Shallow water gravity waves: A note on the particle orbits

When surface gravity waves of small amplitude progress in shallow water of constant mean depth, the fluid particle orbits are observed to be oval, where the longer axis of the oval is parallel to the flat bottom, and at the bottom the orbits are straight lines. Potential flow, upon which the standard wave theory is based, predicts that the oval orbits are ellipses, but by a rather lengthy mathematical procedure that is founded on the questionable assumption of irrotationality. Using a more elementary and physical method, that does not employ the irrotational assumption, the elliptical orbits can be understood much more easily. The elementary method features a balance of two oppositely directed forces on each fluid particle: the outward centrifugal force and the inward pressure force.     

Cyclostrophic adjustment and nonlinear oscillations in the core of an intense atmospheric vortex

Intense atmospheric vortices are characterized by a regime of cyclostrophic balance, i.e., the balance between the pressure gradient and centrifugal force. To describe motions in the core of an axisymmetrical vortex, a class of exact solutions to the equations of gas dynamics with a linear dependence on radius is derived for the velocity components and with a quadratic dependence for temperature. It is shown that small deviations from the balance state give rise to oscillations of the hydrothermodynamic fields in the vortex core with a frequency proportional to the angular velocity of the rotation of the core. For fairly large initial deviations, oscillations are clearly anharmonic and, under the conditions of the prevailing centrifugal force, result in a significant temperature decrease on the vortex axis. The application of this class of solutions to describing the Ranque vortex effect (the intense cooling of gas during rapid rotations) and the acoustic radiation from tornadoes is discussed.     

Shoaling Surface Gravity Waves Cause a Force and a Torque on the Bottom

Freely propagating surface gravity waves are observed to slow down and to stop at a beach when the bottom has a relatively gentle upward slope toward the shore and the frequency range of the waves covers the most energetic wind waves (sea and swell). Essentially no wave reflection can be seen and the measured reflected energy is very small compared to that transmitted shoreward. One consequence of this is that the flux of the wave’s linear momentum decreases in the direction of wave propagation, which is equivalent to a time rate of change of the momentum. It takes a force to cause the time rate of change of the momentum. Therefore, the bottom exerts a force on the waves in order to decrease the momentum flux. By Newton’s third law (action equals reaction) the waves then impart an equal but opposite force to the bottom. In shallow (but finite) water depths the wave force per unit bottom area is calculated, for normal angle of incidence to the beach, to be directly proportional to the square of the wave amplitude and to the bottom slope and inversely proportional to the mean depth; it is independent of the wave frequency. Constants of proportionality are: 1/4, the fluid density and the acceleration of gravity. Swell attenuation near coasts and some characteristics of sand movement in the near-shore region are not inconsistent with the algebraic structure of the wave force formula. Since the force has a depth variation which is significantly faster than that of the dimensions of the particle orbits in the vertical direction, the bottom induces a torque on the fluid particles that decreases the angular momentum flux of the waves. By an extension of Newton’s third law, the waves also exert an equal but opposite torque on the bottom. And because the bottom force on the waves exists over a horizontal distance, it does work on the waves and decreases their energy flux. Thus, theoretically, the fluxes of energy, angular and linear momentum are not conserved for shoaling surface gravity waves. Mass flux, associated with the Stokes drift, is assumed to be conserved, and the wave frequency is constant for a steady medium.     

Gravity torques for surface waves

The effects of the gravity torques acting on the angular momentum of surface gravity waves are calculated theoretically. For short crested waves the gravity torque is caused by the force of gravity on the orbiting fluid particles acting down the slopes of the crests and troughs and in the direction parallel to the crests and troughs. The gravity torque tries to rotate the angular momentum vectors, and thus the waves themselves, counterclockwise in the horizontal plane, as viewed from above, in both hemispheres. The amount of rotation per unit time is computed to be significant assuming reasonable values for the along-crest and trough slopes for waves in a storm area. The gravity torque has a frequency which is double the frequency of the waves. For long crested waves the gravity torque acts in the vertical plane of the orbit and tries to decelerate the particles when they rise and accelerate them when they fall. By disrupting the horizontal cyclostrophic balance of forces on the fluid particles (centrifugal force versus pressure force) the gravity torque accounts qualitatively for the three characteristics of breaking waves: that they break at the surface, that they break at the crest, and that the crest breaks in the direction of wave propagation.     

