# Performance Assessment of Three Turbulence Models Validated through an Experimental Wave Flume under Different Scenarios of Wave Generation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Description of the Wave Flume

#### 2.1. Experimental Wave Flume

#### 2.2. Numerical Wave Flume

#### 2.2.1. The Mesh

_{1}/T)π · sin(2π/T · t)] − [(4A

_{2}/T)π · sin(4π/T · t)],

_{1}[m] is the amplitude of the principal curve and A

_{2}[m] the amplitude of the secondary one. The meshes can be divided in different areas. The air area (number 1 in Figure 3) has no impact in the wave generation and, therefore, has the biggest cells to reduce as much as possible the computational cost. The area below the extinction system (number 2 in Figure 3) has the same cell size than the air area due to its negligible influence in the experiments. The “deep-water” area is the section below the free surface that always contains water (number 3 in Figure 3). This area has smaller cells than the air area because it is affected by the particles’ movement in the waves but is bigger than the free surface and extinction areas also to reduce the computational cost of the simulations. These last areas (number 4 and 5 in Figure 3) have a cell size that depends on the wave generated. The height of the cells is calculated by dividing the wave height (H [mm]) by 20, with this being the biggest cell size that ensures the good definition of the wave. The length of each cell is calculated by having an aspect ratio (AR) of 4, which is the relation between the length and height of the cell:

_{x}/Δ

_{y}.

_{H}[mm]) is calculated depending on the wave height too, with the total height of this area being:

_{H}= H + 20.

#### 2.2.2. Reynolds-Averaged Navier–Stokes Equations (RANS)

_{j}(∂ū

_{i}/∂x

_{j}) = ρf

_{i}+ (∂/∂x

_{j}) [−p∂

_{ij}+ μ((∂ū

_{i}/∂x

_{j}) + (∂ū

_{j}/∂x

_{i})) − ρu

_{i}’u

_{j}’],

_{i}is the time-averaged velocity, p is the pressure field, ρ is the density of the effective flow, μ is the viscosity, and –ρu

_{i}’u

_{j}’ is the Reynolds operator.

#### 2.2.3. Volume of Fluid Method

_{i}= V

_{i}/V.

_{i}(α

_{i}ρ

_{i}),

_{i}(α

_{i}μ

_{i}),

_{i}is the density of each fluid and μ

_{i}is their viscosity. The general conservation equation that describes the transport of volume fractions can be defined as:

_{v}(α

_{i}dV) + ∫

_{s}(α

_{i}(v − v

_{g})da) = ∫

_{v}(S

_{αi}− [(α

_{i}/ρ

_{i})(Dρ

_{i}/Dt)]dV),

_{g}is the grid velocity, and S

_{αi}is the source of the ith phase. It is important to highlight that, in our case, the source term is 0.

## 3. Aims and Methodology

#### 3.1. Experimental Procedures

_{E}, for which the wave will travel forward and backward along the EWF, until it reaches the position of the last probe (x

_{3}) again:

^{2}/2π) tanh(2πh/λ),

_{E}= (2L

_{Tank}+ x

_{3})/c.

_{i}[m]). When the experiment in the EWF ended, the paddle was situated at 0.5 m from the wall to ensure the same starting point for every experiment.

#### 3.2. Data Processing by MATLAB

_{R}corresponds to the instant when the reflection phenomenon is first measured on each probe:

_{R}= (2L

_{Tank}− x

_{i})/c,

_{tank}is the length of the tank and x

_{i}is the position of each sensor. Then, the re-reflection time was calculated with Equation (12) for the position of each probe, respectively. To obtain the parameters of the waves, a specific height condition was created in MATLAB that disregards the data corresponding to the interval in which the wave is still developing. This condition was applied until the height was, at least, 70% of the maximum height before reflection appears. This time value was defined as the start time (t

_{S}[s]). In each analysis, the time before t

_{S}and after t

_{E}were deleted. Then, each run was divided into incident wave and reflection analysis.

_{S}to t

_{R}were selected and fitted according to the corresponding wave theory, first (linear, Airy’s theory) or second (non-linear, Stokes’ theory) order, to obtain the experimental and numerical parameters of that wave (H, T, and λ). The values obtained from this fitting were compared to the theoretical values. It is important to highlight that the fitting avoids the variations from the theory that appear in the free surface displacement data.

