非平整运动海底上n层流动中波浪传播的Hamilton逼近

在海洋水域，界面波对大尺度变化流的作用是一种典型的分层流动现象．考虑一不可压缩、无黏的分层势流运动，建立了一个在非平整运动海底上的n层流体演化系统，并对其进行了Hamilton描述．每层流体具有各自的常密度、均匀流水平速度，其厚度由未扰动和扰动部分构成．相对于顶层流体的自由表面，刚性、运动的海底具有一般地形变化特征．在明确指出n层流体运动的控制方程和各层交界面上的运动学、动力学边界条件(包含各层交界面上张力效应)后，对该分层流动力系统进行了Hamilton构造，即给出其正则方程和其下述的正则变量：各交界面位移和各交界面上的动量势密度差。     

有限深水中骑行波的显式Hamilton描述

小尺度波（扰动波）迭加在大尺度波（未受扰动波）上形成的波动一般之为“骑行波”。研究了有限可变深度的理想不可压缩流体中的骑行波的显式Hamliltn表示，考虑了自由面上流体与空气之间的表面张力。采用自由面高度和自由面上速度势构成的Hamilton正则变量表示骑行波的动能密度，并在未受扰动波的自由面上作一阶展开。运用复变函数论方法处理了二维流动。先用保角变换将物理平面上的流动区域变换到复势平面上的无限长带形区域，然后在复势平面上用Fourier变换解出Laplace方程，最后经Fourier逆变换求出了扰动波速度热所满足的积分方程。作为特例考虑了平坦底部的流动，导出了动能密度的显式表达式。这里给出的积分方程可以替代相当难解的Hamilton正则方程。通过求解积分方程可得出agrange密度的显式表达式。本文提出的方法约研究骑行波的Hamilton描述以及波的相互作用问题提供了新的途径，有助于了解海面的小尺度波的精细结构。     

水平电场下界面波的Hamilton原理与多尺度建模

本文研究水平电场下两层电介质流体间界面波动的多尺度建模. 首先对此系统的Hamilton原理给出详细证明; 然后基于Hamilton结构和Dirichlet–Neumann算子的解析性质, 将Hamilton量中的动能与电势能展开成收敛级数形式并确定截断阶数, 最后通过计算截断后近似总能量的变分导数得到约化模型. 上述过程对该问题给出了一套建立多尺度非线性模型的系统方法. 文章再以“上层深水、下层浅水”为例详细阐述了多尺度建模的全过程, 并利用修正的Petviashvili迭代方法计算了新模型中的非线性相干结构. 本文所发展的渐近分析技巧不同于之前的工作, 其优点在于所导出的约化模型自然保留能量守恒的性质; 同时, 本文亦将原有结果推广至三维情形.      

波浪扰动下河口幂律异重流的动力场分布特性

河口底层浮泥异重流的运动特性对于河口维持以及港口航道泥沙淤积过程具有重要的作用, 是海岸学科研究的关键内容, 也是热点内容之一. 本文首先综述了河口泥沙异重流研究的重要意义, 分析并总结了各家异重流理论模型的不同点和适应条件; 其次, 根据本文研究问题的实际需要, 构建了波浪与底泥相互作用的双层流体理论分析模式, 将上层流体简化为常见的牛顿体, 而将下层流体的流变关系设置为幂律函数, 研究了波浪作用下河口底部幂律异重流的流场特性. 这些特性包括:波浪速度场、底泥运动的流速场、不同密度影响下的压力场以及异重流泥面波与表面 波的波幅比等, 分析了泥层密度、波动圆频率以及底泥幂律指数对流场及界面波的影响. 研究发现, 在波浪扰动下, 两层流体交界处速度分量连续, 压强出现突变. 在下部泥层中, 水平速度幅值曲线存在极大值. 随着波动圆频率增加以及泥层密度与流动指数的减小, 界面处上下压强差值呈现增大的趋势. 本模型与实测波幅比的数据进行对比结果证实了模型的合理性.      

