Vortex shedding and three-dimensional behaviour of flow past a cylinder confined in a channel

A numerical investigation of the flow past a circular cylinder centred in a two-dimensional channel of varying width is presented. For low Reynolds numbers, the flow is steady. For higher Reynolds numbers, vortices begin to shed periodically from the cylinder. In general, the Strouhal frequency of the shedding vortices increases with blockage ratio. In addition, a two-dimensional instability of the periodic vortex shedding is found, both empirically and by means of a Floquet stability analysis. The instability leads to a beating behaviour in the lift and drag coefficients of the cylinder, which occurs at a Reynolds number higher than the critical Reynolds number for the three-dimensional mode A-type instability, but lower than a Reynolds number for any mode B-type instability.     

Linear stability and dynamics of viscoelastic flows using time-dependent numerical simulations

Ultimately, numerical simulation of viscoelastic flows will prove most useful if the calculations can predict the details of steady-state processing conditions as well as the linear stability and non-linear dynamics of these states. We use finite element spatial discretization coupled with a semi-implicit θ-method for time integration to explore the linear and non-linear dynamics of two, two-dimensional viscoelastic flows: plane Couette flow and pressure-driven flow past a linear, periodic array of cylinders in a channel. For the upper convected Maxwell (UCM) fluid, the linear stability analysis for the plane Couette flow can be performed in closed form and the two most dangerous, although always stable, eigenvalues and eigenfunctions are known in closed form. The eigenfunctions are non-orthogonal in the usual inner product and hence, the linear dynamics are expected to exhibit non-normal (non-exponential) behavior at intermediate times. This is demonstrated by numerical integration and by the definition of a suitable growth function based on the eigenvalues and the eigenvectors. Transient growth of the disturbances at intermediate times is predicted by the analysis for the UCM fluid and is demonstrated in linear dynamical simulations for the Oldroyd-B model. Simulations for the fully non-linear equations show the amplification of this transient growth that is caused by non-linear coupling between the non-orthogonal eigenvectors. The finite element analysis of linear stability to two-dimensional disturbances is extended to the two-dimensional flow past a linear, periodic array of cylinders in a channel, where the steady-state motion itself is known only from numerical calculations. For a single cylinder or widely separated cylinders, the flow is stable for the range of Deborah number (De) accessible in the calculations. Moreover, the dependence of the most dangerous eigenvalue on De≡λV/R resembles its behavior in simple shear flow, as does the spatial structure of the associated eigenfunction. However, for closely spaced cylinders, an instability is predicted with the critical Deborah number De_c scaling linearly with the dimensionless separation distance L between the cylinders, that is, the critical Deborah number De_Lc≡λV/L is shown to be an O(1) constant. The unstable eigenfunction appears as a family of two-dimensional vortices close to the channel wall which travel downstream. This instability is possibly caused by the interaction between a shear mode which approaches neutral stability for De ≫ 1 and the periodic modulation caused by the presence of the cylinders. Nonlinear time-dependent simulations show that this secondary flow eventually evolves into a stable limit cycle, indicative of a supercritical Hopf bifurcation from the steady base state.     

Flow past a rotating cylinder translating at different gap heights along a wall

The flow past a rotating circular cylinder translating parallel to a wall at different heights is investigated for Reynolds numbers up to 400 for three discrete rotation rates. In particular, the various wake transitions that occur as a function of gap height are quantified for the three cases examined: non-rotation, and forward and reverse rotations. At low gap heights, only a single steady three-dimensional mode is found to become unstable on the steady base flow. As the gap height is increased, several new three-dimensional modes are observed, of which one attains large amplitudes in the near wake and another preferentially in the far wake. At still larger gap heights, the transition sequence resembles that observed in a rotating cylinder wake, for which the wake first undergoes transition to a periodic state, prior to the onset of three-dimensional flow. Parameter space maps showing the neutral stability curves and regions of instability for each mode are presented for each rotation rate, together with a discussion of the spatio-temporal characteristics and spatial distributions of the new modes. Finally, the force coefficients for the steady and periodic two-dimensional base flows are presented.     

