Internal wave makers
2.3.1 Background and development
The Internal wave maker (IWM) concept is usually implemented by creating a generation line or area inside the domain and adding source terms to the mass or momentum equations to generate waves. This area is expected to be transparent to the wave propagation (i.e., does not reflect waves), therefore, full wave absorption can be achieved as long as passive absorption zones or absorbing boundary conditions are added to inlet/outlet boundaries.
Originally, Larsen and Dancy (1983) proposed an internal generation scheme that is based on adding to or removing some volume of water from the domain by modifying the mass equation of the Boussinesq approach. The source terms were implemented after the equations had been discretised (“numerical forcing”). In their work it was demonstrated that the IWM remained transparent to the propagation of nonlinear regular waves. The proposed implementation was rather rudimentary and required tuning its performance on a case-by-case basis (Wei et al., 1999). It was also demonstrated that it was not suitable for more advanced formulations of the Boussinesq problems (Nwogu, 1993; Wei and Kirby, 1995). The overall concept was nevertheless relatively successful as it was the starting point for further improvements and adaptations in Boussinesq and CFD models.
Wei et al. (1999) proposed a new formulation for IWM, comprising the addition of source functions in the momentum and mass equations before these are discretised (“physical forcing”). The source function is relaxed over a region by using a bell-shaped distribution, rather than being a point source. It seems that the source functions parameters (amplitude, width) require tuning/calibrating against the target wave characteristics. The concept is shown to be transparent to the wave field, as reflected waves can propagate to the other side of the IWM and absorbed at an offshore absorption zone. The physical analogy of this implementation is to use a perforated internal wave maker in a flume/basin, but it is evident that such a concept may be not be fully realisable in a laboratory setting.
In CFD models. Lin and Liu (1999) applied an IWM to their VOF-based model, initiating the concept of a NWT for 2DV numerical flumes. The approach uses a relatively concise source region which is placed internally in the flume and below the free surface. In this region, the continuity equation is locally modified to include inflow and outflow fluxes that cause wave excitation. These terms subsequently drive the momentum equation towards wave generation through the pressure-velocity coupling scheme. The mass source functions
Figure 3: Illustration of internal wave maker in a NWT according to Lin and Liu (1999).
for a monochromatic wave have the following form:
where со, H and c is the wave angular frequency, wave height and celerity, respectively, and A is the area of the IWM. The concept can be extended to cover higher order waves, as well as random and directional waves, by linear superimposition of components. A sketch of the concept used by Lin and Liu (1999) is shown in Figure 3. The physical equivalent of this layout would be installing a submerged outlet in an laboratory wave basin and forcing oscillatory discharge to generate waves. Lin and Liu (1999) performed numerical tests with a rectangular source area placed in the middle of the flume, and absorbing (radiation) boundary conditions at the sides. Their approach performed relatively well for the wave regimes tested, including nonlinear and random waves. It was nevertheless found that the position and size of the source area greatly affects the wave generation performance. It was further demonstrated that the optimal location was at about 1/4-1/5 of the depth below the surface and the optimal size was < 5% of the wavelength, meaning that both the size and location of the source wave maker may have to be optimised depending on the wave conditions. The concept was extensively used in the COBRAS model, in tandem with both absorbing (radiation) boundaries, and relaxation zones for passive absorption (see relevant section). This approach was later proposed for establishing a NWT within the PHOENICS finite element model (Hafsia et al., 2009).
Saincher and Banerjee (2017) further assessed the performance of Lin and Liu (1999) IWM for steep waves in deep, intermediate and shallow waters by placing an IWM in the middle of the tank and using absorption zones at the sides to dissipate outgoing waves. They concluded that steep waves generated in deep or shallow waters may experience excessive wave height damping. It is argued that wave height dissipation is caused by vortices generated by the IWM. In deep water waves, the arrival of these vortices to the free surface was associated with wave height dampening (possibly through artificial incipient breaking), and it was noted that the problem was solved by adjusting the location of the IWM towards deeper water and increasing the size of the source area to reduce velocities. Due to the change of location and size of the IWM, additional calibration of the IWM had to be performed. Using a larger IWM to counter vorticity generated by the local flow gradients developed in IWM similar to the ones proposed in Lin and Liu (1999) was originally conceived by Peric and Abdel-Maksoud (2015) to facilitate deep water wave generation. For shallow water waves, relocating the source to lower depths was not particularly effective, so the authors proposed to increase the source size to cover the full water depth. This modification yielded good results, although it can be argued that this method is converging to the one proposed by Wei et al. (1999), as the source area is extending throughout the whole water column.
