# Comparison of Macro-Scale Porosity Implementations for CFD Modelling of Wave Interaction with Thin Porous Structures

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{®}), this work investigates porosity representation as a porous surface with a pressure-jump condition and as volumetric isotropic and anisotropic porous media. Potential differences between these three types of macro-scale porosity implementations are assessed in terms of qualitative flow visualizations, velocity profiles along the water column, the wave elevation near the structures and the horizontal force on the structures. The comparison shows that all three types of implementation are capable of reproducing large-scale effects of the wave-structure interaction and that the differences between all obtained results are relatively small. It was found that the isotropic porous media implementation is numerically the most stable and requires the shortest computation times. The pressure-jump implementation requires the smallest time steps for stability and thus the longest computation times. This is likely due to the spurious local velocities at the air-water interface as a result of the volume-of-fluid interface capturing method combined with interFoam’s segregated pressure-velocity coupling algorithm. This paper provides useful insights and recommendations for effective macro-scale modelling of thin porous structures.

## 1. Introduction

## 2. Wave Flume Experiments

## 3. Numerical Method

#### 3.1. Theoretical Pressure-Drop Formulation

#### 3.2. Governing Equations

^{®}(The OpenFOAM Foundation v5) [44] is used with its implementation of the incompressible, immiscible, two-phase Navier–Stokes equations and the volume-of-fluid (VOF) interface-capturing method with the multidimensional universal limiter for explicit solution (MULES) for boundedness preservation [45].

^{®}uses a cell-centred finite-volume approach). For the models with porous media, OpenFOAM

^{®}is used with modifications used in both the OlaFlow/IHFoam [48] and waves2Foam [39,49] libraries. These toolboxes provide solvers based on OpenFOAM

^{®}’s standard solver for incompressible, immiscible two-phase flow with the VOF interface-capturing method

`interFoam`and are tuned for free-surface waves and incorporate the use of porous-media modelling. Both use the volume-averaged Reynolds-averaged Navier–Stokes (VARANS) equations based on derivations following [50,51,52]. Their comprehensive derivation can be found in [27,39]. The resulting VARANS equations look the same as the mass conservation Equation (4), and the momentum Equation (5), but are formulated for the intrinsic velocity, $\mathit{U}/n$ (the mean velocity of the fluid inside the perforations), instead of $\mathit{U}$. This takes account of the geometric blockage effect mentioned above. Correspondingly, the pressure-drop formulation (1), and the VOF $\alpha $-Equation (6), are formulated with $\mathit{U}/n$. Inside the porous zone, the macro-scale effects of the porous barrier are applied by means of a momentum source term and with a reduced fluid amount. Outside the porous zone, the VARANS equations are equal to the standard RANS equations (i.e., n = 1).

`interFoam`and the derived solvers of OlaFlow/IHFoam and waves2Foam use the transient PIMPLE algorithm (a combination of the SIMPLE and PISO algorithm) implemented to solve the pressure-velocity coupling in a segregated manner. Further information on those algorithms can, for instance, be found in [53].

#### 3.3. Flow Scales and Turbulence-Free Modelling

## 4. Model Setup

#### 4.1. Domain and Boundary Conditions

#### 4.2. Discretisation

## 5. Results: Comparison of Porosity Implementations

#### 5.1. Minimum Mesh Requirements 2D Sheet Model

#### 5.2. Horizontal Force on the Structures

#### 5.2.1. Force on the 2D Sheet

#### 5.2.2. Force on the Cylinder

#### 5.3. Wave Gauges near the Structures

#### 5.3.1. Wave Gauges near the 2D Sheet

#### 5.3.2. Wave Gauges near the Cylinder

#### 5.4. Velocity Profiles near the Structures

#### 5.4.1. Velocity Profiles near the 2D Sheet

`interFoam`with its standard (MULES) VOF method.

#### 5.4.2. Velocity Profiles near the Cylinder

#### 5.5. Flow Visualization near the Cylinder

#### 5.6. Tabular Summary of the Results

## 6. Discussion

## 7. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Options for porosity representation in computational fluid dynamics (CFD) modeling for thin porous structures.

