A Van Der Pol-Duffing Oscillator Model

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Imran Akhtar1

Interdisciplinary Center for Applied Mathematics,MC 0531,

Virginia Tech,Blacksburg, VA 24061

e-mail: [email protected]

Osama A. Marzouk

Ali H. NayfehUniversity Distinguished Professor

Department of Engineering Science andMechanics,

MC 0219,Virginia Tech,

Blacksburg, VA 24061

A van der Pol–Duffing OscillatorModel of Hydrodynamic Forceson Canonical StructuresNumerical simulations of the flow past elliptic cylinders with different eccentricities havebeen performed using a parallel incompressible computational fluid-dynamics (CFD)solver. The pressure is integrated over the surface to compute the lift and drag forces onthe cylinders. The numerical results of different cases are then used to develop reduced-order models for the lift and drag coefficients. The lift coefficient is modeled with ageneralized van der Pol–Duffing oscillator and the drag coefficient is expressed in termsof the lift coefficient. The parameters in the oscillator model are computed for eachelliptic cylinder. The results of the model match the CFD results not only in the timedomain but also in the spectral domain. �DOI: 10.1115/1.3192127�

IntroductionThe interactive fluid-structure phenomenon of flow separation

nd bluff body wakes has its fundamental significance in flowhysics and its practical importance in aerodynamic and hydrody-amic applications. When flow passes over a bluff body at loweynolds numbers, flow separation may take place over substan-

ial parts of its surface, but the flow around it remains steady. Ashe Reynolds number exceeds a critical value, instability in theeparated shear layers develops, and nonlinear interaction of theseayers with feedback from the wake leads to an organized anderiodic motion of a regular array of concentrated vorticity,nown as the von Kármán vortex street, in the wake. This vortexhedding exerts oscillatory forces on the body, which are oftenecomposed into drag and lift components along the freestreamnd cross-flow directions, respectively. If the body is capable ofexing or moving rigidly, these forces can cause it to oscillate and

he classical vortex-induced vibration �VIV� problem takes place.f the frequency of vortex shedding is close to a natural frequencyf the body, the resulting resonance can generate large-amplitudescillations, which may ultimately cause structural failure. Under-tanding this problem is of great interest in the design and main-enance of offshore structures, such as mooring cables, risers, andpars, and of high-aspect ratio structures subject to air streams,uch as chimneys, high rise buildings, bridges, and cable-uspension systems.

Since Roshko �1� measured the vortex-shedding period behindbluff body, many researchers have investigated this phenomenon

xperimentally and numerically for a wide range of Reynoldsumbers. The most frequently investigated bluff geometry is theircular cylinder. The flow behind a circular cylinder has becomehe canonical problem for studying such external separated flows2–7�. Engineering applications, on the other hand, often involveows over complex bodies, such as wings, submarines, missiles,nd rotor blades, which can hardly be modeled as circular cylin-ers.

Circular cylinders are extensively used in the study of bluffody fluid dynamics due to their geometric simplicity and com-on use in engineering applications. Bishop and Hassan �2� were

1Corresponding author.Contributed by the Design Engineering Division of ASME for publication in the

OURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received January5, 2008; final manuscript received February 12, 2009; published online August 25,009. Review conducted by Harry Dankowicz. Paper presented at the 46th AIAA
erospace Sciences Meeting and Exhibit 2008, Paper No. AIAA-2008-679.
ournal of Computational and Nonlinear DynamicsCopyright © 20

nloaded 10 Nov 2011 to 128.173.39.66. Redistribution subject to ASM

among the earliest to suggest using a self-excited oscillator torepresent the forces over a cylinder due to vortex shedding.Hartlen and Currie �8� formulated a model for elastically re-strained cylinders that are restricted to cross-flow motions. Theyused a Rayleigh oscillator to describe the lift force and coupled itto the cylinder motion by a linear velocity term. Currie and Turn-ball �9� proposed a similar model for the fluctuations in the drag.Skop and Griffin �10� pointed out that the parameters in the modelof Hartlen and Currie �8� lacked clear connection to physical pa-rameters of the problem. They proposed a modified van der Poloscillator to represent the lift coupled to a linear equation of mo-tion for the structure. They also introduced the Skop–Griffin pa-rameter, which is now commonly used in studying VIV problems.Iwan and Blevins �11� considered the fluid mechanics of the vor-tex street and developed a model in terms of a fluid variable thatcaptures the fluid-dynamics effects of the problem. Landl �12�added a nonlinear aerodynamic damping term of fifth order to thevan der Pol oscillator in his two-equation model, suggesting thatthis enables better capturing of some physical characteristics.However, the model involves many constants to be determined.

