pw

ai

y of

Ast

Vortex-induced vibration

Dynamic characteristics

for

iser

ects

dyn

num

al w

enc

leng

at the axial position where the tension is smaller, whereas for higher modes the response amplitude is

larger at the axial position where the bending stiffness is lower. Moreover, vibration wave velocity

increases with increasing modal wave length along the riser length.

deepw

meterstants a constructdynam

dynamic response, an iterative calculation is needed (Bokaian,

beam(Sarpkaya, 2004) is as follows:

m@t2

c@t

@x2

EIx@x2

@x

Tx@x

f x,t

Contents lists available at SciVerse ScienceDirect

lse

Ocean Eng

Ocean Engineering 42 (2012) 712and damping per unit length, respectively, EI(x) and T(x) are theE-mail address: wmchen@imech.ac.cn (W. Chen).1994; Chen et al., 2009; Gei et al., 2007; Lyons and Patel, 1986)1

where y(x,t) is the displacement vertical to the beam axis, x is theaxial position and t is the time; m and c are the structural mass

0029-8018/$ - see front matter & 2011 Elsevier Ltd. All rights reserved.

doi:10.1016/j.oceaneng.2011.12.019

n Corresponding author. Tel.: 86 10 82543891; fax: 86 10 82338527.by a function of the variables related to not only uid but alsostructure. However, when such model is used to calculate the @2yx,t @yx,t @2 @2yx,t

" #@ @yx,t by means of nite element simulation combined with a hydro-dynamic model taking account of the interaction between uidand structure, is developed. Generally, the hydrodynamic forcesuch as vortex-induced lift depends on the ambient uiddynamics and structure motion; therefore, it is usually described

2.1. Hydrodynamic model and nite element simulation

The dynamic equilibrium equation of a tension EulerVIV, of slender riser with varying structural properties has beenfacing new challenges (Grant et al., 2000; Liao and Vandiver,2002; Sarpkaya, 2004).

In this study, rst, a coupling analysis approach in time-domain for dynamic characteristics and VIV of deepwater riser,

by the axial variation of structural properties. The VIV amplitudedepends on the overall effect of multiple factors involving tension,stiffness and modal wave length.

2. Numerical modelingcomplex, e.g. the outer or inner diathe length of riser rather than a conaxial tension is no longer regarded alength of riser due to considerablethe dynamic analysis, such as theof riser gets larger and the structural conguration becomes more tude, of structure is proposed so as to avoid costly iterations, and1. Introduction

As the oil exploitation extends to& 2011 Elsevier Ltd. All rights reserved.

ater ocean, the length

gradually varies alongvalue. Additionally, thestant along the overallure weight. Therefore,ic characteristics and

because the vibration amplitude of structure is unknown beforethe calculation. Here, an alternative lift model expressed bythe instantaneous motion, rather than the nal response ampli-

be more efcient for time domain calculation using nite elementcode. Based on the presented approach, the dynamic character-istics and VIV of the riser with axially varying structural proper-ties are examined. It is observed that the distributions of themodal wave length and displacement are signicantly inuencedDynamic characteristics and VIV of deestructural properties

Weimin Chen a,n, Min Li b, Zhongqin Zheng a, Tianca Key Laboratory of Environmental Mechanics, Institute of Mechanics, Chinese Academb School of Aeronautics Sciences and Engineering, Beijing University of Aeronautics and

a r t i c l e i n f o

Article history:

Received 15 July 2011

Accepted 27 December 2011

Editor-in-Chief: A.I. IncecikAvailable online 25 January 2012

Keywords:

Deepwater riser

Varying tension

a b s t r a c t

A time-domain approach

response of deepwater r

hydrodynamic model. Eff

bending stiffness, on the

presented approach. Our

efciently change the mod

riser VIV response is inu

stiffness and modal wave

journal homepage: www.eSciences, Beijing 100190, China

ronautics, Beijing 100191, China

predicting dynamic characteristics and vortex-induced vibration (VIV)

s is proposed based on nite element simulation combined with a

of axially varying structural parameters, i.e. the effective tension and

amic characteristics and VIV of slender riser are examined using the

erical results indicate that axially varying structural parameters can

ave length as well as the modal displacement. The vibration amplitude of

ed by the complex effect of factors involving the axial tension, bending

th. Generally speaking, for lower modes the response amplitude is largerater riser with axially varying

