Surface and internal waves in a stratified layer of liquid and an analysis of the impedance boundary conditions

The surface and internal waves in a multilayered ideal liquid for specified displacements of the bottom are considered. The upper surface of the liquid is either free or displacements are specified on it. In the long-wave approximation, asymptotically accurate models of the waves propagating along the surface layers of an incompressible liquid are constructed both for intense stratification (the ratio of the densities of neighbouring layers are associated with a small parameter) and for weak stratification. The important case of this problem of the dynamic contact of rigid bodies through a layer of incompressible or compressible liquid is investigated. High-order impedance boundary conditions are constructed and the results of testing them using the exact solutions are presented.     

Scattering of Love waves due to the presence of a rigid barrier of finite depth in the crustal layer of the earth

The problem of scattering of Love waves due to the presence of a rigid barrier of finite depth in the crusfal layer of the earth is studied in the present paper. The barrier is in the slightly dissipative surface layer and the surface of the layer is a free surface. The Wiener-Hopf technique is the method of solution. Evaluation of the integrals along appropriate contours in the complex plane yields the reflected, transmitted and the scattered waves. The scattered waves behave as a decaying cylindrical wave at distant points. Numrical computations for the amplitude of the scattered waves have been made versus the wave number. The amplitude falls off rapidly as the wave number increases very slowly.     

Numerical Analysis of the Interphase Interaction in a Porous Elastic Fluid-Saturated Medium

We consider dynamical processes in a two-phase porous fluid-saturated medium. The equation of the Biot–Frenkel model, which accounts for the influence of the elastic, inertial, and viscous interaction between the liquid and solid phases, is used for modeling the dynamics of the soil layer (flat deformation) with the finite-element method in cases of steady and nonstationary effects. For a layer under a uniaxial stress, we give numerical examples of the displacements of the soil skeleton and interstitial fluid.     

Attenuation of normal waves in a cavity with a liquid in the presence of an elastic shell

The attenuation of normal waves is investigated in a cylindrical cavity of an unbounded viscoelastic medium, in which there is a thin elastic shell filled with a viscous compressible liquid. It is assumed that the absorption of waves in a liquid or solid is small. The effect of the shell thickness on the value of the absorption coefficient of the normal waves is studied.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 25, pp. 49–52, 1987.     

具有微极弹性构架的流体-饱和多孔固体中波在边界面处的传播

通过研究匀质非黏性液体半空间和微极流体饱和多孔固体半空间交界面处平面波的反射与透射,得到不同角度入射波的反射和透射系数.计算出反射波与透射波的振幅比数值,并用图表示出微极性和多孔性对反射和透射的影响.最后根据公式对一些特例进行了推导.     

The relative motion of the core and mantle of a planet in the gravitational field of a point mass

A model of a planet, consisting of two solid bodies – a core and a mantle – between which there is a spherical layer of a viscous incompressible liquid, is considered. The gravitational interaction between the core and the mantle is taken into account. The problem is investigated in a limited formulation, when the mass centre of the planet moves in a fixed elliptical orbit in the gravitational field of a point mass, while the mutual displacements of the core and the mantle are to be determined. The mutual displacements of the core and the mantle of the planet, and also the velocity field of the viscous liquid in the spherical layer, are obtained using multiparameter perturbation theory, where the Reynolds number, the orbit eccentricity and the ratio of the radius of the planet to the distance to its attracting centre are taken as small parameters. In addition, an approximate theory of gyroscopes is used to analyse the equations of motion. The results obtained are illustrated by the example of the motion of the Earth-Moon system.     

弹性固体和充满两种互不相溶粘性流体的多孔固体之间界面的反射和折射及其衰减波

在充满两种互不相溶粘性流体的多孔固体中,研究弹性波的传播.用3个数性的势函数描述3个纵波的传播,用1个矢性的势函数单独描述横波的传播.根据这些势函数,在不同的组合相中,定义出质点的位移.可以看出,可能存在3个纵波和1个横波.在一个弹性固体半空间与一个充满两种互不相溶粘性流体的多孔固体半空间之间,研究其界面上入射纵波和横波所引起的反射和折射现象.由于孔隙流体中有粘性,折射到多孔介质中的波,朝垂直界面方向偏离.将入射波引起的反射波和折射波的波幅比,作为非奇异的线性代数方程组计算.进一步通过这些波幅比,计算出各个被离散波在入射波能量中所占的份额.通过一个特殊的数值模型,计算出波幅比和能量比系数随入射角的变化.超过SV波的临界入射角,反射波P将不再出现.越过界面的能量守恒原理得到了验证.绘出了图形并对不同孔隙饱和度以及频率的变化,讨论它们对能量分配的影响.     

