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High-fidelity viscous computational fluid dynamics (CFD) models coupled to dynamic mooring models is becoming an established tool for marine wave-body-mooring (WBM) interaction problems. The CFD and the mooring solvers most often communicate by exchanging positions and mooring forces at the mooring fairleads. Mooring components such as submerged buoys and clump weights are usually not resolved in the CFD model, but are treated as Morison-type bodies. This paper presents two recent developments in high-fidelity WBM modelling: (i) a one-way fluid-mooring coupling that samples the CFD fluid kinematics to approximate drag and inertia forces in the mooring model; and (ii) support for inter-moored multibody simulations that can resolve fluid dynamics on a mooring component level. The developments are made in the high-order discontinuous Galerkin mooring solver MoodyCore, and in the two-phase incompressible Navier–Stokes finite volume solver OpenFOAM. The fluid-mooring coupling is verified with experimental tests of a mooring cable in steady current. It is also used to model the response of the slack-moored DeepCwind FOWT exposed to regular waves. Minor effects of fluid-mooring coupling were noted, as expected since this a mild wave case. The inter-mooring development is demonstrated on a point-absorbing WEC moored with a hybrid mooring system, fully resolved in CFD-MoodyCore. The WEC (including a quasi-linear PTO) and the submerged buoys are resolved in CFD, while the mooring dynamics include inter-mooring effects and the one-way sampling of the flow. The combined wave-body-mooring model is judged to be very complete and to cover most of the relevant effects for marine WBM problems.

This paper presents ISOPE’s 2020 comparative study on the interaction between focused waves and a fixed cylinder. The paper discusses the qualitative and quantitative comparisons between 20 different numerical solvers from various universities across the world for a fixed cylinder. The moving cylinder cases are reported in a companion paper as part B (Agarwal, Saincher, et al., 2021). The numerical solvers presented in this paper are the recent state of the art in the field, mostly developed in-house by various academic institutes. The majority of the participants used hybrid modeling (i.e., a combination of potential flow and Navier–Stokes solvers). The qualitative comparisons based on the wave probe and pressure probe time histories and spectral components between laminar, turbulent, and potential flow solvers are presented in this paper. Furthermore, the quantitative error analyses based on the overall relative error in peak and phase shifts in the wave probe and pressure probe of all the 20 different solvers are reported. The quantitative errors with respect to different spectral component energy levels (i.e., in primary, sub-, and superharmonic regions) capturing capability are reported. Thus, the paper discusses the maximum, minimum, and median relative errors present in recent solvers as regards application to industrial problems rather than attempting to find the best solver. Furthermore, recommendations are drawn based on the analysis.

The design of wave energy converters should rely on numerical models that are able to estimate accurately the dynamics and loads in extreme wave conditions. A high-fidelity CFD model of a 1:30 scale point-absorber is developed and validated on experimental data. This work constitutes beyond the state-of-the-art validation study as the system is subjected to 50-year return period waves. Additionally, a new methodology that addresses the well-known challenge in CFD codes of mesh deformation is successfully applied and validated. The CFD model is evaluated in different conditions: wave-only, free decay, and wave–structure interaction. The results show that the extreme waves and the experimental setup of the wave energy converter are simulated within an accuracy of 2%. The developed high-fidelity model is able to capture the motion of the system and the force in the mooring line under extreme waves with satisfactory accuracy. The deviation between the numerical and corresponding experimental RAOs is lower than 7% for waves with smaller steepness. In higher waves, the deviation increases up to 10% due to the inevitable wave reflections and complex dynamics. The pitch motion presents a larger deviation, however, the pitch is of secondary importance for a point-absorber wave energy converter.

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 ...

We present a spectral/hp element method for a depth-integrated Boussinsq model for the efficient simulation of nonlinear wave-body interaction. The model exploits a `unified' Boussinesq framework, i.e. the flow under the body is also treated with the depth-integrated approach, initially proposed by (Jiang, 2001) and more recently rigorously analysed by (Lannes, 2016). The choice of the Boussinesq equations allows the elimination of the vertical dimension, resulting in a wave-body model with an 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. In this work we employ flux-based conditions for domain coupling, following the recipes provided by the discontinuous Galerkin spectral/hp element framework. Inside each domain, the continuous spectral/hp element method is used to solve the appropriate flow model. The spectral/hp element method allows to achieve high-order, possibly exponential, convergence for non-breaking waves and account for the nonlinear interaction with fixed and floating bodies. Our main contribution is to include 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 validated against published results for wave-body interaction and confirmed to have excellent accuracy. The proposed nonlinear model is demonstrated to be relevant for the simulation of wave energy devices.

