Presented at the 3rd Int. Offshore & Polar Engng Conf., Singapore, 1993, 3, 715–720.

Two- and Three-Dimensional Simulations of Vortex-Induced

Vibration of a Circular Cylinder

H.M. BlackburnDepartment of Mechanical Engineering, Monash University,

Melbourne, Australia

G.E. KarniadakisDepartment of Mechanical and Aerospace Engineering, Princeton University,

Princeton, New Jersey, USA

AbstractNumerical simulations of the interaction between acircular cylinder and its wake during both forcedand free, vortex-induced, oscillation have been per-formed using a spectral element method in which thecomputational mesh was fixed to the cylinder andthe Navier-Stokes equations solved in this acceler-ating reference frame. Thus far, work has focussedon cross flow, rather than in-line oscillations; withforced oscillation the lock-in phenomenon was ob-served over a range of reduced velocities near critical,while for the freely-vibrating cylinder the amplitude-limiting phenomenon observed in experiments was re-produced. A comparison of free and forced oscilla-tion has been performed in which the forced oscilla-tion amplitude and frequency were set to match thoseachieved in free vibration; the forces exerted on thecylinder by the fluid were similar in each case. Ini-tial simulations were two-dimensional but the methodmay also be applied to three-dimensional flows by em-ploying a spectral element/Fourier representation.

Keywords: Circular cylinder lock-in spectral ele-ment vortex vibration.

1 Introduction

Vortex-induced vibration of slender circular-cylindri-cal structural elements is an important considerationin the design of many offshore structures, for both thefabrication/transport and operational phases. De-spite the relatively fundamental nature of the prob-lem, a comparatively small amount is known aboutthe nature of the fluid-structure interaction, and thedesign rules which exist (e.g., CIRIA 1978) are basedon a limited set of experimental results, such as thosereviewed by King (1977). Partly as a result of thelack of understanding of the phenomenon, apparentlyunrelated sets of design procedures are available for

structures immersed in flows of water (e.g., CIRIA)and air (e.g., ESDU 1985).

Various experimental techniques have been em-ployed in past approaches to the problem of pre-diction of response amplitudes in vortex-inducedvibration, employing both free-vibration testing(Feng 1968, King 1977) and experiments in whichforced oscillations (exclusively in either cross flow orstreamwise directions) were imposed (Bishop & Has-san 1964, Sarpkaya 1978, Blackburn 1992). The con-nection between the two has not been easy to estab-lish, but attempts have been made to predict free-vibration amplitudes on the basis of measurementsof forces obtained in forced oscillation tests (e.g.,Staubli 1983). The outcome of these attempts hasbeen uncertain, since, as noted by Bearman (1984),in no case has the same experimental setup beenused for both the forced oscillation and free-vibrationtests. This rather surprising fact is in part a resultof the difficulty in designing an experimental rig sat-isfactory for both tasks. The measurement of fluidforces acting on a freely-vibrating cylinder is also dif-ficult to arrange, and to our knowledge has only beenattempted once (Reid & Hinwood 1987).

Vortex shedding in flows past circular cylinders ex-hibits a great deal of similarity over a very wide rangeof Reynolds numbers, despite the variety of Reynoldstransitions which affect it. Likewise the basic pat-tern of fluid-structure interaction seems to be similarover the observed range of Reynolds numbers, judg-ing by the fact that the phenomena of frequency-sensitive wake switching in flows with forced cylin-der oscillation (Ongoren & Rockwell 1988) and self-excited vibrations near the critical reduced veloc-ity with amplitude-limiting behaviour as zero mass-damping is approached (Griffin 1992) appear to beuniversal.

Given the difficulty with physical experiments

Figure 1: Computational mesh for domain Ω em-ployed in the calculations, with dimensions conditionsindicated. Layer of elements closest to cylinder havethickness 0.1D.

and the apparent similarity of the phenomena overa range of Reynolds numbers, we have made acomputationally-based approach to the problem, em-ploying numerical simulations in which the cylindercould be forced to oscillate with given amplitudes andfrequencies in the streamwise and cross flow direc-tions or be left to execute free vibration in response tothe forces exerted on it by the fluid. The main bodyof results reported here are for two-dimensional flowat Re = 200, but the extension to three-dimensionalflow has been made, for which we report the numeri-cal formulation. The method is equally applicable tothe study of in-line or cross flow oscillations; in thispaper we concentrate on the latter.