Tan b

vier.com/locate/oceaneng

ineering

bending stiffness and axis tension, respectively. f(x,t) is thehydrodynamic force consisting of the vortex-induced lift forcefv(x,t) and motion-induced uid damping force ff(x,t), whichare given respectively as follows:

f vx,t 1=2CLrV2Dsin ovt 2a

f f z,t 12 CDrDV _y9V _y9CA14 pD2r _V y14pD2r _V 2bwhere CL is the vortex-induced lift coefcient, r and V are thedensity and velocity of uid, respectively, D is the diameter of

and B3 in Fig. 1. A group of equations describing the relationshipbetween the previous curve and the new curve is as follows:

CL0 CLA0 8a

CL0 1oF0

aAL max8o2

3pF0bAL max2

3o3

4F0cAL max3

CL max

8b

oF0

a 16o2

3pF0bAL max

9o3

4F0cAL max2 0 8c

C00L 1oF0

aA08o2

3pF0bA02

3o3

4F0cA03

0 8d

W. Chen et al. / Ocean Engineering 42 (2012) 7128riser and ov is the vortex shedding frequency. CD and CA are thedrag and added mass coefcient, respectively, of which values canbe determined by experiments.

Generally, CL is expressed as a function of the variablesindicating structural motion so as to take account of the interac-tion between uid and structural dynamic. Thus existing liftmodel is presented as a hyperbolical curve of CL against structuralmotion amplitude (see Fig. 1), of which the coefcients such asCL max and CLA0 are based on experimental data. However, beforethe calculation the motion amplitude is not known. Thus, aniterative calculation is used in the existing model, which iscomputationally expensive and unfortunately is unsuitable tothe time domain calculation of dynamic response simulated usingnite element method (FEM) codes. Therefore, here an alternativelift curve expressed by the instantaneous motion of structure isproposed.

The proposed lift coefcient CL, based on which the lift force(Eq. (2a)) inputs the same energy to the structure in a motionperiod as the existing one does, is assumed to be a function ofstructural velocity in the form of

CL _y CL0a _yb _y2c _y3 3where the coefcients CLo, a, b and c will be determined asfollows. The structural motion in lock-in is assumed asy Asin ot. Then the energies input, respectively by the liftforce F F0 cosot with a constant amplitude F0(1/2)2CL0rV2Dand the unsteady lift force F1 in the form of

F1 F0 cosota0 _yb0 _y2c0 _y3 4in a one-fourth period can be written as

W0 Z T=40

F0Aocosot2dtp4F0A 5a

W1 Z T=40

F1Aocosotdtp4F0Aa0A2b0A3c0A4 5b

or

W1=W0 1a00Ab00A2c00A3 6where a00 (o/F0)a, b00 (8o2/3pF0)b and c00 (3o3/4F0)c. There-fore, a new lift coefcient C

00L can be written as

C00L CL01a00Ab00A2c00A3 7The values of a, b, c and CL0 can be resolved by tting the new

curve with the original one at some key points, e.g. points B1, B2Fig. 1. Lift curve.Fig. 2. Comparisons of VIV response between the presented numerical simulationand the existing experimental results of (a) rigid cylinder in uniform ow,(b) exible cylinder in stepped ow and (c) exible cylinder in sheared ow.

where A0 and ALmax is the amplitudes when CL0 and CLCL max,respectively (as shown in Fig. 1).

Based on the presented lift model, nite element simulationcan be carried out for the dynamic response of a riserundergoing VIV.

The riser (shown in Fig. 3) is 1000 m long and the outer and

interesting that the maximum peak of the amplitude moves fromregions of lower tension to regions of higher tension as the modenumber increases. The theoretical explanations about thesephenomena will be discussed later.

3.2. Effect of axially varying stiffness on dynamic characteristics

In practice, the wall thickness of some deepwater risers oftentypically changes along the riser length for reason of material and

1 2 3 4 5 6 7 8 9 100.0

0.1

0.2

0.3Fr

Mode number

Fig. 4. Natural frequencies of riser with varying and constant tensions.

Fig. 5. Modal shape of riser with varying tension dropping from right to left.

W. Chen et al. / Ocean Engineering 42 (2012) 712 9inner diameters are 0.500 m and 0.445 m, respectively. The toptension T is 6.24106 N (at the right end of the riser) and the toptension factor T/W is 1.57 (W is the riser weight). Because of theeffect of the riser weight W (along the negative direction of thex-axis), the tension along the overall riser length decreaseslinearly from the right end to the left end of the riser, or the riserhas an axially varying tension but not a constant tension.

