Non-breaking Waves Research Articles

Experimental particle paths and drift velocity in steep waves at finite water depth

The Lagrangian paths, horizontal Lagrangian drift velocity, $U_{L}$, and the Lagrangian excess period, $T_{L}-T_{0}$, where $T_{L}$ is the Lagrangian period and $T_{0}$ the Eulerian linear period, are obtained by particle tracking velocimetry (PTV) in non-breaking periodic laboratory waves at a finite water depth of $h=0.2~\text{m}$, wave height of $H=0.49h$ and wavenumber of $k=0.785/h$. Both $U_{L}$ and $T_{L}-T_{0}$ are functions of the average vertical position of the paths, $\bar{Y}$, where $-1<\bar{Y}/h<0$. The functional relationships $U_{L}(\bar{Y})$ and $T_{L}-T_{0}=f(\bar{Y})$ are very similar. Comparisons to calculations by the inviscid strongly nonlinear Fenton method and the second-order theory show that the streaming velocities in the boundary layers below the wave surface and above the fluid bottom contribute to a strongly enhanced forward drift velocity and excess period. The experimental drift velocity shear becomes more than twice that obtained by the Fenton method, which again is approximately twice that of the second-order theory close to the surface. There is no mass flux of the periodic experimental waves and no pressure gradient. The results from a total number of 80 000 experimental particle paths in the different phases and vertical positions of the waves show a strong collapse. The particle paths are closed at the two vertical positions where $U_{L}=0$.

Pseudo-dynamic analysis of seawall considering non-breaking wave force

This paper presents the stability determination of seawall, supporting submerged cohesion-less backfill under combined actions of earthquake and non-breaking wave forces. The pseudo-dynamic approach is adopted for the calculation of seawall stability using limit equilibrium method against sliding, overturning and bearing failure. Parametric study shows significant effects of B/H ratio, pore water pressure ratio, horizontal seismic acceleration coefficient, ratio of height of water on seaward side to the wall height, ratio of non-breaking wave height to height of water on seaward side, wall friction angle, ratio of height of water on landward side to the wall height on stability of seawall at both restrained and free water cases, against all failure modes. Present result is compared with results available in literature. The results are presented in non-dimensional forms and design charts are provided to calculate factor of safety for design of stability of seawall against different failures under earthquake forces combined with non-breaking wave forces.

Direct numerical simulation of turbulent heat transfer across a sheared wind-driven gas–liquid interface

Turbulent heat transfer across a sheared wind-driven gas–liquid interface is investigated by means of a direct numerical simulation of gas–liquid two-phase turbulent flows under non-breaking wave conditions. The wind-driven wavy gas–liquid interface is captured using the arbitrary Lagrangian–Eulerian method with boundary-fitted coordinates on moving grids, and the temperature fields on both the gas and liquid sides, and the humidity field on the gas side are solved. The results show that although the distributions of the total, latent, sensible and radiative heat fluxes at the gas–liquid interface exhibit streak features such that low-heat-flux regions correspond to both low-streamwise-velocity regions on the gas side and high-streamwise-velocity regions on the liquid side, the similarity between the heat-flux streak and velocity streak on the gas side is more significant than that on the liquid side. This means that, under the condition of a fully developed wind-driven turbulent field on both the gas and liquid sides, the heat transfer across the sheared wind-driven gas–liquid interface is strongly affected by the turbulent eddies on the gas side, rather than by the turbulent eddies and Langmuir circulations on the liquid side. This trend is quite different from that of the mass transfer (i.e. $\text{CO}_{2}$ gas). This is because the resistance to heat transfer is normally lower than the resistance to mass transfer on the liquid side, and therefore the heat transfer is controlled by the turbulent eddies on the gas side. It is also verified that the predicted total heat, latent heat, sensible heat and enthalpy transfer coefficients agree well with previously measured values in both laboratory and field experiments. To estimate the heat transfer coefficients on both the gas and liquid sides, the surface divergence could be a useful parameter, even when Langmuir circulations exist.

