# The Efficient Application of an Impulse Source Wavemaker to CFD Simulations

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## Abstract

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## 1. Introduction

- Replication of physical wavemakers, such as oscillating flaps, paddles or pistons
- Implementation of numerical/mathematical techniques, introducing source terms or similar into the governing equations

#### 1.1. Impulse Source Wavemaker

#### 1.2. Outline of the Paper

## 2. Implementation

#### 2.1. The Impulse Equation

- $r\rho {\mathbf{a}}_{wm}$: This is the source term used for wave generation, where r is a scalar variable that defines the wavemaker region and ${\mathbf{a}}_{wm}$ is the acceleration input to the wavemaker at each cell centre within r. r is dimensionless, while ${\mathbf{a}}_{wm}$ is given in the units of $\mathrm{m}{\mathrm{s}}^{-2}$.
- $sand\rho \mathbf{U}$: This describes a dissipation term used to implement a numerical beach, where the variable field $sand$ controls the strength of the dissipation, equalling zero in the central regions of the domain where the working wavefield is required and then gradually increasing towards the boundary over the length of the numerical beach [25]. $sand$ is given in units of ${\mathrm{s}}^{-1}$.

#### 2.2. Setting Parameter Values

- r is set to one in the region where the wavemaker is acting and zero everywhere else in the NWT domain. Therefore, the size of the wavemaker and its position within the NWT must also be selected. To offer guidance on the selection of the wavemaker size and position, the case study presented in Section 3 and Section 4 investigates the effect of these parameters on the wavemaker performance.
- $sand$ is initialised using an analytical expression relating the value of $sand$ and the geometric coordinates of the NWT. The simplest expression would be a step function, where the value of $sand$ is constant inside the beach and is zero everywhere else in the NWT. However, such a sharp increase in the dissipation will cause numerical reflections. Instead, the value of $sand$ should be increased gradually from the start to the end of the numerical beach. Equation (3) is used in the current implementation, which has been shown to result in sufficient absorption [25]:$$\begin{array}{cc}\hfill sand\left(x\right)=-2\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}san{d}_{Max}& {\left(\right)}^{\frac{({l}_{beach}-x)}{{l}_{beach}}}3+\hfill \end{array}& 3\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}san{d}_{Max}{\left(\right)}^{\frac{({l}_{beach}-x)}{{l}_{beach}}}2\hfill $$Perić and Abdel-Maksoud [26] recently derived an analytical solution describing the ideal setting for $sand$ and validated the method in a numerical experiment, removing the need for parameter studies in the future.

#### 2.3. Calibration Procedure

- Define target wave series at desired NWT location, ${\eta}_{T}\left(t\right)$, with a signal length $\mathcal{L}$ and $\mathcal{N}$ samples
- Perform a Fast Fourier Transform (FFT) on ${\eta}_{T}\left(t\right)$, to obtain the amplitudes, ${A}_{T}\left({f}_{j}\right)$, and phase components, ${\varphi}_{T}\left({f}_{j}\right)$, for each frequency component, ${f}_{j}$, with $j=\{1,2,\dots ,\frac{\mathcal{N}}{2}\}$, where ${f}_{1}=\frac{0}{\mathcal{L}},{f}_{2}=\frac{1}{\mathcal{L}},\dots ,{f}_{\mathcal{N}}=\frac{\mathcal{N}}{2\mathcal{L}}-\frac{1}{\mathcal{L}}$
- Generate an initial time series for the wavemaker source term, ${\mathbf{a}}_{wm,1}\left(t\right)$ (can be chosen randomly or informed by ${\eta}_{T}\left(t\right)$)
- Perform an FFT on ${\mathbf{a}}_{wm,1}\left(t\right)$, to obtain the amplitudes, ${A}_{a,1}\left({f}_{j}\right)$, and phase components, ${\varphi}_{a,1}\left({f}_{j}\right)$, for each frequency component of the input source term
- Run the simulation, for iteration i, using the wavemaker source term ${\mathbf{a}}_{wm,i}\left(t\right)$, and measure the resulting free surface elevation at the chosen NWT location, ${\eta}_{R,i}\left(t\right)$
- Perform an FFT on ${\eta}_{R,i}\left(t\right)$, to obtain the amplitudes, ${A}_{R,i}\left({f}_{j}\right)$, and phase components, ${\varphi}_{R,i}\left({f}_{j}\right)$, for each frequency component of the generated wave series
- Calculate the new amplitudes for each frequency component of the input source term, ${A}_{a,i+1}\left({f}_{j}\right)$, by scaling the previous amplitudes, ${A}_{a,i}\left({f}_{j}\right)$, with the ratio of target surface elevation amplitude, ${A}_{T}\left({f}_{j}\right)$, and the generated surface elevation amplitude from the previous run, ${A}_{R,i}\left({f}_{j}\right)$:
- ${A}_{a,i+1}\left({f}_{j}\right)=\frac{{A}_{T}\left({f}_{j}\right)}{{A}_{R,i}\left({f}_{j}\right)}{A}_{a,i}\left({f}_{j}\right)$

