An explicit formulation for the evolution of nonlinear surface waves interacting with a submerged body

An explicit formulation to study nonlinear waves interacting with a submerged body in an ideal fluid of infinite depth is presented. The formulation allows one to decompose the nonlinear wave–body interaction problem into body and free‐surface problems. After the decomposition, the body problem satisfies a modified body boundary condition in an unbounded fluid domain, while the free‐surface problem satisfies modified nonlinear free‐surface boundary conditions. It is then shown that the nonlinear free‐surface problem can be further reduced to a closed system of two nonlinear evolution equations expanded in infinite series for the free‐surface elevation and the velocity potential at the free surface. For numerical experiments, the body problem is solved using a distribution of singularities along the body surface and the system of evolution equations, truncated at third order in wave steepness, is then solved using a pseudo‐spectral method based on the fast Fourier transform. A circular cylinder translating steadily near the free surface is considered and it is found that our numerical solutions show excellent agreement with the fully nonlinear solution using a boundary integral method. We further validate our solutions for a submerged circular cylinder oscillating vertically or fixed under incoming nonlinear waves with other analytical and numerical results. Copyright © 2007 John Wiley & Sons, Ltd.     

Fully non‐linear free‐surface simulations by a 3D viscous numerical wave tank

A finite difference scheme using a modified marker‐and‐cell (MAC) method is applied to investigate the characteristics of non‐linear wave motions and their interactions with a stationary three‐dimensional body inside a numerical wave tank (NWT). The Navier–Stokes (NS) equation is solved for two fluid layers, and the boundary values are updated at each time step by a finite difference time marching scheme in the frame of a rectangular co‐ordinate system. The viscous stresses and surface tension are neglected in the dynamic free‐surface condition, and the fully non‐linear kinematic free‐surface condition is satisfied by the density function method developed for two fluid layers. The incident waves are generated from the inflow boundary by prescribing a velocity profile resembling flexible flap wavemaker motions, and the outgoing waves are numerically dissipated inside an artificial damping zone located at the end of the tank. The present NS–MAC NWT simulations for a vertical truncated circular cylinder inside a rectangular wave tank are compared with the experimental results of Mercier and Niedzwecki, an independently developed potential‐based fully non‐linear NWT, and the second‐order diffraction computation. Copyright © 1999 John Wiley & Sons, Ltd.     

Global volume conservation in unsteady free surface flows with energy absorbing far‐end boundaries

A wave absorption filter for the far‐end boundary of semi‐infinite large reservoirs is developed for numerical simulation of unsteady free surface flows. Mathematical model is based on finite volume solution of the Navier–Stokes equations and depth‐integrated continuity equation to track the free surface. The Sommerfeld boundary condition is applied at the far‐end of the truncated computational domain. A dissipation zone is formed by applying artificial pressure on water surface to dissipate the kinetic energy of the outgoing waves. The computational scheme is tested to verify the conservation of total fluid volume in the domain for long simulation durations. Combination of the Sommerfeld boundary and dissipation zone can effectively minimize reflections and prevent cumulative changes in total fluid volume in the domain. Solitary wave, nonlinear periodic waves and irregular waves are simulated to illustrate the numerical developments. Earthquake excited surface waves and nonlinear hydrodynamic pressures in a dam–reservoir are computed. Copyright © 2009 John Wiley & Sons, Ltd.     

同震断层三维裂纹扩展波动时域超奇异积分法

应用波动时域超奇异积分法将P波、S波和磁电热弹多场耦合作用下同震断层任意形状三维裂纹扩展问题转化为求解以广义位移间断率为未知函数的超奇异积分方程组问题;定义了广义应力强度因子,得到裂纹前沿广义奇异应力增量解析表达式;应用波动时域有限部积分概念及体积力法,为超奇异积分方程组建立了数值求解方法,编制了FORTRAN程序,以三维矩形裂纹扩展问题为例,通过典型算例,研究了广义应力强度因子随裂纹位置变化规律;分析了同震断层裂纹扩展中力、磁、电场辐射规律.      

Regular wave integral approach to numerical simulation of radiation and diffraction of surface waves

A regular wave integral method is developed in the discretisation of a linear hydrodynamic problem on radiation and diffraction of surface waves by a floating or submerged body. The velocity potential of the problem is expressed as a solution of a body boundary integral equation involving the pulsating free surface Green function or pulsating free surface sources distributed on the body surface. With the use of a discretisation on the regular wave integral rather than discretisations on the singular wave integral of the Green function as in earlier investigations, the singular wave integral is approximated as an expansion of regular (or nonirregular) wave potentials. Influence coefficients between pulsating free surface source points are computed by the approximate expansion together with Hess–Smith panel integral formulas. Thus the velocity potential solution is evaluated by a boundary element algorithm. The numerical results produced from the proposed method agree well with semi-analytic solution results.     