Choi and Yoon (2009) developed a concept for a momentum source based IWM in ANSYS Fluent. In their model, they adapted the methodology proposed by Wei et al. (1999),
Figure 4: Illustration of internal wave maker in a NWT according to Choi and Yoon (2009).
for application in CFD models and developed a 3D numerical wave tank with capability of generating plane and directional waves. The formulation of the source terms included in the Navier-Stokes equations can be summarised in the following equation:
where x, у are coordinates in the propagation plane, along and perpendicular to the main wave direction, respectively, g is the gravitational acceleration, 0 is a coefficient depending on the width of the generation area and the wavelength and ky and ш are the wavenumber on the vertical (y) axis and angular frequency of the waves, respectively. The concept can be extended to cover higher order waves, as well as random and directional waves, by linear superimposition of components. A sketch of the numerical flume set-up can be shown in Figure 4.
The formulation in Choi and Yoon (2009) did not include the variation of the source terms over the vertical axis as in Wei et al. (1999), since it was found that this did not significantly affect performance. During numerical tests of 2D and 3D regular wave propagation, satisfactory agreement was demonstrated against physical modelling results, analytical solutions and bechmark cases previously simulated by Lin and Liu (1999) model. Choi and Yoon (2009) also observed that the influence of evanescent modes due to the vertical profile at the IWM is limited to a distance of 2-3 water depths from the IWM, however this was observed for intermediate to shallow wave conditions, where the assumption of constant velocity over the depth has some validity, as opposed to deep water waves.
Ha et al. (2013) further improved the approach proposed by Choi and Yoon (2009), by implementing it in the NEWTANK model. They argued that the use of both momentum and mass source in the Navier-Stokes equations improved generation performance. They also note that the method has limitations for generating deep water waves, given that it has been originally developed for Boussinesq equations. The IWM was coupled with passive absorption layers and numerical tests of random and multi-directional waves were performed, showing overall good performance for intermediate and shallow waves, but not for deep water waves. It. could be nevertheless argued that the latter might have been because they used a relatively coarse mesh, as their results demonstrate that the wave height is correctly generated at the IWM location but quickly dissipates during propagation in the NWT.
Liu et al. (2015) implemented the IWM concept of Choi and Yoon (2009) in the framework of an Incompressible Smoothed Particle Dynamics (ISPH) model (particle-based Lan- grangian method). They do not use mass source terms, as these would require adding and removing SPH particles from the domain thus causing difficulties and complications for numerical implementation and stability. They observed that the optimal width of the source region is about 20%-50% of the wavelength and that a buffer zone is required to allow the wave signal to stabilise and reach good agreement with theory. Buffer areas were areas from wavelengths away from the generation zone, for intermediate or shallow water waves. The method is proven not to be very efficient for generating deep water waves, despite the rather refined particle resolution.
2.3.2 Example applications
The IWM is widespread in Boussinesq modelling, as modern Boussinesq models employ variations of the internal generation concept proposed by Wei et al. (1999). In Bouss2D (Nwogu and Demirbilek, 2001) internal wave generation is achieved by adding mass and momentum sources in the form of Dirac equations, which are later discretised over two computational cells. Their scheme is a hybrid of “physical” and “numerical” forcing as the source terms are present in the equations, but the smoothing of the Dirac function is dependent on the grid size rather than other properties (e.g., local wave properties). FUNWAVE (Shi et al., 2016) uses the methodology of Wei et al. (1999) and Mike 21 BW (DHI, 2017) employs a similar scheme.
In CFD models, the IWM has been employed in the COBRAS model, especially during the early days for wave structure interaction applications, such as modelling low-mound or submerged porous structures (Garcia et al., 2004; Lara et al., 2006; Losada et al., 2008), floating breakwaters (Koftis et al., 2006), pressure propagation in soils around structures (Zhang et al., 2018b, 2019) and sediment transport processes (Amoudry et al., 2013), to name a few. Although the COBRAS model may be still used by researchers, it is not as widespread as in the mid-OO's, and it has been only used for 2DV simulations, which are equivalent to numerical wave flume. The NEWTANK (Ha et al., 2013) model is a fully developed 3D model that uses the IWM concept, frequently combining it with Large Eddy Simulation (LES) approach for turbulence. Representative publications include Ha et al. (2014), where a submerged breakwater was studied under tsunami action, and wave interaction with floating structures Zhang et al. (2018a).
