# Wave Propagation Studies in Numerical Wave Tanks with Weakly Compressible Smoothed Particle Hydrodynamics

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## Abstract

**:**

## 1. Introduction

## 2. Smoothed Particle Hydrodynamics Formulation

#### 2.1. WCSPH Numerical Model

#### 2.2. Numerical Dissipation Chemes

#### 2.2.1. Artificial Viscosity

#### 2.2.2. Density Reinitialization

#### 2.2.3. δ-SPH Scheme

#### 2.3. The Smoothing Kernel Functions

#### 2.4. Boundary Handling

## 3. Validation Studies

#### 3.1. Dam-Break Simulations

#### 3.2. Sloshing Simulations

## 4. Numerical Wave Tank Experiments

#### 4.1. Wave Generation and Propagation

#### 4.2. Results and Discussion

#### 4.2.1. Hydrostatic Pressure Initialization

#### 4.2.2. Comparisons with Different Dissipation Schemes

**and**$1\times {10}^{-2}m$

**.**The wave elevation at $x=6\mathrm{m}$ was extracted for the different cases and compared with the analytical solution. The location was chosen since it is sufficiently away from the ends of the wave tank. Thus, there are negligible effects of wave reflection to be anticipated in the results. The L2 errors for the different resolutions were computed for the different resolutions and then interpolated over the range of the particle resolutions chosen by using the MATLAB “fit” function with a smoothing parameter magnitude of 0.995. The L2 error is computed using the formula

#### 4.2.3. Particle Resolution Study

#### 4.2.4. Effect of Kernels on Numerical Dissipation

## 5. Variation of Parameters in WCSPH Simulations

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**left**) Different Wendland kernels used in the study. (

**right**) Derivatives of the kernel functions.

**Figure 3.**Comparison of the interpolated pressure data at the measurement location against the experimental measurements for different numerical dissipation schemes using WC4 kernel. The pressure and the time values have been non-dimensionalized by using $\frac{P}{\rho gH}$ and $t\sqrt{\frac{g}{H}}$, respectively.

**Figure 4.**The computed L2 error for the grid study interpolated over the range of particle resolution shown for the different numerical dissipation schemes used for the dam break simulation. Different order lines have been provided to aid in the interpretation of the convergence rates for the dissipation schemes.

**Figure 5.**Comparison of the interpolated pressure data at the measurement location against the experimental measurements for the chosen numerical scheme for different Wendland kernels.

**Figure 6.**Pressure contour plots at different times for the dam-break simulations by using the Wendland kernels. The pressure and time values have been non-dimensionalized using $\frac{P}{\rho gH}$ and $t{\left(\frac{g}{H}\right)}^{0.5},$respectively. The times are 1.94, 4.52, 5.49, 5.98 and 7.12 units from top to bottom. The plot columns from left to right are for the WC2, WC4 and WC6 kernels, respectively.

**Figure 7.**Schematic of the simulation tank. (

**a**) Lateral sloshing. The pressure is determined at the location depicted on the left wall. (

**b**) Roof sloshing. The pressure is determined at the location depicted on the top of the tank. Sloshing motion is about the axis depicted in red at the bottom of the tank in (

**a**). All dimensions are given in mm.

**Figure 8.**(

**a**) Snapshot of the velocity profile for the lateral sloshing simulation at $t=2.95\mathrm{s}$ showing an overturning wave. (

**b**) Snapshot of the velocity field for the roof sloshing simulation at $t=2.22\mathrm{s}$ with high fluid particle velocities.

**Figure 9.**(

**a**) Comparison of the interpolated pressure data at the measurement location against the experimental measurements for different numerical dissipation schemes with the WC2 kernel. (

**b**) Comparison of the interpolated pressure data using the chosen numerical scheme for different Wendland kernels.

**Figure 10.**(

**a**) Comparison of the interpolated pressure data at the measurement location with the experimental measurements for different numerical dissipation schemes using WC6 kernel. (

**b**) Comparison of the interpolated pressure data using the chosen numerical scheme for different Wendland kernels.

**Figure 11.**Schematic of the numerical wave tank used for the simulations. The height of the piston type wave maker is 1.5 m.

**Figure 12.**The wave conditions for the monochromatic wave used plotted on the Le Mehaute abacus. A MATLAB function designed by Byford [52] has been used to create the plot with the depicted wave condition.

**Figure 13.**Comparison of the wave surface elevation at $x=6\mathrm{m}$ for the different dissipation schemes for the three Wendland kernels with the theoretical second-order Stoke’s solution.

**Figure 14.**Comparisons of errors in wave elevations obtained with the different numerical dissipation schemes for the three different Wendland kernels.

**Figure 15.**The computed L2 error for the grid study interpolated over the range of particle resolution shown for the different numerical dissipation schemes used for the wave tank experiment using Wendland C6 kernel. Different order lines are provided to aid in the interpretation of the convergence rates for the dissipation schemes.

**Figure 16.**(

**a**) Particle grid study with four different values of particle spacing for the three kernels. $\Delta x1=0.005\mathrm{m},\Delta x2=0.01\mathrm{m},\Delta x3=0.015\mathrm{m},\mathrm{and}\Delta x4=0.02\mathrm{m}$ (

**b**) The L2 error for the corresponding grid study interpolated over the range of particle resolution is shown.

**Figure 17.**Comparisons of the wave surface elevations for the different kernel functions at three of the wave probe locations used in the numerical experiments.

