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Wave Propagation Models for Numerical Wave Tanks

Eugeny Buldakov


Most wave-structure interaction models are based on a numerical solution of boundary value problems for partial differential equations. For such models the solution accuracy depends not only on the quality of the numerical approximation of the equations, but also on the accuracy of the boundary conditions. Apart from defining the physical boundaries of a fluid domain, they are used to specify the incoming wave conditions. A common numerical tool for modelling wave-structure interaction is a numerical wave tank (NWT), where the fluid domain is bounded and waves are generated by a numerical wavemaker, e.g., by specifying wave kinematics and elevation at the incoming boundary. Difficulties experienced by NWT users are very similar to difficulties of wave generation in experimental wave facilities. These include the accuracy of incoming wave generation and reflections from the boundaries of the fluid domain.

Effective absorption of reflected waves normally requires large absorbing zones and thus larger computational domains. At the same time, a sufficient distance from the wavemaker to the test section is recommended to allow the natural development of the waves. Depending on the absorption method and type of a wavemaker, this can lead to a considerable increase in the size of the NWT. On the other hand, to achieve higher computational efficiency, it is necessary to minimise the size of the computational domain around a structure. This is particularly important because of the high demands of modern computational fluid dynamics (CFD) models for computing resources. Furthermore, for the direct comparison between experiments and calculations and for the execution of computer-assisted experiments, it is useful to model an entire experimental wave tank with the exact replication of the wavemaker shape and the position of the model. This again leads to a much larger domain size than the region of interest around the structure. Often, the optimal NWT size for accurate wave input is impractical for CFD models in terms of computational efficiency. As can be seen, a numerical wave-structure interaction model should satisfy conflicting requirements and it would be natural to apply different models in different regions of a computational domain or to simulate different aspects of the process. For example, a simpler and faster model can be used to simulate wave evolution in the far field, and the region close to the structure can be modelled by a more sophisticated, but slower, CFD model.

UCL, Department of Civil Engineering, Gower Street, LONDON, WC1E 6BT. UK.

The idea of hybrid models has received considerable attention recently and numerous hybrid models have been developed. The most popular couple for creating a hybrid model are a boundary element model (BEM) as a computationally efficient component and volume of fluid (VoF) as an advanced component (e.g. Lachaume et ah, 2003; Kim et ah, 2010; Guo et ah, 2012). However, coupling of other models has also been attempted, for example BEM with SPH (Landrini et ah, 2012) and a finite element method (FEM) with a meshless Navier- Stokes solver (Sriram et ah, 2014). More examples can be found in the introduction to Sriram et ah (2014). The hybrid methods revive simple but computationally efficient wave propagation models as important elements of wave-structure interaction modelling tools. Over the years, many numerical models of nonlinear water waves have been developed. Descriptions of numerical methods for water wave modelling and reviews of numerical simulation of water waves can be found in Tsai and Yue (1996); Fenton (1999); Kim et ah (1999); Dias and Bridges (2006); Lin (2008); Ma (2010).

In this chapter we introduce inviscid models, starting with a brief historical review of models based on the fully nonlinear potential flow theory (FNPT) given in Section 2. In comparison with CFD models, based on solving Reynolds Averaged Navier-Stokes equations, these methods use much simpler governing equations with smaller number of variables. As a result, FNPT models are more efficient computationally. At the same time, they are able to reproduce the principal physical phenomena important for wave propagation, namely nonlinearity and dispersion. As a result, such models describe propagation of highly nonlinear waves up to breaking with good accuracy, as demonstrated by multiple comparisons with experiments (e.g. Dommermuth et ah, 1988; Skyner, 1996; Seiffert et ah, 2017). We will not consider models based on further simplifications, such as depth-averaged models (shallow water and Boussinesq equations) or models of spectral evolution (nonlinear Schrodinger equation, Zakharov equation). Since we consider the wave propagation problem, we will not review literature related to application of FNPT models to interaction with structures, floating bodies, etc.

FNPT models differ by particular methods of solving Laplace equation for velocity potential in the fluid domain and by methods of specifying a fully nonlinear boundary condition on a moving free surface. There are three main classes of numerical methods used to solve wave problems in potential formulation: boundary element methods (BEM), finite element methods (FEM) and high-order spectral methods (HOS). They are discussed in separate parts of Section 2. We do not consider various finite difference methods, which use a wide range of approaches to discretise a moving domain, including boundary-fitted coordinates and er-transform. They can not be considered the mainstream for inviscid models and an interested reader is referred to the reviews of Tsai and Yue (1996); Fenton (1999) and Kim et al. (1999). Section 2 concludes with a brief review of Lagrangian wave models, which offer certain advantages and can be considered as an alternative to conventional FNPT models.

The rest of the chapter discusses a wave propagation model based on Lagrangian description of fluid motion and is organised as follows. Section 3 gives a detailed description of the Lagrangian wave model, presenting the mathematical and numerical formulation of the model. Then, it introduces a method of numerical treatment for breaking waves, discusses the computational efficiency of the model and validates the model with experimental results. In Section 4 the model is applied to simulate the evolution of steep breaking wave groups in a wave flume. In Section 5 the model formulation for waves over sheared currents is introduced and a numerical wave-current flume is constructed and applied to simulate wave groups over following and opposing currents. Both Section 4 and 5 include comparisons between experimental and numerical results. Finally, brief concluding remarks are given in Section 6, where model coupling for wave-structure interaction problems is discussed.

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