wake oscillator model proposed for the stream-wise vortex-induced vibration of a circular cylinder...

8/11/2019 Wake Oscillator Model Proposed for the Stream-Wise Vortex-Induced Vibration of a Circular Cylinder in the Second

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CHIN. PHYS. LETT. Vol.28, No.12 (2011)124704

Wake Oscillator Model Proposed for the Stream-Wise Vortex-Induced Vibration

of a Circular Cylinder in the Second Excitation Region

XU Wan-Hai(), DU Jie(), YU Jian-Xing(), LI Jing-Cheng()

Key Laboratory of Port and Ocean Engineering, Tianjin University, Tianjin 300072

(Received 30 August 2011)

A wake oscillator model is presented for the stream-wise vortex-induced vibration of a circular cylinder in the

second excitation region. The near wake dynamics related to the fluctuating nature of alternate vortex shedding

is modeled based on the classical van der Pol equation. An appropriate approach used in cross-flow VIV is

developed to estimate the model empirical parameters. The comparison between our calculations and experiments

is carried out to validate the proposed model. It is found that the present model results agree fairly well with

the experimental data.

PACS: 47.32.Cc, 47.85.Dh, 47.11.+j DOI:10.1088/0256-307X/28/12/124704

Vortex-induced vibrations (VIVs) of a circularcylinder occur in many industrial applications, such asthermowells, heat exchangers, underwater cables andrisers. They may cause significant fatigue damage.Thus this phenomenon must be addressed in designof such structures. Some studies on VIVs have beencomprehensively reviewed by Sarpkaya,[1] both exper-imental and theoretical investigations of the funda-mental aspects of vortex-induced vibration of circularcylinders are discussed in some detail.

Many of the works were, however, related to cross-flow oscillations, and few studies were reported onstream-wise (or in-line) oscillations. Flow-inducedstream-wise vibrations of various kinds of cylindrical

and axisymmetric bodies were reviewed in depth byNaudascher.[2] There are two different modes of ex-citation within in-line VIVs, the first excitation re-gion originates from symmetric vortex shedding in thelower reduced velocity region of 1.0< Vr < 2.32.5,while the second excitation region from alternatingvortex shedding in the higher reduced velocity regionof 2.32.5< Vr < 3.8 (Vr is the reduced velocity, de-fined by Vr = U /(fnD), where U is the incident flowvelocity, fn is the natural frequency of the cylinderand D is the cylinder diameter).[3] Flow-induced in-line oscillation of a two-dimensional circular cylindermodel was experimentally investigated in a wind tun-nel using the free-oscillation method in order to un-derstand some of the fundamental characteristics ofthe system by Matsuda et al.[4] Recently, Okajima etal.[5] conducted free oscillation tests in a water tun-nel, instead of a wind tunnel, the rigid cylinder mod-els were elastically supported at both ends, the mass-damping parameter Cn (= 2; is the mass ratio,is the structural damping factor) was varied over a

wide range, in order to evaluate the critical value atwhich the in-line oscillation is suppressed.

A large number of in-line VIV experimental resultshave been reported. However, only a few analyticalmodels have been proposed for the stream-wise oscilla-tions of structures. One mathematical model has beendeveloped by Currie and Turnbull,[6] attempting torepresent a cylinder vibrating in the in-line direction.This mathematical model is based on the Van der Polequation and is similar to the lift wake-oscillator forcross-flow oscillations. They thought that the cylinderoscillations in the second excitation region were a sim-ple harmonic of the velocity driven transverse cylinderoscillations, while those in the first excitation region

might be amplitude driven.For in-line VIVs, very few theoretical studies have

proposed a model governing the drag coefficient, anda practical tool for predicting in-line VIVs is still un-available in industry. The objective of this study is topropose a simple model for the near wake dynamics ofa cylinder with the second excitation region of stream-wise oscillations, a wake oscillator, which is based onthe Van der Pol equation, is used to model the al-ternating vortex shedding behind the cylinder. Theempirical parameters in the wake oscillator model arecalibrated and used. Finally, we compare our calcula-tion results with the experimental data for validatingthe proposed model.

