Non-breaking Waves Research Articles

Study on Flow Fields of Boundary-Layer Separation and Hydraulic Jump during Rundown Motion of Shoaling Solitary Wave

The characteristics of flow fields for a complete evolution of the non-breaking solitary wave, having a wave-height to water-depth ratio of 0.363 and propagating over a 1:5 sloping bottom, are investigated experimentally. This study mainly focuses on the occurrences of both flow separation on the boundary layer under an adverse pressure gradient and subsequent hydraulic jump with the abrupt rising of free surface during rundown motion of the shoaling wave, together with emphasis on the evolution of vortex structures underlying the separated shear layer and hydraulic jump. A flow visualization technique with particle trajectory method and a high-speed particle image velocimetry (HSPIV) system with a high-speed digital camera were used. Based on the instantaneous flow images visualized and/or the ensemble-averaged velocity fields measured, the following interesting features, which are unknown up-to-date, are presented and discussed in this study: (1) Flow bifurcation occurring on both offshore and onshore sides of the explicit demarcation curve and the stagnation point during runup motion; (2) The dependence of the diffuser-like flow field, being changed from the supercritical flow in the shallower region to the subcritical flow in the deeper counterpart, on the Froude number during the early and middle stages of rundown motion; (3) The positions and times for the occurrences of the incipient flow separation and the sudden rising of free surface of the hydraulic jump; (4) The associated movement and evolution of vortex structures under the separated shear layer, the hydraulic jump and/or the high-speed external main stream of the retreated flow; and (5) The entrainment of air bubbles from the free surface into the external main stream of the retreated flow.

Wave spectral response to sudden changes in wind direction in finite-depth waters

The response of a wind-sea spectrum to sudden changes in wind directions of 180° and 90° is investigated. Numerical simulations using the third-generation wave spectral model SWAN have been undertaken at micro timescales of 30 s and fine spatial resolution of less than 10 m. The results have been validated against the wave data collected during the field campaign at Lake George, Australia. The newly implemented ‘ST6’ physics in the SWAN model has been evaluated using a selection of bottom-friction terms and the two available functions for the nonlinear energy transfer: (1) exact solution of the nonlinear term (XNL), and (2) discrete interactions approximation (DIA) that parameterizes the nonlinear term. Good agreement of the modelled data is demonstrated directly with the field data and through the known experimental growth curves obtained from the extensive Lake George data set.The modelling results show that of the various combinations of models tested, the ST6/XNL model provides the most reliable computations of integral and spectral wave parameters. When the winds and waves are opposing (180° wind turn), the XNL is nearly twice as fast in the aligning the young wind-sea with the new wind direction than the DIA. In this case, the young wind-sea gradually decouples from the old waves and forms a new secondary peak. Unlike the 180° wind turn, there is no decoupling in the 90° wind turn and the entire spectrum rotates smoothly in the new direction. In both cases, the young wind-sea starts developing in the new wind direction within 10 min of the wind turn for the ST6 while the directional response of the default physics lags behind with a response time that is nearly double of ST6.The modelling results highlight the differences in source term balance among the different models in SWAN. During high wind speeds, the default settings provide a larger contribution from the bottom-friction dissipation than the whitecapping. In contrast, the whitecapping dissipation is dominant in ST6 while the bottom-friction generated by the new model with ripple formation provides a significant contribution during strong winds only. During low wind speeds and non-breaking wave conditions, a separate swell or nonbreaking dissipation source term continues the decay of waves that cannot be dissipated by the whitecapping dissipation function.

On the numerical simulation of the nonbreaking solitary waves run up on sloping beaches

In this paper, a solitary wave propagation problem running up on a range of relatively steep slopes to gentle slopes is investigated. Boussinesq solitary wave generation method is used in five beach slopes with the still water heights of 15.0–68.6 cm and relative wave heights of 0.046–0.348. The incompressible smoothed particle hydrodynamics method is utilized as the numerical method. Simulation results are compared with analytical results and experimental data in terms of free surface displacement, maximum run up, shoreline movement and fluid particle velocity. Surface profile results agree reasonably with the analytical results at run down and horizontal particle velocities coincide with the experimental data. Furthermore, the results of maximum run up show that for moderate and large wave heights, Lagrangian numerical results pursuit the maximum run up of the experimental data with acceptable accuracy.