A moving boundary problem derived from heat and water transfer processes in frozen and thawed soils and its numerical simulation

The seasonal change in depths of the frozen and thawed soils within their active layer is reduced to a moving boundary problem,which describes the dynamics of the total ice content using an independent mass balance equation and treats the soil frost/thaw depths as moving(sharp)interfaces governed by some Stefan-type moving boundary conditions,and hence simultaneously describes the liquid water and solid ice states as well as the positions of the frost/thaw depths in soil.An adaptive mesh method for the moving boundary problem is adopted to solve the relevant equations and to determine frost/thaw depths,water content and temperature distribution.A series of sensitivity experiments by the numerical model under the periodic sinusoidal upper boundary condition for temperature are conducted to validate the model,and to investigate the effiects of the model soil thickness,ground surface temperature,annual amplitude of ground surface temperature and thermal conductivity on frost/thaw depths and soil temperature.The simulated frost/thaw depths by the model with a periodical change of the upper boundary condition have the same period as that of the upper boundary condition,which shows that it can simulate the frost/thaw depths reasonably for a periodical forcing.     

Modeling the disintegration of modulated liquid jets using volume-of-fluid (VOF) methodology

In this study, we present the numerical investigations on the effect of finite velocity modulations imposed on an otherwise unperturbed cylindrical liquid jet issuing into stagnant gas. Sinusoidal velocity fluctuations of finite frequency and amplitude are imposed at the liquid jet inlet and the resulting liquid jet surface deformation is captured using a volume of fluid (VOF) methodology, utilizing compressive interface capturing scheme for arbitrary meshes (CICSAM) scheme. Variation of the simulation parameters, comprising of the mean liquid jet velocity, modulation amplitude and frequency grouped together using a set of non-dimensional parameters, leads to the formation of a wide gamut of reproducible liquid structures such as waves, upstream/downstream directed bells, chains of droplets similar to those observed in experiments. Elaborate tests on the effect of injection velocity and inlet jet diameter are investigated to characterize the breakup process. The computations efficiently capture the diverse flow structures generated by the evolving modulated liquid jet inclusive of several non-linear dynamics such as growth of surface waves, ligament interaction with shear vortices and its subsequent thinning process. The simulations identify the deterministic behavior of modulated liquid jets to predict liquid disintegration modes under given set of non-dimensional parameters.     

On the Generation of Mean Flows by the Interaction of Görtler Vortices and Tollmien-Schlichting Waves in Curved Channel Flows

There are many fluid flows where the onset of transition can be caused by different instability mechanisms which compete in the nonlinear regime. Here the interaction of a centrifugal instability mechanism with the viscous mechanism which causes Tollmien-Schlichting waves is discussed. The interaction between these modes can be strong enough to drive the mean state; here the interaction is investigated in the context of curved channel flows so as to avoid difficulties associated with boundary layer growth. Essentially it is found that the mean state adjusts itself so that any modes present are neutrally stable even at finite amplitude. In the first instance the mean state driven by a vortex of short wavelength in the absence of a Tollmien-Schlichting wave is considered. It is shown that for a given channel curvature and vortex wavelength there is an upper limit to the mass flow rate which the channel can support as the pressure gradient is increased. When Tollmien-Schlichting waves are present then the nonlinear differential equation to determine the mean state is modified. At sufficiently high Tollmien-Schlichting amplitudes it is found that the vortex flows are destroyed, but there is a range of amplitudes where a fully nonlinear mixed vortex-wave state exists and indeed drives a mean state having little similarity with the flow which occurs without the instability modes. The vortex and Tollmien-Schlichting wave structure in the nonlinear regime has viscous wall layers and internal shear layers; the thickness of the internal layers is found to be a function of the Tollmien-Schlichting wave amplitude.     

Transient response of SH waves in a layered half-space with sub-surface and interface cracks

The transient scattering of SH waves by sub-surface and interface cracks parallel to the free surface in a layered elastic solid is investigated. The problem in frequency domain is solved by using a hybrid method which combines the finite element method of the near field with the boundary integral representation of the far field. The transient responses are then obtained by inverting the spectra via fast Fourier transform with the incident pulse Ricker of wavelet. Numerical results are presented for the surface displacements, dynamic stress intensity factors and wave motion in the layered half-space. Furthermore, the propagations of reflected, diffracted, and direct impact waves at any instant are clearly identified by the present method. To understand the mechanism of elastic wave interaction is very important in the field of ultrasonic non-destructive evaluation (NDE) and fracture mechanics studies.     

