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Discussion

Moving boundary is the wave generation procedure that best matches the well-understood laboratory wavemakers, both for wave generation and absorption. At the same time, thanks to the capabilities of numerical models, some of the shortcomings of laboratory wave gen?eration can be solved. For example, numerical models do not experience mechanical or inertial effects and allow perfect repeatability. Furthermore, the flexibility of numerical frameworks permits testing new and more complex forms of wave generation and absorption, as arbitrarily-deforming boundaries, which are not feasible to develop physically. Other restrictions, such as real time operation, can also be circumvented numerically, offering the possibility to obtaining more efficient wave absorption rates. However, moving wavemaker generation and absorption usually require longer computational times than other techniques, including relaxation zones, because the numerical model performs complex moving mesh steps, which significantly increase the computational cost. In short, moving wavemakers in numerical models offer higher fidelity for reproducing laboratory tests, at the cost of increased model complexity which may result in longer simulation times.

It is not always convenient to generate waves with moving boundaries, as this approach can be complex and quite limited to represent real wave conditions. In this sense, Dirichlet- type wave generation is more flexible, as it offers a way to generate waves using theoretical formulations for the hydrodynamic variables (e.g., free surface elevation, velocities, pressures...). Wave absorption, open BC or AWA. can also be implemented in a similar way and in some cases be applied simultaneously with wave generation. This approach is the most computationally efficient, as it takes place at a fixed boundary, hence, no mesh movement is required, and the domain boundaries can be located close to the area of interest, further minimising the computational costs.

There are several disadvantages for Dirichlet-tvpe wave generation. The most noticeable one is that this method produces a progressive increase in the mean water level, as explained before. Fortunately, linking wave generation with AWA solves this problem without significant increase in computational costs. Another limitation is generating very steep waves, as experience tells that they are more prone to start breaking at the boundary below the expected breaking limit. Regarding wave absorption, both open BCs and AWA work more efficiently for monochromatic waves. Moreover, absorption may be sub-optimal if all the incident wave information is not available beforehand, or in cases of time-varying wave conditions (e.g., irregular wave sea states) and in most cases the performance largely depends on the digital filter that has been designed. AWA, which appears to be the most used fixed- value method presently, is very sensitive to relative water depth conditions. This technique is especially effective for absorbing long waves and it has recently been extended to absorb waves in deep water conditions as efficiently.

Relaxation zones are other effective and widely used methods to generate and absorb waves within a domain, rather than at a boundary, since they are essentially a mean of “diffusing” the boundary condition over a model subdomain, rather than applying the boundary condition locally. The procedure of “diffusing” the boundary conditions can be achieved either by introducing appropriately scaled source or sink terms in the flow equations (“physical forcing”) or by relaxing the flow variables toward the desired boundary condition within the numerical solution algorithm (“numerical forcing”). The inherent property of “diffusing” the boundary condition is responsible for both the most significant advantages and disadvantages of the method. The method is generally considered superior in fidelity of generation for high steepness waves and more efficient in absorbing waves, but it is also more expensive, especially when compared to open BC or AWA. Ongoing research is looking to address computational cost issues and substantial steps have been made towards this direction.

Internal wave makers (IWM) are based on the concept of dedicating internal areas in the mesh for wave generation, while relying on absorbing boundaries or relaxation zones for absorption. This method was proven particularly successful for depth integrated models. The method has been used also in CFD models showing good overall performance. For CFD models, the method requires a substantial calibration procedure and its implementation does not seem as straighforward as other methods and relies on the implementation of Dirichlet- type conditions or the relaxation zone for passive absorption for use in the context of NWT. In addition, IWM can be costly in terms of computational cost and it is very often comparable to the relaxation zone method. Whilst IWMs remain widespread for depth integrated models, it is the authors impression that recent CFD modelling applications increasingly Dirichlet type conditions or the relaxation zone method are more effective IWM for NWTs, due to the aforementioned reasons.

As a conclusion, significant advances have been made in wave generation and absorption techniques for numerical models over the last decades. Developments linked to physical wave tanks have been the main driver pushing the state of the art forward for a long time. This situation has reversed in the last decade, as now the scientific community has wider access to powerful computational clusters. Consequently, future new developments will likely be computational-based, powered by the flexibility that numerical models and other computational tools offer.

In this sense, hybrid modelling is an extremely promising approach. Hybrid models rely on coupling to blend different modelling approaches. For example, wave propagation can be performed with a simplified (e.g., potential flow or Boussinesq-type) model to study the wave transformations before arriving to the area of interest, in which CFD modelling is performed. This approach reduces the overall computational cost but requires a proper link between the models to allow a seamless transition not compromising the wave kinematics. Recent applications include a two-way coupling between the fully nonlinear and dispersive potential flow model 0ceanWave3D (Engsig-Karup et ah, 2009) and the CFD NWT waves2Foam in OpenFOAM® (Jacobsen et ah, 2012), developed with relaxation zones. Another example is the one-way coupling of a Lagrangian model solving the mass and vorticity conservation equations (Buldakov, 2013, 2014) and the CFD NWT olaFlow (Higuera, 2017), also developed in OpenFOAM®, via a boundary condition.

Another area to explore is the combination of existing wave absorption methods to increase the performance of wave absorption while minimising the increase of computational costs. Although this topic has already been explored in the past quite successfully, e.g., Israeli and Orszag (1981); Clement (1996); Grilli and Horrillo (1997) combining open BCs and relaxation zones for absorption, this research line has been discontinued for some time, and new formulations have been introduced and perfected since. For example, some initial tests combining AWA and relaxation zones have been performed in Higuera (2020), showing that this approach can reduce the overall reflections in most cases.

Although to the authors knowledge there has not been an attempt towards this direction, AWA and SWAG could be combined to create a hybrid scheme and improve wave generation at the offshore boundary. For example, the AWA condition could be relaxed over a layer of cells at the generation area, which ideally would be much shorter than the typical length of the generation zone used in the conventional relaxation zone technique. This approach could be combining the better of two worlds, e.g., low computational cost, high absorption efficiency. It could be nevertheless argued that this technique would be solving two equations in the same domain, so challenges for developing this approach may be similar to the ones encountered when developing multi-scale models.

 
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