Nonlinear lifting theory for unsteady WIG in proximity to incident water waves. Part 2: Three-dimension

The present article presents a nonlinear analysis for determining the three-dimensional unsteady potential-flow characteristics about a wing subject to wing-in-ground effect (WIG) operating above progressive water waves. By means of the time-domain Green's function for the three-dimensional dipole moving above the free surface satisfying the dynamic and kinematic boundary conditions on the mean free surface, the influence of the free surface on the vortex ring is considered. Then, the nonlinear unsteady lifting surface theory is developed to study the lifting problem for a three-dimensional wing operating above progressive water waves. Furthermore, the roll-up shed from the wing in the presence of a free surface and water waves is taken into account. With the computed results, the non-dimensional force coefficients (including the lift coefficient, induced drag coefficient and lift-to-drag ratio) are presented with the variation of different geometry and water wave parameters. The data reported in the literature are presented to validate the present approach.     

波浪与带窄缝方箱作用共振现象的数值模拟

By introducing a source term into the Laplace equation, a two-dimensional fully nonlinear time-domain numerical wave flume (NWF) is developed to investigate the resonance induced by the interaction bet...     

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.     

非线性流函数在实验风浪场计算中的应用

风作用于水面产生风浪, 其中由于波流紊动产生的动量和能量的交换机制是一个很复杂的过程。风应力一般用来描述这种能量交换, 可以分为3个部分: 水面的剪切力、波生应力以及紊动应力。采用一种有效的非线性波流分离方法——NSFM(Nonlinear Stream Function Method)对波流运动的动量和能量输移进行定性描述。构造能够有效表达非线性波浪的解析流函数, 摄动求解使其满足拉普拉斯方程、动力边界条件和运动边界条件, 结合实验室风浪数据, 分离出波生速度场。通过交叉谱分析, 得到波生雷诺应力在不同风速下对风应力的贡献。结果表明: NSFM对不同工况条件下的风浪的处理具有较高的精度, 模型适应性良好; 且风速越大, 波生应力沿着水深衰减得越快, 且自由面波生应力在动量输移中的比重会逐渐减弱。     

Viscous free surface effect on coastal embankment hydrodynamics

A finite-differnece method was used to calculate the nonlinear hydrodynamic pressures acting on the coastal embankment faces by seismic-wave actions. The nonlinearity of free surface flow, convective acceleration, viscosity and surface tension of fluid are included in the analysis. The kinematic and dynamic free surface boundary conditions are employed for calculating the horizontal fluid velocity, pressure at the free surface and the surface profile of the fluid. The time-dependent water surface is transformed to the horizontal plane, and the flow field is mapped onto a rectangular, making it convenient to model the complex sea bottom geometry and the wavy water surface by the finite-difference method. Fully nonlinear and weakly nonlinear dynamic free surface conditions are used and compared. The effects of surface tension of fluid are also discussed. The nonslip boundary condition is applied on the most part of the interface between fluid and solid face, except the region near the intersection between free surface and wall face. The numerical results are presented for various water depths and ground motion intensities, and their associate viscous effects on coastal embankment hydrodynamics are discussed.     

Harmonic generation by waves propagating over a submerged step

Harmonic generation by waves propagating over a two-dimensional submerged step is investigated. A nonlinear theory correct to second order is presented for steps of infinite and finite lengths subjected to single harmonic waves.The boundary value problem for the second-order scattered velocity potential is linearly decomposed into two separate boundary value problems, each having only one inhomogeneous boundary condition.Theoretical results indicate that the higher harmonics are generated in the shallow-water region over a step and then are transmitted to the deeper water as free waves.Numerical calculations compare favourably with existing experimental data.     

孤立波与带窄缝双箱相互作用模拟研究

针对孤立波与带窄缝双箱的作用问题,应用时域高阶边界元方法建立了二维数值水槽。其中,自由水面满足完全非线性运动学和动力学边界条件,对瞬时自由表面流体质点采用混合欧拉-拉格朗日法追踪,采用四阶龙格库塔法对下一时刻的自由水面的速度势和波面升高进行更新。采用加速度势法求解物体湿表面的瞬时波浪力。采用推板方法生成孤立波。通过模拟孤立波在直墙上的爬高以及施加在直墙上的波浪力,并与已发表的实验和数值结果对比,验证本数值模型的准确性。通过数值模拟计算研究了窄缝宽度、方箱尺寸对波浪在箱体迎浪侧爬高,窄缝内波面升高,箱体背浪侧透射波高及箱体受波浪荷载的影响。同时研究了有一定时间间隔的双孤立波与带窄缝双箱系统作用问题。     

Nonlinear response of membranes to ocean waves using boundary and finite elements

