Wake-induced Flutter Research Articles

Experiments on flow-induced vibration of four flexible cylinders with large aspect ratio in a square configuration

A system consisting of four cylinders positioned in the quadrilateral arrangement represents the prevalent and fundamental constituent within pipe bundles and tube banks in numerous engineering scenarios. This research presents a laboratorial study on flow-induced vibration (FIV) exhibited by four flexible cylinders positioned within a square geometric arrangement. The uniform flows were produced by a carriage on the top of a towing tank. Considering flow velocities and structural parameters, Reynolds number could reach 16000. The spacing between the cylindrical centres was selected as 6.0 D (D denotes the outer diameter of cylinders), and two typical incidence angles 0° and 45° were examined. For each flexible cylinder model, strain gages at seven distinct measuring positions were utilized to collect the oscillation data in cross-flow (CF) and in-line (IL) directions. Principal vibration modes, response frequencies, response amplitudes and motion trajectories were presented to understand the FIV features. For incidence angles 0°, considering the four flexible cylinders as two sets of tandem cylinders arranged in parallel is not appropriate due to the proximate effect. The behaviours of the two upstream cylinders resembles those of the pair of cylinders positioned in a side-by-side configuration, whereas behaviours for the downstream cylinders are influenced by wake interference generated from leading cylinders. IL response frequencies decrease to the values close to the CF response frequencies, resulting in the elliptical-shaped motion trajectories with large IL displacements. For incidence angles 45°, the upstream cylinder displays characteristics akin to those of a solitary cylinder experiencing vortex-induced vibration (VIV). Reactions for the central pair of cylinders result from the amalgamation of VIV and wake-induced flutter. While, reactions observed for downstream cylinder are characterized by low frequencies, significant IL displacements and more chaotic trajectories.

Flow-induced vibration (FIV) fatigue damage characteristics of two unequal-diameter flexible cylinders in a staggered arrangement

In offshore engineering, fatigue damage caused by flow-induced vibrations (FIVs) presents a considerable and potential threat to flexible cylindrical systems during their service lifetime. Through FIV experiments, this paper estimates and analyzes the fatigue damage of two unequal-diameter cylinders with diameter ratio d/D = 0.5 (where d and D denote the diameter of small and large cylinders) and the center-to-center spacing ratio P/d = 6 (where P denotes the center-to-center distance between two cylinders) through the S–N curve and fatigue damage accumulation methods in the DNV standard. Ten staggered configurations were studied, including two relative positions and five stagger angles (θ = 15°, 30°, 45°, 60°, and 75°). The fatigue damage evaluation results indicate that the FIV fatigue damage is highly affected by the stagger angle and relative position of the two cylinders. The FIV fatigue behavior of the two unequal-diameter cylinders is quite different from that of the two identical cylinders. When the large cylinder is located upstream, its wake shielding effect on the small cylinder is predominant at small stagger angles, so both the IL and CF fatigue damage of the downstream small cylinder is significantly suppressed, especially when Vr ≤ 16.28. But as the stagger angle is increased, the wake-induced flutter can aggravate the fatigue damage of the downstream small cylinder. Moreover, the CF fatigue damage of the large cylinder is decreased at small angles due to the vortex-shedding frequency suppression of the small cylinder and the fatigue damage of the large cylinder in the IL direction is analogous to that of the isolated cylinder and is negligibly affected by the small cylinder. When the small cylinder is located upstream, the IL FIV fatigue damage of the downstream large cylinder is approximately unaffected due to the narrow wake of the small cylinder. However, the CF fatigue damage of the large cylinder is greatly reduced at small stagger angles.

Flow-induced vibrations of two staggered cylinders with a moderate spacing and varying incident angles at subcritical Reynolds numbers

