Combined Mass-damping Parameter Research Articles

Intrinsic features of flow-induced stability of a square cylinder

Vortex-induced vibrations and galloping of an elastically mounted square cylinder are investigated for cylinder mass ratio m* = 2–50, damping ratio ζ = 0–1.0, mass-damping ratio m*ζ = 0–50 and flow reduced velocity Ur = 1–80. We home in on the effects of m*, ζ, m*ζ, $({m^\ast } + m_{a\textrm{0}}^\ast )\zeta$ and $({m^\ast } + m_{ae}^\ast )\zeta$ on the critical reduced velocity Urc marking the onset of galloping, where $m_{a\textrm{0}}^\ast $ is the quiescent-fluid added mass ratio and $m_{ae}^\ast $ is the effective added mass ratio. Vibration responses, forces, vibration frequencies and added mass ratios are studied and discussed. The different branches of vortex-induced vibrations have different dependencies of $m_{ae}^\ast $ on Ur. The $m_{ae}^\ast $ in the initial branch is positive and drops rapidly with Ur, but that in the lower branch is negative and declines gently. In the galloping regime, $m_{ae}^\ast $ jumps from negative to positive at the onset of galloping, declining slightly with increasing Ur. Our results and prediction equations show that when ζ = 0, Urc is independent of m* for m* ≥ 5, albeit slightly higher for m* = 3. The latter is ascribed to mode competition. When ζ > 0, Urc linearly increases with increasing ζ. Detailed analysis substantiates that m*ζ or $({m^\ast } + m_{a\textrm{0}}^\ast )\zeta$ does not serve as the unique criterion to predict the galloping occurrence. Here, we propose a new combined mass-damping parameter $({m^\ast } + m_{ae}^\ast )\zeta$ in the relationship between galloping onsets and structural properties, which successfully scales all data of Urc at different m* and ζ values.

An analytical model for estimating the maximum energy harvesting efficiency from vortex-induced vibration

The present work presents an analytical model (AM) based on a nonlinear reduced-order model (ROM) for estimating the maximum efficiency (ηm) of energy harvesting from vortex-induced vibration (VIV) of a circular cylinder. The proposed AM was validated through comparisons against both the origin ROM and experiments over a wide range of parameters. Of interest, it provided a low-cost analytical approximation relating the maximum efficiency ηm to physical parameters. As inspired by the modified Griffin plot, the use of AM allowed the collapse of the maximum efficiency ηm as a function of the combined mass-damping parameter α. Of significance, the presented AM should be not only effective for estimating the maximum efficiency ηm but also useful to shed light on VIV dynamics.

Free vibration predicted using forced oscillation in the lock-in region

The rationale of using the results from the transverse forced oscillation of a body in the lock-in region to predict the corresponding free vibration is provided based on the mathematical analysis and verified through the numerical results at Re = 106 obtained from the immersed boundary-lattice Boltzmann method. It is also shown through mathematical analysis that when the structural damping is fixed, if the body mass and stiffness vary together following a particular relationship, the free motion will remain the same. With this conclusion, the damping ratio is redefined using the motion frequency of the body instead of the commonly adopted natural frequency of the body. As a result, it is shown that when a cylinder is in periodic free motion, its motion will remain the same if the following two conditions are satisfied: (1) the combined mass-damping parameter remains unchanged and (2) the variations of body mass and stiffness follow a particular pattern. In this sense, the sinusoidal free motion will rely only on the combined mass-damping parameter, even at the low mass region. This is different from the previous result based on the conventionally defined mass-damping parameter. Motion amplitude and frequency contours are plotted against the damping and stiffness components based on the results of forced motion. From these, the sinusoidal free motion results can be predicted and their physics is discussed.

