The document discusses the St. Venant equations which model distributed flow routing in channels. The St. Venant equations consist of the continuity equation and momentum equation. The continuity equation expresses conservation of mass and relates changes in flow over time to changes in flow with distance. The momentum equation applies Newton's second law to a control volume of fluid and relates changes in momentum to forces acting on the control volume such as gravity, friction, pressure, and convective and local acceleration forces. The St. Venant equations can model kinematic, diffusion, or dynamic waves depending on which terms are included or ignored.

1. St Venant Equations
Reading: Sections 9.1 – 9.2

2. 2
Types of flow routing
• Lumped/hydrologic
– Flow is calculated as a function of time alone at a
particular location
– Governed by continuity equation and
flow/storage relationship
• Distributed/hydraulic
– Flow is calculated as a function of space and time
throughout the system
– Governed by continuity and momentum
equations

5. Distributed Flow routing in channels
• Distributed Routing
• St. Venant equations
– Continuity equation
– Momentum Equation
0






t
A
x
Q
What are all these terms, and where are they coming from?
0
)
(
1
1 2



















f
o S
S
g
x
y
g
A
Q
x
A
t
Q
A

6. Assumptions for St. Venant Equations
• Flow is one-dimensional
• Hydrostatic pressure prevails and vertical
accelerations are negligible
• Streamline curvature is small.
• Bottom slope of the channel is small.
• Manning’s equation is used to describe
resistance effects
• The fluid is incompressible

7. Continuity Equation
dx
x
Q
Q



x
Q


t
Adx

 )
(
Q = inflow to the control volume
q = lateral inflow
Elevation View
Plan View
Rate of change of flow
with distance
Outflow from the C.V.
Change in mass
Reynolds transport theorem

 


.
.
.
.
.
0
s
c
v
c
dA
V
d
dt
d



8. Continuity Equation (2)
0






t
A
x
Q
0
)
(






t
y
x
Vy
0









t
y
x
V
y
x
y
V
Conservation form
Non-conservation form (velocity is dependent
variable)

9. Momentum Equation
• From Newton’s 2nd Law:
• Net force = time rate of change of momentum


 


.
.
.
.
.
s
c
v
c
dA
V
V
d
V
dt
d
F 

Sum of forces on
the C.V.
Momentum stored
within the C.V
Momentum flow
across the C. S.

10. Forces acting on the C.V.
Elevation View
Plan View
• Fg = Gravity force due to
weight of water in the C.V.
• Ff = friction force due to shear
stress along the bottom and
sides of the C.V.
• Fe = contraction/expansion
force due to abrupt changes
in the channel cross-section
• Fw = wind shear force due to
frictional resistance of wind at
the water surface
• Fp = unbalanced pressure
forces due to hydrostatic
forces on the left and right
hand side of the C.V. and
pressure force exerted by
banks

11. Momentum Equation


 


.
.
.
.
.
s
c
v
c
dA
V
V
d
V
dt
d
F 

Sum of forces on
the C.V.
Momentum stored
within the C.V
Momentum flow
across the C. S.
0
)
(
1
1 2



















f
o S
S
g
x
y
g
A
Q
x
A
t
Q
A

12. 0
)
( 










f
o S
S
g
x
y
g
x
V
V
t
V
0
)
(
1
1 2



















f
o S
S
g
x
y
g
A
Q
x
A
t
Q
A
Momentum Equation(2)
Local
acceleration
term
Convective
acceleration
term
Pressure
force
term
Gravity
force
term
Friction
force
term
Kinematic Wave
Diffusion Wave
Dynamic Wave

13. Momentum Equation (3)
f
o S
S
x
y
x
V
g
V
t
V
g











1
Steady, uniform flow
Steady, non-uniform flow
Unsteady, non-uniform flow