In-line Dynamic Characteristic of a Circular Cylinder under Vortex-induced Vibration

XU Wan-hai,WU Ying-xiang,HU Song-tao,DU Jie

(1.State Key Laboratory of Hydraulic Engineering Simulation and Safety,School of Civil Engineering,Tianjin University,Tianjin 300072,China;2.Key Laboratory for Hydrodynamics and Ocean Engineering,Institute of Mechanics,Chinese Academy of Science,Beijing 100190,China;3.XinXing Heavy Industries Group Co.,Ltd.,Beijing 100070,China)

1 Introduction

Oscillation of cylinders in the in-line or cross-flow direction relative to an incident flow is a common occurrence in industry.Understanding these phenomena is of great importance in the design of a variety of offshore engineering structures,such as pipelines,cables and risers.Vortex induced vibration of cylinder structures has been the subject of extensive research for several decades.Sarpkaya[1],Gabbai and Benaroya[2],Williamson and Govardhan[3]had reviewed the studies on VIV.Recent experimental and theoretical works were summarized in these review articles.

The response in the in-line(or streamwise)direction has often been neglected in earlier VIV researches,mainly because cross-flow response amplitudes are larger.However,studies have shown that for free spanning pipelines and risers fatigue damage due to in-line response may become significant and even more critical than cross-flow[4].Few studies were reported concerning with in-line VIV.Flow-induced streamwise vibrations of various kinds of cylindrical and axisymmetric bodies were reviewed in depth by Naudascher[5].Two different modes of excitation occur within the in-line VIV,the first excitation region originates from symmetric vortex shedding in the lower reduced velocity region of 1.0＜Vr＜2.3-2.5,while the second excitation region from alternate vortex shedding in the higher reduced velocity region of 2.3-2.5＜Vr＜3.8(Vris the reduced velocity defined by Vr=U/（fnD）),where U is the incident flow velocity,fnis the natural frequency of a cylinder,and D is the cylinder diameter)[6-7].Flow-induced in-line oscillation of a two-dimensional circular cylinder model was experimentally investigated in a wind tunnel using the free-oscillation method in order to understand some of the fundamental characteristics of the system by Matsuda et al[8].Okajima et al[9]conducted on free oscillation tests in the steamwise direction for the circular cylinder elastically supported at both ends and a cantilevered model with aspect ratios from 5 to 21 in a water tunnel,instead of a wind tunnel,and the value of the mass ratio μ(=m/ρD2;m is a mass per unit span length,ρ is fluid density)is small.The mass-damping parameter Cn(=2μξ;ξ is the structural damping factor)was varied over a wide range in order to evaluate the critical value at which the in-line oscillation is suppressed.

Fig.1 Model of coupled cylinder structure and wake oscillator for in-line vortex-induced vibration

Only a few analytical models have been proposed for the streamwise oscillation of structures.Currie and Turnbull[10]developed a wake-oscillator model,which attempts to represent a cylinder vibrating in the in-line direction,they thought that the cylinder oscillations in the second instability region were a simple harmonic of the velocity driven transverse cylinder oscillations,while those in the first instability region might be amplitude driven.An acceleration coupling wake oscillator model for the in-line oscillations in second synchronization region has been presented by Xu et al[11],some results showed the applicability and usefulness of the model for predicting in-line vortex induced vibration of engineering structures.Even fewer analytical models consider the possible coupling of the cross-flow and in-line responses as reported in several literatures.Kim and Perkins[12]adopted a similar approach in adapting the mathematic model in Balasubramanian and Skop[13]for drag,the principal mechanisms of coupled in-line and cross-flow motions of cable suspensions were identified during VIV.A time domain model has been formulated to examine flow induced vibration of cylindrical structures such as risers and free span pipelines by Furnes and Sorensen[14],the in-line and crossflow deflections are coupled,the wake dynamics was described by the classical van der Pol equation.The dynamics of long slender cylinders undergoing vortex-induced vibrations was studied by Ge et al[15],the wake dynamics,including in-line and cross-flow vibrations,was represented using a pair of non-linear oscillators distributed along the cylinder.

