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GALLOPING RESPONSE OF A CYLINDER WITH UPSTREAM WAKE INTERFERENCE

Journal of Fluids and Structures (2001) 15, 503}512
doi:10.1006/j#s.2000.0364, available online at http://www.idealibrary.com onGALLOPING RESPONSE OF A CYLINDER WITHUPSTREAM WAKE INTERFERENCEF. S. HOVER ANDM. S. TRIANTAFYLLOUDepartment of Ocean Engineering, Massachusetts Institute of Technology77 Massachusetts Avenue, Cambridge, MA 02139, U.S.A.(Received 3 September 2000, and in "nal form 15 November 2000)A compliantly-mounted rigid cylinder was towed at Re"3;10, 4)75 diameters behinda stationary leading cylinder of the same size. An in-line con"guration and a 12-degree
staggered arrangement each produced large-amplitude galloping responses, and an upward
extension of the frequency lock-in range to a reduced velocity of at least 17. The frequency
lock-in begins at nearly the same free-stream reduced velocity as a single cylinder, while a large
phase change in the lift force occurs at higher reduced velocities, which can be extrapolated
from the single-cylinder lock-in point. Force spectra indicate that shedding from the upstream
cylinder is completely una!ected by motions of the trailing cylinder. Furthermore, the motion-
coupled peaks suggest that only one lift force cycle and one or two drag force cycles occur per
oscillation, the latter depending on the o!set.2001 Academic Press1. INTRODUCTIONTHE WAKE INTERACTIONof parallel cylinders arises in many applications, including arraysof o!shore risers and moorings, and power transmission lines. For nonoscillating
cylinders, a number of distinct #ow regimes exist, which depend on the separation distance
S (Zdravkovich & Pridden 1977; Igarashi 1981). Small separation distances (S/D(2) limit
the reattachment of the leading cylinder's shear layer to the trailing one, and can lead to
bistable gap #ows. Quasi-steady recirculation cells, with coupled vortex formation, occur
for larger separation distances (2(S/D(4), and, "nally, vortex roll-up from the leading
cylinder occurs for S/D'4, where coupling is diminished. For the case of forced vibrations
of two tandem cylinders, a wake lock-in exists for the extreme motion phase angles of zero
and 1803, and lock-in of the wake to the motion occurs over a dramatically expanded region
of amplitude and frequency, for small S (Mahir & Rockwell 1996).In tests where both cylinders are compliantly-mounted, large-amplitude vibrations ofboth cylinders can occur when the separation is about 5}7D, and at least for lateral o!sets
up to 1)5D (Zdravkovich 1985). The vibrations are limited to a speci"c range of reduced
velocities, typically beginning at lower values than for a single cylinder.Bokaian & Geoola (1984) considered the case of a "xed leading cylinder, and a com-pliantly-mounted trailing cylinder. They report both vortex-resonance regions, i.e., motion
occurring only over a speci"c range of reduced velocity, and galloping instabilities, where
motions persist for high reduced velocities. These two types of responses can occur
independently or coalesce, depending on the separation distance. Vortex resonance occurs
alone for S/D'3, for both the in-line case and with o!sets of one diameter. In other tests at
much higher Reynolds number, galloping is suggested for S/D(7, diminishing as the
vortex-induced vibration is recovered for large S/D (Brika & Laneville 1999).0889}9746/01/040503#10 $35.00/02001 Academic PressFigure 1. Con"gurations for the in-line and o!set VIV tests.The tandem arrangement of two cylinders in mid-proximity (S/D"5}10) creates a re-duced natural shedding frequency, with comparison to a single cylinder. When one cylinder
vibrates freely, however, the wake frequency behind the leading cylinder is una!ected by the
trailing cylinder. The wake of the trailing cylinder still exhibits a lower Strouhal frequency,
consistent with reduced mean #ow calculations, until it reaches lock-in conditions. Hence,
on a reduced-velocity scale employing free-stream velocity, frequency lock-in occurs at
a higher value than for a single cylinder (Brika & Laneville 1999).In order to bridge some of these results, we consider here a compliantly-mounted rigidcircular cylinder in the wake of a stationary "xed cylinder of the same diameter. Locations
for the trailing cylinder are 4)75 diameters downstream (tandem), and 4)75 diameters
downstream, with a lateral o!set of one diameter (relative angle 123); see Figure 1. These
locations are near Zdravkovich's point of maximum response, and near the edge of the
wake interference zone, respectively.2. APPARATUS AND EXPERIMENTAL SETUPTests were conducted at the MIT Testing Tank facility, a 30;2)5;1)2 m still-water towingtank. We used rigid aluminum cylinders with diameter D"7)62 cm and span ¸"200 cm,
