Swirling free surface flow in cylindrical containers

Free surface flow in a cylindrical container with steadily rotating bottom cap is investigated. A regular domain perturbation in terms of the angular velocity of the bottom is used. The flow field is made up of the superposition of azimuthal and meridional fields. The meridional field is solved both by biorthogonal series and a numerical algorithm. The free surface on the liquid is determined at the lowest significant order. The aspect ratio of the cylinder may generate a multiple cell structure in the meridional plane which in turn shapes the free surface.     

Flow Rate Limitation of Steady Convective Dominated Open Capillary Channel Flows Through a Groove

An open capillary channel is a structure that establishes a liquid flow path when the capillary pressure caused by surface tension forces dominates in comparison to the hydrostatic pressure induced by gravitational or residual accelerations. To maintain a steady flow through the channel the capillary pressure of the free surface has to balance the pressure difference between the liquid and the surrounding constant pressure gas phase. Due to convective and viscous momentum transport the pressure along the flow path of the liquid decreases and causes the free surface to bend inwards. The maximum flow rate through the channel is reached when the free surface collapses and gas ingestion occurs near the outlet. This stability limit depends on the geometry of the channel and the properties of the liquid. In this paper we present an experimental setup which is used in the low-gravity environment of the Bremen Drop Tower. Experiments with convective dominated systems have been performed where the flow rate was increased up to the maximum value. In comparison to this we present a one-dimensional theoretical model to determine important characteristics of the flow, such as the free surface shape and the limiting flow rate. Furthermore we present an explanation for the mechanism of flow rate limitation for these flow conditions which is similar to the choking problem for compressible gas flows.     

Two-layer critical flow over a semi-circular obstruction

Steady, trans-critical flow of a two-fluid system over a semi-circular cylinder on the bottom of a channel is considered. Each fluid is assumed to be inviscid and incompressible and to flow irrotationally, but the fluids have different densities, so that one flows on top of the other. Consequently, a sharp interface exists between the fluids, in addition to a free surface at the top of the upper fluid. Trans-critical flow is investigated, in which waves are absent from the system, but the upstream and downstream fluid depths differ in each fluid layer. The problem is formulated using conformal mapping and a system of three integrodifferential equations, and solved numerically with the aid of Newton's method. The free-surface shape and that of the interface are obtained along with the Froude numbers in each fluid layer. Results of computation are presented and discussed.     

Free-surface flow of a fluid body with an inner circular cylinder in impulsive motion

An analytical theory and numerical computations are developed for the two-dimensional free-surface flow of an initially circular layer of inviscid fluid surrounding a rigid circular cylinder. The two cylinders are initially concentric. The fluid packet is released from rest and the flow suddenly starts forced by gravity and by the simultaneous impulsive motion of the inner body. A small-time expansion of the fully nonlinear free-surface problem is developed and a closed-form solution is found up to third order for an arbitrary radius of the rigid cylinder. For the gravitational flow around the body at rest, the solution is extended up to fourth order. Free-surface profiles and hydrodynamic forces on the cylinder are calculated and discussed against numerical solutions of the exact unsteady nonlinear problem. Some basic features, such as the formation of an almost uniform layer surrounding the upstream side of the body, are captured by the theory quite well and only later on in time significant quantitative differences appear. Similarly, the behaviour of hydrodynamic loads is rather well predicted during initial stages preceding larger fluctuations observed on a longer time-scale.     

Linearised solution of a free-surface flow over a trapezoidal obstacle

Summary A theoretical study is made, of an irrotational, inviscid incompressible and steady flow over a two-dimensional trapezoidal obstacle; with disturbing height , lying on the bottom of the running stream, in terms of a linearised theory. Particular attention is given to two cases of the flow, the supercritical and the subcritical. The bottom is represented in integral form using Fourier's double-integral theorem. Following the method suggested by Thomson (1886) and Lamb (1932), we obtain a linearised free-surface profile in series form for the two cases of flow. The linearised solution obtained is based on the assumption that the height of the trapezoidal bump, , is small compared to the channel depth,h. The nature of the free-surface formed depends on whether the flow is subcritical or supercritical.The results are plotted for the two cases of the flow for different shapes of the bottom and different values of Froude number,F. The effect of the Froude number, the bottom height and the shape of the bottom are discussed.     

