Plane Beach Research Articles

Flow characteristics of the rip current system adjacent to a coastal vertical structure for irregular waves

Rip currents are common hazards found on many beaches around the world. The present research examines the rip current flow features adjacent to a coastal vertical structure with total reflection function to the incident waves. The laboratory experiments were conducted over the planar beach for both regular and irregular waves to explore the specific phenomenon for the irregular waves, and a higher-order Boussinesq equation model was adopted in the simulations to analyze the formation mechanism of the flow features. The study results show that, the standing wave pattern for the irregular wave is much weaker than that for the regular wave with inducing only one apparent node rip at the first node line in the reflection wave area. Moreover, the specific “single-vortex” flow pattern was formed in the crossing wave field for the irregular wave tests. This specific flow pattern is found essentially induced by the larger wave reflection rate of the vertical structure, which leads to the strong lateral flow against the downstream longshore current. This specific flow pattern prefers to present for the smaller wave incident angles and larger wave heights and periods, and leads to the feeding flow volume rates discharged more in the form of lateral flow. The C-shaped lateral flow is found mainly induced by the longshore gradient of the wave set-up instead of the non-uniform wave radiation stress nearshore.

Experimental study on wave attenuation and beach profile evolution under the protection of submerged flexible vegetation

Coastal protection methods are evolving from hard engineering techniques to natural enhancement approaches. Aquatic vegetation provides an environmentally friendly solution by effectively dissipating waves and resisting erosion. Laboratory flume experiments were conducted to investigate the effect of submerged flexible vegetation on the regular wave attenuation and cross-shore profiles on the artificial plane beach under varying hydraulic conditions. Variations in the waveform, wave spectrum, and wave height were analysed in the vegetation zone. Results show vegetation distorts the waveform and accelerates the attenuation process of the wave energy, and wave height evolution behaves differently from the theoretically obtained attenuation by Mendez and Losada (2004) due to the underestimating vegetation-induced attenuation and neglecting sediment-induced attenuation. Additionally, the formation process of the winter and summer profiles of the vegetated beach was analysed by the screenshots. Regenerated wave breaking becomes more apparent with increasing wave periods. The critical surf similarity parameter of the vegetated beaches was found to be lower than that of a smooth, impermeable slope because of the energy dissipation. Vegetation did not alter the type of the beach profiles, as compared to the final profiles and their characteristic parameters of the unvegetated beaches. Instead, it reduced the erosion depth and brought the eroded area closer to the shoreline. Dimensionless parameters were formulated to characterize the response of the cross-shore beach profile under the shelter of flexible vegetation, and an empirical equation about maximum erosion depth was developed within the scope of the experiment.

Physical Modelling Of Boulder Transport Under The Influence Of Tsunami Waves

Tsunami events are traditionally represented in the geological record by a sequence of fine-grained sediments, but increasingly coastal boulder deposits are being used as indicators of past tsunami events. The emplacement mechanism of many boulder deposits, however, is heavily debated and determining whether the inundation event was a tsunami or storm remains an unresolved challenge (Cox et al., 2020). Using physical experiments, we aim to achieve a better understanding of how tsunamis move coastal boulders. This knowledge will aid field geomorphologists in the identification of the emplacement mechanism for coastal boulder deposits and allow for the determination of wave parameters. In January 2023, physical experiments using the HR Wallingford Tsunami Simulator were completed as part of the MAKEWAVES collaboration. These experiments investigated the movement of a cuboid and irregular shaped boulder model when impacted by different tsunami waveforms on a plane beach. We propose new empirical formulae to describe relationships between transport distance and different tsunami waves.

An overtopping formula for shallow water vertical seawalls by SWASH

There is now wide evidence that phase-resolving numerical models can capture the general physics of the overtopping process even under complex hydrodynamics. Based on this outcome, an extensive parametric study has been carried out with the model SWASH, to develop a design formula that uniformly predicts the overtopping rate at vertical seawalls for both breaking and non-breaking wave conditions. The use of the numerical model has allowed to vary the experimental conditions smoothly, avoiding the typical limitation of laboratory experiments. Furthermore, particular tests have been performed to assess whether, and up to which extent, the mean overtopping discharge is affected by the low frequency components of the incident wave spectrum.The formula relates the overtopping rate to a new water level statistic, which represents the average of the highest one-fourth wave displacements at the toe of the wall; it has initially been derived for planar beaches and then extended to the case of slope varying foreshores. A comparison with an array of 200 laboratory experiments confirms that the new parametrization can reduce the scatter of data compared to the EurOtop equation.

