Optimum fifth order Stokes wave theory

ＯｐｔｉｍｕｍｆｉｆｔｈｏｒｄｅｒＳｔｏｋｅｓｗａｖｅｔｈｅｏｒｙ￥ＦｕＹｕｈｕａ（ＣｈｉｎａＯｆｆｓｈｏｒｅＯｉｌＤｅｖｅｌｏｐｍｅｎｔ＆ＥｎｇｉｎｅｅｒｉｎｇＣｏｒｐｏｒａｔｉｏｎ，Ｐ．Ｏ．Ｂｏｘ４７０９，Ｂｅｉｊｉｎｇ１０００２７，Ｃｈｉｎａ）...     

Nonlinear Analysis of Nearshore Wave Height Variation Due to Shoaling

Wave formulae derived from the dispersion relation for cnoidal waves are used to find an analytical solution to the problem of nearshore wave height variation on a simple topography, i. e., with an incrementally constant slope. The solution accounts for shoaling, frictional dissipation and will be sufficiently accurate for practical purposes considering the simplified assumptions which are necessary for the treatment of this problem by any method.     

A new approximate solution to the surface wave problem

Recent measurements of wave kinematics, showing that the horizontal velocity under the trough is in absolute value greater than the velocity under the crest, can not be explained by the Stokes theories up to and including the fifth order theory which are normally used by the offshore industry in the design process. This has led to a reconsideration of the surface wave problem. By applying a Green function technique a solution is obtained which reflects these recent experimental results on wave kinematics. This solution and the Stokes higher order solutions represent different approximate solutions to the surface wave problem. It turns out, however, that this new approximate solution gives a better fit to the measurements than do the approximate solutions obtained from the Stokes higher order theories. In the present paper deep water waves are discussed, but the method of solution can just as well be applied to the finite water depth case.     

Towing dynamics of a liferaft and fast rescue craft in a surface wave

The equations of motion for the coupled dynamics of a small liferaft and fast rescue craft in a surface wave are formulated in two dimensions using the methods of Kane and Levinson [1985. Dynamics: Theory and Applications. McGraw-Hill Inc., New York]. It is assumed that the motion normal to the wave surface is small and can be neglected, i.e. the bodies move along the propagating wave profile. The bodies are small so that wave diffraction and reflection are negligible. A Stokes second order wave is used and the wave forces are applied using Morison's equation for a body in accelerated flow. Wind loads are similarly modelled using drag coefficients. The equations are solved numerically using the Runge–Kutta routine “ode45” of MATLAB^®. The numerical model provides guidelines for predicting the tow loads and motions of small craft in severe sea states.     

近岸波浪浅水变形的非线性分析

本文就近岸波浪具有非线性特征提出了应用椭圆余弦波理论来研究波浪浅水变形的非线性问题。本文在椭圆余弦波数值计算的基础上，进一步分析了浅水波浪在ＨＬ~２／Ｄ~３＞２６情形下波高的变化规律，其中考虑了床面底摩擦、底坡和传质水流等因素对波高变化的影响及相应的程度分析。计算结果分析表明，浅水波浪的非线性性质和底部摩擦对波高变化的影响不能忽略，这对确定海岸工程标高有较大的实际意义和经济价值。     

Numerical simulation of wave-induced laminar boundary layers

The three-dimensional numerical model with σ-coordinate transformation in the vertical direction is applied to the simulation of surface water waves and wave-induced laminar boundary layers. Unlike most of the previous investigations that solved the simplified one-dimensional boundary layer equation of motion and neglected the interaction between boundary layer and outside flow, the present model solves the full Navier–Stokes equations (NSE) in the entire domain from bottom to free surface. A non-uniform mesh system is used in the vertical direction to resolve the thin boundary layer. Linear wave, Stokes wave, cnoidal wave and solitary wave are considered. The numerical results are compared to analytical solutions and available experimental data. The numerical results agree favorably to all of the experimental data. It is found that the analytical solutions are accurate for both linear wave and Stokes wave but inadequate for cnoidal wave or solitary wave. The possible reason is that the existing analytical solutions for cnoidal and solitary waves adopt the first-order approximation for free stream velocity and thus overestimate the near bottom velocity. Besides velocity, the present model also provides accurate results for wave-induced bed shear stress.     

