1 Numerical Hydraulics Numerical solution of the St. Venant equation, FD-method Wolfgang Kinzelbach with Marc Wolf and Cornel Beffa

2 Basic equations of open channel flow in variables h and v for rectangular channel Continuity („Flux-conservative form“) Momentum equation

3 Basic equations of open channel flow in variables h and q for rectangular channels Continuity Momentum equation

4 Basic equations of open channel flow for general cross-section in variables A and Q Continuity Momentum equation

5 Boundary conditions At inflow boundary usually the inflow hydrograph should be given At the outflow boundary we can use –water level (also time variable e.g. for tide) –water level-flow rate relation (e.g. weir formula) –slope of water level or energy In supercritical flow it can happen that two boundary conditions are necessary for one boundary (for both v and h)

6 Boundary conditions Number of boundary conditions from number of characteristics In 1D: subcritical flow: IB: 1, OB: 1 supercritical flow: IB: 2, OB: 0 IB = Inflow boundary, OB = Outflow boundary t t

7 Discretized basic equations in variables h and v for rectangular channels Continuity Momentum equation Finite differences: Explicit method („Flux-conservative“ form) i = 2,…,Nx

8 Discretized basic equations in variables h and v for rectangular channels Boundary conditions (i = 1) example Explicit method Boundary conditions (i = Nx+1) example or weir formula

9 Discretized basic equations in variables h and v for rectangular channels Explicit method requires stability condition Courant-Friedrichs-Levy (CFL) criterium must be fulfilled: Explicit method c is the relative wave velocity with respect to average flow

10 Assignment: Determine the wave propagation (water surface profile, maximum water depth, outflow hydrograph) for a rectangular channel with the following data: width b = 10 m, k str = 20 m -1/3 /s length L = 10‘000 m, bottom slope I S =0.002 Inflow before wave, base flow Q 0 = 20 m 3 /s Boundary condition downstream: Weir with water depth 2.2 m Boundary condition upstream: Inflow hydrograph Inflow hydrograph  Q (is added to base flow Q 0 ): Zeit (h)  Q (m 3 /s)

11 Inflow/Outflow hydrographs time steps in 10s Q (m 3 /s) about 4 h