LIU Juan, and HUANG Weiping

1)Shandong Key Laboratory of Ocean Engineering,Ocean University of China,Qingdao266100,P. R. China

2)Institute of Civil Engineering,Agriculture University of Qingdao,Qingdao266009,P. R. China

The Coupling VIV Analysis of SCRs with Rigid Swing

LIU Juan1),2), and HUANG Weiping1),*

1)Shandong Key Laboratory of Ocean Engineering,Ocean University of China,Qingdao266100,P. R. China

2)Institute of Civil Engineering,Agriculture University of Qingdao,Qingdao266009,P. R. China

With the development of deepwater oil and gas exploration, Steel Catenary Risers (SCRs) become preferred risers for resource production, import and export. Vortex induced vibration (VIV) is the key problem encountered in the design of SCRs. In this study, a new model, the rigid swing model, is proposed based on the consideration of large curvature of SCRs. The sag bend of SCRs is assumed as a rigid swing system around the axis from the hanging point to the touch down point (TDP) in the model. The torque, produced by the lift force and the swing vector, provides the driving torque for the swing system, and the weight of SCRs provides the restoring torque. The simulated response of rigid swing is coupled with bending vibration, and then the coupling VIV model of SCRs is studied in consideration of bending vibration and rigid motion. The calculated results indicate that the rigid swing has a magnitude equal to that of bending vibration, and the rigid motion affects the dynamic response of SCRs and can not be neglected in the VIV analysis.

rigid swing; vortex induced vibration; steel catenary riser; dynamic model

1 Introduction

With the development of offshore oil and gas industry, deepwater resource exploration is expanding greatly, and various innovative floating structures, such as TLPs and Spars, are developed for operating under severe deepwater environment. Risers, connecting wells and floating platforms for production, import and export of resources, are the key component in deepwater development. As a new type of deepwater risers, Steel Catenary Risers (SCRs) with the advantages of low costs and no need of top tension, have become the preferred riser systems in recent years (Huanget al., 2009). Vortex induced vibration (VIV) is a core problem in the design of SCRs, and special configuration and complex flow make the VIV analysis of SCRs much more complicated than Top Tensioned Risers (TTRs) (Cunffet al., 2004; Mekha, 2002; Meng and Chen, 2012; Gaoet al., 2011).

Many problems of SCRs have been studied and achievements have been made since the first SCR was installed in 1994 (Nakhaee and Zhang, 2010; Hodder and Byrne, 2010; Lie and Kaasen, 2006; Thethi, 2001; Ioannis, 2010; Holtamet al., 2009). Presently, only bending vibration under lift force is considered in VIV studies of SCRs, while the torque, produced by the lift force and the vector, is often neglected, that is, the swing model around an axisis not considered in dynamic model studies of SCRs. A new model, the rigid swing model, is proposed in this paper. The model is based on the consideration of large curvature of sag bends by taking SCRs as a rigid swing system around the axis from the hanging point to the touch down point (TDP). The swing response is then coupled with bending vibration as inertial force and hydrodynamic damping to study the VIV of SCRs. Numerical simulations show that the rigid swing affects the dynamic response of lower parts greatly and can not be ignored in the VIV study of SCRs.

2 Rigid Swing Model of SCRs

The rigid swing model of SCRs is shown in Fig.1, where ODC is the SCRs system, O is the hanging point, D is the touch-down point, and C is the connection point for wells and risers. The sag bend of SCRs is assumed as a rigid swing system around the axis from O to D in the model, where A is an arbitrary point on the sag bend, B is the intersection point of the swing plane of point A and the swing axis OD,ωis the unit vector of axis OD,sis the vector from the axis to point A, andFLis the lift force due to an angle betweenFLand the SCRs plane (xoyplane). The torque is produced by the forceFLand the vectors, which provides the driving force for the swing.mgis the weight per unit length of riser and is the restoring force for the swing. OAD is in thexoyplane when the swing system is in balance, point A' is the position of point A after a swing, andrαis the swing angle.

