Effect of compressive stress on the dispersion relation of the flexural–gravity waves in a two-layer fluid with a uniform current

The explicitly analytical solution is derived for the dispersion relation of the flexural–gravity waves in a two-layer fluid with a uniform current. The upper fluid is covered by a thin plate with the presence of the elastic, compressive and inertial forces. The density of each of the two immiscible layers is constant. The fluids of finite depth are assumed to be inviscid and incompressible and the motion be irrotational. A linear system is established within the framework of potential theory. A new representation for the dispersion relation of flexural–gravity waves in a two-layer fluid is derived. The critical value for the compressive force is analytically determined. The dispersion relation for the capillary–gravity with an inertial surface in a two-layer fluid can be obtained in parallel. Some known dispersion relations can be recovered from the present solution.     

SURFACE AND INTERFACIAL GRAVITY WAVES INDUCED BY AN IMPULSIVE DISTURBANCE IN A TWO-LAYER INVISCID FLUID

The transient gravity waves due to an impulsive source in a two-layer fluid system are investigated analytically.The fluid is assumed to be inviscid and incompressible.The density of each of the two layers is constant.Five different boundary conditions are considered.The depth of each of the two layers is infinite or finite.The upper fluid of finite depth is covered by a rigid lid or a free surface.Based on the assumption of small-amplitude waves,a linear system is established.The integral solutions for the free-surface and interfacial waves are obtained by means of the Fourier-Laplace transform.The corresponding asymptotic representations are derived for large time with a fixed distance-to-time ratio by the Stokes and Scorer methods of stationary phase.The analytical solutions show that there are two different modes,namely the free-surface and interfacial wave modes.The wave profiles observed depend on the relation between the distance-to-time ratio and the maximal group velocities and on the limiting values of the second derivatives of the frequencies as well.     

Surface and interfacial gravity waves due to a disturbance steadily moving in a two-layer inviscid fluid

The generation and interaction of surface and interfacial gravity waves due to an submerged source moving in a two-layer fluid are investigated analytically for two-dimensional cases. The fluid is assumed to be inviscid and incompressible. The density of each of the two layers is constant. Two different boundary conditions are considered for the upper surface. The upper fluid of finite depth is bounded above by a rigid lid or a free surface. Based on the assumption of small-amplitude waves, a linear system is established. The integral solutions for the free-surface and interfacial elevations are obtained by means of the Fourier transform. Then the corresponding asymptotic representations are derived for far-field waves by the residue theorem. The critical Froude numbers for the existence of far-field waves are derived for the two-layer system bounded above by a rigid lid or a free surface. The effect of different upper boundary conditions on the wave generation are discussed.

     

UNSTEADY WAVES DUE TO AN IMPULSIVE OSEENLET BENEATH THE CAPILLARY SURFACE OF A VISCOUS FLUID

The two-dimensional free-surface waves due to a point force steadily moving beneath the capillary surface of an incompressible viscous fluid of infinite depth were analytically investigated. The unsteady Oseen equations were taken as the governing equations for the viscous flows. The kinematic and dynamic conditions including the combined effects of surface tension and viscosity were linearized for small-amplitude waves on the free-surface. The point force is modeled as an impulsive Oseenlet. The complex dispersion relation for the capillary-gravity waves shows that the wave patterns are characterized by the Weber number and the Reynolds number. The asymptotic expansions for the wave profiles were explicitly derived by means of Lighthill’s theorem for the Fourier transform of a function with a finite number of singularities. Furthermore, it is found that the unsteady wave system consists of four families, that is, the steady-state gravity wave, the steady-state capillary wave, the transient gravity wave, and the transient capillary wave. The effect of viscosity on the capillary-gravity was analytically expressed.     