Run-up on vertical piles due to regular waves: Small-scale model tests and prediction formulae

In wave-structure interaction, one of the most important phenomena clearly identified is wave run-up on offshore structures. In this study, wave run-up on a slender pile due to non-breaking regular waves is investigated by means of small-scale experiments performed in the 2m-wide wave flume of Leichtweiss-Institute for Hydraulic Engineering and Water Resources (LWI) in Braunschweig, Germany. The test programme is designed to generate a comprehensive data set covering a broader range of wave conditions including not only deep and intermediate water conditions but also nearly shallow and shallow water conditions, which are missing in the available laboratory studies on wave run-up on piles. The relative wave height (H/h), relative water depth (h/L) and slenderness of pile (D/L) are identified as the key parameters governing the relative wave run-up (Ru/H). Based on these parameters, new formulae covering the range of tested conditions (0.028≤H/h≤0.593, 0.042≤h/L≤0.861, 0.003≤D/L≤0.206) are developed to predict regular non-breaking wave run-up on single piles using a combination of the M5 model tree and nonlinear regression techniques. Using statistical accuracy metrics such as agreement index Ia, squared correlation coefficient R2 and scatter index SI, the performance of the developed formulae is evaluated. It is shown that the new formulae outperform the current formulae in predicting regular wave run-up on single piles. This success is in part due to the explicit account for the water depth in the new experiments and formulae. The proposed model is valid for a wider range of wave conditions and, therefore, more appealing for engineering practice compared to those available for the estimation of regular wave run-up.

Breaking phase focused wave group loads on offshore wind turbine monopiles

The current method for calculating extreme wave loads on offshore wind turbine structures is based on engineering models for non-breaking regular waves. The present article has the aim of validating previously developed models at DTU, namely the OceanWave3D potential flow wave model and a coupled OceanWave3D-OpenFOAM solver, against measurements of focused wave group impacts on a monopile. The focused 2D and 3D wave groups are reproduced and the free surface elevation and the in-line forces are compared to the experimental results. In addition, the pressure distribution on the monopile is examined at the time of maximum force and discussed in terms of shape and magnitude. Relative pressure time series are also compared between the simulations and experiments and detailed pressure fields for a 2D and 3D impact are discussed in terms of impact type. In general a good match for free surface elevation, in-line force and wave-induced pressures is found.

Modeling wave-structure interactions by an immersed boundary method in a σ-coordinate model

This paper studies wave-structure interactions using an immersed boundary (IB) method in a σ coordinate non-hydrostatic wave model. In contrast to a fixed grid, the σ grid follows the movement of the free surface, which may introduce numerical errors when applying the IB method. When the grid cells located inside the structure at the previous time step move into the flow domain at the current time step, a reconstruction of velocities at the cell center is conducted to accurately estimate the fluxes at the cell faces by enforcing the no-slip boundary condition at the structure boundaries. The model is validated against experimental data in three different problems, including solitary wave interactions with rectangular obstacles, periodic wave interaction with a submerged bar and non-breaking solitary wave interactions with multiple cylinders. These test cases deal with bottom-mounted, floating as well as emergent structures with different shapes. The agreements between simulations and measurements are generally good in terms of free surface elevation, flow velocity, dynamic pressure and wave runup. It is shown that the IB method is a promising tool to simulate wave processes around structures and to predict wave/surge loads on near-coast structures.

Physical modelling of tsunami onshore propagation, peak pressures, and shielding effects in an urban building array

Wave experiments were conducted on a 1:20 length scale to measure water surface elevations and extreme pressures on and around idealized structural elements and arrays of structures. Experiments varied offshore wave characteristics and onshore structural configurations. Conditions in which waves broke on or just before the specimen caused maximum impulsive pressures. Pressures measured under nonbreaking wave conditions agreed with predicted values using design equations suggested by the Japanese Cabinet Office; however bare-earth water surface elevation inputs produced nonconservative estimates in breaking wave trials. Shielded structures experienced pressure reductions of 40–70% under breaking wave conditions. Results indicate that shielding elements constructed nearshore may reduce wave-induced damage. This dataset may be used to validate numerical models of tsunami propagation through urban environments.

Fixed and moored bodies in steep and breaking waves using SPH with the Froude–Krylov approximation

Force prediction on fixed and moored bodies in steep, asymmetric and breaking waves remains a problem of great practical importance. For floating bodies snatch loads on mooring lines are of particular significance. In this paper we present an approximate approach where waves are modelled using incompressible smoothed particle hydrodynamics (SPH) which is well suited for breaking as well as non-breaking waves. For bodies of small size relative to wave length, the total force is assumed to be due to the Froude–Krylov force due to the undisturbed pressure field with additional added mass effects—in effect the Morison assumption. For a fixed vertical column in regular waves on a small slope, breaking wave force magnification is consistent with experiment and for focussed waves peak forces due to initial interaction are in good agreement with experiment; wave asymmetry is the dominant influence on overall force rather than local roller/jet breaker impact. For a taut moored hemispherical buoy in steep focussed waves the loads and motion without snatching are almost independent of added mass coefficient between zero and unity. Without breaking when snatching occurs the motion and loads measured experimentally are well predicted with zero added mass. This close agreement breaks down with wave breaking and the initial snatch load is overestimated by around 30 %. This approach is a fast alternative to fully 3-D simulations which are computationally demanding. Variation of, for example, mooring line properties and buoy position may be efficiently analysed using the same wave field and, as such, the approach has potential to be a useful design tool with further validation.

