Download Flow in River Channels

Distributed Flow Routing
Venkatesh Merwade,
Center for Research in Water Resources
Surface Water Hydrology, Spring 2005
Reading: 9.1, 9.2, 10.1, 10.2
Outline
• Flow routing
• Flow equations for Distributed Flow
Routing (St. Venant equations)
– Continuity
– Momentum
• Dynamic Wave Routing
– Finite difference scheme
Flow Routing
• Definition: procedure to determine the flow
hydrograph at a point on a watercourse
from a known hydrograph(s) upstream
• Why do we route flows?
– Account for changes in flow
hydrograph as a flood
wave passes downstream
• This helps in
– Accounting for storages
– Studying the attenuation of
flood peaks
Flow Routing Types
• Lumped (Hydrologic)
– Flow is calculated as function of time, no
spatial variability
– Governed by continuity equation and
flow/storage relationship
• Distributed (Hydraulic)
– Flow is calculated as a function of space and
time
– Governed by continuity and momentum
equations
Flow routing in channels
• Distributed Routing
• St. Venant equations
– Continuity equation
Q A

0
x t
– Momentum Equation
1 Q 1   Q 2 
y
   g  g (So  S f )  0

A t A x  A 
x
What are all these terms, and where are they coming from?
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
Continuity Equation
Q = inflow to the control volume
q = lateral inflow
Q
x
Q
Rate of change of flow
with distance
Q
dx
x
 ( Adx)
t
Elevation View
Change in mass
Reynolds transport theorem
0
Plan View
Outflow from the C.V.
d
d   V .dA

dt c.v.
c. s .
Continuity Equation (2)
Q A

q 0
x t
 (Vy) y

0
x
t
V
y
V y
y

0
x
x t
Conservation form
Non-conservation form (velocity is dependent
variable)
Momentum Equation
• From Newton’s 2nd Law:
• Net force = time rate of change of momentum
d
 F  dt  Vd   VV .dA
c .v .
c. s .
Sum of forces on
the C.V.
Momentum stored
within the C.V
Momentum flow
across the C. S.
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
Momentum Equation
d
 F  dt  Vd   VV .dA
c .v .
c. s .
Sum of forces on
the C.V.
Momentum stored
within the C.V
Momentum flow
across the C. S.
1 Q 1   Q 2 
y



 g  g ( So  S f )  0


A t A x  A 
x
Momentum Equation(2)
2

1 Q 1  Q 
y
   g  g (So  S f )  0

A t A x  A 
x
Local
acceleration
term
Convective
acceleration
term
Pressure
force
term
Gravity
force
term
Friction
force
term
V
V
y
V
 g  g (So  S f )  0
t
x
x
Kinematic Wave
Diffusion Wave
Dynamic Wave
Momentum Equation (3)
1 V V V y


  So  S f
g t g x x
Steady, uniform flow
Steady, non-uniform flow
Unsteady, non-uniform flow
Applications of different forms of momentum
equation
V
V
y
V
 g  g (So  S f )  0
t
x
x
• Kinematic wave: when gravity forces and friction forces
balance each other (steep slope channels with no back
water effects)
• Diffusion wave: when pressure forces are important in
addition to gravity and frictional forces
• Dynamic wave: when both inertial and pressure forces
are important and backwater effects are not negligible
(mild slope channels with downstream control)
Dynamic Wave Routing
Flow in natural channels is unsteady,
nonuniform with junctions, tributaries, variable
cross-sections, variable resistances, variable
depths, etc etc.
Solving St. Venant equations
• Analytical
– Solved by integrating partial differential equations
– Applicable to only a few special simple cases of kinematic waves
• Numerical
– Finite difference
approximation
– Calculations are performed
on a grid placed over the (x,t)
plane
– Flow and water surface
elevation are obtained for
incremental time and
distances along the channel
x-t plane for finite differences calculations
i-1, j+1
i-1, j+1
i+1, j+1
i, j
i+1, j
∆t
i-1, j
∆x
∆x
Cross-sectional view in x-t plane
x-t plane
∆t
h0, Q0, t1
h1, Q1, t1
h2, Q2, t2
h0, Q0, t0
h1, Q1, t0
h2, Q2, t0
∆x
∆x
Finite Difference Approximations
• Explicit
uij 1 uij 1  uij

t
t
• Implicit
u uij 1  uij11  uij  uij1

t
2t
Temporal derivative
Temporal derivative
uij uij1  uij1

x
2x
Spatial derivative
uij11  uij 1
uij1  uij
u

 (1   )
x
x
x
Spatial derivative
Spatial derivative is written using
terms on known time line
Spatial and temporal derivatives use
unknown time lines for computation
Example