Effects Of Bottom Slope Research Articles

Free Surface Calculations with Bottom Slope Effect

Free Surface Calculations with Bottom Slope Effect

Experimental investigation of breaking regular and irregular waves slamming on an offshore monopile wind turbine

A series of experiments were carried out to investigate the breaking regular and irregular wave forces on a monopile foundation in a wave flume with a model scale factor of 1:40. Breaking regular wave slamming characteristic was firstly analyzed and comparisons were made between typical plunging and spilling breakers. For the irregular wave conditions, the wave slamming processes were recorded to illustrate the concentration and dissipation of wave energy. The effects of bottom slope, water depth, significant wave height and peak period on the wave slamming force and pressure distribution on the monopile were investigated. The results show that the distribution of maximum wave slamming pressure is affected by the water depth and bottom slope. Multiple harmonic components of the wave slamming force are observed and the predominant harmonic forces are first- and second- order components. The significant wave height and turbulence kinetic energy dissipation have significant effects on the wave slamming forces. A Gaussian distribution formula is proposed to fit the exceedance probability distribution of wave slamming force. Modified empirical formulas are proposed to calculate the wave slamming coefficient for the breaking regular and irregular waves on the monopile.

A Modified Mild-Slope Model for the Hydrodynamic Analysis of Arrays of Heaving WECs in Variable Bathymetry Regions

A simplified model based on the Modified Mild-Slope Equation with inclusions is developed for modelling the scattering of waves from multiple heaving point absorbers arranged in an array in general bottom topography. The model is used, in conjunction with a 3D BEM, in order to estimate the parameters modelling the energy extraction of the devices using data obtained from the hydrodynamic responses and performance of the single floating WEC. Subsequently, the present model is used for specific examples to calculate the wave field and the hydrodynamic performance of arrays of heaving WECs in constant depth and variable bathymetry regions and illustrate the effect of bottom slope and variation on the calculated wave field in the domain and in the vicinity of the devices. The present simplified model provides a low-cost first estimation of the wave conditions in the domain, which could be exploited as a supporting tool for best arrangement and park design purposes.

DEVELOPMENT OF A THREE DIMENSIONAL NUMERICAL MODEL OF SEDIMENT TRANSPORT AND MORPHOLOGICAL EVOLUTION ON SANDY BEACH

In recent years, sandy coasts are suffering from erosion. It is of great importance to evaluate the state of coasts and assure the achievement of coastal protection measures. Therefore, a three-dimensional numerical model of sandy beach response was developed based on unstructured grids and with capability of describing nearshore hydrodynamics and sediment transports. A three-dimensional hydrodynamic model was first developed based on a coupled wave-current model system that included the Simulating Waves Nearshore (SWAN) wave model and the Finite Volume Community Ocean Model (FVCOM) circulation model. Information exchange between the two models used Model-Coupling Toolkit (MCT) software following Chen et al. (2018). The new three-dimensional radiation stress including the bottom slope effects was employed (Ji et al. 2017). Based on the hydrodynamic model, a numerical model of sediment transport and morphological evolution on sandy beach was developed.Recorded Presentation from the vICCE (YouTube Link): https://youtu.be/BVVn1kfViH0

Wave resonant scattering mechanism of sinusoidal seabed elucidated by Mathieu Instability theorem

In this article, we shed light on unresolved issues on the physics of the problem, and clarify the effects of slope and other related parameters controlling the wave scattering phase shifting by sinusoidal seabed based on Mathieu Instability theorem. As a starting point, 3D water wave scattering by arbitrarily-varying sea bottom is governed by 2D wave potential and wavenumber coupling equation using Fredholm's Alternative Theorem. Seabed undulating effects are split into different components, including wavenumber and bottom slope and curvature effects. Sinusoidal ripples are used to demonstrate these different seabed undulating effects. The presented model predictions agree very well with lab data. Furthermore, a method with quantitative formulations is constructed and proposed to elucidate the resonance mechanism of wave scattering by sinusoidal seabed using Mathieu instability theorem. General stability chart is generated in a relevant parametric plane to detect the unstable conditions. Quantitively formulations of resonance conditions are presented to predict peak shifting features that beyond the Bragg's perdition. As demonstrated in this work, slope effect is contributed to peak shifting and the calculation formula of the shifting distance is given, though the shifting in single sinusoidal case is small, it is more obvious in double sinusoidal cases.

