Theoretical method for generating solitary waves using plunger-type wavemakers and its Smoothed Particle Hydrodynamics validation

ABSTRACT

Although solitary wave generation using a piston-type wavemaker has been common, the method cannot be applied to all wave tanks because many of them have been or will be equipped with plunger-type wavemakers. How to use a plunger to generate solitary waves as good as those generated by a piston is an unsolved but practically significant problem. Herein, a theoretical method is proposed. New formulae for the precise descent of an arbitrary-geometry plunger are derived and constraints on the produced wave height are given. Taking wedge-shaped, box-shaped, and cylinder-shaped plungers as examples, the generic wave-making theory and constraints are then specified. On their bases a Smoothed Particle Hydrodynamics (SPH) model is applied to simulate the solitary wave generation, and its reliability is validated in advance by reproducing available experiments. The SPH-simulated wave profiles, velocity fields, and pressure fields are examined by comparing SPH results with analytical solutions. In addition, plunger-type and piston-type as well as Rayleigh-based and Boussinesq-based solitary wave generation are compared. The results show that the solitary wave generated by a plunger-type wavemaker can be as accurate as the one generated by a piston-type wavemaker, provided that the plunger does not have a sharp corner or an incompatible surface with the wave profile. It is therefore anticipated that the proposed theory will extend the functionalities of existing plunger-type wavemakers and benefit plunger design in the future.

Introduction

Tsunamis are huge ocean waves triggered by the displacement of a large volume of water, mainly caused by submarine earthquakes [1, 2], volcanic eruptions [3, 4], and landslides [5, 6]. Tsunamis, once reaching coastal areas, can destroy the vast majority of marine and coastal structures and more seriously bring about enormous loss of life. A tsunami is generally represented by a solitary wave, but it should be clear that it is not a thoroughly realistic representation. A tsunami is a wave train, of which the leading wave can be either positive (i.e. surface elevation above the still water level (SWL)) or negative (i.e. surface elevation below the SWL) [7], [8], [9], while a solitary wave is single and always positive. Moreover, there is usually insufficient ocean space for an initial displacement of sea surface evolving into a solitary wave [10, 11]. Notwithstanding the differences, the solitary wave approximation has been widely used for tsunami studies [12], [13], [14], [15], [16], [17], [18] mainly due to its simplicity and high repeatability [19, 20].

It is common practice to generate a solitary wave using the method of Goring [21]. That is to match the velocity of piston-type wavemaker with the depth-averaged horizontal velocity of water particles adjacent to the piston. However, the generated wave exhibits a rapid decrease in height during its propagation, exceeding the theoretical level attributed to the boundary layer effect of the water tank [22]. To enhance the wave invariance, Katell and Eric [23] replaced the Boussinesq assumption [24] adopted by Goring [21] with the Rayleigh one [25]. Malek-Mohammadi and Testik [26] considered the evolving nature of wave generation and derived a time-dependent wave celerity based on the conservation of momentum. Both two measures help to generate a higher-quality wave with a shorter transition distance and less attenuation of wave height. The redundant trailing waves following the main pulse are another focus of the solitary wave generation. Although their heights under Boussinesq assumption are lower than 10% of the height of the main pulse, the ratios can be further suppressed to below 3% if Rayleigh assumption is adopted [23]. Farhadi et al. [27] compared the solitary wave generation under six wave theories, namely Boussinesq assumption [24], 1^st-order and 2^nd-order shallow water equations [28, 29], Rayleigh assumption [25], 3^rd-order Grimshaw solution [30], and 9^th-order Fenton solution [31]. Their results supported Katell and Eric [23] that Rayleigh assumption yields the least attenuation of wave height but also pointed out an unfavorable lead of wave phase. By comparison, the generated waves based on 3^rd- and 9^th-order solutions have better overall performance.

Although substantial theories have been established for piston-type solitary wave generation, they cannot be applied in laboratory tanks equipped with plunger-type wavemakers. However, there are indeed a large number of plunger-type wavemaker being used around the world (see our survey results in Appendix A). A plunger only disturbs near-surface water, thereby performing higher efficiency in a deep-water circumstance than a piston. It also saves horizontal space in the wave direction. Therefore, one cannot rule out the possibility that in the future more plunger-type wavemakers would be installed. Unfortunately, owing to the complex geometry, plunger-type wavemaker is merely designed to generate regular waves and multi-frequency random waves so far [32], [33], [34], [35]. There is hardly any research on plunger-type solitary wave generation. Thus, functionalities of the existing and future plunger-type wavemakers are discounted.

In fact, in the earliest study of solitary wave, Russell [36] has put forward the idea of generating a solitary wave by dropping a heavy object into the water. However, a freely falling object generally results in an unsteady wave train [37, 38]. A desired waveform can only be obtained after a long-distance evolution or it does not emerge at all. Considering the precious laboratory space or computational domain, the generated wave is hoped to achieve its target form in the shortest distance. Moreover, the trailing waves should be as weak as possible to enhance the wave pureness. Therefore, manual control of the movement of the plunger-type wavemaker is necessary. The mechanism of plunger-type solitary wave generation is the real-time equivalence between the immersion volume of the plunger and the volume of the wave. Since a solitary wave lays entirely above the SWL, its volume increases monotonically during its generation and the movement of the plunger should always be downward. A faster or slower descent, however, both results in inappropriate water volume and downgrades the quality of the generated wave. Thus, the primary objective of this study is to propose a new theory that can be used to precisely control the plunger descent so as to produce precise target solitary waves.