Capillary Stability in a Tilted Circular Cylinder

The classical capillary stability problem in a vertical circular cylinder is a special case of the more general problem of the stability of liquid above a capillary surface in a circular cylinder with arbitrary orientation of gravity. This problem can, of course, also be viewed as arbitrary cylinder orientation in a steadily accelerating spacecraft. The general (tilted) circular cylinder capillary stability problem is solved numerically by use of the Surface Evolver code for general tilt and general contact angle. Tens of thousands of combinations of contact angle, tilt angle, and Bond number are solved for with a global volunteer computing network running Surface Evolver. The results appear to be symmetric about 90 degree contact angle, as in the previous vertical cylinder studies, and not symmetric about 45 degree tilt.     

柱面扁壳面畸变的实验研究

为控制覆盖件成形中的面畸变,以矩形柱面扁壳为模型,实验研究了其拉深件及切边后的面畸变,分析了面畸变形成机制及成形条件、切边对面畸变的影响.结果表明:柱面扁壳成形后在柱面方向出现面畸变,面畸变沿柱面方向呈M型且基本成对称的分布,切边后面畸变减小,而且畸变形态转变为U型分布.面畸变随板料厚度的增加、压边力的增加、屈服强度的降低、凸模曲率半径的减小而减小.     

Effects of Gravitational Orientation on Surface Deformation and Weld Pool Geometry During Gas Tungsten Arc Welding

Effects of gravitational orientation on gas tungsten arc (GTA) welding of nickel were studied to determine the impact of free-surface deformation on weld-pool shape. This was accomplished through GTA welding and a numerical study of the welding process. Welding was conducted by varying scan velocity and gravitational orientation, e.g., welding upward opposing gravity (parallel-up weld), welding downward with gravity (parallel-down weld), and welding perpendicular to gravity (perpendicular weld). Slower scan velocity produced more significant free surface deformation. Gravitational orientation caused 21% deeper penetration in the parallel-up weld compared with the parallel-down weld (resulting from 50% or more maximum surface deformation). Weld penetration of the perpendicular weld was between that of parallel-up and parallel-down cases. A model of the welding process, in which an experimentally generated free surface was implemented as a boundary condition, supported the results by showing similar trends.     

Film depth and concentration banding in free-surface Couette flow of a suspension

The film depth of a free-surface suspension flowing in a partially filled horizontal concentric-cylinder, or Couette, device has been studied in order to assess its role in the axial concentration banding observed in this flow. The flow is driven by rotation of the inner cylinder. The banding phenomenon is characterized by particle-rich bands which under flow appear as elevated regions at the free surface separated axially by regions dilute relative to the mean concentration. The concentric cylinders studied had outer radius R(o) = 2.22 cm and inner radii R(i) = 0.64, 0.95 and 1.27 cm; the suspension, of bulk particle volume fraction phi = 0.2 in all experiments described, was composed of particles of either 250-300 microm diameter or less than 106 microm diameter, with the suspending fluid an equal density liquid of viscosity 160 P. The ratio of the maximum to the minimum particle volume fraction along the axis in the segregated condition varies from O(1) to infinite. The latter case implies complete segregation, with bands of clear fluid separating the concentrated bands. The film depth has been varied through variation of the filled fraction, f, of the annular gap between the cylinders and through the rotation rate. Film depth was analysed by edge detection of video images of the free surface under flow, and the time required for band formation was determined for all conditions at which film depth was studied. The film depth increases roughly as the square root of rotation speed for f = 0.5. Band formation is more rapid for thicker films associated with more rapid rotation rates at f = 0.5, whereas slower formation rates are observed with thicker films caused by large f, f > 0.65. It is observed that the film depth over the inner cylinder grows prior to onset of banding, for as yet unknown reasons. A mechanism for segregation of particles and liquid in film flows based upon 'differential drainage' of the particle and liquid phase in the gravity-driven flow within the film over the inner cylinder is formulated to describe the onset of concentration fluctuations. This model predicts that suspension drainage flows lead to growth of fluctuations in phi under regions of negative surface curvature.     

