1. Introduction
The implementation of an armor unit or artificial block [
1] on rubble mound breakwaters is a typical approach for wave mitigation and ocean disaster prevention. It is of great scientific significance and practical engineering value to comprehend the hydrodynamic characteristics of the armor unit under wave force and harness its wave damping effect. Numerical simulation methods, not limited by time and space, to unravel this armor unit’s performance have been widely applied in coastal engineering around the world, and a large number of numerical methods for the interaction between waves and coastal structures have been developed, including the Boundary Element Method [
2,
3], the Finite Volume Method [
4,
5], the Arbitrary Lagrangian Eulerian Method [
6,
7], the Finite Element Method [
8,
9], and also particle-based methods, such as, Moving Particle Semi-implicit method (MPS) [
10] and Smoothed-Particle-Hydrodynamics (SPH) [
11]. Open source code including OpenFOAM [
12], SWASH [
13] and SPHysics [
14], have been extensively investigated and adopted during the last several decades.
Among them, the pure particle method that is based on SPH [
11], not constrained by the fixed grid, is feasible for solving large deformation free surface flow problems [
15] such as wave breaking [
16], liquid splashing [
17], the interaction between waves and porous media [
18], and other fields that the traditional grid simulation methods cannot accurately describe. When SPH was created, the simulation of the real sea state and wave conditions was restricted by limited computer resources. Then, due to algorithm improvements, such as the use of graphics processors, the calculation efficiency has been substantially improved [
19]. It began to be widely used in coastal engineering practices [
20,
21,
22,
23], making it feasible to simulate a wave overtopping scenario with a duration of 1000 irregular wave cycles under real sea conditions [
24].
Since these improvements, the SPH has been applied to a variety of uses. Dominguez et al. [
25] used DualSPHysics and MoorDyn [
26], a coupled model, to simulate the dynamic response of the anchored floating structure under the regular wave forcing, which would been further applied in security assessments of offshore structures such as floating offshore wind turbines, offshore oil platforms, and wave energy converters under real sea conditions. Additionally, SPH has been coupled with the Finite Element Method to investigate the wave-debris-structure interaction for the risk assessment of coastal structures during extreme tsunami events [
27]. Crespo et al. [
28] used SPH to explore the interaction between waves and floating oscillating water column wave energy converters, but due to the lack of an effective wave-making and wave-absorbing systems, it was limited in practical operation [
29]. Meringolo et al. [
30] simulated the interaction between waves and permeable caisson breakwaters based on a two-dimensional SPH model with artificial viscous terms. The simulation results showed the reduction of wave pressure at the sea wall holes, and revealed that the empirical formula [
31] inaccurately estimated the wave pressure previously due to neglect the existence of sea wall holes. Monaghan et al. [
32] and Dalrymple et al. [
21] studied the propagation of solitary waves on the beach, and the climbing and overtopping of waves on breakwaters with different roughness, but they did not consider the effect of an armor block. Altomare et al. [
33] used the DualSPHysics [
19,
34] open-source code for the first time to gain insights for the interaction between waves and a 3D breakwater with armor units, and their numerical model results are in good agreement with the measurement data and empirical formula.
The use of armor blocks, including Accropode
®, Xbloc
®, and Core-Loc™ has been a classical solution for coastal structure construction [
1]. There has been a sustaining effect for new type of armor block development [
35,
36,
37] to achieve better wave damping and low consumption of concrete. Suffice it to say, there is considerable uncertainty in how to effectively protect the coastal structure when facing of longer and larger waves in further deeper coastal waters where more construction has taken place, and this coastal structure is more vulnerable in confronting ocean disasters that are associated with more concern for global warming [
38].
High porosity block is promising in coping with this long period wave. The interactions between the armor blocks and the waves on breakwater could be categorized as the interaction of a wave and solid objects, which has been extensively studied numerically [
39,
40]. The wave run-up and wave overtopping of various blocks under different breakwater structures are quite different, and the empirical formulas that are given by standards could hardly cover all of the structures and blocks, which limits its application. Therefore, it is still necessary to develop new types of amour units and evaluate their function.