_{I}) and the reflected wave (A

_{R}). Then, the reflection coefficient is obtained:

_{R}/A

_{I}.

#### 3.3. Numerical

_{sim}= (VOF

_{water}· 700) − h

_{study},

_{sim}is the position of the free surface in mm, VOF

_{water}is the mean volume fraction of water measure in the study plane, 700 is the total height of the NWF in mm, and h

_{study}is the depth of study in mm. Thus, the value of η

_{sim}at the beginning of each simulation was 0.

#### 3.4. Results Comparison

## 4. Turbulence Models

#### 4.1. The Low Reynolds k-ε Model

_{t}= C

_{μ}f

_{μ}(k

^{2}/ε),

_{μ}is a model constant, f

_{μ}is a damping function, k is the turbulent kinetic energy, and ε is the turbulence dissipation rate. The mathematical modelling can be written in a boundary layer form:

_{t}/σ

_{k})) (∂k/∂y)] + ν

_{t}(∂u/∂y)

^{2}− ε

ū(∂ε/∂x) + v(∂ε/∂y) = (∂/∂y)[(ν + (ν

_{t}/σ

_{k})) (∂ε/∂y)] + C

_{ε1}f

_{1}(ε/k)ν

_{t}(∂u/∂y)

^{2}− C

_{ε}

_{2}f

_{2}(ε

^{2}/k) + Ε.

_{j}(∂k/∂x

_{j}) = ∂/∂x

_{j}[ν + ν

_{t}/σ

_{k}∂k/∂x

_{j}] − ε + τ

_{ij}∂ū

_{i}/∂x

_{j}.

_{j}(∂k/∂x

_{j}) = ∂/∂x

_{j}[ν + ν

_{t}/σ

_{k}∂ε/∂x

_{j}] − C

_{ε1}(ε/k)τ

_{ij}(∂ū

_{i}/∂x

_{j}) − C

_{ε2}(ε

^{2}/k),

_{k}= 1 and σ

_{ε}= 1.3 are the Prandtl numbers for k and ε. The damping functions in the standard k-ε low-Re are f

_{2}:

_{2}= 1 − C·exp(−Re

_{t}

^{2}),

_{μ}:

_{μ}= 1 − exp[−(C

_{d0}√(Re

_{d}) + C

_{dl}Re

_{d}+ C

_{d2}Re

_{d}

^{2})].

#### 4.2. The Shear Stress Transport (SST) Model

_{t}= (k/ω),

_{j}(∂k/∂x

_{j}) = ∂/∂x

_{j}[(ν + σ* k/ω) ∂k/∂x

_{j}]β* kω + τ

_{ij}(∂ū

_{i}/∂x

_{j}).

_{j}(∂ω/∂x

_{j}) = ∂/∂x

_{j}[(ν + σk/ω) ∂ω/∂x

_{j}] − β kω

^{2}+ (σ

_{d}/ω)(∂k/∂x

_{j})(∂ω/∂x

_{j}) + a (ω/k)τ

_{ij}(∂ū

_{i}/∂x

_{j}).

#### 4.3. Large-Eddy Simulation (LES)

_{i}/∂x

_{i}= 0

∂ρu

_{i}/∂t + ∂ρu

_{i}u

_{j}/∂x

_{j}+ ∂p/∂x

_{i}− ∂σ

_{ij}/∂x

_{j}≈ − ∂τ

_{ij}/∂x

_{j}

∂ρE/∂t + ∂(ρE + p)u

_{j}/∂x

_{j}− ∂u

_{i}σ

_{ij}/∂x

_{i}− ∂q

_{j}/∂x

_{j}≈ − 1/γ−1 ∂(pu

_{j}− pu

_{j})/∂x

_{j}− u

_{j}(∂τ

_{ij}/∂x

_{j}).

_{t}

_{=}ρΔ

^{2}S,

_{v}C

_{s}V

^{1/3},

_{S}is the model coefficient and f

_{v}is the Van Driest damping function, which in the software can be computed as:

_{v}= 1 − exp(−y

^{+}/A),

^{+}is the dimensionless wall distance.