Two-degree-of-freedom (2DOF) flow-induced vibrations (FIV) of a stagger-arranged circular cylinder pair with a constant centre-to-centre spacing ratio P/D=6.0 and varying incident angles (α) are numerically studied. The two cylinders can oscillate freely in both the in-line and cross-flow directions. The Reynolds number range Re=1470–10320 belongs to the subcritical flow regime. Two identical cylinders with mass ratios of m∗=2.6 and mass-damping parameters of m∗+Caζ=0.013 are adopted in the present simulation. Four incident angles ranging from 0° to 90° with an increment of 30° are considered to investigate the effect of α on the FIV responses, hydrodynamic characteristics and wake patterns of the two cylinders at this specific P/D. For small incident angles (i.e., α=0° and 30°), the influence of Cylinder II on the FIV of Cylinder I is almost negligible. In contrast, owing to the complicated wake–cylinder and wake–wake interactions, large-amplitude vibrations are found for Cylinder II at high reduced velocities (Vr). The wake patterns suggest that the periodic vortex coupling arisen from the synchronous movements between the upstream vortices and Cylinder II contributes to maintaining a vibration mode that resembles wake-induced flutter (WIF). For large incident angles of α=60° and 90°, the two cylinders oscillate similarly and their response amplitudes experience decline at lower Vr values. The analyses of the wake characteristics demonstrate that the restrained wake widths and the stronger proximity interference between the two cylinders lead to the vibration attenuation.

Flow-induced vibration analysis of two unequal-diameter flexible cylinders in staggered configuration from a hydrodynamic force perspective

This paper presents an experimental study on the hydrodynamic features of two unequal-diameter flexible cylinders in staggered arrangement subject to flow-induced vibration (FIV). Five position angles (α = 15°, 30°, 45°, 60°, and 75°) are defined between the tandem (α = 0°) and side-by-side (α = 90°) cases with the centre-to-centre spacing being fixed at P = 6.0d, where d is the diameter of the small cylinder. The time-varying strains obtained from the experimental tests are converted into displacements, and then the hydrodynamic force coefficients of the two cylinders are derived using an inverse analysis method. It is found that the variation trends of the cross-flow (CF) hydrodynamic coefficients with the reduced velocity do not demonstrate qualitative differences from those of their isolated counterparts except in the mode transition regions. The downstream cylinder experiences stronger influence from the upstream cylinder on its CF hydrodynamic characteristics especially at small α. The more frequent mode transitions in the in-line (IL) direction lead to the multi-frequency responses associated with changes in the IL hydrodynamic coefficients. The typical wake-induced flutter (WIF) phenomenon is observed when Vrs = 5.01, 8.77 and 16.28–25.05 for the downstream small cylinder in Case I and α = 75°. The influence of the characteristics of the WIF phenomenon on the hydrodynamic force is evident because of the nonexistence of the high-frequency components in the oscillations.

A study on the FIV hydrodynamic force coefficients of two staggered flexible cylinders via an inverse method

The flow-induced vibration (FIV) of two cylindrical structures may cause serious fatigue damage, which is a major concern in engineering fields. Few studies have focused on the FIV hydrodynamic characteristics of two staggered flexible cylinders. In this paper, the hydrodynamic coefficients of two staggered flexible cylinders are calculated using the structural responses acquired from the model tests. The cross-flow (CF) spacing ratios are set as 2, 3, 4 and 6, and the in-line (IL) spacing ratios are set as 4, 6, 8; thus, a total of twelve cases are considered. The fluctuating forces in the CF and IL directions are reconstructed using a finite element model. The forgetting factor least squares method is employed to decompose the CF/IL fluctuating force into lift/varying drag and CF/IL added mass force. In the CF direction, the hydrodynamic coefficients of the two staggered cylinders are significantly different from those of the single cylinder in the mode switch regions. The CF vibrations of the two staggered cylinders enter the higher-order mode earlier, resulting in an increase in the CF added mass coefficients. The downstream cylinder has larger lift coefficients due to the vortex shedding effect of the upstream cylinders. In the IL direction, the upstream cylinder has lower IL added mass coefficients due to the higher-order harmonics in the IL direction. The varying drag coefficients of the two staggered cylinders significantly differ from those of the single cylinder owing to the subharmonic frequency components. A hybrid aspect of wake-induced flutter and vortex-induced vibration is observed in the vibration of the downstream cylinder.

Response of Two Unequal-Diameter Flexible Cylinders in A Side-by-Side Arrangement: Characteristics of FIV

