Waves and singularities in nonlinear capillary free-surface flows

The problem is a free-surface flow of a fluid, emerging from a semi-infinite container. The fluid is assumed to be inviscid, incompressible and the flow to be two dimensional and irrotational. When surface tension is neglected the free surface leaves the wall of the container tangentially. We show that when surface tension is taken into account, there is, in general, a train of waves on the free surface and a discontinuity in slope where the free surface separates from the wall of the container. These new solutions include, as particular cases, previously obtained solutions for which the free surface is waveless in the far field. Although the calculations are presented for a special flow configuration, the results are general and apply to other potential free surface flows where a free-surface intersects a rigid wall.     

A comparison of linear and nonlinear computations of waves made by slender submerged bodies

Potential flow about a slender spheroid beneath a free surface is considered in order to determine the ability of thin-ship theory to reproduce the free-surface elevation accurately. A fully nonlinear code involving interior Rankine sources is used, enabling comparisons between exact (`Neumann-Stokes') outputs, outputs with exact body condition but linearised free-surface conditions (`Neumann-Kelvin'), and a consistent thin-ship approximation (`Michell-Kelvin'). In general, these computations agree to within a few percent, except when the body is so close to the free surface that the nonlinear computation suggests that breaking is imminent at one point above the body, and even then thin-ship theory still compares well except very near to that isolated point. The thin-ship theory has also been implemented in a separate general-purpose code using Havelock sources, and detailed free-surface contours computed by this linear method are shown for spheroids that are too close to the surface for the nonlinear code to converge.     

Viscous and inviscid matching of three-dimensional free-surface flows utilizing shell functions

A methodology is presented for matching a solution to a three-dimensional free-surface viscous flow in an interior region to an inviscid free-surface flow in an outer region. The outer solution is solved in a general manner in terms of integrals in time and space of a time-dependent free-surface Green function. A cylindrical matching geometry and orthogonal basis functions are exploited to reduce the number of integrals required to characterize the general solution and to eliminate computational difficulties in evaluating singular and highly oscillatory integrals associated with the free-surface Green-function kernel. The resulting outer flow is matched to a solution of the Navier?CStokes equations in the interior region and the matching interface is demonstrated to be transparent to both incoming and outgoing free-surface waves.     

Parallel finite element computation of free-surface flows

In this paper we present parallel 2D and 3D finite element computation of unsteady, incompressible free-surface flows. The computations are based on the Deformable-Spatial-Domain/Stabilized Space-Time (DSD/SST) finite element formulation, which takes automatically into account the motion of the free surface. The free-surface height is governed by a kinematic free-surface condition, which is also solved with a stabilized formulation. The meshes consist of triangles in 2D and triangular-based prism elements in 3D. The mesh update is achieved with general or special-purpose mesh moving schemes. As examples, 2D flow past spillway of a dam and 3D flow past a surface-piercing circular cylinder are presented.     

Extension of physical component BFC method for the analysis of free-surface flows coupled with moving boundaries

A newly developed “physical component boundary fitted coordinate (PCBFC) method” is extended for the analysis of free-surface flows coupled with moving boundaries. Extra techniques are employed to deal with the coupling movement of the free surface and moving boundaries. After the validation of the extension by several benchmark problems, the method is successfully applied for the first time to the simulation of overflow-induced vibration of the weir coupled with sloshing of the free-surface liquid.     

薄壁筒形件粘性介质外压缩径失稳起皱分析

缩径成形受筒坯稳定性的影响,容易产生失稳起皱,很难成形出缩径量较大的零件.提出了采用粘性介质外压缩径工艺成形直径变化量较大的薄壁异形曲面筒形零件新方法.针对坯料端部约束和自由条件下的失稳起皱问题进行了试验和有限元模拟,分析了试件失稳起皱过程中几何形状和应力的变化规律.研究表明,约束坯料端部可以改善试件失稳起皱后的应力分布规律,有利于消除褶皱,在较高粘性介质压力条件下可以直接成形出缩径量为15%的试件.     

Linear sloshing frequencies in the annular region of a circular cylindrical container in the presence of a rigid baffle

Sloshing in any type of container may invite instability to it. If some part of the free liquid surface in the annular region of a specially designed circular cylindrical container is covered with an annular baffle, the natural frequencies and the response of the liquid in the container undergo a drastic change. A partly covered free surface shifts the natural frequency above and away from the control frequency of the vehicle, in which the liquid-filled container is placed, which results in the reduction of sloshing mass participating in the dynamic motion of the system. The fundamental natural frequency of an inviscid and incompressible liquid is determined for increasing width of the baffle that is attached to the outer tank wall on the free surface. It is observed that by increasing the width of the baffle, natural frequencies can be significantly increased. Investigations are also carried out for different values of Bond number, which depicts different states of surface tension, and for varying values of the part of the radius in the fluid region. It is also observed that by increasing the fluid height inside the container, the natural frequencies can be increased, which results in reduction of sloshing.     