2 Problem Formulation

The computational domain Ω employed for the two-dimensional simulations is shown in Fig. 1; the do-main extended a distance of 12.5D upstream andcross flow from the cylinder centreline, and a dis-tance of 25D downstream. A large cross flow meshdimension was employed to minimize blockage effects.Prescribed-velocity boundary conditions were usedon the inflow, upper and lower boundaries while atthe downstream end of the domain, outflow veloc-ity and pressure boundary conditions were used, asindicated on the figure. For three-dimensional simu-lations the x-y planar projection of the mesh wouldbe the same, with a number of planes spaced equallyalong the cylinder axis, in which direction the flow isassumed to be periodic.

Rather than arrange a method in which the com-putational grid deformed to allow cylinder motion,we have used a fixed-configuration grid attached tothe cylinder. By applying appropriate boundary con-

ditions and adding a ficticious forcing term to theNavier-Stokes equations, the flow simulation may becarried out in this accelerating reference frame. Theincompressible Navier-Stokes equations, written inrotational form for an inertial reference frame are

∂u

∂t= −∇Π+ u× ω + ν∇2u, (1)

where Π = p/ρ+u ·u/2 and ω = ∇×u. If the refer-ence frame in which the Navier-Stokes equations aresolved has a rectilinear acceleration a, the equationsbecome

∂u

∂t= −∇Π+ u× ω + ν∇2u− a. (2)

The appropriate velocity boundary conditions atprescribed-velocity boundaries are

u = −v, (3)

where v is the velocity of the reference frame. Thepressure boundary condition is obtained by takingthe normal component of (2), generated by dottingthe unit normal boundary vector n into (2) to make

∂Π

∂n= n · [u× ω − ν∇× ω] , (4)

where the form of the viscous term follows the sug-gestion of Orszag, Israeli and Deville (1986).

The force per unit length exerted by the pressurefield on the cylinder is given by

fp =

∮

pn ds =

∮

ρΠn ds, (5)

where n is again the unit outward normal of the fluiddomain and the integration is performed around thecircumference of the cylinder, while the viscous forceper unit length is

fv = −

∮

µn · [∇u + (∇u)T ] ds. (6)

If the cylinder undergoes forced oscillation, thevelocity and acceleration of the reference frame aregiven, however in the case of free vibration the mass,stiffness and damping of the cylinder must be consid-ered and second order ODEs of the form

x + 2ζωnx + ω2nx = f/m (7)

must be solved for each axis of motion, where fortwo-dimensional calculations m is the cylinder massper unit length and f = fp + fv is the total force perunit length exerted by the fluid. Following normalpractice these are decomposed into a set of first orderODEs with v = x, giving

v = f/m− 2ζωnv− ω2nx,

x = v. (8)

Here v is identical with the reference frame velocitysince it is fixed to the cylinder.

The equations (2) are solved together with the in-compressibility constraint

∇ · u = 0 in Ω. (9)

This completes the problem formulation and de-scribes the form used for the two-dimensional calcu-lations. In the three-dimensional representation wehave used, all variables are assumed to have period-icity in the z-direction (along the cylinder axis) andmotion is constrained to the x and y directions. Thecylinder is assumed prismatic, i.e., does not deformwithin its length. This approximates conditions nearthe centre-span of a slender elastic structural elementin its first mode of free vibration. The dependentvariables are represented in the three-dimensional for-mulation by

u(x, y, z, t)v(x, y, z, t)w(x, y, z, t)p(x, y, z, t)

=K−1∑

k=0

u(x, y, t)v(x, y, t)w(x, y, t)p(x, y, t)

eiβkz (10)

where the spanwise wavenumber β = 2π/Lz. TheNavier-Stokes equations (2) and the incompressibilityconstraint (9) then become sets of equations to besolved for each Fourier mode k;

∂uk∂t

= −∇Πk + [u× ω]k +

ν[

∇2xy − k2β2

]

uk − ak in Ωk, (11)

∇ · uk = 0 in Ωk, (12)

where the operators ∇ and ∇2xy are defined as

∇ =

(

∂

∂x,∂

∂y, ikβ

)

, ∇2xy =

∂2

∂x2+

∂2

∂y2. (13)

The computational domain Ωk is a projection ontothe x-y plane of Ω, hence is the same for each mode.Since the reference frame acceleration a has compo-nents only in the x-y plane, ak = 0, k 6= 0.