The natural frequencies of the riser with axially varyingtension are shown and compared with that of the riser with aconstant tension (as the effect of structural weight is neglected) inFig. 4. The selected mode shapes are presented in Fig. 5. Fig. 4indicates that the frequency of the riser with linearly decreasingtension along the riser length gets smaller than the frequency ofthe riser with a constant tension. Observing the modal shapes inFig. 5, we can see that the wave length of the modal shape axiallyvaries with the variation of riser tension, or it becomes longer inregions of higher tension along the riser length; also, the ampli-tude of each peak axially varies with the riser tension, it is2.2. Validation against experimental results

To validate the proposed model, the numerical results arecompared to experimental results (see Fig. 2), i.e. a rigid cylinderundergoing uniform ow by Khalak and Williamson (1999), andthe exible cylinders, respectively undergoing a stepped ow byChaplin et al. (2005) and a sheared ow by Trim et al. (2005).

In the numerical simulations, the hydrodynamic coefcientsare set to be CA1.0, Cd1.2, which were determined based onthe corresponding experiments (Sarpkaya, 2004; Lyons and Patel,1986,). Considering that the lift force for a rigid cylinder isgenerally higher than a exible cylinder, we set the lift coef-cients to be CL0.51.82A1.29A20.707A3 and CL0.221.62A2.31A20.754A3 for rigid and exible cylinder, respec-tively (Liao and Vandiver, 2002; Zhang and Chen, 2010). Fig. 2indicates that the calculated amplitudes agree with experimentalresults for both rigid and exible cylinders in various ow elds.

3. Numerical results and discussions

In practice, especially for deepwater riser, the structuralconguration is usually complex, e.g. the outer or inner diametermay gradually vary along the riser length rather than keepinga constant value and the axial tension is no longer regardedas a constant along the overall length of riser due to theconsiderable structure weight. Therefore the structural propertyalong the overall length of riser is not uniform yet. In this section,the dynamic characteristics and VIV of the riser with axiallyvarying structural properties, i.e. axial tension and bendingstiffness, are examined.

3.1. Effect of axially varying tension on dynamic characteristicsFig. 3. Sketch of the riser with varying tension.0.4

0.5

0.6

0.7

eaue

ncy

Hz

Constant tensionVarying tensionstructural efciencies. Here, the variation of bending stiffness

EI(x) is considered by varying the wall thickness. The riser isdivided into ve parts with the same outer diameter but differentinner diameters, i.e. 0.365 m, 0.385 m, 0.405 m, 0.425 m and0.445 m, where the stiffness is higher at the bottom of the riser.The gravity is neglected here, or the tension has a constant value,6.24106 N, along overall riser length.

Selected mode shapes are presented in Fig. 6. It shows thatbecause of the axial decrease in stiffness, both wave length andamplitude axially increase. The distribution of the amplitudealong the riser length is approximately inversely proportional tothe distribution of the stiffness EI(x) that becomes more evidentfor higher mode, e.g. mode 43 (the last plot in Fig. 6).

3.3. VIV of riser with axially varying structural properties

The parameters of the riser are same with those in Section 3.2.Additionally, the gravity is taken into account. Two cases of toptensions T1.91107 N and T6.24106 N are examined. Aswe know, the motion of top end of ocean platform induced by theenvironmental wave and current may introduce pronouncedinuence on submarine structure (Huse et al., 1999; Lie andKaasen, 2006; Chen et al., 2006). For example, the vertical motion(the heave) of the platform may make the riser tension changeremarkably, or even, in extreme case, into negative value atcertain positions along the riser length.

In our FEM simulations, the hydrodynamic forces involving thevortex-induced lift fv(x,t) and the uid damping force ff(x,t) areexerted on the riser. The vortex-induced lift force fv(x,t) is exertedat the centers of modal shape, and the uid damping force ff(x,t) isuniformly distributed along the riser. The frequencies of the liftforce are consistent with the natural frequencies of the risermodes ranging from mode 1 to 35. Then, the VIV responses arecalculated numerically by the FEM model presented above.

For case of higher tension, T1.91107 N, selected results,the modal shape and VIV displacement of mode 27, are presentedin Fig. 7 where the vortex-induced lift force is loaded at theseventh trough from the left end of the riser, i.e. x/L0.47. Themode shape is shown in Fig. 7a, as well the temporalspatial

W. Chen et al. / Ocean Engineering 42 (2012) 71210Fig. 6. Modal shape of riser with varying stiffness increasing from right to left.Fig. 7. Mode shape and VIV response of riser with axially varying structuralproperties (mode 27, tension T1.91107 N): (a) mode shape of mode 27,

(b) temporalspatial evolution of VIV displacement and (c) rms of displacement.

evolution and rms of displacement are plotted in Fig. 7b and c,respectively.

