Coaxial interactions of two vortex rings or of a ring with a body

Inviscid coaxial interactions of two vortex rings, including head-on collisions and leapfrogging motions, are considered using a contour dynamics technique. Interactions of vortex rings with solid bodies are also investigated by combining the contour dynamics technique with a boundary integral equation method. Numerical results show that a clean, successful passage motion is possible for two vortex rings with not too thick cores. In both cases of head-on collisions and leapfrogging motions, very large core deformations are observed when two vortex rings get close to each other. A head-tail structure is formed in the later stage of a head-on collision of two fat vortices. Numerical results also show that a vortex ring will stretch and slow down when it moves toward a solid boundary, will shrink and speed up when it moves away from a solid boundary, and will either translate steadily or approach an oscillating asymptotic state when it is far away from any boundaries. The project supported by The National Education Commission of China and NASA under cooperative grant agreement #NCC5-34.     

涡激诱导并列双圆柱碰撞数值模拟研究

圆柱类结构物的涡激振动是工程中较为常见的一种现象,如果圆柱结构物之间的距离较小, 就会产生涡激诱导碰撞现象,而涡激碰撞会比涡激振动对结构物疲劳破坏产生更严重的威胁.采用浸入边界法模拟流体中的动边界问题,避免了传统贴体网格方法在求解流体中存在固体间碰撞问题时出现数值求解不稳定问题,采用有限元方法对圆柱的运动和碰撞进行求解,通过数据回归方法建立了流体流动条件下的润滑模型,对不同间隙比下涡激诱导并列双圆柱振动及碰撞过程进行了数值模拟, 数值结果表明,如果两圆柱产生了碰撞将会有连续的碰撞发生, 碰撞时出现了多阶频率,振动主频率要比无碰撞时大, 两圆柱碰撞时的相对速度比自由来流速度小;当两圆柱相互接近时, 随着涡环分离角度的逐渐倾斜, 横向流体力先逐渐减小,当两圆柱间涡环开始相互影响发生挤压时, 横向流体力开始逐渐增大;当两圆柱开始反弹时, 两圆柱间形成了低压区, 改变了横向流体阻力的方向,使两圆柱又产生了接近运动,如此反复从而产生了碰撞后横向流体力和圆柱速度的振荡现象.      

Secondary flow of Rivlin-Ericksen fluids between two concentric spheres oscillating about a fixed diameter

Summary The paper deals with the study of the nature of secondary flow of aRivlin-Ericksen fluid, contained between two concentric spheres, which perform oscillations about a fixed diameter. The steady part of the secondary flow is discussed in detail in the following three cases (i) the outer sphere at rest, the inner oscillating, (ii) the two spheres oscillating with the same angular velocity in the same sense and (iii) the spheres oscillating with the same angular velocity in opposite sense. In a previous paper, a similar problem was discussed for theOldroyd fluids. We find that the secondary flow is strongly dependent on the common frequency of oscillation of the two spheres and on the rotational nature of the motion for the present investigation also. Certain contrasting features of interest between the secondary flow field of the two fluids are also noted.     

Molecular orientation in non-Newtonian flow of dilute polymer solutions around spheres

A novel approach is presented to study the benchmark problem of flow around spheres in model dilute solutions of monodisperse samples of atactic polystyrene in di-octyl phthalate. Spheres are held stationary on flexible cantilevers of known spring-constant, k, while the polymer solutions are pumped past at controlled flow rates, allowing access to a wide range of Deborah number. In this way the non-Newtonian forces experienced by the spheres can be measured as a function of Deborah number, while detailed observations and measurements of birefringence are made, enabling assessment of macromolecular strain and orientation. In addition the flow field around a sphere has been measured in an a-PS solution. Experiments have been performed on a single sphere and on two spheres axially aligned in the direction of flow. The extensional flow around the downstream stagnation point of the single sphere is found to play a pivotal role in the development of molecular strain and stress, resulting in flow modification and subsequent non-Newtonian behaviour. The flow birefringence in the wake is found to modify severely the flow around a second, downstream, sphere, affecting the non-Newtonian forces encountered by the second sphere. This provides an explanation for the time interval dependent terminal velocity often observed when two spheres follow the same path through viscoelastic liquids.     