波浪扰动下河口幂律异重流的动力场分布特性

河口底层浮泥异重流的运动特性对于河口维持以及港口航道泥沙淤积过程具有重要的作用,是海岸学科研究的关键内容,也是热点内容之一.本文首先综述了河口泥沙异重流研究的重要意义,分析并总结了各家异重流理论模型的不同点和适应条件;其次,根据本文研究问题的实际需要,构建了波浪与底泥相互作用的双层流体理论分析模式,将上层流体简化为常见的牛顿体,而将下层流体的流变关系设置为幂律函数,研究了波浪作用下河口底部幂律异重流的流场特性.这些特性包括:波浪速度场、底泥运动的流速场、不同密度影响下的压力场以及异重流泥面波与表面波的波幅比等,分析了泥层密度、波动圆频率以及底泥幂律指数对流场及界面波的影响.研究发现,在波浪扰动下,两层流体交界处速度分量连续,压强出现突变.在下部泥层中,水平速度幅值曲线存在极大值.随着波动圆频率增加以及泥层密度与流动指数的减小,界面处上下压强差值呈现增大的趋势.本模型与实测波幅比的数据进行对比结果证实了模型的合理性.     

骑行波的非线性演化方程

从能量的角度出发,采用Ｈａｍｉｌｔｏｎ描述交结合变分原理和摄动分析,并借助于符号运算导出了骑行在大波上的小波的Ｈａｍｉｔｏｎ密度函数和非线性动力学方程。这里的大流和小波是对波高而言的。在Ｈａｍｉｌｔｏｎ描述中,正则变量取为波高和速度势。本文导出了描述小波演化的二阶方程,在一阶近下的方程与Ｈｅｎｙｅｙ等人（１９８８）的结果一致。     

位移浅水内孤立波

研究两层浅水系统中的内孤立波，该系统由两层常密度不可压缩无黏性水组成。利用Lagrange坐标和Hamilton原理，推导了两层浅水系统的位移浅水内波方程，并进一步导出了两层浅水系统的位移内孤立波解。数值实验表明，位移内孤立波与经典的KdV内孤立波吻合很好，说明Lagrange坐标和Hamilton方法适用于内波分析，可以为构造内波分析的保辛方法提供一种途径。     

位移浅水内孤立波

研究两层浅水系统中的内孤立波,该系统由两层常密度不可压缩无黏性水组成。利用Lagrange坐标和Hamilton原理,推导了两层浅水系统的位移浅水内波方程,并进一步导出了两层浅水系统的位移内孤立波解。数值实验表明,位移内孤立波与经典的KdV内孤立波吻合很好,说明Lagrange坐标和Hamilton方法适用于内波分析,可以为构造内波分析的保辛方法提供一种途径。     

无黏流体与固体界面的漏瑞利波

漏瑞利波存在于半无限无黏性流体和半无限固体媒质的界面处. 首先推导流固无限各向同性介质界面处漏瑞利波的特征方程和位移及应力的解析计算公式. 然后结合典型结构通过数值计算研究了漏瑞利波特性以及位移和应力在流体和固体中的分布规律. 数值计算结果表明漏瑞利波的相速度和衰减随流固密度比的增大而增大, 在流固界面上法向位移连续而切向位移不连续. 流固密度比对固体媒质中沿垂直于漏瑞利波的传播方向的位移、正应力和剪应力有比较大的影响,而对沿漏瑞利波的传播方向的正应力几乎没影响. 为利用漏瑞利波的无损检测与评价提供了理论基础.      

阴影法和差分干涉法在分层流密度测量中的应用

本文讨论了用阴影法和差分干涉法测量水-盐两层分层界面上的密度分布。此外,还对一任意密度分布的分层流体作了测量和计算。     

Hamiltonian formulation of nonlinear water waves in a two-fluid system

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The Compressible Viscous Surface-Internal Wave Problem: Nonlinear Rayleigh–Taylor Instability

This paper concerns the dynamics of two layers of compressible, barotropic, viscous fluid lying atop one another. The lower fluid is bounded below by a rigid bottom, and t he upper fluid is bounded above by a trivial fluid of constant pressure. This is a free boundary problem: the interfaces between the fluids and above the upper fluid are free to move. The fluids are acted on by gravity in the bulk, and at the free interfaces we consider both the case of surface tension and the case of no surface forces.We are concerned with the Rayleigh–Taylor instability when the upper fluid is heavier than the lower fluid along the equilibrium interface. When the surface tension at the free internal interface is below the critical value, we prove that the problem is nonlinear unstable.     

