Elsevier

Applied Ocean Research

Volume 81, December 2018, Pages 15-33
Applied Ocean Research

Improved relaxation zone method in SPH-based model for coastal engineering applications

https://doi.org/10.1016/j.apor.2018.09.013Get rights and content

Highlights

  • An improved algorithm for wave generation and absorption for meshless Lagrangian methods is proposed.

  • The algorithms, based on the Relaxation Zone method (RZ), can be extended to model coupling.

  • A detailed validation using Smoothed Particle Hydrodynamics and NLSW solvers is provided.

  • A detail methodology to design the setup of RZ in SPH is proposed and optimized for wave generation and abortion.

  • The results prove that the RZ method is convenient from the point of view of both accuracy and computing time.

Abstract

An improved Relaxation Zone (RZ) method has been implemented in the meshless SPH-based DualSPHysics model. Final purpose of this work is to have a general wave generation scheme that allows coupling SPH-based models to other models, e.g. Eulerian based wave models, besides employing the RZ as alternative wave generation in SPH as a stand-alone scheme. Using RZ in SPH, the movement of the fluid particles is controlled by correcting their orbital velocity by means of a weighting function in a specified generation area. In the present work, the new technique is used to couple DualSPHysics to the non-hydrostatic wave-flow model SWASH. The results of RZ employed both as stand-alone wave generation technique and as coupling framework with SWASH model are validated for wave generation and wave reflection for monochromatic waves. Then, the method is tested successfully for generation and absorption of irregular waves. Finally, the coupling between DualSPHysics and SWASH using RZ is validated against experimental data concerning the wave flow impacts on vertical walls. A procedure for a proper design of the RZ (i.e. shape of the weighting function, size of the RZ) is described in the present work. Overall, the results indicate that the proposed improved RZ technique is among the most effective alternatives for wave generation in SPH-based models for coastal engineering application.

Introduction

Meshless methods are getting popular in coastal engineering as result of the developments of numerical techniques and computation technologies in the last decades and are more and more often employed to study free-surface flows and wave-structure interaction phenomena [1]. Models based on the so-called Smoothed Particle Hydrodynamics (SPH) [2,3] method are becoming very popular among researchers for their capacity to simulate highly nonlinear free-surface flows (e.g [4]). However the computational cost of the SPH models is still huge, even compared with one of the most expensive wave models, e.g. RANS-VOF models [5]. In particular, one of the SPH models, namely DualSPHysics [6], significantly improved the computational time by using GPU (Graphics Processing Unit) and multi-GPU techniques [7,8], although the computational cost is still high and thus it is not always easy to apply it to realistic coastal engineering projects.

In order to overcome this limitation, a coupling to computationally less demanding wave models was introduced and proved to be useful for coastal engineering applications, as for example in [9]. On the one hand, less demanding computational models (e.g. Boussinesq models, Non-Linear Shallow Water equation models) are accurate enough for wave transformation at a reasonable computational cost but not very accurate for wave-structure interaction due to their assumptions (e.g. depth integrated feature). On the other hand, SPH models are capable of generating accurate results for both wave transformation and wave-structure interaction [10,11] while these are computationally too expensive to cover a large domain at affordable time duration.

Coupling to other models is actually identified as one of the SPHERIC Grand Challenges from the Ercoftac Special Interest Group for SPH (SPHERIC, http://spheric-sph.org/). More and more attempts to enhance the SPH capability of modelling real engineering problems by means of coupling techniques have been presented during the last years (e.g. [12,13]). Previous attempts to couple DualSPHysics with wave propagation models can be found in [14,15]. In [14] SWASH model is hybridized with DualSPHysics. SWASH is a model based on layer-averaged Non-Linear Shallow Water equations with non-hydrostatic pressure and is much less demanding computationally. The wave propagation is calculated by SWASH and wave-structure interaction is calculated by DualSPHysics. In this way, the drawbacks of each model are overcome. Waves are generated in DualSPHysics by means of moving boundaries that move accordingly to the velocity input from SWASH. However, the displacement of the moving boundary in DualSPHysics does not take into account any compensation for the reflected waves that might reach the generation zone: therefore, this method does not allow wave generation over longer time periods. This problem is solved by activating AWAS (Active Wave Absorption System) at the wave generation [16] in DualSPHysics used as stand-alone model, but there is no similar implementation for the coupled model. In [15] a two-way coupling methodology between a fully nonlinear potential flow wave propagation model and DualSPHysics was introduced. Boundary moving blocks are used in DualSPHysics to generate and absorb waves. However, there are two main limitations in the methodology proposed by [15]: i) the vertical orbital velocities are not coupled; ii) the horizontally moving boundaries slowly drift away from each other, limiting the simulation time. All the three approaches ([14,15,16]) implement moving boundaries (hereafter MB) mimicking piston-type wavemakers for generating waves in SPH. Therefore, the implementations are rather different with respect to the RZ method here presented, which prevents the use of moving boundaries. The differences are following described:

  • In [14] the MB real time displacement is reconstructed on the basis of the velocity time series calculated by SWASH model in a specific point of the fluid domain: the method, even though very efficient and accurate to study phenomena such as wave run-up on gentle beaches, has a strong limitation for highly reflective cases. In fact, re-reflection at the MB is not prevented, limiting the application of the technique to low-reflective cases or to a very short time series. Instead, RZ technique allows a coupling where the reflected waves are compensated inside the relaxation zone. In this way, both highly reflective cases and long wave time series can be generated, without biasing the results for extra spurious wave energy introduced in the system by wave re-reflection.

