Unsteady Forces Research Articles

Flow-induced vibrations of elastically-mounted C- and D-section cylinders

We numerically study the flow-induced vibration (FIV) response of an elastically-mounted cylinder in two-dimensional coordinates at Reynolds number of 100. In particular, the effects of shape of frontbody and afterbody are systematically investigated by varying the cylinder’s cross-section, keeping the mass ratio of 5 as constant. The following cross-sections are considered — circular, C-section, inverted D-section, D-section, and inverted C-section. We employ an in-house flow solver based on the sharp-interface immersed boundary method and the solver is one-way coupled with a forced harmonic oscillator equation with a single degree of freedom. We explain the FIV characteristics using displacement amplitude, spectral characteristics of displacement and force signals, and wake modes. Considering a circular cylinder as a baseline case, a modification in the shape of the frontbody from convex to flat to concave causes large amplitude vibrations. A D-section cylinder, which corresponds to the flat frontbody, shows a significant increase in the amplitude for a wide range of reduced velocity compared to the circular cylinder, explained by combined Vortex-induced vibration (VIV) galloping response. By contrast, a variation in the shape of the afterbody from convex to flat to concave results in reducing amplitude, implying VIV suppression. The suppression is explained by the reduction of unsteady pressure forcing during vortex shedding on the cylinder. We discuss wake structures and vortex shedding patterns as a function of reduced velocity for the cylinders and explain these signatures in terms of the respective FIV response. The fundamental insights reported here are potentially helpful for structural health monitoring and energy harvesting applications.

Experimental Investigation and Validation on Suppressing the Unsteady Aerodynamic Force and Flow Structure of Single Box Girder by Trailing Edge Jets

In the present investigation, a wind tunnel experiment was performed to evaluate the effectiveness of the trailing edge jets control scheme to mitigate the unsteady aerodynamic force and flow structure of a single box girder (SBG) model. The flow control scheme uses four isolated circular holes for forming the jet flow to modify the periodic vortex shedding behind the SBG model and then alleviate the fluctuation of the aerodynamic force acting on the test model. The Reynolds number is calculated as 2.08 × 104 based on the incoming velocity and the height of the test model. A digital pressure measurement system was utilized to obtain and record the surface pressure that was distributed around the SBG model. The surface pressure results show that the fluctuating amplitude of the aerodynamic forces was attenuated in the controlled case at a specific range of the non-dimensional jet momentum coefficient. The Strouhal number of the controlled case also deviates from that of the original SBG model. Except for the pressure measurement experiment, a high-resolution digital particle image velocimetry system was applied to investigate the detailed flow structure behind the SBG model to uncover the unsteady vortex motion process from the SBG model with and without the trailing edge jets flow control. As the jet flow blows into the wake, the alternating vortex shedding mode is switched into a symmetrical shedding mode and the width of the wake flow is narrowed. The proper orthogonal decomposition was used to identify the energy of the different modes and obtain its corresponding flow structures. Moreover, the linear stability analysis of the flow field behind the SBG model shows that the scheme of trailing edge jets can dramatically suppress the area of unsteady flow.

The Effect of Inflow Distortion on the Rotordynamic Characteristics of a 1400-MW Reactor Coolant Pump Annular Seal

The annular seal between stator and rotor substantively acts as a bearing that affects the rotordynamic characteristic of the turbomachinery rotor system. The rotor wake turbulence in a canned motor Reactor Coolant Pump (RCP) will lead to inflow pressure distortion at the annular seal entrance, thus further affecting the seal rotordynamic characteristics and threatening the stable operation of RCP. In order to obtain the seal rotordynamic coefficients, a transient numerical method applies the mesh deformation technique to simulate the multiple-frequency elliptical rotor whirling orbit model. The transient solutions were proposed to solve the unsteady reaction forces of annular seals at five excitation frequencies for each case. The inflow pressure distortion patterns were simplified as harmonic functions, including two important influence parameters that are impeller blades number m and pressure fluctuation ratio λ. The numerical results showed that with nonuniform time-averaged pressure distribution at the entrance of the annular seal in Case 2, the inflow distortion significantly affects the seal rotordynamic coefficients, while the rotational spinning speed in Case 3 can weaken the time-averaged nonuniformity and accordingly make a dent in the influence. Increasing impeller blades number m and pressure fluctuation λ both result in a sharp diminution of the negative stiffness Keff, as well as an obvious increase in the effective damping Ceff, which will strengthen rotor misalignment and system stability. In addition, the larger impeller blades number m and higher pressure fluctuation λ will make the effective damping Ceff more independent of the whirling frequency. These results provide theoretical guidance for the operation safety of RCP.