We present a Spectral Element Fully Nonlinear Potential Flow (FNPF-SEM) model developed for the simulation of wave-body interactions between nonlinear free surface waves and impermeable structures. The solver is accelerated using an iterative p-multigrid algorithm. Two cases are considered: (i) a surface piercing box forced into vertical motion creating radiated waves and (ii) a rectangular box released above its equilibrium resulting in freely decaying heave motion. The FNPF-SEM model is validated by comparing the computed hydrodynamic forces against those obtained by a Navier-Stokes solver. Although not perfect agreement is observed the results are promising, a significant speedup due to the iterative algorithm is however seen.

We present a Spectral Element Fully Nonlinear Potential Flow (FNPF- SEM) model developed for the simulation of wave-body interactions between nonlinear free surface waves and impermeable structures. The solver is accelerated using an iterative p-multigrid algorithm. Two cases are considered: (i) a surface piercing box forced into vertical motion creating radiated waves and (ii) a rectangular box released above its equilibrium resulting in freely decaying heave motion. The FNPF-SEM model is validated by comparing the computed hydrodynamic forces against those obtained by a Navier-Stokes solver. Although not perfect agreement is observed the results are promising, a significant speedup due to the iterative algorithm is however seen.

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 C 0 - 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.

This paper focuses on modelling snap loads in mooring cables. Snap loads are a known problem for the established oil and gas industry, and they pose a major challenge to robust mooring design for the growing industry of wave energy conversion. We present a discontinuous Galerkin formulation using a local Lax-Friedrich Riemann solver to capture snap loads in mooring cables with high accuracy. An hp−adaptive scheme is used to dynamically change the mesh size h and the polynomial order p, based on the local solution quality. We implement an error indicator and a shock identifier to capture shocks with slope-limited linear elements, while using high-order Legendre polynomials for smooth solution regions. The results show exponential error convergence of order p + 1∕2 for smooth solutions. Efficient and accurate computations of idealised shock waves in both linear and nonlinear materials were achieved using hp−adaptivity. Comparison with experimental data gives excellent results, including snap load propagation in a mooring chain. Application on a wave energy device using coupled simulations highlights the importance of the touch-down region in catenary moorings. We conclude that the formulation is able to handle snap loads with good accuracy, with implications for both maximum peak load and fatigue load estimates of mooring cables.

Mooring systems often make use of submerged buoys (SBs) in order to make the moorings compliant. In this paper we present the dynamic effects of changing the buoy geometry or the buoy model fidelity on the mooring system response. Three cylindrical SBs with increasing slenderness (height/diameter) are studied for a mooring leg with two polyester ropes and a SB. The results show a large impact of SB geometry on the mooring dynamics. A larger height/diameter ratio (with preserved mass and buoyancy) is shown to be beneficial as it gives both smaller tension force magnitudes and, more importantly, avoids slack-snap occurrence in the upper cable. We further present a comparison between four numerical methods for SB dynamics: (i) a high-fidelity model using computational fluid dynamics (CFD); (ii) the Morison equation with slender body drag force approximation using numerical quadrature; (iii) the Morison equation with an independent evaluation of the fluid drag due to translation and rotation; and (iv) a translating Morison model which simulates a vertical cylinder in three degrees of freedom with no rotation. All methods are used together with a high-order finite element mooring dynamics solver. The results show that the translating method is inadequate to model this mooring configuration. The remaining three methods agree moderately well, but the Morison formulations give larger motions and higher tensions compared to the CFD results. We show that the quadrature drag model is better suited to model the drag moment on SBs than the independent model, and that the improvement increases with increasing slenderness of the buoy. The uncertainty, sensitivity and importance of the hydrodynamic coefficients of the buoy are discussed and examined by a regression analysis from the CFD data.

We present a high-order nodal spectral element method for the two-dimensional simulation of nonlinear water waves. The model is based on the mixed Eulerian–Lagrangian (MEL) method. Wave interaction with fixed truncated structures is handled using unstructured meshes consisting of high-order iso-parametric quadrilateral/triangular elements to represent the body surfaces as well as the free surface elevation. A numerical eigenvalue analysis highlights that using a thin top layer of quadrilateral elements circumvents the general instability problem associated with the use of asymmetric mesh topology. We demonstrate how to obtain a robust MEL scheme for highly nonlinear waves using an efficient combination of (i) global $$L^2$$ projection without quadrature errors, (ii) mild modal filtering and (iii) a combination of local and global re-meshing techniques. Numerical experiments for strongly nonlinear waves are presented. The experiments demonstrate that the spectral element model provides excellent accuracy in prediction of nonlinear and dispersive wave propagation. The model is also shown to accurately capture the interaction between solitary waves and fixed submerged and surface-piercing bodies. The wave motion and the wave-induced loads compare well to experimental and computational results from the literature.

