A circulating gravity wave in a cylindrical tank

A circular cylindrical vertical tank is partially filled with a liquid (Water) and a gas (air) above it. The top lid of the container rotates around the cylinder axis and induces a flow in the gas and the liquid. Above a critical rotational speed a large amplitude circulating gravity wave forms. The two phase flow problem will be reduced to a free surface single phase flow problem and the critical parameters, when the axis-symmetric flow becomes unstable with respect to the circulating gravity wave will be determined. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Resonant Flow of a Rotating Fluid Past an Obstacle: The General Case

We consider the flow of a rotating fluid past an antisymmetric obstacle placed on the axis of a cylindrical tube, for the case when the upstream flow is nearly resonant, or critical, so that the speed of a free linear long wave is nearly zero in the frame of reference of the obstacle. The perturbed flow is dominated by the resonant mode, whose amplitude satisfies a forced Korteweg—de Vries equation in this general case when the upstream flow contains radial shear andjor radially dependent angular velocity.     

参数激励圆柱形容器中的非线性Faraday波

在柱坐标系下,通过奇异摄动理论的多尺度展开法求解势流方程,研究了垂直强迫激励圆柱形容器中的单一模式水表面驻波模式。假设流体是无粘、不可压且运动是无旋的,在忽略了表面张力的影响下,用两变量时间展开法得到一个具有立方项以及底部驱动项影响的非线性振幅方程。对上述方程进行了数值计算,计算的结果显示了在不同驱动振幅和驱动频率下,会激发不同自由水表面驻波模式,从等高线的图像来看,和以往的实验结果相当吻合。     

圆柱形容器中竖直激励表面波的毛细影响

在竖直振动的圆柱形容器中,利用理想流体中两时间尺度奇摄动展开法,研究了包括表面张力影响的自由面单一表面驻波的运动．通过求解势流方程,获得了一个包含三阶非线性项、外激励及表面张力影响的非线性振幅方程．结果表明当驱动频率较低时,表面张力对表面波模式选择不重要;然而,当驱动频率较高时,表面张力的影响是不可忽略的．说明表面张力具有使得自由面返回到平衡位置的作用．另外,由于考虑了表面张力的影响,使得理论结果比无表面张力时更加接近先前的实验结果．     

The movement of bubbles in a vibrating liquid

The problem of movement of bubbles in a liquid in a rigid vertical cylindrical container with a rigid base is considered. The container vibrates vertically with a displacement amplitude Δ and an angular frequency ω with the acceleration g due to gravity.     

Rotation of an axisymmetric particle in a plane flow

The rotation of an inertialess ellipsoidal particle in a shear flow of a Newtonian fluid has been firstly analyzed by Jeffery [1]. He found that the particle rotates such that the end of its symmetry axis describes a closed periodic orbit. Based on the balance equation of the angular momentum we derived the equation of rotational motion of a cylindrical particle, that is suspended in a plane shear flow field of a viscous fluid, and solved numerically. The rotary inertia is taken into account. The solution is compared with the rotation of a slender particle. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Exact Free Surface Flows for Shallow Water Equations I: The Incompressible Case

A comprehensive exact treatment of free surface flows governed by shallow water equations (in sigma variables) is given. Several new families of exact solutions of the governing PDEs are found and are shown to embed the well-known self-similar or traveling wave solutions which themselves are governed by reduced ODEs. The classes of solutions found here are explicit in contrast to those found earlier in an implicit form. The height of the free surface for each family of solutions is found explicitly. For the traveling or simple wave, the free surface is governed by a nonlinear wave equation, but is arbitrary otherwise. For other types of solutions, the height of the free surface is constant either on lines of constant acceleration or on lines of constant speed; in another case, the free surface is a horizontal plane while the flow underneath is a sine wave. The existence of simple waves on shear flows is analytically proved. The interaction of large amplitude progressive waves with shear flow is also studied.     

Measurements of transient flow fields driven by a discontinuously applied rotating magnetic field

This experimental study considers the transient liquid metal flow which is generated inside a cylindrical container by discontinuously applying a rotating magnetic field (RMF). The focus is on the fluid motion arising from the impulsive spin–up from the resting state. The ultrasonic Doppler velocimetry (UDV) has been used to determine profiles of the fluid velocity in the ternary alloy GaInSn. The azimuthal and vertical velocity components have been measured allowing for an analysis of both the primary, swirling flow and the secondary flow in the radial–meridional plane. The experimental results show an excellent agreement with recently published numerical results. The investigations reveal that the recirculating flow in the radial–meridional plane undergoes characteristic oscillations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Exact Free Surface Flows for Shallow Water Equations II: The Compressible Case

Exact free surface flows with shear in a compressible barotropic medium are found, extending the authors’ earlier work for the incompressible medium. The barotropic medium is of finite extent in the vertical direction, while it is infinite in the horizontal direction. The “shallow water” equations for a compressible barotropic medium, subject to boundary conditions at the free surface and at the bottom, are solved in terms of double psi-series. Simple wave and time-dependent solutions are found; for the former the free surface is of arbitrary shape while for the latter it is a damping traveling wave in the horizontal direction. For other types of solutions, the height of the free surface is constant either on lines of constant acceleration or on lines of constant speed. In the case of an isothermal medium, when γ = 1, we again find simple wave and time-dependent solutions.     

