(Download full test case data files here: SPHERIC_TestCase13.zip)

Introduction

Perforated wall-caisson breakwaters are coastal structures widely used in harbours and port areas with the aim to dissipate incident wave energy, allowing for safe navigation conditions during sea storms. In recent years, the increasing number of installation of perforated breakwaters in port areas addressed the interest of the scientific community to a deeper understanding of the dissipation process occurring inside the chamber of the breakwaters and the related analysis of wave reflection. Moreover, the analysis of the dynamic pressures at the walls is fundamental in terms of stability analysis. The SPH offers a further insight for an engineering problem whose dimensioning is nowadays essentially based on approximated empirical formulas.

Flow phenomena

The dissipation process occurring in the non-reflective cells of the perforated breakwater depends on the geometrical configuration of the caisson and on the incident wave characteristics. The fluid dynamics inside the chamber of the caisson, induced by the wave action, is a complex phenomenon in which the pressure gradients between the outside and the inside of the structure regulates the flow evolution.

Geometry

Two geometries of perforated breakwaters, characterized by a top cover plate over the free surface on an internal chamber (see Fig. 1), are analysed. Both analyses are performed considering water depth d = 0.4 m, while d’ = 0.2 m. The length of the numerical wave flume is L = 4 m. Breakwater n°1 The first breakwater is perforated by two holes at the front face, with B = 0.2 m, b’ = 0.08 m, s = 0.038 m, h = 0.024 m, b = 0.096 m. The incident wave train is regular with H = 0.08 m and T = 1.2 s. Breakwater n°2 The second breakwater is perforated by three holes at the front face, with B = 0.15 m, b’ = 0.16 m, s = 0.038 m, h = 0.024 m, b = 0.096 m. The incident wave train is regular with H = 0.1 m and T = 1 s.

Figure 1. Breakwaters sketch (Figure only).

Boundary conditions

No-slip solid boundary conditions are enforced. The wave generation is obtained by implementing the left wall of the wave flume with a sinusoidal time law.

Initial conditions

The fluid particles are initialized with a hydrostatic pressure distribution, while the initial velocity field is set equal to zero everywhere in the domain.

Discretisation

A spatial resolution dx = 0.0035 m is adopted. The specification of the initial position of the fluid and solid particles is given in the files: “Initial_particles_distribution_breakwater_n1.dat” “Initial_particles_distribution_breakwater_n2.dat” Column 1: X position (m) Column 2: Z position (m) Column 3: horizontal velocity (m/s) Column 4: vertical velocity (m/s) Column 5: density (kg/m^3) Column 6: pressure (Pa) Column 7: def = 1 for fluid particles, def = 2 for solid particles

Results specification

Results to be returned for this test are the spatial dynamic pressure distribution at the walls of the breakwater (front face, front inner wall, rear inner wall). The wave considered for the analysis is the first wave impacting on the wall (after the initial wave if the wavemaker motion presents a transient initial law to avoid shock waves in the channel). The spatial distributions of dynamic pressures at the front wall and at the internal walls of the chamber, are analyzed when the maximum pressure induced by the wave crest within the regular wave train appears in correspondence to the SWL. Is noticed that the pressure peaks at the three reference walls appear at different time instants. Their phase shift is dependent on the wave celerity and the width of the chamber.

Results format

The experimental result of the spatial distribution of the dynamic pressures (obtained by Chen et al., 2007) are presented in the files: “Delta_P_EXPERIMENTAL_breakwater_n1.dat” “Delta_P_EXPERIMENTAL_breakwater_n2.dat” Column 1: Z position of the pressure gauge (m) Column 2: Dynamic pressure (kPa) The SPH results of the spatial distribution of the dynamic pressures, measured in this case for all the points along the depth, equispaced with a distance dx, are given in the files “Delta_P_SPH_breakwater_n1.txt” and “Delta_P_SPH_breakwater_n2.txt”. The results refer to the coupled diffusive formulation (green line in Fig. 2, for more information see Aristodemo et al., 2015): Column 1: Z position of the pressure gauge (m) Column 2: Dynamic pressure (kPa)

Benchmark results

For more specification on the benchmark results see Chen et al., 2007. For more specifications on the numerical results see Aristodemo et al., 2015. Figs. 2 and 3 the frames of the simulations with pressure field and velocity vectors are shown for the case of the breakwater no. 1 and no. 2, respectively (Figures only).

Figure 2. Frames of the simulation with pressure distribution and velocity vectors for the breakwater geometry no. 1. Left plot: wave crest at the front wall. Right plot: wave trough at the front wall.

Figure 3. Frames of the simulation with pressure distribution and velocity vectors for the breakwater geometry no. 2. Left plot: wave crest at the front wall. Right plot: wave trough at the front wall.

The spatial distributions of dynamic pressures at the front wall and at the internal walls of the chamber, are shown in Fig. 4 and 5. Positive dynamic pressures are displayed on the external side of the walls and negative ones on their internal side.

Figure 4. Spatial distribution of wave pressures at perforated breakwater no. 1: Experiments vs SPH.

Figure 5. Spatial distribution of wave pressures at perforated breakwater no. 2: Experiments vs SPH.

Publications using this test case as a benchmark

• Aristodemo, F., D.D. Meringolo, P. Groenenboom, A. Lo Schiavo, P. Veltri, M. Veltri, “Assessment of dynamic pressures at vertical and perforated breakwaters through diffusive SPH schemes”. Mathematical Problems in Engineering, Article ID 305028, vol. 10 pages, 2015.

• Meringolo, D.D., 2016. Weakly-Compressible SPH Modeling of Fluid-Structure Interaction Problems (Ph.D. thesis). Università della Calabria, Cosenza, Italy.

References

• Chen, X. F., Y. C. Li, and B. L. Teng, “Numerical and simplified methods for the calculation of the total horizontal wave force on a perforated caisson with a top cover,” Coastal Engineering, vol. 54, no. 1, pp. 67–75, 2007.

• Meringolo, D.D. F. Aristodemo, P. Veltri, “SPH numerical modeling of wave-perforated breakwater interaction”, Coastal Engineering vol. 101 pgs. 48-68, 2015.

• Antuono, M., Colagrossi, A., Marrone, S., Molteni, D. Free-surface flows solved by means of SPH schemes with numerical diffusive terms. Computer Physics Communications, 181:532–549, 2010.

• Molteni, D., Colagrossi, A. A simple procedure to improve the pressure evaluation in hydrodynamic context using the SPH. Computer Physics Communications, 180:861–872, 2009.