Results from Blind Test Series 1, part of the Collaborative Computational Project in Wave Structure Interaction (CCP-WSI), are presented. Participants, with a range of numerical methods, simulate blindly the interaction between a fixed structure and focused waves ranging in steepness and direction. Numerical results are compared against corresponding physical data. The predictive capability of each method is assessed based on pressure and run-up measurements. In general, all methods perform well in the cases considered, however, there is notable variation in the results (even between similar methods). Recommendations are made for appropriate considerations and analysis in future comparative studies.

Floating wave energy converters (WECs) operating in the resonance region are strongly affected by non-linearities arising from the interaction between the waves, the WEC motion and the mooring restraints. To compute the restrained WEC motion thus requires a method which readily accounts for these effects. This paper presents a method for coupled mooring analysis using a two-phase Navier–Stokes (VOF–RANS) model and a high-order finite element model of mooring cables. The method is validated against experimental measurements of a cylindrical buoy in regular waves, slack-moored with three catenary mooring cables. There is overall a good agreement between experimental and computational results with respect to buoy motions and mooring forces. Most importantly, the coupled numerical model accurately recreates the strong wave height dependence of the response amplitude operators seen in the experiments.

Mooring systems are required to keep floating wave energy converters (WECs) on station. The mooring concept chosen for this has an impact on the performance of the WEC, on its cost and on its integrity. To clarify the pros and cons of different mooring designs, we present the results of experiments on physical models of three different mooring concepts in regular and irregular waves, including operational and survival conditions. The parameters evaluated are the tension in the cables, the motions of the device in the different degrees of freedom and the seabed footprint in each case. We can see that the mooring system affects the wave energy converters, but the magnitude of the impact depends on the parameter analysed, on the mode of motion studied and on the conditions of the sea. Moreover, different configurations have similar performances in some situations and the choice of one over another might come down to factors such as the type of soil of the seabed, the spacing desired between devices, or environmental impacts. The results of our experiments provide information for a better selection of the mooring system for a wave energy converter when several constraints are taken into account (power production, maximum displacements, extreme tensions, etc).

A coupling between a dynamic mooring solver based on high-order finite element techniques (MooDy) and a radiation-diffraction based hydrodynamic solver (WEC-Sim) is presented. The high-order scheme gives fast convergence resulting in high-resolution simulations at a lower computational cost. The model is compared against a lumped mass mooring code (MoorDyn) that has an existing coupling to WEC-Sim. The two models are compared for a standard test case and the results are similar, giving confidence in the new WEC-Sim-MooDy coupling. Finally, the coupled model is validated using experimental data of a spread moored cylinder with good agreement.

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 nonlinear potential flow spectral element model. The spectral element model can accurately handle very steep waves as illustrated by modelling solitary waves up to limiting steepness. The analytic 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 presented in literature. We present simulations of breathers over variable bathymetry and 3D simulations of a breather impinging on a mono-pile.

We present recent progress on the development of a new fully nonlinear potential flow (FNPF) model for estimation of nonlinear wave-body interactions based on a stabilised unstructured spectral element method (SEM). We introduce new proof-of-concepts for forced nonlinear wave-body interaction in two spatial dimensions to establish the methodology in the SEM setting utilising dynamically adapted unstructured meshes. The numerical method behind the proposed methodology is described in some detail and numerical experiments on the forced motion of (i) surface piercing and (ii) submerged bodies are presented.

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 bodies using an efficient spectral/hp element depth-integrated unified Boussinesq model. The unified Boussinesq model treats also the fluid below the body in a depth-integrated approach. We illustrate the versatility of the model by predicting the reflection and transmission of solitary waves passing truncated bodies. We also use the model to simulate the motion of a latched heaving box. In both cases the unified Boussinesq model show acceptable agreement with CFD results – if applied within the underlying assumptions of dispersion and nonlinearity – but with a significant reduction in computational effort.