The swing equation per unit length of risers is derived based on the theory of momentum moment：

where,mandmaare the mass and the added-mass per unit length of risers,cais the added-damping coefficient,fLzandfLxare the projections of lift force onz- andx-axis, respectively, andrα,, andare the angular displacement, the velocity, and the acceleration of a rigid swing, respectively.sis the vector from the swing axis to a unit of the riser,.sis the norm of the vectors,s1,s2ands3are the projections ofsonx-,y- andz-axis, respectively,

where (xA,yA,zA) and (xB,yB,zB) are the coordinates of points A and B, respectively, ands3=0 when the system is in balance (Fig.1).

wheredis the length of axis OD,

and (xD,yD,zD) and (xO,yO,zO) are the coordinates of points D and O, respectively.

Fig.1 Rigid swing model of SCRs.

In the finite element analysis of the rigid motion, an unit of the riser is shown in Fig.2, in whichiandjare the nodes of the unit,x'-axis is the local axis of the unit and the origin is set at nodei.siandsjare the swing vectors of nodesiandj, respectively, and the unit vectorsis expressed by node vectors：

By substituting the node coordinates and corresponding vectors into Eq. (2), the linear equations about parameteracan be obtained：

Eq.(3) can be solved as：

By substituting Eq. (4) into Eq. (2), the unit vector expressed by node vectors is derived as：

and,

By bringsands2into Eq. (1), and integrating the equation along the unit and then the riser, the equation of rigid swing of SCRs is derived as：

whereIis the moment of inertia,Cis the hydrodynamic damping coefficient,Kis the restoring-force coefficient, andMαis the external force moment for rigid swing. Solving Eq. (7) by time-domain method, the swing response of SCRs can be obtained.

Fig.2 Swing unit of the SCR.

3 Coupling Analysis of VIV of SCRs

By considering both bending vibration and rigid swing, VIV of SCRs can be expressed as follows：

whererb,andare the bending displacement, the velocity and the acceleration of risers, respectively;andare the linear velocity and the acceleration of rigid swing,,, respectively;cis the structure damping andcais the hydrodynamic damping,kis the bending stiffness andfLis the lift force per unit length of risers.

By moving the terms associated with rigid swing to the right hand side of Eq. (8), the following equation can be obtained：

Eq. (9) is the popular form of the vibration equation of risers, which shows that the rigid motion is coupled with the bending vibration as inertial force and hydrodynamic damping.

Eq. (9) can be expressed as coordinate components：

where (ub,vb,wb),, andare the projections of the bending displacement, the velocity and the acceleration onx-, y-andz-axis, respectively;andare the projections of the linear velocity and the acceleration of rigid swing, where

whereandare the rigid swing response obtained from Eq. (1).

Eq. (10) can be solved by the time-domain method and the solutions are the VIV response of SCRs with both bending vibration and rigid swing.

4 Numerical Simulations

Based on the nonlinear FEM, a dynamic code of SCRs, Cable3D, was developed (Chen, 2002). The flexible cable theory is applied to simulate risers in the code. By using small extensible slender rods with bending stiffness for SCR modeling, the accurate static configuration and vibration response of SCRs can be obtained with the code (Bai, 2009).

In this study, the static configuration and the VIV response of a SCR are first simulated using Cable3D, then a dynamic program, named RT_Res, is developed based on the rigid swing model of SCRs. The obtained response is further coupled with bending vibration, a new program, named VRT_Cable, is developed to simulate the VIV of the SCR, and the obtained results are compared with the prediction of Cable3D.

The SCR is 2500 m long with an outer diameter of 0.355 m and a thickness of 0.025 m. The depth of water is 1100 m and the horizontal distance from the hanging point to the wellhead is 1846 m. The restraint of the hanging point is by hinged joint and the TDP is by elastic restraint. Other main parameters are listed in Table 1. With a top tension of 2100 kN, the static configuration of the SCR is shown in Fig.3; the angle between the tangent direction of the hanging point of the SCR and the vertical is 16°

Table 1 Key parameters of the SCR and flow

Fig.3 Static configuration of the SCR.

The incoming flow is alongx-axis with a speed of 0.20 m s-1and Strouhal constantSt=0.2 (see Table 1 for other flow parameters). The VIV response of the riser is simulated using Cable3D. The history curves of the 18thand 225thnodes are shown in Figs.4-5 and the frequency of the curves is the Strouhal frequency of 0.0769 (see Fig.3 for the location of both nodes).