THE RADIATION AND DIFFRACTION OF WATER WAVES BY A BOTTOM-MOUNTED CIRCULAR CYLINDER IN A TWO-LAYER FLUID

An analytical method was proposed to analyze the radiation and diffraction of water waves by a bottom-mounted circular cylinder in a two-layer fluid. Analytical expressions for added mass and damping coefficients, as well as the wave excitation forces of the circular cylinder were obtained by an eigenfunction expansion method. The hydrodynamic forces on the bottom-mounted circular cylinder in a two-layer fluid include not only the added mass and damping coefficients, but also the wave forces of the surface and internal-wave modes. This is different from the case of a homogenous fluid. Some examples were given, showing that density stratification can have a relative large effect on these hydrodynamic forces over a wide range of frequencies.     

THE WAVE-MAKING CHARACTERISTICS OF A MOVING BODY IN A TWOLAYER FLUID

The Wave-making characteristics of a moving body in a two-layer fluid with free surface is investigated numerically and experimentally. The numerical analysis is based on the modified layered boundary integral equation system. The wave characteristics on the free surface and interface generated by a moving sphere and an ellipsoid is numerically simulated in both finite depth and infinite depth of lower layer model. The numerical results of the sphere are compared with the analytical results for a dipole with the same velocity in a two-layer fluid of finite depth. The dependence of the wave systems and structures on the characteristic quantities is discussed. Three kinds of measurement techniques are used in model experiments on the internal waves generated by a sphere advancing in a two-layer fluid. The effects of the varying velocity and stratification on the wavelength, wave amplitudes and the maximum half angles of internal waves are analyzed qualitatively.     

有限深两层流体中水波与水平圆柱潜体的相互作用

研究了有限深两层流体中表面波模态和内波模态的水波与水平圆柱潜体的相互作用问题。在线性势流理论框架内,建立了两种模态入射波作用下,水平圆柱体绕射势的多极展开求解理论。基于所建立的求解模型,对作用在圆柱体上的波浪力进行了数值计算分析,并且与均匀流中情况进行了比较。结果表明了,在入射波的某个频率范围内,流体的分层效应对圆柱体波浪力的影响是不可忽略的。同时,对流域内的绕射速度矢量场进行了数值分析。     

Homotopy-based analytical approximation to nonlinear short-crested waves in a fluid of finite depth

A nonlinear short-crested wave system, consisting of two progressive waves propagating at an oblique angle to each other in a fluid of finite depth, is investigated by means of an analytical approach named the homotopy analysis method(HAM). Highly convergent series solutions are explicitly derived for the velocity potential and the surface wave elevation. We find that, at every value of water depth, there is little difference between the kinetic energy and the potential energy for nonlinear waves. The nonlinear short-crested waves with a larger angle of incidence always contain the more potential wave energy. With the aid of the HAM, we obtain the dispersion relation for nonlinear short-crested waves. Furthermore, it is shown that the wave elevation tends to be smoothened at the crest and be sharpened at the trough as the water depth increases, and the wave pressure crests and troughs become steeper with increasing incident wave steepness.     

NONLINEAR INTERFACIAL CAPILLARY-GRAVITY WAVES IN SHEAR FLOW

In this paper,the capillary-gravity solitary waves in a two-fluid system withshear flow are theoretically investigated.Three cases are considered.For the first case(the nor-mal case),the KdV equation,with the effect of shear flow being taken into account,is derivedand it is found that there may exist single mode travelling solitary waves,slowly travelling soli-tary waves and even more slowly travelling solitary waves at some shear flow conditions.Thesecond case considered is that the fluid system is at the first kind critical condition.The modifiedKdV equation in which the cubic nonlinearity balances the dispersion is obtained and the solutionof mKdV solitary wave in shear flow is put forward.The third case is that the fluid system is atthe second kind critical condition.The generalized KdV equation with the effect of shear flowconsidered is given.It is shown that there exist normal and oscillatory solitary waves and thecondition of existence of oscillatory solitary waves are also presented.     