- Calculate the new phase components, ${\varphi}_{a,i+1}\left({f}_{j}\right)$, by summing ${\varphi}_{a,i}\left({f}_{j}\right)$ with the difference between the target elevation phase, ${\varphi}_{T}\left({f}_{j}\right)$, and the measured surface elevation phase from the previous run, ${\varphi}_{R,i}\left({f}_{j}\right)$:
- ${\varphi}_{a,i+1}\left({f}_{j}\right)={\varphi}_{a,i}\left({f}_{j}\right)+\left(\right)open="["\; close="]">{\varphi}_{T}\left({f}_{j}\right)-{\varphi}_{R,i}\left({f}_{j}\right)$

- Generate the new time series for the wavemaker source term, ${\mathbf{a}}_{wm,i+1}\left(t\right)$, by performing an inverse Fourier transform on ${A}_{a,i+1}\left({f}_{j}\right)$ and ${\varphi}_{a,i+1}\left({f}_{j}\right)$
- Repeat Steps 5–9 until either a maximum number of iterations or a threshold for the Mean-Squared Error (MSE) between the target and resulting surface elevation is reached.

#### 2.4. Calibration Procedure for Regular Waves

- The source term is set to oscillate in the horizontal direction with the desired wave frequency.
- The amplitude ${A}_{a,i}$ is initialised with an arbitrary value.
- After the initial, and each subsequent run, the surface elevation is analysed in the time domain. The mean is removed from the surface elevation. The mean wave height, ${H}_{R,i}$, is then obtained as the difference between the mean of the positive and mean of the negative peaks.
- A new wave maker amplitude, ${A}_{a,i+1}$, is obtained by linearly scaling the previous value with the ratio of target ${H}_{T}$ and resulting wave height, ${H}_{R}$, as follows ${A}_{a,i+1}=\frac{{H}_{T}}{{H}_{R,i}}{A}_{a,i}$

## 3. Case Study

#### 3.1. Target Waves

#### 3.1.1. Multi-Frequency Wave Packet

#### 3.1.2. Regular Waves

#### 3.2. Source Region

#### 3.3. Simulation Platform

#### 3.4. Numerical Wave Tank Setup

#### 3.4.1. Geometry

#### 3.4.2. Boundary Conditions

#### 3.4.3. Mesh

#### 3.4.4. Calibration of the Numerical Beach

#### 3.5. Source Region Shape and Position

## 4. Results and Discussion

#### 4.1. Example Result

#### 4.2. Wave Packet: Deep Water Waves

#### 4.3. Wave Packet: Intermediate Water Waves

#### 4.4. Wave Packet: Shallow Water Waves

#### 4.5. Regular Waves

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Source Region Parameters and MSE Values

Height/Length | 1L | 0.75L | 0.5L | 0.25L | 0.125L |
---|---|---|---|---|---|

1.25d | 1.87$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 7.87$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 1.84$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 2.88$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 7.09$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ |

1d | 9.05$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 1.11$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 1.16$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 3.47$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 4.48$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ |

0.75d | 1.13$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 1.41$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 1.41$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 3.46$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 3.99$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ |

0.5d | 3.60$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 8.57$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 1.21$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 4.74$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 3.10$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ |

0.25d | 1.23$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 6.86$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 4.96$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 2.80$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 4.75$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ |

0.125d | 4.53$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 4.59$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 6.66$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.72$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 1.32$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ |

**Table A2.**MSE values for different source heights and lengths, intermediate water case. Centre at $\frac{d}{3}$.

Height/Length | 1L | 0.75L | 0.5L | 0.25L | 0.125L |
---|---|---|---|---|---|

1.25d | 1.80$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.22$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 4.50$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 4.43$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 6.59$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ |

1d | 1.24$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 1.06$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 4.92$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 4.16$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 6.22$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ |

0.75d | 2.01$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 1.46$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 5.03$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 4.36$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 6.64$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ |

0.5d | 1.34$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 1.40$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 4.66$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 6.81$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 8.24$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ |

0.25d | 1.39$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 2.94$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 5.83$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 9.33$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 1.19$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ |

0.125d | 1.67$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 6.89$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 8.06$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 1.38$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 2.82$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ |

**Table A3.**MSE values for different source heights and lengths, intermediate water case. Centre at $\frac{d}{2}$.

Height/Length | 1L | 0.75L | 0.5L | 0.25L | 0.125L |
---|---|---|---|---|---|

1.25d | 1.26$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 3.15$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 9.38$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 4.46$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 6.69$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ |

1d | 9.63$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 1.11$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 6.26$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 5.12$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 7.29$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ |

0.75d | 9.85$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.75$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 6.84$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 6.67$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 8.26$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ |

0.5d | 3.97$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 1.75$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 5.35$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 8.43$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 1.04$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ |

0.25d | 2.82$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 1.99$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 5.74$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 1.25$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 1.14$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ |

0.125d | 2.21$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 3.45$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 8.16$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 9.14$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 2.05$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ |

**Table A4.**MSE values for different source heights and lengths, intermediate water case. Centre at $\frac{2d}{3}$.

Height/Length | 1L | 0.75L | 0.5L | 0.25L | 0.125L |
---|---|---|---|---|---|

1.25d | 8.46$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 1.41$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 9.18$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 5.89$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 7.15$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ |

1d | 1.16$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 3.65$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 1.19$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 7.13$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 8.33$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ |

0.75d | 5.96$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-4}$ | 5.08$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 9.64$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 7.92$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 9.07$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ |

0.5d | 1.03$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 1.84$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 7.52$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 1.00$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 9.65$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ |

0.25d | 4.45$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 1.64$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 7.79$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 7.71$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 1.10$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ |

0.125d | 2.27$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 2.69$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-5}$ | 1.03$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 1.09$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 1.13$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ |

Height/Length | 0.75L | 0.5L | 0.25L |
---|---|---|---|

1.25d | 7.47$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-6}$ | 5.38$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ | 6.78$\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{-7}$ |

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**Figure 2.**Target wave packet for the impulse wave maker to produce in the case study: surface elevation (

**a**,

**c**,

**e**) and spectral density (

**b**,

**d**,

**f**) for the deep (top), intermediate (middle) and shallow water case (bottom).

**Figure 3.**Schematic of the NWT including the main dimensions (schematic not at scale). The function of the gradual increase of $sand$ in the numerical beach is included in purple. The source region is marked in red. The spatial discretisation, with a fine resolution around the free surface interface, is shown in the background. For the shallow water cases, the water depth d is set to $0.74\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ for the intermediate water case to $0.025\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ and for the deep water case to $0.74\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$.

**Figure 5.**Example of the target and resulting surface elevation, at different calibration iterations, for the same case as Figure 4.

**Figure 6.**Deep water case: minimal error ($2.8\times {10}^{-7}$ m${}^{2}$) for a source height of $\frac{d}{4}$ and length $\frac{1}{4}{\lambda}_{p}$.

**Figure 7.**Intermediate water case. Source centre at $\frac{d}{3}$: minimal error ($4.5\times {10}^{-7}$ m${}^{2}$) for source height of $1d$ and source length $\frac{1}{4}{\lambda}_{p}$.

**Figure 8.**Intermediate water case. Source centre at $\frac{d}{2}$: minimal error ($4.2\times {10}^{-7}$ m${}^{2}$) for source height of $1.25d$ and source length $\frac{1}{4}{\lambda}_{p}$.

**Figure 9.**Intermediate water case. Source centre at $\frac{2d}{3}$: minimal error ($5.9\times {10}^{-7}$ m${}^{2}$) for source height of $1.25d$ and source length $\frac{1}{2}{\lambda}_{p}$.

**Figure 10.**Surface elevation time series for intermediate water experiments that produced the best results for the three different source centre locations.

**Figure 12.**Target and resulting surface elevation time traces after nine calibration iterations, for the shallow water experiments. Source length $0.5\lambda $.

**Figure 13.**Surface elevation time series for regular waves. Numbers indicate the calibration iteration.

d (m) | ${\mathit{A}}_{0}$ (m) | ${\mathit{T}}_{\mathit{p}}$ (s) | $\mathit{\lambda}$ (m) | $\mathit{d}\phantom{\rule{-1.111pt}{0ex}}/\phantom{\rule{-0.55542pt}{0ex}}\mathit{\lambda}$ (–) | $\mathit{kd}$ (–) | |
---|---|---|---|---|---|---|

Deep water | 0.74 | 0.02 | 0.975 | 1.48 | 0.5 | 3.13 |

Intermediate water | 0.25 | 0.02 | 1.1 | 1.48 | 0.17 | 1.06 |

Shallow water | 0.74 | 0.015 | 6.0 | 15.9 | 0.05 | 0.29 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Schmitt, P.; Windt, C.; Davidson, J.; Ringwood, J.V.; Whittaker, T.
The Efficient Application of an Impulse Source Wavemaker to CFD Simulations. *J. Mar. Sci. Eng.* **2019**, *7*, 71.
https://doi.org/10.3390/jmse7030071

**AMA Style**

Schmitt P, Windt C, Davidson J, Ringwood JV, Whittaker T.
The Efficient Application of an Impulse Source Wavemaker to CFD Simulations. *Journal of Marine Science and Engineering*. 2019; 7(3):71.
https://doi.org/10.3390/jmse7030071

**Chicago/Turabian Style**

Schmitt, Pál, Christian Windt, Josh Davidson, John V. Ringwood, and Trevor Whittaker.
2019. "The Efficient Application of an Impulse Source Wavemaker to CFD Simulations" *Journal of Marine Science and Engineering* 7, no. 3: 71.
https://doi.org/10.3390/jmse7030071