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Interaction of Two Solitary Waves of Large Amplitude

SCSTW-2008, Shanghai, China. Interaction of Two Solitary Waves of Large Amplitude. Hua Liu Benlong Wang Shanghai Jiao Tong University hliu@sjtu.edu.cn. Outline. Motivation A high order Boussinesq equation Propagation and reflection of a solitary wave

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Interaction of Two Solitary Waves of Large Amplitude

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  1. SCSTW-2008, Shanghai, China Interaction of Two Solitary Waves of Large Amplitude Hua Liu Benlong Wang Shanghai Jiao Tong University hliu@sjtu.edu.cn

  2. Outline • Motivation • A high order Boussinesq equation • Propagation and reflection of a solitary wave • Head on collision of two solitary waves • Overtaking of two solitary waves • Concluding remarks

  3. Motivation • Validation of the high order Boussinesq equations check the flow field of a solitary wave of large amplitude and the force acting on a vertical wall • Overtaking of two solitary waves check if the critical ratio of wave amplitude varies with wave amplitude?

  4. A high order Boussinesq equation • Definition of velocity variables Madsen, Bingham & Liu (2002)

  5. Irrotational flows ——Zakharov(1968) , Witting(1984), Dommermuth & Yue (1987)

  6. 4 equations, 6 unknowns

  7. Exact solution of Laplace equation ——L. Rayleigh 1876 On waves

  8. Velocity solution formulation in terms as the velocity defined at an arbitrary level of depth

  9. Taylor expansion • Series expansions

  10. Pade expansion • Series expansions

  11. Linear dispersion • Nonlinearity

  12. Numerical aspects • Spatial discrectization: 7 point central difference scheme • Time stepping：5 order Cash-Karp-Runge-Kutta scheme • Smoothing：Savitsky-Golay smoothing method • Relaxed analytic approach for wave generation and absorbing

  13. Propagation of a solitary wave

  14. End-wall reflection of a solitary wave

  15. Head-on collision of two solitary waves

  16. Overtaking of two solitary waves

  17. Wang, Zhang & Liu (2007, PRE)

  18. KdV • mKdV • Full potential theory

  19. Concluding Remarks • The high order Boussinesq model is applied to numerical simulation of a solitary wave reflected by a vertical wall. • Among the three patterns of overtaking of two solitary waves, the critical condition for the flat peak pattern is related with the incoming wave amplitude. • For extremely small wave, the critical relative amplitude approaches to 3, which indicates the various KdV models or bidirectional long wave models give reasonable correct predictions. • With increasing of the wave amplitude, the critical relative amplitude increases and is apparently different from 3. For the incoming solitary wave of extremely large amplitude, e.g. a= 0.6, the critical condition reaches the magnitude of 4.

  20. Thank you for your attention.

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