On waves propagating over poro-elastic seabed

In this study, an analytical solution is developed for the problem of periodic waves propagating over a poro-elastic seabed of infinite depth. Water waves above the seabed are described using the linear wave theory. The poro-elastic seabed is modelled based on the Biot theory in which the inertia effect and Darcy's friction are added. Continuity of dynamic pressure and flow flux at the interfacial seabed surface are considered. Adopting an approach similar to Hsu et al. (1993), the governing equations for the pore pressure and displacements of the poro-elastic medium are derived. The present analytic solution compares favorably well with experimental results by Yamamoto et al. (1978), and analytical results by Song (1993) for the case of fine sand. Using the present theory, variations of the wavelength and fluid pressure caused by coupling of waves and the poro-elastic seabed are discussed. Results show that higher elasticity of the poro-elastic seabed induces larger interface pressure, but higher permeability causes smaller pressure on the seabed interface. The wave length is affected by the poro-elastic seabed and becomes shorter for softer seabed and shallower water depth.     

The calculation of zig-zag maneuver in regular waves with use of the impulse response functions

The traditional governing equations for sway–yaw maneuvering motion are a set of ordinary differential equations with constant coefficients. But, as is well known, integro–differential equations with impulse response functions are more strict governing equations that can handle the frequency dependence of hydrodynamic forces.In this paper, the two types of equation are compared and used to calculate the 10°–10° zig-zag maneuver in waves. Differences between the solutions are discussed.     

On waves propagating over a submerged poro-elastic structure

In this paper, the problem of incident waves propagating over a submerged poro-elastic structure is studied theoretically. A linear wave theory is used to describe the wave motion. The submerged poro-elastic structure is modeled based on Biot's theory, in which the fluid motion is described using the potential wave theory of Sollitt and Cross (1972). In the present approach, the problem domain is divided into four subregions. Using general solutions for each region and matching dynamic and kinematic conditions for neighboring regions, analytic solutions are derived for the wave fields and poro-elastic structure. The present analytic solutions compare very well with simplified cases of impermeable, rigid structures, and with those of porous structures. Using the present analytic solution, the effects of a poro-elastic submerged structure on waves are studied. The results show that softer poro-elastic structures can induce higher reflection and lower transmission from incident waves. For low permeability conditions, the elasticity of the structure can induce resonance, while higher permeability can depress the resonant effects.     

On response of structures to random waves

The solution to the non-linear, stochastic, ordinary differential equation governing the dynamic response of offshore structures to random waves, hitherto achieved by the statistical equivalent linearization method, is approached in a different manner. It is shown that the correlation function of the response is governed by a non-linear integral equation which can be solved successively. To the first order of approximation, the solution coincides with that of the statistical equivalent linearization method. This equation is solved up to the second order.     

Mild-slope equation for water waves propagating over non-uniform currents and uneven bottoms

I~IOXThe interaction between surface waves and ambient currents and nearshore topography lies atone of the heat of morphological medelling. Accurate predictions of how wave propagates overcurrents and topography, and of the consequent erosion and dePOSition of sand on a beach or tidalflat are vital when assessing how a coastline may be affected by changing conditions.The mild-slope equation was introduced by Berkhoff (1972) as a way of approximating therefraction-diffraction of linearized s…     

A coupled-mode system with application to nonlinear water waves propagating in finite water depth and in variable bathymetry regions

A non-linear coupled-mode system of horizontal equations is presented, modelling the evolution of nonlinear water waves in finite depth over a general bottom topography. The vertical structure of the wave field is represented by means of a local-mode series expansion of the wave potential. This series contains the usual propagating and evanescent modes, plus two additional terms, the free-surface mode and the sloping-bottom mode, enabling to consistently treat the non-vertical end-conditions at the free-surface and the bottom boundaries. The present coupled-mode system fully accounts for the effects of non-linearity and dispersion, and the local-mode series exhibits fast convergence. Thus, a small number of modes (up to 5–6) are usually enough for precise numerical solution. In the present work, the coupled-mode system is applied to the numerical investigation of families of steady travelling wave solutions in constant depth, corresponding to a wide range of water depths, ranging from intermediate depth to shallow-water wave conditions, and its results are compared vs. Stokes and cnoidal wave theories, as well as with fully nonlinear Fourier methods. Furthermore, numerical results are presented for waves propagating over variable bathymetry regions and compared with nonlinear methods based on boundary integral formulation and experimental data, showing good agreement.     

Wave deformation and vortex generation in water waves propagating over a submerged dike

The Navier–Stokes equations and the exact free surface boundary conditions are solved to simulate wave deformation and vortex generation in water waves propagating over a submerged dike. Incident waves are generated by a piston-type wavemaker set up in the computational domain. Numerical results are compared with experimental data in order to confirm the validity of the numerical model. The fast Fourier transform and a wave resolution technique are applied to decompose the transformed waves and the higher harmonics. Effects of different parameters on wave transformation and vortex generation are studied systematically. These parameters include the Ursell number, the Keulegan–Carpenter number, the water depth ratio, the Reynolds number, the length aspect ratio of the dike, and the type of dike.     

