三层流体中的捕获模态分析（英文）

In this work, trapped mode frequencies are computed for a submerged horizontal circular cylinder with the hydrodynamic set-up involving an infinite depth three-layer incompressible fluid with layer-wise different densities. The impermeable cylinder is fully immersed in either the bottom layer or the upper layer. The effect of surface tension at the surface of separation is neglected. In this set-up, there exist three wave numbers: the lowest one on the free surface and the other two on the internal interfaces. For each wave number, there exist two modes for which trapped waves exist. The existence of these trapped modes is shown by numerical evidence. We investigate the variation of these trapped modes subject to change in the depth of the middle layer as well as the submergence depth. We show numerically that two-layer and single-layer results cannot be recovered in the double and single limiting cases of the density ratios tending to unity. The existence of trapped modes shows that in general, a radiation condition for the waves at infinity is insufficient for the uniqueness of the solution of the scattering problem.     

Trapped waves within the blocking frequency under compressed sea ice and two-dimensional current

The existence of trapped modes due to a horizontal submerged rigid cylinder placed below a compressed ice-covered surface is shown to exist within the frequency band of wave blocking by employing linearized water wave theory and with the inclusion of the effects of a two-dimensional current and the obliqueness of the propagating wave. A new modified multipole expansion method catering to the multiple propagating modes is proposed. The impact of different choices of propagating modes at a fixed frequency within the above-mentioned band on the trapped modes is detailed. Multiple trapped modes whose number depends on the direction and magnitude of the current exist within the frequency band. Out of the three choices of current types considered, while two of them result in the generation of trapped modes for all three possible choices of the propagating wave, the remaining one shows that only one propagating mode can generate a trapped wave. A break in the continuous spectrum of the trapped mode frequency due to the wave blocking and at the point of inflexion is presented. There exists a region of angle of propagation within which an abundance of the trapped mode is shown. The surface profiles for different angles of propagation and choices of propagating modes are illustrated graphically. The trapped modes associated with a higher angle of propagation decay at a faster rate.     

Wave scattering by undulating bed topography in a two-layer ocean

The problem of wave scattering by undulating bed topography in a two-layer ocean is investigated on the basis of linear theory. In a two-layer fluid with the upper layer having a free surface, there exist two modes of waves propagating at both the free surface of the upper layer and the interface between the two layers. Due to a wave train of a particular mode incident on an obstacle which is bottom-standing on the lower layer, reflected and transmitted waves of both modes are created by the obstacle. For small undulations on the bottom of the lower layer, a perturbation method is employed to obtain first-order reflection and transmission coefficients of both modes for incident wave trains of again both modes in terms of integrals involving the bed-shape function. For sinusoidal undulations, numerical results are presented graphically to illustrate the energy transfer between the waves of different modes by the undulating bed. U. BASU was born in 1949. She is a professor in the Department Applied Mathematics, Calcutta University, India. Her current research interests include water wave problems, continuum mechanics, etc.     

冰盖下两层流体水波与可渗透起伏海床的作用(英文)

The scattering problem involving water waves by small undulation on the porous ocean-bed in a two-layer fluid,is investigated within the framework of the two-dimensional linear water wave theory where the upper layer is covered by a thin uniform sheet of ice modeled as a thin elastic plate.In such a two-layer fluid there exist waves with two different modes,one with a lower wave number propagate along the ice-cover whilst those with a higher wave number propagate along the interface.An incident wave of a particular wave number gets reflected and transmitted over the bottom undulation into waves of both modes.Perturbation analysis in conjunction with the Fourier transform technique is used to derive the first-order corrections of reflection and transmission coefficients for both the modes due to incident waves of two different modes.One special type of bottom topography is considered as an example to evaluate the related coefficients in detail.These coefficients are depicted in graphical forms to demonstrate the transformation of wave energy between the two modes and also to illustrate the effects of the ice sheet and the porosity of the undulating bed.     