Evangelinos et al. �13� used direct numerical simulation �DNS�of the incompressible Navier–Stokes equations to solve, at a Rey-nolds number 1000, for the fluid-structure problem of rigid andflexible cylinders allowed to move freely in the cross-flow direc-tion. The flexible cylinder �or beam� was represented by a simpli-fied linear model with a forcing term proportional to the lift coef-ficient and no structural damping. The governing equations weretransformed into the Fourier domain and solved with a spectralelements method that employs a hybrid grid in the xy-plane andFourier complex modes in the z-direction. They considered shortand long cylinders with length-to-diameter ratios of 4� and 378,respectively. They reported that the often-used empirical formulaproposed by Skop et al. �14� overpredicts the drag coefficient.

Norberg �15� presented an overview of the fluctuating pressureand lift acting on a circular cylinder, taking special considerationof the influence of the Reynolds number and the relation betweenthe fluctuating lift and flow features �e.g., laminar shedding, waketransition, and turbulent shedding� in the near-wake region. Hecompiled the then available data from the diverse experimentaland numerical approaches, including two-dimensional and three-dimensional simulations, and reviewed the different measurementmethods such as the force-element method, total force method,electromagnetic method, and the family of momentum pressuremethods. These results show significant changes in the fluctuating
pressure distribution over the cylinder for Reynolds numbers
OCTOBER 2009, Vol. 4 / 041006-109 by ASME

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anging from 47 to 200,000. A spanwise correlation length ofbout 30 cylinder diameters was observed for flows near the sub-ritical regime Re=30,000.

Williamson and Govardhan �16� gave a detailed review of thexperimental and computational work on the VIV problem up to004. The primary concern was with the free and forced oscilla-ions of elastically mounted rigid cylinders and bodies having twoegrees of freedom and the dynamics of cantilevers, pivoted cyl-nders, cables, and tethered bodies. They underlined some of theundamental questions in VIV and touched upon some debatableoncepts, such as the added mass and the effect of combinedass-damping parameter on the peak-amplitude data in the Griffin

lot.Gabbai and Benaroya �17� presented a comprehensive review

f the studies �experimental, semi-experimental, and numerical�elated to VIV problems. They covered a variety of issues relatedo the physics of the problem, such as the dynamics of a vibratingylinder in a flow, the vortex-shedding modes, and the three-imensionality effects. They also reviewed different models �e.g.,ake models and force-decomposition models� and approaches

e.g., variational approach, vortex-in-cell �VIC� approach, DNS,nd the finite-element method� used to simulate the problem ofIV of both rigid and elastic cylinders.In the present paper, we present a numerical methodology ca-

able of solving the flow past elliptic cylinders. We vary the cyl-nder eccentricity from �=0.5 to �=1.0 with an increment of 0.1nd validate/verify our simulation results by comparing them withther experimental and numerical results. We compute the lift andrag coefficients and analyze them using higher-order spectraloments to obtain the dominant frequencies and their relative

hases. Using the spectral data, we identify the parameters ineduced-order models for the lift and drag. One of the potentialpplications would be to build a database of the model parametersor different Reynolds numbers and eccentricities and use it toredict the induced loads on cylinderlike and ellipselike struc-ures.

Numerical MethodologyThe Navier–Stokes and continuity equations are the governing

quations for the present problem. For incompressible flows, theyan be represented as follows:

�ui = 0 �1�

Lx

U� Ly

��

�=2�

�=�/2

�=�

�=3�/2

X

Y

�=0

(a)

Fig. 1 „a… Geometry of an elliptic cylinder ansors in the �-direction

�xi

41006-2 / Vol. 4, OCTOBER 2009


�ui

�t+

�

�xj�ujui� = −

1

�

�p

�xi+ �

�2ui

�xj � xj�2�

where i , j=1,2 ,3; the ui represent the Cartesian velocity compo-nents �u ,v ,w�; p is the pressure; � is the fluid density; and � is thefluid kinematic viscosity. Equations �1� and �2� are nondimension-alized using the cylinder diameter �D� as the length scale and thefreestream velocity �U�� as the velocity scale. Thus, the Reynoldsnumber is given by ReD=DU� /�. The other important geometricquantity, spanwise length of the body, is nondimensionalized us-ing the cylinder diameter D and is denoted by Lz.

We use a body conformal “O” type grid to simulate the flowover a body and employ curvilinear coordinates �� ,� ,�� in anEulerian reference frame; a planar view is shown in Fig. 1�a�.Here, X and Y indicate the Cartesian coordinates and Lx and Ly arethe major and minor axes, respectively. The thickness ratio of theellipse is given by �=Ly /Lx. This choice of coordinate systemallows us to simulate the flow past any arbitrary closed shape,such as circular and elliptic cylinders and airfoils.

Equations �1� and �2� are transformed into curvilinear coordi-nates in a strong conservative form as follows:

�Um

��m= 0 �3�

��J−1ui��t

+�Fim

��m= 0 �4�

where the flux is defined as

Fim = Umui + J−1��m

�xip −

1

ReDGmn �ui

��n�5�

Here, J−1=det��xi /�� j� is the inverse of the Jacobian or the vol-ume of the cell, Um=J−1��m /�xj�uj is the volume flux �contra-variant velocity multiplied by J−1� normal to the surface of con-stant �m, and Gmn=J−1��m /�xj��n /�xj� is the “mesh skewnesstensor.”

