# Calibration of CFD Numerical Model for the Analysis of a Combined Caisson

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## Abstract

**:**

## 1. Introduction

^{®}which has a large user base across most engineering and science areas. In particular, to simulate the interaction between wave and the porous media into composite caisson, in this study the toolbox olaflow was used [8]. This toolbox is based on the Volume-average Reynolds-averaged Navier–Stokes (VARANS) equations [9] using the version proposed by del Jesus et al. [10]. In order to take into account the effects due to phenomena that cannot be dealt when the volume-average method is used (i.e., frictional forces, pressure force, and added mass), the calibration of some coefficients is required. Following Darcy [11], Forchheimer [12], and Polubarinova-Koch [13], the closure terms, which represent the drag and the inertia forces due to the porous media, can be defined by the following relationship known as the extended Forchheimer equation [14]:

## 2. Experimental Campaign

## 3. Numerical Model

^{®}was used for the CFD simulation. Such software, developed by OpenCFD Ltd since 2004, has a large user base across most engineering and science areas. In particular, it is widely used to simulate a great number of physical processes in coastal engineering [28,29,30,31]. Generally, Reynolds-Averaged Navier–Stokes (RANS) equations are used which represent the continuum characteristics of the fluid both in space and time.

^{®}, some adjustments of Equation (5) were proposed [14]:

^{®}, the model SST-$k-\omega $ was used in the present study. Such a model has the advantage to combine the $k-\omega $ and $k-\u03f5$ turbulence models. The $k-\omega $ model is used in the boundary layer’s inner region, while the $k-\u03f5$ model is used in the free shear flow. Indeed, this allows a good performance of the numerical model also near the walls, within the boundary layer region [10,32]. In the present study, such a model allows the proper simulation of turbulence effects both during the interaction of the waves and the frontal wall and when the airflow penetrates through the holes of the top cover.

#### 3.1. Computational Domain and Boundary Conditions

^{®}mesh generation tools: blockMesh and snappyHexMesh.

^{®}’s tool blockMesh was generated. Then, using the tool snappyHexMesh, each cell was split according to the refinement level defined by the user. For example, fixing a refinement level of 1, which means the refinement factor is equal to ${2}^{1}$, a cell of the background mesh will be cut in two along in each direction (creating 4 sub-cells for a 2D mesh). Such a refinement level can assume different values within the computational domain.

#### 3.2. Benchmark Test

## 4. Calibration of the Porous Media Parameters

- Configuration 1 (C1): $\alpha $ = 0.0 and $\beta $ = 0.0;
- Configuration 2 (C2): $\alpha $ = 200 and $\beta $ = 1.1;
- Configuration 3 (C3): $\alpha $ = 1000 and $\beta $ = 1.1;
- Configuration 4 (C4): $\alpha $ = 200 and $\beta $ = 2.0;
- Configuration 5 (C5): $\alpha $ = 1000 and $\beta $ = 2.0.

