Transient waves generated by a vertical flexible wavemaker plate with a general ramp function

Wavemaker problem is a fundamental and important issue in the study of marine and coastal engineering. An accurate incident wave field provided by a wavemaker is crucial to explore the problems of wave and wave-structure interaction. Generally, wavemakers start working from a rest state both in physical and numerical tanks. Excessive acceleration induced by sudden motions of wavemaker will result in generating wave instability even wave breaking unexpectedly and has potential harm to wavemaker facilities in physical tanks. It might cause numerical instability in numerical tanks as well. Although different ramp functions can be constructed and introduced to mitigate this issue, theoretical analysis on the performance of different ramp functions is still lacking. In this study, based on the new elementary solution on transient waves given by Chen and Li (Journal of Engineering Mathematics, 2019, 115(1): 121-140), a time-domain analytical model on transient waves with a general ramp function is developed for a vertical flexible wavemaker plate. Two special ramp functions, namely Eatock Taylor's and Lighthill's ramp function, are used to investigate their effects on the spatial-temporal evolution of transient waves. Moreover, based on the open source two-phase flow model Basilisk, nonlinear effect on ramp functions' performance is investigated by increasing wave steepness of incident waves. Under weakly nonlinear conditions, the numerical solution is consistent with the present analytical solution, which further confirms the validity of the present analytical solution. And it is found that computational cost for the Lighthill's ramp function is less than that of the Eatock Taylor's ramp function in numerical simulations. Finally, it should be pointed out that the present theoretical solution provides an effective way to obtain new insights on the performance of those potential ramp functions which might be useful and applied in future studies.