Wave action on a movable plate immersed into a fluid. Physical and numerical experiments

The effect of the motion of a rigid inclined plate immersed into a fluid counter incident waves is studied both experimentally and numerically. In the wave water channel experiments the velocity of a trolley with a plate running freely on rails is determined as a function of the wave parameters, the immersion depth, the angle of inclination, and the plate dimensions. The interaction between the traveling waves and the plate having a single translational degree of freedom along the horizontal axis is numerically calculated in the time-dependent, two-dimensional formulation. The dependence of the upwave motion effect on the parameters varied in the full-scale experiment is analyzed. In the numerical experiment a regime of the downwave plate motion at a constant high velocity is found to exist. The channel bottom effect is estimated and the behavior of the plate with a flap is studied.     

Testing and calculations of the motion of a model ship with a direct-flow wave propulsor

A new design of the wave propulsor is presented. In this design the thrust mechanism is due to the interaction between the waves and the ship structure elements rather than to ship’s motions. To verify the possibility of using a rigidly fixed inclined plate as the ship wave propulsor a model catamaran was constructed in the Institute of Mechanics of Moscow State University. The effect of the upwave motion of the ship, whose mean velocity is a nonmonotonic function of the wavelength, is studied. As the plate edge pierces the water surface, the ship starts to move in the opposite direction, that is, downwave. The experimentally observable effects are also revealed in the numerical simulation using the XFlow software package which involves the meshless lattice Boltzmann method. On the basis of the calculated results it is shown that the upwave motion effect is due to a variation in the hydrostatic force component in the case of wave breakdown.     

The oblique-plate impact experiment

An experimental method is presented for the study of one-dimensional plane waves corresponding to combined pressure and shear. The experiment involves the impact of two skewed flat plates. A projectile plate is accelerated using a gas gun and made to impact a target plate in a vacuum chamber. The projectile and target plates are parallel, but inclined relative to the axis of the gun so that the particle velocity in the target has components both normal and parallel to the plane of impact. The particle velocity at the target rear (free) surface is recorded as a function of time. The normal velocity component is monitored using a laser velocity interferometer; the transverse motion is monitored using a shadow technique. The measured wave profiles can be compared to theoretical predictions based on one-dimensional-wave theory.     

Wave-induced dynamics of a particle on a thin circular plate

In this paper, the dynamics of a particle placed on a thin circular plate carrying circumferential harmonic travelling wave is studied. Coulomb friction is used to model the particle–surface interaction. Distinct regions on the plate surface are identified where either of the three phases of particle motion, namely jumping, sliding and sticking, occurs. Also, the effect of wave frequency and the plate geometry on these regions is studied. Interestingly, there exists an optimum plate thickness for which the region of sliding is maximum. At certain wave frequencies, from the numerical simulations within sticking and sliding regions, it is observed that the average particle motion spirals inwards towards the plate centre. Such an average motion is observed whenever the particle is placed initially with a zero velocity relative to the plate surface. The Gedanken experiments discussed herein provide cogent explanations to all the observed average (slow) dynamics and are also found to be useful in predicting the slow dynamics of the particle a priori, that is, before the actual numerical simulations. The particle’s velocity couples the radial and tangential sliding friction components and is found to be the key physical feature that explains the observed behaviour. Also, it is observed that the plate surface excited by circumferential travelling waves can provide acoustic lubrication to a particle by reducing the limiting force required to move it relative to the surface. The methods discussed in this paper can be extended to study the dynamics of a group of particles (granular materials) and extended rigid bodies, interacting with such surface waves.

     

Rayleigh-type waves in a water-saturated porous half-space with an elastic plate on its surface

The paper deals with the plane problem of steady-state time harmonic vibrations of an infinite elastic plate resting on a water-saturated porous solid. The displacements of the plate are described by means of the linear theory of small elastic oscillations. The motion of the two-phase medium is studied within the framework of Biot's linear theory of consolidation. The main interest is focused on the investigation of properties of the Rayleigh-type waves propagating alongside of the contact surface between the plate and the porous half-space. In particular, the dependence of the phase velocity and attenuation of the waves on the plate stiffness, mass coupling coefficient, and degree of saturation of the medium is studied. Besides, for the limiting case of an infinitely thin plate, the comparison of the wave characteristics is carried out with those of the pure Rayleigh waves.     

Flexural gravity wave motion over poroelastic bed

Using Biot’s consolidation theory, effect of poroelastic bed on flexural gravity wave motion is analyzed in both the cases of single-layer and two-layer fluids. The model for the flexural gravity waves is developed using linear water wave theory and small amplitude structural response in finite water depth. The effects of permeability and shear modulus of poroelastic bed and time period on flexural gravity wave motion are studied by analyzing the dispersion relation, phase speed, plate deflection, interface elevation and pressure distribution along water depth. Various results for surface gravity waves are analyzed as special cases. The study reveals that bed permeability retards the hydrodynamic pressure distribution along the water depth significantly compared to shear modulus whilst, floating plate deflection decreases significantly with change in shear modulus compared to permeability of the poroelastic bed. The present study can be generalized to analyze various wave–structure interaction problems over poroelastic bed.     