**Figure 18.**Construction of the strip for calculation of energies of the particles. The length of the strip has been exaggerated to aid in visualization.

**Figure 19.**Comparisons of the RMS values at the different probe locations with increasing distance from the wave-maker for the different Wendland kernels: (

**a**) potential energy, (

**b**) kinetic energy, (

**c**) internal energy, and (

**d**) total energy.

**Figure 20.**Comparisons over $x=2\mathrm{m}$ to $x=10\mathrm{m}$ of the wave tank with the implementations of the three Wendland kernels: (

**a**) potential energy, (

**b**) kinetic energy and (

**c**) total energy profiles.

$\mathbf{Fluid}\mathbf{Particle}\mathbf{Spacing}\left(\mathbf{\Delta}\mathit{x}\right)$ | H/100 |
---|---|

Boundary particle spacing | 0.6$\Delta x$ |

Smoothing length $h$ | 1.5$\Delta x$ |

${\rho}_{ref}$ | 1000 |

Artificial viscosity coefficient $\alpha $ | $\in \left[0,0.2\right]$ |

$\beta $ used in momentum equation | $\in \left[0,6.0\right]$ |

$\delta $—SPH coefficient | $\in \left[0,0.2\right]$ |

${v}_{max}$ | $\sqrt{gH}$ |

${c}_{0}$ | $20{v}_{max}$ |

$\nu $ used in Morris viscosity factor | 8.97 × 10^{−7} |

$\mathbf{Fluid}\mathbf{Particle}\mathbf{Spacing}\left(\mathbf{\Delta}\mathit{x}\right)$ | W/450 |
---|---|

Boundary particle spacing | 0.6$\Delta x$ |

Smoothing length $h$ | 1.5$\Delta x$ |

${\rho}_{ref}$ | 998 |

Artificial viscosity coefficient $\alpha $ | $\in \left[0,0.2\right]$ |

$\beta $ used in the momentum equation | $\in \left[0,4.5\right]$ |

$\delta $—SPH coefficient | $\in \left[0,0.3\right]$ |

${v}_{max}$ | $1.0$ |

${c}_{0}$ | $20{v}_{max}$ |

$\nu $ used in Morris viscosity factor | 8.97 × 10^{−7} |

Height H | 0.12 m |

Depth d | 0.66 m |

Wavelength L | 4.52 m |

Time Period T | 2 s |

Initial phase δ | 0 |

Steepness s | 0.026 |

Relative depth | 0.146 |

$\mathbf{Fluid}\mathbf{Particle}\mathbf{Spacing}\left(\mathbf{\Delta}\mathit{x}\right)$ | $\in \left\{2\times {10}^{-2}\mathbf{m},1.5\times {10}^{-2}\mathbf{m},1\times {10}^{-2}\mathbf{m},5\times {10}^{-3}\mathbf{m}\right\}$ |
---|---|

Boundary particle spacing | $\in \left\{0.7\Delta x,0.5\Delta x\right\}$ |

Smoothing length $h$ | $1.5\Delta x$ |

${\rho}_{ref}$ | 1000 |

Artificial viscosity coefficient $\alpha $ | $\in \left[0,0.01\right]$ |

$\beta $ used in momentum equation | $\in \left[0,2.0\right]$ |

$\delta $—SPH coefficient | $\in \left[0,0.1\right]$ |

${v}_{max}$ | $3.0$ |

${c}_{0}$ | $10{v}_{max}$ |

$\nu $ used in Morris viscosity factor |
8.97 × 10^{−7} |

Time step (Fixed time integration) | $1\times {10}^{-5}\mathrm{s}$ |

$\mathbf{Fluid}\mathbf{Particle}\mathbf{Spacing}(\mathbf{\Delta}\mathit{x}\mathbf{)}$ | $\left\{2\times {10}^{-2}\mathbf{m},1.5\times {10}^{-2}\mathbf{m},1\times {10}^{-2}\mathbf{m},5\times {10}^{-3}\mathbf{m}\right\}$ |
---|---|

Wendland C2 errors | {0.0176, 0.0162, 0.0156, 0.0155} |

Wendland C4 errors | {0.0177, 0.0155, 0.0160, 0.0156} |

Wendland C6 errors | {0.0195, 0.0165, 0.0164, 0.0159} |

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**MDPI and ACS Style**

Chakraborty, S.; Balachandran, B.
Wave Propagation Studies in Numerical Wave Tanks with Weakly Compressible Smoothed Particle Hydrodynamics. *J. Mar. Sci. Eng.* **2021**, *9*, 233.
https://doi.org/10.3390/jmse9020233

**AMA Style**

Chakraborty S, Balachandran B.
Wave Propagation Studies in Numerical Wave Tanks with Weakly Compressible Smoothed Particle Hydrodynamics. *Journal of Marine Science and Engineering*. 2021; 9(2):233.
https://doi.org/10.3390/jmse9020233

**Chicago/Turabian Style**

Chakraborty, Samarpan, and Balakumar Balachandran.
2021. "Wave Propagation Studies in Numerical Wave Tanks with Weakly Compressible Smoothed Particle Hydrodynamics" *Journal of Marine Science and Engineering* 9, no. 2: 233.
https://doi.org/10.3390/jmse9020233