For practical reasons,one cannot rely on numerical or experimental techniques for the long cylinder structures VIV prediction,rather one needs to rely on reduced-order models that take into consideration all of the physical aspects,are validated by a combination of numerical simulation and experiments,and are capable of reliably predicting and simulating VIV.The objective of this study is to investigate some aspect of the in-line VIV dynamics,which is hardly observed by experiments and numerical simulation.A simple wake oscillator model proposed by Xu et al[11]for the near wake dynamics of a cylinder in streamwise direction,was used to model the vortex shedding behind a structure.

2 Model description

A cylinder subjected to vortex-induced oscillations in the streamwise direction(Fig.1)is described as a damped mass-spring oscillator,the instantaneous drag in the second region is assumed to be associated with a vortex shedding frequency corresponding to two times the Strouhal number,and satisfies a van der Pol equation.Thus,the dimensionless structure and wake model can be written as[11]:

where the dot represents the derivative with respect to dimensionless time t,y is space coordinate in the in-line direction,δ=Ωs/Ωfis the reduced angular frequency of the structure,Ωsis the structure angular frequency and the vortex shedding angular frequency Ωf=2πStU/D,where St is the Strouhal number,γ is a stall parameter,defined asis the mean drag coefficient.The dimensionless wake variable q is associated to the fluctuating drag coefficient on the cylinder,it is defined as q=2CD/CD0,where CD0could be interpreted as the fluctuating drag coefficient of a stationary cylinder,CDis the instantaneous drag coefficient.s is the action of the fluid near wake on the structure,in dimensionless variables,and it can be expressed in the following form:

The action of the structure on the fluid wakeε and A are the model empirical parameters.

In this paper,the fluctuating drag coefficient of a stationary cylinder CD0is taken equal to 0.2[16].Strouhal number St depends on Reynolds number,for the sake of simplicity,assuming St=0.17[17],the mean drag coefficientthe value of A as well as that of ε were determined from the experimental results on forced and free VIV,we had used a similar approach in the determination of wake oscillator model’s empirical coefficients in streamwise VIV[11].The parameter A can be expressed as:

where ω is angular frequency,define as ω=1/（VrSt）.We assumed that the maximum structure displacement amplitude was the same as the experimental results presented by y2nd_max=0.172e-0.949Cn,the specification of the empirical parameter ε was given.The readers are referred to Ref.[11]for details of the calibration of model parameters.

3 Results and discussion

The reduced velocity Vrused in present calculation was varied from 2.3 to 3.8,which was defined as the second in-line VIV region.The influences of mass-damping parameter,mass ratio and structural damping factor on cylinder in-line VIV were investigated.

The response amplitudes are sensitive to the mass-damping parameter Cnin the second excitation region with alternate vortices;it is fundamental and important to evaluate the critical values of Cnfor a two-dimensional cylinder like a heat exchanger and underwater riser,until now,there is almost no experiments observing the mechanisms of in-line VIV while Cnis much small.Fig.2 shows the response curves of in-line oscillation of a circular cylinder at three really small mass-damping parameters Cn=0.2,0.4 and 0.6.Keeping the mass ratio μ=10.5[18],it can be seen that the response amplitude of the in-line oscillation damps with increasing Cnvalue in the second excitation regions.There is an increase in streamwise amplitude from Vr=2.3 to 2.9,and a decrease from Vr=3.1 to 4.0,with the maximum value obtained at Vr=3.0.

Fig.2 The response amplitude of a twodimensional circular cylinder with different values of Cn

Fig.3 The response amplitude of a two-dimensional circular cylinder with different values of mass ratio μ

Fig.4 The response amplitude of a two-dimensional circular cylinder with different values of structural damping factor ξ

Although the maximum in-line structure displacement determined by the combined massdamping parameters Cn,the range of lock-in is known to be a function of both,the mass ratio μ and structural damping factor ξ,separately.The lock-in domain is here considered at Cn=0.4,the response amplitude of in-line oscillation of a circular cylinder was calculated at three small mass ratio μ=2.0,4.0 and 6.0.Cross-flow VIV experiments had shown the existence of a critical mass ratio makes lock-in domain unbound at higher reduced velocity,while there is no experiments concerned the in-line VIV characters of a circular cylinder at really small mass ratio μ;this is the reason why we investigate the mass ratio μ influence on the in-line VIV,here.It can be seen that the maximum response amplitude nearly remains the same while mass ratio is different in Fig.3.It is noted that the phenomenon of persistent lock-in remained at μ=2 and a critical mass ratio may exist.This conclusion should be further confirmed by extra in-line VIV experiments.

