Sloshing dynamics modulated cryogenic helium fluids driven by gravity gradient or jitter accelerations associated with slew motion in microgravity

The mathematical formulation of the Dewar container sloshing dynamics for a partially filled liquid of cryogenic superfluid helium II driven by the gravity gradient or jitter accelerations associated with slew motion in a microgravity environment are studied. The numerical computation of sloshing dynamics is based on the non-inertia container bounded frame and the solution of time-dependent, three-dimensional partial differential equations subjected to the initial and boundary conditions. This study discloses that the capillary effect of sloshing dynamics governs the liquid-vapor interface fluctuations driven by the gravity gradient or jitter accelerations associated with slew motion in a microgravity environment. The peculiar behavior of superfluid helium in response to sloshing dynamics is also investigated.     

“Leapfrogging” solitons in a system of coupled KdV equations

A system of two KdV equations coupled by small linear dispersive terms is considered. This system describes, for example, resonant interaction of two transverse gravity internal wave modes in a shallow stratified liquid. In the framework of an approach based on Hamilton's equations of motion, evolution equations for parameters of two solitons belonging to different wave modes are obtained in the adiabatic approximation. It is demonstrated that when the solitons' velocities are sufficiently lose, the solitons may form a breather-like oscillatory bound state, which provides a natural explanation for recent numerical experiments demonstrating “leapfrogging” motion of the two solitons. The frequency and the maximum amplitude of the “breather”'s internal oscillations are obtained. For the case when the relative velocity of the solitons is not small, perturbation-induced phase shifts of the two colliding free solitons are calculated. Then emission of radiation (small-amplitude quasilinear waves) by an oscillating “breather,” also detected in the numerical experiments, is investigated in the framework of the perturbation theory based on the inverse scattering transform. The intensity of the emission is calculated. Radiative effects accompanying collision of the free solitons are also investigated.     

Water surface waves in a horizontally oscillating container

This paper presents the results of experimental verification of Sretenskii’s linear theory of gravity waves in a container partially filled with water and oscillating horizontally according to the harmonic law. It has been shown that this theory predicts the existence of an infinite ordered countable set of generation modes of unstable waves. It has been experimentally confirmed that the waves are unstable if the container oscillation frequency is equal to the frequency of any odd standing-wave mode. At even eigenfrequencies of container oscillations, the theory predicts wave amplitudes up to a constant term. Experiment has shown that, in this case, the waves are stable and have minimum amplitudes.     

Internal gravity waves in critical generation modes and in the vicinity of trajectories of motion of perturbation sources

The field of internal gravity waves in a layer of an arbitrary stratified fluid is studied for critical generation modes and in the vicinity of trajectories of motion of the perturbation sources. The exact solutions describing the structure of a separate mode of the wave field in the vicinity of the perturbation source in the critical generation modes are investigated, and expressions for the total field representing the sum of all wave modes are obtained. In the vicinity of the trajectories of the perturbation sources, asymptotic representations of the eigenfunctions and eigenvalues of the basic vertical spectral problem of internal waves are constructed in the approximation of large wave numbers and asymptotic expressions for a separate mode of the wave field are obtained that describe the spatial structure and features of the fields of internal gravity waves. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 70–79, September–October, 2008.     

