Reflection of regular and irregular waves from a partially perforated caisson breakwater with a rock-filled core

The reflection of regular and irregular waves from a partially perforated caisson breakwater with a rock-filled core is examined. The present mathematical model is developed by means of the matched eigenfunction method. Numerical results of the present model are compared with the experimental data of different researchers. Numerical examples are given to examine the effect of rock fill on the reflection coefficient. The differences between regular and irregular waves are also investigated by means of theoretical and experimental results. It is found that the minimum reflection coefficient of irregular waves is larger than that of corresponding regular waves, but the contrary is the case for the maximum reflection coefficient.

Experimental Investigation of Wave Load and Run-up on the Composite Bucket Foundation Influenced by Regular Waves

In the design of wind turbine foundations for offshore wind farms, the wave load and run-up slamming on the supporting structure are the quantities that need to be considered. Because of a special arc transition, the interaction between the wave field and the composite bucket foundation(CBF) becomes complicated. In this study, the hydrodynamic characteristics, including wave pressure, load, upwelling, and run-up, around the arc transition of a CBF influenced by regular waves are investigated through physical tests at Shandong Provincial Key Laboratory of Ocean Engineering, Ocean University of China. The distributions of the wave pressures and upwelling ratios around the CBF are described, and the relationship between the wave load and the wave parameters is discussed. New formulae based on the velocity stagnation head theory with linear wave theory and the second-order Stokes wave theory for wave kinematics are proposed to estimate the wave run-up. Moreover, the multiple regression method with nonlinear technology is employed to deduce an empirical formula for predicting run-up heights. Results show that the non-dimensional wave load increases with the increase in the values of the wave scattering parameter and relative wave height. The wave upwelling height is high in front of the CBF and has the lowest value at an angle of 135? with the incoming wave direction. The performance of the new formulae proposed in this study is compared using statistical indices to demonstrate that a good fit is obtained by the multiple regression method and the analytical model based on the velocity stagnation head theory is underdeveloped.

FORCES ON CYLINDERS IN COEXISTING FIELD OF REGULAR WAVES AND CURRENTS

Based on the linear wave theory, the lift force on the cylinder under the action of both regular waves and currents related to inline force and the resultant force has been investigated. The relationship between the hydrodynamic coefficients of resultant force Cf, of drag force Cd, of inertia force Cm and of lift force CL and the redefined KC number is reported. It is indicated that in a certain region of KC number, the influence of lift force on the resultant force can not be ignored.

Hydrodynamic Coefficient Investigation on a Partial Permeable Stepped Breakwater Under Regular Waves

Traditional breakwater takes the advantage of high protection performance and has been widely used.However,it contributes to high wave reflection in the seaside direction and poor water exchange capacity between open seawater and an inside harbor.Consequently,a partially permeable stepped breakwater(PPSB)is proposed to ensure safety and good water exchange capacity for an inside harbor,and a 3-D computational fluid dynamics(CFD)mathematical model was used to investigate the hydrodynamic coefficients using Reynolds-Averaged Navier-Stokes equations,Re-Normalization Group(RNG)k-εequations,and the VOF technique.A series of experiments are conducted to measure the wave heights for validating the mathematical model,and a series of dimensionless parameters considering wave and PPSB effects were presented to assess their relationships with hydrodynamic coefficients,respectively.With the increase in the reciprocal value of PPSB slope,incident wave steepness and permeable ratio below still water level(SWL),the wave reflection coefficient decreases.The wave transmission coefficient decreases with an increase in the reciprocal value of the PPSB slope and incident wave steepness;however,it increases with the increase in the permeable ratio below SWL.With increases in the reciprocal value of the PPSB slope,permeable ratio below SWL and incident wave steepness for relatively high wave period scenarios,the wave energy dissipation coefficient increases;however,it decreases slightly with increases in the incident wave steepness for the smallest wave period scenarios.Furthermore,simple prediction formulas are conducted for predicting the hydrodynamic coefficients and they are well validated with the related data.

Experimental Researches on Reflective and Transmitting Performances of Quarter Circular Breakwater Under Regular and Irregular Waves

A series of regular and irregular wave experiments are conducted to study the reflective and transmitting performances of quarter circular breakwater （QCB） in comparison with those of semi-circular breakwater （SCB）. Based on regular wave tests, the reflection and transmission characteristics of QCB are analyzed and a few influencing factors are investigated. Then, the wave energy dissipation as wave passing over the breakwater is discussed based on the hydraulic coefficients of QCB and SCB. In irregular wave experiments, the reflection coefficients of QCB and their spectrums are studied. Finally, the comparisons between the experimental results and numerical simulations for QCB under regular and irregular wave conditions are presented.