## 5. Results and Discussion

_{S}and t

_{R}time values. This fitting, shown in Figure 4, matches almost perfectly with the free surface variation data obtained from the experimental runs. This adjustment constantly achieves a quality of overlapping over 95% in the 3 probes for every wave.

^{−5}for the turbulent kinetic energy and 1 × 10

^{−4}for the turbulence dissipation rate. These changes help to improve the fitting between experimental and numerical values by defining the crests of the waves better, although troughs are yet to be better defined.

_{S}= 0.1, C

_{t}= 3.5, A = 25, and the von Karman constant κ = 0.41. Table 5 shows the values and relative errors of each simulation using this LES model. The adjustments of wave height are worse compared to the two-equation models, but the definition of the wavelength is in the same error interval as the two other models. Because of this, it can be concluded that this model reproduces significantly well the experimentally generated waves.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

A | Amplitude |

A_{I} | Amplitude of the incident wave |

A_{R} | Amplitude of the reflected wave |

AR | Aspect Ratio |

c | Wave celerity |

CFD | Computational Fluid Dynamics |

C_{μ} | k-ε model constant |

EWF | Experimental Wave Flume |

FOWT | Floating Offshore Wind Turbine |

FS_{H} | Free Surface Height |

f_{μ} | Damping function |

H | Wave Height |

K | Reflection Coefficient |

k | Kinetic energy |

L_{Tank} | Tank Length |

LES | Large Eddy Simulation |

NWT | Numerical Wave Tank |

OWC | Oscillating Water Column |

p | Pressure |

RANS | Reynolds-Average Navier Stokes |

SST | Shear Stress Transport |

u | Fluid velocity |

V | Volume of the cell |

v | velocity |

v_{g} | Grid velocity |

VOF | Volume of Fluid |

x_{i} | Position of the probe |

αi | Volume Fraction of a fluid |

Δ | Length Scale |

ε | Turbulence dissipation rate |

η | Free Surface Position |

μ | Dynamic viscosity |

ν_{t} | Kinematic viscosity |

ω | Specific dissipation rate |

σ_{ε} | Prandtl constant for turbulence dissipation rate |

S_{ij} | Mean strain-rate tensor |

σ_{k} | Prandtl constant for kinematic energy |

T | Period |

t | time |

τ_{ij} | Favre_averaged specific Reynolds-stress tensor |

τ_{xy} | Favre-specific Reynolds shear stress |

μ_{T} | Subgrid viscosity |

NWF | Numerical Wave Flume |

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**Figure 1.**(

**a**) The general view of the EWF from the piston-type wavemaker (left) to the extinction system (right). (

**b**) Picture of the self-designed extinction system installed in the wave flume and the highlights of the join (A) and the position of the cranks (B and C) to modify the angle

**Figure 2.**Disposition of the resistive probes in the EWF. The arrows highlight the join with the structure that allows them to move along the flume.

**Figure 3.**(

**a**) Mesh corresponding to a depth of 50 cm, (

**b**) focus on the prism layers around the generation paddle. (

**c**) The different areas of the mesh: (1) air, (2) below the paddle, (3) “deep-water”, (4) free surface, and (5) extinction system.

**Figure 4.**Fitting of the free surface displacement for the wave 3: fitting (orange) and experimental (blue) measured by the first probe of the EWF.

**Figure 5.**Comparison of the free surface displacement for the turbulence model of (

**a**) k-ε, (

**b**) SST, and (

**c**) LES with the experimental data (blue) obtained in the EWF for the wave 3.

**Figure 7.**Comparison of the incident and reflected amplitudes of the waves by the reflection method developed by Mansard and Funke for the experimental and numerical runs.

h (m) | H (mm) | T (s) | λ (m) | c (m/s) | |
---|---|---|---|---|---|

Wave 1 | 0.3 | 40 | 1.25 | 1.88 | 1.54 |

Wave 2 | 0.3 | 30 | 1.42 | 2.20 | 1.54 |

Wave 3 | 0.4 | 39 | 0.81 | 1.01 | 1.24 |

Wave 4 | 0.4 | 64 | 1.27 | 2.11 | 1.65 |

Wave 5 | 0.5 | 29 | 1.20 | 2.06 | 1.71 |

Wave 6 | 0.5 | 60 | 1.42 | 2.64 | 1.85 |

**Table 2.**Experimental results and relative errors of the incident wave with respect to the theoretically intended values.