Unlike the piston-type wavemaker which has a simple geometry and can displace an arbitrary quantity of water by adjusting the piston stroke, the complex plunger-type wavemaker may encounter an incompatibility problem at the interface between plunger and wave (see Figs. 3 and 6). It also displaces limited water quantity due to the finite water depth or the finite volume of the plunger (see Fig. 2). Thus, to obtain the constraints on the generated wave and understand the consequences of exceeding the constraints are the secondary objective of this work.

There are two steps in validating the proposed theory: first, to generate solitary waves based on the proposed theory; second, to examine the accuracy of the generated wave. As for the first step, conducting a physical experiment is a reliable approach, however, it depends on the financial resources, laboratory space, facility availability, etc. Then, numerical simulation can be used alternatively due to its increased accuracy and popularity. The Smoothed Particle Hydrodynamics (SPH) [39], [40], [41], [42] is a Lagrangian meshfree numerical method, which has been successfully applied in various fields of coastal and ocean engineering, including wave/current-structure interaction [43], [44], [45], [46], wave propagation over topography [47], [48], [49], [50], [51], fluid resonance [52, 53], renewable energy utilization [54], [55], [56], sediment dynamics [57, 58], and many others as mentioned by Gotoh and Khayyer [59, 60]. The SPH method is adopted here to simulate the plunger water entry and solitary wave generation because it is naturally good at simulating highly nonlinear wave motion, its Lagrangian characteristic facilitates the treatment of the water-plunger interface, and the latest SPH method is able to capture the vortex dynamics [61, 62, 49] in the water entry process of a plunger with sharp corners. Over the past two decades, there have already been many SPH works on the water entry and the consequent wave generation. For example, Monaghan and Kos [63] and Monaghan et al. [64] simulated the vertical sinking and inclined slide of a box respectively, and established relations between the water entry speed and the wave height. Lin et al. [65] modelled the entire process of the wedge-shaped landslide, surge wave generation, propagation, and overtopping of a dam. By analysing subaerial landslides into various water body geometries, Heller et al. [66] provided a novel insight into the slide and wave kinematics. Capone et al. [67] and Xenakis et al. [6] treated sliding masses as non-Newtonian fluid so as to investigate the deformable landslide process. These representative studies are all circumstantial evidence that the SPH method is applicable to the present study.

Based on the SPH method, several open-source packages have been launched such as the classic SPHysics [68] and the advanced DualSPHysics [69]. SPHysics is a FORTRAN code that includes both serial and parallel versions and both 2D and 3D versions. It is easy to read and modify but is generally applicable to small and low-resolution domains. The code we use here is developed from the 2D serial SPHysics of version 2.2 but incorporating several beneficial functionalities, for instance, the OpenMP parallelization, modified dynamic boundary condition [70], δ-SPH method [71, 72], and particle shifting technique [73]. DualSPHysics is a set of C++, CUDA, and Java codes that can be executed either on a CPU or a GPU. It is characterized by high computational efficiency, which makes large-domain and high-resolution simulations ordinary. Moreover, it constantly incorporates the latest SPH achievements. Owing to the above features, DualSPHysics has been attracting a rapidly growing number of users.

All the simulations in this work are performed in 2D space based on the following considerations. In the laboratory, the width of a plunger-type wavemaker is generally close to that of the water tank. Thus, fluid motion is dominated by the wave propagation parallel to the sidewalls of the tank. There can be some fluid separation and wave diffraction due to the narrow gaps between the plunger and side walls or among multi-segment plungers, however, they are relatively weaker than the dominated wave motion. In addition, although 3D vortex motion can be triggered during the fluid-plunger interaction, their effects on the wave motion should also be small.

As for the second step in validation of the proposed theory, namely, to examine the accuracy of the generated wave, the SPH-simulated wave profile, velocity field, and pressure field can be examined by either comparing with experimental data or analytical solutions. Herein, the latter is preferred mainly due to three reasons. First, existing experiments usually focus on partial wave information [74], [75], [76], while analytical solutions are more comprehensive. Second, existing experiments only involve a limited number of wave conditions and are unlikely to cover all the conditions of this study, while analytical solutions are unlimited, provided their prerequisites are satisfied. Third, solitary wave theory has been almost mature. The experimental data are not necessarily more reliable than the analytical solutions if considering the instrument and measurement errors.

This paper is organized in the following manner. After the introduction, Sec. 2 derives the generic theory of plunger-type solitary wave generation and the generic constraints on the generated wave. Sec. 3 introduces the SPH method. In Sec. 4 to Sec. 6, three representative plungers, namely wedge, box, and cylinder are tested, respectively. In each section, formulae for the plunger descent and constraints on the produced wave height are first specified. Then, the reliability of the SPH model is validated by reproducing an available experiment. Finally, the validated SPH model is used to simulate the solitary wave generation based on the derived formulae, and physical quantities of the generated wave are examined by comparing SPH results with the analytical solutions. In Sec. 7, plunger-type and piston-type solitary wave generation and Rayleigh-based and Boussinesq-based solitary wave generation are compared. The main conclusions are drawn in Sec. 8, together with our perspectives on future research.