Non-breaking Waves Research Articles

Improved guidance on roughness and crest width in overtopping of rubble mound structures along EurOtop

In this paper existing guidelines to predict wave overtopping on rubble mound breakwaters and coastal structures are modified and improved with respect to the influence of the roughness and crest width. Data from recently made model tests and existing data are combined to demonstrate the need for modifying these formulations in EurOtop. A new reduction factor γ cw for the crest width is established and is an improvement of the method by Besley. The influence of the roughness of the slope normally include also an influence of the breaker parameter when it is larger than a certain limit (EurOtop suggest ξ m-1,0 > 5). The present study shows that the breaker parameter is not the ideal dimensionless parameter describing the influence of the wave period for breakwaters with steep slopes, as for such structures the front slope has much less influence on the overtopping than the wave steepness. Thus slope angle and wave steepness have been uncoupled to describe the influence of the armour roughness on wave overtopping. The improvement in the overtopping prediction compared to EurOtop is significant, specifically for the new data sets that have data outside the range of the calibration data used for influence of roughness in EurOtop. The proposed improved methods enlarge the range of applicability with respect to crest width and wave steepness. • Influence of crest width and wave steepness on wave overtopping at rubble mound is investigated. • New influence factors are established for crest width and roughness. • EurOtop formula for non-breaking waves is modified to include those factors. • The influence of wave steepnees and front slope is included in the new roughness factor for rock and other armour units. • Predictions are significantly improved with the modified formula for the new data.

Experimental investigation on the characteristics of directional focusing waves

The analysis on fixed or floating structures exposed to directional extreme waves has been a topic of keen interest in ocean engineering. In this context, an attempt is made to study the characteristics of laboratory generated directional rogue waves, termed also as focusing waves. The first-order wave paddle displacements were generated using the linear superimposition principle by applying a constant steepness spectrum associated with directional components. The phase angles are set to be constant at a predefined location to create an extreme event. The experimental test cases span from narrow to broadband spectra having two central frequencies, namely, 0.68 and 0.85 Hz. The bandwidth ratios of 0.5 and 0.75 were considered with five discrete directions ranging from −25° to 25°. The tests include spilling breakers and non-breaking waves. The investigation involves studying the variations in the Atiltness parameter (ATP), Crest height ratio, and the nonlinearity parameter provided by Hastings, an index to measure the effect of nonlinearity. In addition, directional studies using the maximum likelihood method (MLM) are also presented. The critical combination of frequency spectrum and bandwidth is identified from the focusing wave characteristics and discussed in this paper for the tested conditions. The shift in directional spectrum due to nonlinearity was noticed in the present study.

Sobre la disipación de energía cinética turbulenta asociada con olas que aún no rompen

The turbulent kinetic energy dissipation is an essential quantity in the study of turbulence in fluids. In particular, on the transference of turbulent energy from large to small scales and determining its state of equilibrium and stationarity. This work has the purpose of understanding turbulence generation by non-breaking waves from turbulent kinetic energy dissipation analysis. Different groups of nonbreaking monochromatic waves with different slopes were mechanically constructed in a laboratory, whereby an acoustic device, the wave orbitals velocities were measure in various depths. Considering the inertial subrange in the power spectrum of components of turbulence velocity, a turbulent kinetic energy dissipation rate was quantified. It was detected that the magnitude of the turbulent kinetic energy dissipation rate increases with the wave slope and that deeper is invariant to axis rotations. It was distinguished that most of the profiles of the turbulent kinetic energy dissipation agree with an atypical logarithmic layer. Finally, a term of turbulence production was introduced, relative to nonbreaking wave orbital velocities. Unlike other turbulence production terms or approximations, it adequately reproduces the values of the turbulent kinetic energy dissipation rate regardless of wave slope, which established that the wave orbitals velocities shear is the generator mechanism of turbulence in a fluid under nonbreaking waves.

