Ursell Number Research Articles

Influence of artificial sandbar on nonlinear wave transformation: Experimental investigation and parameterizations

Wave nonlinearity is a main driving force for near-bed sediment transport, and thus affects the beach evolution. Accurate prediction of wave nonlinearity is an essential need for the design of beach nourishment and restoration projects. So far, wave nonlinearity evolution patterns on the nourished beaches with abrupt changes of bottom slopes and water depth are not fully understood. This paper presents experimental results of nonlinear wave transformation across the submerged artificial sandbar. It can be found that the artificial sandbar intensifies wave shoaling processes and increases the magnitudes of the wave nonlinearity. Wave skewness and asymmetry are parameterized and both dependent on Ursell number and bottom slopes. By taking bottom slopes into consideration, the prediction accuracy for wave skewness is improved 17% with dataset measured in this study and 21%–65% with an independent dataset, compared with a common-used formula also calibrated on a steep low-crest breakwater without considering the bottom slope. Wave skewness decreases with increasing Ursell number in the inner surf zone, indicating the artificial sandbar influences its shoreward wave nonlinearities through the wave history. Finally, it is found that wave triads interactions can be enhanced by the steep seaward slope and small water depth of the artificial sandbar.

Local scour characteristics of monopile foundation and scour protection of cement-improved soil in marine environment - Laboratory and site investigation

Monopile foundations are the most common foundation type for offshore wind turbines. Local scour and scour protection around piles in the marine environment are major concerns for the engineering community. Physical simulations of local scour around monopiles without scour protection and edge scour around monopiles with cement-improved soil protection under the combined action of wave and current were performed using a flume. The effects of Ucw, the KC number and (θ/θcr)0.5 on the local scour were investigated. The empirical expressions of the relative scour depth S/D based on the average-velocity based Froude number (Fra) and modified Ursell number Urp’ under the combined wave and current conditions were proposed. Moreover, the scour protection performance of cement-improved soil was revealed, and the optimal protection range was 7D and 5D for small- and large-diameter piles, respectively. The effects of the ratio of protection range to pile diameter (Ds/D) and aspect ratio Ar on edge scour were investigated. Finally, the feasibility and effectiveness of the new scour protection method for cement-improved soil were verified by site testing.

Hydrodynamic Characteristics of Idealized Flexible Vegetation under Regular Waves: Experimental Investigations and Analysis

Wang, H.; Yin, Z.; Luan, Y.; Wang, Y., and Liu, D., 2022. Hydrodynamic characteristics of idealized flexible vegetation under regular waves: Experimental investigations and analysis. Journal of Coastal Research, 38(3), 673–680. Coconut Creek (Florida), ISSN 0749-0208. Salt marshes protect coastal communities from erosion and damage by damping incoming waves. A series of physical experiments were conducted to investigate the wave attenuation characteristics of idealized flexible vegetation, and the parameters of submergence, stem density, wave height, and wave length were considered. It was observed that the wave transmission coefficient decreased and the wave damping factor increased with increasing submergence ratio, relative wave height, and relative stem density and decreasing relative wave length. Moreover, the wave transmission coefficient decreased with increasing Ursell number. In addition, with an increase in the submergence ratio, relative wave height, Reynolds number, Keulegan-Carpenter number, and Ursell number, the bulk drag coefficient decreased. The bulk drag coefficient was more strongly related to the Ursell number than the other parameters. A simple equation for the wave transmission coefficient was constructed using the dimensional analysis method, the least squares method, and randomly selected experimental data, and it was effectively validated based on the other experimental data.

