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VORTEX INDUCED OSCILLATIONS OFCYLINDERS AT LOW AND INTERMEDIATEREYNOLDS NUMBERS

Roberto Camassa, Bong Jae Chung, Philip Howard, Richard McLaughlin, AshwinVaidya

1 Department of Mathematics, University of North Carolina, ,Chapel Hill, NC 27599, USA,camassa@email.unc.edu

2 Department of Marine Sciences, University of North Carolina, Chapel Hill, NC 27599,USA, bjchung@email.unc.edu

3 Department of Statistics, University of North Carolina, Chapel Hill, NC 27599, USA,pdhoward@email.unc.edu

4 Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599, USA,rmm@email.unc.edu

5 Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599, USA,avaidya@email.unc.edu

Summary. We study the orientational behavior of a hinged cylinder suspended in a watertunnel in the presence of an incompressible flow with Reynolds number (Re), based on par-ticle dimensions, ranging between 100-6000 and non-dimensional inertia of the body(I∗) inthe range 0.1-0.6. The cylinder displays four unique features, which include: steady orienta-tion, random oscillations, periodic oscillations and autorotation. We illustrate these featuresdisplayed by the cylinder using a phase diagram which captures the observed phenomena asa function ofRe andI∗. We identify criticalRe andI∗ to distinguish the different behaviorsof the cylinders. We used the hydrogen bubble flow visualization technique to show vortexshedding structure in the cylinder’s wake which results in these oscillations.

Key words: Vortex Induced Oscillation, Autorotation, Vortex Shedding.

1.1 Introduction

This paper deals with a fundamental question of orientationof a symmetric rigid body in afluid. The orientation behavior of cylinder, for instance, shows several transitions, dependingupon the inertia of the fluid in which it is immersed and the inertia of the body. These include(i) steady state orientation, (ii) random oscillations, (iii) periodic oscillations and in some ex-treme cases, even (iv) autorotation.

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The question of orientation bodies in fluids dates back to Kirchoff [8] who examined thedynamics of falling paper. The steady state orientation of bodies with certain classes of sym-metries has been long known. A sedimenting cylinder, for instance, is known to fall with itsaxis of symmetry perpendicular to gravity in a Newtonian fluid when its length exceeds itsdiameterd. However, in the case of a disk, when the length of the cylinder is less than the di-ameter, the disk falls with its axis of rotation parallel to gravity. Several more recent systematicstudies have experimentally, theoretically and numerically explored the steady state dynamicsof falling bodies (see [5, 7, 9, 15] for instance) in the Stokes and low inertial regimes. Theorientational dynamics becomes even more interesting in the unsteady regime, vortex shed-ding effects become significant and gives rise to oscillations of the body. In the context ofsedimentation, several relevant studies have been conducted, both experimental and numeri-cal to document the highly nonlinear dynamics of disk like bodies, i.e. bodies whose aspectratios ( length to diameter ratio, denotedτ ) are much less than 1, which are typically repre-sent disks or flat plates. See for instance [2, 4, 17, 20] and references therein. It is seen that asedimenting disk or flat plate can exhibit (i) fluttering, (ii) tumbling and (iii) chaotic motions.Attempts have been made to classify these different phenomena by means of non-dimensionalparameters such as particle aspect ratio, reduced inertia,Froude number, Strouhal number andReynolds number.

The bulk of previous studies have focused upon translational motions of disks, vibrational mo-tions of bodies or response of disks in aerodynamic flows[2, 6, 22, 20, 19]. The literature on thefirst two of these studies is far too vast to be discussed here.The more relevant of these studiesis the last one. Willmarth et al. [20] have studied the free and forced oscillations of disks ofdiameters ranging from 15cms to about 30 cms in a wind tunnel with Reynolds numbers inthe range 68,000 - 636,000. The aspect ratio in these experiments were 0.014< τ < 0.125.The particles were seen to oscillate periodically with increasing fluid inertia and eventuallydisplaying autorotation at sufficiently large wind speeds.Perhaps the work that comes closestto our study is due to Mittal et al. [13] who perform some interesting two dimensional numer-ical simulations for uniform flow past a hinged plate which isfree to rotate about its centralaxis and compare it to the dynamics of falling bodies. They examine the effects of varyingReandI∗. The Reynolds number is defined here asRe =

U lν , whereU is the centerline velocity

in the absence of the body andl is the maximum of the length or diameter of the cylinder. Thesecond non-dimensional parameter, namely the reduced inertia, is define byI∗ =

Iρ f d5 , where

I is the moment of inertia of the body with respect to the symmetry axis andρ f stands for thefluid density. TheRe achieved in this experiment ranges from about 100-6000 while the valuesof I∗ typically range from 0.1-0.6. In the previous literature, the appearance ofautorotation isidentified in terms of critical values ofRe andI∗.