Observations and Parametrization of the Turbulent Energy Dissipation Beneath Non-Breaking Waves

Here, for non-breaking short surface waves, we have experimentally determined the value of the turbulent eddy viscosity νT or its ratio νT*≡νT/ν, where ν is the water kinematic viscosity. The non-breaking wave-generated turbulent eddy viscosity νT was found to depend on the ratio of the wave period, T, to the microscale Kolmogorov time scale, τη. Our observations were consistent with νT*=1.46·(T/τη)−2.6 when (T/τη)<0.9. That implied that the νT*∝ϵ−1.3, where ϵ is the background turbulent energy dissipation rate. The near-surface turbulent flow associated with non-breaking waves was characterized by a short inertial subrange. The background turbulence appears to modulate the amount of energy the non-breaking waves dissipate locally and, consequently, the wave’s decay rate. Our results imply that the background turbulent flow acts as a lubricant, permitting waves to propagate further when traveling over a more energetic turbulent background flow. Our results have implications for the modeling of oceanic wave propagation or the air–sea exchange processes.

Physical and numerical modelling of representative tsunami waves propagating and overtopping in converging channels

Tsunami waves pose a threat to coastal communities through overland inundation and overtopping of coastal defence structures and river walls. Tsunami wave overtopping in rivers and in straight or converging channels is yet to be studied in detail. The present study evaluates the performance of current numerical models that are employed in various tsunami warning centres across the world, in terms of their accuracy for the prediction of tsunami wave characteristics in converging channels. The study evaluates each model performance against a benchmark dataset obtained from novel physical model experiments of representative tsunami waves (solitary waves and bores) overtopping a river wall in a converging channel. The complex process of simultaneous steepening and overtopping was observed to be different for solitary waves and bores. The experimental data were then used to compare results from a suite of different numerical models to predict wave heights and overtopping volumes in straight and converging channels. Different numerical models ranging from nonlinear/nondispersive in two-dimensions (2D) to fully nonlinear/dispersive in three-dimensions (3D) were used. The widely used models based on the nonlinear shallow water equations ( ANUGA , BASEMENT , and TELEMAC 2D) fail to capture the wave profiles and heights, and consequently do not accurately predict the overtopping volumes in channels for both breaking and non-breaking waves. A Boussinesq model ( TELEMAC 2D) was found to produce results consistent with experimental measurements for non-breaking waves only, with an error of up to 8% and 20%, for wave heights and overtopping volumes, respectively. On the other hand, the wave heights and overtopping volumes are underpredicted and overpredicted, for solitary waves and bores, respectively, using a three-dimensional nonhydrostatic model ( TELEMAC 3D). When the overtopping occurs in the converging channel, reflection of the incident waves is minimal in both the numerical and the physical models. However, in the absence of overtopping, both the numerical models and the classical analytical long-wave model fail to correctly predict the observed minimal reflection of non-breaking solitary waves in the converging channel. The reasons for this remain to be investigated. • Novel experiments of representative tsunami waves overtopping river wall were conducted. • 2D and 3D numerical modelling results were compared with experimental data. • Nonlinear shallow water models fail to predict solitary wave/bore heights, shapes and overtopping volumes in channels. • Boussinesq model produces satisfactory results for non-breaking solitary waves only. • Numerical models and classical analytical long-wave model fail to predict solitary wave reflection in a converging channel.

Effects of the bed roughness and beach slope on the non-breaking solitary wave runup height

In this study, a series of laboratory experiments were carried out to investigate the effects of bed roughness and beach slope on the non-breaking solitary wave runup height. Experimental measurements reveal that the bottom friction plays different roles in the solitary wave runup process for different beach slopes. Under the same incident wave condition, the maximum runup height on a smooth steel bed was found to be larger than that on a rough sandpaper bed over a 10° beach slope, whereas the same runup heights were observed for the smooth and rough beds over a 45° beach slope. By introducing the runup similarity parameter, quantitative relationships between the maximum runup height and the bed roughness, as well as the beach slope, were analyzed. On the basis of Synolakis (1987), a new empirical model for estimating the non-breaking solitary wave induced runup height was proposed, after incorporating a slope-related enhancement coefficient and a friction-related reduction coefficient. It is found that both the enhancement coefficient (being always larger than unity) and the reduction coefficient (being always smaller than unity) are large under the small relative incident wave height and steep beach slope condition. Influence of the bed roughness is more significant under the small runup similarity parameter condition, under which the runup height monotonically decreases with the increase of the runup similarity parameter for the small bed roughness, whereas it first increases then decreases with the increase of the runup similarity parameter for the large bed roughness. Nevertheless, influence of the bed roughness could be neglected for the large runup similarity parameter condition. These results are consistent with experimental observations. Finally, the present model was validated using a large amount of existing solitary wave runup data with different bed frictions and beach slopes. The good agreement between the model predictions and measured data indicates the reliability and accuracy of the present model.