moving at constant speed horizontally; the downstream cylinder oscillates transversely to
the #ow (in the vertical direction). A view of the device from inside the tank is given in
Figure 2. The cylinders terminate with 0)2 cm gaps onto 31 cm diameter end-plates at each
end. The downstream cylinder is supported by a pair of three-axis piezoelectric load cells,
which in turn attach to a heaving structure that also supports the end-plates. This assembly
is positioned using a lead screw with 30 cm travel, driven by a brushless servomotor. The
uniform tow velocity for the tests corresponds with Re"3)05;10.We employed a robotic force-feedback loop as described in Hover et al. (1998). In thissystem, dynamic lift force measurements are injected into a real-time simulation of a com-
pliant structure, whose output drives the servomotor reference trajectory, and ultimately
the physical cylinder. The functional result is a cylinder that appears to be compliantly
mounted, even though its position is controlled very accurately with a servomotor. In the
present experiments, the simulated compliance consisted of a simple mass and spring.
The simulation mass M and sti!ness K can be arbitrarily speci"ed by the user, thus allowing
the variation of nominal reduced velocity <PL"2 ;/ LD at constant Reynolds number.Here, ; represents the steady towing speed of the carriage, andL"(K/M is theundamped natural frequency of the virtual structure.504F. S. HOVER AND M. S. TRIANTAFYLLOUFigure 2. The testing apparatus installed at the MIT Towing Tank, showing one 2 m cylinder, with end-platesand lower yoke assembly. Photo viewpoint is inside the tank.Intrinsic in the feedback loop is a correction for the component of measured force thatis due to the inertia in the material test cylinder; this mass is e!ectively replaced with M.
The data in the current work were obtained with the nondimensional mass ratio mH"
4M/D¸"3)0, and an e!ective damping ratio of about 4%. The nonzero damping ratiois an artifact of the closed-loop control system, which imposes some phase loss to achieve
smooth operation.The following coe$cients are calculated for each test: (i) average 1/10th-highest ampli-tude/diameter ratio A/D, with excursions taken from the mean position; (ii) mean dragcoe$cient, mean(CB); (iii) #uctuating drag coe$cient std(CB), the standard deviation of thedrag signal; (iv) total lift coe$cient amplitude CJ"2"F"/ D¸;, constructed as the Euclid-ean norm of components in phase with vibration velocity and position; (v) phase angle,between the oscillating lift force and the imposed motion, computed as an arctangent of the
lift coe$cients in phase with velocity and position.3. AMPLITUDE, DRAG, AND LIFT COEFFICIENTSWe give the amplitudes and force coe$cients in Figure 3, and the phase angle in Figure 4. In
each "gure, the calculated value is plotted against nominal reduced velocity for three
con"gurations: (a) in-line, (b) o!set, and (c) single cylinder.The case of a single cylinder is provided for comparison with previously published results,and re#ects a number of typical characteristics. First, as <PL increases, the amplitude ratioapproaches unity, then drops to a short plateau with A/D+0)75, and then a longer plateau
at A/D+0)50, before dropping again in the range of <PL"9}11. This type of A/D responseenvelope, i.e. localized on the <PL-axis, is termed vortex resonance in the sequel. Thehighest-amplitude part of the curve correlates with a steady drag coe$cient of about 3)0,
and the main plateau matches the zero-motion drag coe$cient of 1)25. A similar depend-
ence can be observed for the #uctuating component of drag, as well as lift coe$cient
magnitude.GALLOPING OF A CYLINDER WITH INTERFERENCE505Figure 3. Average one-tenth highest amplitude of motion (upper left), mean drag coe$cient (lower left),#uctuating lift coe$cient (upper right), and #uctuating drag coe$cient (lower right), as functions of <PL: (a) in-line;(b) o!set; (c) single cylinder.The phase angle for the single cylinder in free-vibration undergoes a rapid transition fromnear zero to near 1803, near <PL"6)0; this event generally marks a mode change from &&2S''- to &&2P''-type vortex shedding, evident in forced and free vibrations (Williamson & Roshko
1988; Brika & Laneville 1993). The phase points near 903, amid the transition, result from
averaging the phase calculated at each end of the cylinder. In this regime, we often observe
one large-amplitude end force with 03 phasing, and one small-amplitude end force with 1803
phasing.For both con"gurations involving interference, galloping occurs, without any clearsignatures of vortex resonance; the growth of motion with <PL is largely monotonic.Amplitude ratios become very high, reaching 1)9 in the in-line case and 1)3 in the o!set case.
