Non-linear wave generation and absorption using open boundaries within DualSPHysics

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Abstract

The present work introduces the implementation of wave generation and wave absorption of non-linear, long-crested regular and irregular waves in the WCSPH-based (Weakly Compressible Smoothed Particle Hydrodynamics) DualSPHysics solver. Open boundaries are applied here for both wave generation and absorption. These boundaries consist of buffer zones, on which physical quantities are imposed, or extrapolated from the fluid domain using ghost nodes. Several layers of buffer particles are used to create an inlet and an outlet, where the horizontal component of the orbital velocities, surface elevation and pressure can be imposed from any external source or extrapolated from the fluid domain. This allows the creation of a numerical wave flume with a length of one wavelength. Reflections within the fluid domain are successfully mitigated using a velocity correction term at both inlet and outlet. The implementation is validated with theoretical solutions, in terms of water surface elevation, wave orbital velocities, and dynamic pressure. The model proves to be capable of propagating waves with less than 5% reflection, and RMSE errors on physical quantities lower than 4.3%. The application of open boundaries proves to be an accurate method to generate and absorb non-linear waves within a restricted domain.

Introduction

In recent years, the use of Smoothed Particle Hydrodynamics (SPH) for numerical modelling of both fundamental and engineering problems has known a steady increase. The Smoothed Particle Hydrodynamics method is a flexible Lagrangian and mesh-less technique for Computational Fluid Dynamics (CFD). The Lagrangian reference frame of SPH makes it useful in solving problems with large deformations and distorted free surfaces. In comparison with other numerical methods, the SPH formulation is simple and robust [1], [2].

This research focuses on wave propagation, wave–structure interactions and the application of SPH modelling to real engineering problems within coastal and offshore engineering. In this regard, SPH has been successfully applied to a number of free-surface problems that involve wave propagation, wave breaking and splashing [3], [4]. The impact between a rigid body and water has been studied in [5]. Wave–body interactions using an incompressible SPH (ISPH) solver have been studied in [6]. In [7], wave impact on coastal structures has been investigated. A fixed cylinder in a wave train and forced motion of cylinders generating waves are mentioned in [8], while floating bodies in waves have been successfully studied in 2-D [9]. The feasibility of applying SPH for modelling wave energy converters has been studied in [10], [11]. 3-D problems of wave generation by a heaving cone and a floating body in waves undergoing predominantly heave motion are investigated in [12]. The latter have also indicated that simulations using a variable particle mass distribution can be beneficial.

Typically, SPH schemes are computationally intensive. However, recent advances using High Performance Computing (HPC) and Graphical Processing Units (GPU) have strongly contributed to significant gains in computational effort [2]. Despite the use of HPC and GPUs, it is still challenging to model real engineering problems, which are usually multi-scale. An alternative to optimizing SPH for powerful computing hardware, is to study possible reduction of the computational domain. Importantly, this requires accurate and stable boundary conditions, which is one of the SPHERIC Grand Challenges (http://spheric-sph.org/grand-challenges). This research focuses on applying open boundary conditions within a small computational domain to accurately model wave generation and wave absorption of non-linear regular and irregular waves, with a high accuracy.

In numerical modelling, three main types of wave generation can be distinguished [13]:

  • moving boundary generation;

  • internal generation;

  • static boundary generation;