Wave-averaged properties for non-breaking waves in a canopy: Viscous boundary layer and vertical shear stress distribution

The wave-averaged contributions to the vertical shear stress distribution in both submerged and emerging canopies has been derived. Additionally, the contribution from a viscous boundary layer on top of a submerged canopy on both the vertical shear stress distribution and the Lagrangian Stokes drift velocity are quantified. The analytical model is in principal restricted to a description of the local behaviour, but it is shown, how the model can be used to describe the wave damping and the velocity field through an entire canopy. The theoretical findings are validated with laboratory experimental data on the wave attenuation across a canopy and the in-canopy wave-induced velocities. The combined analytical and experimental findings are synthesised to provide recommendations of how to incorporate effect of waves in large-scale practical engineering models for Eulerian-mean hydrodynamics. • An analytical shear stress distribution due to wave dissipation in canopies. • A constant-viscosity boundary layer solution in a submerged canopy is proposed. • Viscous effects on Lagrangian Stokes drift is quantified. • Wave attenuation is predictable with reduced velocities and fixed force coefficients. • Procedure to link spectral wave models and hydrodynamic models is outlined.

Step approximation on oblique water wave scattering and breaking by variable porous breakwaters over uneven bottoms

In this study, the scattering of oblique water surface gravity waves by variable porous breakwaters over uneven bottoms is investigated using the eigenfunction matching method (EMM). In the EMM procedure, the variable breakwaters and bottom profiles were sliced into shelves separated by steps. The solutions on the shelves are composed of eigenfunctions. Then, by applying the conservations of mass and momentum, a system of linear equations is obtained and can be solved with a sparse-matrix solver. The effects of wave breaking and dissipation were also included in the EMM formulation using energy-dissipation factors. In order to validate the proposed EMM method, the numerical solutions were compared with the results in the literatures for non-breaking wave scattering by rectangular and parabolic porous breakwaters. The convergence of the proposed solutions was studied. Then some discussions were provided when the EMM with energy-dissipation factors was applied to evaluate the surface elevations of breaking and/or dissipated waves. The numerical results illustrated the potential application of the EMM for problems with wave breaking and dissipation if the empirical formulas for the energy-dissipation factors can be well calibrated in future researches. Some further applications were provided, including wave scattering by multiple porous breakwaters.

A Two‐Part Model for Wave‐Sea Ice Interaction: Attenuation and Break‐Up

AbstractWaves and sea ice form a closely coupled system: waves govern sea ice through stress, floe break‐up, and wave‐induced currents, while sea ice affects waves through attenuation and reflection. Wave‐induced sea ice break‐up is particularly important as it can regulate air‐sea interaction and consequently also regulate the growth and melt of sea ice. This coupled nature is complex and generally, especially at the large scale, neglected in modeling of the polar climate system. Here, we explore a novel way of coupling through wave‐induced ice break‐up, and conduct a case study for the Antarctic summer of 2019/2020. Our modeling approach builds upon previous investigations as follows: (a) sea ice takes a binary form, either “broken” or ”unbroken,” (b) waves may break sea ice, transitioning it from unbroken to broken, (c) a threshold separating breaking and nonbreaking wave fields is used to identify when this occurs, (c) two modes of attenuation for waves in ice (dependent upon the ice state), representing the observed on/off switch in wave attenuation. By characterizing wave attenuation and sea ice break‐up as described above, we achieve two‐way wave‐sea ice coupling, thereby allowing wave‐sea ice feedbacks. This model is limited to the ice‐melt season as refreeze is not represented here. We demonstrate that our model can simulate both the wavefield in and the evolution of the Marginal Ice Zone. Our results show the validity of empirically derived wave‐induced sea ice break‐up threshold, and substantiate that waves have a critical influence on the morphology of the Marginal Ice Zone.