竖直振动管中颗粒毛细效应的离散元模拟

将一根细管插入填充有颗粒的静止容器中并对管施加竖直振动,颗粒将在管内发生上升运动,并最终稳定在一定高度,这一现象与液体毛细效应类似,被称为颗粒毛细效应.为探究颗粒毛细效应过程中伴随的颗粒尺度动力学行为及机理,基于离散元方法建立颗粒运动模型,对颗粒毛细效应动力学过程和特性开展数值模拟研究.模拟再现了文献中实验得到的颗粒毛细效应全过程,给出了管内颗粒柱高度随时间的演变规律,结果表明,受到颗粒系统参数的影响,本模拟条件下颗粒毛细效应过程呈现单周期上升、倍周期上升和倍周期稳定三个阶段,在倍周期上升阶段颗粒柱上升速度逐渐减小,平缓过渡到稳定阶段.在此基础上,分析了管内颗粒速度场和填充率分布随时间的演变特性,揭示了颗粒毛细效应过程中由容器传输到管内的颗粒的占比分布.研究发现,管内不同高度位置颗粒的运动并不同步,随着管的振动,管内出现速度波,速度波的传播引起管内颗粒出现膨胀和压缩交替的情况,从而管内颗粒填充率随时间发生周期性波动;在上升阶段,越接近管壁由容器传输到管内的颗粒占比越大,在稳定阶段,管内上层颗粒的对流引起容器传输到管内的颗粒占比发生反转.      

Fluid particle dynamics and stokes drift in gravity and capillary waves generated by the faraday instability

In this paper we study particle motions in nearly square containers due to gravity and capillary waves generated by vertical, periodic oscillation of the container. The method of second order partial averaging is used to decompose the particle motions into periodic oscillations and a slow Stokes drift. In the case of gravity waves, it is shown that long distance (several wavelengths) particle transport is possible. In the case of capillary waves, it is shown that, in agreement with experimental observations of Ramshankaret al., particle trajectories can be chaotic even when the wave pattern is regular so long as the pattern is spatially modulated.Dedicated to Professor P. R. Sethna on the Occasion of His 70th BirthdayThis research was partially supported by an NSF Presidential Young Investigator Award and an ONR Grant No. N00014-89-J-3023.     

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.     

Transition to instability of a phase interface in a porous medium in the isothermal approximation

The stability of the phase interface in geothermal systems is considered in the isothermal approximation with allowance for capillary effects. The dispersion relation is obtained and the domains of stability and instability of steady-state vertical flows are found. Possible types of transition to instability, namely, transitions with the most unstable mode corresponding to zero and infinite wavenumbers or to all wavenumbers simultaneously, are described. In the first case the nonlinear Kolmogorov-Petrovskii-Piskunov equation describing the evolution of a narrow strip of weakly unstable modes on the stability threshold is derived. The effect of the parameters of the system on its stability is investigated.     

Nonlinear modulation of the interfacial waves of two superposed fluids

The slow modulation of the interfacial capillary — gravity waves of two superposed fluids with uniform depths and solid walls is investigated by using the method of multiple scales. The evolution of a packet is described by the nonlinear Schrödinger equation, and then the stability of the so-called Stokes wave train is discussed.     

滚波演化中聚合过程的数值模拟研究

滚波是一种重力作用下自由液面失稳诱发的水面波动现象, 通常可分为具有相对稳定波形和波速的周期性滚波与波形和波速不断变化的不规则滚波(自然滚波). 不规则滚波的相互作用和发展演化过程十分复杂, 至今对其认识尚不成熟. 本文采用基于雷诺平均Navier-Stokes方程的立面二维数值模型, 对不规则滚波发展过程中的吸收聚合和追赶聚合现象进行了数值模拟研究. 分析了两种聚合模式的演化过程, 给出了滚波聚合过程中完整的波形、波速、速度剖面以及湍流黏性等重要信息. 结果表明滚波的聚合过程是不规则滚波演化和增长的重要机制, 在特定条件下滚波增长由自然增长模式转变为以吸收聚合和追赶聚合为主的增长模式. 滚波聚合过程中, 依次经历后波追赶、爬升、与前波合并、内部流场调制等多个步骤, 最终形成一个具有更大波长和波高的滚波. 本文发现了在3个滚波间距较近的情况下, 会发生二重聚合现象, 即后两个滚波首先聚合, 然后与前波进一步聚合形成一个新的滚波.      