Numerical simulations of inviscid and viscous free‐surface flows with application to nonlinear water waves

The purpose of the present study is to establish a numerical model appropriate for solving inviscid/viscous free‐surface flows related to nonlinear water wave propagation. The viscous model presented herein is based on the Navier–Stokes equations, and the free‐surface is calculated through an arbitrary Lagrangian–Eulerian streamfunction‐vorticity formulation. The streamfunction field is governed by the Poisson equation, and the vorticity is obtained on the basis of the vorticity transport equation. For computing the inviscid flow the Laplace streamfunction equation is used. These equations together with the respective (appropriate) fully nonlinear free‐surface boundary conditions are solved using a finite difference method. To demonstrate the model feasibility, in the present study we first simulate collision processes of two solitary waves of different amplitudes, and compute the phenomenon of overtaking of such solitary waves. The developed model is subsequently applied to calculate (both inviscid and the viscous) flow field, as induced by passing of a solitary wave over submerged rectangular structures and rigid ripple beds. Our study provides a reasonably good understanding of the behavior of (inviscid/viscous) free‐surface flows, within the framework of streamfunction‐vorticity formulation. The successful simulation of the above‐mentioned test cases seems to suggest that the arbitrary Lagrangian–Eulerian/streamfunction‐vorticity formulation is a potentially powerful approach, capable of effectively solving the fully nonlinear inviscid/viscous free‐surface flow interactions. Copyright © 2009 John Wiley & Sons, Ltd.     

A meshless numerical wave tank for simulation of nonlinear irregular waves in shallow water

Time domain simulation of the interaction between offshore structures and irregular waves in shallow water becomes a focus due to significant increase of liquefied natural gas (LNG) terminals. To obtain the time series of irregular waves in shallow water, a numerical wave tank is developed by using the meshless method for simulation of 2D nonlinear irregular waves propagating from deep water to shallow water. Using the fundamental solution of Laplace equation as the radial basis function (RBF) and locating the source points outside the computational domain, the problem of water wave propagation is solved by collocation of boundary points. In order to improve the computation stability, both the incident wave elevation and velocity potential are applied to the wave generation. A sponge damping layer combined with the Sommerfeld radiation condition is used on the radiation boundary. The present model is applied to simulate the propagation of regular and irregular waves. The numerical results are validated by analytical solutions and experimental data and good agreements are observed. Copyright © 2008 John Wiley & Sons, Ltd.     

Meshless numerical simulation for fully nonlinear water waves

A meshless numerical model for nonlinear free surface water wave is presented in this paper. An approach of handling the moving free surface boundary is proposed. Using the fundamental solution of the Laplace equation as the radial basis functions and locating the source points outside the computational domain, the problem is solved by collocation of only a few boundary points. Present model is first applied to simulate the generation of periodic finite‐amplitude waves with high wave‐steepness and then is employed to simulate the modulation of monochromatic waves passing over a submerged obstacle. Good agreements are observed as compared with experimental data and other numerical models. Copyright © 2005 John Wiley & Sons, Ltd.     

Simulation of wave–body interaction: A desingularized method coupled with acceleration potential

A two-dimensional, potential-theory based, fully nonlinear numerical wave tank (NWT) is developed for the simulation of wave–body interaction. In this NWT, the concept of acceleration potential is adopted in addition to the velocity potential. Both potentials are solved using the desingularized boundary integral equation method (DBIEM). By tapping the strength of the DBIEM, a new acceleration-potential solving method is proposed, which turns the originally implicit kinematic boundary condition on the surface of a passively moving body into an explicit one. Unlike the other existing methods such as the mode decomposition method, the indirect method and the iterative method, the present method requires solving of only one boundary value problem to determine the acceleration potential, and hence significantly enhances the computational efficiency. Using this NWT, the nonlinear interaction between a freely floating barge and various incident waves is investigated. It is confirmed that the new acceleration potential solving method outperforms other existing methods, saving at least 45% of the computational time.     