It is likely that larger amplitudes could also occur at higher velocities.Steady drag coe$cients for the in-line and o!set cases with no vibration, i.e. low <PL, are0)35 and 0)80, respectively. These values are in reasonable agreement with Zdravkovich506F. S. HOVER AND M. S. TRIANTAFYLLOUFigure 4. Lift coe$cient phase as a function of <PL: (a) in-line; (b) o!set; (c) single cylinder.& Pridden (1977), and also with the mean wake analysis described by Huse (1992), for which
trailing cylinder CB values of 0)49 and 0)83 are generated from a base CB of 1)2. For thein-line case, the mean drag jumps from 0)35 to about 1)2, at <PL+6, although some scatterexists. In contrast, the o!set case has CB increasing to about 1)8 at <PLK5, before a gradualdescent to about 1)0 at the higher <PL. Roughly speaking, these upward jumps in meanCB for both con"gurations correspond to regimes of highly irregular amplitude ratios.Fluctuating drag for the o!set case reaches the same maximum value as the singlecylinder, although at a slightly higher <PL, and then declines gradually to a value near 0)4,markedly higher than the single cylinder. When the trailing cylinder is in-line, values are
much lower overall, and a local minimum at <PL+5)5 is in the same area of scatteredamplitudes noted above.Peak lift coe$cients CJ for the interference cases are much lower than for the singlecylinder, and both curves have an area of low value, again roughly in the regime of scattered
amplitudes. Noteworthy is the fact that each lift coe$cient has the nature of a vortex
resonance, in the sense that it decays at high velocities. Phase angle, however, indicates that
the main transition occurs well away from the usual range of <PL. The in-line con"gurationchanges phase at <PL"9}11, while it changes over a larger range, <PL"7}10, for the o!setcase. Several e!ects are likely. The in-line cylinder arrangement imposes a reduced mean#ow on the trailing cylinder, while in the o!set case, the trailing cylinder can additionallyemerge from the wake periodically, and is therefore exposed to higher velocities. Further
discussion of phase variation is given in Section 5.Interestingly, no unique features in any of the other coe$cient plots (Figure 3) signal thephase change with wake interference. For the single cylinder, amplitude and mean drag
both drop dramatically through the phase change.4. SPECTRAL DESCRIPTION OF RESPONSESIn Figures 5}7 are plotted the spectral content of the displacement, and lift and drag forces.
Each set of Fourier transform magnitudes has arbitrary scaling, and these are overlaid onGALLOPING OF A CYLINDER WITH INTERFERENCE507Figure 5. (a) Displacement and (b, c) force spectra of a cylinder centered 4)75D behind a stationary leadingcylinder.a vertical <PL-axis. The horizontal axis carries normalized frequency, scaled so that unitycorresponds with the undamped natural frequency of the structure. Dashed lines are also
given in each subplot, which follow the evolution of various peaks. Spectral peaks widen at
the higher reduced velocities, but this is only a remnant of the frequency nondimensionali-
zation, since all of the tests were performed at the same physical velocity.In Figure 7 (single cylinder), the lines classify features in the following ways. Thedisplacement (y) and lift (FW) peaks follow the single-cylinder shedding frequency1 (St"0)185) at low <PL, and lock on to the structural mode L at around <PL"6)5. Thelocked-in nondimensional frequencyW/ LK1)15 is typical for tests with mH"3, wherenegative added mass signi"cantly increases the net natural frequency. Lift force spectra
track the motion spectra closely, becoming quite small for <PL'10. The #uctuating drag508F. S. HOVER AND M. S. TRIANTAFYLLOUFigure 6. (a) Displacement and (b, c) force spectra of a cylinder located 4)75D behind a stationary leadingcylinder, with a one-diameter lateral o!set.force (FV) is signi"cant only near the onset of lock-in, and the peak here follows the secondharmonic ofW. The lift and drag forces thus indicate the simplest modes of VIV, involving2S and 2P types of wake.For the in-line experiments of Figure 5, oscillations are narrow-banded, and occur nearand just aboveL; the amplitude A/D grows throughout the range of reduced velocity.Lift has very signi"cant components at both the frequency of motionW and at 1,especially for high <PL. The shedding frequency plotted here is the same as that for the singlecylinder, and therefore likely relates to the incoming wake. Below the phase change at
