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Moving wavemakers with active absorption

Using moving wavemakers to generate and absorb waves in numerical models presents the advantage that they can replicate wave modelling operations in a laboratory environment, i.e., performing the exact movement of the physical wavemakers. This behaviour may pose significant benefits in seiche or harbour resonance studies, as long waves are especially sensitive to changes in the length of the domain, which varies due to the displacement of the wavemaker (Higuera et ah, 2015).

2.4.1 Moving wavemaker types

Normally, the most popular devices simulated in the literature are of piston or flap (including variable draft) type. However, devices that are not that common, as double-, triple-flap (Halfiani et al., 2015), mixed type (piston + flap) or plunging wavemakers (typically wedge- shaped) (Gadelho et ah, 2015) can also be replicated. Even arbitrarily-deforming boundaries (Koshizuka et ah, 1998) have been tested.

Piston-tvpe wavemakers (e.g., Huang et ah (1998)) are probably the most extended in coastal engineering studies, both in experimental and numerical modelling references. The velocity profile generated with this category of wavemakers is constant throughout the water depth, thus it matches well the kinematics of waves in shallow waters. Alternatively, flap wavemakers (e.g., Antuono et ah (2011)) are better suited to replicate the hydrodynamic conditions of waves in deep waters, thus, they are widely used for offshore engineering applications. In both cases, piston and flap wavemakers support second order wave generation for irregular waves, a breakthrough feature that suppresses the generation of spurious free waves. Second order generation was developed in Schaffer (1996) in 2D and later extended to 3D in Schaffer and Steenberg (2003).

A number of procedures for performing active wave absorption (AWA) with moving wavemakers are available in literature. In all cases a displacement correction to the original wavemaker signal is calculated using some hydrodynamic feedback as input. The new calculated movement absorbs the incoming waves to the boundary, preventing a high degree of reflections. Furthermore, AWA can be connected to perform independently in purely absorbing boundaries or simultaneously to the target wave generation.

As mentioned before, wave absorption in CFD models was adopted from systems initially developed for laboratory wavemakers. The two main AWA programmes AWACS and AwaSvs, already introduced in Section 2.1, are straightforward to apply on numerical moving wavemakers. The procedure is equivalent to AWA on static boundaries, but the correction velocity is used to correct the movement of the wavemaker. For example, for a piston-type wavemaker, considering that velocity is the time derivative of displacement, the displacement correction is calculated as:

The simplest version of AWACS (Schaffer and Klopman, 2000), introduced in Equation 3, has been applied successfully to absorb waves in Lara et al. (2011); Higuera et al. (2015). Reasonable results are reported outside the range of application (shallow water wave regime) of this simplified approach.

The way to generate directional waves with moving wavemakers is based on what is usually called “snake movement”. This means that each individual paddle of the wavemaker moves with a lag with respect to its neighbours. This method is also applicable to absorb oblique waves, but it cannot absorb the wave components propagating tangentially to the wavemaker, though.

AWA has also been developed for wedge-type wavemakers. Bullock and Murton (1989) presented a system for a physical wavemaker in which the solid wedge moved obliquely up and down. The feedback came from a gauge located at the front wall of the wavemaker, mounted on rails so that it was always at the same level, and was converted via an analog recursive filter. The authors are not aware of an implementation of this method in CFD models, however, plunging wedge-shaped wavemakers (without AWA) have been modelled in Gadelho et al. (2015).

Another completely different approach for AWA is presented in Spinneken and Swan (2009a,b). In these works the feedback is not obtained from free surface elevation gauges, but from forces measured at the wavemaker. The absorption correction is then calculated via an infinite impulse response (HR) filter. This method presents the advantage that the feedback signal is not obtained from point measurements, but from an integrated magnitude instead (force is calculated by integrating fluid pressures at the wavemaker), therefore, the output is less sensitive to local disturbances. The force-feedback AWA is applied numerically in Spinneken et al. (2014). Even though the numerical model is a nonlinear Boundary Element Method (BEM) and the method is applied for first order wavemaker theory, there are no limitations for porting it to any CFD package and with the second order formulation. However, some shortcomings are pointed out in Spinneken et al. (2014). The most important is the limitation to absorb high-frequency or high-nonlinearity waves because of the extremely high accelerations required (i.e., “excessive amount of added mass”), although this should not pose a major challenge in numerical models.

