Wave transformation through flushing culverts operating at seawater level in coastal structures

Ocean Engineering 89 (2014) 211–229

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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng

Wave transformation through flushing culverts operating at seawater level in coastal structures V.K. Tsoukala a,n, V. Katsardi a,1, K.A. Belibassakis b a b

Laboratory of Harbor Works, School of Civil Engineering, National Technical University of Athens, Zografos, Athens, Greece School of Naval Architecture and Marine Engineering, National Technical University of Athens, Athens, Greece

art ic l e i nf o

a b s t r a c t

Article history: Received 30 November 2013 Accepted 13 August 2014 Available online 6 September 2014

The placement of flushing culverts in breakwaters is a simple way to counteract a decline in water quality in harbour basins. A series of 63 experiments, which were conducted in a physical model of a breakwater in a 2D wave flume, are used to investigate the effect of the wave properties as well as the geometrical characteristics of the flushing culvert placed on breakwaters, in the temporal water surface profiles, the harmonic generation and the transmission coefficient. It is shown (i) that the harmonic generation downwave of the structure is more intense when wave nonlinearity increases; (ii) the harmonic generation and the transmission coefficient is mostly affected by the culvert's dimensions, especially the culvert's width as it is associated with the energy transmitted to the lee side of the structure and with diffraction. A 2D coupled-mode system (CMS) model is applied for the numerical simulation of waves propagating through flushing culverts at seawater level that are never perfectly filled with water. Good comparisons with the experimental results for linear and weakly non-linear waves and wider culverts are shown; proving the usefulness of the CMS model in calculating the effectiveness of a flushing culvert in an every-day basis of harbour function. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Flushing culverts Experimental studies Coupled-mode system model Wave transformations Transmission coefficient

1. Introduction In harbour design water quality within a basin must be considered being particularly important for health and environmental purposes, especially in warmer climates where biological processes are accelerated. Successful control of water quality is usually dependent upon periodic exchange of the harbour basin water with the sea water of the open sea (Dunham and Finn, 2002; U. S. Army Corps of Engineers, 2002). A common and economic method to improve the water quality in harbour basins is the construction of flushing culverts (hereafter referred as FCs) at characteristic sites of external harbour structures, which enhance water circulation through the amplification of the velocity field inside the basin and consequently contribute to the reduction of the renewal times and water quality improvement (see e.g. Stamou et al., 2004). They are mainly constructed in protective structures, i.e., breakwaters (Papaioannou et al., 1999; Tsoukala et al., 2006). Underwater placement of culverts is standard internationally, where tidal hydrodynamics is the main mechanism for enhanced flushing. However, in regions, as the Mediterranean, n

Corresponding author. Tel.: þ 30 2107722372. E-mail addresses: [email protected] (V.K. Tsoukala), [email protected] (V. Katsardi), kbel@fluid.mech.ntua.gr (K.A. Belibassakis). 1 Now in Civil Engineering Department, University of Thessaly, Volos, Greece. http://dx.doi.org/10.1016/j.oceaneng.2014.08.009 0029-8018/& 2014 Elsevier Ltd. All rights reserved.

where the ranges of the tides are low, it is preferred to construct the culverts with their longitudinal axis at sea water level (Tsoukala and Moutzouris, 2009). The evolution of a surface water wave that propagates up a breakwater slope and through an FC at the seawater level (Fig. 1) is an interesting yet very complex physical phenomenon. In the vertical plane, it involves very sudden and large changes in the water depth leading to severe wave transformations including energy dissipation through friction as the wave propagates on the breakwater slope as well as inside the FC, wave reflection in both sides of the FC, wave shoaling, possible wave breaking and thus possible wave induced flow, wave diffraction at the exit of the FC due to the sudden change in water depth, harmonic generation as the surface wave passes through the FC, and additional complex phenomena when the wave's maximum crest elevation hits the top border of the FC. On the other hand, in the transverse direction, when the surface wave meets the FC, it is forced to propagate through a finite width as opposed to the open sea. This leads to less proportion of wave energy being transferred in the lee side of the breakwater, to more energy dissipation inside the FC and to wave diffraction at the exit on re-entering deep water; the magnitude of all the above being proportional to the culvert's width. This paper focuses on the effective design of FCs in breakwaters constructed in low tide areas. The FC design has to be associated with the everyday function of the harbour and hence

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Fig. 1. Sketch of the breakwater's side and 3D detail of the flushing culvert showing basic notation.

naturally with waves of small amplitude and not with the low probability of occurrence of large waves incident to the breakwater. In effect, the waves considered in this study do not interact with the top border of the FC, providing no additional effects due to relevant reflections. One possible measure concerning the effectiveness of a flushing culvert is the wave transmission coefficient, Kt, and it is used as one of the main non-dimensional parameters concerning the water quality within the harbour basin (Tsoukala and Moutzouris, 2009). It is defined as the ratio of the transmitted wave height through the FC to the incident wave height. It is correlated with the wave characteristics (wave height and wave period), the geometrical characteristics (height, hf, width bf and length lf) of the FC and the water depth, h (see notation defined in Fig. 1). Dimensional, parametric, and regression analysis has been used in order to define the wave transmission coefficient Kt (Tsoukala and Moutzouris, 2008) investigating a number of experimental results in a 3D wave basin (Tsoukala et al., 2006) as well as in a 2D wave flume (Tsoukala et al., 2010) involving regular incident waves that propagate normal to trapezoidal breakwaters with FCs. In the above studies it is clear that increased transmission coefficients are associated with large ratios of culvert height and width with respect to the wave height H (hf =H and bf =H respectively), related to the increased proportion of energy being transferred in the lee side of the openings, and small ratios of culvert length with respect to the wave height ðlf =HÞ, related to small energy losses within the FC. In contrary, regular waves with increasing wave steepness (H=λ, with λ representing the wavelength) lead to smaller transmission coefficients, mainly due to wave breaking. Furthermore, several numerical methods have been examined in Stamou et al. (2004) and Michalopoulou et al. (2008) to simulate the wave transmission through FCs, based on the linear threedimensional mass continuity and momentum equations where the field is divided in layers of constant width. This method is adequate for problems of constant flow such as wave-induced circulation and hence is not the most appropriate in describing the wave characteristics downstream of a flushing culvert. In the case of a wide FC with normal wave incidence, the system could be considered to be an extreme type of a submerged breakwater characterised by severely abrupt changes in water depth. The latter is because the ratio of the water depth in the FC to the water depth on the sea side hs =h, with hs ¼ hf =2 representing the depth over the breakwater crest (Fig. 1), is smaller than the usual relevant ratios in studies about submerged breakwaters. Moreover, wave propagation through FCs involve sudden changes in the available width of propagation, finding similarities in propagation through

entrances in harbours and/or through series of breakwaters entailing wave diffraction. In this connection, Dattatri et al. (1978) made a series of experimental measurements and found that the most important parameters affecting the transmission coefficient are the relative crest length ðlf =λÞ, where the longer crests are associated with smaller transmission coefficients, Kt, and the ratio of the water depth on the breakwater crest to the incident water depth ðhs =hÞ, where larger ratios are associated with increased transmission, as it is naturally expected. On the topic of harmonic generation the interruption of the wave propagation process by a coastal structure, such as a submerged breakwater, produces flow perturbation, and consequently harmonic generation and energy transfers, which have important effects on the hydrodynamics at the lee side of the structure (Johnson et al., 1951; Mei and Black, 1969; Longuet-Higgins, 1977; Ohyama et al., 1995; Losada et al., 1997; Goda et al., 1999). When waves approach the breakwater containing the FC, part of the wave energy is reflected by the structure and this process is mainly linear regardless of the steepness of the wave field (Driscoll et al., 1992; Christou et al., 2008). Also, a part of the wave energy is dissipated (through friction and/or wave breaking). However, at the breakwater crest, which is the equivalent bottom of the culvert, harmonic generation with energy transfers to higher harmonics is observed and is transmitted to deeper water behind the breakwater as free waves. A second-order theory was developed by Massel (1983) where the second-order potential was linearly decomposed by using the wave steepness as the perturbation parameter. This theory has been extended by Belibassakis and Athanassoulis (2002) to treat weakly non-linear waves propagating over general bottom profiles, and is shown to accurately predict harmonic generation up to second order. This paper is part of an ongoing research relating to wave propagation through flushing culverts. In effect, motivation for this study was given by the experimental results in a 3D wave basin (Tsoukala and Moutzouris, 2009) that involved physical models of existing harbours. Although the latter lead to very interesting results, the parameters associated with the problem were many and could not be easily isolated. Moreover, the experimental scale was relatively small leading to correspondingly small openings and questions have risen concerning the increased viscous effects in the culverts walls and whether these lead to underestimated transmission coefficients. Similar issues were pointed out in the experimental set-up presented in Lara et al. (2006). Therefore, standard Froude scaling has been applied to the experiments presented in this work, paying special attention to the minimum dimensions of the FC in order to avoid excessive dissipation effects (Figs. 1 and 2).

