numerical simulation of the vortex-induced vibration of a

Numerical Simulation of the Vortex-Induced Vibration of A CurvedFlexible Riser in Shear FlowZHU Hong-juna, b, LIN Peng-zhib, *aState Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu610500, China

bState Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu 610065, China

Received October 27, 2017; revised March 1, 2018; accepted March 27, 2018

©2018 Chinese Ocean Engineering Society and Springer-Verlag GmbH Germany, part of Springer Nature

AbstractA series of fully three-dimensional (3D) numerical simulations of flow past a free-to-oscillate curved flexible riser inshear flow were conducted at Reynolds number of 185–1015. The numerical results obtained by the two-wayfluid–structure interaction (FSI) simulations are in good agreement with the experimental results reported in theearlier study. It is further found that the frequency transition is out of phase not only in the inline (IL) and crossflow(CF) directions but also along the span direction. The mode competition leads to the non-zero nodes of the root-mean-square (RMS) amplitude and the relatively chaotic trajectories. The fluid–structure interaction is to someextent reflected by the transverse velocity of the ambient fluid, which reaches the maximum value when the riserreaches the equilibrium position. Moreover, the local maximum transverse velocities occur at the peak CFamplitudes, and the values are relatively large when the vibration is in the resonance regions. The 3D vortex columnsare shed nearly parallel to the axis of the curved flexible riser. As the local Reynolds number increases from 0 at thebottom of the riser to the maximum value at the top, the wake undergoes a transition from a two-dimensionalstructure to a 3D one. More irregular small-scale vortices appeared at the wake region of the riser, undergoing largeamplitude responses.Key words: vortex-induced vibration, flexible riser, shear flow, fluid–structure interaction, computational fluid

dynamics

Citation: Zhu, H. J., Lin, P. Z., 2018. Numerical simulation of the vortex-induced vibration of a curved flexible riser in shear flow. China OceanEng., 32(3): 301–311, doi: https://doi.org/10.1007/s13344-018-0031-z

1 IntroductionLong flexible risers are employed in the offshore in-

dustry to convey fluids from the seabed to sea level. Suchslender circular cylinders are vulnerable to undergoing thevortex-induced vibration (VIV) in complex waves and cur-rents. Fatigue damage of risers caused by the VIV may res-ult in huge loss and environmental catastrophe. Therefore,detailed understanding and efficient prediction of the vibra-tion response of risers have been increasingly urgent and hotissues for ocean engineers and researchers.

Considerable efforts have been dedicated to understand-ing the VIV of rigid cylinders over the past few decades.One may refer to the comprehensive reviews by Sarpkaya(2004), Williamson and Govardhan (2004, 2008) and Bear-man (2011). Until the past two decades, the VIV of flexiblerisers with a large length-to-diameter (L/D) has been invest-igated experimentally by many researchers (Trim et al.,

2005; Song et al., 2011; Xu et al., 2017). Trim et al. (2005)conducted VIV tests of a pipe model with an L/D of 1400 ina towing tank of the Norwegian Marine Technology Re-search Institute. In the similar experiments performed bySong et al. (2011), the aspect ratio was up to 1750 and multi-mode response of a bare cylinder was observed. More re-cently, the response feature of an inclined flexible bare cyl-inder with an L/D of 350 has been investigated by Xu et al.(2017) through a series of experiments in a towing tank.These experiments provide better insights into some VIVcharacteristics such as the dominant modes, response amp-litudes and orbital trajectories versus the reduced velocity.Since the cylinder models were towed in the experiments,only uniform flow or linearly sheared flow was generatedand considered. In our earlier experimental study (Zhu et al.,2016), the VIV of a flexible riser was tested in log-lawshear flow generated in a water flume, which better mod-

China Ocean Eng., 2018, Vol. 32, No. 3, P. 301–311DOI: https://doi.org/10.1007/s13344-018-0031-z, ISSN 0890-5487http://www.chinaoceanengin.cn/ E-mail: [email protected]