2.4.2 Mesh movement algorithms

Replicating moving wavemakers in CFD models requires mesh-movement algorithms, as waves are generated by prescribing a deterministic displacement on a solid, given by the methods presented in the previous subsection. However, we would like to stress that the techniques that are going to be reviewed next can also be applied for a wide variety of other cases, as discussed later.

Generally speaking, wavemakers can be emulated in numerical models by two procedures: Moving Boundaries (MB) and Immersed Boundary Methods (IBM). A complete diagram showing the most relevant algorithms and examples of applications is presented in Figure 5.

The MB approach is able to represent the prescribed movement of a solid in which one or more of its faces correspond to the mesh boundaries. This can be the case of a piston or flap wavemaker (a single boundary moving to replicate the device) or an embedded body which has been meshed around (e.g., a plunger wavemaker with several faces), similar to the one portrayed in Figure 5. Usually, the MB procedure involves a set of boundary conditions (BCs) to replicate the action of the body movement, which will deform the mesh. The procedure can be outlined as follows. First, for each time step the BCs set the displacement of the individual nodes at the moving boundaries and keep the rest of the nodes at the static boundaries in the same position. Then, the mesh is deformed accordingly. This step can be performed in a number of ways, some of which are introduced in .lasak and Tukovic (2006). For example, the displacement of the internal nodes can be calculated by solving a Laplacian equation:

where V = щ-; к is the diffusivity, i.e., a scalar field, either constant or variable in space, which helps redistribute the node displacements across the mesh: Ad is the displacement vector; and At is the time step. Further details on this mesh movement technique can be found in Higuera et al. (2015). Next, the system needs to re-distribute the quantities (e.g., VOF, velocities...) and fluxes to adjust to the new cell volumes and face areas from the deformed mesh. Finally, the BCs are imposed to the fluid to solve the Navier-Stokes equations as usual. The moving boundaries are represented as moving walls in this step, with a no-slip boundary condition and no-flux across them, therefore, the fluid in contact with a wall will have the same velocity as the wall.

The MB procedure discussed is only valid for small displacements. The larger the displacements experienced, the further the mesh quality will decrease, until the case will fail for violating the mesh quality requirements for the finite volume discretization. If displacements are large, the mesh evolution stage might require topological changes (TC), such as

Moving mesh algorithms available for wave generation with moving boundaries

Fi gure 5: Moving mesh algorithms available for wave generation with moving boundaries.

remeshing, to maintain the mesh quality. This approach includes an interesting technique which might be useful for simulating long-stroke wavemakers: to insert or remove layers of cells in front of the piston as it moves back and forth. This procedure is applied already when simulating pistons in turbomachinery (Montorfano et ah, 2014), however, the authors are not aware of any applications to piston wavemakers published yet.

Methodologies for mitigating issues with mesh quality over large mesh deformations are also discussed in de Lataillade (2019) in the context of floating structures. These methodologies may be used to maintain an acceptable mesh quality without the need of introducing TC in the mesh. In this work, the application of Arbitrary Eulerian Langragian (ALE) techniques (Hilt et ah, 1974) are discussed for mesh motion, namely the linear elastostatics model (LEM, e.g., Dwight (2009)) and using mesh adaptivity (without TCs) with monitoring functions for mesh quality (e.g., Grajewski et ah (2005)). Both methods perform well in the context of float ing structures, but it is further commented that for long simulations involving nonlinear oscillations, the LEM could allow mesh quality to deteriorate after several dozens of cycles. The authors are not aware of any applications to piston wave makers published yet, but it could be argued that such an application could be straightforward, due to the proximity of the moving/floating structures with the numerical paddle problem.

Another very important technique that enables the simulation of large displacements of solids, this time on a fixed grid, is the Immersed Boundary Method (IBM) (Mittal and Iaccarino, 2005). Although there are countless IBM implementations in literature, the underlying idea is to apply a momentum source within the mesh cells to simulate an interface between the fluid and the solid. This enables representing a complex solid moving inside a mesh that is not body-fitted (Kim and Choi, 2019). This is the main reason why the IBM is widely applied to solve Fluid-Structure Interaction (FSI) problems, as for example ship hydrodynamics (Yang and Stern, 2009), rigid floating objects (Calderer et al., 2014; Bihs and Kamath, 2017) or with flexible elements mimicking vegetation (Chen and Zou, 2019).