## 5. Conclusions

- the behaviour of the composite caisson can be adequately simulated using VARANS equations;
- the reflection coefficient and the force in the front wall are well reproduced when a value of $\alpha $ = 1000 and $\beta $ = 2.0 are used;
- the force at the rear wall is underestimated when the terms related to drag force are considered (i.e., $\alpha $ > 0 and/or $\beta $ > 0); and
- if the drag forces due to the rubble mound is not considered, a precautionary estimate of the forces at the rear wall is obtained.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Zanuttigh, B.; van der Meer, J.W. Wave reflection from coastal structures in design conditions. Coast. Eng.
**2008**, 55, 771–779. [Google Scholar] [CrossRef] - Altomare, C.; Gironella, X.; Sanchez-Arcilla, A.; Sospedra, J. Wave reflection on dissipative quay walls: An experimental study. In Proceedings of the International Conference of the Application of Physical Modeling to Port and Coastal Protection, Ghent, Belgium, 17–20 September 2012; pp. 779–786. [Google Scholar]
- Faraci, C.; Scandura, P.; Foti, E. Reflection of sea waves by combined caissons. J. Waterw. Port Coast. Ocean Eng.
**2015**, 141, 04014036. [Google Scholar] [CrossRef] - Altomare, C.; Gironella, X. An experimental study on scale effects in wave reflection of low-reflective quay walls with internal rubble mound for regular and random waves. Coast. Eng.
**2014**, 90, 51–63. [Google Scholar] [CrossRef] - Faraci, C.; Cammaroto, B.; Cavallaro, L.; Foti, E. Wave reflection generated by caissons with internal rubble mound of variable slope. In Proceedings of the 33rd International Conference on Coastal Engineering, Santander, Spain, 1–6 July 2012; Volume 16. [Google Scholar]
- Liu, Y.; Faraci, C. Analysis of orthogonal wave reflection by a caisson with open front chamber filled with sloping rubble mound. Coast. Eng.
**2014**, 91, 151–163. [Google Scholar] [CrossRef] - Faraci, C.; Liu, Y. Analysis of wave forces acting on combined caissons with inner slope rubble mound. Coast. Eng.
**2014**, 34. [Google Scholar] [CrossRef][Green Version] - Higuera, P. olaFlow: CFD for Waves [Software]. 2017. Available online: https://doi.org/10.5281/zenodo.1297013 (accessed on 7 October 2021).
- Hsu, T.J.; Sakakiyama, T.; Liu, P.L.F. A numerical model for wave motions and turbulence flows in front of a composite breakwater. Coast. Eng.
**2002**, 46, 25–50. [Google Scholar] [CrossRef] - del Jesus, M.; Lara, J.L.; Losada, I.J. Three-dimensional interaction of waves and porous coastal structures: Part I: Numerical model formulation. Coast. Eng.
**2012**, 64, 57–72. [Google Scholar] [CrossRef] - Darcy, H. Les Fontaines Publiques de la ville de Dijon: Exposition et Application...; Victor Dalmont: Paris, France, 1856. [Google Scholar]
- Forchheimer, P. Wasserbewegung durch boden. Z. Ver. Deutsch. Ing.
**1901**, 45, 1782–1788. [Google Scholar] - Polubarinova-Koch, P.I. Theory of Ground Water Movement; Princeton University Press: Princeton, NJ, USA, 2015. [Google Scholar]
- Higuera, P.; Lara, J.L.; Losada, I.J. Three-dimensional interaction of waves and porous coastal structures using OpenFOAM
^{®}. Part I: Formulation and validation. Coast. Eng.**2014**, 83, 243–258. [Google Scholar] [CrossRef] - Feichtner, A.; Mackay, E.; Tabor, G.; Thies, P.R.; Johanning, L.; Ning, D. Using a porous-media approach for CFD modelling of wave interaction with thin perforated structures. J. Ocean Eng. Mar. Energy
**2021**, 7, 1–23. [Google Scholar] [CrossRef] - Engelund, F. On the Laminar and Turbulent Flows of Ground Water through Homogeneous Sand; Akademiet for de Tekniske Videnskaber: Copenhagen, Denmark, 1953. [Google Scholar]
- Van Gent, M.R.A. Wave interaction with permeable coastal structures. Int. J. Rock Mechan. Min. Sci. Geomechan. Abstracts
**1996**, 6, 277A. [Google Scholar] - Jensen, B.; Jacobsen, N.G.; Christensen, E.D. Investigations on the porous media equations and resistance coefficients for coastal structures. Coast. Eng.
**2014**, 84, 56–72. [Google Scholar] [CrossRef] - Losada, I.J.; Lara, J.L.; del Jesus, M. Modeling the interaction of water waves with porous coastal structures. J. Waterw. Port Coast. Ocean Eng.
**2016**, 142, 03116003. [Google Scholar] [CrossRef] - Van Gent, M. Wave Interaction with Permeable Coastal Structures. Ph.D. Thesis, Delft University of Technology, Delft, The Netherlands, 1995. [Google Scholar]
- Liu, P.L.F.; Lin, P.; Chang, K.A.; Sakakiyama, T. Numerical modeling of wave interaction with porous structures. J. Waterw. Port Coast. Ocean Eng.