Impact of extreme waves on a vertical wall

This study addresses the impact of nonlinear wave evolution on the resulting wave force values on a vertical wall. To this end, the problem of interaction between non-breaking water waves and a vertical wall over constant depth is investigated. The investigation is performed using a two-dimensional wave flume model which is based on the high-order spectral method. Wave generation is simulated at the flume entrance by means of the additional potential concept. This model enables to preserve full dispersivity. Therefore, the model enables to examine the role of nonlinear evolution in the formation of extreme wave force values on a vertical wall for a wide range of water depths. The results for the force exerted on a vertical wall are presented for shallow and deep water conditions. In shallow water, extreme wave force values occur due to the formation of an undular bore. In deep water, extreme wave forces have been obtained as a result of disintegration process of incident wave groups into envelope solitons. Multiple maximum force values have been detected for each of the highest run-up peaks. This phenomenon has been introduced in shallow water conditions and is extended here for deep water conditions.

Extreme run-up events on a vertical wall due to nonlinear evolution of incident wave groups

Nonlinear evolution of long-crested wave groups can lead to extreme interactions with coastal and marine structures. In the present study the role of nonlinear evolution in the formation of extreme run-up events on a vertical wall is investigated. To this end, the fundamental problem of interaction between non-breaking water waves and a vertical wall over constant water depth is considered. In order to simulate nonlinear wave–wall interactions, the high-order spectral method is applied to a computational domain which aims to represent a two-dimensional wave flume. Wave generation is simulated at the flume entrance by means of the additional potential concept. Through this concept, the implementation of a numerical wavemaker is applicable. In addition to computational efficiency, the adopted numerical approach enables one to consider the evolution of nonlinear waves while preserving full dispersivity. Utilizing these properties, the influence of the nonlinear wave evolution on the wave run-up can be examined for a wide range of water depths. In shallow water, it is known that nonlinear evolution of incident waves may result in extreme run-up events due to the formation of an undular bore. The present study reveals the influence of the nonlinear evolution on the wave run-up in deep-water conditions. The results suggest that extreme run-up events in deep water may occur as a result of the disintegration of incident wave groups into envelope solitons.

Vertical oil dispersion profile under non-breaking regular waves

An experimental program was conducted to investigate vertical oil dispersion of surface oil spills under non-breaking regular waves. The variation in oil concentration caused by oil dispersion in a water column was studied to determine the vertical oil dispersion profile. The experiments were performed using different waves characteristics for different volumes of oil spill to evaluate the variation in oil concentration at three depths at two sampling stations. The correlations between oil concentration and the main parameters of wave characteristics, oil spill volume, sampling depth, and distance of sampling stations to spill location were assessed. The results revealed that the trend of variation in oil concentration versus wave steepness is linear. The results obtained from experimental measurements indicated that the oil concentrations at mid-depth were 44–77 % and the concentrations near the flume bed were 12–33 % of the concentration near the water surface.

Surface waves affect frontogenesis

AbstractThis paper provides a detailed analysis of momentum, angular momentum, vorticity, and energy budgets of a submesoscale front undergoing frontogenesis driven by an upper‐ocean, submesoscale eddy field in a Large Eddy Simulation (LES). The LES solves the wave‐averaged, or Craik‐Leibovich, equations in order to account for the Stokes forces that result from interactions between nonbreaking surface waves and currents, and resolves both submesoscale eddies and boundary layer turbulence down to 4.9 m × 4.9 m × 1.25 m grid scales. It is found that submesoscale frontogenesis differs from traditional frontogenesis theory due to four effects: Stokes forces, momentum and kinetic energy transfer from submesoscale eddies to frontal secondary circulations, resolved turbulent stresses, and unbalanced torque. In the energy, momentum, angular momentum, and vorticity budgets for the frontal overturning circulation, the Stokes shear force is a leading‐order contributor, typically either the second or third largest source of frontal overturning. These effects violate hydrostatic and thermal wind balances during submesoscale frontogenesis. The effect of the Stokes shear force becomes stronger with increasing alignment of the front and Stokes shear and with a nondimensional scaling. The Stokes shear force and momentum transfer from submesoscale eddies significantly energize the frontal secondary circulation along with the buoyancy.