There are two main types of IBM depending on how the forcing is applied. Continuous forcing implies that the IB force term is included in the Navier-Stokes equations before the discretization, whereas discrete forcing involves applying the IB force to the discretized equations (Mittal and Iaccarino, 2005). Most of the modern implementations if IBM, such as the Fractional Area-Volume Obstacle Representation (FAVOR) (Wei, 2005), the Cartesian cut-cell method with openness coefficients (Lara et ah, 2011) or the ghost-fluid (Bihs and Kamath, 2017) methods are discrete forcing methods. One of the common points in most methods is the way to represent the solids, by means of the area and volume fractions that each solid occupies for each face and cell, respectively.

The IBM technique presents advantages for large displacements, thus it has been applied to simulate long-stroke piston-type wavemakers (Usui et ah, 2016), which are capable of simulating long tsunami-like waves. The main disadvantage is that it is not straightforward to acquire flow variables at the boundary, necessary to feed the active absorption algorithm because these are likely to be retrieved by interpolation, which will lead to increasing numerical errors.

The last mesh movement algorithm reviewed in this chapter is overset (or Chimera) grids (Panahi and Shafieefar, 2009; English et ah, 2013). In overset grids (OG), the flow is solved in independent but overlapping meshes simultaneously. There are no restrictions on any of the coexisting meshes, i.e., they can be moving or static, structured or unstructured, deforming or fixed. Generally, the background mesh is static, significantly larger than others and usually simple (i.e., structured). Additional meshes can have different applications, such as to provide a finer resolution than the base mesh locally, or to simulate fluid-solid interactions. Such meshes are usually body-fitted to internal solids/obstacles and may employ curvilinear grids, allowing having a proper boundary layer discretization if required. Similar to IBM grids, obstacles can be given a prescribed movement, or response to the action of the flow (6 degrees of freedom).

The peculiarity of the OG method is that the moving mesh does not necessarily undergo any deformations, but it is generally subjected to rigid movements and rotations only, which is ideal for simulating relative motion of different solid components. In this sense OG has been employed to simulate very complex problems such as Wave Energy Converters (WECs) (Windt et ah, 2018) or ship self-propulsion and manoeuvring (Shen et ah, 2015).

Due to the multi-mesh approach, all the individual meshes need to be coupled. Coupling takes place over the regions of overlap, generally near the border of the smaller meshes. Therefore, the CFD code requires a specific library or routines and additional steps to calculate the connectivity between meshes, perform 3D interpolation to map and exchange the flow fields data and establish locations that can be excluded from the solution, to avoid duplicating calculations.

Although OG have been extensively applied in the marine industry, they are still a very novel approach and the authors are not aware of any works in which OG are applied to simulate moving wavemakers. Possibly this is because OG methods require a more significant effort in development and implementation than the other MB methods discussed here, and the moving paddle problem is relatively simple, so such an effort may not be justified.

In short, all the techniques reviewed in this section present advantages and disadvantages, therefore, the user needs to decide which method to apply in case several are applicable. Moving boundaries present a clear advantage, since no topological changes are required for small displacements, the mesh deformation takes relatively less time and mass conservation is easily achievable. The possible complexity to create the mesh and the limitation to small displacements are the main drawbacks. If large displacements are required, additional algorithms need to be applied to monitor the mesh quality and adapt the mesh (with or without topological changes). In case of topological changes, the remeshing steps would take longer time, errors can be introduced due the grid-to-grid interpolation and mass conservation might be violated because remeshing changes the topology (e.g., introducing or deleting cells). When the algorithms involve monitoring quality and adapting the mesh, these particular errors may not be present, but other errors or computational overheads may be present, depending on the complexity and the efficiency of the monitoring algorithm. As the main advantage, moving mesh methods with a scheme for monitoring quality offer more flexibility and can preserve a better mesh quality.

