Benchmarking of Numerical Models for Wave Overtopping at Dikes with Shallow Mildly Sloping Foreshores: Accuracy versus Speed
Christopher Lashley, Barbara Zanuttigh, Jeremy Bricker, Jentsje van der Meer, Corrado Altomare, Tomohiro Suzuki, Volker Roeber, Patrick Oosterlo
11 BENCHMARKING OF NUMERICAL MODELS FOR WAVE OVERTOPPING AT DIKES WITH MILDLY SLOPING SHALLOW FORESHORES: ACCURACY VS. SPEED Christopher H. Lashley a, * 2. Barbara Zanuttigh b Jeremy D. Bricker a, c Jentsje van der Meer d,e Corrado Altomare f, g Tomohiro Suzuki a,h Volker Roeber i a) Dept. of Hydraulic Engineering, Delft University of Technology, Stevinweg 1, 2628 CN Delft, the Netherlands. b)
Dept. Civil, Chemical, Environmental and Materials Engineering, University of Bologna, Via Zamboni, 33, 40126 Bologna BO, Italy. c)
Dept. of Civil and Environmental Engineering, University of Michigan, 2350 Hayward St. Ann Arbor, MI 48109 USA. d)
Water Science and Engineering Department, IHE Delft, Westvest 7, 2611 AX Delft, the Netherlands. e)
Van der Meer Consulting, P.O. Box 11, 8490 AA Akkrum, the Netherlands. f)
Universitat Politecnica de Catalunya โ BarcelonaTech, carrer Jordi Girona 1-3, 08034, Barcelona, Spain; g)
Ghent University, Department of Civil Engineering, Technologiepark 60, 9052 Gent, Belgium. h)
Flanders Hydraulics Research, Berchemlei 115, 2140 Antwerp, Belgium. i)
Assistant Professor, E2S chair HPC-Waves Laboratoire SIAME, Pau, France: [email protected] * Corresponding author; Telephone: +31684097811; Email: [email protected]
Abstract
To accurately predict the consequences of nearshore waves, coastal engineers often employ numerical models. A variety of these models, broadly classified as either phase-resolving or phase-averaged, exist; each with strengths and limitations owing to the physical schematization of processes within them. Models which resolve the vertical flow structure or the full wave spectrum (i.e. sea-swell (SS) and infragravity (IG) waves) are considered more accurate, but also more computationally demanding than those with approximations. Here, we assess the speed-accuracy trade-off of six well-known wave models for overtopping ( ๐๐๏ฟฝ ), under shallow foreshore conditions. The results demonstrate that: i) ๐๐๏ฟฝ is underestimated by an order of magnitude when IG waves are neglected; ii) using more computationally-demanding models does not guarantee more accurate results; and iii) with empirical corrections to account for IG waves, phase-averaged models like SWAN can perform on par, if not better than, phase-resolving models but with far less computational effort. Keywords: Infragravity wave, OpenFOAM, BOSZ, XBeach, SWASH, SWAN
Software availability โข โข SWASH โ developed at Delft University of Technology, the model is available freely from: http://swash.sourceforge.net/download/download.htm โข BOSZ โ developed at the University of Hawaiโi at Manoa, the model is freely available under request from: [email protected] โข XBeach Non-hydrostatic and XBeach Surfbeat โ developed by IHE Delft, Deltares, Delft University of Technology and the University of Miami, both models are freely available from: https://oss.deltares.nl/web/xbeach/download โข SWAN โ developed at Delft University of Technology, the third-generation wave model is freely available from: http://swanmodel.sourceforge.net/download/download.htm Introduction
Background
Coastal engineers often employ numerical modelling in the design, assessment and rehabilitation of coastal structures to accurately forecast nearshore waves and currents, sometimes including the consequences (Akbar and Aliabadi, 2013, Sierra, et al., 2010, Smith, et al., 2012, Suzuki, et al., 2017). Of particular interest is the extent to which waves reach and pass over the crest of a structure, referred to as wave overtopping. Extreme overtopping events are characterized by considerable flow velocities which impose serious hazards to both people and infrastructure; with flooding or coastal inundation as the most critical consequence. The integration of numerical modelling in estimating wave overtopping and the design of coastal structures is becoming increasingly more attractive given the progress in available computing power and the limitations of traditional empirical approaches which are typically limited to the number of simplified structure configurations and the range of environmental conditions applied in their derivation. Furthermore, as many of the empirical models (e.g. EurOtop, (2018)) require the incident significant wave height ( ๐ป๐ป ๐๐0 , ๐๐ , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐ ) and spectral wave period ( ๐๐ ๐๐โ1 , , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐ ) at the toe of the structure as input, numerical models are often needed to accurately capture the nonlinear effects associated with the shoaling and breaking of high-frequency sea-swell (SS) waves in shallow water (Altomare, et al., 2016, Mase, et al., 2013). Such effects include a rise in mean water levelโknown as wave-induced setupโand the growth of low-frequency infragravity (IG) waves (Figure 1) which not only contribute to ๐ป๐ป ๐๐0 , ๐๐ , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐ but also result in higher values of ๐๐ ๐๐โ1 , , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐ (Hofland, et al., 2017). Figure 1 Schematic representation of wave transformation over a shallow foreshore (from an XBeach model simulation), showing a) the growth of IG waves; b) the increase MWL at the dike toe; and c) the shift in the peak in energy density to lower frequencies from offshore (P1) to the dike toe (P2). Vertical line in panel โcโ indicates the separation between SS and IG frequencies.
A variety of numerical models, which may be broadly classified as phase-resolving or phase-averaged, have been developed for such applications; each with strengths and limitations owing to the physical parameterization of processes and the numerical schemes incorporated within them (Cavaleri, et al., 2007, Vyzikas and Greaves, 2018). Models which attempt to resolve the vertical flow structure and those that consider the full frequency range of nearshore waves (i.e. both SS and IG waves) are considered not only more accurate, but also more computationally demanding than those which make use of approximations. Within the phase-resolving class of wave models, those that resolve the vertical flow structure and solve the fully nonlinear, time-averaged Navier-Stokes (NS) equationsโoften referred to as Computational Fluid Dynamics (CFD) or depth-resolving modelsโhave the least theoretical limitations and are generally considered the most accurate. CFD models, such as the mesh-based Eulerian approach OpenFOAM (Jasak, et al., 2007) or mesh-less Lagrangian approach DualSPHysics (Crespo, et al., 2015), are able to simulate complex wave problems, such as: nonbreaking and breaking waves, wave-current interaction and wave-structure interaction from deep to shallow water conditions, including the overturning (Lowe, et al., 2019) and roller formation of breaking waves (Higuera, et al., 2013). However, these models require a significant amount of computational effort (unless a coupling method is applied (Altomare, et al., 2015, Altomare, et al., 2018, Verbrugghe, et al., 2018)); thus, limiting their application so far to very local phenomenaโfor example, wave overtopping. As depth-resolved (fully 3D or 2DV) models are generally considered too computationally expensive for operational use, the problem may be further simplified by depth-averaging. These models, in which the vertical structure is not directly resolved but only modelled parametrically, are referred to as two-dimensional in the horizontal (2DH), or 1DH where only a cross-shore transect is simulated (Brocchini and Dodd, 2008). As a result of depth-averaging, processes such as wave overturning, air-entrainment and wave generated turbulence are not directly solved. Those that simulate the amplitude and phase variation of SS waves are often referred to as phase-resolving. Within this type of model, there are generally two main sets of governing equations: i) the Non-linear shallow water (NLSW) equations; and ii) the Boussinesq type. While the Boussinesq-type models (e.g. FUNWAVE (Kirby, et al., 1998), MIKE21-BOUSS (Warren and Bach, 1992) and BOSZ (Roeber and Cheung, 2012)) directly account for the dispersive properties of waves in deeper water, the NLSW models assume that waves are non-dispersive and are therefore limited to shallow-water applications (Brocchini and Dodd, 2008, Zijlema and Stelling, 2008). This limitation can be removed by taking a SS-wave averaged approach; however, at the cost of decreased accuracy (due to exclusion of SS-wave motions). The high-frequency waves are averaged, resulting in only motions at the scale of the wave group; thus, reducing the computational demand (e.g. XBeach Surfbeat (XB-SB) model (Roelvink and Costas, 2019, Roelvink, et al., 2009)). In order to use the NSLW equations for phase-resolving simulation of SS-wave motions, Stelling and Zijlema (2003) proposed another method to account for dispersion (a result of non-hydrostatic pressure) whereby the pressure is decomposed into non-hydrostatic and hydrostatic pressure components (e.g. SWASH (Zijlema, et al., 2011), NHWAVE (Ma, et al., 2012) and XBeach Non-hydrostatic (XB-NH) (Smit, et al., 2010) numerical models). This approach improves the dispersive properties without neglecting the higher-frequency motions; however, at the expense of more computational demand. The accuracy and range of applicability of the non-hydrostatic models may be further enhanced by coarsely dividing the model domain into a fixed number of vertical layers (
๐พ๐พ โค ); thereby, improving the frequency dispersion (e.g. SWASH, NHWAVE or XB-NH in multi-layered mode (De Ridder, 2018)). By further increasing the number of vertical layers ( ๐พ๐พ โฅ ), models like SWASH may be extended to the depth-resolving class. This approach increases the computational demand but allows processes, such as undertow and the shoreward flow near the surface, to be resolved. Given that phase-resolving models require a grid resolution high enough to resolve the individual SS-wave components, they are generally computationally feasible only for areas of limited size. For large-scale modelling of wave motion, a phase-averaged approach is most commonly used. This type of model is constructed on the assumption that a random sea-state is composed of a superposition of linear waves whose height is a function of their frequency and direction of propagation. For an individual wave train the rate of change of wave energy (or action) flux is balanced by the wave energy transfer among different wave components in different directions and different frequencies, as well as energy input and dissipation. With the phase information filtered out, these models can use much courser computational grids and therefore be applied to large areas. However, as individual waves are not resolved, these models must be combined with empirical formulae to estimate wave run-up and overtopping (Oosterlo, et al., 2018, Sierra, et al., 2010). Commonly used spectral models in nearshore applications include SWAN (Booij, et al., 1999) and STWAVE (Smith, et al., 2001). These models are generally able to accurately reproduce higher harmonics (SS waves); however, they do not account for the interactions that force IG-wave motions (Cavaleri, et al., 2007), which tend to dominate in shallow water. With respect to previous model comparisons in shallow coastal environments, Buckley, et al. (2014) assessed the performance of SWASH, SWAN and XB-SB in predicting SS wave heights ( ๐ป๐ป ๐๐0 , SS ), IG wave heights ( ๐ป๐ป ๐๐0 , IG ) and setup ( ๐๐ฬ ) across a steep laboratory fringing reef profile (varying from 1:5 to 1:18.8). Results showed that each model was capable of accurately predicting ๐ป๐ป ๐๐0 , SS ; however, SWAN failed to simulate the transformation of energy to lower frequencies and thus, failed to predict ๐ป๐ป ๐๐0 , IG . Likewise, SWAN showed considerably more error in its prediction of ๐๐ฬ compared to SWASH and XB-SB. On the other hand, XB-SB performed comparably well to its phase-resolving counterparts in the prediction of nearshore wave heights; and surprisingly the extent of wave run-up, particularly when IG-waves dominated at the shoreline (Lashley, et al., 2018). From these previous studies, the points of discussion that naturally arise are: i) Can phase-averaged models like SWAN be accurately applied under very shallow conditions, where IG waves dominate and ๐๐ฬ is significant? ii)
Given that IG waves dominate, are models of increasing complexity needed or is a short-wave averaged but IG-wave resolving approach all that is required? and iii)
While attempts at model comparisons for wave overtopping have been made (St-Germain, et al., 2014, Vanneste, et al., 2014), no study to date has the full range of model complexity (from depth-resolving to phase-averaged) or successfully quantified the accuracy versus speed of these models under irregular wave forcing .