Low Reynolds number stall flutter of a linear cascade with large amplitudes at low reduced frequencies: Unsteady loads and flow features

In this paper, we report on an experimental investigation of stall flutter of a linear cascade undergoing large amplitude heave oscillations (≈ 10% of chord) at low reduced frequencies (k<0.1) and at low Reynolds number (Re≈50,000). Three blade incidence cases are studied representing light, moderate, and deep stall conditions of the cascade. Simultaneous measurements of energy transfer to the blade using a load cell, and flow field measurements using Particle Image Velocimetry (PIV) are carried out while the blades in the cascade are allowed to perform large amplitude heaving oscillations at the prescribed reduced frequency (k) and Inter Blade Phase angle (σ). The resulting normalized energy transfer values (CE¯) indicate that the cascade becomes unstable at low k values for a range of σ values with the range of σ and k within the unstable region increasing as the cascade is pushed deeper into stall. The energy transfer variation with k and σ, for a given blade incidence, are observed to fit the empirical model CE¯=a(k)+b(k)sin(σ+c(k)) with the terms a and c corresponding to quadratic and linear functions of k for all the incidence cases. At very low k values (k∼0.03), the quadratic term is small, and the energy transfer curves are observed to be broadly consistent with the trends reported by Corral and Vega (2016a), with the terms a(k) and c(k) being linear and b(k) being a constant. As the cascade is pushed gradually deeper into stall, the term b(k), which represents the influence of adjacent blades, is observed to go from being nearly a constant value with k for light stall, to a quadratic function of k at deep stall. More importantly, the deviation of b from being a constant is observed to occur at lower k values as the cascade goes deeper into stall. These results indicate that the additional quadratic terms need to be considered in the classical flutter based linear models for the case with deep stall and large oscillation amplitudes. The flow field measurements using PIV indicate the presence of a completely separated shear layer that is alternating between a state of being attached and detached to the blade surface during the oscillation cycle. The results show the shear layer phase (ϕs) with respect to the blade displacement to be close to the phase of the unsteady force (ϕ), thus helping to link the flutter behavior with the unsteady shear layer dynamics for stall flutter of a cascade at low Re.

A Modified Van Der Pol Oscillator Model for the Unsteady Lift Produced by a Flapping Flat Plate for Different Positions of the Rotation Axis

To understand the nonlinear interaction between unsteady aerodynamic forces and the kinematics of structures, we theoretically and numerically investigated the characteristics of lift coefficients produced by a flapping thin flat plate controlled by the rotation axis position. The flat plate was placed in a 2-D incompressible flow at a very low Reynolds number (Re = 300). We showed that the behavior of the unsteady aerodynamic forces suggests the existence of a limit cycle. In this context, we developed a Reduced Order Model (ROM) by resolving the modified van der Pol oscillator using the Taylor development method and computational fluid dynamics (CFD) solutions. A numerical solution was obtained by integrating the differential equation of the modified van der Pol oscillator using the fourth-order Runge–Kutta method (RK4). The model was validated by comparing this solution with the reformulated equation of the added mass lift coefficient. Using CFD and ROM solutions, we analyzed the dependency of the unsteady lift coefficient generation on the kinematics of the flapping flat plate. We showed that the evolution of the lift coefficient is influenced by the importance of the rotation motion of the Leading Edge (LE) or Trailing Edge (TE), according to the position of the rotation axis. Indeed, when the rotation axis is moved towards the LE, the maximum and the minimum values of the lift coefficient are proportional to the downward and upward motions respectively of the TE and the rotation axis. However, when the rotation axis is moved towards the TE, the maximum and the minimum values of the lift coefficient are proportional to the downward and upward motions respectively of the LE and the rotation axis.

