Application of the VOF method to the sloshing of a fluid in a partially filled cylindrical container

In a previous work we solved numerically the steady-state motion of an ideal fluid that fills a moving cylindrical container with partitions, and were able to compute the equivalent moments of inertia. Here we extend this work in two steps. First we introduce time dependence and then free surfaces, and are able to compute the transient motion of the fluid not filling the container. The main body of the work has to do with the treatment of free surfaces. Our approach is an extention to three dimensions of the volume of fluid method of Hirt and Nichols. The solution algorithm is outlined, and two examples that demonstrate its capability are presented.     

Influence of convective acceleration in the stokes equation on the pressure in a fluid

High rates of slip at points of friction, and also the use of water as a lubricant lead one to consider the problem of the influence of inertial forces on the development of the load capacity of a film of lubricant. A number of authors [1–6] have taken into account the convective terms in the Navier-Stokes equation in determining the pressure in a fluid film of lubricant in a bearing. An increase in the load capacity of the lubricant film by 20% at Re = 5000 was noted in [6]. In the present paper, we show that the convective inertia terms in the Navier-Stokes equation are equivalent to the square of the pressure, the tangential stresses, and the vorticity. The derived equation is used to determine in the first approximation the contribution of the inertial forces to the load capacity of the lubricant film of a high-speed bearing.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 148–152, September–October, 1979.I thank A. K. D'yachkov for supervising the work, M. V. Korovchinskii for helpful comments, and also E. I. Poddubnaya for her computer calculations.     

填隙幂率流体下两刚性圆球相对错移时的粘性阻力

湿颗粒离散元模型以两球作用时填隙流体定常流动解为基础,其中切向作用是难点,国外仅有Goldman的牛顿流体渐近解.基于Reynolds润滑理论导出了两刚性球切向错动时填隙幂律流体的压力方程,并利用傅立叶级数展开简化,通过数值解法得到相应的压力分布、黏性阻力及阻力矩.该方程的解较之作者先前对速度场附加假定的结果精确,而当幂指数为1时等价于Goldman的牛顿流体渐近解.     

Impact of a rectangular plate on a liquid half-space

The potential flow of an ideal incompressible fluid occupying a half-space resulting from the impact of a rectangular plate on its surface is considered. Outside the plate the surface of the fluid is free. An integral equation of the first kind is obtained for the impulsive pressure beneath a flexible plate. It is solved on a computer by the power series method for the particular case of a rigid nondeformable plate. The accuracy of the method is estimated. The theoretical dependence of the virtual mass and virtual moment of inertia coefficients of a rigid nondeformable plate on the plate geometry is constructed and compared with the experimental data and with empirical formulas [1-3] not directly related with the solution of the Laplace equation.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.5, pp. 120–126, September–October, 1992.     

变刚度可瘪乳胶管的参数激振

当脉动流经过乳胶管时,如果跨壁压力为负,则乳胶管就被压瘪,其截面积和截面惯距都将减小。本文首先导出了上述问题的控制方程——有阻尼 Mathieu 方程,然后再用摄动法求解。证实了截面积和截面惯距的周期变化加强了脉动流的激励、容易使乳胶管发生位移振荡,这与实验结果在定性上是吻合的。     

Instantaneous Squeeze-Film Force Between a Heat Exchanger Tube and a Support Plate for Arbitrary Tube Motion

The instantaneous squeeze-film force between a heat exchanger tube and a support plate is studied. Based on a two-dimensional rectangular plate model, a short-sleeve squeeze-film model for arbitrary tube motion is developed. The instantaneous squeeze-film force is expressed in normal and tangential directions. The normal squeeze-film force consists of four nonlinear terms, the viscous, unsteady inertia, convective inertia and centripetal inertia terms. Three nonlinear terms, the viscous, unsteady inertia and Coriolis inertia terms, make up the tangential squeeze-film force. An experimental apparatus was developed in order to evaluate the theoretical models against measurements of a finite length squeeze film. A modified model based on the experimental data is obtained where the viscous terms for both directions are multiplied by the instantaneous Reynolds number. All the inertia terms are multiplied by constant coefficients. The modified model is in good agreement with most experimental cases for unsymmetrical linear motion, approximate circular motion and elliptical motion. The form of the modified model is suitable for predicting instantaneous squeeze-film forces in the simulation of heat exchanger tube vibration. Further work using different sized components and fluid properties is required in order to finalize coefficient values.     

