Ursell Number Research Articles

Parametrizing nonlinearity in orbital velocity at fetch-limited, low-energy beaches

Wave nonlinearity plays an important role in cross-shore beach morphodynamics and is often parameterized in engineering-type morphodynamic models through a nonlinear relationship with the Ursell number. It is not evident that the relationship established in previous studies also holds for sheltered sites with fetch-limited seas as they are more prone to effects of local winds and currents, the waves are generally steeper, and the beaches are typically reflective. This study investigates near-bed orbital velocity nonlinearity from wave records collected at two sheltered beaches in The Netherlands and contrasts them to earlier observations made along the exposed, wave-dominated North Sea coast. Our observations at sheltered beaches show that the Ursell number has comparable skill in predicting wave nonlinearity as it has on previously studied exposed coasts. However, the orbital velocities at sheltered coasts are more asymmetric for the same Ursell number than on exposed coasts. When exposed coast data were examined for moments with comparable high-steepness waves, a similar effect on asymmetry was observed. In addition, following and opposing winds were found to have a clear relationship with total nonlinearity, while they did not affect the phase between skewness and asymmetry at the sheltered beaches. Refitting the free parameters of an Ursell-based predictor improved the bias for the asymmetry parameterization. Whether this has implications for modeling of the magnitude of wave-nonlinearity-driven sediment transport using engineering type models is strongly dependent on the sediment transport formulation used, as these formulations depend on additional calibration coefficients too.

Parameterisation and evolution of non-breaking wave nonlinearity over flexible vegetation

Wave nonlinearity significantly influences wave attenuation and can cause sediment transport imbalances in vegetated zones, impacting shoreline stability and ecosystem-based coastal defences. Despite its importance, the parameterisation and mechanisms of wave nonlinearity evolution over flexible vegetation remain unclear. This study examines the mechanisms behind non-breaking wave nonlinearity evolution over flexible vegetation and provides parameterisation of wave nonlinearity, based on experimental data collected by Anderson and Smith (2014). The study derives a new set of empirical formulae for predicting wave nonlinearity in Spartina alterniflora, which considers Ursell number, relative vegetation width, vegetation submergence, and vegetation density. The predictions made using these formulae are in good agreement with the measurements. The results indicate a decrease in wave skewness across the span of vegetation, whereas wave asymmetry increases until it reaches a maximum before decreasing. The findings suggest a preferential dissipation of high harmonics compared to low harmonics in the vegetation zone. Additionally, bispectrum analysis reveals that vegetation enhances the difference interaction while suppressing the sum interaction between wave components. The findings enable improvements in our understanding of wave nonlinearity in vegetation, providing better predictive capabilities for wave attenuation and sediment transport, which are crucial for the design and optimization of coastal defence systems.

Laboratory observation of nonlinear wave shapes due to spatial varying opposing currents

Physical experiments were conducted to examine the influence of adverse currents on the propagation of shallower water waves and their impact on the evolution of nonlinear wave shapes. Irregular waves, characterized by varying initial peak periods and amplitudes, were generated in a physical wave flume equipped with a bottom slope of 1/20. Three groups of spatially varying opposing currents were generated in the flume and interacted with the aforementioned wave trains. Experimental results confirm that strong opposing currents can significantly intensify local wave nonlinearity, thereby further influencing the deformation characteristics of waves. In contrast, a weak opposing current (without wave-blocking) has a negligible effect. Bicoherence analysis revealed that the degree of phase coupling among triads of waves increased with an elevated current velocity before wave-blocking. Nevertheless, during partial wave-blocking, the degree of phase coupling was seemingly weakened by an increase in opposing current. Moreover, the influence of a co-existing current in the formation of extreme waves was recognized. The study confirms that adverse currents notably affect extreme wave steepness and extreme wave skewness, with a negligible impact on extreme wave asymmetry. As a result, empirical formulas modified by current effects are presented, describing certain key nonlinear wave shapes as functions of the local Ursell number.

