Interaction of current and flexural gravity waves

The interaction between current and flexural gravity waves generated due to a floating elastic plate is analyzed in two dimensions under the assumptions of linearized theory. For plane flexural gravity waves, explicit expressions for the water particle dynamics and trajectory are derived. The effect of current on the wavelength, phase velocity and group velocity of the flexural gravity waves is analyzed. Variations in wavelength and wave height due to the changes in current speed and direction are analyzed. Effects of structural rigidity and water depth on wavelength are discussed in brief. Simple numerical computations are performed and presented graphically to explain most of the theoretical findings in a lucid manner.     

Interactions of nonlinear gravity waves and uniform current in Lagrangian system

The particle trajectory on a weakly nonlinear progressive surface wave obliquely interacting with a uniform current is studied by using an EulerLagrange transformation. The thirdorder asymptotic solution is a periodic bounded function of Lagrangian labels and time, which imply that the entire solution is uniformlyvalid. The explicit parametric solution highlights the trajectory of a water particle and mass transport associated with a particle displacement can now be obtained directly in Lagrangian form. The angular frequency and Lagrangian mean level of the particle motion in Lagrangian form differ from those of the Eulerian. The variations in the water particle orbits resulting from the oblique interaction with a steady uniform current of different magnitudes are also investigated.     

The wave transformation and breaking phenomena in shallow water

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地形与流对水平无旋浅水波的影响

本文在水平无旋及Boussinesq假设之下,导出了水面变化与水平速度场耦合方程组,以及相应的压力与垂向速度解析表示式。通过数值方法求得水面变化及某一特定水深的水平速度分布之后,压力分布及其余水深的速度分布即可由简单计算得到。由色散关系式可知,不同水深的长波色散关系在O(ε)近似之下是等同的。粘性的存在会使波高随时间的增加而衰减,但粘性与底斜率的耦合又可能使波高增长,形成不稳定;计算及分析说明,当同向流增大时,波速增大,波长增大,波高会增加;而水深减少,使波速减少,波长缩短,振幅增加。     

An extension of the Airy theory for linear waves into shallow water

As a fully developed (Airy) wave propagates from deep into shallow water, its crest becomes more peaked while the trough flattens out. The median crest diameter MCD, defined as the distance between the wave flanks under the crest at a level halfway between the crest and trough, therefore decreases relative to the similarly defined median trough diameter MTD, which remains constant up to the breaking point. The MCD is directly related to other wave characteristics, which enables water particle velocities to be calculated for any water depth without having to recur to more complex, higher-order Stokes, cnoidal or Fenton theories. Over a nearly horizontal bottom, most fully developed wave characteristics can be expressed as functions of the wave period T_w. It is shown that the horizontal particle velocity at the bottom under the breaker crest is at least 9 times faster than under the breaker trough, which explains why sediment is transported landward under fair weather conditions. The proposed equations also shed new light on the formation of spilling, plunging and surging/collapsing breakers.     

Forces and moment on a horizontal plate due to regular and irregular waves in the presence of current

This work presents an experimental study of a submerged plate used as a breakwater for coastal areas protection. Questions addressed concern the influence of current on the reflective power of the plate, and its influence on the hydrodynamic loads exerted on it. Results concern both monochromatic and irregular waves. Generally speaking, an influence of the current is found, changing the reflecting power of the structure up to 50%. A homogenized behavior of the loads and moments is found in the presence of currents, meaning that the load values become less sensitive to the frequency. Furthermore, the influence of waves reflected by the wave absorber, representing partially reflective conditions at the shore, is found to be of same order in the absence of current. In any case, the linear behavior of the breakwater is emphasized through the irregular waves approach.     