Hydroelastic instabilities in viscoelastic flow past a cylinder confined in a channel

Experimental investigation of viscoelastic flow past a cylinder is conducted for a polyisobutylene-based polymer solution. High-image-density particle image velocimetry is utilized to quantitatively determine the spatial features of elastic wake instabilities. The viscoelastic flow bifurcates from steady two-dimensional flow to steady three-dimensional flow for values of the Deborah number (dimensionless flow rate) greater than a critical value. These hydroelastic flow transitions are manifested in the form of three-dimensional cells spaced periodically along the axis of the cylinder. The elastic flow structures do not exist in the Newtonian counterpart of creeping flow past a cylinder. Received: 7 October 1998 / Accepted: 22 April 1999     

An experimental evaluation of the behaviour of mono and polydisperse polystyrenes in Cross-Slot flow

The behaviour of a number of mono and polydisperse polystyrenes are probed experimentally in complex extensional flow within a Cross-Slot geometry using flow-induced birefringence. Polystyrenes with similar molecular weight (M _w) and increasing polydispersity (PD) illustrated the effect of PD on the principal stress difference (PSD) pattern in extensional flow. Monodisperse materials exhibited only slight asymmetry at moderate flowrates, although increased asymmetry and cusping was observed at high flowrates. The response of monodisperse materials of different M _w at various flowrates is presented and characterised by Weissenberg numbers for both chain stretch and orientation using a theory for linear entangled polymers. The comparison of stress profiles against Weissenberg number for each process is used to determine whether the PSD pattern observed is independent of M _w and elucidate which relaxation mechanism is dominant in the flow regimes probed. For monodisperse materials, at equivalent chain orientation Weissenberg number (We _τd), different molecular weight materials were seen to exhibit similar steady state PSD patterns independent of We _τR (chain stretch We). Whilst no obvious critical Weissenberg number (We) was found for the onset of increased asymmetry and cusping, it was found to occur in the “orientating flow without chain stretch” regime.     

Effect of inertia on thermoelastic flow instability

Thermal effects induced by viscous heating cause thermoelastic flow instabilities in curvilinear shear flows of viscoelastic polymer solutions. These instabilities could be tracked experimentally by changing the fluid temperature T₀ to span the parameter space. In this work, the influence of T₀ on the stability boundary of the Taylor–Couette flow of an Oldroyd-B fluid is studied. The upper bound of the stability boundary in the Weissenberg number (We)–Nahme number (Na) space is given by the critical conditions corresponding to the extension of the time-dependent isothermal eigensolution. Initially, as T₀ is increased, the critical Weissenberg number, We_c, associated with this upper branch increases. Increasing T₀ beyond a certain value T^* causes the thermoelastic mode of instability to manifest. This occurs in the limit as We/Pe → 0, where Pe denotes the Péclet number. In this limit, the fluid relaxation time is much smaller than the time scale of thermal diffusion. T₀ = T^* represents a turning point in the We_c–Na_c curve. Consequently, the stability boundary is multi-valued for a wide range of Na values. Since the relaxation time and viscosity of the fluid decrease with increasing T₀, the elasticity number, defined as the ratio of the fluid relaxation time to the time scale of viscous diffusion, also decreases. Hence, O(10) values of the Reynolds number could be realized at the onset of instability if T₀ is sufficiently large. This sets limits for the temperature range that can be used in experiments if inertial effects are to be minimized.     