The swash of solitary waves on a plane beach: flow evolution, bed shear stress and run-up

The swash of solitary waves on a plane beach is studied using large-scale experiments. Ten wave cases are examined which range from non-breaking waves to plunging breakers. The focus of this study is on the influence of breaker type on flow evolution, spatiotemporal variations of bed shear stresses and run-up. Measurements are made of the local water depths, flow velocities and bed shear stresses (using a shear plate sensor) at various locations in the swash zone. The bed shear stress is significant near the tip of the swash during uprush and in the shallow flow during the later stages of downrush. In between, the flow evolution is dominated by gravity and follows an explicit solution to the nonlinear shallow water equations, i.e. the flow due to a dam break on a slope. The controlling scale of the flow evolution is the initial velocity of the shoreline immediately following waveform collapse, which can be predicted by measurements of wave height prior to breaking, but also shows an additional dependence on breaker type. The maximum onshore-directed bed shear stress increases significantly onshore of the stillwater shoreline for non-breaking waves and onshore of the waveform collapse point for breaking waves. A new normalization for the bed shear stress which uses the initial shoreline velocity is presented. Under this normalization, the variation of the maximum magnitudes of the bed shear stress with distance along the beach, which is normalized using the run-up, follows the same trend for different breaker types. For the uprush, the maximum dimensionless bed shear stress is approximately 0.01, whereas for the downrush, it is approximately 0.002.

Volumetric velocity measurements of turbulent coherent structures induced by plunging regular waves

The instantaneous turbulent velocity fields induced by the breaking of plunging regular waves on a 1 in 40 plane slope were measured using a volumetric three-component velocimetry (V3V) system. The measurement volume was located in the outer surf zone to capture the plunger vortex generated at incipient breaking and the splash-up vortex generated at the second plunge point. The measurement volume extended from a few millimeters above the bottom to just below the wave trough level. The aerated water was seeded with fluorescent particles and long-pass filters were used to block scattered light from entrained air bubbles. The volumetric velocity measurements revealed the three-dimensional (3D) structure of breaking-wave-generated vortices previously captured in two-dimensional (2D) planes using a particle image velocimetry (PIV) system. The evolution of the vortices was also captured. The most common turbulent coherent structure observed was a vortex loop with counter-rotating vorticity; two longitudinal vortices oriented obliquely upward in the direction of wave propagation are connected at the base by a transverse vortex. The vortex loop was carried in a downburst of turbulent fluid, which impinged on the bottom around the instant of maximum positive (onshore) wave-induced velocity. Downbursts carried large amount of turbulent kinetic energy and induced large apparent shear stress over the bottom. The distributions of vorticity, turbulent kinetic energy, turbulent kinetic energy flux and turbulent shear stress in the transverse segment of a vortex loop were examined. In a separate experiment, water particle velocities associated with non-breaking waves were measured at different heights above a plane slope using an acoustic Doppler velocimeter (ADV) to verify the V3V measurements.

TTP로 피복된 경사식구조물의 처오름높이 산정식: 사면경사 1:1.5 조건

The runup height is an important design parameter to determine the crest elevation of coastal structures and seawalls. In this study, two dimensional wave runup tests for rubble-mound structure covered by tetrapods were conducted. Incident waves at the toe include nonbreaking, breaking and broken random wave conditions. A empirical formula to predict runup elevation of rubble-mound structure with 1:1.5 front slope was proposed on the basis of physical model test results using a surf similarity parameter. The test results from this study were compared with those from van der Meer and Stam(1992).