Till now, little information is available on the flow-induced vibration (FIV) of multiple flexible cylinders with unequal diameters. Some FIV characteristics of unequal-diameter cylinders can be predicted based on the knowledge of equal-diameter cylinders, while there are still other features remaining unrevealed. In this paper, the FIV characteristics of two flexible cylinders with unequal diameters arranged side-by-side are experimentally investigated. The diameter ratio of the small cylinder (Small Cyl.) to the large cylinder (Large Cyl.) is nearly 0.5. The aspect ratios and mass ratios of the two flexible cylinders are 350/181 and 1.90/1.47, respectively. The centre-to-centre spacing ratio in the cross-flow (CF) direction is kept constant as 6.0 and the two cylinders can oscillate freely in both the CF and in-line (IL) directions. The towing velocity varies from 0.05 m/s to 1.00 m/s. The dominant modes and frequencies, CF and IL displacement amplitudes and response trajectories are discussed. Compared with the case of two identical cylinders in our previous study, the FIV responses demonstrate some similarities and differences. The similarities are as follows. Both cylinders exhibit multi-mode vibration features and they interact with each other. Meanwhile, the IL FIV shows a more complex behaviour than that in the CF direction. The difference is that as the diameter of one cylinder is increased, the effect on the smaller cylinder becomes more significant. For Large Cyl., the FIV response is similar to its isolated counterpart, which indicates that Small Cyl. has a negligible effect on the FIV of the larger one. Whereas Large Cyl. perplexes the FIV of Small Cyl. during the vibration process. The spacing would change when both cylinders are oscillating. Proximity interference between the two cylinders and wake shielding effect of the Large Cyl. may occur. The dominant frequencies of Small Cyl. are reduced and the wake-induced flutter of Small Cyl. is observed from the response trajectories at different measuring points.

Flow-induced vibration of two staggered flexible cylinders with unequal diameters

The flow-induced vibration (FIV) of multiple flexible cylinders typically involves multi-mode and complex flow-structure interactions. The FIV characteristics of two staggered flexible cylinders with unequal diameters are experimentally studied in this paper. The two flexible cylinders have aspect ratios of 350 and 181 and mass ratios of 1.90 and 1.47 and are free to oscillate in both the cross-flow (CF) and in-line (IL) directions. According to the relative positions of the two cylinders, 10 different staggered arrangements are studied (P/d = 6, where P is the centre-to-centre spacing and d is the small cylinder diameter; α = 15°, 30°, 45°, 60°, and 75°, where α is the position angle of the downstream cylinder). The multi-mode FIV behaviours of the two staggered flexible cylinders with unequal diameters were investigated with respect to various aspects, e.g., strain and displacement responses, dominant frequencies and dominant modes. Due to the presence of the small cylinder, the CF vibration of the large cylinder was suppressed at α = 15° and 30° when the large cylinder was upstream or downstream. Furthermore, the CF and IL response amplitudes of the small cylinder were significantly enhanced by flow interactions at various position angles. When the small cylinder was located downstream, the FIV behaviours were most greatly affected at small and large position angles (α = 15° or 75°). In addition, the wake-induced flutter phenomenon was observed in the IL FIV response when the small cylinder was downstream. However, this phenomenon was not observed when the large cylinder was located downstream.

Flow-induced vibration of two identical long flexible cylinders in a staggered arrangement

The effects of spatially staggered arrangements on the flow-induced vibration (FIV) characteristics of two identical flexible cylinders were experimentally studied in a towing tank. Flexible cylinders with an aspect ratio of 350 and a mass ratio of 1.90 were free to vibrate in both the cross-flow (CF) direction and the in-line (IL) directions. During the experiments, the towing velocity varied from 0.05 m/s to 1.0 m/s with an interval of 0.05 m/s, corresponding to Reynolds numbers in the range of 800–16,000. A total of twelve different staggered arrangements for two-cylinder models were selected (S/D=2, 3, 4, 6 and T/D=4, 6, 8, where S and T are the center-to-center separation distances between the two cylinders in the CF and IL directions and D is the cylinder diameter). The experimental results show that the multi-mode vibrations of both flexible cylinders are influenced by the complicated flow interference. The Strouhal number St is sensitive to the CF spacing ratio (S/D) for both flexible cylinders. The displacements of the upstream cylinder are larger than those of the isolated cylinder due to proximity interference. The dynamic characteristics of the downstream cylinder are significantly affected by the wake interference. Wake-induced flutter can act on the downstream cylinder, leading to a large IL displacement and an elliptical motion trajectory. Two prominent frequency components are observed in the IL displacement response. The higher frequency component is related to the vortex shedding of the downstream cylinder. The lower frequency component, which approximates to the CF dominant frequency, is typical of wake-induced flutter.