The BCI criterion for the initiation of breaking process in Boussinesq-type equations wave models

The Breaking Celerity Index (BCI) is proposed as a new wave breaking criterion for Boussinesq-type equations wave propagation models (BTE).The BCI effectiveness in determining the breaking initiation location has been verified against data from different experimental investigations conducted with incident regular and irregular waves propagating along uniform slope [Utku, M. (1999). “The Relative Trough Froude Number. A New Criteria for Wave Breaking”. Ph.D. Dissertation, Dept. of Civil and Enviromental Engineering, Old Dominion University, Norfolk, VA; Gonsalves Veloso dos Reis, M.T.L. (1992). “Characteristics of waves in the surf zone”. MS Thesis, Department of Civil Engineering, University of Liverpool., Liverpool; Lara, J.L., Losada, I.J., and Liu, P.L.-F. (2006). “Breaking waves over a mild gravel slope: experimental and numerical analysis”. Journal of Geophysical Research, VOL 111, C11019] and barred beaches [Tomasicchio, G.R., and Sancho, F. (2002). “On wave induced undertow at a barred beach”. Proceedings of 28th International Conference on Coastal Engineering, ASCE, New York, 557–569]. The considered experiments were carried out in small-scale and large-scale facilities. In addition, one set of data has been obtained by the use of the COBRAS model based upon the Reynolds Averaged Navier Stokes (RANS) equations [Liu, P.L.-F., Lin, P., Hsu, T., Chang, K., Losada, I.J., Vidal, C., and Sakakiyama, T. (2000). “A Reynolds averaged Navier–Stokes equation model for nonlinear water wave and structure interactions”. Proceedings of Coastal Structures ‘99, Balkema, Rotterdam, 169–174; Losada, I.J., Lara, J.L., and Liu, P.L.-F. (2005). “Numerical simulation based on a RANS model of wave groups on an impermeable slope”. Proceedings of Fifth International Symposium WAVES 2005, Madrid].Numerical simulations have been performed with the 1D-FUNWAVE model [Kirby, J.T., Wei, G., Chen, Q., Kennedy, A.B., and Dalrymple, R.A. (1998). “FUNWAVE 1.0 Fully Nonlinear Boussinesq Wave Model Documentation and User's Manual”. Research Report No CACR-98-06, Center for Applied Coastal Research, University of Delaware, Newark]. With regard to the adopted experimental conditions, the breaking location has been calculated for different trigger mechanisms [Zelt, J.A. (1991). “The run-up of nonbreaking and breaking solitary waves”. Coastal Engineering, 15, 205–246; Kennedy, A.B., Chen, Q., Kirby, J.T., and Dalrymple, R.A. (2000). “Boussinesq modeling of wave transformation, breaking and run-up. I: 1D”. Journal of Waterway, Port, Coastal and Ocean Engineering, 126, 39–47; Utku, M., and Basco, D.R. (2002). “A new criteria for wave breaking based on the Relative Trough Froude Number”. Proceedings of 28th International Conference on Coastal Engineering, ASCE, New York, 258–268] including the proposed BCI.The calculations have shown that BCI gives a better agreement with the physical data with respect to the other trigger criteria, both for spilling and plunging breaking events, with a not negligible reduction of the calculation time.     

A Time-Domain Coupled Model for Nonlinear Wave Forces on A Fixed Box-Shaped Ship in A Harbor

A 2-D time-domain numerical coupled model is developed to obtain an efficient method for nonlinear wave forces on a fixed box-shaped ship in a harbor.The domain is divided into an inner domain and an outer domain.The inner domain is the area beneath the ship and the flow is described by the simplified Euler equations.The other area is the outer domain and the flow is defined by the higher-order Boussinesq equations in order to consider the nonlinearity of the wave motions.Along the interface boundaries between the inner domain and the outer domain,the volume flux is assumed to be continuous and the wave pressures are equal.Relevant physical experiment is conducted to validate the present model.It is shown that the numerical results agree with the experimental data.Compared with the coupled model with the flow in the inner domain governed by the Laplace equation,the present coupled model is more efficient and its solution procedure is more simple,which is particularly useful for the study on the effect of the nonlinear wave forces on a fixed box-shaped ship in a large harbor.     

A Finite Element Solution of Wave Forces on Submerged Horizontal Circular Cylinders

In this paper, a numerical model is established for estimating the wave forces on a submerged horizontal circular cylinder. For predicting the wave motion, a set of two-dimensional Navier-Stokes equations is solved numerically with a finite element method. In order to track the moving non-linear wave surface boundary, the Navier-Stokes equations are discretized in a moving mesh system. After each computational time step, the mesh is modified according to the changed wave surface boundary. In order to stabilize the numerical procedure, a three-step finite element method is applied in the time integration. The water sloshing in a tank and wave propagation over a submerged bar are simulated for the first time to validate the present model. The computational results agree well with the analytical solution and the experimental data.Finally, the model is applied to the simulation of interaction between waves and a submerged horizontal circular cylinder.The effects of the KC number and the cylinder depth on the wave forces are studied.     