Our study is carried out in a horizontal water tunnel, whereby the effects of gravity do notmatter by hinging the cylinder. The advantage of this experiment is that it allows for verylong observation times when compared to the case of sedimentation where a falling body isrestricted by the height of the tank. We also wish to explore if, neglecting gravity and alsotranslational motions, makes the problem under investigation any different from the freefallexperiments. A rigorous quantification of the varying dynamics of the body is missing in theliterature, which explores for the most part, merely qualitative aspects of the phenomena. Weuse a wider range of aspect ratios in our study which has not previously been examined leadingus further into the three dimensional effects of vortex induced wake flows. One of our centralcontribution in this paper lies in the discussion of the effect of autorotation. Autorotation is a

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phenomena, primarily observed in aerodynamics and defined by Lugt [11] as:

...any continuous rotation of a body in a parallel flow without external sources ofenergy...

Autorotation is rather well studied in aerodynamic applications, it has not been documented inthe literature for three dimensional liquid flows. In this article, we fill this gap in the literature;we capture the phenomena of autorotation for symmetric, cylindrical bodies and also discusstheir dependence uponτ, Re andI∗.

Our specific objectives include: (i) construction of a phasediagram to classify the particlebehavior, (ii) obtaining dominant frequency of periodic oscillations and (iii) visualization ofthe vortex shedding structure. The paper begins with a briefdescription of the experimentalsetup and methodology. This is followed in section 3 with an analysis of the particle motionand also a discussion of the wake structure. The concluding section 4 summarizes the centralcontributions of this paper to the field of vortex induced vibrations and also briefly recountsthe additional work being carried out.

It must be said at the outset that the study is still in its preliminary stages. Several questionsregarding the experimental setup still need to be addressedwhich may contribute to errors inour observations. These include a more elaborate study of: (i) friction caused by the suspen-sion, (ii) the pump frequency of the water tunnel, (iii) the effect of varying the tension of thesuspension, (iv) flow profile in the water tunnel as a functionof increasing flow velocity and(v) the disturbance to the flow in the tank due to the presence on the hydrogen bubble setup.We are continuing to address these issues currently.

1.2 Experimental setup

The experimental setup consists of plastic cylinders of diameter (d) 0.635 cm and lengths rang-ing from 0.32cms-1.27cms. The aspect ratio,τ, therefore ranged from 0.5−2.0. The cylinderswere made ofABS, Lexan and Delrin (plastics) with densities 1.05g/cc, 1.18g/cc and 1.4g/cc,respectively. The cylinders were held at the center of a water tunnel (Engineering LaboratoryDesign Inc., Model 502) with range of flow rates between 0.1-1.0 fps at 0.5HP. The dimen-sions of the test section of the water tunnel are 6×6×18 inches. The cylinder was suspendedby means of a stainless steel wire of thickness 0.023cms passing through a hole of diameter0.04cms at the center of the tank. The cylinder was suspendedin such a way as to allow itto oscillate freely along the flow direction alone. Figure 1 shows a schematic of the generalsetup. The dynamics of the cylinder were recorded using a 72mm SONY HDR FX1 camerafitted with +6 diopter zoom lens (Hoya Inc.). The operationalspeed of the camera was set at30 frames per second and the resulting videos were analyzed using the Video Spot TrackerV.5.20 program [18].

Previous studies suggest that the specific motions displayed by the body depend upon thevortex shedding process in the wake of the body which is visualized using the hydrogen bubbletechnique. The setup involves a copper wire of diameter 0.01cms mounted vertically at adistance of 12cms from the cylinder as shown in figure 1. A copper rod was used as the anode.Four 9V batteries in parallel were used to generate the bubbles.

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1.3 Results

This section focuses on the analysis of the particle motion.Our analysis reveals that the cylin-drical particle displays the following types of motions:

• Steady Orientation (S): the particles position remains unchanged with time.

• Oscillations (O): defined to represent particle motion which displays time dependent fluc-tuations in orientation, but not in a systematic manner. Ourcriterion for defining a motionas being an oscillation is that the average fluctuation with respect to time for the particlebe at least 0.1 radians.

• Periodic Oscillations (P): particles oscillate in a steady periodic manner.

• Autorotation (A): represents particles that rotate completely (by an angle of 2π) and peri-odically around the axis of suspension.

