1. Numerical and Physical Experiments of Wave Focusing in Short- Crested Seas Félicien Bonnefoy, Pierre Roux de Reilhac, David Le Touzé and Pierre Ferrant Ecole Centrale de Nantes, France Rogue Waves’2004, Brest

2. Topic of the talk Generation of deterministic wave packets in a numerical or physical wave basin Wavemaker motion Fully-nonlinear waves In 2 and 3 dimensions (mono- and multidirectional) Numerical tool: Spectral method with high-order technique Non-periodic Specific treatment for wavemaking Wave elevation record Time in s.

3. Theoretical Framework Potential flow theory and Free surface potential Fully-nonlinear free surface conditions on

4. unknown: the only non surfacic quantity Time-marching Runge-Kutta 4 Nonlinear Free surface equations Time evolution strategy on Separately approximated by an High-Order technique

5. Standard Higher-Order Techniques The two main methods available are: –Higher-Order Spectral HOS ( West et al 1987, Dommermuth and Yue 1987 ) Formal and Taylor series expansion of the potential only (not for equations) to obtain the vertical velocity. –Dirichlet to Neumann Operator DNO ( Craig and Sulem 1993 ) Formal and Taylor series expansion of the DNO only (not for equations) ie the normal velocity. Decomposition in recursive Dirichlet problems solved by Fourier spectral method and collocation nodes

6. High-Order Method Advantages : –Fast solvers with computational costs in O(NlogN) thanks to the use of Fast Fourier Transforms.  Large number of wave components for random seas or steep wave fields. –High accuracy of the spectral methods Limitations of the HOS method: –Non-breaking cases Steep wave field involves high order nonlinearities –increase the number of modes –dealiase carefully –Sawtooth instabilities for very steep wave calculations  Standard five-point smoothing applied regularly through the steepest simulations or decrease of the number of modes Standard Higher-Order Simulations –Periodic boundary conditions on the free surface –Initial stage: Free surface elevation and potential specified at t=0 –Pneumatic wave generation F(x+L x, y) = F(x, y) F(x, y+L y ) = F(x, y)

7. A High-Order Approach for Wave Basin Basin with rigid walls By simply changing the basis functions on which we expand our solution : The natural modes of the basin Cosine functions Still possible to use Fast Fourier Transforms A wavemaker to generate the waves starting from rest (no initial wave description required) The concept of additionnal potential (Agnon and Bingham 1999 ) Inlet flux condition solved in an extended basin

8. Resolution of the additional potential in a extended basin Extensively validated in a previous second order model ( Bonnefoy et al ISOPE’02, Bonnefoy et al OMAE’04 ) In 3D: segmented wavemaker Improved control laws (Dalrymple method for large wave angles) Wavemaker modelling Inlet Flux Condition Also solved by spectral method

9. Applications Improved deterministic reproduction technique in 2D Deterministic reproduction of directional focused wave packets in 3D Wave elevation record Time in s. Wavemaker motion

10. Deterministic reproduction in 2D Wavemaker motion to reproduce this wave field –Control in the frequency domain with a set of components: amplitude, phase (+ angle in case of 3D generation) Wave probe Wavemaker Wave elevation record Time in s. Characteristics: Steep wave packet: k p A l = 0.26 Asymetric in time Basin dimensions: 50m long 5m deep

11. Analytical methods Linear backward propagation: reverse phase method ( e.g. Mansard and Funke 1982 ) Second-order bound correction of amplitudes ( e.g. Duncan and Drake 1995 ) Wave probe Wavemaker Wave elevation record Time in s. Wavemaker motion Wavemaker Transfer Function

12. Iterative corrections Target Before iteration After 5 iterations Initial guess Non-linear simulation Correction on amplitudes and/or phases Corrected motion n=1 h n (t) Comparison to the target after Fourier analysis Target wave Time in s. Target First order input Second order input Time in s. Without iteration With iterations Elevation in m.

13. A first step towards higher order control of nonlinearities In litterature: analytical-empirical approach Clauss et al (OMAE’04) Crest and trough focusing Johannessen et Swan (PRSL 2001) Zang et al (OMAE 2004) Bateman (PhD Thesis 2001) Separation in odd and even orders Phase modification by third order effects is present in odd and even elevation

14. Validation with a small amplitude wave packet Second order effects 10 cm amplitude wave packet (at the focusing point) for 5 m mean wavelength Nonlinear effects reduced to second order Good agreement between first order and odd elevation, and between second order and even elevation First order Odd elevation Second order Even elevation Target wave packet Crest focusing Trough focusing Measured elevations Odd-even decomposition Time in s.