Communication of information between modes isrequired only during the formulation of the [u× ω]kterms, when the products are formed in physicalspace, so that computations for each mode may berun in parallel except for the communication requiredduring this step. The calculations performed for eachmode are very similar to those for a two-dimensionalcalculation.

In three dimensions the integrals in (5) and (6)are extended to include the spanwise dimension; mbecomes the total mass of the cylinder and f becomesthe total force exerted by the fluid on the cylinder,but due to the use of Fourier expansions in the zdirection, the form of the computation is exactly thesame as in two dimensions, since the integrals needto be evaluated only for the k = 0 mode.

2.1 Discretization

A three-level high-order time-split scheme with astiffly-stable semi-implicit integration method, de-scribed by Karniadakis, Israeli and Orzsag (1991),was employed for time integration of the Navier-Stokes equations. We give the three-dimensionalFourier-mode formulation; restriction to two dimen-sions is straightforward. The first sub-step deals withthe nonlinear terms and the frame acceleration

uk −∑J−1

q=0αqu

n−qk

∆t=

J−1∑

q=0

βq [u× ω − a]n−qk ; (14)

αq and βq are implicit/explicit weights for a stiffly-stable scheme of order J . The next substep recognizesthe influence of the gradient of total pressure and theincompressibility constraint:

ˆuk − uk

∆t= −∇Πn+1

k

∇ · ˆuk = 0. (15)

The incompressibility constraint is applied to the firstof this set of equations to produce a Helmholtz equa-tion which must be solved for the total pressure, sub-ject to the mode-equivalent of (4). The last sub-step incorporates the viscous correction and velocityboundary conditions;

γ0un+1

k − ˆuk∆t

= ν[

∇2xy − k2β2

]

un+1 (16)

with boundary velocities −v, k = 0 and identicallyzero for the higher modes. Here γ0 is a weight inthe backward differentiation employed in the stiffly-stable scheme.

The stiffly-stable scheme was also used for inte-gration of the body-motion ODEs (8), with explicitextrapolation used to approximate displacements, liftand drag forces at the new time level;

γ0vn+1 −

∑J−1

q=0αqv

n−q

∆t= −2ζωnv

n+1 −

J−1∑

q=0

βq

[

ω2nx

n−q −fn−q

m

]

(17)

γ0xn+1 −

∑J−1

q=0αqx

n−q

∆t= vn+1. (18)

Minimal additional computation was required forthe ODEs. In most cases second-order integration(J = 2) was used with dimensionless time steps∆tU/D = 0.005. At high amplitudes of forced oscilla-tion (∼ 0.5D) and for free-vibration with low valuesof the density ratio m/ρD2 it was sometimes neces-sary to use first-order integration with time steps of0.002 in order to maintain stability. Tests showedthat the results were well-resolved in time. Since for

all Fourier modes other than the zeroth the integra-tions in (5) and (6) are zero when extended to threedimensions, the calculations needed for integrationof the ODEs (8) may be restricted to the processorwhich carries the zeroth Fourier mode in a parallelcomputation.

A spectral element scheme (Karniadakis 1990,Henderson 1993), formed the basis for discretizationin the x and y directions. 86 macro-elements wereused, as shown in Fig. 1; eighth-order Legendre-Lagrangian polynomial tensor-product interpolantswere used within each element (81 collocation pointsper element). In trials with tenth-order interpolants(121 points per element) agreement to within 1% inStrouhal number, lift and drag forces was obtainedin two-dimensional computations with a stationarycylinder; hence it was decided that the eighth-orderinterpolant gave sufficient spatial resolution.

3 Forced Oscillation Results

The results in this and the two following sections re-late to the two-dimensional simulations at Re = 200.The Strouhal number St = fvU/D was 0.200, wherefv is the vortex shedding frequency.

A series of tests were carried out in which the cylin-der was forced to oscillate cross flow at a range ofreduced frequencies fr = fmU/D near “critical”, de-fined as that reduced frequency at which the forcedoscillation or mechanical frequency fm matches fv.Three oscillation amplitudes were chosen: 0.1D,0.2D, and 0.5D (the peak-to-peak amplitudes aretwice these values). Five frequencies were used at theamplitudes 0.1D and 0.5D, while 13 were employedto obtain a more detailed survey at amplitude 0.2D,as shown in Fig. 2, where approximate frequency lim-its for which lock-in (frequency coalescence) occurredare illustrated also.