Variable energy blast wave through rotating gas

The aim of the present paper is to study the propagation of a variable energy blast wave (cylindrical) through a gas having solid-body rotation. The modified similarity method has been used to obtain the first, second and third approximation solutions. The analysis shows that the effects of solid-body rotation on the flow are of third order. Variation of pressure, density and radial velocity distributions with initial angular velocity has been discussed.     

Modeling and control of surface gravity waves in a model of a copper converter

Undesirable splashing appears in copper converters when air is injected into the molten matte to trigger the conversion process. We consider here a cylindrical container horizontally placed and containing water, where gravity waves on the liquid surface are generated due to water injection through a lateral submerged nozzle. The fluid dynamics in a transversal section of the converter is modeled by a 2-D inviscid potential flow involving a gravity wave equation with local damping on the liquid surface. Once the model is established, using a finite element method, the corresponding natural frequencies and normal modes are numerically computed in the absence of injection, and the solution of the system with injection is obtained using the spectrum. If a finite number of modes is considered, this approximation leads to a system of ordinary differential equations where the input is represented by the fluid injection. The dynamics is simulated as perturbations around a constant fluid injection solution, which is the desired operating state of the system, considering that the conversion process does not have to be stopped or seriously affected by the control. The solution is naturally unstable without control and the resulting increase of amplitude of the surface waves are assimilable to the splashing inside the converter. We show numerically that a variable flow around the operating injection is able to sensibly reduce these waves. This control is obtained by a LQG feedback law by measuring the elevation of the free surface at the point corresponding to the opposite extreme to where the nozzle injection is placed.     

Retracted: Unsteady flow and heat transfer on a rotating disk in a porous medium

This article has been retracted. See retraction notice DOI: 10.1002/mma.850 . An unsteady flow and heat transfer in a porous medium of a viscous incompressible fluid over a rotating disk in an otherwise ambient fluid are studied. The unsteadiness in the flow field is caused by the angular velocity of the disk which varies with time. The new self‐similar solution of the Navier–Stokes and energy equations is obtained numerically. The solution obtained here is not only the solution of the Navier–Stokes equations, but also of the boundary layer equations. Also, for a simple scaling factor, it represents the solution of the flow and heat transfer in the forward stagnation‐point region of a rotating sphere or over a rotating cone. The asymptotic behaviour of the solution for a large porosity or for a large independent variable is also examined. The surface shear stresses in the radial and tangential directions and the surface heat transfer increase as the acceleration parameter increases. Also, the surface shear stress in the radial direction and the surface heat transfer decrease with increasing porosity, but the surface shear stress in the tangential direction increases. Copyright © 2006 John Wiley & Sons, Ltd.     

弱粘性流体中垂直激励表面波的阻尼效应

在竖直振动的圆柱形容器中,将Navier-Stokes方程线性化,利用两时间尺度奇异摄动展开法研究了弱粘性流体的单一自由面驻波运动．整个流场被分为外部势流区和内部边界层区两部分,对两部分区域分别求解,得到包含阻尼项和外驱动影响的线性振幅方程．利用稳定性分析,得到形成稳定表面波的条件,给出了临界曲线．此外,还获得了阻尼系数的解析表达式．最后,将线性阻尼加到理想流体条件下所得到的色散关系中对其进行修正,理论结果证明修正后的驱动频率更加接近实验的结果．通过计算发现,当驱动的频率较低时,流体的粘性对表面波模式选择有重要影响,而表面张力的影响不明显;但当驱动频率较高时,流体的表面张力起主要作用,而流体的粘性影响甚小．     

用时变雷诺方程模型模拟孤立波与半圆型防波堤的相互作用

以时变雷诺方程为控制方程,用k-ε模型封闭该方程,采用体积函数(VOF)方法来跟踪波动自由表面,建立了二维垂向波浪数学模型,并用已有的实验资料进行了验证．随后用该模型模拟了半圆型防波堤与孤立波在淹没、平顶水位、完全露顶且不越浪3种典型工况下的相互作用过程．得到了半圆堤附近的流场、压强场和波面的变形过程．结果表明,在淹没状态下,半圆堤背浪面的底部会产生涡旋;平顶水位时,由于越浪的冲击作用,在半圆堤的背浪面将逐渐形成一对较大的涡旋,而半圆堤背浪面的底部,速度始终相对较小;而在露顶不越浪时,半圆堤的迎浪面会出现波浪的二次爬升的现象．为进一步研究结构物附近的污染物的输移扩散和泥沙运动提供基础．     