A study on the energy transfer of a square prism under fluid-elastic galloping

In this paper, power transfer of an elastically mounted body under the influence of fluid-elastic galloping is analysed.The quasi-steady state model equations are first analysed to find suitable governing parameters. It is shown that, as well as Re, the system is a function of three dimensionless groups: a combined mass-stiffness parameter, Π1; a combined mass-damping parameter, Π2; and mass ratio, m⁎.Data obtained by numerically integrating the quasi-steady state equations show that for high values of Π1, the power extracted from the flow is a function of Π2 only. For low values of Π1, the power extracted is still a strong function of Π2, but is also a weak function of Π1. For all the cases tested, the power extracted was independent of the value of m⁎.These results are then compared to results of direct numerical simulations. It is found that Π1 has a much stronger impact on the power extracted than predicted by the quasi-steady state model. The error is shown to be an inverse function of Π1. The failure of the quasi-steady state model at low Π1 is hypothesised to be due to the stronger influence of vortex shedding, which is not accounted for in the quasi-steady model. Spectral analysis of the DNS cases at low Π1 shows a significant response at the vortex shedding frequency. The strength of the vortex shedding response is also shown to be an inverse function of Π1.Even though the quasi-steady state model does not accurately predict the power extracted, it does predict the parameter values at which maximum power transfer occurs reasonably well, and both the quasi-steady model and the direct numerical simulations show that this value is basically independent of Π1.

Experimental investigation of the forces and motion responses of two interfering VIV circular cylinders at various tandem and staggered positions

Experimental results are presented in the paper of two elastically supported rigid circular cylinders subjected to steady flows in a flume. The two cylinders were initially placed at various tandem and staggered positions with one in the wake of the other when subject to the steady flows. The in-line centre-to-centre distance varied from 2 to 5 diameters whilst the cross-flow distance from 0 to 2 diameters. The nominal Reynolds numbers were in the sub-critical regime and ranged from 1.12 × 104 to 5.52 × 104, and the nominal reduced velocities from 1.78 to 8.77. The damping ratio of the test set-up is low at 0.003 which gives a combined mass-damping parameter of 0.0046. Both the cylinders were free to respond in both the in-line and the cross-flow directions. The cylinder motion was measured simultaneously with the hydrodynamic loading in the two directions. It was found that the motion trajectories of the downstream cylinder show qualitative difference depending upon whether it is in tandem with the upstream cylinder or in the wake with a transverse offset. The VIV response of the downstream cylinder is dependent upon the reduced velocity of the upstream cylinder and its own reduced velocity based upon the actual mean wake velocity. The drag amplification of the downstream cylinder in the wake appears to be fundamentally different from that of a single VIV cylinder in isolation. Furthermore, unlike the two fixed cylinders in cross flow, the downstream cylinder undergoing VIV no longer experiences a marked non-zero mean lift. The upstream cylinder is largely unaffected by the downstream cylinder when the initial spacing is greater than 3 diameters. On the other hand, the motion response of and the fluid loading on the downstream cylinder are strongly influenced by the upstream cylinder in the spacing range tested.

Interference Between Two Stationary or Elastically Supported Rigid Circular Cylinders of Unequal Diameters in Tandem and Staggered Arrangements

Analysis of model test results was carried out to investigate the hydrodynamic interaction between pairs of fixed or elastically supported rigid cylinders of dissimilar diameters in a water flume. The two cylinders are placed with one situated in the wake of the other. The spacing between the cylinders ranges from 1 to 15 times the larger cylinder diameter. The Reynolds numbers are within the subcritical range. For the vibrating cylinders which are free to oscillate in both the in-line and the cross-flow directions, the reduced velocity ranges from 1 to 13 and the low damping ratio of the test setup at 0.006 gives a combined mass-damping parameter of 0.02. For the fixed cylinders, the downstream cylinder experiences a drag reduction and it was found that this drag reduction also depends upon the diameter ratio. The lift on the fixed downstream cylinder has the frequency components derived from the upstream cylinder's vortex shedding as well as from its own vortex shedding, and the relative importance of the two sources is influenced by the spacing between the two cylinders. This is reflected in the downstream cylinder's vortex induced vibration (VIV) response which appears to be dependent upon the actual reduced velocities of both the cylinders.