Analysis of a plate containing an elliptic inclusion with eigencurvatures

Summary An infinite plate containing an elliptic subregion in which a uniform eigencurvature is prescribed is analyzed. The problem is formulated by using the classical plate theory. Employing the Maysel's relation, an integral-type solution to the equilibrium equation is expressed in terms of the eigencurvature. Closed-form solutions of the displacement and corresponding resultant moment are obtained for interior points as well as for exterior points of the ellipse. An infinite plate containing an elliptic inhomogeneity in which a uniform eigencurvature is prescribed is also considered. The disturbance of the displacement and corresponding resultant moment due to the inhomogeneity is determined by the equivalent eigencurvature method. Solutions of a circular finite plate with uniform eigencurvature in a circular zone are also obtained analytically. Received 30 September 1997; accepted for publication 3 February 1998     

The PROSPER General Circulation Model,an adapted form of the Sandia Ocean Modelling method: A numerical description

The PROSPER General Circulation Model (PGCM) is a three-dimensional model based on the incompressible Navier-Stokes equations, an equation of state and the heat equation. The hydrostatic approximation and the rigid lid approximation are used. The system of equations is converted into an equivalent form in which the surface pressure is more directly expressed in terms of a two-dimensional Poisson equation. The finite difference method is described and analysed. In particular, the iteration method within every time step to determine the new surface pressure and velocity components, and numerical diffusion aspects due to the use of the staggered Arakawa-C grid are looked at. Since part of the development of the PGCM code is a result of studying the Sandia Ocean Modelling System (SOMS), a comparison is made with respect to the concepts used in both models.     

Motion of a permeable shell in a spherical container filled with non-Newtonian fluid

This paper presents an analytical study of creeping motion of a permeable sphere in a spherical container filled with a micro-polar fluid. The drag experienced by the permeable sphere when it passes through the center of the spherical container is studied.Stream function solutions for the flow fields are obtained in terms of modified Bessel functions and Gegenbauer functions. The pressure fields, the micro-rotation components,the drag experienced by a permeable sphere, the wall correction factor, and the flow rate through the permeable surface are obtained for the frictionless impermeable spherical container and the zero shear stress at the impermeable spherical container. Variations of the drag force and the wall correction factor with respect to different fluid parameters are studied. It is observed that the drag force, the wall correction factor, and the flow rate are greater for the frictionless impermeable spherical container than the zero shear stress at the impermeable spherical container. Several cases of interest are deduced from the present analysis.     

Stability of aneurysm solutions in a fluid-filled elastic membrane tube

When a hyperelastic membrane tube is inflatedby an internal pressure, a localized bulge will form when thepressure reaches a critical value. As inflation continues thebulge will grow until it reaches a maximum size after whichit will then propagate in both directions to form a hat-likeprofile. The stability of such bulging solutions has recentlybeen studied by neglecting the inertia of the inflating fluidand it was shown that such bulging solutions are unstableunder pressure control. In this paper we extend this recentstudy by assuming that the inflation is by an inviscid fluidwhose inertia we take into account in the stability analysis.This reflects more closely the situation of aneurysm forma-tion in human arteries which motivates the current series ofstudies. It is shown that fluid inertia would significantly re-duce the growth rate of the unstable mode and thus it has astrong stabilizing effect.     

Investigation of the effect of the Coriolis force on a thin fluid film on a rotating disk

The effect of the Coriolis force on the evolution of a thin film of Newtonian fluid on a rotating disk is investigated. The thin-film approximation is made in which inertia terms in the Navier–Stokes equation are neglected. This requires that the thickness of the thin film be less than the thickness of the Ekman boundary layer in a rotating fluid of the same kinematic viscosity. A new first-order quasi-linear partial differential equation for the thickness of the thin film, which describes viscous, centrifugal and Coriolis-force effects, is derived. It extends an equation due to Emslie et al. [J. Appl. Phys. 29, 858 (1958)] which was obtained neglecting the Coriolis force. The problem is formulated as a Cauchy initial-value problem. As time increases the surface profile flattens and, if the initial profile is sufficiently negative, it develops a breaking wave. Numerical solutions of the new equation, obtained by integrating along its characteristic curves, are compared with analytical solutions of the equation of Emslie et al. to determine the effect of the Coriolis force on the surface flattening, the wave breaking and the streamlines when inertia terms are neglected.     