Non-intrusive measurements of wave nonlinearity over movable bed of sediment with different beach slopes

A series of laboratory experiments were undertaken to qualitatively investigate the evolution of wave nonlinearity over a movable bed of sediment with five different beach slopes under regular waves in a medium-scale wave flume. An innovative non-intrusive data collecting system, which mainly consists of three side-looking high-speed cameras, was developed to collect high-resolution and synchronous data on free-surface water elevation of waves and bed level changes without causing any disturbances to wave motions and the movable bed of sediment. On analyzing the collected experimental data, it is found that regular waves become nonlinear when they propagate to the shoaling zone and start breaking, and linear wave theory is quite accurate for calculating wave parameters such as orbital velocity with the correlation coefficient r2 = 0.8–0.95 before the waves break, but becomes less accurate after the waves break or are in the breaking zone with the smaller correlation coefficient r2 = 0.4–0.6. Four parameters, wave skewness Skη, asymmetry Ayη, Ursell number Ur, and Rocha number NP0, are introduced to describe the wave nonlinearity, of which Skη and Ayη are found to be of largest amplitudes at the wave breaking point and then start to decrease in the breaking zone and are almost unchanged for different beach slopes, while Ur further increases in the breaking zone and exceed the first larger value as waves approach to shoreline, but NP0 is almost linearly proportional to wave orbital velocity amplitude and quite sensitive to beach slope. The location of sandbar is found close to the wave breaking point in the wave flume and may be also considered as the point where wave nonlinearity becomes important for sediment transport in the surf zone, and the linear wave theory becomes less accurate.

A guide for selecting periodic water wave theories - Le Méhauté (1976)’s graph revisited

This note provides guidelines for selecting appropriate analytical periodic water wave solutions for applications, based on two physical parameters, namely, frequency dispersion parameter (water depth divided by wavelength) and the nonlinearity parameter (wave height divided by wavelength). The guidelines are summarized in a graphic format in the two-parameters space. The new graph can be viewed as the quantification of the well-known Le Méhauté (1976)’s graph with quantitative demarcations between applicable analytical periodic water wave solutions. In the deep water and intermediate water regimes the fifth-order Stokes wave theories (Zhao and Liu, 2022) are employed for the construction of the graph, while in the shallower water regime, Fenton (1999)’s higher order cnoidal wave theories are used. The dividing lines between the applicable ranges of Stokes wave and cnoidal wave theories are determined by the values of Ursell number, a ratio between the nonlinearity and frequency dispersion.

Experimental and numerical investigation of wave-induced dynamics of emergent flexible vegetation

Understanding and modeling the dynamics of flexible vegetation is fundamental to investigating wave attenuation by flexible vegetation, which is fast becoming an essential issue in nature-based coastal defense. Existing studies are mainly restricted to submerged vegetation and linear wave assumptions. Hence, a new mechanical model for simulating wave-induced emergent flexible vegetation dynamics is developed by considering the dynamic variation of the submergence ratio. A set of wave flume experiments on emergent flexible vegetation dynamics driven by waves are conducted. The reliability of the developed model is revealed based on a series of model validations. Experimental data indicates that emergent flexible vegetation dynamics are influenced by the Ursell number, Cauchy number, Excursion ratio, stiffness-to-force ratio, Keulegan–Carpenter number, and Reynolds number. Simulation results also demonstrate the usefulness of the emergent flexible vegetation dynamic model. There is a considerable difference in simulated horizontal force and stem postures regarding emergent flexible vegetation between the previous submerged flexible vegetation dynamic model and the developed new emergent flexible vegetation dynamic model. The results from this study optimize the simulation theory and method of wave-induced flexible vegetation dynamics, and generate new insights into the interaction between coastal flexible vegetation and waves.

