Joint statistical distribution of two-point sea surface elevations in finite water depth

Based on the second-order random wave solutions of water wave equations in finite water depth, a joint statistical distribution of two-point sea surface elevations is derived by using the characteristic function expansion method. It is found that the joint distribution depends on five parameters. These five parameters can all be determined by the water depth, the relative position of two points and the wave-number spectrum of ocean waves. As an illustrative example, for fully developed wind-generated sea, the parameters that appeared in the joint distribution are calculated for various wind speeds, water depths and relative positions of two points by using the Donelan and Pierson spectrum and the nonlinear effects of sea waves on the joint distribution are studied.     

一种计算有限水深波能功率的新方法

海洋波浪能平均功率的准确计算是波浪能开发和利用的基础。实践中,波浪能转换装置一般安装在有限水深区域。对于随机波,只有当详尽的波浪谱已知的时候,有限水深区的波能功率才能被准确计算出来。由于种种原因,实践中波浪的实测数据大多以散点图或有义波高和统计波周期的形式给出,而波浪谱信息有时则很难获得。基于这种情况,传统上人们利用无限水深条件下的相关公式来估算有限水深区域的波能功率,但这种做法会造成较大的误差。本研究显示,对于50 m水深的理论波谱JONSWAP谱来说该误差高达14.6%。为了提高波能功率计算的准确性,本文提出了一种基于能量频率的一阶和二阶近似算法,可以在未知波浪谱的情况下较为准确地计算不同水深时的波能功率。针对两种理论波浪谱的计算结果表明,本方法在计算有限带宽内的波能功率时计算误差低于2.8%。     

Phase velocities of surface waves in a sea of finite depth

Dispersion relations and phase velocities of surface gravity waves, with their non-linearity being considered, have been derived numerically from an equation for a nonvortical incompressible fluid of finite depth. For all the depths being considered, the dispersion relations are readily realized for the wavenumbers smaller than the wavenumber of the basic harmonic. Then the acquired relations tend to increasingly deviate from the linear dispersion relations. This is the case for all the depths and wave steepnesses under discussion. When the depth of the sea diminishes, the deviations dramatically increase in both cases, when waves grow steeper and when the form of the wave spectrum changes from wide to narrow. This also holds for the phase velocity of waves. Numerical results are matched up with the experimentally derived data. For calculations, JONSWAP spectra of various width have been used. Translated by Vladimir A. Puchkin.     

Nonlinear wave profiles of wave–wave interaction in a finite water depth by fixed point approach

In this paper, a superposition of two periodic wave profiles in a finite water depth was investigated. This paper is focused on the improvement of a wave profile on the linear superposition of two waves. This improvement was realized by introducing an iterative method, which was based on a fixed point approach. Application of the fixed point approach to the wave superposition made it possible to obtain a wave profile of wave–wave interaction. The improved result of the wave profile was in good agreement with that of the nonlinear perturbation solution of the second order. It was interesting that the improved result revealed the higher-order nonlinear frequencies for two interacting Stokes waves while Dalzell's solution by a perturbation method could not predict them.     

Parameterization of ocean wave-induced mixing processes for finite water depth

Three dimensional wave-induced mixing plays an important role in shallow water area. A quite direct approach through the Reynolds average upon characteristic length scale is proposed to parameterize the horizontal and vertical shallow water mixing. Comparison of finite depth case with infinite depth results indicates that the difference of the wave-induced mixing strength is evident. In the shallow water condition, the infinite water depth approximation overestimates the mixing strength in the lower layers. The nonzero horizontal wave-induced mixing presents anisotropic property near the shore. The Prandtl’s mixing length theory underestimated the wave-induced mixing in the previous studies.     