H (mm) | T (s) | λ (m) | c (m/s) | Error H (%) | Error T (%) | Error λ (%) | Error c (%) | |
---|---|---|---|---|---|---|---|---|

Wave 1 | 42.07 | 1.25 | 1.78 | 1.42 | 5.41 | −0.63 | −7.08 | −6.48 |

Wave 2 | 30.04 | 1.41 | 2.10 | 1.49 | 0.14 | 1.19 | 4.59 | 3.40 |

Wave 3 | 36.89 | 0.79 | 0.99 | 1.24 | 5.39 | 1.36 | 1.47 | 0.10 |

Wave 4 | 66.29 | 1.27 | 2.11 | 1.65 | 3.58 | 0.15 | 0.15 | 0.01 |

Wave 5 | 28.34 | 1.18 | 2.02 | 1.71 | 2.26 | 1.80 | 1.83 | 0.03 |

Wave 6 | 56.50 | 1.41 | 2.68 | 1.90 | 5.82 | 1.23 | 1.78 | 3.06 |

**Table 3.**Results obtained from the simulations carried out using the k-ε turbulence model and relative errors with respect to the experimental values of the incident wave.

H (mm) | T (s) | λ (m) | c (m/s) | Error H (%) | Error T (%) | Error λ (%) | Error c (%) | |
---|---|---|---|---|---|---|---|---|

Wave 1 | 39.15 | 1.25 | 1.84 | 1.46 | −6.96 | −0.09 | 3.30 | 2.54 |

Wave 2 | 29.42 | 1.42 | 2.20 | 1.54 | −2.08 | 0.64 | 4.48 | 3.19 |

Wave 3 | 36.90 | 0.80 | 1.00 | 1.24 | −5.39 | −1.36 | −1.24 | −0.69 |

Wave 4 | 63.65 | 1.27 | 2.14 | 1.67 | −3.99 | −0.70 | 1.06 | 0.86 |

Wave 5 | 28.31 | 1.20 | 2.18 | 1.81 | −0.12 | 1.25 | 7.47 | 5.76 |

Wave 6 | 60.56 | 1.42 | 2.68 | 1.87 | 7.17 | 0.69 | −0.37 | −1.97 |

**Table 4.**Results obtained from the simulations carried out using the SST turbulence model and relative errors with respect to the experimental values of the incident wave.

H (mm) | T (s) | λ (m) | c (m/s) | Error H (%) | Error T (%) | Error λ (%) | Error c (%) | |
---|---|---|---|---|---|---|---|---|

Wave 1 | 39.28 | 1.25 | 1.88 | 1.49 | −6.65 | −0.09 | 5.54 | 4.65 |

Wave 2 | 29.52 | 1.42 | 2.22 | 1.55 | −1.74 | 0.64 | 5.43 | 3.86 |

Wave 3 | 36.89 | 0.80 | 1.00 | 1.24 | −5.39 | −1.36 | −1.24 | −0.69 |

Wave 4 | 63.12 | 1.28 | 2.03 | 1.59 | −4.79 | 0.09 | −4.13 | −3.97 |

Wave 5 | 28.29 | 1.19 | 1.97 | 1.65 | −0.19 | 0.40 | −2.88 | −3.59 |

Wave 6 | 59.64 | 1.42 | 2.84 | 1.99 | 5.54 | 0.69 | 5.58 | 4.33 |

**Table 5.**Results obtained from the simulations carried out using the LES turbulence model and relative errors with respect to the experimental values of the incident wave.

H (mm) | T (s) | λ (m) | c (m/s) | Error H (%) | Error T (%) | Error λ (%) | Error c (%) | |
---|---|---|---|---|---|---|---|---|

Wave 1 | 37.73 | 1.26 | 1.88 | 1.49 | −10.34 | 0.55 | 5.51 | 4.93 |

Wave 2 | 28.31 | 1.43 | 2.21 | 1.55 | −5.76 | 1.10 | 5.01 | 3.87 |

Wave 3 | 36.89 | 0.81 | 1.01 | 1.24 | −5.39 | −0.06 | −0.49 | −0.43 |

Wave 4 | 62.44 | 1.28 | 2.03 | 1.59 | −5.82 | 0.27 | −3.91 | −4.16 |

Wave 5 | 28.99 | 1.20 | 1.97 | 1.65 | 2.27 | 1.08 | −2.84 | −3.88 |

Wave 6 | 58.91 | 1.42 | 2.84 | 2.00 | 4.25 | 0.84 | 5.57 | 4.68 |

**Table 6.**Comparison of the wave height and period of the resultant wave between the experimental and k-ε model, and the error between them.