On waves due to rolling of a ship in water of finite depth

Summary The problem of the generation of waves due to small rolling oscillations of a thin vertical plate partially immersed in uniform finite-depth water is investigated here by utilizing two mathematical methods assuming the linearised theory of water waves. In the first method, the use of eigenfunction expansion of the velocity potentials on the two sides of the plate produces the amplitude of wave motion at infinity in terms of an integral involving the unknown horizontal velocity across the gap, and also in terms of another integral involving the unknown difference of the potential across the plate. These unknown functions satisfy two integral equations. Any one of these, when solved numerically, can be used to compute the amplitude of the wave motion set up at either infinity on the two sides of the plate for various values of the wave number.In the second method, the problem is formulated in terms of a hypersingular integral equation involving the difference of the potential function across the plate. The hypersingular integral equation is solved numerically, and its numerical solution is used to compute the wave amplitude at infinity. The two methods produce almost the same numerical results. The results are illustrated graphically, and a comparison is made with the deep-water result. It is observed that the deep-water result effectively holds good if the plate is partially immersed to the order of one-tenth of the bottom depth.This work was initiated when the first Author was visiting Mathematics Department, Indian Institute of Science, Bangalore. It was partially supported by DST, and by CSIR. The authors take this opportunity to thank the Managing Editor for his suggestions to improve the paper in the present form.     

Acoustic waves emitted by a vortex ring passing near a wedge-like plate

Acoustic waves emitted by a vortex ring moving near a thin wedge-like plate of finite width have been studied. The experiments are performed for three configurations: the plate (A) is held edgeways to the direction of the vortex motion, (B) is held sideways to the direction, and (C) is held edgeways at an angle of 45° against the vortex motion. The observed sound wave is of dipole radiation type, and the magnitude of the pressure is large in the direction of the normal to the plate plane and small in parallel. The observed pressure is proportional to the third power of the vortex speed. The instantaneous force exerted on the plate by the vortex motion has also been examined. The force vector is mainly normal to the plate plane. The observed profiles agree within a reasonable degree of accuracy with the theoretical ones calculated for the vortex ring interacting with the flat plate of thickness zero.     

Transient bending waves in plates studied by hologram interferometry

Propagating bending waves are studied in plates made of aluminum and wood. The waves are generated by the impact of a ballistic pendulum. Hologram interferometry, with a double pulsed ruby laser as the light source, is used to record the out of plane motion of the waves. Elliptic-like fringes visualize differences in wave speed for different directions in the anisotropic plate and circular ones are obtained for the isotropic plate. The experimental data for the isotropic plate compare favorably with analytical results derived from the Kirchhoff-plate equation with a point impact of finite duration. A similarity variable is found when starting conditions are modeled as a Dirac pulse in space and time, that brings new understanding to the importance of specific parameters for wave propagation in plates. A formal solution is obtained for a point force with an arbitrary time dependence. For times much larger than the contact time, the plate deflection is shown to be identical to that from a Dirac pulse applied at the mean contact time. A method for determining material parameters, and the mean contact time, from the interferograms is hence developed.     

Plotnikov——Toland板模型中水弹性孤立波的迎撞

通过奇异摄动方法研究了在薄冰层覆盖的不可压缩理想流体表面上传播的两个水弹性孤立波之间的迎面碰撞.借助特殊的 Cosserat 超弹性壳 理论以及Kirchhoff--Love 板理论,冰层由 Plotnikov--Toland板模型描述.流体运动采用浅水假设和Boussinesq 近似. 应用Poincaré--Lighthill--Kuo 方法进行坐标变形,进而渐近求解控制方程及边界条件, 给出了三阶解的显式表达. 可以观察到碰撞后的孤立波不会改变它们的形状和振幅. 波浪轮廓在碰撞之前是对称的, 而在碰撞之后变成不对称的并且在波传播方向上向后倾斜. 弹性板和流体表面张力减小了波幅. 图示比 较了本文与已有结果可知线性板模型可作为本文的一个特例.      

Application of PIV to velocity measurements in a liquid film flowing down an inclined cylinder

PIV technique is applied for measurements of instant velocity distributions in a liquid film flowing down an inclined tube in the form of a wavy rivulet. An application of special optical calibration is applied to correct distortion effects caused by the curvature of the interface. A vortex flow of liquid is observed inside a wave hump in the reference system moving with wave phase velocity. Conditionally averaged profiles of longitudinal and transverse components of liquid velocity are obtained for different cross-sections of developed non-linear waves. It is shown that the increase in wave amplitude slightly changes the location of the vortex center. The analysis of modification of vortex motion character due to wavy flow conditions, such as tube inclination angle, film Reynolds number, wave excitation frequency, is fulfilled.     