Figure 2 displays some sample plots of angle versus time showing each of the cases discussedabove for differentI∗ andRe. The plots in figure 2 were made based upon analysis performedon the Spot Tracker program. The program follows the motion of a dark spot marked on thebody and computes thex andy coordinates of the dark spot in each frame. From this data, wecan evaluate the angle versus time. The Spot Tracker analysis was applied to all the cases (i.e.recorded movies) except the steady and autorotation case. The former of these cases was notanalyzed since there is no motion to speak of while focusing issues due the rapid speed of theautorotating bodies caused difficulties in the Spot Trackeranalysis.

Using the phase diagram adopted by Fields et al. [4], we create a diagram ofI∗ versusRefor the four different features observed in our experiments. The figure 1.3 shows the four fea-tures plotted as distinct points marked by the symbols X, filled circle, open circle and filledtriangle which stand for S,O,P and A respectively. To identify clear patterns in the phase dia-gram, we make a contour diagram using four different color schemes to distinguish clusteringof the various features (see figure 1.4). The plot shows four distinctive regions dependingupon the values ofRe andI∗. Our data indicates: (i) the steady behavior falls along regionsof lower valuesRe (ii) the oscillations predominantly fall in the regions above I∗ = 0.29 and1800< Re < 4000 (and possibly largerRe corresponding to the higherI∗ values), (iii) the fea-tures of periodic behavior is shown to occur approximately for 1600< Re and in an expandingI∗ region contained between 0.15−0.35 and finally, (iv) autorotation is densely packed in re-gions of largeRe, exceeding 3000 and for some intermediate values of 0.19< I∗ < 0.29. Tocharacterize the patterns more clearly, we would like to have a larger range of data with higherRe. Interestingly, the particles can display periodic oscillations even at relatively lowRe, aslong asI∗ is limited to a certain value.

Based upon figure 1.4, we can analyze how the frequency of the periodic oscillations are af-fected byRe andI∗. This analysis is performed by the Power Spectral Density method usingthe Matlab software which plots part of the power of the signal within some frequency bins.The results of this analysis are shown in the figure 1.6,1.5. The frequency chosen in these plotscorrespond to the maximum power in the spectrum and only for particles that exhibit periodicmotion. These dominant frequencies, denotedf , seem to lie in the range 0.7-5.2 Hz. In ourgraphs, we chose to work with the non dimensional parameter,given by f L

U , which can also

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be interpreted as the particle Strouhal number.

The figure 1.5 shows the dimensionless frequency versusRe while the figure 1.6 shows thedimensionless frequency versusI∗. A linear fit to the non-dimensional frequency is also dis-played on the plots as a dashed line but shows a very lowR2 value making it difficult todiscern any explicit correlations between the frequency and Re or I∗ at this stage. We needseveral more data at higherI∗ andRe values to be able to capture any possible trends. The nondimensional frequency values in figure 1.5 for all of these cases lie between 0.03 and 0.14 for1000< Re < 6000.

The shedding of vortices in the wake of bodies is a very well studied problem. The classicalKarman vortex patterns have been previously documented [21, 22]. It is also well establishedthat in the case of free falling bodies (or hinged bodies as inour case), the asymmetric vortexshedding gives rise to oscillatory behavior [2, 13, 4]. The shedding behavior merits some atten-tion although our observations in this regard remains very preliminary. The Strouhal numberSr =

f LU is used to quantify the vortex shedding phenomena, where,f represents the frequency

of the vortex shedding,L is the characteristic length which we take to be the maximum of thelength or diameter of the cylinder andU is the velocity of the fluid. The dependence ofSruponRe is well known for the case of spheres and circular cylinders [1, 14] and also knownto be sensitive to the geometry of the body. In the lowRe range, namely for 40< Re < 6000,the value of 0.17< Sr < 0.21.

Visualizations using hydrogen bubble were performed, bothfor the case of a fixed cylinderand also for a body free to oscillate, for sake of comparison.A visual comparison of the twocases is made in the figures 1.7,1.8 below (Re ≈ 3000,I∗ ≈ 0.3) for six different times. Thefirst of these, figure 1.7 shows a sequence of images showing variations in the wake structurewith time when the cylinder oscillates. Although details ofthe vortex shedding patterns dueto the largeness of theRe and the three dimensional effects are difficult to visualize, one canclearly see the variations in the wake. In the second figure 1.8 for a fixed cylinder, the wakestructure shows no dramatic changes over time in the field of view. A quantitative comparisonhas also been made. Using our flow visualization images, we are able to visually track theemergence of vortex patterns in the wake of a fixed cylinder and then determine its frequency.Similarly, we were, in a few cases, able to determine the vortex shedding frequency past anoscillating cylinder. Our objective in this case was to verify if the Sr number for the two cases(fixed and oscillating) was significantly different. Our estimates are summarized in the table1, below.