**1999**, 125, 322–330. [Google Scholar] [CrossRef] - Wu, Y.T.; Hsiao, S.C. Propagation of solitary waves over a submerged permeable breakwater. Coast. Eng.
**2013**, 81, 1–18. [Google Scholar] [CrossRef] - Lara, J.L.; Losada, I.J.; Maza, M.; Guanche, R. Breaking solitary wave evolution over a porous underwater step. Coast. Eng.
**2011**, 58, 837–850. [Google Scholar] [CrossRef] - Vieira, F.; Taveira-Pinto, F.; Rosa-Santos, P. Novel time-efficient approach to calibrate VARANS-VOF models for simulation of wave interaction with porous structures using Artificial Neural Networks. Ocean Eng.
**2021**, 235, 109375. [Google Scholar] [CrossRef] - Faraci, C. Experimental investigation of the hydro-morphodynamic performances of a geocontainer submerged reef. J. Waterw. Port Coast. Ocean Eng.
**2018**, 144, 04017045. [Google Scholar] [CrossRef] - Faraci, C.; Scandura, P.; Petrotta, C.; Foti, E. Wave-induced oscillatory flow over a sloping rippled bed. Water
**2019**, 11, 1618. [Google Scholar] [CrossRef][Green Version] - Mansard, E.P.; Funke, E. The measurement of incident and reflected spectra using a least squares method. In Proceedings of the 17th International Conference on Coastal Engineering, Sydney, Australia, 23–28 March 1980; pp. 154–172. [Google Scholar]
- Higuera, P.; Lara, J.L.; Losada, I.J. Simulating coastal engineering processes with OpenFOAM
^{®}. Coast. Eng.**2013**, 71, 119–134. [Google Scholar] [CrossRef] - Higuera, P.; Lara, J.L.; Losada, I.J. Three-dimensional interaction of waves and porous coastal structures using OpenFOAM
^{®}. Part II: Application. Coast. Eng.**2014**, 83, 259–270. [Google Scholar] [CrossRef] - Devolder, B.; Rauwoens, P.; Troch, P. Application of a buoyancy-modified k-ω SST turbulence model to simulate wave run-up around a monopile subjected to regular waves using OpenFOAM
^{®}. Coast. Eng.**2017**, 125, 81–94. [Google Scholar] [CrossRef][Green Version] - Di Paolo, B.; Lara, J.L.; Barajas, G.; Losada, Í.J. Waves and structure interaction using multi-domain couplings for Navier-Stokes solvers in OpenFOAM
^{®}. Part II: Validation and application to complex cases. Coast. Eng.**2021**, 164, 103818. [Google Scholar] [CrossRef] - Higuera, P.; Lara, J.L.; Losada, I.J. Realistic wave generation and active wave absorption for Navier–Stokes models: Application to OpenFOAM
^{®}. Coast. Eng.**2013**, 71, 102–118. [Google Scholar] [CrossRef] - ITTC. Practical guidelines for ship CFD applications. In Recommended Procedures and Guidelines; 2011; p. 18. Available online: https://ittc.info/media/1357/75-03-02-03.pdf (accessed on 7 October 2021).
- Vanneste, D.; Troch, P. 2D numerical simulation of large-scale physical model tests of wave interaction with a rubble-mound breakwater. Coast. Eng.
**2015**, 103, 22–41. [Google Scholar] [CrossRef] - Elhanafi, A.; Macfarlane, G.; Fleming, A.; Leong, Z. Experimental and numerical measurements of wave forces on a 3D offshore stationary OWC wave energy converter. Ocean Eng.
**2017**, 144, 98–117. [Google Scholar] [CrossRef] - López, I.; Rosa-Santos, P.; Moreira, C.; Taveira-Pinto, F. RANS-VOF modelling of the hydraulic performance of the LOWREB caisson. Coast. Eng.
**2018**, 140, 161–174. [Google Scholar] [CrossRef] - Cavallaro, L.; Iuppa, C.; Musumeci, R.E.; Scandura, P.; Foti, E. Wave loads on a navigation lock sliding gate: Non-linear effects. Coast. Eng. Proc.
**2020**, 8. [Google Scholar] [CrossRef] - Wang, D.X.; Dong, S.; Sun, J.W. Numerical modeling of the interactions between waves and a Jarlan-type caisson breakwater using OpenFOAM. Ocean Eng.
**2019**, 188, 106230. [Google Scholar] [CrossRef] - Liu, X.; Liu, Y.; Lin, P.; Li, A.j. Numerical simulation of wave overtopping above perforated caisson breakwaters. Coast. Eng.
**2021**, 163, 103795. [Google Scholar] [CrossRef] - Jacobsen, N.G.; van Gent, M.R.; Wolters, G. Numerical analysis of the interaction of irregular waves with two dimensional permeable coastal structures. Coast. Eng.
**2015**, 102, 13–29. [Google Scholar] [CrossRef]