The forced oscillation had a substantial effect onlift and drag forces within the lock-in regime, as maybe seen in Fig. 3 (a) where mean and standard de-viation coefficients of drag (Cd, C ′

d) and standarddeviation coefficients of lift (C ′

l) are presented asfunctions of the frequency ratio fm/fv for an 0.2Damplitude of oscillation. Values found for the fixedcylinder are presented for comparison. Long-termmotion-correlated lift forces were computed usingleast-squares techniques (Blackburn 1992) and fromthese the phase angle with which motion-correlatedforces led cylinder velocity was computed as shownin Fig. 3 (b). The phase angle calculations show thata change of sign of power transfer from the fluidto the cylinder occurred at the critical frequency,as previously demonstrated in physical experiments.This phenomenon is associated with the rapid changein phase with respect to cylinder motion of vor-tex shedding observed near the critical frequency(Zdravkovich 1982, Ongoren & Rockwell 1988).

Figure 2: Frequency-amplitude envelope employedfor forced-oscillation computations. Approximatelimits within which lock-in was observed are shownshaded.

Figure 3: Results for forced oscillation at amplitude0.2D. (a); Mean and standard deviation coefficientsof drag, Cd, C

′

d and standard deviation of coefficientof lift, C ′

l , as functions of frequency ratio fm/fv.Fixed-cylinder values shown dashed for comparison.(b); Phase angle with which motion-correlated forcesled cylinder velocity.

Figure 4: Free vibration cylinder centreline displace-ment phase portrait for ζ = 0.01, m/ρD2 = 1.

4 Free Vibration

In the free vibration tests, the cylinder was releasedfrom rest in an established (steady amplitude, time-varying) flow field. The natural frequency of thecylinder, ωn, was selected to match the vortex shed-ding frequency for the fixed cylinder. After release,cylinder oscillations grew in amplitude, with the timetaken to reach steady state oscillation and maxi-mum amplitudes being functions of the density ra-tio m/ρD2 and damping ratio ζ; transient timesand maximum amplitudes increased with decreas-ing m/ρD2 and ζ. Steady state displacement, ve-locity and acceleration phase portraits had the gen-eral form of Lissajous figures generated by pairs ofharmonic signals, one (streamwise direction) twicethe frequency of the other, as illustrated for examplein Fig. 4. Note the mean streamwise displacement,which is correct for the mean drag force and cylinderspring rate.

It is a well-established experimental result thatsteady-state amplitudes of cross flow and streamwiseoscillation in free vibration increase toward maximumvalues in the limit ζ → 0. This is in contrast towhat would be expected if there was no influence ofcylinder motion on lift and drag forces, in which caseit would be expected that amplitudes of oscillationwould increase without bounds as damping reduced.To see if the computations would reproduce this be-haviour, damping was reduced to low values, start-ing above the critical damping ratio, for two fixedvalues of density ratio, m/ρD2 = 1 and 10. Themaximum steady-state peak-to-peak amplitudes are

Figure 5: Maximum values of steady-state peak-to-peak free vibration oscillation amplitudes asfunctions of the mass-damping product ζs/µ =8π2St2mζ/ρD2. A comparison of computed valuesfor density ratios m/ρD2 = 1 and 10 with a compi-lation of experimental values reproduced from Grif-fin (1992).

shown as functions of the mass-damping product inFig. 5, superimposed on a range of experimental re-sults compiled by Griffin (1992). The nomenclaturein his figure is slightly different to that which we haveused; YEFF,MAX corresponds to ymax/D in the presentpaper, while ζs/µ corresponds to 8π2St2mζ/ρD2.

The first point of note is that the general trendof the experimental values, namely the amplitude-limiting effect as ζ → 0, was reproduced by thecomputations, and that there is reasonable agree-ment between experimental and computed maximumamplitudes, considering the variation in Reynoldsnumber between computations (Re = 200) and ex-periments 300 < Re < 106. At high values ofmass-damping the computed values tended to fallabove the experimental values; we suspect this is re-lated to three-dimensional wake flows in the physicalexperiments when oscillation amplitudes were low,which would have acted to reduce spanwise correla-tion and hence overall lift on a finite length of cylin-der and thereby the amplitude of oscillation. Ob-viously this behaviour cannot be reproduced by atwo-dimensional calculation. Finally it will be notedthat variation of density ratio m/ρD2 had an observ-able effect, particularly at low damping ratios; lowervalues of density ratio gave rise to increased oscil-lation amplitudes. The justification for coalescenceof density and damping ratios into a single parame-ter (e.g., 8π2St2mζ/ρD2) is based in part on the as-sumption of simple harmonic motion; at lower valuesof m/ρD2 significant higher frequency componentscould be inferred from examination of accelerationtimeseries and the characterization of amplitude bythe single combined parameter becomes less valid.