Influence of Mild Bottom Slopes on the Overtopping Flow over Mound Breakwaters under Depth-Limited Breaking Wave Conditions

The crest elevation of mound breakwaters is usually designed considering a tolerable mean wave overtopping discharge. However, pedestrian safety, characterized by the overtopping layer thickness (OLT) and the overtopping flow velocity (OFV), is becoming more relevant due to the reduction of the crest freeboards of coastal structures. Studies in the literature focusing on OLT and OFV do not consider the bottom slope effect, even if it has a remarkable impact on mound breakwater design under depth-limited breaking wave conditions. Therefore, this research focuses on the influence of the bottom slope on OLT and OFV exceeded by 2% of incoming waves, hc,2% and uc,2%. A total of 235 2D physical tests were conducted on conventional mound breakwaters with a single-layer Cubipod® and double-layer rock and cube armors with 2% and 4% bottom slopes. Neural networks were used to determine the optimum point to estimate wave characteristics for hc,2% and uc,2% calculation; that point was located at a distance from the model toe of three times the water depth at the toe (hs) of the structure. The influence of the bottom slope is studied using trained neural networks with fixed wave conditions in the wave generation zone; hc,2% slightly decreases and uc,2% increases as the gradient of the bottom slope increases.

On Baroclinic Instability over Continental Shelves: Testing the Utility of Eady-Type Models

AbstractThis study examines the utility of Eady-type theories as applied to understanding baroclinic instability in coastal flows where depth variations and bottom drag are important. The focus is on the effects of nongeostrophy, boundary dissipation, and bottom slope. The approach compares theoretically derived instability properties against numerical model calculations, for experiments designed to isolate the individual effects and justified to have Eady-like basic states. For the nongeostrophic effect, the theory of Stone (1966) is shown to give reasonable predictions for the most unstable growth rate and wavelength. It is also shown that the growing instability in a fully nonlinear model can be interpreted as boundary-trapped Rossby wave interactions—that is, wave phase locking and westward phase tilt allow waves to be mutually amplified. The analyses demonstrate that both the boundary dissipative and bottom slope effects can be represented by vertical velocities at the lower boundary of the unstable interior, via inducing Ekman pumping and slope-parallel flow, respectively, as proposed by the theories of Williams and Robinson (1974; referred to as the Eady–Ekman problem) and Blumsack and Gierasch (1972). The vertical velocities, characterized by a friction parameter and a slope ratio, modify the bottom wave and thus the scale selection. However, the theories have inherent quantitative limitations. Eady–Ekman neglects boundary layer responses that limit the increase of bottom stress, thereby overestimating the Ekman pumping and growth rate reduction at large drag. Blumsack and Gierasch’s (1972) model ignores slope-induced horizontal shear in the mean flow that tilts the eddies to favor converting energy back to the mean, thus having limited utility over steep slopes.

Analytical Solution of Shallow Water Equations for Ideal Dam-Break Flood along a Wet-Bed Slope

AbstractThe existing analytical solutions of dam-break flow do not consider simultaneously the effects of wet downstream bottom and bed slope on the dam-break wave propagation. In this study, a new...

The Evolution of Undular Bore in Coastal Zone: Effect of Bottom Slope, Friction and Special Topography

When tsunami waves propagate into shallow and gentle continental shelves, the effects of submarine topography and wave dispersion lead to the appearance of undular bore. This kind of wave is quite different from the initial tsunami wave in the waveform, and its amplitude is enhanced significantly. The undular bore will cause great and repeated runups on the beach, which brings larger destruction to the coastal region. In order to investigate the characteristics of the propagation of undular bore, this paper simulates the evolution from the sinusoidal long wave to the undular bore and analyzes the influencing factors. The beach slope has an important effect on the development of undular bores in terms of the undulation form, amplitude, wave length and occurrence time. The influence of bottom friction and submarine topography on the undulations and soliton fissions are discussed. These bottom conditions will lead to great change of the waveform and amplitude of undular bores.