Impact Pressure of Incident Regular Waves and Irregular Waves on the Subface of Open-Piled Structures

This paper presents the results of comparison of impact pressures on open-plied structures induced by regular waves and irregular waves in a laboratory channel. Regular waves with wave heights ranging from 0.1～0.2 m and periods ranging from 1.0～2.0 s are tested. The target spectrum for the irregular wave is JONSWAP spectrum. Irregular waves with significant wave heights in the range of 0.10～0.25 m and peak periods in the range of 1.0～2.0 s are tested. The relative clearance s/H 1/3(H) is between -0.1 and 0.4, s being the subface level of structure model above the still water level. Time series of impact pressure are analyzed to indicate whether the property of impact pressures induced by the regular wave significantly deviates from that by the irregular wave. The distribution of the impact pressure along the underside of the structure is compared for different types of incident waves. The effects of wave parameters, structure dimension and structure clearance on the impact pressure are also discussed.

COMPARISON OF SCOURS IN FRONT OF VERTICAL BREAKWATERS DUE TO REGULAR AND IRREGULAR WAVES

In this paper the scours of sand bed due to regular and irregular waves in front of vertical breakwaters, such as scouring patterns, scouring mechanisms, etc., are investigated. Thereby, the scours under the action of the two kinds of waves are compared connected.

NUMERICAL SIMULATION OF TWO-DIMENSIONAL OVERTOPPING AGAINST SEAWALLS ARMORED WITH ARTIFICIAL UNITS IN REGULAR WAVES

A new mathematical model for the overtopping against seawalls armored with artificial units in regular waves was established. The 2-D numerical wave flume, based on the Reynolds Averaged Navier-Stokes （RANS） equations and the standard κ-ε turbulence model, was developed to simulate the turbulent flows with the free surface, in which the Volume Of Fluid （VOF） method was used to handle the large deformation of the free surface and the relaxation approach of combined wave generation and absorbing was implemented. In order to consider the effects of energy dissipation due to the armors on a slope seawall, a porous media model was proposed and implemented in the numerical wave flume. A series of physical model experiments were carried out in the same condition of the numerical simulation to determine the drag coefficient in the porous media model in terms of the overtopping discharge. Compared the computational value of overtopping over the seawall with the experimental data, the values of the effective drag coefficient was calibrated for the layers of blocks at different locations along the seawalls.

Oxygen transfer characteristics of bubbly jet in regular waves

The dissolved oxygen level is an important index of the water environment,and in this paper,the oxygen transfer of the bubbly jet in regular waves is investigated numerically and experimentally.The Reynolds-averaged Navier-Stokes equations,the re-normalisation group k-e equations,and the volume of fluid(VOF)technique are used along with a 2-D CFD model to simulate the wave and bubble motions as well as the turbulence,and a dissolved oxygen transport equation is used to model the oxygen transfer behavior both through the bubbly interface and the wave surface.A series of experiments are conducted to validate the mathematical model,with good agreement.In addition,a group of dimensionless parameters are defined from the wave parameter and the aeration parameter,and their relationships with the total oxygen transfer coefficient are explored.Furthermore,the dimensional analysis and the least squares methods are used to derive simple prediction formulas for the total oxygen transfer coefficient,and they are validated with the related experimental data.

Laboratory Study on the Interaction Between Regular Obliquely Incident Waves and Vertical Walls

The characteristics of wave forces are studied based on physical model tests with regular waves. The ratio of obliquely incident wave forces to normally incident wave forces on unit length of a vertical wall is related with various factors. A linear reduction of the mean force of obliquely incident waves is confirmed with an increase in the relative caisson length. Also the characteristics of reflection coefficient of diagonal waves are discussed.

Separation Method of Bi-directional Reflected Waves and Oblique Incident Regular Waves

Wave reflection is one of the key problems afecting wave simulation quality in ocean engineering basin. The deep ocean engineering basin is equipped with two-sided segmented wavemakers and two wave absorbing beaches, which are located opposite to wave generators to reduce wave reflection efects. When an oblique long- crested wave is made by two-sided segmented wavemakers in a wave basin, two bi-directional reflected waves with the same azimuth but opposite propagation directions are generated. According to this feature, based on the two-point approach developed by Goda, a method to separate an incident regular wave from two bi-directional reflected waves using three wave gauges is proposed. The validity of this method is proved by numerical composite waves. The results indicate that the method can separate incident wave from reflected waves efectively. The method can be used to determine the reflection coefcient and verify the capacity of wave absorbing beaches in deep ocean engineering basin.

Experimental observation on water entry of a sphere in regular wave

This paper presents a novel experiment to observe the whole water entry process of a free-falling sphere into a regular wave.A time-accurate synchronizing system modulates the moment elaborately to ensure the sphere impacting onto the water surface at the desirable wave phase.Four high-speed cameras focus locally to measure the high-precision size of the cavity evolution.Meanwhile,the aggregated field view of the camera array covers both the splash above the free surface and the entire cavity in the wave.The detailed methodologies are described and verified for the hardware set-up and the image post-processing.The theoretical maximum deviation is 1.7%on the space scale.The integral morphology of the cavity is captured precisely in the coordinate system during the sphere penetrates through the water at four representative wave phases and the still water.The result shows that the horizontal velocity of the fluid particle in the wave impels the cavity and changes the shape distinctly.Notably,the wave motion causes the cavity to pinch offearlier at the wave trough phase and later at the wave crest phase than in the still water.The wave motion influences the falling process of the sphere slightly in the present parameters.

Experimental Study of Hydrodynamic Loading on Submarine Composite Pipelines under the Action of Waves and Current

This article is aimed at the experimental study of the hydrodynamic loading induced by waves and current acting on the JZ 20-2 submarine composite pipelines. The corresponding hydrodynamic coefficients obtained in this experimental study can be applied to engineering projects.

Meshless Method with Domain Decomposition for Submerged Porous Breakwaters in Waves

Based on the improved version of the meshless singular boundary method(ISBM)in multi domain(MD),a numerical method is proposed in this paper to study the interaction of submerged permeable breakwaters and regular waves at normal incidence.To account for fluid flow inside the porous breakwaters,the conventional model of Sollitt and Cross for porous media is adopted.Both single and dual trapezoidal breakwaters are examined.The physical problem is formulated in the context of the linear potential wave theory.The domain decomposition method(DDM)is employed,in which the full computational domain is decomposed into separate domains,that is,the fluid domain and the domains of the breakwaters.Respectively,appropriate mixed type boundary and continuity conditions are applied for each subdomain and at the interfaces between domains.The solution is approximated in each subdomain by the ISBM.The discretized algebraic equations are combined,resulting in an overdetermined full system that is solved using a least-square solution procedure.The numerical results are presented in terms of the hydrodynamic quantities of reflection,transmission,and wave-energy dissipation.The relevance of the results of the present numerical procedure is first validated against data of previous studies,and then selected computations are discussed for various structural conditions.The proposed method is demonstrated to be highly accurate and computationally efficient.

Numerical Study of Submerged Vertical Plane Jets Under Progressive Water Surface Waves

When wastewater is discharged into a coastal area through an ouffall system, it will always be subjected to the action of waves. It is important to study and quantify the mixing of the discharge with the ambient water so that accurate environmental impact assessment can be made for such discharge conditions. The present work aims to study the phenomenon of a plane jet discharged into water environment with regular waves. A 3D numerical model based on the full Navier- Stokes equations （NSE） in the a-coordinate is developed to study the present problem. Turbulence effects are modeled by a subgrid-scale （SGS） model using the concept of large eddy simulation （LES）. The operator splitting method is used to solve the modified NSE. The model has been applied to the simulation of three different eases of submerged plane jets with surface waves： jet with strong waves, jet with weak waves and jet without waves. Numerical results show that the waves enhance the mixing of the jet with the ambient fluid, and cause a periodic deflection of the jet. The size of the recirculation is about 1.5- 2.4 h （water depth） . The velocity profile of the jet is serf-similar in the zone of established flow for both the pure jet and jet in wave circumstances. The spreading characteristic constant a is 0. 100 and 0. 105 for pure momentum jets with Re numbers 1025 and 2050. The value of a increases from 0. 130 to 0. 147 for a jet in weak and strong wave circumstances, showing that waves have an obvious effect on the mixing and dilution properties of jets. Numerical results are in good agreement with the experimental data for the cases of pure jets and jets with waves.

Numerical solutions to regularized long wave equation based on mixed covolume method

The mixed covolume method for the regularized long wave equation is devel- oped and studied. By introducing a transfer operator γh, which maps the trial function space into the test function space, and combining the mixed finite element with the finite volume method, the nonlinear and linear Euler fully discrete mixed covolume schemes are constructed, and the existence and uniqueness of the solutions are proved. The optimal error estimates for these schemes are obtained. Finally, a numerical example is provided to examine the efficiency of the proposed schemes.

Numerical Prediction of Regular Wave Breaking on Very Gentle Slopes

Regular wave deformation and breaking on very gentle slopes is calculated by Mixed-Eulerian-Lagrangian procedure. The velocity potentials and their normal derivatives on the boundary are calculated through the mixed 0-1 boundary element method. The wave elevation and the potentials of Lime-stepping integration are determined by the 2nd-order Taylor expansion at the nodes of free surface boundary elements. During calculation the x-coordinates of the free surface element nodes are supposed to remain unchanged, i.e. the partial derivatives of wave elevation and potentials with respect to x are considered as zero. The numerical results of asymmetric parameters of breaking waves are verified by experimental study. It is shown that when the wave asymmetry is weak, the maximum horizontal velocity of water particales occurs at the wave peak and, the average ratio of this maximum velocity to wave celerity is 0.96. However, when the wave asymmetry is strong, the maximum horizontal velocity of water particles occurs just before the wave crest, and the average ratio of the maximum velocity to wave celerity is about 0.98. The numerical results also show that the asymmetry of wave profiles affects the value of the wave breaking index (H/d) (b), that is, when the asymmetric characteristics are weak, the value of wave breaking index coincides with that given by Goda; on the contrary, when the asymmetry of wave profiles is notable, the value of wave breaking index is close to Nelson's result. The experimental study gives the same conclusions.

EXPONENTIAL ATTRACTOR FOR THE GENERALIZED SYMMETRIC REGULARIZED LONG WAVE EQUATION WITH DAMPING TERM

The global fast dynamics for the generalized symmetric regularized long wave equation with damping term is considered. The squeezing property of the nonlinear semi_group associated with this equation and the existence of exponential attractor are proved. The upper bounds of its fractal dimension are also estimated.

Experimental Investigation of the Response of Monopiles in Silty Seabed to Regular Wave Action

Excessive displacement responses of monopiles affect the serviceability of offshore structures.Related to complicated pile−seabed−wave interactions,the actual behavior of monopiles in silty seabed under periodic wave action remains unclear,and relevant studies in the literature are limited.A series of experiments were conducted in a wave flume containing single piles in silty seabed with relative density of 0.77 subjected to regular waves.Two stages of wave loading were applied successively,accompanied by data recording which included pore water pressure,water surface elevation,pile head displacement,and pile strain.Development of pile-head displacement and pore pressure in silty seabed was the main focus,but the effects of pile diameter,pile type,and pile stiffness were also investigated.The experimental results indicate that,in silty seabed,piles of large diameter or with fins accelerate soil liquefaction,resulting in strengthened soil which allows a higher upper boundary of pore pressure.Using fins at deeper locations led to a quick failure of the piles,but the opposite result was observed with an increase in fin dimensions.Once pile-head displacement entered its rapid development period,the wave load calculated via the pile moment was an overestimation,especially for the piles of large diameter.

An available formula of the sandy beach state induced by plunging waves

Laboratory experiments are performed to explore the response rule of a sandy beach profile under plunging wave on a non-uniform sediment-bed slope. The initial beach slope of combination of 1/10 and 1/20 is exposed to regular waves and cnoidal waves respectively. The free surface elevation, process of wave propagation, wave breaking, uprush and backwash and the change of a cross-shore beach profile are measured and recorded. The beach profile under the regular waves action exhibits two parts： a sandbar profile and a beach berm profile, and only one typical profile transformation under the cnoidal waves action is obtained, which is the beach berm profile. In the laboratory experiments, it is found that the beach states under wave action related to the previous factors. In addition, they are related to the characteristic of breaking waves such as the breaking intensity of the plunging wave. A concept about the characteristic angle of the plunging wave has been put forward through the observation and analysis of the phenomenon of the laboratory experiment. A qualitative analysis about the sediment transport carrying by currents generated from the plunging wave and the state of beach profile under the wave action has been done. The quantitative analysis about the relationship between the characteristic angle and Irribarren number has been done. An available formula of equilibrium states for the sandy beach induced by the plunging wave has been established based on the relationship between Irribarren number and the beach profile. By fitting these experimental results and others＇ experimental results to three lines, the three fitting coefficients can be calculated in their formula respectively. The recommended empirical formulas can divide three states of a beach morphology profile obviously, which include a depositive beach, an erosive beach and an intermediate beach.