5 Comparison of Free and

Forced Oscillation

As mentioned in the Introduction, a comparison oflift and drag forces experienced by a cylinder in freevibration and forced oscillation at the same motionamplitude has never been conducted experimentally,however the present computational method providesan ideal vehicle for such a comparison. To performthis test, one set of results was selected as a basisfor comparison; in this run the cylinder was free tovibrate in each axis of motion and the density anddamping ratios were m/ρD2 = 10 and ζ = 0.01.Steady-state oscillation amplitudes were 0.00475 and0.434 in the streamwise and cross flow directions re-spectively, and the reduced frequency of cross flowoscillation was 0.199. Three comparisons were per-formed; in the first, the cylinder was also free to vi-brate, but motion was restricted to the cross flowdirection only; in the second, the cylinder was drivencross flow only, at the same frequency and amplitudeas achieved in the base runs; finally the cylinder wasdriven in both the streamwise and cross flow direc-tions with amplitudes and frequencies which matchedthose in base runs, while phase angle between thestreamwise and cross flow motion was adjusted to bethe same as for the base runs (−45). The first casecorresponds to free vibration tests in many experi-ments (as, e.g., Feng 1968), while the second corre-sponds to tests in most forced-oscillation experiments(e.g., Sarpkaya 1978).

A comparison of lift and drag force timeseries ispresented in Fig. 6, together with the timeseries of liftand drag for a stationary cylinder; the magnitudes offluctuating lift in all the cases for which the cylin-der was in motion were significantly reduced fromthose observed for the fixed cylinder. This accordswith the amplitude-limiting effect, as seen in Fig. 5;if lift forces did not reduce as amplitude of motionincreased, motion amplitude in free vibration wouldgrow without limit as ζ → 0, as noted above. Themagnitudes of lift and drag were similar for all thecases in which the cylinder was in motion, as might beexpected considering the comparatively small magni-tude of streamwise oscillation and the almost pureharmonic motion experienced on each axis of motionin free vibration for the base case.

6 Conclusions

From the range of results presented here, we concludethat the computational method described is able toreproduce many of the phenomena observed exper-imentally in tests on cylinders in oscillation, bothfree and forced, in cross flow. It is planned to carryout a number of three-dimensional simulations forReynolds numbers at which the wake flow is three-

Figure 6: Timeseries of coefficients of lift and drag;the upper (dashed) line in each section is Cd(t) andthe lower is Cl(t). (a); Base case, free vibration onboth axes of motion with m/ρD2 = 10 and ζ = 0.01,(b); Free vibration only on cross flow axis, (c); Forcedoscillation in cross flow direction only, amplitude andfrequency chosen to match base case, (d); Forced os-cillation on both axes, with amplitudes, frequenciesand phase angle chosen to match base case, (e); Re-sults for fixed cylinder presented for comparison.

dimensional in order to assess the effect of cylindermotion on the flow, since it is known that the nearwake tends to become more nearly two-dimensionalwhen there is significant motion. Further work willextend the range of Reynolds number by introducinglarge eddy simulation, and extend the formulation todeal with oscillatory, rather than steady flow.

Besides observing the phenomena of coupling be-tween cylinder motion and wake flows in both two andthree dimensions it is planned to examine the phys-ical mechanisms which account for the coupling andthe suppression of three-dimensionality in the nearwake region.

Acknowledgements

We would like to acknowledge the support given thisproject by Ron Henderson, Department of Mechani-cal and Aerospace Engineering, Princeton University.

The first author would like to thank the Depart-ment of Mechanical and Aerospace Engineering andthe Program in Applied and Computational Mathe-matics at Princeton University for their hospitalityduring a period of secondment from Monash Univer-sity.

This work was supported by the Office of NavalResearch under contract N00014-90-J-1315.

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