**Figure 2.**Cross section of the adopted wave flume belonging to Hydraulics Laboratory at the University of Messina (dimension in m).

**Figure 4.**Computational domain used for the numerical simulation of the composite caisson: (

**a**) cross section of the entire domain; (

**b**) detail of the grid close to the model; (

**c**) detail of the grid in the transition zone. The symbols indicate $B1$, $B2$, and $B3$ the name of the three blocks; $SB1$ and $SB2$ the name of the two sub-blocks; $hc$ the height of the numerical flume; h the mean water level; and H the height of the incident regular waves.

**Figure 5.**Reflection coefficient: (

**a**) reflection coefficient measured in the experimental campaign and that estimate by the numerical model as a function of the parameter $kd$; (

**b**) comparison between the reflection coefficient measured in the experimental campaign and that estimated by the numerical model. The dashed lines represent the limits of 10% and 20% usually assumed in previous studies for this kind of comparison.

**Figure 6.**Comparison between the pressure measured in the experimental campaign by the three transducers located on the rear wall and that estimated by the numerical model. The data refer to a wave with $kd$ = 0.62.

**Figure 7.**Comparison between the pressure measured in the experimental campaign by the three transducers located on the rear wall and that estimated by the numerical model. The data refer to a wave with $kd$ = 0.86.

**Figure 8.**Comparison between the pressure measured in the experimental campaign by the three transducers located on the rear wall and that estimated by the numerical model. The data refers to a wave with $kd$ = 1.93.

**Figure 9.**Comparison between the wave loading measured in the experimental campaign on the rear wall of the caisson and that estimated by the numerical model. ${F}_{rw}^{-}$ (

**a**) and ${F}_{rw}^{+}$ (

**b**) indicate the minimum and maximum wave loading respectively.

**Figure 10.**Comparison between the reflection coefficient estimated from the experimental data and that estimated from the numerical model: (

**a**) reflection coefficient as a of function $kd$; (

**b**) experimental data vs. numerical data.

**Figure 11.**Comparison between the wave loading at the front wall estimated from the experimental data and that estimated from the numerical model: (

**a**) dimensionless wave loading as a function of $kd$; (

**b**) ${F}_{fw,exp}$ vs. ${F}_{fw,num}$.

**Figure 12.**Comparison between the wave loading at the rear wall estimated from the experimental data and that estimated from the numerical model: (

**a**) ${F}_{rw}$ as a function of $kd$; (

**b**) ${F}_{rw,exp}$ vs. ${F}_{rw,num}$.

**Figure 13.**Performance of the configurations tested expressed through $rmse$ (

**a**), $si$ (

**b**) and $slope$ (

**c**).

**Table 1.**Wave characteristics close to the model (where the symbols indicate: H wave height, f wave frequency, T wave period, L wavelength, $k\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\frac{2\pi}{L}$ wave number, and d water depth). The wavelength was estimated through the dispersion relation.

N. | H [m] | f [Hz] | T [s] | L [m] | $\mathit{kd}$ [−] |
---|---|---|---|---|---|

1 | 0.01 | 0.60 | 1.67 | 2.40 | 0.62 |

2 | 0.03 | 0.80 | 1.25 | 1.71 | 0.86 |

3 | 0.06 | 1.00 | 1.00 | 1.28 | 1.15 |

4 | 0.05 | 1.40 | 0.71 | 0.76 | 1.93 |

5 | 0.02 | 1.80 | 0.56 | 0.48 | 3.08 |

Zone | Factor |
---|---|

${x}_{b2}<x<{L}_{w}$ and $0<z<h-1.5H$ | 1 |

${x}_{b2}<x<{L}_{w}$ and $h+1.5H<z<{h}_{c}$ | 1 |

$0<x<{x}_{b2}$ and $0<z<h-1.5H$ | 2 |

$0<x<{x}_{b2}$ and $h+1.5H<z<{h}_{c}$ | 2 |

$0<x<{L}_{w}$ and $h-1.5H<z<h+1.5H$ | 3 |

**Table 3.**Performances parameters of the numerical model: the root mean square discrepancy between the two sets of data (rmse), the scatter index (si), and the slope.

Variables | rmes [−] | si [−] | slope [−] |
---|---|---|---|

$\frac{{F}_{rw}^{+}}{\left(\right)}$ | 0.06 | 0.15 | 1.02 |

$\frac{|{F}_{rw}^{-}|}{\left(\right)}$ | 0.08 | 0.25 | 1.07 |

${k}_{r}$ | 0.05 | 0.10 | 1.01 |

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**MDPI and ACS Style**

Iuppa, C.; Carlo, L.; Foti, E.; Faraci, C.
Calibration of CFD Numerical Model for the Analysis of a Combined Caisson. *Water* **2021**, *13*, 2862.
https://doi.org/10.3390/w13202862

**AMA Style**

Iuppa C, Carlo L, Foti E, Faraci C.
Calibration of CFD Numerical Model for the Analysis of a Combined Caisson. *Water*. 2021; 13(20):2862.
https://doi.org/10.3390/w13202862

**Chicago/Turabian Style**

Iuppa, Claudio, Lilia Carlo, Enrico Foti, and Carla Faraci.
2021. "Calibration of CFD Numerical Model for the Analysis of a Combined Caisson" *Water* 13, no. 20: 2862.
https://doi.org/10.3390/w13202862