Numerical computations of two-dimensional solitary waves generated by moving disturbances

Two-dimensional solitary waves generated by disturbances moving near the critical speed in shallow water are computed by a time-stepping procedure combined with a desingularized boundary integral method for irrotational flow. The fully non-linear kinematic and dynamic free-surface boundary conditions and the exact rigid body surface condition are employed. Three types of moving disturbances are considered: a pressure on the free surface, a change in bottom topography and a submerged cylinder. The results for the free surface pressure are compared to the results computed using a lower-dimensional model, i.e. the forced Korteweg–de Vries (fKdV) equation. The fully non-linear model predicts the upstream runaway solitons for all three types of disturbances moving near the critical speed. The predictions agree with those by the fKdV equation for a weak pressure disturbance. For a strong disturbance, the fully non-linear model predicts larger solitons than the fKdV equation. The fully non-linear calculations show that a free surface pressure generates significantly larger waves than that for a bottom bump with an identical non-dimensional forcing function in the fKdV equation. These waves can be very steep and break either upstream or downstream of the disturbance.     

推广的多次透射边界：标量波清形

利用空间外推将多次透射边界条件推广以提高这一边界在时域计算中的精度。对全波场仅由入射波构成及由入射波和充分发育的反射波构成两种情况导出了边界反射系数的频域公式。从时域计算精度角度分析了边界反射系数以阐明在透射大角度入射行进波动和消减运动方面推广边界的优点。     

A lattice Boltzmann method for viscous free surface waves in two dimensions

We propose a new method based on the combination of the lattice Boltzmann equation (LBE) and the kinematic boundary condition (KBC) method to simulate viscous free surface wave in two dimensions. In our method, the flow field is modeled by LBE, whereas the free surface is explicitly tracked by the local height function, which is calculated by the KBC method. The free surface boundary condition (FSBC) for LBE is revised from previous researches. Interpolation‐supplemented lattice Boltzmann (ISLB) method is introduced, which enables our approach to be applied on arbitrary, nonuniform mesh grids. Five cases are simulated respectively to validate the LBE–KBC method: the stationary flow and the solitary waves simulated by the revised‐FSBC are more accurate than the one obtained by the former‐FSBC; numerical results of standing waves show that our method is compatible to the existing two‐dimensional finite‐volume scheme; cases of small amplitude Stokes wave and waves traveling over a submerged bar show good agreement on wave celerity, wavelength, wave amplitude and wave period between numerical results and corresponding analytical solutions and/or experiment data.Copyright © 2012 John Wiley & Sons, Ltd.     

Fully nonlinear modeling of radiated waves generated by floating flared structures

The nonlinear radiated waves generated by a structure in forced motion, are simulated numerically based on the potential theory. A fully nonlinear numerical model is developed by using a higher-order boundary element method （HOBEM）. In this model, the instantaneous body position and the transient free surface are updated at each time step. A Lagrangian technique is employed as the time marching scheme on the free surface. The mesh regridding and interpolation methods are adopted to deal with the possible numerical instability. Several auxiliary functions are proposed to calculate the wave loads indirectly, instead of directly predicting the temporal derivative of the velocity potential. Numerical experiments are carried out to simulate the heave motions of a submerged sphere in infinite water depth, the heave and pitch motions of a truncated flared cylinder in finite depth. The results are verified against the published numerical results to ensure the effectiveness of the proposed model. Moreover, a series of higher harmonic waves and force components are obtained by the Fourier transformation to investigate the nonlinear effect of oscillation frequency. The difference among fully nonlinear, body-nonlinear and linear results is analyzed. It is found that the nonlinearity due to free surface and body surface has significant influences on the numerical results of the radiated waves and forces.     

Reflection of a beam of elastic waves by a periodic surface profile

Reflection of elastic waves from a traction-free solid-air boundary of periodic saw-tooth profile is investigated analytically and experimentally. For an incident plane wave the surface displacements on the profile are computed as the solution of a singular integral equation. The reflected field is subsequently obtained by using an integral representation. Incident beams of finite width are represented by Fourier superpositions of plane waves. The dependence of the reflected signal spectra on the incident beam width is examined closely near the fundamental surface resonance frequency. Experimental spectra which were obtained using two different diameter transducers, are compared to the corresponding theoretical spectra. It is found that the depth of the spectral minima depends on the incident beam width. Both analytical and experimental results exhibit the splitting of an incident beam of elastic waves into two reflected beams. The beam splitting is more pronounced for a narrower incident beam and for frequencies close to a resonance frequency of the profile.     

