Wave Steepness Parameter Research Articles

Hydrodynamic behaviour of composite bucket foundation with random waves

In real sea conditions, the effect of nonlinear phenomena, such as wave run-up and wave pressure, on the foundations of offshore wind structures may threaten the safety of the platforms. This study investigated the hydrodynamic behaviour of a composite bucket foundation (CBF) under random waves. A hydrodynamic simulation model of the interaction between random waves and the CBF was developed using the renormalisation group (RNG) k-ε model, which was verified through experiments. The relationship between wave run-up and wave parameters was studied, and the distributions of wave pressure around the CBF was described. The flow field and velocity vector distribution characteristics around the CBF were analysed. The results show that two types of wave fields appear around the CBF. The maximum run-up value of the CBF is 100%–150% higher than the minimum value, and the run-up value at the back side is 50%–70% higher than the minimum value. A larger wave steepness parameter leads to a smaller wave run-up value at the location of the smallest wave run-up value. The results can provide a reference for the practical application of a CBF.

New insights in the probability distributions of wave-by-wave overtopping volumes at vertical breakwaters

Advances in the development of prediction tools for wave overtopping allow now for overtopping volumes to be estimated with good accuracy, with the combined use of mean overtopping rates and maximum wave by wave overtopping volumes in a sequence of wave overtopping events. While previous literature has tended to focus on mean overtopping rates at coastal structures, limited studies have investigated the wave by wave overtopping volumes at coastal sea defences; in particular, a paucity of studies have focussed on the prediction of the shape parameter in the Weibull distribution (i.e., Weibull b) of overtopping volumes. This study provides new insights on the probability distribution of individual wave overtopping volumes at plain vertical seawalls by analysing the measured Weibull b values derived from a series of laboratory experiments on seawalls performed on a wide range of wave conditions and crest freeboards. The influence of wave conditions (wave steepness, significant wave height), structural parameters (crest freeboard, toe water depth), impulsiveness, probability of overtopping waves, and overtopping discharge on Weibull b parameter were examined, and then compared with the well-established empirical formulae. For the conditions covered within this study, it was found that the probability distribution of wave-by-wave overtopping volumes follow a 2-parameter Weibull distribution. No apparent differences in Weibull b values were reported with the variation of incident wave steepness and impulsiveness parameter. Results of this study revealed that Weibull b values at vertical walls, subjected to non-impulsive wave conditions, can be predicted reasonably well using relative freeboard and relative overtopping rates. A new unified formula is proposed for the estimation of Weibull b values at vertical walls under impulsive and non-impulsive wave attack.

STUDY OF MODIFIED PERFORATED BREAKWATER AS RENEWABLE ENERGY DEVICE

Aim: This study investigates to determine the influence of wave steepness, relative freeboard, and breaker parameters on overtopping discharge at a perforated breakwater. Methodology and results: The research method used was using both a numerical model simulations on three-dimensional computational fluid dynamics (CFD) modelling software namely FLOW-3D; and empirical equation computation. The evaluation of both approaches were performed for understanding the characteristics of wave discharge that overtopping the perforated breakwater. The experimental results of modified perforated breakwater revealed that the lowest slope possible with the highest porosity possible can generate the highest value of dimensionless overtopping discharge for wave energy harvesting. Conclusion, significance and impact study: The findings of this study formulated the optimum slope and porosity to the highest wave energy harvested. Further studies recommend that data collection from onsite trials of modified perforated breakwater are performed.

Direct and fast probabilistic assessment of long term monopile load distribution from combined metocean data and fully nonlinear wave kinematics

A novel method is introduced to determine the long-term extreme value distribution of variables with an associated short-term distribution. The method is applicable to e.g. the extreme value with a return period of 10.000 years for wave height, crest height of waves as well as wave-induced response and loads on offshore wind turbine monopiles. The extreme values are determined by a combination of a joint probabilistic description of the met-ocean environment and of a recently developed database of fully nonlinear wave kinematics computations. For two benchmark monopile structures, representative for the conditions at Dogger Bank and the German Bight in the North Sea, the one-hour max value of crest height, inline force and overturning moment are computed via the Rainey slender-body force model. Convolution with the joint probability of the sea state parameters in the nondimensional space of wave steepness and Ursell parameter leads to the marginal probability distribution for these parameters up to a 10.000 year return period and properly accounts for the uncertainty of the extreme value parameters in also the short-term distributions. It is found that the best extreme value fit is generally obtained with a third-order polynomial fit within the space of steepness and Ursell parameter. Compared to the Forristall crest height distribution, the inclusion of full nonlinearity leads to larger crest heights at low return period levels. However, due to its realistic treatment of the breaking limitation, lower crests are predicted at the 10.000 year level for the two positions considered in the North Sea. With further verification of the wave kinematics in combination with load models and a thorough comparison to present engineering practice, the proposed methodology provides a robust future solution for directly estimating extreme value distributions of loads and response of offshore wind turbine monopiles.