Objective
In the present study, it is our primary aim to quantify the accuracy versus speed of computation of six commonly-used nearshore wave models (Table 1) in their prediction of irregular wave overtopping of a dike with very shallow foreshore conditionsโwhere IG waves and setup contribute significantly.
Outline
This report is organized as follows: Section 2 provides descriptions of the physical and numerical models applied, followed by descriptions of key parameters and empirical formulae used in the analysis. It ends with a description of the metrics used to quantify model accuracy. In Section 3, the results of the model-data comparisons and the overall influence of IG waves on overtopping are presented and discussed. Section 4 concludes the report by summarising the findings, acknowledging the present studyโs limitations and identifying areas for future work.
Table 1 Overview of the numerical models considered for comparative analysis.
Model Model Type Wave Propagation Overtopping SS Waves IG Waves
OpenFOAM Phase-resolving Depth-resolving Directly Directly SWASH a BOSZ Depth-averaged XB-NH XB-SB IG-wave resolving; SS-wave averaged Action-balance Directly Directly for IG waves b SWAN Phase-averaged Action-balance Excluded Empirically a Does not resolve wave overturning or wave roller formation. b Does not include SS-wave overtopping. Methods
This section begins with a description of the physical model tests under consideration. After which it describes the five numerical models under evaluation, including their governing equations and setup details. A description on the parameters and metrics used to assess model accuracy and computation speed is then provided. Finally, the additional numerical simulations for comparative analysis are described.
Description of the Physical Models
In the present study, we consider two specific test cases that were both performed at Flanders Hydraulics Research in a smooth, 1-m wide section of their 70-m long and 1.45-m deep wave flume (Altomare, et al., 2016) with different deep water wave heights ( ๐ป๐ป ๐๐0 , ๐๐ , ๐๐๐ก๐ก๐ก๐ก๐๐ ), peak periods ( ๐๐ ๐๐ ), foreshore slopes ( ๐๐ ), initial water depths at the toe ( โ ๐ก๐ก๐ก๐ก๐ก๐ก ), dike slopes ( ๐ผ๐ผ ) and dike freeboards ( ๐ ๐ ๐๐ ) (Table 2). These cases were selected to cover a wide range of deep-water wave steepness ( ๐ ๐ ), from very mild ( ๐ ๐ = 0.007, typical of swell conditions) to very steep ( ๐ ๐ = 0.047, typical of wind-sea conditions). With relative water depths ( โ ๐ก๐ก๐ก๐ก๐ก๐ก ๐ป๐ป ๐๐0 , ๐๐ , ๐๐๐ก๐ก๐ก๐ก๐๐ โ ) < 1, these conditions are considered very shallow (Hofland, et al., 2017). Both experiments simulated irregular spilling waves (with breaker parameter based on ๐๐ , ๐๐ , ๐๐๐ก๐ก๐๐๐ก๐ก < 0.5) with a duration approximately equal to 500 waves to obtain accurate and comparable estimates of the mean overtopping discharge ( ๐๐๏ฟฝ ) (Romano, et al., 2015). Table 2 Summary of test conditions for both the mild- and steep-wave cases.
Case ๐ป๐ป ๐๐0 , ๐๐ , ๐๐๐ก๐ก๐ก๐ก๐๐ (m) ๐๐ ๐๐ (s) cot ๐๐ ๐ ๐ ๐๐โ โ ๐ก๐ก๐ก๐ก๐ก๐ก (m) โ ๐ก๐ก๐ก๐ก๐ก๐ก ๐ป๐ป ๐๐0 , ๐๐๐ก๐ก๐ก๐ก๐๐ cot ๐ผ๐ผ ๐ ๐ ๐๐ (m) Mild swell 0.06 2.29 50 0.007 0.98 0.032 0.53 2 0.06 Steep wind-wave 0.21 1.70 35 0.047 1.45 0.025 0.12 3 0.08 For the mild swell-wave case, the variations of water-surface elevations were measured using 10 resistance-type gauges, all synchronously sampling at 50 Hz (Figure 2a); while 6 gauges with a sample frequency of 20 Hz were used in the steep-wave case (Figure 2b). In the analysis to follow, the term โoffshoreโ is used to refer to gauges 1 to 7 and 1 to 3 of the mild swell and steep-wind wave cases, respectively; and the term โnearshoreโ to refer to gauges 8 to 10 and 4 to 6, respectively. In either case, the term โtoeโ refers to the last wave gauge (gauge 10 and gauge 6 of the mild swell and steep wind-wave cases, respectively). In both cases, the instantaneous overtopping was measured using two Balluff โMicropulseโ water sensors situated inside the overtopping box; and ๐๐๏ฟฝ was then obtained by dividing the total volume of water collected at the end during the test by the total test duration.
Figure 2 Physical model setups showing gauge locations for both the: a) mild swell; and b) steep wind-wave cases.