Numerical solutions of three dimensional dynamic crack problems and simulation of dynamic fracture phenomena by a “non-standard” finite difference method

A brief presentation of the novel features of a non-standard finite difference method (HEMP and HEMP 3D computer codes) is given. To demonstrate the capability of the HEMP 3D code with the extrapolation sub-routine for solving dynamic problems in linear fracture mechanics, through-thc-thickness cracks and semielliptic surface cracks of a section of cylindrical pipe under Heaviside-function time dependence pressure applied to the interior surface are considered and their dynamic stress intensity factors are calculated. The dynamic fracture phenomenon caused by the penetration of a rigid cylindrical punch into the center of a cylindrical wafer of linear-elastic brittle material is simulated numerically by using the HEMP code with the incorporation of the method of equivalent free-surface boundary conditions and the method of artificial velocity. A simple physical model that crack will initiate and propagate in any region where a critical stress has been reached and the crack will propagate in a direction perpendicular to the maximum principal stress is chosen here.     

Applicability of the method of fundamental solutions to 3-D wave–body interaction with fully nonlinear free surface

A numerical model for three-dimensional fully nonlinear free-surface waves is developed by applying a boundary-type meshless approach with a leap-frog time-marching scheme. Adopting Gaussian Radial Basis Functions to fit the free surface, a non-iterative approach to discretize the nonlinear free-surface boundary is formulated. Using the fundamental solutions of the Laplace equation as the solution form of the velocity potential, free-surface wave problems can be solved by collocations at only a few boundary points since the governing equation is automatically satisfied. The accuracy of the present method is verified by comparing the simulated propagation of a solitary wave with an exact solution. The applicability of the present model is illustrated by applying it to the problem of a solitary wave running up on a vertical surface-piercing cylinder and the problem of wave generation in infinite water depth by a submerged moving object.     

Nonlinear waves on a rotating viscous fluid with a cylindrical free surface

Summary An asymptotic theory is developed for the study of nonlinear wave motion of a rotating viscous fluid with a cylindrical free surface. The method used here is based upon a multiple-parameter singular perturbation scheme within the framework of long-wave approximation. Wave speed and a set of asymptotic evolution equations are derived, and a criterion for the instability of the wave motion is defined.     

Effect of inertia on electrified film flow over a wavy wall

The effect of inertia on the steady flow of a liquid layer down a wavy wall in the presence of an electric field is investigated. Both the liquid film and the region above it are assumed to act as perfect dielectrics. A linearised perturbation analysis is performed for flow down a wall with small-amplitude sinusoidal corrugations, and the free-surface amplitude and phase shift are computed numerically for a broad range of flow conditions. It is shown that the electric field can be used to manipulate the phase shift between the free surface and the wall. In particular, when the Reynolds number lies below a threshold value, an electric field of sufficient strength will bring the free surface precisely into phase with the wall. An electric field can also be used to mitigate the resonance effect identified by previous workers, in which the free surface suffers significant amplification in comparison to the height of the wall corrugations at a particular Reynolds number. Working on the basis of the lubrication approximation, a nonlinear equation for the film thickness is derived featuring a non-local term due to the electric field. Numerical solutions for flow over a wavy wall of finite amplitude reveal that the effect of inertia on the free-surface characteristics depends on the electrical properties of the fluid layer and the strength of the imposed electric field.     

Viscosity and Surface-Tension Effects on Wave Generation by a Translating Body

Various theoretical and experimental studies have been carried out to examine the generation of waves ahead of a translating body. Not all issues pertaining to this wave-motion problem are, however, fully resolved. In particular, mechanisms pertaining to generation of white-water instability and inception of vortices in the bow region are not fully understood. In this paper, the two-dimensional, unsteady, nonlinear, viscous-flow problem associated with a translating surface-piercing body is solved by means of finite-difference algorithm based on boundary-fitted coordinates. Effects of surface tension and surfactants are examined. Results of this work resolve certain classic issues pertaining to bow flows. A continuous generation of short and steepening bow waves is observed at low (draft) Froude number, a nonlinear phenomenon uncovered recently in the case of inviscid fluid also. This indicates that, steady-state nonlinear bow-flow solution may not exist, even at low speed. It is postulated that these short bow waves are responsible for the white-water instability commonly observed ahead of a full-scale ship. The amplitudes of these short bow waves are suppressed by surface tension, which is, possibly, the reason why white-water instability is not distinctly observed in laboratory-scale experiments. The presence of surfactants on the free surface is found to intensify the generation of free-surface vorticity, thus resulting in the formation of bow vortices. The accumulation of surface-active contaminants at the bow is hence responsible for the generation of bow vortices observed in laboratory experiments at low Froude number. At high Froude number, an impulsive starting motion of the body results in the generation of a jet-like splash at the bow and a gentle start an overturning bow wave, as previously observed in the case of inviscid bow flow.     