<PLK10, the lift force is primarily at W; above the phase change, 1 dominates for a shortwhile, before a peak atW grows to similar magnitude. The phase transition occurs ata <PL where the lift force peaks are broad-banded and small.GALLOPING OF A CYLINDER WITH INTERFERENCE509Figure 7. Displacement and force spectra of a single cylinder.The drag signals present at least four traceable frequency peaks. The dominant forceoccurs at 2 W, but there are also harmonics at 3 W and 4 W. Additionally, we see signi"cantenergy atW# 1, for <PL'11, above the phase change. Other harmonics may be presentas well, although they do not appear to be as repeatable.When the trailing cylinder is moved to the o!set position, the main frequency of motionis still centered just above the structural mode. Lift has a strong component at the shedd-
ing rate throughout the range of <PL, although a component at W intensi"es at high <PL, asfor the in-line case. Drag has a dominant component atW, and a signi"cant secondharmonic. We observe #uctuating drag at1 and W# 1 also, the latter for the highestrange of <PL.5. DISCUSSIONBoth of the wake-interference systems considered show strong galloping, in the sense that
vibrations occur for a wide range of reduced velocities, and seem to increase with <PL. Theresult pertains to a separation distance smaller than that of Brika & Laneville (1999), whose
data arguably points to galloping below S/D"7. With regard to the other studies, we
employed a much higher Reynolds number than Bokaian & Geoola (1984). High wake
sensitivity to Re has in fact been noted for tandem stationary cylinders, but in somewhat510F. S. HOVER AND M. S. TRIANTAFYLLOUTABLE1Physical parameters of some wake-interference VIV studiesBlockageReferenceResponseRe/10¸/DratioSAmHZdravkovich (1985)*VR1}811)70)042231126Bokaian & Geoola (1984)-VR0)06}0)618)10)0531)078)42Brika & Laneville (1999)-VR, G0)51}2)7552)70)0180)78 K1000Current study-G3)226)30)0450)953)0* Leading cylinder compliantly mounted. VR: vortex resonance; G: galloping.-Leading cylinder stationary.closer proximity (Igarashi 1981). Compliance of the leading cylinder remains a dominant
factor in the response; Zdravkovich (1985) observed vortex resonance in the same geometry.
Table 1 lists these references, along with some of the physical parameters from the
experiments for comparison.Contrasting with Brika & Laneville (1999), the traceable e!ect of lower mean velocity onthe rear cylinder is not so much an increase in the free-stream <PL at which motion starts,but rather on the location of the phase change. An extrapolation of phase change location
from single-cylinder tests can be made using the same mean wake analysis as for the drag
coe$cient in Section 3 (Huse 1992). In terms of the drag coe$cients, the corrected reduced
velocity for a feature occurring at <HPL, in free-stream conditions, is<HPL [mean(CB)  /mean(CB)  ],where the superscript &&free'' indicates exposure to the free-stream velocity, and &&wake''
indicates wake #ow conditions. Since phase passes through 903 at <PLK6)5 for the singlecylinder, we then have the estimates <PLK10)2 for the in-line wake, and for the o!set wake<PLK7)8. These values are in good agreement with Figure 4, and suggest that the primarymechanism for <PL-dependent phase change stays remarkably intact in the wake. On theother hand, the trailing cylinders begin to oscillate at the same free-stream <PL as a singlecylinder, and at identical frequencies. With respect to a local <PL scale, the trailing cylinderlocks to the structural mode quite early, by <PL"3)8 for the in-line case. Thus, a phasechange and frequency locking to the structural mode cannot both de"ne lock-in in the usual
sense: wake interference causes these events to occur independently.Despite the variation of reduced velocity that marks the lift phase change, the componentfrequencies of lift evolve largely as expected. There are two main peaks at high <PL: 1,associated with the stationary leading cylinder, and anotherWK L, associated with theprimary motion near the structural mode. First, we may observe that no reduction of1 isevident; the same value, St"0)185, matches peaks throughout, for every con"guration.