A non-staggered-grid layout is employed to solve the trans-formed Navier–Stokes equations. The Cartesian velocity compo-nents �u ,v ,w� and pressure �p� are defined at the center of thecontrol volume in the computational space and the volume fluxes�U ,V ,W� are defined at the midpoints of its corresponding faces.All of the spatial derivatives are approximated with second-orderaccurate central differences except for the convective terms. Usingthe same central differencing for the convection terms may lead to

b)

b… “O”-grid distributed among eight proces-

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spurious oscillations in the coarser regions of the grid, thereby

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eading to erroneous results. In the present formulation, we dis-retize the convective terms using a variation of quadratic up-tream interpolation for convective kinematics �QUICK�; that is,e calculate the face values of the velocity variables �ui� from theodal values using a quadratic upwinding interpolation. The up-inding of QUICK is carried out by computing the positive andegative volume fluxes ��Um+ �Um�� /2� and ��Um− �Um�� /2�, re-pectively, and using the generic stencil.

A semi-implicit scheme is employed to advance the solution inime. The diagonal viscous terms are advanced implicitly usinghe second-order accurate Crank–Nicolson method, whereas all ofhe other terms are advanced using the second-order accuratedams–Bashforth method. The Adams–Bashforth scheme was

hosen because of its computational efficiency when coupled withhe fractional-step method. The discretized equations are

�Um

��m= 0 �6�

J−1uik+1 − ui

k

t=

3

2�Ci

k + DE�uik�� −

1

2�Ci

k−1 + DE�uik−1�� + Ri�pk+1�

+1

2�DI�ui

k+1 + uik�� 7�

here � /��m represent the discrete finite-difference operators inhe computational space, the superscripts represent the time step,

i represent the convective terms, Ri are the discrete operators forhe pressure gradient terms, DI is the discrete operator represent-ng the implicitly treated diagonal viscous terms, and DE is theiscrete operator for the explicitly treated off-diagonal viscouserms. Mathematically, these terms are defined as follows:

Ci = −�

��m�Umui� �8�

Ri = −�

��m�J−1��m

�xi� �9�

DI =�

��m��Gmn �

��n� for m = n �10�

DE =�

��m��Gmn �

��n� for m � n �11�

t is important to note that, due to the orthogonality property forhe specific case of a cylinder, the cross terms for the mesh skew-ess tensor Gmn are zero, that is, when m�n. Therefore, the termsn Eq. �11� are identically zero and the stencil for the problem is aeven-point stencil.

In the present formulation, we apply a fractional-step method18� to advance the solution in time. The fractional-step methodplits the momentum equation into �a� an advection-diffusionquation—momentum equation solved without the pressure termnd �b� a pressure Poisson equation—constructed by implicit cou-ling between the continuity equation and the pressure in the mo-entum equation, thus satisfying the constraint of mass conserva-

ion.The governing equations are solved using a methodology simi-

ar to that employed by Zang et al. �19�. However, the algorithm isxtended to parallel computing platforms and the 2D domain de-omposition technique is employed to distribute the problemmong different processors. To implement the algorithm on aistributed-memory, message-passing parallel computer, we use awo-dimensional domain decomposition technique such that eachrocessor gets a “slice” of the grid, as shown in Fig. 1�b�. In thisgure, a two-dimensional view of 192256 grid is shown, which

s divided among eight processors such that each processor has a
omputational load of 19232 grid points. This technique allows
ournal of Computational and Nonlinear Dynamics


for a simple way to implement boundary conditions and keeps anequal load distribution for each processor. Moreover, the datapoints are exchanged only in one direction �i.e., �-direction�,thereby reducing the interprocessor communication cost. For athree-dimensional simulation, the grid is divided in the �- and�-directions. Details of the solution algorithm and parallel imple-mentation can be found in Refs. �20,21�.

Because an application of suitable and well-posed boundaryconditions is crucial for any simulation, we apply appropriateboundary conditions on different sections of the domain boundary.For the inflow boundary condition, we use a Dirichlet boundarycondition. For the outflow boundary condition, we use a Neumannboundary condition. No-slip and no-penetration boundary condi-tions are applied on the cylinder surface. Moreover, periodicboundary conditions are applied in the �- and �-directions.

The fluid force on a cylinder is the manifestation of the pressureand shear stresses acting on its surface. The net force can bedecomposed into two components: lift and drag forces. Theseforces are nondimensionalized with respect to the dynamic pres-sure. For the two-dimensional case, the coefficients of lift anddrag can thus be written in terms of the dimensional pressure andshear stresses as follows:

CL = −1

�U�2�

0

2� �p sin � −1

Re�z cos ��d� �12a�

CD = −1

�U�2�

0

2� �p cos � +1

Re�z sin ��d� �12b�

where �z is the spanwise vorticity component on the cylindersurface.

