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Review of wave generation and absorption methods

In this section we will review the most important references on wave generation and absorption applied in numerical modelling. Since in most of coastal engineering cases the dynamics of interest are wave-driven, the accuracy of the wave kinematics will have a major impact in the solution, therefore, producing accurate waves starting from wave generation will minimise error propagation and second order effects. Additionally, in most cases we would like to simulate open-sea conditions, in which waves will travel away from the domain of interest.

However, since it is impractical to simulate large domains in CFD modelling, the role of the absorbing methods is to simulate the correct conditions for the wave to propagate outside of the domain, minimising the reflections.

There are several wave generation and absorption approaches that are worth mentioning, thus, this section will be split into four subsections. Taking into consideration practical space limitations, the review presented in this section is not complete, therefore, before reviewing individual techniques it is worth mentioning several recent review papers which can help the reader gather further details on wave generation and absorption methods.

The first reference worth mentioning is Schaffer and Klopman (2000), which is a classical reference in this field. Although this paper is focussed on active wave absorption for laboratory wavemakers, it has been extensively applied in numerical models afterwards, both for static and dynamic wave absorbing conditions.

The second highlighted reference is Miquel et al. (2018). In this paper different combinations of wave generation and absorption methods are tested in terms of performance (reflection coefficient) and the differences between them are reported. The model used in this work is REEF3D, and the results are compared with previous benchmarks published for OpenFOAM®.

Finally, Windt et al. (2019) is the most recent reference. This work includes an extensive assessment of all the wave generation methods available in the OpenFOAM® framework presently, considering multiple implementations of boundary condit ions, relaxation zones and moving boundaries. Comparisons are made in terms of accuracy, comput ational requirements and features available. Differences point out that none of the methods are consistently superior in all three key performance indicators, therefore, Windt et al. (2019) can be used as a guide to select the most suitable wave generation and absorption methods for a particular problem.

Static boundary wave generation and absorption boundary conditions

Static boundary wave generation is the most straightforward technique that can be implemented in most Eulerian numerical models. In it, the theoretical expressions for free surface elevation and velocities are applied on static boundaries as fixed value (Dirichlet-tvpe) BCs. There are numerous wave theories available with which to calculate the necessary wave kinematics, ranging from the most simple ones (Stokes first order, Stokes (1847)), only applicable to a limited range of conditions, to universal theories of high order (e.g., streamfunction, Dean (1965)) that can be used for nearly-breaking waves. The most suitable theory for a particular wave condition can be obtained with Le Mehaute (1976). Wave absorption can also be performed at the boundaries, either independently or, in some cases, simultaneously with wave generation. The most relevant wave absorption techniques are reviewed in next subsection.

2.1.1 Wave absorption boundary conditions

Probably the simplest wave absorption boundary condition is Sommerfeld radiation condition (Sommerfeld, 1964), also called wave-transmissive or open boundary condition. This technique is based on the analytical solution of radiation problems:

in which coul is the phase velocity and Фш" is the potential of the target wave component to absorb. As noted in Dongeren and Svendsen (1997), perfect absorption can be achieved, in principle, for monochromatic waves perpendicular to the boundary and in which the wave celerity is known. Since Equation 1 is a first-order approximation, Engquist and Majda (1977) extended this BC to higher order and derived local approximations, decreasing the reflections for angles of incidence that deviate from the normal. Higdon (1986, 1987) further developed this boundary condition, introducing the incident angle in the formulation. The reader is referred to Givoli (1991); Dongeren and Svendsen (1997) for a comprehensive review of additional developments.

In this chapter we will focus only on one of the most recent advances available in literature. The work by Wellens et al. (2009) and Wellens (2012) aimed to apply open boundary conditions to real offshore problems, in which dispersive irregular wave sea states are required. Since wave components with wave celerities (c°ul) different from the target in Equation 1 will produce noticeable reflections, Wellens (2012) proposed a rational approximation for them with the form of a digital filter (DF):

in which a, and bj are the coefficients of the filter. The digital filter needs to be designed accordingly to the wave celerity range expected, with the goal of minimising the overall reflections. In this sense, Wellens (2012) recommended using the wave celerity associated to the peak spectral frequency in irregular sea states. Extensive work was performed to obtain a set of weights which achieved high performance for real cases, yielding reflections generally below 5%.

Active Wave Absorption (AWA) is another Dirichlet-tvpe BC that can absorb waves at a boundary with minimum reflections. AWA was originally developed for laboratory wavemakers, nevertheless, it is directly applicable to numerical models, both for Dirichlet- type and moving boundaries (later introduced in Section 2.4).

The concept behind AWA is that the boundary responds actively to the incident waves, unlike passive absorption (later introduced in Section 2.2), in which damping does not change with time. In order to do so, the AWA technique requires a hydrodynamic feedback, with which to estimate the waves incident to the boundary, and react accordingly.

