A coupled numerical model of wave interaction with porous medium

A new coupling model of wave interaction with porous medium is established in which the wave field solver is based on the two dimensional Reynolds Averaged Navier-Stokes (RANS) equations with a kε closure. Incident waves, which could be linear waves, cnoidal waves or solitary waves, are produced by a piston-type wave maker in the computational domain and the free surface is traced through the Piecewise Linear Interface Construction-Volume of Fluid (PLIC-VOF) method. Nonlinear Forchheimer equations are adopted to calculate the flow field within the porous media. By introducing a velocity–pressure correction equation, the wave field and the porous flow field are highly and efficiently coupled. The two fields are solved simultaneously and no boundary condition is needed at the interface of the internal porous flow and the external wave. The newly developed numerical model is used to simulate wave interaction with porous seabed and the numerical results agree well with the experimental data. The additional numerical tests are also conducted to study the effects of seabed thickness, porosity and permeability coefficient on wave damping and the pore water pressure responses.     

Comparisons of internal solitary wave and surface wave actions on marine structures and their responses

In this paper, a time-domain numerical model is established for computing the action of internal solitary wave on marine structures and structure motion responses. For a cylindrical structure, its side and bottom are discretized by pole and surface elements, respectively. The drag and inertial forces in the perpendicular direction of the structure are computed by the Morison equation from the pole elements, and the Froude–Krylov force in the axial direction of the structure due to internal wave motion is computed by integration of the dynamic pressure over the surface elements. The catenary theory is used to analyze the reaction force due to mooring lines, and the motion equation of the marine structure is solved by the fourth-order Runge–Kutta method in the time domain. The model is used to calculate the interaction of the internal solitary wave with a Spar platform with mooring system, and the surface wave action with the platform has also been computed by a frequency-domain boundary element method for comparison. Through the comparison based on a practical internal wave and surface wave states, it can be concluded that the internal wave force on the structure is only 9% of the one due to surface waves. However, the motion response due to the internal wave is much greater than the one due to the surface waves. It shows that the low-frequency effect of internal solitary waves is a great threat to the safety of marine structures.     

Generation and propagation of transient nonlinear waves in a wave flume

A semi-analytical nonlinear wavemaker model is derived to predict the generation and propagation of transient nonlinear waves in a wave flume. The solution is very efficient and is achieved by applying eigenfunction expansions and FFT. The model is applied to study the effect of the wavemaker and its motion on the generation and propagation of nonlinear waves. The results indicate that the linear wavemaker theory may be applied to predict only the generation of waves of low steepness for which the nonlinear terms in the kinematic wavemaker boundary condition and free-surface boundary conditions are of secondary importance. For waves of moderate steepness and steep waves these nonlinear terms have substantial effects on wave profile and wave spectrum just after the wavemaker. A wave spectrum corresponding to a sinusoidally moving wavemaker possesses a multi-peak form with substantial nonlinear components, which disturbs or may even exclude physical modeling in wave flumes. The analysis shows that the widely recognized weakly nonlinear wavemaker theory may only be applied to describe the generation and propagation of waves of low steepness. This is subject to further restrictions in shallow and deep waters because the kinematic wavemaker boundary condition as well as the nonlinear interaction of wave components and the evolution of wave energy spectrum is not properly described by weakly nonlinear wavemaker theory. Laboratory experiments were conducted in a wave flume to verify the nonlinear wavemaker model. The comparisons show a reasonable agreement between predicted and measured free-surface elevation and the corresponding amplitudes of Fourier series. A reasonable agreement between theoretical results and experimental data is observed even for fairly steep waves.     

Free-surface evolution and wave kinematics for nonlinear uni-directional focused wave groups

This paper concerns the propagation of transient wave groups, focused at a point in time and space to produce locally large waves having a range of steepness. The experimental study was carried out in a wave flume at Dalian University of Technology. The numerical simulations were based on a nonlinear boundary integral equation solved by a higher-order boundary element method (HOBEM). Rather than simulate the whole experimental tank, local surface elevation measurements were used to drive the numerical solution from a point less than two wavelengths upstream of the focus position, leading to significant savings in computational time. Excellent agreement is achieved between the water surface elevations and the water particle kinematics measured in the experiments and those predicted numerically at wave group focus, even for near-breaking waves up to a steepness of kA=0.405 for which even locally matched 2nd-order theory is inadequate. Results based on the linear and 2nd-order theory are also presented in the comparisons. When compared with the first- and 2nd-order solutions, the fully nonlinear wave–wave interactions produce a steeper wave envelope in which the central wave crest is higher and narrower, while the adjacent wave troughs are broader and less deep.     