**Figure 2.**Schematics of the investigated porosity implementations, showing mesh cells and arrows that represent the velocity vectors close to a porous sheet; bold lines along the cell faces indicate the components of the velocity that are subject to a pressure-drop; shades represent porous-media zones. (

**a**) isotropic (same resistance in x-, y- and z-direction) (

**b**) orthotropic (resistance in x-direction) (

**c**) baffle (resistance in normal direction to a surface).

**Figure 3.**CFD model setup: (

**a**) sketch of the model configuration for both the 2D and 3D numerical flume, (

**b**–

**d**) sections of the mesh of the 3D model with the cylinder represented by porous media where (

**b**,

**c**) show sections in plan view and (

**d**) the side view of a clip across the vertical x − z symmetry plane. The colours of the rectangles in (

**a**) correspond to the colours of the frames of (

**b**–

**d**). (

**a**) Sketch of the model configuration for both the 2D (side view only) and 3D (plan- and side view) numerical flume, all dimensions in [m]. The positions for the WGs A2 and A5 vary between the 2D sheet model and cylinder—the dimensions for the 2D model are stated in brackets (

**b**) Mesh close-up in plan view (

**c**) Mesh section in plan view (

**d**) Clipped mesh section in side view.

**Figure 4.**Snapshot of the 3D model where the porous cylinder is represented by orthotropic porous media. An animation of the wave-structure interaction is available as supplementary material.

**Figure 5.**Mesh independence study for the normalized horizontal force on the 2D sheet, F/Fexp, separately for the number of (

**a**) cells per sheet thickness (where d = 10 mm) in horizontal x-direction (Nx/d) and (

**b**) cells per wave height (where H = 177.5 mm) in vertical z-direction (N

_{z}/H), including the force results from models with all investigated porosity implementations. (

**a**) refinement in horizontal x-direction (

**b**) refinement in vertical z-direction.

**Figure 6.**Comparison of the horizontal force on the porous 2D sheet, f(t), for the investigated types of porosity implementation. (

**a**) whole time series (

**b**) section of the time series (

**c**) close-up of the time series.

**Figure 7.**Comparison of the horizontal force on the cylinder, f(t), for the investigated types of porosity implementation. (

**a**) whole time series (

**b**) section of the time series (

**c**) close-up of the time series.

**Figure 8.**CFD results of the wave elevation, $\eta \left(t\right)$, at selected WGs close to the porous sheet for all types of porosity implementation, including the whole time series and selected close-ups.

**Figure 9.**Experimental and CFD results of the wave elevation, $\eta \left(t\right)$, at selected WGs close to the porous cylinder for all types of porosity implementation, including the whole time series and selected close-ups. (

**a**) comparison of the CFD and experimental results further away from the cylinder (

**b**) comparison of the CFD results close to the cylinder (no experimental WG results available).

**Figure 10.**CFD and experimental results of the normalized mean wave amplitudes, $A/{A}_{input}$, for all WGs before and after the cylinder. The position of the center of the cylinder is indicated with a dashed vertical line and the cylinder front and back are indicated with solid vertical lines.

**Figure 11.**Velocity profiles in horizontal, u

_{x}, and vertical, u

_{z}, direction 0.1 m before (WG C1) and 0.1 m after (WG C2) the sheet for a wave trough and crest at the sheet. (

**a**) profiles for a wave trough at the sheet at t = 36 s (

**b**) profiles for a wave crest at the sheet at t = 43.4 s.

**Figure 12.**Velocity profiles in horizontal, ${u}_{x}$, and vertical, ${u}_{z}$, direction at the cylinder centre (WG B2), 0.5 m before (WG B1) and 0.5 m after (WG B3) the axis for a wave trough at the cylinder centre.

**Figure 13.**Velocity profiles in horizontal, ${u}_{x}$, and vertical, ${u}_{z}$, direction at the cylinder centre (WG B2), 0.5 m before (WG B1) and 0.5 m after (WG B3) the axis for a wave crest at the cylinder centre.

