1. Free surface flows in Code Saturne Results 21/09/2009 Olivier Cozzi

2. Equations of the problem  Mass Conservation Law  Momentum Conservation Law  Scalar Conservation Law +  Space Conservation Law  respected when the mesh just moves vertically +  Kinetic boundary condition  on the free surface, that is to say:  Dynamic boundary condition  (because, on the free surface, sheer stress, normal stress, and effect of the surface tension can be neglected)

3. Test cases From “Application du prototype de module ALE du solveur commun a des cas de surface libre” (H2000H400170), by F. Archambeau EDF : closed tank (and solitary wave, almost ready...)  Wave amplitude A = 2m  Wavelength λ = 0.5L  Mesh: 105*20*1  Initial shape and 2 nd order theoretical solution (Chabert d'Hieres formula):  Airy's formula:  T = 6s period in this case

4. Computation of the free surface ALE method used to move the mesh verticaly, according to the free surface speeds calculated by:  1) Non iterative explicit Euler scheme  2) Non iterative RK4 scheme  3) Iterative explicit Euler scheme  4) Iterative Crank-Nicolson scheme

5. Non iterative explicit Euler scheme On the free surface, the mesh vertical speeds at time step n+1, i.e. w n+1 are calculated with the values of time step n  Results:  Increasing waveheight  Real period bigger than theoretical one

6. Non iterative RK4 scheme On the free surface, the mesh vertical speeds at time step n+1, i.e. w n+1 are calculated with the values of time step n and predictions where  Results better than Euler time scheme but still:  Increasing waveheight  Real period bigger than theoretical one

7. Iterative explicit Euler scheme On the free surface, the mesh vertical speeds at time step n+1, i.e. w n+1 are calculated with iteration moving the mesh before solving the Navier-Stokes equations Start of time step tn+1 Calculation of wn+1 from values of time step tn Moving of the mesh according to wn+1 values Solution of NS equations Calculation of wn+1 thanks to new values Moving of the mesh according to new wn+1 values Solution of NS equations Etc…Final wn+1 values

8. Iterative explicit Euler scheme In the end, the iterative scheme converge to a final solution of w n+1, which respects the Navier Stokes equations in the new geometry  Results different:  Damping of waveheight  Real period bigger than theoretical one

9. Iterative Crank-Nicolson scheme where n+1* values are calculated iteratively and converge to final values, and so to a final solution of w n+1  Results better than Euler time scheme:  Damping of waveheight reduced  Real period bigger than theoretical one

10. And now ?  New test case of the solitary wave  Verification of the mass flow values  Iterative RK4 scheme ?  Any ideas ?  …