Hamiltonian Formulation for the Motion of a Two-Fluid System with Free Surface

In this work, a theoretical investigation is performed on modeling interfacial and surface waves in a layered fluid system. The physical system consists of two immiscible liquid layers of different densities ₁ > ₂ with an interfacial surface and a free surface, inside a prismatic-section tank. On the basis of the potential formulation of the fluid motion, we derive a nonlinear system of partial differential equations using the Hamiltonian formulation for irrotational flow of the two fluids of different density subject to conservative force. As a consequence of the assumption of potential velocity, the dynamics of the system can be described in terms of variables evaluated only at the boundary of the fluid system, namely the separation surface and the free surface. This Hamiltonian formulation enables one to define the evolution equations of the system in a canonical form by using the functional derivatives.     

Stability of the wave motion on the charged interface between two immiscible fluids in the presence of a tangential velocity field discontinuity

In the second-order approximation in the dimensionless wave amplitude, the problem of nonlinear periodic capillary-gravity wave motion of the uniformly charged interface between two immiscible ideal incompressible fluids, the lower of which is perfectly electroconductive and the upper, dielectric, moves translationally at a constant velocity parallel to the interface, is solved analytically. It is shown that on the uniformly charged surface of an electroconductive ideal incompressible fluid the positions of internal nonlinear degenerate resonances depend of the medium density ratio but are independent of the upper medium velocity and the surface charge density on the interface. All resonances are realized at densities of the upper medium smaller than the density of the lower medium. In the region of Rayleigh-Taylor instability with respect to density there is no resonant wave interaction.     

Analytical solutions of singularities moving with an arbitrary path when two fluids are present

The derivations are carried out for the velocity potentials of singularities moving withan arbitrary path either in the upper fluid or in the lower fluid with or without a horizontalbottom when two fluids are present.In such a case,the pressure distribution is no longerequal to a constant or zero at the free interface.Taking the influence of an upper fluid uponthe lower fluid into consideration,a series of fundamental solutions in closed forms arepresented in this paper.     

Numerical simulation and experimental verification of silicone oil flow over magnetic fluid under applied magnetic field

Two-layer flow of magnetic fluid and non-magnetic silicone oil was simulated numerically. The continuity equation, momentum equations, kinematic equation, and magnetic potential equation were solved in two-dimensional Cartesian coordinate. PLIC （piecewise linear integration calculation） VOF （volume of fluid） scheme was employed to track the free interface. Surface tension was treated via a continuous surface force （CSF） model that ensures robustness and accuracy. The influences of applied magnetic field, inlet velocity profile, initial surface disturbance of interface and surface tension were analyzed. The computed interface shapes at different conditions were compared with experimental observation.     

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。     

Wave scattering by flexible porous vertical membrane barrier in a two-layer fluid

The scattering of water waves by a flexible porous membrane barrier in a two-layer fluid having a free surface is analysed in two dimensions. The membrane barrier is extended over the entire water depth in a two-layer fluid, each fluid being of finite depth. In the present analysis, linear wave theory and small amplitude membrane response are assumed. The porous membrane barrier is tensioned and pinned at both the free surface and the seabed. The associated mixed boundary value problem is reduced to a linear system of equations by utilizing a general orthogonality relation along with least-squares approximation method. Because of the flow discontinuity at the interface, the eigenfunctions involved have a discontinuity at the interface and the orthogonality relation used is a generalization of the classical one corresponding to a single-layer fluid. The reflection and transmission coefficients for the surface and internal modes, the free surface and interface elevations and the nondimensional membrane deflection are computed for various physical parameters like the nondimensional tension parameter, porous-effect parameter, fluid density ratio, ratio of water depths of the two fluids to analyse the efficiency of a porous membrane as a wave barrier in the two-layer fluid.