3 Validation and VerificationWe validate the CFD results by comparing them with other

experimental and numerical studies. We also perform grid anddomain dependence study to verify the numerical results. We thenperform numerical simulations of the flow past elliptic cylinderswith thickness ratios varying from �=0.6 to �=1.0 with an incre-ment of 0.1. The flow is simulated over a 192256 �� gridwith an outer domain size of 30D. In the present case, the grid isdivided into eight domains/processors, such that each processorhas a load of 19232 grid points, as shown in Fig. 1�b�. UsingEqs. �12a� and �12b�, we compute the lift and drag coefficients,respectively, for each cylinder and analyze them using higher-order spectral moments to obtain the dominant frequencies andtheir relative phases. We use these data to compute the parametersof the van der Pol–Duffing oscillator model.

3.1 Validation. To validate the CFD results, we simulate theflow past elliptic cylinders. We compare the peak-to-valley lift

coefficient CLp−v, the mean drag coefficient CD, the Strouhal num-

ber St, and the mean surface pressure coefficient CP on the cylin-der for the following two cases:

• Case I: �=0.5 �elliptic cylinder�• Case II: �=1.0 �circular cylinder�

In Case I, we simulate the flow past an elliptic cylinder with�=0.5 and a Reynolds number of 525 based on the projectedlength �Ly� of the elliptic cylinder. The numerical results are pre-

sented in Table 1. Our results for CLp−v, CD, and St are in good

agreement with those of Mittal and Balachandran �22�. Also, weplot in Fig. 2�a� the mean pressure coefficients CP over the sur-face of the elliptic cylinder. The pressure distribution is normal-ized such that CP=1.0 at the stagnation point; that is, �=180°.The CP distribution compares well with that of Mittal and Bal-achandran �22�.

In Case II, we also simulate the flow past a circular cylinder at

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e=525 on a 192256 grid. In the simulation, the two-imensional domain has a diameter of 30D. For the specific casef a circular cylinder Lx=Ly =D. Similar to Case I, we computehe physical parameters associated with the flow and compare theesults with those of Mittal and Balachandran �22� in Table 1. Welso compare the CP distribution over the cylinder surface in Fig.�b� and observe good agreement between the two numerical re-ults.

3.2 Grid and Domain Dependence Studies. Grid and do-ain independence studies are critical for verifying the accuracy

f the computational results. Therefore, we perform a grid/domaintudy for the flow past a circular cylinder at ReD=525 and com-

are some physical parameters, such as CD and St.We simulate the flow over a computational domain of 30D on

ight processors with 192256 and 288384 grid points. Theesults from these simulations are given in Table 2. We observe

hat the percentage difference for CD and St is less than 5%, whichlearly indicates that the flow in the vicinity of the structure isirtually grid independent.

To establish domain independence, we increase the domain sizey 60% to 50D and use 216256 grid points. The slight increasen the grid points in the �-direction, on the larger domain, is in-entional to maintain comparable grid spacing in the two grids.he results from these simulations given in Table 2 indicate that

he domain size of 30D is quite adequate for the current study.

Reduced-Order ModelFor the flow past a stationary bluff body, vortex shedding de-

ends on many parameters, such as the Reynolds number, bodyeometry, and angle of attack. However, for a given set of param-ters, the numerical and experimental results show that the longime history of the lift on the body is periodic and independent ofhe initial conditions. In other words, the flow is self-excited.ence, when viewed as a dynamical system, the lift force can be

epresented by a self-excited oscillator. Using center manifoldheory, one can reduce the dynamical system to an autonomouswo-dimensional dynamical system �or a single-degree-of-reedom oscillator� that exhibits a Hopf bifurcation as one of its

Table 1 Validation for the two cases

ase Study CLp−v CD St

Mittal and Balachandran �22� 1.21 0.78 0.2a

Present study 1.05 0.766 0.216

I Mittal and Balachandran �22� 2.42 1.44 0.22Present study 2.366 1.415 0.225

Re=1000.

η

CP

0 90 180 270 3-1

-0.5

0

0.5

1

(a)

Fig. 2 Mean surface CP distribution
„solid… and Mittal and Balachandran †22
41006-4 / Vol. 4, OCTOBER 2009


parameters that varies across a critical value. Prototype self-excited oscillators include the van der Pol and Rayleigh oscilla-tors, which are often employed to model self-excitations in elec-trical and mechanical engineering applications. It is important tonote that no autonomous single-degree-of-freedom equation,which does not include negative linear damping and positive non-linear damping, can produce limit cycles.

We showed �23� that the van der Pol–Duffing oscillator modelis applicable over a wide range of Reynolds number for the flowpast a circular cylinder for the steady-state as well as transient liftcoefficient and match the phase between dominant frequencies. Inthis paper, we extend the application of van der Pol–Duffing os-cillator to elliptical bodies with varying eccentricity.