Liquid sloshing in a horizontally forced vessel with bottom topography

This paper presents a numerical study of the free-surface evolution for inviscid, incompressible, irrotational, horizontally forced sloshing in a two-dimensional rectangular vessel with an inhomogeneous bottom topography. The numerical scheme uses a time-dependent conformal mapping to map the physical fluid domain to a rectangle in the computational domain with a time-dependent aspect ratio Q(t), known as the conformal modulus. The advantage of this approach over conventional potential flow solvers is the solution automatically satisfies Laplace's equation for all time, hence only the integration of the two free-surface boundary conditions is required. This makes the scheme computationally fast, and as grid points are required only along the free-surface, high resolution simulations can be performed which allows for simulations for mean fluid depths close to the shallow water water regime. The scheme is robust and can simulate both resonate and non-resonate cases, where in the former, the large amplitude waves are well predicted.Results of nonlinear simulations are presented in the case of non-breaking waves for both an asymmetrical ‘step’ and a symmetric ‘hump’ bottom topography. The natural free-sloshing mode frequencies are compared with the small topography asymptotic results of Faltinsen and Timokha (2009) (Sloshing, Cambridge University Press (Cambridge)), and are found to be lower than this asymptotic prediction for moderate and large topography magnitudes. For forced periodic oscillations it is shown that the hump profile is the most effective topography for minimizing the nonlinear response of the fluid, and hence this topography would reduce the stresses on the vessel walls generated by the fluid. Results also show that varying the width of the step or hump has a less significant effect than varying its magnitude.

Validation of a fully nonlinear and dispersive wave model with laboratory non-breaking experiments

With the objective of modeling coastal wave dynamics taking into account nonlinear and dispersive effects, a highly accurate nonlinear potential flow model was developed. The model is based on the time evolution of two surface quantities: the free surface position and the free surface velocity potential. A spectral approach is used to resolve vertically the velocity potential in the domain, by decomposing the potential using an orthogonal basis of Chebyshev polynomials. With this approach, a wide range of relative water depths can be simulated, as demonstrated here with the propagation of nonlinear regular waves over a flat bottom with kh=2π and 4π (where k is the wave number and h the water depth). The model is then validated by comparing the simulation results to experimental data for four non-breaking wave test cases: (1) nonlinear dynamics of a wave train generated by a piston-type wavemaker in constant water depth, (2) shoaling of a regular wave train on beach with constant slope up to the breaking point, (3) propagation of regular waves over a submerged bar, and (4) propagation of nonlinear irregular waves over a barred beach. The test cases show the ability of the model to reproduce well nonlinear wave interactions and the dynamics of higher-order bound and free harmonics. The simulation results agree well with the experimental data, confirming the model's ability to simulate accurately nonlinear and dispersive effects for non-breaking waves.

Observing and estimating of intensive triad interaction occurrence in very shallow water

In this study, a series of field measurements were carried out to investigate triad interactions of spectral peak in near shore. The water level fluctuations were recorded at 5 stations with depths varying from 0.8 to 5m along a shore-perpendicular transect at sandy coasts of Nowshahr, located in the southern Caspian Sea coast. Two storms with significant wave height of approximately 1.4m were observed during the measurement period.Using bispectral analysis, a new quantitative index is proposed to investigate temporal and spatial intensity of nonlinear interaction between spectral peak and other harmonics. The proposed index was evaluated for time series of water level data and compared with the bicoherence value of self-spectral peak triad interaction (SSPT); b2(fp,fp). Comparing to SSPT, the proposed new index includes all positive and negative triad interactions with spectral peak. The relative depth, kpd, of non-breaking waves varied from 0.25 to 2.00 along the transect during the study period, where kp is the wave-number and d is the water depth. In general, SSPT increased by decreasing kpd; however, the results showed that in two shallow stations the maximum SSPT did not correspond to the lowest values of kpd. A considerable time lag was observed between occurrence of the most intensive triad interactions and termination of wave breaking in post storm condition. Furthermore, the intensive triad interactions happened several hours after the largest Ursell numbers of non-breaking waves.

