Numerical Investigation of the Physics of Higher Order Effects Generated by Wave Paddles

Abstract

When free-surface waves are generated using wave paddles to produce the desired waves, higher order effects might be inevitable for some cases. These can be due to the mismatch in the wave paddle displacement and non-linear free-surface wave kinematics, as well as the moving boundary of wave paddles. Such higher order effects are often manifested as higher harmonic waves, which can propagate independently (or free waves). The presence of such waves will contaminate the quality of the tank test, and together with effects due to scaling and finite size of tank, it is important to reduce or mitigate such effects as much as possible in a wave tank in order to simulate a more realistic scenario. This study investigates the above problem in a systematic manner by using a fully-nonlinear numerical wave tank based on the three-dimensional time-domain Harmonic Polynomial Cell (HPC) method. Wave is generated by flap-type wave paddles on one end of the tank, and is damped on the other end. The paddle boundary conditions are satisfied on the instantaneous paddles surfaces, and the free surface is tracked by the generalized semi-Lagrangian scheme. In this study, first order paddle signal is used to generate regular waves, and the focus is on characterising the behaviour of the generated free higher harmonic waves. We first look into a rectangular wave tank where the paddles are distributed at one side of the tank. Upon the generation of an oblique regular wave (primary wave), it is observed that the generated free waves propagate at a different angle/direction. An explicit analytical expression is derived for the direction of the free waves, which agrees with the numerical observation. Besides propagating at a different direction, the free waves also interact with the primary waves resulting in additional bound waves of the first and third harmonics. Next, we consider a circular wave tank, where paddles along half of the circumference are used to generate planar regular wave, while paddles at the other half are assumed to be able to fully absorb the wave. The generated free waves are observed to focus at a particular region in the tank due to constructive interference. To eliminate or at least mitigate such undesired waves, correction to first order paddle signal is required. Second order correction scheme based on Schaffer (1996) is implemented for such purpose. Preliminary results seem to suggest that second order correction to the paddle signal can only mitigate but cannot completely eliminate the existence of free higher harmonic waves.

A 3D fully‐nonlinear potential‐flow solver for efficient simulations of large‐scale free‐surface waves

“…Note that this normalized error is only nondimensional if 𝜒 = 𝜑. The absolute errors from equation (15) in the cell node with local index 27 for 𝜑 and the components of ∇𝜑 are plotted in Figure 11 as a function of the cell's aspect ratios. Similar to the observations by Ma et al 7 in 2D, 𝜖 𝜑 (x 27 ) is always small for cubic cells with Δx∕Δz = Δy∕Δz = 1.…”

“…Only wave‐propagation scenarios are considered here, but the NWT is developed with future extensions to include wave‐body interactions in mind. Several additional authors have used the HPC method in 2D studies considering various problems in marine hydrodynamics, including Liang et al, 11 Strand, 12 Robaux and Benoit 13 and Shen et al 14 3D applications are more limited, where the most relevant examples are the work by Shao and Faltinsen 4 for wave propagation in a narrow NWT and wave run‐up around bottom‐mounted cylinders using a vertically stretched free‐surface fitted grid, and the work by Law et al 15 simulating multi‐directional waves generated by wave paddles using an overset grid. In contrast, the present work documents the first time that the HPC method is combined with an immersed‐boundary technique and AGR in 3D.…”

“…The analytical solution in Equation (13) is enforced in all boundary nodes with the following parameters: k = 0.5236 m −1 corresponding to a wavelength 𝜆 = 12 m, 𝛽 = 45 • , 𝜁 A = 1.0 m and h = 0.5𝜆. From the deep-water dispersion relation we have 𝜔 = 2.2664 s −1 , corresponding to a wave period T = 2.77 s. The time instant is taken as t = 0.2 T.The absolute errors from equation(15) in the cell node with local index 27 for 𝜑 and the components of ∇𝜑 are plotted in Figure11as a function of the cell's aspect ratios. Similar to the observations by Ma et al7 in 2D, 𝜖 𝜑 (x 27 ) is always small for cubic cells with Δx∕Δz = Δy∕Δz = 1.…”

Accurate and efficient hydrodynamic analysis of structures with sharp edges by the Extended Finite Element Method (XFEM): 2D studies

“…Shao and Faltinsen (2012) and Shao and Faltinsen (2014) proposed high-order Harmonic Polynomial Cell (HPC) methods in both 2D and 3D respectively to study water waves and their interaction with structures. Some recent extensions were made to utilize immersed boundary strategies and overset meshes to achieve better accuracy and stability (e.g., see Hanssen et al, 2018;Tong et al, 2019Tong et al, , 2021Law et al, 2020;Liang et al, 2020). Compared to the BEMs, field solvers deal with spare matrices, and the computational costs are roughly linearly dependent on the number of unknowns.…”