Some observations of two interfering VIV circular cylinders of unequal diameters in tandem

Analysis of model test results was carried out to investigate the hydrodynamic interaction between a pair of elastically-supported rigid cylinders of dissimilar diameters in a water flume. The two cylinders are placed in tandem with one situated in the wake of the other. The diameter of the upstream cylinder is twice as large as that of the downstream cylinder. The spacing between the two cylinders ranges from 1 to 10 times the larger cylinder diameter. The Reynolds numbers are within the sub-critical range. The cylinders are free to oscillate in both the in-line and the cross-flow directions. The reduced velocity ranges from 1 to 10 and the low damping ratio of the model test set-up at 0.006 gives a combined mass-damping parameter of 0.02. It is found that the lift on and the cross-flow motion of the downstream cylinder have the frequency components derived from the upstream cylinder's vortex shedding as well as from its own vortex shedding, and the relative importance of the two sources of excitation is influenced by the spacing between the two cylinders. The downstream cylinder's VIV response appears to be largely dependent upon the actual reduced velocity of the cylinder.

Vortex and wake-induced vibrations of a tandem arrangement of two flexible circular cylinders with near wake interference

Results showing the dynamic response of a tandem arrangement of two vertical high aspect ratio (length over diameter) and low mass ratio (mass over mass of displaced fluid) flexible cylinders vibrating at low mode number are presented in this paper. Two circular cylinder models were aligned with the flow, so the downstream or trailing cylinder was immersed in the wake of the leading one. Centre-to-centre distances from 2 to 4 diameters were studied. The models were very similar in design, with external diameters of 16 mm and a total length of 1.5 m. Reynolds numbers up to 12 000 were achieved with reduced velocities, based on the fundamental natural frequency of the downstream cylinder in still water, up to 16. The trailing model had a mass ratio of 1.8 with a combined mass-damping parameter of 0.049, whilst the corresponding figures for the leading cylinder were 1.45 and 0.043, respectively. The dynamic response of the trailing model has been analysed by studying cross-flow and in-line amplitudes, dominant frequencies and modal amplitudes. The dynamic response of the leading one is analysed by means of its cross-flow amplitudes and dominant frequencies and it is also related to the motion of the trailing cylinder by studying the synchronisation between their instantaneous cross-flow motions. Planar digital particle image velocimetry (DPIV) was used to visualise the wake. Different response regimes have been identified based on the type of oscillations exhibited by the cylinders: vortex-induced (VIV), wake-induced (WIV) or combinations of both.

Wake structures and vortex-induced vibrations of a long flexible cylinder—Part 2: Drag coefficients and vortex modes

Drag coefficients and vortex structures in the wake of a vertical long flexible cylinder vibrating at low mode numbers are presented in this paper. A model with an external diameter of 16 mm and a total length of 1.5 m giving an aspect ratio of about 94 was used to perform more than 100 runs in which Reynolds numbers ranged between 1200 and 12 000. Only the lower 40% of its length was exposed to the water current in the flume and applied top tensions varied from 15 to 110 N giving fundamental natural frequencies in the range from 3 to 7.1 Hz. Reduced velocities based on the fundamental natural frequency up to 16 were reached. The mass ratio was 1.8 and the combined mass-damping parameter about 0.05. The largest drag coefficients were found related to the largest x–y synchronised motions. Digital particle image velocimetry was used to investigate the vortex structures in the wake of the cylinder model. Two modes of vortex shedding were observed, depending on the response branch and the position along the length of the model at which the interrogations were performed.