Consistent expressions for the free-surface Green function in finite water depth

Fast and accurate computation of the free-surface Green function is of key importance for the numerical solution of linear and second-order wave-structure interaction problems in three dimensions. Integral and series expressions for the Green function are derived for which the limiting values for zero and infinite frequency are consistent with the zero and infinite frequency Green function defined in terms of infinite series of Rankine image sources. The integral expressions presented here have the advantage that they are slowly varying with the non-dimensional wave frequency, making them more efficient to approximate compared with previous expressions.     

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.     

On diffraction and radiation problem for two cylinders in water of finite depth

The hydrodynamic problem arising form the interaction of linear water waves with a wave energy device consisting of two coaxial vertical cylinders of different radii is investigated. One cylinder is riding in waves, while another is submerged in fluid. By use of the method of separation of variables and the method of matched eigenfunction expansion, analytical expressions for the potentials are obtained. Using the expressions for the potentials, analytical expressions for the hydrodynamic coefficients and exciting forces/moments on the device are obtained. Numerical results of the hydrodynamic coefficients and exciting forces/moments are presented for some ratios of the radius of the submerged cylinder to that of the riding one. It is found that the radius of the submerged cylinder has a significant influence on the hydrodynamic coefficients and exciting forces/moments for relatively bigger radius of the submerged cylinder at low frequencies.     

Nonlinear progressive waves in water of finite depth — an analytic approximation

An analytical solution using homotopy analysis method is developed to describe the nonlinear progressive waves in water of finite depth. The velocity potential of the wave is expressed by Fourier series and the nonlinear free surface boundary conditions are satisfied by continuous mapping. Unlike the perturbation method, the present approach is not dependent on small parameters. Thus solutions are possible for steep waves. Furthermore, a significant improvement of the convergence rate and region is achieved by applying Homotopy-Padé Approximants. The calculated wave characteristics of the present solution agree well with previous numerical and experimental results.     

Numerical and experimental study on seakeeping performance of ship in finite water depth

Vessels operating in shallow waters require careful observation of the finite-depth effect. In present study, a Rankine source method that includes the shallow water effect and double body steady flow effect is developed in frequency domain. In order to verify present numerical methods, two experiments were carried out respectively to measure the wave loads and free motions for ship advancing with forward speed in head regular waves. Numerical results are systematically compared with experiments and other solutions using the double body basis flow approach, the Neumann-Kelvin approach with simplified m-terms, and linearized free surface boundary conditions with double-body m-terms. Furthermore, the influence of water depths on added mass and damping coefficients, wave excitation forces, motions and unsteady wave patterns are deeply investigated. It is found that finite-depth effect is important and unsteady wave pattern in shallow water is dependent on both of the Brard number τ and depth Froude number F_h.     

Hydrodynamic coefficients for a thick-walled bottomless cylindrical body floating in water of finite depth

The paper deals with the linearized hydrodynamic forces acting on a thick-walled, bottomless cylindrical body having vertical symmetry axis and oscillating in water of finite depth. For the solution of the radiation problem, the flow field around the structure is subdivided into ring-shaped fluid regions, in each of which an axisymmetric eigenfunction expansion for the velocity potential is made. By implementing Galerkin's method the various potential solutions are then matched and numerical results concerning the hydrodynamic coefficients for heave, surge and pitch motions, as well as the coupling terms between the last two modes are obtained.     

On the local equilibrium of winds and wind-waves in relation to surface drag

Local equilibrium of winds and wind-waves is discussed as a basis for research of the drag coefficient of the water surface as well as for the spectral growth of wind-waves. This hypothesis assumes, in a narrow sense, that statistical properties are determined from four physical quantities, which represent winds and wind-waves: the friction velocityu _*, the gravitational accelerationg, the powerE of the surface displacement, and the peak frequency _p of a wind-wave spectrum. Then one has only one nondimensionalcontrol parameter, which may be either the wave age or wave nonlinearity (slope) of dominant waves. In a wide-sense, one can take into account viscosity and surface tension in terms of one more additional parameter by virtue of the virtual invariance of those material constants; that parameter describes the scale ratio between dominant waves and the short waves for which viscosity or surface tension is important. A unified expression for the roughness height according to this hypothesis turns out to include Charnock's and Toba's formulas as special cases. On the basis of a preliminary analysis of the experimental data, a new empirical formula is proposed.     