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.     

The study of wave pressure and uplift force on a submerged plate

Boundary element method (BEM) is used to study wave pressure acting on a vertical quay, as well as the uplift forces on a submerged plate. The plate, placed in front of the quay, is bored with holes. The results are then compared with the case of a plate without any perforation.The results showed that: for q/h = 0.45, where q is the depth of submergence of the plate and h is water depth, wave pressure acting on the vertical quay is very similar for plates with or without borings. However, with q/h = 0.20, wave pressure on the quay is changed according to wave periods, widths of the plate, the distance of the plate away from the quay, as well as the ratio of the holes to the plates. Uplift forces exerted on the plate are indeed smaller for plates with borings than without, and decrease with increasing porous ratio. Furthermore, the uplift forces will be the same for the same porous ratios, regardless of these ratios being caused by a single or a double perforation.     

Hydrodynamic coefficients for inclined cylinders

The wave induced dynamic pressures around a circular cylinder of diameter 0.2 m due to regular waves were measured in a wave flume in a water depth of 1 m and in a wave basin in a water depth of 3 m. The experimental investigations were carried out with the cylinder inclined along and against the direction of wave propagation. The least-squares technique was employed to evaluate the coefficients of drag (C_D) and inertia (C_M) from the sectional force time histories obtained by integrating the measured circumferential pressure distribution. The variation of drag and inertia coefficients are presented as a function of Keulegan-Carpenter number (KC) for different inclinations of the cylinder. The comparison between the measured and the theoretical force derived from the evaluated hydrodynamic coefficients is found to be good.     

Calculating the Wave Force on Partially Immersed Large-Scale Horizontal Cylinders

Large-scale interceptors constitute the main structure of offshore self-driven floating marine litter collection devices,and the structural stability of such interceptors under the action of waves directly influences the overall safety of the device.When the ratio of the diameter of a horizontal cylinder in such interceptors to the incident wavelength is larger than 0.25,the wave force can be calculated by using the diffraction theory,by considering the problem as that of the interaction between the waves and a partially immersed large-scale horizontal cylinder.In this study,an analytical approach to calculate the wave force on a partially immersed large-scale horizontal cylinder was formulated by using the stepwise approximation method.Physical model tests were conducted to investigate the effects of different factors(wave height,period,and immersion depth)on the wave force on a large-scale horizontal cylinder under conditions involving short-period waves.The results show that both horizontal and vertical wave forces on the cylinder increase as the wave height(immersion depth)increases in most cases.The vertical wave force decreases with the decrease of the period.For the horizontal wave force,it increases with the decrease of the period when the wavelength is larger than the diameter of the cylinder and decreases with the decrease of the period when the wavelength is smaller than the diameter of the cylinder.     

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.     

Wave interaction with a submarine pipeline in clayey soil due to random waves

The wave pressure and uplift force due to random waves on a submarine pipeline (resting on bed, partially buried and fully buried) in clayey soil are measured. The influence of various parameters viz., wave period, wave height, water depth, burial depth and consistency index of the soil on wave pressures around and uplift force on the submarine pipeline was investigated. The wave pressures were measured at three locations around the submarine pipeline (each at 120° to the adjacent one). It is found that the wave pressure and uplift force spectrum at high consistency index of the soil is smaller compared to that of low consistency index. Just burying the pipeline (e/D=1.0) in clayey soil reduces the uplift force to less than 60% of the force experienced by a pipeline resting on the seabed (e/D=0.0) for I_c=0.33.     