Plane strain dynamics of elastic solids with intrinsic boundary elasticity, with application to surface wave propagation

In this paper, in a development of the static theory derived by Steigmann and Ogden (Proc. Roy. Soc. London A 453 (1997) 853), we establish the equations of motion for a non-linearly elastic body in plane strain with an elastic surface coating on part or all of its boundary. The equations of (linearized) incremental motions superposed on a finite static deformation are then obtained and applied to the problem of (time-harmonic) surface wave propagation on a pre-stressed incompressible isotropic elastic half-space with a thin coating on its plane boundary. The secular equation for (dispersive) wave speeds is then obtained in respect of a general form of incompressible isotropic elastic strain-energy function for the bulk material and a general energy function for the coating material. Specialization of the form of strain-energy function enables the secular equation to be cast as a quartic equation and we therefore focus on this for illustrative purposes. An explicit form for the secular equation is thereby obtained. This involves a number of material parameters, including residual stress and moment in the properties of the coating. It is shown how this equation relates to previous work on waves in a half-space with an overlying thin layer set in the classical theory of isotropic elasticity and, in particular, the significant effect of omission of the rotatory inertia term, even at small wave numbers, is emphasized. Corresponding results for a membrane-type coating, for which the bending moment, inertia and residual moment terms are absent, are also obtained. Asymptotic formulas for the wave speed at large wave number (high frequency) are derived and it is shown how these results influence the character of the wave speed throughout the range of wave number values. A bifurcation criterion is obtained from the secular equation by setting the wave speed to zero, thereby generalizing the bifurcation results of Steigmann and Ogden (Proc. Roy. Soc. London A 453 (1997) 853) to the situation in which residual stress and moment are present in the coating. Numerical results which show the dependence of the wave speed on the various material parameters and the finite deformation are then described graphically. In particular, features which differ from those arising in the classical theory are highlighted.     

Coupled free vibration analysis of a fluid-filled rectangular container with a sagged bottom membrane

In the present paper, two-dimensional coupled free vibrations of a fluid-filled rectangular container with a sagged bottom membrane are investigated. This system consists of two rigid walls and a membrane anchored along two rigid vertical walls. It is filled with incompressible and inviscid fluid. The membrane material is assumed to act like an inextensible material with no bending resistance. First, the nonlinear equilibrium equation is solved and the equilibrium shape of the membrane is obtained using an analytical formulation neglecting the membrane weight. The small vibrations about the equilibrium configuration are then investigated. Along the contact surface between the bottom membrane and the fluid, the compatibility requirement is applied for the fluid–structure interactions and the finite element method is used to calculate the natural frequencies and mode shapes of the fluid–membrane system. The vibration analysis of the coupled system is accomplished by using the displacement finite element for the membrane and the pressure fluid-finite element for the fluid domain. The variations of natural frequencies with the pressure head, the membrane length, the membrane weight and the distance between two rigid walls are examined. Moreover, the mode shapes of system are investigated.     

Slosh dynamics of inviscid fluids in two‐dimensional tanks of various geometry using finite element method

This paper brings into focus some of the interesting effects arising from the motion of the liquid free surface due to sloshing in partially filled containers of several geometrical shapes in two dimensions. The slosh characteristics that include frequencies, free surface profiles and the hydrodynamic pressure over the container walls have been reported in this study. The equations of motion of the fluid, considered inviscid, are expressed in terms of the pressure variable alone. It is assumed that the frequency of the exciting oscillation is not in the immediate neighborhood of the natural slosh frequency, so that the slope of the free surface is small. Simple harmonic oscillation and earthquake excitations are used as the prescribed boundary conditions. A finite difference‐based iterative time‐stepping technique is employed to advance the solution in the time domain. The paper presents numerical solutions for rectangular, vertically mounted annular cylindrical, trapezoidal and horizontal circular cylindrical containers. Numerical results obtained are compared with the available existing solutions to validate the code developed. The parametric study of the slosh dynamic systems shows the importance of the nature of excitation, fluid height and the geometry of the container. Copyright © 2007 John Wiley & Sons, Ltd.     

Extensions to the Macroscopic Navier–Stokes Equation

A development is provided showing that for any phase, by not neglecting the macroscopic terms of the deviation from the intensive momentum and of the dispersive momentum, we obtain a macroscopic secondary momentum balance equation coupled with a macroscopic dominant momentum balance equation that is valid at a larger spatial scale. The macroscopic secondary momentum balance equation is in the form of a wave equation that propagates the deviation from the intensive momentum while concurrently, in the case of a Newtonian fluid and under certain assumptions, the macroscopic dominant momentum balance equation may be approximated by Darcys equation to address drag dominant flow. We then develop extensions to the dominant macroscopic Navier–Stokes (NS) equation for saturated porous matrices, to account for the pressure gradient at the microscopic solid-fluid interfaces. At the microscopic interfaces we introduce the exchange of inertia between the phases, accounting for the relative fluid square velocities and the rate of these velocities, interpreted as Forchheimer terms. Conditions are provided to approximate the extended dominant NS equation by Forchheimer quadratic momentum law or by Darcys linear momentum law. We also show that the dominant NS equation can conform into a nonlinear wave equation. The one-dimensional numerical solution of this nonlinear wave equation demonstrates good qualitative agreement with experiments for the case of a highly deformable elasto-plastic matrix.     