Foundation item: The research work was financially supported by the National Natural Science Foundation of China (Grant Nos. 11502220 and51479126), the Youth Science and Technology Foundation of Sichuan Province (Grant No. 2017JQ0055), and the Youth Scientific and TechnologicalInnovation Team of the Safety of Deep-Water Pipe Strings of Southwest Petroleum University (Grant No. 2017CXTD06).*Corresponding author. E-mail: [email protected]

elled the real marine environment in the continental shelfarea. Additionally, the riser model was deployed as thecatenary type with a curved shape. The response amplitude,dominant frequency and wake mode of scattered markingswere observed in the experiments. Though the informationobtained by the experiments is limited, it provides somegood benchmarks for verifying numerical models.

A number of numerical studies have been carried out onthe VIV of flexible cylinders, in order to capture more de-tailed flow characteristics and vibration features. Multi-striptechnique, as a quasi-three-dimensional method, was previ-ously employed to simulate the riser response (Willden andGraham, 2001; Yamamot et al., 2004). The transverse vibra-tion of a flexible cylinder with an aspect ratio of 100 wassimulated by Willden and Graham (2001) in a linearlysheared flow at a low Reynolds number. Cellular sheddingwas observed and the vibration shown significant spanwisecorrelation over a large length of the cylinder. Yamamot etal. (2004) adopted the same method to simulate the oscilla-tion of a flexible riser subjected to a uniform flow. In theirsimulations, a 2S (two single vortices are shed per cycle,similar to a von Karman vortex street) mode was found inthe regions of low amplitudes, which changed to a 2P (twopairs of counter-rotating vortices are shed per cycle) modein the regions of larger amplitudes. Since the cylinder is dis-cretized in several panels in such multi-strip method, thethree-dimensional (3D) vortex structure and instantaneousshape of the cylinder cannot be treated accurately. Thus,fully 3D numerical simulations have been carried out byseveral scholars. Kamble and Chen (2016) conducted 3Dsimulation of the VIV of flexible cylinders using computa-tional fluid dynamics (CFD) approach with an overset gridsystem, and the results were in good agreement with the ex-perimental data by Trim et al. (2005). Wang and Xiao(2016) carried out a numerical study on the vibration re-sponse of a flexible riser using the commercial softwareANSYS CFX, and the numerical approach was validatedwith the experimental data by Lehn (2003). Further under-standings of the VIV of vertical flexible cylinders have been

obtained through such fully 3D simulations, but which arestill quite limited. Moreover, few studies concerned about acurved flexible cylinder in a nonlinear shear flow.

In this paper, the VIV of a curved flexible riser model inlog-law shear flow is investigated using a fully 3Dfluid–structure interaction (FSI) numerical approach. Thenumerical results of the flexible riser under such a specificarrangement and flow conditions are validated with the ex-perimental results reported in the earlier study (Zhu et al.,2016). Apart from the vibration responses, the 3D wakestructure and instantaneous shape of the flexible riser arediscussed in detail.

2 Problem descriptionThe curved flexible riser model tested in our earlier

study (Zhu et al., 2016) is considered in simulations. Theflexible riser is fixed in two ends and is deployed as thecatenary type in a vertical plane with its concave facing theincoming flow. The main parameters of the riser model aresummarized in Table 1. As shown in Fig. 1, a cuboid com-putational domain is adopted to simulate the flow past sucha curved riser. The height (H) equals the experimental wa-ter depth of 0.5 m. The width is 20D (D is the external dia-meter of the riser) and the two lateral boundaries are 10Daway from and parallel to the xoz plane at which the risermodel is located. The horizontal distance between the inletand the lower end of the riser is 10D while the distance is

Table 1 Main properties of the riser model.Parameter Value

Riser model length, L 0.65 mStill water depth, H 0.50 mExternal diameter, D 0.006 mInternal diameter, Di 0.004 mAspect ratio, L/D 108

Modulus of elasticity, E 6.22×106 N/m2

Linear mass, m 0.018 kg/mMass ratio, m* 1.08

Damping ratio in water, ζ 0.1018Mass-damping ratio, m*ζ 0.1099

Fig. 1. Sketch of flow past a curved flexible riser.

