# Efficient Nonlinear Hydrodynamic Models for Wave Energy Converter Design—A Scoping Study

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## Abstract

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## 1. Introduction

#### 1.1. R&D Context for WEC Simulation

**A**in Figure 1b). Conversely, the alternative Readiness before Performance approach (depicted by trajectory

**B**in Figure 1b), reduces the probability of market entry and, in wave energy, virtually guarantees bankruptcy [1].

#### 1.2. Requirements of WEC Simulation

- (a)
- Fidelity requirement—be reliable for all possible trial vectors that an optimization algorithm might generate for evaluation,
- (b)
- Flexibility requirement—be general enough to be applicable to a wide variety of candidate WEC concepts, and
- (c)
- Computational requirement—be fast/affordable enough to allow sufficient generations or iterations to be completed in practical time scales on available and affordable hardware.

#### 1.3. Objective of Scoping Study

- At the low-fidelity/high-speed end of the spectrum are methods based on LPF, including frequency-domain methods [6], Cummins equation time-domain methods [7], and extensions of Cummins methods, such as the Nonlinear Froude–Krylov (NLFK) approach (see Section 3.6.2).
- At the high-fidelity/low-speed end of the spectrum are approaches based on RANS CFD codes (see review in [8]). Schmitt et al. [9] present the challenges and advantages for the application of RANS-based methods in the design process of a WEC, concluding the major drawback is the significant computational power required.

#### 1.4. Previous Reviews

#### 1.5. Outline

- Section 3 reviews the CFD theories intermediate to RANS and LPF, which simplify the NSEs yet retain the potential to simulate nonlinear WEC hydrodynamics.
- Section 4 reviews the use of Domain Decomposition, which aims to reduce the required computational overhead of high fidelity CFD codes, by simulating the bulk of the computational domain with a lower fidelity model with fast computational speed, while only simulating the area in the immediate vicinity of the WEC with computationally expensive, higher fidelity models.

## 2. Efficient Nonlinear Hydrodynamic Models for WECs

#### 2.1. Hydrodynamic Nonlinearities—Types

#### 2.1.1. Viscosity

#### 2.1.2. Nonlinear Ocean Waves

#### 2.1.3. Time-Varying Wetted Body Surface

#### 2.2. Hydrodynamic Nonlinearities—Dependences

#### 2.2.1. Device Dependence

#### 2.2.2. Sea State Dependence

#### 2.2.3. Control Dependence

#### 2.2.4. Scale Dependence

#### 2.3. The Navier–Stokes Equations

#### 2.3.1. Solving

- Mesh: A common approach, which is the main focus of the models reviewed in Section 3, discretizes the domain into nodes/cells.
- Particles: Discretizes the domain into lagrangian particles, via methods such as smoothed particle hydrodynamics (SPH) (discussed in Section 4.4).
- Particle distribution field: Discretizes the domain into a field giving a statistical representation of the particle distribution using methods such as Lattice Boltzman (LB) (discussed in Section 4.5).

#### 2.3.2. Methods

#### 2.3.3. A Note on Turbulence

## 3. Simplifying the Navier–Stokes Solutions

#### 3.1. Simplifications

#### 3.1.1. Simplifying the Equations

- (a)
- Incompressibility: Considering the fluid to be incompressible is a very good assumption for water, resulting in very little loss in accuracy. This provides a major simplification, as the density becomes a known constant, allowing the NSE to be solved as a system of four equations with four unknowns, without the need of including the additional energy equation and equation of state for closure (as discussed in Section 2.3.1). However, one case in wave energy where this does not hold is for OWCs when considering the dynamics of the air chamber. To accurately model the entire OWC system, the compressibility of air is important, requiring thermodynamic considerations, as described in the reviews [58,59] which give a good overview of the modeling of OWC systems. Falco and Henriques [60] explicitly review “the spring-like effect” of the air compressibility in OWC chambers and Lopez et al. [61] and Elhanafi et al. [62] show that neglecting air compressibility may lead to significant errors. However, as the scope of this review pertains to the hydrodynamics of WECs, rather than the thermodynamics, all methods herein consider the assumption of incompressibility.
- (b)
- Inviscid: Setting the viscosity to zero eliminates the stress tensor from the momentum equation, (Equation (2)), yielding the Euler equations, which are reviewed in Section 3.2.
- (c)
- Irrotational: An irrotational flow implies that the velocity can be described as the gradient of a potential, which allows the continuity equation for the velocity to be transformed into Laplace’s equation for the potential. This is termed potential flow (PF), which greatly simplifies the solution to the NSE, as the potential is a scalar value that can be obtained from a single equation, compared to solving for the three components of the velocity vector. PF based methods are reviewed in Section 3.3, Section 3.4, Section 3.5 and Section 3.6.