Flow of wormlike micelle solutions past a confined circular cylinder

In this paper, we present the results of an investigation into the flow of a series of viscoelastic wormlike micelle solutions past a confined circular cylinder. Although this benchmark flow has been studied in great detail for polymer solutions, this paper reports the first experiments to use a viscoelastic wormlike micelle solution as the test fluid. The flow kinematics, stability and pressure drop were examined for two different wormlike micelle solutions over a wide range of Deborah numbers and cylinder to channel aspect ratios. A combination of particle image velocimetry and pressure drop measurements were used to characterize the flow kinematics, while flow-induced birefringence measurements were used to measure the micelle deformation and alignment in the flow. The pressure drop was found to decrease initially due to the shear thinning of the test fluid before increasing at higher flow rates as elastic effects begin to dominate the flow. Above a critical Deborah number, an elastic instability was observed for just one of the test fluids studied, the other remained stable for all Deborah number tested. Flow-induced birefringence and velocimetry measurements showed that observed instability originates in the extensional flow in the wake of the cylinder and appears not as periodic counter-rotating vortices as has been observed in the flow of polymer solutions past circular cylinders, but as a chaotic rupture event in the wake of the cylinder that propagates axially along the cylinder. Reducing the cylinder to channel aspect ratio and the degree of shearing introduced by the channel walls had a weak impact on the stability of the flow. These measurements, when taken in conjunction with previous work on flow of wormlike micelle solutions through a periodic array of cylinders, definitively show that the instability can be attributed to a breakdown of the wormlike micelle solutions in the extensional flow in the wake of the cylinder.     

The stability of two-dimensional linear flows of an Oldroyd-type fluid

A linear stability analysis is made for an Oldroyd-type fluid undergoing steady two-dimensional flows in which the velocity field is a linear function of position throughout an unbounded region. This class of basic flows is characterized by a parameter λ which ranges from λ = 0 for simple shear flow to λ = 1 for pure extensional flow. The time derivatives in the constitutive equation can be varied continuously from co-rotational to co-deformational as a parameter β varies from 0 to 1. The linearized disturbance equations are analyzed to determine the asymptotic behavior as time t → ∞ of a spatially periodic initial disturbance. It is found that unbounded flows in the range 0 < λ ? 1 are unconditionally unstable with respect to periodic initial disturbances which have lines of constant phase parallel to the inlet streamline in the plane of the basic flow. When the Weissenberg number is sufficiently small, only disturbances with sufficiently small wavenumber α₃ in the direction normal to the basic flow plane are unstable. However, for certain values of β, critical Weissenberg numbers are found above which flows are unstable for all values of the wavenumber α₃.     

Forward roll coating flows of viscoelastic liquids

Roll coating is distinguished by the use of one or more gaps between rotating cylinders to meter and apply a liquid layer to a substrate. Except at low speed, the two-dimensional film splitting flow that occurs in forward roll coating is unstable; a three-dimensional steady flow sets in, resulting in more or less regular stripes in the machine direction. For Newtonian liquids, the stability of the two-dimensional flow is determined by the competition of capillary and viscous forces: the onset of meniscus nonuniformity is marked by a critical value of the capillary number. Although most of the liquids coated industrially are non-Newtonian polymeric solutions and dispersions, most of the theoretical analyses of film splitting flows relied on the Newtonian model. Non-Newtonian behavior can drastically change the nature of the flow near the free surface; when minute amounts of flexible polymer are present, the onset of the three-dimensional instability occurs at much lower speeds than in the Newtonian case.Forward roll coating flow is analyzed here with two differential constitutive models, the Oldroyd-B and the FENE-P equations. The results show that the elastic stresses change the flow near the film splitting meniscus by reducing and eventually eliminating the recirculation present at low capillary number. When the recirculation disappears, the difference of the tangential and normal stresses (i.e., the hoop stress) at the free surface becomes positive and grows dramatically with fluid elasticity, which explains how viscoelasticity destabilizes the flow in terms of the analysis of Graham [M.D. Graham, Interfacial hoop stress and instability of viscoelastic free surface flows, Phys. Fluids 15 (2003) 1702–1710].     