  • In [15] the MB is discretized in moving blocks that move only horizontally and the movement of which can generate a drift. The drift will cause detachment of one block with the contiguous one which is only prevented by limiting the simulation in time.

  • In [16] the MB mimics an experimental piston-type wavemaker with reflection compensation employing the correction proposed by [17] based on shallow-water linear theory. It is a pure wave generation and absorption technique, to be used only in DualSPHysics as stand-alone model. Instead, the RZ method can be employed as coupling framework between the DualSPHysics model and a wave propagation model (e.g. SWASH).

To summarize, [16] presented a wave generation technique in SPH with no coupling, meanwhile [14] described the implementation of a coupling based on MB which lacks of active wave absorption system to prevent wave re-reflection. Instead, a coupled model between DualSPHysics and SWASH with reflection compensation seems to be a promising model for coastal engineering application but still there is a limitation for shallow foreshore applications (e.g [18,19].). In principle the coupling point should be outside the breaking zone since AWAS is based on the linear theory and thus it could only be applied to non-breaking area where non-linear effects are limited. Ideally the coupling point is placed as close to the target structure as possible. This indicates that the coupling point for the shallow foreshore case (where the distance between coastal structures and breaking zone is big) needs to be in the breaking zone, otherwise the efficiency of the coupling model is limited.

As seen above for a specific hybridization between DualSPHysics and SWASH, the coupling between the two numerical models relies on the method used to transfer information from one model to another and on the technique that each model implements to generate sea waves but also to absorb wave reflection. In particular, three kinds of wave generation techniques can be identified for numerical wave tanks [20]: MB wave generation, static-boundary and internal wave generation. The first mechanism mimics the wave generation as in experimental facilities, implementing a moving boundary as a numerical wave-maker that generates and absorbs waves, e.g. [21]. This type of wave generation is ideal for SPH models because of their Lagrangian nature and, hence, has been implemented in DualSPHysics [16]. The second mechanism corresponds to static wave generation and absorption (i.e. Dirichlet-type boundary conditions). Examples include RANS [22,23], NLSW models (e.g. SWASH [24]) and potential flow models [25]. Examples of internal wave generation exist for RANS [26] and potential flow models [27]. In SPH, Liu et al. [28] proposed a non-reflective internal wavemaker algorithm, where a momentum source term derived from the Boussinesq equations is employed and added into the Lagrangian form of Navier–Stokes equations for an Incompressible SPH model. For Weakly Compressible SPH [29], presented a momentum source function derived from the linear wave theory. Recently a wave generation method based on the so-called Relaxation Zone (RZ) technique has been introduced in SPH [30]. The authors refer to this approach as “source generation” in SPH. However, this sounds not appropriate since it is basically an application of the RZ to a meshless method. Some very early works having a similar kind of wave generation method can be found in literature, for example for solitary wave generation as presented by [31], where the authors specify the initial wave conditions by applying the analytical solution for wave profile and horizontal wave velocity to the SPH fluid particles. Note that what presented in [30] is different from the method of [32], which actually uses source generation with the boundary integral equation method (BIEM) to generate nonlinear gravity waves in Eulerian methods. Therefore, the approach presented in [30] will be redefined as RZ hereafter. Unlike the source generation with BIEM and MB wave generators such as piston-type wave-makers, RZ is a numerical method in which the movement of the particles is controlled by a weighting function (i.e. C function) in a specified generation domain. Since RZ is a pure numerical method, it might be applied to cases in which MB cannot be used (e.g., sea dikes with very shallow and mild foreshore slopes). A part from the aforementioned wave generation techniques, it is worth to mention the open boundary conditions (OBCs) as a very specific implementation for SPH and meshless methods in general [33,34]. Very recently [35], proposed a SPH numerical wave flume with wave generation and wave absorption techniques based on the non-reflective open boundary conditions, capable of generating solitary waves, linear and second-order regular waves. Yet, the second-order bound long wave generation and active wave absorption of irregular waves was not considered in [35].