Description of Numerical Models
In this study, six widely-used open-source numerical wave models are considered for comparative analysis. Each model is forced at its boundary with still water levels and parametric spectra (JONSWAP) to match those observed at the most offshore wave gauge during each physical experiment. Likewise, the smooth flume bottom was represented as either a Manning coefficient ( ๐๐ ) of 0.01 s/m or a Nikuradse geometrical roughness ( ๐๐ ๐ ๐ ) of 0.3 x 10 -3 m (in the case of SWAN). A general description of each model is provided in the sections that follow. As we investigate two extremes: very mild swell and very steep wind waves, it is reasonable that some calibration was required for the depth-averaged models (BOSZ, XB-NH and XB-SB). Therefore, a description of the main calibration parameters, their optimum values and impact on model results is also provided. In general, calibration was aimed at reducing the error in ๐๐ ๏ฟฝ and ๐ป๐ป ๐๐0 , ๐๐ . -1-0.5 0 Bed LevelOffshore GaugesNearshore Gauges OpenFOAM
The software OpenFOAM is an Open Source object-oriented library, composed by solvers and utilities (Jasak, et al., 2007). The formers are designed to numerically solve continuum mechanics problems, while the latter perform tasks involving data manipulation. For the present study, the library waves2Foam , a toolbox capable of generating and absorbing free surface water waves, has been adopted. Currently, the method applies the relaxation zone technique (active sponge layers) and supports a large range of wave theories (Jacobsen et al.,
Reynolds Averaged NavierโStokes equations (Equations 1 and 2): ๐๐๐๐๐๐๐๐๐๐ + โ ยท [ ๐๐๐๐๐๐ ๐๐ ] = โ โ๐๐ โ โ ๐๐ ยท ๐ฅ๐ฅโ๐๐ + โ ยท [ ๐๐โ๐๐ + ๐๐๐๐ ] + ๐๐ ๐๐ ๐ ๐ ๐พ๐พ โ๐พ๐พ , (1) coupled with the continuity equation (2) for incompressible ๏ฌow: โ ยท ๐๐ = 0, (2) where ๐๐ is the velocity ๏ฌeld, ๐๐ โ is the dynamic pressure component, ๐๐ is the density, g is the acceleration due to gravity and ๐๐ is the dynamic molecular viscosity. The Reynolds stress tensor ๐๐ is defined as: ๐๐ = 2 ๐๐ ๐๐ ๐ก๐ก ๐๐ โ ๐๐๐๐ , (3) where ๐๐ ๐ก๐ก is the dynamic eddy viscosity, ๐๐ is the strain rate tensor, ๐๐ is the turbulent kinetic energy per unit mass and ๐๐ is the identity matrix. The last term in Equation 1 is the effect of surface tension, where ๐๐ ๐๐ is the surface tension coef๏ฌcient and ๐ ๐ ๐พ๐พ is the surface curvature (Jacobsen, et al., 2012). The track of the free surface is performed by using the VOF method (Hirt and Nichols, 1981). For the mild and the steep cases, two regular slightly different meshes have been generated, to account for the differences between the two wave conditions. The numerical domains of the mild and steep cases are respectively composed by 49021 and by 70316 cells, with a graded mesh both in the x (0.3-0.005 m for the mild, 0.1-0.01 m for the steep) and in the y (0.05-0.005 m for the mild, 0.1-0.01 m for the steep) directions. In both cases, the selected regular and constant mesh allowed for a fair compromise between the computational effort and the accuracy of the results. 1 SWASH
SWASH is a time domain model for simulating non-hydrostatic, free-surface and rotational flow. It solves the NLSW equations with an added non-hydrostatic pressure correction term (Smit, et al., 2013): ๐๐๐๐๐๐๐๐ + ๐๐๐๐โ๐๐๐ฅ๐ฅ = 0, (4) ๐๐๐๐๐๐๐๐ + ๐๐๐๐๐๐๐๐๐ฅ๐ฅ + ๐๐๐๐๐๐๐๐๐๐ = โ ๐๐ ๐๐ ( ๐๐ โ + ๐๐ ๐๐โ ) ๐๐๐ฅ๐ฅ + ๐๐๐๐ ๐ฅ๐ฅ๐ฅ๐ฅ ๐๐๐ฅ๐ฅ + ๐๐๐๐ ๐ฅ๐ฅ๐ฅ๐ฅ ๐๐๐๐ , (5) ๐๐๐๐๐๐๐๐ + ๐๐๐๐๐๐๐๐๐ฅ๐ฅ + ๐๐๐๐๐๐๐๐๐๐ = โ ๐๐ ๐๐๐๐ ๐๐โ ๐๐๐๐ + ๐๐๐๐ ๐ฅ๐ฅ๐ฅ๐ฅ ๐๐๐๐ + ๐๐๐๐ ๐ฅ๐ฅ๐ฅ๐ฅ ๐๐๐ฅ๐ฅ , (6) ๐๐๐๐๐๐๐ฅ๐ฅ + ๐๐๐๐๐๐๐๐ = 0, (7) where ๐๐ is the free surface elevation; ๐๐ ( ๐ฅ๐ฅ , ๐๐ , ๐๐ ) and ๐๐ ( ๐ฅ๐ฅ , ๐๐ , ๐๐ ) are the horizontal and vertical velocities, respectively; โ is the water depth; ๐๐ is the density of water; ๐๐ โ and ๐๐ ๐๐โ are the hydrostatic and non-hydrostatic pressures, respectively; and ๐๐ ๐ฅ๐ฅ๐ฅ๐ฅ , ๐๐ ๐ฅ๐ฅ๐ฅ๐ฅ , ๐๐ ๐ฅ๐ฅ๐ฅ๐ฅ and ๐๐ ๐ฅ๐ฅ๐ฅ๐ฅ are the turbulent stresses. The model exhibits good linear dispersion up to ๐๐โ โ
8 and ๐๐โ โ
16 with two and three equidistant (sigma) vertical layers ( ๐พ๐พ ), respectively; its frequency dispersion is further improved by increasing ๐พ๐พ . Here, the model was applied with ๐พ๐พ = 20, which is sufficient for the phase velocity at the breaking wave front to be computed accurately. As such, no additional control is required to initiate or terminate wave breaking. The vertical pressure gradient was discretized by the standard central differencing scheme with the ILU pre-conditioner. The standard k- ฮต turbulence model is applied to take into account vertical mixing. A cross-shore grid spacing ( โ๐ฅ๐ฅ ) of 0.04 m was specified for both the mild- and steep-wave cases. This resulted in approximately 200 and 110 grid cells per deep-water wavelength ( ๐ฟ๐ฟ โ๐ฅ๐ฅโ ) for the mild- and steep-wave cases, respectively. For phase-resolving models, ๐ฟ๐ฟ โ๐ฅ๐ฅโ is typically kept between 50 and 100 (by rule of thumb) to ensure that the wave components are accurately resolved; however, as waves propagate in very shallow water, the local wavelength becomes much shorter than ๐ฟ๐ฟ . Thus, in order to maintain a reasonable number of grid cells per local wave length, these higher-than-typical grid resolutions ( ๐ฟ๐ฟ โ๐ฅ๐ฅโ = 200 and 110) were specified. 2 Rijnsdorp, et al. (2017) proposed a sub-grid approach to improve model efficiency, where vertical accelerations and non-hydrostatic pressures are resolved on a relative course grid while the horizontal velocities and turbulent stresses are resolved on a much finer sub-grid. This approach was attempted here, however, the simulations failed due to instabilities. BOSZ
The BOSZ wave modelโwhich is freely-available upon request from the developersโcomputes hazardous free surface flow problems ranging from near-field tsunamis to extreme swell ranges generated by hurricanes. It solves the following re-formulated, depth-integrated Boussinesq equations of Nwogu (1993), in vector notation: ๐๐ ๐ก๐ก + โ [( โ + ๐๐ )U] + โ ยท ๏ฟฝ๏ฟฝ๐๐ฬ ๐ผ๐ผ2 โ โ ๏ฟฝ โโ ( โ ยท U) + ๏ฟฝ๐๐ฬ ๐ผ๐ผ + โ ๏ฟฝ โโ [ โ ยท ( โ U)] ๏ฟฝ = 0 (8) U ๐ก๐ก + U( โ ยท U) + ๐๐๐ป๐ป๐๐ + ๏ฟฝ๐๐ฬ ๐ผ๐ผ2 โ ( โ ยท U) + ๐๐ฬ ๐ผ๐ผ โ [ โ ยท ( โ U)] ๏ฟฝ ๐ก๐ก = 0, (9) where U is the horizontal flow velocity defined at a reference depth ๐๐ฬ ๐ผ๐ผ = -0.55502 โ (Simarro, et al., 2013). The governing equations exhibit good dispersion accuracy up to ๐๐โ โ ๐๐ . Given the difficulty of Boussinesq equations in handling flow discontinuities (such as with breaking waves), the model deactivates the dispersion terms during wave breaking and makes use of the underlying NLSW equations where the breaking wave is then approximated as a bore or hydraulic jump. Wave breakingโand the deactivation of the dispersion termsโoccurs in the model based on the momentum gradient: ( โ + ๐๐ ) ๐๐๐๐๐๐๐ฅ๐ฅ > ๐ต๐ต๏ฟฝ๐๐ ( โ + ๐๐ ), (10) where ๐ต๐ต is a calibration coefficient (by default = 0.5). Here, ๐ต๐ต = 0.8 produced the best agreement between model and observations for both cases. This suggests that under these particularly shallow conditions, the wave face becomes very steep prior to breaking. For a detailed overview of the modelโs sensitivity to this parameter, the reader is pointed to (Roeber, et al., 2010). All other model parameters were kept at their default values. The grid resolution ( ๐ฟ๐ฟ ๐๐๐ฅ๐ฅโ ) was set as 200 for the mild swell-wave case but was reduced to 60 for the steep wind-wave case to ensure model stability. For the steep-wave case, higher grid resolutions and lower ๐ต๐ต values led to instabilities in the form of strong oscillations in surface elevation in the breaking region. This phenomenon, explored extensively by Kazolea and 3 Ricchiuto (2018), is due to the modelโs hybrid approach to handling wave breaking; that is, where the Boussinesq equations are reduced to the NLSW equations during wave breaking. It should be noted that Boussinesq wave models which take a different (eddy viscosity) approach to wave breaking reportedly show less sensitivity to the grid size (Kazolea and Ricchiuto, 2018); however, this was not evaluated here. XBeach Non-hydrostatic