#### 3.1.2. Simplifying the Boundary Conditions and Computational Domain

- Considering the entire 3D computational domain, whose boundaries change position in time in-line with the evolution of the free surface and WEC body surface, fully nonlinear PF (FNPF) is the most accurate method, but also requires the most computation of the PF methods. FNPF is discussed in Section 3.3.
- Assuming shallow-water conditions, the vertical component of the velocity can be neglected, which reduces the computational domain to 2D, and the various methods applying the shallow-water equations are reviewed in Section 3.4.
- Assuming the scattered (diffracted and radiated) waves are small, the free surface boundary conditions can be linearized on the incident wave elevation, using the weak scatterer method detailed in Section 3.5.1.
- Employing a Taylor series expansion of the free surface and body boundary conditions about their mean positions enables the computational domain to remain static and is the basis of weakly nonlinear PF (WNPF) reviewed in Section 3.5.2.
- Considering the free surface boundary at its means position, but applying the WEC body boundary conditions on the instantaneous wetted surface at each time-step, is the partially nonlinear PF presented in Section 3.6.
- At the extreme end of the simplifications to the NSE, is LPF, which linearizes the free surface and WEC body boundary conditions around their mean positions.

#### 3.2. Euler Equations

#### 3.3. Fully Nonlinear Potential Flow

- Projection methods: such as the boundary element method (BEM), virtual source method or spectral method, that involve projecting the problem onto some portion of the fluid boundary and thereby reducing the computational dimension of the problem by one.
- Field solvers: such as the finite difference method (FDM), finite element method (FEM) and harmonic polynomial cell (HPC) method, that discretize the whole computational domain.

#### 3.3.1. Boundary Element Method

#### 3.3.2. Spectral Methods

#### 3.3.3. Virtual Source Method

#### 3.3.4. Finite Element Method

#### 3.3.5. Spectral Element Method

#### 3.3.6. Finite Differences Method

#### 3.3.7. Hybrid Spectral Method

#### 3.3.8. Harmonic Polynomial Cell Method

#### 3.3.9. FNPF-Summary

#### 3.4. Shallow-Water Equations

#### 3.4.1. Boussinesq Equations

#### 3.4.2. Non-Hydrostatic Flow

#### 3.4.3. Congested Shallow Water

#### 3.5. Weakly Nonlinear Models

#### 3.5.1. Weak-Scatterer

#### 3.5.2. Weakly Nonlinear Potential Flow

#### 3.6. Partially Nonlinear

#### 3.6.1. Time-Varying Parameters

#### 3.6.2. Nonlinear Froude–Krylov

#### 3.7. Including Viscous Effects

#### 3.7.1. Viscous Force Term

#### 3.7.2. Potential Flow

#### 3.7.3. Euler Equations

#### 3.7.4. Domain Decomposition

## 4. Domain Decomposition Methods

#### 4.1. Wave Model to Potential Flow

#### 4.2. Wave Model to RANS

#### 4.3. Potential Flow to RANS

#### 4.3.1. SWENSE

#### 4.3.2. Grid2Grid

#### 4.3.3. OceanWave3D

#### 4.3.4. qaleFOAM

#### 4.3.5. PVC3D

#### 4.3.6. Others

#### 4.4. Potential Flow to SPH

#### 4.5. Potential Flow to Lattice Boltzman

#### 4.6. Incompressible to Compressible

#### 4.7. Temporal Domain Decomposition

## 5. Computationally Efficient Modeling Techniques

#### 5.1. Parametric Models Identified From Data

#### 5.1.1. Model Structures

#### 5.1.2. Model Parameterizations

#### 5.1.3. Identification Data

- Scale: NWTs offer the significant advantage of being able to test at full scale. The scaling issue is a major drawback of using PWTs for SID of WEC models, since nonlinear effects may not upscale correctly from PWT to full scale, as discussed in Section 2.2.4 and Section 3.7.1, as well as in Cruz et al. [315]. However, numerically resolving some high fidelity models, such as RANS, at full-scale, can be computationally expensive (see [67]). Therefore, it is vital the SID experiments are designed to provide the maximum information in the minimum time (as discussed in [314]), otherwise NLPF models or domain decomposition techniques may be required for longer duration SID experiments in NWTs at full-scale.
- Reflections: NWTs are superior in eliminating undesired reflections from the tank walls contaminating the SID experiments, with numerical absorption zones able to limit reflections below 1% [316], whereas world-class PWTs can incur reflection coefficients of around 10% [315,317] in the wave propagation direction, and often have no side wall absorption.
- Constraints and restraints: NWTs allow the WEC to be easily constrained to single DoFs, allowing SID of each DoF separately if desired, whereas PWTs require complex mechanical restraints to achieve this task, which introduces friction and alters device dynamics. The same is true for external forces, which can be applied exactly to the WEC in an NWT, but require physical actuators in a PWT which introduce some level of inaccuracy.
- Measurements: NWTs allow non-intrusive measurement of as many variables as desired, with zero measurement noise, without requiring physical measuring devices to be added to the system. Schmitt et al. [9] discuss a significant challenge of PWT experiments is to ensure that the measurement instrumentation is as non-intrusive as possible and does not contaminate any of the results. NWTs also allow easy measurement of some useful variables which are extremely difficult/impossible to measure in a PWT, such as the exact pressure everywhere on the WEC surface, or the fluid velocity and vorticity around the WEC.
- Cost: In the design phase of a WEC development, varying the WEC geometry may be necessary for optimization studies, which can easily be implemented in an NWT through a few lines of code, whereas a physical prototype needs to be manufactured for each geometry tested in a PWT. Schmitt et al. [9] also discuss that a significant investment of resource and money often goes into the design, manufacture, installation, and calibration of specialized pieces of PWT measurement equipment often custom made for a particular WEC design and scale, whereas in an NWT, sensors, actuators, and constraints can be arbitrarily added through a few lines of code. Furthermore, time at a PWT facility can range from hundreds to tens of thousands of euros per day.
- Availability: Testing time in PWT facilities must be organized months in advance and is kept to a tight schedule, whereas with the rise of cloud computing, NWT resources are always available, multiple experiments can be run in parallel and testing time can be increased on the fly.

#### 5.2. Probabilistic Models

#### 5.2.1. Spectral Domain

#### 5.2.2. Polynomial Chaos

#### 5.3. Nonlinear Frequency Domain

## 6. Discussion

#### 6.1. Applications

#### 6.1.1. WEC Design—Productivity, Loading, and Survival Characterizations in Operational Sea States

- Wave excitation forces are overestimated in all but the smallest waves.
- When body rotations are large there is no justifiable method for determining whether to apply the calculated wave forces in a reference frame that moves with the body or a reference frame that is fixed to the body’s mean position. Neither approach is correct and both approaches result in incorrect results.
- There is no automatic way to enforce the Budal limit [347], this leads to simulation results that are invalid due to unrealistically large amplitudes of motion.
- Because the Budal limit is not enforced estimated performance of designs that would violate this limit is not properly penalized so geometry optimizations that converge on these designs are not reliable. This is a particular problem for objective functions that favor smaller devices (e.g., energy yield per surface area or energy yield per cubic displacement). In the worst instances, this issue leads to convergence on physically meaningless solutions such as WEC devices with zero area/volume/stroke.

#### 6.1.2. WEC Design—Loading and Survival Characterizations in Extreme Sea States

- Simulation length—the creation of statistically derived “design waves” to represent entire sea states in limited time lengths has been a subject of many studies in naval architecture, with detailed literature reviews presented in [351,352]. For example, the NewWave approach is a popular method, which results in a focused wave representing the average shape of the largest wave derived from a given wave spectrum. The usage of NewWaves for RANS simulations of WEC survivability is demonstrated in Ransley et al. [353] and is also employed as the input waves for the blind tests in [125,247] which compare the performance of a range modeling approaches.
- Sea state selection—One common method used to estimate extreme conditions employs environmental contours of extreme conditions, as reviewed in Edwards and Coe [354]. However, it is not always the largest wave that causes the largest load. For example, Harnois et al. [355] use field test measurements to investigate the extreme load analysis of WEC mooring systems and observe that that peak mooring loads do not occur for the sea states on the external contour line of the measured sea states, but for the sea states inside the scatter diagram. This agrees with the finding of Yu et al. [356], who discuss that the extreme wave load does not always occur at the largest wave, but instead, it is often a series of specific wave trains that cause extreme loads due to the resulting combination of the instantaneous WEC position and wave elevation. In addition, Coe and Neary [357] discuss that for different WEC components, the largest load may happen at different wave environments. Therefore, it is essential to develop a systematic approach to identify the critical sea states that are likely to cause an extreme wave load. Such a systematic approach necessitates utilizing a combination of tools, as suggested in the review by Coe and Neary [19] and later demonstrated in Van Rij et al. [358].

#### 6.1.3. Wave Farm Design—Productivity Characterization of a Farm Using Already Very Well Characterized WEC Technology

#### 6.2. Device Type

- Wave activated body/OWC/overtopping device
- Fixed/floating
- Surface piercing/submerged
- Point absorber/larger terminator or attenuator
- Near shore/offshore
- Rigid body/flexible membrane
- Smooth outputs/latching control or end-stop collisions

#### 6.3. Higher-Order Waves

#### 6.4. Open-Source Software

#### 6.5. Hardware

#### 6.5.1. GPUs

#### 6.5.2. A Note on Parallel Processing

#### 6.6. Summary of Methods

## 7. Conclusions

- A variety of nonlinear PF methods, with increasing levels of complexity, are available or under development. However, a general means of robustly including viscous effects to the PF-based simulations does not appear present and is likely to be important for a broad range of WEC simulation cases.
- A particularly promising approach, gaining increased attention and popularity, is domain decomposition, where lower-fidelity models are used in the bulk of the simulation and are coupled to more computationally expensive high fidelity models that are localized to the domains of interest. In particular, the nesting of RANS models inside of NLPF models, yields a rigorous method of including the effects of viscosity, without incurring the computational overhead of a full RANS simulation.
- Surrogate modeling appears to have good potential, allowing computationally efficient models to be identified which can encapsulate nonlinear behavior with similar fidelity to data they are trained upon.
- The best choice of simulation tool can depend on the particular application or device type, but for most applications, a combination of tools will be most likely required. Although this could potentially put strain on developers/engineers, due to the requirement of purchasing and learning to use multiple tools, a push towards development and sharing of open-source WEC simulation software can be noted throughout the community, which would reduce the aforementioned strain on developers/engineers.
- Although this review focused on simulation methods more computationally efficient than RANS, it can be stated that RANS modelling is still required as an integral part for many of these more efficient simulation methods. For example, to provide system identification data for viscous effects. RANS and other high fidelity models will still need to play an important role and be used for model identification, inner domains for domain decomposition methods, and for extreme/survival testing.
- Judging computational feasibility should be done in line with respect to modern and future computing hardware architecture, with a paradigm shift in high-performance computing to include heterogeneous CPU-multi GPU architectures. Practical run-times may be achievable depending on the computing hardware at hand, investment to a particular method should ensure its ability to leverage such facilities in the future.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ALE | Arbitrary Lagrangian–Eulerian |

ANN | Artificial Neural-Network |

ARX | AutoRegressive with eXongenous inputs |

BEM | Boundary Element Method |

CFD | Computational Fluid Dynamics |

CPU | Central Processing Unit |

DoF | Degrees of Freedom |

ECN | Ecole Centrale de Nantes |

FD | Frequency Domain |

FDM | Finite Difference Method |

FEM | Finite Element Method |

FK | Froude–Krylov |

FMM | Fast Multipole Method |

FNPF | Fully Nonlinear Potential Flow |

FPSO | Floating Production Storage and Offloading |

FVM | Finite Volume Method |

GMRES | Generalized Minimal Residual |

GPU | Graphical Processor Unit |

HOS | High Order Spectral |

HPA | Heaving Point Absorber |

HPC | Harmonic Polynomial Cell |

KC | Keulegan–Carpenter |

KGP | Kolmogorov–Gabor Polynomial |

LB | Lattice Boltzmann |

LPF | Linear Potential Flow |

MEL | Mixed Eulerian-Lagrangian |

NLFD | Nonlinear Frequency Domain |

NLFK | Nonlinear Froude–Krylov |

NURBS | Non-Uniform Rational B-Splines |

NWT | Numerical Wave Tank |

OSC | Oscillating Surge Converter |

OWC | Oscillating Water Column |

PF | Potential Flow |

PC | Polynomial Chaos |

PTO | Power Take-Off |

PWT | Physical Wave Tank |

QALE | Quasi Arbitrary Lagrangian–Eulerian |

RANS | Reynolds Averaged Navier–Stokes |

SD | Spectral Domain |

SID | System Identification |

SEM | Spectral Element Method |

SPH | Smoothed Particle Hydrodynamics |

SWENSE | Spectral Wave Explicit Navier-Stokes Equations |

TD | Time Domain |

TPL | Technology Performance Level |

TRL | Technology Readiness Level |

WNPF | Weakly Nonlinear Potential Flow |

WSA | Weak-Scatterer Approximation |

WSI | Wave-Structure Interaction |

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**Figure 1.**(

**a**) The Technology Readiness Level (TRL)—Technology Performance Level (TPL) matrix: wave energy converter (WEC) technology value map. (

**b**) The TRL-TPL matrix, highlighting two different development trajectories to commercial viability: Path A represents the Performance before Readiness approach, and Path B is a more expensive trajectory (adapated from Weber et al. [1]).