Whole-processes modeling of flow movement and sediment transport during the period of water-sediment regulation in Xiaolangdi Reservoir

A one-dimensional coupled model has been proposed to consider the flow exchanges between main stream and tributaries during the alternative calculations between open-channel flow and turbidity currents, which is capable of simulating the whole process of flow and sediment transport covering these three different physical factors during the period of water-sediment regulation in the Xiaolangdi Reservoir. The modified storage-cell method was adopted for a complex tributary system to calculate the net flow exchanges caused by the water surface variation, and the discharge formula considering the effect of bottom slope of tributaries was applied to calculate the turbidity currents intrusion. Meanwhile, the plunge criterion of turbidity is used to connect the calculations of open-channel flow and turbidity currents, the static water surface assumption was no longer used so as to couple the interaction between the upper clear water and lower turbidity currents. The whole event of water-sediment regulation conducted in the Xiaolangdi Reservoir in 2002 was used for parameter calibration and the event of 2012 was simulated using both the proposed coupled model (Model I) and the model without considering the flow exchanges (Model II). The results show that the reservoir can be divided into three regions: the upper region with no tributary, the middle region with just open-channel flow and the lower region with turbidity currents, where both contain tributaries. Model I presented the results to closely agree with the measured processes in three regions. Among which the upper region is the only region the results calculated by Model II are as precise as Model I, However, with effects of the flow exchanges on the sediment transport and the conservation of water mass and flow momentum being omitted, Model II presented incorrect predictions in the lower two regions where there are tributaries: the calculated water level, discharge and sediment concentration in the middle region are rather high during the occasion of water intrusion and rather small during the occasion of water recharge; in the lower region, the location and occurrence of the plunge point were predicted to be delayed, and these turbidity factors were overestimated, covering interface level, discharge, flow velocity and sediment concentration, correspondingly, the sediment delivery ratio are rather large. Therefore, the effect caused by the tributary system is so enormous that it should not be neglected in the simulation. In addition, a detailed sensitivity analysis of coefficients in the turbidity module was also conducted in order to investigate the effects of these coefficients on the model predictions, with the proper variation range being determined for each coefficient. It indicates that the effects on the calculated factors caused by the drag coefficient at interface, the resistance coefficient of intrusion and the mixing coefficient between clear water and turbidity decreased in turn. Besides, with an increase in drag coefficient, the model response shows a time lag at a certain extent; with the bottom slope of tributaries not being constant, the effect of resistance coefficient requires a further discussion based on a specific slope.

Experimental study of sheet-flow sediment transport under nonlinear oscillatory flow over a sloping bed

The seabed is sloped in shallow coastal regions, where strong nonlinear shoaling waves can generate sheet-flow sediment transport. Both wave nonlinearity and bottom slope can lead to a wave-averaged (net) sediment transport rate, but the bottom slope effect is often overlooked in existing sediment transport models. A full-scale experimental study of sheet-flow sediment transport under the oscillatory flows with nonlinear features (skewness or asymmetry) was conducted in an inclinable oscillatory water tunnel, so the bottom-slope-induced net transport rate in the context of nonlinear waves was experimentally investigated. The tests cover a variety of nonlinear wave shape, bottom slope (0 to 2.5∘) and two sediment diameters (0.24 and 0.51 mm). Since the downslope direction is the “offshore” direction in our tests, it is found that bottom slope reduces the net “onshore” transport rate due to wave nonlinearity. The slope contribution to the net transport rate is evaluated by taking the difference between the measured net transport rate from a sloping-bed experiment and the measured net transport rate from the corresponding horizontal-bed experiment. The results suggest that slope-induced net transport rate is insensitive to wave nonlinearity, and its variation with bottom slope can be considered linear in the context of predicting net transport rate. Comparisons between this study and a preceding sinusoidal-flow study of Yuan et al. (2017b) suggest that nonlinear waves can be approximated as sinusoidal flows for estimating the slope contribution to the net transport rate. Subsequently, an empirical model for predicting net sheet-flow transport rate due to bottom slope is established based on dimensional analysis. This model can be simply added to existing sheet-flow sediment transport models to improve their applicability in coastal regions.

Baroclinic Instability of Nonzonal Flows and Bottom Slope Effects on the Propagation of the Most Unstable Wave

AbstractA linear, two-layer potential vorticity (PV) equation model on the β plane is employed to study the baroclinic instability of nonzonal basic currents flowing over a uniform bottom slope. Criteria for the instability, phase speed, and growth rate of unstable waves are all given as functions of the basic shear velocity, β, and bottom slope. The study suggests two kinds of long wave cutoff, one induced by the slope and the other by β; the first one exists in all directions, while the second requires at least a slight deviation of the wave vector from the meridians. Subtle differences between configurations of the PV gradient lead to completely different characteristics of unstable perturbations, such as propagation and scale. In the case of a positive slope (the bottom slope in the same direction as the isopycnal tilt), the fastest-growing wave is capable of propagating across the basic flow streamlines. By contrast, in the case of a negative slope (the bottom slope opposed to the isopycnal tilt), the most unstable wave always propagates along the streamlines. In addition, the spatial scale of the most unstable mode can be heavily reduced by a negative slope.