Observing eddy dye patches induced by shear instabilities in the surf zone on a plane beach

Observing eddy dye patches induced by shear instabilities in the surf zone on a plane beach

Exact solutions for the initial stage of dam-break flow on a plane hillside or beach

Inviscid, incompressible liquid is released from rest by a sudden dam break, accelerating under gravity over a uniformly sloping impermeable plane bed. The liquid flows downhill or up a beach. A linearised model is derived from Euler's equations for the early stage of motion, of duration $2\sqrt {H/g}$ , where H is the depth scale and $g$ is the acceleration due to gravity. Initial pressure and acceleration fields are calculated in closed form, first for an isosceles right-angled triangle on a slope of $45^{\circ }$ . Second, the triangle belongs to a class of finite-domain solutions with a curved front face. Third, an unbounded domain is treated, with a curved face resembling a steep-fronted breaking water wave flowing up a beach. The fluid goes uphill due to a nearshore pressure gradient. In all cases the free-surface-bed contact point is the most accelerated particle, exceeding the acceleration due to gravity. Physical consequences are discussed, and the pressure approximation of shallow water theory is found poor during this early stage, near the steep free surface exposed by a dam break.

SWASH FLOWS GENERATED BY A TRAIN OF SOLITARY WAVES ON A PLANAR SLOPE

The objective of this study is to gain insights of the swash flows generated by a train of solitary waves on a planar beach and the corresponding turbulence structure. Two dimensional velocity measurements are made using a 1000-Hz high speed particle image velocimetry (HSPIV). The turbulence characteristics in terms of the temporal and spatial evolution of the turbulent kinetic energy (TKE), TKE spatial spectra, and turbulence length scales are examined.

The polar-Cartesian mild-slope equation

A novel mild-slope equation is derived based on a manipulation of the cylindrical and Cartesian coordinate reference systems. The vertical profile of the velocity field is constructed by solving an approximate problem in cylindrical coordinates. This allows us to address the local derivatives on the bottom profile along a constant-slope line. This formulation is as opposed to the Cartesian-based mild-slope equations in terms of which the profile is constructed by assuming a constant depth. An angular profile is derived for the three-dimensional case on a sloping plane beach. For the two-dimensional case, a mild-slope polar-Cartesian equation is derived, for which an improved linear dispersion relation is reconstructed. This is accomplished due to the inclusion of first-order derivatives of the local bottom profile. The coefficients of the polar-Cartesian mild-slope equation contain the derivatives of the bottom profile up to third order as opposed to second-order derivatives in the Cartesian-based equations. The equation is derived by applying the variational principle to the Cartesian Lagrangian when formulated as a function of the profile in polar coordinates. It is then compared with existing models of the mild-slope equation for simulations of two-dimensional test cases and a quasi-three-dimensional case, which have known analytical solutions. Our modified equation exhibits better matching to the exact solutions for a majority of the investigated cases.

Inundation, runup and flow velocity of wavemaker generated bores on a planar beach

Undulating and breaking bores are generated in the laboratory using a programmable long-stroke wavemaker. By changing the stroke length and the speed of the wavemaker, both non-decaying and decaying bores are generated and studied. Bore strength, height and duration are measured and compared with the solutions derived by using the method of characteristics, with excellent agreement. The measurements for inundation depth, runup height and flood duration are checked with the formulas presented in Barranco & Liu (J. Fluid Mech., vol. 915, 2021). The comparisons show that the formulas are also accurate for the non-decaying bores generated by the wavemaker. The maximum inundation depth predicted by the formula for zero bore length at the beach toe agrees with the laboratory observations for decaying bores. Using a high-speed particle image velocimetry system, the ensemble-averaged velocities and fluctuating velocities under undulating bores and breaking bores are measured in constant water depth and in the vicinity of the still water shoreline. Detailed analyses of the velocity fields are presented and discussed. For the undulating bore a long quiescent flood duration is observed, while for the breaking bore the up-rush flow changes into down-rush flow almost linearly.

