Generating Second-Order Stokes Waves Using a Cylinder-Shaped Plunger—Theoretical Derivations and SPH Simulations

Plunger-type wavemakers have been and will be utilized in numerous wave tanks around the world, mainly due to their high efficiency in relatively deep water, space-saving in the wave direction, etc. Although theoretical methods for generating second-order Stokes waves based on a piston-type wavemaker are nearly mature, they cannot be extended to the plunger-type wavemaker. How to control the plunger motion to produce high-quality second-order Stokes waves is still an unsolved but practically meaningful issue. In this study, new formulae that relate the vertical motion of a cylinder-shaped plunger to the target second-order Stokes waves are derived, based on the conservation between the change in the immersion volume of the cylinder and the volume of the generated waves. Constraints on the generation of second-order Stokes waves are also proposed. Then, a numerical wave flume is established based on the Smoothed Particle Hydrodynamics (SPH) method. After examining its accuracy and analyzing its convergence by reproducing a physical experiment reported in the literature, the flume is used to simulate the second-order Stokes wave generation. The computed wave profiles, flow velocities, and pressures are compared with analytical solutions to verify the reliability of the proposed wave-making theory. The results show that the second-order Stokes waves produced by a plunger-type wavemaker are both precise and stable. Besides, high-quality second-order Stokes waves can be generated within a short distance, saving the precious laboratory space and computational domain between the wavemaker and the structure if exists.