Propagation of torsional surface waves in a homogeneous layer of finite thickness over an initially stressed heterogeneous half-space

In the present paper, the dispersion equation which determines the velocity of torsional surface waves in a homogeneous layer of finite thickness over an initially stressed heterogeneous half-space has been obtained. The dispersion equation obtained is in agreement with the classical result of Love wave when the initial stresses and inhomogeneity parameters are neglected. Numerical results analyzing the dispersion equation are discussed and presented graphically. The result shows that the initial stresses have a pronounced influence on the propagation of torsional surface waves. It has also been shown that the effect of density, directional rigidities and non-homogeneity parameter on the propagation of torsional surface waves is prominent.     

Streamwise oscillations of a cylinder beneath a free surface: Free surface effects on vortex formation modes

A computational study of a viscous incompressible two-fluid model with an oscillating cylinder is investigated at a Reynolds number of 200 and at a dimensionless displacement amplitude of A=0.13 and for the dimensionless forcing cylinder oscillation frequency-to-natural vortex shedding frequency ratios, f/f₀=1.5,2.5,3.5. Specifically, two-dimensional flow past a circular cylinder subject to forced in-line oscillations beneath a free surface is considered. The method is based on a finite volume discretization of the two-dimensional continuity and unsteady Navier-Stokes equations (when a solid body is present) on a fixed Cartesian grid. Two-fluid model based on improved volume-of-fluid method is used to discretize the free surface interface. The study focuses on the laminar asymmetric flow structure in the near wake region and lock-on phenomena at a Froude number of 0.2 and for the dimensionless cylinder submergence depths, h=0.25, 0.5 and 0.75. The equivorticity patterns and pressure distribution contours are used for the numerical flow visualization. The code validations in special cases show good comparisons with previous numerical results.     

On the behaviour of thin liquid layers at curved solid edges

The knowledge of the behaviour of thin liquid layers on solid surfaces is of fundamental interest as far as basic research and applications like technical covering or lubrication processes are concerned. The subject is currently of increasing importance, particularly due to stronger requirements of thin liquid layer functionalities. We therefore studied the development of the liquid layer thickness distribution field as well as the liquid layer disturbance propagation on a plane solid surface in the vicinity of a curved solid edge experimentally using LIF technique. The investigation focusses on the influence of the initial layer thickness as well as on liquid properties like viscosity and surface tension on the liquid film behaviour, especially close to solid edges. Selected experimental results are shown highlighting relevant quantities with respect to disturbance propagation and validation. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

四段式有限元法分析热耦合的流体-固体相互作用问题

给出了一种流(体)-热-结构综合的分析方法,固体中的热传导耦合了粘性流体中的热对流,因而在固体中产生热应力.应用四段式有限元法和流线逆风Petrov-Galerkin法分析热粘性流动,应用Galerkin法分析固体中的热传导和热应力.应用二阶半隐式Crank-Nicolson格式对时间积分,提高了非线性方程线性化后的计算效率.为了简化所有有限元公式,采用3节点的三角形单元,对所有的变量:流体的速度分量、压力、温度和固体的位移,使用同阶次的插值函数.这样做的主要优点是,使流体-固体介面处的热传导连接成一体.数个测试问题的结果表明,这种有限元法是有效的,且能加深对流(体)-热-结构相互作用现象的理解.     

Influence of Finite Amplitude Disturbances on the Nonstationary Modes of a Compressible Boundary Layer Flow

A weakly nonlinear stability analysis is performed to search for the effects of compressibility on a mode of instability of the three-dimensional boundary layer flow due to a rotating disk. The motivation is to extend the stationary work of [ 1 ] (hereafter referred to as S90) to incorporate into the nonstationary mode so that it will be investigated whether the finite amplitude destabilization of the boundary layer is owing to this mode or the mode of S90. Therefore, the basic compressible flow obtained in the large Reynolds number limit is perturbed by disturbances that are nonlinear and also time dependent. In this connection, the effects of nonlinearity are explored allowing the finite amplitude growth of a disturbance close to the neutral location and thus, a finite amplitude equation governing the evolution of the nonlinear lower branch modes is obtained. The coefficients of this evolution equation clearly demonstrate that the nonlinearity is destabilizing for all the modes, the effect of which is higher for the nonstationary waves as compared to the stationary waves. Some modes particularly having positive frequency, regardless of the adiabatic or wall heating/cooling conditions, are always found to be unstable, which are apparently more important than those stationary modes determined in S90. The solution of the asymptotic amplitude equation reveals that compressibility as the local Mach number increases, has the influence of stabilization by requiring smaller initial amplitude of the disturbance for the laminar rotating disk boundary layer flow to become unstable. Apart from the already unstable positive frequency waves, perturbations with positive frequency are always seen to compete to lead the solution to unstable state before the negative frequency waves do. Also, cooling the surface of the disk will be apparently ineffective to suppress the instability mechanisms operating in this boundary layer flow.     

