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Numerical Hydraulics Open channel flow 1

Numerical Hydraulics Open channel flow 1. Wolfgang Kinzelbach with Marc Wolf and Cornel Beffa. Saint Venant equations in 1D. continuity momentum equation. b(h). l b. A(h). h. z. Saint Venant equations in 1D. continuity (for section without inflow)

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Numerical Hydraulics Open channel flow 1

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  1. Numerical Hydraulics Open channel flow 1 Wolfgang Kinzelbach with Marc Wolf and Cornel Beffa

  2. Saint Venant equations in 1D • continuity • momentum equation b(h) lb A(h) h z

  3. Saint Venant equations in 1D • continuity (for section without inflow) • Momentum equation from integration of Navier-Stokes/Reynolds equations over the channel cross-section:

  4. Saint Venant equations in 1D • In the following we use: • and

  5. Saint Venant equations in 1D The friction can be expressed as energy loss per flow distance: Using friction slope and channel slope Alternative: Strickler/Manning equation for IR

  6. Saint Venant equations in 1D we finally obtain

  7. Approximations and solutions • Steady state solution • Kinematic wave • Diffusive wave • Full equations

  8. Steady state solution(rectangular channel) Solution: 1) approximately, 2) full

  9. Steady state solution(rectangular channel) Approximation: Neglect advective acceleration Normal flow Full solution (insert second equation into first): yields water surface profiles

  10. Classification of profiles hgr = water depth at critical flow hN = water depth at uniform flow Is = slope of channel bottom Igr = critical slope Horizontal channel bottom Is = 0 H2: h > hgr H3: h < hgr

  11. Classification of profiles Mild slope: hN > hgr Is < Igr M1: hN <h > hgr M2: hN > h > hgr M3: hN > h < hgr Steep slope: hN < hgr Is > Igr S1: hN <h > hgr S2: hN < h < hgr S3: hN > h < hgr

  12. Classification of profiles Critical slope hN = hgr IS = Igr C1: hN < h C3: hN > h Negative slope IS < 0 N2: h > hgr N3: h < hgr

  13. Numerical solution(explicit FD method) Subcritical flow: Computation in upstream direction Solve for h(x) Supercritical flow: Computation in downstream direction Solve for h(x+Dx)

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