Unsteady Behavior of an Elastic Beam Floating on the Surface of an Infinitely Deep Fluid

The effect of initial disturbances and unsteady external loading on an elastic beam of finite length which floats freely on the surface of an ideal incompressible fluid is studied in a linear treatment. The fluid flow is considered potential. The beam deflection is sought in the form of an expansion in the eigenfunctions of beam vibrations in vacuum with time-dependent amplitudes. The problem reduces to solving an infinite system of integrodifferential equations for unknown amplitudes. The memory functions entering this system are determined by solving the radiation problem. The beam behavior is studied for various loads with and without allowance for the weight of the fluid. The effect of fluid depth on beam deformation was determined by comparing with the previously obtained solutions of the unsteady problem for a beam floating in shallow water. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 1, pp. 85–94, January–February, 2006.     

Nonlinear Capillary-Gravity Waves on the Charged Surface of an Ideal Fluid

A second-order asymptotic expression for the profile of a capillary-gravity wave traveling over the charged surface of an ideal incompressible fluid is calculated analytically. Two types of steady-state profiles of nonlinear periodic capillary-gravity waves are found. For a certain fixed dimensionless surface charge the shape of the tops of the nonlinear waves changes: from blunt to pointed for short waves and from pointed to blunt for long waves.     

Gravity Waves due to Discontinuity Decay Over an Open-Channel Bottom Drop

Experimental data are given on wave shapes and propagation speeds and characteristic headwater and tailwater depths after removal of a shield producing an initial free-surface level drop and located above a bottom drop in a rectangular open channel. Check is performed of self-similar solutions of the problem obtained earlier using a hydraulic approximation. It has been established that in certain ranges of time, longitudinal coordinate, and problem parameters, these solutions are supported by experimental results.     

Gravity and Capillary-Gravity Periodic Travelling Waves for Two Superposed Fluid Layers,One Being of Infinite Depth

The mathematical study of 2D travelling waves in the potential flow of two superposed layers of perfect fluid, with free surface and interfaces (with or without surface tensions) and with the bottom layer of infinite depth, is set as an ill-posed reversible evolution problem, where the horizontal space variable plays the role of a "time". We give the structure of the spectrum of the linearized operator near equilibrium. This spectrum contains a set of isolated eigenvalues of finite multiplicities, a small number of which lie near or on the imaginary axis, and the entire real axis constitutes the essential spectrum, where there is no eigenvalue, except 0 in some cases. We give a general constructive proof of bifurcating periodic waves, adapting the Lyapunov-Schmidt method to the present (reversible) case where 0 (which is "resonant") belongs to the continuous spectrum. In particular we give the results for the generic case and for the 1 : 1 resonance case.     

Waves on the Surface of an Ideal Incompressible Heavy Fluid under Wind Loads

The gravity-forced motion of an ideal incompressible fluid of infinite depth is studied when a periodic pressure is applied to the surface of the fluid. This problem is solved on the basis of the small amplitude wave theory. The analytical solutions for the velocity potential, the velocity field, and the shape of the free surface are found. An expression for the horizontal force is obtained in the case of a traveling wave.     

Generation of Transient Waves by Impulsive Disturbances in an Inviscid Fluid with an Ice-Cover

The dynamic responses of an ice-covered fluid to impulsive disturbances are analytically investigated for two- and three-dimensional cases. The initially quiescent fluid of infinite depth is assumed to be inviscid, incompressible and homogenous. The thin ice-cover is modelled as a homogenous elastic plate with negligible inertia. Four types of impulsive concentrated disturbances are considered, namely an instantaneous mass source immersed in the fluid, an instantaneously dynamic load on the plate, an initial impulse on the surface of the fluid, and an initial displacement of the ice plate. The linearized initial-boundary-value problem is formulated within the framework of potential flow. The solutions in integral form for the vertical deflexions at the ice-water interface are obtained by means of a joint Laplace-Fourier transform. The asymptotic representations of the wave motions for large time with a fixed distance-to-time ratio are derived by making use of the method of stationary phase. It is found that there exists a minimal group velocity and the wave system observed depends on the moving speed of the observer. For an observer moving with the speed larger than the minimal group velocity, there exist two trains of waves, namely the long gravity waves and the short flexural waves, the latter riding on the former. Moreover, the deflexions of the ice-plate for an observer moving with a speed near the minimal group velocity are expressed in terms of the Airy functions. The effects of the presence of an ice-cover on the resultant wave amplitudes, the wavelengths and periods are discussed in detail. The explicit expressions for the free-surface gravity waves can readily be recovered by the present results as the thickness of ice-plate tends to zero.     