A Framework for Flutter Clearance of Aeroengine Blades

A framework for flutter operability assessment, based upon a new set of similarity parameters, has been developed. This set consists of four parameters which embrace both the performance characteristics in terms of corrected mass flow and corrected speed, and the flight condition in terms of inlet temperature and density (or, equivalently, inlet pressure). It is shown that a combined mass-damping parameter, g/ρ*, novel in the field of turbomachinery aeroelasticity, can summarize the individual effects of mechanical damping, g, and blade mass ratio, μ. A particular selection of four nondimensional parameters, including g/ρ* and a compressible reduced frequency parameter, K*, allows for a decoupling of corrected performance effects from purely aeroelastic effects, for a given machine and a specific modeshape. This view of flutter operability is applied to the analysis of full-scale engine data. The data exhibits the trend that increasing K* and increasing g/ρ* have stabilizing effects, which is consistent with previous work in flutter stability. We propose that these trends hold generally, and apply the trends towards constructing a flutter clearance methodology, a test procedure which satisfies the requirements for comprehensive flutter stability testing.

MOTIONS, FORCES AND MODE TRANSITIONS IN VORTEX-INDUCED VIBRATIONS AT LOW MASS-DAMPING

These experiments, involving the transverse oscillations of an elastically mounted rigid cylinder at very low mass and damping, have shown that there exist two distinct types of response in such systems, depending on whether one has a low combined mass-damping parameter (low m*ζ), or a high mass-damping (highm*ζ ). For our low m*ζ, we find three modes of response, which are denoted as an initial amplitude branch, an upper branch and a lower branch. For the classical Feng-type response, at highm*ζ , there exist only two response branches, namely the initial and lower branches. The peak amplitude of these vibrating systems is principally dependent on the mass-damping (m*ζ), whereas the regime of synchronization (measured by the range of velocity U*) is dependent primarily on the mass ratio, m*ζ. At low (m*ζ), the transition between initial and upper response branches involves a hysteresis, which contrasts with the intermittent switching of modes found, using the Hilbert transform, for the transition between upper–lower branches. A 180° jump in phase angle φ is found only when the flow jumps between the upper–lower branches of response. The good collapse of peak-amplitude data, over a wide range of mass ratios (m*=1–20), when plotted against (m*+CA) ζ in the “Griffin” plot, demonstrates that the use of a combined parameter is valid down to at least (m*+CA)ζ ∼0·006. This is two orders of magnitude below the “limit” that had previously been stipulated in the literature, (m*+CA) ζ>0·4. Using the actual oscillating frequency (f) rather than the still-water natural frequency (fN), to form a normalized velocity (U*/f*), also called “true” reduced velocity in recent studies, we find an excellent collapse of data for a set of response amplitude plots, over a wide range of mass ratiosm* . Such a collapse of response plots cannot be predicted a priori, and appears to be the first time such a collapse of data sets has been made in free vibration. The response branches match very well the Williamson–Roshko (Williamson & Roshko 1988) map of vortex wake patterns from forced vibration studies. Visualization of the modes indicates that the initial branch is associated with the 2S mode of vortex formation, while the Lower branch corresponds with the 2P mode. Simultaneous measurements of lift and drag have been made with the displacement, and show a large amplification of maximum, mean and fluctuating forces on the body, which is not unexpected. It is possible to simply estimate the lift force and phase using the displacement amplitude and frequency. This approach is reasonable only for very low m*.

Forced-vibration study of the aeroelastic instability of a square-section cylinder near vortex resonance

Force measurements made on a square-section cylinder forced to oscillate in a direction normal to a stream are presented. These measurements are used to estimate the values that the combined mass-damping parameter of an equivalent spring-mounted cylinder should take if it is to perform oscillation at prescribed values of amplitude ratio and reduced velocity. The mass-damping parameter appears to be predicted well by the quasisteady theory of galloping when the reduced velocity is well above lock-in. Below resonance, the parameter is negative, contrary to the predictions of quasisteady theory, and this indicates why galloping is not usually observed. It is found from the forced-vibration measurements that at a given value of reduced velocity the combined mass-damping parameter may have the same value for more than one amplitude of oscillation, and this observation explains why a spring-mounted cylinder can vibrate at more than one amplitude. An attempt is made to fit the measurements of force and phase angle around vortex resonance with predictions of the nonlinear lift oscillator model proposed by Hartlen and Currie [5]. It is found that the phase angle between the lift and the cylinder displacement is not predicted well by the theory.