Combined action of transverse oscillations and uniform cross-flow on vortex formation and pattern of a circular cylinder

This paper attempts to study the roles of lateral cylinder oscillations and a uniform cross-flow in the vortex formation and wake modes of an oscillating circular cylinder. A circular cylinder is given lateral oscillations of varying amplitudes (between 0.28 and 1.42 cylinder-diameters) in a slow uniform flow stream (Reynolds number=284) to produce the 2S, 2P and P+S wake modes. Detailed flow information is obtained with time-resolved particle-image velocimetry and the phase-locked averaging techniques. In the 2S and 2P mode, the flow speeds relative to the cylinder movement are less than the uniform flow velocity and it is found that initial formation of a vortex is caused by shear-layer separation of the uniform flow on the cylinder. Subsequent development of the shear-layer vortices is affected by the lateral cylinder movement. At small cylinder oscillation amplitudes, vortices are shed in synchronization with the cylinder movement, resulting in the 2S mode. The 2P mode occurs at larger cylinder oscillation amplitudes at which each shear-layer vortex is found to undergo intense stretching and eventual bifurcation into two separate vortices. The P+S mode occurs when the cylinder moving speeds are, for most of the time, higher than the speed of the uniform flow. These situations are found at fast and large-amplitude cylinder oscillations in which the flow relative to the cylinder movement takes over the uniform flow in governing the initial vortex formation. The formation stages of vortices from the cylinder are found to bear close resemblance to those of a vortex street pattern of a cylinder oscillating in an otherwise quiescent fluid at Keulegan–Carpenter numbers around 16. Vortices in the inclined vortex street pattern so formed are then convected downstream by the uniform flow as the vortex pairs in the 2P mode.     

Oscillations of a viscous fluid in a thin layer between two coaxial disks rotating together

A study is made of the problem of the motion of an incompressible viscous fluid in the space between two coaxial disks rotating together with constant angular velocity under the assumption that the pressure changes in time in accordance with a harmonic law. The problem is solved using the equations of unsteady motion of an incompressible viscous fluid in a thin layer. It is shown that the velocity field in this case is a superposition on a steady field of damped oscillations with cyclic frequency equal to twice the angular velocity of the disks and forced oscillations with cyclic frequency equal to the cyclic frequency of the oscillations of the pressure field. It is shown that the amplitude of the forced oscillations of the velocity field depends strongly on the ratio of the cyclic frequency of the oscillations of the pressure field to the angular velocity of the disks. It is shown that there is a certain value of the ratio at which the amplitude of the forced oscillations has a maximal value (resonance). It is shown that even for very small amplitudes of the pressure oscillations the amplitude of the oscillations of the relative velocity at resonance may reach values comparable with the mean velocity of the main flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 166–169, January–February, 1984.     

MHD viscous flow between torsionally oscillating disks

The unsteady motion of an incompressible, viscous, stratified fluid between two parallel infinite disks maintained at different temperatures is studied under the influence of a uniform transverse magnetic field. The whole system is under rigid rotation in the initial state and perturbations are created by the small amplitude torsional oscillations of the disks. The time required for the transient velocity and temperature to decay is found for various ranges of the values of the forcing frequency of the disks. The steady state velocity and temperature distributions represent boundary layers on the disks and an interior flow. The interplay between the Hartmann number and the Ekman number in determining the boundary layers on the disks is discussed.     

Motion of a cylinder in a viscous electroconductive medium with a magnetic field

The method of force sources is used to consider the planar problem of the motion of a circular cylinder in a viscous electroconductive medium with a magnetic field. The conventional and magnetic Reynolds numbers are assumed to be small. Expressions are obtained for the hydrodynamic reaction forces of the medium, acting on the moving cylinder. It is shown that as a result of the flow anisotropy in the medium, caused by the magnetic field, in addition to the resistance forces on bodies moving at an angle to the field, there are deflecting forces perpendicular to the velocity vector. The velocity field disturbances at great distances from the moving cylinder are determined.The problems of viscous electroconductive flow about solid bodies in the presence of a magnetic field constitute one of the divisions of magnetohydrodynamics. Motion of an electroconductive medium in a magnetic field gives rise to inductive electromagnetic fields and currents which interact with the velocity and pressure hydrodynamic fields in the medium [1, 2]. Under conditions of sufficiently strong interaction, the number of independent flow similarity parameters in MHD is considerably greater than in conventional hydrodynamics. This circumstance complicates the theoretical analysis of MHD flow about bodies, and therefore we must limit ourselves to consideration of individual particular flow cases.Here we consider the linear problem of the motion of an infinite circular cylinder in a viscous incompressible medium with finite electroconductivity located in a uniform magnetic field.There are many studies devoted to the flow of a viscous electroconductive medium with a magnetic field about solid bodies (see, for example, [3–5]). Because of this, some of the results obtained here include previously known results, which will be indicated below. In contrast to the cited studies, the examination is made by the method of force sources, suggested in [6]. This method permits obtaining integral equations for the distribution of the forces acting on the surface of the moving body. Their solution is obtained for small Reynolds and Hartmann numbers. Then the nature of the velocity disturbances at great distances from the body are determined. These results are compared with conventional viscous flow about a cylinder in the Oseen approximation.     