**Figure 2.**Comparison of the displacement-velocity operational space for a heavy point absorber (HPA) in a JONSWAP sea spectrum, with and without an energy maximizing controller, predicted using a linear Cummins equation model and a Reynolds averaged Navier–Stokes (RANS) simulation, from the case study presented in Davidson et al. [26].

**Figure 3.**The validity limits of different wave theories dependent on the water depth, the wave height and the wave period. The region in which linear theory is valid is shown in white (Adapted from [31]).

**Figure 5.**Schematic comparison of the accuracy versus computational requirements for solving the NSE. The more parameters and the smaller the timestep, thus the smaller the cross-sectional area of the pyramid at a given height, the greater the computational requirements. Here the vertical scale is likely logarithmic, the horizontal scales exponential and the cut-off positions between methods are not exact (Image inspired from [49,50]). The white region, “Efficient nonlinear hydrodynamic models,” represents the focus of the scoping study. (Insert: Example simulation resolution from the different methods, for airflow past a dimpled sphere, adapted from [51]).

**Figure 6.**Hierarchy of simplifying assumptions applied to the NSE, yielding lower accuracy and more computationally efficient equations for hydrodynamic analysis (adapted from [63]).

**Figure 7.**Schematic of the finite volume discretization and the one fluid/two-phase (air and water) modeling for the Euler equations.

**Figure 8.**Schematic of the mixed Eulerian–Lagrangian (MEL) scheme. (

**a**) Equilibrium position with no waves and (

**b**) dynamic motion of the mesh to capture time-varying free surface.

**Figure 9.**Schematic of unified Boussinesq approach in Jiang and Henn [155], showing the outer and inner domains, the single cell in the vertical dimension discretization and the staggered finite difference method (FDM), where velocity and pressure control volumes do not coincide.

**Figure 10.**Schematic of the computational domains for (

**a**) the weak scatter and (

**b**) the FNPF methods.

**Figure 11.**Schematic of domain decomposition using nonlinear potential flow (NLPF) in the far-field and Reynolds averaged Navier–Stokes (RANS) in the near field.

**Figure 12.**Domain decomposition to capture flow separation around a ship in [272].

**Figure 13.**Schematic of the system identification (SID) approach to obtain computationally efficient parametric models. The real system to be modeled generates the input and output data which are utilized by the identification algorithm to estimate the parameters of the model.

