Secondary instabilities of the in-phase synchronized wakes past two circular cylinders in side-by-side arrangement

In the present study we investigate the secondary instability of the in-phase synchronized vortex shedding from two side-by-side circular cylinders at low Reynolds numbers. Two distinct Floquet modes become unstable for different values of the Reynolds number and of the non-dimensional gap spacing, leading to the onset of the well-known flip-flop instability of the two cylinder wakes. In both cases the two-dimensional Floquet analysis reveals that at very low Reynolds numbers, a pair of complex-conjugate multipliers crosses the unit circle, showing the same frequency as the biased gap-flow flip-over. In the past literature this behaviour has been often ascribed to a bistability of the flow. On the contrary, the present DNS and stability results provide evidence that at low Reynolds numbers, the flip-flopping behaviour originates from a Neimark–Sacker bifurcation of the in-phase shedding cycle.     

Stability of plane poiseuille flow of a highly elastic liquid

The stability of fully developed pressure driven plane laminar flow of a Maxwell fluid has been studied using linear hydrodynamic stability theory. Elasticity is destabilizing in the inertial regime, but the flow is found to be stable to infinitesimal disturbances at low Reynolds numbers. This result contradicts previous calculations, which predicted a low Reynolds number flow instability at a critical recoverable shear of order unity. The previous calculations were carried out using less accurate numerical methods; the eigenvalue problem which must be solved is a delicate one, requiring sophisticated umerical techniques in order to avoid the calculation of spurious unstable modes.This work has direct bearing on the question of the mechanism of a low Reynolds number extrusion instability known as “melt fracture”. It is observed that the intensity of melt fracture increases with increasing die length for high density polyethylene, and it is therfore believed by some experimentalists that fully-developed die flow is unstable for this polymer above a critical recoverable shear. The analysis appears to be at variance with this interpretation of the experimental results.     

Flow of a shear-thinning liquid in a cylindrical container with a rotating end wall

The flow patterns produced by rotating one end wall of a circular cylinder completely filled with a strongly shear-thinning viscoelastic liquid have been investigated using the laser-induced fluorescence flow visualization technique. An intense toroidal vortex is produced in the vicinity of the rotating end wall with outward spiraling flow over the end wall itself. This vortex drives a second countercirculating vortex of low intensity in the region of the stationary end wall. Under some circumstances an axial jet of fluid is observed moving away from the rotating end wall. This jet showed evidence of instability, whereas all flows were otherwise completely steady. The double-vortex structure is different from those recently observed in either a Newtonian or slightly shear-thinning liquid or in the low Reynolds number flow of an elastic liquid. There are, however, similarities with older work for a viscoelastic liquid at relatively high Reynolds numbers. The observations highlight the suitability of the cylinder/rotating end wall configuration as a sensitive test case for computational work.     

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.     

Numerical simulation of three-dimensional instability of flow past a short cylinder

The plane-parallel flow past an infinitely long circular cylinder becomes three-dimensional starting with Reynolds numbers Re ≈ 190. The corresponding instability mode is called mode A. When Re ≈ 260, vortex structures with a smaller cross scale are formed in the wake as a result of a secondary three-dimensional instability (mode B). The transition to three-dimensionality for a short cylinder bounded by planes is considered. The length of the cylinder is chosen to eliminate the unstable perturbations of mode A. Two instability modes similar to modes A and B modified under the effect of the bounding lateral planes are found. The problems of three-dimensional flow are numerically solved using the Navier-Stokes equations.     

弹性支撑圆柱绕流稳定性分析

基于CFD 技术,采用系统辨识方法,建立了亚临界雷诺数(Re < 47) 下绕圆柱流动的非定常气动力模型(reduced order model, ROM). 耦合结构运动方程和降阶气动力模型,建立了弹性支撑圆柱绕流的稳定性分析模型. 算例分析了亚临界雷诺数下,结构固有频率、质量比等参数以及支撑方式对弹性系统稳定性的影响. 对于单自由度横向支撑圆柱,当结构固有频率趋近流动最不稳定模态频率时,弹性系统会在一定频率范围内失稳,这种现象最低可在Re~20 时出现. 旋转自由度的释放能够进一步降低系统的稳定性,可将临界雷诺数进一步降低至18 左右. ROM 方法不仅具有很高的效率,而且清晰地指出了弹性系统失稳的根本原因:流动模态和结构模态耦合作用导致结构模态失稳所致. 因此,失稳状态下系统振荡频率锁定于结构固有频率. 基于ROM 技术预测的失稳边界与直接CFD/CSD 仿真结果吻合,证明了该方法的正确性和精度.      