规则波在浅水中运动及破碎的数值计算

利用一维Boussinesq方程描述了在浅水中的波浪运动以及破碎情况。在方程中引入了表面翻滚的概念，认为翻滚的水体是以波速运动的，翻滚的作用表现在水平速度的垂直分布上，产生了附加迁移动量项。通过对Airy波在浅水中运动以及破碎情况的研究，得出的一些结论与Schaeffer和合田的成果吻合良好。     

Spectral characteristics of high shallow water waves

The spectral characteristics of shallow water waves with significant wave height more than 2 m based on the data collected along the Indian coast is examined. It was found that the value of Joint North Sea Wave Project (JONSWAP) parameters (α and γ) increases with significant wave height and mean wave period and decreases with spectral peak period. The estimated average value (0.0027 and 1.63) of the JONSWAP parameters, α and γ were less than the generally recommended values of 0.0081 and 3.3, respectively. By carrying out a multi-regression analysis, an empirical equation is arrived relating the JONSWAP parameters with significant wave height, peak wave period and mean wave period. It was found that the Scott spectra underestimate the maximum spectral energy of high waves. The study shows that the measured wave spectra can be represented by JONSWAP spectra with the JONSWAP parameters estimated based on the equation proposed in this paper.     

Effects of following and opposing vertical current shear on nonlinear wave interactions

A Navier-Stokes solver in OpenFOAM^® is combined with the Volume of Fluid (VOF) surface capturing method to investigate the wave interaction with depth-varying currents in intermediate and shallow waters. A special attention is paid to the separate effect of vertical current shear on near resonant triad wave interactions. It was found that in the presence of following vertical current shear, the wave exhibits a sharper crest and flatter trough, and the opposite is true in the presence of opposing vertical current shear. Our model results indicate that the wave steepness at which the current shear starts to affect the crest elevation is greater in deeper water than in shallower water. We found that adding vertical current shear to the uniform current further enhances the relative harmonic wave energy and the extent of triad interaction in the following current while weakens them in the opposing current. As a result, following and opposing current shear may cause wave to break at a lower and higher sea state respectively. Due to the increased wave nonlinearity in the presence of a following current shear, a linear superposition of the individual wave and current velocities is no longer adequate to represent the total horizontal velocity close to the free surface.     

Momentum balance in shoaling gravity waves: Comment on ’shoaling surface gravity waves cause a force and a torque on the bottom’ by K. E. Kenyon

In a recent paper, Kenyon (2004) proposed that the wave-induced energy flux is generally not conserved, and that shoaling waves cause a mean force and torque on the bottom. That force was equated to the divergence of the wave momentum flux estimated from the assumption that the wave-induced mass flux is conserved. This assumption and conclusions are contrary to a wide body of observations and theory. Most importantly, waves propagate in water, so that the momentum balance generally involves the mean water flow. Although the expression for the non-hydrostatic bottom force given by Kenyon is not supported by observations, a consistent review of existing theory shows that a smaller mean wave-induced force must be present in cases with bottom friction or wave reflection. That force exactly balances the change in wave momentum flux due to bottom friction and the exchange of wave momentum between incident and reflected wave components. The remainder of the wave momentum flux divergence, due to shoaling or wave breaking, is compensated by the mean flow, with a balance involving hydrostatic pressure forces that arise from a change in mean surface elevation that is very well verified by observations.     

Particle trajectories of nonlinear gravity waves in deep water

The paper presents a numerical method for calculating the particle trajectories of nonlinear gravity waves in deep water. Particle trajectories, mass-transport velocity and Lagrangian wave period can be accurately determined by the proposed method. The high success rate of the proposed method is examined by comparing the present results with those of (a) Longuet-Higgins, M.S., 1986, 1987. Eulerian and Lagrangian aspects of surface waves. Journal of Fluid Mechanics 173, 683-707 and (b) Lagrangian moments and mass transport in Stokes waves. Journal of Fluid Mechanics 179, 547-555. It is shown that the dimensionless mass-transport velocity can exceed 10% for large waves, and the Lagrangian wave period is much larger than the Eulerian wave period for large waves.     

Wave force on vertically submerged circular thin plate in shallow water

The paper presented is a solution of shallow water wave force, using small amplitude linear wave theory on two-dimensional vertically submerged circular thin plates under three different configurations: (1) a surface-piercing circular thin plate, (2) a submerged circular thin plate, and (3) a bottom-standing circular thin plate. Finally Morison's equation is used for the determination of wave force which is based on the linear wave theory. The plate is submerged in water near the shore on uniformly sloping bottom. The solution method is confined in a finite domain, which contains both the region of different depth of water and the plate. Laplace's equation and boundary value problems are solved in a finite domain, by the method of separation of variables and the small amplitude linear wave theory. The variation of horizontal force by single particle, total horizontal force and moment with respect to the wave amplitude are obtained at different depth of water and at different wave period. It is observed that the force and moment are converging with the increase of wave period and the gradients of force and moment with respect to the wave amplitude are extremely high for lower wave period.     