Immersed boundary methods offer the same capabilities to simulate large displacements without the need to perform mesh deformation or deal with topology changes. The main advantage of IBM is that the base mesh can often be very simple and, at the same time, the method can deal easily with very complex geometries. However, the simple treatment of the solids uses a castellated (saw-tooth) approach, which is not suitable to simulate near wall dynamics correctly. This issue can be overcome to some extent by increasing the resolution locally. Nevertheless, as outlined in Kalitzin and Iaccarino (2003) and Kim and Choi (2019), the application of IBM to high Reynolds number cases still poses challenges, because gradients are large in turbulent boundary layer flows and body-fitted meshes are desirable as compared to castellated meshes.

Overset grids are probably the most advanced technique to simulate moving objects. The advantages of this method are very similar to those of the IBM: extreme flexibility to simulate long displacements, plus enabling the simulation of relative motion between solids. Furthermore, “overhead grids” do not pose problems to resolving the boundary layer flow if required, as IBM does, since proper body-fitted meshes can be used around solids. The main disadvantage of OG method is the additional complexity in setting up the simulations, and the additional computational costs associated with the steps required to couple the meshes. Nevertheless, as remarked by Windt et al. (2018), “overhead grids” are a suitable option when moving boundaries would present numerical instabilities and the results are not needed as fast as with other methods.

2.4.3 Examples and applications

Moving wavemaker procedures have been widely applied in literature for different CFD modelling techniques, for commercial and research models alike.

Piston wavemakers can be traced back to (Monaghan, 1994), the first paper to present the SPH method applied to model inviscid and incompressible free surface flows. Some years later a special moving wavemaker with arbitrary deformation was presented in Koshizuka et al. (1998). In this paper the numerical model is implemented with the Moving Particle Semi-Implicit (MPS) method, and the wavemaker displacement amplitude varies along the water depth to mimic the velocity profile of cnoiclal waves.

The increase in adoption of moving wavemakers is linked to the general adoption of Eulerian Navier-Stokes models in the last 10 years. Lai and Elangovan (2008) simulated a numerical wave tank with a flap-type wavemaker in ANSYS CFX as a moving boundary. They studied different combinations of water depths, oscillation periods and amplitudes, and found good agreement with theory. In Lara et al. (2011), a piston wavemaker was replicated in the model IH2VOF using the IBM and openness coefficients. The goal of this paper was to simulate the propagation of infragravity waves generated by a short focused wave group breaking over a sloping beach. Waves were generated with a second order theory and active wave absorption was implemented following the method in Schaffer and Klopman (2000) and comparisons with experimental results were excellent .

Piston-type wavemakers have also been implemented in Flow3D (Vanneste and Troch, 2015). In this solver the paddle is represented by a moving solid IBM using the FAVOR technique (Wei, 2005). Vanneste and Troch (2015) simulated regular waves interacting with a porous breakwater. Results showed good accuracy when compared to experimental measurements. Long time series were simulated in this work, therefore, active wave absorption has been reported to play an important role in achieving stable simulations. The AWA methodology is the same applied in Troch and De Rouck (1999), in which the feedback is obtained from the horizontal and vertical velocities at a fixed location in front of the wavemaker, but applied to the moving wavemaker.

All the references presented so far in this section were 2D. However, Wu et al. (2016) performed a 3D simulation of a piston wavemaker to investigate the effects of the gap between the moving paddle and the walls of the flume. Solitary waves were generated with the classical method developed by Goring (1978) and a previous formulation (Wu et al., 2014), which yields a cleaner solitary wave. The flow leaks around the edges of the wavemaker was found to impact the wave height of the wave generated, decreasing significantly the wave amplitude, as reported in Madsen (1970) and later by Chwang (1983).

Higuera et al. (2015) presented an implementation in OpenFOAM® to replicate piston wavemakers with the moving boundary approach. The first case presented in this paper replicates an unpublished experiment from the set referenced in Lara et al. (2011) as a basic validation test. Similarly to the work in Lara et al. (2011), the AWA procedure implemented in Hi guera et al. (2015) is based on Schaffer and Klopman (2000). The major novelty introduced in this work is an implementation of a multi-paddle piston wavemaker. This solver used the time series extracted from a laboratory wavemaker with 10 paddles as input and was able to successfully replicate a 3D wave focussing. Several enhancements to the original developments in Higuera et al. (2015) have been added to olaFlow model (Higuera, 2017), which now includes the capability to reproduce flap-type wavemakers too. Most recently, olaFlow has been applied to simulate the evolution of breaking and near-breaking wave groups (Stagonas et al., 2018; Buldakov et al., 2019), demonstrating excellent capabilities to simulate accurate wave kinematics.