15. Third order effects for higher wave amplitude Resonant Interactions No instabilities detected (in contrast with Johannessen and Swan (PRSL 2001) Phase velocity Non resonant Interactions Bound terms Example with a 30 cm wave packet Odd and linear elevationEven and second order elevation linear phase velocity nonlinear phase velocity even elevation linear phase velocity nonlinear phase velocity odd elevation Time in s. Elevation in m. To build the linear elevation

16. Application to deterministic reproduction Initial decomposition : linear Initial decomposition : second order The main features of the focused target wave packet are well reproduced with only one correction of the wavemaker motion (no iteration so far) Central crest and lateral troughs are close to the target both in amplitudes and phases Central crest amplitude is correctly estimated Better control of the high-frequency waves Time in s. Elevation in m. Wavemaker motion corrected with the phase shift due to nonlinear phase speed modification

17. Focused wave packet reproduction in 3D Directional irregular wave field S(f, q) = S(f) D( q,f) Modified Pierson-Moskowitz spectrum (f peak =0.5Hz, H s = 4 cm) Directional spreading with s=10 Focusing time t=45 s Elevation recorded in 5 locations (probe array used for short-crested seas analysis) t = 25 s.t = 45 s.

18. Reproduction of a directional focused wave field Analysis in the frequency domain (for the 5 probes) Three unknowns at each frequency : A set of nonlinear equations solved with a nonlinear least squares method (local minima are expected) and different initial guesses We obtain a set of solutions of the nonlinear equations: we choose the one that minimises

19. Directional focused wave field Simulated wave packets with the HOS model of the wave basin for both the focused target and the reproduced wave packet Target wave field f p = 0.5 Hz, H s = 4 cm Directional spreading s=10 Focusing time t = 45 s Reproduced wave field Prescribed snake-like wavemaker motion Large waves angles generated with the Dalrymple method

20. Directional focused wave field View of the wave field before the focusing event at t = 33.5 s Target wave fieldReproduced wave field The main features of the focusing packet are correctly reproduced The high-fequency range is underestimated in the predicted wavemaker motion

21. Directional focused wave field View of the wave field at the focusing event t = 45 s Target wave field Reproduced wave field Underestimation of the wave crest Overestimation of the width of the crest

22. Conclusion High-Order Spectral method applied to the wave generation in a wave basin Improvement of the wavemaker motion for the generation of deterministic wave packets Part of third order effects (phase velocity) taken into account in 2D Attempt of deterministic reproduction in 3D Future work Phase velocity correction applied iteratively Application to different kinds of wave packets (narrow- banded, broad-banded…)

23. Comparison between numerical simulations and experiments Amplitude 40 cm Amplitude 30 cm

24. Nonlinear simulation results for linear and second-order wavemaker motion Difference with the target signal: –Time shift –Peak amplitude –Through amplitude Improvement of the second order model: –Reduced time shift Basin dimensions:50m long 5m deep Number of modes: Nx=512 Nz=64 Order of decomposition: M=5 Target First order input Second order input Time in s. Analytical methods Elevation in m.

25. Wavemaker control Successive corrections Initial guess: second-order analytical model Initial guess Non-linear simulation Correction on amplitudes and/or phases Corrected motion n=1 h n (t) Comparison to the target after Fourier analysis Target wave

26. Numerical checkings Number of modes Order of decomposition for Time interval between successive smoothing (dt = 0.014s) Number of modesPeak heightRelative error 2560.44592.1 % 5120.45160.08 % 10240.4520reference Order MPeak heightRelative error 30.45340.40 % 40.45240.18 % 50. 4516reference Number of time steps Peak heightRelative error 600.4523Reference 400.45220.02 % 300.45190.09 % 200. 45160.15 %

27. Iterative corrections 1-Phase lag refinement Montrer l’amélioration pic très nette Creux pas assez Wave elevation obtained at iteration « n » with motion Correction on the phase f n+1 = f n + Df with Df = f - y n Simulation result after 5 iterations Target Before iteration After 5 iterations Time in s. Elevation in m. Target phase

28. Iterative corrections 2-Amplitude and phase Correction of the phase f n+1 = f n + Df with Df = f - y n Correction of the amplitude with D a = Simulation result after 5 iterations Montrer l’amélioration des creux Oscillations qui croissent avec les itérations Target Before iteration After 5 iterations Target amplitude Elevation in m. Time in s.

29. Iterative Corrections CaseFirst order First plus second order Iterations on the phase Iterations on both the phase and amplitudes Error (%) 37351614 Wave elevation record Time in s. Comparison interval Elevation in m. Time in s. Discrepancies for the peak amplitude Spurious oscillations before the focused event