The wave loads and the resulting motions of floating wave energy converters are traditionally computed using linear radiation–diffraction methods. Yet for certain cases such as survival conditions, phase control and wave energy conver- ters operating in the resonance region, more complete mathematical models such as computational fluid dynamics are preferred and over the last 5 years, computational fluid dynamics has become more frequently used in the wave energy field. However, rigorous estimation of numerical errors, convergence rates and uncertainties associated with computa- tional fluid dynamics simulations have largely been overlooked in the wave energy sector. In this article, we apply formal verification and validation techniques to computational fluid dynamics simulations of a passively controlled point absor- ber. The phase control causes the motion response to be highly nonlinear even for almost linear incident waves. First, we show that the computational fluid dynamics simulations have acceptable agreement to experimental data. We then present a verification and validation study focusing on the solution verification covering spatial and temporal discretiza- tion, iterative and domain modelling errors. It is shown that the dominating source of errors is, as expected, the spatial discretization, but temporal and iterative errors cannot be neglected. Using hexahedral cells with low aspect ratio and 30 cells per wave height, we obtain results with less than 5% uncertainty in motion response (except for surge) and restraining forces for the buoy without phase control. The amplified nonlinear response due to phase control caused a large increase in numerical uncertainty, illustrating the difficulty to obtain reliable solutions for highly nonlinear responses, and that much denser meshes are required for such cases.

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 bodies using an efficient spectral/hp element depth-integrated unified Boussinesq model. The unified Boussinesq model treats also the fluid below the body in a depth-integrated approach. We illustrate the versatility of the model by predicting the reflection and transmission of solitary waves passing truncated bodies. We also use the model to simulate the motion of a latched heaving box. In both cases the unified Boussinesq model show acceptable agreement with CFD results – if applied within the underlying assumptions of dispersion and nonlinearity – but with a significant reduction in computational effort.

Time-stepping algorithms and their implementations are a critical component within the solution of time-dependent partial differential equations (PDEs). In this article, we present a generic framework-both in terms of algorithms and implementations-that allows an almost seamless switch between various explicit, implicit and implicit-explicit (IMEX) time-stepping methods. We put particular emphasis on how to incorporate time-dependent boundary conditions, an issue that goes beyond classical ODE theory but which plays an important role in the time-stepping of the PDEs arising in computational fluid dynamics. Our algorithm is based upon J.C. Butcher's unifying concept of general linear methods that we have extended to accommodate the family of IMEX schemes that are often used in engineering practice. In the article, we discuss design considerations and present an object-oriented implementation. Finally, we illustrate the use of the framework by applications to a model problem as well as to more complex fluid problems.

An adaptive spectral/hp discontinuous Galerkin method for the two-dimensional shallow water equations is presented. The model uses an orthogonal modal basis of arbitrary polynomial order p defined on unstructured, possibly non-conforming, triangular elements for the spatial discretization. Based on a simple error indicator constructed by the solutions of approximation order p and p−1, we allow both for the mesh size, h, and polynomial approximation order to dynamically change during the simulation. For the h-type refinement, the parent element is subdivided into four similar sibling elements. The time-stepping is performed using a third-order Runge–Kutta scheme. The performance of the hp-adaptivity is illustrated for several test cases. It is found that for the case of smooth flows, p-adaptivity is more efficient than h-adaptivity with respect to degrees of freedom and computational time.

Boussinesq-type equations are used to describe the propagation and transformation of free-surface waves in the nearshore region. The nonlinear and dispersive performance of the equations are determined by tunable parameters. Recently the authors presented conditions on the free parameters under which a Nwogu-type equations would yield bounded eigenspectra (Eskilsson and Engsig-Karup, J Comput Phys 271:261–280, 2014). This leads to a global conditional CFL time-step restriction which is shown to not be affected by the discretisation method and in this sense the CFL condition is tamed to impose a minimal constraint. In this paper we extend the previous study and provide numerical experiments which confirms the theoretical results also is valid in two horizontal dimensions.

The Wave Dragon is a floating Wave Energy Converter (WEC) working by the overtopping principle. The overtopping discharge has been determined by model scale experiments in wave basins. In the present study we numerically simulate the overtopping behavior of the Wave Dragon device using a VOF based incompressible Euler/Navier-Stokes solver in the OpenFOAM® framework. We present simulations of: (i) a complete sea state for different crest heights, and (ii) regular waves for different wave conditions and crest heights. The simulations compare reasonably well with the experimental data, albeit the irregular wave simulations predict a larger overtopping discharge than observed in the experiments.