The results of displacement response (shown in Fig.7b and c)indicate that the vibration of the riser is dominated by anintermediate wave (Zhang and Chen, 2010; Vandiver, 1993)between standing and travelling wave. The displacement attenu-ates slowly at the right part of the riser and quickly at the left partdue to the increase of structural bending stiffness from the rightto left of the riser, or the displacement at the right part of the riseris larger than the displacement at the left part, this amplitudedistribution is similar with that of the mode shape. The explana-tion of above phenomenon might be that the amplitude ofdisplacement depends not only on the attenuation velocity,but also the propagation distance because of the inuence of

travelling wave. Additionally, Fig. 7b and c show that the wavevelocity becomes faster as the wave length increases.

For case of lower tension, T6.24106 N, less than structuralweight, the mode shapes for selected modes, i.e. modes 3, 5, 6, 7,9, 11 and 29, are presented in Fig. 8. It shows that a sinusoidalapproximation is only appropriate for higher modes (higher thanmode 7). For lower modes there is almost no displacement in theregions of higher tension.

Among the calculated VIV results, the abrupt displacementaugmentation at the position with negative tension is observed,as shown in Fig. 9. The maximum augmentation is around 1020times, which may introduce instability and even consequentdamage of structure, thus this unfavorable case should be noted.

3.4. Discussion

For a tension beam written as Eq. (1), if the displacement isassumed as y yxeiot and the right-hand side of Eq. (1) is set tobe zero, the governing equation can be rewritten as

EIx d4y

dx4 dEI

dx

d3y

dx3 d

2EIdx2

d2y

dx2T d

2y

dx2dTdx

dy

dxmxo2yicxoy 0

9

W. Chen et al. / Ocean Engineering 42 (2012) 712 11Fig. 8. Selected modal shapes of riser with varying stiffness for case of lower

tension T6.24106 N.Fig. 9. VIV response of riser with varying stiffness for case of lower tensionBy virtue of the WKB method (Liao and Vandiver, 2002), we let

yx eiyxfa0xa1xa2xg 10

where a1 and a2 are higher order corrections. Also, we let

dyx=dx gx 11

where g(x) is the local wave number. Substituting Eqs. (10) and(11) into Eq. (9) and remaining the leading order terms yields thedispersion relation as follows:

EIxg4xTxg2xmxo2icxo 0 12

Remaining the next order terms yields

EI6ig0g2a0g4a14ig3a002EI0ig3a0Tig0a0g2a12iga00

T 0iga0mo2a1icoa1 0 13T6.24106 N.

Eliminating all terms with a1 by means of Eq. (12) yields thatthe amplitude a0(x) along the riser axis can be written as

a20xEIxg3x1=2Txgx cons: 14According to Eq. (14), for the case of beam with axially varying

tension, wave number g(x) would become smaller or wave lengthl(x) (l(x)(2p/g(x))) would become larger with increasingtension T(x). That is consistent with the modal shapes shown inFig. 5. The value of amplitude a0(x) depends on the value of the

towibrato

ment response is observed. Effects of axially varying structural

The displacement augmentation, around 1020 times, at theposition with lower tension may introduce instability and con-sequent damage of the riser structure that should be paidattention in practical engineering. It is also noted that thepresented approach cannot fully capture the physics of VIV suchas the wake-jumping and hysteresis since it is virtually based on asemi-empirical lift curve.

Acknowledgments

This work is supported by the Intellectual Innovation Project of

(Grant no. YWF-10-01-B05).

W. Chen et al. / Ocean Engineering 42 (2012) 71212properties are examined. Based on our numerical results, we drawthe following conclusions:

1) Along the riser length, modal wave length increases withincreasing tension or decreasing bending stiffness.

2) Modal amplitude is inuenced by axially varying tension,stiffness and wave number (or wave length) as described inEq. (14). Generally, for lower modes the amplitude gets largerat the axis position where tension is smaller, whereas forhigher modes the amplitude gets larger at the axis positionwhere bending stiffness is lower.

3) For the case of VIV response, vibration wave velocity wouldincrease, i.e. the vibration would propagate faster along theriser, due to the increasing modal wave length.from the position with larger stiffness EI(x) to the position withsmaller EI(x) as the mode number increases (see Fig. 6).