Like SWASH, XB-NH solves the NLSW equations with a non-hydrostatic pressure correction term (Equations 4 to 7). Here, XBeach version 1.235527 (also known as the โXBeachXโ release) is applied in reduced (simplified) two-layer mode, where the non-hydrostatic pressure is assumed constant in the lower (first) layer (De Ridder, 2018). The water depth is divided into two layers with heights ๐๐ = ๐ผ๐ผโ and ๐๐ = (1 โ ๐ผ๐ผ ) โ , where ๐ผ๐ผ is the layer distribution. The resulting layer-averaged velocities ( ๐๐ and ๐๐ ) are transformed to a depth-averaged velocity ( ๐๐ ) and a velocity difference ( ฮ u ). Due to the simplified non-hydrostatic pressure in the lower layer, the vertical velocity between layers is neglected. Therefore, only the continuity relation for the upper (second) layer is required: ๐๐๐๐๐ฅ๐ฅ [(1 + ๐ผ๐ผ ) โ๐๐ + (1 โ ๐ผ๐ผ ) โ๐ผ๐ผฮ u] + 2 ๐๐ โ ๐๐ ๐๐๐๐๐๐๐ฅ๐ฅ โ ๐๐ ๐๐๐๐ ๐๐๐ฅ๐ฅ = 0, (11) To determine the water elevation, the global continuity equation is applied: ๐๐๐๐๐๐๐๐ + ๐๐โ๐๐๐๐๐ฅ๐ฅ = 0, (12) In order to control the computed location and magnitude of depth-limited wave breaking, a hydrostatic front approximation is applied. With this, the pressure distribution under breaking waves is considered hydrostatic when the local surface steepness exceeds a maximum prescribed value ( ๐๐ = 0.5, by default): ๐๐๐๐๐๐๐๐ > ๐๐ , (13) Here, ๐๐ = 0.9 and 0.7 produced the best agreement between the model and observations for the mild- and steep-wave cases, respectively. This further supports the statement that for very shallow foreshores, the waves become particularly steep before breaking. All other model parameters were kept at their default values. Additionally, the grid resolution ( ๐ฟ๐ฟ โ๐ฅ๐ฅโ ) was set to ~200 and ~180 for the two respective cases. 4 XBeach Surfbeat
XB-SB solves SS-wave motions using the wave-action equation with time-dependent forcing, similar to that of the HISWA model (Holthuijsen, et al., 1989). The model represents the SS-wave frequency spectrum by a single frequency ( ๐๐ ๐๐๐ก๐ก๐๐ ) and the wave-action equation is applied at the timescale of the wave group: ๐๐๐๐๐๐๐๐ + ๐๐๐๐ ๐๐๐ฅ๐ฅ ๐๐๐๐๐ฅ๐ฅ = โ ๐ท๐ท ๐ค๐ค ๐๐ , (14) ๐๐ ( ๐ฅ๐ฅ , ๐๐ ) = ๐๐ ๐ค๐ค ( ๐ฅ๐ฅ , ๐๐ ) ๐๐ ( ๐ฅ๐ฅ , ๐๐ ) (15) ๐๐ = ๏ฟฝ๐๐๐๐ tanh ๐๐โ (16) where ๐๐ is the wave action, ๐๐ ๐ค๐ค is the wave energy density, ๐๐ is the intrinsic wave frequency, ๐๐ is the wave number, ๐ท๐ท ๐ค๐ค is a dissipation term to account for wave breaking and ๐๐ ๐๐๐ฅ๐ฅ is the wave-action propagation speed in the cross-shore direction. To simulate wave breaking, XB-SB applies a dissipation model (Roelvink, 1993), by default, for use with SS-wave groups; and a roller model (Nairn, et al., 1991, Svendsen, 1984) to represent momentum stored in surface rollers which results in a shoreward delay in wave forcing. The radiation stress gradients that result from these variations in wave action exert forces on the water column and drive IG waves and unsteady currents which are solved by the NLSW equations (Equations 4 to 7). Therefore, the model directly simulates wave-driven currents and the run-up and overtopping of IG waves. ๐ท๐ท๏ฟฝ ๐ค๐ค = 2 ๐ผ๐ผ๐๐ ๐๐๐ก๐ก๐๐ ๐๐ ๐๐ ๐ธ๐ธ ๐ค๐ค ๐ป๐ป ๐๐๐๐๐ ๐ โ , (17) ๐๐ ๐๐ = 1 โ exp ๏ฟฝโ ๏ฟฝ ๐ป๐ป ๐๐๐๐๐ ๐ ๐ป๐ป ๐๐๐๐๐ฅ๐ฅ ๏ฟฝ ๏ฟฝ (18) where ๐ท๐ท๏ฟฝ ๐ค๐ค is the total (directionally-integrated) wave energy dissipation due to breaking, ๐๐ ๐๐๐ก๐ก๐๐ =1 ๐๐ ๐๐๐ก๐ก๐๐ โ is the representative wave period and ๐๐ ๐๐ is the fraction of breaking waves; the root-mean-square SS-wave height , ๐ป๐ป ๐๐๐๐๐ ๐ = ๏ฟฝ ๐ธ๐ธ ๐ค๐ค ๐๐๐๐โ ; the maximum wave height, ๐ป๐ป ๐๐๐๐๐ฅ๐ฅ = ๐พ๐พ ๐๐ โ ; ๐ธ๐ธ ๐ค๐ค is the wave-group varying SS-wave energy; ๐ผ๐ผ is a dissipation (by default = 1) and ๐พ๐พ ๐๐ is the ratio of breaking waves to local water depth (by default = 0.55 but typically used for calibration). Here, ๐พ๐พ ๐๐ = 0.45 and 0.65 provided the best agreement between the model and observations for the mild swell and steep wind-wave cases, respectively. XB-SB does not directly produce the SS-wave component of the energy density spectrum, instead it computes the change in SS-wave energy as a change in the bulk ๐ป๐ป ๐๐๐๐๐ ๐ parameter, as 5 described above. In order to produce a complete energy density ( ๐ถ๐ถ ๐๐๐๐ ) spectrum at each gauge location, a JONSWAP distribution was assumed around the peak-frequency ( ๐๐ ๐๐ ), where ๏ฟฝ โซ ๐ถ๐ถ ๐๐๐๐ ๐๐๐๐ ๐๐ / = ๐ป๐ป ๐๐๐๐๐ ๐ . This SS-wave spectrum (Figure 3b) was then combined with the IG-wave spectrum (Figure 3a)โobtained directly from the computed surface elevationโto produce the complete spectrum (Figure 3c). Figure 3 Example of a) the IG-wave spectrum based on the computed surface elevation; b) an assumed SS-wave spectrum (JONSWAP shape) based on the computed root-mean-square SS-wave height ( ๐ฏ๐ฏ ๐๐๐๐๐๐ ); and c) the total combined spectrum, for XB-SB at the dike toe (steep wind-wave case). For the mild swell-wave case, the grid resolution was varied such that it increased shoreward. This reduced computation time while ensuring that the steep dike slope was accurately capture. As such, ๐ฟ๐ฟ โ๐ฅ๐ฅโ varied from ~25 (offshore) to ~160 (at the dike) in the mild-wave case; and from ~45 to ~90 in the steep-wave case. SWAN
SWAN is a third-generation, phase-averaged wave model used to estimate the generation (by wind), propagation and dissipation (by depth-induced breaking and bottom friction) of waves from deep water to the surf zone. This includes wave-wave interactions, in both deep and shallow water, and wave-induced setup; but neglects wave-induced currents and the generation or propagation of IG waves. Like XB-SB, SWAN computes the spectral evolution of ๐๐ in space and time. This is done in a manner similar to Equation 14; however, unlike XB-SB which makes -3 -3 -3
6 use of a single representative frequency, SWAN takes the frequency distribution of action density into account. To simulate wave breaking, SWAN uses the following parametric dissipation model (Battjes and Janssen, 1978):
๐ท๐ท๏ฟฝ ๐ค๐ค = ๐ผ๐ผ ๐๐๐๐๐๐ ๐๐๐ก๐ก๐๐๐๐ ๐๐ ๐๐ ๐ป๐ป ๐๐๐๐๐ฅ๐ฅ2 , (19) and ๐๐ ๐๐ is estimated as: โ ๐๐ ๐๐ ln ๐๐ ๐๐ = โ ๐ธ๐ธ ๐ก๐ก๐ก๐ก๐ก๐ก ๐ป๐ป ๐๐๐๐๐ฅ๐ฅ2 , (20) where ๐๐ ๐๐๐ก๐ก๐๐๐๐ is the mean wave frequency, ๐ป๐ป ๐๐๐๐๐ฅ๐ฅ = ๐พ๐พ ๐๐๐๐ โ and ๐ธ๐ธ ๐ก๐ก๐ก๐ก๐ก๐ก is the total wave-energy variance. Here, ๐พ๐พ ๐๐๐๐ = 0.73 (default value) provided good agreement between the model and observations for both the mild swell and steep wind-wave cases. For both wave cases, a constant grid spacing of 0.25 m was applied. This corresponded to ๐ฟ๐ฟ โ๐ฅ๐ฅโ โ
30 for the mild-wave case and ๐ฟ๐ฟ โ๐ฅ๐ฅโ โ
20 for the steep-wave case.