欧拉-伯努利梁运动场的正交性及其能量传导特性分析

从无阻尼欧拉—伯努利梁振动方程解析解出发, 推导了有限长梁的关于谱系数的时间—空间平均能量和功率流表达式. 在此基础上, 从泛函分析观点, 探讨了弯曲运动场: 衰减振动、行波模式分解关于能量、功率泛函的正交性. 结果表明: 弯曲衰减振动模式和行波模式关于功率流、机械能时间—空间平均是相互独立的, 即关于场能和场功率互不干涉, 满足叠加原理; 衰减振动场导能与行波场导能的重要区别在于功率流关于右、左衰振动模式分解不满足叠加原理, 即弯曲衰减振动场间的相互"干涉"是使其具有能量传导能力的内在原因. 通过右端集中阻尼器支撑的梁的稳态功率流仿真分析计算, 表明低频区振动导能不可忽略, 同时, 衰减振动场和行波场间存在一定的能量交换现象, 但随着频率升高, 振动场传导能量不断下降, 同时能量传导效率也不断下降.      

The phase configuration of the waves around an accelerating disturbance in a rotating stratified fluid

Ray theory is extended to consider the case of an accelerating disturbance which is producing waves in a rotating stratified fluid. Starting from the equations of motion, dispersion relations are derived for surface gravity waves, capillary waves, Rossby waves and internal-inertial waves. The wave system is studied in each case for the problem of a body starting impulsively from rest and for a body starting from rest and moving with constant acceleration.     

三层流体中斜入射波作用下半无限板的水弹性响应

解析地研究了在三层流体中斜入射波浪与半无限弹性板的相互作用引起的波散射和板的水弹性响应. 三层流体在界面处的密度发生阶跃, 各层为一常数. 假设流体不可压缩、无黏、流体运动无旋. 在线性势流理论框架下, 使用本征函数展开法和内积式给出波板相互作用的半解析解. 根据色散关系分析, 得到了表面波模态和界面波模态入射时的临界入射角. 随着物理参数的变化, 临界角将随之发生变化. 临界角决定了当由开阔水域向板覆盖水域传播的表面波或界面波的存在性: (1)板覆盖水域入射界面上, 透射波能否存在; (2)入射界面之上界面中, 板覆盖水域中的透射波以及开阔水域中的反射波能否存在. 当下界面波入射时并且入射角足够大时, 开阔水域中的下界面波模态是整个流体域中唯一存在的模态.      

DAMPING OF VERTICALLY EXCITED SURFACE WAVE IN WEAKLY VISCOUS FLUID

In a vertically oscillating circular cylindrical container, singular perturbation theory of two-time scale expansions is developed in weakly viscous fluids to investigate the motion of single free surface standing wave by linearizing the Navier-Stokes equation. The fluid field is divided into an outer potential flow region and an inner boundary layer region. The solutions of both two regions are obtained and a linear amplitude equation incorporating damping term and external excitation is derived. The condition to appear stable surface wave is obtained and the critical curve is determined. In addition, an analytical expression of damping coefficient is determined. Finally, the dispersion relation, which has been derived from the inviscid fluid approximation, is modified by adding linear damping. It is found that the modified results are reasonably closer to experimental results than former theory. Result shows that when forcing frequency is low, the viscosity of the fluid is prominent for the mode selection. However, when forcing frequency is high, the surface tension of the fluid is prominent.     

Numerical simulation of rarefied suspension sedimentation in a container

The evolution of the large-scale velocity perturbations in a homogeneous suspension sedimenting in a rectangular container with rigid horizontal walls and periodic conditions on the vertical boundaries is considered. Numerical simulation of the point-particle motion showed that the density and velocity fluctuations decrease with time. The perturbations are damped due to reshaping of the sedimentation front and the nonlinear interaction of the different modes.     

