The document discusses the fundamental principles of fluid mechanics - conservation of mass, energy, and momentum - and how they are applied to derive equations for open channel flow. It specifically covers the continuity, energy, and momentum equations. The energy equation relates changes in energy within a control volume, while the momentum equation relates the overall forces on the control volume boundaries. The document also discusses topics like specific energy, critical flow, hydraulic jumps, and how these concepts are used to analyze channel transitions and design channel flows.

1. The equations which describe the flow of fluid are derived from
three fundamental laws of physics (r/ship of fluid motion):
Conservation of matter (or mass)
Conservation of energy
Conservation of momentum
Assumptions have been done with in the control volume in each
principles.
1.4. Energy & Momentum Principles in Open Channel Flow

2. A. Continuity equation
For any control volume during the small time interval dt the principle of
conservation of mass implies that the mass of flow entering the control
volume minus the mass of flow leaving the control volume equals the
change of mass within the control volume.
If the flow is steady and the fluid incompressible the mass entering is
equal to the mass leaving, so there is no change of mass within the
control volume.

3. B. Energy Equation
oConsider the forms of energy available for the above control volume. If the
fluid moves from the upstream face 1, to the downstream face 2 in time dt
over the length L.
Similarly the total energy per unit weight of section two also computed and consider no energy
is supplied between the inlet and outlet of the control volume, energy leaving equal to energy
entering.

4. Bernoulli's Equation

5. C. Momentum Equation
The law of conservation of momentum says that a moving body cannot gain or
lose momentum unless acted upon by an external force.
The resultant force acting on a free body of fluid in any direction is equal to the
time rate of change of momentum in that direction
The flow may be compressible or incompressible, real (with friction) or ideal
(frictionless), steady or unsteady moreover, the equation is not only valid along a
streamline

6. Energy Equation Momentum Equation
 Relates the change in energy within the
control volume
 Relates the overall forces on the boundary
of the control volume
 Applicable to only steady flows in which
the energy changes are negligible
 Applicable to steady and unsteady flows
 The fluid is ideal, incompressible, one
dimensional
 Conditions within the control volume is not
taken in to consideration.
 The theorem is useful when energy
changes are known and velocity and
pressure distribution are required
 The theorem is useful when energy
changes are unknown and only overall
knowledge of the flow is required
 Used to determine velocity distribution or
pressure distribution
 Useful to determine the resultant force
acting on the boundary of flow passage
 To determine the characteristics of flow
when there is abrupt change of flow
section ( sudden enlargement in pipe,
hydraulic jump..)
 Useful when detailed information of the
flow condition inside the control volume is
not known

7. Application of Bernoulli's Equation for Uniform Flow interrupted by raised
humbs
velocity and depth of flow over the raised hump.

8. Specific Energy Equation
We have to understand
the flow condition

9. In order to adjust the water level in open channel “Channel Transitions” are
incorporated
 Rise in bed elevation
 Drop in bed elevation
 Sudden enlargement in width
 Sudden Contraction in width
The objective here in designing is to minimize the energy loss due to such
channel transitions
Concept of momentum , continuity and specific energy is used to solve such
flow problems

10. Specific Energy
Considering the energy correction factor= 1 and slope is insignificant
The head attained by a fluid element per unit weight wrt channel bed
as a datum:-

11. The two alternate depths represent two totally different flow regimes: slow & deep on the upper limb of the
curve (sub- critical flow) & fast & shallow on the lower limb of the curve, (Super critical flow)
Check this

13. Salient feature of critical flow
Specific energy for a given discharge is minimum.
The discharge for a given specific energy is maximum.
The Froude number is equal to unity
The velocity head is equal to one half of the hydraulic depth

14. Example
A channel of a rectangular section, 7 m wide, discharges water at a rate of 18 m3/s with
an average velocity of 3 m/s. Find: (10 points)
A. Specific-energy head of the flowing water,
B. Depth of water, when specific energy is minimum,
C. Velocity of water, when specific energy is minimum,
D. Minimum specific-energy head of the flowing water,
E. Type of flow.

17. Home Study & Assignment Work
Case 2…….When Specific energy is constant Y= f (Q)
 Critical depth in rectangular channels
 Critical depth for Non rectangular channels
 Computation of critical flow

18. 1.5. Hydraulic Jump
Topics
• Definition
• Impulse momentum equation
• Advantage of Hydraulic jump
• Assumptions made for analysis of hydraulic jump

19. The hydraulic jump is an important feature in open channel flow and is an example of
rapidly varied flow.
A hydraulic jump occurs when a super-critical flow and a sub-critical flow meet. The jump
is the mechanism for the two surfaces to join.
They join in an extremely turbulent manner which causes large energy losses. Because of
the large energy losses the energy or specific energy equation cannot be use in analysis, the
momentum equation is used instead.
If energy actually leaks from the system via frictional head loss the Bernoulli equation will
overstate the energy available to the flow and the related predictions of velocity and
depth will proportionately be in error. To recall our earlier strategy, we minimize this error
by considering only short reaches of channel and only gradual transitions.
In certain flow phenomena, however, we simply can no longer ignore the energy losses
and we must look to alternative ways of describing the flow.

20. When subcritical flow accelerates into the supercritical state the transition often is
smooth with gradually increasing velocity and decreasing depth bringing about a
smooth drop in the water surface until the alternate depth is achieved. Any disturbance
to the water surface is smoothed out by the surface or gravity wave propagation
mechanism discussed earlier.
In these circumstances energy losses are not great and the Bernoulli equation does a
credible job of describing the changes to the flow. When supercritical flow changes to
subcritical flow, however, there is no smoothing of the water surface upstream of the
transition because the high downstream velocity prevents upstream diffusion of the
water-surface deformation.
As a result the transition to subcritical flow is sudden and marked by an abrupt
discontinuity, or hydraulic jump, in the water.

22. Flow over weir ( Look The Linked Video)

23. Purposes of hydraulic jump
To increase the water level on the d/s of the hydraulic structures
To reduce the net up lift force by increasing the downward force due to the
increased depth of water,
To increase the discharge from a sluice gate by increasing the effective head
causing flow
For removing air pockets in a pipe line.

24. Analysis of Hydraulic Jump
Assumptions
The length of the hydraulic jump is small, consequently, the loss of head due to
friction is negligible,
The channel is horizontal as it has a very small longitudinal slope. The weight
component in the direction of flow is negligible.
The portion of channel in which the hydraulic jump occurs is taken as a control
volume & it is assumed the just before & after the control volume, the flow is
uniform & pressure distribution is hydrostatic.
Objective: To describe (drive) geometry of channel undertaking
hydraulic jump
Let us consider a small reach of a channel in which the hydraulic jump occurs.
The momentum of water passing through section (1) per unit time is
given as:

25. The momentum of water passing through section (1) per unit time is given as: