# Intercomparison of Three Open-Source Numerical Flumes for the Surface Dynamics of Steep Focused Wave Groups

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

^{nd}order harmonics and the analytical 2

^{nd}order theory solution is also included in the Appendix A. General conclusions and recommendations are drawn at the last section of the paper.

## 2. Methods and Testing Conditions

#### 2.1. Experimental Conditions

#### 2.2. Phase Decomposition

^{th}order harmonic can be found from the envelope of the linear harmonics by raising the latter to the n

^{th}order [22]. To decompose the recorded wave signal to its harmonics, a number of wave groups with different phase shifts should be obtained, and their appropriate algebraic combination returns the harmonics of the signal. Generally speaking, a larger number of wave groups with different phases shifts guarantee greater accuracy in the extraction of harmonics by reducing the need for frequency filtering to the signal.

^{rd}order and 5

^{th}order harmonics, while the extracted even harmonics contain the 2

^{nd}sum, 4

^{th}and 6

^{th}order harmonics. The linear harmonics are thus perturbed by the 3

^{rd}and the 5

^{th}order harmonics and cannot be readily isolated, unless the spectrum is sufficiently narrowbanded, resulting in no overlap between the linear and 3

^{rd}order harmonics. Under such conditions only, frequency filtering can be employed. However, this is rarely the case for realistic broadbanded spectra or for steep wave groups, whose free-wave spectrum may broaden considerably during focusing. As such, discrepancies are common in the literature, due to the limitation of the two-wave decomposition [22,23,24,25].

^{rd}order harmonics can be separated algebraically without the need for frequency filtering. Filtering is only needed to separate the linear from the 5

^{th}order terms, which is trivial, since these harmonics occupy distinctively different frequency bands. Moreover, the contribution of the 5

^{th}order harmonics is orders of magnitudes smaller than that of the linear and the 3

^{rd}order harmonics. The 2

^{nd}order super-harmonics, aka 2

^{nd}sum, can be readily extracted, while, for the 2

^{nd}order sub-harmonics, aka 2

^{nd}difference, trivial filtering from the 4

^{th}order harmonics is required.

^{nd}difference terms ${f}_{20}$, which are important and they can be separated by frequency filtering from the 4

^{th}order terms (${f}_{44}$) [26]. Here, the cut-off frequency for the 2

^{nd}difference harmonics is approximately taken at 2.5${f}_{p}$.

^{th}order harmonics are readily separated from the 2

^{nd}difference harmonics and the 5

^{th}order harmonics are calculated separately. Nevertheless, for the wave group examined by Hann et al. [30], the performance of the twelve-wave decomposition was similar to the four-wave decomposition regarding the extracted surface elevation. Therefore, the use of the twelve-wave decomposition is not deemed necessary, especially because the simulation of eight extra groups at every iteration step of the focusing methodology requires considerable additional computational resources.

#### 2.3. Correction Methodology

- The target amplitude spectrum is defined and the desired locations for the amplitude and phase corrections, namely AM and PF, respectively, are determined. Moreover, the focal time is selected, usually as half of the repeat time of the periodic signal.
- Wave groups of different phase shifts are generated at the wave paddle. For a four-wave decomposition, four wave groups with phase shifts of 0, $\pi /2$, $\pi $ and $3\pi /2$ are used to generate CF, positive slope, TF and negative slope focused waves at the PF location, respectively. For the first run, the linear dispersion relation can be used to backwards propagate the signal from PF to the wavemaker, as a best guess. An example is given in Figure 2, where the contraction of the wave group towards focusing is also evident.
- The linear harmonics are extracted using a suitable linear combination of the four wave groups measured at PF, according to the four-wave decomposition (see Equation (4)) in the frequency domain after performing a Fast Fourier Transform (FFT) of the measured signals.
- The phases and amplitudes of the wave components of the linear harmonic are corrected using Equations (5).$$\begin{array}{c}\hfill {\alpha}_{in}^{i+1}={\alpha}_{in}^{i}\times {\alpha}_{trg}/{\alpha}_{out}^{i}\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{\varphi}_{in}^{i+1}={\varphi}_{in}^{i}-({\varphi}_{trg}-{\varphi}_{out}^{i})\end{array}$$
- The corrected signal for the wavemaker can then be calculated: (
**a**) the phases of wave components of the corrected linear spectrum are found by propagating backwards the signal from PF to the wavemaker using the linear dispersion relation, while (**b**) the corrected amplitudes of the components are not altered according to linear theory, being the same at AM and the wavemaker. - The process is repeated iteratively from step 2 to 5 until the target values for $\alpha $ and $\varphi $ match the target values within the desired accuracy.

## 3. The Numerical Models

#### 3.1. RANS: OpenFOAM

#### 3.1.1. Description of the Solver

#### 3.1.2. Design of the NWT

#### 3.1.3. Convergence Tests

^{rd}+ higher order harmonics and deeper 2

^{nd}order wave troughs when the focusing methodology is employed. Here, it is shown that the overall increase of the crest amplitude for the coarser resolutions reaches 20%. Coarser resolutions than the R10-C0.2 cause local distortions of the crest, which can be associated with premature breaking. The total number of cells in the computational domain is 2.48 millions for the high resolution NWT that was finally chosen (R2.5-C0.1), resulting to a computational time of approximately 50 h on a 16-core Intel Xeon E5-2650 @ 2.6 GHz.

#### 3.2. NLSWE: SWASH

#### 3.2.1. Description of the Solver

#### 3.2.2. Design of the NWT

#### 3.2.3. Convergence Tests

#### 3.3. PFT: HOS-NWT

#### 3.3.1. Description of the Solver

#### 3.3.2. Design of the NWT

^{th}order scheme for the integration in time, with a tolerance at ${10}^{-4}$ is selected after tests. The input signal for the simulation is given by an amplitude-frequency spectrum, using $icase=3$ of HOS-NWT as the initial point for the present set-up.

#### 3.3.3. Convergence Tests

#### 3.4. Summary of NWTs

## 4. Results and Comparisons

^{th}order.

#### 4.1. Numerical Dispersion

#### 4.2. Evolution of the Linear Harmonics

#### 4.3. Evolution of the 2^{nd} Sum Harmonics

^{nd}order sum harmonic is presented in Figure 9. It can be seen that the numerical models produce practically identical results everywhere in the NWT. Discrepancies exist only near the wavemaker at WG2 due to the spurious free waves created by the linear wave generation shown between −3 s and 1 s. Although, all wavemakers operate linearly, their function is not identical, especially when compared to the physical motion of the experimental wave paddle, which introduces greater spurious waves. From WG3 and downstream the agreement between the models and the experiment is almost excellent, since the spurious 2

^{nd}order free waves have separated from the wave group.

#### 4.4. Evolution of the 2^{nd} Difference Harmonics

^{nd}order difference harmonics refer to the long bound wave components that appear in the form of a set-down of the MWL under unidirectional wave groups. However, in directional seas, the long bound wave can take also the form of a set-up, for example when two wave groups cross at a specific angle, as discussed for the Draupner wave [61]. The results in Figure 10 show that there can be non-negligible discrepancies among the models and the experiments at the reproduction of the 2

^{nd}order difference harmonics.

^{nd}order difference harmonics are practically zero for the experiment, SWASH and HOS-NWT, while OpenFOAM gives a spurious set-up before the main set-down of the wave group, which, precedes the main wave group [5]. Also, the experimental results show an artificial elevation at WG2, which appears to be a spurious local effect. From WG3 and downstream, the agreement between the models and the experiment improves considerably, especially for reproducing the main trough. SWASH has the shallowest trough at almost all locations, but at PF shows the best agreement with the experiment. At the last two WGs, the experimental results show a second trough at later times, which is the reflected long wave. At the examined time window, reflections do not appear in the SWASH and HOS-NWT, because the outlet boundary was placed further downstream and the reflected waves take longer time to return. They neither appear in OpenFOAM, where the wave dissipation of IHFOAM follows the shallow water approximation, and it is thus more effective for long waves than the beach of the physical experiment, which usually performs best for short waves.

^{nd}order wave generation can be effective in at least decreasing the spurious preceding crest of the long bound waves [19]. To further examine the 2

^{nd}order harmonics, the present results are compared with the analytical solution in the Appendix A.

#### 4.5. Evolution of the 3^{rd} Order Harmonics

^{th}order harmonics. The 4

^{th}order harmonics have similar behaviour to the 3

^{rd}order harmonics [5], and here, only the evolution of the latter is presented.

^{rd}order harmonics is shown in Figure 11. It can be seen that close to the wavemaker (Figure 11a,b), where the wave group is dispersed and not very steep, the magnitude of the 3

^{rd}order harmonics is negligible. Moreover, close to the boundary at AM, the agreement among the models and the experiment is not very good due to the spurious high order free waves. However, at downstream locations and towards the focusing of the wave group, the comparison among the models and the experiment improves considerably, with only some minor discrepancies being noticeable at the central crest and adjacent troughs.

#### 4.6. Wave Group Evolution

^{rd}order, here, the evolution of the wave group is examined from the AM to the PF location. This refers to the free surface elevation as measured in the flumes without any processing and it is presented in Figure 12. It can be seen that all the models are in good agreement with the experiment at all locations. The worst agreement is observed at WG2 in Figure 12b, where the spurious free waves have started separating from the main wave group. After the separation of the spurious waves, the comparison between the models is immediately improved. The best agreement among all the models seems to be at the middle of the tank, namely at WG3 and WG4, where influence of the boundary fades (spurious waves) and the steepness is not yet very high.

#### 4.7. Comparison at the Focal Point

^{th}and 5

^{th}order harmonics and quantitative comparison at wave crest.

^{th}order is examined in Figure 14 at the PF location. It is noted that the 4

^{th}and 5

^{th}order harmonics are not directly obtained by the four-wave harmonic decomposition, but the former are included in 2

^{nd}difference harmonics and the latter in linear and 3

^{rd}order harmonics, as shown in Equation (4). The 4

^{th}order harmonics can be trivially separated with frequency filtering from the 2

^{nd}difference harmonics, since they occupy non-overlapping frequency bands. The 5

^{th}order harmonics are taken only from the linear harmonics using a similar frequency filtering at approximately $4{f}_{p}$. For simplicity, the 5

^{th}order harmonics within the 3

^{rd}order harmonics are ignored, because the magnitude of the 5

^{th}order harmonics within the extracted linear harmonic is much greater than that included in the 3

^{rd}harmonics.

^{nd}difference harmonics and the 5

^{th}order harmonics. Discrepancies are observed also at the crest of the linear harmonics, with the experiment having the highest crest elevation. It is very interesting to observe that OpenFOAM appears to overestimate all the nonlinear harmonics, but yet to give the best overall result (see Figure 13). At the same time, SWASH and HOS-NWT seem to give a very good agreement with the experiment for all the nonlinear harmonics.

^{nd}difference harmonics) is presented in Table 5. It can be seen that OpenFOAM has practically an identical crest elevation to the experiment with a difference of only 0.1%, which is less that the accuracy of the experimental WGs ($\pm 1$ mm). The performance of SWASH and HOS-NWT is also impressive, since they achieve a small error of approximately 5% at the crest of nearly breaking wave groups. However, Table 5 also demonstrates that the excellent agreement between OpenFOAM and the experiment is a result of intercancellation of the overestimation of the nonlinear harmonics. In fact, for almost all of the individual harmonics, with the exception of the 3

^{rd}order harmonics, SWASH and HOS-NWT provide the best agreement with the experiment.

## 5. Conclusions

^{th}order are captured accurately. Especially impressive was the performance of the weakly nonlinear solvers, SWASH and HOS-NWT, clearly demonstrating that they can simulate high order nonlinear wave-wave interactions with accuracy. The best comparison with the experimental results was observed for OpenFOAM, achieving practically absolute agreement with the experiment at the crest elevation. However, it is interesting to observe that this excellent agreement stems to an extent from intercancellation of the individual nonlinear harmonics that are all overpredicted by OpenFOAM. In fact, it was shown that SWASH and HOS-NWT may provide a more accurate simulation of the individual harmonics, which, for certain types of studies, e.g., overtopping [64,65], may be important. Thus, the present results show that, for wave propagation, CFD two-phase models is not necessarily the gold standard in numerical modelling. This is a very important aspect to take into account when using CFD models to validate other numerical models. A possible explanation for this observation is that SWASH and HOS-NWT are numerical models specifically developed to simulate waves, while OpenFOAM is a more general modelling CFD platform that was only adapted to simulate waves using appropriate boundary conditions. Also, the complication of solving for two-phase fluid flows in OpenFOAM may be a source of error that should not be factored out. To confirm these findings and be able to identify credibly the sources of the errors, other suitable solvers can be used that employ similar equations and modelling approaches as these of the present work [66,67].

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

AM | Amplitude matching location |

CF | Crest-focused wave group |

CFD | Computational Fluid Dynamics |

FFT | Fast Fourier Transform |

FVM | Finite Volume Method |

HOS | High Order Spectral method |

MWL | Mean water level |

NSE | Navier-Stokes Equations |

NSWE | Nonlinear Shallow Water Equations |

NWT | Numerical Wave Tank |

PF | Phase focal location |

PFT | Potential Flow Theory |

PM | Pierson-Moskowitz Spectrum |

RANS | Reynolds Averaged Navier-Stokes equations |

SWL | Still water level |

TF | Trough-focused wave group |

VoF | Volume of Fluid method |

WG | Wave gauges |

## Appendix A. Comparison with 2^{nd} Order Theory

^{nd}order harmonics are further analysed and compared with the analytical “exact” solution. In particular, based on the analysis in Section 4.4, the 2

^{nd}order difference harmonics exhibited the greatest discrepancies between the numerical models and the experiments, due to spurious effects from the wave generation. The comparison with the analytical solution can indicate which wave generation method and model is more accurate for the 2

^{nd}order harmonics.

^{nd}order solution of an irregular wave signal was given for two waves by Dalzell [72]. Here, the expressions presented refer to an arbitrary number of linear wave components (N), propagating in a single direction on finite water depth. Dalzell’s approach follows the same potential flow theory assumptions as [73], but it is based on symbolic computations. Its application is straightforward and it was used for the 2

^{nd}order wave generation boundary conditions in OpenFOAM [74].

^{nd}order theory, the surface elevation $\eta $ is given from the first harmonic and the matrix of the 2

^{nd}order interactions with each free wave with all the other free waves, including the self-interaction, as shown in Equation (A1). The 2

^{nd}order harmonics are calculated by Equations (A2) and (A3). The respective coefficients are given in Equations (A4) and (A5) for any possible combination of any wave component i with a component j. Since, a unidirectional wave propagation is assumed for the present case, the angle between the components is zero and the $cos({\varphi}_{i}-{\varphi}_{j})=1$.

^{nd}order harmonics analytically using the previous formulas, the linear harmonics should be known. In the present study, this can be extracted accurately at any location in the flume, thanks to the four-wave decomposition. As shown in [5,75], which followed a similar principle to that of [7], for the better estimation of the 2

^{nd}order harmonics, the evolved (locally broadened) linear harmonic should be considered. Using the harmonic from the original spectrum (see Table 2) would result in considerably higher discrepancies in the comparisons since the underlying spectra are different. Here, the comparisons will be performed for the wave group at PF, where the nonlinear harmonics reach their maximum energy content, and the spurious bound waves have separated from the main wave group, as well as any reflections have not returned in the examined time window. For simplicity, and since the extracted linear harmonics are almost identical among the numerical models and the experiment, the extracted harmonic from HOS-NWT is used, which also had the best agreement with the target spectrum at AM, as shown in Figure 6a.

^{nd}order harmonics are compared with the extracted 2

^{nd}order harmonics at PF from the nonlinear models and the experiment in Figure A1. The quantitative comparison of the crests and adjacent troughs of the 2

^{nd}sum harmonics, as well as the trough of the 2

^{nd}difference harmonics, is presented as (%) difference between the corresponding values of the analytical solution and the extracted harmonics ($\frac{\left|\mathrm{measured}-\mathrm{theory}\right|}{\mathrm{theory}}$) in Table A1. It is noted that the adjacent troughs of the 2

^{nd}sum harmonics are not identical, and for the comparisons of Table A1, their mean value is considered.

^{nd}sum harmonics, Figure A1a shows that the extracted harmonics of the experiment and the models have a crest which is similar, but higher than that of the analytical 2

^{nd}order solution. Moreover, the extracted harmonics of the models and the experiment produce deeper lateral troughs compared to analytical solution.

^{nd}difference harmonics, Figure A1b the analytical solution predicts a shallower trough than that of the extracted harmonics. The extracted harmonics that are the closest to the theoretical results are reproduced by the experiment and SWASH. OpenFOAM predicts the deepest trough, followed by HOS-NWT. It can be also observed that, in contrast to the experimental and the numerical harmonics, the analytical solution of the 2

^{nd}order difference harmonics do not include the spurious preceding set-up, which can be spotted between the times −2 s to −1 s in Figure A1b.

**Figure A1.**Comparison of the free surface elevation of the calculated and extracted 2

^{nd}order harmonics at the PF location: (

**a**) 2

^{nd}sum; (

**b**) 2

^{nd}diff.

^{nd}order sum harmonics and the trough of the 2

^{nd}order difference harmonics can be reasonably predicted by 2

^{nd}order theory, when the evolved extracted linear amplitude spectrum is used. It can be seen that OpenFOAM gives the greatest overprediction of the 2

^{nd}order harmonics compared to the analytical solution, while SWASH is the model that gives the best agreement with the analytical 2

^{nd}order solution. The better performance of SWASH and HOS-NWT compared to OpenFOAM can be justified from the fact that the two former models are specifically designed for the propagation of non-breaking waves, as those examined in the present study, while the VoF can introduce complications and numerical artefacts [62].

**Table A1.**Comparison between the extracted 2

^{nd}sum and diff harmonics with 2

^{nd}order theory solution using the evolved extracted linear amplitude spectrum at PF.

Sum Crest | Sum Trough | Diff Trough | |
---|---|---|---|

2^{nd} order theory calculated (mm) | +40.3 | −24.2 | −22.9 |

Experiment | +12.7% | −37.3% | −13.2% |

OpenFOAM | +17.8% | −41.5% | −30.0% |

SWASH | +8.8% | −33.7% | −15.3% |

HOS-NWT | +11.0% | −33.8% | −24.3% |

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**Figure 2.**Timeseries of the normalized free surface elevation of four wave groups of different phases at the wavemaker (

**a**) and at the PF location (

**b**) calculated by linear theory.

**Figure 8.**Comparison of the free surface elevation of the linear harmonics between the numerical models and the experiment at different locations (

**a**–

**f**).

**Figure 9.**Comparison of the free surface elevation of the 2

^{nd}order sum harmonics between the numerical models and the experiment at different locations (

**a**–

**f**).

**Figure 10.**Comparison of the free surface elevation of the 2

^{nd}order difference harmonics between the numerical models and the experiment at different locations (

**a**–

**f**).

**Figure 11.**Comparison of the free surface elevation of the 3

^{rd}order harmonics between the numerical models and the experiment at different locations (

**a**–

**f**).

**Figure 12.**Comparison of the measured surface elevation between the numerical models and the experiment at different locations (

**a**–

**f**).

**Figure 13.**Comparison of the free surface elevation between the numerical models and the experiment at the PF location.

**Figure 14.**Comparison of the free surface elevation of the harmonics between the numerical models and the experiment at the PF location.

**Table 1.**Location of the wave gauges, as distance in m from the wavemaker (AM: amplitude matching; PF: phase focal).

WG1 (AM) | WG2 | WG3 | WG4 | WG5 | WG6 | WG7 (PF) |
---|---|---|---|---|---|---|

1.63 | 5.17 | 9.40 | 11.50 | 13.80 | 13.90 | 14.10 |

Gaussian Spectrum | |
---|---|

Peak frequency (f_{p}) | 0.64 Hz |

Standard Deviation (σ) | 0.13 |

k_{p}d | 1.75 |

Linear crest amplitude A_{Th} (m) | 0.154 |

**Table 3.**Boundary condition for the NWT in IHFOAM [45].

Boundary | ${\mathit{\gamma}}_{\mathit{i}}$ | $\mathit{Pressure}$ | $\mathit{Velocity}$ |
---|---|---|---|

Inlet | IH_Waves_InletAlpha | buoyantPressure | IH_Waves_InletVelocity |

Outlet | zeroGradient | buoyantPressure | IH_3D_2DAbsorbtion_InletVelocity |

Top | inletOutlet | totalPressure | presureInletOutletVelocity |

Bottom | zeroGradient | buoyantPressure | fixedValue |

Lateral walls | empty | empty | empty |

NWT Parameters | OpenFOAM | SWASH | HOS-NWT |
---|---|---|---|

Equations | RANS | NLSWE | PFT |

Mesh | quasi 3D static | 2D moving | 1D spectral |

Wave generation | vel. distribution | vel. distribution | piston |

Wavemaker motion | stationary | stationary | moving |

Wave absorption | active | passive | passive |

Length of NWT | 20 m | 30 m | 50 m |

No. cells/nodes | 2.48 × 10${}^{6}$ | 6 × 10${}^{3}$ | 0.5 × 10${}^{3}$ |

Comp. cost (core hours) | 800 | 0.4 | 0.05 |

**Table 5.**Intercomparison of phase-resolving models at the PF location at the crest and through (2

^{nd}diff). The experiment is used as the benchmark and the differences are expressed as absolute (mm) and percentage (%).

Harmonics | Experiment | OpenFOAM | SWASH | HOS-NWT | |||
---|---|---|---|---|---|---|---|

Total (measured) | 217.5 | 0.2 | 0.1% | −10.5 | −4.8% | −12.5 | −5.7% |

Linear | 158.5 | −3.3 | −2.1% | −4.1 | −2.6% | −5.3 | −3.3% |

2^{nd} sum | 45.4 | 2.1 | 4.5% | −1.6 | −3.4% | −0.7 | −1.5% |

2^{nd} difference | −25.9 | −3.8 | 14.8% | −0.5 | 1.8% | −2.5 | 9.7% |

3^{rd} order | 21.7 | 1.5 | 6.7% | −1.6 | −7.3% | −1.8 | −8.1% |

4^{th} order | 8.8 | 1.8 | 20.2% | 0.2 | 2.1% | 0.1 | 1.6% |

5^{th} order | 5.3 | 1.4 | 26.1% | −0.2 | −3.8% | −0.3 | −4.8% |

Sum of harmonics | 213.9 | −0.4 | 0.2% | −7.7 | −3.6% | −10.3 | −4.8% |

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**MDPI and ACS Style**

Vyzikas, T.; Stagonas, D.; Maisondieu, C.; Greaves, D. Intercomparison of Three Open-Source Numerical Flumes for the Surface Dynamics of Steep Focused Wave Groups. *Fluids* **2021**, *6*, 9.
https://doi.org/10.3390/fluids6010009

**AMA Style**

Vyzikas T, Stagonas D, Maisondieu C, Greaves D. Intercomparison of Three Open-Source Numerical Flumes for the Surface Dynamics of Steep Focused Wave Groups. *Fluids*. 2021; 6(1):9.
https://doi.org/10.3390/fluids6010009

**Chicago/Turabian Style**

Vyzikas, Thomas, Dimitris Stagonas, Christophe Maisondieu, and Deborah Greaves. 2021. "Intercomparison of Three Open-Source Numerical Flumes for the Surface Dynamics of Steep Focused Wave Groups" *Fluids* 6, no. 1: 9.
https://doi.org/10.3390/fluids6010009