302 ZHU Hong-jun, LIN Peng-zhi China Ocean Eng., 2018, Vol. 32, No. 3, P. 301–311

30D between the outlet and the upper end.

U Ur Ur = U/(

f1yD)

In the experiments, stable log-law shear flows with amean velocity of 0–0.25 m/s were generated in the flumewith the water depth fixed at 0.5 m, and the flow velocityprofiles were obtained by an Acoustic Doppler Velocimetry(ADV). Among them, seven representative flow velocityprofiles are adopted in simulations, as listed in Table 2. Inthe table, q is the flow rate per width, U is the free-streamvelocity, and ( ) are the mean velocityand the mean reduced velocity, respectively, and Re is thecorresponding Reynolds number. The first natural frequen-cies in the in-line (IL) and cross-flow (CF) directions aref1x=2.43 Hz and f1y=1.16 Hz, respectively, which were ob-tained through the decay tests. The second and third naturalfrequencies were calculated using the ANSYS modal, whichare f2x=5.13 Hz, f2y=3.01 Hz, f3x=9.24 Hz and f3y=5.96 Hz.The upstream boundary is imposed with a velocity inletboundary condition defined as the free-stream velocity pro-file. The downstream boundary is imposed with a pressureoutlet boundary condition, which meets a linear static pres-sure distribution. Apart from the inlet and outlet boundaryconditions, the opening boundary condition is applied on thetop boundary as p=0 Pa, the symmetry condition is spe-cified on the two lateral boundaries, and the no-slip bound-ary condition is assigned on the surface of the riser and thebottom wall.

As shown in Fig. 1, 31 black rings were marked alongthe flexible riser model in the experiments, which representdifferent locations along the span. The number of markingsand the corresponding height (Z/H) are employed in thecomparison between the numerical and experimental resultsin this paper.

3 Numerical methodsThe vibration response of a flexible riser is a typical flu-

id-structure interaction issue. It is still a challenging task toconduct fully coupled calculation (He et al., 2012). Thus,the fluid and solid fields are usually solved separately, i.e.,there are no iterations between the fluid and solid fieldswithin one time step (Wang and Xiao, 2016). In the presentwork, a two-way FSI approach is utilized through the multi-field solver of the software package ANSYS MFX, in whichthe CFX is employed to solve the fluid field while the tran-sient structural module is adopted to calculate the structure

response. The flow chart of the solution procedure is plot-ted in Fig. 2.

The unsteady Reynolds-Averaged Navier–Stokes (UR-ANS) equations coupled with a shear stress transport (SST)k–ω turbulence model (Menter, 1994) are used to capturethe flow past the curved flexible riser. Additionally, the Ar-bitrary-Lagrangian–Eulerian (ALE) scheme (Hughes et al.,1981) is adopted to follow the moving boundary of the riser.Thus, the flow governing equations are expressed as:∂ui

∂xi= 0; (1)

∂ρui

∂t+∂ρui

(u j− u j

)∂x j

= − ∂p∂xi+μ∇2ui−

∂ρu′iu′j

∂x j, (2)

−ρu′iu′jwhere is the Reynolds stresses defined by

−ρu′iu′j = μt

(∂ui

∂x j+∂u j

∂xi

)− 2

3ρktδij, (3)

u′i

u′iu j

where t is the time; p is the pressure; ρ is the fluid density; xiis the Cartesian coordinate in the i direction; ui and rep-resent the instantaneous Cartesian velocity component andthe fluctuation velocity component, respectively; ui or with a top bar represents its time average one; is the velo-city of the mesh movement in the xj direction; μ and μt are