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.     

三维物体上二阶波浪绕射的实时模拟

采用二阶时域理论对非线性波浪在任意三维物体周围的绕射问题进行了研究，对自由表面边界条件进行Taylor级数展开，应用摄动展开可以建立相应的边值问题，而且此边值问题的计算域不随时间变化，运用基于B－样条的边界元方法求解每一时刻的波浪场，二阶自由表面边界条件在时间上进行数值积分，在自由表面加了一个人工阻尼层以避免波浪的反射，速度势分解为已知的入射势和未知的散射势，初始条件采用二阶Stokes波浪场，通过加入物体表面边界条件，得到散射势在时间和空间上的发展，本文对圆柱所受规则波的二阶波浪力和波浪爬高进行了计算，数值结果表明此理论计算准确，效率高，数值稳定。     

二阶Boussinesq方程的孤立波解

基于同量阶迭代法，在保留同阶面的前提下，对林建国等（1998a)得到的二阶Boussinesq类方程进行了求解，得到了与其量阶相对应的取立波解，并春与Euler方程的二阶孤立波解进行了比较，结果显示，本文解比传统Boussinesq方程的孤立波解有明显的改善，扩大了孤立的适用范围。     

具有精确色散性的非线性波浪水平二维数学模型

建立具有色散性的水平二维非线性波浪方程,方程的非线性近似到了三阶。方程以波面升高和自由表面速度势表达的微分-积分型数学方程,给出方程的数值求解方法和算例,对方程积分项的处理给出了计算方法。计算结果与Boussinesq方程模型和缓坡方程模型的对应计算结果进行了对比。     

Fixed-frequency Stokes wave expansion

The study describes a new fixed-frequency Stokes wave theory that differs from previous Stokes wave theories that fix the wave number. The present wave expansion analytically reveals that the wavelength increases with wave height and exceeds than the wavelength obtained by linear wave theory. A method proposed to comparably transform the wave celerity of Fenton's [Fenton, J.D., 1985. A fifth-order Stokes theory for steady waves. Journal of Waterway, Port, Coastal and Ocean Engineering 111, 216–234.] wave theory to the present one. A direct calculation of the wavelength is introduced for practical solutions, avoiding the need to solve a nonlinear equation using an iterative numerical method.     

Moving particle method for modeling wave interaction with porous structures

This paper presents a numerical model for simulating wave interaction with porous structures. Incompressible smoothed particle hydrodynamics in porous media (ISPHP) method is introduced in this study as a mesh free particle approach that is capable of efficiently tracking the large deformation of free surfaces in a Lagrangian coordinate system. The developed model solves two porous and pure fluid flows simultaneously by means of one equation that is equivalent to the unsteady 2D Navier–Stokes (NS) equations for the flows outside the porous media and the extended Forchheimer equation for the flows inside the porous media. Interface boundary between pure fluid and porous media is effectively modeled by the SPH integration technique. A two-step semi-implicit scheme is also used to solve the fluid pressure satisfying the fluid incompressibility criterion.The developed ISPHP model is then validated via different experimental and numerical data. Fluid flow pattern through porous dam with different porosities is studied and regular wave attenuation over porous seabed is investigated. As a practical case, wave running up and overtopping on a caisson breakwater protected by a porous armor layer are modeled. The results show good agreements between numerical and laboratory data in terms of free surface displacement, overtopping rate and pressure distribution. Based on this study, ISPHP model is an efficient method for simulating the coastal applications with porous structures.     