The unsteady force response of an accelerating flat plate with controlled spanwise bending

The unsteady force response of an accelerating flat plate, subjected to controlled spanwise bending, is investigated experimentally. The flat plate was held normal to the flow (at an angle of attack of $90^{\circ }$ ), and it was dynamically bent along the spanwise direction with the help of internal actuation. Two bending directions were tested. In one case, part of the plate (denoted by flexion ratio) was bent into the incoming flow (the bend-down configuration). In another case, the plate was bent away from the flow (the bend-up configuration). We used two different aspect ratio ( $AR$ ) plates, namely $AR = 2$ and 3. Three acceleration numbers, namely $A_c = 0.57$ , 1.6 and 3.2 (corresponding to dimensional acceleration of 0.036, 0.1 and 0.2 m s $^{-2}$ , respectively) were tested with a fixed terminal Reynolds number (Re) of 18 000. For each acceleration number, three bending durations, namely 1.2, 2.4 and 3.6 s were implemented. The results indicate that the highest impulse was imparted by the highest bending rate (duration 1.2 s) during all three accelerations tested. We show that controlled spanwise bending can significantly change the unsteady force response by manipulating the inertial forces during a start-up manoeuvre. The unsteady forces depend on the vector sum of the forward acceleration and the bending acceleration of the plate. The unsteady drag was augmented when the plate was bent towards the incoming flow. The initial force peaks were significantly reduced when the bending direction was reversed. The development of the edge vortices from the flat plate was measured with the help of particle image velocimetry (PIV) at the 70 % and the 90 % span locations. The PIV measurements were also carried out at the midchord plane closer to the tip region to capture the growth of the tip vortex. The vorticity field calculated from these PIV measurements revealed that controlled bending contributed to a variation in the circulation growth of the edge vortices. During the bend-down case, the circulation growth was faster and the tip vortices stayed closer to the plate. This resulted in increased interaction with the edge vortex at the 90 % span. This interaction was more severe for $AR = 2$ . During the bend-up case, the growth of the edge vortex was delayed, but the vortex grew for a longer time compared with the bend-down case. Finally, a mathematical model is presented which correctly captured the trend of the force histories measured experimentally during both the bend-up and bend-down cases.

Modification of subcritical cylinder flow with an upstream rod

In the current research, a small circular rod is placed upstream of the main cylinder. This is used to change the incoming flow and, thus, impact the vortex shedding process of the wake as well as the resultant aerodynamic forces. The experiments are carried out in a wind tunnel with a subcritical Reynolds number of Re = 32 000. The sectional pressure distribution around the main cylindrical model is obtained to calculate unsteady aerodynamic forces. The experimental results show that the gap ratio markedly affects the aerodynamic forces exerting on the main cylinder. The optimal gap ratios for drag reduction are found to be G/D = 1.5 and 2.0, where the average drag and unsteady lift forces exerting on the main cylinder are suppressed. In addition to pressure measurements, we also apply the high-speed particle image velocimetry system to obtain flow characteristics of the cylinder wake as well as the gap flow. Two different modes of gap flow are found, i.e., the cavity mode and the wake impingement mode, and the transition value between them is around G/D = 1.5. Moreover, at the appropriate gap ratio, the wake vortex shedding pattern of the main cylindrical model changes from antisymmetric to symmetric.

Vortex Ring Formed by a Butterfly Wing and Dynamic Forces

Butterfly generates unsteady dynamic fluid force with a pair of vortex rings. In order to elucidate the flight mechanism of butterflies, it is necessary not only to clarify the vortex rings formed around the wings but also elucidate the characteristics of the unsteady fluid force generated by the flapping motion. By applying the momentum theory to the vortex rings formed during flapping motion, it is possible to estimate the force generated by flapping motion. In this study, we reconstruct a pair of vortex rings formed by flapping motion based on a small number of visualization results and elucidate dynamic fluid force characteristics. A part of estimation result of Lift estimated by the rough shape vortex rings generated by downstroke and upstroke do not match with the direct measurement result. But not only the peak value of estimation result of Lift is near with the direct measurement result quantitatively but also the value of estimation result between upstroke is a small negative as well as the direct measurement result. Although there is a phase difference between the estimation result and direct measurement result, there is a qualitative and quantitative agreement between estimation results and direct measurement result. From those results, butterfly generate the unsteady dynamic Lift between downstroke, especially at butterfly spread wing maximum (t/T = 0.40). After that, the unsteady dynamic Lift decrease and become 0 at upstroke.