An application of the finite difference‐based lattice Boltzmann model to simulating flow‐induced noise

This paper presents a novel approach to simulate aerodynamically generated sounds by modifying the finite difference‐based lattice BGK compressible fluid model for the purpose of speeding up the calculation and also stabilizing the numerical scheme. With the model, aerodynamic sounds generated by a uniform flow around a two‐dimensional circular cylinder at Re = 150 are simulated. The third‐order‐accurate up‐wind scheme is used for the spatial derivatives, and the second‐order‐accurate Runge–Kutta method is applied for the time marching. The results show that we successively capture very small acoustic pressure fluctuations, with the same frequency of the Karman vortex street, much smaller than the whole pressure fluctuation around a circular cylinder. The propagation velocity of the acoustic waves shows that the points of peak pressure are biased upstream owing to the Doppler effect in the uniform flow. For the downstream, on the other hand, it is faster. It is also apparent that the amplitude of sound pressure is proportional to r^?1/2, r being the distance from the centre of the circular cylinder. Moreover, the edgetone generated by a two‐dimensional jet impinging on a wedge to predict the frequency characteristics of the discrete oscillations of a jet‐edge feedback cycle is investigated. The jet is chosen long enough to guarantee the parabolic velocity profile of the jet at the outlet, and the edge is of an angle of α = 23°. At a stand‐off distance w, the edge is inserted along the centreline of the jet, and a sinuous instability wave with real frequency is assumed to be created in the vicinity of the nozzle exit and to propagate towards the downstream. We have succeeded in capturing small pressure fluctuations resulting from periodic oscillation of jet around the edge. Copyright © 2006 John Wiley & Sons, Ltd.     

Application of Differing Forcing Function Models on Simulated Flow Past an Oscillating Cylinder in a Uniform Low Reynolds Number Flow

A Computational Fluid Dynamics (CFD) model is presented for the uniform viscous two dimensional flow past an oscillating cylinder at low Reynolds number. Numerical simulations are made to study the effect of differing forced induced oscillation mechanisms with a large range of cylinder forcing frequencies. In the first case sinusoidal velocity slip boundary conditions are adopted for the cylinder surface to simulate cylinder oscillation. The implication suggests that no modification or additional term need to be added to the Navier-Stokes equations. In the second case this time extra body force terms which are assumed to account for velocity effects due to cylinder movement are included in the Navier-Stokes equations with the imposition of same boundary conditions. Drag and lift coefficients are extracted from present numerical results and other detailed computations of these coefficients are made at a Reynolds number of 80 and an amplitude-to diameter ratio 0.14. The results are found to be in agreement with each other at low force driving frequencies below and near lock-in. However, differences are found at higher frequencies above lock-in. Agreement are also found with experimental results at some frequency ranges.     

Flow-induced vibrations of a tethered circular cylinder

One of the most basic examples of fluid-structure interaction is provided by a tethered body in a fluid flow. The tendency of a tethered buoy to oscillate when excited by waves is a well-known phenomenon; however, it has only recently been found that a submerged buoy will act in a similar fashion when exposed to a uniform flow at moderate Reynolds numbers, with a transverse peak-to-peak amplitude of approximately two diameters over a wide range of velocities. This paper presents results for the related problem of two-dimensional simulations of the flow past a tethered cylinder. The coupled Navier–Stokes equations and the equations of motion of the cylinder are solved using a spectral-element method. The response of the tethered cylinder system was found to be strongly influenced by the mean layover angle as this parameter determined if the oscillations would be dominated by in-line oscillations, transverse oscillations or a combination of the two. Three branches of oscillation are noted, an in-line branch, a transition branch and a transverse branch. Within the transition branch, the cylinder oscillates at the shedding frequency and modulates the drag force such that the drag signal is dominated by the lift frequency. It is found that the mean amplitude response is greatest at high reduced velocities, i.e., when the cylinder is oscillating predominantly transverse to the fluid flow. Furthermore, the oscillation frequency is synchronized to the vortex shedding frequency of a stationary cylinder, except at very high reduced velocities. Visualizations of the pressure and vorticity in the wake reveal the mechanisms behind the motion of the cylinder.     