有限水深下独立波群的能量变化试验研究

为了研究独立高斯型波群在中等水深条件下的非线性演化情况,进行了多种波况的物理试验,重点分析了波浪各要素对波群非线性演化的影响。试验结果表明,波陡在波群演化中占主导作用,其次是波群的谱宽,前者引起的非线性主要体现在高阶谐波的能量变化而后者引起的非线性主要体现在自由波能量的变化。频带下移现象只在大波陡情况下才会发生,随着波陡的增加,频带下移发生的相对更早,随着波群的谱变窄,频带下移相对越明显。波陡的增加会使得波群更早的开始分裂。大波陡的波群在演化中会发生破碎,破碎的类型以崩破为主,并且崩破是间歇发生的,连续崩破发生的距离约为1.0～1.5倍波长。此外,破碎引起的能量损耗主要来源于主频和高频成分波,耗散率约10%～18%,平均每个崩破能耗约为初始能量的2%。     

A field study of the development of wind-waves

This paper presents the results of observation on the development of wind-waves which were generated in a lake water about 420 cm deep with a fetch 12 km long. Measurements of surface elevation were carried out at the end of an observational pier where the water depth was 80 cm. The wave momentum flux, i.e., the growth rate of the wave momentum, was estimated from both significant waves and power spectral densities for the wave records. The values obtained by the two ways accorded fairly well and they were 57 % as large as the wind stress measured simultaneously. The exponential growth rate of spectral densities for a frequency component was in good accord with that observed bySnyder andCox (1966) and by others. If these growth rates are applied to all the components of the spectrum, the wave momentum flux must exceed the wind stress. This cannot explain the experimental results nor can be physically accepted. The difference of spectral densities between the two successive runs showed that the increase of spectral densities was. limited in several bands of frequency. The phenomena are discussed in relation with the overshoot-undershoot effects studied byBarnett andSutherland (1968).Observational results suggest that the spectral growth of a certain component is closely related to the spectral densities of other components. Energy exchange among componented waves has not been considered in the theories for generation and development of wind-waves established by Phillips, Miles and others.New generation mechanism suggested byLonguet-Higgins (1969) was found to be able to describe the observed growth rates of the form(f)={(1/2)(t–t_1/2)}²: the spectral density(f) was proportional to the square of durationt. However, the mechanism can not explain the overshoot-undershoot effects peculiar to the equilibrium spectrum of windwaves.Three frequencies characterizing the discrete distributions of frequency bands where spectral densities increased were examined and three waves corresponding to these frequencies were found to be satisfying the resonance conditions for the wave-wave interactions among three sinusoidal wave trains as studied byPhillips (1960),Longuet-Higgins (1962) andBenny (1962). The interactions are suggested to predict well both the spectral growth proportional to squares of duration and the ceaseless oscillations of spectral densities in an equilibrium spectrum.     

Statistical distribution of wave-surface elevation for second-order random directional ocean waves in finite water depth

Based on the second-order random wave solutions of water wave equations in finite water depth, a statistical distribution of the wave-surface elevation is derived by using the characteristic function expansion method. It is found that the distribution, after normalization of the wave-surface elevation, depends only on two parameters. One parameter describes the small mean bias of the surface produced by the second-order wave-wave interactions. Another one is approximately proportional to the skewness of the distribution. Both of these two parameters can be determined by the water depth and the wave-number spectrum of ocean waves. As an illustrative example, we consider a fully developed wind-generated sea and the parameters are calculated for various wind speeds and water depths by using Donelan and Pierson spectrum. It is also found that, for deep water, the dimensionless distribution reduces to the third-order Gram–Charlier series obtained by Longuet-Higgins [J. Fluid Mech. 17 (1963) 459]. The newly proposed distribution is compared with the data of Bitner [Appl. Ocean Res. 2 (1980) 63], Gaussian distribution and the fourth-order Gram–Charlier series, and found our distribution gives a more reasonable fit to the data.     