Obstacle-induced spiral vortex breakdown

An experimental investigation on vortex breakdown dynamics is performed. An adverse pressure gradient is created along the axis of a wing-tip vortex by introducing a sphere downstream of an elliptical hydrofoil. The instrumentation involves high-speed visualizations with air bubbles used as tracers and 2D Laser Doppler Velocimeter (LDV). Two key parameters are identified and varied to control the onset of vortex breakdown: the swirl number, defined as the maximum azimuthal velocity divided by the free-stream velocity, and the adverse pressure gradient. They were controlled through the incidence angle of the elliptical hydrofoil, the free-stream velocity and the sphere diameter. A single helical breakdown of the vortex was systematically observed over a wide range of experimental parameters. The helical breakdown coiled around the sphere in the direction opposite to the vortex but rotated along the vortex direction. We have observed that the location of vortex breakdown moved upstream as the swirl number or the sphere diameter was increased. LDV measurements were corrected using a reconstruction procedure taking into account the so-called vortex wandering and the size of the LDV measurement volume. This allows us to investigate the spatio-temporal linear stability properties of the flow and demonstrate that the flow transition from columnar to single helical shape is due to a transition from convective to absolute instability.     

Adaptive panel representation for oblique collision of two vortex rings

In this article, we use a hierarchical panel method for representing vortex sheet surface motion in 3D flow to investigate the oblique collision of two vortex rings. The particles representing the sheet are advected by a regularized Biot-Savart integral with smoothed Rosenhead-Moore kernel. The particle velocities are evaluated by an adaptive treecode algorithm based on Taylor expansions in Cartesian coordinates. The method allowed us to consider late stages of a vortex rings collision, producing a funnel region. Vorticity iso-surfaces evolution is also investigated.     

An experimental study of viscous vortex rings

The work focuses on the problem of stability and viscous decay of single vortex rings. A tentative classification scheme is proposed for vortex rings which is based on extensive hot-wire measurements of velocity in the ring core and wake, and flow visualization, viz. laminar, wavy, turbulence-producing, and turbulent. Prediction of vortex ring type is shown to be possible, at least approximately, based on the vortex ring Reynolds number alone. Linear growth rates of ring diameter with time are observed for all types of vortex rings, with different growth rates occurring for laminar and turbulent vortex rings. Data on the viscous decay of vortex rings are used to provide experimental confirmation of the accuracy of Saffman's equation for the velocity of propagation of a vortex ring.The work reported herein is supported through a grant of the Natural Sciences and Engineering Research Council of Canada. Special thanks are due to CAPES (Brazil) for the award of a scholarship to the senior author.     

Flow Bifurcations in a Thin Gap Between Two Rotating Spheres

This paper presents the use of a parameter continuation method and a test function to solve the steady, axisymmetric incompressible Navier–Stokes equations for spherical Couette flow in a thin gap between two concentric, differentially rotating spheres. The study focuses principally on the prediction of multiple steady flow patterns and the construction of bifurcation diagrams. Linear stability analysis is conducted to determine whether or not the computed steady flow solutions are stable. In the case of a rotating inner sphere and a stationary outer sphere, a new unstable solution branch with two asymmetric vortex pairs is identified near the point of a symmetry-breaking pitchfork bifurcation which occurs at a Reynolds number equal to 789. This solution transforms smoothly into an unstable asymmetric 1-vortex solution as the Reynolds number increases. Another new pair of unstable 2-vortex flow modes whose solution branches are unconnected to previously known branches is calculated by the present two-parameter continuation method. In the case of two rotating spheres, the range of existence in the (Re ₁ , Re ₂ ) plane of the one and two vortex states, the vortex sizes as a function of both Reynolds numbers are identified. Bifurcation theory is used to discuss the origin of the calculated flow modes. Parameter continuation indicates that the stable states are accompanied by certain unstable states. Received 26 November 2001 and accepted 10 May 2002 Published online 30 October 2002 Communicated by M.Y. Hussaini     