Of these, only the first two have been applied to SPH models [14]. Internal wave generation with a non-reflective internal wavemaker algorithm has been proposed in [15] where the Boussinesq equations are used to derive a momentum source term, which is added into an Incompressible SPH model using the Lagrangian Navier–Stokes equations. The most common wave generation method in SPH is the moving boundary generation. This mechanism tries to translate the mechanical wave generation techniques of experimental facilities directly into the numerical model. A moving boundary is implemented as a numerical wavemaker that generates and absorbs waves. Examples of this generation method can be found in [9], [14], [16], [17]. A 2-D numerical wave tank based on the open-source SPH-based DualSPHysics solver [18] was presented in [19] using analytical relaxation approach. In that work water particles inside the source generation zone move according to periodical velocities obtained from the Stokes wave theory. However, only regular wave cases were validated using this approach. Neither irregular waves nor second-order bound long waves were simulated. Additionally, this type of wave generation technique has a higher computational cost than wave generation with moving boundaries. This is due to the large number of water particles needed in both the generation zone and the sponge layers [20]. In [21], the relaxation zone method was successfully implemented into DualSPHysics, acting as an internal wave maker and allowing coupling to other models or analytical solutions. Wen et al. [22] developed an absorbing wavemaker using the SPHysics model [23]. However, only linear wave theory was applied to generate the waves and only regular wave tests were considered. Omidvar et al. [24] used an irregular wave generation based on the linear wave theory to generate focused waves but neither super nor subharmonic components were considered in their approach. Recently, [25] presented a wave generation and absorption technique with non-reflective open boundaries, similar to the work presented in this paper. Non-reflective boundaries have been also presented in [26], [27], but focus more on confined flows rather than free-surface waves. Similarly, open boundary formulations for confined flow using an ISPH solver were introduced by [28], [29] presented open boundary conditions for an SPH Shallow Water Equations model. In [30], an open boundary formulation using Riemann invariants to calculate the flow properties is introduced and applied to regular wave generation. Alternatively, wave generation in SPH can also be achieved by coupling the SPH solver to a wave propagation model. This was first demonstrated by [31], where the Boussinesq model FUNWAVE was coupled to an SPH solver to model coastal wave propagation. In [32], an ISPH solver is coupled to a quasi-arbitrary Lagrange–Euler finite element method to generate sinusoidal waves. Similarly, the SPH solver DualSPHysics was coupled to the wave propagation model SWASH in [33] and to the fully non-linear potential flow model OceanWave3D in [34].

In addition to wave generation, wave absorption is equally important in any physical or numerical model within coastal engineering. Wave absorption is specifically necessary to damp the wave energy and reduce the reflections generated by the domain boundaries. This can be done using passive wave absorber systems, which can be established by placing a gentle slope, porous material or screens in front of the boundaries. Like this, a large amount of the incident wave energy can be dissipated. Dimensions of passive wave absorption systems typically depend on the specific wave conditions. An exponential wave damping zone was applied in [35]. In DualSPHysics, a similar algorithm is implemented and introduces a damping region in the fluid domain [14]. The numerical algorithm is very similar to application of sponge areas or porous materials in physical model tests.

Passive absorption is not sufficient when waves interact with structures; active wave absorption system is then needed. In active absorption, the wave generation method is corrected in order to remove the reflected waves present in the domain and to damp the re-reflection phenomenon. With moving wave generators, such as paddles and flaps, the corrected wavemaker displacement in function of time is obtained by transforming the original wave signal, to which an appropriate filter is applied. This filter can be a time-domain or frequency-domain filter. In literature, there are differences noticeable in the type of feedback correction signal used. In [14], [36], the free-surface elevation at the wavemaker is used, while free-surface elevation and/or orbital velocities at a fixed position in the fluid domain were used by [37]. In [38], forces acting on the wavemaker were measured and used to calculate the correction signal. The active wave absorption algorithm developed in this work applies velocity corrections to the wave generator and wave absorber, based on the measured surface elevations within the fluid domain, and thus relates the most to [14], [36].

Typically, SPH domains for wave propagation modelling are at least 3–4 wavelengths long [33]. Combined with a required small particle size to accurately reproduce the surface elevation, this leads to computationally intensive simulations. This research is aimed at reducing the necessary fluid domain to a length of only one wavelength, and provide accurate boundary conditions capable of active wave generation and absorption. In this manner, real open sea conditions can be simulated where waves enter at the left-hand-side of the fluid domain and exit freely at the right-hand-side. The WCSPH model DualSPHysics will be employed to demonstrate these new wave generation and wave absorption techniques, using the recently developed open boundaries [39]. The applied open boundary formulation is based on the use of buffer layers adjacent to the fluid domain. Buffer particles are used to enforce certain conditions in presence of fluid inlets and outlets. Particularly, the physical information of buffer particles is either assigned a priori or extrapolated from the fluid domain using a first order consistent procedure. The major benefits of this method are:

  • Using open boundaries for wave generation and absorption is meant to cover those cases where classical wave generation techniques can fail or are very computationally expensive, e.g. open sea states, simulating floating devices, wave breaking conditions, etc.