Free-surface flow over a polygonal and smooth topography

Summary The study of two-dimensional, irrotational, inviscid, incompressible steady state motion generated by a polygonal and a smooth obstruction, is made in terms of the linearized theory. The bottom is represented in integral form using Fourier's double integral theorem. Then following Lamb [1], a linear free-surface profile is obtained for the supercritical and subcritical cases. The results are plotted and discussed for the two cases of the flow for different shapes of the bottom and different values of Froude number,F.     

On small-time expansion of nonlinear free surface problems

The small-time expansion of the 2-D problem of a heaving semi-submerged circular cylinder, starting from rest, in analyzed. The first three terms of the small-time expansion of the outer solution (away from the cylinder) are developed. A logarithmic singularity is found for the second derivative in time of the surface elevation at the intersection point. This singularity is of second order in the heaving velocity of the cylinder, and the inner expansion developed for the linear wavemaker problem therefore does not cover this case. The problem of the horizontal motion of the semi-submerged cylinder is analyzed as well. For the case with the initial condition for the velocity potential chosen as zero on the free surface, the logarithmic singularity found for the wavemaker is recovered. Selecting the initial condition as a vanishing normal velocity on the free surface leads to a solution where the singularity appears in the fifth derivative in time of the free-surface elevation. The solution of this problem shows the characteristic behaviour of formation of asymmetry, which finally would lead to formation of lee waves. The inner solution of the problem of the heaving cylinder is discussed, showing that the oscillatory behaviour reported for the wavemaker problem appears in the present solution as well.     

Influence of Bond Number on Behaviors of Liquid Drops Deposited onto Solid Substrates

The problem of spreading behaviors of pendant and sessile drops was studied experimentally and numerically under the action of gravity force and surface tension. Bond number was considered to be a main factor of the influence on shape behaviors of liquid drops. This study was performed in the framework of an experimental investigation of drop behaviors in microgravity onboard a Chinese satellite in future. The experiments were carried out in the Drop Tower of Beijing, which could supply about 3.6?s of microgravity (free-fall) time. The surface shape change of liquid drops was investigated and the contact angle variety in sessile and pendant drops were measured from normal gravity to microgravity. A sharp decrease and oscillatory variation of the contact angle for both sessile and pendant drops were found with the sudden decrease of Bond number. The succedent comparison between experimental and numerical results suggests that Bond number has a significant influence on the drop contact angle. Additionally, the drop shapes and the bulk flows inside sessile and pendant drops were analyzed numerically, and it was found that the bulk flows could affect the free-surface shape of liquid drops apparently. Comparison of the moving velocity of contact line between sessile and pendant drops indicated that the pendant drops had a faster response to Bond number.     

Film flow of a suspension down an inclined plane

A method is developed for simulating the film flow of a suspension of rigid particles with arbitrary shapes down an inclined plane in the limit of vanishing Reynolds number. The problem is formulated in terms of a system of integral equations of the first and second kind for the free-surface velocity and the traction distribution along the particle surfaces involving the a priori unknown particle linear velocity of translation and angular velocity of rotation about designated centres. The problem statement is completed by introducing scalar constraints that specify the force and torque exerted on the individual particles. A boundary-element method is implemented for solving the governing equations for the case of a two-dimensional periodic suspension. The system of linear equations arising from numerical discretization is solved using a preconditioner based on a particle-cluster iterative method recently developed by Pozrikidis (2000 Engng Analysis Bound. Elem. 25, 19-30). Numerical investigations show that the generalized minimal residual (GMRES) method with this preconditioner is significantly more efficient than the plain GMRES method used routinely in boundary-element implementations. Extensive numerical simulations for solitary particles and random suspensions illustrate the effect of the particle shape, size and aspect ratio in semi-finite shear flow, and the effect of free-surface deformability in film flow.     