非周期波浪与直墙作用的非线性数值研究

基于时域高阶边界元方法,建立了完全非线性二维数值波浪水槽,对非周期波浪与直墙的相互作用问题进行了模拟和研究.自由表面满足完全非线性自由水面运动学和动力学边界条件,采用混合欧拉-拉格朗日方法追踪瞬时自由面流体质点,采用四阶Runge-Kutta法对下一时间步的波面和自由面速度势进行更新.采用加速度式法求解直墙表面速度势的时间导数,对瞬时物体湿表面上的水动力压强积分,得到作用在物体上的瞬时波浪力.首先,将全非线性与Serre-Green-Naghdi(SGN)模型的结果进行了对比分析,发现对于大幅值双入射波问题,仅满足弱色散关系的SGN模型大大低估了最大波浪爬高;其次,研究了双入射波与直墙的非线性作用问题,发现线性预报对波浪最大爬高有较大低估,而波浪的非线性成分不只导致了自由面爬高的异常增大,也引起了局部自由面的高频振荡,该物理过程中,直墙所受的波浪载荷,也展示出了与波浪爬高相似的非线性特性;最后,对波浪爬升和波浪力的时间历程进行了频谱分析,发现入射主频波的部分能量传递给了更高频的波浪成分,反映出该问题具有典型的非线性特性.     

3D free‐surface flow computation using a RANSE/Fourier–Kochin coupling

A coupling method for numerical calculations of steady free‐surface flows around a body is presented. The fluid domain in the neighbourhood of the hull is divided into two overlapping zones. Viscous effects are taken in account near the hull using Reynolds‐averaged Navier–Stokes equations (RANSE), whereas potential flow provides the flow away from the hull. In the internal domain, RANSE are solved by a fully coupled velocity, pressure and free‐surface elevation method. In the external domain, potential‐flow theory with linearized free‐surface condition is used to provide boundary conditions to the RANSE solver. The Fourier–Kochin method based on the Fourier–Kochin formulation, which defines the velocity field in a potential‐flow region in terms of the velocity distribution at a boundary surface, is used for that purpose. Moreover, the free‐surface Green function satisfying this linearized free‐surface condition is used. Calculations have been successfully performed for steady ship‐waves past a serie 60 and then have demonstrated abilities of the present coupling algorithm. Copyright © 2003 John Wiley & Sons, Ltd.     

有限散射信号下二维缺陷形状识别的罚函数方法

研究在有限照射角度和频带宽度下二维缺陷的形状识别问题。首先,通过引进介质参数扰动函数,建立介质参数扰动函数和弹性波散射场之间的非线性关系,并将所关心的缺陷的形状识别问题转化为关于扰动函数的反演;然后,利用变分技术和优化方法求解,为了弥补散射数据的不足,在总的目标函数中,采用附加度量函数作为罚函数;最后,对后场散射远场测量时有限照射角度和频带宽度下几种典型缺陷进行了模拟识别,表明了;表明了罚函数法的有效性。     

A numerical method to solve the m‐terms of a submerged body with forward speed

To model mathematically the problem of a rigid body moving below the free surface, a control surface surrounding the body is introduced. The linear free surface condition of the steady waves created by the moving body is satisfied. To describe the fluid flow outside this surface a potential integral equation is constructed using the Kelvin wave Green function whereas inside the surface, a source integral equation is developed adopting a simple Green function. Source strengths are determined by matching the two integral equations through continuity conditions applied to velocity potential and its normal derivatives along the control surface. After solving for the induced fluid velocity on the body surface and the control surface, an integral equation is derived involving a mixed distribution of sources and dipoles using a simple Green function and one component of the fluid velocity. The normal derivatives of the fluid velocity on the body surface, namely the m‐terms, are then solved by this matching integral equation method (MIEM). Numerical results are presented for two elliptical sections moving at a prescribed Froude number and submerged depth and a sensitivity analysis undertaken to assess the influence of these parameters. Furthermore, comparisons are performed to analyse the impact of different assumptions adopted in the derivation of the m‐terms. It is found that the present method is easy to use in a panel method with satisfactory numerical precision. Copyright © 2002 John Wiley & Sons, Ltd.     

An efficient curvilinear non‐hydrostatic model for simulating surface water waves

An efficient curvilinear non‐hydrostatic free surface model is developed to simulate surface water waves in horizontally curved boundaries. The generalized curvilinear governing equations are solved by a fractional step method on a rectangular transformed domain. Of importance is to employ a higher order (either quadratic or cubic spline function) integral method for the top‐layer non‐hydrostatic pressure under a staggered grid framework. Model accuracy and efficiency, in terms of required vertical layers, are critically examined on a linear progressive wave case. The model is then applied to simulate waves propagating in a canal with variable widths, cnoidal wave runup around a circular cylinder, and three‐dimensional wave transformation in a circular channel. Overall the results show that the curvilinear non‐hydrostatic model using a few, e.g. 2–4, vertical layers is capable of simulating wave dispersion, diffraction, and reflection due to curved sidewalls. Copyright © 2010 John Wiley & Sons, Ltd.