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Due to the complexity of the problem, as explained earlier, the various parameters that involve the transformations of the waves propagating through FCs are treated in a series of separate papers. In the present paper, the parameters that are isolated are specifically the different geometrical characteristics of the FC and the wave characteristics. This is accomplished by gathering several experimental measurements that have been conducted in a 2D flume in the Laboratory of Harbour Works of National Technical University of Athens (LHW-NTUA) involving regular waves. Harmonic generation is discussed and compared with some of the aforementioned studies that involve submerged breakwaters pointing out the differences that occur due to the change in the propagation's width. However, this paper does not aim to give answers to all the physics involved in propagation through FCs. The results presented here are based on a 2D simplified approach of an ongoing numerical and experimental research, and corresponding 3D aspects have been recently presented in Belibassakis et al. (2014a) and Chondros et al. (2014), respectively, involving effects of normal and oblique wave incidence. Other important issues, such as underwater openings, waves hitting the top border of the FC, wave-induced currents, including wave-current interactions, and random incident waves, involving a broad frequency band, are subject of a future study. The present study brings a first insight in the engineering design of FCs in breakwaters especially concerning the effectiveness of the FCs in water renewal. In effect, special emphasis has been given in how the transmission coefficient is affected by the geometric characteristics of the FC as well as the incident regular wave conditions. Moreover, a numerical model to estimate the function and effectiveness of FCs for design purposes in the case of normal wave incidence and propagation through an FC that is never filled with water and the still water level is at its central axis. Keeping in mind that the performance of the FC is maximised for low steepness normally incident waves, especially when transmission coefficient is concerned, which is a critical parameter for the design of the FC, the coupled-mode model by Athanassoulis and Belibassakis (1999) and Belibassakis and Athanassoulis (2002) has been selected for numerical simulations. The latter model has been proven able to successfully handle the effects of large and sudden depth variations all over the water column on the vertical plane, up to second-order, in a computationally very efficient manner. Therefore, this paper provides a rapid way of calculating the transmission coefficients through an FC and hence, in terms of design purposes, the efficiency of an FC. This paper continues in Section 2 with a brief review of the background on numerical methods valid for such problems on the vertical plane. This leads into a short summary of the coupledmode model in Section 3. The experimental set-up as well as some experimental results are presented in Section 4. Comparisons between experimental observations and the model predictions are provided in Section 5 and discussion and conclusions in Section 6.

2. Background on numerical methods There are a number of numerical methods for ideal fluids that have been applied in order to describe the evolution of wave trains that propagate over sudden changes in bottom topography. These methods only partly describe the propagation through an FC as they involve abrupt changes in water depths but not threedimensional effects due to variations in the transverse direction. The first group of wave models is based on Boussinesq equations as derived by Peregrine (1967); very common extensions are the ones by Madsen et al. (1991) and Madsen and Sorensen (1992) which are widely used in coastal engineering

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applications (MIKE21BW DHI, 2005). Fountoulis and Memos (2005) used the MIKE21-BW model and successfully simulated openings with no depth variation between the open sea and the harbour. However, these types of models do not allow the simulation of very steep bottom slopes, hence the sudden changes in water depth that occur when the incident waves meet the FCs and when they are transmitted into the harbour. This is because the Boussinesq models assume a depth-average velocity profile which does not allow the detailed description of the evanescent modes that arise due to the abrupt changes in water depths. Other Boussinesq approaches include Beji and Battjes (1994) and Eldeberky and Battjes (1994) who simulated the propagation of waves over a submerged trapezoidal breakwater with a 1/20 slope on the sea side of the breakwater and 1/10 on the lee side. These examples involve much less sudden changes in water depths than the one presented in this study, where the bed slope is m ¼ 1 : 1:5 in the front side of the structure and infinite in the back side (see Fig. 1) rendering the Boussinesq models rather inappropriate. Eldeberky and Battjes (1994) also found that the bound harmonics that are generated on the crest of the breakwater propagate with the linear phase speed on the lee side implying that it is then that they are fully released as free waves. Moreover, Ohyama et al. (1995) compared three different models with experimental measurements of weakly nonlinear regular waves propagating over a trapezoidal submerged breakwater with very steep slopes in both sides: a Boussinesq-type model (Nwogu, 1993), a Stokes second-order theory and a fully nonlinear boundary element model (BEM) (Ohyama and Nadaoka, 1991). They pointed that the Boussinesq model provides good descriptions of the wave profile over the breakwater crest in highly nonlinear cases and incident height-to-depth ratio up to H=hs o 0:6. However, the energy transfer from bound components to free waves is overestimated in the lee side of the structure. The second-order Stokes theory provides satisfactory predictions for waves of small nonlinearity and incident height-to-depth ratio H=hs o 0:2. The boundary element approach proved the best in describing the harmonic generation of the waves. In the present study the above ratio varies 0:28 o H=hs o1:77. Driscoll et al. (1992) also compared experimental results using the fully nonlinear BEM method of Grilli et al. (1989). Similarly, Christou et al. (2008) compared 2.5 and 10 times steeper waves than those of Beji and Battjes (1993) and Driscoll et al. (1992), respectively, using a multiple-flux BEM method by Hague and Swan (2009) that provided very good agreement of the wave profiles for various values of the breakwater crest length (lf). Also, the BEM method proved very effective in the isolation of each harmonic. The harmonic generation is more accurately predicted for lf =λ Z 0:97 which is in the range of this paper's study. Appropriate numerical methods accounting for effects of turbulent wave flows for real fluids are based on the Reynolds Averaged Navier–Stokes (RANS) equations. Indeed, Garcia et al. (2004) and Lara et al. (2006) used such methods to examine wave propagation over low crested porous structures for regular and irregular waves respectively in a 2D domain. The surface elevation is described by a Volume of Fluid (VOF) method and the velocity and pressure field described in the COrnell BReaking waves And Structures (COBRAS) model (Lin and Liu, 1998) by the VolumeAveraged Reynolds Averaged Navier–Stokes (VARANS) equations, obtained by integration of the RANS equations. An improved Navier–Stokes model that solves the instantaneous Navier–Stokes equations in three dimensions was presented by Losada et al. (2005) based on the Large Eddy Simulation (LES) approach. By the experimental verification of the models they concluded that such models include many processes neglected previously by the existing models, such as wave breaking and turbulence and changes in mean water level.

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Nevertheless, the latter effective methods (BEM and RANS) are computationally expensive making them more appropriate for the understanding of the physics underlying the phenomenon of wave transformation through the FCs, rather than using it to investigate the effectiveness of FCs in harbours. Finally, another very attractive family of models for ideal fluids is obtained by reformulating the problem as a system of equations on the horizontal plane with variable coefficients. Berkhoff (1972) derived an one-equation model for gentle bottom slopes, called the mild-slope equation, in which the vertical distribution of the wave potential has been prescribed; with deviations and extended versions given by many authors (Mei, 1989; Massel, 1989; Porter and Chamberlain, 1989; Dingemans, 1997; Massel, 1992; Porter and Staziker, 1995). However, the expansion in the extended version has been found to be inconsistent with the Neumann condition on a sloping bottom. This fact has two important consequences: (i) the velocity field in the vicinity of the bottom is poorly represented, and (ii) the wave energy is not generally conserved. This problem has been remedied by the coupledmode theory developed by Athanassoulis and Belibassakis (1999), in which the standard local-mode representation has been enhanced by including an additional term, called the slopingbottom mode, leading to a consistent coupled-mode system of equations. This model is free of any simplifications concerning the vertical structure of the wave field and of any assumptions concerning the smallness of the bottom slope and curvature. Therefore, the coupled-mode system is able to model waves propagating over almost vertical changes of the bottom topography and therefore, accounts for the main problem in describing waves propagating through FCs (Katsardi et al., 2012). Also, the coupled-mode system accounts for the effects of dispersion, and the local-mode series exhibits fast convergence enabling long-range propagation and/or three-dimensional applications. The present study is interested in determining the way the FCs work favourably in an every-day function of the ports or the marinas on the breakwater of which they are placed; hence the design wave for the construction of FCs is certainly linear and/or weakly nonlinear, in contrast to the highly nonlinear waves that are associated with the design of the breakwater itself. For all of the above reasons, numerical results based on the 2D version of the coupled-mode system of equations (briefly presented in Section 3), involving both linear (Athanassoulis and Belibassakis, 1999) and weakly nonlinear results (Belibassakis and Athanassoulis, 2002), are presented for waves propagating over a region with a submerged breakwater, simulating FCs that are never perfectly filled with water. These are compared with experimental results, involving different wave conditions and/or geometrical characteristics of the FC.