A note on the cause of rebound in the head-on collision of a vortex ring with a wall

It is well documented that a trailing vortex pair approaching the ground, and a vortex ring colliding head-on with a rigid plane, experience a reversal in axial velocity which is commonly referred to as rebound. One explanation of this phenomenon suggests that it is essentially an inviscid process due to the effect of the finite core-size, whereas another and more widely accepted explanation attributes it to the influence of a secondary vortex which is generated at the surface by viscous effects. The aim of this paper is to assess experimentally the validity of these competing explanations. To achieve this, flow visualization studies of the collision of a vortex ring with a wall are compared with those of the head-on collision of two identical rings. The head-on collision is designed to mimic the inviscid, free-slip case of a ring/wall interaction. This paper describes the experimental findings.A version of this paper was presented at the 10th Australasian Fluid Mechanics Conference, University of Melbourne, Australia, 11–15 December 1989     

Modeling of softsphere normal collisions with characteristic of coefficient of restitution dependent on impact velocity

This letter presents a theoretical model of the normal (head-on) collisions between two soft spheres for predicting the experimental characteristic of the coefficient of restitution dependent on impact velocity. After the contact force law between the contacted spheres during a collision is phenomenologically formulated in terms of the compression or overlap displacement under consideration of an elastic—plastic loading and a plastic unloading subprocesses, the coefficient of restitution is gained by the dynamic equation of the contact process once an initial impact velocity is input. It is found that the theoretical predictions of the coefficient of restitution varying with the impact velocity are well in agreement with the existing experimental characteristics which are fitted by the explicit formula.     

Vortex particle‐in‐cell method for three‐dimensional viscous unbounded flow computations

A new vortex particle‐in‐cell (PIC) method is developed for the computation of three‐dimensional unsteady, incompressible viscous flow in an unbounded domain. The method combines the advantages of the Lagrangian particle methods for convection and the use of an Eulerian grid to compute the diffusion and vortex stretching. The velocity boundary conditions used in the method are of Dirichlet‐type, and can be calculated using the vorticity field on the grid by the Biot–Savart equation. The present results for the propagation speed of the single vortex ring are in good agreement with the Saffman's model. The applications of the method to the head‐on and head‐off collisions of the two vortex rings show good agreement with the experimental and numerical literature. Copyright © 2000 John Wiley & Sons, Ltd.     

Specified discharge velocity models for numerical simulations of laminar vortex rings

We numerically and theoretically investigate the flow generated at the exit section of a piston/cylinder arrangement that is generally used in experiments to produce vortex rings. Accurate models for the velocity profile in this section (also called specified discharge velocity, SDV models) are necessary in (i) numerical simulations of laminar vortex rings that do not compute the flow inside the cylinder and (ii) in slug-models that provide a formula for the total circulation of the flow. Based on the theoretical and numerical analysis of the flow evolution in the entrance region of a pipe, we derive two new and easy to implement SDV models. A first model takes into account the unsteady evolution of the centerline velocity, while the second model also includes the time variation of the characteristics of the boundary layer at the exit plane of the vortex generator. The models are tested in axisymmetric direct numerical simulations of vortex rings. As distinguished from classical SDV model, the new models allow to accurately reproduce the characteristics of the flow. In particular, the time evolution of the total circulation is in good agreement with experimental results and previous numerical simulations including the vortex generator. The second model also provides a more realistic time evolution of the vortex ring circulation. Using the classical slug-model and the new correction for the centerline velocity, we finally derive a new and accurate analytical expression for the total circulation of the flow.     