Impact of nearshore vegetation on coastal dune erosion: assessment through laboratory experiments

The primary objective of this study was to delineate the impact of vegetation on dune erosion under the influence of storm waves. This was achieved through laboratory flume experiments. Specifically, physical model experiments were used to identify the features responsible for coastal dune erosion generated by storm waves under the protection and non-protection of vegetation cover. Observations were then formulated based on the effective variables such as wave height and period, vegetation and sediment characteristics through Buckingham pi theorem to obtain dimensionless parameters like fall speed, vegetation length, depth and wave steepness parameters. Among the dimensionless parameters, the vegetated length parameter was effective in defining, measuring and comparing the coastal dune erosion. Based on vegetated length parameter, the change in erosion volume per unit width was calculated as 9.7–74% for regular wave conditions and 2.9–70.1% for irregular wave conditions. Furthermore, as the vegetated length parameter reduced wave damping ratio was not significantly affected, whereas the increase in the vegetated length parameter increases the wave damping ratio up to 0.5. As a result, the wave attenuation was recorded between 45 and 50%. An approach to correlate the wave energy coming out of the vegetation field with the energy necessary to erode the coastal dune profile depicted a linear correlation. As a result, empirically based equations have been developed to be used in soft coastal defense approaches which predict the volume of dune erosion and the wave attenuation under the protection of emergent vegetation. In addition, the energy required for the evolution of dune erosion is formulated in terms of attenuated wave energy.

Response of a slender cylindrical member under Breaking wave impact

A slender cylindrical member is the fundamental component of many coastal and offshore structures. It is essential to understand the action of breaking waves on structural members since, when these members are subjected to wave impact load, the structural damages and failures may be catastrophic. In the present study, an experimental investigation has been carried out to measure the response of the slender vertical cylinder subjected to breaking wave impact. Pressure measurements were made in order to estimate the total force acting on the structure. An empirical relation is established between total impact force and the wave steepness parameter which dictates the intensity of the breaking. Pressure rise time is an important parameter dictating the impact event. Simultaneous acceleration and strain measurements were made to evaluate the response of the cylinder. Deflection has been derived from structure acceleration. The deflection due to the moderate plunging differs by two order more than severe plunging events. Strain rate is another important consideration for the impact loading. The higher strain rate increases the resistance of the material. Under the breaking wave impact on the cylinder, the strain rate varies from2 to 10−4.

Nonhydrostatic and surfbeat model predictions of extreme wave run-up in fringing reef environments

The accurate prediction of extreme wave run-up is important for effective coastal engineering design and coastal hazard management. While run-up processes on open sandy coasts have been reasonably well-studied, very few studies have focused on understanding and predicting wave run-up at coral reef-fronted coastlines. This paper applies the short-wave resolving, Nonhydrostatic (XB-NH) and short-wave averaged, Surfbeat (XB-SB) modes of the XBeach numerical model to validate run-up using data from two 1D (alongshore uniform) fringing-reef profiles without roughness elements, with two objectives: i) to provide insight into the physical processes governing run-up in such environments; and ii) to evaluate the performance of both modes in accurately predicting run-up over a wide range of conditions. XBeach was calibrated by optimizing the maximum wave steepness parameter (maxbrsteep) in XB-NH and the dissipation coefficient (alpha) in XB-SB) using the first dataset; and then applied to the second dataset for validation. XB-NH and XB-SB predictions of extreme wave run-up (Rmax and R2%) and its components, infragravity- and sea-swell band swash (SIG and SSS) and shoreline setup (<η>), were compared to observations. XB-NH more accurately simulated wave transformation but under-predicted shoreline setup due to its exclusion of parameterized wave-roller dynamics. XB-SB under-predicted sea-swell band swash but overestimated shoreline setup due to an over-prediction of wave heights on the reef flat. Run-up (swash) spectra were dominated by infragravity motions, allowing the short-wave (but not wave group) averaged model (XB-SB) to perform comparably well to its more complete, short-wave resolving (XB-NH) counterpart. Despite their respective limitations, both modes were able to accurately predict Rmax and R2%.

