Module 6
5 Lectures
Equations Governing Hydrologic and
Hydraulic Routing
Prof. Subhankar Karmakar
IIT Bombay
Objectives of this module is to understand the physical
phenomena behind the Reynolds transport theorem and
Saint Venant equations.
Module 6
Topics to be covered
 Reynolds Transport Theorem
 Control Volume Concept
 Open Channel Flow
 Saint Venant Equations
 Continuity Equation
 Momentum Equation
 Energy Equation
Module 6
Module 6
Lecture 1: Reynolds transport theorem and
open channel flow
Fluids Problems-Approaches
1. Experimental Analysis
2. Differential Analysis
3. Control Volume Analysis
 Looks at specific regions, rather than specific masses
Module 6
Reynolds Transport Theorem
Osborne Reynolds
Reynolds' transport theorem (Leibniz-Reynolds' transport theorem) is a 3-D
generalization of the Leibniz integral rule. The theorem is named after Osborne
Reynolds (1842–1912).
 Control volume: A definite volume specified in space. Matter in a control volume
can change with time as matter enters and leaves its control surface.
 Extensive properties (B) : Properties depend on the mass contained in a fluid,
 Intensive properties (β) : Properties do not depend on the mass.
=
β
dB
=
or dB β dm
dm
d ( m)
= 1 momentum ( mv )
mass (m)  β =
dm
P.E. (mgh)  β = gh
K.E. (1/2 mv2) 
d ( mv )
β =
=v
dm
β = 12 v 2
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Reynolds Transport Theorem
Contd…
Reynolds Transport Theorem:
The total rate of change of any extensive property B (=βdm = βρd∀) of a system
occupying a control volume C.V. at time ‘t’ is equal to the sum of:
a) the temporal rate of change of B within the C.V.
b) the net flux of B through the control surface C.S. that surrounds the C.V.
dB ∂
= ∫∫∫ βρd∀ + ∫∫ βρ Vrel .dA
dt ∂t c .v .
c.s.
This theorem applies to any transportable property,
including mass, momentum and energy.
Module 6
Reynolds Transport Theorem
Contd…
 Reynolds Transport Theorem can be applied to a control volume of finite size
 No flow details within the control volume is required
 Flow details at the control surfaces is required
Here, control volume is the sum
of I & II
fluid particles at time ‘t’
o fluid particles at time ‘t+∆t’
We will use Reynolds Transport Theorem to solve
many practical fluids problems
Module 6
Let us consider the following system,
I
II
III
Fixed frame in space
(upper and lower boundaries are impervious)
- Control volume position occupied by fluid at time ‘t’
- Position occupied by fluid at time ‘t+∆t’
I
II
- Position occupied by fluid at time ‘t’, but not at ‘t+∆t’
- Position occupied by fluid at both the time ‘t’ and ‘t+∆t’
III - Position occupied by fluid at time ‘t+∆t’, but not at ‘t’
Finally, control volume at time ‘t’  I&II and at time ‘t+∆t’ II&III
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1
dB
[(BII + BIII )t +∆t − (BI + BII )t ]
= Lt
dt sys ∆t →0 ∆t
...(6.1)
Here, dB = βdm = β (ρd∀)
dm
where ρ =
d∀
d∀ = elemental volume
dm = mass of the fluid contained in d∀
From eq.(6.1)
1
dB
{[(BII )t +∆t − (BII )t ] − [(BIII )t +∆t − (BI )t ]}
= Lt
dt sys ∆t →0 ∆t
1st term
2nd term
1st term is equivalent to the change in extensive property stored in control
volume (as ∆t 0)
Module 6
Extensive property in the control volume,
Here, area vector is always normally outward to the surface
θ

dA

V
∆l
control surface
d A = Area vector of magnitude dA
v = velocity vector
Volume of
fluid pas sin g through " dA" in time " ∆t"
= dA(− ∆l cos θ )
= −∆l cos θ .dA
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Total volume of
fluid pas sin g through the control surface
= − ∫∫ dA(∆l cos θ )
c.s.
Note :
∂
(B ) ⇒ Change in extensive properties only
∂t
d
(B ) ⇒ Change in extensive properties + Shape change
dt
⇒
dB
= Total rate of change of extensive property of
dt sys
fluid
= ( Rate of change of extensive properties in the control volume) +
( Net outflow of extensive property through control surface)
Module 6
Consider B to be mass. Now, as per law of conservation of mass,
dBsys
=
d (mass )
=0
dt
dt
d
∴ ∫∫∫ ρd∀ + ∫∫ ρV .dA = 0
......(6.2)
dt c .v .
c.s.
This is the equation of var iable density unsteady flow.
d
For steady − state flow,
ρd∀ = 0
∫∫∫
dt c .v .
(i ) ρ is cons tan t
Assumptions :
d
d∀ + ∫∫ V .dA = 0
∫∫∫
dt c .v .
c.s.
∴ ∫∫∫ d∀ = S and
c .v .
∫∫ V .dA = Q(t ) − I (t ), where
c.s.
S = storage, Q(t ) = outflow(+ ve) and I (t ) = inf low(−ve)
Module 6
dS
+ Q(t ) − I (t ) = 0
dt
dS
or
= I (t ) − Q(t ) = 0, hence proved .
dt
i.e
Reynolds transport
equation becomes,
Steady State
(ii ) ρ is cons tan t and steady flow
dS
= 0, then I (t ) = Q(t )
dt
∴ dS = I (t )dt − Q(t )dt
dBsys
∴
Sj
j∆t
j∆t
S j −1
( j −1) ∆t
( j −1) ∆t
∴ ∫ dS =
∫ I (t )dt − ∫ Q(t )dt
( S j − S j −1 ) = I j − Q j
where S j = storage at time j ,
dt
 