Nayfeh et al. �24� investigated two wake-oscillator models torepresent the lift on a stationary circular cylinder. Analyzing thelift obtained with CFD using higher-order spectral moments, theyfound that the phase angle 13 between the two dominant liftcomponents at fs and 3fs is around 90 deg. Based on this finding,they concluded that the van der Pol oscillator

CL + �2CL = �CL − �CL2CL �13�

is more suitable for modeling the steady-state lift coefficient. Theangular frequency � in Eq. �13� is related �but not equal� to theangular shedding frequency �s=2�fs, the positive parameter �represents negative damping, and the positive parameter � repre-sents positive nonlinear damping. The values of � and � are takenpositive so that the linear damping is destabilizing while the non-linear damping is stabilizing. As a consequence, small distur-bances grow and large ones decay, both eventually approaching astable limit cycle. The values of the parameters in Eq. �13� dependon the Reynolds number.

Nayfeh et al. �25� found that the analytical model �24� repro-duces the steady-state and the transient lift obtained with the CFD.They integrated Eq. �13� with different initial conditions and ob-served that the closer the initial conditions to the steady-state CFDvalues are, the better the matching between the results of the CFDand lift model is. Their results show that matching the transientlift also depends on the Reynolds number. At low Reynolds num-bers, the transients due to the numerical simulation �e.g., impulseor zero initial conditions� decay much slower and the physicaltransient behavior of the dynamical system dominates later. How-ever, at higher Reynolds numbers, the numerical transients decay

η

CP

0 90 180 270 360

-1.5

-1

-0.5

0

0.5

1

(b)

r „a… Case I and „b… Case II: present

Table 2 Grid and domain dependence studies

Grid size �� Domain size CD St

192256 30D 1.415 0.2246288384 30D 1.429 0.2246216256 50D 1.381 0.2246

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uch faster and the van der Pol model predicts both of the tran-ient and steady-state responses accurately. As an example, theondimensional decay times of the numerical transients are ap-roximately 50 and 16 for the Reynolds numbers 200 and00,000, respectively.

Nayfeh et al. �24� examined the spectra of the drag and liftbtained with CFD and found that the major peaks in the dragccur at 2fs and 4fs and that the phase between the drag peak atfs and the lift peak at fs is near 270 deg. Hence, they inferred that

he periodic component of the drag is proportional to −CLCL androposed the following drag model:

CD�t� = CD − 2a2

�sa12CL�t�CL�t� �14�

here a2 is the amplitude of the drag component at 2fs and CDs the mean drag value. In the steady state, the mean component ofhe drag CD= CDss is constant, while in the transient state, CDs a monotonically increasing function of time. The constant valueCDss is determined from the CFD steady-state time history of therag and the value of a2 is determined from its spectral analysis.in �26� proposed that the quadratic term in the drag model

hould be of the form CL2 instead of CLCL. He also found linear

oherence between the drag and lift components at fs and 3fs andntroduced a linear lift term in the drag model to account for it.

Marzouk et al. �23� examined the phase angles 13 between theift peaks at 3fs and fs more closely and found that it can vary byp to �25 deg around 90 deg, depending on the Reynolds num-er. To match this phase, they added a Duffing-type cubic term tohe van der Pol oscillator model in Eq. �13� and obtained

CL + �2CL = �CL − �CL2CL − �CL

3 �15�

oreover, they examined the phase 12 between the peak in therag at 2fs and the peak in the lift at fs and found that it can varyp to �85 deg around 270 deg, again depending on the Reynoldsumber. To match this phase, they modified the drag model to

CD�t� = CD + 2k1a2

a12 �CL

2 − CL2� + 2k2

a2

�sa12CLCL �16�

In this paper, we investigate whether Eqs. �15� and �16� canodel the lift and drag on elliptic cylinders. To this end, we ex-

ress the solution of Eq. �15� using the method of harmonic bal-nce as

CL�t� = c1 cos��st� + c2 sin��st� + c3 cos�3�st� + c4 sin�3�st��17�

ubstituting Eq. �17� into Eq. �15�, separating the terms, and mul-iplying the different sine and cosine functions yield the fourth-rder linear system

Ay = b �18�

here y= ��2 ,� ,� ,��, b= �c1 ,c2 ,9c3 ,9c4��s2, and A=A�ci ,�s�.

Next, we determine the ci by numerically simulating the flowast elliptic cylinders with different thickness ratios for ReLx525. We consider five elliptic cylinders: �=0.6 �Case 1� to �1.0 �Case 5�, with an increment of 0.1. All cases are simulatedsing eight processors and are run long enough to reach steadytate. In Fig. 3, we plot the instantaneous spanwise vorticity for allases. We observe a similar vortex-shedding pattern; however, thehedding frequency is different in each case. Moreover, the widthf the wake increases with increasing �, thereby increasing therag. In other words, the size of the vortices being shed is of therder of Ly and the vorticity increases with �. This is also evidentrom the fact that the projected area of the cylinders, as “seen” byhe flow, increases with increasing �.