A cylinder subjected to vortex-induced oscillationsin the stream-wise direction (Fig. 1) is described as a

damped mass-spring oscillator,[6]

Y +

2s+

f

Y + 2s Y =

S

m, (1)

where Y is the coordinate of the cylinder axis in

Supported by the Specialized Research Fund for the Doctoral Program of Higher Education of China (SRFDP)(20100032120047), the Independent Innovation Fund of Tianjin University (2010XJ-0098), and the National Natural Science Foun-

dation of China (10902112).Correspondence author. Email: [email protected]; [email protected] 2011Chinese Physical Societyand IOP Publishing Ltd

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the stream-wise direction, noting that the overdot de-notes the derivative with respect to the dimensionaltime T; is the structure reduced damping, s isthe structure angular frequency and the vortex shed-ding angular frequency f = 2StU/D, where St

is the Strouhal number, is a stall parameter, de-fined as = CD/4St,[7] CD is the mean drag co-

efficient. The dimensionless mass ratio = m/D2

with m being the cylinder mass, the fluid density,D the cylinder diameter. Both the mass of the struc-ture and the fluid-added mass are taken into account.

S= U2DCD/2 is the forcing term induced by vortex

shedding with CD is the fluctuating drag coefficient.

Z

Y

q

U

Fig. 1. Model of coupled cylinder structure and wake forin-line vortex-induced vibrations.

In the present study, the second excitation regionis defined by the reduced velocity range from 2.3 to3.8. The mean drag force term does not contribute

to the cylinder oscillations, the instantaneous drag inthis region is assumed to be associated with a vor-tex shedding frequency corresponding to two times theStrouhal number, and thus the instantaneous drag co-efficient CD satisfies the Van der Pol equation

[8]

q+ f(q2 1)q+ 42fq= F. (2)

The dimensionless wake variable may be associatedwith the fluctuating drag coefficient on the cylinder,it is defined as q = 2CD/CD0, where CD0 could beinterpreted to the fluctuating drag coefficient of a sta-tionary cylinder. Fis the forcing term, which models

the effects of the cylinder motion on the near wake.Through the forcing terms to the Van der Pol equa-

tion for the instantaneous fluctuating drag coefficientand the forcing term induced by vortex shedding toa damped mass-spring oscillator, the wake and struc-ture are coupled. Introducing the dimensionless timet= Tf and space coordinate y = Y /D, Eqs.(1)and(2) can be rewritten in dimensionless forms:

y+

2+

y+ 2y= s, (3)

q+ (q2 1)q+ 4q= f, (4)

where = s/f is the reduced angular frequencyof the structure, s is the action of the fluid near the

wake structure, in dimensionless variables, and it canbe expressed as

s= Mq, M = CD0

162St2. (5)

The action fof the structure on the fluid wake oscil-lator has several choices, named as displacement cou-plingf=Ay, velocity coupling f=Ay and accelera-tion coupling f =Ay,[7] whereA and are empiricalcoefficients to be determined. Because a static in-linedisplacementy of the structure in a uniform flow doesnot modify the fluctuating nature of the near wake,an in-line displacement of the structure at a constantvelocity only changes the mean drag force. Therefore,it is advisable that the coupling term of structure andwake is acceleration.

Fig. 2. Lock-in bands in the (, y0) plane for synchro-nization of vortex shedding with in-line cylinder vibration.Diamond: numerical result forRe= 300.[12] Plus: exper-imental data for Re= 200.[13] Triangle: for Re= 190.[14]

Solid line: for model parameter A = 12. Dashed line:A= 8.

Next, we calibrate all the parameters describedabove. The fluctuating drag coefficient of the station-ary cylinderCD0taken is equal to 0.2.

[9] The Strouhalnumber St depends on the Reynolds number, for thesake of simplicity, assuming St = 0.17,[10] the meandrag coefficient CD = 1.2, the value of A as well

as that of is determined from the experimental re-sults on forced and free VIVs, we have used a simi-lar approach in the determination of a wake oscillatormodels empirical coefficients in cross-flow VIVs.[11]

In-line oscillation of the cylinder and wake is definedas

y= y0cos(t), (6)

q= q0cos(t ), (7)

where y0 and q0 are the dimensionless amplitudesof the cylinder and wake, respectively, is angular

frequency, is phase. The action f of the struc-ture on the fluid wake oscillator could be written as

f = A2 cos(t). A reduced velocity Vr = 1/(St)

