SCSTW-2008, Shanghai, China. Interaction of Two Solitary Waves of Large Amplitude. Hua Liu Benlong Wang Shanghai Jiao Tong University firstname.lastname@example.org. Outline. Motivation A high order Boussinesq equation Propagation and reflection of a solitary wave
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SCSTW-2008, Shanghai, China Interaction of Two Solitary Waves of Large Amplitude Hua Liu Benlong Wang Shanghai Jiao Tong University email@example.com
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
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?
A high order Boussinesq equation • Definition of velocity variables Madsen, Bingham & Liu (2002)
Irrotational flows ——Zakharov(1968) , Witting(1984), Dommermuth & Yue (1987)
Exact solution of Laplace equation ——L. Rayleigh 1876 On waves
Velocity solution formulation in terms as the velocity defined at an arbitrary level of depth
Taylor expansion • Series expansions
Pade expansion • Series expansions
Linear dispersion • Nonlinearity
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
KdV • mKdV • Full potential theory
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.