Decomposition model for naval hydrodynamic applications, Part II: Verification and validation

This paper presents the verification and validation of the decomposition computational model presented in the accompanying paper. The decomposition model is primarily targeted to simulating incompressible, free surface flows in naval hydrodynamics related to wave-like phenomena. The model is implemented in foam-extend-3.1, a community driven fork of the OpenFOAM CFD software. The first set of tests considers regular wave propagation in a 2-D tank. Studies regarding Level Set parameters and reflection are carried out. Mass conservation during a long simulation is also considered. The solution is compared with non-linear potential flow solution obtained with stream-function wave theory for a range of wave steepness parameters. Higher-order wave loads on a vertical circular cylinder are calculated and compared to well-established non-linear potential flow results. Time refinement, mesh refinement and wave frequency studies are carried out and presented.

Characteristics of bolus formation and propagation from breaking internal waves on shelf slopes

A series of laboratory experiments was conducted to study the formation of internal boluses through the run up of periodic internal wave trains on a uniform slope/shelf topography in a two-layer stratified fluid system. In the experiments, the forcing parameters of the incident waves (wave amplitude and frequency) are varied for constant slope angle and layer depths. Simultaneous particle image velocimetry (PIV) and planar laser-induced fluorescence (PLIF) measurements are used to calculate high resolution, two-dimensional velocity and density fields. Over the range of wave forcing conditions, four bolus formation types were observed: backward overturning into a coherent bolus, top breaking into a turbulent bolus, top breaking into a turbulent surge and forward breaking into a turbulent surge. Wave forcing parameters, including a wave Froude number $Fr$, a wave Reynolds number $Re$ and a wave steepness parameter $ka_{0}$, are used to relate initial wave forcing to a dominant bolus formation mechanism. Bolus characteristics, including the bolus propagation speed and turbulent components, are also related to wave forcing. Results indicate that for $Fr&gt;0.20$ and $ka_{0}&gt;0.40$, the generated boluses become more turbulent in nature. As wave forcing continues to increase further, boluses are no longer able to form.

An experimental study on the inline wave force on a truncated vertical cylinder

ABSTRACTCylindrical structures are widely used in offshore engineering, such as the hard tank of a spar platform, columns of a semi-submersible, jackets, drilling pipes, risers and so on. Interactions between the waves and offshore structures are of great importance for engineering applications. This paper focuses on the inline wave force acting on a truncated vertical circular cylinder. Experimental studies are carried out in the wave tank at Shanghai Jiao Tong University in China. The diameter of the cylinder (D) is 10 cm and five submergence depths (hs), i.e., hs= 0.5D, 1D, 2D, 3D, 4D, are considered, respectively. The inline force is found to be influenced by the submerged depth of the cylinder, wave steepness and scatter parameter. High-harmonic wave forces and the second-load cycle are also discussed in the paper.

Excess energy approach for wave energy dissipation at submerged structures

Using the excess energy approach, it is found that the net energy dissipation rate on submerged structures can be defined in terms of submerged depth, transmission coefficient and transmitted wave height. In order to generate a definition for the net energy dissipation on submerged structures, the difference between the wave energy dissipation rates per unit volume at break and the wave energy dissipation rate during the wave transformation is incorporated. The analytical solution is tested by laboratory study and the computed solutions are validated with the results of previously published studies. Based on laboratory results an empirical relationship representing the wave transmission coefficient is also proposed based on spilling and plunging type of breaks. Both the relations were linear and were defined in terms of relative submergence parameter and wave steepness parameter. The dominant effect of plunging type of breaks on wave energy dissipation rates is validated for submerged structures while smaller effect of spilling type of break is approved. It is shown that the proposed definition for energy dissipation is robust and accurate, and can easily demonstrate the net energy dissipation rates at submerged structures.