3. The coupled-mode system The wave field is excited by a harmonic incident wave, with direction of propagation normal to the depth-contours. In the framework of linearised water wave (Wehausen and Laitone, 1960; Mei, 1989), the fluid motion is described by the 2D wave potential
 igH ð1Þ Φðx; z; tÞ ¼ Re  φðx; z; μÞe  iωt 2ω where H is the incident wave height, g is the acceleration pffiffiffiffiffiffiffiffidue to gravity, μ ¼ ω2 =g is the frequency parameter, and i ¼  1. The wave potential φðx; zÞ in the variable bathymetry region is represented by the following enhanced local-mode representation

introduced by Athanassoulis and Belibassakis (1999): 1

φðx; zÞ ¼ φ  1 ðxÞZ  1 ðz; xÞ þ φ0 ðxÞZ 0 ðz; xÞ þ ∑ φn ðxÞZ n ðz; xÞ:

ð2Þ

n¼1

In the above expansion the term φ0 Z 0 ðz; xÞ denotes the propagating mode and the remaining terms φn Z n ðz; xÞ; n ¼ 1; 2; … are the evanescent modes. The additional term φ  1 Z  1 ðz; xÞ is a correction term, called the sloping-bottom mode, which properly accounts for the satisfaction of the Neumann bottom boundary condition on the non-horizontal parts of the bottom. The functions Z n ðz; xÞ, n ¼ 0; 1; 2; …, are obtained as the eigenfunctions of local vertical Sturm–Liouville problems, and are given by Z 0 ðz; xÞ ¼

cosh½k0 ðz þ hÞ ; coshðk0 hÞ

Z n ðz; xÞ ¼

cos ½kn ðz þ hÞ ; cos ðkn hÞ

n ¼ 1; 2; …; ð3Þ

where the x-dependent eigenvalues fik0 ðxÞ, ikn ðxÞg are obtained as the roots of the local dispersion relation

μhðxÞ ¼ kðxÞhðxÞ tan ½kðxÞhðxÞ;

a r x r b;

ð4Þ

where a and b denote the ends of variable bathymetry domain. Specific convenient forms of the function Z  1 ðz; xÞ are defined in Athanassoulis and Belibassakis (1999). By introducing the above expansion in a variational principle, the coupled-mode system of horizontal equations for the amplitudes of the wave potential is obtained. An important feature of the enhanced representation is that it exhibits an improved rate of decay of the modal amplitudes jφn j of the order Oðn  4 Þ. Thus, only a few modes suffice to obtain a convergent solution in this region, even for bottom slopes of the order of 1:1, or higher. Furthermore, the dissipation of wave energy due to bottom friction and wave breaking can be indirectly taken into account, by including an appropriate imaginary part in the wavenumber, similarly as in the case of the mild-slope and the extended mildslope models (see, e.g. Dingemans, 1997; Massel, 1992). Since the propagating mode φ0 ðxÞ is the term that essentially controls the wave propagation features, in order to model energy dissipation, the coupled-mode system is modified as follows: 1

∑ famn ðxÞφ″n ðxÞ þ bmn ðxÞφ0n ðxÞ þðcmn ðxÞ þ iγ ðxÞk0 ðxÞδ0n Þφn ðxÞg ¼ 0;

n ¼ 1

m ¼  1; 0; 1; …; where δ0n is Kronecker's delta and coefficient (Massel, 1992),

γ ðxÞ ¼ γ f ðxÞ þ γ b ðxÞ:

ð5Þ

γ denotes the dissipation ð6Þ

The x-dependent coefficients of the above system are defined at each horizontal position in terms of the vertical functions Z n ðz; xÞ and are provided in detail in Table 1 of Athanassoulis and Belibassakis (1999) and in Sec. 7.1 of Belibassakis and Athanassoulis (2002) as concerns the second-order problem. The dissipation coefficient γ, Eq. (6), combines the effects of seabed friction and wave breaking on wave propagation; see, e.g., Massel (1992). The bottom friction dissipation coefficient γ f ðxÞ is obtained in terms of the local wave-velocity at the bottom, as follows (Massel and Gourlay, 2000; Massel, 2013):

γ f ðxÞ ¼

16f w jub ðxÞj3 ; 3π gC g ðxÞH 2 ðxÞ

ð7Þ

where ub denotes the bottom orbital velocity calculated from the derivatives of the wave potential on the bottom surface z ¼  hðxÞ, and H(x) is the local wave height. In Eq. (7), Cg is the local wave group velocity, and fw is the bottom friction coefficient for turbulent flow, which is dependent on the bottom surface

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roughness, taking values in the interval 0:1 o f w o 0:2 in natural conditions (Nelson, 1996; Dingemans, 1997). Following Massel and Gourlay (2000), the wave breaking dissipation coefficient γ b ðxÞ is modelled as pffiffiffiffiffiffiffiffiffiffiffi αω ghðxÞHðxÞ HðxÞ H m ðxÞ 4 ; ð8Þ when γ b ðxÞ ¼ hðxÞ hðxÞ π CðxÞC g ðxÞhðxÞ and 0 otherwise. In the above equation C ¼ ω=k is the local phase velocity, Hm(x) is the local maximum allowable wave height, and α ¼ Oð1Þ is an empirical coefficient. The effect of bottom friction (γf) and wave breaking (γb) dissipation parameters is very important, and especially the latter when wave breaking occurs over sloping seabeds. Details can be found in Belibassakis et al. (2007, 2014b) where the above effects are demonstrated and predictions by the present model are compared against experimental data. Moreover, the expressions suggested by Massel and Gourlay (2000) have been used to determine the subregions where depthinduced breaking occurs. The present wave theory fully models diffraction phenomena due to seabed variation at first and second order, including reflection and backscattering of wave energy in the offshore side, as well as reflection and diffraction of waves due to sudden increase of depth, as it happens at the downwave end of the FC. In fact, from the solution of the coupled-mode system (5), the reflection (AR) and transmission coefficients (AT) in the examined domain are calculated, ð1Þ

ð1Þ

AR ¼ jðφ0 ðaÞ  eik0 a Þeik0 a j;

ð3Þ

AT ¼ jφ0 ðbÞe  ik0 b j

ð9Þ

as well as the local value of the transmission coefficient (Kt) K t ðxÞ ¼ jφðx; z ¼ 0Þj ¼ j∑φn ðxÞj:

ð10Þ

n

Finally, the latter model has been extended to treat wave propagation and diffraction in general three-dimensional environments (Belibassakis et al., 2001) to predict second-order and fully non-linear waves in variable bathymetry (Belibassakis and Athanassoulis, 2002, 2011), and it has been applied to the transformation of offshore spectra to nearshore over realistic 3D bottom topographies (Gerostathis et al., 2005) as well as wave scattering by non-homogeneous currents in general 3D bathymetry (Belibassakis et al., 2011).

4. The experiments 4.1. Experimental set-up and wave conditions In this paper 63 experimental cases are presented which were conducted in a wave flume at the LHW of NTUA and involve physical model experiments. The flume (Fig. 2) is 27 m long with a rectangular cross-section that is 0.60 m wide and 1.52 m high. The still water depth was kept h ¼0.60 m for all the experiments. The tested model was a rubble mound breakwater (Fig. 2) with a length of 1.80 m, a height of 0.90 m, and a width of 0.60 m, which is equal to that of the flume. The slope of the breakwater at the offshore face is m ¼ 1 : 1:5. The lee side of the breakwater was constructed by a steel vertical plate. The flushing culvert is of

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rectangular shape with variable width, bf, and height, hf, but constant length, lf ð ¼ 1 mÞ, and their central axis is always located in still water level. A piston-type wave generator is fixed at the one end of the flume and is capable of producing monochromatic, sinusoidal waves which are adequately absorbed by a coarse composed wave absorber at the other end of the flume. Water surface elevation time series were measured using five resistance-type wave probes. The probes were placed on stands at specified positions along the flume in locations shown in Fig. 2. For monitoring the generated wave conditions the wave probe P1 was placed in a distance of 1.5 m away from the wave paddle, P2 at the toe of the breakwater, P3 measures the waves exactly before they enter the culvert, while probes P4 and P5 were used to evaluate the transmitted wave characteristics. The characteristics of the experimental cases that are chosen and presented here are shown in Table 1. All generated wave cases describe monochromatic waves ranging from almost linear to highly nonlinear, none of which allow any overtopping. The generated wave steepness is expressed in Table 1 as the ratio of the generated wave height to the corresponding wavelength ðH 1 =λ1 Þ. The heightto-depth ratio in probe position P3 ðH 3 =ðhf =2Þ ¼ H 3 =hs Þ, which is used to investigate whether the incident waves break before entering the culvert, is also given in Table 1. Probe P3 is positioned 0.05 m before the entrance to the culvert. The water depth at this position is the water depth in the FC hs ¼ hf =2. The experiments examined in the present study were chosen so that the wave heights in both positions P1 and P3 are smaller than the culvert height (H1 and H 3 ohf ); hence the culvert is never completely filled with water, as is also described in the Introduction. Moreover, in Table 1 the experimental and numerical calculations of the wave transmission coefficient, Kt, defined as the ratio of the transmitted wave height H4 and H5 in probe positions P4 and P5 respectively, to the incident wave height H1 are also presented. More details about the sample rate, possible reflections and wave generation, as well as the full set of experimental results can be found in Michalopoulou et al. (2008) and Rizos (2012). 4.2. Experimental results A number of cases from Table 1 are presented in Figs. 3–8 and more specifically the surface elevation and the corresponding normalised amplitude spectra in front (sea side) and behind (leeward) of the FC. With regard to the front of the breakwater, both probes P2 and P3 are situated at the breakwater slope. In order to associate H1 with the incident wave heights only the appropriate first part of records has been considered with no contamination by the reflected components. However, H3 measurements, although certainty affected by partial reflection, are used here in order to take into account the evolved wave in the entrance of the FC after shoaling, run-up and dissipation due to breaking or bottom friction on the breakwater. Indeed, an increase in surface elevations is observed in probe P3 and it is also reflected into the corresponding normalised spectra, where the growth of 2nd, 3rd and 4th order harmonics is now apparent (Figs. 3b–8b). This increase is due to shoaling, while partial reflection may lead

Fig. 2. Experimental setup and wave probe locations.

