st venant equations
DESCRIPTION
St Venant Equations. Reading: Sections 9.1 – 9.2. 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 - PowerPoint PPT PresentationTRANSCRIPT
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St Venant Equations
Reading: Sections 9.1 – 9.2
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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
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Distributed Flow routing in channels
• Distributed Routing• St. Venant equations
– Continuity equation
– Momentum Equation
0
tA
xQ
What are all these terms, and where are they coming from?
0)(11 2
fo SSgxyg
AQ
xAtQ
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
dxxQQ
xQ
tAdx
)(
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
dAVddtd
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Continuity Equation (2)
0
tA
xQ
0)(
ty
xVy
0
ty
xVy
xyV
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
dAVVdVdtdF
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
dAVVdVdtdF
Sum of forces on the C.V.
Momentum stored within the C.V
Momentum flow across the C. S.
0)(11 2
fo SSgxyg
AQ
xAtQ
A
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0)(
fo SSgxyg
xVV
tV
0)(11 2
fo SSgxyg
AQ
xAtQ
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 SSxy
xV
gV
tV
g
1
Steady, uniform flow
Steady, non-uniform flow
Unsteady, non-uniform flow