Estimating tsunami runup with fault plane parameters

Forecasting the maximum runup height and inundation area caused by a tsunami event has often been done by solving a numerical model describing the physical process. This approach requires data of fault plane parameters and bathymetry. The time required to prepare and execute the numerical model could be substantial. Therefore, it might not be suitable for the purpose of warning tsunami coastal flooding. In this paper we offer an alternative method that provides analytical relationship between the runup height and the fault plane parameters and the characteristic slope of coastal bathymetry. The method uses Okada's linear elastic dislocation model (Okada, 1985) to estimate the coseismic seafloor deformation and the corresponding (initial) sea surface displacement for a given set of fault plane parameters. Once the tsunami waves are generated, Carrier & Greenspan's solution (1958) is adopted to yield analytical expressions for the time histories of shoreline elevation and velocity. Two types of problems are investigated. In the first scenario, the bathymetry is modeled as a constant slope that is connected to a constant depth region, where a seismic event occurs. In the second scenario, the bathymetry is further simplified as a constant slope, on which a seismic event occurs. As far as the Carrier–Greenspan's solution for shoreline movement is concerned, the former is treated as a boundary-value-problem (BVP) and the latter as an initial-value-problem (IVP). For both problems the relationships between dimensionless runup height and dimensionless fault plane parameters are developed and discussed. In addition, since Carrier–Greenspan's approach is valid only for non-breaking waves, expressions limiting the applicability of the present solutions are obtained. Application of the present solutions is demonstrated with two examples: the 2004 Sumatra and 2010 Chile earthquakes. Acceptable agreements are obtained with the field measurements of these events. The present solutions offer a fast way to estimate the runup heights with the simplified beach geometry.

Can the Rayleigh distribution be used to determine extreme wave heights in non-breaking swell conditions?

A reliable set of tools for prediction of low-exceedance design waves is of high importance when designing coastal protection structures. The significant wave parameters are typically obtained from buoys or numerical wave propagation models and design values are found by extreme analysis. Statistical wave height distributions are used to transform the significant wave height to lower exceedance wave heights. These extreme single waves will cause the highest loads and wave overtopping volumes on structures and thereby represent the design conditions. An under-prediction of the design maximum wave height causes unsafe designs, while an over-prediction causes too conservative and thus expensive designs. The wave height distribution by Longuet-Higgins (1952) (Rayleigh-distribution) for deep-water non-breaking waves is in the present paper evaluated against data from numerical tests with long period and long-crested swell waves. The numerical model is validated against data from physical model tests. Generally, it is concluded that the Rayleigh-distribution is under-predicting the low-exceedance wave heights in irregular swell waves. This is expected to be caused by wave non-linearity and thus a new modified wave height distribution is suggested, where the shape parameter in the distribution is dependent on the wave non-linearity, represented by the Ursell-number. The new proposed wave height distribution for non-linear and non-breaking waves is highly applicable for practical engineering design of both near-shore and offshore structures under influence of swell-waves.

Influence of non-breaking wave force on seismic stability of seawall for passive condition

Proper design of seawall in earthquake prone region is one of the major concerns in geotechnical earthquake engineering. This paper presents the stability analysis of seawall under the combined action of earthquake forces, non-breaking wave force, hydrostatic and hydrodynamic forces and uplift force. Stability of seawall is assessed in terms of its factor of safety against landward sliding and landward overturning modes of failures. Seismic passive earth resistance has been calculated using pseudo-static approach. A detailed parametric study has been conducted to study the effect of non-breaking wave height, depth of water on seaward and land ward sides, soil and wall friction angles, and horizontal and vertical seismic accelerations. The factor of safety against overturning mode of failure decreases by about 52%, for a change in the ratio of non-breaking wave height to the depth of water on seaward side from 0 to 0.60. Present study shows that the seismic stability of seawall is more sensitive to non-breaking wave height, soil friction angle, wall friction angle and horizontal seismic acceleration. Proposed closed-form solutions and design charts can be used for the seismic design of seawall for passive case under the combined action of earthquake and non-breaking wave forces.

Modeling the Effect of Waves on the Diurnal Temperature Stratification of a Shallow Lake

In this paper a three-dimensional circulation model has been applied to determine the spatial distribution of the temperature and mixed-layer depth (MLD) of a medium-sized shallow lake. In order to reproduce the vertical thermal structure of the lake, the impact of waves has been incorporated using several recently published schemes. These schemes approximate the additional mixing due to the orbital motion of waves, the turbulent kinetic energy produced by waves, or both. The reference solver is the Reynolds-Averaged Navier-Stokes equations with two-equation Mellor-Yamada 2.5 turbulence closure. The wave field is characterized by bulk parameters, e.g. significant wave height. The sensitivity to the choice of the wave mixing scheme is analyzed by means of an academic test case. The accuracy of the scheme is explored through data collected in Lake Balaton. The modeled temperature profiles and MLDs match measurements much better when the effect of non-breaking surface waves is included.

Sliding and overturning stability of seawalls subjected to non-breaking waves

Sliding and overturning stability of seawalls subjected to non-breaking waves