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is defined based on the angular frequency,[11] enforcingthe hypothesis of harmonicity and frequency synchro-nization. Substituting Eqs. (6) and (7) in the wakeoscillator, Eq. (4), only the main harmonic contribu-tion of the nonlinearities is considered, the amplitude

of the transfer function of the wake oscillator is yieldedusing elementary algebra, it reads

q608q40+ 16

[1 +

2 4

2]q20 = 16

A2y0

2. (8)

Fig. 3. The normalized maximum amplitude versus themass-damping parameter Cn and the proposed curve fit-ting. Diamond: cantilevered cylinder response amplitudein a water tunnel.[15] Circle: two-dimensional circularcylinder response amplitude in a wind tunnel. [16] Trian-gle: two-dimensional circular cylinder response amplitudein a water tunnel. [17]

The free wake oscillator response q0 = 2 is sup-posed to prevail on the forced response, a lock-out

state is defined, looking for the boundary from poly-nomial (8) in the (, y0)plane. In Fig. 2,the points ofnumerical simulation[12] and bounds of the lock-in re-gion, adapted from the experimental results[13,14] areplotted. The parameterAmay be chosen by matchingthe model response(8) to experimental and numericaldata on lock-in extension in previous literature. It canbe expressed as

A= 12 for 0 2, A = 8 for >2. (9)

For a given set of model parameters, the amplitudeof cylinder vibration is a function of the structuraldamping factor , the mass ratio and the ratio ofthe nominal vortex shedding frequency to the natu-

ral frequency of the cylinder structure. SubstitutingEqs.(6) and (7) in the dimensionless structure model,Eq.(3), the amplitude y0 is found in the form:

y0 = 2M[(

2 2)2

+

2+

22]1/2

[1 +

A2M (2 2)(2 4)

2(2+ /)]1/2

.(10)

Considering the appearance of the lock-in phe-nomenon, the basic resonance state displayed =

= 2, the mass-damping parameter Cn is used inthe stream-wise direction instead ofSG (= 2

3St2)in cross-flow direction. The maximum structure dis-placement amplitude reads

ymax= CD0162St2(2Cn+ )

1+ A

CD0

162St2(2Cn+).

(11)Experimental measurements of the modally normal-ized maximum amplitude versus the response param-

eter Cn is plotted in Fig. 3. It can be found that theproposed curve-fit formula could be written as

y2nd_max= 0.172e0.949Cn . (12)

Assuming that the maximum structure displacementamplitude is the same as the experimental results pre-

sented by Eq. (12), according to Eqs. (9), (11) and(12), we can obtain the specification of the empiricalparameter.

Fig. 4. The response amplitude of the circular cylinderwith= 16.0, Cn = 0.77.

Fig. 5. The response amplitude of the circular cylinderwith= 10.5, Cn = 1.58.

We validated the simulation results using the wakeoscillator model described in the previous section by

comparing them with some experimental data. Thereduced velocity Vr is varied from 2.3 to 3.8. Thereaders can refer to Refs. [5,15,17] for details of the

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experiments. Okajima et al.[5,15,17] studied the flow-induced in-line oscillation of a circular cylinder; theexperiments were carried out by free-oscillation testsin a water tunnel at subcritical Reynolds numbers.The relationship between movement of a cylinder and

flow of a near field during an in-line oscillation wasinvestigated. We simulate the two cases of the ex-periments, = 16.0, Cn = 0.77 and = 10.5,Cn = 1.58. The rms displacement response ampli-tudeyrms is calculated. The comparison between ourcalculation and experiment with = 16.0, Cn = 0.77is shown in Fig. 4. It can be seen that the simulation

response amplitude of the in-line oscillation increasesfrom Vr = 2.3 to 3.0, and decreases from Vr = 3.0 to3.8, with the maximum value obtained at Vr = 3.0.We plot the numerical results and the experimentaldata with = 10.5, Cn = 1.58 in Fig. 5. It can be

shown that there is an increase in the stream-wiseamplitude from Vr = 2.3 to 2.9, and a decrease fromVr = 2.9 to 3.8, with the maximum value obtainedat Vr = 2.9. The maximum response amplitude ob-tained by the present model and experiment is nearlythe same. From Figs. 4 and 5, we can find that thepresent model results agree fairly well with the exper-imental data, some aspect of the dynamics observedwith experiments can be reproduced.