On steady non-breaking downstream waves and the wave resistance

In this work we have obtained exact analytical formulae expressing the wave resistance of a two-dimensional body by the parameters of the downstream non-breaking waves. The body moves horizontally at a constant speed $c$in a channel of finite depth $h$. We have analysed the relationships between the parameters of the upstream flow and the downstream waves. Making use of some results by Keady & Norbury (J. Fluid Mech., vol. 70, 1975, pp. 663–671) and Benjamin (J. Fluid Mech., vol. 295, 1995, pp. 337–356), we have rigorously proved that realistic steady free-surface flows with a positive wave resistance exist only if the upstream flow is subcritical, i.e. the Froude number$\mathit{Fr}=c/\sqrt{gh}<1$. For all solutions with downstream waves obtained by a perturbation of a supercritical upstream uniform flow the wave resistance is negative. Applying a numerical technique, we have calculated accurate values of the wave resistance depending on the wavelength, amplitude and mean depth.

2D numerical simulation of large-scale physical model tests of wave interaction with a rubble-mound breakwater

Experimental measurements on a large-scale, multi-layered breakwater model are used to extensively validate a numerical model for wave interaction with permeable coastal structures, built in a generic multiphysics CFD code, FLOW-3D. A novel contribution to this code is the implementation of the piston-type wave-maker with active wave absorption. The use of active absorption is crucial to maintain an accurate and stable incident wave field in the validation test series with long duration. Due to the nonbreaking wave conditions, the model setup is simplified to two-dimensions and single-phase laminar flow in the clear-fluid region. The validation includes a verification of the incident wave field and a detailed study of the hydrodynamics of the flow field within the breakwater. The evolution of experimentally measured free-surface elevations in and outside the breakwater shows a good agreement with numerical predictions. Numerical results of pore pressure height throughout the breakwater core are generally corresponding well with experimental values, except for a limited zone near the free surface where a clear reduction of the pore pressure is measured, due to the effect of air entrainment. This validation study, using large-scale model tests, proves that this model, and by extension the nowadays standard branch of VOF-based Navier–Stokes models, are a useful tool in predicting wave interaction with permeable coastal structures. The study also reveals that it is possible to achieve a level of accuracy that can be considered sufficient for engineering applications, even when using some model simplifications, in particular those related to the treatment of the aerated turbulent flow on and inside the armor layer.

Crest modifications to reduce wave overtopping of non-breaking waves over a smooth dike slope

The formula to quantify the average wave overtopping discharge of non-breaking waves over a dike according to the TAW-report (2002) and included in the EurOtop Manual (2007), only contains the influence of the roughness of the dike slope and the obliqueness of the waves on wave overtopping. Unlike the formula for breaking waves, the reductive effect of a storm wall or a berm is not included in this formula. Over 1000 scale model tests with non-breaking waves on a wide variety of structures have been carried out for this paper. The crest of the dike slope has been modified with a storm wall, a storm wall and parapet, a stilling wave basin, a promenade, and a combination of a promenade and storm wall with or without parapet. For all these modified crests, a reduction factor has been deducted to include in the wave overtopping formula of non-breaking waves. This paper presents the results of all investigated measures to reduce wave overtopping over a smooth dike slope in an easy and logical way to serve as a guidance for use by designers. An example is given where all presented measures are compared to each other, and to other well-known structures.

Capillary effects on wave breaking

We investigate the influence of capillary effects on wave breaking through direct numerical simulations of the Navier–Stokes equations for a two-phase air–water flow. A parametric study in terms of the Bond number, $\mathit{Bo}$, and the initial wave steepness, ${\it\epsilon}$, is performed at a relatively high Reynolds number. The onset of wave breaking as a function of these two parameters is determined and a phase diagram in terms of $({\it\epsilon},\mathit{Bo})$ is presented that distinguishes between non-breaking gravity waves, parasitic capillaries on a gravity wave, spilling breakers and plunging breakers. At high Bond number, a critical steepness ${\it\epsilon}_{c}$ defines the onset of wave breaking. At low Bond number, the influence of surface tension is quantified through two boundaries separating, first gravity–capillary waves and breakers, and second spilling and plunging breakers; both boundaries scaling as ${\it\epsilon}\sim (1+\mathit{Bo})^{-1/3}$. Finally the wave energy dissipation is estimated for each wave regime and the influence of steepness and surface tension effects on the total wave dissipation is discussed. The breaking parameter $b$ is estimated and is found to be in good agreement with experimental results for breaking waves. Moreover, the enhanced dissipation by parasitic capillaries is consistent with the dissipation due to breaking waves.