Table 2 Simulation cases (z is the water height calculated from the bottom).Case q (m2/s) U (m/s) U (m/s) Ur ReC2 0.01746 U=0.00878+0.004ln(1661.13z) 0.0309 4.441 185C5 0.03081 U=0.01396+0.006ln(2638.52z) 0.0551 7.921 330C6 0.03674 U=0.01581+0.007ln(2994.01z) 0.0616 8.853 370C8 0.04139 U=0.02062+0.009ln(3906.25z) 0.0828 11.894 497C10 0.05828 U=0.02808+0.013ln(6172.84z) 0.1166 16.749 700C12 0.06894 U=0.03263+0.015ln(5319.15z) 0.1379 19.812 827C16 0.08452 U=0.04035+0.019ln(6172.84z) 0.1690 24.286 1015

Fig. 2. Flow chart of the two-way FSI solution procedure.

ZHU Hong-jun, LIN Peng-zhi China Ocean Eng., 2018, Vol. 32, No. 3, P. 301–311 303

the dynamic viscosity and turbulent viscosity, respectively;kt is the turbulent energy and δij is the Kronecker delta func-tion (δij = 1 for i = j). The SST k–ω turbulence model pro-posed by Menter (1994) has been demonstrated good per-formance in capturing turbulent vortices (Zhu and Yao,2015), and is applied to closure the flow governing equa-tions. One may refer to the literatures by Menter (1994) formore detailed information about this model.

The flow governing equations are discretized with the fi-nite volume method. The Rhie–Chow interpolation (Rhieand Chow, 1982) is used to deal with the coupling betweenthe pressure and the flow velocity. A second-order back-wind Euler scheme and a high resolution scheme are em-ployed for the temporal discretization and the discretizationof the convective terms, respectively.

In the software package ANSYS MFX, a finite elementmethod is used and the riser is discretized into n elements.For each element, the governing equation is described as:

M{∂2Xi

∂t2

}+C

{∂Xi

∂t

}+K {Xi} = {Fi (t)} , (4)

where M is the mass matrix with the added mass being con-sidered, C is the damping matrix, K is the equivalent stiff-ness matrix which is converted from the bending stiffness(EI), Xi donates the displacement in the i direction and Fi(t)is the hydrodynamic force in the i direction, which is com-puted by performing an integration involving both the pres-sure (p) and viscous stress (τ) on the micro element of ther i s e r . Th e mo t i on e qu a t i o n s a r e s o l v ed by t h eHilber–Hughes–Taylor method (Hilber et al., 1977).

∇ · (km∇S i) = 0

In one time step, the hydrodynamic forces are firstly ob-tained by solving the flow governing equations. Then, themotion equations of the riser are solved to find a new posi-tion for the structure. Simultaneously, the mesh deformedfollowing the motion of the riser, which is prepared for theflow field calculation at the next time step. The displace-ment diffusion model is employed for the mesh deforma-tion, which meets , where Si is the displace-ment of the mesh points in the xi direction and km is themesh stiffness. In the simulation, the mesh stiffness is set as10.

The time step Δt adopted for each case meets the cri-terion that the Courant number Co is kept below 1.0 in theentire computational domain, where Co is defined as:

Co =|u|∆t∆l, (5)

|u|U∆t/D

where Δl is the size of a cell in the direction of the flow ve-locity and is the magnitude of the velocity through thecell. This results in a non-dimensional time-step ( ) of0.001.

As shown in Fig. 3, a structured hexahedron mesh isconstructed for the fluid domain. The regions around theriser model and near the bottom wall are tessellated with afine mesh. In a xoy plane, the riser perimeter is equally dis-

cretized with 116 nodes. The radial size of the first layer ofmesh next to the riser is 0.01D (y+<5). The element expan-sion ratio in the O-ring region around the riser is kept be-low 1.2 while it is 1.0 for the rest regions. A close-up viewof the mesh around the riser in the xoy plane is shown inFig. 3c. Apart from the fluid domain, the solid domain istessellated with a two-layer structured mesh, in which theriser perimeter and length are equally discretized with 24nodes and 60 segments, respectively. The two ends of theriser are fixed, and the outer surface of the riser is set to be afluid–solid interface for the data transfer.