Data Processing and Analysis
Mean water level
The mean water level ( ๐๐ฬ ) was calculated by taking the average of the surface elevation, ๐๐ ( ๐๐ ) , at each gauge location, relative to the elevation of the dike toe. The wave-induced setup, < ๐๐ >, was then obtained as the difference between ๐๐ฬ at each gauge location and ๐๐ฬ at the most offshore gauge. Separation of infragravity and sea-swell waves
The time series of ๐๐ ( ๐๐ ) were further analysed using the Welchโs average periodogram method and a Hann filter with a 50% maximum overlap. The resulting one-dimensional spectra of wave energy density, ๐ถ๐ถ ๐๐๐๐ ( ๐๐ ) โwith ~43 degrees of freedom and a frequency resolution of ~0.008 Hzโwere then used to determine ๐ป๐ป ๐๐0 , ๐๐ , ๐ป๐ป ๐๐0 , ๐๐๐๐ and ๐ป๐ป ๐๐0 , ๐ผ๐ผ๐ผ๐ผ , as follows: ๐ป๐ป ๐๐0 , ๐๐ = 4 ๏ฟฝ ๏ฟฝ ๐ถ๐ถ ๐๐๐๐ ๐๐๐๐ . , (21) ๐ป๐ป ๐๐0 , ๐๐๐๐ = 4 ๏ฟฝ ๏ฟฝ ๐ถ๐ถ ๐๐๐๐ ๐๐๐๐ ๐๐ / , (22) and 7 ๐ป๐ป ๐๐0 , ๐ผ๐ผ๐ผ๐ผ = 4 ๏ฟฝ ๏ฟฝ ๐ถ๐ถ ๐๐๐๐ ๐๐๐๐ , ๐๐ ๐๐ / . (23) where half the peak frequency ( ๐๐ ๐๐ โ = 1/2 ๐๐ ๐๐ ) is taken as the cut-off to separate SS and IG motions (Roelvink and Stive, 1989). This choice of cut-off frequency is based on the tendency that, in deep water, the majority of SS-wave energy is found at frequencies > ๐๐ ๐๐ โ while the majority of IG-wave energy lies at frequencies < ๐๐ ๐๐ โ . Spectral Wave Period
In addition to wave heights, the spectral wave period ( ๐๐ ๐๐โ1 , ) at each gauge location was calculated as follows: ๐๐ ๐๐โ1 , = ๐๐ โ1 ๐๐ , (24) where, ๐๐ ๐๐ = ๏ฟฝ ๏ฟฝ ๐ถ๐ถ ๐๐๐๐ ยท ๐๐ ๐๐ ๐๐๐๐ . . (25) Empirical Estimate of the Incident Infragravity Waves
As SWAN neglects the contribution of IG waves to the total wave incident height at the dike toe ( ๐ป๐ป ๐๐0 , ๐๐ , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐ ) we apply an empirical correction, proposed by (Lashley, et al., Forthcoming ). Using a dataset of 672 XBeach simulations, an empirical formula for the relative magnitude of the IG waves (
๐ป๐ป๏ฟฝ
๐ผ๐ผ๐ผ๐ผ ) was derived with influence factors to account for variations in offshore wave directional spreading ( ๐พ๐พฬ ๐๐ ), โ ๐ก๐ก๐ก๐ก๐ก๐ก ( ๐พ๐พฬ โ ), cot ๐๐ ( ๐พ๐พฬ ๐๐ ), vegetation ( ๐พ๐พฬ ๐ฃ๐ฃ ) and cot ๐ผ๐ผ ( ๐พ๐พฬ ๐๐ ): ๐ป๐ป๏ฟฝ
๐ผ๐ผ๐ผ๐ผ = 0.36 ยท ๐ป๐ป ๐๐0 , ๐๐ , ๐๐๐ก๐ก๐ก๐ก๐๐0 . โ ๐พ๐พฬ ๐๐ โ ๐พ๐พฬ โ โ ๐พ๐พฬ ๐๐ โ ๐พ๐พฬ ๐ฃ๐ฃ โ ๐พ๐พฬ ๐๐ , (26) For an incident waves analysis (i.e. without the influence of the dike slope) with no directional spreading (1D flume conditions) or vegetation, ๐พ๐พฬ ๐๐ , ๐พ๐พฬ ๐ฃ๐ฃ and ๐พ๐พฬ ๐๐ = 1; while, ๐พ๐พฬ โ = 1.04 โ exp( โ โ โ ๐ก๐ก๐ก๐ก๐ก๐ก ) + 0.9 โ exp( โ โ โ ๐ก๐ก๐ก๐ก๐ก๐ก ) (27) and ๐พ๐พฬ ๐๐ = 1.56 โ ๐ผ๐ผ ๐๐๐ก๐ก๐๐๐ก๐กโ0 . . (28) As ๐ป๐ป๏ฟฝ
๐ผ๐ผ๐ผ๐ผ represents the ratio of IG to SS waves, ๐ป๐ป ๐๐0 , ๐ผ๐ผ๐ผ๐ผ , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐ can be obtained from a SWAN estimate of ๐ป๐ป ๐๐0 , ๐๐๐๐ , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐ : 8 ๐ป๐ป ๐๐0 , ๐ผ๐ผ๐ผ๐ผ , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐ = ๐ป๐ป๏ฟฝ
๐ผ๐ผ๐ผ๐ผ ยท ๐ป๐ป ๐๐0 , ๐๐๐๐ , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐ . (29) Finally, a corrected estimate of ๐ป๐ป ๐๐0 , ๐๐ , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐ was obtained as follows: ๐ป๐ป ๐๐0 , ๐๐ , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐ = ๏ฟฝ๐ป๐ป ๐๐0 , ๐๐๐๐ , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐2 + ๐ป๐ป ๐๐0 , ๐ผ๐ผ๐ผ๐ผ , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐2 . (30) Empirical Wave Overtopping
While the fully phase-averaged models like SWAN areโto some extentโable to estimate nearshore wave conditions, they cannot directly simulate wave overtopping, as this requires that the individual waves be resolved. In order to estimate wave overtopping, these models can be (and are often) combined with well-established empirical models that require wave parameters at the dike toe as input. In the present study, the EurOtop (2018) formulae based on the work of Van Gent (1999) and Altomare, et al. (2016) for (very) shallow foreshores are applied in combination with SWAN. For smooth dikes under perpendicular wave attack with โ ๐ก๐ก๐ก๐ก๐ก๐ก ๐ป๐ป ๐๐0 , ๐๐ , ๐๐๐ก๐ก๐ก๐ก๐๐ โ < 1.5: ๐๐๏ฟฝ๏ฟฝ๐๐ ยท ๐ป๐ป ๐๐0 , ๐๐ , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐3 = 10 โ0 . ยท exp ๏ฟฝโ ๐ ๐ ๐๐ ๐ป๐ป ๐๐0 , ๐๐ , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐ ยท ๏ฟฝ ๐๐ ๐๐โ1 , ๏ฟฝ๏ฟฝ , (31) with ๐๐ ๐๐โ1 , = tan ฮฑ ๐ ๐ ๐๐ ๏ฟฝ๐ป๐ป ๐๐0 , ๐๐ , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐ ๐ฟ๐ฟ ๐๐โ1 , ๏ฟฝ , (32) ๐ฟ๐ฟ ๐๐โ1 , = ๐๐ ยท ๐๐ ๐๐โ1 , , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐2 ๐๐ , (33) tan ฮฑ ๐ ๐ ๐๐ = 1.5 ๐ป๐ป ๐๐0 , ๐๐ , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐ + ๐ ๐ ๐ข๐ข2% ๏ฟฝ ๐ป๐ป ๐๐0 , ๐๐ , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐ โ โ ๐ก๐ก๐ก๐ก๐ก๐ก ๏ฟฝ ยท ๐๐ + ( โ ๐ก๐ก๐ก๐ก๐ก๐ก + ๐ ๐ ๐ข๐ข2% ) ยท cot ๐ผ๐ผ , (34) ๐ ๐ ๐ข๐ข2% ๐ป๐ป ๐๐0 , ๐๐ , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐ = 4 โ ๏ฟฝ๐๐ ๐๐โ1 , , (35) where ๐๐ is the gravitational constant of acceleration, ฮฑ ๐ ๐ ๐๐ is an equivalent slope (to account for waves breaking on the foreshore) and ๐๐ ๐๐โ1 , , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐ is the spectral wave period at the dike toe based on the incident waves (i.e. without the influence of waves reflected at the dike). It should be noted that ๐๐ ๐๐โ1 , and ๐ ๐ ๐ข๐ข2% are obtained iteratively (until ๐ ๐ ๐ข๐ข2% converges), with a first estimate of ๐ ๐ ๐ข๐ข2% = 1.5 ๐ป๐ป ๐๐0 , ๐๐ , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐ . 9 Additionally, as SWAN excludes the contribution of IG waves, corrected estimates of ๐๐ ๐๐โ1 , , ๐ก๐ก๐ก๐ก๐ก๐ก are typically obtained using Equations 36 and 37 (Hofland, et al., 2017), as outlined in the EurOtop (2018) manual: ๐๐ ๐๐โ1 , , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐ ๐๐ ๐๐โ1 , , ๐๐๐ก๐ก๐ก๐ก๐๐ โ ๏ฟฝโ โ๏ฟฝ๏ฟฝ + exp ๏ฟฝโโ๏ฟฝ๏ฟฝ , (36) where, โ๏ฟฝ = โ ๐ก๐ก๐ก๐ก๐ก๐ก ๐ป๐ป ๐๐0 , ๐๐ , ๐๐๐ก๐ก๐ก๐ก๐๐ ๏ฟฝ cot ๐๐ ๏ฟฝ . . (37) Error Metrics
In order to compare the performance of the numerical models, we assess the mean relative accuracy in an approach similar to that of Lynett, et al. (2017):
๐๐๐๐๐๐๐๐
๐ ๐ ๐๐๐๐๐ ๐ ๐ ๐ ๐น๐น = 1 ๐๐ ๏ฟฝ ๐น๐น ๐๐๐ก๐ก๐๐๐๐ ๐น๐น ๐ก๐ก๐๐๐ ๐ ๐๐๐๐๐๐=1 , (38) where ๐น๐น is a stand-in for the parameter under consideration ( ๐๐ฬ , ๐ป๐ป ๐๐0 , ๐๐ , ๐ป๐ป ๐๐0 , ๐๐๐๐ , ๐ป๐ป ๐๐0 , ๐ผ๐ผ๐ผ๐ผ and ๐๐ ๐๐โ1 , ) for the ๐๐ wave-gauge locations; and subscripts ๐๐๐ ๐ ๐๐ and ๐ ๐ ๐๐๐ ๐ refer to model predictions and observations made during the physical experiment, respectively. Here we make a distinction between gauges offshore and nearshore (Figure 2) A mean ratio of 1 suggests perfect agreement between the model and observations, while values higher or lower than one indicate over- or under-predictions, respectively. It should be noted that all wave gauges (offshore and at the dike toe, see Figure 2) are considered in Equation 38. While the focus of this study is primarily at the dike toe, it is important to assess the model performance offshore to ensure that: i) the boundary conditions are correctly modelled; and ii) that no (significant) numerical dissipation occurs in deep water, as a result of a coarse grid resolution for example. Finally, the performance of each model for wave overtopping was also assessed by comparing the absolute relative error in the prediction of mean overtopping discharge: ๐๐๐๐๐ ๐ ๐ ๐ ๐ด๐ด๐๐๐๐๐๐
๐ ๐ ๐๐๐ด๐ด๐๐๐๐๐ ๐ ๐ ๐ ๐๐
๐ธ๐ธ๐ธ๐ธ๐ธ๐ธ๐ ๐ ๐ธ๐ธ ๐๐๏ฟฝ = ๏ฟฝ โ ๐๐๏ฟฝ ๐๐๐ก๐ก๐๐ ๐๐๏ฟฝ ๐ก๐ก๐๐๐ ๐ ๏ฟฝ . (39) Computation Speed
Two work stations (WS) were used to carry out this research (Table 3). Given the required computational effort, the OpenFOAM simulations were performed on WS-A, while the other 0 models were run on WS-B. To assess computation speed, the duration of each simulation (in wall clock time) was recorded.