Vortex generation and evolution in water waves propagating over a submerged rectangular obstacle: Part II: Cnoidal waves

Vortex generation and evolution due to flow separation around a submerged rectangular obstacle under incoming cnoidal waves is investigated both experimentally and numerically. The Particle Image Velocimetry (PIV) technique is used in the measurement. Based on the PIV data, a characteristic velocity, phrased in terms of incoming wave height, phase speed, dimension of the obstacle, and a local Reynolds number are proposed to describe the intensity of vortex. The numerical model, which solves the two dimensional Reynolds Averaged Navier Stokes (RANS) equations, is used to further study the effects of wave period on the vortex intensity. Measurements for the mean and turbulent velocity fields further indicate that the time history of the intensity of fluid turbulence is closely related to that of the vortex intensity.     

Vortex generation and evolution in water waves propagating over a submerged rectangular obstacle: Part I. Solitary waves

Interactions between a solitary wave and a submerged rectangular obstacle are investigated both experimentally and numerically. The Particle Image Velocimetry (PIV) technique is used to measure the velocity field in the vicinity of the obstacle. The generation and evolution of vortices due to flow separation at the corners of the obstacle are recorded and analyzed. It is found that although the size of the vortex at the weatherside of the obstacle is smaller than that at the leeside, the turbulence intensity is, however, stronger. A numerical model, based on the Reynolds Averaged Navier–Stokes (RANS) equations with a k–ϵ turbulence model, is first verified with the measurements. Overall, the agreement between the numerical results and laboratory velocity measurements is good. Using the RANS model, a series of additional numerical experiments with different wave heights and different heights of the rectangular obstacle are then performed to test the importance of the energy dissipation due to the generation of vortices. The corresponding wave transmission coefficient, the wave reflection coefficient and the energy dissipation coefficient are calculated and compared with solutions based on the potential flow theory. As the height of the obstacle increases to D/h=0.7, the energy dissipation inside the vortices can reach nearly 15% of the incoming wave energy.     

On some uncertainties related to the short term stochastic modelling of ocean waves

A short-term model for the representation of ocean waves is considered. The model is described and the associated uncertainties are briefly reviewed. Emphasis is placed on a discussion of the validity of the basic assumptions regarding stationarity and Gaussianity, together with an elaboration on uncertainties related to the shape of the spectral density function. The basis for the present investigation is 4586 time series for the sea surface elevation, representing a rather wide range of different sea states. Conclusions presented subsequently rest to some extent on the assumption that the actual time series reproduce the true time fluctuations of the surface with a sufficient accuracy. A verification of this assumption requires a very extensive investigation, and could not be included within the scope of this work.     

稳恒水波的Fourier近似解研究

A computational method for steady water waves is presented on the basis of potential theory in the physical plane with spatial variables as independent quantities. The finite Fourier series are applied to approximating the free surface and potential function. A set of nonlinear algebraic equations for the Fourier coefficients are derived from the free surface kinetic and dynamic boundary conditions. These algebraic equations are numerically solved through Newton＇s iterative method, and the iterative stability is further improved by a relaxation technology. The integral properties of steady water waves are numerically analyzed, showing that （1） the set-up and the set-down are both non-monotonic quantities with the wave steepness, and （2） the Fourier spectrum of the free surface is broader than that of the potential function. The latter further leads us to explore a modification for the present method by approximating the free surface and potential function through different Fourier series, with the truncation of the former higher than that of the latter. Numerical tests show that this modification is effective, and can notably reduce the errors of the free surface boundary conditions.     

On spectral and statistical characteristics of shallow water waves

Spectral, zero up-crossing and Tucker's methods of analysis are examined for shallow water wave records. Among the wave height parameters H_s and are most reliable and consistent irrespective of the analysis technique. Tucker analysis, though simple, seems versatile for estimating these wave height parameters. The standard period parameters are less consistent. For practical purposes every period that might reasonably occur has to be considered along with their corresponding height estimate. Joint distribution of heights and periods is in agreement with the function proposed by CNEXO.     

On energy transport and group velocity of water waves

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An analytic solution to the mild slope equation for waves propagating over an axi-symmetric pit

An analytic solution to the mild slope equation is derived for waves propagating over an axi-symmetric pit located in an otherwise constant depth region. The water depth inside the pit decreases in proportion to an integer power of radial distance from the pit center. The mild slope equation in cylindrical coordinates is transformed into ordinary differential equations by using the method of separation of variables, and the coefficients of the equation in radial direction are transformed into explicit forms by using the direct solution for the wave dispersion equation by Hunt (Hunt, J.N., 1979. Direct solution of wave dispersion equation. J. Waterw., Port, Coast., Ocean Div., Proc. ASCE, 105, 457–459). Finally, the Frobenius series is used to obtain the analytic solution. Due to the feature of the Hunt's solution, the present analytic solution is accurate in shallow and deep waters, while it is less accurate in intermediate depth waters. The validity of the analytic solution is demonstrated by comparison with numerical solutions of the hyperbolic mild slope equations. The analytic solution is also used to examine the effects of the pit geometry and relative depth on wave transformation. Finally, wave attenuation in the region over the pit is discussed.     