3D numerical modeling of wave forces on tandem fixed cylinders using the BEM

In this paper a 3D numerical model was developed to study the complicated interaction between waves and a set of tandem fixed cylinders.The fluid was considered to be inviscid and irrotational.Therefore,the Helmholtz equation was used as a governing equation.The boundary element method（BEM） was adopted to discretize the relevant equations.Open boundaries were used in far fields of the study domain.Linear waves were generated and propagated towards tandem fixed cylinders to estimate the forces applied on them.Special attention was paid to consideration of the effect on varying non-dimensional cylinder radius and distance between cylinders,ka and kd on forces and trapped modes.The middle cylinder wave forces and trapped modes in a set of nine tandem cylinders were validated utilizing analytical data.The comparisons confirm the accuracy of the model.The results of the inline wave force estimation on n tandem cylinders show that the critical cylinder in the row is the middle one for odd numbers of cylinders.Furthermore the results show that the critical trapped mode effect occurs for normalized cylinder radiuses close to 0.5 and 1.0.Finally the force estimation for n tandem cylinders confirms that force amplitude of the middle cylinder versus normalized separation distance fluctuates about that of a single cylinder.     

An analytical investigation for oscillations in a harbor of a parabolic bottom

Based on the linear shallow water approximation, longitudinal and transverse oscillations in a rectangular harbor with a parabolic bottom are analyzed. The longitudinal ones are combinations of the Legendre functions of the first and second kinds and the transverse ones are expressed with modified Bessel equations. Analytic results for longitudinal oscillations show that the augmentation of rapidity of variation of the water depth shifts the resonant wave frequencies to larger values and slightly changes the positions of the nodes for the resonant modes. For the transverse oscillations trapped within the harbor which are typically standing edge waves, the dispersion relationship is derived and the spatial structures of the first four modes are presented. The solutions illustrate that all the trapped modes are affected by the varying water depth parameters, especially for the higher modes whose profiles extend farther and the distribution of the energy of transverse oscillations is influenced by the rapidity of variation of the bottom within the harbor.     

球体波能转换装置捕获能量的理论研究

球体波能转换装置在波浪的激励下可以同时在水平和竖直两个模式振荡。基于线性微幅波浪和势流理论,将波浪与球体相互作用简化为球体固定不动,波浪绕过球体和球体在静止流体中运动。线性叠加以后得到总的速度势,计算出球体波能转换装置的平均功率和能量俘获宽度因子的解析表达式。理论研究表明:装置与波浪运动的相位差是影响波能的最重要的参数,最优相位差为-π/2,并且给出了最优相位差调节的理论表达式。另外,对波长、波幅、球体半径、水深和球心距静水面的距离等因素也进行了分析。     

不同间距下大直径圆筒防波堤波吸力试验研究

防波堤的波浪力计算对防波堤的设计和稳定具有重要意义。针对目前圆筒防波堤波吸力的设计计算方法尚未成熟,通过开展物理模型试验,进行不规则波作用下不同间距下大直径圆筒结构波吸力的分布规律、影响因素和计算方法研究。结果表明:波谷作用下,大直径圆筒结构波吸力的横向分布规律与圆筒的间距有关,纵向分布规律表现为随着水深增加先线性增大然后线性减小,且最大波吸力的位置出现在静水位一倍波高以下。圆筒的相对间距、相对水深以及波陡对不同间距大直径圆筒防波堤的波吸力影响较大。基于直立墙结构波吸力公式给出折减系数的拟合公式,用以计算圆筒周身迎浪面的波吸力。     

线性水波理论中远场无波势的构建(英文)

Various water wave problems involving an infinitely long horizontal cylinder floating on the surface water were investigated in the literature of linearized theory of water waves employing a general multipole expansion for the wave potential. This expansion involves a general combination of a regular wave, a wave source, a wave dipole and a regular wave-free part. The wave-free part can be further expanded in terms of wave-free multipoles which are termed as wave-free potentials. These are singular solutions of Laplace’s equation (for non-oblique waves in two dimensions) or two-dimensional Helmholz equation (for oblique waves) satisfying the free surface condition and decaying rapidly away from the point of singularity. The method of constructing these wave-free potentials is presented here in a systematic manner for a number of situations such as deep water with a free surface, neglecting or taking into account the effect of surface tension, or with an ice-cover modelled as a thin elastic plate floating on water.     