Flow in rotating radial channels at small Rossby and Ekman numbers

The problem of flow of a viscous fluid in a rotating channel is considered in the region of very small Rossby and Ekman numbers and moderately large Reynolds numbers. Asymptotic expressions with respect to the Ekman number are found for the velocity components and the longitudinal pressure gradient by solving a system of linear differential equations using Fourier series. The stability limits of such flow are predicted. Attention is drawn to a similarity between the velocity profiles of these flows and flows of a magnetic fluid and a fluid executing longitudinal oscillations in a fixed channel.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 11–15, January–February, 1984.     

VORTEX SHEDDING RESONANCE FROM A ROTATIONALLY OSCILLATING CYLINDER

Vortex shedding resonance of a circular cylinder wake to a forced rotational oscillation has been investigated experimentally by measuring the velocity fluctuations in the wake, pressure distributions over the cylinder surface, and visualizing the flow field with respect to cylinder oscillations. The vortex shedding resonance occurs near the natural shedding frequency at small amplitude of cylinder oscillations, while the peak resonance frequency shifts to a lower value with an increase in oscillation amplitude. The drag and lift forces acting on the cylinder at fixed forcing Strouhal number indicate that the phase lag of fluid forces to the cylinder oscillations increases with an increase in oscillation amplitude, supporting the variation of resonance frequency with oscillation amplitude. The comparative study of the measured pressure distributions and the simultaneous flow visualizations with respect to cylinder rotation shows the mechanisms of phase lag, which is due to the strengthened vortex formation and the modification of the surface pressure distributions.     

A circular cylinder undergoing large-amplitude transverse oscillations in a slow uniform cross flow

This study explores the vortex patterns formed by a circular cylinder undergoing lateral cylinder oscillations with large amplitudes and in the presence of a slow uniform cross flow. It is an extension of our previous study (Lam et al., 2010b) in which formation of the 2S, 2P and P+S vortex modes were discussed from the viewpoint of interaction of a uniform cross-flow with the vortex street patterns of a cylinder oscillating in an otherwise quiescent fluid at Keulegan–Carpenter numbers up to KC=8.9. The present paper reports three additional experimental sets in which the amplitudes of cylinder oscillations have even larger values, at A/D>2.5, and lie beyond the vortex mode map usually quoted from Williamson and Roshko (1988). It is found that the slow uniform cross-flow at λ/D≈3 and Reynolds number based on cross-flow velocity at 232 acts to convect the corresponding vortex patterns in the absence of cross-flow downstream across the line of cylinder oscillation. Vortex–vortex interaction and vortex–cylinder interaction are observed to affect the subsequent development of vortices. The P+S vortex mode is found to occur up to KC=16. At KC between 16 and 24, a new vortex mode is observed in which only one vortex pair can be convected downstream every cylinder oscillation cycle. Another new vortex mode with two vortex pairs and two stationary vortices are found at KC>24.     

Steady flow around,and drag on a circular cylinder moving at low speeds in a viscous liquid between two parallel planes 1

The present paper deals with the two-dimensional viscous flow around a circular cylinder moving at low speeds along the central plane parallel to two parallel side planes. It is semi-analytically solved by applying the collocation method for the stream function. The present results for velocity, pressure and viscous stress agree well with other numerical results. The pressure drag and the friction drag on a cylinder are obtained from the pressure distribution and the viscous stress distribution at the surface of a circular cylinder, respectively. It is shown that the pressure drag is larger than the friction drag and the ratio of the former to the latter decreases with increase of the distance between the two parallel side planes. An empirical formula for the drag which fits the present results is obtained by modifying Faxen's analytical formula. The new formula covers a wider range of the distance between the two side planes than Faxen's one.     