Table 3 Overview of work stations used to carry out the numerical simulations.
Work Station (WS) A B Operating System Ubuntu 14.04 LTS Windows 10 Memory 31.2 GB 16 GB Processor Intel Xeonยฎ CPU ES-2690 v3 @ 2.60 GHz x 16 Intelยฎ Coreโข i7-6600 CPU @ 2.60GHz, 2.81 GHz x 4 Graphics Gallium 0.4 on NVE7 Intelยฎ HD Graphics 520 Type 64-bit 64-bit Disk 1.9 TB 239 GB Results and Discussions
In this section, the results of the model-data comparisons are presented and discussed. As wave overtopping is the end result of wave propagation, the performance of each model for the prediction of mean water levels, wave heights and periods is first assessed. For the models where calibration was carried out (BOSZ, XB-NH and XB-SB), both default and calibrated results are presented. Note that no parameter tuning was done for the depth-resolving models (OpenFOAM and SWASH) as wave breaking is intrinsically resolved. Likewise, SWAN with default settings showed reasonable agreement and was therefore not calibrated. Lastly, it should be noted that the BOSZ simulation of the steep wind-wave case with default settings resulted in instabilities (see Section 2.2.3) and is therefore not included in the analysis.
Mean Water Level
Each model, excluding OpenFOAM, is able to accurately (within 15% error) and consistently reproduce ๐๐ ๏ฟฝ for both the mild swell (Figure 4a) and steep wind-wave (Figure 4b) cases. This includes the increase in ๐๐ ๏ฟฝ nearshore, referred to as wave-induced setup (< ๐๐ >), highlighted in Figure 5 with the XB-NH results representing the general behaviour of the numerical models. While OpenFOAM agrees well with the observations for the mild swell case, it overestimates < ๐๐ > offshore (gauges 2 and 3) and underestimates < ๐๐ > nearshore (gauges 4 to 6) for the steep wind-wave case (Figure 5). This may be indicative of premature wave breaking in OpenFOAM. The satisfactory performance of BOSZ and XB-NH observed here (Figure 4) is in contrast with previous studies (Lashley, et al., 2018, Zhang, et al., 2019), which found that depth-averaged 1 models were unable to accurately estimate < ๐๐ > due to their lack of vertical resolution and exclusion of wave roller dynamics. However, the difference in model performance here is likely due to the spilling nature of the waves ( ๐๐ , ๐๐๐ก๐ก๐๐๐ก๐ก = 0.23 and 0.13 for the mild swell and steep wind-wave cases, respectively) compared to the plunging waves and steep fore-reef slopes assessed by Lashley, et al. (2018) ( ๐๐ , ๐๐๐ก๐ก๐๐๐ก๐ก > 1.1) and Zhang, et al. (2019) ( ๐๐ , ๐๐๐ก๐ก๐๐๐ก๐ก = 1.29). That is, while resolving the vertical structure of flow may be critical for plunging breakers, depth-averaged models are able to perform well under spilling waves. Figure 4 Mean ratio of modelled to observed ๐ผ๐ผ๏ฟฝ (markers) for both the a) mild swell and b) steep wind-wave cases, with error bars representing the standard deviation. Solid horizontal lines represent perfect agreement between model and observations. Dashed lines correspond to +/- 15% error.
Default OffshoreDefault NearshoreCalibrated OffshoreCalibrated Nearshore
OpenFOAM SWASH BOSZ XB-NH XB-SB SWAN00.511.5 Figure 5 Cross-shore profiles of modelled (XB-NH and OpenFOAM) and observed < ๐ผ๐ผ > for both the mild swell and steep wind-wave cases. There is also a notable difference in the observed maximum < ๐๐ > between the mild swell (< ๐๐ > = 0.004 m) and steep wind-wave (< ๐๐ > = 0.015 m) cases (Figure 5). This substantial increase in < ๐๐ > as โ ๐ก๐ก๐ก๐ก๐ก๐ก ๐ป๐ป ๐๐0 , ๐๐ , ๐๐๐ก๐ก๐ก๐ก๐๐ โ decreases agrees with the findings of Gourlay (1996) on shallow reefs, and suggests that < ๐๐ >โwhich contributes to wave run-up (Stockdon, et al., 2006) and, by extension, overtoppingโincreases proportionally as foreshores become more shallow, or as deep water wave conditions become more energetic. Significant Wave Height
SWASH, BOSZ, XB-NH and XB-SB are able to reproduce ๐ป๐ป ๐๐0 , ๐๐ , both offshore and nearshore, within 15% error for two cases (Figure 6). On the other hand, OpenFOAM and SWAN both show notable differences; with SWAN consistently and considerably underestimating ๐ป๐ป ๐๐0 , ๐๐ nearshore. While SWAN is able to accurately simulate the propagation of high-frequency waves ( ๐ป๐ป ๐๐0 , ๐๐๐๐ , Figure 7), it does not compute the low-frequency waves ( ๐ป๐ป ๐๐0 , ๐ผ๐ผ๐ผ๐ผ , Figure 8) and therefore underestimates ๐ป๐ป ๐๐0 , ๐๐ nearshore, where the contribution of IG waves is significant (Figure 6). The relatively high standard deviation associated with SWANโs nearshore ๐ป๐ป ๐๐0 , ๐๐๐๐ estimates is due to its exclusion of wave reflection. In the physical model, the superposition of the incident and reflected waves results in a nodal/anti-nodal pattern with a maximum at the dike (outsets in Figure 9a and Figure 10a). As SWAN excludes the reflected component, the model -40 -35 -30 -25 -20 -15 -10 -5 0-5051015 10 -3 Observed - MildXB-NH - MildOpenFOAM - MildObserved - SteepXB-NH - SteepOpenFOAM - Steep
3 underestimates ๐ป๐ป ๐๐0 , ๐๐๐๐ immediately in front of the dike, where the incident and reflected waves add up. On the other hand, this shortcoming makes SWAN especially suitable for use with the empirical overtopping models that require incident-wave conditions as input. SWAN also predicts a higher and lower maxima in ๐ป๐ป ๐๐0 , ๐๐๐๐ (just before breaking) than XB-NH, for the mild swell (Figure 9a) and steep wind-wave (Figure 10a) cases, respectively. This is likely due to the dissipation model employed by SWAN (Equation 19). Tuning ๐พ๐พ ๐๐๐๐ โthe parameter which controls the maximum wave height to water depth ratio in SWANโwould yield better agreement between the two models; however as there were no wave gauges in this region it is difficult to ascertain which model is correct here. With respect to OpenFOAM, the model shows inconsistent results between the two cases. Under the mild swell conditions, OpenFOAM underestimates ๐ป๐ป ๐๐0 , ๐๐ nearshore (Figure 6a); however for the steep wind-wave case, the model overestimates ๐ป๐ป ๐๐0 , ๐๐ nearshore (Figure 6b). In both cases, the model appears to be too dissipative, resulting in a reduction in ๐ป๐ป ๐๐0 , ๐๐๐๐ offshore. For the mild swell case, this dissipation is minor resulting in a consistent under-prediction of ๐ป๐ป ๐๐0 , ๐๐๐๐ (Figure 7a and Figure 9a) and ๐ป๐ป ๐๐0 , ๐ผ๐ผ๐ผ๐ผ (Figure 8a and Figure 9b). Under the steep wind-wave conditions, however, the dissipation is significant. This observation, combined with the overestimation of < ๐๐ > offshore (Figure 5), indicates premature wave breaking in OpenFOAM. This reduction in ๐ป๐ป ๐๐0 , ๐๐๐๐ offshore results in unbroken SS-waves reaching the dike and the overestimation of ๐ป๐ป ๐๐0 , ๐๐๐๐ nearshore (Figure 7b and Figure 10a). XB-NH also shows some numerical dissipation offshore but this is negligible compared to that of OpenFOAM (Figure 10a). As a reduction in grid size did not significantly improve the OpenFOAM model results, the observed dissipation is possibly due to an over-production of turbulence leading to premature wave decay (Larsen and Fuhrman, 2018). Though SWASH is able to accurately predict ๐ป๐ป ๐๐0 , ๐ผ๐ผ๐ผ๐ผ it underestimated ๐ป๐ป ๐๐0 , ๐๐๐๐ nearshore in both cases. This is possibly due to the standard k- ฮต turbulence model applied in multi-layered mode, which may overestimate the turbulent (vertical) viscosity. Similar to OpenFOAM, a reduction in grid size from 0.04 m to 0.025 m did not significantly improve the estimates (~3% change in ๐ป๐ป ๐๐0 , ๐๐๐๐ ). While calibration generally improved model performance for ๐๐ ๏ฟฝ (Figure 4), ๐ป๐ป ๐๐0 , ๐๐ (Figure 6) and ๐ป๐ป ๐๐0 , ๐ผ๐ผ๐ผ๐ผ (Figure 8), it resulted in the overestimation of ๐ป๐ป ๐๐0 , ๐๐๐๐ nearshore by XB-SB (Figure 7). This is as a result of tuning ๐พ๐พ ๐๐ (Equation 18) which affects both the maximum ๐ป๐ป ๐๐0 , ๐๐๐๐ and ๐ป๐ป ๐๐0 , ๐ผ๐ผ๐ผ๐ผ . Perhaps a different approach, where ๐ผ๐ผ (Equation 17)โthe parameter that controls the magnitude of dissipationโis calibrated (Lashley, et al., 2018) would 4 yield better results. However, as XB-SB predicts IG-wave overtopping only, the loss in accuracy for ๐ป๐ป ๐๐0 , ๐๐๐๐ to improve ๐ป๐ป ๐๐0 , ๐ผ๐ผ๐ผ๐ผ predictions was considered acceptable.
