Distributed Flow Routing
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Transcript of Distributed Flow Routing
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Distributed Flow Routing
Reading: Sections 9.1 – 9.4,
10.1-10.2
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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)(11 2
fo SSgx
yg
A
Q
xAt
Q
A
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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
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Continuity Equation
dxx
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
....
.0scvc
dAVddt
d
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Continuity Equation (2)
0
t
A
x
Q
0)(
t
y
x
Vy
0
t
y
x
Vy
x
yV
Conservation form
Non-conservation form (velocity is dependent variable)
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Momentum Equation
• From Newton’s 2nd Law: • Net force = time rate of change of momentum
....
.scvc
dAVVdVdt
dF
Sum of forces on the C.V.
Momentum stored within the C.V
Momentum flow across the C. S.
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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
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Momentum Equation
....
.scvc
dAVVdVdt
dF
Sum of forces on the C.V.
Momentum stored within the C.V
Momentum flow across the C. S.
0)(11 2
fo SSgx
yg
A
Q
xAt
Q
A
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0)(
fo SSgx
yg
x
VV
t
V
0)(11 2
fo SSgx
yg
A
Q
xAt
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
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Momentum Equation (3)
fo SSx
y
x
V
g
V
t
V
g
1
Steady, uniform flow
Steady, non-uniform flow
Unsteady, non-uniform flow
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Dynamic Wave Routing
Flow in natural channels is unsteady, non-uniform with junctions, tributaries, variable cross-sections, variable resistances, variable depths, etc etc.
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Obtaining river cross-sections
Traditional methods
Depth sounder and GPS
Cross-sections are also extracted from a contour map, DEM, and TIN
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Triangulated Irregular Network
Node
Edge
Face
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3D Structure of a TIN
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Real TIN in 3D!
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TIN for UT campus
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TIN as a source of cross-sections
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CrossSections
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Channel and Cross-Section
Direction of Flow
Cross-Section
Channel
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HEC GeoRAS
• A set of ArcGIS tools for processing of geospatial data for – Export of geometry HEC-RAS – Import of HEC-RAS output for display in GIS
• Available from HEC at• http://www.hec.usace.army.mil/software/hec-ras/hec-georas_downloads.html
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Hydraulic Modeling with Geo-RAS
GIS data HEC-RAS Geometry
HEC-RAS Flood Profiles
Flood display in GIS
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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
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Finite Difference Approximations
• Explicit• Implicit
t
uu
t
u ji
ji
ji
11
x
uu
x
u ji
ji
ji
211
Temporal derivative
Spatial derivative
t
uuuu
t
u ji
ji
ji
ji
21
11
1
Temporal derivative
x
uu
x
uu
x
u ji
ji
ji
ji
1
111 )1(
Spatial derivative
Spatial derivative is written using terms on known time line
Spatial and temporal derivatives use unknown time lines for computation
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Example