Numerical predictions for unsteady viscous flow past an array of cylinders

The unsteady two-dimensional flow around an array of circular cylinders submerged in a uniform onset flow is analysed. The fluid is taken to be viscous and incompressible. The array of cylinders consists of two horizontal rows extending to infinity in the upstream and downstream directions. The centre-to-centre distance between adjacent cylinders is fixed at three diameters, and the rows are staggered. Advantage is taken of spatially periodic boundary conditions in the flow direction. This reduces the computational domain to a rectangular region surrounding a single circular cylinder. Two cases, for Reynolds numbers of 1000 and 10,000, are presented.     

Flow structure around two finite circular cylinders located in an atmospheric boundary layer: side-by-side arrangement

The flow structure around the free-end region of two adjacent finite circular cylinders embedded in an atmospheric boundary layer (ABL) was investigated experimentally. The experiments were carried out in a closed-return-type subsonic wind tunnel, in which two finite cylinders with an aspect ratio of 6 were mounted vertically on a flat plate in a side-by-side arrangement. The Reynolds number based on the cylinder diameter was about Re=2×10⁴. Systems with gap ratios (i.e., center-to-center distance/cylinder diameter) in the range 1.0–2.0 were investigated. A hot-wire anemometer was employed to measure the wake velocity, and the mean pressure distribution on the cylinder surface was also measured. The flow past two finite cylinders was found to have a complicated three-dimensional wake structure in the region near the free ends. As the gap ratio increases, regular vortex-shedding becomes dominant, but the length of the vortex formation region decreases. The pressure distribution and flow structure around two cylinders were found to differ substantially from the behavior of a two-dimensional circular cylinder due to mutual interference. The three-dimensional flow structure seems to originate from the strong entrainment of irrotational fluids caused by the downwash counter-rotating vortices separated from the finite cylinder (FC) free ends.     

Effects of an axial flow component on the Honji instability

This paper is concerned with a numerical study of the three-dimensional Honji instability that can arise in an oscillatory flow impinging on a circular cylinder. It is well known that when the fluid motion far from the cylinder is perpendicular to its axis then the flow is liable to a three-dimensional breakdown via this instability which initially appears as an axially periodic mushroom-like structure attached to the surface of the cylinder. Here the focus is on examining the Honji instability under an oblique inflow. The obliqueness of the free stream is represented by an angle of attack through introducing an axial flow component. It is found that the Honji mode is suppressed by increasing the axial flow component, and when this component is sufficiently large the instability mechanism is no longer operative so that all that remains is a featureless two-dimensional columnar flow. At smaller values of angle of attack, though the Honji structure remains, it is deformed by the axial flow component. The developed two-layer near-cylinder vortical structures can be related to the energy and momentum transfer between the two layers.     

Vortex induced vibration and vortex shedding characteristics of two side-by-side circular cylinders of different diameters in close proximity in steady flow

This paper presents results of a numerical study of vortex-induced vibrations of two side-by-side circular cylinders of different diameters in steady incompressible flow. The two-dimensional Reynolds-averaged Navier–Stokes equations with a SST k–ω turbulence model are solved using the Petrov–Galerkin finite element method and the Arbitrary-Lagrangian–Eulerian scheme. The diameter ratio of the two cylinders is fixed at 0.1 and the mass ratio of both cylinders is 5.0. Both cylinders are constrained to oscillate in the transverse direction only. The Reynolds number based on the large cylinder diameter and free stream velocity is fixed at 5000. The effects of the reduced velocities of the cylinders on the vibration amplitude and vortex shedding regimes are investigated. It is found that for the range of parameters considered, collision between the two cylinders is dependent on the difference of the reduced velocities of the cylinders. Presence of the small cylinder in the proximity of the large one appears to have significant effects on the vortex shedding regime and vibration amplitude of the large cylinder.     