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Table 1 Experimental conditions and results. Wave case Culvert dimensions

Experimental wave conditions

Results

hf (m)

bf (m)

T (s) H 1 ðmÞ H1 =λ1 H3 =ðhf =2Þ Kt4

Kt5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

0.18

0.30

1.47

0.027 0.041 1.06 0.127 0.156 0.9 0.092 0.102 1.47 0.028 0.043 0.056 0.069 1.06 0.159 0.9 0.100 1.47 0.029 0.042 0.055 0.069 1.06 0.139 0.9 0.098 1.47 0.029 0.044 0.058 0.072 0.9 0.098

0.009 0.014 0.074 0.091 0.073 0.082 0.010 0.015 0.019 0.024 0.092 0.080 0.010 0.014 0.019 0.024 0.081 0.078 0.010 0.015 0.020 0.025 0.079

0.40 0.56 1.22 1.47 0.75 1.00 0.42 0.61 0.73 0.95 1.55 0.87 0.41 0.62 0.74 0.93 1.39 0.86 0.37 0.57 0.79 0.98 1.23

0.99 1.07 0.38 0.3 0.36 0.35 0.79 0.82 0.77 0.49 0.27 0.31 0.51 0.69 0.49 0.34 0.25 0.23 0.30 0.41 0.40 0.27 0.27

0.84 0.80 0.25 0.28 0.64 0.45 0.56 0.46 0.52 0.34 0.21 0.33 0.37 0.41 0.36 0.32 0.24 0.28 0.21 0.25 0.35 0.24 0.33

24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49

0.15

1.47

0.028 0.041 0.053 0.063 1.26 0.021 0.037 0.071 0.090 0.9 0.108 1.47 0.027 0.039 0.055 0.069 1.47 0.027 0.042 0.056 0.067 1.26 0.025 0.040 0.076 0.100 0.123 1.47 0.029 0.045 0.059 0.069

0.010 0.014 0.018 0.022 0.014 0.016 0.031 0.040 0.087 0.009 0.013 0.019 0.024 0.009 0.014 0.019 0.023 0.011 0.018 0.033 0.044 0.054 0.010 0.015 0.020 0.024

0.48 0.58 0.77 0.97 0.27 0.59 1.01 1.41 1.36 0.47 0.70 0.86 1.00 0.46 0.69 0.86 1.01 0.47 0.62 1.07 1.40 1.82 0.47 0.69 0.89 1.04

1.42 0.73 0.59 0.53 1.28 0.90 0.44 0.59 0.35 0.92 1.16 0.61 0.52 1.03 0.79 0.51 0.49 1.26 1.12 0.55 0.72 0.69 0.76 0.53 0.42 0.35

0.71 0.48 0.40 0.38 0.55 0.52 0.48 0.59 0.36 0.54 0.64 0.38 0.33 0.48 0.43 0.32 0.29 0.37 0.23 0.24 0.57 0.43 0.26 0.23 0.20 0.21

50 51 52 53 54 55 56 57 58 59 60 61 62 63

0.12

1.47

0.008 0.013 0.018 0.022 0.078 0.008 0.012 0.018 0.023 0.008 0.016 0.028 0.046 0.074

0.52 0.72 0.99 1.22 1.31 0.57 0.83 1.08 1.31 0.45 0.89 1.41 1.18 1.40

0.96 0.74 0.53 0.46 0.26 0.95 0.61 0.42 0.41 1.02 0.74 0.76 0.40 0.31

0.62 0.36 0.32 0.30 0.32 0.38 0.29 0.25 0.20 0.26 0.18 0.19 – –

0.24

0.18

0.12

0.30

0.24

0.18

0.12

0.30

0.18

0.024 0.037 0.052 0.063 0.9 0.098 1.47 0.023 0.035 0.051 0.066 1.26 0.019 0.036 0.065 0.9 0.058 0.092

to either additional increase or decrease in the surface elevation. Notwithstanding, the increase in surface elevation at probe P3 is also verified by the numerical model; characteristic results presented in Figs. 17 and 18. Moreover, the present study focuses on the wave characteristics on the lee side of the FC. As waves propagate over the

shallow water region inside the FC, part of the wave energy is transferred from a primary wave component to its higher harmonics because of the nonlinearity increase, as similarly observed by Mei and Ünlüata (1972) for waves propagating in water depth of kh o 0:6, where k ¼ 2π =λ the wavenumber. In the present study the relative water depth inside the FC is khs o0:45. Decomposition into shorter waves occurs immediately after they pass through the leeward side of the structure since the phase celerity differs significantly between the primary wave and its higher harmonics, and bound waves can no longer exist. The transmitted wave propagates as non-permanent waves composed of multiplefrequency components. Similar observations have been made by Ohyama et al. (1995) and Losada et al. (1997). Although the depth variation in the conducted experiments is larger than these previous studies (hs=h r0:15 whereas e.g. in Ohyama et al., 1995, it is hs=h ¼ 0:3), the general flow separation is similar, however more intense. Waves are transmitted onshore and include higher bound and free harmonic waves that are generated over the bottom of the FC and transmitted to the deeper water behind the breakwater as free waves. Therefore, the wave height behind the FC varies spatially and depends on the number of significant harmonics generated and transmitted inside the culvert. This phenomenon was clearly observed throughout testing. The timeseries of surface elevation of the transmitted waves was recorded, as already mentioned, using the wave probes P4 and P5. The generation and growth of harmonics and their progression leeward were obtained by analysing the probe records. All the acquired time series show strongly nonlinear wave forms that begin to split into a series of smaller waves at the lee of the FC and further down to the left end of the flume. The transmitted spectra are as expected different from the incident spectra; all being normalised with respect to the amplitude that corresponds to the peak frequency of the incident spectrum. Waves transmitted through the FC generate two or more transmitted waves on the lee side. The transmission coefficient in the lee side of the FC is calculated in two positions, as the ratio of the transmitted wave height immediately after the culvert (probe P4) and further down the flume (probe P5) with the incident wave height (probe P1). The transmission coefficient in P4 is larger than the one in P5 (see Table 1). The explanation for this lies in the fact that as the evolved waves exit the FC, they are reflected and diffracted due to the sudden increase of the water depth (see for example Massel, 1989). As P4 is very close to the culverts exit (5 cm), these evanescent mode effects are more present than when the wave reaches P5, 2 m downwave, where they have dissipated as expected. Additional diffraction takes place due to the presence of the FCs finite width. In any case, the FC operates as a generation source for other waves. The composition of these waves constitutes the wave profile at the lee side of the breakwater. These phenomena are consistently observed in the experimental measurements presented in the paper, leading to increased wave height measurements in position P4. This is further verified with the numerical model in Section 5 after extensive examination of all cases; examples being illustrated in Figs. 17 and 18. In the following, the evolution of the waves on both the incident and the lee side of the FC is examined in association with parameters due both to wave and culverts characteristics. First, the effect of wave steepness which is directly relevant to nonlinearity is presented with added discussion on the possible effect of wave breaking. Keeping the wave steepness constant, the effect of wave period related to the effective water depth is also discussed. Second, the effect of the FC's height (hf) and width (bf) is analysed; with the latter being the key parameter that differs this research from other researches on submerged breakwaters.

V.K. Tsoukala et al. / Ocean Engineering 89 (2014) 211–229

P1, x = 1.5m

0.02

1

0

0.5 0

−0.02

P2, x = 11.2713m

0.02

1 Normalised amplitude

0 −0.02 P3, x = 11.95m

η [m]

0.02 0 −0.02

P4, x = 13.05m

0.02 0

0.5 0 1 0.5 0 1 0.5 0

−0.02

P5, x = 15m

0.02

1

0 −0.02

217

0.5 0

2

4

6

8

10

0

12

0

1

2

3

4

f [Hz]

t [s]

Fig. 3. Case 7: T ¼ 1.47 s, H¼0.028 m, hf ¼0.18 m, bf ¼0.24 m (a) measured surface elevations and (b) corresponding normalised amplitude spectra at different positions of probes.

P1, x = 1.5m

1

0.02 0 −0.02

0.5 0 P2, x = 11.2713m

1 Normalised amplitude

0.02 0 −0.02 η [m]

P3, x = 11.95m 0.02 0 −0.02 P4, x = 13.05m 0.02 0 −0.02

0.5 0 1 0.5 0 1 0.5 0

P5, x = 15m

1

0.02 0 −0.02

0.5 0

2

4

6

8

10

0

12

0

1

t [s]

2

3

4

f [Hz]

Fig. 4. Case 8: T ¼ 1.47 s, H¼ 0.043 m, hf ¼ 0.18 m, bf ¼0.24 m (a) measured surface elevations and (b) corresponding normalised amplitude spectra at different positions of probes.

P1, x = 1.5m

0.04

1

0

0.5 0

−0.04 P2, x = 11.2713m

0.04

1 Normalised amplitude

0 −0.04 P3, x = 11.95m

η [m]

0.04 0 −0.04

P4, x = 13.05m

0.04 0

0 1 0.5 0 1 0.5 0

−0.04 P5, x = 15m

0.04

1

0 −0.04

0.5

0.5 0

2

4

6 t [s]

8

10

12

0

0

1

2

3

4

f [Hz]

Fig. 5. Case 9: T ¼1.47 s, H¼0.056 m, hf ¼ 0.18 m, bf ¼ 0.24 m (a) measured surface elevations and (b) corresponding normalised amplitude spectra at different positions of probes.