Longshore Current Profiles and Instabilities on Plane Beaches with Mild Slopes

The accurate determination of cross-shore longshore current profiles in the surf zone is essentially important in understanding of coastal physical processes and modelling of longshore sediment transport. In this study, a comprehensive laboratory study was undertaken to directly measure longshore current profiles over plane beaches with two mild slopes, 1:40 and 1:100, in a wave basin 55 m long, 34 m wide, and 0.7 m deep. Different wave conditions with an incidence angle of 30° were generated by piston-type wave makers consisting of 70 individual paddles, and two arrays of 29 Acoustic Doppler Velocimeters (ADVs) were used to measure longshore currents in the surf zone. Based on the experimental data collected in this study, three types of cross-shore longshore current profiles were found on the two plane beaches under different testing wave conditions, namely quasi-Rayleigh, quasi-Gaussian, and M-shape profiles. The quasi-Rayleigh profile was found on the beach slope of 1:40, and the other two types were found on the beach slope of 1:100. Analytical formulae were proposed to describe these profiles and agreed well with the laboratory data. The fluctuations of longshore currents observed in this study were attributed to their shear instabilities based on linear instability analysis. The results of the linear instability analysis and the spectra of measured velocities also showed that the three types of velocity profiles led to different instability characteristics.

An ISPH with modified k–ε closure for simulating breaking periodic waves

The Incompressible Smoothed Particle Hydrodynamics (ISPH) method, solving the 2D RANS (Reynolds Averaged Navier-Stokes) equations with the modified k–ε turbulence closure developed by Wang and Liu (2020) [ Coastal Engineering , 157, 1–28], is further extended to periodic waves. The capability of this numerical model is demonstrated by applying it to three laboratory experiments: (1) Dynamics of surf-zone turbulence generated by a spilling breaker and a plunging breaker over a plane slope in small-scale wave flume experiments, (2) breaking wave generated turbulence over a stationary barred beach in large-scale wave flume experiments, and (3) spatial and temporal distributions of turbulence generated by bi-chromatic breaking waves over a submerged stationary barred slope in medium-scale wave flume experiments. In this new ISPH model, the gradient correction_ dynamically stabilized scheme is adopted to reduce numerical dissipation. The experimental data for the spilling breaker on a plane beach is used to calibrate the model. Once the model is calibrated, all the empirical coefficients in the turbulence closure model, including the stress limiter coefficient λ 3 , which is required in modifying the eddy viscosity, are fixed for the validation processes with other experimental data sets. All model-data comparisons are made in terms of free surface profile, mean velocity field and turbulent kinetic energy. The differences between experimental observations and numerical simulations are quantified by the percentage error. Overall, good agreement is observed for all experiments. This paper presents the first comprehensively validated 2D ISPH model with the modified k–ε turbulence closure, which can be applied to periodic wave breaking problems. • The Incompressible Smoothed Particle Hydrodynamics (ISPH) method is coupled with the modified k–ε turbulence closure equations to solve the 2D RANS (Reynolds Averaged Navier-Stokes) equations. The gradient correction_ dynamically stabilized scheme is adopted to reduce numerical dissipation. • The model has been checked with three laboratory experiments for periodic breaking waves. (1) Spilling and plunging cnoidal waves over a plane slope in small-scale wave flume experiments, (2) plunging periodic breaking wave over a stationary barred beach in large-scale wave flume experiments, and (3) bi-chromatic breaking waves over a submerged stationary barred slope in medium-scale wave flume experiments. • All model-data comparisons are made in terms of free surface profile, mean velocity field and turbulent kinetic energy. • The differences between experimental observations and numerical simulations are quantified by the percentage error. • Overall, good agreement is observed for all experiments. Good agreements are observed in terms of free surface profile, mean velocity field, vorticity field, turbulent kinetic energy and turbulent shear stress.