In Usui et al. (2016), a piston wavemaker is simulated in the model CADMAS-SURF 3D. Since the goal of this work was to replicate tsunamis and storm surge, the wavemaker presents a relatively long stroke, thus it required being simulated with the immersed boundary method. Zhang et al. (2017) presented an application to replicate a piston wavemaker with the moving boundary technique for ANSYS Fluent. His implementation includes second order wave generation, which is found advantageous with respect to first order in terms of preventing the development of spurious free waves, as expected. Chella et al. (2017) presented an application of flap wavemakers for the model REEF3D to simulate focused waves breaking over a slope. Comparisons of free surface elevations and the overall breaking wave shape were compared with experimental data successfully.

Finally, a piston-tvpe wavemaker has been implemented recently in a Lattice-Boltzmann method model (Davarpanah et al., 2018), combined with nested grids to obtain higher mesh resolutions locally to the free surface.

2.4.4 Advantages and disadvantages

The application of moving wavemakers in numerical models presents several advantages with respect to the laboratory systems. As previously discussed in Section 2.1, the ease of access to all the variables and not having the requirement to operate in real-time offer a higher degree of flexibility for AWA. Such conditions enable testing formulations, as complex as desired, which can be more efficient than their laboratory counterparts.

Within the methods, the moving boundary technique conserves the mass of the system. This has important implications, as in Dirichlet-tvpe wave generation or other flux-based wave generation methods, the mass imbalance between crests and troughs can accumulate and make the mean water level increase in time, requiring absorption to be rectified. In a BM NWT, the only variations in the mean water level are induced by the shortening/enlarging of the tank itself due to the wavemaker movements, as happens in physical wave tanks.

Another advantage of numerical models is that they can mimic even the smallest and detailed features of the experimental facilities. For example, models can simulate any degree of leakage of water between the edges of the wavemaker and the flume walls, including the watertight condition too. Such fine details can be proven important to ensure fidelity between the modelled and simulated results, as shown in Wu et al. (2016).

Finally, numerical wavemakers present no mechanical or inertial effects, hence, they respond instantly to the prescribed displacement and/or velocity. Furthermore, numerical wavemakers always move in the same way, irrespectively of the environmental conditions, unlike physical systems in which factors such as the temperature of the system can have a significant impact. This means that the numerical models produce perfectly repeatable simulations, and they can replicate in an exact way a time series of displacements provided.

There are several disadvantages of applying moving wavemakers. Two of them are caused by mechanical limitations, therefore they can be circumvented in the numerical model. First is the limitation to generate or to absorb long waves, as such will require long strokes, i.e., on the order of magnitude of the target wavelength, and generally, the total stroke of physical wavemakers is quite limited. Another issue is the limitation to generate and to absorb large steepness waves, which usually requires large accelerations that the wave generation machines often cannot provide.

Another issue is inherent to the method itself, therefore, it appears for physical and numerical experiments alike. This problem arises because waves are seldom perfect sinusoids, and small but systematic deviations from a zero-mean wavemaker movement will produce a constant drift of the wavemaker, as noted in Schaffer and Klopnmn (2000). Once the movement of the wavemaker has reached either of the limits, waves cannot continue to be generated or absorbed normally, since the wavemaker will stop at the minimum or maximum displacement positions. This behaviour is called saturation and can be prevented by forcing the wavemaker to return automatically to its initial position. The time frame for this additional movement needs to be short enough to prevent saturation, but long enough at the same time to be able to absorb longer waves to some extent and prevent the development of seiches in the flume (Schaffer and Klopnmn, 2000).

Finally, another of the disadvantages associated with moving wavemakers in numerical models is the larger computational costs associated with the moving-mesh techniques. Reports available in literature state that longer simulation times, between 20% to 40%, can be expected when comparing the moving wavemaker generation mode versus Dirichlet-tvpe wave generation (Higuera et al., 2015).

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