A generic point-absorbing wave energy converter is modeled in CFD as a vertical cylinder, moored with a single catenary chain that is fully coupled through a dynamic mooring code. The method of choice is very complete and takes much of the non-linearities in the highly coupled system of the moored body into account. The paper presents numerical results compared with experimental data for surge, heave and pitch motion in both decay tests and regular waves. Further, the wave motion response of the cylinder is computed using both a viscous and a non-viscous formulation as a first attempt to quantify viscous effects. Results show a good match between numerical and experimental results in heave, while the surge and pitch motion are more difficult to reproduce. The mooring load cycle appearance compares well with the experiments in shape but gives higher peak values. Although made at low Keulegan-Carpenter numbers, the simulations show vortical structures due to the heave motion, and the resulting motions are clearly affected by the inclusion or exclusion of viscosity. More test-cases and detailed experimental results are needed for further quantification of the viscous impact on floating point absorbers.

The wave energy sector relies heavily on the use of linear hydrodynamic models for the assessment of motions, loads and power production. The linear codes are computationally efficient and produce good results if applied within their application window. However, recent studies using two-phase VOF-RANS simulations of point-absorbers close to resonance have indicated that there might be significant differences between the power production using linear hydrodynamics and VOF-RANS. At present VOF-RANS simulations are too computational expensive to be used in the design cycle. In shallow and intermediate waters a possible middle way between the highly simplified and fast linear hydrodynamics and the very complete but slow VOF-RANS simulations is to use nonlinear, dispersive wave equations of Boussinesq-type. Jiang (2001) presented a unified approach for including bodies into the Boussinesq framework and solved the system with finite differences. In the unified approach the pressure working on the body are solved for using the instantaneous draft. In this study we will outline how to implement the approach of Jiang in a spectral/hp element setting, and simulate the heave motion of a body using different asymptotic wave equations. We will especially focus on the stabilization of the coupled system.

A spectral/hp element method for solving enhanced Boussinesq-type equations in two horizontal dimensions is introduced. The numerical model is based on the discontinuous Galerkin method on unstructured meshes with expansions of arbitrary order. Numerical computations are used to illustrate that the computational efficiency of the model increases with increasing (i) expansion polynomial order, (ii) integration time and (iii) relative depth. Thus, the spectral/hp element technique appears to offers potentially significant savings in computational time for a fixed numerical error, compared to low-order numerical methods, for large-scale and long-time simulations of dispersive wave propagation. The practical applicability of the model is illustrated by several test cases.

We present a spectral/hp element discontinuous Galerkin model for simulating shallow water flows on unstructured triangular meshes. The model uses an orthogonal modal expansion basis of arbitrary order for the spatial discretization and a third-order Runge–Kutta scheme to advance in time. The local elements are coupled together by numerical fluxes, evaluated using the HLLC Riemann solver. We apply the model to test cases involving smooth flows and demonstrate the exponentially fast convergence with regard to polynomial order. We also illustrate that even for results of ‘engineering accuracy’ the computational efficiency increases with increasing order of the model and time of integration. The model is found to be robust in the presence of shocks where Gibbs oscillations can be suppressed by slope limiting. Copyright 2004 John Wiley & Sons, Ltd.

A major challenge in next-generation industrial applications is to improve numerical analysis by quantifying uncertainties in predictions. In this work we present a formulation of a fully nonlinear and dispersive potential flow water wave model with random inputs for the probabilistic description of the evolution of waves. The model is analyzed using random sampling techniques and non-intrusive methods based on generalized Polynomial Chaos (PC). These methods allow to accurately and efficiently estimate the probability distribution of the solution and require only the computation of the solution in different points in the parameter space, allowing for the reuse of existing simulation software. The choice of the applied methods is driven by the number of uncertain input parameters and by the fact that finding the solution of the considered model is computationally intensive. We revisit experimental benchmarks often used for validation of deterministic water wave models. Based on numerical experiments and assumed uncertainties in boundary data, our analysis reveals that some of the known discrepancies from deterministic simulation in comparison with experimental measurements could be partially explained by the variability in the model input. We finally present a synthetic experiment studying the variance based sensitivity of the wave load on an off-shore structure to a number of input uncertainties. In the numerical examples presented the PC methods have exhibited fast convergence, suggesting that the problem is amenable to being analyzed with such methods.

A numerical model (MOODY) for the study of the dynamics of cables is presented in Palm et al. (2013), which was developed for the design of mooring systems for floating wave energy converters. But how does it behave when it is employed together with the tools used to model floating bodies? To answer this question, MOODY was coupled to a linear potential theory code and to a computational fluid dynamics code (OpenFOAM), to model small scale experiments with a moored buoy in linear waves. The experiments are well reproduced in the simulations, with the exception of second order effects when linear potential theory is used and of the small overestimation of the surge drift when computational fluid dynamics is used. The results suggest that MOODY can be used to successfully model moored floating wave energy converters.