Figure 6 Mean ratio of modelled to observed ๐ฏ๐ฏ ๐๐๐๐ , ๐ป๐ป (markers) for both the a) mild swell and b) steep wind-wave cases, with error bars representing the standard deviation. Solid horizontal lines represent perfect agreement between model and observations. Dashed lines correspond to +/- 15% error. Default OffshoreDefault NearshoreCalibrated OffshoreCalibrated Nearshore
OpenFOAM SWASH BOSZ XB-NH XB-SB SWAN00.511.5 Figure 7 Mean ratio of modelled to observed ๐ฏ๐ฏ ๐๐๐๐ , ๐บ๐บ๐บ๐บ (markers) for both the a) mild swell and b) steep wind-wave cases, with error bars representing the standard deviation. Solid horizontal lines represent perfect agreement between model and observations. Dashed lines correspond to +/- 15% error.
Default OffshoreDefault NearshoreCalibrated OffshoreCalibrated Nearshore
OpenFOAM SWASH BOSZ XB-NH XB-SB SWAN00.511.5 Figure 8 Mean ratio of modelled to observed ๐ฏ๐ฏ ๐๐๐๐ , ๐ฐ๐ฐ๐ฐ๐ฐ (markers) for both the a) mild- and b) steep-wave cases, with error bars representing the standard deviation. Solid horizontal lines represent perfect agreement between model and observations. Dashed lines correspond to +/- 15% error.
Default OffshoreDefault NearshoreCalibrated OffshoreCalibrated Nearshore
OpenFOAM SWASH BOSZ XB-NH XB-SB SWAN00.511.5 Figure 9 Cross-shore profiles of modelled and observed: a) ๐ฏ๐ฏ ๐๐๐๐ , ๐บ๐บ๐บ๐บ and b) ๐ฏ๐ฏ ๐๐๐๐ , ๐ฐ๐ฐ๐ฐ๐ฐ for the mild-swell case; with c) bed level, for reference. Outset in panel โaโ magnifies the plot area between -1 and 0 m away from the dike.
ObservedOpenFOAMSWANXB-NH -1 -0.5 00.010.020.030.040.05 Figure 10 Cross-shore profiles of modelled and observed: a) ๐ฏ๐ฏ ๐๐๐๐ , ๐บ๐บ๐บ๐บ and b) ๐ฏ๐ฏ ๐๐๐๐ , ๐ฐ๐ฐ๐ฐ๐ฐ for the steep wind-wave case; with c) bed level, for reference. Outset in panel โaโ magnifies the plot area between -1 and 0 m away from the dike.
Spectral Wave Period
SWASH, XB-NH and XB-SB show good agreement between modelled and observed ๐๐ ๐๐โ1 , predictions; while SWAN, OpenFOAM and BOSZ show notable deviations. As the accurate prediction of ๐๐ ๐๐โ1 , requires the models to correctly represent the distribution of wave energy by frequency (Equation 25), we assess the modelled versus observed wave spectra (Figure 12). SWASH, BOSZ, XB-NH and XB-SB correctly capture the shift in peak energy density ( ๐ถ๐ถ ๐๐๐๐ ) from the SS-wave (Figure 12a and b) to the IG-wave band (Figure 12c and d); however BOSZ overestimates the magnitude of the IG peak and shows it at slightly lower frequencies than observed. This, coupled with a minor underestimation of the SS-wave energyโmost evident for the mild swell case (Figure 12c)โresults in an overestimation of ๐๐ ๐๐โ1 , . The consistent underestimation of ๐๐ ๐๐โ1 , by SWAN is expected due to its exclusion of ๐ถ๐ถ ๐๐๐๐ at IG frequencies (Figure 12c and d). OpenFOAM, on the other hand, does show a shift in energy ObservedOpenFOAMSWANXB-NH -1 -0.5 00.020.040.06
9 from offshore to the dike toe; however, it shows two distinct IG peaks (Figure 12c and d), not present in the observations. In the mild swell case, this misrepresentation of ๐ถ๐ถ ๐๐๐๐ at IG frequencies couple with the underestimation of ๐ถ๐ถ ๐๐๐๐ in the SS-wave band resulted in the significant overestimation of ๐๐ ๐๐โ1 , nearshore (Figure 11a). Under the steep wind-wave conditions, OpenFOAM also shows considerable ๐ถ๐ถ ๐๐๐๐ in the SS-wave band (nearshore) while the observed spectra shows very little (Figure 12d). This further supports the argument that due to premature wave decay in the model, some unbroken SS waves are able to reach the dike. Figure 11 Mean ratio of modelled to observed ๐ป๐ป ๐๐โ๐๐ , ๐๐ (markers) for both the a) mild swell and b) steep wind-wave cases, with error bars representing the standard deviation. Solid horizontal lines represent perfect agreement between model and observations. Dashed lines correspond to +/- 15% error. Default OffshoreDefault NearshoreCalibrated OffshoreCalibrated Nearshore
OpenFOAM SWASH BOSZ XB-NH XB-SB SWAN00.511.5 Figure 12 Model-data comparison of wave spectra: a) offshore (at gauge 1) and c) nearshore (at gauge 10) for the mild swell case; and b) offshore (at gauge 1) and d) nearshore (at gauge 6) for the steep wind-wave case. Dashed vertical lines indicate the frequency separating SS- and IG-wave motions.
Wave Overtopping
General
An important remark is the difference in the observed ๐๐๏ฟฝ between the mild swell (0.094 l/s per m) and steep wind-wave (0.205 l/s per m) cases. Despite having similar ๐ป๐ป ๐๐0 , ๐๐๐๐ , ๐ก๐ก๐ก๐ก๐ก๐ก valuesโ0.039 m (Figure 9) and 0.038 m (Figure 10a) for the mild swell and steep wind-wave cases, respectivelyโthe steep-wave case with a higher ๐ ๐ ๐๐ and lower โ ๐ก๐ก๐ก๐ก๐ก๐ก (Table 2), produces double the ๐๐๏ฟฝ . This observation suggests that the nonlinear effects of wave breakingโthat is, the generation of IG-waves and wave-induced setupโcontribute significantly to the resulting overtopping discharge. While the effects of vegetation are not considered here, this observation highlights a potential limitation in studies that assess the effectiveness of shallow foreshores but focus only on the attenuation of ๐ป๐ป ๐๐0 , ๐๐๐๐ and neglect the contribution of ๐ป๐ป ๐๐0 , ๐ผ๐ผ๐ผ๐ผ (Vuik, et al., 2016, Yang, et al., 2012). To further investigate the influence of the IG-waves, we compare the overtopping estimated using SWAN and EurOtop with and without the corrections to ๐ป๐ป ๐๐0 , ๐ผ๐ผ๐ผ๐ผ , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐ and ๐๐ ๐๐โ1 , , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐ obtained through Equations 29 and 36, respectively: -3 -3 -3 ObservedOpenFOAMSWASHBOSZXB-NHXB-SBSWAN Table 4 SWAN results with and without the empirial corrections for ๐ฏ๐ฏ ๐๐๐๐ , ๐ฐ๐ฐ๐ฐ๐ฐ , ๐๐๐๐๐๐ , ๐๐๐๐ (Equation 29) and ๐ป๐ป ๐๐โ๐๐ , ๐๐ , ๐๐๐๐๐๐ , ๐๐๐๐ (Equation 36). Case
SWAN ๐ป๐ป ๐๐0 , ๐๐๐๐ , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐ (m) ๐ป๐ป ๐๐0 , ๐ผ๐ผ๐ผ๐ผ , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐ (m) ๐ป๐ป ๐๐0 , ๐๐ , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐ (m) ๐๐ ๐๐โ1 , , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐ (s) ๐๐๏ฟฝ (l/s per m) Modelled Observed Mild swell Original 0.029 0 0.029 2.17 0.014 0.094 Corrected 0.029 0.014 0.032 5.38 0.053 Steep wind wave Original 0.033 0 Under the mild swell-wave conditions, the ratio
๐ป๐ป๏ฟฝ
๐ผ๐ผ๐ผ๐ผ = 0.5 (Equation 29) and the contribution of ๐ป๐ป ๐๐0 , ๐ผ๐ผ๐ผ๐ผ , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐ to ๐ป๐ป ๐๐0 , ๐๐ , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐ is minor (Table 4). On the other hand, including the IG waves resulted in a 2.5-fold increase in ๐๐ ๐๐โ1 , , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐ and magnitude 4-fold increase in the predicted ๐๐๏ฟฝ , compared to the original SWAN estimates. The difference is more striking for the steep-wave case where the IG waves dominate at the dike toe ( ๐ป๐ป๏ฟฝ
๐ผ๐ผ๐ผ๐ผ = 0.92). The inclusion of the IG waves resulted in 36% increase in ๐ป๐ป ๐๐0 , ๐๐ , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐ , a 5-fold increase in ๐๐ ๐๐โ1 , , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐ and an order of magnitude increase in the predicted ๐๐๏ฟฝ . Furthermore, the original SWAN estimatesโwithout any corrections to ๐๐ ๐๐โ1 , , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐ and ๐ป๐ป ๐๐0 , ๐๐ , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐ โerroneously show a decrease in ๐๐๏ฟฝ between the mild swell and steep wind-wave cases, while the observations show a notable increase. These results further emphasize the danger of neglecting the IG-wave contributionโdemonstrated here by the correction of input to the empirical formulaeโin the design and assessment of coastal structures with very shallow foreshores. 2 Figure 13 Mean ratio of modelled to observed ๐๐๏ฟฝ (markers) for both the a) mild swell and b) steep wind-wave cases, with error bars representing the standard deviation. Solid horizontal lines represent perfect agreement between model and observations. Dashed lines correspond to a factor of 4 larger and lower than the observations.