We then compute the time histories of the pressure distributions
n the surfaces of the cylinders. Using Eqs. �12a� and �12b�, we


compute the lift and drag coefficients for each case. It followsfrom Figs. 4�a�–4�e� that the amplitudes of the fluctuating forcesincrease as � increases.

Using spectral analysis, we compute the dominant frequenciesand the phases 12 and 13. The results are given in Table 3. InFig. 5, we plot the shedding frequency and the amplitudes a1, a2,and a3 of the dominant frequencies fs, 2fs, and 3fs, respectively.We note that odd harmonics appear in the lift spectrum, whereaseven harmonics appear in the drag spectrum. In Fig. 5�a�, weobserve that the shedding frequency fs decreases with increasing�, whereas the corresponding amplitude a1 increases as � in-creases, as shown in Fig. 5�b�. We observe approximately a five-fold increase in a1 as � increases from 0.6 to 1.0. Likewise, thereis a similar trend in a2 and a3; however, the harmonics are muchstronger in Case 5 than in Case 1 when compared with the respec-tive a1. For example, in Case 1, a2 and a3 are 3.4% and 0.23% ofa1, respectively, whereas in Case 5, they are 10.9% and 3%, re-spectively. Thus, the power distribution in the frequency spectrumis related to the projected area of the cylinder. We also note asteady increase in the mean drag, a direct consequence of theincrease in the projected length of the cylinder, as shown in Fig.5�c�.

Next, we determine ci by matching Eq. �17� with the steady-state CFD solution, which is expressed as

CL�t� = a1 cos��st� + a3 cos�3�st + 13� �19�

Comparing Eq. �19� with Eq. �17�, we have

c1 = a1, c2 = 0, c3 = a3 cos� 13�, c4 = − sin� 13� �20�

Substituting Eq. �20� into Eq. �18� and solving the resulting equa-tion, we obtain y= ��2 ,� ,� ,�� and hence all of the parameters inEq. �15�. Next, we integrate the identified van der Pol–Duffingequation to predict the lift on the elliptic cylinders. In Fig. 6, wecompare the time histories of the lift coefficient obtained using themodel with those obtained with CFD for all of the cases. Weobserve good agreement between the CFD and model results.

Next, we turn to the drag. The drag coefficient obtained withCFD can be expressed as

CD�t� = CD + a2 cos�2�st + 12� + ¯ �21�

Fig. 3 Instantaneous spanwise vorticity contours

Moreover, substituting CL=a1 cos��st� into Eq. �16� yields

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CD�t� = CDss + a2�k1 cos�2�st� − k2 sin�2�st�� + ¯ �22�

omparing Eqs. �21� and �22�, we conclude that k1=cos 12 and2=sin 12. Having determined k1 and k2, we compute the timeistories of the drag coefficients using Eqs. �16� and �15�. Theredicted time histories of the drag coefficient are compared withhose obtained with CFD in Fig. 7. Again the agreement is excel-ent.

300 305 310 315 320 325 330 335 340

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Time

CL,C

D

(a)

300 305 310 315 320 325 330 335 340

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time

CL,C

D

(c)

300 305 310 315 320 325 330 335 340

−1

−0.5

0

0.5

1

1.5

Time

CL,C

D

(e)

Fig. 4 Time histories of the lift „s

Table 3 Spectral analysis o

Case 1 Case 2

fs 0.318 0.273a1 0.238 0.425a2 0.0081 0.0202a3 0.00055 0.0028CDss 0.51 0.66 12 �30 deg �25 deg 13 81 deg 71 deg�s 1.998 1.715

41006-6 / Vol. 4, OCTOBER 2009


We list in Table 4 the model parameters for the five cylinders.We observe that the angular frequency � of the van der Pol–Duffing oscillator model is close �but not equal� to the angularshedding frequency �s. In Fig. 8�a�, we plot the lift-model coef-ficients �, �, and � as functions of �. The parameters � and � arepositive, indicating the presence of a limit cycle. It is clear fromthe values of � and � that the influence of the Duffing term,

300 305 310 315 320 325 330 335 340

−0.4

−0.2

0

0.2

0.4

0.6

Time

CL,C

D

b)

300 305 310 315 320 325 330 335 340

−0.5

0

0.5

1

TimeC

L,C

Dd)

d… and drag „dashed… coefficients

e lift and drag coefficients

Case 3 Case 4 Case 5

0.248 0.231 0.2250.643 0.891 1.1780.0437 0.0836 0.1290.0087 0.0182 0.0360.85 1.08 1.42

