Exploring Dissipation Terms in the SPH Momentum Equation for Wave Breaking on a Vertical Pile
Abstract
1. Introduction
2. Methodology: Smoothed Particle Hydrodynamics (SPH)
2.1. Fundamentals of SPH
2.1.1. SPH Governing Equations
2.1.2. Dissipation and Turbulence Modeling
2.1.3. Equation of State
2.2. Boundary Conditions
2.2.1. Modified Dynamic Boundary Conditions
2.2.2. Open Boundary Conditions
3. Case Study: Large-Scale Experiments by Irschik et al. [46]
3.1. Experimental Setup and Wave Conditions
3.2. Numerical Setup
4. Model Validation Against Experimental Data
4.1. Evolution of Wave Breaking and Breaking Point
4.2. Free-Surface Elevation
4.3. Slamming Force Prediction
Comparison of Slamming Forces with Li & Fuhrman [15]
5. Velocity Field and Turbulence Characteristics
5.1. Velocity Field Decomposition
- is the time-averaged velocity over several wave periods;
- is the phase-averaged velocity, representing the coherent, repeatable component associated with the wave motion;
- is the turbulent (incoherent) fluctuation, defined as the deviation from the phase-averaged field.
5.2. Turbulent Kinetic Energy Distribution
5.3. Coherent Structures
6. Discussion of Model Performance and Limitations
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations and Nomenclature
Abbreviations
AV | Artificial Viscosity |
CFL | Courant Number |
DBC | Dynamic Boundary Conditions |
DNS | Direct Numerical Simulation |
EARSM | Explicit Algebraic Reynolds Stress Models |
EMD | Empirical Mode Decomposition |
ISPH | Incompressible Smoothed Particle Hydrodynamics |
LANS | Lagrangian Averaged Navier–Stokes |
LES | Large Eddy Simulation |
mDBC | Modified Dynamic Boundary Conditions |
OWT | Offshore Wind Turbines |
RANS | Reynolds-Averaged Navier–Stokes |
SPH | Smoothed Particle Hydrodynamics |
SPS | Sub-Particle Scale |
Nomenclature
Symbol | Definition |
Coefficient for artificial viscosity | |
Gravitational acceleration vector | |
h | Smoothing length in SPH |
k | Turbulent kinetic energy |
Q | Q-criterion value () |
Particle position vector | |
Strain-rate tensor | |
Instantaneous velocity field | |
Turbulent (incoherent) velocity fluctuation | |
Phase-averaged (coherent) velocity component | |
Time-averaged (mean) velocity | |
Vorticity tensor (antisymmetric part of velocity gradient) | |
Kinematic (molecular) viscosity | |
Eddy (turbulent) viscosity | |
Fluid density | |
Reference fluid density | |
t | Time (s) |
T | Wave period (s) |
Initial particle spacing in SPH (m) | |
N | Number of wave cycles used for phase-averaging |
Filter size or sub-particle length scale | |
Spatial coordinates |
Appendix A. Influence of Artificial Viscosity
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Altomare, C.; Li, Y.P.; Tafuni, A. Exploring Dissipation Terms in the SPH Momentum Equation for Wave Breaking on a Vertical Pile. J. Mar. Sci. Eng. 2025, 13, 1005. https://doi.org/10.3390/jmse13061005
Altomare C, Li YP, Tafuni A. Exploring Dissipation Terms in the SPH Momentum Equation for Wave Breaking on a Vertical Pile. Journal of Marine Science and Engineering. 2025; 13(6):1005. https://doi.org/10.3390/jmse13061005
Chicago/Turabian StyleAltomare, Corrado, Yuzhu Pearl Li, and Angelantonio Tafuni. 2025. "Exploring Dissipation Terms in the SPH Momentum Equation for Wave Breaking on a Vertical Pile" Journal of Marine Science and Engineering 13, no. 6: 1005. https://doi.org/10.3390/jmse13061005
APA StyleAltomare, C., Li, Y. P., & Tafuni, A. (2025). Exploring Dissipation Terms in the SPH Momentum Equation for Wave Breaking on a Vertical Pile. Journal of Marine Science and Engineering, 13(6), 1005. https://doi.org/10.3390/jmse13061005