Massive physical experiment scenarios would be required for the size parametric selection before the generation of a new type of armor unit, which is largely inefficient. Meanwhile, through the establishment of a numerical model that describes the wave-breakwater-armor-unit interaction could be a solid, low-cost, and efficient tool for abundant parametric selection scenarios under various wave input conditions.
The chief objective of this paper is to optimize the size of a new type of armor block and to gain insight into its wave damping effects. To achieve that, a wave-breakwater-block numerical model that was based on the open-source code DualSPHysics was established to simulate the climbing and overtopping of regular waves on a rubble mound breakwater. First, the numerical results are verified with the results of the physical model test. After that, with this verified model, sensitivity studies are conducted to evaluate the effect of blocks on wave damping with a diversity of design sizes to provide the optimal size. After optimization, the hydrodynamic characteristics of the new artificial block are further explored and a wave run-up empirical formula for this block is provided. Finally, the conclusion and research limitations are given.
5. Conclusions
Based on the open-source code DualSPHysics, a numerical investigation has been conducted for the parameter design and the understanding of the wave damping effects and hydrodynamic characteristics of one new type of armor block, TB-CUBE. The main conclusions are summarized as follows:
(1) The three-dimensional, SPH-based active absorption wave-making flume has been constructed to reproduce wave evolution on rubble mound breakwaters and to obtain the optimal wave damping effects under various design combinations. The results have proven that when the radius of the sphere is constant, both the wave run-up and reflection coefficient shrink with the increase of the cylinder size, and the overtopping rises. The amount of wave run-up and overtopping decreases with the increase of the size of the sphere under invariant radius of the sphere.
Given the factors of wave run-up, overtopping, wave reflection, porosity, and concrete consumption, the TB-CUBE block with R1 and R2 values of 0.475 h and 0.315 h, respectively, outperforms others, and thus has been selected for further study.
(2) The mean roughness coefficient that was given by numerical simulations for this optimized TB-CUBE block is 0.462, indicating a promising capability of wave damping. It could also be viewed from the three-dimensional visualization that the structural characteristics of the TB-CUBE block itself have exerted large impacts on the interaction between the wave and the breakwater. It can accommodate the water body and form a turbulent flow effect. The threshold effect makes the quantity of overtopping decrease.
(3) The results from the wave-block-breakwater model demonstrate that the wave run-up value increases along with the increase of wave height, displaying a unimodal distribution trend with the growth of slope, and rises with the increase of inverse wave steepness. The empirical formula for wave run-up on this TB-CUBE has been provided, which would be further employed in engineering practice.
(4) This study has paid attention to the interaction of non-breaking waves on slope breakwater with new type of armor unit, the TB-CUBE, in a 2D wave flume experiment and numerical investigation. Previous studies on unbroken and broken solitary waves highlighted that the interaction of these two types of waves with a coastal deck and the associated hydrodynamic effects were totally different, with the broken waves being more turbulent and introducing more impulsive effects [
46]. It is reported that the “aleatory uncertainty” that is involved in the wave breaking and the wave loads on coastal structures, since a minor change in the wave height offshore (e.g., 1%) can lead to a variability of 35% in the pressures and forces that are applied to coastal structures [
47]. In fact, this uncertainty is one of the main parameters that is related to the impact of broken waves on structures that is hard to capture via numerical simulations [
47]. Further study should consider the effects of wave breaking effects on the stability of the armor unit on slope breakwater. In addition, kinetic energy and velocity or momentum flux of the flow that overtops the breakwater variations are valuable information for this process.
This study used normal incidence of waves. However, the dynamic process of the interaction between breakwater and incoming oblique waves is inherently a three-dimensional problem. Previous research also reported the reduction in overtopping when wave is oblique [
48,
49]. However, it is still necessary to evaluate the inundation process of the structure and the applied hydrodynamic loads, which includes also yaw and roll moments that further increases the risk for structural damage [
9].
Our research is from a small scale (1:40) wave flume experiment which is a common experimental scale as other studies [
50,
51,
52], and scale effects exist. It is well-accepted that solitary waves normally decay faster in the model than in the prototype due to boundary layer effects and fluid viscosity [
53]. Full scale research that is not limited in wave-breakwater-armor-unit interaction has been reported in literature [
54,
55,
56], and more numerical studies that are based on that would be feasible to be performed in the future.