Numerical and Experimental Study on Vertical Takeoff and Separation of the Compound Aircraft

The traditional technical solution to achieve vertical take-off of the fixed-wing aircraft is adding power components, which will inevitably increase the weight of the aircraft. This paper proposes a compound flight scheme to achieve vertical take-off for the fixed-wing aircraft, using the carrier aircraft with vertical take-off capability to carry the target aircraft. Compared with the vertical take-off performance of a single aircraft, the present scheme enables the target aircraft to have a larger carrying capacity and longer flight distance. In order to verify the feasibility of the current scheme, this paper first investigates the steady aerodynamic performance of the compound aircraft and the dynamical separation process by computational fluid dynamics simulation. The steady lift coefficient of the compound aircraft during the acceleration stage and the unsteady aerodynamic force during the separation stage are obtained and analyzed. Next, an unmanned aerial vehicle (with vertical take-off and horizontal acceleration capabilities) is designed specifically, and the flight experiments of the vertical take-off and separation processes are conducted. Successful separations are achieved many times in the flight tests, which validate the feasibility of the present scheme.

Aeroelastic Analysis of a Supercritical Airfoil with Free-Play in Transonic Flow

An investigation has been made into the nonlinear aeroelastic behavior of a supercritical airfoil (NLR 7301) considering free-play in transonic flow. Computational Fluid Dynamics (CFD) based on NavierStokes equations is implemented to calculate unsteady aerodynamic forces. A loosely coupled scheme with steady CFD and a graphic method are developed to obtain the static aeroelastic position. Frequency domain flutter solution with transonic aerodynamic influence coefficient is used to capture the linear flutter characteristic at different angle of attack (AoA). Time marching approach based on CFD is used to calculate the nonlinear aeroelastic response. The bifurcation diagram of pitching motion shows that as the airspeed increases, the limit cycle oscillation (LCO) appears then quenches, forming the first LCO branch; LCO occurs again and sustains until the divergence of the response, forming the second LCO branch. This phenomenon is essentially induced by the combined action of the static aeroelastic position caused by the camber effect of the airfoil and the S shape flutter boundary with respect to AoA.

Neural ODE to model and prognose thermoacoustic instability.

Thermoacoustic instability in a reacting flow field is characterized by high amplitude pressure fluctuations driven by a positive coupling between the unsteady heat release rate and the acoustic field of the combustor. In a turbulent flow, the transition of a thermoacoustic system from a state of chaos to periodic oscillations occurs via a state of intermittency. During the transition to periodic oscillations, the unsteady heat release rate synchronizes with the acoustic pressure fluctuations. Thermoacoustic systems are traditionally modeled by coupling the model for the heat source and the acoustic subsystem, each estimated independently. The response of the unsteady heat source, i.e., the flame, to acoustic fluctuations is characterized by introducing unsteady external forcing. The forced response of the flame need not be the same in the presence of an acoustic field due to their nonlinear coupling. Instead of characterizing individual subsystems, we introduce a neural ordinary differential equation (neural ODE) framework to model the thermoacoustic system as a whole. The neural ODE model for the thermoacoustic system uses time series of the heat release rate and the pressure fluctuations, measured simultaneously without introducing any external perturbations, to model their coupled interaction. Furthermore, we use the parameters of neural ODE to define an anomaly measure that represents the proximity of system dynamics to limit cycle oscillations and thus provide an early warning signal for the onset of thermoacoustic instability.

Unsteady forcing of turbulence by a randomly actuated impeller array

We investigate the unsteady forcing of turbulent flow in a well-stirred reactor using opposing arrays of pitched-blade impellers which randomly and independently reverse rotation. We systematically explore the dependence of the large-scale motions and the homogeneity and isotropy of the turbulence upon the forcing. We identify three dimensionless control parameters: the source fraction (the fraction of time spent in clockwise motion), the dimensionless forcing period and an impeller Reynolds number. We find the timescale of unsteady motion corresponds to the forcing period T, the average period of impeller reversal, independently of the impeller angular speed varOmega and source fraction. As in jet-stirred tanks, unsteady forcing substantially increases the unsteady kinetic energy, energy dissipation, integral length scale and Taylor microscale Reynolds number (R_lambda) and improves the homogeneity and isotropy of the flow, provided the source fraction is chosen optimally and the forcing period is sufficiently large (varOmega T > 10^3); impeller Reynolds number has a relatively small influence. The forcing period must be matched to angular speed: decreasing the forcing period below this threshold results in a less intense, more inhomogeneous turbulent flow. Spectra of two-point velocity increments demonstrate that unsteady energy injection is dominated by axial shear generated across impellers and becomes less prominent at smaller scales. However, even at R_lambda approx 354, the signature of this unsteady forcing can still be detected in near-dissipation-range statistics. These observations provide insight into optimisation of forcing and the mechanism of energy transfer when using unsteady forcing to generate turbulence in confined vessels.Graphical abstract