Modelling of gas–solid turbulent channel flow with non-spherical particles with large Stokes numbers

This paper describes a complete framework to predict the behaviour of interacting non-spherical particles with large Stokes numbers in a turbulent flow. A summary of the rigid body dynamics of particles and particle collisions is presented in the framework of Quaternions. A particle-rough wall interaction model to describe the collisions between non-spherical particles and a rough wall is put forward as well. The framework is coupled with a DNS-LES approach to simulate the behaviour of horizontal turbulent channel flow with 5 differently shaped particles: a sphere, two types of ellipsoids, a disc, and a fibre. The drag and lift forces and the torque on the particles are computed from correlations which are derived using true DNS.The simulation results show that non-spherical particles tend to locally maximise the drag force, by aligning their longest axis perpendicular to the local flow direction. This phenomenon is further explained by performing resolved direct numerical simulations of an ellipsoid in a flow. These simulations show that the high pressure region on the acute sides of a non-spherical particle result in a torque if an axis of the non-spherical particle is not aligned with the flow. This torque is only zero if the axis of the particle is perpendicular to the local direction of the flow. Moreover, the particle is most stable when the longest axis is aligned perpendicular to the flow.The alignment of the longest axis of a non-spherical particle perpendicular to the local flow leads to non-spherical particles having a larger average velocity compared to spherical particles with the same equivalent diameter. It is also shown that disc-shaped particles flow in a more steady trajectory compared to elongated particles, such as elongated ellipsoids and fibres. This is related to the magnitude of the pressure gradient on the acute side of the non-spherical particles. Finally, it is shown that the effect of wall roughness affects non-spherical particles differently than spherical particles. Particularly, a collision of a non-spherical particle with a rough wall induces a significant amount of rotational energy, whereas a corresponding collision with a spherical particle results in mostly a change in translational motion.     

Unsteady flow around impacting bluff bodies

The flow resulting from the collision without rebound of generic bluff bodies with a wall in a still viscous fluid is investigated both computationally and experimentally. Emphasis is on the case of a circular cylinder impact (two-dimensional geometry), but comparisons with the flow generated by the impact of a sphere (axisymmetric geometry) are included. For normal cylinder impacts, the two counter-rotating vortices forming behind the body during its motion continue their trajectory towards the wall after the collision, leading to the generation of opposite-signed secondary vorticity at the cylinder and wall surfaces. Secondary vortices forming from this vorticity at higher Reynolds numbers exhibit a short-wavelength three-dimensional instability. Comparison with the sphere impact reveals significant differences in the scales of the vortices after the collision, due to the additional vortex stretching acting in the axisymmetric geometry. This leads to a delay in the onset of three-dimensionality and to a different instability mechanism. Oblique cylinder impacts are also considered. For increasing impact angles, the wall effect is gradually reduced on one side of the cylinder, which favours the roll-up of the secondary vorticity and increases the rebound height of the vortex system.     

A new model for rotating gas molecules

It is assumed that upon collision of two easily deformed rotating bodies the shock impulses are distributed over a small area near the initial point of contact and lead to a main vector and a main moment. The sphere surfaces are considered absolutely elastic, if after collision the difference of the contact point velocity vectors and the difference of the angular velocity of rotation vectors change only in sign. For this case, the law of conservation of the sum of sphere kinetic energies upon collision is satisfied, and the volume element in 12-dimensional velocity space is conserved, serving as the basis for introducing this model of rotating molecules into the kinetic theory of gases together with the Bryan model.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 149–151, May–June, 1976.     