Model of the evolution of mounds placed in the nearshore

A one-dimensional mathematical model is presented that describes the cross-shore evolution of a long linear mound of noncohesive sediment placed in the nearshore and exposed to non-breaking waves. The equation for the transport rate accounts for wave asymmetry and gravity, and the net rate is expressed by reference to an equilibrium profile shape. Simplifications reduce the governing equation for mound evolution to the diffusion equation, for which analytical solutions are available for various initial shapes of the mound. Temporal and spatial dependencies governing mound evolution are obtained from the analytic solutions; for example, a doubling of the wave height implies a certain mound response in 1/8 of the time compared to the original conditions. The governing equation is also solved numerically in order to avoid schematization of the forcing, initial, and boundary conditions. Both the analytical and numerical models are compared with data on mound evolution from several sites around the world. Model predictions agree with trends in measurements of four mounds at widely different sites. An example is given concerning the application of the analytical model for preliminary mound design. The formulation presented also applies to infilling of dredged trenches with lengths much greater than their widths. (Less)

Extreme wave loads on submerged water intakes in shallow water

This paper provides new guidance concerning the hydrodynamic loads on submerged intake structures located in shallow water under breaking and non-breaking waves. Results from a series of experiments conducted in a large wave flume at 1:15 scale to study the hydrodynamic forces exerted on a generic intake structure located on a sloping seabed in shallow water below breaking and non-breaking irregular waves are presented. Based on analysis of the experimental data, empirical relationships are developed to describe the peak loads in terms of characteristic wave parameters such as significant wave height and peak wave period. The distribution of the peak loads across different parts of the intake structure is also described. Drag and inertia force coefficients for the horizontal forcing on the intake structure and for the main structural sub-components are derived and presented. It is shown that the well-known Morison equation, with appropriate drag and inertia force coefficients, can provide reasonable estimates of the moderate horizontal loads, but the peak loads are less well predicted.

Investigation of Spectra Variations of Wave Groups in Finite Depth

A series of physical experiments were conducted to investigate the spectra evolution of single wave packets. Chirped-typed wave groups were selected in this study. Nine wave packets varying initial amplitudes and wave components were generated in the wave flume. Hence both non-breaking and breaking wave groups can be performed. It is found that the evolution process is far more than the linear superposition, and that it experiences significant nonlinear process. The modulational instability is found in the evolution of wave groups. For non-breaking groups, it is found that the most non-breaking energy dissipation occurs in the vicinity of the spectral peak (f/fp = 0.9-1.1). The non-breaking dissipation energy for spectral peak depends on depend on BFI (Benjamin-Feir Index). For breaking cases, the energy level of the above-peak frequency region (f/fp = 1.1-2.0) first increase slightly, and decrease during breaking occurring, but the overall change is limited. However, the below-peak frequency region (f/fp = 0.5-0.9) gains with the expense energy of the peak and above-peak frequency regions and the growth depends on the initial spectra width.

Pressure Acting on Underside of Bridge Deck

Pressure acting on a bridge superstructure with a π-shaped and a rectangular cross-sections are measured. The characteristics of the pressure acting on the underside of the bridge deck are investigated. The pressure caused by air compression on the underside of the bridge deck is also discussed. The author shows that the pressure at the landward corner inside the bridge superstructure with the π-shaped cross-section was caused by the air compression. The author also shows that the pressure acting on the seaward point on the underside of the model bridge deck with the π-shaped cross-section was smaller than that acting on the landward point on the underside under just breaking and non-breaking wave conditions.