Waves generated by a source below a free surface in water of finite depth

The free-surface flow due to a submerged source in water of finite depth is considered. The fluid is assumed to be inviscid and incompressible. The problem is solved numerically by using a boundary integral equation formulation due to Hocking and Forbes [6]. The numerical results show that there is a train of waves on the free surface in accordance with the results of Mekias and Vanden-Broeck [5]. For small values of the Froude number, the amplitude of the waves is so small that the free surface is essentially flat in the far field. These waveless profiles agree with the calculations of Hocking and Forbes [6].     

Radiation and Diffraction Analysis of the McIver Toroid

A hydrodynamic analysis is performed of a special toroidal body which is known to have a nontrivial solution of the homogeneous linearized free-surface boundary-value problem with oscillatory time- dependence. This solution corresponds physically to unbounded resonant motion of the fluid in the 'moon pool' at the center of the toroid. The added mass, damping, and elevation of the free surface in the moon pool are computed for a range of wavenumbers, with singular results in the resonant regime.     

Influences of shear stresses on the dynamic instability of exponentially graded sandwich cylindrical shells

The aim of present study is to investigate the dynamic instability of exponentially graded (EG) sandwich cylindrical shells under static and time dependent periodic axial loadings using the shear deformation theory (SDT). The modified Donnell-type dynamic instability equations of EG sandwich cylindrical shells based on the SDT are deduced. Then are reduced to Mathieu-Hill equation and by solving the expressions for the boundaries of instability regions of EG sandwich cylindrical shells are obtained. The similar expressions for EG single-layer shell, ceramic-rich shell and metal coated sandwich cylindrical shell on the basis of SDT and classical shell theory (CST) are obtained in a special case. The numerical illustrations concern the influences of compositional profiles of coating layers, shear stresses and geometrical parameters of sandwich cylindrical shells on the boundaries of instability regions. As a check on the accuracy of the present study, the values of the lower and upper boundaries of instability regions are compared with those in the literature.     

An adaptive boundary-element approach to 3D transient free-surface flow of viscous fluids

An adaptive (Lagrangian) boundary-element approach is proposed for the general three-dimensional simulation of confined free-surface flow of viscous incompressible fluids. The method is stable as it includes remeshing capabilities of the deforming free surface, and thus can handle large deformations. A simple algorithm is developed for mesh refinement of the deforming free surface mesh. Smooth transition between large and small elements is achieved without significant degradation of the aspect ratio of the elements in the mesh. Several flow problems are presented to illustrate the utility of the approach.     

A coupled PFEM–Eulerian approach for the solution of porous FSI problems

This paper aims to present a coupled solution strategy for the problem of seepage through a rockfill dam taking into account the free-surface flow within the solid as well as in its vicinity. A combination of a Lagrangian model for the structural behavior and an Eulerian approach for the fluid is used. The particle finite element method is adopted for the evaluation of the structural response, whereas an Eulerian fixed-mesh approach is employed for the fluid. The free surface is tracked by the use of a level set technique. The numerical results are validated with experiments on scale models rockfill dams.     

Transient behavior of granular materials with symmetric conditions for tumbler shapes and fill fractions

The purpose of this research is to continue to find the mechanism behind transient granular behavior. The granular motion in circular tumblers can be described as steady-state, since there is no feature to disrupt the consistency of the top free-surface flowing layer. In polygon tumblers however, the available space for the free surface is perpetually changing, which creates unsteady flow behaviors. The effect of tumbler size and speed were studied previously in a triangular tumbler whereas this follow-up research investigates the effect of both the tumbler shape and fill fraction. For the experiment, various amounts of zirconium silicate beads generate a fluctuating flowing layer inside quasi-2D rotating tumblers of triangle and square shapes. Symmetric fill fractions generate the same flowing layer dimensions but results indicate the rate of increasing or decreasing space at the free surface affects the dynamics of the angle of repose. The balance between the potential and kinetic energy causes the flowing layer and angles of repose to appear as an optimized functional form of a catenary, which is much more apparent in the triangular tumbler with upstream and downstream dimensions changing out of phase compared to the square tumbler where the parallel sides act in-phase with one another.     

Nonlinear capillary free-surface flows

Capillary free-surface flows are considered. The fluid is taken to be inviscid and incompressible and the flow to be irrotational. Particular attention is devoted to two-dimensional flows for which the free surfaces intersect rigid walls. These include cavitating flows and local flows at the front of a small object (probe or insect) moving at the surface of a fluid. A general study of the effect of surface tension on the possible singularities which can occur at the separation points is presented. The results confirm and generalise previous findings on the subject.     

On the effects of non-linearity in free-surface flow about a submerged point vortex

Summary Two-dimensional free-surface flow about a point vortex in a stream of infinite depth is investigated. The non-linear problem is formulated in terms of an integrodifferential equation on the exact, unknown location of the free surface, and this equation is then solved numerically. The non-linear results are compared with the predictions of linearized theory and, for positive circulation, it is found that the latter may under-estimate the drag force significantly. For negative circulation, the linearized theory grossly over-predicts the value of the wave resistance, which apparently even becomes zero in a limiting configuration.