Mild-slope Equation Research Articles

Application of a Consistent Nonlinear Mild-Slope Equation Model to Random Wave Propagation and Dissipation

Application of a Consistent Nonlinear Mild-Slope Equation Model to Random Wave Propagation and Dissipation

Modeling Rip Current Systems around Multiple Submerged Breakwaters

Multiple submerged breakwaters (MSBWs) are commonly used coastal protection structures due to their specific advantages over the emerged ones. Rip currents, as the inevitable natural hazard in the gaps of these constructions, are investigated numerically in the present study. A fully nonlinear mild-slope equation (NMSE) model possessing both fully nonlinear and fully dispersive properties is validated and adopted in the simulations. With four monochromatic wave conditions of different wave heights, periods and incidences representing low-energy, typical, storm and oblique waves tested, the flow patterns and the low-frequency oscillations of the rip currents are studied. For the convenience of risk assessment, the rip risk level is divided into three degrees according to the maximum rip flow speed. The effects of the configurations of the MSBWs on the rip current system as well as the rip risk level are examined, considering different breakwater widths, heights, forms, gap widths and gap numbers. Simulation results suggest that the cross-shore configurations of MSBWs influence the rip risk level by inducing different wave energy dissipations but the longshore configurations of MSBWs by changing flow field patterns.

Derivation and analysis of lattice Boltzmann form of the mild slope equation

The present study investigates the performance of the Lattice Boltzmann Method (LBM) as an alternative numerical method for simulating wave propagation in coastal zones. While formulations like the Mild-Slope Equation (MSE) remain efficient for coastal engineering applications due to their simplicity, LBM offers an explicit numerical solver for partial differential equations (PDEs). LBM utilizes local operations, making it well-suited for Graphics Processing Unit (GPU) acceleration and consequently efficient simulations. This study examines LBM's performance in several benchmark cases, including wave propagation behind a single breakwater, harbor resonance, refraction of long waves over a parabolic shoal, and wave propagation over an elliptic shoal. The Chapman-Enskog multi-scale expansion was used within a popular 2D LBM scheme to derive the relaxation time and equilibrium distribution function. The results showed a good agreement between LBM predictions and benchmark data. Furthermore, LBM demonstrated efficiency in reducing the computation time, achieving a notable 30–50% decrease in CPU time and an 85% reduction when utilizing a GPU.

Wave blocking performance of the symmetrical double-wing floating breakwater

In this paper, a double-wing floating breakwater (DWFB) is proposed, that is, the wing with a curved upper surface and flat lower surface is installed on both sides of the box-type floating breakwater (FB). The analytical model of the DWFB under wave action is established to solve the wave diffraction problem. The modified mild slope equation (MMSE) is used to deal with the unsteady water depth problem above the wing, and the reflection coefficient, transmission coefficient, and the wave excitation force in the surge, heave, and pitch are obtained. After the analytical model is verified, the effect of the wing shape and wing size (height and length) on the wave blocking performance of the DWFB is analyzed, and then the single-wing FB and DWFB are compared. The results show that the increase in wing length is obvious to reduce the transmission coefficient. The single-wing floating breakwater (SWFB) with a longer wing length but similar cross-sectional area to the DWFB has a smaller transmission coefficient when the wave number is between 0.45 and 0.85. However, as the wave number continues to increase, the wave blocking performance of the DWFB is relatively better.

Improving the Amount of Captured Energy of a Point-Absorber WEC on the Mexican Coast

Although there are constant improvements in wave energy converter (WEC) technology, it is crucial to investigate site-specific sea conditions for optimal power absorption and efficiency. This study compares the efficiency of a floating buoy-type WEC device, with three differently shaped floats: a semi-sphere, a cylinder considered suitable for a location near Ensenada, on the Baja California peninsula, and a novel, rounded, semi-rectangular float. A statistical analysis of the wave climate of the last 42 years was performed to define the conditions to which the device is subjected. The WEC location was chosen for shallow waters, using a computational model that solves the modified mild slope equation. The hydrodynamic response of the three float designs was then analyzed in the frequency and time domains, using the software ANSYS AQWA 19.2, to assess the dynamics of the floating body, the forces exerted, and the power absorbed, as well as the suitability of the proposed power take-off (PTO) system. The findings show that the proposed float design absorbs the most energy, with an annual power of 135.11 MW, and that the PTO mechanism is appropriate.