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V.K. Tsoukala et al. / Ocean Engineering 89 (2014) 211–229

P1, x = 1.5m

0.02

1

0

0.5

−0.02

0

P2, x = 11.2713m

0.02

1 Normalised amplitude

0 −0.02 P3, x = 11.95m

η [m]

0.02 0 −0.02

P4, x = 13.05m

0.02 0

0 1 0.5 0 1 0.5

−0.02

0

P5, x = 15m

0.02

1

0 −0.02

0.5

0.5 0

2

4

6

8

10

0

12

0

1

2

3

4

f [Hz]

t [s]

Fig. 6. Case 37: T ¼1.47 s, H¼0.027 m, hf ¼0.15 m, bf ¼ 0.18 m (a) measured surface elevations and (b) corresponding normalised amplitude spectra at different positions of probes.

P1, x = 1.5m

0.02

1

0

0.5 0

−0.02 P2, x = 11.4279m

0.02

1 Normalised amplitude

0 −0.02 η [m]

P3, x = 11.95m 0.02 0 −0.02 P4, x = 13.05m

0.02 0

0 1 0.5 0 1 0.5 0

−0.02 P5, x = 15m

0.02

1

0 −0.02

0.5

0.5 0

2

4

6

8

10

0

12

0

1

2

3

4

f [Hz]

t [s]

Fig. 7. Case 41: T ¼1.26 s, H¼ 0.025 m, hf ¼ 0.15 m, bf ¼0.18 m (a) measured surface elevations and (b) corresponding normalised amplitude spectra at different positions of probes.

P1, x = 1.5m

0.04

1

0

0.5

−0.04

0

P2, x = 11.2713m

0.04

1 Normalised amplitude

0 −0.04 P3, x = 11.95m

η [m]

0.04 0 −0.04

P4, x = 13.05m

0.04 0

0 1 0.5 0 1 0.5

−0.04

0

P5, x = 15m

0.04

1

0 −0.04

0.5

0.5 0

2

4

6 t [s]

8

10

12

0

0

1

2

3

4

f [Hz]

Fig. 8. Case 33: T ¼1.47 s, H¼0.027 m, hf ¼ 0.15 m, bf ¼ 0.24 m (a) measured surface elevations and (b) corresponding normalised amplitude spectra at different positions of probes.

V.K. Tsoukala et al. / Ocean Engineering 89 (2014) 211–229

Finally, the experimental analysis examines the effect of the above parameters in transmission coefficients because, as already explained, is one of the popular measures concerning the effectiveness of a flushing culvert in terms of retaining water quality within a harbour. 4.2.1. Temporal water surface profiles and harmonic generation through the flushing culvert In order to understand how the various aspects of the problem affect the wave transmission, this section examines the effects of the wave characteristics and more specifically the increasing nonlinearity, by investigating the effects of wave steepness and wave period, and then, the effects of the culvert's dimensions. The latter are limited to the culvert height, hf, and width, bf, since the culvert length, lf, in the above experiments remains constant. 4.2.1.1. Effect of wave steepness, H 1 =λ1 . The effect of wave nonlinearity is firstly approached in Figs. 3–5 (cases 7–9, Table 1) with respect to wave steepness. In these cases the culvert height and width are hf ¼0.18 m and bf ¼ 0.24 m respectively. The generated (at probe P1) wave period is T¼ 1.47 s and the corresponding wave heights are H1 ¼0.028 m, 0.043 m and 0.056 m associated with wave steepnesses that varies as 0:01 o H 1 =λ1 o0:019; case 7 being weakly nonlinear and cases 8 and 9 being of increasing nonlinearity. When the wave steepness and hence the nonlinearity increases, the transmission of energy in higher harmonics is more intense with 4th and even 5th harmonics generated in cases 8 and 9 (probe P4 in Figs. 3b–5b). Although some discrepancies are apparent between the linear (case 7) and the two nonlinear cases (8 and 9), among the two nonlinear cases the differences are hardly visible; with the increased underlying energy spectrum being only proportionally larger in case 9 but not different to the one associated with case 8. Furthermore, increased wave steepness may lead to wave breaking and it is examined below whether this affects the phenomena described above. All of the 63 waves are far from wave breaking on the constant depth region of the flume (h1 ¼ 0.60 m). Specifically, the maximum wave steepness ðH 1 =λ1 Þmax C0:07 o 0:147, which is the limit of wave steepness for monochromatic waves and the maximum height-to-depth ratio ðH 1 =h1 Þmax C 0:05{0:78, which is the classic breaking limit for solitary waves by McCowan (1894). However, the above do not exclude wave breaking while the wave propagates on the breakwater slope and/or inside the FC. Indeed, shoaling increases wave steepness and also the height-to-depth ratio increases, hence the probability of wave breaking also increases. Wave breaking will naturally lead to decreased reflection from the offshore side of the breakwater and intense wave breaking will lead to induced flow and hence will modify the mean water levels (see for example Losada et al., 1998; Méndez et al., 2001; Garcia et al., 2004) affecting indirectly all the other wave transformations further downwave, such as the wave reflection and diffraction in the exit of the FC. As already mentioned, with the records of probe P3 alone, it is difficult to identify the largest wave height at the entrance to the culvert. Nevertheless, it is informative to use these records together with the part of the records in probe P1 that are not affected by wave reflection. Taken the above things in mind, none of the three cases 7, 8 and 9 break on the slope due to increased wave steepness as ðH 1 =λ3 Þmax C0:04 o 0:147 and ðH 3 =λ3 Þmax C 0:05 o 0:147, where λ3 is the calculated wave length at probe position P3. On the other hand, the height-to-depth ratio at the entrance of the FC, as shown at probe position P3, is H 3 =ðhf =2Þ ¼ H 3 =hs ¼ 0:42 for case 7 and H 3 =hs ¼ 0:61 and 0.73 for cases 8 and 9 respectively (Table 1). Again, these waves, according to the conservative criterium by McCowan (1894) for solitary waves, do not break (as H 3 =hs o 0:78).

219

However, for regular and irregular waves this criterium is estimated to lie within the range 0:55 o H=h o 0:78 (Kamphuis, 1991; Massel, 1998; Nelson, 1994); hence case 7 is a certainly nonbreaking case while cases 8 and 9 are likely breaking inside the FC. Yet, as it is clearly shown, both from the crest elevation profiles and the corresponding spectra (Figs. 3–5), wave breaking does not significantly affect the energy transfers associated with harmonic generation; the energy being dissipated from all frequencies in an average sense. The minor effect of wave breaking in harmonic generation was also noted by Beji and Battjes (1993) with waves propagating in a much milder slope ðm ¼ 1=20Þ. This study verifies the above findings involving a much steeper slope (m C1=1:5 – Fig. 1). 4.2.1.2. Effect of wave period, T. Increasing wave heights that lead to increasing wave steepness cannot fully describe the effect of wave nonlinearity. This is because deviations in wave nonlinearity are also associated with the wave period, T. For example, the propagation of two cases, with the same wave steepness, differs when it takes place over different relative water depths; with the shallower depth leading to an increase of wave nonlinearity as it is observed in Figs. 6 and 7 through the comparison of waves with the same wave steepness and different wave period and consequently different relative water depth. In the above figures cases 37 and 41 propagate through a flushing culvert of bf ¼0.18 m width and of hf ¼0.15 m height. Both waves have nearly the same generated wave steepness ðH 1 =λ1 C 0:01Þ, although different wave periods, T¼1.47 s and T ¼1.26 s respectively. The height-to-depth ratio is also similar inside the FC ðH 3 =hs C 0:46 o 0:55Þ, hence both waves are non-breaking. Therefore, the effective difference between the two cases is the relative water depth that for cases 37 and 41 is respectively k1 h1 ¼ 1:30 and k1 h1 ¼ 1:64, for the horizontal section of the wave flume and k3 hs ¼ 0:38 and k3 hs ¼ 0:45, for the full length of the FC. Therefore, case 37 (T ¼1.47 s) propagates as expected in relatively shallower water depth than case 41 (T ¼1.26 s). The effect of this difference is shown in both the surface elevations and in the corresponding amplitude spectra. The most exploitable results are those of probes P4–P5, while the difference in surface elevations recorded in wave probes P2 and P3 for both cases (Figs. 6a and 7a) cannot be safely commented. In wave case 41 (T ¼1.26 s), counter to case 37 (T¼ 1.47 s), the growth of a 2nd order harmonic is dominant in all the positions in the flume apart from probe P5, where the diffraction effects and the evanescent modes have been dissipated (Fig. 7b). An increase in crest elevations is again observed in probe P3 due to shoaling for both cases, and this is reflected in the growth of first order harmonic in Figs. 6b and 7b. Moreover, looking at the amplitude spectra of the shorter wave that correspond to measurements in probes P2, P3 and P4 (Fig. 7b), it is obvious that although the proportion of energy of the first order harmonic, that is transmitted downstream of the breakwater, is larger than in the longer case (Fig. 6b), the generation of the 2nd order harmonic dominates the phenomenon and is thus responsible for the large waves recorded in probe P4. The wave decomposition in the deeper region is not as drastic as that of the long waves. The waves are vertically symmetric at the exit of the FC (P4) and appear as higher order Stokes waves; the latter being also observed for the shorter waves examined in Beji and Battjes (1993). However, unlike in Beji and Battjes (1993), where the wave cases examined are also of different wave steepness, in both cases discussed here the generated wave steepness is the same, and hence the effect of relative water depth can be isolated. Although the diffracted wave components just at the exit of the culvert probably affect the results, as already discussed previously, it is the difference in the relative water depth (k3 hs ¼ 0:38 for T ¼1.47 s as opposed to k3 hs ¼ 0:45 for

220

V.K. Tsoukala et al. / Ocean Engineering 89 (2014) 211–229

T ¼1.26 s) that makes the evolution of the longer waves more nonlinear (Mei and Ünlüata, 1972, kh o 0:6).

change in the culvert height (hf) examined earlier. Indeed, the surface elevation is larger for the case that involves the larger culvert width (case 1, Fig. 9a, bf ¼0.30 m) and gradually becomes smaller with a reduction in the culvert width (Figs. 3a and 10a). However, the waveforms continue to show some nonlinearity and this is also reflected in the associated amplitude spectra where higher order harmonics (at least up to 4th order) are still present albeit much reduced in amplitude. This is associated with (i) less energy being transmitted to the lee side of the breakwater, (ii) with the fact that in reduced culvert widths more energy is being dissipated through friction due to the side walls (see also, Tsoukala and Moutzouris, 2009) while (iii) the effect of diffraction is more significant when the culvert width is smaller.