2.3.3 Advantages and disadvantages
The IWM concept has been very successful in its use for depth-integrated and in particular Boussinesq models. In these models, the free surface elevation variable is an integral part of the equations and it is, therefore, straightforward to develop source terms associated with a particular forcing of the free surface elevation. This is not the case in two-phase flow CFD models, where the free surface elevation is implicitly calculated by the air/water interface transport equations.
The IWM is demonstrated to have a good overall accuracy, but often requiring additional treatment for steep waves in deep and shallow waters. As the IWM does not force an analytical solution of a wave theory, buffer areas have to be introduced in the domain to allow smooth development of the waves and to dampen evanescent modes. These areas, depending on the approach and the wave regime, could be from 2-3 times the water depth according to (Choi and Yoon, 2009) up to 2-3 wavelengths (Liu et al., 2015) and can be implicitly associated with increased computational cost.
Due to these discrepancies, it could be argued that in VOF based models, the application of the internal wave maker concept is less advantageous, often requiring adaptation of the size and position of the IWM or calibration of the source terms to achieve the target wave height. These adaptation and calibration procedures on a case-by-case basis would require multiple simulations to achieve a target wave height, thus increasing computational cost. In Schmitt et al. (2019), a calibration procedure is designed for a momentum source wave maker, similar but perhaps simpler to the one proposed by Choi and Yoon (2009). Schmitt et al. (2019) argue that their procedure can be performed in 2DV tanks with a relatively low computational cost. However, their method is assessed for regular waves or wave groups, rather than for random wave series that correspond to full storms, for which much longer simulations would be expected. Taking into account the need for buffer areas and the calibration, it could be argued that on some occasions, the IWM method could be as costly as the relaxation zone, which is generally considered an expensive method. It should be nevertheless stated that on some occasions, wave calibration could be desired regardless of the wave generation method, for, e.g., quality assurance purposes or for direct comparison with experimental data. With respect to computational cost for calculations of trigonometric functions (see discussion in the relaxation zone section), these should not be as important because the extent of the IWM remains relatively small compared to the wavelength and the overall size of the flume. Optimisation methods such as the ones discussed in (Dimakopoulos et al., 2016, 2019) could nevertheless benefit the methods and help maintain efficiency in relatively refined meshes.
Regarding absorption efficiency, this will depend on the wave absorption scheme. Researchers have employed both radiation boundary conditions and absorption zones, with a preference to the latter, as it is more aligned with the theoretical background of the IWM (adding source/sink terms to momentum equations). Using an absorption zone in tandem with an IWM could be both efficient and relatively cost-effective, as long as the mesh is coarser in the absorption zone, following recommendations from Dimakopoulos et al. (2016); Saincher and Banerjee (2017).
To summarise, whilst the IWM method can be conveniently implemented through the mass or momentum equations, its overall use is not as straightforward as, e.g., the relaxation zone method or radiation conditions, which require little or no wave calibration. For highly nonlinear and deep water waves in particular, the IWM may require more intensive calibration procedures and treatment (e.g., resizing the source area, or scaling source terms). Absorption efficiency can be in par with the relaxation zone method. In terms of computational cost, although the IWM will increase the size of the computational domain per se, addition of an absoprtion zone at the upstream boundary, and the establishment of buffer areas between the generation zone and the area of interest, may reach up to 2-3 wavelengths.
As a conclusion, it could be stipulated that whilst the IWM method is the norm for Boussinesq models, its use is not as well established in CFD models, whereas the relaxation zone and the radiation boundary conditions are increasingly becoming widespread. Notwithstanding the disadvantages of the method, this could be because the IWG method needs to be complemented either with an absorption technique based on “competing” techniques, such as the radiation condition or the relaxation zone method. In terms of implementation, it is sometimes more convenient and efficient for CFD developers, who often deal with highly complex software codes, to convert these “competing” techniques to accommodate simultaneous wave generation, rather than design a different technique for generation from scratch.