To study how this works we will analyse the simplest implementation of AWA, which is based on the assumption of linear wave theory in shallow waters and it has been widely applied in the literature with outstanding results. The starting point is Equation 1 in Schaffer and Klopman (2000),

which is a very simple digital filter to calculate the velocity correction (Uc) necessary to absorb a disturbance (i.e., incident wave) of magnitude rj, only depending on gravity (jj itself. The negative sign indicates that positive velocity (i.e., inflow) is required to absorb a negative elevation disturbance (i.e., wave trough), and that outflow will absorb wave crests.

Several more complex commercial implementations of AWA exist. In these, the incident- reflected wave separation is often performed via digital filters (see Antoniou (2006)) from one or several hydrodynamic feedback magnitudes measured inside the wave tank. For example, nonrecursive DF (Finite Impulse Response, FIR) involves a convolution, which introduces a time delay, whereas recursive digital filters (Infinite Impulse Response, HR), which are composed of decaying exponentials, can be designed to be delay-free. As a result, FIR requires the hydrodynamic feedback to be collected far enough from the wavemaker to allow the system to react in time, e.g., in Christensen and Frigaard (1994) two free surface elevation gauges are placed away from the wavemaker. Alternatively, in Schaffer et al. (1994) the hydrodynamic feedback is produced by a free surface elevation gauge mounted directly on the wavemaker paddle.

The AWA procedure is suitable to absorb not only normal long-crested waves, but also oblique waves too. This is done by splitting the boundary into smaller pieces, similar to the individual paddles of a laboratory wavemaker, each of which applies the Equation 3 independently. One of the limitations is that this procedure cannot absorb the components of the waves that propagate tangentially to the boundary which are also difficult to absorb in the laboratory.

The simple DF in Schaffer and Klopman (2000) has been applied successfully in literature (Torres-Freyermuth et ah, 2007; Higuera et ah, 2013; Miquel et ah, 2018), and is quite effective, especially for shallow water conditions. AWA has also been applied outside the shallow water regime, with acceptable but decreasing performance as the wave conditions approach deep waters. The reflection coefficients obtained for deep water conditions are generally larger than 20%, thus not acceptable for practical simulations. This is caused by two simplifications that the shallow water conditions assume: a constant velocity profile along the water depth and non-dispersive waves (c = y/g h), whereas in deep waters, the velocity profile decreases to almost zero throughout the water column and wave celerity is calculated by solving the general dispersion relation.

In the recent work of Higuera (2020), AWA absorption has been extended to work efficiently in deep water conditions. Two important shortcomings have been corrected by introducing wave dispersivity and vertically-varying velocity correction profiles. An additional input parameter to the model, the wave period (T), is required to calculate the precise wave celerity according to the dispersion relation. Moreover, the velocity correction profile is adapted to the depth conditions based on the general linear wave theory. Finally, a combination of AWA and passive wave absorption (relaxation zone) has been tested, indicating that each technique can benefit from the other to increase the overall wave absorption performance, as will be discussed in Section 3.

2.1.2 Examples and applications

Sommerfeld (open) boundary conditions have been widely applied in nonlinear shallow water and Boussinesq models (Israeli and Orszag, 1981; Larsen and Dancy, 1983). As Navier-Stokes equations models were introduced, Open BCs were extended to work with free surface flows via the Volume Of Fluid (VOF) technique. Examples of initial applications include the SKYLLA (van Gent et al., 1994), SOLA-VOF (Iwata et ah, 1996) and the COBRAS (Lin and Liu, 1999) models. Lin and Liu (1999) report a good performance of the open BC, even for deviations of the celerity around 20%. More recently Wellens et al. (2009); Wellens (2012) have perfected the open BC in ComFLOW. Their approach is only applicable in numerical modelling, because it uses the pressure and velocities and their gradients throughout the water column as input variables, but extends the applicability of open BCs to irregular sea states.

Open boundary conditions have also been recently developed for weakly compressible SPH solvers in Verbrugghe et al. (2019), claiming that it can be extended to the incompressible SPH method too. It can be argued that this approach corresponds to a relaxation zone rather than to a boundary condition, because strictly speaking wave generation and absorption are performed in an area of buffer particles serving as ghost nodes. However, it must also be noted that these areas are much smaller (8 particles wide) than the typical lengths required in classical relaxation zone techniques, on the order of magnitude of a wavelength, therefore being closer to the application characteristics of boundary conditions.