不规则波作用下沿岸流流速分布规律分析

基于MIKE21 PMS,对1/10、1/20、1/40、1/100四种不同坡度地形下不规则波浪产生的沿岸流进行数值模拟。将模拟得到的流速随着离开静水岸线距离的变化曲线进行无量纲化处理,并选用高斯分布函数对流速曲线进行拟合,进而得到高斯分布函数中流速最大值与其出现位置以及流速分布宽度等参数随着入射波浪要素和地形坡度的变化关系。研究结果表明,文中给出的高斯分布函数适用于描述不规则波浪作用生成的沿岸流流速分布规律。     

Measurement and modeling of bed shear stress under solitary waves

Direct measurements of bed shear stresses (using a shear cell apparatus) generated by non-breaking solitary waves are presented. The measurements were carried out over a smooth bed in laminar and transitional flow regimes (~ 10⁴ < R_e < ~ 10⁵). Measurements were carried out where the wave height to water depth (h/d) ratio varied between 0.12 and 0.68; maximum near bed velocity varied between 0.16 m/s and 0.51 m/s and the maximum total shear stress (sum of skin shear stress and Froude–Krylov force) varied between 0.386 Pa and 2.06 Pa. The total stress is important in determining the stability of submarine sediment and in sheet flow regimes. Analytical modeling was carried out to predict total and skin shear stresses using convolution integration methods forced with the free stream velocity and incorporating a range of eddy viscosity models. Wave friction factors were estimated from skin shear stress at different instances over the wave (viz., time of maximum positive total shear stress, maximum skin shear stress and at the time of maximum velocity) using both the maximum velocity and the instantaneous velocity at that phase of the wave cycle. Similarly, force coefficients obtained from total stress were estimated at time of maximum positive and negative total stress and at maximum velocity. Maximum positive total shear stress was approximately 1.5 times larger than minimum negative total stress. Modeled and measured positive bed shear stresses are well correlated using the best convolution model, but the model underestimates the data by about 4%. Friction factors are dependent on the choice of normalizing using the maximum velocity, as is conventional, or the instantaneous velocity. These differ because the stress is not in phase with the velocity in general. Friction factors are consistent with previous data for monochromatic waves, and vary inversely with the square-root of the Reynolds number. The total shear stress leads the free stream fluid velocity by approximately 50°, whereas the skin friction shear stress leads by about 30°, which is similar to that reported by earlier researchers.     

Application of a nonlinear frequency domain wave–current interaction model to shallow water recurrence effects in random waves

The effect of ambient currents on nearshore nonlinear wave–wave energy transfer in random waves is studied with the use of a nonlinear frequency domain wave–current interaction model. We focus on the phenomenon of wave recurrence as a classical nonlinear phenomenon whose characteristics are well established for systems truncated to small numbers of frequency modes. The model used for this study is first extended to enhance accuracy; comparisons of permanent form solutions to analytical forms confirm the model accuracy. Application of the model to a highly truncated system confirmed the model’s consistency with published results for both positive (following) and negative (adverse) currents. Propagation of random wave spectra over a flat bottom was performed with the model, with the intent of determining the prevalence of recurrence between the spectral peak and its harmonics. For spectra of moderate Ursell number, it was found that positive currents extended the length scale of recurrence relative to the case with no currents; conversely, negative currents reduced the recurrence lengths. However, beyond a propagation distance of ≈40 wavelengths of the spectral peak, recurrence becomes almost completely damped as the spectra becomes broad and the spectral energies equilibrate. For spectra of high Ursell number, in contrast, recurrence is almost immediately damped, suggesting that the nonlinearity is sufficient to allow immediate spectral broadening and equilibration and overwhelming any preferential interactions among the spectral peak and its harmonics, regardless of current magnitude or direction.     