4.2.1.3. Effect of culvert height, hf. To enhance the above argument, the effect of the culvert height, hf, is examined below. The effect of the culvert height hf in the wave evolution is equivalent with the effect of the water depth that the waves evolve inside the FC, hs ¼ hf =2. Specifically, case 33 is presented in Fig. 8 in order to make comparisons with case 7 (Fig. 3). These two cases have similar generated wave steepness (H 1 =λ1 C 0:01 with T¼ 1.47 s); both being weakly nonlinear. They both propagate through a flushing culvert of bf ¼0.24m width but different culvert heights hf. Waves in case 7 and case 33 propagate through a culvert height of hf ¼0.18 m and hf ¼0.15 m respectively. Comparing the two figures it is obvious that although the evolution of the wave train on the sea side of the breakwater is identical, the transformed wave train on the lee side is not; with case 33 (Fig. 8, hf ¼0.15 m) being more nonlinear. All the high order harmonics (4 1st order) are larger in case 33 (Fig. 8b) implying the fact that, inside the FC with smaller height (case 33, hf ¼0.15 m), the nonlinearity that has been introduced is more significant than in case 7 (hf ¼0.18 m). This is consistent with Eldeberky and Battjes (1994) who showed that significant energy transfers take place over the shallow water regions of their submerged breakwater amplifying the bound harmonics; the latter being then released as free waves downstream involving more nonlinear interactions as the water depth reduces and nonlinearity increases.

4.2.2. Transmission coefficient The calculation of the wave transmission coefficient (Kt) and how it is affected by the wave properties as well as the geometrical characteristics of the flushing culvert is presented. Kt, as already mentioned, is calculated as the ratio of the transmitted wave height measured at probe positions P4 and P5 respectively, to the generated wave height measured at probe position P1. In a previous experimental study by Tsoukala and Moutzouris (2009) the effect of different dimensional and non-dimensional variables in the transmission coefficient was examined. In effect, the transmission coefficient increased with submergence depth, FC's width and wave period, whereas FC's length and incident wave height had inverse effects on Kt. The present paper, using non-dimensional variables and isolating some of the above parameters, tries to give further physical explanation for the above variations. In Figs. 11 and 12 the transmission coefficient in both P4 and P5 probe positions is plotted against the critical factor of wave steepness measured in probe position P1 (H 1 =λ1 , left-hand column) and the wave steepness in probe position P3 (H 3 =λ3 , right-hand column); the former corresponding to the generated wave conditions at the paddle, with no contamination from reflected waves from the rubble mound breakwater, and the latter expressing the wave conditions, exactly before the entrance in the FC, formulated due to shoaling, wave breaking, run-up, partial reflection etc. associated with the presence of the breakwater slope. In order to understand better the effect of each parameter the data is separated in three sub-figures corresponding to the three different culvert heights hf ¼0.18 m, 0.15 m and 0.12 m

4.2.1.4. Effect of culvert width, bf. Unlike other studies, some mentioned previously, that have examined the effect of the relative depth of submergence, hs =h, the effect of the difference in the width of propagation is not equally examined. In order to address this effect cases 1 and 19 are introduced in Figs. 9 and 10 and are compared with case 7 (Fig. 3). Again, all three generated wave cases are weakly nonlinear with wave steepness H 1 =λ1 C 0:01 and T ¼1.47 s. The culvert height is the same for all three cases (hf ¼0.18 m) and the culvert width is bf ¼0.30 m, 0.24 m and 0.12 m for cases 1, 7 and 19 respectively. As expected, the wave evolution presents nearly no alteration in the positions upstream of the breakwater. However, on the lee side of the breakwater, the effect of the change in the culvert width (bf) is dominant and undoubtedly more significant than the

P1, x = 1.5m

0.02

1

0

0.5 0

−0.02 P2, x = 11.2713m

0.02

1 Normalised amplitude

0 −0.02 P3, x = 11.95m

η [m]

0.02 0 −0.02

P4, x = 13.05m

0.02 0

0 1 0.5 0 1 0.5 0

−0.02 P5, x = 15m

0.02

1

0 −0.02

0.5

0.5 0

2

4

6 t [s]

8

10

12

0

0

1

2

3

4

f [Hz]

Fig. 9. Case 1: T ¼1.47 s, H¼0.027 m, hf ¼0.18 m, bf ¼0.30 m (a) measured surface elevations and (b) corresponding normalised amplitude spectra at different positions of probes.

V.K. Tsoukala et al. / Ocean Engineering 89 (2014) 211–229

P1, x = 1.5m

0.02

1

0

0.5

−0.02

0

P2, x = 11.2713m

0.02

1 Normalised amplitude

0 −0.02 P3, x = 11.95m

η [m]

0.02 0 −0.02

P4, x = 13.05m

0.02 0

0.5 0 1 0.5 0 1 0.5

−0.02

0

P5, x = 15m

0.02

1

0 −0.02

221

0.5 0

2

4

6

8

t [s]

10

12

0

0

1

2

3

4

f [Hz]

Fig. 10. Case 19: T¼ 1.47 s, H¼ 0.0292 m, hf ¼0.18 m, bf ¼ 0.12 m (a) measured surface elevations and (b) corresponding normalised amplitude spectra at different positions of probes.

respectively, while in each sub-figure the four different culvert widths, bf ¼0.30 m, 0.24 m, 0.18 m and 0.12 m, are identified with different colours. For waves propagating through FCs of hf ¼0.18 m height (Figs. 11(a, b) and 12(a, b)), waves of relatively small steepness, hence weakly nonlinear ðH=λ r 0:03Þ, show a large variation in the corresponding transmission coefficient due to the difference in culvert width; the larger culvert width being associated with large transmission coefficients and thereafter reducing as the culvert width reduces. This is consistent with the 3D experimental analysis of Tsoukala and Moutzouris (2009) as well as the observations above concerning the crest elevations and the corresponding energy spectra; with more energy being transferred through the FCs of increased culvert widths. However, for waves of larger steepness, thus highly nonlinear ðH=λ 4 0:03Þ, the transmission coefficient shows much less variation and dependence on the culvert width and it reduces between 0:3 oK t4 o0:5 for probe P4 and 0:2 o K t5 o 0:5 for probe P5. The only exception is one nonlinear case that propagates through a bf ¼0.30 m culvert width that retains a value of transmission coefficient K t5 4 0:6 at probe P5. The above ranges are consistent with the findings of Dattatri et al. (1978) that performed a large range of laboratory tests on submerged breakwaters (hence without the effect of the culvert width) and found that for H=λ Z 0:03, the transmission coefficient varied as 0:3 oK t4 o0:5. Nevertheless, as the culvert height reduces, the effect of the wave steepness of the generated and incident wave is more profound in the estimation of the transmission coefficient. Indeed, for hf ¼0.15 m and hf ¼0.12 m (Figs. 11(c–f) and 12(c–f)), as wave steepness increases the transmission coefficient gradually reduces; though abruptly for small increases of nonlinearity (wave steepness) ðH=λ o 0:03Þ. Furthermore, wave steepness is related both to wave heights and wave lengths. It is known theoretically that whether longer waves that propagate over a submerged bar lead to larger transmission coefficients depends on the given frequency band (see, e.g. Massel, 1989). Indeed, in the experimental findings of Tsoukala and Moutzouris (2009), longer waves were associated with larger Kt through an FC. However, this is not necessarily the case here, as the increased nonlinearity is naturally related to dissipation due to wave-breaking and hence leads to fundamental energy losses before the entrance and/or inside the FC. Finally, the effect of culvert width in the transmission coefficient is again important; with small culvert widths generally associated with reduced transmission coefficients.