Significance of relative velocity in the estimation of drag force of a tethered spherical float

In the present study, a tethered spherical float undergoing oscillatory motion in regular waves is analysed. Float velocity is computed through dynamical equation of motion and particle velocity using linear wave theory. The results show that variation of particle velocity with respect to wave period, wave height or water depth is small compared to the variation of float velocity with respect to the same parameters. The results further indicate that the difference between float velocity and particle velocity is considerable and, in such cases, the relative velocity instead of the particle velocity has to be considered for drag force or drag power estimation. However, it is suggested that float velocity must be higher than the particle velocity in order to use relative velocity in drag force or drag power estimation.     

Effects of Topography and Current on Horizontal Irrotational Waves in Shallow Water

Based on the Boussinesq assumption,derived are couple equations of free surface elevationand horizontal velocities for horizontal irrotational flow,and analytical expressions of the correspondingpressure and vertical velocity.After the free surface elevation and horizontal velocity at a certain depth areobtained by numerical method,the pressure and vertical velocity distributions can be obtained by simplecalculation.The dispersion at different depths is the same at the O(ε)approximation.The waveamplitude will decrease with increasing time due to viscosity,but it will increase due to the matching ofviscosity and the bed slope.thus,flow is unstable.Numerical or analytical results show that the waveamplitude.velocity and length will increase as the current increases along the wave direction.but theamplitude will increase.and the wave velocity and length will decrease as the water depth decreases.     

Experimental Study on Ocean Internal Wave Force on Vertical Cylinders in Different Depths

A series of experimental studies about the force of internal solitary wave and internal periodic wave on vertical cylinders have been carried out in a two-dimensional layered internal wave flume. The internal solitary waves are produced by means of gravitational collapse at the layer thickness ratio of 0.2, and the internal periodic waves are produced with rocker-flap wave maker at the layer thickness ratio of 0.93. The wave parameters are obtained through dyeing photography. The vertical cylinders of the same size are arranged in different depths. The horizontal force on each cylinder is measured and the vertical distribution rules are researched. The internal wave heights are changed to study the impact of wave heights on the force. The results show that the horizontal force of concave type internal solitary wave on vertical cylinder in the upper-layer fluid has the same direction as the wave propagating, while it has an opposite direction in the lower-layer. The horizontal force is not evenly distributed in the lower fluid. And the force at different depths increases along with wave height. Internal solitary wave can produce an impact load on the entire pile. The horizontal force of internal periodic waves on the vertical cylinders is periodically changed at the frequency of waves. The direction of the force is opposite in the upper and lower layers, and the value is close. In the upper layer except the depth close to the interface, the force is evenly distributed; but it tends to decrease with the deeper depth in the lower layer. A periodic shear load can be produced on the entire pile by internal periodic waves, and it may cause fatigue damage to structures.     

方形网箱水平波浪力的迭加计算和实验验证

采用小直径圆柱体绕流理论和网渔具理论为基础的经验水动力计算法，分别考虑网箱框架是刚性体、网衣和绳索是柔性体的特征，结合正弦波理论对方形网箱及其构件受到的水平波浪力特性进行了分析研究，理论给出了波浪力迭加计算法，并把计算结果与小尺度的网箱水槽实验进行对比验证。结果表明，计算数值与试验数据较接近，平均误差率在15％以内。网箱水平波浪力呈周期性、不对称变化，在波浪周期小于0.72s时，主要以框架的波浪力为主，反之以网衣的为主。在波长为0.8m、周期为0.72s、水深为0.7m、波高为68.3mm情形下，计算结果显示网箱的框架尺寸和形状及其系泊、布局的选择应根据实际海况而定，网箱高度可适当增加，当设置水层下降深度相对于网箱高度比率为20％-40％，时，波浪力峰值下降相对率达68.03％-88.20％，表明可以较大程度地降低其波浪力。     