In summary, we have proposed a wake oscillatormodel for the stream-wise VIV of a circular cylinder

in the second synchronization region. The predicted

results are compared with the experimental data tovalidate the present model. Good agreement is ob-tained, and some aspects of the cylinder in-line VIVcould be reproduced. These comparisons show the ap-plicability and usefulness of the present model for pre-dicting the in-line VIV of engineering structures. Such

a model becomes really useful when computationallimits arise for flow-field numerical simulations, par-ticularly at high Reynolds numbers. Moreover, phe-nomenological models based on wake oscillators allowaccessible analytical considerations and thus help the

understanding of the physics of the stream-wise VIV.

References

[1] Sarpkaya T 2004 J. Fluids Structures 19 389[2] Naudascher E 1987 J. Fluids Structures 1 265[3] Okajima A, Kosugi T and Nakamura A 2002 ASME J. Pres-

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gin. Industrial Aerodynamics 91 83[5] Okajima A, Nakamura A, Kosugi T, Uchida H and Tamaki

R 2004 Eur. J. Mech. B Fluids 23 115[6] Currie I G and Turnbull D H 1987 J. Fluids Structures 1

185

[7] Facchinetti M L, de Langre E and Biolley F 2004 J. FluidsStructures 19 123

[8] Furnes G K and Sorensen K 2007 Proceedings of the 17thInternational Offshore and Polar Engineering Conference

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[10] Finn L, Lambrakos K and Maher J 1999 Proceedings of theFourth International Conference on Advances (AberdeenScotland: Riser Technologies)

[11] Xu W H, Wu Y X, Zeng X H, ZHONG X F and YU J X2010 J. Hydrodyn. 22 381

[12] Nobari M R H and Naderan H 2006 Computers and Fluids35 393

[13] Hall MS and Griffin OM 1993 Trans ASME, J. Fluids En-gin. 115 283

[14] Griffin O M and Ramberg S E 1976 J. Fluid Mech. 75 257[15] Nakamura A, Okajima A and Kosugi T 2001 JSME Int. J.

B 44 705[16] Matsuda K, Uejima H and Sugimoto T 2003 J. Wind En-

gin. Industrial Aerodynamics 91 83[17] Okajima A, Kosugi T and Nakamura A 2001 JSME Int. J.

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http://cpl.iphy.ac.cn/http://dx.doi.org/10.1016/j.jfluidstructs.2004.02.005http://dx.doi.org/10.1016/0889-9746(87)90243-Xhttp://dx.doi.org/10.1115/1.1430670http://dx.doi.org/10.1016/S0167-6105(02)00336-7http://dx.doi.org/10.1016/j.euromechflu.2003.09.009http://dx.doi.org/10.1016/S0889-9746(87)90331-8http://dx.doi.org/10.1016/j.jfluidstructs.2003.12.004http://dx.doi.org/10.1016/0029-8018(77)90002-6http://dx.doi.org/10.1016/S1001-6058(09)60068-8http://dx.doi.org/10.1016/j.compfluid.2005.02.004http://dx.doi.org/10.1115/1.2910137http://dx.doi.org/10.1299/jsmeb.44.705http://dx.doi.org/10.1016/S0167-6105(02)00336-7http://dx.doi.org/10.1299/jsmeb.44.695http://dx.doi.org/10.1299/jsmeb.44.695http://dx.doi.org/10.1016/S0167-6105(02)00336-7http://dx.doi.org/10.1299/jsmeb.44.705http://dx.doi.org/10.1115/1.2910137http://dx.doi.org/10.1016/j.compfluid.2005.02.004http://dx.doi.org/10.1016/S1001-6058(09)60068-8http://dx.doi.org/10.1016/0029-8018(77)90002-6http://dx.doi.org/10.1016/j.jfluidstructs.2003.12.004http://dx.doi.org/10.1016/S0889-9746(87)90331-8http://dx.doi.org/10.1016/j.euromechflu.2003.09.009http://dx.doi.org/10.1016/S0167-6105(02)00336-7http://dx.doi.org/10.1115/1.1430670http://dx.doi.org/10.1016/0889-9746(87)90243-Xhttp://dx.doi.org/10.1016/j.jfluidstructs.2004.02.005http://cpl.iphy.ac.cn/

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