The dependence of wave height probability distributions on physical parameters from measurements near Sakhalin Island

The data of long-term surface waves measurements with bottom sensors near Sakhalin Island were used to build instrumental probability distributions for exceedance of wave heights. Waves with heights exceeding the significant wave height by more than three times were recorded. Specific features of the observations conducted during the periods of open and ice-covered sea surfaces are discussed. The subsets of statistically homogeneous data are arranged through selection considering the natural physical dimensionless parameters of the task, that control the effects of finite depth and nonlinearity (namely, the wave steepness, the wave amplitude to water depth ratio, the Ursell parameter). The manifestation of these and composite parameters in theoretical probability distributions of wave heights is discussed. The effects of nonlinearity and the measurement point depth on probability distributions are estimated with a focus on abnormally high waves. In particular, it is shown that for the place of registration an increase in the wave amplitude to water depth ratio parameter leads to a decrease in the probability of abnormally high waves. This behavior is consistent with the Glukhovsky theoretical distribution. The waves characterized by relatively large dimensionless depth parameter exhibit a higher probability of substantial exceedance of the significant height and are better described by the Rayleigh distribution.

Evolution and Statistical Analysis of Internal Random Wave Fields within the Benjamin–Ono Equation

This study investigates the numerical evolution of an initially internal random wave field characterized by a Gaussian spectrum shape using the Benjamin–Ono (BO) equation. The research focuses on analyzing various properties associated with the random wave field, including the transition to a steady state of the spectra, statistical moments, and the distribution functions of wave amplitudes. Numerical simulations are conducted across different Ursell parameters, revealing intriguing findings. Notably, it is observed that the spectra of the wave field converge to a stationary state in a statistical sense, while exhibiting statistical characteristics that deviate from a Gaussian distribution. Moreover, as the Ursell parameter increases, the positive skewness of the wave field intensifies, and the kurtosis increases. The investigation also involves the computation of the probability of rogue wave formation, revealing deviations from the Rayleigh distribution. Notably, the study uncovers distinct types of internal rogue waves, specifically referred to as the “two sisters” and “three sisters” phenomena.

MAXIMUM MOMENTUM FLUX FOR STABILITY ANALYSIS OF MODEL AND PROTOTYPE BREAKWATERS

None of the established formulae for breakwater armour stone stability, Hudson (1958) or van der Meer (1987), explicitly account for the water depth at the toe of the structure. More recently, Hughes (2004), Melby and Hughes (2004), Melby and Kobayashi (2011) have developed equations for stability and runup utilising the concept of wave momentum flux which explicitly accounts for the water depth of the wave probe(s) in close proximity to the structure. The equations are adapted to three forms; namely (1) linear, (2) extended-linear and (3) non-linear. In the paper linearity is assessed by using the Ursell number at each probe depth. Also, due to the placement of probes at various depths in MHL’s 2D wave flume it is possible to correlate the linearity of the wave measurement for the same time series and subsequently test the appropriateness of the momentum flux equation applied for assessment of stability and runup. The stability and runup data from 43 2D physical model tests where stability was previously assessed using van der Meer’s and Hudson’s equations are assessed using the momentum flux equations and an evaluation of the results has been made. It was found that the estimation of notional permeability and selection of the use of the plunging or surging formulae was critical to obtaining a closer match between measurement and prediction. The equations were also utilised in conjunction with numerical models to evaluate the armour size for repair of two breakwater heads in South Camden Haven and Bellambi. The maximum momentum flux equations were found to perform satisfactorily at these locations where the Ursell numbers were found to be high and the waves non-linear.