Table 1.1. A comparison ofSr versusRe for the case of fixed versus oscillating cylinders.

Re 568 946 13251514 3029 3408Sr f ixed 0.21 0.25 – 0.21 0.26 0.28Srosc – – 0.12 – 0.1050.108

Our analysis thus far reveals a significant difference in thevalues ofSr versusRe for thecase of the fixed cylinder when compared to the oscillating one. TheSr for the former case

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lie in the neighborhood of 0.21-0.28 as observed in the literature, however for the oscillatingcylinders, theSr is almost half of the previous values and seems to be much lesssensitive toRe. Further, theSr values, based on particle oscillation frequencies, reported in figure 1.6, liein the same range asSr due to the vortex shedding past the oscillating cylinders. To concludethis discussion, while our study of the vortex shedding process deserves further attention, thereseems enough evidence to suggest that the oscillatory behavior displayed by the particles aredriven by vortex shedding.

1.4 Discussion

Our experiments on hinged cylindrical bodies in a water tunnel reveals four different behav-ioral characteristics. We have characterized these orientational features of the body as a func-tion of the Reynolds number and reduced inertia. We present aphase diagram which localizesthe orientational features of the cylinders in distinct zones. Our analysis indicates that in therange explored, the transition from oscillation to autorotation is highly sensitive to the valueof Re andI∗. Autorotation lies in the approximate range 0.19< I∗ < 0.29 and forRe > 3000.Similarly, periodic oscillation can be said to effectivelylie in some region enclosed withinRe > 1000 and 0.15 < I∗ < 0.35. We make comparisons with the work of Willmarth et al.They report the oscillation amplitude increases withRe until the body eventually begins toautorotate. As in our experiments (see figure 1.4 ), they alsonote that cylinders with aspectratio in the neighborhood ofτ = 1, have a tendency to oscillate periodically or even autoro-tate sooner than others. Our experiments also reveal that inthe presence of an obstacle placeddownstream and close to the particle, the cylinder has a tendency to autorotate for values ofRe andI∗ at which it otherwise does not. This phenomena also merits future study.

The critical non dimensional parameters that we report herevary from those in the literature[2, 4, 12] since these studies are primarily for much smallerdisks or for thin filaments. Asummary of the different critical values in the literature can be found in the paper by Mittalet al. [13]. The two dimensional numerical results of Mittalet al. are performed for cylinderswith 0 < τ < 0.5, I∗ > 0.17 Re < 600 in order to keep the vortex structure essentially twodimensional. Their investigation also reveals a distinctive separation of the oscillation versusautorotation data based on some critical values ofI∗ andRe.

Our calculation of the Strouhal number indicates a marked difference in the values ofSr forfixed versus oscillating cylinders;Sr in the case of oscillations dropping to almost half of thefixed case. Further the ratioSrp/Srs, whereSrp is the particle Strouhal number andSrs is theshedding Strouhal number is nearly 1, indicating alock-on phenomena.

Our work at this stage being the only three dimensional studywith aspect ratios exceeding 1cannot be compared effectively to any other study in the literature. It needs to be pointed outthat our study is marred by some drawbacks. Our phase diagram, though contains several datapoints, is lacking in data in regions where perhaps some effective comparisons to the literaturecan be made, especially forτ << 1 andRe > 6000. More data for the power spectrum analysiswould yield better possibilities of discerning a correlation between the oscillating frequencyandRe or I∗. Errors, frictional for instance, coming from the suspension mechanism must bereduced. Also a quantification of the flow in the tank and also the vortex shedding is needed.We provide no quantitative data on the vortex shedding mechanism at this stage. However,a quantification of the shedding frequency is currently being looked into using a constant

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temperature anemometer. We hope to report these results anda detailed comparison of theshedding frequency to that of the oscillating bodies in a soon to follow article. It has beenpointed out [10] that bodies with sharp corners such as cylinders, display noticeably differentvortex shedding patterns compared to smooth bodies. A next step would therefore to repeatthese experiments for spheroidal bodies to study the effectof changing geometry. We are alsocarrying out three dimensional numerical simulations of this phenomena using the Chimeragrid method which is based on a finite volume approach.