Aerodynamic interference between the cables of the suspension bridge hanger

The hangers of long-span suspension bridges are significantly prone to wind-induced vibrations due to their light mass, low frequency, and small structural damping. However, the underlying mechanism of the hanger vibration is not clearly clarified yet. To study the aerodynamic interference between the cables of the hanger, which is a possible mechanism for the hanger vibration, a series of wind tunnel tests were carried out to measure the mean aerodynamic drag and lift coefficients of a leeward cylinder. Then, the motion equations governing the vibration of leeward cable were derived based on the quasi-steady assumption. The numerical results show that large-amplitude vibrations of the leeward cable will occur in the region of 1 ≤ | Y| ≤ 3, where Y is a non-dimensional vertical coordinate normalized with the diameter of the cylinder. It appears that the stable trajectory of the leeward cable is ellipse, and trajectory is clockwise above the center line of the wake, whereas anti-clockwise below the center line of the wake. An important finding is that the frequency of the stable vibration of the leeward cable is slightly smaller than its natural frequency, which implies that a negative aerodynamic stiffness might arise. The time histories of the aerodynamic stiffness and damping forces on the leeward cable were identified from the numerical results. It seems that there is always a positive work done within a period by the aerodynamic stiffness force, whereas a negative work by the aerodynamic damping force. The response characteristics of the leeward cable of the hanger of suspension bridge obtained in this study are identical with those of the wake-induced flutter widely discussed for the power transmission line. This implies that wake-induced flutter theory could well illustrate the underlying mechanism of the aerodynamic interference effects on the hangers of a suspension bridge.

Numerical Investigation of Flow-pattern and Flow-induced Noise for Two Staggered Circular Cylinders in Cross-flow by LBM

The flowfield behind two cylinders and flow-induced noise generated from the cylinders in various arrangement are numerically investigated based on the finite difference lattice Boltzmann model with 21 velocity bits, which is introduced a flexible specific heat γ to simulate diatomic gases like air. In an isolated cylinder with two type of mesh, some flow parameters such as Strouhal number S t and acoustic pressure ? p simulated from the solution are given and quantitatively compared with those provided the previous works. The effects of the center-to-center pitch ratio L cc /d=2.0 in staggered circular cylinders as shown in Fig. 1 and angles of incidence α=30°(T cc /d=0.5), 45°(T cc /d=0.707) and 60° (T cc /d=0.866), respectively, are studied. Our analysis focuses on the small-scale instabilities of vortex shedding, which occurs in staggered arrangement. With the results of drag C d and lift C l coefficients and vorticity contours, the mechanisms of the interference phenomenon and its interaction with the two-dimensional vortical structures are present in the flowfields under Re ≤ 200. The results show that we successively capture very small pressure fluctuations, with the same frequency of vortex shedding, much smaller than the whole pressure fluctuation around pairs of circular cylinders. The upstream cylinder behaves like an isolated single cylinder, while the downstream one experiences wake-induced flutter. It is expected that, therefore, the relative position of the downstream cylinder has significant effects on the flow-induce noise, hydrodynamic force and vortex shedding characteristics of the cylinders.

FLOW-INDUCED OSCILLATIONS OF TWO CYLINDERS IN TANDEM AND STAGGERED ARRANGEMENTS

A stabilized finite element formulation is employed to study flow-induced oscillations of a pair of equal-sized cylinders in tandem and staggered arrangement placed in uniform incompressible flow. Computations, restricted to 2-D, are carried out for Reynolds numbers 100 for various values of the structural frequency of the oscillator. The cylinders are separated by 5·5 times the cylinder diameter in the streamwise direction. For the staggered arrangement, the cross-flow spacing between the two cylinders is 0·7 times the cylinder diameter. In this arrangement, the downstream cylinder lies in the wake of the upstream one and therefore experiences an unsteady in-flow. Since the spacing between the two cylinders is beyond the critical value for proximity interference, it is expected that the upstream cylinder behaves like an isolated single cylinder, while the downstream one experiences wake-induced flutter. The Re=100 flow leads to a very organized wake and large amplitude motion is observed for the downstream cylinder. The trajectory of the upstream cylinder resembles a figure-of-eight. The downstream cylinder shows a similar behaviour for the tandem arrangement. However, for the staggered arrangement, the trajectory of the rear cylinder resembles a tilted oval. Soft lock-in is observed in almost all the cases.

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.

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.

