We present a depth-integrated Boussinesq model for the efficient simulation of nonlinear wave–body interaction. The model exploits a ‘unified’ Boussinesq framework, i.e. the fluid under the body is also treated with the depth-integrated approach. The unified Boussinesq approach was initially proposed by Jiang (2001) and recently analyzed by Lannes (2017). The choice of Boussinesq-type equations removes the vertical dimension of the problem, resulting in a wave–body model with adequate precision for weakly nonlinear and dispersive waves expressed in horizontal dimensions only. The framework involves the coupling of two different domains with different flow characteristics. Inside each domain, the continuous spectral/hp element method is used to solve the appropriate flow model since it allows to achieve high-order, possibly exponential, convergence for non-breaking waves. Flux-based conditions for the domain coupling are used, following the recipes provided by the discontinuous Galerkin framework. The main contribution of this work is the inclusion of floating surface-piercing bodies in the conventional depth-integrated Boussinesq framework and the use of a spectral/hp element method for high-order accurate numerical discretization in space. The model is verified using manufactured solutions and validated against published results for wave–body interaction. The model is shown to have excellent accuracy and is relevant for applications of waves interacting with wave energy devices.

The paper discusses the use of CFD simulations to analyse the parametric excitation of moored, full scale wave energy converters in six degrees of freedom. We present results of VOF- RANS and VOF-Euler simulations in Open FOAM® for two body shapes: (i) a truncated cylinder; and (ii) a cylinder with a smooth hemispherical bottom. Flow characteristics show large differences in smoothness of flow between the hull shapes, where the smoother shape results in a larger heave response. However the increased amplitude makes it unstable and parametric pitch excitation occurs with amplitudes up to 30°. The responses in surge, heave and pitch (including the transition to parametric motion) are found to be insensitive to the viscous effects. This is notable as the converters are working in resonance. The effect of viscous damping was visible in the roll motion, where the RANS simulations showed a smaller roll. However, the roll motion was found to be triggered not by wave-body interaction with the incident wave, but by reflections from the side walls. This highlights the importance of controlling the reflections in numerical wave tanks for simulations with WEC motion in six degrees of freedom.

CFD simulations of floating wave energy converters are computationally very heavy. This paper deals with a straightforward attempt to cut down on the computational effort by using adaptive mesh refinement (AMR). We investigate the use of AMR for simulations involving floating bodies inside the open-source finite volume framework OpenFOAM. A simple error indicator based on the pressure jump over cell faces is used to drive the AMR. First the use of the error indicator is illustrated for propagation of a very steep stream function wave. Then the AMR technique is applied to two cases of floating bodies: (i) a floating box and (ii) a bottom reacting point-absorber. As expected the AMR significantly reduce the number of cells in the computational meshes and subsequently lower the  computational effort.

A 3D fully nonlinear potential flow (FNPF) model based on an Eulerian formulation is presented. The model is discretizedusing high-order prismatic – possibly curvilinear – elements using a spectral element method (SEM) that has support foradaptive unstructured meshes. The paper presents details of the FNPF-SEM development, and a model is illustrated toexhibit exponential convergence for steep stream function waves to serve as validation. The model is then applied to the caseof focused waves impacting on a surface-piercing, fixed FPSO-like structure. Good agreement is found between numericaland experimental wave elevations and pressures.

Nonlinear wave-body problems are important in renewable energy, especially in case of wave energy converters operating in the near-shore region. In this paper we simulate nonlinear interaction between waves and truncated bodiesusing an efficient spectral/hp element depth-integrated unified Boussinesq model. The unified Boussinesq model treatsalso the fluid below the body in a depth-integrated approach. We illustrate the versatility of the model by predictingthe reflection and transmission of solitary waves passing truncated bodies. We also use the model to simulate themotion of a latched heaving box. In both cases the unified Boussinesq model show acceptable agreement with CFDresults – if applied within the underlying assumptions of dispersion and nonlinearity – but with a significant reductionin computational effort.

We present recent progress on the development of a newfully nonlinear potential flow (FNPF) model for estimation ofnonlinear wave-body interactions based on a stabilised unstructuredspectral element method (SEM). We introduce new proofof-concepts for forced nonlinear wave-body interaction in twospatial dimensions to establish the methodology in the SEM settingutilising dynamically adapted unstructured meshes. The numericalmethod behind the proposed methodology is describedin some detail and numerical experiments on the forced motionof (i) surface piercing and (ii) submerged bodies are presented.

Breather solutions to the nonlinear Schrödinger equation have been put forward as a possible prototype for rouge waves and have been studied both experimentally and numerically. In the present study, we perform high resolution simulations of the evolution of Peregrine breathers in finite depth using a fully non- linear potential flow spectral element model. The spectral ele- ment model can accurately handle very steep waves as illustrated by modelling solitary waves up to limiting steepness. The an- alytic breather solution is introduced through relaxation zones. The numerical solution obtained by the spectral element model is shown to compare in large to the analytic solution as well as to CFD simulations of a Peregrine breather in finite depth pre- sented in literature. We present simulations of breathers over variable bathymetry and 3D simulations of a breather impinging on a mono-pile.

In this paper we present and discuss the use of CFD for coupled mooring analysis of floating wave energy converters. We use the two-phase Navier-Stokes finite volume solver in OpenFOAM and a high-order finite element model for the cable dynamics. The implementation of the coupling is described in some detail and we show validation of the scheme against laboratory data. A comparison between RANS and Euler simulations isolates effects of viscosity and geometric scale for moored point-absorbers, and parametric pitch excitation is demonstrated. The inclusion of power take-off/control follow the same blueprint as the mooring restraint and we illustrate the use of phase control.

For  the assessment of experimental measurements of focused wave groups impacting a surface-piecing fixed structure, we present a new Fully Nonlinear Potential Flow (FNPF) model for simulation of unsteady water waves. The FNPF model is discretized in three  spatial dimensions (3D) using high-order prismatic - possibly curvilinear - elements using a  spectral  element  method (SEM) that has support for adaptive unstructured meshes. This  SEM-FNPF model is based on an Eulerian formulation and deviates from past works in that a  direct discretization of the Laplace problem is used making it straightforward to handle  accurately floating structural bodies of arbitrary shape. Our objectives are; i) present detail of a new SEM modelling developments and ii) to consider its application to address a wave-body interaction problem for nonlinear design waves and their interaction with a model-scale fixed Floating Production, Storage and Offloading vessel (FPSO).  We first reproduce  experimental measurements for focused design waves that represent a probably extreme  wave event for a sea state represented by a wave spectrum and seek to reproduce these measurements in a numerical wave tank. The validated input signal based on measurements is then generated in a NWT setup that includes the FPSO and differences in the signal caused by nonlinear diffraction is reported.

The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials, modified to accommodate a C0 - continuous expansion. Computationally and theoretically, by increasing the polynomial order p , high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use of the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order´to use the spectral/hp element method in more complex science and engineering applications are discussed