The behavior of a highly deformable membrane to ocean waves was studied by coupling a nonlinear boundary element model of the fluid domain to a nonlinear finite element model of the membrane. The hydrodynamic loadings induced by water waves are computed assuming large body hydrodynamics and ideal fluid flow and then solving the transient diffraction/radiation problem. Either linear waves or finite amplitude waves can be assumed in the model and thus the nonlinear kinematic and dynamic free surface boundary conditions are solved iteratively. The nonlinear nature of the boundary condition requires a time domain solution. To implicitly include time in the governing field equation, Volterra's method was used. The approach is the same as the typical boundary element method for a fluid domain where the governing field equation is the starting point. The difference is that in Volterra's method the time derivative of the governing field equation becomes the starting point.The boundary element model was then coupled through an iterative process to a finite element model of membrane structures. The coupled model predicts the nonlinear interaction of nonlinear water waves with highly deformable bodies. To verify the coupled model a large scale test was conducted in the OH Hinsdale wave Research Laboratory at Oregon State University on a 3-ft-diameter fabric cylinder submerged in the wave tank. The model data verified the numerical prediction of the structure displacements and of the changes in the wave field.The boundary element model is an ideal modeling technique for modeling the fluid domain when the governing field equations is the Laplace equation. In this case the nonlinear boundary element model was coupled with a finite element model of membrane structures, but the model could have been coupled with other finite element models of more rigid structures, such as a pontoon floating breakwater.     

Irregular wave kinematics from a kinematic boundary condition fit (KBCF)

Calculation of the kinematics of random waves above the mean water line presents great difficulties. The kinematic boundary condition fit (KBCF) method approximates the solution through the numerical calculation of a potential function which fits the kinematic boundary condition on a specified surface. Comparisons with a high order regular wave show that the method converges to the true solution when the surface is accurately specified. Tests of the method for irregular waves were made with measurements from a laser-Doppler current meter in the Delft wave tank. These tests showed good agreement between theory and measurement when the surface evolution was calculated correct to second order. Stretched linear theory was also compared to the measurements. The stretched velocities were reasonably good when the phases of the component wavelets were measured but somewhat low when the phases were selected from a uniform distribution.     

Nonlinear numerical simulation on extreme-wave kinematics

A fully nonlinear numerical model based on a time-domain higher-order boundary element method (HOBEM) is founded to simulate the kinematics of extreme waves. In the model, the fully nonlinear free surface boundary conditions are satisfied and a semi-mixed Euler-Lagrange method is used to track free surface; a fourth-order Runga-Kutta technique is adopted to refresh the wave elevation and velocity potential on the free surface at each time step; an image Green function is used in the numerical wave tank so that the integrations on the lateral surfaces and bottom are excluded. The extreme waves are generated by the method of wave focusing. The physical experiments are carried out in a wave flume. On the horizontal velocity of the measured point, numerical solutions agree well with experimental results. The characteristics of the nonlinear extreme-wave kinematics and the velocity distribution are studied here.     

Simulating nonlinear waves on the free surface in surf zones with two-dimensional sloping beach

Unsteady nonlinear wave motions on the free surface in shallow water and over slopes of various geometries are numerically simulated using a finite difference method in rectangular grid system. Two-dimensional Navier–Stokes equations and the continuity equation are used for the computations. Irregular leg lengths and stars are employed near the boundaries of body and free surface to satisfy the boundary conditions. Also, the free surface which consists of markers or segments is determined every time step with the satisfaction of kinematic and dynamic free surface conditions. Moreover, marker-density method is also adopted to allow plunging jets impinging on the free surface. Either linear or Stokes wave theory is employed for the generation of waves on the inflow boundary. For the simulation of wave breaking phenomena, the computations are carried out with various wave periods and sea bottom slopes in surf zone. The results are compared with other existing computational and experimental results. Agreement between the experimental data and the computation results is good.     

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.     

Time domain three-dimensional fully nonlinear computations of steady body–wave interaction problem

Three-dimensional fully nonlinear waves generated by moving disturbances with steady forward speed without motions are solved using a mixed Eulerian–Lagrangian method in terms of an indirect boundary integral method and a Runge–Kutta time marching approach which integrates the fully nonlinear free surface boundary conditions with respect to time.A moving computational window is used in the computations by truncating the fluid domain (the free surface) into a computational domain. The computational window maintains the computational domain and tracks the free surface profile by a node-shifting scheme applied within it. An implicit implement of far field condition is enforced automatically at the truncation boundary of the computational window.Numerical computations are applied to free surface waves generated by Wigley and Series 60 hulls for the steady problem. The present numerical results are presented and compared with existing linear theory, experimental measurements, and other numerical nonlinear computations. The comparisons show satisfactory agreements for these hydrodynamic problems.     

极限波浪运动特性的非线性数值模拟

利用时域高阶边界元方法建立了模拟极限波浪运动的完全非线性数值模型,其中自由水面满足完全非线性自由水面条件.采用半混合欧拉-拉格朗日方法追踪流体瞬时水面,运用四阶Runge-Kutta方法更新下一时间步的波面和速度势,同时应用镜像格林函数消除水槽两个侧面和底面上的积分.研究中利用波浪聚焦的方法产生极限波浪,并且在水槽中开展了物理模型实验,将测点试验数据与数值结果进行了对比,两者吻合得很好.对极限波浪运动的非线性和流域内速度分布进行了研究.     

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.     