TOPOGRAPHICAL EFFECTS ON WAVE SCATTERING BY AN ELASTIC PLATE FLOATING ON TWO-LAYER FLUID

This article illustrates the hydroelastic interactions between surface gravity waves and a floating elastic plate in a two-layer liquid with variable bottom topography under the assumptions of small amplitude waves and potential flow theory. In this study, semi-infinite and finite-length plates are considered. The eigenfunction expansion method is applied in the fluid region with uniform bottom topography. A system of differential equations (mild-slope equations) is solved in the fluid region with variable bottom topography. From the matching and jump conditions, the solution is expressed as a linear algebraic system from which all the unknown constants are computed. We explored the effects of density ratio, depth ratio, and bottom topography on the bending moment, shear force, and the deflection of the elastic plate. Results show that when the density ratio becomes closer to one, the occurred bending moment and shear forces to the elastic plates tend to diminish. The bending moment and shear forces to the pates are higher and lower at a smaller depth ratiofor the incident surface wave and interfacial waves, respectively. The variations in the bending moment, shear force, and plate deflection, caused by surface and interfacial waves, are observed to be in opposite trends, respectively. Bottom profiles similarly affect semi-infinite and finite-length plates when they undergo free-edge conditions. These effects, however, are substantial when the plate is simply supported at the edges. Elastic plate with free edges experiences lower deflection for concave-up and plane-sloping bottoms for incident surface and interfacial waves, respectively.

A set of fully nonlinear mild slope equations

A set of time-dependent fully nonlinear mild slope equations is established in terms of velocity vector at still water level and free surface elevation, which is derived from the Euler equations without approximation in nonlinearity and only with the potential flow assumption adopted for bottom slope terms. Apart from the fully nonlinear property, the model is also different from previous nonlinear mild slope equations by adopting the velocity expression accurate to the first-order in bottom slope, which allows the new expressions of higher-order bottom slope terms to be established and the fully nonlinear terms related to bottom slope to be included in the equations. The model can be reduced to the classical mild slope equation for linear waves and the fully nonlinear Boussinesq equations for nonlinear waves. The advantages of the present model are validated against the related laboratory experimental results and demonstrated by comparison with the numerical results of other relevant models.

Enhanced Mild-Slope Wave Model with Parallel Implementation and Artificial Neural Network Support for Simulation of Wave Disturbance and Resonance in Ports

Ensuring sea surface tranquility within port basins is of paramount importance for safe and efficient port operations and vessels’ accommodation. The present study aims to introduce a robust numerical model based on mild-slope equations, capable of accurately simulating wave disturbance and resonance in ports. The model is further enhanced by the integration of an artificial neural network (ANN) to address partial reflection, and its efficiency is optimized by developing a parallel algorithm based on OpenMP, allowing for a reduction in the required simulation times for real port areas spanning several kilometers horizontally. Numerous numerical experiments focusing on wave reflection against a vertical wall were conducted to develop the ANN. This neural network was designed to determine the appropriate value of the eddy viscosity coefficient, a crucial parameter in the momentum equation of the mild-slope model, tailored to incident wave characteristics. The model’s validity was confirmed through rigorous validation against experimental measurements, covering wave disturbance, rectangular harbor resonance, and Bragg resonance. The model consistently demonstrated a more than satisfactory performance across all considered scenarios. In a practical application, the model was deployed in the Port of Rethymno, Crete Island, Greece, effectively capturing and describing dominant phenomena within the port area. The implementation of a parallel algorithm significantly reduced the simulation times by ~92%, compared to the serial algorithm, thereby enhancing the model’s efficiency and applicability in real-world port environments.

