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Numerical Hydraulics Open channel flow 2

Numerical Hydraulics Open channel flow 2. Wolfgang Kinzelbach with Marc Wolf and Cornel Beffa. Approximations. Kinematic wave Diffusive wave Dynamic wave (Full equations). Example: Rectangular channel. Approximations.

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Numerical Hydraulics Open channel flow 2

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  1. Numerical Hydraulics Open channel flow 2 Wolfgang Kinzelbach with Marc Wolf and Cornel Beffa

  2. Approximations • Kinematic wave • Diffusive wave • Dynamic wave (Full equations) Example:Rectangular channel

  3. Approximations In the different approximations different terms in the equation of motion are neglected against the term gIS: 1. Approximation: Kinematic wave 2. Approximation: Diffusive wave Complete solution: Dynamic wave

  4. Kinematic wave • Normal flow depth. Energy slope is equal to channel bottom slope. Therefore Q is only a function of water depth. E.g. using the Strickler/Manning equation: • Inserting into the continuity equation yields • This is the form of a wave equation (see pressure surge) with wave velocity w = v+c Instead of using Q=Q(h) the equation can be derived using v=v(h)

  5. Kinematic wave • With the Strickler/Manning equation we get: • For a broad channel the hydraulic radius is approximately equal to the water depth. The wave velocity then becomes and

  6. Kinematic wave • The wave velocity is not constant as v is a function of water depth h. • Varying velocities for different water depth lead to self-sharpening of wave front • Pressure propagates faster than the average flow. • Advantage of approximation: PDE of first order, only one upstream boundary condition required. • Disadvantage of approximation: Not applicable for bottom slope 0. No backwater feasible as there is no downstream boundary condition.

  7. Diffusive wave • Now Q is not only a function of h but also of h/ x. • Insertion into the continuity equation yields: with IR = IR(Q/A) from Strickler or Darcy-Weisbach

  8. Diffusive wave • This equation has the form of an advection-diffusion equation with a wave velocity w=v+c and a diffusion coefficient D: with

  9. Diffusive wave • Using the Strickler/Manning equation and assuming a broad rectangular channel (h = Rhy) one obtains: and Insertion into the continuity equation yields with and

  10. Diffusive wave • D is always positive, as the energy slope is always positive in flow direction. • The wave moves downstream and flattens out diffusively. A lower boundary condition is necessary because of the second derivative. This allows the implementation of a backwater effect.

  11. St. Venant equation as wave equation Linear combinations: Multiply second equation with ± l and add to first equation

  12. St. Venant equation as wave equation Write derivatives of h and v as total derivatives along a characteristic line: the two characteristics have the same relative wave velocity c (with respect to average water velocity v). Choosing

  13. St. Venant equation as wave equation The characteristics are therefore: and the relative wave velocity for shallow water waves is In contrast to the surge in pipes, v cannot be neglected in comparison to c!

  14. St. Venant equation as wave equation: alternative view Continuity channel flow Continuity pipe flow using h=p/rg: Comparison with the pipe flow case shows the equivalence and with

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