Experimental study on the performance of twin plate breakwater

The wave transmission characteristics and wave induced pressures on twin plate breakwater are investigated experimentally in regular and random waves.A total of twenty pressure transducers are fixed on four surfaces of twin plate to measure the wave induced dynamic pressures.The spatial distribution of dynamic wave pressure is given along the surface of the twin plate.The uplift wave force obtained by integrating the hydrodynamic pressure along the structure is presented.Discussed are the influence of different incident wave parameters including the relative plate width B /L,relative wave height /i H a and relative submergence depth s /a on the non-dimensional dynamic wave pressures and total wave forces.From the investigation,it is found that the optimum transmission coefficient,t K occurs around B /L 0.41 ~ 0.43,and the twin plate breakwater is more effective in different water depths.The maximum of pressure ratio decreases from 1.8 to 1.1 when the relative submergence depth of top plate is increased from 0.8to +0.8.     

Inertial wave loads on horizontal cylinders: A field experiment

A long submerged horizontal circular cylinder of .90 m diameter was assembled off the beach at Reggio Calabria where the wind waves typically have significant height ranging within 0.20 and 0.40 m and dominant period within 1.8 and 2.6 s. Three ultrasonic probes recorded the waves, and two sets of pressure transducers, the first one at the cylinder and the second one in the undisturbed wave field, enabled to compare the force amplitude on the cylinder to the force amplitude on an equivalent mass of water in the undisturbed wave field (Froude-Krylov F-K force). After ten days of measurements, the experiment was repeated with a cylinder of .45 m diameter. The Keulegan-Carpenter number was within 2.5, and the wave forces proved to be inertial. The following general features emerged: (i) the force spectrum is usually very narrow even if the wave spectrum is broad; (ii) the vertical diffraction coefficient is somewhat smaller than the horizontal diffraction coefficient; (iii) the positive extremes of F_z (vertical force referred to the buoyancy force) markedly exceed the negative extremes; (iv) the pressure fluctuations induced by the highest waves at the cylinder are very similar to the measured pressure-surface displacement covariances. In each of the 580 records obtained in the course of the experiment it was found that the propagation speed reduces to about a half at the cylinder, and the amplitude of the pressure fluctuations increases of 10–15% at the upper half of the cylinder and decreases of about the same percentage at the lower half. These phenomena fully explain why the force amplitude on the cylinder is larger than the F-K force amplitude.     

Wave transmission and reflection characteristics of a rigid surface and submerged horizontal plate

The wave transmission and reflection characteristics of a rigidly fixed surface and submerged horizontal plate were investigated experimentally in detail for a wide range of incident wave steepnesses and for different depths of submerge of the plate in deep water conditions in regular water wave fields. The experiments were conducted at the Ocean Engineering Centre, Indian Institute of Technology, Madras, India, in a wave flume 10 m long, 0.3 m wide and in a constant water depth of 0.8 m. The horizontal plate is 0.22 m thick and 1.2 m in length, covering the enrire width of the flume. From the present investigation, it is found that for a rigid surface plate, the coefficient of transmission is a minimum and the coefficient of reflection is a maximum, but the maximum value of the coefficient of energy loss occurs for plates submerged closer to the still-water level (SWL) and not for the surface plate. It is also found that the value of the coefficient of reflection increases with the increase in the value of the Reddy-Neelamani (RN) number, the ratio of horizontal water particle excursion at the bottom of the plate in its absence to the length of the plate. The coefficient of transmission is found to decrease rapidly with increase in the value of RN number up to 0.1. The wave transmission is only 5% for RN from 0.1 to 0.2. It is also found that for RN number greater than 0.04, the minimum energy dissipation is consistently about 60% of the incident wave energy.     