Regarding active wave absorption, this technique was first developed based on the Sommerfeld equations in Van der Meer et al. (1993) and Sabeur et al. (1997). Later, the VOFbreak model (Troch and De Rouck, 1999) was implemented for fixed boundaries, based on a commercial AWA system available for laboratory wavemakers. Some differences were introduced in the numerical model version, taking advantage of the flexibility that numerical models offer. First, the feedback was changed from free surface elevation to velocities, which are directly retrievable from the model and do not reduce the absorption performance (Hald and Frigaard, 1996). Second, arbitrarily long filters could be used because the numerical model, unlike the physical system, does not require real-time performance, thus enhancing the overall absorption rates.

A simpler system introduced in Schaffer and Klopman (2000), which does not require designing complex digital filters and only uses the free surface elevation at the wavemaker as input, was implemented and applied for the IH2VOF model in Torres-Fieyermuth et al. (2010). The same approach was later extended to 3D simulations in Higuera et al. (2013) for the IHFOAM model based on OpenFOAM®. The 2D version has been implemented in the REEF3D model (Miquel et al., 2018). As mentioned before, the aim of their paper was to explore the performance of different wave generation and absorption methods and their combinations.

The limitations of this method, derived from the initial assumptions of linear waves in shallow waters, have been recently revisited and extended to work at any water depth regime in Higuera (2020). The new method, called extended range active wave absorption (ER-AWA) offers a higher overall performance and has been implemented and released in the open source olaFlow model (Higuera, 2017), also based on OpenFOAM®.

2.1.3 Advantages and disadvantages

Dirichlet-type wave generation presents multiple advantages. These include the simplicity of implementing them, simply as a fixed-value boundary condition, and the low computational cost of this procedure, which is often negligible and it is the lowest among the other wave generation techniques reviewed in this chapter. Nevertheless, the main drawback of this method is that it needs to be coupled with wave absorption necessarily, either at the same boundary or elsewhere. This is so because there is an imbalance between the amount of mass (water) introduced in the domain to produce a wave crest and the mass extracted to generate a wave trough. The mass differences accumulate wave by wave, producing a progressive increase of the mean water level in the domain, which needs to be absorbed (Torres-Freyermuth et al., 2007). Many wave absorption methods are suitable to deal with this effect, except for specific passive wave absorption methods which damp momentum and not mass such as momentum damping zones or dissipative beaches. As demonstrated next, AWA is the natural selection to mitigate this effect.

Regarding wave absorption, the main advantage of Sommerfeld BCs is their simplicity, which generally does not produce any significant increase in the computational cost of the model. Moreover, the absorption performance can be extremely high, provided the wave celerity and wave incidence direction is known in advance. The main disadvantages of traditional radiation conditions are that they were formulated to absorb monochromatic waves, therefore, they are not able to absorb effectively irregular (multi-chromatic) sea states. Furthermore, these techniques were conceived to work in purely absorbing boundaries rather than combined with wave generation. Nevertheless, the work by Wellens (2012) corrects these two factors.

The application of AWA in numerical models presents several advantages with respect to the laboratory systems. One convenient feature of the numerical models is the ease of obtaining flow measurements, without disturbing the flow and including variables that cannot be measured directly in the lab. New variables can provide valuable feedback for AWA systems, which by accepting additional features as input can improve their absorption performance or provide additional functionalities. For example, Troch and De Rouck (1999) uses velocities instead of free surface elevation, which are easy to obtain in the numerical model. Moreover, depth-averaged velocity components at the wavemaker boundaries were used in Higuera et al. (2013) to discriminate waves parallel and perpendicular to them, preventing spurious wave generation for non-incident waves.

Another important factor is that, AWA systems in numerical models do not require to operate in real-time, unlike AWA in laboratories. This constraint limits the complexity of the digital filters that can be applied in physical wavemakers, whereas arbitrarily complex digital Biters can be used in numerical models, because they would just delay the start of the next time step calculations. The main consequence is that higher absorption performance can be achieved in numerical models, as reported in Troch and De Rouck (1999). Furthermore, eliminating the requirement of having a real-time response permits having a more flexible testing framework in which to develop new ideas.

As mentioned earlier, AWA may be required even if no incident waves to the boundary are expected, in order to prevent the increase in water level produced by the Dirichlet type wave generation. Fortunately, AWA is perfect for inhibiting such mean water level rise, which can be assimilated with a long period wave, because AWA is extremely efficient absorbing long waves. Moreover, AWA can work at the same boundary as wave generation and without significant additional computational cost. Finally, unlike the traditional radiation BCs, AWA can be applied to absorb irregular waves, not only monochromatic waves.

The main disadvantage of AWA is that it usually depends on complex digital filters, which might need to be designed ad-hoc for specific wave conditions. The simplest version of AWA (Schaffer and Klopman, 2000) presents an additional limitation, its application is typically constrained to shallow water conditions. However, the range of applicability has been extended recently in Higuera (2020).

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