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

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 I R = I R (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 = R hy ) one obtains: and Insertion into the continuity equation yields withand

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 ± 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: Choosing the two characteristics have the same relative wave velocity c (with respect to average water velocity v).

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

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