Experimental investigation of wave attenuation and bulk drag coefficient in mangrove forest with complex root morphology

Mangrove forests play a crucial role in coastal protection. Numerous experimental studies have been performed to examine the interaction between wave flows and mangrove forests, while the mangrove forests are commonly modeled as cylinder arrays, and the effect of the complex root system on wave attenuation is still unclear. A series of physical experiments were conducted to study the hydrodynamic characteristics induced by a 10-m-long mangrove forest model, which was built to reproduce a mature Rhizophora forest. Wave-attenuation analysis was performed to highlight the importance of considering the wave damping produced by mangrove roots under different water depths. In addition, the wave-attenuation coefficient β increases linearly with an increase in the relative wave height and wave steepness to varying degrees. In addition, the bulk drag coefficient (CD), a key parameter for quantifying the mean vegetation resistance, was obtained based on the classical model proposed by Dalrymple et al. (1984). The influences of several dimensionless hydrodynamic parameters on the CD are discussed in detail. The results show that the Ursell number and Keulegan–Carpenter number (KCd) have stronger correlations with CD for vegetation models without roots. The KCLe number calculated using effective characteristic length Le is suitable for describing the CD of the mangrove model for given water depth. Then, a new parameter, KCrv, was defined using the average pore velocity and vegetation-related hydraulic radius by considering the bulk characteristics of the wave and vegetation. Finally, a generic empirical formula was proposed for the CD to reveal the drag coefficient characteristics of vegetation with a relatively high correlation.

Numerical modelling of wave transformation and runup over rough fringing reefs using VARANS equations

Computational Fluid Dynamics (CFD) models are gaining popularity recently in efforts to understand the complex dynamics of wave interaction with coral reef profiles. To provide an alternative approach to the current CFD models to account for the reef surface with large bottom roughness, a 3D numerical wave tank based on the CFD tool OpenFOAM® is developed in this study. The VARANS equations are solved for two-phase incompressible flow with the k-ω SST model for the turbulence closure and VOF method for tracking the free surface. The reef surface with high friction is modeled by using a porous media model in the VARANS equations. The numerical model is first validated with laboratory experiments. The model is then applied to evaluate the impacts of forcing condition (wave and water level), reef morphology (reef-flat width, fore-reef slope, and back-reef slope), and reef roughness on wave transformation and wave runup. Moreover, an empirical formula is derived from the numerical results to predict wave runup. Subsequently, the simulated cross-reef variations of energy dissipation, wave spectrum, wave shape parameters (skewness and asymmetry) and nonlinearity parameter (Ursell number) are discussed for both the smooth and rough reefs. Finally, the cross-reef variations of the TKE and its dissipation rate are also examined for both reefs via the numerical simulations.

Effects of wave nonlinearity on submerged flexible vegetation dynamics and wave attenuation

Dynamic characteristics and wave attenuation mechanism of flexible vegetation have propelled to the forefront in investigations of coastal engineering. Given that existing studies of wave nonlinearity are mainly restricted to rigid vegetation assumptions, this study was undertaken to quantitatively determine the effect of wave nonlinearity on submerged flexible vegetation dynamics as well as wave attenuation based on flexible vegetation dynamic model and experimental results. Meanwhile, the influence of different wave theories along with different determination methods of characteristic wave velocity was evaluated quantitatively. Results reveal that wave nonlinearity exerts substantial nonlinear and nonmonotonic effects upon flexible vegetation dynamics and wave attenuation rate. The maximum horizontal force, stem tip excursion, and wave height attenuation rate increase as the dimensionless Ursell number increases on the whole under the present investigation conditions. Moreover, it is suggested that different wave theories and determination methods of characteristic wave velocity can result in slight discrepancies of flexible vegetation dynamics, which may further affect wave attenuation by flexible vegetation. Determining the impacts of wave nonlinearity, wave theory, and characteristic wave velocity provides clearer insight into the wave attenuation mechanism of flexible vegetation, which is of great benefit to the flexible vegetation dynamics and related wave attenuation model improvement.

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.