Nonlinear wave evolution with data-driven breaking

Wave breaking is the main mechanism that dissipates energy input into ocean waves by wind and transferred across the spectrum by nonlinearity. It determines the properties of a sea state and plays a crucial role in ocean-atmosphere interaction, ocean pollution, and rogue waves. Owing to its turbulent nature, wave breaking remains too computationally demanding to solve using direct numerical simulations except in simple, short-duration circumstances. To overcome this challenge, we present a blended machine learning framework in which a physics-based nonlinear evolution model for deep-water, non-breaking waves and a recurrent neural network are combined to predict the evolution of breaking waves. We use wave tank measurements rather than simulations to provide training data and use a long short-term memory neural network to apply a finite-domain correction to the evolution model. Our blended machine learning framework gives excellent predictions of breaking and its effects on wave evolution, including for external data.

Numerical investigation of inline wave force on a truncated vertical cylinder with different cross-sections in regular head waves

The non-linear wave amplification around a truncated vertical cylinder, the high-order horizontal wave loading from non-breaking steep waves and the associated physical process are studied numerically. Considering the high-order free-surface deformation due to high-frequency wave scattering Type-1 and Type-2, and lateral edge waves, the simulations are conducted for a stationary, surface-piercing cylinder with different cross-sections subject to the short and long waves, propagating in deep water. A numerical wave tank based on the Unsteady Navier-Stokes/VOF model is established, using OpenFOAM combined with the olaFlow toolbox for wave generation and absorption. The capabilities of the present numerical model are assessed in conjunction with the ITTC (OEC) benchmark, where the numerical results are in good agreement. Overall, the results indicate the high potential of the present numerical model for accurate modeling of nonlinear wave interaction with cylindrical columns. Variation of wavelength together with the corner ratio significantly influences the magnitude of high-order harmonic wave loading. Moreover, the numerical results indicate the important contribution of the cross-section shape on the non-linear wave field surrounding each cylinder. As the geometrical presence of corner becomes apparent from circular to the square cross-section, there is a systematic increase in the mean value, 1st, 2nd and 3rd harmonics of the inline force subject to both short and long incident waves. Furthermore, interestingly the formation of slightly developed wave scattering of Type-2 at the front of the cylinder in short waves is observed for cross-sections with corner ratios of (rc/R=0.25&0). The development of a secondary load cycle is also observed which is found to be closely related to the occurrence of wave scattering Type-2 and the lateral edge waves where the important contribution from, mean value, 3rd, and 4th harmonics of the inline force are apparent.

Wave breaking probabilities under wind forcing in open sea and laboratory

Water wave breaking represents one of the most arduous problems in fluid mechanics. Understanding the process of wave breaking and developing an ability to quantify the associated energy losses and redistribution are critical across a wide range of coastal oceanic applications, maritime navigation, and climate and hydrodynamic research. Naturally, waves become steeper toward the inception of breaking; however, there is still a lack of unanimity regarding the relationship between breaking probability statistics and wave steepness. Here, we present a detailed investigation of breaking vs non-breaking statistics estimated using a recently developed method for accurate detection of breaking waves, based on the phase-time approach to identify breaking-associated patterns in the instantaneous frequency variations of surface elevation fluctuations. The findings are based on data collected both in the open sea and in a laboratory wind wave flume. An in-depth examination of celerities and steepnesses of breaking and non-breaking waves is presented. The analysis, which involved wave-by-wave examination, produced skewed Gaussian-like steepness histograms, revealing that non-breaking waves and breaking waves can reach steeper profiles, above the Stokes limit. All extreme steepness values were investigated and are presented here.

Energy Dissipation and Nonpotential Effects in Wave Breaking

This paper presents a comparative numerical study of the energy dissipation process in the breaking of focused waves by using a potential flow model and a coupled potential/viscous flow model. An empirical eddy viscosity term is introduced to the fully-nonlinear potential (FNP) flow model to account for the breaking wave energy dissipation. The FNP model is further coupled with an incompressible two-phase NavierStokes (NS) flow solver through a one-way linkage to generate and propagate focused waves in the domain. Numerical absorbing regions are placed in front of the outlet boundaries to dampen wave reflection. The standalone FNP model and the coupled FNP+NS model are applied to deal with each scenario comparatively. This enables the accurate quantification and comparison of the energy loss of breaking waves calculated by the two numerical models. The velocity field is decomposed into potential component, which is reconstructed from the corresponding free surface elevation computed in the coupled model by using the weakly-nonlinear wave theory, and non-potential rotational component. Detailed analysis of the numerical results shows that: (1) energy loss is closely related to wave steepness, (2) mild rotational motion produced by a non-breaking wave is local in time with a short life-span, (3) strong non-potential motion triggered by wave breaking is not local in time but persists in the flow for dozens of or even many more wave periods.