**Figure 14.**Schematic of the various inputs and outputs to the models. (

**a**) Input waves only. (

**b**) Input waves and subsystem forces. (

**c,d**) Input waves with modelling of the subsystems.

**Figure 15.**General structure of a spectral domain model (reproduced from [322]).

**Table 1.**Review of publications utilizing the Euler equations for wave energy converter (WEC) simulation.

Ref. | Software | WEC Type | Analysis |
---|---|---|---|

[64] | AMAZON-3D | Manchester Bobber | Extreme wave loadings |

[65] | AMAZON-3D | Manchester Bobber | Slamming |

[66] | AMAZON-3D | Manchester Bobber | Survivability design scenarios |

[67] | OpenFOAM | Wave-Dragon | Over-topping |

[34] | OpenFOAM | HPA | Investigation of parametric resonance |

[47] | OpenFOAM | HPA | Investigation of scale effects |

Ref. | Method | WEC Type | Analysis |
---|---|---|---|

[69] | Cauchy’s Eqn | Salter’s Duck (2D) | Capsizing in extreme waves |

[70] | Cauchy’s Eqn | Submerged wave lens | Focusing of wave energy |

[72] | BEM | OWC (2D) | Geometry analysis |

[73] | BEM | Land-based OWC (2D) | Validation against RANS and experiments |

[74] | HOBEM | Bristol Cylinder (2D) | Proof of modeling concept |

[75] | BEM | Floating OWC (2D) | Using the acceleration potential method |

[76] | BEM | Floating OWC (2D) | Including viscosity to NLPF model |

[77] | BEM | Floating OWC (2D) | Evaluating sharp versus round corner geometry |

[29] | HOBEM | Fixed OWC (2D) | Investigating hydrodynamic performance |

[78] | HOBEM | Submerged cylinder (2D) | Proof of modeling concept |

[79] | BEM | Rolling-cams (2D) | Including viscosity to NLPF model |

[80] | HOBEM | Bristol Cylinder (2D) | Validation against experiments |

[81] | BEM | Fixed OWC | Investigation of stepped-bottom OWC |

[82] | BEM | HPA (Cylinder) | Effect of wave steepness |

[43] | HOBEM | Fixed OWC (2D) | Including viscosity to NLPF model |

[83] | BEM | OSC (2D) | Hydrodynamic performance for large motion |

[84] | HOBEM | OSC | Performance in waves and currents |

[85,86] | BEM | HPA (Cylinder) | Latching control/Wave-body-PTO interaction |

Ref. | Method | WEC Type | Analysis |
---|---|---|---|

[143] | Boussinesq (SEM) | HPA | Comparison against linear and RANS models |

[144,145,146] | Boussinesq (SEM) | HPA (2D) | Verification and validation of model |

[147,148] | Congested shallow water | HPA (2D) | Verification and validation of model |

[149,150] | Non-hydrostatic flow | Submerged PA | Simulate interactions between nonlinear wave field and WEC over coastal scale regions |

[151] | Non-hydrostatic flow | Submerged PA | Parametric instability |

[152] | Non-hydrostatic flow | Submerged PA | Geotechnical stability of the moorings |

Ref. | Method | WEC Type | Analysis |
---|---|---|---|

[163] | Perturbation | OWC | Nonlinear free surface effects |

[164] | 2nd Order PF | Tight moored piston-like WEC | Performance of 2-body system with moonpool |

[165] | WSA | PA with pendulum PTO | Powering small scale data buoys |

[166] | 2nd Order PF | 2 × HPA (Hemisphere) | Array interaction |

[167] | 2nd Order PF | Fixed cylindrical OWC | Effect of weakly nonlinear waves on efficiency |

[168] | WSA | HPA (Sphere) | Comparison against linear and NLFK models |

[169] | 2nd Order PF | 4 × HPA (Cylinder) | Trapped modes in arrays |

[170] | WSA | CETO | Coupling with multi-body mechanical solver |

[171] | WSA | Wavestar | Comparison against linear and NLFK models |

[172] | WSA | CETO and WaveRoller | Comparison against linear models |

[173] | 2nd Order PF | OSC array | Sub-harmonic and higher-order effects |

[174] | 2nd Order PF | Curved gate (2D) | Nonlinear effects of the curved surface |

[175] | 2nd Order PF | Curved OSC array | Nonlinear effects of the curved surface |

[176,177] | 3rd Order PF | Curved gate array | Nonlinear effects of the curved surface |

**Table 5.**Review of publications utilizing partially nonlinear models for WEC simulation (Note: NLFK method not collated because of the large number of publications).

Ref. | WEC Type | Analysis |
---|---|---|

[188] | OSC | Improved model accuracy for large pitch angles |

[189] | OSC | Improved model accuracy for large pitch angles |

[190] | Submerged HPA | Demonstration of method |

[191] | Reconfigurable OSC | Modeling change in hydrodynamics for moving control surfaces |

[35] | HPA | Modeling the occurrence of parametric pitch resonance |

[192] | OSC | Improved model accuracy for large pitch angles |

Ref. | Method | WEC Type | Analysis |
---|---|---|---|

[228,229] | FD to TD PF | Pico OWC | Efficient TD simulations |

[230] | Wave model to PF | Array (HPA) | Wave interactions with WEC array |

[231] | Wave model to RANS | OSC | Modeling real bathymetry |

[232] | PF to I.RANS to C.RANS | OSC | Slamming loads |

[233,234] | Wave model to PF | Array | Multiple WECs in variable bathymetry |

[235,236] | Wave model to PF | Array (OSC) | Wake effects in arrays |

[237,238] | Wave model to PF | Multiple Arrays (HPAs) | Impact of WEC array separation distance |

[239] | Wave model to PF | Arrays (HPAs and OSCs) | Comparison of far-field effects |

[240,241] | Wave model to PF | Array (HPAs) | Validation of model |

[242,243,244] | NLPF to SPH | HPA | Proof of concept |

[245] | NLPF to SPH | OWC | Demonstration of method |

[246] | NLPF to RANS | HPA | Blind comparison test [247] |

**Table 7.**Review of publications utilizing system ID for WEC simulation, detailing the parameterization employed, the data source used, the type of WEC investigated, and the analysis performed.