Measurements of free-surface profile in transient flow with a simple light-slicing method

A simple light-slicing technique for the direct measurement of the free-surface shape h(x) in onedirectional transient liquid flows is studied. The method is used to determine h(x) with a precision of the order of 10(-2) mm along a line several centimeters long, allowing for height differences of some millimeters. It is useful for transient flows in rectangular channels in which capillarity and liquid adhesion at walls perturb lateral observations; the method can also be used for nondiffusive fluids in cases in which it is not advisable to add diffusive particles.     

Some dynamical characteristics of a non-spherical bubble in proximity to a free surface

The motions of a gas bubble in proximity to a free surface with and without buoyancy force, as well as in shallow water are simulated based on a numerical time integration coupled with three-dimensional boundary integral spatial solution. The fluid is assumed to be inviscid, incompressible, and the flow irrotational. The unsteady Bernoulli equation is applied on the free surface and bubble surface as one of the boundary conditions of the Laplace equation for the potential. Improvements have been made in the mesh generation of the free surface and rigid boundary, the modeling of the toroidal bubble after the jet impact and the investigation into the combined effects on the motion of a bubble in the presence of the rigid bottom and free surface. The growth and collapse of a gas bubble together with the formation of the toroidal bubble after the jet impact are simulated. The shapes and positions of the bubble, the trajectories and velocities of the poles of the bubble as well as the pressure distributions in the fluid under different standoff distances and buoyancy parameters are obtained to better illuminate the mechanism underlying the motions of gas bubble and free surface. When a bubble is initiated sufficiently close to a free surface, the free surface spike and the second accelerating phenomenon of the free surface during the collapse phase can be observed. The buoyancy force has significant effects on the jet formation and development within the bubble and it may reverse the direction of the liquid jet when exceeding the effect of the Bjerknes force induced by the free surface. The large contortions in the shallow water and the formation of the high-pressure region between the bubble and the free surface are captured when the bubble is close enough to the rigid bottom and the free surface.     

Two-Dimensional Numerical Simulation for Flow Pattern Transition of Thermal-Solutal Capillary Convection in an Annular Pool

In order to understand the transition characteristics of the thermal-solutal capillary convection in an annular pool, a series of two-dimensional numerical simulations were conducted. The bottom of the pool is adiabatic rigid wall and the top is adiabatic and non-deformable free surface. The inner and outer cylindrical walls maintain at constant temperature and solute concentration, respectively. The thermo-capillary force is supposed to equal to the solute-capillary force, but their directions are contrary. Results show that the thermal-solutal capillary convection is steady at a small Reylonds number. When the capillary Reynolds number exceeds a critical value, the steady flow transits into unstable thermal-solutal capillary convection. The transition from the steady to oscillatory flow undergoes a Hopf bifurcation. Furthermore, the effects of the liquid layer aspect ratio, the radius ratio, the Prandtl number and the Lewis number on the onset of flow pattern transition are discussed. The physical mechanism of the unstable thermal-solutal capillary convection is also analyzed.     