Wake response downstream of a spanwise-oscillating hemispherical turret

An experimental Pressure Sensitive Paint (PSP) study of the unsteady pressure fields downstream of and on an oscillating hemispherical turret at subsonic freestream Mach numbers between 0.3 and 0.6 and Reynolds numbers of about 2 million is presented. The oscillating turret consisted of a hemispheric shell, mounted on an aluminum rectangular plate and is designed to oscillate in the spanwise direction at a single frequency, which lies within the frequency range of the dominant wake motion. The resonant-based aero-elastic response of the turret to the flow resulted in intermittent turret oscillations. Pressure fluctuations in the wake were found to be less energetic for the oscillating turret and the size of the recirculating region was increased, compared to the stationary turret. POD-based modal analysis of the pressure fields revealed that the wake dynamics is characterized by global anti-symmetric and symmetric shedding and showed that the oscillating turret primarily suppressed the global anti-symmetric shedding. The analysis of the unsteady forces acting on the turret demonstrated that the oscillating turret reduces the spanwise component of the unsteady force, while keeping other force components mostly unchanged. Intermittent nature of the spanwise force, which is responsible for the intermittent motion of the oscillating turret, was observed and discussed.

Lock-in prediction for vortex-induced vibrations of a long hanged and weighted riser in internal fluid flow and external currents

The occurrence of lock-in, a local synchronization between the vortex shedding and cross-flow (CF) structural vibration frequencies, is investigated for a long flexible riser, of which the top is hanged and bottom is weighted and moveable in both the in-line (IL) and CF directions, conveying internal fluid flow (IF) and immersed in uniform, and linear shear currents (flows). Our established models are used to conduct a numerical simulation, in which unsteady hydrodynamic forces associated with wake dynamics are described using two van der Pol wake oscillators in both the IL and CF directions that are coupled with the riser. The riser is discretized into a string of spatial slender beam elements, and the wake oscillators are distributed on every immersed element nodes. MATLAB (R2014a) is used to establish a computational program, which is verified using existing experimental data. The lock-in frequency and location are captured by comparing the CF oscillation frequency of the wake model with the CF vibration frequency of the riser at every node of the finite element system when both coincide with each other at the frequency as well as location. The influences of the oncoming flow velocity (OFV), IF velocity (IFV) and bottom hanging weight (BHW) on the lock-in frequency, as well as the lock-in distribution along the riser axis are discussed. In UF, a high IFV (4.5 m/s) makes lock-in occurrence in riser’s “whole oncoming velocity region” in a single-frequency mode but a zero or low IFV (1.5 m/s) makes it null. In linear shear flow, a zero or low IFV (1.5 m/s) makes the lock-in occurrence in riser’s “low and middle oncoming velocity region” in a single-frequency mode and a high IFV (4.5 m/s) makes it in the “whole oncoming velocity region” in the multi-frequency modes, the lock-in frequency mode (but not the lock-in amount) is regardless of OFV and BHW in the context of this paper.

Investigation of aerodynamic instability vibration of rectangular cylinder based on energy transfer

Aerodynamic instability vibration in orthogonal direction of rectangular cylinder of depth to breadth ratio of B/D = 2 was calculated by fluid-structure interaction. The energy transferred from the fluid to the vibrating rectangular cylinder was characterized, and the aerodynamic instability vibration was investigated using energy map obtained from forced vibration calculation.The energy map represents the energy of various frequencies and amplitudes. The energy map was discussed comparing with unsteady aerodynamic forces, phase differences and energy of structural damping. The energy map distribution on the high wind velocity side of vortex induced vibration region is complicated. This is caused by the formation of different vortex patterns depending on the vibration amplitude, which indicates the so-called hysteresis of vortex induced vibration.The zero value on the energy map represents the amplitude of the aerodynamic instability vibration at various wind velocities. The free vibration of a rectangular cylinder was also calculated and the energy map was used to predict the amplitude of the aerodynamic instability vibration. The results showed that the amplitudes of the zero value in the energy map and the vibration amplitudes of the free vibration were in good agreement for both vortex induced vibration and galloping.