Wave‐Driven Hydrodynamic Processes Over Fringing Reefs With Varying Slopes, Depths, and Roughness: Implications for Coastal Protection

AbstractWave breaking on the steep fore‐reef slopes of shallow fringing reefs can be effective at dissipating incident sea‐swell waves prior to reaching reef shorelines. However, wave setup and free infragravity waves generated during the sea‐swell breaking process are often the largest contributors to wave‐driven water levels (wave runup) at the shoreline. Laboratory flume experiments and a two‐dimensional vertical phase‐resolving nonhydrostatic wave‐flow model, which includes a canopy model to predict drag forces generated by roughness elements, were used to investigate wave‐driven water levels for along‐shore uniform fringing reefs. In contrast to many previous studies, both the laboratory experiment and the numerical model account for the effects of large bottom roughness. The numerical model reproduced the observations of the wave transformation and runup over both smooth and rough reef profiles. The numerical model was then extended to quantify the influence of reef geometry and compared to simulations of plane beaches lacking a reef. For a fixed offshore forcing condition, the fore‐reef slope controlled wave runup on reef‐fronted beaches, whereas the beach slope controlled wave runup on plane beaches. As a result, the coastal protection utility of reefs is dependent on these slopes. For our examples, with a fore‐reef slope of 1/5 and a 500 m prototype reef flat length, a beach slope of ∼1/30 marked the transition between the reef providing runup reduction for steeper beach slopes and enhancing wave runup for milder slopes. Roughness coverage, spacing, dimensions, and drag coefficient were investigated, with results indicating the greatest runup reductions were due to tall roughness elements on the reef flat.

Statistical modeling of undertow on a natural beach

ABSTRACT To better understand the physical mechanism of undertow, field observations were conducted during barred and planar beach states on the Hasaki coast in Japan. The two observation periods spanned 21 days from May 13 to June 2 in 2016, and 13 days from May 9 to May 22 in 2017, respectively. A horizontal acoustic Doppler current profiler was used to measure the undertow velocity and water level. The observed data were statistically analyzed to investigate the characteristics of the spatial and wave-energetic distributions of the undertow exceedance probability, PE . The results reveal that when the wave energy flux level was high, the undertow velocity increased, as did its PE . By contrast, the undertow PE during a low-wave-energy level decreased with greater water depths. Here, a Weibull distribution was applied to explain PE of the undertow. The Weibull parameters: scale and shape were estimated by the combination of normalized values of the wave energy flux, relative surf zone locations, and normalized water depths, and this generalized equation was considered a statistical model for estimating undertow PE . A comparison of the statistical model against the measurement demonstrates that the model accurately predicted the undertow PE with a small error.

Phase‐Resolved Modeling of Wave Interference and Its Effects on Nearshore Circulation in a Large Ebb Shoal‐Beach System

AbstractA time‐domain Boussinesq model was applied to modeling wave interference and its effects on nearshore circulation in San Francisco Bar and the adjacent Ocean Beach, CA. The model predicted the wave interference phenomena caused by the ebb shoal, with interference scales consistent with the radar observation and are persistent with nodal lines unchanged with time. Nearshore circulation predicted by the model shows small‐scale flow structures tied with the wave modulation patterns. However, the small‐scale modulation in the wave field seems not to generate alongshore variation in wave setup at similar scales. Therefore, in a large‐scale view, the alongshore currents predicted by the Boussinesq model still keep the general features shown in a wave‐averaged model, such as the flow divergence caused by the pressure gradient force associated with the alongshore variation of wave setup. The further analysis based on idealized short crest waves on a plane beach suggests that the alongshore pressure gradient is mainly balanced by the gradient of radiation stresses in the antinodal region, while by the flow advection in the nodal region. The rip current behavior is forced locally near breakers, rather than by gradients in wave setup close to shore. The time‐domain Boussinesq model predicted the spatial variability of wave‐induced processes. The alongshore‐varying wave breakers caused by wave interference are the source of the vorticity generation, inducing energetic vortex eddies nearshore.

Modeling solitary wave propagation and transformation over complex bathymetries using a two-layer Boussinesq model

Solitary gravity waves play a key role in the field of coastal and ocean engineering. The two-layer Boussinesq model developed by the authors has excellent linear and nonlinear properties. However, the permanent solitary wave solution and the numerical performance of the model in describing the solitary waves over complex bathymetries remain unknown. This research addresses this issue by conducting numerical simulations. The test cases include high amplitude solitary wave propagation over a constant water depth, shoaling of a solitary wave over a slope, and reflection of a solitary wave from a truncated composite plane beach. New experiments for solitary wave propagation and transformation over two submerged breakwaters are also presented. The computed results are compared against the analytical solutions and experimental data, with more attention given to the interior kinematics. Excellent performance is obtained, and the model shows significant improvement over traditional Boussinesq-type models. A detailed comparison between model predictions and laboratory data/analytical solutions is given for each case.