Analysis of wave induced drift forces acting on a submerged sphere in finite water depth

A solution is presented for the wave induced drift forces acting on a submerged sphere in a finite water depth based on linearised velocity potential theory. In order to obtain the velocity potential, use has been made of multipole expansions in terms of an infinite series of Legendre functions with unknown coefficients. The series expression for the second order mean forces (drift forces) is provided by integrating the fluid pressure over the body surface. The horizontal drift force is also expressed by a series solution obtained using the far-field method.     

On natural frequencies and modal shapes in two-dimensional asymmetric and symmetric moonpools in finite water depth

In this paper theoretical models are proposed for computing the natural frequencies and modal shapes of two-dimensional asymmetric and symmetric moonpools in the finite water depth. The boundary value problem is solved by using a domain decomposition approach. On the outer vertical boundary bounded by the beam of the two bodies, linearized velocity potential is assumed to be nil. Eigenvalue problem is formulated by matching the velocity potential and fluid flux on the common boundaries to obtain the natural frequencies and modal shapes of the free surface elevation. In the symmetric moonpool cases, so-called single mode approximations (SMA) have been derived and can be adopted for rapid estimation of the natural frequencies for both piston and sloshing modes. The present results have been extensively compared with the solutions using the two-dimensional infinite water depth model developed by Molin [1], the numerical solutions and experimental data by Faltinsen et al. [2]. It is found that the solutions have been improved from the infinite water depth model. It is demonstrated that the proposed models can well predict the resonance frequencies and modal shapes for the two-dimensional asymmetric and symmetric moonpools.     

Third order contribution to the wave-making resistance of a ship at finite depth of water

This paper presents a potential based boundary element method for solving a nonlinear free surface flow problem for a ship moving with a uniform speed in finite depth of water. The free surface boundary condition is linearized by the systematic method of perturbation in terms of a small parameter up to third order. The surfaces are discretized into flat quadrilateral elements and the influence coefficients are calculated by Morino's analytical formula. Dawson's upstream finite difference operator is used in order to satisfy the radiation condition. The second order solution gives better result than the first or third order solution. So the present method with the second order solution can be adopted as a powerful tool for the hydrodynamic analysis of the thin ship in finite depth of water.     

浅水环境下波浪能能流密度计算方法研究

本文列举了3种波浪能能流密度计算方法,通过实测资料和数值模拟的实验,分析了3种方法在浅水环境中的适用性。分析得知,定义方法对浅水环境中的海浪特征考虑最为周全,计算结果最准确,但强烈依赖海浪谱的存在,适用面较窄;经验方法仅依赖常规海浪参数,对各种基础资料几乎都适用,但对波能能流密度的估计往往偏低,且不能明显反应出地形抬升对波能能流密度的汇聚效应;修正方法考虑了浅水的影响,通过适当的选取参数,可以给出较准确的结果,但对于大的波能能流密度值存在过高估计的问题。3种计算方法各有优劣,可以根据实际需要适当选择。     

Experimental investigation of pressure distribution on a cylinder due to the wave diffraction in a finite water depth

The experimental investigations on the pressure distribution around a large vertical cylinder fixed on a wave channel, piercing the free surface and subjected to regular waves have been carried out. Recently, considerable advances have been made in the development of analytical techniques for studying non-linear wave loading phenomena on large structures. However, there is a lack of high quality experimental data that may be used for validating the analytical and numerical solutions obtained. This paper describes the design and execution of an experiment conducted at the Hydraulics Laboratory in the Civil Engineering Department of the Istanbul Technical University. Experimental measurements of pressures at different locations are presented including comparisons with the computational results. This study showed that the experimental and computational results generally exhibited better correlation, however, the measured pressure values diverged from the computational results while approaching the free surface.