Effects of the stator prewhirl angle on the unsteady force under uniform and turbulent inflow for a pump-jet propulsor: a numerical study

ABSTRACT The spatial non-uniformity and temporal non-steadiness of the rotor inflow condition impact the unsteady force differently. The interaction between the rotor and the stator of a pump-jet propulsor can induce strong pressure pulsations and vibrations. Therefore, the effects of the stator prewhirl angle on the propulsion performance and the unsteady force of a pump-jet propulsor under uniform and turbulent inflow were numerically investigated in this study. The Shear-Stress Transport (SST) k–ω turbulence model was employed in conjunction with the sliding mesh technique to determine the propulsion performance and unsteady force under uniform inflow. The turbulence-generating grid and the Fourier synthesis methods were employed to produce inflow turbulence with a spatial flow structure and temporal pulsation. These methods were combined with Large Eddy Simulation (LES) to obtain the unsteady force under turbulent inflow. This research could help improve the stator design of pump-jet in terms of unsteady force spectra.

Modeling and Structural Analysis of Morphing Wing in UAVs

Morphing structure enables an aircraft to alter its configuration in maneuver to operate effectively in a given flight reign. A continuous wing is utilized to reduce drag as the wing deformation is approximated more closely. With such wing morphing of skin supported with underlying truss structure, they are able to control the unsteady aerodynamic forces. This project seeks to expand and unravel the capability of an aircraft through the application of a reconfigurable “morphing” wing. In this project, the structural model of the morphing wing consists of primarily wing components designed in solid works. Structural analysis of wing components has been written to minimize wing deformation under aerodynamic load using ANSYS. With fixed geometry, the wing is allowed to analyze Fluid Structure Interaction (FSI) in Fluent. Smart materials which carry aerodynamic loads such as shape memory alloys, piezoelectric actuators, and shape memory polymers have been discussed briefly. Along these lines, this paper fills in as an abstract for scientists and designers endeavoring to get to know the current progressions and advancements in morphing wing technology

Unsteady aerodynamic forces on a tapered prism during the combined vibration of VIV and galloping

Unsteady aerodynamic forces on a tapered prism during the combined vibration of VIV and galloping

Nonlinear dynamic response and bifurcation of asymmetric rotor system supported in axial-grooved gas-lubricated bearings

Dynamic characteristics of the asymmetric rotor system supported in axial-grooved gas-lubricated bearings are studied. In order to solve nonlinear dynamic response of rotor system effectively, a hybrid numerical model is established by coupling the motion equation of rotor with the rational function model of the gas film forces. The rational function model of the gas film forces of gas-lubricated bearing is established based on vector fitting theory. By using the hybrid numerical model, the repeated calculations of the unsteady Reynolds equation and gas film forces are avoided; the continuous rotor trajectory and the dynamic gas film forces can be calculated simultaneously; and for the rotor system supported in the same bearings, the computing cost can be saved effectively. The nonlinear dynamic responses of asymmetric rotor system supported in axial-grooved gas-lubricated bearings are investigated by trajectory diagrams, frequency spectrum, Poincaré maps, and time series. The bifurcations are analyzed by the bifurcation diagrams with different rotating speeds and mass eccentricities. The dynamic behaviors of the asymmetric rotor system appear complex nonlinear dynamic phenomenon and specific bifurcation characteristics.

Numerical Simulation and Modeling Convention of Unsteady Fluidelastic Forces of Tube Arrays

Abstract This paper presents a numerical investigation on the unsteady fluidelastic forces of tube arrays. The key focus is on the consistency between the unsteady fluidelastic force model and the quasi-steady model for tube arrays at large reduced flow velocities, as well as comparing two well-known conventions for the unsteady model. Two-dimensional unsteady Reynolds-averaged Navier–Stokes (URANS) simulations are used to prove that the viscous damping coefficients of Tanaka's convention approach their quasi-steady values as the reduced flow velocity approaches infinity, whereas the hysteretic damping coefficients of Chen's modified convention always approach zero and hence result in low-resolution data plots as the reduced flow velocity becomes large. The nonconstant viscous damping coefficients of Tanaka's experimental data at high reduced flow velocities (which motivated the introduction of Chen's modified convention) might be induced by a systematic identification error in the phase of the fluidelastic force. A row of three flexible cylinders is used as a numerical example to analyze the effect of systematic phase error on the predicted stability boundary of the fluidelastic instability. Although identical fluidelastic forces are simulated by using the two conventions, Tanaka's convention is recommended due to its compatibility with the quasi-steady theory and optimal resolutions of data plots over any range of reduced flow velocities.