Wave scattering by a three-dimensional submerged horizontal rectangular plate in a channel: experiments and numerical computations

Experiments are conducted in a wave tank to investigate wave scattering by a three-dimensional submerged horizontal rectangular plate in a channel. The free-surface elevation around the plate is presented for various water depths and depths of submergence of the plate. The wave forces and moments are obtained using an underwater load measuring system. The numerical simulations are performed with a parallelized three-dimensional boundary element method. The numerical set-up follows closely the set-up of the laboratory experiments. The numerical results are compared with the experimental results for non-breaking waves. In most cases a good agreement is found for the free-surface elevation, the vertical force and the moment. A physical interpretation of the flow around the plate is provided. The pressure distribution on the lower surface of the plate differs from the linear distribution from the leading edge to the trailing edge that is obtained in the equivalent two-dimensional problem. The reflection by the lateral walls is investigated.

Assessment of wave overtopping risk for pedestrian visiting the crest area of coastal structure

Wave overtopping can be a great threat to the pedestrians visiting the coastal structure’s crest area, but a good tool for risk assessment is lacking. In this study, a quantitative risk assessment framework was developed, which is based on considering the overtopping impact force acting on a pedestrian’s body. The centerpiece of the framework is a model for the likelihood of overtopping accident, which is defined as being mobilized by overtopping flow at least once during a visit. The largest single wave of a given sea state at the structure’s toe is first evaluated, and the maximum overtopping force of this largest wave is then calculated using a force predictor (assuming non-breaking waves before the structure), in which the human body is approximated as a cylinder. A probabilistic instability model, which was calibrated based on real-human experiments in flood flows, was developed in this study. This model converts an instability number, i.e., the ratio of maximum impact force to ground friction, into the probability of instability, so the likelihood of overtopping accident can be quantified. By applying our model to an adult standing on a 1-on-3 sloped revetment, it was found that the model gives thresholds of tolerable overtopping conditions (in terms of discharge rate and maximum overtopping volume) very close to those suggested in EurOtop (2018). A risk assessment framework, which quantifies risk as the sum of likelihood and severity, was proposed. The merit of this framework is that it can account for different pedestrian’s body size, location of the pedestrian, structure’s slope and etc. As a demonstration, the risk assessment framework was applied to Singapore’s coastal structures. Limitations of this framework are given at the end of the paper.

Smoothed Particle Hydrodynamics simulations of reef surf zone processes driven by plunging irregular waves

As waves interact with the slopes of coral reefs and other steep bathymetry profiles, plunging breaking usually occurs where the free surface overturns and violent water motion is triggered. Resolving these surf zone processes pose significant challenges for conventional mesh-based hydrodynamic models, due to the rapidly-deforming nature of the free surface and associated flows. Yet the accurate prediction of these surf zone hydrodynamics is critical for predicting a wide range of nearshore processes driven by wave breaking (e.g., wave dissipation and energy transfers; mean water levels and currents; and wave runup). In this study we assess the ability of the mesh-free, Lagrangian particle-based numerical modelling approach Smoothed Particle Hydrodynamics (SPH) based on DualSPHysics, to simulate the fine-scale hydrodynamic processes driven by irregular wave transformation over a fringing reef profile, by comparing results against detailed experimental observations from a physical modelling study. To greatly improve the computational efficiency, the SPH model was coupled to the mesh-based multi-layer nonhydrostatic wave-flow model SWASH. With this coupled approach, SWASH was used to efficiently simulate the evolution of non-breaking waves from the wavemaker up to the fore reef slope, with the SPH model then used to simulate the detailed hydrodynamic processes over the reef from just offshore of the breakpoint to the shoreline. The SPH model was able to accurately reproduce the complex free surface deformations during plunging breaking, the spectral evolution of waves across the reef flat (including nonlinear wave shape), the mean water levels and currents, and wave runup at the shoreline. Using the long duration simulations (>400 wave periods), the model was able to reproduce the full range of wave motions over the reef (from sea-swell to infragravity frequencies), including the increasing dominance of low frequency waves towards the shoreline and the large cross-reef standing wave motions excited by the reef geometry.

