Numerical Simulation of 2-D Solitary Wave Run-Up over Various Slopes Using a Particle-Based Method
Abstract
:1. Introduction
2. Numerical Methods
2.1. Governing Equations
2.2. Kernel Function
2.3. Gradient Model
2.4. Laplacian Model
2.5. Incompressibility Model with a Staggered Divergence-Free Model
2.6. Boundary Conditions
2.6.1. Free-Surface Boundary Condition
2.6.2. Wall Boundary Condition
2.7. Turbulence Model
3. Numerical Simulations and Discussion
3.1. Hydrostatic Pressure Test
3.2. Solitary Wave Run-Up (Constant Slope Angles)
3.3. Solitary Wave Run-Up (Varying Slope Angle)
4. Conclusions
Author Contributions
Funding
Conflicts of Interest
Nomenclature
, , , | coefficients for making target wave (-) |
wave-velocity (ms−1) | |
corrective matrix for gradient model (-) | |
Smagorinsky constant (-) | |
number of space dimensions (-) | |
external force for unit mass (ms−2) | |
, , , | centers of the east, west, north, and south control surfaces of particle (-) |
gravitational acceleration (ms−2) | |
water depth (m) | |
maximum wave height (m) | |
-direction component in the Cartesian coordination system (-) | |
turbulent kinetic energy (m2s−2) | |
wave-number (m−1) | |
or | particle index (-) |
particle size (m) | |
time step index (-) | |
fixed particle number density for the incompressibility (-) | |
particle number density of particle (-) | |
maximum number of neighboring particles in the initial distribution (-) | |
number of neighboring particles of particle (-) | |
pressure (Pa) | |
position vector between two particles (m) | |
effective radius for particle interactions (m) | |
, | position vector of particle and (m) |
relative position vector between particle and (m) | |
distance between particle and (m) | |
wave run-up (m) | |
strain rate (-) | |
stroke of wavemaker’s paddle (cm) | |
time (s) | |
velocity vector (ms−1) | |
temporary velocity components of particle (ms−1) | |
velocity correction of particle (ms−1) | |
Reynolds stress term for turbulent flows (m2s−2) | |
statistical volume of particle (-) | |
speed of wavemaker’s paddle (cms−1) | |
kernel function (-) | |
kernel function between particle and (-) | |
distance between particle and in the x-direction (m) | |
position of wave-maker at time (m) | |
distance between particle and in the y-direction (m) | |
height of wavemaker’s paddle (m) | |
, | parameters for detecting free surface (-) |
Kronecker’s delta (-) | |
particle-to-particle spacing (m) | |
difference of scalar quantities between particle and (m) | |
, | scalar quantities at particles and (-) |
blending parameter for incompressibility model (-) | |
parameter that makes the increase in variance equal to that of the analytical solution (-) | |
slope angle of bottom (°) | |
density (kgm−3) | |
kinematic viscosity (m2s−1) | |
eddy viscosity (m2/s−1) |
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L (m) | Z (m) | h (m) | θ (°) | l0 (m) | H/h (-) |
---|---|---|---|---|---|
1.5 | 0.3 | 0.1 | 90 | 0.00125, 0.0025, 0.005, 0.010 | 0.1, 0.2, 0.3, 0.4, 0.5,0.6 |
45 | 0.0025 | 0.05, 0.1, 0.2, 0.3, 0.4, 0.5 | |||
26 | 0.15, 0.163, 0.2, 0.25, 0.3, 0.35 |
Case | Breaking | Target Wave Height (-) | A1 (-) | A2 (-) | A3 (-) | A4 (-) |
---|---|---|---|---|---|---|
1 | ▲ | 0.3 | 12.85 | 2.93 | 9.42 | −12.85 |
2 | O | 0.7 | 19.63 | 4.368 | 12.25 | −19.63 |
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Jeong, S.-M.; Park, J.-I.; Park, J.-C. Numerical Simulation of 2-D Solitary Wave Run-Up over Various Slopes Using a Particle-Based Method. Water 2019, 11, 462. https://doi.org/10.3390/w11030462
Jeong S-M, Park J-I, Park J-C. Numerical Simulation of 2-D Solitary Wave Run-Up over Various Slopes Using a Particle-Based Method. Water. 2019; 11(3):462. https://doi.org/10.3390/w11030462
Chicago/Turabian StyleJeong, Se-Min, Ji-In Park, and Jong-Chun Park. 2019. "Numerical Simulation of 2-D Solitary Wave Run-Up over Various Slopes Using a Particle-Based Method" Water 11, no. 3: 462. https://doi.org/10.3390/w11030462