A mesh independence test is conducted to ensure thatthe numerical results are independent of the grid size, i.e.,the grid size is decreased until the results have no change.Six mesh systems with different sizes are used to simulatethe case C12 and the results are compared in Fig. 4. Thetotal number of the cells increases from 263712 (M1) to1524808 (M6). The cross-flow vibration displacements atfive different locations along the span direction are con-sidered, and the locations are distinguished with the mark-ing numbers used in the experiments. As the number of thecells increases, the difference of the numerical results be-

Fig. 3. Computational mesh.

Fig. 4. Results of the mesh independence test (C12).


comes smaller, which reduces to only 0.98% between M5and M6. The displacements predicted by M5 and M6 are inagreement with the experimental results (Zhu et al., 2016)though a small deviation presents at some locations. By tak-ing into account the computational efforts, M5 is selected asthe final mesh to carry out the simulations.

4 Results and discussion

4.1 Vibration responseFig. 5 compares the IL and CF root-mean-square (RMS)

amplitudes between the present simulations and the earlierexperiments (Zhu et al., 2016). The envelopes of the vibra-tion amplitudes along the riser model at different Reynoldsnumbers illustrate that the dominant mode varies with theflow velocity. When the Reynolds number is smaller than370, both the IL and CF oscillations are dominated by thefirst mode. The dominant mode changes to the second modewhen the Reynolds number is in the range of 497–827. Thisphenomenon of the dominant mode increasing with thegrowth of the flow velocity is observed in many experi-mental studies on VIV of flexible cylinders, not only in thecases of uniform flows (Trim et al., 2005; Song et al., 2011)but also in the cases of linearly sheared flows (Huang et al.,2011; Gao et al., 2015). At Re=1015, the IL dominant mode

shifts to the third one while the CF dominant mode keepsthe second one. It illustrates that the dominant modes areout of phase in the streamwise and transverse directions.Such asynchronous phenomenon was observed by Wu et al.(2016) and Seyed-Aghazadeh and Modarres-Sadeghi (2016)in their studies on the vibration response of a straight flex-ible cylinder. In terms of the dominant modes and the envel-opes of amplitudes, the simulation results are in good agree-ment with the experimental data. However, there are somedeviations in the amplitudes along the riser model. The loca-tions presenting the peak and trough exhibit relatively largedeviations. Both the envelopes of the IL and CF amplitudesare asymmetry along the span direction, i.e., the peak pointof the first-mode dominated vibration and the trough pointof the second-mode dominated vibration do not locate at thehalf of the water depth. The nonlinear flow velocity profileand the nonlinear deployment of the riser contribute to thisresult. For the CF amplitude, it is clearly seen that thetrough point of the second-mode dominated vibration, as anode exhibiting the minima of the response envelope, loc-ates at Marking #17 (Z/H = 0.37), which is just the mid-point of the riser model. It indicates that the midspan of aflexible cylinder is a characteristic point for vibration re-sponse. Additionally, the CF RMS amplitude of the node

Fig. 5. Comparison of the RMS amplitudes and orbital trajectories at seven typical positions.


does not equal zero, illustrating that more than one modeparticipate in the vibration and the second mode is the dom-inant one. It is confirmed in Fig. 6 that the time histories ofthe vibration amplitudes exhibit non-constant values sincemultiple frequencies take part in the dynamic response. Thecontributions of different frequencies are variable along thespan. This mode competition phenomenon was observed byGao et al. (2016) for a straight flexible cylinder in uniformand linearly sheared flows, and by Han et al. (2017) for aninclined flexible cylinder in uniform flow. However, thenode of the IL RMS amplitude is not located at the midspanof the riser, which is different from the straight flexible cyl-inders. For this curved flexible riser, the IL motion belongsto an in-plane vibration while the CF motion is an out-of-plane vibration. Therefore, there are some different charac-teristics in the vibration responses between the two differ-ent directions, which are reported by Meng and Chen (2012)and Quéau et al. (2013).