Experimental and asymptotic investigation of a rotary wave in a cylindrical container

Rotary gravity waves in a partially filled vertical cylindrical container excited by a rotating disc at the top of the cylinder are investigated. Analytical results for the growth rate of the waves are reported. Moreover, the development of the wave is shown in an experiment. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sloshing characteristics in half-full horizontal elliptical tanks with vertical baffles

The effects of surface-piercing or bottom-mounted vertical baffles on two-dimensional liquid sloshing characteristics in a half-full non-deformable horizontal cylindrical container of elliptical cross section is investigated. The problem solution is achieved by employing the linear potential theory in conjunction with the successive conformal transformation technique, leading to matrix eigenvalue problems on simple (rectangular and semi-infinite strip) regions. Plots of the resonant eigen-frequencies as a function of the baffle extension at selected container aspect ratios are presented and discussed for the three lowest antisymmetric and symmetric transverse oscillation modes. Also, the effects of vertical baffles on the hydrodynamic pressure mode shapes and sloshing flows are examined through appropriate 2D images. The surface-piercing vertical baffle is demonstrated to be an effective tool in reducing the antisymmetric sloshing frequencies, especially for lower aspect ratio tanks and higher modes. On the other hand, the bottom-mounted baffle is shown to have a great influence on the higher antisymmetric slosh modes only when its tip approaches the liquid free surface. Limiting cases are considered and good agreements with available analytic and numerical solutions as well as experimental data are obtained.     

Instability of Three Dimensional Waves in Shear Flows

The instability of a shear flow is greatly affected by the presence of one or more preexisting waves. This problem is considered for an oblique wave on a parallel shear flow with a free surface. The analysis uses a mean flow first harmonic theory.     

低重环境下液体非线性晃动的稳态响应

在俯仰激励作用下,圆柱贮箱中液体晃动存在平面运动、旋转运动和平面运动中的旋转运动等,而这些运动的稳定、不稳定区间的分界线与贮箱的半径、充液深度、重力强度、表面张力系数和晃动阻尼等基本系统参数有关．据此,首先建立了液体非线性晃动的微分方程组,并借助变分原理建立了液体压力体积分形式的Lagrange函数;然后将速度势函数在自由液面处作波高函数的级数展开,通过变分从而导出自由液面运动学和动力学边界条件非线性方程组;最后用多尺度法求解非线性方程组,就重力强度对圆柱形贮箱中液体非线性晃动的全局稳态响应的影响进行了详细的理论分析,并发现系统软硬特性的变化、跳跃和滞后等非线性现象．     

Sur l'existence des petites oscillations d'une colonne liquide cylindrique en apesanteur

Résumé L'auteur considère une colonne de liquide parfait incompressible en apesanteur dans un container cylindrique infiniment long de section droite arbitraire.Quand le container est fixe, il y a une position d'équilibre de liquide pour laquelle la surface libre est un cylindre circulaire. En utilisant une méthode classique de l'analyse fonctionnelle, il prouve l'existence de petites oscillations du liquide autour de cette position.Quand le container est en rotation uniforme, il est nécessaire d'appliquer une nouvelle méthode. A partir d'une formulation variationnelle du problème, l'auteur démontre, au moyen de la théorie des semi-groupes, l'existence des petites oscillations du liquide, quelle que soit la vitesse angulaire de rotation.

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The author considers a column of inviscid incompressible liquid under zero gravity contained in a infinitely long cylindrical container, with an arbitrary cross-section.When the container is fixed, there is an equilibrium position of the liquid, the free surface being a circular cylinder. Using a classical method of the functional analysis, he proves the existence of small oscillations of the liquid about this position.When the container is in uniform rotation, it is necessary to apply a new method. From a variational formulation of the problem, the author proves, by means of semi-groups theory, the existence of small oscillations of the liquid, for an arbitrary angular velocity.

     

Kelvin–Helmholtz instability with a free surface

The well-posedness of the hydrostatic equations is linked to long wave stability criteria for parallel shear flows. We revisit the Kelvin--Helmholtz instability with a free surface. In the wall-bounded case, the flow is unstable to all wave lengths. Short wave instabilities are localized and independent of boundary conditions. On the other hand, long waves are shown to be stable if the upper boundary is a free surface and gravity is sufficiently small. We also consider smooth velocity profiles of the base flow rather than a velocity jump. We show that stability of long waves for small gravity generally holds for monotone profiles U(y). On the other hand, this need not be the case if U is not monotone.