Stability and modal analysis of shock/boundary layer interactions

The dynamics of oblique shock wave/turbulent boundary layer interactions is analyzed by mining a large-eddy simulation (LES) database for various strengths of the incoming shock. The flow dynamics is first analyzed by means of dynamic mode decomposition (DMD), which highlights the simultaneous occurrence of two types of flow modes, namely a low-frequency type associated with breathing motion of the separation bubble, accompanied by flapping motion of the reflected shock, and a high-frequency type associated with the propagation of instability waves past the interaction zone. Global linear stability analysis performed on the mean LES flow fields yields a single unstable zero-frequency mode, plus a variety of marginally stable low-frequency modes whose stability margin decreases with the strength of the interaction. The least stable linear modes are grouped into two classes, one of which bears striking resemblance to the breathing mode recovered from DMD and another class associated with revolving motion within the separation bubble. The results of the modal and linear stability analysis support the notion that low-frequency dynamics is intrinsic to the interaction zone, but some continuous forcing from the upstream boundary layer may be required to keep the system near a limit cycle. This can be modeled as a weakly damped oscillator with forcing, as in the early empirical model by Plotkin (AIAA J 13:1036–1040, 1975).     

ACTUATION OF SLOSHING MODULATED FORCE AND MOMENT ON LIQUID CONTAINER DRIVEN BY JITTER ACCELERATIONS ASSOCIATED WITH SIEW MOTION IN MICROGRAVITY

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Wave propagation in circular cylindrical shells with periodic axial curvature

The propagation of longitudinal and flexural waves in axisymmetric circular cylindrical shells with periodic circular axial curvature is studied using a finite element method previously developed by the authors. Of primary interest is the coupling of these wave modes due to the periodic axial curvature which results in the generation of two types of stop bands not present in straight circular cylinders. The first type is related to the periodic spacing and occurs independently for longitudinal and flexural wave modes without coupling. However, the second type is caused by longitudinal and flexural wave mode coupling due to the axial curvature. A parametric study is conducted where the effects of cylinder radius, degree of axial curvature, and periodic spacing on wave propagation characteristics are investigated. It is shown that even a small degree of periodic axial curvature results in significant stop bands associated with wave mode coupling. These stop bands are broad and conceivably could be tuned to a specific frequency range by judicious choice of the shell parameters. Forced harmonic analyses performed on finite periodic structures show that strong attenuation of longitudinal and flexural motion occurs in the frequency ranges associated with the stop bands of the infinite periodic structure.     

双自由面溶质?热毛细液层的不稳定性

溶质?热毛细对流是流体界面的浓度和温度分布不均导致的表面张力梯度驱动的流动, 它主要存在于空间微重力环境、小尺度流动等表面张力占主导的情况中, 例如晶体生长、微流控、合金浇筑凝固、有机薄液膜生长等. 对其流动进行稳定性分析具有重要意义. 本文采用线性稳定性理论研究了双自由面溶质?热毛细液层对流的不稳定性, 得到了两种负毛细力比(η)下的临界Marangoni数与Prandtl数(Pr)的函数关系, 并分析了临界模态的流场和能量机制. 研究发现: 溶质?热毛细对流和纯热毛细对流的临界模态有较大的差别, 前者是同向流向波、逆向流向波、展向稳态模态和逆向斜波, 后者是逆向斜波和逆向流向波. 在Pr较大时, Pr增加会降低流动稳定性; 在其他参数下, Pr增加会增强流动稳定性. 在中低Pr, 溶质毛细力使流动更加不稳定; 在大Pr时, 溶质毛细力的出现可能使流动更加稳定; 在其他参数下, 溶质毛细力会减弱流动稳定性. 流动稳定性不随η单调变化. 在多数情况下, 扰动浓度场与扰动温度场都是相似的. 能量分析表明: 扰动动能的主要能量来源是表面张力做功, 但其中溶质毛细力和热毛细力做功的正负性与参数有关.      

On time-dependent,two-layer flow over topography: II. Evolution and propagation of solitary waves

The evolution of topographically generated interfacial motion is considered in a two-layer model. A system of two non-linear equations, similar to the Boussinesq equations for shallow water waves, is derived. The consequences of the cubic non-linearity of these equations on the nature of the solitary wave solutions are explored. A dispersion relation for solitary waves implies the existence of maxima for speed and displacement in a wave. The limiting values are shown to agree with other studies. The growth of solitary and/or cnoidal waves is studied for finite pulses of displacement and for internal bores.