�19 deg �21 deg �20 deg86 deg 98 deg 97 deg1.558 1.451 1.414

(

(

oli

f th



J

Dow

τf s

0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

τ

a 1,a 2,

a 3

0.6 0.7 0.8 0.9 1

0

0.2

0.4

0.6

0.8

1

1.2

τ

CD

0.6 0.7 0.8 0.9 10

0.5

1

1.5

τφ 12

,φ 13

0.6 0.7 0.8 0.9 1-180

-90

0

90

180

(b)(a)

(c) (d)

Fig. 5 Spectral analysis parameters from the CFD simulations

200 205 210 215 220−0.4

−0.2

0

0.2

0.4

CL

Time

ModelCFD

(a)200 205 210 215 220

−0.8

−0.4

0

0.4

0.8

CL

Time

ModelCFD

(b)

200 205 210 215 220

−1

−0.5

0

0.5

1

CL

Time

ModelCFD

(c)200 205 210 215 220

−1.5

−1

−0.5

0

0.5

1

1.5

CL

Time

ModelCFD

(d)

200 205 210 215 220−1.8

−0.8

0.2

1.2

CL

Time

ModelCFD

(e)

Fig. 6 Comparison between the time histories obtained by the lift model with those ob-
tained with CFD
ournal of Computational and Nonlinear Dynamics OCTOBER 2009, Vol. 4 / 041006-7

nloaded 10 Nov 2011 to 128.173.39.66. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

0

Dow

Table 4 Parameters of the reduced-order model

Case 1 Case 2 Case 3 Case 4 Case 5

� 1.992 1.671 1.528 1.484 1.436� 0.037 0.084 0.169 0.236 0.35� 2.582 1.831 1.623 1.198 1.012� 0.855 1.146 0.279 �0.137 �0.042k1 0.866 0.906 0.946 0.934 0.94k2 �0.5 �0.423 �0.326 �0.358 �0.342

200 205 210 215 2200.504

0.512

0.52

0.528

CD

Time

ModelCFD

(a)200 205 210 215 220

0.63

0.65

0.67

0.69

CD

Time

ModelCFD

(b)

200 205 210 215 2200.78

0.82

0.86

0.9

0.94

CD

Time

ModelCFD

(c)200 205 210 215 220

0.95

1

1.05

1.1

1.15

1.2

1.25

CD

Time

ModelCFD

(d)

200 205 210 215 2201.2

1.4

1.6

CD

Time

ModelCFD

(e)

Fig. 7 Comparison between the time histories obtained by the drag model with those ob-tained with CFD

τ

µ,α

,γ

0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

τ

k 1,k 2

0.6 0.7 0.8 0.9 1

-1

-0.5

0

0.5

1

(b)(a)

Fig. 8 Parameters of the van der Pol–Duffing oscillator and drag models

41006-8 / Vol. 4, OCTOBER 2009 Transactions of the ASME

nloaded 10 Nov 2011 to 128.173.39.66. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

aim

bm

5

esbdoClhdrwtTpa

A

vsIs

R

J

Dow

lthough lesser in magnitude, cannot be neglected in the model-ng. Likewise, we compute the coefficients k1 and k2 in the dragodel, as shown in Fig. 8�b�. The values of k1 and k2 suggest that

oth of the terms CL2 and CLCL contribute significantly to the

odeling of the drag.

Conclusion and Future WorkThe lift and drag coefficients for two-dimensional flows over

lliptic cylinders were examined and modeled as a preliminarytep to understanding the complex problem of vortex-induced vi-rations. We extended the circular cylinder model for the lift andrag to elliptic cylinders. We performed numerical simulationsver elliptic cylinders with varying eccentricities using a parallelFD code. We studied five cases of elliptic cylinders and calcu-

ated the hydrodynamic forces acting on them. We performedigher-order spectral analysis of the time histories of the lift andrag coefficients and computed the dominant frequencies and theirelative phases. Using these data for different elliptic cylinders,e identified the coefficients in a van der Pol–Duffing model for

he lift and a drag model that depends quadratically on the lift.he model results predict well the CFD data. In the future, we willerform more simulations on elliptic cylinders with incoming flowt different angles of attack.

cknowledgmentNumerical simulations were performed on Virginia Tech Ad-

anced Research Computing-System X. The allocation grant andupport provided by the staff is also gratefully acknowledged.mran Akhtar would like to thank the Government of Pakistan forupport during his graduate studies.

eferences�1� Roshko, A., 1955, “On the Wake and Drag Of Bluff Bodies,” J. Aero. Sci.,

22�2�, pp. 124–132.�2� Bishop, R., and Hassan, A., 1964, “The Lift and Drag Forces on a Circular

Cylinder in Flowing Fluid,” Proc. R. Soc. London, Ser. A, 277, pp. 32–50.�3� Karniadakis, G. E., and Triantafyllou, G. S., 1992, “Three-Dimensional Dy-

namics and Transition to Turbulence in the Wake of Bluff Objects,” J. FluidMech., 238, pp. 1–30.