Acknowledgement. The article is dedicated to Prof. Giovanni Paolo Galdi on theoscassionof his 60th birthday. We acknowledge the help of several folks in this project. Among them,Angele Freeman, Dr. Arvind Santhanakrishnan, Neal Johnson, Dr. Laura Miller and VivekMenon. RMM is partially supported by NSF DMS-030868 and RC ispartially supported byDMS-0509423 and NSF DMS-0620687. PH and AV are supported by an NSF Research Train-ing Grant, RTG NSF DMS-0502266.

References

1. Aachenbach, E., Vortex Shedding from a Sphere, J. Fluid Mech., 62 (2), 209-221, 1974.

2. Belmonte, A., Eisenberg, H. and Moses, E., From flutter to tumble: Inertial drag andFroude similarity in falling paper, Phys. Rev. Letters, 81,2, 345-349, 1998.

3. Clayton, B.R. and Massey, B.S., Flow visualization in water: a review of techniques, J.Sci. Intrum., 44, 2-11, 1967.

4. Fields, S.B., Klaus, M., Moore, M.G. and Nori, F., Chaoticdynamics of falling disks,Nature, 388, 252-254, 1997.

5. Galdi, G., P., On the Motion of a Rigid Body in a Viscous Fluid: A MathematicalAnalysis with Applications, Handbook of Mathematical Fluid Mechanics, ElsevierScience, Amsterdam, 653-791, 2002.

6. Hover, F.S., Tvedt, H. and Triantafyllou, M.S., Vortex induced vibrations of a cylinderwith tripping wires, J. Fluid Mech., 448, 175-195, 2001.

7. Howard, H.H., Joseph, D.D., and Fortes, A.F., Experiments and Direct Simulations ofFluid Particle Motions, 1992, Int. Video J. Eng. Research, 2, 17-24.

8. Kirchoff, G., 1869, Uber die Bewegung enines Rotationskorpers in einer flussigkeit, J.Reine Ang. Math. Soc., 71, 237-281.

9. Leal, L.G., 1980, Particle Motion in a Viscous Fluid, Ann.Rev. Fluid Mech., 12,435-476.

10. Lugt, H., Autorotation of an elliptic cylinder about an axis perpendicular to the flow, J.Fluid Mech., 99, 817-840, 1980.

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11. Lugt, H., Autorotation, Annu. Rev. Fluid Mech., 15, 123-147, 1983.

12. Seshadri, V., Mittal, R. and Udaykumar, H.S., Vortex induced oscillations of a hingedplate: a computational study, Proceedings of ASME, 4th ASMEJSME Joint FluidsEngineering Conference, July 6-10, 2003.

13. Mittal, R., Seshadri, V. and Udaykumar, H.S., Flutter, Tumble and Vortex InducedOscillations, Theoret. Comput. Fluid Dynamics, 17(3),165-170, 2004.

14. Okajima, A., Strouhal numbers of rectangular cylinders, J. Fluid Mech., 123, 379-398,1982.

15. T.W. Pan, R. Glowinski, Galdi G.P., 2002, Direct Simulation of a settling ellipsoid in aNewtonian fluid, J. Comp. Appl. Math., 149, 71-82.

16. Skews, B.W., Autorotation of rectangular plates, J. Fluid Mech., 217, 33-40, 1990.

17. Tanabe, Y. and Kaneko, K., Behavior of falling paper, Physical Review Letters, 73, 10,1372-1376, 1994.

18. Taylor, R.M., www.cs.unc.edu/cismm/download/spottracker/videos pottracker.html,2005.

19. Vandenberghe, N., Zhang, J. and Childress, S., Symmetrybreaking leads to forwardflapping flight, J. Flud Mech., 506, 147-155, 2004.

20. Willmarth, W.W., Hawk, N.E., Galloway, A.J. and Roos, F.W., J. Fluid Mech., 27, 1,177-207, 1967.

21. Williamson, C.H.K., Vortex dynamics in the cylinder wake, Annual Review of FluidMech., 28, 477-539, 1996.

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Fig. 1.1. A schematic of the experimental setup.

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Fig. 1.2. Angle versus time variations showing representative sample data for the cases of (a)oscillation and (b) periodic behavior.

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Fig. 1.3. A phase diagram showing where the different features displayed by the particles lieas a function ofI∗ andRe.

Fig. 1.4. A contour plot of the phase diagram showing where the different features displayedby the particles lie as a function ofI∗ andRe.

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Fig. 1.5. The non-dimensional frequency versus Reynolds number.

Fig. 1.6. The domainant non-dimensional frequency versus reduced inertia.

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Fig. 1.7. Flow visualization for vortex shedding past an oscillatingcylinder.

Fig. 1.8. Flow visualization for vortex shedding past a fixed cylinder.