Motion response of floating structures to regular waves

Linearized and non-linear methods for analysing motion response of floating structures to regular waves are discussed in this paper. The linearized method is based upon a traditional frequency domain approach, whereas the non-linear method is based upon the time integration method as proposed by Newmark. In addition, a new approximate method is proposed. This method, which employs a Newton-Raphson solution of the equations of motion, is an extension of the linearized method and is therefore valid for near linear systems. These methods are applied to a taut line moored structure; the Aker Tethered Production Platform and to a conventional semi-submersible, the Aker H3.     

Stochastic response of compliant platforms to irregular waves

Response of a compliant platform to irregular waves is determined using finite element method. The tower is idealized by 2-D beam elements with an elastic support at the guy lines location. The flexural characteristics of the beam correspond to the four corner members of the trusses. The guying system is modelled by an axial element with linear load deformation characteristics. A computer program based on the linearized Morison's equation and the linear (Airy) wave theory, is developed to calculate the total force based on the storm wave height data at different levels of the structure. The response of the structure to random waves is based on the spectral approach. The direct and cross spectral densities of the generalized wave forces are determined and used to obtain the spectral densities of the generalized modal coordinates and mean square response at each level. Possible extension of the method is indicated to compute the evolutionary response to nonstationary wave forces.     

Moored floating breakwater response to regular and irregular waves

The simple, yet versatile numerical technique particularly suitable for investigating the problem of the wave attenuation by moored floating breakwater was recently developed by the author. In order to verift the theory, nearly full scale model tests were conducted in a large wave tank ($\text{3.6}\text{m wide}\text{× 4.5}\text{m high}\text{× 106}\text{m long}$). Both random waves and monochromatic waves were used to compare the results. A breakwater with a rectangular cross-section and a hydrodynamically shaped «three-cycle cylinderå breakwater were tested. Incident wave spectra were successfully decomposed from the multi-reflected sea spectra. Frequency response functions of transmitted wave, sway, heave and roll motions of the breakwater as well as mooring forces were all experimentally determined and compared with the theory. Generally, excellent agreements between the theory, the random wave tests and the monochromatic wave tests were obtained for the hydrodynamically shaped breakwater. Except near the modal frequencies of body motion generally good agreement between theory and experiment was obtained for the rectangular breakwater. Near the modal frequencies, the body motion was damped by the flow separation at the sharp corners of the rectangular breakwater. Generation of higher harmonics in wave, body motion and mooring forces was observed and measured, but was generally small. The slow drift oscillation and its effects on the performance of the spring moored breakwaters were also small. From the comaprisons of the small scale test and the large scale tests, it was found that the scale effects were negligibly small on the performance of the spring-moored breakwaters.     

On the radiation and diffraction of water waves by a rectangular buoy

An analytical method is presented to analyze the radiation and diffraction of water waves by a rectangular buoy in an infinite fluid domain of finite water depth. Analytical expressions for the radiated potentials and the diffracted potentials are obtained by use of the method of separation of variables. The unknown coefficients in the expressions are determined by use of the eigenfunction expansion matching method. The added masses and damping coefficients for the buoy heaving, swaying and rolling in calm water are obtained by use of the corresponding radiated potentials. Wave excitation forces are calculated by two different approaches, one is by use of the radiated potentials through Haskind’s theorem and the other is by the diffracted potential. It can be seen that the latter approach for wave forces on a rectangular buoy is much simpler than the former. To verify the correctness of the method, two specific examples in the past references are recomputed and the obtained results are in good agreement with those by use of other methods, which shows that the present method is correct.     

On pressure forcing techniques for numerical modelling of unidirectional nonlinear water waves

A method of incorporating pressure forcing into a nonlinear potential flow wave model is presented. A semi-analytical pseudo-spectral method is used to calculate dynamic response of a water body exposed to evolving local pressure distribution. Surface slope coherent and slope proportional pressure functions are directly applied through a pressure term appearing in the dynamic free-surface boundary condition of a formulated initial boundary-value problem. First, a monochromatic pressure distribution is used to generate steady regular waves of permanent form. The pressure-induced wave motion exhibits stable harmonic structure for deepwater, transitional water and shallow water waves. In the next step, a more complex pressure system is used to initiate multi-component wave propagation. It is demonstrated that the proposed method provides well-posed initial conditions for studying various water wave scenarios within a framework of nonlinear potential flow solutions.