The equilibrium depth of wave scour around both slender and large piles

The equilibrium stage of the scour caused by the wave effect around the marine piles was investigated experimentally for both slender and large piles. Unlike the experimental studies in the literature, some of the experiments were carried out under clear water scour condition, since the coarse bed material was used in the present study. While the effective parameter on scour for slender piles is only the Keulegan-Carpenter number, it is understood that the diffraction parameter is also effective for large piles. It has been determined that when the steady-streaming structure around a large pile is effective, the scour increases with the increase in the Keulegan-Carpenter number and the diffraction parameter. For the estimation of the relative equilibrium scour depth around the slender piles, the results were evaluated together with the previous experimental study results and a new empirical equations based on the Keulegan-Carpenter number (KC > 4) was proposed. A new empirical correlation is derived, which includes the Keulegan-Carpenter number and the diffraction parameter (10−3<D/L < 3·10−1) for both slender and large piles. In addition, alternatively, a new equation is proposed using the modified Ursell parameter (Urp<105). The nominal limit of the relative scour depth according to the diffraction parameter was determined as 0.0082·(D/L)−1.

Experimental study of bound triad interactions across a dissipative surf zone under different wave breaking conditions

This study experimentally investigates the development and decay of bound energy fractions due to positive and negative interactions (b2p and b2n, respectively) applying bispectral analysis. It was revealed that the Ursell number is the most successful parameter to estimate the variation patterns of b2p and b2n. The experimental waves included 27 waves with three offshore wave heights, three periods, and three offshore water depths propagating over a mild slope of 1:30. A wide variety of wave nonlinearity and bound coupling intensities were observed across the experimental dissipative surf zone. For a wave propagating toward the shore, the corresponding Ursell numbers for the relative maximum values of b2p and b2n are 0.27 and 0.22, respectively. The ratio of significant wave height to water depth corresponding to these extreme values for positive and negative triads are ~0.4 and ~0.32, respectively. The highest values of b2p and b2n for a progressive wave can be predicted as a function of the Ursell number using two new proposed experimental formulations. The sub-harmonics are more persistent during wave-breaking conditions than super-harmonics due to lower wave steepness. The biphase values of self-coupling of the spectral peak are attained negative for waves with Ursell numbers greater than 0.35.

Real-world rogue wave probabilities

Rogue waves are dangerous ocean waves at least twice as high as the surrounding waves. Despite an abundance of studies conducting simulations or wave tank experiments, there is so far no reliable forecast for them. In this study, we use data mining and interpretable machine learning to analyze large amounts of observational data instead (more than 1 billion waves). This reveals how rogue wave occurrence depends on the sea state. We find that traditionally favored parameters such as surface elevation kurtosis, steepness, and Benjamin–Feir index are weak predictors for real-world rogue wave risk. In the studied regime, kurtosis is only informative within a single wave group, and is not useful for forecasting. Instead, crest-trough correlation is the dominating parameter in all studied conditions, water depths, and locations, explaining about a factor of 10 in rogue wave risk variation. For rogue crests, where bandwidth effects are unimportant, we find that skewness, steepness, and Ursell number are the strongest predictors, in line with second-order theory. Our results suggest that linear superposition in bandwidth-limited seas is the main pathway to “everyday” rogue waves, with nonlinear contributions providing a minor correction. This casts some doubt whether the common rogue wave definition as any wave exceeding a certain height threshold is meaningful in practice.

Predicting the bulk drag coefficient of flexible vegetation in wave flows based on a genetic programming algorithm

The prediction of the bulk drag coefficient (CD) for aquatic vegetation is of great significance for evaluating the influence of vegetation on the hydrodynamic processes in wave environments. Different CD empirical formulas have been mostly proposed as functions of either Reynolds (Re) number or Keulegan–Carpenter (KC) number in the literature, and the influences of other wave and vegetation parameters on CD were often ignored. The difference in formulas is largely attributable to inconsistent uses of characteristic velocity and length scales in the definitions of Re and KC. By considering the vegetation and hydrodynamic characteristics in this study, new Re and KC numbers were redefined using the mean pore velocity and vegetation-related hydraulic radius. Besides, a genetic programming algorithm was adopted to develop a robust relationship between CD and possible dimensionless variables based on extensive experimental data. Ultimately, a new CD predictor that has a similar form to that of the classical expression was obtained without any prespecified forms before searching. It turns out that the new predictor depends on not only the new KC number but also the submergence ratio and Ursell number. Compared with the existing predictors, the proposed CD predictor exhibits a considerable improvement in predictive ability for a wider parameter space.