Extreme Wave Loading on a Vertical Circular Cylinder

Extreme wave loading on a marine structure, consisting of a quasi-static and a dynamically slamming component, often drives the design of such a structure. Their accurate predictions remain challenging tasks. This paper examines the slamming contribution to the force–time histories from a series of experiments in which the crest of a wave is forced to hit a truncated cylinder suspended from above. A range of inundation levels, representing the breaker heights, are considered. The work also provides insights into nonlinear load characteristics on a vertical truncated cylinder. A simple analytical model based on the Newtonian momentum analysis is extended to describe the scaling of the horizontal peak force with the inundation level, i.e., the relationship between the wave slamming loads and the breaker height. More specially, it is found that the peak horizontal impact force is proportional to the inundation level and the square of the linear wave amplitude. In addition, the horizontal and the vertical impact forces on a truncated cylinder are found to increase with the increasing inundation level, while the effect from the wave steepness is relatively small. Furthermore, the higher-order wave components driving nonlinear (quasi-static) loading on a structure are separated by applying a phase-based separation method assuming a Stokes-like approximation. The separation method is found to work well even for long shallow-water waves that have strong nonlinearities. The results suggested that the relative contribution from the fundamental linear wave and higher-order wave components decreases and increases with the increasing nonlinearity of long waves, respectively, characterized by the Ursell number. Finally, this increase in the higher-order wave components is found to be saturated at large Ursell numbers.

Optimization of the drag coefficient in wave attenuation by submerged rigid and flexible vegetation based on experimental and numerical studies

This study investigates wave attenuation due to submerged rigid and flexible vegetation through a combination of experimental and numerical approaches. Accurate reproduction of physical measurements was achieved by adjusting the drag coefficients within the numerical model to account for the influence of wave conditions and vegetation characteristics on wave attenuation. The decay of wave height in both rigid and flexible vegetation fields is closely linked to wave nonlinearity, which can be quantified using the Ursell number. Wave nonlinearity has a more pronounced impact on wave attenuation when rigid vegetation is present compared to flexible vegetation. According to the relationship between the drag coefficient and the Keulegan–Carpenter number, a new drag coefficient formula was proposed based on a revised Keulegan–Carpenter number that considers wave nonlinearity. The deformation of flexible vegetation in response to waves was characterized by introducing the effective vegetation height, which can be correlated with the wave Cauchy number and the stem height ratio. By identifying the transition between rigid and flexible vegetation, an empirical expression encompassing multiple vegetation types was developed by incorporating vegetation flexibility. The derived empirical equation can be implemented in wave simulations with applicability to wave attenuation by vegetation and optimal flood response strategies.

Effects of currents on nonlinear wave transformation in a reef-lagoon-channel system

Wave nonlinearity can drive the near-bed sediment transport and nutrient delivery on coral reefs. Previous studies have primarily concentrated on the wave nonlinearity evolution for a specific cross-shore profile and have scarcely taken the effects of currents into consideration. To better understand the nonlinear wave transformation on reef coasts, the wave height distribution and wave nonlinearity evolution are analyzed based on 3D reef-lagoon-channel experiments in a wave basin. By taking the current velocity into consideration, the Glukhovskiy distribution function is improved in calculating the probability of large wave height in the sequence. Wave skewness and asymmetry on the reef and in the channel are parameterized with the Ursell number considering the Doppler effect induced by opposing or following currents. The accuracies for estimating wave skewness and asymmetry are improved by 39% and 66%, respectively. The influence of Doppler effect on wave shapes on the reef flat and in the channel varies significantly with the incident wave heights and reef submergences. Furthermore, it is also noticed that the reduction of local water depth caused by the propagation of infragravity wave troughs increases short-wave skewness at individual wave scale on the middle of reef flat, while its effect on short-wave asymmetry is not significant.

A new expression for free stream velocity with skewness and asymmetry

A new expression for skewed and asymmetric free stream velocities (near-bed wave orbital velocities at the edge of the wave boundary layer) is established in this paper. A comparison of the new expression with others is examined, and the new expression is found to be more satisfactorily fitted to wave orbital velocities with comparatively strong skewness and weak asymmetry than those presented by Abreu et al. (2010) and Malarkey and Davies (2012). Methods for determining the parameters in the new expression based on the Ursell number Ur are presented, and consequently, two empirical formulae are recommended for estimating bedload and suspended load sand transport rates related to wave skewness, respectively. Reasonability of the new expression for irregular waves is discussed by comparing the predicted time averaged quantities to those calculated by measured data, and good agreement is found.