Transition phenomena in the wake of a square cylinder

The transition phenomena in the wake of a square cylinder were investigated. The existence of mode A and mode B instabilities in the wake of a square cylinder was demonstrated. The critical Reynolds numbers for the inception of these instability modes were identified through the determination of discontinuities in the St–Re curves, and were found to have mean values of 160 and 204 for the onset of mode A and B instabilities, respectively. The spectra and time traces of the wake streamwise velocity component were found to display three distinct patterns in laminar, mode A and mode B flow regimes. Streamwise vortices with different wavelength at various Reynolds numbers were observed through different measures. The symmetries and evolution of the secondary vortices were observed using laser-induced-fluorescent dye. It was found that, just like the case of a circular cylinder, the secondary vortices from the top and bottom rows were out-of-phase with each other in the mode A regime, but in-phase with each other in the mode B regime. From the flow visualization, it was qualitatively proven that there is stronger interaction between braid regions in the mode B regime. At the same time, analysis of PIV measurements quantitatively demonstrated the presence of the stronger cross flow in mode B regime when compared to the mode A regime. It suggests that the in-phase symmetry of the mode B instability is the result of strong interaction between the top and bottom vortex rows. It was also observed that although the vorticity of the secondary vortices in the mode A regime was smaller, its circulation was more than twice that of mode B instability. Compared to primary vortices, the circulations of both mode A and mode B vortices were much smaller, which indicates that the secondary vortices most likely originate from the primary vortices. The wavelengths of the streamwise vortices in the mode A and B regimes were measured using the auto-correlation method, and were found to be 5.1 (±0.1)D, 1.3 (±0.1)D, and 1.1 (±0.1)D at Re=183 (mode A), 228 and 377 (both mode B), respectively. From the present investigation, mode A instability was likely to be due to the joint-effects of the deformation of primary vortex cores and the stretching of vortex sheets in the braid region. On the other hand, mode B instability was thought to originate from the “imprinting” process.     

旋转振动圆柱绕流周期解和Floquet稳定性

对低雷诺数旋转振动圆柱绕流问题运用低维Galerkin方法将N－S方程约化为一组非线性常微分方程组。运用打靶法数值求解了这组方程的周期解,并用Tloquet理论对周期解的稳定性进行了分析,确定了流动失稳的机制。     

The turbulent flow field around a circular cylinder

The flow field around a circular cylinder mounted vertically on a flat bottom has been investigated experimentally. This type of flow occurs in several technical applications, e.g. local scouring around bridge piers. Hydrogen bubble flow visualization was carried out for Reynolds numbers ranging from 6,600 to 65,000. The main flow characteristic upstream of the cylinder is a system of horse-shoe vortices which are shed quasi-periodically. The number of vortices depends on Reynolds number. The vortex system was found to be independent of the vortices that are shed in the wake of the cylinder. The topology of the separated flow contains several separation and attachment lines which are Reynolds number dependent. In the wake region different flow patterns exist for each constant Reynolds number.     

Experiments on elongational flow of dilute polymer solutions: Part II: velocity field for the flow through small apertures

Following Part I which reported on the jet reaction and the excess pressure drop, this paper describes the results of flow velocities measured upstream of small apertures for dilute polymer solutions. The results suggest the existence of the two flow regimes reorted by Boger and Cable: the vortex growth regime and the divergent flow regime. The former corresponds to the regime of zero jet reaction and the latter to the regime of positive jet reaction. The axial velocity development for PEO solutions along the center line upstream of the aperture is found to yield an extremely high velocity rise at the onset of the vortex region and to give a nearly constant elongational rate within the vortex region. The constant elongational rate takes values such that the product with the relaxation time is about 0.5. Separan solutions do not provide so high a velocity rise at the onset of the vortex region but show an exponential rate of elongation in the vortex region.It is shown, by using the upper convected Maxwell model, that the steep velocity rise at the onset of the vortex region is given in the simplified flow model and the constant elongational rate within the vortex region holds the elongational stress nearly constant.     