Considering the wider model comparison, each modelโwith the exception of BOSZโfails to reproduce the overtopping for the mild swell case. This is particularly evident for SWASH and XB-SB which significantly underestimate ๐๐๏ฟฝ for both wave cases, with the calibrated XB-SB model producing zero overtopping. This suggests that while XB-SB may estimate wave run-up accurately in IG-wave dominant environments (Lashley, et al., 2018), itโs exclusion of the SS-wave component considerably limits its performance for wave overtopping. The poor performance of SWASH here for wave overtopping is surprising, since it performed reasonably well in the prediction of ๐๐ ๏ฟฝ , ๐ป๐ป ๐๐0 and ๐๐ ๐๐โ1 , in both cases here and has been previously successful in one-layered mode (Suzuki, et al., 2017). However, Suzuki, et al. (2017) focused on obtaining good agreement at the toe (the last wave gauge only) and the resulting ๐๐๏ฟฝ -4 -2 DefaultCalibratedCorrected via Equations 29 and 36
OpenFOAM SWASH BOSZ XB-NH XB-SB SWAN10 -4 -2
3 in their tuning of SWASH; therefore ๐๐๏ฟฝ was not assessed unless wave heights and periods at the toe were within a certain accuracy range, regardless of the input (offshore) conditions. Whereas here, we assess the modelโs general performance for wave propagation (both offshore and nearshore), in addition to ๐๐๏ฟฝ . It should be noted that a finer grid resolution had little impact on SWASH predictions of ๐๐ ๏ฟฝ , ๐ป๐ป ๐๐0 and ๐๐ ๐๐โ1 , (~3%), it increased ๐๐๏ฟฝ by a factor of 7โthough still significantly underestimated (not shown)โfor the mild swell case, with significantly increased computational demand. The models do, however, perform considerably better for the steep-wave case. This is consistent with the findings of Roelvink, et al. (2018) and Suzuki, et al. (2017) who showed that XB-NH and SWASH, respectively, were more accurate for higher overtopping rates, but suffered for rates below 0.08 โ 0.16 l/s per m (in model scale). The improvement in SWAN with the corrections is most evident for the steep-wave case, with the estimated ๐๐๏ฟฝ now on par with that of BOSZ and outperforming the other more physically-complex models. Figure 14 shows the modelled relative overtopping discharge ( ๐๐๏ฟฝ ๏ฟฝ๐๐๐ป๐ป ๐๐0 , ๐๐ , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐3 ๏ฟฝ ) versus the relative freeboard ( ๐ ๐ ๐๐ ( โ ๐ป๐ป ๐๐0 , ๐๐ , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐ (0.33+0.022 ๐๐ ๐๐โ1 , , ๐ ๐ ๐๐ )) where ๐ป๐ป ๐๐0 , ๐๐ , ๐ก๐ก๐ก๐ก๐ก๐ก , ๐๐๐๐ and ๐๐ ๐๐โ1 , (to compute ๐๐ ๐๐โ1 , , ๐ ๐ ๐๐ ) are taken from Table 4. If we take the +/5 % exceedance lines of the EurOtop formula (Equation 31, Figure 14) as the general range of acceptable overtopping predictions, OpenFOAM, BOSZ, XB-NH and SWAN (with corrections) are all reasonable. SWASH and XB-SB, on the other hand, underestimate ๐๐๏ฟฝ and fall outside this acceptable range. As most of the models performed reasonably well for wave propagation, the excellent agreement between BOSZ and the observed ๐๐๏ฟฝ is likely not dependent on underlying governing equations (Boussinesq versus NLSW) but more to do with how the shoreline and wave run-up are treated numerically. However, an in-depth analysis of the various numerical schemes implemented in each numerical model was beyond the scope of this study. 4 Figure 14 Relative overtopping discharge versus relative freeboard. Solid line corresponds to Equation 31 with dashed lines representing +/- 5% exceedance. Outset magnifies plot area between ๐๐๏ฟฝ ๏ฟฝ๐๐๐ฏ๐ฏ ๐๐๐๐ , ๐ป๐ป , ๐๐๐๐๐๐ , ๐๐๐๐๐๐ ๏ฟฝ = 5 x 10 -3 and 2 x 10 -2 . Accuracy versus Speed
In contrast with the general assumption that models of increasing physical complexity produce more accurate results, Figure 15 shows no clear relationship between computational demand (simulation time) and the absolute relative error in overtopping. Furthermore, the depth-resolving models (SWASH and OpenFOAM), which have significantly higher simulation times show larger errors than the depth-averaged models (XB-NH and BOSZ). The phase-averaged models (XB-SB and SWAN (original)), despite their considerable speed advantage, significantly underestimated the overtopping discharge due to their exclusion of higher- and lower-frequency wave components, respectively. However, by including the IG-waves empirically, SWANโs performance improved significantly; now within acceptable limits and on par with those of XB-NH and BOSZ but at little to no computational cost (Figure 15). It should be noted that the use of SWAN with Equation 36 is already the recommended approach in EurOtop (2018); the novelty here is the further improvement in results offered by Equation 29. -6 -5 -4 -3 -2 -1 OpenFOAMSWASHBOSZXB-NHXB-SBSWAN (Original)SWAN (Corrected via Eqns. 29 and 36)Observed -2 Figure 15 Accuracy versus speed of the numerical wave models for wave overtopping with triangles and circles representing the mild swell and steep wind-wave cases, respectively. Conclusion
In the present study we assess the ability of 6 widely used numerical models to simulate waves overtopping steep dikes with mildly-sloping shallow foreshores. However, with the exception of OpenFOAM and to some extent SWASH (multi-layered mode) the above (phase-resolving) models were originally developed to simulate wave evolution over mildly-sloping foreshores; and not specifically for wave run-up and overtopping of steep structure slopes. Since their development, the phase-resolving models have each been successfully applied to wave propagation over steep reefs and run-up of relatively steep beaches. Likewise, depth-resolving models like OpenFOAM and SWASH (multi-layered) were originally developed to simulate wave-structure interaction and not specifically for wave propagation. In the present study we test the ability of these models in both applications: i) wave evolution over a shallow mildly-sloping foreshore; and ii) the resulting overtopping discharge. Overall, BOSZ and XB-NH (under steep wind-waves) showed high skill in both applications with a reasonable computational demand; while OpenFOAM, with a much higher computational demandโshowed difficulty in performing both functions. The broad implication of the present work is that higher-resolution, more computationally-demanding wave models OpenFOAMSWASHBOSZXB-NHXB-SBSWAN (Original)SWAN (Corrected via Equations 29 and 36)
6 may simply not be needed; specifically where the analysis is focused on bulk, time-averaged physical quantities ( ๐ป๐ป ๐๐0 , ๐๐ ๐๐โ1 , and ๐๐๏ฟฝ ), as shown here. Should more detail be requiredโfor example, estimates of the vertical velocity profile or turbulenceโthen a depth-resolving model such as SWASH (multi-layer) or OpenFOAM should be applied. Moreover, SWASH and OpenFOAM are likely to perform well if the computational domain begins at the dike toe and ends at the overtopping box; i.e., where simulating wave propagation over a large domain is not required. In addition, our results showed that with simple empirical corrections, phase-averaged models like SWAN can perform on parโif not better thanโphase-resolving models, with much less computational effort. Importantly, our work emphasizes the importance of including IG waves in the design and assessment of coastal dikes; as neglecting their contribution to ๐ป๐ป ๐๐0 , ๐๐ , ๐ก๐ก๐ก๐ก๐ก๐ก and ๐๐ ๐๐โ1 , , ๐ก๐ก๐ก๐ก๐ก๐ก can lead to under-predictions in ๐๐๏ฟฝ of up to two orders of magnitude. Given the scope of the model comparison, including both phase-resolving and phase-averaged, a detailed wave-by-wave comparison of the higher-resolution models was not carried out. Future work should address this and investigate the influence of the various numerical schemes implemented in the respective numerical models, as this was not within the scope of the present work. Additionally, Equations 31 and 36 were developed (in part) using the wider dataset from which these cases were taken; therefore their performance under different conditions is still to be confirmed. Despite these limitations, the findings here can aid practitioners in their decision making; specifically in deciding which numerical model should be applied based on the level of accuracy required. Acknowledgements
This work is part of the
Perspectief research programme
All-Risk with project number B2 which is (partly) financed by NWO Domain Applied and Engineering Sciences, in collaboration with the following private and public partners: the Dutch Ministry of Infrastructure and Water Management (RWS); Deltares; STOWA; the regional water authority, Noorderzijlvest; the regional water authority, Vechtstromen; It Fryske Gea; HKV consultants; Natuurmonumenten; and waterboard HHNK. Dr. Corrado Altomare acknowledges funding from the European Unionโs Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No.: 792370. Volker Roeber acknowledges financial support from the Isite 7 program Energy Environment Solutions (E2S), the Communautรฉ d'Agglomรฉration Pays Basque (CAPB) and the Communautรฉ Rรฉgion Nouvelle Aquitaine (CRNA) for the chair position HPCWaves and the European Unionโs Horizon 2020 Research and Innovation Programme. The authors also acknowledge Akshay Patil for his incite and efforts towards the success of the research.
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