圆柱绕流三维不稳定性的低维Galerkin法分析

利用低维Ｇａｌｅｒｋｉｎ方法及Ｆｌｏｑｕｅｔ稳定性分析理论,计算分析了圆柱绕流的三维线性不稳定性．分析中构造了能较好描述尾迹区流动的周向基函数,建立了完备的合理的基函数组,改进了计算机算法．结果证实圆柱二维周期流对展向小扰动为不稳定的,正确地预计了出现三维长波不稳定性的临界雷诺数Ｒｅｃ＝１９０;扰动展向波长为λｃ一３．６ｄ．对雷诺数Ｒｅ为１８０,１９０两种工况下的圆柱三维绕流流场的计算进一步证实了这种流动的整体不稳定性．本文所预计的临界值比Ｎｏａｃｋ等人的结果更为精确,与Ｂａｒｋｌｅｙ等人的ＤＮＳ解一致,与Ｗｉｌｌｉａｍｓｏｎ的实验相符．     

Unsteady incompressible flows past two cylinders in tandem and staggered arrangements

A stabilized finite element formulation is employed to study incompressible flows past a pair of cylinders at Reynolds numbers 100 and 1000 in tandem and staggered arrangements. Computations are carried out for three sets of cylinder arrangements. In the first two cases the cylinders are arranged in tandem and the distance between their centres is 2·5 and 5·5 diameters. The third case involves the two cylinders in staggered arrangement. The distance between their centres along the flow direction is 5·5 diameters, while it is 0·7 diameter in the transverse direction. The results are compared with flows past a single cylinder at corresponding Reynolds numbers and with experimental observations by other researchers. It is observed that the qualitative nature of the flow depends strongly on the arrangement of cylinders and the Reynolds number. In all cases, when the flow becomes unsteady, the downstream cylinder, which lies in the wake of the upstream one, experiences very large unsteady forces that may lead to wake-induced flutter. The Strouhal number, based on the dominant frequency in the time history of the lift coefficient, for both cylinders attains the same value. In some cases, even though the near wake of the two cylinders shows temporal periodicity, the far wake does not. © 1997 John Wiley & Sons, Ltd.     

Purely tangential flow of a PTT-viscoelastic fluid within a concentric annulus

An analytical solution is presented for the steady state, purely tangential flow of a viscoelastic fluid obeying the Phan-Thien–Tanner (PTT) constitutive equation in a concentric annulus with relative rotation of the inner and outer cylinders. The influence on the velocity distribution within the annulus and on fRe of the Weissenberg number, aspect ratio and an elongational parameter are investigated. The results show that the differences between the radial location of the minimum velocity and of the critical angular velocity compared with their Newtonian counterparts increase as the fluid elasticity increases. The results also show that fRe decreases with increasing Weissenberg number, radius ratio and the elongational parameter in the case of inner-cylinder rotation. In contrast, fRe increases with increasing radius ratio when the outer cylinder is rotating while the inner cylinder is at rest.     

Flow-induced vibration of a circular cylinder subjected to wake interference at low Reynolds number

Two- and three-dimensional numerical simulations of the flow around two circular cylinders in tandem arrangements are performed. The upstream cylinder is fixed and the downstream cylinder is free to oscillate in the transverse direction, in response to the fluid loads. The Reynolds number is kept constant at 150 for the two-dimensional simulations and at 300 for the three-dimensional simulations, and the reduced velocity is varied by changing the structural stiffness. The in-line centre-to-centre distance is varied from 1.5 to 8.0 diameters, and the results are compared to that of a single isolated flexible cylinder with the same structural characteristics, m^?=2.0 and ζ=0.007. The calculations show that significant changes occur in the dynamic behaviour of the cylinders, when comparing the flow around the tandem arrangements to that around an isolated cylinder: for the tandem arrangements, the lock-in boundaries are wider, the maximum displacement amplitudes are greater and the amplitudes of vibration for high reduced velocities, outside the lock-in, are very significant. The main responsible for these changes appears to be the oscillatory flow in the gap between the cylinders.     

