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Distributed Flow Routing

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

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Distributed Flow Routing

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  1. 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

  2. Outline • Flow routing • Flow equations for Distributed Flow Routing (St. Venant equations) • Continuity • Momentum • Dynamic Wave Routing • Finite difference scheme

  3. 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

  4. 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

  5. Flow routing in channels • Distributed Routing • St. Venant equations • Continuity equation • Momentum Equation What are all these terms, and where are they coming from?

  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 Q = inflow to the control volume q = lateral inflow Rate of change of flow with distance Outflow from the C.V. Change in mass Reynolds transport theorem Elevation View Plan View

  8. Continuity Equation (2) 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 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. • 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 Elevation View Plan View

  11. Momentum Equation Sum of forces on the C.V. Momentum stored within the C.V Momentum flow across the C. S.

  12. 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) Steady, uniform flow Steady, non-uniform flow Unsteady, non-uniform flow

  14. Applications of different forms of momentum equation • 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)

  15. Dynamic Wave Routing Flow in natural channels is unsteady, nonuniform with junctions, tributaries, variable cross-sections, variable resistances, variable depths, etc etc.

  16. 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

  17. i-1, j+1 i-1, j+1 i+1, j+1 ∆t i-1, j i+1, j i, j ∆x ∆x Cross-sectional view in x-t plane x-t plane h0, Q0, t1 h1, Q1, t1 h2, Q2, t2 ∆t h0, Q0, t0 h1, Q1, t0 h2, Q2, t0 ∆x ∆x

  18. Explicit Implicit Finite Difference Approximations Temporal derivative Temporal derivative Spatial derivative Spatial derivative Spatial and temporal derivatives use unknown time lines for computation Spatial derivative is written using terms on known time line

  19. Example

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