Validation of the three-wave quasi-kinetic approximation for the spectral evolution in shallow water

This paper aims at validating the three-wave quasi-kinetic approximation for the spectral evolution of weakly nonlinear gravity waves in shallow water. The problem is investigated using a one-dimensional numerical wave propagation model, formulated in the spectral representation. This model includes both a nonlinear triad interactions term and a wave breaking dissipation term. Some numerical tests were carried out in order to show the importance of using the triad nonlinear term in wave propagation spectral models, particularly to describe both behavior of the spectral integral parameters and of the spectral shape evolution in shallow water depth. Furthermore; a comparison against different set of experimental observations was carried out. Comparing the numerical results with the experimental observations made it possible to show the modeling efficiency of the three-wave quasi-kinetic approximation.     

浅水极限波浪几何特征的实验研究

该文通过物理模型实验,对浅水区域内的波浪在破碎前极限状态下的几何特征进行了研究。实验基于JONSWAP谱对不规则波浪进行模拟,通过对波群中出现的单体极限波浪进行捕捉并对波形进行测量而得到研究样本。为了考察底坡因素对极限波浪几何特征的影响,实验共考虑了3组大小分别为β=1/15、1/30以及1/45的地形坡度。统计结果表明,在实验所采用的坡度范围内,当地波高与水深对近岸极限波浪的影响最为显著,随着水深与波高因素变化,极限波浪的几何特征也出现明显的改变。坡度因素对极限波陡和偏度的影响很小,可以被忽略,但是对不对称度参数的影响相对比较明显,坡度越陡,不对称程度越剧烈。最后,通过参数化,本文给出了极限波浪几何特征变化的经验公式。     

Interaction of water waves with vertical cylinders using null-field integral equations

The scattering of water waves by bottom-mounted vertical circular cylinders is solved by using the null-field integral equations in conjunction with degenerate kernels and Fourier series to avoid calculating the Cauchy and Hadamard principal values. In the implementation, the null-field point can be exactly located on the real boundary owing to the introduction of degenerate kernels for fundamental solutions. An adaptive observer system of polar coordinates is considered to fully employ the properties of degenerate kernels. For the hypersingular equation, vector decomposition for the radial and tangential gradients is carefully considered. This method can be seen as a semi-analytical approach since errors attribute from the truncation of Fourier series. Neither hypersingularity in the Burton and Miller approach nor the CHIEF concept was required to deal with the problem of irregular frequencies. Five advantages of free of calculating principal value, well-posed algebraic system, convergence rate of exponential order, meshfree and elimination of boundary-layer effect, are achieved by using the present approach. Numerical results are given for the forces and free-surface elevation around the circular boundaries. Also, the near-trapped behavior arisen from the physical resonance is detected. A general-purpose program for water wave impinging several circular cylinders with arbitrary number, radii, and positions was developed. Several examples of water wave structure interaction by vertical circular cylinders were demonstrated to see the validity of the present formulation.     

Interaction of cnoidal waves with an array of vertical concentric porous cylinders

An array of large concentric porous cylinder arrays is mounted in shallow water exposed to cnoidal waves. The interactions between waves and cylinders are studied theoretically using an eigenfunction expansion approach. Semi-analytical solutions of hydrodynamic loads and wave run-up on each cylinder are obtained using first approximation to cnoidal waves. The square array configuration of four-legged identical concentric porous cylinder is investigated in present study. Numerical results reveal the variation of dimensionless wave force and wave run-up on individual cylinder with angle of incidence, porosity parameter, spacing between outer and inner cylinders, spacing between concentric porous cylinders and wave parameter. Different mechanism of wave force is found under different range of scattering parameter.     

Separation of incident and reflected waves in wave–current flumes

A technique is developed to separate the incident and reflected waves propagating on a known current in a laboratory wave–current flume by analyzing wave records measured at two or more locations using a least squares method. It can be applied to both regular and irregular waves. To examine its performance, numerical tests are made for waves propagating on quiescent or flowing water. In some cases, to represent the signal noise and measurement error, white noise is superimposed on the numerically generated wave signal. For all the cases, good agreement is observed between target and estimation.