On the origin of wake-induced vibration in two tandem circular cylinders at low Reynolds number

We numerically investigate flow-induced vibrations of circular cylinders arranged in a tandem configuration at low Reynolds number. Results on the coupled force dynamics are presented for an isolated cylinder and a pair of rigid cylinders in a tandem configuration where the downstream cylinder is elastically mounted and free to vibrate transversely. Contrary to turbulent flows at high Reynolds number, low frequency component with respect to shedding frequency is absent in laminar flows. Appearance and disappearance of the vorticity regions due to reverse flow on the aft part of the vibrating cylinder is characterized by a higher harmonic in transverse load, which is nearly three times of the shedding frequency. We next analyze the significance of pressure and viscous forces in the composition of lift and their phase relations with respect to the structural velocity. For both the isolated and tandem vibrating cylinders, the pressure force supplies energy to the moving cylinder, whereas the viscous force dissipates the energy. Close to the excitation frequency ratio of one, the ratio of transverse viscous force to pressure force is found to be maximum. In addition, movement of stagnation point plays a major role on the force dynamics of both configurations. In the case of isolated cylinder, displacement of the stagnation point is nearly in-phase with the velocity. During vortex-body interaction, the phase difference between the transverse pressure force and velocity and the location of stagnation point determines the loads acting on the cylinder. When the transverse pressure force is in-phase with velocity, the stagnation point moves to higher suction region of the cylinder. In the case of the tandem cylinder arrangement, upstream vortex shifts the stagnation point on the downstream cylinder to the low suction region. Thus a larger lift force is observed for the downstream cylinder as compared to the vibrating isolated cylinder. Phase difference between the transverse load and the velocity of the downstream cylinder determines the extent of upstream wake interaction with the downstream cylinder. When the cylinder velocity is in-phase with the transverse pressure load component, interaction of wake vortex with the downstream cylinder is lower compared to other cases considered in this study. We extend our parametric study of tandem cylinders for the longitudinal center-to-center spacing ranging from 4 to 10 diameter.     

Motion of a circular cylinder in a viscous liquid between parallel plates

The two-dimensional motion of a cylinder in a viscous fluid between two parallel walls of a vertical channel is studied. It is found that when the cylinder moves very closely along one of the channel walls, it always rotates in the direction opposite to that of contact rolling along the nearest wall. When the cylinder is away from the walls, its rotation depends on the Reynolds number of the flow. In this study two numerical methods were used. One is for the unsteady motion of a sedimenting cylinder initially released from a position close to one of the channel walls, where the Navier-Stokes equations are solved for the fluid and Newton's equations of motion are solved for the rigid cylinder. The other method is for the steady flow in which a cylinder is fixed in a uniform flow field where the channel walls are sliding past the cylinder at the speed of the approaching flow, or equivalently a cylinder is moving with a constant velocity in a quiescent fluid. The flow field, the drag, the side force (lift), and the torque experienced by the cylinder are studied in detail. The effects of the cylinder location in the channel, the size of the channel relative to the cylinder diameter, and the Reynolds number of the flow are examined. In the limit when the cylinder is translating very closely along one of the walls, the flow in the gap between the cylinder and the wall is solved analytically using lubrication theory, and the numerical solution in the other region is used to piece together the whole flow field.This research was supported by NSF DMR91-20668 through the Laboratory for Research on the Structure of Matter at the University of Pennsylvania and from the Research Foundation of the University of Pennsylvania.     

FEEDBACK CONTROL OF VORTEX SHEDDING FROM A CIRCULAR CYLINDER BY ROTATIONAL OSCILLATIONS

The present paper describes a new active method for controlling vortex shedding from a circular cylinder in a uniform flow at medium Reynolds numbers. It uses rotary cylinder oscillations controlled by the feedback signal of a reference velocity in the cylinder wake. The effectiveness of this feedback control is evaluated by measuring the response of mean and fluctuating velocities in the cylinder wake, the spanwise correlation, the power spectrum, and the fluid forces acting on the cylinder. It is found that the velocity fluctuations and the fluid forces are both reduced by the feedback control with optimum values of the phase lag and feedback gain. The simultaneous flow visualization synchronized with the cylinder oscillation indicates the attenuation as well as the mechanisms of vortex shedding under the feedback control, which is due to the dynamic effect of cylinder oscillation on the vortex formation.     

Numerical investigation of a laminar pulsating flow in a rectangular duct

A pulsating laminar flow of a viscous, incompressible liquid in a rectangular duct has been studied. The motion is induced under an imposed pulsating pressure difference. The problem is solved numerically. Different flow regimes are characterized by a non‐dimensional parameter based on the frequency (ω) of the imposed pressure gradient oscillations and the width of the duct (h). This, in fact, is the Reynolds number of the problem at hand. The induced velocity has a phase lag (shift) with respect to the imposed pressure oscillations, which varies from zero at very slow oscillations, to 90° at fast oscillations. The influence of the aspect ratio of the rectangular duct and the pulsating pressure gradient frequency on the phase lag, the amplitude of the induced oscillating velocity, and the wall shear were analyzed. Copyright © 1999 John Wiley & Sons, Ltd.     