PIV measurements of the near-wake behind a sinusoidal cylinder

The three-dimensional near-wake structures behind a sinusoidal cylinder have been investigated using a particle image velocimetry (PIV) measurement technique at Re=3,000. The mean velocity fields and spatial distributions of ensemble-averaged turbulence statistics for flows around the sinusoidal and corresponding smooth cylinders were compared. The near-wake behind the sinusoidal cylinder exhibited pronounced spanwise periodic variations in the flow structure. Well-organized streamwise vortices with alternating positive and negative vorticity were observed along the span of the sinusoidal cylinder. They suppress the formation of the large-scale spanwise vortices and decrease the overall turbulent kinetic energy in the near-wake of the sinusoidal cylinder. The sinusoidal surface geometry significantly modifies the near-wake structure and strongly controls the three-dimensional vortices formed in the near-wake.     

CONTROL OF FLOW PAST BLUFF BODIES USING ROTATING CONTROL CYLINDERS

Computational results for control of flow past a circular cylinder using small rotating cylinders are presented. A well-proven stabilized finite-element method, that has been applied to various flow problems earlier, is utilized to solve the incompressible Navier–Stokes equations in the primitive variables formulation. The formulation is first applied to study flow past an isolated rotating cylinder. Excellent match with experimental results, reported earlier, is observed. It is found that in purely two-dimensional flows, very high lift coefficients can be realized. However, it is observed, via three-dimensional Navier–Stokes simulations, that the end-effects and centrifugal instabilities along the cylinder span lead to a loss of lift and increase in drag. The aspect ratio of the cylinder plays an important role. The flow past a bluff body with two rotating control cylinders is studied using 2-D numerical simulations. The effect of the Reynolds number is studied by carrying out simulations for Re=10²and 10⁴. Finite element meshes with an adequate number of grid points are employed to resolve the flow in the gap between the main and control cylinders. Two values of the gap are considered: 0·01D and 0·075 D, where D is the diameter of the main cylinder. It is observed that when the control cylinders rotate at high speed, such that the tip speed is 5 times the free-stream speed, the flow at Re=100 achieves a steady state. For Re=10⁴, even though the flow remains unsteady, the wake is highly organized and narrower compared to the one without control. The results are in good agreement with the flow-visualization studies conducted by other researchers for bluff bodies using similar control concepts. In all the cases, a significant reduction in the overall drag coefficient and the unsteady aerodynamic forces acting on the main cylinder is observed. Results are also presented for the power requirements of the system for translation and rotation. It is found that the coefficient of power required for the rotation of control cylinders is significant for Re=100 but negligible for Re=10⁴flow. The size of the gap is found to be more critical for the Re=10⁴flows. This study brings out the relevance of the gap as a design parameter for such flow control devices.     

Modes of vortex formation and transition to three-dimensionality in the wake of a freely vibrating cylinder

The present work is aimed to give some insight into the relation between vortex shedding modes and transition to three-dimensionality in the wake of a freely vibrating cylinder by establishing a numerical model and analyzing the relevant results of two- and three-dimensional simulations. The compressible flow past an elastically-mounted cylinder is solved by using the immersed boundary method (IB method). The cylinder is free to vibrate in the transverse direction with zero structure damping. The response of displacement amplitude is studied with the variation of reduced velocity. Whether P+S mode exists in three-dimensional flow and the occurrence of 2P mode is caused by flow transition from two-dimensional to three-dimensional are problems of concern. Both 2P and P+S wake modes are observed in two- and three-dimensional simulations. The numerical results indicate that the flow transition from two-dimensional to three-dimensional is coupled with the cylinder vibration in the synchronization/lock-in regime. The wake formation given by three-dimensional simulations suggests that the P+S mode might exist in reality when the flow is reverted to two-dimensional by vortex induced vibration (VIV) at Re=300–350. When Reynolds number increases to 425, the wake formation undergoes transition to three-dimensionality and 2P mode is observed. The effect of mass ratio on the flow transition to three-dimensionality is studied. The relationship between wake modes and aerodynamic forces is discussed.