Given that the transmission coefficient is similarly affected by the parameters shown in Figs. 11 and 12, in both positions P4 and P5, hereafter the transmission coefficient will be examined only in position P4 for brevity purposes. In order to make the relative effect of culvert height and width clearer, in Fig. 13 the transmission coefficient in P4 position is plotted against the wave steepness measured in probe position P1 (H 1 =λ1 , left-hand column) and the wave steepness in probe position P3 (H 3 =λ3 , right-hand column), but the data are here separated in four sub-figures with respect to the four different culvert widths, while in each sub-figure the three different culvert heights are identified with different colours. For waves propagating through FCs of bf ¼0.30 m, again the increase in wave steepness is associated with a decrease in the transmission coefficient; the relevance with the incident wave steepness (H 3 =λ3 , Fig. 13b) being more gradual than the one with the generated wave steepness (H 1 =λ1 , Fig. 13a). Nevertheless, for H=λ 4 0:03, Kt hardly changes and stays at the order of K t C 0:35. The effect of the culvert height hf is equivalent to the effect of relative depth of submergence hs =h ¼ ðhf =2Þ=h, as in all experiments the water depth is h ¼ 0:60 m. Although it is obvious that small culvert heights lead to smaller transmission coefficients, this effect is less important than the corresponding effect of reduced culvert widths (Fig. 12(a, b)). As the culvert width reduces, the transmission coefficients also reduce, and the effect of the culvert height is less evident (Fig. 13 (c–h)); the latter data appearing much more scattered. This also implies that the effect of the changes in the culvert width is more important than the changes in the culvert height. The reason for this is threefold. First, as also seen in the comparisons of the energy spectra of cases 1, 7 and 19 (Figs. 9, 3 and 10), less energy is being transferred in the lee side of the breakwater through smaller FCs and second, the side walls of the FC reduces the incident wave energy through friction and this effect increases with the reduction of the culvert width. Third, as already mentioned earlier, probe P4 is located in the centre of the contribution of the diffracted components due to the side walls and this certainly affects the transmission coefficient; the smaller the culvert width, the more important the diffraction and thus the more significant the effect on the transmission coefficient. Earlier in this section it was shown that wave breaking does not significantly affect the energy transfers associated with harmonic generation (Figs. 3–5). In Fig. 14a the transmission coefficient for probe P4 against the height-to-depth ratio at the

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h =0.18m

h =0.18m

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0

0.5

0

0.02

0.04

0.06

0.08

0.1

H1/λ1

0

0

0.05 H3/λ3

Fig. 11. Transmission coefficient K t4 ¼ H 4 =H 1 against (a, c, e) generated wave steepness H 1 =λ1 (corresponding to probe P1) and (b, d, f) incident wave steepness H 3 =λ3 (corresponding to probe P3) for all experimental results separated with respect to culvert width bf. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)

FC ðH 3 =ðhf =2Þ ¼ H 3 =hs Þ is plotted. Very similar trends, only dissipated, appear again for the probe position P5 and hence are not plotted here. In Fig. 14 whether depth limited wave breaking inside the FC affects the transmission coefficient on the lee side of the breakwater is examined. The data are sorted according to wave breaking index of Kamphuis (1991) and Massel (1998), H=hC 0:6 and are indicated with red when they are breaking and with black when they are not breaking. It is obvious that for nonbreaking waves the transmission coefficient has a large variation ð0:3 oK t4 o1:4Þ, while for waves with H 3 =hs 40:6, thus certainly breaking, the corresponding variation is much reduced ð0:3 o K t4 o 0:7Þ showing the profound effect of wave breaking. Consistently with the above, the experimental data presented

by Gu and Wang (1992), on the basis of waves propagating over porous submerged breakwaters, showed that once wavebreaking is exceeded, the transmission coefficient hardly varies. Although, this is also obvious in the case of propagation through flushing culverts, the data appear more scattered. Trying to give an explanation for this, in Fig. 14b, Kt4 in probe position P4 is plotted against the dimensionless culvert width with respect to wave height ðbf =H 3 Þ, while the indication of wave breaking is still apparent. It is seen in Fig. 14b that non-breaking waves, which propagate through relatively large culvert widths, are certainly associated with larger transmission coefficients and this is due to the fact that more energy is being transferred through a larger opening and less energy is being dissipated through breaking and due to the side-walls of the FC.

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h =0.18m

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1

Kt5

Kt5

b =0.24m

K

K

t5

0.8

1

0

0

0.05 H /λ 3

3

Fig. 12. Transmission coefficient K t5 ¼ H 5 =H 1 against (a, c, e) generated wave steepness H 1 =λ1 (corresponding to probe P1) and (b, d, f) incident wave steepness H 3 =λ3 (corresponding to probe P3) for all experimental results separated with respect to culvert width bf. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)

Nevertheless, the transmission coefficient of breaking waves still vary with respect to the relative culvert width and this again verifies that the phenomenon is governed by the effect of the culvert width; the transmission coefficients being proportional to the relative culvert width.

5. Numerical results and comparisons As discussed previously, the problem of a flushing culvert, since it is never entirely filled with water, will be approached numerically by the more common problem of a submerged breakwater. The geometry of the flume (Fig. 15) is kept almost the same with

the experiments. The water depth either side of the breakwater is again 0.60 m and on the breakwater crest is half the height of the FC hs ¼ hf =2, measured from still water level to the bottom of the culvert. The coupled-mode system described in Section 3 has been used for simulations by keeping five terms in the local-mode series (Eq. (2)), the propagating mode (n ¼0), the sloping-bottom mode (n ¼  1) and the three first evanescent modes (n ¼ 1; 2; 3), which were found to be enough for numerical convergence. Representative numerical results are presented in Figs. 16–18 as obtained by the CMS, using a (constant) value of fw ¼0.2 for energy dissipation due to bottom friction. In particular, in Fig. 16, concerning case 1, the real parts of the wave field, which is

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bf=0.30m

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Kt4

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1 0.5 0

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Fig. 13. Transmission coefficient K t4 ¼ H4 =H 1 against (a, c, e, g) generated wave steepness H1 =λ1 (corresponding to probe P1) and (b, d, f, h) incident wave steepness H 3 =λ3 (corresponding to probe P3) for all experimental results separated with respect to relative depth of submergence, hs =h ¼ ðhf =2Þ=h. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)

proportional to the pressure field in the water column, are illustrated using contour lines both concerning the first-order solution (Fig. 16a, b) and the second-order field (Fig. 16c, d). In the same plots the calculated free-surface elevation is also shown by using red lines. To obtain numerical results, in this example, a total number of 5 modes and N ¼3000 segments subdividing the horizontal interval have been used, which was found enough for

numerical convergence. It is clearly observed that the equipotential lines intersect the bottom profile perpendicularly (see closeups in Fig. 16b and d), which is evidence of fulfillment of the bottom boundary condition, and especially in the front and back faces of the shoal where the bottom slope attains great values. In Figs. 17 and 18, concerning cases 1 and 33 respectively, the ratio of the calculated wave height to the incident wave height, in

V.K. Tsoukala et al. / Ocean Engineering 89 (2014) 211–229

1.5

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breaking non−breaking

0

0

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1 H /(h /2) 3

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10 b /H

f

f

15

3

Fig. 14. Transmission coefficient K t4 ¼ H 4 =H 1 against (a) height-to-depth ratio H3 =hs ¼ H 3 =ðhf =2Þ and (b) ratio of culvert width to incident wave height bf =H 3 (corresponding to probe P3) for all experimental results. The effect of wave breaking is indicated. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)

z, η [m]

0.2

P1

P2 P3

P4

P5

0 −0.2 −0.4 −0.6 0

5

10

15

x [m] Fig. 15. Sketch of numerical setup and wave probe locations. The dashed line indicates the top border of the FC, which is not presented in the numerical solution. Note that the horizontal scale is different than the vertical scale.

all positions down the tank is shown using the linear and 2ndorder CMS respectively. Both cases presented in Figs. 17 and 18 have the same generated wave steepness ðH 1 =λ1 Þ but case 1 propagates through culverts of larger height hf making it less nonlinear and hence it converges quicker than case 33 (i.e. needs fewer repeat runs to converge). Regarding case 1, in both P4 and P5 probes the numerically calculated transmission coefficient is in very good agreement with the experiments. On the other hand, in wave case 33 although in Probe P4 the agreement is still good, the numerical model predicts smaller transmission coefficients in probe P5, compared to the experiments. However, it reflects the fact that the wave surface elevation is balanced in a smaller wave height 2 m further down the flume where the strong evanescent modes have been dissipated. Clearly, predictions by the present second-order model are expected to be valid up to the point where energy transfer to higher harmonics, results into significant overestimation of the second-harmonic, as presented and discussed in Belibassakis and Athanassoulis (2002, Sec. 8.3). Additionally, analysing the generation and transmission of 2nd order harmonics (Belibassakis and Athanassoulis, 2002) through the FC (Figs. 17b and 18b), it is obvious that the model predicts a large generation/growth of the 2nd order harmonic on the top and on the lee side of the submerged breakwater. Specifically, the growth of the second harmonic, in the corresponding experimental cases (Figs. 9b and 8b) is less larger than the one calculated by the CMS in the position of the probes P4 and P5. However, the model satisfactory predicts that the 2nd harmonic is larger than the 1st harmonic in position P5. Indeed, in case 1 the second harmonic in the experiments is shown 137% larger than the first harmonic and the model predicts that it is 133% larger. The prediction is less good in case 33 with the above values being