Estimation of harbor and bay resonances by MMS-FEM model with application to the Bay of Toulon, France

Bay and harbor resonances are investigated in this work, taking into account the variable bathymetry of the semi-enclosed basin. The Modified Mild-Slope (MMS) equation is implemented for the description of combined refraction-diffraction effects, from which the eigenperiods and eigenmodes are calculated by means of a low-order Finite Element Method (FEM scheme). The model is first applied to a coastal-port region of Toulon, France, illustrating the versatility of the model to easily include coastal structures such as detached breakwater. Next, the present model is applied to the extended nearshore area of Toulon including the Gulf of Giens showing the applicability of the developed MMS-FEM model for the estimation of harbor and bay resonances, as well as more extended nearshore regions where variable bottom topography effects become important. The calculated resonant frequency depends on the domain characteristics and the size of the open sea boundary and accurately reproduces the measurements within Toulon Bay. On the other hand, for open bays such as the Gulf of Giens, a discrepancy is observed between calculated and measured eigenperiods which is due to a very wide opening of the sea boundary that cannot accurately describe the seiching. This underlines the difficulty of accurately calculating the resonance frequency for open bays, in contrast to the classic studies carried out for ports, which are considered virtually closed basins, and confirms the complementary nature of long-term water level measurements and numerical calculations, for better quantification of the risks associated with energetic meteorological and/or oceanographic events.

Finite Difference Solution of the Modified Mild Slope Equation for Wave Reflection by a Slope With Superimposed Undulating Seabed

Finite Difference Solution of the Modified Mild Slope Equation for Wave Reflection by a Slope With Superimposed Undulating Seabed

A simplified consistent nonlinear mild-slope equation model for random waves propagation and dissipation

The model for ocean surface wave propagation can be formulated either in the form of deterministic models or stochastic models. The stochastic models appear to be particularly attractive in the global domain due to their computational efficiency. However, in the nearshore region, the phase becomes highly correlated, and the phase information therefore becomes critical. Therefore, a simplified consistent nonlinear mild-slope equation model has been developed in order to take advantage of the deterministic model for handling phase information, as well as the stochastic model for numerical simplicity. We demonstrate the advanced performance of the present model for random waves by comparing it with laboratory data and previous models.

Effect of Sloping Bottom on Wave Interaction With Multiple Flexible Moored Breakwaters

Abstract An analytical framework is developed to analyze the interaction of oblique waves with multiple flexible porous breakwaters under the consideration of bottom undulation. The mathematical problem is tackled using the small amplitude water-wave theory, with Darcy’s law being applied to account for wave interaction with porous media. The bottom topography is considered to have a finite length, flanked by two semi-infinite sections of uniform bottom. The solution to the boundary value problem is approached by employing the eigenfunction expansion method within the uniform bottom regions. For the varying bottom topography, a modified mild-slope equation (MMSE) is utilized. To address the solution at the slope discontinuity at the bottom, a mass-conserving jump condition is applied. By matching solutions at the interfaces, a set of equations is derived. This system of equations encapsulates the behavior of reflection and transmission coefficients, as well as the force exerted on the breakwaters. These parameters are then investigated across various factors such as the length of the varying bottom, depth ratio, angle of the mooring line, angle of incidence, and flexural rigidity. Graphical representations of the reflection and transmission coefficients, along with the breakwater force, provide insights into the system’s behavior under different conditions. The water wave energy can be dissipated for the optimum values of flexural rigidity. The transmission coefficient is observed to be least for higher mooring angle.

Study on Numerical Modeling for Pollutants Movement Based on the Wave Parabolic Mild Slope Equation in Curvilinear Coordinates

Nearshore waves and currents induced by breaking of obliquely incident waves are important dynamic factors that significantly affect pollutants movement at offshore zones. A combined numerical model in orthogonal curvilinear coordinates is developed to simulate pollutants movement in surf zones, including a wave transformation model based on the parabolic mild slope equation, a wave-induced current model, and a pollutant transport model driven by waves and currents. This combined model has been applied to pollutants movement laboratory cases, and comparison of the simulations with experimental measurements shows good agreement. The model has also been applied to simulation of the Gourlay experiment and it shows consistent results with the distribution of waves and offshore currents. This coupled numerical model has preliminarily improved the situation of rare numerical research on the nearshore pollutant transport in coastal wave and wave-induced current based on wave radiation stress theory in orthogonal curvilinear coordinates.

GENERATION OF LINEAR WAVES WITH BOTTOM WAVE MAKERS IN A FLUME: AN EFFICIENT WAY TO ABSORB REFLECTED WAVES

When waves are generated in a flume by the conventional piston or flap wave maker, waves reflected from structures will arrive at the wave maker and experience re-reflection. These re-reflected waves will propagate to the structures so that the target wave energy cannot be accurately obtained. Since 1980s, methods of internally generating waves inside a computational domain and putting sponge layers at the domain end have been developed in numerical modelling. Internally generated waves will propagate both directions from the wave generation zone and waves reflected from the structures will pass through the wave generation zone without any reflection by the wave maker and be absorbed in the sponge layers. Using the idea of internal generation of waves, Jung et al. (2018) developed bottom wave makers in a flume which have paddles moving up and down from the bottom (see also Lu et al. (2017)). In this study, following Jung et al.’s (2018) techniques, we numerically develop both rectangular and triangular bottom wave makers in a flume using the analytical method and the extended mild-slope equations.

LANDSLIDE TSUNAMI HAZARD ASSESSMENT: A NUMERICAL MODEL FOR THE SIMULATION OF MULTIPLE LANDSLIDE-INDUCED TSUNAMIS SCENARIOS IN A MONTE CARLO FRAMEWORK

Submarine landslides can pose serious tsunami hazard to coastal communities. However, performing a comprehensive landslide tsunami hazard assessment for a given area is in general difficult in view of the large uncertainties associated with tsunamigenic source parameters. These are often known only approximately, on the basis of estimates of the landslide geometry, slide material properties, and kinematics. Here, we present an efficient model to perform such hazard analyses, based on solving the linear mildslope equation with a time-dependent source term representing the seafloor motion as detailed in Iorio et al. (2021). This approach allows carrying out many computations, for a large number of landslide scenarios, in a Monte Carlo (MC) approach framework, at a reduced computational cost compared to other available methods, while still providing physically accurate simulations of most landslide tsunami generation and propagation.

Generation of Linear Waves with Bottom Wave Makers Using Analytical Solutions and Extended Mild-Slope Equations

Generation of Linear Waves with Bottom Wave Makers Using Analytical Solutions and Extended Mild-Slope Equations

The polar-Cartesian mild-slope equation

A novel mild-slope equation is derived based on a manipulation of the cylindrical and Cartesian coordinate reference systems. The vertical profile of the velocity field is constructed by solving an approximate problem in cylindrical coordinates. This allows us to address the local derivatives on the bottom profile along a constant-slope line. This formulation is as opposed to the Cartesian-based mild-slope equations in terms of which the profile is constructed by assuming a constant depth. An angular profile is derived for the three-dimensional case on a sloping plane beach. For the two-dimensional case, a mild-slope polar-Cartesian equation is derived, for which an improved linear dispersion relation is reconstructed. This is accomplished due to the inclusion of first-order derivatives of the local bottom profile. The coefficients of the polar-Cartesian mild-slope equation contain the derivatives of the bottom profile up to third order as opposed to second-order derivatives in the Cartesian-based equations. The equation is derived by applying the variational principle to the Cartesian Lagrangian when formulated as a function of the profile in polar coordinates. It is then compared with existing models of the mild-slope equation for simulations of two-dimensional test cases and a quasi-three-dimensional case, which have known analytical solutions. Our modified equation exhibits better matching to the exact solutions for a majority of the investigated cases.

Probabilistic landslide tsunami modeling of the 2018 Palu Bay event

On September 28, 2018, a Mw 7.5 earthquake triggered near Central Sulawesi generated a highly destructive tsunami within Paul Bay (Indonesia). Field surveys and various studies conducted after the event showed that, as a result of the earthquake, several large submarine landslides were triggered along the shores of the bay, that significantly contributed to tsunami generation. The estimated geometry and other parameters for these slides, however, were affected by a large uncertainty. Here, we present a probabilistic tsunami hazard analysis of this event, based on Monte Carlo simulations using a linear Mild Slope Equation (MSE) model combined with a Green’s function approach, that allow efficiently simulating a large number of stochastic landslide tsunami generation and propagation scenarios within Palu Bay, for each of the identified landslides. In the MSE model, a space and time-dependent source term is used to represent the seafloor motion associated with each landslide scenario. In the Green’s function approach, a large database of elementary solutions and their tsunami elevation at a large number of coastal save points is pre-computed for tsunamis generated by a unit seafloor acceleration specified over a small area. Then, given an actual submarine landslide scenario, with a specific acceleration function, the tsunami elevation at the save points is simply and efficiently computed as a weighed linear superposition of the elementary solutions. Tsunami runup is finally obtained using a semi-empirical method, based on results computed at save points for each landslide scenario. The model is applied to the 2018 Palu Bay tsunami event, allowing to investigate how the uncertainty in landslide parameters affects tsunami hazard. A comparison with observed runups is made, which shows that these fall within the range of uncertainty of the simulated probabilistic runups.

Numerical investigation on the mitigation of harbor oscillations by periodic undulating topography

Mitigation of harbor oscillations is a crucial concern in coastal engineering. It is more economical and accessible for an established harbor to reduce the intensity of oscillations by changing the incident wave conditions rather than modifying the harbor layout. Periodic sandbars and sand ripples, which are common in nearshore zones, can be modified into submerged breakwaters. When the wavelength of the bottom undulations is approximately half of the incident wavelength, the resonant reflection (i.e., Bragg reflection) appears, leading to significant enhancements in wave reflection, which further results in the reduction of the incident wave energy.The present study evaluates the effectiveness of Bragg reflection in inhibiting harbor oscillations using an extended mild-slope equation. The results show that undulating seabed topography outside the harbor can significantly weaken the intensity of the harbor oscillations. Subsequently, a possible mitigation mechanism is proposed. The influences of the bar wavelength, bar number, bar height, and the slope of seabed on the mitigation effect of harbor oscillations are studied. Furthermore, it is found that the optimal mitigation effect for different resonant modes is not constant, and the reason for this is explained.

Analytical study of Bragg resonances by a finite periodic array of congruent trapezoidal bars or trenches on a sloping seabed

For water waves propagating over a finite periodic array of congruent trapezoidal bars on a sloping seabed or trenches dug on a sloping seabed, an analytical solution in terms of Taylor series to the modified mild-slope equation (MMSE) is constructed. Based on the analytical solution, influences of various topographic parameters (e.g., the slope of seabed, bar/trench number, bar height or trench depth, and bar/trench width) on the peak value, peak phase and zero reflection of Bragg resonances are investigated. Especially, starting from the original definition of Bragg resonances, a modified Bragg's law for characterizing the excitation condition of the nth order Bragg resonance is derived, which shows that the excitation of Bragg resonances depends entirely on the average phase velocity of the water wave within the bar/trench field. When the influence of the bar/trench field on the average phase velocity is quite small, the modified Bragg's law degenerates into the traditional Bragg’s law borrowed from X-ray crystallography, which also reflects the essential difference between the water wave as a mechanical wave and the non-mechanical X-ray wave. In addition, for N bars/trenches on a flat seabed, there are at least N−1 zero reflections between any two adjacent Bragg resonance peaks. When the bars are very low or the trenches are very shallow, the phase of the N−1 zero reflections in those zero reflections is exactly the position of the N−1 zero points of the Chebyshev polynomial of the second kind, UN−1(cosk0d), where k0 is the wavenumber on the flat seabed and d is the bar/trench spacing. However, once a sloping seabed is added, all these N−1 zero reflections disappear possibly due to the destruction of the symmetry of the background bar/trench field. Finally, regardless of the number of bars/trenches, when the bar/trench width varies, the peak value of the nth order Bragg resonance oscillates periodically, leading to n local maxima. A comparative study shows that these characteristics are consistent with the band structure shown in the Bloch periodic gap map (PGM).