The aim of the present work is to validate a revised RZ technique which moves beyond [30], by implementing the concept in DualSPHysics with relevant improvements. In fact, even though [30] introduced the RZ in the SPH-based models, it is limited only to regular waves, whereas random waves (i.e. real sea states) are not modelled. Furthermore, the accuracy of the method is not assessed in case of highly reflective wave conditions, which always represent the most critical situation to judge the performance of a wave generation and absorption techniques, either in a numerical scheme or an experimental set up. Hence, the RZ has been improved and optimized with respect to the one proposed in [30] as follows:

  • i)

    the weighting function C has been re-defined in a more general way, assuming a hyperbolic functional, the shape of which can be varied depending on the case study;

  • ii)

    the effectiveness of the C function has been analyzed in depth with reference to wave generation and wave absorption;

  • iii)

    a method to design the RZ has been established and it is proposed in this work; a drift correction is introduced to compensate the mass transport as predicted by Stokes’ 2nd order solution in order to avoid the unphysical decrease of the water level in the RZ;

  • iv)

    the RZ has been extended to study test cases characterised by highly reflective conditions;

  • v)

    the RZ has been validated not only for monochromatic waves as in [30] but also for irregular waves;

  • vi)

    finally, the RZ technique has been applied for coupling DualSPHysics with a less computational-expensive wave propagation model, such as SWASH.

In this work the applicability and the limitations of this method for wave transformation and wave reflection compensation are discussed. To this end, we define a geometrical layout that includes a highly reflective condition, namely a vertical wall. Once the RZ is validated for DualSPHysics as stand-alone model, the method is extended to the coupling case to SWASH, and the basic behavior is investigated. Final aim of the work is, in fact, to employ RZ not only as internal wavemaker in DualSPHysics where other wave generation techniques fails, but also and most of all as a mean to couple SPH models to Eulerian grid-based methods. Despite some attempts to enhance the SPH capability of modelling engineering problems by means of coupling techniques, the present work comprehensively analyze the performance of the coupling technique for real engineering cases and different wave conditions.

Section snippets

The SPH-based DualSPHysics model

Smoothed Particle Hydrodynamics is a fully Lagrangian and meshless method [3], where the fluid is discretised into a set of particles or nodal points. Physical quantities of each particle, such as position, velocity, density and pressure, are computed as an interpolation of the values of the neighbouring particles. The contribution of the nearest particles is computed depending on the distance between particles and using a weighted kernel function (W). The area of influence of the kernel

Relaxation Zone (RZ) method

Relaxation zone (RZ) methods are typically implemented in mesh-based models (e.g. [47]) for wave generation/absorption. Only recently [30], extended this methodologies to SPH. The method presented in [30] lays the groundwork for the implementation of a more generic wave generation and absorption method in DualSPHysics model. We refer here to implementation in 2DV or quasi-3D, which corresponds to the generation of long-crested waves in experimental wave facilities. A generation zone is arranged

Applicability of RZ for wave generation and reflection compensation for regular waves

Applicability of RZ to a highly reflective condition is investigated here as one of the benchmark tests. The highly reflective condition is practically the most critical case for the reflection absorption. Therefore, once a model is validated for wave generation and reflection absorption under such a condition, it guarantees that the model works properly in most of the coastal engineering applications.

Extension to irregular wave generation for real sea states

So far, the application and accuracy of RZ has been discussed for cases with regular waves only. Extension to irregular wave cases are discussed in the present section. A JONSWAP spectrum has been used to generate an irregular wave train with significant wave height Hm0 = 0.10 m, peak period Tp = 1.60 s for a water depth d = 0.80 m. A stretched algorithm as described in [16] is used to define the bandwidth. As for the parameters, it has been used the following set WRZ=Lp, ψ=0.50 and β=6 and

Case study

The coupling between SWASH and DualSPHysics by means of the RZ method is here applied to a case study consisting of wave overtopping flow impacts on sea dikes with vertical walls in shallow water conditions. RZ-SWASH is validated against experimental data for post-overtopping processes. The case of study consists of a multi-functional sea dike with gentle and very shallow foreshore [49,50]. Physical model tests were carried out in a 4.0 m wide, 1.4 m deep and 70.0 m long wave flume at Flanders

Conclusions

Relaxation Zone (RZ) technique has been implemented in DualSPHysics and tested the performance of wave generation and wave reflection compensation for 2nd-order Stokes’ waves. In order to optimize the performance of RZ, an hyperbolic C function which generalizes and extends the one proposed by [30] is implemented. The model performance of RZ depends on the shape of the C function and the parameters related to the width of the RZ. The optimum values for RZ width and the ψ and β parameters that

Acknowledgements

We thank Prof. S. Aoki from Osaka University, Japan to promote the internship at Flanders Hydraulics Research of Akihiro Usui, who contributed to develop the hyperbolic weighting function that has been implemented in DualSPHysics. The physical model tests carried out at Flanders Hydraulics Research were sponsored by the STW-programme on integral and sustainable design of multifunctional flood defences, Project No. 12760, Flanders Hydraulic Research, and the WTI 2017 project (1,209,437)

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