Determination of Natural Frequencies of Internal Waves in an Essentially Inhomogeneous Fluid

The two-dimensional problem of the natural wave motions of a heavy inhomogeneous fluid confined between two absolutely rigid plane-parallel boundary surfaces is investigated. It is assumed that the fluid is stably and continuously stratified in the vertical. The motions are considered in the class of waves moving in the horizontal direction. An efficient numerical-analytic method of solving and analyzing the corresponding Sturm-Liouville problem for a coordinate function depending on the depth is proposed. A technique for constructing high-accuracy two-sided estimates of the natural frequencies of the traveling waves is developed. A recurrent accelerated-convergence scheme for the refinement of the eigenvalues (natural frequencies) and eigenfunctions (forms) of the wave motion of the fluid is proposed and tested. Two examples of stratification are calculated and the dispersion curves and the oscillation shapes are constructed.     

Internal Gravity Waves Generated by an Oscillating Source of Perturbations Moving with Subcritical Velocity

This paper considers the problem of constructing far-field asymptotics of internal gravity waves generated by an oscillating local source of perturbations moving in a stratified flow of finite depth. The velocity of the perturbation source does not exceed the maximum group velocity of an individual wave mode. The wave pattern consists of waves of two types: annular and wedge-shaped. Solutions expressed in terms of the Hankel function are obtained for the asymptotics of annular waves. The asymptotics of wedge-shaped waves are expressed in terms of the Airy function and its derivative.     

Bounds for Arbitrary Steady Gravity Waves on Water of Finite Depth

Necessary conditions for the existence of arbitrary bounded steady waves are proved (earlier, these conditions, that have the form of bounds on the Bernoulli constant and other wave characteristics, were established only for Stokes waves). It is also shown that there exists an exact upper bound such that if the free-surface profile is less than this bound at infinity (positive, negative, or both), then the profile asymptotes the constant level corresponding to a unform stream (supercritical or subcritical). Finally, an integral property of arbitrary steady waves is obtained. A new technique is proposed for proving these results; it is based on modified Bernoulli’s equation that along with the free surface profile involves the difference between the potential and its vertical average.     

Experimental Study of the Bottom Structure Effect on the Damping of Standing Surface Waves in a Rectangular Vessel

The damping coefficient is estimated for standing surface waves in a rectangular vessel: (1) with a smooth horizontal rigid bottom, (2) with developed sandy bottom structures, (3) with a profiled rigid bottom, and with thin bottom layers of (4) fine-grained sand and (5) glass spheres. The results obtained are compared with available theoretical models.     

Rayleigh Waves on an Exponentially Graded Poroelastic Half Space

In this paper we consider the propagation of seismic waves in isotropic poroelastic half spaces with continuously varying elastic properties, namely with an exponentially decaying depth profile. The present paper shows that the problem leads naturally to a bicubic equation. We obtain explicit inhomogeneous plane wave solutions in an exponential evanescent form with respect to the depth of half space. Further, these solutions are used to solve the boundary value problem of a Rayleigh surface wave and the secular equation is established. The results obtained theoretically are exemplified for numerical data and represented graphically for a representative poroelastic material.     

On the Existence and Conditional Energetic Stability of Solitary Gravity-Capillary Surface Waves on Deep Water

This paper presents an existence and stability theory for gravity-capillary solitary waves on the surface of a body of water of infinite depth. Exploiting a classical variational principle, we prove the existence of a minimiser of the wave energy E{{\mathcal E}} subject to the constraint I=?2m{{\mathcal I}=\sqrt{2}\mu}, where I{{\mathcal I}} is the wave momentum and 0 < m << 1{0 < \mu \ll 1} . Since E{{\mathcal E}} and I{{\mathcal I}} are both conserved quantities a standard argument asserts the stability of the set D _μ of minimisers: solutions starting near D _μ remain close to D _μ in a suitably defined energy space over their interval of existence. In the applied mathematics literature solitary water waves of the present kind are modelled as solutions of the nonlinear Schr?dinger equation with cubic focussing nonlinearity. We show that the waves detected by our variational method converge (after an appropriate rescaling) to solutions of this model equation as mˉ 0{\mu \downarrow 0} .     