Vortex shedding and structural loading characteristics of finned cylinders

The flow development and structural loading characteristics of cylinders with equispaced circular fins were studied experimentally for a range of fin pitches with constant fin thickness and diameter. The experiments were performed for a range of Reynolds numbers, corresponding to the shear layer transition turbulent shedding regime. Time-resolved planar Particle Image Velocimetry and direct mean drag and fluctuating lift measurements are employed to relate spatio-temporal flow development to structural loading. The results show that wake development is dominated by vortex shedding for all the cases examined. However, the fin pitch ratio has a significant effect on vortex shedding characteristics. The addition of fins increases the characteristic spatial and temporal scales of the main spanwise vortices forming in the near wake. As the fin pitch is decreased to a critical value, the coalescence of boundary layers between the adjacent fins leads to a significant enlargement of the vortex formation region. A modified vortex shedding frequency scaling is proposed, based on the effective diameter, that incorporates a Reynolds number dependence associated with the lateral boundary layers developing on the fin surfaces. A detailed analysis is conducted to characterize the strength of the vortical structures forming in the near wake. The addition of the fins is shown to produce a stabilizing effect on the roll-up process, associated with a reduction in the generation of smaller scale, three-dimensional structures. The results demonstrate that the addition of fins leads to an increase in the mean drag, which is driven primarily by the associated increase in skin friction. The significant effect of the fin pitch ratio on the characteristics of the shed vortices as well as the size of the vortex formation region is shown to lead to substantial variations in the fluctuating loads.     

Resonance in flow past oscillating cylinder under subcritical conditions

Flow around an oscillating cylinder in a subcritical region are numerically studied with a lattice Boltzmann method(LBM). The effects of the Reynolds number,oscillation amplitude and frequency on the vortex wake modes and hydrodynamics forces on the cylinder surface are systematically investigated. Special attention is paid to the phenomenon of resonance induced by the cylinder oscillation. The results demonstrate that vortex shedding can be excited extensively under subcritical conditions, and the response region of vibration frequency broadens with increasing Reynolds number and oscillation amplitude. Two distinct types of vortex shedding regimes are observed. The first type of vortex shedding regime(VSR I) is excited at low frequencies close to the intrinsic frequency of flow, and the second type of vortex shedding regime(VSR II)occurs at high frequencies with the Reynolds number close to the critical value. In the VSR I, a pair of alternately rotating vortices are shed in the wake per oscillation cycle,and lock-in/synchronization occurs, while in the VSR II, two alternately rotating vortices are shed for several oscillation cycles, and the vortex shedding frequency is close to that of a stationary cylinder under the critical condition. The excitation mechanisms of the two types of vortex shedding modes are analyzed separately.     

湍流涡致振动频率特性

用数值模拟方法对固定圆柱湍流涡脱落频率与弹性圆柱湍流涡致振动频率特性进行了研究,湍流计算模型采用标准κ-ε模型,压力泊松方程提法基于非交错网格系统.研究结果表明:固定圆柱湍流绕流涡脱落频率基本不随雷诺数而变,对于同一固有频率弹性圆柱,涡振频率基本不随雷诺数而变;对于某一固定雷诺数流动涡振频率在一定范围内与系统固有频率有关.     

Hydrodynamic instability of separated viscous flow around a circular cylinder

A uniform viscous flow around a circular cylinder is studied numerically in the Reynolds number range from 0 to 500. It is shown that the existence and the basic properties of self-oscillating regimes are specified by the evolution of their hydrodynamic instability. It is found that the vortex formation in a near wake is associated with the separation zone dynamics in the main flow. The values of critical Reynolds numbers for the four successive bifurcations of the self-oscillating regimes of flow are obtained. An interpretation of experimental data on the vortices in the near wake is discussed.     