= ∑ βρV ⋅ A
CS
Unsteady State
 
d
= ∫ βρd∀ + ∑ βρV ⋅ A
CS
dt
dt CV
dBsys
S j −1 = storage at time j − 1,
I j = Inflow int o the system between the time t j −1 and t j ,
Q j = Outflow from the system between the time t j −1 and t j .
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If
j = 1,
( S1 − S 0 ) = I1 − Q1
......(6.3)
j = 2,
( S 2 − S1 ) = I 2 − Q2
......(6.4)
From Eq.(3), S1 = S 0 − Q1 + I 1
......(6.5)
Substituting value of
S1 in (6.4),
S1 − S 0 + Q1 − I 1 = I 2 − Q2
or S 2 = S 0 + ( I 1 − Q1 ) + ( I 2 − Q2 )
Thus,
j
S j = S 0 + ∑ ( I t − Qt )
t =1
Module 6
Open channel flow
 An open channel is a waterway, canal or conduit in which a liquid flows
with a free surface.
 In most applications, the liquid is water and the air above the flow is
usually at rest and at standard atmospheric pressure.
udel.edu/~inamdar/EGTE215/Open_channel.pdf
Module 6
Pipe flow vs. Open channel flow
Flow driven by
Pipe flow
Open channel flow
Pressure work
Gravity(i.e. Potential Energy)
Flow cross-section Known (Fixed by
Varies based on the depth of flow
geometry)
Characteristic flow Velocity deduced from
Flow depth and velocity deduced by
parameters
solving simultaneously the
continuity equation
continuity and momentum
equations
Specific boundary
Atmospheric pressure at the water
conditions
surface
(Source:www.uq.edu.au/~e2hchans/reprints/b32_chap01.pdf)
Different flow conditions in an open channel
Section 1 – rapidly varying flow
Section 2 – gradually varying flow
Section 3 – hydraulic jump
Section 4 – weir and waterfall
Section 5 – gradually varying
Section 6 – hydraulic drop due to
change in channel slope
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Open Channel Flow
Unsteady
Steady
Varied
Uniform
Varied
Gradually
Gradually
Rapidly
Rapidly
(NPTEL, Computational Hydraulics)
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Different Types of flow in an open channel
1.
Steady Uniform flow
2.
Steady Gradually-varied flow
3.
Steady Rapidly-varied flow
4.
Unsteady flow
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Case (1) – Steady uniform flow:
 Steady flow is where there is no change with time, ∂/∂t = 0.
 Distant from control structures, gravity and friction are in balance, and if the
cross-section is constant, the flow is uniform, ∂/∂x = 0
Case (2) – Steady gradually-varied flow:
 Gravity and friction are in balance here too, but when a control is introduced
which imposes a water level at a certain point, the height of the surface varies
along the channel for some distance.
i.e. ∂/∂t = 0, ∂/∂x ‡ 0.
Case (3) – Steady rapidly-varied flow:
 Here depth change is rapid.
Case (4) – Unsteady flow:
 Here conditions vary with time and position as a wave traverses the waterway.
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Types of Open Channel Flows
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Hydraulic Routing
Momentum
Equation
Physics of water
movement
Hydraulic
routing
Hydraulic routing is intended to describe the dynamics of the water or flood wave
movement more accurately
Hydrological routing
Continuity equation + f (storage, outflow, and possibly inflow) relationships 
assumed, empirical, or analytical in nature. Eg: stage-discharge relationship.
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Dynamic Routing- Advantages
 Higher degree of accuracy when modeling flood situations because it
includes parameters that other methods neglect.
 Relies less on previous flood data and more on the physical properties of
the storm. This is extremely important when record rainfalls occur or other
extreme events.
 Provides more hydraulic information about the event, which can be used to
determine the transportation of sediment along the waterway.
Module 6