A comprehensive analysis of solitary wave run-up at sloping beaches using an extended experimental dataset

In this work, the phenomenon of wave run-up induced by solitary waves in breaking and nonbreaking conditions has been investigated. Despite several studies have been carried out to study this kind of physical process, a robust determination of solitary wave run-up on plane beaches over a wide range of wave conditions and beach slopes is still lacking. In this context, 99 laboratory tests were undertaken in the wave channel of the University of Calabria by adopting three beach slopes and three water depths. The experimental set-up was made up of four wave gauges, a pressure transducer, a velocimeter and an adjustable waterproof ramp to reproduce the plane sloping beach. Using graduated markers along the beach and a video camera, the run-up was deduced. The present experiments cover a range not extensively analyzed in literature, and specifically related to wave run-up comparable or greater than the water depth. The laboratory run-up, velocity and pressure data were validated using the IH-2VOF numerical model. Two different approaches, both for breaking and nonbreaking conditions, were used to obtain semi-analytical formulas for the wave run-up. The first approach is conventional, being based on wave non-linearity and beach slope, while the second is based on the wave energy.

Prediction of landslide tsunami run-up on a plane beach through feature selected MLP-based model

We proposed new prediction models based on multilayer perceptron (MLP) which successfully predict the maximum run-up of landslide-generated tsunami waves and assess the role of parameters affecting it. The input is approximately 55,000 rows of data generated through an analytical solution employing slide’s cross section, initial submergence, vertical thickness, horizontal length, beach slope angle and the maximum run-up itself, along with its occurrence time. The parameters are first ranked through a feature selection algorithm and six models are constructed for a 9,000-row randomly sampled dataset. These MLP-based models led predictions with a minimum Mean Absolute Percentage Error of 1.1% and revealed that vertical slide thickness has the largest impact on the maximum tsunami run-up, whereas beach slope angle has minimal effect. Comparison with existing literature showed the reliability and applicability of the offered models. The methodology introduced here can be suggested as fast and flexible method for prediction of landslide-induced tsunami run-up.

On the Oval Shapes of Beach Stones

This article introduces a new stochastic non-isotropic frictional abrasion model, in the form of a single short partial integro-differential equation, to show how frictional abrasion alone of a stone on a planar beach might lead to the oval shapes observed empirically. The underlying idea in this theory is the intuitive observation that the rate of ablation at a point on the surface of the stone is proportional to the product of the curvature of the stone at that point and the likelihood the stone is in contact with the beach at that point. Specifically, key roles in this new model are played by both the random wave process and the global (non-local) shape of the stone, i.e., its shape away from the point of contact with the beach. The underlying physical mechanism for this process is the conversion of energy from the wave process into the potential energy of the stone. No closed-form or even asymptotic solution is known for the basic equation, which is both non-linear and non-local. On the other hand, preliminary numerical experiments are presented in both the deterministic continuous-time setting using standard curve-shortening algorithms and a stochastic discrete-time polyhedral-slicing setting using Monte Carlo simulation.

Physical and Numerical Modeling of Wave-by-Wave Overtopping along a Truncated Plane Beach

Abstract Wave-by-wave and cumulative overtopping data from fixed planar impermeable smooth beaches will be presented from random wave experiments and compared with predictions from two nonlinear sh...

Numerical Simulation of Propagation and Run-Up of Long Waves in U-Shaped Bays

Wave propagation and run-up in U-shaped channel bays are studied here in the framework of the quasi-1D Saint-Venant equations. Our approach is numerical, using the momentum conserving staggered-grid (MCS) scheme, as a consistent approximation of the Saint-Venant equations. We carried out simulations regarding wave focusing and run-ups in U-shaped bays. We obtained good agreement with the existing analytical results on several aspects: the moving shoreline, wave shoaling, and run-up heights. Our findings also confirm that the run-up height is significantly higher in the parabolic bay than on a plane beach. This assessment shows the merit of the MCS scheme in describing wave focusing and run-up in U-shaped bays. Moreover, the MCS scheme is also efficient because it is based on the quasi-1D Saint-Venant equations.