固—液耦合Timoshenko管道的稳定性分析

根据Ｈａｍｉｌｔｏｎ原理的固－液耦合振同分方程用幂级数法计算了Ｔｉｍｏｓｈｅｎｋｏ管道的固有频率和临界流速。给出了管道前三阶固有频率－流速的关系曲线,分析了转动惯量对该输流管道的稳定特性的影响。计算结果表明,转动一对两端简支的固－液合Ｔｉｍｏｓｈｅｎｋｏ管道的静力失稳没有影响,但对其频率特性和动力失稳有影响。     

A SIMPLIFIED LINEARIZED MODEL FOR THE FLUID-COUPLED VIBRATIONS OF SPENT NUCLEAR FUEL RACKS

Fluid effects can induce strong coupling between immersed spent nuclear fuel racks, when they are subjected to earthquake excitations. Broc and co-workers found that such system can display two-dimensional vibratory responses, even when external excitations are applied along a symmetry axis of the system. Their analysis was supported by numerical computations, using a finite element code developed at CEA (Saclay) which is well suited for fluid–structure-coupled systems. The present paper was inspired by their work as a possible lighter computational alternative. Here we develop a theoretical model which enables the computation of fluid-coupling effects, subject to some simplifying assumptions: (i) three-dimensional flow effects are neglected, (ii) gaps between the fuel assemblies (and between these and the container) are small when compared with longitudinal length-scales. From these assumptions, we postulate a simplified flow inside the channels, such that gap-averaged velocity and pressure fields are described in terms of a single space coordinate for each fluid channel. Using this approach, the flow can be formulated in analytical terms, enabling effective computation of the dynamical response, of a multi-rack fluid-coupled system.     

Dewar瓶内液体晃动的近似计算方法

本文研究Dewar瓶内液体的晃动特征问题，将液体晃动的微分方程边值问题转换为具有积分形式的泛函极值问题，在此基础上建立了旋转对称容器液体晃动特征问题的有限元数值计算方法和通用程序，并计算了Dewar球，球腔，圆柱腔液体的晃动固有频率，结果显示了Dewar瓶综合了球腔和圆柱腔液体晃动的特征，它将液体晃动频率控制在一窄带范围之内。利用该程序，本文又计算了Dewar瓶液体晃动的单摆模型，为了便于工程应用，本文提出了一种对Dewar瓶建立等效圆柱容器液体晃动模型的方法，给出了模型的参数变换公式，从而可以通过对圆柱窝器液体晃动的解析计算得到Dewar瓶的液体晃动结果。利用数值计算结果，本文验算了近似计算方法的有效性和近似程序。     

The indentation of a Newtonian fluid in a finite cylindrical container by a right circular cylinder

The indentation of the free surface of a Newtonian fluid in a finite cylindrical container by a right circular cylinder is considered. It is assumed that weight and inertia effects are negligible compared to viscous effects. A finite difference technique is used to obtain approximate values for initial velocities, pressures, and stresses at any point in the fluid as well as an estimate of the force required to indent the fluid with a given velocity. The solution obtained forms the basis for a primary indenter viscometer for very viscous fluids which have viscosities in the range of 10⁴–10¹⁰ poises.     

A numerical study of flows driven by a rotating magnetic field in a square container

The magnetically induced fluid flow in a square container is investigated by means of numerical simulations. Low frequency/ low induction conditions are assumed. The effect of the rotating magnetic field gives rise to a time-independent magnetic body force, computed via the electrical potential equation and Ohm's law and a time-dependent part that is neglected due to the low-interaction parameter. The magnetic body force calculation is verified successfully by comparison with the exact solution. The behavior of the fluid flow in the square container reveals similar features to the flow in the cylindrical container, for instance, in the dependence on the intensities of the magnetic field. However, we did find differences in the velocity field distribution. Particularly, in the finite as well as infinite geometry, the velocity field is influenced by the corner of the container and remains non-axisymmetric in a wide range of Taylor numbers.