  • The buffer zones in the open boundaries accept physical information extrapolated from fluid domain or imposed from any external source: e.g. linear wave theory, non-linear wave theories, external numerical models such as CFD models, or even measurement data.

Although a weakly compressible SPH scheme has been adopted in the present work, it is worth noting that the key elements of the proposed procedure can be extended also to other types of SPH models such as Incompressible SPH [35] and/or ones based on the Arbitrarily Eulerian–Lagrangian formulation originally proposed by Vila [40].

The structure of this paper is as follows: in Section 2, the applied SPH model is described in detail. Next, the methodology of wave generation and absorption applying open boundaries is discussed in Section 3. This includes a general description of the methodology and the applied non-linear wave theories. A specification of the benchmark tests used to validate the introduced methodology is given in Section 4. In Section 5, the results of the benchmark tests are shown and discussed. In Section 6, the conclusions and future work are finally presented.

Section snippets

Smoothed particle hydrodynamics model

The solver used for the detailed modelling of the wave–structure interactions is DualSPHysics [18]. This section explains the theory behind the software, and is strongly based on the DualSPHysics User Guide v4.2 [41]. DualSPHysics applies the SPH formulation, a meshless method that describes the fluid as a set of discrete elements, named particles. The physical properties of a particle a, determined by the Navier–Stokes equations, can be calculated by interpolation of the values of the nearest

General description

In this work, generation and absorption of non-linear regular and irregular waves is performed within the Weakly Compressible SPH (WCSPH) solver DualSPHysics, by applying the open boundary formulation from [39], as described in Section 2.3. The open boundaries are implemented as a zone of buffer particles. Physical quantities such as velocity, surface elevation and pressure can either be imposed on the buffer particles or extrapolated from within the fluid domain. The imposed physical

Test program

The application of wave generation and absorption using open boundaries is hereby validated with theoretical and experimental results from the literature. Firstly, a stable non-linear standing wave is simulated by considering the outlet as a fixed wall. Secondly, simple wave propagation of regular and irregular waves is studied. Thirdly, a number of tests is performed to investigate the correct reproduction of wave transmission and wave reflection.

Standing wave test

The accuracy of wave propagation with open boundaries is assessed by comparing SPH surface elevation and orbital velocities with the corresponding theoretical results, as illustrated in Fig. 9. The first graph shows the comparison of the surface elevation, the second graph the horizontal orbital velocities and the third graph the vertical orbital velocities. The theoretical standing wave was calculated as a Stokes 2nd order standing wave. For a perfectly linear standing wave pattern, the

Conclusions

In this paper, a novel wave generation and absorption method using open boundaries was introduced. A fluid domain of one wavelength long is selected, with an inlet and outlet composed of 8 buffer particle layers at both sides. At the inlet, theoretical horizontal velocity and surface elevation are imposed to the buffer particles, while the pressure is extrapolated from ghost nodes placed in the fluid domain. At the outlet, only theoretical horizontal velocity is imposed, while pressure and

Acknowledgements

This work was supported by the Agency for Innovation by Science and Technology in Flanders (IWT/Vlaio), Belgium with the scholarship 141402. This work was also partially financed by Xunta de Galicia (Spain) under project ED431C 2017/64 Programa de Consolidación e Estructuración de Unidades de Investigaciósn Competitivas (Grupos de Referencia Competitiva) cofunded by European Regional Development Fund (FEDER) and by Xunta de Galicia (Spain) postdoctoral grant ED481B-2018/020. The work was also

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