Plane elastic standing waves of finite amplitude

A plane standing wave solution is obtained for homogeneous, isotropic, incompressible nonlinearly elastic solids. The motion describes the nonlinear interaction of two oppositely propagating, circularly polarized waves. It is used to obtain exact steady state solutions for nonlinear vibrations of a plate and for reflection and transmission of finite amplitude, circularly polarized waves at a plane interface.     

Roll waves in a gas–liquid medium

The onevelocity motion of a gas–liquid medium with a variable mass fraction of the gas phase, which is equilibrium in terms of phase pressures, is considered. The existence conditions of nonlinear periodic wave packets similar in structure to roll waves in open inclined channels are found. The structure of travelling waves in the medium with continuous addition of energy to the gas phase is studied.     

Permanent Waves in Slow Free-Surface Flow of a Herschel–Bulkley fluid

Unsteady flow of a viscoplastic fluid on an inclined plane is examined. The fluid is described by the three-parameter Herschel–Bulkley constitutive equation. The set of equations governing the flow is presented, recovering earlier results for a Bingham fluid and steady uniform motion. A permanent wave solution is then derived, and the relation between wave speed and flow depth is discussed. It is shown that more types of gravity currents are possible than in a Newtonian fluid; these include some cases of flows propagating up a slope. The speed of permanent waves is derived and the possible surface profiles are illustrated as functions of the flow behavior index.     

Propagation of Lamb waves in transversely isotropic thermoelastic diffusive plate

The constitutive relations and field equations for anisotropic generalized thermoelastic diffusion are derived and deduced for a particular type of anisotropy, i.e. transverse isotropy. Green and Lindsay (GL) theory, in which, thermodiffusion and thermodiffusion–mechanical relaxations are governed by four different time constants, is selected for study. The propagation of plane harmonic thermoelastic diffusive waves in a homogeneous, transversely isotropic, elastic plate of finite width is studied, in the context of generalized theory of thermoelastic diffusion. According to the characteristic equation, three quasi-longitudinal waves namely, quasi-elastodiffusive (QED-mode), quasi-massdiffusive (QMD-mode) and quasi-thermodiffusive (QTD-mode) can propagate in addition to quasi-transverse waves (QSV-mode) and the purely quasi-transverse motion (QSH-mode), which is not affected by thermal and diffusion vibrations, gets decoupled from the rest of the motion of wave propagation. The secular equations corresponding to the symmetric and skew symmetric modes of the plate are derived. The amplitudes of displacements, temperature change and concentration for symmetric and skew symmetric modes of vibration of plate are computed numerically. Anisotropy and diffusion effects on the phase velocity, attenuation coefficient and amplitudes of wave propagation are presented graphically in order to illustrate and compare the analytically results. Some special cases of frequency equation are also deduced from the existing results.     

The non-linear,dynamic response of an impulsively loaded circular plate with a central hole

A numerical method, called Direct Analysis, is described and applied to solve the problem of a plate undergoing a large impulsive load. For generality, an expanded, non-linear form of the equations of motion is used and shear correction and rotatory inertia are considered. The wave speeds are calculated from the non-linear equations and appropriate boundary conditions are applied so that reflected waves are included. The results for two types of step loading pulses are presented and compared with previously presented solutions. The response of the plate is discussed and conclusions as to the effects of the non-linearities are given.     

Effects of SH waves in a functionally graded plate

A computational method is presented to investigate SH waves in functionally graded material (FGM) plates. The FGM plate is first divided into quadratic layer elements (QLEs), in which the material properties are assumed as a quadratic function in the thickness direction. A general solution for the equation of motion governing the QLE has been derived. The general solution is then used together with the boundary and continuity conditions to obtain the displacement and stress in the wave number domain for an arbitrary FGM plate. The displacements and stresses in the frequency domain and time domain are obtained using inverse Fourier integration. Furthermore, a simple integral technique is also proposed for evaluating modified Bessel functions with complex valued order. Numerical examples are presented to demonstrate this numerical technique for SH waves propagating in FGM plates.     

Influence of a longitudinally compressed elastic plate on the nonstationary wave motion of a homogeneous liquid

In a linear formulation, a study is made of the influence of a longitudinally compressed elastic isotropic plate on the nonstationary wave motion of a stream of homogeneous liquid of finite depth on which the plate floats. The waves are generated by periodic (in time) normal stresses applied to a restricted region of the plate surface and beginning at a certain initial time.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 68–75, September–October, 1980.     

倾斜正交异性矩形板热振动分岔

研究了倾斜正交异性矩形板在热状态下的振动分岔,讨论分析了温度、长宽比、板厚、倾斜角对正交异性矩形板发生混沌运动区域的影响。