4. Conclusions

An approach, for predicting dynamic characteristics and VIV ofdeepwater riser, by nite element simulation combined with acoupling hydrodynamic model in time domain is proposed.Compared with the experiments, satised agreement of displace-stifmber can be seen in Fig. 5.Similarly, for the case of a beam with axially varying bendingfness EI(x), amplitude peak would move along the beam lengthThenucket, i.e. EI(x)g3(x), would mainly dominate the total value duea stronger inuence of the bending stiffness. So at position xth larger tension T(x) (or smaller g(x)), the value of the squarecket is smaller because of the drop of EI(x)g3(x) value. That issay, the amplitude a0(x) would be larger as shown in Eq. (14).movement of the amplitude peak with increasing modeEq.braxwith larger tension T(x), the value of the square bracket is larger,and then the amplitude a0(x) would be smaller because theproduct value of the two terms is constant as written in

(14). However, for higher modes, the rst term in the square References

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Chaplin, J.R., Bearman, P.W., Huera, F.J., et al., 2005. Laboratory measurements ofvortex-induced vibrations of a vertical tension riser in a stepped current. J.Fluid Struct. 21, 324.

Chen, W.M., Zhang, L.W., Li, M., 2009. Prediction of vortex-induced vibration ofexible riser using an improved wake-oscillator model. In: Proceedings of theASME 28th International Conference on Ocean, Offshore and Arctic Engineer-ing OMAE 2009, May 2009, Honolulu, Hawaii.

Chen, X.H., Ding, Y., Zhang, J., et al., 2006. Coupled dynamic analysis of a mini TLP:comparison with measurements. Ocean Eng. 33, 93117.

Gei, F., Hui, L., Hong, Y.S., 2007. Vortex-induced vibration of submarine oatingtunnel undergoing shear ow. J. Grad. Sch. Chin. Acad. Sci. 24 (3), 352356. (inChinese).

Grant, R., Litton, R., Finn, L., Maher, J., Lambrakos, K., 2000. Highly compliant rigidrisers: Field test benchmarking a time domain VIV algorithm. In: Proceedingsof the Offshore Technology Conference, Houston, Texas, Offshore TechnologyConference, OTC 11995.

Huse, E., Kleiven, G., Nielsen, F.G., 1999. VIV-induced axial vibration on deep searisers. In: Proceedings of the Offshore Technology Conference, Houston, Texas,Offshore Technology Conference, OTC 10932.

Khalak, A., Williamson, C.H.K., 1999. Motions, forces and mode transitions invortex-induced vibrations at low mass-damping. J. Fluid Struct. 13, 813851.

Liao, J.C., Vandiver, J.M., 2002. Vortex-Induced Vibration of Slender Structure inUnsteady Flow. Ph.D. Dissertation, Massachusetts Institute of Technology.

Lie, H., Kaasen, H.K., 2006. Modal analysis of measurements from a large-scale VIVmodel test of a riser in linearly sheared ow. J. Fluid Struct. 22, 557575.

Lyons, G.J., Patel, M.H., 1986. A prediction technique for vortex induced transverseresponse of marine risers and tethers. J. Sound Vib. 111 (3), 467487.

Sarpkaya, T., 2004. A critical review of the intrinsic nature of vortex-inducedvibration. J. Fluids Struct. Mech. 46, 389447.

Trim, A.D., Braaten, H., Lie, H., et al., 2005. Experimental investigation of vortex-induced vibration of long marine risers. J. Fluid Struct. 21, 335361.

Vandiver, J.K., 1993. Dimensionless parameters important to the prediction ofvortex-induced vibration of long, exible cylinders in ocean currents. J. FluidStruct. 7 (5), 423455.

Zhang, L.W., Chen, W.M., 2010. Study on the parameters for determining responsetypes of long exible riser undergoing VIV in deepwater. China Offshore Oiland Gas 22 (3), 202206. (in Chinese).inuence of the tension than the bending stiffness. So at position the Chinese Academy of Sciences (Grant no. KJCX2-YW-L07) andthe Fundamental Research Funds for the Central Universitiesterms in the square bracket of Eq. (14). For modal shape of lowermode, the second term in the square bracket, (1/2)T(x)g(x),dominates the total value of the square bracket due to a stronger

Dynamic characteristics and VIV of deepwater riser with axially varying structural propertiesIntroductionNumerical modelingHydrodynamic model and finite element simulationValidation against experimental results

Numerical results and discussionsEffect of axially varying tension on dynamic characteristicsEffect of axially varying stiffness on dynamic characteristicsVIV of riser with axially varying structural propertiesDiscussion

ConclusionsAcknowledgmentsReferences