Remarks on a Strongly Nonlinear Model for Two‐Layer Flows with a Top Free Surface

We revisit in this paper the strongly nonlinear long wave model for large amplitude internal waves in two‐layer flows with a free surface proposed by Choi and Camassa [1] and Barros et al. [2]. Its solitary‐wave solutions were the object of the work by Barros and Gavrilyuk [3], who proved that such solutions are governed by a Hamiltonian system with two degrees of freedom. A detailed analysis of the critical points of the system is presented here, leading to some new results. It is shown that conjugate states for the long wave model are the same as those predicted by the fully nonlinear Euler equations. Some emphasis will be given to the baroclinic mode, where interfacial waves are known to change polarity according to different values of density and depth ratios. A critical depth ratio separates these two regimes and its analytical expression is derived directly from the model. In addition, we prove that such waves cannot exist throughout the whole range of speeds.     

Lattice Boltzmann simulation for high-speed compressible viscous flows with a boundary layer

In this study, the lattice Boltzmann method is employed for simulating high-speed compressible viscous flows with a boundary layer. The coupled double-distribution-function lattice Boltzmann method proposed by Li et al. (2007) is employed because of its good numerical stability and non-free-parameter feature. The non-uniform mesh construction near the wall boundary in fine grids is combined with an appropriate wall boundary treatment for the finite difference method in order to obtain accurate spatial resolution in the boundary layer problem. Three typical problems in high-speed viscous flows are solved in the lattice Boltzmann simulation, i.e., the compressible boundary layer problem, shock wave problem, and shock boundary layer interaction problem. In addition, in-depth comparisons are made with the non-oscillatory and non-free-parameter dissipation (NND) scheme and second order upwind scheme in the present lattice Boltzmann model. Our simulation results indicate the great potential of the lattice Boltzmann method for simulating high-speed compressible viscous flows with a boundary layer. Further research is needed (e.g., better numerical models and appropriate finite difference schemes) because the lattice Boltzmann method is still immature for high-speed compressible viscous flow applications.     

Energy Spectrum Width and Nonequilibrium Energy of Rayleigh Waves Scattered on a Two-Dimensional Near-Surface Electron Layer

We obtain expressions for the energy spectrum widths of Rayleigh waves corresponding to their deformational coupling to Fermi and Boltzmann electrons in a two-dimensional layer near the surface of a semibounded solid. We evaluate the nonequilibrium energy of Rayleigh waves that depends on these widths and is caused by the same coupling to the corresponding hot electrons. We show that this energy is independent of the degeneracy degree of the electrons and is given by the mean energy of free Rayleigh waves heated up to temperature of the electrons. We find conditions under which the thermodynamics is determined by this nonequilibrium energy of Rayleigh waves in films of a certain thickness with Fermi electrons near the surface and by the equilibrium energy of bulk phonons in thicker samples. All the results are obtained using the Keldysh diagram technique applied to the case of semibounded media.     

张力腿平台有限振幅运动的方程和数值解

论证了张力腿平台（TLP）在波浪作用下发生有限振幅运动时,所受惯性力、粘性力、浮力等载荷不仅与波浪场有关,还与瞬时响应有关,是响应的非线性函数;张力腿拉力也是各自由度位移的非线性函数．所以分析TLP受力时必须考虑平台的瞬时加速度、速度和位移,在瞬时位置建立运动方程．据此推导出TLP发生有限振幅运动时的外力计算公式,建立了TLP 6自由度有限振幅运动非线性控制方程．其中考虑了由6自由度有限位移引起的多种非线性因素,如各自由度之间的耦合、瞬时湿表面、瞬时位置等;还包括自由表面效应、粘性力等因素引起的非线性．用数值方法求解所得到的非线性运动方程．对典型平台ISSC TLP进行了数值分析,求得该平台在规则波作用下的6自由度运动响应．用退化到线性范围的解与已有解进行了对比,吻合良好．数值结果表明,综合考虑非线性因素后响应有明显改变．