Laboratory Study on the Characteristics of Deep-water Breaking Waves

In this paper, deep-water breaking waves are generated by the method of energy focusing in a wave flume and the intensity of wave breaking is toned by changing input wave steepness. In the experiment, the time series of surface elevation fluctuation along the flume are obtained utilizing 22 wave probes which are mounted along the mid-stream of the flume. The characteristics of wave surface are analyzed. The spectrum are computed for surface elevations by a fast Fourier transform (FFT). It is concluded that the energy keeps stable in low frequencies part and spreads toward the higher frequencies of the first harmonic band as the wave approaching the breaking zone. After the breaking, the spectrum restores almost its initial shape, but the spectrum energy is lost in the high-frequency end of the first harmonic band, which is more appreciable when the wave breaking happens and is stronger. As the energy is spread to higher frequencies for non-breaking wave, the “spectrally weighted wave frequency” fs becomes bigger and the “spectrally weighted group velocity” Cgs become smaller after the wave focusing. When the wave breaking occurs, the loss in energy is obvious leading to fs decreasing and Cgs increasing after wave breaking, which is more appreciable for plunging wave.

Run-up of non-breaking double solitary waves with equal wave heights on a plane beach

The evolution and run-up of double solitary waves on a plane beach were studied numerically using the nonlinear shallow water equations (NSWEs) and the Godunov scheme. The numerical model was validated through comparing the present numerical results with analytical solutions and laboratory measurements available for propagation and run-up of single solitary wave. Two successive solitary waves with equal wave heights and variable separation distance of two crests were used as the incoming wave on the open boundary at the toe of a slope beach. The run-ups of the first wave and the second wave with different separation distances were investigated. It is found that the run-up of the first wave does not change with the separation distance and the run-up of the second wave is affected slightly by the separation distance when the separation distance is gradually shortening. The ratio of the maximum run-up of the second wave to one of the first wave is related to the separation distance as well as wave height and slope. The run-ups of double solitary waves were compared with the linearly superposed results of two individual solitary-wave run-ups. The comparison reveals that linear superposition gives reasonable prediction when the separation distance is large, but it may overestimate the actual run-up when two waves are close.

Analysis of mean velocity and turbulence measurements with ADCPs

The present study examines the vertical structure of the coastal current in the inner part of the Gulf of Taranto, located in the Ionian Sea (Southern Italy), including both the Mar Grande and Mar Piccolo basins. To this aim, different measuring stations investigated by both a Vessel Mounted Acoustic Doppler Current Profiler (VM-ADCP) and a bottom fixed ADCP were taken into consideration. Two surveys were carried out in the target area on 29.12.2006 and on 11.06.2007 by the research unit of the Technical University of Bari (DICATECh Department), using a VM-ADCP to acquire the three velocity components along the water column in selected stationing points. The measurements were taken in shallow waters, under non-breaking wave conditions, offshore the surf zone. Due to the recording frequency of the instrument time-averaged vertical velocity profiles could be evaluated in these measuring stations. Water temperature and salinity were also measured at the same time and locations by means of a CTD recorder. A rigidly mounted ADCP, located on the seabed in the North-Eastern area of the Mar Grande basin, provided current data relative to the period 10–20 February 2014. Set to acquire the three velocity components with higher frequency with respect to the VM-ADCP, it allowed us to estimate the turbulent quantities such as Reynolds stresses and turbulent kinetic energy by means of the variance method.Therefore, the present research is made up of two parts. The first part examines the current pattern measured by the VM-ADCP and verifies that, for each station, the classical log law reproduces well the vertical profile of the experimental streamwise velocities extending beyond its typical limit of validity up to the surface i.e. reaching great heights above the sea bed. This behavior is quite new and not always to be expected, being generally limited to boundary layers. It has been convincingly observed in only few limited experimental works. In the present study this occurred when two conditions were met: (i) the flow was mainly unidirectional along the vertical; (ii) the interested layer was non-stratified.The second part of the research studies the turbulent statistics derived from the beam signals of the fixed ADCP by means of the variance method. This technique had the advantage of being able to measure the time evolution of the turbulent mixing throughout the entire water column, thus making it possible to perform a detailed study on momentum transfer and turbulence. The deduced vertical profiles of the Reynolds stresses and of the turbulent kinetic energy TKE showed an increasing trend toward the surface, in agreement with previous results in literature.New data-sets of mean velocities and shear stresses, coming from field measurements, are always needed. In fact they represent the first step to derive reliable reference values of coefficients and parameters for the implementation and calibration of the used mathematical hydrodynamic models.Consequently, an effort was made to evaluate consistent bottom drag and wind drag coefficients, on the basis of the calculated bottom and surface shear stresses, respectively.