Effect of Thermal/Gravity Modulation on the Onset of Convection in a Maxwell Fluid Saturated Porous Layer

The effect of thermal/gravity modulation on the onset of convection in a Maxwell fluid saturated porous layer is investigated by a linear stability analysis. Modified Darcy–Maxwell model is used to describe the fluid motion. The regular perturbation method based on the small amplitude of modulation is employed to compute the critical Rayleigh number and the corresponding wavenumber. The stability of the system characterized by a correction Rayleigh number is calculated as a function of the viscoelastic parameter, Darcy–Prandtl number, normalized porosity, and the frequency of modulation. It is found that the low frequency symmetric thermal modulation is destabilizing while moderate and high frequency symmetric modulation is always stabilizing. The asymmetric modulation and lower wall temperature modulations are, in general, stabilizing while the system becomes unstable for large values of Darcy–Prandtl number and for small frequencies. It is shown that in general the gravity modulation produces a stabilizing effect on the onset of convection for moderate and high frequency. The small frequency gravity modulation is found to have destabilizing effect on the stability of the system.     

Waves in a viscous layer of fluid on an elastic half-space

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 26, No. 4, pp. 3–7, April, 1990.     

Fluid experimental flow estimation based on an optical-flow scheme

We present in this paper a novel approach dedicated to the measurement of velocity in fluid experimental flows through image sequences. Unlike most of the methods based on particle image velocimetry (PIV) approaches used in that context, the proposed technique is an extension of “optical-flow” schemes used in the computer vision community, which includes a specific enhancement for fluid mechanics applications. The method we propose enables to provide accurate dense motion fields. It includes an image based integrated version of the continuity equation. This model is associated to a regularization functional, which preserve divergence and vorticity blobs of the motion field. The method was applied on synthetic images and on real experiments carried out to allow a thorough comparison with a state-of-the-art PIV method in conditions of strong local free shear.     

具有小密度差的两层流体中运动点源的二阶内波解

在具有自由面的两层流体中，运动点源生成的Kelvin船波存在两种模式，即表面波模式和内波模式。当上、下层流体密度比趋于1时，由内波模式计算的界面波幅趋于无穷大，这与实验事实相违背。为克服此困难，在自由面和界面作小波幅运动的假设，引入一个小密度差参数。研究了运动点源在无粘、不可压且具有小密度差的两层有限深流体中生成的高阶波动。首先利用摄动方法推导了各阶小参数满足的边值问题；其次，给出了小密度差情形下的可解性条件。证明了在密度比趋于1的极限情形，不存在导致界面波幅无穷大的内波模式；最后，利用Phillips的非线性共振相互作用理论，构造了具有自由面的两层有限深流体中Kelvin船波系的二阶一致有效波动解，并证明了该解在深水情形下退化为Newman关于均匀流体中自由面的二阶波动解。     

Interaction of Internal Gravity Current with an Obstacle on the Channel Bottom

The problem of hydrodynamic loads arising from the interaction of gravity currents with an obstacle on the channel bottom was studied experimentally. The gravity-current structure was visualized at the stage of formation and at the stage of interaction with the obstacle. The dependence of the propagation velocity of the gravity-current front on the nondimensional current depth and the Archimedes number was studied. In the region of self-similarity in the Archimedes number, the behavior of hydrodynamic-load coefficients was studied as a function of the nondimensional gravity current depth.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 4, pp. 39–46, July–August, 2005.     

Waves generated on an ice cover by a source pulsating in fluid

A point source of variable intensity located at rest in a plane infinitely deep fluid layer under an ice cover is considered. The general expression for perturbations of the fluid-ice interface is obtained. In the case of the long operation in the pulsating regime the wave established on the ice cover is found.