Calculated by the program RT_Res in the rigid swing model, the history response of the angular displacement of the riser is shown in Fig.6. It can be seen that the swing angle of the riser is 3×10-4rad. If converted to a linear displacement using the equationrr=ar×s, the linear displacement of the 225thnode is 0.03 m and is equal to that of the bending vibration as shown in Fig.5. The rigid swing of the riser is a forced motion, and therefore the frequency of the swing is the Strouhal frequency 0.0769, which is the same as that of the bending vibration.

Fig.4 The VIV response history of the 18thnode.

Fig.5 The VIV response history of the 225thnode.

Fig.6 The history response of rigid swing of the SCR.

Figs.7-8 are the VIV history curves of the 18thand 225thnodes with rigid swing simulated by VRT_Cable. It can be seen that the swing response has a small effect on the 18thnode but a great effect on the 225thnode. Because the swing vector at the lower part of the riser is larger than that at the upper part, the linear displacement at the lower part is greater for the case of the same swing angle, and, therefore, the swing responses affect the lower part of risers more significantly and should be considered as an important factor in the VIV analysis of SCRs.

Fig.7 The response history of the 18thnode with rigid swing.

Fig.8 The response history of the 225thnode with rigid swing.

Huanget al.(2011, 2012) studied the VIV of flexible cylinders based on model experiments, their results indicated that when VIV is beyond the lock-in district, the vortex frequency of the model is not equal to Strouhal frequency and it changes with different natural frequencies of the model; besides, the VIV of a cross flow is a strong random vibration and the response frequency is in a wide range, that is, the vortex shedding mode of the wake is very unstable due to the vibration of cylinders. Based on the results, Liu and Huang (2013) developed a lift force model by considering structure vibration and its coupling with fluid, and this revised model describes the stochastic force with varying frequencies and amplitudes.

Based on the revised lift force model the program CPVRT_Cable is updated to simulate the VIV of a SCR, and the response of the riser is a stochastic vibration with varying frequencies and amplitudes. Figs.9 and 10 are the responses of the 84thand 276thnodes of the riser; the dashed line and the solid line are history curves with and without rigid swing, respectively. The location of both nodes is shown in Fig.3, and Figs.9 and 10 also show that the rigid swing affects the upper part of the riser slightly but has a great effect on the lower part.

Fig.9 The VIV history of the 84thnode with rigid swing.

Fig.10 The VIV history of the 276thnode with rigid swing.

Fig.11 Response spectrum of the 276thnode without rigid swing.

Fig.12 Response spectrum of the 276thnode with rigid swing.

Fig.11 is the VIV response spectrum of the 276thnode without the rigid swing, and the peak frequency of the curve is slightly lower than the Strouhal frequency of 0.0769. Fig.12 is the spectrum of the node with the rigid swing. Comparing the two figures, it can be seen that the curve of Fig.12 contains more frequency components, and the peak and bandwidth are rather large. The comparison also indicates that the response of the riser and the corresponding added damping of the fluid have increased significantly with the rigid swing of the SCR.

5 Conclusions

SCRs, connecting floating platforms and wellheads as both flow lines and risers, have become the popular riser system in marine resource exploration. Many studies have been conducted in the past several decades, such as dynamic characteristics of SCRs, SCR coupled with platforms and VIV models of SCRs.

In this study, the dynamic modeling results of the vortex induced vibration of SCRs are examined. By introducing the concept of rigid swing, a new dynamic model in the VIV study of SCRs is developed. The model can be used to simulate the dynamic characteristics of SCRs and to reproduce the out-of-plane motion of SCRs reasonably well. Simulation results show that the swing responses have the same order of magnitude as the bending vibration, and affect the lower part of risers more significantly and should not be ignored in the VIV analysis of SCRs.

The assumption that the touch down point is a fixed point may introduce errors to model results, and further research is necessary on the rigid swing model of SCRs.

Acknowledgements

The study is funded by the National Natural Science Foundation of China (51079136, 51179179, 51239008), and our thanks also go to Professor Jun Zhang of Texas A&M University, the copyright owner of Cable3D.

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(Edited by Xie Jun)

(Received April 7, 2013; revised May 9, 2013; accepted January 6, 2015)

? Ocean University of China, Science Press and Springer-Verlag Berlin Heidelberg 2015

* Corresponding author. Tel： 0086-532-66781850 E-mail： wphuang@ouc.edu.cn