A METHODOLOGY FOR MEASURING THE VELOCITY AND THICKNESS OF WAVE-INDUCED UP-RUSHING JETS ON VERTICAL SEAWALLS AND SUPERSTRUCTURE

Waves breaking on vertical seawalls result in up-rushing water jets with high velocities, which in turn lead to large overtopping rates and pose a serious threat to pedestrians, vehicles or any infrastructure in the vicinity of the seawall. Nevertheless, knowledge on the characteristics, namely the velocity and thickness, of such jets is scarce. Based upon recent experiments in the Large Wave Flume (Groser Wellenkanal, GWK) of Forschungszentrum Kuste (FZK) the current paper proposes a semi-automated methodology for measuring velocity and thickness of up-rushing jets. The methodology was applied on cases considering waves ranging from non-breaking to nearly and strongly breaking and yielded encouraging results. With regards to the velocities a good repeatability is reported for non-breaking waves, while for breaking waves velocities are, as anticipated, observed to depend on the intensity of breaking. When the thickness is concerned, user based decisions are required as the selection process is drastically affected by the increased air content of the up-rushing jet.

ATTENUATION OF SOLITARY WAVE BY PARAMETERIZED FLEXIBLE MANGROVE MODELS

The systematic laboratory investigation on tsunami attenuation by flexible mangrove models was performed in order to improve the knowledge on tsunami-coastal forest interaction. A sophisticated parameterization method, based on structural and bio-mechanical properties of a mature mangrove (Rhizophora sp.), was developed for the construction of the mangrove models under assumption of stiff and flexible structure. The forest model examined in the laboratory experiments consisted of the selected flexible mangrove models, arranged in different configurations, which was impacted by a tsunami-like solitary wave of varying height, propagating in different water depths. Based on the envelopes of max. wave height and wave forces induced on single tree models, wave evolution modes were determined to identify the source of wave attenuation. The results indicate the dependence of wave transmission on the observed wave evolution modes and relative forest width: the highest transmission coefficient is attributed to nonbreaking waves (ca. 0.78 and 0.55 for forest width of 0.75 and 3.0 m, respectively), while the lowest transmission coefficient corresponds to wave breaking in front of/in the forest model (ca. 0.5 and 0.3 for forest width of 0.75 and 3.0 m, respectively).

Soft shore protection methods: The use of advanced numerical models in the evaluation of beach nourishment

Beach nourishment is one of the worldwide most common soft shore protection methods. However, the design of these projects is usually based on empirical equations and rules, leaving large margins of error regarding their expected efficiency. In the present work, an advanced wave and sediment transport numerical model is developed and tested in the evaluation of beach nourishment. Non-linear wave transformation in the surf and swash zone is computed by a non-linear breaking wave model based on the higher order Boussinesq equations, for breaking and non-breaking waves. The new Camenen and Larson (2007), transport rate formula for non-cohesive sediments (involving unsteady aspects of the sand transport phenomenon) is adopted for estimating the sheet flow sediment transport rates, as well as the bed load and suspended load over ripples. Suspended sediment transport rate is incorporated by solving the 2DH depth-integrated transport equation. Model results are compared with experimental data of both profile (cross-shore) and planform morphology evolution; the agreement between the two is considered to be quite satisfactory.