Second-order wave run-up on a vertical cylinder adjacent to a plane wall based on the application of quadratic transfer function in bi-directional waves

A numerical model was developed in this study to simulate the wave diffraction caused by an arbitrarily shaped structure in the presence of bi-directional incident waves based on a higher-order boundary element method (HOBEM). Based on the developed model, the wave elevation quadratic transfer function (QTF) in bi-directional waves, which is defined as the second-order wave elevation caused by two incident waves of unit amplitude from two directions, can be determined. The developed model was subsequently used to investigate the wave interaction with a cylinder situated near a vertical wall. The image principle was applied to transform the original problem into an equivalent one of wave diffraction caused by two symmetrical cylinders in open seas exposed to bi-directional incident waves. The second-order wave run-up on the cylinder can then be determined using the wave elevation QTF obtained from an analysis of the equivalent problem. A detailed numerical analysis was then conducted. Numerical results revealed that the presence of the vertical wall can apparently disturb the wave diffraction process from the cylinder, and lead to significantly amplified second-order wave elevation within the region between the wall and cylinder. In addition, the respective contribution from the first- and second-order components to the overall wave elevation around the cylinder was discussed.     

二维Oseen流中潜体产生的非定常表面波

本文考虑无限深不可压粘性流体中潜体产生的非定常表面波.受扰流动由非定常Oseen方程控制,对自由面上的波动则采用线性化边界条件,物体用具有周期振荡强度的Oseen极子模拟.通过Fourier变换,自由表面波的精确解表示为带复变色散函数的积分式,波动力学性质由Reynolds数和Strouhal数刻画.运用Lighthill定理,导出了亚临界和超临界Strouhal数时表面波的远场解的渐近式.结果显示,振荡Oseen极子生成的波动包含稳态和瞬态响应.对于亚临界Strouhal数粘性流动,存在四列波,三列在下游区域传播,一列在上游区域;对于超临界Strouhal数粘性流动,仅有在下游区域传播的二列波.     

波浪在珊瑚礁坪上传播的水槽试验研究

为研究波浪在珊瑚礁上的传播规律,分析波浪传至礁缘后破碎及传播情况,为设计波要素的确定提供参考依据。试验模拟了常年平均波浪和重现期50 a两种波浪条件。试验结果表明:波浪沿传播方向在礁坪边缘附近破碎后继续向坪内传播,并达到波面稳定状态,礁坪上的波浪传播特性表现为礁缘处波浪易破碎和礁坪上波浪衰减较快;波浪破碎表现为行进波破碎,且小于波浪在一般缓坡地形上的破碎指标,破碎波高水深比(Hb/d)约为0.5附近,重现期50 a大浪时礁缘处最大可达0.76。     

Numerical simulation with a macroscopic CFD method and experimental analysis of wave interaction with fixed porous cylinder structures

Porous structures have been widely applied in the coastal and ocean engineering due to their wave energy dissipation mechanism. The macroscopic computational fluid dynamics (CFD) approach where the quadratic pressure drop condition of porous surface is introduced to model the wave interaction with porous cylinders. A series of CFD simulations of waves interacting with a single porous cylinder and the combined structure of a porous cylinder with a concentric inner solid column are performed, with corresponding tank tests conducted. The CFD method is compared with experiments, linear potential model, and the quadratic BEM (boundary element method) model. The effects of porosity and porous cylinder radius on wave force and wave heights inside porous cylinder are analyzed to evaluate the performance of porous shell reducing wave loads and wave surface elevation, and the wave force variation with incident wave amplitudes are also investigated. The results demonstrate that the established CFD model is reliable for engineering analysis and thereby being of great significance for reference purpose in the CFD simulations of waves interacting with porous structures.     

不规则波浪在陡坡上非线性传播变形的试验研究

通过试验研究了不规则波浪在一陡坡(1:10)上的传播过程中的非线性特征。为达到研究目的,在试验水槽中以JONSWAP谱为靶谱生成了两组随机波浪。试验结果显示,在坡前常水深区域和坡顶,两种波况下波高分布均符合瑞利分布;但是在变浅区域两种波况的波高分布却不尽相同。应用基于小波变换的二阶相位谱来分析波浪在传播过程中的非线性相位耦合特征,结果表明：随着水深的变浅,波浪的非线性逐渐增强并且参与非线性相互耦合的频率成份也越来越多。通过二阶相位谱发现：随着水深的减小,参与最强的非线性相位耦合频率向高频处移动。另外通过傅立叶频谱、二阶谱以及二阶频谱共同分析陡坡上两组波况下低频波浪的演化情况。     

Hydrodynamics of a body floating in a two-layer fluid of finite depth. Part 2. Diffraction problem and wave-induced motions

