2D water-wave interaction with permeable and impermeable slopes: Dimensional analysis and experimental overview
Introduction
The main function of breakwaters is to protect harbors and coastal structures from wave action. They are an important type of coastal and port infrastructure because of their functionality as well as their cost, design complexity, and environmental and socioeconomic impacts. The conception, design, and verification of a breakwater mainly depends on the slope of the sea bottom, water depth at the breakwater toe, h, and the characteristic values of incident waves H, T, θ (height, period, and incidence angle). They also depend on the available materials, construction and repair techniques, and the consequences that ensue when and if objectives are not attained. The performance of the breakwater against wind waves is mainly determined by the slope on both sides of the breakwater, shape and weight of the unit pieces, number of armor layers, thickness and emplacement of the main layer/secondary layers and the width, crest elevation, and size of the core materials (ROM 0.0, 2001; ROM 1.1, 2019).
Battjes (1974) proposed using the Iribarren number (Iribarren and Nogales, 1949), , as the dynamic similarity parameter to analyze wave train behavior on an indefinite impermeable flat slope, slope angle α, where L is the characteristic wavelength. He also conjectured that the value of Ir identified breaker type as spilling, plunging, collapsing, or surging (Iversen, 1952; Galvin, 1968). Furthermore, he anticipated its capacity to determine the phase difference and wavebreaking index, wave run-up and run-down, mean level, and the reflection and dissipation (absorption) of the waves on the breakwater slope.
In the field of harbor and maritime structures (and also beach morphodynamics), the seminal work of Battjes (1974) led to research whose objective was to determine the transformation of incident energy when waves interacted with the breakwater by means of the reflected energy coefficient, KR2, transmitted energy coefficient, KT2, and the bulk dissipation rate, D*. Still another objective was to develop formulas for wave run-up, run-down, overtopping and stability of the main armor layer in the domain of interest, Ir > 1.5, as reflected in the following references, among others: Bruun and Günbak (1976); Losada and Gimenez-Curto (1981); Seelig and Ahrens (1981); Allsop and Channell (1989); Martin et al. (1999); Zanuttigh and van der Meer (2008); Burcharth et al. (2010); Van Der Meer (2011); Gómez-Martín and Medina (2014); Vílchez et al. (2016a).
These studies show that in the domain, Ir > 1.5, the Iribarren number reveals the general tendency of coefficients [KR2, KT2]. However, the values tend to scatter as the value of Ir increases, depending on the slope angle. Energy transmission at a non-overtopped breakwater is usually small, KT2 < 0.15, but this information is necessary in order to evaluate the bulk dissipation at the structure. Nonetheless, there are relatively few articles on the calculation of wave dissipation and is still an open quention. Such studies include the following, among others: Seelig and Ahrens (1981); Kobayashi and Wurjanto (1992); Pérez-Romero et al. (2009); Van Gent (2013); and Vílchez et al. (2016b).
Forty years after Battjes (1974), physical experiments on breakwaters are still based on the working hypothesis that the Iribarren number is a dynamic similarity parameter between model and prototype, and that, generally speaking, the Iribarren number is the main variable in formulas that determine the wave energy transformation coefficients [KR2, KT2, D*] for a breakwater and related hydrodynamic performance.
Over the last thirty years, research studies have questioned the dependence of KR2 = f(Ir). Hughes and Fowler, 1995, and Sutherland and O'Donoghue (1998) applied the parameter xm/L (where xm = h/) to also quantify the phase of the reflected wave train. Davidson et al. (1996) defined a reflection number that includes Ir and the characteristic diameter of the armor layer. Van der Meer (1988, 1992) incorporated a permeability parameter and fit the exponents of Ir by means of a multiple regression analysis. Benedicto (2004) analyzed wave reflection, depending on h/L and grain size. Finally, Vílchez et al. (2016a) modifies the Iribarren number to incorporate, grain diameter, and the width and depth of the breakwater core in a single parameter.
The main objectives of this research were the following: (1) to collate the dependence of wave energy transformation processes (reflection, transmission, and bulk dissipation rate) with Iribarren number; (2) to apply dimensional analysis to the design of experiments for both a permeable and impermeable slope; and (3) to analyze the variability of the results and identify those characterized by the hydrodynamic performance of the breakwater. This study involved numerical experiments using an undefined, impermeable, rigid slope and the application of the IH-2VOF model (Lara et al., 2008). These were combined with laboratory experiments in the 2D flume of a non-overtoppable mound breakwater with a cube layer and porous core of finite width. Linear wave theory was applied to separate the incident, reflected, and transmitted time series of the data records of the vertical displacement of the free surface at different points in the experimental setup. The wave energy conservation equation was applied to obtain the bulk dissipation rate on the breakwater.