**Figure 14.**Velocity vectors for two points in time, t = 36.0 s at a wave trough (on the left) and t = 43.4 s at a wave crest (on the right) at the cylinder center. The waves propagate from the left to the right. The velocity magnitude and color of the vectors respectively ranges between 0.0–0.5 m/s. (

**a**) isotropic porous-media implementation (

**b**) orthotropic porous-media implementation (

**c**) baffle implementation.

**Table 1.**Water depth, h, and wave parameters used in the present study (input and target parameters).

h [m] | T [s] | $\mathit{\lambda}$ [m] | H [m] | A [m] | $\mathit{kh}$ [-] | $\mathit{kA}$ [-] | ${\mathit{c}}_{\mathit{g}}$ [m/s] | |
---|---|---|---|---|---|---|---|---|

1.00 | 2.1 | 5.58 | 0.1775 | 0.08875 | 1.13 | 0.10 | 1.96 |

**Table 2.**Summary of the normalized mean force amplitude results, $F/{F}_{exp}$, and the normalized mean wave amplitude results, $A/{A}_{input}$ at the WGs, including mesh cell number and execution times for all models.

Model | 2D Sheet | 3D Cylinder | ||||
---|---|---|---|---|---|---|

Porosity Impl. | Isotropic | Orthotropic | Baffle | Isotropic | Orthotropic | Baffle |

Number of Cells | 68,890 | 68,890 | 68,060 | 9,840,065 | 9,840,065 | 7,192,448 |

Max. $CFL$ | 0.3 | 0.3 | 0.05 | 0.3 | 0.3 | 0.05 |

Used CPUs | 1 | 1 | 1 | 28 | 28 | 28 |

Execution Time | 6.6 h | 7.8 h | 1 d 9.9 h | 8 d 14.1 h | 13 d 2.7 h | 72 d 17 h |

$F/Fexp$ | 1.0860 | 1.0659 | 1.0242 | 1.0388 | 0.9567 | 1.1078 |

$A/{A}_{input}$ WG A1 | 0.6541 | 0.6556 | 0.6867 | 0.9573 | 0.9282 | 0.9307 |

$A/{A}_{input}$ WG A2 | 0.6395 | 0.6413 | 0.6596 | 0.9783 | 0.9406 | 0.9408 |

$A/{A}_{input}$ WG A3 | 1.3073 | 1.3117 | 1.3054 | 1.0494 | 1.0630 | 1.0552 |

$A/{A}_{input}$ WG A4 | 1.2073 | 1.2028 | 1.2066 | 1.0202 | 1.0184 | 1.0130 |

$A/{A}_{input}$ WG A5 | 0.6334 | 0.6370 | 0.6570 | 0.9412 | 0.9406 | 0.9408 |

$A/{A}_{input}$ WG B1 | - | - | - | 1.1201 | 1.1125 | 1.1109 |

$A/{A}_{input}$ WG B2 | - | - | - | 0.9985 | 1.0216 | 0.9897 |

$A/{A}_{input}$ WG B3 | - | - | - | 1.0877 | 1.1102 | 1.0975 |

$A/{A}_{input}$ WG C1 | 1.3476 | 1.3521 | 1.3397 | - | - | - |

$A/{A}_{input}$ WG C2 | 0.6569 | 0.6535 | 0.6704 | - | - | - |

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**MDPI and ACS Style**

Feichtner, A.; Mackay, E.; Tabor, G.; Thies, P.R.; Johanning, L. Comparison of Macro-Scale Porosity Implementations for CFD Modelling of Wave Interaction with Thin Porous Structures. *J. Mar. Sci. Eng.* **2021**, *9*, 150.
https://doi.org/10.3390/jmse9020150

**AMA Style**

Feichtner A, Mackay E, Tabor G, Thies PR, Johanning L. Comparison of Macro-Scale Porosity Implementations for CFD Modelling of Wave Interaction with Thin Porous Structures. *Journal of Marine Science and Engineering*. 2021; 9(2):150.
https://doi.org/10.3390/jmse9020150

**Chicago/Turabian Style**

Feichtner, Anna, Ed Mackay, Gavin Tabor, Philipp R. Thies, and Lars Johanning. 2021. "Comparison of Macro-Scale Porosity Implementations for CFD Modelling of Wave Interaction with Thin Porous Structures" *Journal of Marine Science and Engineering* 9, no. 2: 150.
https://doi.org/10.3390/jmse9020150