Shared-Memory parallelization of consistent particle method for violent wave impact problems

A shared-memory parallelization is implemented to the recently developed Consistent Particle. Method (CPM) for violent wave impact problems. The advantages of this relatively new particle method lie in four key aspects: (1) accurate computation of Laplacian and gradient operators based on Taylor series expansion, alleviating spurious pressure fluctuation and being able to model two-phase flows characterized by large density difference, (2) a thermodynamics-based compressible solver for modelling compressible air that eliminates the need of determining artificial sound speed, (3) seamless coupling of the compressible air solver and incompressible water solver, and (4) parallelization of the numerical model based on Open Multi-Processing (OpenMP) and a parallel direct sparse solver (Pardiso) to significantly improve computational efficiency. Strong and weak scaling analyses of the parallelized CPM are conducted, showing an efficiency speedup of 100 times or more depending on the size of simulated problem. To demonstrate the accuracy of the developed numerical model, three numerical examples are studied including the benchmark study of wave impact on seawall, and our experimental studies of violent water sloshing under rotational excitations and sloshing impact with entrapped air pocket. CPM is shown to accurately capture highly deformed breaking waves and violent wave impact pressure including pressure oscillation induced by air cushion effect.     

Wave interaction with a wave absorbing double curtain-wall breakwater

This study examines the hydrodynamic performance of a wave absorbing double curtain-wall breakwater. The breakwater consists of a seaward perforated wall and a shoreward impermeable wall. Both walls extend from above the seawater to some distance above the seabed. Then the below gap allows the seawater exchange, the sediment transport and the fish passage. By means of the eigenfunction expansion method and a least square approach, a linear analytical solution is developed for the interaction of water waves with the breakwater. Then the reflection coefficient, the transmission coefficient and the wave forces acting on the walls are calculated. The numerical results obtained for limiting cases agree very well with previous predictions for a single partially immersed impermeable wall, the double partially immersed impermeable walls and the bottom-standing Jarlan-type breakwater. The predicted reflection coefficients for the present breakwater also agree reasonable with previous experimental results. Numerical results show that with appropriate structure parameters, the reflection and transmission coefficients of the breakwater may be both below 0.5 at a wide range of the relative water depth. At the same time, the magnitude of wave force acting on each wall is small. This is significant for practical engineering.     

Deterministic combination of numerical and physical coastal wave models

A deterministic combination of numerical and physical models for coastal waves is developed. In the combined model, a Boussinesq model MIKE 21 BW is applied for the numerical wave computations. A piston-type 2D or 3D wavemaker and the associated control system with active wave absorption provides the interface between the numerical and physical models. The link between numerical and physical models is given by an ad hoc unified wave generation theory which is devised in the study. This wave generation theory accounts for linear dispersion and shallow water non-linearity. Local wave phenomena (evanescent modes) near the wavemaker are taken into account. With this approach, the data transfer between the two models is thus on a deterministic level with detailed wave information transmitted along the wavemaker.     

Virtual wave gauges based upon stereo imaging for measuring surface wave characteristics

A virtual wave gauge (VWG) technique based on stereo imaging is developed to remotely measure water wave height, period, and direction. VWG minimizes computational costs by directly tracking the elevation of the water surface at selected points of interest using a Eulerian based dynamic searching algorithm. Results show that the VWG technique developed in this paper dramatically improves efficiency by two orders of magnitude compared to the traditional Lagrangian–Eulerian based point cloud method of stereo image processing. VWG is tested against traditional wave wire gauges to within 98% accuracy for significant wave height. Furthermore, the flexibility of the VWG is demonstrated in two field applications. First in an offshore breaking wave case, an array of VWGs is used to efficiently measure wave directionality. Second to investigate the reflection coefficient of a rock-mounted structure interacting with nearshore waves, linear and spatial VWG arrays are designed and implemented based on a priori information of the wave field from a preliminary VWG measurement. Overall, we demonstrate that the flexible and computational efficient VWG technique has the potential to make real-time remote stereo imaging wave measurements a reality.     

A comparison of several wave spectra for the random wave diffraction by a semi-infinite breakwater

A comparison of the diffraction of multidirectional random waves using several selected wave spectrum models is presented in this paper. Six wave spectrum models, Bretschneider, Pierson–Moskowitz, ISSC, ITTC, Mitsuyasu, and JONSWAP spectrum, are considered. A discrete form for each of the given spectrum models is used to specify the incident wave conditions. Analytical solutions based on both the Fresnel integrals and polynomial approximations of the Fresnel integrals and numerical solutions of a boundary integral approach have been used to obtain the two-dimensional wave diffraction by a semi-infinite breakwater at uniform water depth. The diffraction of random waves is based on the cumulative superposition of linear diffraction solution. The results of predicted random wave diffraction for each of the given spectrum models are compared with those of the published physical model presented by Briggs et al. [1995. Wave diffraction around breakwater. Journal of Waterway, Port, Coastal and Ocean Engineering—ASCE 121(1), 23–35]. Reasonable agreement is obtained in all cases. The effect of the directional spreading function is also examined from the results of the random wave diffraction. Based on these comparisons, the present model for the analysis of various wave spectra is found to be an accurate and efficient tool for predicting the random wave field around a semi-infinite breakwater or inside a harbor of arbitrary geometry in practical applications.     

Comparative study on the accuracy of solitary wave generations in an ISPH-based numerical wave flume

In the present study, six solitary wave generations by different mathematical approximations are investigated using a piston type wave maker at dimensionless amplitudes ranging from 0.1 to 0.6 and two water depths. Incompressible smoothed particle hydrodynamics is used to simulate solitary wave propagation along the fixed depth channel. The present numerical results are compared with analytical results and experimental data in terms of free surface displacements, fluid particle velocity, phase speed, paddle motion, etc. The present mesh-free numerical results of wave profile variations over time proved that “Rayleigh” has the lowest relative wave height variation. However, its solitary wave has notable phase lead, while “Third order” and “Ninth order” have the least wave lags. Furthermore, the record of present numerical free surface elevation at different distances and the loss of amplitude of the main pulse showed that regarding both of them, “Ninth order” has supremacy over five others. Considering the numerical velocity components of generated solitary wave, “Third order” and “Ninth order” trace analytical results more accurately than other four ones, whereas “Rayleigh” is the most accurate one in predicting the maximum runup. Finally, the paddle motion, its velocity, and displacement, as well as phase speed and outskirts decay coefficient are also compared and discussed intensely.     

Numerical simulation of random wave slamming on structures in the splash zone

The numerical investigation of random wave slamming on superstructures of marine structures in the splash zone is presented in this paper. The impact pressures on the underside of the structure are computed based on the improved volume of fluid method (VOF). The governing equations are Reynolds time-averaged equations and the two equation k– model. The third order upwind difference scheme is applied to the convection term to reduce the effect of numerical viscosity. The numerical wave flume with random wave-maker suitable for VOF is established. Appropriate moving contact-line boundary conditions are introduced to the model wave in contact with and separated from the underside of structure. Parametric studies have been carried out for different incident waves, structure dimensions and structure clearance. The numerical results are verified by the experimental results.     

Surface gravity wave interaction with elastic bottom

In the present paper, a hydroelastic model is developed to deal with surface gravity wave interaction with an elastic bed based on the small amplitude water wave theory and plate deflection in finite water depth. The elastic bottom bed is modelled as a thin elastic plate and is based on the Euler-Bernoulli beam equation. The wave characteristics in the presence of the elastic bed is analyzed in both the cases of deep and shallow water waves. Further, the linearized long wave equation is generalized to include bottom flexibility. A generalized expansion formula for the velocity potential is derived to deal with the boundary value problems associated with surface gravity waves having an elastic bed. The utility of the expansion formula is illustrated by demonstrating specific physical problems which will play significant role in the analysis of wave structure interaction problems. Behavior of the wave spectra are discussed in the case of closed basin having a free surface and an elastic bottom topography.     

Hydrodynamic characteristics of pile supported skirt breakwater models

Facilities for handling large draft vessels, modern container ships and tankers have become essential due to the rapid growth in marine traffic. A pile supported skirt breakwater (PSSB) is one of the most promising concepts that could fulfill this requirement as PSSBs are environment friendly and economical for locations where tidal fluctuations are large and soil conditions are poor compared to other types of conventional breakwaters. The structure consists of an impermeable barrier piercing the free surface and extending up to a certain depth of submergence. The barrier which is responsible for attenuating the incident waves through partial reflection is supported on closely spaced concrete or steel piles. The barrier would consist of pre-cast elements that are connected to the piles on site. A numerical model based on the Eigen function expansion theory for linear waves to investigate the reflection and transmission characteristics of a PSSB consisting of single and double rows has been developed. The wave run-up on the skirt of the front row as well as the oscillation of the water surface in between the two rows was also computed. The results on the above stated parameters are reported as a function of wave and structural parameters in a dimensionless form. The numerical results are compared with experimental results and the agreement in general is found to be good.     

An active wave generating–absorbing boundary condition for VOF type numerical model

The objective of the present work is to discuss the implementation of an active wave generating–absorbing boundary condition for a numerical model based on the Volume Of Fluid (VOF) method for tracking free surfaces. First an overview of the development of VOF type models with special emphasis in the field of coastal engineering is given. A new type of numerical boundary condition for combined wave generation and absorption in the numerical model VOFbreak² is presented. The numerical boundary condition is based on an active wave absorption system that was first developed in the context of physical wave flume experiments, using a wave paddle. The method applies to regular and irregular waves. Velocities are measured at one location inside the computational domain. The reflected wave train is separated from the incident wave field in front of a structure by means of digital filtering and subsequent superposition of the measured velocity signals. The incident wave signal is corrected, so that the reflected wave is effectively absorbed at the boundary. The digital filters are derived theoretically and their practical design is discussed. The practical use of this numerical boundary condition is compared to the use of the absorption system in a physical wave flume. The effectiveness of the active wave generating–absorbing boundary condition finally is proved using analytical tests and numerical simulations with VOFbreak².     

RANS modelling applied to random wave interaction with submerged permeable structures

This paper is the second part of the work presented by Garcia et al. [Garcia, N., Lara, J.L., Losada, I.J., 2004. 2-D numerical analysis of near-field flow at low-crested breakwaters. Coastal Engineering 51 (10), 991–1020]. In the mentioned paper, flow conditions at low-crested rubble-mound breakwaters under regular wave attack were examined, using a combination of measured data of free surface, bottom pressure and fluid velocities from small-scale experiments and numerical results provided by a VOF-type model (COBRAS) based on the Reynolds-Averaged Navier–Stokes (RANS) equations. This paper demonstrates the capability of the COBRAS model to reproduce irregular wave interaction with submerged permeable breakwaters. Data provided by the numerical model are compared to experimental data of laboratory tests, and the main processes of wave–structure interaction are examined using both experimental and numerical results. The numerical model validation is carried out in two steps. First, the procedure of irregular wave generation is verified to work properly, comparing experimental and numerical data of different cases of irregular wave trains propagating over a flat bottom. Next, the validation of the numerical model for wave interaction with submerged rubble-mound breakwaters is performed through the simulation of small-scale laboratory tests on different incident wave spectra. Results show that the numerical model adequately reproduces the main aspects of the interaction of random waves with submerged porous breakwaters, especially the spectral energy decay at the structure and the spectrum broadening past the structure. The simulations give good results in terms of height envelopes, mean level, spectral shape, root-mean-square height for both free surface displacement and dynamic pressure inside the breakwater. Moreover, large-scale simulations have been conducted, on both regular and irregular incident wave conditions. The overall pattern of the wave interaction with a large-scale submerged breakwater is adequately reproduced by the numerical model. The processes of wave reflection, shoaling and breaking are correctly captured. The good results achieved at a near prototype scale are promising regarding the use of the numerical model for design purposes.     

Water wave interaction with a floating porous cylinder

The interaction of water waves with a freely floating circular cylinder possessing a side-wall that is porous over a portion of its draft is investigated theoretically. The porous side-wall region is bounded top and bottom by impermeable end caps thereby resulting in an enclosed fluid region within the structure. The problem is formulated based on potential flow and linear wave theory and assuming small-amplitude structural oscillations. An eigenfunction expansion approach is then used to obtain semi-analytical expressions for the hydrodynamic excitation and reaction loads on the structure. Numerical results are presented which illustrate the effects of the various wave and structural parameters on these quantities. It is found that the permeability, size and location of the porous region may have a significant influence on the horizontal components of the hydrodynamic excitation and reaction loads, while its influence on the vertical components in most cases is relatively minor.     

Wave–ice interactions in the marginal ice zone. Part 2: Numerical implementation and sensitivity studies along 1D transects of the ocean surface

The theoretical foundation of a wave–ice interaction model is reported in Part 1 of this study. The model incorporates attenuation of ocean surface waves by sea ice floes and the concomitant breaking of the floes by waves that determines the structure of the marginal ice zone (MIZ). A numerical implementation of the method is presented here. Convergence of the numerical method is demonstrated, as temporal and spatial grids are refined. A semi-analytical method, which does not require time-stepping, is also developed to validate the numerical results, when dispersion is neglected. The wave energy lost during ice breakage is parameterized, as part of the numerical method. Sensitivity studies are conducted in relation to the energy loss and also dispersive effects, the choice of the attenuation model, the properties of the wave field, and sea ice properties such as concentration, thickness and breaking strain. Example simulations intended to represent conditions in the Fram Strait in 2007, which exploit reanalyzed wave and ice model data, are shown to conclude the results section. These are compared to estimates of MIZ widths based on a concentration criteria, and obtained from remotely-sensed passive microwave images.     

Interaction of oblique waves with infinite number of perforated caissons

An analytic solution based on the division of the fluid domain is developed for the interaction of obliquely incident waves with infinite number of perforated caissons. The whole fluid domain is firstly divided into infinite sub-domains according to the division of structures, and subsequently eigenfunction expansion is employed to represent the velocity potential in each domain. A phase relation is utilized for the analysis of wave oscillation in each caisson, and the character of structure geometry is considered in setting up the mathematical model of reflection waves. The reflection waves from the present analysis include many propagation waves traveling in different directions when the incident wave frequency is high. Benchmark examinations show that the continuous condition of water particle velocity is satisfied at the front walls of caissons, and the reflection coefficients keep agreement with the energy conservation relation very well when porous effect parameter is infinite. Numerical results show that the reflection coefficients of obliquely incident waves are smaller when the length of caissons is shorter at low frequency. The wave reflection coefficients and the wave forces normal to caissons decrease and the wave forces along caissons increase with the increase of the wave incident angle.     

Exact asymmetric slope distributions in stochastic Gauss–Lagrange ocean waves

The stochastic Lagrange wave model is a realistic alternative to the Gaussian linear wave model, which has been successfully used in ocean engineering for more than half a century. This paper presents exact slope distributions and other characteristic distributions at level crossings for symmetric and asymmetric Lagrange space and time waves. These distributions are given as expectations in a multivariate normal distribution, and they have to be evaluated by simulation or numerical integration. Interesting characteristic variables are: slopes obtained by asynchronous sampling in space or time, slopes in space or time, and horizontal particle velocity, when waves are observed when the water level crosses a predetermined level.     

Numerical study for vegetation effects on coastal wave propagation by using nonlinear Boussinesq model

The vegetation has important impacts on coastal wave propagation. In the paper, the sensitivities of coastal wave attenuation due to vegetation to incident wave height, wave period and water depth, as well as vegetation configurations are numerically studied by using the fully nonlinear Boussinesq model. The model is based on the implementation of drag resistances due to vegetation in the fully nonlinear Boussinesq equation where the drag resistance is provided by the Morison’s formulation for rigid structure induced drag stresses. The model is firstly validated by comparing with the experimental results for wave propagation in vegetation zones. Subsequently, the model is used to simulate waves with different height, period propagating on vegetation zones with different water depth and vegetation configurations. The sensitivities of wave attenuation to incident wave height, wave period, water depth, as well as vegetation configurations are investigated based on the numerical results. The numerical results indicate that wave height attenuation due to vegetation is sensitive to incident wave height, wave period, water depth, as well as vegetation configurations, and attenuation ratio of wave height is increased monotonically with increases of incident wave height and decreases of water depth, while it is complex for wave period. Moreover, more vegetation segments can strengthen the interaction of vegetation and wave in a certain range.