Thus, the normal shedding mechanism from the leading cylinder is completely una!ected by
even the large-scale motions of the trailing cylinder.With regard to the second component of lift, the drag spectra in the in-line con"gurationindicate motion-coupled forcing consistent with two or four symmetric vortices per cycle.
The existence of a smaller third harmonic ofW probably pertains to an odd symmetry suchthat some forcing cycles may be sporadic. Nonetheless, the typical mechanisms for motion-
coupled shedding seem to be present. For the o!set case, drag peaks occurring at1 corre-late with one out of two shed vortices from the leading cylinder reaching the rear cylinder,
or at least a signi"cant imbalance in the pressure force from the pair. Note that, in theGALLOPING OF A CYLINDER WITH INTERFERENCE511corresponding lift spectra, we cannot discern directly whether one or two incoming vortices
per cycle cause lift, since they may now act on the same side of the trailing, o!set cylinder.
The dominant motion-coupled peak in drag nearW similarly indicates that the loading hasa large asymmetry, especially at high <PL.6. CONCLUSIONSLightly-damped cylinders in free vibrations 4)75D behind a stationary cylinder are capable
of large-scale galloping, helping "ll-in a gap between similar tests at larger spacing ratios,
and the case of dually compliant cylinders. Frequency lock-in occurs at a low reduced
velocity and remains through <PL"17, but the phase change, which typically accompaniesfrequency lock-in, occurs at higher speeds. This phase change is of the same nature as for
single-cylinder tests, and suggests the same fundamental mode transition.The spectra of the rear cylinder lift and drag forces allow a plausible description ofa simple motion-coupled forcing superimposed with a stable wake from the leading
cylinder. The former component in the in-line case indicates two or four symmetric vortices
per cycle throughout the range of <PL; for the o!set cylinder, a one-sided loading is evident.We plan DPIV and anemometry tests to verify these observations.ACKNOWLEDGEMENTSWe thank Mr F. Gillebo for helping to perform some of the tests and processing. This work
is funded by the O$ce of Naval Research (Ocean Engineering Division), under grant no.
N00014-95-1-0106, monitored by Dr T. F. Swean, Jr.REFERENCESBOKAIAN, A. & GEOOLA, F. 1984 Wake-induced galloping of two interfering circular cylinders. Journalof Fluid Mechanics 146, 383}415.BRIKA, D. & LANEVILLE, A. 1993 Vortex-induced vibrations of a long #exible cylinder. Journal of FluidMechanics 250, 481}508.BRIKA, D. & LANEVILLE, A. 1999 The #ow interaction between a stationary cylinder and a downstream#exible cylinder.Journal of Fluids and Structures 13, 579}606.HOVER, F. S., TECHET, A. H. & TRIANTAFYLLOU, M. S. 1998 Forces on oscillating uniform and taperedcylinders in cross#ow. Journal of Fluid Mechanics 363, 97}114.HUSE, E. 1992 Current force on individual elements of riser arrays. In Proceedings second InternationalOwshore and Polar Engineering Conference (ed. J. Chung). San Francisco: ISOPE.IGARASHI, T. 1981 Characteristics of the #ow around two circular cylinders arranged in tandem (Firstreport). Bulletin of the Japanese Society of Mechanical Engineers 24, 323}331.MAHIR, N. & ROCKWELL, D. 1996 Vortex formation from a forced system of two cylinders. Part 1:tandem arrangement. Journal of Fluids and Structures 10, 473}489.WILLIAMSON, C. H. K. & ROSHKO, A. 1988 Vortex formation in the wake of an oscillating cylinder.Journal of Fluids and Structures 2, 355}381.ZDRAVKOVICH, M. M. 1985 Flow induced oscillations of two interfering circular cylinders. Journal ofSound and Vibration 101, 511}521.ZDRAVKOVICH, M. M. & PRIDDEN, D. L. 1977 Interference between two circular cylinders; series ofunexpected discontinuities. Journal of Industrial Aerodynamics 2, 255}270.512F. S. HOVER AND M. S. TRIANTAFYLLOU

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