At low values of the mass-damping parameter,Cn=0.4,the in-line VIV dynamical behavior of a cylinder is discussed.The root mean square vibration amplitude is presented in Fig.4 for the ξ=0.03,0.06 and 0.1 cases.It can be found that the response amplitude reaches maximum value while reduced velocity Vris nearly at the same value 3.0.

4 Conclusions

In this paper,some aspect of the streamwise vortex-induced vibration characteristics of a circular cylinder,which is hardly observed by experiments,was investigated.A wake oscillator model,which has been proved the applicability and usefulness for predicting in-line VIV of engineering structures,was used.It can be found that the response amplitudes are sensitive to mass ratio and structural damping factor.There is a critical mass ratio,just like the cross-flow VIV of a cylinder.This needs confirming in further in-line VIV experiments.

[1]Sarpkaya T.A critical review of the intrinsic nature of vortex-induced vibrations[J].Journal of Fluids and Structures,2004,19:389-447.

[2]Gabbai R D,Benaroya H.An overview of modeling and experiments of vortex-induced vibration of circular cylinders[J].Journal of Sound and Vibration,2005,282:575-616.

[3]Williamson C H K,Govardhan R.A brief review of recent results in vortex-induced vibrations[J].Journal of Wind Engineering and Industrial Aerodynamics,2008,96(6-7):713-735.

[4]Baarholma G S,Larsenb C M,Lie H.On fatigue damage accumulation from in-line and cross-flow vortex-induced vibrations on risers[J].Journal of Fluids and Structures,2006,22:109-127.

[5]Naudascher E.Flow-induced streamwise vibrations of structures[J].Journal of Fluids and Structures,1987,1:265-298.

[6]King R,Prosser M J,Johns D J.On vortex excitation of model piles in water[J].Journal of Sound and Vibration,1973,29:169-188.

[7]Okajima A,Kosugi T,Nakamura A.Flow-induced in-line oscillation of a circular cylinder in a water tunnel[J].ASME Journal of Pressure Vessel Technology,2002,124:89-96.

[8]Matsuda K,Uejima H,Sugimoto T.Wind tunnel tests on in-line oscillation of a two-dimensional circular cylinder[J].Journal of Wind Engineering and Industrial Aerodynamics,2003,91:83-90.

[9]Okajima A,Nakamura A,Kosugi T,Uchida H,Tamaki R.Flow-induced in-line oscillation of a circular cylinder[J].European Journal of Mechanics B/Fluids,2004,23:115-125.

[10]Currie I G,Turnbull D H.Streamwise oscillations of cylinders near the critical Reynolds number[J].Journal of Fluids and Structures,1987,1:185-196.

[11]Xu Wanhai,Du Jie,Yu Jianxing,Li Jingcheng.Wake oscillator model proposed for the stream-wise Vortex-Induced Vibration of a circular cylinder in the second excitation region[J].Chinese Physics Letters,2011,28(12):124704.

[12]Kim W J,Perkins N C.Two-dimensional vortex-induced vibration of cable suspensions[J].Journal of Fluids and Structures,2002,16(2):229-245.

[13]Balasubramanian S,Skop R A.A new twist on an old model for vortex-excited vibrations[J].Journal of Fluids and Structures,1997,11:395-412.

[14]Furnes G K,Sorensen K.Flow induced vibrations modeled by coupled non-linear oscillators[C]//Proceedings of the 17th International Offshore and Polar Engineering Conference.Lisbon,Portugal,ISOPE,2007:2781-2787.

[15]Ge Fei,Long Xu,Wang Lei,Hong Youshi.Flow-induced vibrations of long circular cylinders modeled by coupled nonlinear oscillators[J].Science in China Series G:Physics,Mechanics and Astronomy,2009,53(7):1086-1093.

[16]King R.A review of vortex shedding research and its application[J].Ocean Engineering,1977,4:141-171.

[17]Finn L,Lambrakos K,Maher J.Time domain prediction of riser VIV[C]//Proceedings of the Fourth International Conference on Advances in Riser Technologies.Aberdeen,Scotland,1999.

[18]Okajima A,Kosugi T,Nakamura A.Experiments on flow-induced in-line oscillation of a circular cylinder in a water tunnel(1st report,the difference of the response characteristics when a cylinder is elastically supported at both ends and cantilevered)[J].JSME International Journal B,2001,44(4):695-704.