Effect of ventilation on the flowfield around a sphere

The flowfield around a sphere with and without ventilation was investigated in a wind tunnel over a range of Reynolds numbers in an incompressible flow. At supercritical Re, the pressure drag of a sphere can be nearly nullified by venting only 2% of the frontal area of the sphere to the base through a smooth internal duct. The drag reduction is achieved by increased pressures in the separated flow region close to the base. At high Re, the vent flow breaks through the near wake and brings about symmetry in the global flowfield. When the internal shear is increased by using a rough internal duct, the base pressure is unchanged, but the external flow is accelerated to velocities beyond that achieved by the potential flow around the basic sphere. The findings can be explained by a flow model in which the near wake is aerodynamically streamlined by a pair of counterrotating vortex rings at the base. A roughness element can be made to partially destroy the vortex system at the base and result in a steady asymmetric wake. A 1.2 mm diameter wire placed at 70° was found to overtrip the boundary layer and completely destroy the vortex system. Simultaneously, the turbulent separation is advanced and the drag increased.At subcritical Re, ventilation marginally increases static pressures all over the surface. Since the large pressure differential between the windward and leeward sides is not reduced, the internal flow has a rapid acceleration to a velocity close to that of the free stream. The reverse flow associated with the near wake forces the vent flow to rest within itself and the wake profile is unchanged. The main features of subcritical flow around the basic sphere are retained in spite of ventilation. The upstream effects of ventilation are greater for subcritical flow than for supercritical flow.The work reported was carried out under a study grant from the German Academic Exchange Service (DAAD) in Bonn. The authors wish to thank the Director of DAAD in Bonn for the same. Thanks are due to Dr. F. R. Grosche and colleagues at DLR in Göttingen who assisted in the design, fabrication and wind tunnel testing of the sphere model. Thanks are also due to Prof. D. G. Mabey, visiting Professor, Imperial College, London for useful discussions. The many useful discussions with the research advisors of the first author viz., Dr. P. R. Viswanath of National Aerospace Laboratories and Prof. A. Prabhu of Indian Institute of Science, Bangalore are acknowledged with thanks. The support given by the Head, Experimental Aerodynamics Division, National Aerospace Laboratories is thankfully acknowledged.     

Vortex shedding behind rings and discs

The wake structure of discs and bluff rings has been investigated experimentally in a wind tunnel. The rings have an inner diameter d_i, and an outer diameter d_o and are classified according to the parameter (d_o + d_i)/(d_o − d_i) = d/w. the ratio of mean diameter to ring width. As d/w → ∞ the flow approaches that around a two dimensional bluff body whereas as d/w tends to unity the body approaches a solid disc. A distinct change in the vortex shedding pattern is found around d/w = 5. Below this critical value velocity fluctuations in the wake have a weak periodic component which is 180° out of phase across a diameter of the body. Above d/w = 5. regular and coherent axisymmetric vortex ring shedding is observed with shedding occurring alternately from the inner and outer circumferences of the bluff body. Flow visualization and conditional averaging of hot-wire data are used to investigate the vortex structure.     

Pipe flow with solid particles fixed in space

A group of solid particles were hung by slender rods in a pipe to make a model of two-phase flow of coarse particles. Pressure gradient and velocities were measured for different types of the models. The drag on the particles (spheres) were obtained from measurements of pressure gradient with some assumptions. The results are summarized as follows. (1) Mean velocities of fluid are lower in the central part of the pipe than in the circumferential part. Turbulence is remarkably increased by particles. The spectrum distribution of turbulent velocity becomes flatter. These results are similar to the gas-solid flow of coarse particles in a vertical pipe. (2) At a large Reynolds number, the drag coefficient per one sphere in the group is larger than that of a single isolated sphere in a uniform flow. When the spheres are arranged along the same line in the longitudinal direction, the drag coefficient becomes smaller as the longitudinal distance between the spheres is shortened.     

Dynamic interaction of fixed dual spheres for several configurations and inflow conditions

The changes in force characteristics as well as the shedding patterns for various dual sphere configurations are studied. The Reynolds numbers considered are 300, 600 and two different inflow conditions are used: steady and pulsating. The sphere formations are defined by the separation distance D₀ between the spheres and the angle between the line connecting the centres of the spheres and the main flow direction, γ. The position of one of the spheres is varied in the range 0°–90° using a 15° increment. Two separation distances are studied; 1.5D and 3D. The method used for the simulations is the Volume of Solid (VOS) approach, a method based on Volume of Fluid (VOF). A major conclusion from this work is that the sphere interaction alters the wake dynamics by obstructing the vortex shedding (generating a steady wake or a wake with lower Strouhal number) and by changing the direction of the lift force so that it in most cases is directed in the plane containing the sphere centres. The results also show that changing the inflow condition gives the same relative change in drag and lift as for a single sphere. The drag is substantially reduced by placing the sphere downstream in a tandem arrangement and slightly increased in a side-by-side arrangement. However, the effect is decreased by increasing separation distance and increasing Reynolds number.