波浪作用下方块石护面稳定性研究

护面是海堤和护岸的重要结构,直接抵御波浪作用,可采用人工块体、块石等,种类繁多。采用紧密排列方块石作为护面结构是一种景观性较好的型式,依据方块石厚度不同能抵御不同大小的波浪作用。干砌条石及干砌块石护面曾有一些规范给出过计算方法,但现行规范没有相关内容可供设计参考,已有计算方法的理论分析还存在不足。当波浪与斜坡堤相互作用时,方块石护面出现位移或脱落可能发生在波浪回落最低阶段、波浪破碎打击阶段及破后爬高水流作用阶段,通过研究得到了不同阶段波浪对方块石护面作用力的计算方法。在波浪回落最低阶段,考虑了护面及其下方垫层渗透性影响,通过理论分析建立了低渗透护面浮托压强计算模型,采用物模试验将计算结果与试验测量值进行了对比分析,结果表明总体趋势符合,量值接近;在波浪破碎冲击阶段,基于射流冲击作用原理,提出了波浪在斜坡面破碎冲击压强计算方法,通过试验分析了波浪破碎水深波高比与破波相似参数的关系,利用浅水波理论计算了波浪破碎冲击水流流速;在爬高水流作用阶段,提出了水流引起的方块石护面垂直浮托力及水平拖曳力的计算方法,通过试验结果拟合了浮托力系数和拖曳力系数,验证了水流作用下护面的受力特征。最后,针对方块...     

杰氏风暴潮剖面预报方法在我国的应用

近年来，在美国风暴潮数值预报中，Jelesnianski提出了一个剖面预报模式(以下简称杰氏模式)，在作业上是很独特的。这一模式在某种合理规定的限度内，充分显示出预报基本风暴潮的技巧，并已成功地对美国东海岸及墨西哥湾沿岸的风暴潮进行了预报。因此，目前已被美国国家天气服务局确定为风暴潮的业务预报工具。 杰氏模式源于风暴潮的流体动力学方程组的数值解。与其他研究者不同的是：它在业务预报实践中，只要利用事先算好的一个在“标准海域”中、由“标准风暴”引起的内边界增水剖面，然后，通过预报海域的风场订正因子和深度订正因子进行修正，使其与实际海域和实际风暴相一致，就可作出预报。由于用这种方法所给出的结果是一条沿海域内边界的风暴潮剖面图，故称为风暴潮剖面预报方法。应用这种方法所需的基本资料是：风暴移行速度、风暴相对于海岸线的移动方向、最大风速半径、风暴中心气压及风暴登陆点附近的海域水深。因此，只要深度剖面订正值为已知，台风强度及登陆点预报得准确，利用杰氏模式就可以很快作出离风暴登陆点不同距离的沿岸增水高度及最大增水出现位置的预报。为此，我们进行了杰氏风暴潮剖面预报方法在我国的应用试验和研究。应该指出，这种方法不仅对台风暴潮的预报有用，而且对确定港工建筑，防潮工程等极值水位的设计也有其重要的应用价值。 杰氏模式能否应用到我国台风暴潮预报业务中的问题，关键在于如何确定最大风速半径以及我国沿岸的深度订正因子。实践表明：当我们确定了这两个参数后，应用杰氏方法对我国东南沿海登陆型台风进行后报，获得满意的结果。本文中，我们将着重讨论如何确定最大风速半径和给出深度订正因子问题。并简单介绍其他参数的选取方法及怎样预报台风暴潮剖面，对杰氏模式则不拟过多涉及。     

A study of Coriolis effects on long waves propagating in an unbounded ocean, channel and basin

The effects of Coriolis force on long waves have been discussed based on gravity waves propagating in an unbounded ocean, channel and basin. In case of ocean, results show that the Coriolis effect will be significant and negligible, when the wave period is comparable to 2π/f and much shorter, respectively. Results also show in a channel, the wave amplitude and water particle velocity decrease exponentially in the positive y direction in the northern hemisphere (where f is positive). Moreover, in a basin, the Cotidal lines have been found as curves and rotate counterclockwise around the origin.