Apart from the RMS amplitudes, the orbital trajectoriesof the riser model at seven typical positions are also com-pared in Fig. 5. The trajectories are plotted in the xoy planewith the same scales in the x and y axes. As seen from Fig.5, some of the trajectories exhibit eight-shape figures indic-ating the 2:1 ratio on the IL to CF excited frequencies.However, C-shape and line-shape trajectories appear atsome positions where the IL amplitudes are much smaller

than the CF amplitudes. These trajectories mainly occur atsmall flow velocities or at the positions near the two ends ofthe riser model. For the cases dominated by the secondmode, some of the trajectories become a bit chaotic, whichis attributed to the frequency competition.

The vibration frequencies of the riser model at seventypical positions are shown in Fig. 7. In the figure, the fre-quencies are normalized with the first natural frequencies(f1x and f1y for the IL and CF directions, respectively), and fsis the Strouhal vortex shedding frequency defined as:

fs =UStD, (6)

where St is the Strouhal number taken as 0.18. The obliquedashed lines in Fig. 7 describe the normalized Strouhal vor-tex shedding frequency which increases linearly with the re-duced velocity.

Ur

It is clearly seen that only one vibration frequency ap-pears at low reduced velocities ( ≤7.921) while morethan one frequency take part in the vibration when the re-duced velocity is larger than 8.853. There are two frequen-cies participating in the CF oscillation for different posi-tions along the riser in the range of Re=370–1015. It is in-teresting to find the occurrence of the frequency competi-tion at Re=370 (C6), which is dominated by the first mode.The reason is that it undergoes the mode transition from thefirst mode to the second one. When the vibration is domin-

Fig. 6. Time histories of the vibration amplitudes at seven typical positions at Re = 827 (C12).


ated by the second mode (the dominant frequency becomeshigher), the low frequency corresponding to the first modestill appears although its contribution is lower. Additionally,the frequencies at different positions do not keep the samevalue in the mode-transition cases such as C6, C8 and C16.It indicates that the transition of the vibration frequencies isout of phase along the span direction. Though some part ofthe riser keeps the dominant frequency of the first mode(such as the midspan in C8), the dominant mode of the risershifts to the second one when most parts vibrate in a domin-ant frequency close to the second natural frequency.

The most obvious difference between the IL and CF vi-bration frequencies is that there are three frequencies parti-cipating in the IL vibration at Re=1015 (C16), which isdominated by the third mode. It further illustrates that thelower modes still take part in the vibration though it is dom-inated by a higher mode. Besides, the difference in the ILfrequencies at different locations is more obvious than that

in the CF direction, indicating that the frequency competi-tion is more intense in the IL direction.

4.2 Instantaneous shape of the riserFig. 8 depicts the instantaneous shapes of the flexible

riser and its response envelopes at different flow velocities.It is clearly seen that the instantaneous shape of the risermodel varies over time with different amplitudes along thespan. Since the IL displacements are much smaller com-pared with the horizontal span, the instantaneous shape ofthe riser presented on the left column of the figure is seenfrom the upstream along the flow direction, which mainlyshows the CF displacements. In the figure, t0 is the timewhen the riser reaches its equilibrium position and t0–T/4and t0+T/4 are two other typical moments during a vibra-tion cycle, where T is the CF vibration cycle. Besides, thesurface of the riser is colored by the transverse velocity (v)of the fluid around the riser, which may reflect thefluid–structure interaction. It is seen from Fig. 8 that the

Fig. 7. Comparison of the vibration frequencies of the riser at seven typical positions versus the average reduced velocity.