180% and 129% respectively but, as already mentioned, this latter case is more nonlinear than case 1 since the waves propagate over a shallower region through the culvert. In effect, these deviations can be explained by the fact that the present model can only predict energy transfers up to second order and hence such higher order phenomena cannot be captured. As already discussed in Section 3, the CMS system that is applied in this paper can describe waves that are linear or weakly nonlinear (up to second order) and that they propagate over very sudden changes of bottom topography. These changes are only two-dimensional and hence the CMS system cannot describe the changes in the width of the propagation that the waves meet when they approach the breakwater through the FC; the latter being associated with less energy being transferred in the lee side, with wave energy dissipation and with wave diffraction as discussed previously in the paper. However, about  1=3 of the experiments involve a relatively small wave steepness ðH 1 =λ1 r 0:024Þ and large culvert widths (bf ¼ 0.30 m and 0.24 m). In these cases, which are also interesting for the design of FCs, the CMS application becomes more realistic providing predictions that could be compared against experimental data. In Table 2, the transmission coefficients, in probe positions P4 and P5, based on the CMS numerical results, are presented as comparisons to the selected experimental results. Although the wave height is increased within the FC, as regularly observed at the crown of submerged breakwaters, the wave conditions examined in the present work are such that the largest calculated wave height to culvert height ratio is H=hf ¼ 0:70 o 1 which means that the wave does not hit the top border of the FC and hence the FC is never entirely filled with water. In Fig. 19 the measured transmission coefficients Kt4 and Kt5 against the predicted by the CMS system, shown in Table 2, are presented. The points plotted with grey are not considered as Kt4 for these points is larger than 1. This transmission coefficient can be further explained by the fact that the FC operates not only as a generation source for other waves, as discussed in Section 4, but also as a generation of flow due to the abrupt changes in the propagation width caused by the presence of the culvert. This flow is sometimes captured by probe P4 because it is too close to the culvert's exit. This is consistent with the findings of Tsoukala and Moutzouris (2009) in a 3D wave basin, who monitored the above flow generation in some cases. The 2D CMS system will not describe this phenomenon. Therefore, if the rest of the points are considered, the model predicts very well the transmission coefficient at position

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Fig. 16. Vertical contour plot of the solution and surface elevations for (a) linear and (c) second order solutions. In (b) and (d) a close-up of the solution on the breakwater is presented. Case 1: T ¼ 1.47 s, H¼0.027 m, hf ¼0.18 m. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)

2

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Fig. 17. Ratio of incident to local wave height numerical calculations. Case 1: T ¼ 1.47 s, H¼ 0.027 m, hf ¼ 0.18 m (a) linear CMS (b) harmonic generation. The red line is the convergent line and the dashed lines indicate the position of the wave probes. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)

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x [m]

Fig. 18. Ratio of incident to local wave height numerical calculations. Case 33: T ¼1.47 s, H¼ 0.027 m, hf ¼ 0.15 m (a) linear CMS (b) harmonic generation. The red line is the convergent line and the dashed lines indicate the position of the wave probes. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)

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Table 2 Transmission coefficients from selected experimental and numerical results. Wave case

Experimental results Kt4

Kt

1 2 7 8 9

0.99 1.07 0.79 0.82 0.77

24 25 26 27 28 29 33 34 35 36 50 51 52 53

Numerical results Kt4

Kt

0.84 0.80 0.56 0.46 0.52

0.93 0.83 0.93 0.80 0.69

0.86 0.77 0.86 0.74 0.64

1.42 0.73 0.59 0.53 1.28 0.90 0.92 1.16 0.61 0.52

0.71 0.48 0.40 0.38 0.55 0.52 0.54 0.64 0.38 0.33

0.86 0.70 0.60 0.54 0.88 0.80 0.88 0.72 0.59 0.50

0.78 0.63 0.54 0.48 0.82 0.74 0.79 0.65 0.53 0.45

0.96 0.74 0.53 0.46

0.62 0.36 0.32 0.30

0.84 0.62 0.47 0.41

0.75 0.55 0.42 0.36

5

5

1

Kt − CMS

0.8 0.6 0.4

Kt4 Kt5 y=0.96x y=1.24x y=x

0.2 0

0

0.5

1

1.5

K − experiments t

Fig. 19. Comparison of measured and calculated (CMS model) transmission coefficients Kt 4 and Kt 5 along with their trend-lines. The y¼ x grey line is plotted to facilitate comparisons.

Kt4 with the trend-line that best describes the comparison being exp very close to y ¼ x ðK cms t4 ¼ K t4 Þ, verifying that the CMS successfully describes the physics of wave propagating over large and sudden deviations of water depth. Simultaneously, the predictions in probe position P5 are also fairly good; the trend-line of the comparison being less than 24% larger than 1. Although the numerical calculations have taken into account the energy dissipation due to breaking and bottom friction (see Section 3), the viscous decay due to the flume's side walls, that is also shown in Figs. 3b–8b, is not realistic and is not calculated by the model. Furthermore, additional effects such as wave-induced currents and interaction of those currents with the waves are also neglected (Belibassakis et al., 2007). A consistent difference of 20–30% is therefore expected in these comparisons.

6. Discussion and conclusions The evolution of waves that propagate through flushing culverts in rubble mound breakwaters, constructed in low tide areas, is examined in this paper. The effective design of FCs, and hence the maintenance of water quality inside the harbour, is critically dependent upon the transmission coefficient of the wave energy on the lee side of the breakwater. This paper examines how the

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wave evolution and the transmission coefficient are affected by the geometric dimensions of the culvert as well as the characteristics of the incident wave. Wave transformation through such openings is difficult to describe, because it involves, not only an abrupt change in the water depth, but also a sudden change in the propagation width, as the waves meet the entrance. Therefore, the physics underlying the wave evolution may involve many different phenomena, such as wave shoaling; run-up; wave reflection in both side of the opening; diffraction due to the change in the water depth and the change in the propagation width; dissipation due to wave breaking and turbulence, due to interaction with the rubble mound slope and with the side walls of the FC; additional reflections when the maximum crest elevation hits the top border of the FC. Currents will be generated as a consequence of wave breaking both before the entrance but also inside the FC and because of the change of the propagation width. These currents will affect the mean water level at the offshore side of the structure (see for example Losada et al., 1998; Méndez et al., 2001; Garcia et al., 2004; Belibassakis et al., 2007) and may affect the transmission of wave energy at the exit of the FC. The interaction of the currents with the waves may lead to modifications in the amplitude of the harmonics generated and measured at the lee side of the structure. Moreover, real sea-states are unsteady and directional, hence associated with wide frequency spectra; leading to additional more complex and nonlinear wave–wave interactions and energy transfers. Finally, the main direction of the waves may be either normal or oblique to the breakwater's longitudinal axis again affecting the proportion of energy being reflected by the breakwater and transmitted through the FC. All of the above are associated with harmonic generation and certainly affect the wave transmission in the lee side of the structure and hence the FC design. As this paper does not aim to give answers to all the physics involved in propagation through FCs and it is a part of a parallel and on-going study, only some of the above features are isolated and examined. In effect, a wide range of laboratory measurements in a 2D flume has been gathered and it was attempted to explain the evolution of regular waves through FCs placed on breakwaters on the seawater level. In general, it was shown that the culvert operates as a generation source for other waves. The composition of these waves constitutes the wave profile at the lee side of the breakwater. First, in relation with the wave characteristics, it was shown that when the wave steepness, and hence the nonlinearity, increases, the transmission of energy in higher harmonics is more intense with wave breaking limiting the extend of this phenomenon. In effect, wave breaking does not significantly affect the energy transfers associated with harmonic generation; the energy being dissipated from all frequencies in an average sense. Also, keeping the wave steepness constant (but corresponding to nonbreaking waves), it was shown that the effect of wave period is dominant showing that the wave decomposition in the lee side of the FC is more drastic for long waves. This is also related by the fact that the effective water depth is smaller for long generated waves; the latter being enhanced when longer waves propagate up the breakwater slope and then as they pass through the FC, and hence related to an increase in nonlinearity compared with the shorter waves. Second, regarding the FC's dimensions, a reduction in culvert's height amplifies the bound harmonics; the latter being then released as free waves downstream involving again more nonlinear interactions as the water depth reduces. Moreover, the paper highlights the effect of the culvert width, which appears more significant than the change in the culvert height. With a gradual decrease in culvert width, the wave-forms continue to show some nonlinearity where higher order harmonics are still present albeit much reduced in amplitude.

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Simultaneously, as mentioned earlier, apart of the above qualitative analysis performed in this research, the quantitative determination of Kt was held and it was shown that (a) increased wave steepness leads to reduced values of Kt, mainly because of dissipation due to wave breaking and (b) Kt it is mostly affected by the geometric characteristics of the opening; with the culvert width playing a more important role than the culvert height, as also observed in the discussion relating to harmonic generation. The effect of culvert width is related to (i) the proportion of energy being transmitted to the lee side of the breakwater, (ii) the energy being dissipated through friction due to the side walls and (iii) the effect of diffraction which is more significant when the culvert width is smaller. Comparisons with a 2D coupled-mode numerical model were also provided and it was shown that it gives results in relatively good agreement with the experimental measurements, providing however, some limitations concerning the nonlinearity of the wave field and the width of the culvert. The success is mainly due to the effective way with which the CMS model describes the effects by the sudden changes in water depth in the vicinity of the structure/opening. This is important for estimating the efficiency of the FC concerning the harbour's water renewal, for the standard every-day function of the harbour that involves waves of small amplitude, hence less nonlinear and this is where the presented version of the CMS model can be used as a tool that very quickly and efficiently will give an estimation on the appropriateness of the designed FC. Nevertheless, when simulating steeper waves, generation of higher than second harmonics is of increasing significance and should be taken into account. Corresponding 3D aspects have been recently presented in Belibassakis et al. (2014a) and Chondros et al. (2014), respectively, involving effects of normal and oblique wave incidence. Other important issues, such as underwater openings, waves hitting the top border of the FC and random incident waves, involving a broad frequency band, as explained in detail in the beginning of this section, are subject of a future study, while the present paper brings a first insight in the engineering design of flushing culverts.

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