Viscous Flow Fields Induced by the Run-Up of Periodic Waves on Vertical and Sloping Seawalls

Unsteady two-dimensional Reynolds Averaged Navier-Stokes (RANS) equations coupled with the k−ε turbulence model was developed to simulate the viscous flow field near seawalls. The complex free-surface configuration was captured using the particle level set method (PLSM). An innovative solid-fluid coupling method was employed to mimic the solid-fluid interaction on fixed Cartesian grids. The proposed numerical model was applied to investigate the viscous flow fields induced by periodic waves propagating on three different seawalls, including vertical and steep seawalls, which are commonly designed in Taiwan with slopes of 1:2 and 1:5. The results reveal that the wave run-up height is closely related to Iribarren number and breaking wave. Partial standing wave or asymmetric recirculating cells in front of a steep seawall lose symmetry with increasing slope. The wave pressure on the seawall face increases as the Ursell number or seawall slope increases. The undertow caused by breaking waves has a significant effect on the bottom shear stress in front of the seawall.

Numerical study of cross-shore variations of bound super-harmonics over a single sandbar system

This study primarily investigates the capability of the one-dimensional mode of the SWASH model to reproduce bound super-harmonics resulted from self-coupling of the spectral peak. The model results were compared with an array of wave measurements along a straight perpendicular transect in west of Nowshahr Port in the Southern Caspian Sea. The wave model was calibrated during moderate conditions and the bispectral analysis was applied for model results and field measurements.It was revealed that the model can properly reproduce nonlinear waves experiencing significant shoaling as well as disappearing the couplings for breaking waves within stormy periods. Secondary and tertiary local peaks as results of self-coupling of the spectral peak could be found in the simulated wave spectra; however, erroneous intensive couplings were also represented by the model for very low energy conditions with no cross-shore wave transformation.The validated model results were used to investigate the variations of the bound triad interactions across the shore. It was found that the corresponding Ursell numbers for the most intensive positive couplings were in the range of 0.2–0.3. Also, it was concluded that the couplings may be drastically weakened before intensive wave breaking.

Effects of barotropic tides and ocean stratification on the generation and propagation of internal solitary waves in the deep south Andaman Sea

The Andaman Sea is one of the sea areas with the most internal solitary waves (ISWs) observed in the world and an ideal experimental site for studying the generation and propagation characteristics of ISWs. In this paper the generation and propagation characteristics of the ISWs in the deep south Andaman Sea are investigated based on the results of Massachusetts Institute of Technology General Circulation Model (MITgcm) numerical models. The reliability of the model is verified. The sea surface current gradient of 3D model is consistent with Terra MODIS and Sentinel-1A SAR remote sensing images. The turbulent kinetic energy dissipation rate of 2D model is also consistent with in-situ mooring measurements in other sea areas. The variation of temperature, energy and current field caused by barotropic tides and oceanic stratification are analyzed and compared. The results show that ISWs cannot be generated directly by the individual diurnal tides in our 2D model, and ISWs are mainly generated by the semidiurnal tides in the deep south Andaman Sea. Ocean stratification affects the waveform, the velocity and the generation position of ISWs. ISWs are easier to form in the hot season because of a larger Ursell number and stronger nonlinearity.

Solitary Wave Interaction with a Floating Pontoon Based on Boussinesq Model and CFD-Based Simulations

A mathematical model of solitary wave interaction with a pontoon-type rigid floating structure over a flat bottom is formulated based on Boussinesq-type equations under weakly nonlinear dispersive waves. Based on the higher-order Boussinesq equations, the solitary wave equation is derived, and a semi-analytical solution is obtained using the perturbation technique. On the other hand, brief descriptions of the application of wave2Foam and OceanWave3D on the aforementioned problem are presented. The analytical solitary wave profiles in the outer region are compared with Computational Fluid Dynamics (CFD) and OceanWave 3D model simulations in different cases. The comparison shows a good level of agreement between analytical, wave2Foam, and OceanWave3D. In addition, based on the wave2Foam and coupled OceanWave3D model, the horizontal, vertical wave forces, and the pressure distributions around the pontoon are analysed. Further, the effect of the Ursell number, pontoon length, and water depth on the solitary wave profiles are analysed based on the analytical solution. The paper validates each of the three models and performs intercomparison among them to assess their fidelity and computational burden.