Influence of lateral acceleration on capillary interfaces between parallel plates

The paper considers limits for the refill of capillary liquid vanes of propellant management systems for low gravity applications under the influence of residual acceleration. The limit of existence of a connected capillary liquid interfaces in the vanes is analyzed. A mathematical approach to describe the characteristics of a liquid meniscus in the vane is derived under the assumptions of a quasi static interface and negligible viscous and inertia forces. Analytical considerations allow to determine the maximum width of the liquid column within the vane that can be stabilized by capillary forces. If the width of the capillary vanes exceeds this limit, the vane will be only partially filled. The consecutive equation is solved analytically with the approximation that the lateral acceleration does not distort the asymptotic shape of the free surface extended along the edges of the vanes from a circular cylindrical shap. The resulting relations can readily be analyzed and provide very clear insight into the di.erent limits for the meniscus to exist. The results of the approximate solution are compared with numerical solutions considering the actual shape of the distorted free surface. Although the global shape of the existence chart and the lower limits of existence are rather well predicted by the approximate solution, considerable deviations from the numerical solution can be observed for the upper limits with repect to Bond number and maximum width of the vanes. The numerical solution therefore should be preferred. Additionally, the numerical solution does not provide mere existence criteria but real static (geometric) stability criteria. Experimentally, the behaviour of liquid interfaces in vanes was studied under microgravity conditions in a drop tower, using a micro-g centrifuge to impose variable lateral accelerations on the liquid column between the vanes. The results of the experiment are shown to be in very good agreement with the analytical predictions. The analytical approach presented thus yields a powerful means to determine static stability limits of interface configurations subject to lateral forces for arbitrary geometries of the supporting capillary.     

A novel methodology to study shape and surface tension of drops in Electric Fields

A novel methodology is introduced that can be used to study the behavior of conducting drops in electrostatic fields, when gravity effects are negligible. This methodology, called Axisymmetric Drop Shape Analysis — Electric Field (ADSA-EF), generates numerical drop profiles in the electrostatic field, for a given surface tension. Then, it calculates the true value of the surface tension by matching the theoretical profiles with the shape of the experimental drops, with the surface tension as an adjustable parameter. ADSA-EF can be employed for simulating drop shapes in the electric field, detecting the effect of an electric field on liquid surface tensions, and measuring surface tensions in microgravity, where current drop-shape techniques are not applicable. The predicted drop shapes in the electric field were compared with experimental images, indicating good agreement. Preliminary experiments according to ADSA-EF methodology suggested that the surface tension of water increases by about one percent in the electric field.     

Use of an extremal functional in solving for an unknown bottom surface given a free surface profile

In this paper the free surface fluid flow is induced by an obstacle on the bottom of the channel whose exact shape and location are unknown a priori. The inviscid fluid flow is assumed to be steady, incompressible and irrotational under the influence of gravity. A boundary integral technique, which is based on the combination of the boundary integral method (BIM), the variational principle technique (VPT) and the application of a minimization technique (MT) is developed. Two minimization techniques, namely the extremal pressure method (EPM) and the extremal energy method (EEM), are extensively used in identifying the unknown bottom surfaces. To illustrate these techniques the free surface profile to be applied in the inverse analysis has been generated following a direct formulation when the solid bottom boundary is monotonically decreasing, monotonically increasing, when it possesses a single hump and when it possesses a single depression. It is found that in all four cases both extremum techniques can be very accurately used to identify the shape and position of the solid obstacle.     

Simulation of particle deposition at the bottom surface in a room-scale chamber with particle injection

An Euler-type gas/particle two-phase flow numerical model is presented for simulating the deposition of particles of tens of microns. At this size range of particles, the dominant effect considered for deposition on the bottom surface is gravitational settling. The mass conservation equation for the particulate phase is developed and simulated to include convection, diffusion, and gravitational settling effects. Experiments were conducted in a room-scale chamber by injecting nano-particle aggregates, and the deposition rates were measured using a specially designed sequential sampler. The measured deposition-rate data are compared with the simulation results for validations. Distributions of particle-number density at different times are plotted in several viewing planes to facilitate discussion of the particle-distribution patterns. The comparisons show that the agreement between the modeling and the measurement is best at the intermediate particle-size range.