Effect of bottom slope on the nonlinear triad interactions in shallow water

This paper aims at investigating the effect of bottom slope to the nonlinear triad interactions for irregular waves propagating in shallow water. The physical experiments are conducted in a wave flume with respect to the transformation of waves propagating on three bottom slopes (β = 1/15, 1/30, and 1/45). Irregular waves with different type of breaking that are mechanically generated based on JONSWAP spectra are used for the test. The obviously different variations of spectra measured on each bottom reveal a crucial role of slope effect in the energy transfer between harmonics. The wavelet-based bispectrum were used to examine the bottom slope effect on the nonlinear triad interactions. Results show that the different bottom slopes which waves are propagated on will cause a significant discrepancy of triad interactions. Then, the discussions on the summed bicoherence which denote the distribution of phase coupling on each frequency further clarify the effect of bottom slope. Furthermore, the summed of the real and imaginary parts of bispectrum which could reflect the intensity of frequency components participating in the wave skewness and asymmetry were also investigated. Results indicate that the value of these parameters will increase as the bottom slope gets steeper.

Modeling net sheet-flow sediment transport rate under skewed and asymmetric oscillatory flows over a sloping bed

A one-dimensional-vertical model is developed for net sheet-flow sediment transport rate under oscillatory boundary layer flows. This model couples the boundary-layer model proposed by Yuan and Madsen (2015) with new predictors for bedload and suspended-load transport rates. Sediment concentration is modeled by solving the advection-diffusion equation with a parameterized turbulence diffusivity and a bottom pick-up rate. The slope effect is included in the predictions of instantaneous bedload transport rate and sediment pick-up rate, so the model can rigorously account for wave nonlinearity (velocity skewness and asymmetry) together with a mild bottom slope. The model performance on net sediment transport rate is excellent for large grain sizes, except that the velocity-asymmetry effect seems to be moderately underestimated. For small grain sizes, the model performance deteriorates, but is still acceptable for many cases. This is possibly because the model does not consider turbulence damping due to stratification and underestimates the phase-lag effect. Using the validated model, it is found that a boundary layer streaming due to velocity skewness and/or asymmetry can be a major contributor to net transport rate, when significant sediment suspension occurs. A simple case study also suggests that a mild bottom slope can be equally important as the wave nonlinearity for producing net sediment transport rate. Model predictions show that wave-nonlinear and bottom-slope effects can be linearly superimposed, which provides a simple way for incorporating bottom slope into existing transport-rate formulas.

Predicting net cross-shore total load transport: A phase-averaging, quasi-steady approach incorporating undertow contribution

Wave shapes that induce velocity skewness and acceleration asymmetry are usually responsible for onshore sediment transport, whereas undertow and bottom slope effect normally contribute to offshore sediment transport. By incorporating these counteracting driving forces in a phase-averaged manner, the theoretically-based quasi-steady formula of Wang (2007) is modified to predict the magnitude and direction of net cross-shore total load transport under the coaction of wave and current. The predictions show an excellent agreement with the measurement data on medium and fine sand collected by Dohmen-Janssen and Hanes (2002) and Schretlen (2012) in a full-scale wave flume at the Coastal Research Centre in Hannover, Germany. The modified formula can predict the net onshore transport of fine sand in sheet flows. In particular, it can predict the net offshore transport of medium sand in rippled beds through enlarged bed roughness, as well as the net offshore transport of fine-to-coarse sand in sheet flows with the aid of a new criterion to judge the occurrence of net offshore transport.

Effects of basin bottom slope on jet hydrodynamics and river mouth bar formation

River mouth bars are strategic morphological units primarily responsible for the development of entire deltaic systems. This paper addresses the role of receiving basin slope in the hydrodynamics of an exiting sediment-laden turbulent jet and in resulting mouth bar morphodynamics. We use Delft3D, a coupled hydrodynamic and morphodynamic numerical model, along with a theoretical formulation to reproduce the physics of the problem, characterized by a fluvially dominated inlet free of waves and tides. We propose an updated theoretical model with a slope-dependent entrainment coefficient, showing that the rate at which ambient fluid is incorporated into a jet increases with higher basin slopes. Transient results reveal that the magnitude of a basin slope can alter the stability of a jet, favoring the formation of an unstable meandering jet. While a stable jet gives rise to “middle-ground” bars accompanied by diverging channels, a “lunate” mouth bar results from unstable jets. Additional morphodynamic simulations demonstrate that the time required for mouth bar stagnation in its final position increases linearly with the basin slope. In contrast, the distance at which the mouth bar eventually forms decreases until reaching an asymptotic value for slopes higher than 2%. Moreover, the basin slope highly influences sedimentary processes responsible for bar formation: for milder slopes, progradation processes prevail, while in steeper basins aggradation is more relevant. Finally, the minimum relative water depth over a bar crest that forces the flow to bifurcate around a fully developed bar decreases with the basin slope.

3D hydroelastic analysis of very large floating bodies over variable bathymetry regions

The coupled-mode model developed by Belibassakis and Athanassoulis (J Fluid Mech 531:221–249, 2005) is extended and applied to the hydroelastic analysis of three-dimensional large floating bodies of shallow draft lying over variable bathymetry regions. The method is also applicable to the problem of wave interaction with ice sheets of small thickness. A general bathymetry is assumed, characterized by a continuous depth function, joining two regions of constant, but possibly different, depth. We consider the scattering problem of harmonic incident surface waves, under the combined effects of variable bathymetry and a floating elastic plate of orthogonal planform shape. Under the assumption of small-amplitude waves and plate deflections, the hydroelastic problem is formulated within the context of linearized water-wave and thin elastic-plate theory. To consistently treat the wave field beneath the elastic floating plate, down to the sloping bottom boundary, a complete, local, hydroelastic-mode series expansion of the wave field is used, enhanced by an appropriate sloping-bottom mode. The latter enables the consistent satisfaction of the Neumann bottom-boundary condition on a general topography. Numerical results concerning floating structures over flat and inhomogeneous seabed are presented, and the effects of wave direction, bottom slope and bottom corrugations on the hydroelastic responses are discussed.

Wave–wave interactions on a sloping bottom

This study derived an analytic solution for two progressive waves propagating on a sloping bottom. The solution of flow field in explicit form is obtained as a three-parameter function of rendering the two wave ordering parameter ε1,ε2 and the bottom slope perturbed to the third order. The results can be simplified to a single propagating wave on sloping beach and wave–wave interaction in uniform depth. The wave number and angular frequency are related to water depth, bottom slope, and wave steepness and are also affected by another propagating wave, the increasing wave steepness or wave length of which will enhance the influence of the dispersion relation. The solutions include the wave–wave interaction, wave shoaling, nonlinearity, and the effect of bottom slope. Those are observed by means of the discussion of the free-surface displacement. The resonance caused by wave–wave interaction will occur in a specified water depth and the ratio of two wave numbers, leading to the singularity of free-surface displacement.

Parameterization of nonlinear shallow water waves over sloping bottoms

Investigation of the bottom slope effects on the nonlinear transformation of irregular waves, which are generated based on JONSWAP spectra, is carried out in a physical wave flume with three slopes (β=1/15, 1/30, 1/45). The slope effects on the estimation of representative wave height are examined first. To obtain a better estimation of wave height, the slope effect should be considered when slope is larger than 1/30. The nonlinear parameters (bicoherence, skewness and asymmetry) are estimated by using the wavelet-based bispectrum, and the empirical formulae regarding these nonlinear parameters as a function of the local Ursell number are derived based on the present data measured on each slope. The results indicate that the slopes have a negligible effect on the variations of the skewness. The fitted coefficients of the formulae for the other parameters on slope β=1/15 are clearly different from the results on the slopes β=1/30 and 1/45, indicating that slope influence on the parameterization cannot be ignored when β>1/30. Hence, new formulae considering the slope effect are presented. Furthermore, the empirical formulae for the data in surf zone are recommended.

Study of Energy Dissipation on Inclined Rectangular Contracted Chute

Hydraulic jump primarily serves as an energy dissipater to dissipate excess energy of flowing water downstream of hydraulic structures. This study shows the results of an experimental investigation on relative energy dissipation conducted at chutes of different slopes with different contractions. So the effects of contraction and bottom slope on relative energy dissipation were investigated. Experiments have been carried out in channels, where chute slope was varied between 1V:10H and 1V:6H, and chute contraction varying between 0° and 6°. It was found that the energy dissipation increases with increasing slope and decreases with increasing contraction and ratio between the critical water depth (hc) at section 0 and the chute height (H). The amount of energy loss of such structures is essentially a function of the hc/H ratio, the chute slope and the chute contraction. Relationships were proposed to evaluate relative energy dissipation as a function of chute slope, chute contraction and hc/H ratio.