Numerical Simulation of Wave Interaction With a Pair of Fixed Large Tandem Cylinders Subjected to Regular, Non-Breaking Waves

Abstract This paper presents the application of a two-phase computational fluid dynamics (CFD) model to carry out a detailed investigation of the nonlinear wave field surrounding a pair of columns placed in the tandem arrangement in the direction of wave propagation and corresponding harmonics. The numerical analysis is conducted using the unsteady Reynolds-averaged-Navier-Stokes/volume of fluid (VOF) model based on the openfoam framework combined with the olaFlow toolbox for wave generation and absorption. For the simulations, the truncated cylinders are assumed vertical and surface piercing with a circular cross section subjected to regular, non-breaking fifth-order Stokes waves propagating with moderate steepness in deep water. Primarily, the numerical model is validated with experimental data for a single cylinder. Future, the given simulations are conducted for different center-to-center distances between the tandem large cylinders. The results show the evolution of a strong wave diffraction pattern, and consequently, high wave amplification harmonics around cylinders are apparent.

Experimental investigation of hydrodynamic loading induced by regular, steep non-breaking and breaking focused waves on a fixed and moving cylinder

Monopiles are commonly adopted in marine structures and subject to combined loading from waves and currents. The nature of superposition of wave and current loads needs to be known during the design stage. In the present paper, combined hydrodynamic loading induced by nonlinear waves and uniform currents on a cylinder is experimentally investigated. The current is represented by towing the cylinder along the flume. By this, nonlinear wave–current interactions are excluded physically, but the loading of a proportional current following or opposing a wave group is captured and analyzed. It is argued that towing makes provision for analyzing the nature of superposition of wave and current loads using Morison theory (which is not applicable to true combined wave–current fields) and also facilitates experimentation of a wide range of nonlinear wave and uniform current loading combinations onto the structure. Accordingly, regular, steep non-breaking and breaking focused waves interacting with the cylinder towed along and in opposition to the wave-field at different speeds have been investigated. The non-breaking wave–structure interactions have been analyzed within the framework of Morison theory using Fully Nonlinear Potential Theory (FNPT) based kinematics. Breaking wave–cylinder interactions have been analyzed through a spectral approach. The experiments demonstrate that wave and locally-acting current loads on the structure can be linearly superimposed, irrespective of the nature of waves and towing speed. Hence, provided wave–current interactions are excluded, steep breaking wave and uniform current loads can be linearly superimposed, despite focused wave generation itself being inherently nonlinear.

Experimental study on a breaking-enforcing floating breakwater

Floating breakwaters are moored structures that attenuate wave energy through a combination of reflection and dissipation. Studies into floating breakwaters have been generally restricted to optimising the attenuation performance. This study presents a novel floating breakwater type that was developed to have good attenuation performance while keeping wave drift loads as small as possible. The floating breakwater was designed as a submerged parabolic beach that enforces wave energy dissipation through breaking. The design was tested in a 3D shallow-water wave basin in captive and moored setups for regular and irregular wave conditions. Results are presented in terms of attenuation performance, motions, and (mooring) loads. The results show that the breaking of waves improves the attenuation performance of the floater in captive setup. However, in moored setup, the attenuation performance was dominated by diffraction and radiation of the wave field, with breaking being of secondary importance. This shows that breaking-enforcing floating breakwaters have potential, but require a high vertical hydrostatic and/or mooring stiffness in order to enforce intense breaking. Mean wave drift loads on the object showed significant difference between breaking and non-breaking waves in both setups, with breaking waves leading to lower normalized loads. This is attributed to breaking-induced set-up and set-down of the water level. As a result, the new breakwater design has a more favourable balance between wave attenuation and drift loads than common (i.e., box-, pontoon-, or mat-type) floating breakwater designs. Tests with varying surface roughness showed that floating breakwaters may benefit from dual-use functions that naturally increase the roughness (e.g., shellfish, vegetation), which have a marginal effect on the attenuation performance, but increase the added mass and hydrodynamic damping and as such, reduce mooring line loads.

Droplet Size Distribution Effect on the Vertical Dispersion of Spilled Oil Due to Regular Non-Breaking Waves

Droplet Size Distribution Effect on the Vertical Dispersion of Spilled Oil Due to Regular Non-Breaking Waves