Stability analysis of the onset of vortex shedding for wakes behind flat plates

Above a critical Reynolds number, wake flows behind flat plates become globally unstable, the leading modal instability in this case is known as Kelvin–Helmholtz mechanism. In this article, both local and BiGlobal linear instability analyses are performed numerically to study the onset of the shedding process. Flat plates with different base shapes are considered to assess geometry effects, and the relation between the critical shedding Reynolds number, \(Re_\mathrm{cr}\), and the boundary layer thickness is studied. Three types of base shapes are used: square, triangular and elliptic. It is found that the base shape has a great impact on the growth rate of least stable disturbance mode, thus would influence \(Re_\mathrm{cr}\) greatly, but it has little effect on the vortex shedding frequency. The shedding frequency is determined mainly by boundary layer thickness and has little dependence on the Reynolds number and base shape. We find that for a fixed Reynolds number, increasing boundary layer thickness acted in two ways to modify the global stability characteristics: It increases the length of the absolute unstable region and it makes the flow less locally absolutely unstable in the near-wake region, and these two effects work against each other to destabilize or stabilize the flow.     

Instability of a vertical columnar vortex in a stratified fluid

In this article, we study experimentally the evolution of a vertical columnar vortex in a stratified fluid. Three different measurement techniques are used. Particle image velocimetry allows us to monitor the time evolution of the characteristics of the vortex (Froude and Reynolds numbers). Dye visualizations reveal the existence of an instability for Froude numbers smaller than one, which creates an undulation of the vortex centerline. Synthetic schlieren visualization shows that the density structure of the unstable mode is very similar to the structure found recently numerically for the radiative instability of a Lamb–Oseen vortex (Riedinger et al. in J Fluid Mech, 2010). The experimental stability diagram and unstable wavelengths are compared with these numerical results. A secondary instability associated with the presence of critical layers is also observed for Froude numbers larger than one.     

Topology of vortex creation in the cylinder wake

We analyze the topology of the two-dimensional flow around a circular cylinder at moderate Reynolds numbers in the regime where the vortex wake is created. A normal form for the stream function close to the cylinder is presented and used to predict the streamline pattern both in the steady and the periodic regime, where two different vortex shedding scenarios are identified. The theoretical predictions are verified numerically. For the vorticity, a very different topology occurs with infinite nested sequences of iso-curves moving downstream. General equations of motion for critical points are derived.     

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     

Numerical Study of the Flow Behind a Rotary Oscillating Circular Cylinder

A numerical study is performed of flow behind a rotationally oscillating circular cylinder in a uniform flow by solving the two-dimensional incompressible Navier-Stokes equations. The flow behavior in lock-on regime and the timing of vortex formation from the oscillating cylinder are studied. When the frequency of excitation of the cylinder is in the vicinity of the natural vortex formation frequency, a lock-on vortex formation regime appears. As the excitation frequency being increased relative to the natural frequency the initially formed vorticity concentration switches to the opposite side of the cylinder. The effects of oscillating frequency and amplitude on the vortex structures formed in the near wake of the cylinder are also investigated. Based on the present calculated results, some complicated vortex patterns are identified and are consistent with the previous experimental visualizations.     

Effect of span length and temperature on the 3-D confined flow around a vortex promoter

This article presents a numerical study on the influence of span length and wall temperature on the 3-D flow pattern around a square section vortex promoter located inside a micro-channel in the low Reynolds number regime. The first objective of the work is to quantify the critical Reynolds number that defines the onset of vortex shedding and to identify the different regimes that appear as a function of the channel aspect ratio (span to height ratio). We found that the critical Reynolds number for the onset of the Karman street regime increases as the aspect ratio decreases. In particular, for the aspect ratio of 1/2 the critical Reynolds number is nearly six times the critical Reynolds number of the 2-D problem. An intermediate oscillating regime between the steady and the Karman street solutions was also found to exist within a rather wide range of Reynolds numbers for small channel aspect ratios. The second objective was to investigate the influence of the vortex promoter wall temperature on both vortex shedding and flow pattern. This has practical engineering implications because the working fluid considered in the article is water that has a viscosity that depends significantly on temperature and promotes a strong coupling between the momentum and energy equations that influences the system behaviour. Results indicate that high surface temperature on the prism promotes the onset of the Karman street, suggesting design guidelines for micro-channel based heat sinks that make use of vortex promoters.