The rest of this paper is organized as follows. Section 2 describes the theoretical background to understand the main aspects of the physical processes that intervene in the water-wave interaction with the breakwater. Section 3 presents the experimental design, analysis and setups (numerical and physical) performed in this study and presented in terms of Ir. Section 4 presents the numerical and physical results obtained. The results are initially represented, depending on the Iribarren number, after which they are displayed, depending on the dimensionless quantities obtained when the dimensional analysis was applied. Section 5 discusses the results, evaluates their validity in reference to the hypothesis and the experimental desviation obtained. Section 6 presents the conclusions derived from this research. Finally, the revised dimensional analysis of this study is outlined in Appendix A.
Section snippets
Background
The dissipation of a wave train on a breakwater slope is mainly caused by the generation, transport, and dissipation of turbulence during the following processes: (i) wave evolution and eventually wavebreaking on the free surface of the slope; (ii) interaction (circulation and friction) with the main armor layer; (iii) wave propagation through the secondary layers and porous core; and (iv) wave transmission leewards of the structure.
Experimental design and setup
This research study is based on the following: (a) numerical experiments, using the IH-2VOF model (Lara et al., 2008), on a mound breakwater with an undefined, impermeable, rigid slope; (b) physical experiments using a wave flume, on a mound breakwater with a non-overtoppable permeable constant slope, a cube armor layer and a porous core of constant finite width, B*, and grain size, D50,p. The flume bottom is horizontal and the water depth in the wave generation zone and in the flume up to the
Results
This section presents the results of the wave transformation coefficients [] for the two configurations: (impermeable and permeable) non-overtoppable mound breakwater.
Discussion
This section discusses the aspects of the function fitted to the data and the experimental deviation technique that could have influenced or conditioned the results presented in Section 3.
Conclusions
The main objective of this research is to characterize and quantify the prevalent physical processes in the energy transformation of a regular wave train while interacting with permeable or impermeable breakwaters. Two sources of experimental data are considered: (a) numerical experiments on an undefined impermeable rigid slope with the numerical model (IH–2VOF), and (b) physical experiments on a non-overtoppable permeable breakwater with a cube armor layer and a porous core of finite width in
CRediT authorship contribution statement
Pilar Díaz-Carrasco: Conceptualization, Funding acquisition, Formal analysis, Data curation, Writing - original draft, Writing - review & editing. Mª Victoria Moragues: Conceptualization, Funding acquisition, Formal analysis, Data curation, Writing - original draft, Writing - review & editing. María Clavero: Conceptualization, Funding acquisition, Formal analysis, Data curation, Writing - original draft, Writing - review & editing. Miguel Á. Losada: Conceptualization, Funding acquisition,
Acknowledgements
This work was supported by the research group TEP-209 (Junta de Andalucía) and by two projects: (1) “Protection of coastal urban fronts against global warming – PROTOCOL” (917PTE0538) and (2) “Integrated verification of the hydrodynamic and structural behavior of a breakwater and its implications on the investment project – VIVALDI” (BIA2015-65598-P). The work of the first author was funded by the Spanish Ministry of Education, Culture and Sports (Research Contract FPU14/03570). The second
References (51)
- et al.
Numerical analysis of run-up oscillations under dissipative conditions
Coast. Eng.
(2014) - et al.
On the one-dimensional steady and unsteady porous flow equations
Coast. Eng.
(1995) - et al.
Large eddy simulation of breaking waves
Coast. Eng.
(2001) - et al.
A new non-dimensional number for the analysis of wave reflection from rubble mound Breakwaters
Coast. Eng.
(1996) - et al.
Flow under standing waves: Part 1. Shear stress distribution, energy flux and steady streaming
Coast. Eng.
(2009) - et al.
Pressure-induced forces and shear stresses on rubble mound breakwater armour layers in regular waves
Coast. Eng.
(2014) - et al.
Investigations on the porous media equations and resistance coefficients for coastal structures
Coast. Eng.
(2014) - et al.
Wave interaction with low-mound breakwaters using a RANS model
Ocean Eng.
(2008) - et al.
Flow characteristics on rough, permeable slopes under wave action
Coast. Eng.
(1981) - et al.
Run-up of tsunamis and long waves in terms of surf-similarity
Coast. Eng.
(2008)