Solution of the nonlinear equations for wake-induced flutter via the Krylov and Bogoliubov method of averaging

Using the Krylov & Bogoliubov (1949, Introduction to Non-Linear Mechanics, Princeton: Princeton University Press) method of averaging, solutions have been obtained to the previously developed nonlinear equations of motion for wake-induced flutter of one circular cylinder in the wake of a fixed similar cylinder. The results obtained have been compared with previously obtained ones via numerical integration of the same equations; the agreement is excellent, but there is a considerable saving in computational time using the Krylov and Bogoliubov method. It has been reconfirmed that, when the windward cylinder is held fixed, increasing mechanical damping is not a very efficient way of decreasing the amplitude of oscillation of the leeward cylinder. Provided the same set of aerodynamic force coefficients are employed the nonlinear analysis predicts essentially the same lower flutter boundary as that obtained via a linearized analysis. There is, however, evidence that the leeward cylinder acts as a hard oscillator, resulting in a considerable difference in the upper flutter boundary obtained via the nonlinear analysis vis-a-vis the linearized theory.

On the efficacy of mechanical damping and frequency detuning in alleviating wake-induced flutter of overhead power conductors

A nonlinear, quasi-steady analysis of the wake-induced instability of a leeward conductor in the wake of a second identical windward conductor has been developed. In addition to the nonlinear analysis, the aerodynamic forces acting on the conductors are linearized, and a linear form of the equations of motion formulated. Both the linear and nonlinear analyses account for motion of the windward conductor. Although the quasi-steady assumption is employed, the time delay between flow leaving the windward conductor and influencing the aerodynamic forces on the leeward conductor is accounted for, as is the retardation of wake flow approaching the leeward conductor. Accounting for the above “unsteady effects” results in considerable differences in the equations of motion vis-a-vis the fixed windward conductor analysis; in particular, they change significantly the stability of the two conductors and induce a new form of instability—a negative damping instability. This new instability mechanism results in some configurations of the two conductors, which the fixed windward conductor analysis predicts to be stable, being unstable. Using both the linear and nonlinear analyses, the individual effects of mechanical damping and frequency detuning are investigated, the particular areas of interest being the threshold wind speed at which instability will occur and the limit cycle amplitudes once instability has occurred. In general, increasing mechanical damping, in both the in-plane and out-of-plane modes, is the most effective way of enhancing the stability of the system; it can, in some circumstances, raise the threshold velocity necessary to cause instability, and reduce the limit cycle amplitude of an unstable system. For the frequency coalescence instability mechanism, detuning the modes can be beneficial, but there are specific combinations of modes for which frequency detuning has little effect on either the minimum wind speed for instability to occur, or on the limit cycle amplitudes. Thus, for transmission line bundles, in which a multitude of modes may coalesce, it is unlikely that frequency detuning will be particularly beneficial. Also, not surprisingly, frequency detuning has almost no effect on the negative damping instabilities.

An investigation of the effect of mechanical damping to alleviate wake-induced flutter of overhead power conductors

A non-linear version of a previously developed linearized quasi-static aeroelastic analysis of a leeward conductor in the wake of a fixed windward conductor is developed. Using both this new non-linear analysis and the original linearised analysis, the effect of mechanical damping on the aeroelastic stability of the leeward conductor is investigated. The linearized analysis is used to investigate the effect of mechanical damping on the minimum wind speed necessary to cause instability; the non-linear analysis is used to investigate the effect of mechanical damping on the limit cycle amplitudes of the leeward conductor once it has become unstable. In general, it is found that mechanical damping does not have a significant effect on the stability of the leeward conductor, at least for the range of mechanical damping typically measured on transmission lines. This is so, irrespective of whether or not the damping is present in both the in-plane and out-of-plane modes which coalesce to form the unstable mode. However, damping was found to be particularly ineffective if it is present in the out-of-plane mode only, which is usually the case for twin conductor bundles. These results suggest that the damping provided by spacer-dampers is not particularly beneficial to the stability of overhead transmission lines, at least as far as wake-induced flutter is concerned.

On Wake-Induced Flutter of a Circular Conductor in the Wake of Another

This paper, which is an extension of [1], treats two-dimensional aeroelastic stability of two coupled conductors. It is found that the wake-induced flutter is symmetric with respect to the horizontal axis of the wake for all cases provided that the sign of the static coupling coefficient, ε = kxy/kxx, is changed. It appears that the spacer coupling ratio, K/kxx = Ω/ωx, is the most important factor in determining stability. For practical purposes, the system is almost always stable for K/kxx = Ω/ωx = 0.8, because the frequency ratio, κ = ωy/ωx, deviates less than ten percent from unity for a typical transmission line. On the other hand, within our range of interest, damping has little or no effect on the stability of coupled conductors. When the windward conductor is fixed, i.e., K = 0, then damping does influence the stability of the leeward conductor.