Interaction of finite-amplitude waves with vertically sheared current fields

A computationally efficient numerical method is developed to investigate nonlinear interactions between steep surface gravity waves and depth-varying ocean currents. The free-surface boundary conditions are used to derive a coupled set of equations that are integrated in time for the evolution of the free-surface elevation and tangential component of the fluid velocity at the free surface. The vector form of Green's second identity is used to close the system of equations. The closure relationship is consistent with Helmholtz's decomposition of the velocity field into rotational and irrotational components. The rotational component of the flow field is given by the Biot–Savart integral, while the irrotational component is obtained from an integral of a mixed distribution of sources and vortices over the free surface. Wave-induced changes to the vorticity field are modelled using the vorticity transport equation. For weak currents, an explicit expression is derived for the wave-induced vorticity field in Fourier space that negates the need to numerically solve the vorticity transport equation. The computational efficiency of the numerical scheme is further improved by expanding the kernels of the boundary and volume integrals in the closure relationship as a power series in a wave steepness parameter and using the fast Fourier transform method to evaluate the leading-order contribution to the convolution integrals. This reduces the number of operations at each time step from O(N2) to O(NlogN) for the boundary integrals and O[(NM)2] to O(NlogN) for the volume integrals, where N is the number of horizontal grid points and M is the number of vertical layers, making the model an order of magnitude faster than traditional boundary/volume integral methods. The numerical model is used to investigate nonlinear wave–current interaction in depth-uniform current fields and the modulational instability of gravity waves in an exponentially sheared current in deep water. The numerical results demonstrate that the mean flow vorticity can significantly affect the growth rate of extreme waves in narrowband sea states.

Variation Assimilating of Synthetic Radar Data into a Pseudo-spectral Wave Model

In this paper, a computationally-efficient scheme is developed to assimilate radar images into a pseudo-spectral wave evolution model. The nonlinear wave model uses a fourth-order Runge-Kutta scheme to integrate in time a coupled set of equations for the evolution of the free surface elevation and velocity potential at the free surface. An asymptotic expansion of the model variables in terms of wave steepness parameter is used to close the system of equations and determine the vertical velocity at the free surface from the velocity potential at the free surface. A variational data assimilation scheme is then developed to find an optimal initial wave field that minimizes a cost function defined as the squared difference between model predictions and radar observations over an assimilation interval. The conjugate gradient method is used for the minimization scheme with the adjoint technique used to calculate the gradient of the cost function with respect to the initial condition. Numerical experiments have been conducted with one dimensional pseudo-observations that have been generated by the forward model by adding uncorrelated noise, as well as synthetic radar data to validate the proposed assimilation scheme.

Effects of Wave Non-Linearity on Residual Pore Pressures in Marine Sediments

A better understanding of the wave-induced pore pressure accumulations (i.e., residual pore pressure) is a key factor in the analysis of the wave-induced liquefaction in marine sediments. In this paper, the residual mechanism of nonlinear wave-induced pore water pressure accumulation in marine sediments is examined. Unlike previous investigations, the second-order Stokes wave theory is considered in this study. The new model is verified with experimental data and provides a better prediction of pore pressure accumulation than the previous solution with linear wave theory. The parametric study concludes that the influences of wave non-linearity increase as the following parameters increase (i) wave steepness (H/L) and (ii) residual parameter (β). However, an opposite trend is found for (i) wave period (T), (ii) relative water depth (d/L), (iii) seabed thickness (h/L) and (iv) another residual parameter (α). Furthermore, the effect of wave non-linearity becomes more significant in soft seabed (Soil C).

On the effects of wave steepness on higher order Stokes waves

The effect of wave steepness on higher order finite amplitude Stokes waves is investigated analytically and numerically. It is shown that the phase speed increases as the wave steepness increases thereby initiating the wave instabilities. As the order increases, the phase speed also increases .However, the impact of wave steepness on phase speed is more pronounced for the wave steepness parameter &#949&#88050.09 where 0.02 &#8805 &#9490 &#8805 0.33 Beyond the upper bound, instability – wave breaking comes in. JONAMP Vol. 11 2007: pp. 369-374

Discussion of “Distributions of Wave Steepness and Surf Parameter” by M. Aziz Tayfun