Fig. 8. Instantaneous shapes of the flexible riser seen from the upstream (along the flow direction) and the response envelopes of the riser (the IL dis-placements of the envelopes on the right column are magnified by a factor of 120).


transverse velocity of the ambient fluid reaches its maxim-um value when the riser reaches the equilibrium position.For the first-mode dominated cases (C5), the red surfacearound the middle part of the riser indicates that the ambi-ent fluid has the maximum velocity along the positive y ax-is. For the second-mode dominated cases, both the red sur-face and the dark blue surface of the riser illustrate that theambient fluid reach local maximum velocities, though one isalong the positive axis and the other one is along the negat-ive axis. The local maximum transverse velocities of theambient fluid occur at the peak CF vibration amplitudes.Additionally, these local maximum transverse velocitieshave relatively large values when the vibration is in the firstand second resonance regions (C5 and C12, respectively). Itimplies that the transverse velocity of the ambient fluid ismainly induced by the CF vibration of the riser. When theriser reaches the place where the CF displacement reachesthe largest value (t0±T/4), the transverse velocity of the am-bient fluid becomes smaller.

As shown in Fig. 8, the response envelopes more clearlypresent the dominant modes both in the IL and CF direc-tions. When the vibration is dominated by a high mode, it isseen that the nodes of the envelopes deviate from the origin-al position of the riser model, which is more obvious in theIL direction. It further confirms that multiple frequenciestake part in the response and the frequency competition inthe IL direction is more intense.

4.3 Flow field and wake structureFig. 9 shows the flow velocity and vorticity around the

flexible riser in one vibration cycle at Re=827. In the figure,the bold black line represents the riser, whose location var-ies over time. It is seen that the vortices are shed alternat-ively from the two sides of the riser, and the streamwiseflow velocities in the core of the vortices are negative. Theminimum flow velocity is –0.03 m/s, appearing in the coreof the vortex just shed from the riser surface. In addition,the velocity contours at different levels reflect the shearflow profile. The maximum flow velocity occurs at two

sides of the riser near the water surface. However, the velo-city difference is not obvious at the lowest plane close to thebottom, which is attributed to the relatively small incomingflow velocity. Therefore, the corresponding vortices in thewake region are also relatively small.

The flow wakes at six different Reynolds numbers arecompared in Fig.10. The 2S, 2P and P+S (a pair of the vor-tices are shed on one side and a single vortex is shed on theother side) vortex shedding mode are captured in the simu-lations. As seen from the contours of ωz at different xoyplanes, different vortex shedding patterns are appeared atdifferent positions along the span, where ωz is defined as:

ωz =12

(∂v∂x− ∂u∂y

), (7)

|ωz|

where u and v are the IL and CF flow velocities, respect-ively. In the first-mode dominated cases, the majority of thevortex shedding exhibits a clear 2S pattern, whereas a P+Spattern is observed in the wake of the midspan where themaximum amplitude appears. In the second-mode domin-ated cases, a P+S mode appears at the positions correspond-ing to the peak amplitudes while a 2S mode mainly presentsat the position corresponding to the trough. Such observa-tions are in agreement with the conclusions of Yamamoto etal. (2004), Meneghini et al. (2004) and Wang and Xiao(2016) for the vibration response of flexible risers that a 2Smode appeared behind the parts of small amplitudes.Moreover, the asymmetric P+S mode at most positionsalong the riser is attributed to the multi-frequency re-sponses. Apart from the wake structure, the numerical res-ults provide some quantitative information about the vortexintensity and size. As shown in Fig. 10, becomes largeras the flow velocity increases, indicating the riser extractsmore energy from the fluid flowing in a higher velocity. Ad-ditionally, the vortex size is relatively large behind the partsof the riser undergoing large amplitudes. It implies that themotion of the riser, in turn, affects the vortex shedding.Therefore, the numerical simulations give some comple-ments for further understanding the experimental results.