UNSTEADY INTERFACIAL WAVES DUE TO SINGULARITIES IN TWO SEMI-INFINITE INVISCID FLUIDS

1.INTRODUCTION ThelinearCauchy Poissonproblems,which areconcernedwiththegenerationandpropagation offree surfacewavesduetoaninitialelevationor apointpressureactingonthefreesurfaceofanin compressibleinviscidfluid,havebeeninvestigated bymanyresearchersinviewofitstheoreticalinterestandpracticalimportance[15].Thetransient developmentoftwo dimensionalgravitywavesdue toanoscillatingsurfacepressurewasconsideredby Miles[6]whoproposedawell posedinitialvalue problemtocorrectStoker'sformulation[2].A…     

具表面张力（I.Bond数大于1／3）的流体流过地形的共振流动

本文讨论了具表面张力作用下不可压缩无粘流体流过地形障碍物的共振流动。采用摄动方法，本文导出了一个新的方程，具负色散的ｆｋｄｖ方程，数值结果表明该情形可激发出下游下凹孤立波串，这个结果有待进一步的实验验证。     

WAVES GENERATED BY A SUBMERGED BODY MOVING IN STRATIFIED FLUIDS AND VERTICAL STRUCTURES OF INTERNAL WAVES

This dissertation deals with the internal waves generated by a submerged moving body in stratified fluids by combining theoretical and experimental methods. Our purpose is to provide some scientific evidences for non-acoustic detection of underwater moving bodies based on the principles of dynamics of the internal waves. An approach to velocity potentials obtained by superposing Green‘s functions of sources and sinks was proposed for Kelvin waves at the free surface or interface in a two-layer fluid. The effects of interacting surface and internal wave modes induced by a dipole on the surface divergence field were investigated. A new theoretical model formulating the interaction of a two dimensional submerged moving body with the conjugate flow in a three-layer fluid was established. An exact solution satisfying the two dimensional Benjamin Ono equation was obtained and the vertically propagating properties of the weakly nonlinear long waves were studied hy means of the ray theory and WKB method. The above theoretical results are qualitatively consistent with those obtained in the experiments conducted by the author.     

两层模式下内波K-dv方程的频散关系

为了更好地了解内孤立波的特性，分别利用椭圆余弦波方法和摄动展开方法，对两层模型下内孤立波K-dv方程的频散关系进行了推导。结果显示：利用后一种方法得到的两层模式下内孤立波K-dv方程的频散关系是一种方便、实用的解析式。据此解析式可进一步得到内孤立波半波宽度与内波波数间的关系，从而可将内孤立波理论与内波合成孔径雷达（SAR）图像联系起来。用此解析式计算得到的内孤立波频率和相速度与实测结果具有很好的一致性，因而为进一步进行分析和利用数据估计内孤立波的部分参数提供了理论依据。对解析式中频散项和非线性项量级的进一步分析认为，非线性项的作用在从SAR图像中提取内波振幅等参数时最好不予忽略。     

WAVE ATTENUATION OVER MUD BED: A PSEUDO-PLASTIC MODEL

A two-layer model, with the upper layer being the perfect fluid and the lower layer being the pseudo-plastic fluid describing water wave attenuation over mud bed, was established. A simplified method based on the principle of e-quivalent work was applied to solve the boundary value problems. The computational results of the model show that the two-layer perfect fluid model and the perfect-viscous fluid model are all special cases of the present model. The complex nonlinear properties of wave attenuation over mud bed, can be explained by the present model, e. g. , the wave dissipation rate decreases with the wave height in certain cases, while the small wave propagates over mud bed with less energy dissipation and large wave attenuates rapidly in other cases. Other factors influencing the wave attenuation were also discussed.     