A linearized two-dimensional diffraction problem in a two-layer fluid of finite depth was solved for a general floating body and relevant wave-induced motions were studied. In a two-layer fluid, for a prescribed frequency, incident waves propagate with two different wave modes. Thus the wave-exciting forces and resulting motions must be computed separately for each mode of the incident wave. The boundary integral equation method developed by the authors in the Part-1 article was applied to directly obtain the diffraction potential (pressure) on the body surface. With the computed results, an investigation was carried out on the effects of the fluid density ratio and the interface position on the wave-exciting forces on the body and the motions of the body, including the case in which the body intersects the interface. By a systematic derivation using Green's theorem, all the possible reciprocity relations were derived theoretically in explicit forms for a system of finite depth; these relations were confirmed to be satisfied numerically with very good accuracy. Experiments were also carried out using water and isoparaffin oil as the two fluids and a Lewis-form body. Measured results for the sway- and heave-exciting forces and the heave motion were compared with the computed results, and a favorable agreement was found.     

Hydrodynamic forces acting on a circular cylinder oscillating in waves and a small current

A complete theory to obtain semianalytical solutions of the wave drift damping for a circular cylinder freely oscillating in waves is developed. The wave drift damping can be significantly increased by heave and pitch motions. Effects of the draft of the cylinder and effects of the water depth are shown. The effective evaluation of the free-surface integral and the corner problem are also discussed, and the computed results of the far and near field formulations are compared. These semianalytical solutions may prove to be important in providing validation of results obtained by numerical techniques such as the higher-order boundary element method.     

Construction of wave-free potentials and multipoles in a two-layer fluid having free-surface boundary condition with higher-order derivatives

There is a large class of problems in the field of fluid structure interaction where higher-order boundary conditions arise for a second-order partial differential equation. Various methods are being used to tackle these kind of mixed boundary-value problems associated with the Laplace’s equation (or Helmholtz equation) arising in the study of waves propagating through solids or fluids. One of the widely used methods in wave structure interaction is the multipole expansion method. This expansion involves a general combination of a regular wave, a wave source, a wave dipole and a regular wave-free part. The wave-free part can be further expanded in terms of wave-free multipoles which are termed as wave-free potentials. These are singular solutions of Laplace’s equation or two-dimensional Helmholz equation. Construction of these wave-free potentials and multipoles are presented here in a systematic manner for a number of situations such as two-dimensional non-oblique and oblique waves, three dimensional waves in two-layer fluid with free surface condition with higher order partial derivative are considered. In particular, these are obtained taking into account of the effect of the presence of surface tension at the free surface and also in the presence of an ice-cover modelled as a thin elastic plate. Also for limiting case, it can be shown that the multipoles and wave-free potential functions go over to the single layer multipoles and wave-free potential.     

具有高阶导数自由表面边界条件的两层流体无波势和多级的构建（英文）

There is a large class of problems in the field of fluid structure interaction where higher-order boundary conditions arise for a second-order partial differential equation. Various methods are being used to tackle these kind of mixed boundary-value problems associated with the Laplace's equation(or Helmholtz equation) arising in the study of waves propagating through solids or fluids. One of the widely used methods in wave structure interaction is the multipole expansion method. This expansion involves a general combination of a regular wave, a wave source, a wave dipole and a regular wave-free part. The wave-free part can be further expanded in terms of wave-free multipoles which are termed as wave-free potentials. These are singular solutions of Laplace's equation or two-dimensional Helmholz equation. Construction of these wave-free potentials and multipoles are presented here in a systematic manner for a number of situations such as two-dimensional non-oblique and oblique waves, three dimensional waves in two-layer fluid with free surface condition with higher order partial derivative are considered. In particular, these are obtained taking into account of the effect of the presence of surface tension at the free surface and also in the presence of an ice-cover modelled as a thin elastic plate. Also for limiting case, it can be shown that the multipoles and wave-free potential functions go over to the single layer multipoles and wave-free potential.     

Experimental research on friction-reduction with super-hydrophobic surfaces

Many recent studies have confirmed the existence of liquid slip over particular types of solid surfaces, and these so-called super-hydrophobic surfaces have been shown to generate effective liquid slip because of the air trapped between the surface structures. In this paper, based on boundary layer theory, the microscopic structure of the super-hydrophobic surface is analyzed. The liquid slip effect on friction-reduction over super-hydrophobic surfaces under various flow conditions is investigated by experiments with a flume and water tunnel. The experimental results show that the greatest amount of drag-reduction that can be achieved is 8.76% at a low Re.