Ref. | Parameterization | Data | WEC Type | Analysis |
---|---|---|---|---|

[300] [193] | State-space | RANS | HPA | ID of amplitude dependant equivalent linear models |

[301] | State-space | RANS | OWC | Heave motion in OWC chamber |

[204] | Cubic polynomial | PWT | Wavestar | Nonlinear restoring force |

[302] [303] | Hammerstein- Weiner Volterra | Thermo- dynamics | OWC | Air pressure fluctuations inside OWC chamber |

[304] | NARX | RANS | HPA | Nonlinear hydrostatic restoring force |

[305] | ANN | RANS | HPA | Optimization of input/out data for SID |

[306] | Volterra theory | PWT | OWSC | ID of accurate models from PWT data |

[307] | ARX, Hammerstein, KGP | RANS | HPA | Nonlinear excitation force |

[308] | ARX, KGP, ANN | RANS | HPA | SID of nonlinear discrete-time models |

[309] | Grey- and black-box | PWT | HPA | Investigation of SID techniques |

[310] [221] | Pseudo-spectral | RANS | HPA | Online ID of time-varying equivalent linear models for adaptive control |

[311] | Multi regression model | CFD + PWT | OWC | Empirical model design tool |

[312] | ARX, KGP | PWT | Wavestar | Nonlinear model ID from PWT data |

Ref. | Method | WEC Type | Analysis |
---|---|---|---|

[318] | FD | OWC | Optimization of turbine size and rotational speed |

[220] | SD | OSWC | Modeling vortex shedding and large amplitude effects |

[321] | FD | HPA | Geometry, mass distribution, and mooring design optimization |

[323] | SD | HPA (Array) | Modeling arrays and Coulomb friction |

[324] | SD | OWC | Validation of model against tank data |

[325] | SD | OSWC | Accommodate nonlinear WEC dynamics in an optimal controller |

[326] | SD | OSWC | Investigation of controllable surfaces |

[319] | FD | Spar buoy OWC | Statistical values of device dynamics and power output for an array |

[327] | SD | HPA (Sphere) | Modeling nonlinear hydrostatic stiffness |

[328] | SD | Submerged PA | Include nonlinear viscous drag and compare with other methods |

[329] | PC | HPA | Predicting long term extreme loads |

[330] | PC | HPA | Uncertainty quantification |

[331,332] | SD | HPA/OWC | Statistical linearization of nonlinearities |

[320] | FD | Spar buoy OWC | Design optimization |

Ref. | WEC Type | Analysis |
---|---|---|

[340] | OSC + HPA (Sphere) | Demonstration of modeling approach |

[341] | HPA (Sphere) | Model’s ability to handle different nonlinearities |

[343] | OSC | Computationally efficient control |

[346] | OSC + HPA (Sphere) | Effect of spectral shape of wave spectrum on power output estimates |

[342] | ISWEC | Comparison against MDoF time-domain models |

[344] | Ocean Grazer | PTO system performance of a single device |

[345] | Ocean Grazer | PTO system performance for full array |

Application | Time-Span | Spatial Domain | Hydrodynamic Resolution |
---|---|---|---|

Section 6.1.1 WEC operation | Vast—interested in characterizing and optimizing the device performance across all sea states at a given location | Small—local effects only | High—accurate modeling required for assessment of device performance |

Section 6.1.2 WEC survival | Small—interested in the subset of all sea states, and wave series within those sea states, which give rise to maximum loads | Small—local effects only | Very high—must be able to capture the very nonlinear waves and WSI associated with extreme wave conditions |

Section 6.1.3 WEC farms | Medium—time averaged interactions between the wave farm and the environment | Very large—must simulate the entire domain of the wave farm | Medium—interested in the average effects of the interactions between fully designed WECs in arrays/farms |

Software | References | Software | References |
---|---|---|---|

OpenFOAM | [9,26,27,34,35,45,46,47,54,56,57,61,67,125,139,143,146,159,190,193,217,221,232,246,247,258,259,260,261,262,265,266,267,268,269,270,272,273,274,275,277,284,300,301,304,305,307,308,310,314,353,364] | SPHysics | [242,243,244,245] |

Nemoh | [172,179,189,190,192,205,206,216,235,236,237,238,239,240,241,249,250,344,346] | Gmsh | [123,180] |

OceanWave3D | [129,131,132,134,242,243,244,245,250,266,268] | Grid2Grid | [264,265] |

WEC-Sim | [192,195,330,358,359] | ExaFMM | [91] |

SWASH | [149,150,151,152,162] | Palabos | [53] |

HOS Ocean/NWT | [100,103,132,201] | Nektar++ | [120] |

Strengths | Limitations | |
---|---|---|

Euler Equations | • Can simulate wave breaking, green water etc. • Simulates both air and water phases • Can include drag and vortex effects • Implementable in CFD solver frameworks e.g., OpenFOAM | • Computationally expensive • Waves must be solved and propagated to WEC location |