The author presents a study on the development of approximate theoretical forms of the distributions of wave steepness and surf parameter. The approach is based on assuming the random wave process to be long crested and narrow banded. The results are compared with data from measurements at sea representing two severe storms. Both parameters are lognormal distributed. The resulting statistics for the surf parameter are applied to breakers at normal incidence on sloping beaches. As stated by the author, the joint statistics of wave steepness with wave heights or crest heights, or the wave steepness with wave heights or crest heights above a specified threshold may be more appropriate in formulating risks of capsizing of vessels, overtopping, and slamming forces on seawalls, etc. However, the wave steepness is of interest by itself, particularly in relation with many of the surf-zone processes. This discussion points out that other data sets may result in other distributions. This is exemplified by using the data referred to by the author—the data used by Myrhaug and Kjeldsen 1984, 1987 , Myrhaug and Rue 1993 , and Myrhaug and Kvalsvold 1995 . These papers discuss various aspects of wave steepness statistics using data from a large population of waves obtained by waverider buoys at three different deep water locations on the Norwegian continental shelf. Fig. 1 here shows the cdf of the normalized wave steepness s=S /Srms in Weibull scale, where Srms is as defined in Eq. 20 of the original paper. The pluses represent the Myrhaug and Kjeldsen 1984 data. The full line represents the combined lognormal and Weibull distribution obtained as a best fit by eye to the data; the pdf of the lognormal distribution is given by Eq. 17 of the original paper with , = −0.257,0.523 for s 1.2; the pdf of the Weibull distribution is given by Eq. 14 with , = 1.40,0.84 for s 1.2. It appears that this model gives a good representation of the data. Myrhaug and Rue 1993 used the Weibull model to study the statistics of two successive wave steepness parameters with the focus on steep waves S 0.10 . Myrhaug and Kjeldsen 1984 , Myrhaug and Kvalsvold 1995 , and Myrhaug 1994 discussed the joint distribution of wave height and wave steepness by taking f h ,s = f s h f h where f s h is given by the lognormal distribution in Myrhaug and Kjeldsen 1984, Eq. 4 , with x=s, Eq. 18 , and Eq. 19 , and f h is given by the Weibull distribution in Myrhaug and Kjeldsen

Distributions of Wave Steepness and Surf Parameter

Predictions of wave action on slopes, and various wave- and depth-induced processes in the surf zone rely on two principal parameters: Wave steepness and surf parameter. For irregular waves, the theoretical distributions of these parameters are not known. Some studies suggest that wave steepness is described well by the Weibull and lognormal probability distributions. These distributions and thus their predictive utility depend on two key statistics, namely the mean and root-mean-square values of wave steepness. This study develops approximate theoretical forms of these statistics for narrow-band long-crested waves. For waves with steepness less than a Miche-Stokes type upper limit, comparisons with measurements from two severe storms show that the theoretical forms so derived and the resulting lognormal law in particular describes empirical data well. These results then permit the distribution of surf parameter to be approximated in a lognormal form also. Consequently, the nature and implications of such an approximation, resulting statistics and their plausible applications to breaker heights and types on slopes are explored.

Nonlinear Evolution of Surface Gravity Waves over Highly Variable Depth

New nonlinear evolution equations are derived that generalize those presented in a Letter by Matsuno [Phys. Rev. Lett. 69, 609 (1992)]] and a terrain-following Boussinesq system recently deduced by Nachbin [SIAM J Appl. Math. 63, 905 (2003)]]. The regime considers finite-amplitude surface gravity waves on a two-dimensional incompressible and inviscid fluid of, highly variable, finite depth. A Fourier-type operator is expanded in a wave steepness parameter. The novelty is that the topography can vary on a broad range of scales. It can also have a complex profile including that of a multiply valued function. The resulting evolution equations are variable coefficient Boussinesq-type equations. The formulation is over a periodically extended domain so that, as an application, it produces efficient Fourier (fast-Fourier-transform algorithm) solvers.

Asymptotic nonlinear wave modeling through the Dirichlet-to-Neumann operator

New nonlinear evolution equations are derived that generalize the system by Matsuno [16] and a terrain-following Boussinesq system by Nachbin [23]. The regime considers finite-amplitude surface gravity waves on a two-dimensional incompressible and inviscid fluid of, highly variable, finite depth. The asymptotic simplification of the nonlinear potential theory equations is performed through a perturbation anaylsis of the Dirichlet-to-Neumann operator on a highly corrugated strip. This is achieved through the use of a curvilinear coordinate system. Rather than doing a long wave expansion for the velocity potential, a Fourier-type operator is expanded in a wave steepness parameter. The novelty is that the topography can vary on a broad range of scales. It can also have a complex profile including that of a multiply-valued function. The resulting evolution equations are variable coefficient Boussinesq-type equations. These equations represent a fully dispersive system in the sense that the original (hyperbolic tangent) dispersion relation is not truncated. The formulation is done over a periodically extended domain so that, as an application, it produces efficient Fourier (FFT) solvers. A preliminary communication of this work has been published in the Physical Review Letters [1].

A Two-Dimensional Weibull Distribution and Its Application to Rolling

Abstract A two-dimensional Weibull distribution is presented which is obtained by transforming a two-dimensional Rayleigh distribution. The emphasis herein is on its application to describe the statistics of successive wave periods, and the results are illustrated by discussing the rolling of smaller vessels in beam seas in a simplified manner. Reference is also made to the related subjects on the joint correlation of successive high waves and on the joint statistics of wave steepness parameters for two successive waves, where the wave steepness parameters considered are the crest front steepness and the total wave steepness.