H_{exp} (mm) | T_{exp} (s) | H_{k-ε} (mm) | T_{k-ε} (s) | Error H (%) | Error T (%) | |
---|---|---|---|---|---|---|

Wave 1 | 40.34 | 1.26 | 38.83 | 1.25 | −3.73 | −0.79 |

Wave 2 | 31.02 | 1.45 | 31.72 | 1.42 | 2.26 | −1.66 |

Wave 3 | 37.13 | 0.81 | 34.46 | 0.81 | −7.19 | 0.41 |

Wave 4 | 65.27 | 1.25 | 63.76 | 1.25 | −2.31 | 0.50 |

Wave 5 | 28.50 | 1.18 | 28.05 | 1.20 | −1.61 | 1.88 |

Wave 6 | 56.78 | 1.43 | 59.42 | 1.41 | 4.66 | −1.50 |

**Table 7.**Comparison of the wave height and period of the resultant wave between the experimental and SST model, and the error between them.

H_{exp} (mm) | T_{exp} (s) | H_{SST} (mm) | T_{SST} (s) | Error H (%) | Error T (%) | |
---|---|---|---|---|---|---|

Wave 1 | 40.34 | 1.26 | 38.69 | 1.25 | −4.08 | −0.78 |

Wave 2 | 31.02 | 1.45 | 31.59 | 1.42 | 1.85 | −1.70 |

Wave 3 | 37.13 | 0.81 | 34.80 | 0.81 | −6.28 | 0.30 |

Wave 4 | 65.27 | 1.25 | 63.76 | 1.25 | −2.31 | 0.50 |

Wave 5 | 28.50 | 1.18 | 28.16 | 1.20 | −1.21 | 1.87 |

Wave 6 | 56.78 | 1.43 | 59.40 | 1.41 | 4.61 | −1.45 |

**Table 8.**Comparison of the wave height and period of the resultant wave between the experimental and LES model, and the error between them.

H_{exp} (mm) | T_{exp} (s) | H_{LES} (mm) | T_{LES} (s) | Error H (%) | Error T (%) | |
---|---|---|---|---|---|---|

Wave 1 | 40.34 | 1.26 | 37.10 | 1.26 | −8.03 | 0.05 |

Wave 2 | 31.02 | 1.45 | 30.29 | 1.42 | −2.35 | −1.66 |

Wave 3 | 37.13 | 0.81 | 36.47 | 0.81 | −1.76 | 0.68 |

Wave 4 | 65.27 | 1.25 | 64.56 | 1.25 | −1.08 | 0.54 |

Wave 5 | 28.50 | 1.18 | 28.82 | 1.20 | 1.12 | 1.89 |

Wave 6 | 56.78 | 1.43 | 59.43 | 1.41 | 4.66 | −1.46 |

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**MDPI and ACS Style**

Galera-Calero, L.; Blanco, J.M.; Izquierdo, U.; Esteban, G.A.
Performance Assessment of Three Turbulence Models Validated through an Experimental Wave Flume under Different Scenarios of Wave Generation. *J. Mar. Sci. Eng.* **2020**, *8*, 881.
https://doi.org/10.3390/jmse8110881

**AMA Style**

Galera-Calero L, Blanco JM, Izquierdo U, Esteban GA.
Performance Assessment of Three Turbulence Models Validated through an Experimental Wave Flume under Different Scenarios of Wave Generation. *Journal of Marine Science and Engineering*. 2020; 8(11):881.
https://doi.org/10.3390/jmse8110881

**Chicago/Turabian Style**

Galera-Calero, Lander, Jesús María Blanco, Urko Izquierdo, and Gustavo Adolfo Esteban.
2020. "Performance Assessment of Three Turbulence Models Validated through an Experimental Wave Flume under Different Scenarios of Wave Generation" *Journal of Marine Science and Engineering* 8, no. 11: 881.
https://doi.org/10.3390/jmse8110881