Smoothed Particle Hydrodynamics (SPH), a mesh-less method that describes the fluid as a set of discrete elements, named particles, is typically computationally very intensive. However, recent advances using High Performance Computing (HPC) and Graphical Processing Units (GPU) have strongly contributed to significant gains in computational effort [1
]. Despite the use of HPC and GPUs, it is still challenging to model realistic engineering problems, which are usually multi-scale problems. This research tries to mitigate this problem statement by studying the possible reduction of the required SPH computational domain. This can be done by coupling SPH to a faster external numerical model which can deal easily with large computational domains. This requires accurate and stable boundary conditions. Both the development of accurate boundary conditions and the coupling of SPH to external models are part of the SPHERIC Grand Challenges [2
], which list the key issues to be addressed in order to make SPH a mature method. In literature, there are several research examples where coupling was applied involving SPH methods. A general algorithm for one-way (exchange of information in only one direction) coupling of SPH with an external solution has been proposed in Bouscasse et al. [3
]. The interaction between the SPH solver and the external solution is achieved through an interface region containing a so-called “ghost fluid”, used to impose any external boundary condition. In Fourtakas et al. [4
], a hybrid Eulerian-Lagrangian incompressible SPH formulation is introduced, where two different SPH formulations are coupled, rather than two completely different solvers. The SPH solver DualSPHyics has been coupled in Altomare et al. [5
] and Altomare et al. [6
], where a one-way coupling was realized with the wave propagation model SWASH [7
]. A numerical wave flume has been created to simulate wave impact and run-up on a breakwater. The first part of the used numerical flume is simulated using the faster SWASH model, while the wave impact and run-up are calculated using DualSPHysics. Here, a one-way coupling is sufficient, since there is only interest in the impact of waves on the breakwater. In Kassiotis et al. [8
], a similar approach has been adopted, where a 1D Boussinesq-type model is applied for wave propagation in the largest part of the spatial domain, and SPH computations focus on the shoreline or close to off-shore structures, where a complex description of the free-surface is required. In Narayanaswamy et al. [9
], the Boussinesq model FUNWAVE [10
] was coupled to DualSPHysics, where the key development was the definition of boundary conditions for both models in the overlap zone. A wave generator in SPH moved according to the velocities from the adjacent Boussinesq nodes. Similarly, an incompressible SPH solver has been coupled to a nonlinear potential flow solver QALE-FEM [11
] in Fourtakas et al. [12
]. In Chicheportiche et al. [13
], a one-way coupling between a potential Eulerian model and an SPH solver is realised, applying a non-overlapping method using the unsteady Bernoulli equation at the interface. These studies applied coupling to speed up the simulation time by minimizing the computationally intensive SPH domain. Other studies apply coupling to combine both the benefits of mesh-based and mesh-less Computational Fluid Dynamics (CFD) methods. In Didier et al. [14
], the wave propagation model FLUINCO [15
] is coupled to an SPH code, and validated with experimental data of wave impact on a porous breakwater. A hybrid multiphase OpenFOAM-SPH model is presented in Kumar et al. [16
], where the SPH method is used on free surfaces or near deformable boundaries, whereas OpenFOAM is used for the larger fluid domain. A similar coupling is used, where breaking waves are modelled with SPH and the deeper wave kinematics are modelled with a Finite Volume (FV) method. This has been demonstrated in Marrone et al. [17
] for a Weakly-Compressible SPH (WCSPH) solver and in Napoli et al. [18
] for an Incompressible SPH (ISPH) solver. This research focuses on applying open boundaries in a two-way coupling methodology (exchange of information in two directions between the coupled models) between the Fully Nonlinear Potential Flow (FNPF) wave propagation solver OceanWave3D [19
] and the WCSPH solver DualSPHysics [20
]. A first version of this coupling has been introduced in Verbrugghe et al. [21
]. However, instead of open boundaries, moving boundaries were applied to transfer the orbital velocities from the wave propagation model to the SPH solver.
Typically, the numerical domain for wave propagation modelling in DualSPHysics is at least 3–4 wavelengths long [5
]. 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, and providing accurate boundary conditions capable of active wave generation and wave absorption by applying a coupling with a wave propagation model. In this manner, realistic open sea conditions can be simulated where waves enter on the left-hand-side of the fluid domain and exit freely on the right-hand-side of the fluid domain. The WCSPH solver DualSPHysics and the FNPF solver OceanWave3D are used here to demonstrate the coupling methodology, using the recently developed open boundaries [22
]. The open boundary formulation applies so-called “buffer zones” containing layers of buffer particles, positioned adjacent to the fluid domain. Buffer particles are used to enforce certain conditions in the presence of fluid inlets and outlets. Particularly, the physical information of buffer particles can be imposed by the user a priori or can be extrapolated from the fluid domain with a procedure, which is first-order consistent.
Although these open boundaries are similar to what was presented by Ni et al. [25
], there are some key differences making the formulation used here by Tafuni et al. [24
] more flexible. Firstly, flow reversion problems can not be simulated with the method by Ni et al. [25
]. Secondly, there is no possibility to extrapolate flow quantities using ghost nodes. Thirdly, the method to impose free surface elevation is different. Fourthly, the applied velocity profiles and corrections are depth-averaged. Lastly, only 2nd order wave generation is possible, where the method introduced here is compatible with up to 5th order generation.
Applying open boundaries for wave generation and wave 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 structures and energy devices, wave breaking conditions, etc. Additionally, the buffer zones in the open boundaries accept physical information from any source: e.g., linear wave theory, nonlinear wave theories, external numerical models such as CFD models, or even measurement data.
The presented two-way coupling methodology expands the current SPH state-of-the art with the following features: