LINEAR WAVES IN THE ROTATIONAL HOMOGENEOUS FLUID WITH UNIFORM WATER DEPTH (Ⅰ)

Linear wave equations for incompressible ideal homogeneous fluid are derived without making the assumptions of irrotation and hydrostatic pressure. The obtained equations are suitable for arbitrary bottom topography. Unified solutions of the existing waves in uniform-depth waters are found from those equations. Discussions about the above assumptions are made. Magnitude order of the error caused by the assumption of hydrostatic pressure is given.     

A Unique Solvable Higher Order BEM for Wave Diffraction and Radiation

- For the discretization of higher order elements, this paper presents a modified integral domain method to remove the irregular frequencies inherited in the integral equation of wave diffraction and radiation from a surface-piercing body. The set of over-determined linear equations obtained from the method is modified into a normal set of linear equations by superposing a set of linear equations with zero solutions. Numerical experiments have also been carried out to find the optimum choice of the size of the auxiliary domain and the discretization on it.     

Nonlinear tidal waves in a kind of estuary with gradually varying cross-section

-Nonlinear tidal waves in a kind of estuary are studied in the paper using one-dimensional nonlinear hydrody-namic equations with friction. The estuary has exponentially varying width B=B0 e-bx and uniform depth h. The one-dimensional hydrodynamic equations are solved by perturbation method. It was found that our solution included two special cases, Pelisenpeki's solution and Airy's solution. The former can be got by letting b=0 in our solutions, and the latter by setting 6 = 0 and f= 0 (f is linear frictional coefficient). In terms of the second-order solution, the physical mechanism of nonlinear tidal waves in estuaries with gradually varying cross-section is explored. It is shown that, under the assumption of linear friction coefficient, shallow water constituent waves consist of two parts, one is produced by shallow water nonlinear effect outside the estuary, the other is generated by shallow water nonlinear effect inside estuary. In addition, the physical mechanism of the residual tidal current and     

Hydrodynamic Analysis of Elastic Floating Collars in Random Waves

As the main load-bearing component of fish cages, the floating collar supports the whole cage and undergoes large deformations. In this paper, a mathematical method is developed to study the motions and elastic deformations of elastic floating collars in random waves. The irregular wave is simulated by the random phase method and the statistical approach and Fourier transfer are applied to analyze the elastic response in both time and frequency domains. The governing equations of motions are established by Newton’s second law, and the governing equations of deformations are obtained based on curved beam theory and modal superposition method. In order to validate the numerical model of the floating collar attacked by random waves, a series of physical model tests are conducted. Good relationship between numerical simulation and experimental observations is obtained. The numerical results indicate that the transfer function of out-of-plane and in-plane deformations increase with the increasing of wave frequency. In the frequency range between 0.6 Hz and 1.1 Hz, a linear relationship exists between the wave elevations and the deformations. The average phase difference between the wave elevation and out-of-plane deformation is 60° with waves leading and the phase between the wave elevation and in-plane deformation is 10° with waves lagging. In addition, the effect of fish net on the elastic response is analyzed. The results suggest that the deformation of the floating collar with fish net is a little larger than that without net.     

Numerical Wave Channel with Absorbing Wave-Maker

The numerical wave channel has been developed based on the volume of fluid method (VOF) in conjunction with the Navier-Stokes equations. The absorbing wave-maker boundary on the left side of the channel is presented by prescribing velocity reference to linear wave-maker theory. The principle of which is that the numerical wave-maker is designed to move in a way that generates the required incident wave and cancels out any reflected wave that reach it at the same time. On the right side of the channel, the open boundary is set to permit incident waves to be transmitted freely. The parametric studies have been carried out at a range of ratios of water depth to wave length d/ L from 0.124 to 0.219, with wave height in the front of paddle/water depth ratio (H0 / d) from 0.1 to 0.3. Wave height, wave pressure distribution along the channel and velocity field are obtained for both open boundary condition and reflective boundary condition at the other end of the channel. For a reflective case, it is shown that     

Analytical Research on Wave Diffraction on Arc-Shaped Bottom-Mounted Perforated Breakwaters

An analytical method is developed to study wave diffraction on arc-shaped and bottom-mounted perforated breakwaters.The breakwater is assumed to be rigid,thin,vertical,immovable and located in water of constant depth.The fluid domain is divided into two regions by imaginary interface.The velocity potential in each region is expanded by eigenfunctions.By satisfying the continuity of pressure and normal velocity across the imaginary fluid interface,a set of linear algebraic equations can be obtained to determine the unknown coefficients of eigenfunctions.Numerical results,in the form of contour maps of the relative wave amplitude around the breakwater,are presented for a range of wave and breakwater parameters.Results show that the wave diffraction on the arc-shaped and bottom-mounted perforated breakwater is related to the incident wavelength and the porosity of the breakwater.The porosity of the perforated breakwater may have great effect on the diffracted field.     

Influence of the earth rotation on the interfacial wave solutions and wave-induced tangential stress in a two-layer fluid system

Interfacial waves and wave-induced tangential stress are studied for geostrophic small amplitude waves of two-layer .uid with a top free surface and a .at bottom. The solutions were deduced from the general form of linear .uid dynamic equations of two-layer .uid under the f -plane approximation, and wave-induced tangential stress were estimated based on the solutions obtained. As expected, the solutions derived from the present work include as special cases those obtained by Sun et al. (2004. Science in China, Ser. D, 47(12): 1147–1154) for geostrophic small amplitude surface wave solutions and wave-induced tangential stress if the density of the upper layer is much smaller than that of the lower layer. The results show that the interface and the surface will oscillate synchronously, and the in.uence of the earth’s rotation both on the surface wave solutions and the interfacial wave solutions should be considered.     

PROPAGATION AND TRANSFORMATION OF NON-LINEAR WAVES ON UNIFORMLY SLOPING BEACHES Ⅰ. A SOLUTION FOR NON-LINEAR PROGRESSING WAVES

Two-dimensional non-linear hydrodynamical equations are solved by using perturbation method and treating slopping beaches as bottom boundary conditions so that a kind of solution for nonlinear progressing waves is obtained. The first order of approximation is the same potential function as used by Biesel, and the second order is calculated numerically. Based on the solution, wave characteristics before breaking, especially the wave set-down, are discussed. It turns out that for the whole course of waves propagating from deep to shallow waters the theory proposed in this paper has a wider valid range of application than others.     

Influence of the earth rotation on the interfacial wave solutions and wave-induced tangential stress in a twolayer fluid system

Interfacial waves and wave-induced tangential stress are studied for geostrophic small amplitude waves of two-layer fluid with a top free surface and a flat bottom. The solutions were deduced from the general form of linear fluid dynamic equations of two-layer fluid under the f-plane approximation, and wave-induced tangential stress were estimated based on the solutions obtained. As expected, the solutions derived from the present work include as special cases those obtained by Sun et al. (2004. Science in China, Ser. D, 47(12):1147-1154) for geostrophic small amplitude surface wave solutions and wave-induced tangential stress if the density of the upper layer is much smaller than that of the lower layer. The results show that the interface and the surface will oscillate synchronously, and the influence of the earth''s rotation both on the surface wave solutions and the interfacial wave solutions should be considered.     

Dynamic Stability Analysis of Caisson Breakwater in Lifetime Considering the Annual Frequency of Severe Storm

In the dynamic stability analysis of a caisson breakwater, most of current studies pay attention to the motion characteristics of caisson breakwaters under a single periodical breaking wave excitation. And in the lifetime stability analysis of caisson breakwater, it is assumed that the caisson breakwater suffers storm wave excitation once annually in the design lifetime. However, the number of annual severe storm occurrence is a random variable. In this paper, a series of random waves are generated by the Wen Sheng-chang wave spectrum, and the histories of successive and long-term random wave forces are built up by using the improved Goda wave force model. It is assumed that the number of annual severe storm occurrence is in the Poisson distribution over the 50-year design lifetime, and the history of random wave excitation is generated for each storm by the wave spectrum. The response histories of the caisson breakwater to the random waves over 50-year design lifetime are calculated and taken as a set of samples. On the basis of the Monte Carlo simulation technique, a large number of samples can be obtained, and the probability assessment of the safety of the breakwater during the complete design lifetime is obtained by statistical analysis of a large number of samples. Finally, the procedure of probability assessment of the breakwater safety is illustrated by an example.     

Wave diffraction on arc-shaped floating perforated breakwaters

An analytical method is developed to study the sheltering effects on arc-shaped floating perforated breakwaters. In the process of analysis, the floating breakwater is assumed to be rigid, thin, vertical, and immovable and located in water with constant depth. The fluid domain is divided into two regions by imaginary interface. The velocity potential in each region is expanded by eigenfunction in the context of linear theory. By satisfying continuity of pressure and normal velocity across the imaginary fluid interface, a set of linear algebraic equations can be obtained to determine the unknown coefficients for eigenfunction expansions. The accuracy of the present model was verified by a comparison with existing results for the case of arc-shaped floating breakwater. Numerical results, in the form of contour maps of the non-dimensional wave amplitude around the breakwater and diffracted wave amplitude at typical sections, are presented for a range of wave and breakwater parameters. Results show that the sheltering effects on the arc-shaped floating perforated breakwater are closely related to the incident wavelength, the draft and the porosity of the breakwater.     

Numerical Method for Wave Forces Acting on Partially Perforated Caisson

The perforated caisson is widely applied to practical engineering because of its great advantages in effectively wave energy consumption and cost reduction. The attentions of many scientists were paid to the fluid–structure interaction between wave and perforated caisson studies, but until now, most concerns have been put on theoretical analysis and experimental model set up. In this paper, interaction between the wave and the partial perforated caisson in a 2D numerical wave flume is investigated by means of the renewed SPH algorithm, and the mathematical equations are in the form of SPH numerical approximation based on Navier–Stokes equations. The validity of the SPH mathematical method is examined and the simulated results are compared with the results of theoretical models, meanwhile the complex hydrodynamic characteristics when the water particles flow in or out of a wave absorbing chamber are analyzed and the wave pressure distribution of the perforated caisson is also addressed here. The relationship between the ratio of total horizontal force acting on caisson under regular waves and its influence factors is examined. The data show that the numerical calculation of the ratio of total horizontal force meets the empirical regression equation very well. The simulations of SPH about the wave nonlinearity and breaking are briefly depicted in the paper, suggesting that the advantages and great potentiality of the SPH method is significant compared with traditional methods.     

Nonlinear waves in a fluid-filled thin viscoelastic tube

In the present paper the propagation property of nonlinear waves in a thin viscoelastic tube filled with incompressible inviscid fluid is studied. The tube is considered to be made of an incompressible isotropic viscoelastic material described by Kelvin--Voigt model. Using the mass conservation and the momentum theorem of the fluid and radial dynamic equilibrium of an element of the tube wall, a set of nonlinear partial differential equations governing the propagation of nonlinear pressure wave in the solid--liquid coupled system is obtained. In the long-wave approximation the nonlinear far-field equations can be derived employing the reductive perturbation technique (RPT). Selecting the exponent α of the perturbation parameter in Gardner--Morikawa transformation according to the order of viscous coefficient η, three kinds of evolution equations with soliton solution, i.e. Korteweg--de Vries (KdV)--Burgers, KdV and Burgers equations are deduced. By means of the method of traveling-wave solution and numerical calculation, the propagation properties of solitary waves corresponding with these evolution equations are analysed in detail. Finally, as a example of practical application, the propagation of pressure pulses in large blood vessels is discussed.     

ANALYTICAL SOLUTION FOR THE VELOCITY POTENTIAL OF LINEAR WAVES OVER SLOPING BEACHES

An analytical solution for the velocity potential of linear waves traveling over sloping beaches is obtained in the present paper, the restriction to the solution, i. e. the cube of bottom slope a being negligible. When the terms of order of a2 are neglected, the solution is the same as that presented by Biesel in 1951[1]. When the terms of order of a2 are retained, the wave dispersion relation is corrected. Forthermore, the solution corrected to any higher order of a can be obtained without difficulty by means of the disturbation method given by this paper".     

Generation and Properties of Freak Waves in A Numerical Wave Tank

Freak waves are generated based on the mechanism of wave focusing in a 2D numerical wave tank. To set up the nonlinear numerical wave tank, the Boundary Element Method is used to solve potential flow equations incorporated with fully nonlinear free surface boundary conditions. The nonlinear properties of freak waves, such as high frequency components and wave profile asymmetry, are discussed. The kinematic data, which can be useful for the evaluation of the wave forces exerted on structures to avoid underestimation of linear predictions, are obtained, and discussed, from the simulated results of freak waves.     

Fully Coupled Simulation of Interactions Among Waves,Permeable Breakwaters and Seabeds Based on N−S Equations

Interstitial flows in breakwater cores and seabeds are a key consideration in coastal and marine engineering designs and have a direct impact on their structural safety.In this paper,a unified fully coupled model for wave?permeable breakwater?porous seabed interactions is built based on an improved N?S equation.A numerical wave flume is constructed,and numerical studies are carried out by applying the finite difference method.In combination with a physical model test,the accuracy of the numerical simulation results is verified by comparing the calculated and measured values of wave height at measurement points and the seepage pressure within the breakwater and seabed.On this basis,the characteristics of the surrounding wave field and the internal flow field of the pore structure,as well as the evolution process of the fluctuating pore water pressure inside the breakwater and seabed,are further analyzed.The spatial distribution of the maximum fluctuating pore water pressure in the breakwater is compared between two cases by considering whether the seabed is permeable,and then the effect of seabed permeability on the dynamic pore water pressure in the breakwater is clarified.This study attempts to provide a reference for breakwater design and the protection of nearby seabeds.     

VIBRATIONAL ANALYSIS OF FREE FLOATING CYLINDRICAL OFFSHORE PLATFORMS

- Based on small body assumption, linear and nonlinear vibrational analyses have been done for the free floating cylindrical offshore platform under the action of regular waves. The analytical solutions to the coupled linear equations" of motion are deduced which provide an elementary tool for the preliminary engineering design. In the nonlinear case, the exceptional frequencies associated with large amplitude subharmonic oscillation are studied. By means of the method presented in this paper, the results of calculation for a big cylindrical offshore platform show good agreement with the model test data obtained at the Netherlands Ship Model Basin.     

Numerical Model of Cnoidal Wave Flume

The volume of fluid(VOF)method is used to set up a wave flume with an absorbing wavemaker of cnoidal waves.Based on the transfer function between wave surface and paddle velocity obtained bythe shallow water wave theory,the velocity boundary condition of an absorbing wave maker is introduced toabsorb reflected waves that reach the numerical wave maker.For H/d ranging from 0.1 to 0.59 and T(g/d)~(1/2)from 7.9 to 18.3,the parametric studies have been carried out and compared with experiments.     

One-Dimensional Horizontal Boussinesq Model Enhanced for Non-Breaking and Breaking Waves

Based on a set of fully nonlinear Boussinesq equations up to the order of O（μ^2, ε^3μ^2） （where ε is the ratio of wave amplitude to water depth and ,μ is the ratio of water depth to wave length） a numerical wave model is formulated. The model＇s linear dispersion is acceptably accurate to μ ≌ 1.0, which is confirmed by comparisons between the simulat- ed and measured time series of the regular waves propagating on a submerged bar. The moving shoreline is treated numer- ically by replacing the solid beach with a permeable beach. Run-up of nonbreaking waves is verified against the analytical solution for nonlinear shallow water waves. The inclusion of wave breaking is fulfilled by introducing an eddy term in the momentum equation to serve as the breaking wave force term to dissipate wave energy in the surf zone. The model is applied to cross-shore motions of regular waves including various types of breaking on plane sloping beaches. Comparisons of the model test results comprising spatial distribution of wave height and mean water level with experimental data are presented.