Two dimensional stability analysis of two coupled conductors with one in the wake of the other

For some years, two outstanding problems in wake-induced flutter, namely, flutter asymmetry and frequency coalescence, mystified both researchers and practicing engineers. These problems were clarified by using the Routh-Hurwitz stability criteria to study the flutter of the leeward conductor in the wake of the fixed windward conductor within the framework of quasi-steady aerodynamic theory in two previous papers [1,2]. This paper, which is a sequel to the previous ones, deals with the two dimensional stability of two coupled conductors with and without damping. It is found that, as in the case of a fixed windward conductor, the symmetry principle of wake-induced flutter still holds. In other words, the wake flutter is symmetrical with respect to the horizontal axis of the wake if there is a change of sign of the static coupling coefficient, ε = k xy k xx . It is also found that, within our range of interest, the system is always unstable when the spacer spring constant, K = 1·2 k xx and 1·4 k xx , where k xx is the horizontal spring constant of the conductor suspension system. It appears that the spacer coupling ratio, K k xx is the dominant factor in determining stability. For practical cases, the system is always stable for a spacer coupling ratio in the neighborhood of 0·64, because the frequency ratio, κ = ω y ω x , deviates less than ten percent from unity. On the other hand, within the range of interest here, damping has little or no effect on the stability of the coupled conductors. Nevertheless, it does have an influence on the stability of the leeward conductor when the windward conductor is fixed.

Two-dimensional stability analyses of a circular conductor in the wake of another

Routh-Hurwitz stability criteria are employed to study the flutter of the leeward conductor in the wake of the fixed windward conductor within the framework of quasi-steady aerodynamic theory. Two important points of wake flutter are clarified. (1) The symmetry principle of wake-induced flutter: in other words, the wake flutter is symmetrical with respect to the horizontal line if there is no static coupling in the x and y directions for the suspension of the leeward conductor. When coupled, the flutter symmetry is preserved, provided the sign of the static coupling is changed. This finding is supported by recent data and is contrary to previous results. (2) For the uncoupled case, vertical and horizontal frequency coalescence does not necessarily imply no oscillation. This is at variance with previous studies. Some highlights of the numerical results are as follows: (a) The region of instability roughly lies between 0 < |e| < 0.1 and 0.8 < κ < 1.2, where e and κ are the static coupling coefficient and frequency ratio, respectively. (b) The region of instability enlarges as |e| increases from zero to 0.08 when everything else is kept fixed. (c) For both subcritical and supercritical flow, the region of instability shrinks as d/c varies from 10 to 16, when all other factors are fixed. (d) The region of instability shrinks as the flow changes from subcritical to supercritical when everything else remains unchanged.

On Wake-Induced Flutter of a Circular Cylinder in the Wake of Another

Two-dimensional stability of leaward cylinder in the wake of fixed windward cylinder is studied within the framework of quasi-static aerodynamic theory at both subcritical and supercritical flow region. Routh-Hurwitz stability criterion is employed. Two important points are clarified: 1. The wake-induced flutter is symmetric with respect to the horizontal line if there is no xy (refer to Fig. 1) static coupling. When coupled, the wake symmetry preserves provided the sign of static coupling is changed. This finding is supported by recent experimental data [7] and is in contrary with previous results [4–6]. 2. For the uncoupled case, vertical and horizontal frequency coalescence does not necessarily imply no oscillation. This is in variance with previous studies [2, 3, 34]. The region of instability is related to four physical variables: vertical to horizontal natural frequency ratio κ = ωy/ωx = (Kyy/Kxx)1/2, static coupling coefficient ε = Kxy/Kxx, cylinder spacing to diameter ratio d/c and flow characteristics. Some highlights of numerical results [1] are as follows: (a) The region of instability roughly lies between 0 &lt; |ε| &lt; 0.1 and 0.8 ≤ κ ≤ 1.2. (b) The region of instability enlarges as |ε| increases from zero to 0.8 when everything else is kept fixed. (c) For both subcritical and supercritical flow, the region of instability shrinks as d/c varies from 10–16 when keeping all other factors fixed. (d) The region of instability shrinks as the flow changes from subcritical to supercritical when everything else remains unchanged.