2-D numerical analysis of near-field flow at low-crested permeable breakwaters

This paper describes the capability of a numerical model named COrnell BReaking waves And Structures (COBRAS) [Lin, P., Liu, P.L.-F., 1998. A numerical study of breaking waves in the surf zone. Journal of Fluid Mechanics 359, 239–264; Liu, P.L.-F., Lin, P., Chang, K.A., Sakakiyama, T., 1999. Numerical modeling of wave interaction with porous structures. Journal of Waterway, Port, Coastal and Ocean Engineering 125, 322–330, Liu, P.L.-F., Lin, P., Hsu, T., Chang, K., Losada, I.J., Vidal, C., Sakakiyama, T., 2000. A Reynolds averaged Navier–Stokes equation model for nonlinear water wave and structure interactions. Proc. Coastal Structures '99, 169–174] based on the Reynolds Averaged Navier–Stokes (RANS) equations to simulate the most relevant hydrodynamic near-field processes that take place in the interaction between waves and low-crested breakwaters. The model considers wave reflection, transmission, overtopping and breaking due to transient nonlinear waves including turbulence in the fluid domain and in the permeable regions for any kind of geometry and number of layers. Small-scale laboratory tests were conducted in order to validate the model, with different wave conditions and breakwater configurations. In the present study, regular waves of different heights and periods impinging on a wide-crested structure are considered. Three different water depths were tested in order to examine the influence of the structure freeboard. The experimental set-up includes a flow recirculation system aimed at preventing water piling-up at the lee of the breakwater due to overtopping. The applicability and validity of the model are examined by comparing the results of the numerical computations with experimental data. The model is proved to simulate with a high degree of agreement all the studied magnitudes, free surface displacement, pressure inside the porous structure and velocity field. The results obtained show that this model represents a substantial improvement in the numerical modelling of low-crested structures (LCS) since it includes many processes neglected previously by existing models. The information provided by the model can be useful to analyse structure functionality, structure stability, scour and many other hydrodynamic processes of interest.     

Deterministic and stochastic evolution equations for fully dispersive and weakly nonlinear waves

This paper presents a new and more accurate set of deterministic evolution equations for the propagation of fully dispersive, weakly nonlinear, irregular, multidirectional waves. The equations are derived directly from the Laplace equation with leading order nonlinearity in the surface boundary conditions. It is demonstrated that previous fully dispersive formulations from the literature have used an inconsistent linear relation between the velocity potential and the surface elevation. As a consequence these formulations are accurate only in shallow water, while nonlinear transfer of energy is significantly underestimated for larger wave numbers. In the present work we correct this inconsistency. In addition to the improved deterministic formulation, we present improved stochastic evolution equations in terms of the energy spectrum and the bispectrum for multidirectional waves. The deterministic and stochastic formulations are solved numerically for the case of cross shore motion of unidirectional waves and the results are verified against laboratory data for wave propagation over submerged bars and over a plane slope. Outside the surf zone the two model predictions are generally in good agreement with the measurements, and it is found that the accuracy of e.g., the energy spectrum and of the third-order statistics is considerably improved by the new formulations, particularly outside the shallow-water range.     

Incompressible SPH flow model for wave interactions with porous media

The paper presents an Incompressible Smoothed Particle Hydrodynamics (ISPH) method to simulate wave interactions with a porous medium. The SPH method is a mesh free particle modeling approach that is capable of tracking the large deformation of free surfaces in an easy and accurate manner. The ISPH method employs a strict incompressible hydrodynamic formulation to solve the fluid pressure and the numerical solution is obtained by using a two-step semi-implicit scheme. The ISPH flow model solves the unsteady 2D Navier–Stokes (NS) equations for the flows outside the porous media and the NS type model equations for the flows inside the porous media. The presence of porous media is considered by including additional friction forces into the equations. The developed ISPH model is first validated by the solitary and regular waves damping over a porous bed and the solitary wave interacting with a submerged porous breakwater. The convergence of the method and the sensitivity of relevant model parameters are discussed. Then the model is applied to the breaking wave interacting with a breakwater covered with a layer of porous materials. The computational results demonstrate that the ISPH flow model could provide a promising simulation tool in coastal hydrodynamic applications.     

A 3-D time-domain coupled model for nonlinear waves acting on a box-shaped ship fixed in a harbor

A 3-D time-domain numerical coupled model is developed to obtain an efficient method for nonlinear waves acting on a box-shaped ship fixed in a harbor. The domain is divided into the inner domain and the outer domain. The inner domain is the area beneath the ship and the flow is described by the simplified Euler equations. The remaining area is the outer domain and the flow is defined by the higher-order Boussinesq equations in order to consider the nonlinearity of the wave motions. Along the interface boundaries between the inner domain and the outer domain, the volume flux is assumed to be continuous and the wave pressures are equal. Relevant physical experiment is conducted to validate the present model and it is shown that the numerical results agree with the experimental data. Compared the coupled model with the flow in the inner domain governed by the Laplace equation, the present coupled model is more efficient and its solution procedure is simpler, which is particularly useful for the study on the effect of the nonlinear waves acting on a fixed box-shaped ship in a large harbor.