Generation and propagation of transient nonlinear waves in a wave flume

A semi-analytical nonlinear wavemaker model is derived to predict the generation and propagation of transient nonlinear waves in a wave flume. The solution is very efficient and is achieved by applying eigenfunction expansions and FFT. The model is applied to study the effect of the wavemaker and its motion on the generation and propagation of nonlinear waves. The results indicate that the linear wavemaker theory may be applied to predict only the generation of waves of low steepness for which the nonlinear terms in the kinematic wavemaker boundary condition and free-surface boundary conditions are of secondary importance. For waves of moderate steepness and steep waves these nonlinear terms have substantial effects on wave profile and wave spectrum just after the wavemaker. A wave spectrum corresponding to a sinusoidally moving wavemaker possesses a multi-peak form with substantial nonlinear components, which disturbs or may even exclude physical modeling in wave flumes. The analysis shows that the widely recognized weakly nonlinear wavemaker theory may only be applied to describe the generation and propagation of waves of low steepness. This is subject to further restrictions in shallow and deep waters because the kinematic wavemaker boundary condition as well as the nonlinear interaction of wave components and the evolution of wave energy spectrum is not properly described by weakly nonlinear wavemaker theory. Laboratory experiments were conducted in a wave flume to verify the nonlinear wavemaker model. The comparisons show a reasonable agreement between predicted and measured free-surface elevation and the corresponding amplitudes of Fourier series. A reasonable agreement between theoretical results and experimental data is observed even for fairly steep waves.     

Numerical study on cnoidal wave run-up around a vertical circular cylinder

A finite element model of Boussinesq-type equations was set up, and a direct numerical method is proposed so that the full reflection boundary condition is exactly satisfied at a curved wall surface. The accuracy of the model was verified in tests. The present model was used to further examine cnoidal wave propagation and run-up around the cylinder. The results showed that the Ursell number is a nonlinear parameter that indicates the normalized profile of cnoidal waves and has a significant effect on the wave run-up. Cnoidal waves with the same Ursell number have the same normalized profile, but a difference in the relative wave height can still cause differences in the wave run-up between these waves. The maximum dimensionless run-up was predicted under various conditions. Cnoidal waves hold entirely distinct properties from Stokes waves under the influence of the water depth, and the nonlinearity of cnoidal waves enhances rather than weakens with increasing wavelength. Thus, the variations in the maximum run-up with the wavelength for cnoidal waves are completely different from those for Stokes waves, and there are even significant differences in the variation between different cnoidal waves.     

Gravity wave interaction with floating membrane due to abrupt change in water depth

Using the recently developed expansion formulae for wave structure interaction problems, the scattering of surface water waves by a semi-infinite floating membrane due to abrupt change in bottom topography is analyzed. Both the cases of finite and infinite steps are analyzed. In the present paper, the analysis is based on the linearized theory of water waves and small amplitude membrane response. Combining the linearized kinematic and dynamic surface conditions on the water surface with the dynamic pressure condition on the membrane, a third order differential equation is derived to describe the membrane covered free surface condition. General wave energy relation for wave scattering by floating horizontal membrane is derived by the application of law of conservation of energy flux and alternately by the direct application of Green's second identity. In the floating membrane covered region, the wave energy density is a combination of the kinetic and potential energy density due to the surface gravity waves, and the surface energy density which is due to the existence of the floating membrane on the free surface. Gravity wave transformations due to an abrupt change in bottom topography in the presence of a floating membrane in finite water depth are analyzed based on shallow water approximation. Numerical results are computed and analyzed to understand the wave transformation due to the floating membrane when there is an abrupt change in topography in different cases.     

深水波中进动共振的数值模拟研究

进动(precession)共振是一种非线性共振相互作用,2016年才有学者对这一现象进行研究。采用非静压二维自由表面流模型模拟了深水条件下重力波的进动共振现象。通过边界造波的方法产生双色波,分析了触发进动共振的初始条件;探讨了进动共振在小振幅前提条件下发生的简化初始条件。数值模拟分析两组对称测点,对不同测点的波面、能量谱进行对比分析。数值结果表明:非静压二维自由表面流模型可以模拟进动共振现象,并且可以采用双色波作为条件来研究深水五波进动共振现象,进动共振需要一定的能量转化时间,进动共振发生的条件是三波组合的进动频率等于一个系统存在的非线性频率。     

Second-Order Wave Diffraction Around 3-D Bodies by A Time-Domain Method

A time-domain method is applied to simulate nonlinear wave diffraction around a surface piercing 3-D arbitrary body. The method involves the application of Taylor series expansions and the use of perturbation procedure to establish the corresponding boundary value problems with respect to a time-independent fluid domain. A boundary element method based on B-spline expansion is used to calculate the wave field at each time step, and the free surface boundary condition is satisfied to the second order of wave steepness by a numerical integration in time. An artificial damping layer is adopted on the free surface for the removal of wave reflection from the outer boundary. As an illustration, the method is used to compute the second-order wave forces and run-up on a surface-piercing circular cylinder. The present method is found to be accurate, computationally efficient, and numerically stable.