Mooring observed mode-2 internal solitary waves in the northern South China Sea

The mode-2 internal solitary waves (ISWs) generated by mode-2 internal tide (IT) are identified by mooring observations in the northern South China Sea (SCS) from 2016 to 2017. Two mode-2 ISWs with a re-appearance period of 24.9 h observed on 29 and 30 July 2016 are characterized by type-b ISWs. They occurred when the isotherms compressed obviously in the vertical direction. Modal decomposition of IT horizontal currents shows that the vertical compression of the isotherms is mainly caused by diurnal mode-2 IT. The analysis of the role of the density stratification reveals that a deeper and thinner pycnocline is favorable for generation of mode-2 ISWs rather than pycnocline intensity. By comparing the mode-2 nonlinear, dispersion coefficients and the Ursell numbers calculated based on the stratification associated with different kinds of ITs with the observation results, it is shown that the diurnal mode-2 IT plays a crucial role in the generation of the mode-2 ISWs. When the diurnal mode-2 IT interacts with the semidiurnal IT and causes a deeper and thinner pycnocline, the mode-2 ISWs are easily excited.

On the distribution of maximum crest and wave height at intermediate water depths

We report new descriptions for the (probability) distributions of hourly maximum crest and wave height of water surface gravity waves for intermediate water depths. Estimated distributions are based on analysis of laboratory-scale measurements at the DHI wave basin. For a given sea state, the distribution of both hourly maximum crest and hourly maximum wave height, normalised by sea state significant wave height, is found to follow a generalised extreme value (GEV) distribution. Variation of the three parameters of the GEV distribution across sea states, is expressed in terms of a response surface model as a function of non-dimensional sea state Ursell number and wave steepness, and wave directional spreading angle. For inference, conventional Monte Carlo wave basin measurements are supplemented with measurements selected by means of a novel “pre-selection” sampling scheme using numerical simulations. This scheme effectively guarantees that extreme events from tails of distributions are produced, and reduces uncertainties associated with the estimated distributions. Estimation is performed using Bayesian inference, allowing uncertainties to be quantified, and providing estimates of posterior predictive tail distributions for sea states with arbitrary characteristics within the domain of sea state characteristics covered by the model.

Concerted nonlinear mild-slope wave models for enhanced simulation of coastal processes

• One-way coupling of parabolic and hyperbolic models. • Introducing exact nonlinear wave effects in parabolic models. • Handling efficiently the principal wave propagation direction in parabolic models. • Models verification against demanding experiments. The present paper aims to incorporate nonlinear amplitude dispersion effects in parabolic and hyperbolic approximation models. First, an explicit and analytical method for considering nonlinearities in parabolic approximation models is investigated. This method follows the concept of calculating spatially and temporally varying wave phase celerities within the simulation. The nonlinear dispersion relation to be applied is dependent on the local Ursell number and wave steepness, in relation to valid regions of analytical wave theories. Furthermore, nonlinearities are introduced in a hyperbolic approximation model by proposing a novel method which combines a parabolic and a hyperbolic model without a significant loss in accuracy, leading to a substantial reduction in the required simulation time. This is achieved by inputting initial boundary conditions into the hyperbolic model based on the output of the parabolic model, which is used for an initial calculation of the spatial distribution of nonlinear dispersion effects over the entire numerical domain. Numerical results were compared with measurements obtained via demanding experimental setups and illustrated satisfactory performance of the models. These concerted nonlinear models offer ease in implementation, short simulation times, and accurate results while incorporating the majority of wave transformation processes including shoaling, refraction, diffraction, reflection, bottom friction and depth-induced wave breaking.