行进波在两层流体中引起的泥层质量输运

本文考虑了两层不可压流体系统中的质量输运问题上，上层为理想流体，下层为粘性流体，导出了它们的色散关系，可以计算波浪在淤泥床上的衰减。导出了下层粘性流体中的质量输运速度，尤拉平均速度和司托克司漂流速度，都用显示表示。理论模型与实验结果进行了比较，两者基本相符，最后对某些质量规律进行了讨论。     

分层流体中的椭圆余弦波

本文采用Friedrichs展开法导出了椭圆余弦波在分层流体中传播问题的解答,精度为σ的一次方。所得的结果表明:同分层流体中线性波传播一样,分层流体中的椭圆余弦波存在着两种可能的模态—快模态和慢模态,流体波动的水平流速与z无关,垂直流速为零,压力服从静压分布规律。另外,由本文结果还可求得两种极限情形—微幅波和孤立波在分层流体中传播的解答。     

BASIC EQUATIONS AND WAVE THEORY OF GAS-LIQUID TWO-PHASE TRANSIENT FLOW

This paper is aimed at a systematic discussion on the basic equations for two-phasetransient flow and the theory of pressure waves and interfacial waves.The characters of the ba-sic equations have been analyzed.The formulas for wave speed of pressure and interfacial waveshave been derived.It has been proved that various mathematical models can be unified by theproposed integrating model and various wave speed formulas can be unified by the derived uni-versal formula.The basic equations for the separated flow model are ill-conditioned because ofthe unreasonable neglect of the pressure differences between the phases and the interfaces.Driftflux model can improve the accuracy if some additional equations are provided.However theymust be carefully chosen.The basic equations for homogeneous flow model are well-conditionedand they can be used in two-phase transient flow,especially in the cases of no velocity slips be-tween the phases.     

RAYLEIGH-TAYLOR INSTABILITIES BY OVERTURNING EXPERIMENTS IN TANK

A laboratory study of the turbulence front initiated by the Rayleigh-Taylor instabilities was conducted by overturning a two-layer stratified flow in a tank. Dye was introduced to the upper layer and the concentration of the dye was determined using a video imaging method. The mass center of the heavier upper layer moved continuously to a lower elevation because of the intrusion of the heavier fluid into the lower layer and the lighter fluid into the upper layer. The downward movement of the mass center was accurately determined from the dye concentration profile. It is concluded that the initial advancement of the mass center follows a quadratic relation with time and the final advancement obeys a linear relation with time.     

Ship waves in rivers: Environmental criteria and analysis methods for measurements

A literature review shows that the most important physical quantities determining environmental impact of ship waves in a waterway are the fluid velocities, maximum and minimum water levels, and size of drawdown events. Fluid velocity can vary strongly in the vertical so that the usual measurements at a single point are not enough unless made near where the effects are most important, often the bed. Customary use of wave height as a measure of impact has been misleading, because the all-important fluid velocity is of a scale given by wave height divided by wave period. A good and simple estimate of the surface velocity as a disturbance scale is shown to be given by the time derivative of the free surface height. The most important role of linear wave theory is to explain and understand the physics and measurement procedures, such as done here in several places. Its use for obtaining numerical results is criticised. Instead, three integral measures of impact are proposed, all of which use surface elevation measurements and which require no essential mathematical approximations or wave-by-wave analysis. The methods are applied to a study of ship waves on the Danube River. A number of results are presented, including examination of the effects of measurement frequency. After a ship passage, due to repeated shoreline reflections of the wake waves, the river is brought into a long-lasting state of short-crested disturbances, with finite fluid velocities. The environmental consequences of this might be important. After the primary and secondary ship waves it could be called the tertiary wave system.     

REFLECTION OF OBLIQUE INCIDENT WAVES BY PERFORATED CAISSONS WITH TRAVERSE WALL

The interaction of oblique incident waves with infinite number of perforated caissons is investigated. The fluid domain is divided into infinite sub-domains by the caissons, and eigen-function expansion is applied to expand velocity potentials in each domain. A phase relation is introduced for wave oscillation in each caisson, and the structure geometry is considered in constructing the models of reflection waves. The reflected waves with the present analysis include all of the waves traveling in different directions when incident wave period is short. Numerical examinations show that velocities at the inner and outer sides of the front walls of caissons are close to each other, and reflection coefficients satisfy the energy conservation relation very well when porous effect parameter is infinite. Numerical results show that the reflection coefficients of oblique incident waves are smaller for shorter caissons at low frequency, and decrease with the increase of wave incident angle.