Analysis of viscous wave generation at low Ursell number in an optimal numerical wave tank using inlet velocity method

The present study analyzes the viscous wave generation at a low Ursell number in an optimal numerical wave tank. Commercial computational fluid dynamics code ANSYS FLUENT software is used in a two-dimensional numerical wave tank in this model. The finite volume method is a discretization technique to solve the Reynolds-average Navier–Stokes equations. The inlet velocity method is used to generate the desired waves. The standard k–ε turbulent model is used for the verification. Viscous models were tested at a constant water depth ( d) under 12 cases to complete the analysis. Surface elevation, wave velocities, velocity contour, and streamlined motions are important studies in this problem. Simulating result shows that the wave velocities ( u, w) at the free surface ( z = 0) increase with the wave steepness ( H/L) at a particular time period ( T) and gradually decrease from the free surface ( z = 0) toward the bottom of the water ( z = − d). Many practical systems, such as wave energy converters, offshore marine constructions, and other ocean engineering challenges, may benefit significantly from this research's realistic wave creation. This analysis of different periods gives the knowledge of flow behavior which provides the previous test of the scale model.

An empirical model for predicting wave attenuation inside vegetation domain

An empirical model for wave attenuation predictions when periodic waves propagate through vegetation domain is presented based on the dimensional analysis. A new nondimensional parameter KaL which represents the averaged wave attenuation per wavelength in the whole vegetation domain is introduced to denote the wave energy dissipation due to the presence of vegetation stems. The wave attenuation can be predicted by the multiplications of several nondimensional terms containing both wave and vegetation parameters without CD value. The model was firstly calibrated with emergent vegetation cases, and the applicability of Dalrymple et al. (1984)'s model was studied with the same datasets. The comparisons showed the empirical model performed better than Dalrymple et al. (1984)'s one, especially for cases with non-cylindrical stems, incident irregular waves and relatively dense vegetation domain. Then, the empirical model was validated with submerged cases in the literature and also floating cases conducted in the present study. After that, the influence of vegetation domain vertical positions on wave attenuation is discussed based on the validated model, and the non-dimensional vegetation diameter term in the model is changed and analyzed. Moreover, the simplification of the model which substitutes the Ursell number for wave nonlinearity and relative water depth is implemented.

Modeling wave attenuation by submerged flexible vegetation with XBeach phase-averaged model

Understanding and modeling wave attenuation by flexible vegetation have emerged as essential issues in coastal and ocean engineering. Given that the existing XBeach phase-averaged wave model can only simulate wave attenuation by rigid vegetation, and previous studies of wave energy dissipation by vegetation are mainly restricted to linear wave theory. This study aims to extend the XBeach phase-averaged wave model to simulate wave attenuation by both homogeneous and heterogeneous submerged flexible vegetation, and determine wave attenuation simulation deviation based on different wave theories. Hence, the existing flexible vegetation dynamic model is incorporated into the XBeach phase-averaged wave model. The reliability of the coupled model in simulating wave attenuation by both homogeneous and heterogeneous submerged flexible vegetation is revealed. Simulation results also demonstrate that the computed wave attenuation rate based on Stokes second-order wave theory is larger than that based on linear wave theory within the parameters used in this investigation. Meanwhile, wave attenuation simulation deviation between these two wave theories enlarges as the Ursell number increases within the parameters used in this investigation. This study can broaden the application of the XBeach model and lay the technical foundation for future engineering practice of nature-based coastal defence.