Hydromagnetic Stokes flow in a rotating fluid with suspended small particles

An initial value investigation is made of the motion of an incompressible, viscous conducting fluid with embedded small spherical particles bounded by an infinite rigid non-conducting plate. Both the plate and the fluid are in a state of solid body rotation with constant angular velocity about an axis normal to the plate. The flow is generated in the fluid-particle system due to non-torsional oscillations of a given frequency superimposed on the plate in the presence of a transverse magnetic field. The operational method is used to derive exact solutions for the fluid and the particle velocities, and the wall shear stress. The small and the large time behaviour of the solutions is discussed in some detail. The ultimate steady-state solutions and the structure of the associated boundary layers are determined with physical implications. It is shown that rotation and magnetic field affect the motion of the fluid relatively earlier than that of the particles when the time is small. The motion for large times is set up through inertial oscillations of frequency equal to twice the angular velocity of rotation. The ultimate boundary layers are established through inertial oscillations. The shear stress at the plate is calculated for all values of the frequency parameter. The small and large-time behaviour of the shear stress is discussed. The exact solutions for the velocity of fluid and the wall shear stress are evaluated numerically for the case of an impulsively moved plate. It is found that the drag and the lateral stress on the plate fluctuate during the non-equilibrium process of relaxation if the rotation is large. The present analysis is very general in the sense that many known results in various configurations are found to follow as special cases.     

Effect of ion slip on the time-varying Hartmann flow of a non-Newtonian viscoelastic fluid with heat transfer

Ion slip in a time-varying Hartmann flow of a conducting incompressible non-Newtonian viscoelastic fluid between two parallel horizontal insulating porous plates is studied with allowance for heat transfer. A uniform and constant pressure gradient is applied in the axial direction. An external uniform magnetic field and uniform suction and injection through the surface of the plates are applied in the normal direction. The two plates are maintained at different but constant temperatures; the Joule and viscous dissipations are taken into consideration. Numerical solutions for the governing momentum and energy equations are obtained with the use of finite differences, and the effect of various physical parameters on both the velocity and temperature fields is discussed.     

Laminar viscous flow through regular arrays of parallel solid cylinders

Solutions are found for the Stokes equations of motion for a viscous fluid flowing either parallel or perpendicular to the axes of cylinders in square, rectangular, triangular and hexagonal arrays. This is done by matching a solution outside one cylinder to a sum of solutions with equal singularities inside every cylinder of an infinite array. Some of the terms in the solution are indeterminate but these indeterminacies are resolved. The resulting solutions are several terms of a power series in the density.High density approximations are found for the longitudinal case when the cylinders overlap.For low densities the mean velocity for transverse flow found to be independent of orientation of the array and is half the mean velocity for parallel flow in the same pressure field to several orders of magnitude of the volume concentration of cylinders.     

Stability of thin film flowing down the outer surface of a rotating non-uniformly heated vertical cylinder

Nonlinear Dynamics - We investigated an incompressible viscous liquid film flow over a rotating vertical cylinder of radius R and of infinite length rotating with a uniform angular velocity...     

Potential flow around a deforming bubble in a Venturi

The differential pressure between the entrance and throat of a Venturi will fluctuate if the liquid flowing through the Venturi contains bubbles. This paper reports computations of the pressure fluctuation due to the passage of a single bubble. The liquid is assumed inviscid and its velocity, assumed irrotational, is computed by means of a boundary integral technique. The liquid velocity at the entrance to the Venturi is assumed constant and uniform across the pipe, as is the pressure at the outlet. The bubble is initially far upstream of the Venturi and moves with velocity equal to that of the liquid. Buoyancy is neglected. If the bubble is sufficiently small that interactions with the Venturi walls may be neglected, a simple one-dimensional model for the bubble velocity is in good agreement with the full boundary integral computations. The differential pressure (taken to be positive) decreases when the bubble enters the converging section of the Venturi, and then increases to a value higher than for liquid alone as the bubble passes the pressure measurement position within the throat. The changes can be estimated using the one-dimensional model, if the bubble is small. The bubble is initially spherical (due to surface tension) but is perturbed by the low pressure within the Venturi throat. In the absence of viscosity, the bubble oscillates after leaving the Venturi. The quadrupole oscillations of the bubble are similar in frequency to those of a bubble in unbounded fluid; the frequency of the monopole oscillations is modified by the presence of the pipe walls. Numerical results for the frequency of monopole oscillations of a bubble in a uniform tube of finite length are in good agreement with analytic predictions, as is the computed drift of the oscillating bubble.