Fig. 9. Flow field around the riser in one vibration cycle at Re=827 (C12).


|ωz||ωx|

In addition, the 3D wake structure that is hard to cap-ture in experiments is obtained by the simulations, as shownin Fig. 11. The left column of the figure exhibits the iso-sur-faces of , while the right column presents the iso-sur-faces of (streamwise vorticity) which is defined as:

ωx =12

(∂w∂y− ∂v∂z

), (8)

where w is the vertical flow velocity. As seen from Fig. 11,the vortex columns are shed nearly parallel to the axis of the

riser. It indicates that the vortex shedding is related to theyaw angle of the riser. This phenomenon was observed byMiliou et al. (2007) in their simulations of flow past acurved cylinder and by Assi et al. (2014) and Seyed-Aghazadeh et al. (2015) in their experiments of flow over afixed concave cylinder. Besides, the wake of the lower partof the riser near the bottom wall is well organized with noobvious shedding, while the wake flow of the remainderpart is entirely 3D. The nonlinear shear flow profile contrib-utes to this variation, since the depth-dependent incoming

Fig. 10. Contours of ωz at seven different xoy planes (the flow is from the left to the right).

|ωz| |ωx |Fig. 11. Iso-surfaces of and for various Re values (the flow is from the left to the right), in which the red and blue colors denote positive and neg-

ative values of ωz, respectively, and the green and yellow colors denote positive and negative values of ωx respectively.


flow velocity decreases quickly as Z/H is reduced. From thebottom of the riser to its top, the local Reynolds number in-creases from 0 to the maximum value. Therefore, the wakestructure undergoes a transition from 2D to 3D. It is similarto the wake transition of a straight circular cylinder as theincoming flow velocity increases (Williamson, 1992; Jianget al., 2016). The wake at Re=827 (C12) is dominated bymore irregular small-scale vortices, which agrees with theobservation of Zhang et al. (2017) for a straight circular cyl-inder undergoing the VIV in the upper branch. It indicatesthat the riser experiencing a strong vibration obviously af-fects its 3D vortex structure.

5 ConclusionsA series of fully 3D numerical simulations have been

carried out to investigate the VIV of a curved flexible risersubject to shear flow. The two-way FSI simulations are ac-complished by using the multi-field solver of the softwarepackage ANSYS MFX. The numerical results are in goodagreement with the experimental results, indicating that thepresent numerical method is reliable and capable of predict-ing vibration responses of nonlinear-deployed flexible cyl-inders in nonlinear shear flow. The main findings of thepresent work are summarized as follows.

(1) The dominant mode increases with the growth of theflow velocity, but it is not synchronous in the streamwiseand transverse directions. Moreover, the transition of the vi-bration frequencies is out of phase along the span direction.The lower modes still participate in the vibration though it isdominated by a higher mode. Such mode or frequency com-petition results in the non-zero nodes of the RMS amplitudeand the relatively chaotic motion trajectories of the riser.

(2) The transverse velocity of the ambient fluid reachesthe maximum value when the riser reaches the equilibriumposition, while it becomes smaller when the riser leaves itsequilibrium position. Moreover, the local maximum trans-verse velocities occur at the peak CF amplitudes, and thevalues are relatively large when the vibration is in the reson-ance regions. It reveals that the transverse velocity of theambient fluid is mainly induced by the CF oscillation.

|ωz|

(3) Apart from the wake structure, the numerical resultsprovide quantitative information about the vortex size andthe vorticity. As the flow velocity increases, becomeslarger, illustrating that the riser extracts more energy fromthe flow with a higher velocity. In addition, the vortex sizeis relatively large for the parts undergoing large amplitudes,indicating that the motion of the riser, in turn, also affectsthe vortex shedding.

(4) The vortex columns are shed nearly parallel to theaxis of the curved flexible riser. The wake structure under-goes a transition from 2D to 3D as the local Reynolds num-ber increases from 0 at the bottom of the riser to the maxim-um value at the top. For the riser undergoing VIV in thelock-in region, the wake is dominated by more irregular

small-scale vortices.

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