = A2V2
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.

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.

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. therefore, d( ) in momentum of area
about FLS corrosponding to change 'dy' in depth y.
Az Az
Now, change in momentm = New moment of area abt FLS with depth (y+dy)
- Old moment of area with depth y
OR
 
( )
2
dy
d Az a z dy Tdy Az
 
 
   
 
 
 
 

■ Neglecting higher power of dy,
( ̅) = = ( ̅) =
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)

Fluid Mechanics Unit 4 Notes Flow in Open channel

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Fluid Mechanics Unit 4 Notes Flow in Open channel