�4� Roshko, A., 1954, “On the Development of Turbulent Wakes From VortexStreets,” NACA Report No. 1191.

�5� Tomboulides, A. G., Israeli, M., and Karniadakis, G. E., 1989, “Efficient Re-moval of Boundary Divergence Errors in Time-Splitting Methods,” J. Sci.
Comput., 4�3�, pp. 291–308.


�6� Williamson, C. H. K., 1996, “Vortex Dynamics in the Cylinder Wake,” Annu.Rev. Fluid Mech., 28, pp. 477–539.

�7� Wu, J., Sheridan, J., and Welsh, M. C., 1994, “An Experimental Investigationof the Streamwise Vortices in the Wake of a Bluff Body,” J. Fluids Struct., 8,pp. 621–635.

�8� Hartlen, R., and Currie, I., 1970, “Lift-Oscillator Model of Vortex-InducedVibration,” J. Engrg. Mech. Div., 96, pp. 577–591.

�9� Currie, I., and Turnball, D., 1987, “Streamwise Oscillations of Cylinders Nearthe Critical Reynolds Number,” J. Fluids Struct., 1�2�, pp. 185–196.

�10� Skop, R., and Griffin, O., 1973, “A Model for the Vortex-Excited ResonantResponse of Bluff Cylinders,” J. Sound Vib., 27�2�, pp. 225–233.

�11� Iwan, W., and Blevins, R., 1974, “A Model for Vortex-Induced Oscillation ofStructures,” ASME J. Appl. Mech., 41�3�, pp. 581–586.

�12� Landl, R., 1975, “A Mathematical Model for Vortex-Excited Vibration ofCable Suspensions,” J. Sound Vib., 42�2�, pp. 219–234.

�13� Evangelinos, C., Lucor, D., and Karniadakis, G. E., 2000, “DNS-DerivedForce Distribution on Flexible Cylinders Subject to Vortex-Induced Vibra-tions,” J. Fluids Struct., 14, pp. 429–440.

�14� Skop, R. A., and Balasubramanian, S., 1997, “A New Twist on an Old Modelfor Vortex-Excited Vibrations,” J. Fluids Struct., 11�4�, pp. 395–412.

�15� Norberg, C., 2003, “Fluctuating Lift on a Cylinder: Review and New Measure-ments,” J. Fluids Struct., 17, pp. 57–96.

�16� Williamson, C. H. K., and Govardhan, R., 2004, “Vortex-Induced Vibrations,”Annu. Rev. Fluid Mech., 36, pp. 413–455.

�17� Gabbai, R. D., and Benaroya, H., 2005, “An Overview of Modeling and Ex-periments of Vortex-Induced Vibration of Circular Cylinder,” J. Sound Vib.,282, pp. 575–616.

�18� Kim, J., and Moin, P., 1985, “Application of a Fractional-Step Method toIncompressible Navier-Stokes,” J. Comput. Phys., 59, pp. 308–323.

�19� Zang, Y., Street, R., and Koseff, J., 1994, “A Non-Staggered Grid, FractionalStep Method for Time-Dependent Incompressible Navier-Stokes Equations inCurvilinear Coordinates,” J. Comput. Phys., 114, pp. 18–33.

�20� Akhtar, I., 2008, “Parallel Simulations, Reduced-Order Modeling, and Feed-back Control of Vortex Shedding Using Fluidic Actuators,” Ph.D. thesis, Vir-ginia Tech, Blacksburg, VA.

�21� Akhtar, I., Nayfeh, A. H., and Ribbens, C. J., 2009, “On the Stability andExtension of Reduced-Order Galerkin Models in Incompressible Flows: ANumerical Study of Vortex Shedding,” Theor. Comput. Fluid Dyn., 23�3�, pp.213–237.

�22� Mittal, R., and Balachandar, S., 1995, “Effect of Three-Dimensionality on theLift and Drag of Nominally Two-Dimensional Cylinders,” Phys. Fluids, 7�8�,pp. 1841–1865.

�23� Marzouk, O. A., Nayfeh, A. H., Arafat, H. N., and Akhtar, I., 2007, “ModelingSteadystate and Transient Forces on a Cylinder,” J. Vib. Control, 13�7�, pp.1065–1091.

�24� Nayfeh, A. H., Owis, F., and Hajj, M. R., 2003, “A Model for the Coupled Liftand Drag on a Circular Cylinder,” ASME Paper No. DETC2003-VIB48455.

�25� Nayfeh, A. H., Marzouk, O. A., Haider, H. N., and Akhtar, I., 2005, “Model-ing the Transient and Steady-State Flow Over a Stationary Cylinder,” ASMEPaper No. DETC2005-86376.

�26� Qin, L., 2004, “Development of Reduced-Order Models for Lift and Drag onOscillating Cylinders With Higher-Order Spectral Moments,” Ph.D. thesis,
Virginia Tech, Blacksburg, VA.
OCTOBER 2009, Vol. 4 / 041006-9


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