Are all KdV-type shallow water wave equations the same with uniform solutions?

ABSTRACT Cnoidal wave and its extreme case, the solitary wave, can be described by the KdV equation, which was first derived by Korteweg and de Vires with the first-order accuracy. Subsequently, different authors proposed their derivations and claimed that their equations, sharing similar expressions but different corresponding coefficients, were the same as the original one. After introducing a unified dimensionless frame, this study re-derived the KdV equation with respect to seven existing methods and confirmed that KdV equation indeed refers to a type of first-order equations, rather than a specified one. Differences in equations come from the influence of the second-order quantities associated with the derivation process. Regarding the cnoidal wave and solitary wave, the KdV-type equations obtained using different methods present the same first-order solution for wave profile. Nevertheless, in their directly derived results, different wave celerities and water particle velocities are presented due to the influence of second-order quantities. Additionally, comparing with the second-order solutions, all directly derived wave celerity solutions predict well for the Ursell number between 20 and 100. As for the first-order solution of the water particle velocity, all methods present the same result except Dean’s expression which contains a different coefficient.

Evolution of internal solitary waves on the slope-shelf topography in the northern South China Sea

Based on a non-hydrostatic two-dimensional and high-resolution model, evolution of internal solitary waves (ISWs) on the typical slope-shelf topography in the northern South China Sea is investigated numerically, and the influences of the initial amplitude, seasonal stratification and topographic characteristics are analyzed with a series of sensitivity runs. The results indicate that the initial amplitude affects the fission of ISW, resulting in three wave groups for large ISW and two wave groups for small ISW. In addition, the generation of mode-2 waves is influenced since energetic beams are engendered by large initial ISW, which impact the pycnocline and generate the mode-2 ISWs. Seasonal stratification has significant impacts on the evolution of the ISW. In winter, the changing sign of the nonlinearity coefficient at the bump near the shelf break implies the inversion of polarity of the ISW. Therefore, the transmitted and fissioned waves behave differently from those in summer and annual stratifications. Furthermore, the speed and energy of the leading wave are minimal in winter but maximal in summer. The bump near the continental shelf has two impacts: promoting the fission of the incident ISW and generating mode-2 ISWs by increasing the Ursell number (the ratio of nonlinear coefficient to dispersion coefficient). However, the formation of the trailing nonlinear wave packet is not affected by these factors, despite of some differences in waveforms.

Bound infragravity wave observations at the Nowshahr beaches, southern Caspian Sea

This paper presents the results of bound infragravity observation at the Nowshahr beach located in the southern part of the Caspian Sea. The water level data were measured at 6 stations along a transect perpendicular to the coastline. The bispectral method was utilized to show that the extreme values for the bound infragravity energy fraction (bii) were occurring when dimensionless values of wave height to depth ratio, relative depth, and wave steepness were in the range of 0.18‒0.22, 0.4‒0.5 and 0.015‒0.017, respectively. These detailed results could be equivalently summarized with the condition of having the Ursell number of peak waves within the range of 30‒40. The corresponding short wave groupiness factor (G.F.) values for waves with high values of bii were between 0.5 and 0.55. Also, it was revealed that the wave groupiness envelope had a positive correlation with infragravity wave energy level, but its correlation with the bound infragravity wave was negative when G.F. > 0.55. On the other hand, the low values of bii against the spectrum peakedness parameter for breaking waves followed an exponential relationship. The correlation of bound and free infragravity waves with wind waves was also studied. For non-breaking condition, the correlations of bound infragravity waves with swell and sea waves were considerable. The correlation of free waves with swells was also significant. In contrast, for breaking waves, a negligible correlation was observed between bound infragravity waves and wind waves.