FNPF | • Large amplitude waves and body motions permitted • Currently the focus of much research and development • Flexible/deformable bodies can be easily handled by FEM/SEM • Well suited for use in domain decomposition with RANS | • Computationally feasible with adequate hardware • Waves must be solved and propagated to WEC location • Design of efficient and flexible methods to handle both the nonlinear wave propagation and wave-body interactions is non-trivial, as FNPF NWTs have traditionally been designed to focus on one or the other. |

Shallow water | • Efficiency is gained by eliminating/reducing the vertical dimension and by representing the free-surface as a single-valued function • Valid in the nearshore | • While this method has been shown to work effectively for floating bodies, it is unclear if this could be applicable to submerged bodies • Loses validity in the deep water • Waves must be solved and propagated to WEC location |

Weak scatterer | • Large body motions • Nonlinear steep incident waves • Incident wave does not need to be solved and propagated • Mesh only needs to be refined close to the WEC and large cells can be used elsewhere | • Valid for small perturbed waves • Computational expense in updating the computational domain at each time step, like FNPF |

WNPF | • FD possible • Efficient time-domain simulations since the hydrodynamic coefficient matrices are the same for every time-step, thus only need to be set-up and inverted once. • Incident wave does not need to be solved and propagated | • Although an improvement upon LPF, this method still contains limitations on the wave steepness and amplitude of body motion |

Time-varying parameters | • Computationally efficient • Can precompute the required hydrodynamic coefficients offline beforehand and use look-up tables/interpolation during simulation • Can input wave signal directly, rather than generated at boundary and propagated to the WEC | • Body nonlinear only, with linear assumptions still applied on the input waves • Might only be suitable for NLFK forces, since implementation of time-varying linear models for radiation and diffraction forces is unclear due to memory effects and convolution integrals in time. |

NLFK | • Computationally efficient • Can consider nonlinear/steep waves • Can input wave signal directly, rather than generated at boundary and propagated to the WEC | • Only useful for WEC geometries for which the NLFK forces are significant • Not applicable for fully submerged devices • Linear wave radiation and diffraction forces |

Domain decomposition | • Can provide high fidelity WSI equivalent to RANS and better wave propagation modeling, for less computational effort than RANS • Can be used for extreme and survival conditions • Can be used for large spatial scales considering WEC arrays and wave farms | • Computationally expensive • Two-way coupling methods not yet mature/widely demonstrated |

Parametric models identified from data | • Computationally efficient time-domain simulations with fidelity equivalent to the data used to identify the model • Can input wave signal directly, rather than generated at boundary and propagated to the WEC | • Requires high fidelity training data set spanning the frequency and amplitude ranges • Models may not accurately extrapolate to input conditions outside the range which they were trained upon |

Spectral Domain | • Extremely fast calculation of statistical values, such as mean annual power output etc. | • No temporal responses • Unable to model monochromatic waves • Only valid for Gaussian inputs • The accuracy of the SD model decreases with the deviation from a Gaussian response, therefore unsuited to some nonlinear forces, such as abrupt end-stop collisions, latching control etc. |

Polynomial chaos | • Can test wide ranges of design variables, without simplifying the physics involved. | • Restricted to relatively low number of dimensions in the parameter space (curse of dimensionality) • Smoothness restrictions on the quantity of interest |

NLFD | • Can produce time-domain results an order of magnitude faster than traditional Cummins equation methods • Can input wave signal directly, rather than generated at boundary and propagated to the WEC | • Linear assumption on input waves • WEC outputs must remain smooth, therefore cannot handle abrupt end-stop collisions, latching control etc • Periodic steady-state responses only |

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**MDPI and ACS Style**

Davidson, J.; Costello, R. Efficient Nonlinear Hydrodynamic Models for Wave Energy Converter Design—A Scoping Study. *J. Mar. Sci. Eng.* **2020**, *8*, 35.
https://doi.org/10.3390/jmse8010035

**AMA Style**

Davidson J, Costello R. Efficient Nonlinear Hydrodynamic Models for Wave Energy Converter Design—A Scoping Study. *Journal of Marine Science and Engineering*. 2020; 8(1):35.
https://doi.org/10.3390/jmse8010035

**Chicago/Turabian Style**

Davidson, Josh, and Ronan Costello. 2020. "Efficient Nonlinear Hydrodynamic Models for Wave Energy Converter Design—A Scoping Study" *Journal of Marine Science and Engineering* 8, no. 1: 35.
https://doi.org/10.3390/jmse8010035