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

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

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 I R

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

10 Classification of profiles h gr = water depth at critical flow h N = water depth at uniform flow I s = slope of channel bottom I gr = critical slope Horizontal channel bottom I s = 0 H 2 : h > h gr H 3 : h < h gr

11 Classification of profiles Mild slope: h N > h gr I s < I gr M1: h N h gr M2: h N > h > h gr M3: h N > h < h gr Steep slope: h N < h gr I s > I gr S1: h N h gr S2: h N < h < h gr S3: h N > h < h gr

12 Classification of profiles Critical slope h N = h gr I S = I gr C1: h N < h C3: h N > h Negative slope I S < 0 gr N2: h > h gr N3: h < h gr

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