Improved Calculation of Nonlinear Near-Bed Wave Orbital Velocity in Shallow Water: Validation against Laboratory and Field Data

A new parameterization for calculating the nonlinear near-bed wave orbital velocity in the shallow water was presented. The equations proposed by Isobe and Horikawa (1982) were modified in order to achieve more accurate predictions of the peak orbital velocities. Based on field data from Egmond Beach in the Netherlands, the correction coefficient and maximum skewness were determined as functions of the Ursell number. The obtained equations were validated against measurements from Egmond Beach, and with laboratory data from small-scale wave flume experiments at Delft University of Technology and from large-scale wave flume experiments at Delft Hydraulics. Inter-comparisons with other previously developed parameterizations were also carried out. The model simulations by the present study were in good agreement with the measurements and have been improved compared to the previous ones. For Egmond Beach, the root-mean-square errors for the peak onshore (uc) and offshore (ut) orbital velocities were approximately 21%. The relative biases were small, approximately 0.013 for uc and −0.068 for ut. The coefficient of determination was in the range between 0.64 and 0.68. For laboratory experiments, the root-mean-square errors in a range of 7.2%–24% for uc, and 7.9%–15% for ut.

Field observation of wave reflection from plunging cliff coasts of Chabahar

In this study, a linear formulation is found to estimate the wave reflection coefficient (Kr) from plunging cliff coasts of Chabahar city according to field data measurements. The field measurement included directional wave data via three bursts for each hour and in total 1345 bursts during summer Southern-West monsoon using an ADCP gague. The significant wave heights reached 2.5 m and the corresponding peak periods were greater than 11 s.It was observed that the wave reflection coefficients are fully tidally modulated. The maximum values of Kr reached 0.7 in a flood period of spring tide, while the minimum values were greater than 0.3 and corresponded to ebb periods in spring tide. The corresponding dimensionless parameters of wave height to depth, wave steepness and Ursell number to the highest values of Kr were obtained as 0.17, 0.014, and 22, respectively. The variation ranges of reflection coefficient pertaining to the effect of four main tidal constituents as M2, S2, K1, and O1 were achieved as ∼ 9.6, 4.1, 7.1, and 3.6%, respectively. Finally, a linear formulation as a function of the tidal level was resulted to estimate the most accurate values of Kr in the studied area. The averaged inaccuracy of resulted formulation was approximately 6.5%.The frequency reflection coefficients Kr(f) was also investigated in this study. The results indicated the values close to 1 for long infragravity waves, because of occurrence of no significant dissipation. However, the similar values for short waves (fr > 0.25 Hz) were greater than 1 due to wind wave breaking as a result of the cliff toe collision. The temporal histogram of Kr(f) revealed the tidally controlled variations of this parameter.

Evolution of irregular wave shape over a fringing reef flat

The main objective of this paper was to examine the influence of forereef bottom slope and the effects of coral degradation on the transformation of wave shapes over a fringing reef flat. Based on verification from a physical experiment, the numerical model FUNWAVE 2.0, based on the fully nonlinear Boussinesq equations, was employed to simulate irregular waves propagating over fringing reefs of different topographies. Empirical formulae relating wave skewness and asymmetry to local Ursell numbers were extended to three dimensions, including the effect of forereef slope and roughness of reef surface. Results demonstrated that the gentle forereef slope will lead to a larger wave asymmetry on a reef flat. Otherwise, wave asymmetry on a smooth reef bed was slightly larger than that on a rough bed, indicating that coral degradation may increase wave asymmetry over the reef flat. It was found that the different evolution of wave asymmetry on the reef flat could be primarily attributed to the different infragravity wave motions, which were induced by different forereef bottom slopes. Moreover, coral degradation would increase the asymmetry of infragravity waves, leading to some differences in wave asymmetry on the reef flat. However, wave skewness from different types of fringing reefs was almost identical, implying that the aforementioned factors have a negligible effect on wave skewness.