HUANG Shan

Department of Naval Architecture and Marine Engineering, University of Strathclyde, Glasgow G4 0LZ, UK, E-mail: shan.huang@strath.ac.uk

SWORN Andy

BP Exploration Operating Company Ltd., Middlesex TW 16 7LN, UK

SOME OBSERVATIONS OF TWO INTERFERING VIV CIRCULAR CYLINDERS OF UNEQUAL DIAMETERS IN TANDEM*

HUANG Shan

Department of Naval Architecture and Marine Engineering, University of Strathclyde, Glasgow G4 0LZ, UK, E-mail: shan.huang@strath.ac.uk

SWORN Andy

BP Exploration Operating Company Ltd., Middlesex TW 16 7LN, UK

Analysis ofmodel test results was carried out to investigate the hydrodynamic interaction between a pair of elastically-supported rigid cylinders of dissimilar diameters in a water flume. The two cylinders are placed in tandem with one situated in the wake of the other. The diameter of the upstream cylinder is twice as large as that of the downstream cylinder. The spacing between the two cylinders ranges from 1 to 10 times the larger cylinder diameter. The Reynolds numbers are within the sub-critical range. The cylinders are free to oscillate in both the in-line and the cross-flow directions. The reduced velocity ranges from 1 to 10 and the low damping ratio of themodel test set-up at 0.006 gives a combinedmass-damping parameter of 0.02. It is found that the lift on and the cross-flowmotion of the downstream cylinder have the frequency components derived from the upstream cylinder’s vortex shedding as well as from its own vortex shedding, and the relative importance of the two sources of excitation is influenced by the spacing between the two cylinders. The downstream cylinder’s VIV response appears to be largely dependent upon the actual reduced velocity of the cylinder.

vortex-induced vibration, circular cylinder, drag, lift, flow interference

Introduction

Deepwater offshore engineering systems often involve clusters of long flexible cylindrical structures running vertically in parallel or nearly in parallel across the water depth. These risers can be close to each other in the horizontal planes. Examples include the vertical risers of Tension Leg Platforms (TLPs) where in themean position the typical centre-to-centre spacing to diameter ratio is around 15. When subject to horizontal ocean currents, some risersmay therefore be situated in the wake of other upstream risers, and due the wake shielding effects the riser spacingmay become smaller than themean value when the currents are absent[1]. Moreover, risers of different diameters are often used in a same cluster for different operational functions, e.g., drilling and production risers. The diameter ratio can typically vary from 1 to 4.

On the topic of Vortex Induced Vibration (VIV), there has been a plethora of technical papers on this specific and complex aspect of riser design over the past decade[2-4]. However, in spite of this heavy research effort, a high level of uncertainty in themarine riser VIV prediction still remains. This is reflected in the continuing publication of a large amount of analytical, numerical, empirical and experimental works on this topic, as well as extremely large safety factors typically applied for this aspect of riser design.

Interference between two ormore stationary circular cylinders in various relative positions in cross flows is one of the classic topics of fluidmechanics and has been studied bymany researchers. These studies were typically for clusters of cylinders with an identical diameter and the investigationmethod wasmainly experimental tomeasure the fluid loading[5]. In the recent years, CFD has increasingly been applied formore detailed flow structure analyses[6,7]. On the effects of interference on the vortex-induced vibration response, however, far less research work has been carried out in comparison. Hover and Triantafyllou, and Assi et al. both considered the effects of the upstream cylinder wake on the VIV response of the downstream cylinder[8,9]. In both the investigations, the upstream cylinder was stationary while the down-stream cylinder was constrained in the in-line direction in order to have the cross flow VIV only. The upstream and the downstream cylinders have a same diameter, and the cylinder centre-to-centre spacing was at 4.75 diameters in the first paper and varied between 2 to 5.6 diameters in the second paper. The results of the investigations show that the lock-in VIV extends to high values of the nom inal reduced velocity based upon the free stream velocity. Hover and Triantafyllou also observed that, if a corrected reduced velocity is used which takes into account the wake flow conditions, some features of the downstream cylinder VIV can be explained as in single isolated cylinder VIV tests. Their results also indicate that the lift force on the downstream cylinder has a component at the vortex shedding frequency of the upstream cylinder[8].

The number of studies on two cylinders both undergoing VIV is very lim ited indeed[10-12]. According to these studies, the VIV results of the downstream cylinder are farmore complex to interpret. The resulting oscillations induced by the vortex shedding are considerablymodified by and strongly depend on the arrangement of the two cylinders. For example, in the last paper where two tandem long and flexible cylinders partially submerged and subject to cross flows with the centre-to-centre distances varying from 2 to 4 diameters, Huera-Huarte and Bearman found that the upstream cylinder VIV is strongly influenced by the gap, particularly when the gap is small. The downstream cylindermay ormay not exhibit wakeinduced oscillations at high reduced velocities, depending upon the spacing between the two cylinders.

Fig.1 Schematic of themodel test setup

The present study primarily focuses on the wake interference regime where the two cylinders have dissimilar diameters with the larger cylinder upstream. The streamw ise spacing between the two cylinders varies between 1 to 10 diameters of the larger upstream cylinder. The diameter of the upstream cylinder is used as the reference diameter, and the spacing-to-diameter ratios investigated reflect the realistic spacing-to-diameter ratios ofmarine riser clusters in the wake interference regime.

1. Model test set-up

A series of tests were carried out at Danish Hydraulic Institute involving pairs of cylinders of different diameters with one in the wake of another. Tests were carried out for two elastically supported rigid cylinders undergoing VIV response. A variety ofmass parameters and natural frequency ratios, as well as diameter ratios, of the cylinder pairs were investigated. The tests were carried out in a current flume and themodel setup is illustrated in Fig.1. For each pairing, the two cylinders were initially placed at different locations relatively to each other and then subjected to the incom ing flows. These relative positions weremainly in tandem and staggered arrangements in order to investigate the proxim ity-wake and the wake interferences. The flume is 35m long and 3m w ide, and the water depth used for themodel testing was 0.79m. Themaximum flow velocity is around 0.7m/s. The tests were carried out for amean inflow velocity between 0.1m/s and 0.6m/s. The turbulence intensity,iT, of the inflow is relatively high. Typically, Ti=3.8% for themean inflow velocity at 0.4m/s. The turbulence intensity is defined as the standard deviation of the inflow velocity non-dimen- sionalised by themean inflow velocity.

The cylinders were suspended from vertical elastic rods above them. The cylindermodel as well as its elastic supporting system were carefully designed to ensure that the cylinder is free tomove in any direction perpendicular to its vertical axis. The suspension system was designed to allow the cylinder to vibrate in both the in-line and the cross-flow directions without tilting. In addition, the elastic suspension system was adjustable in order to give different natural frequency ratios between the two cylinders. If the elastic suspension system above the cylinder was blocked, the cylinders became rigidly fixed and hydrodynamic interaction between two stationary cylinders could also be investigated. Themodel cylinders weremanufactured in aluminium and the surface of the cylinder was regarded as hydrodynam ically smooth. The cylinder’s upper end was 0.02m below the calm water surface, and its lower end 0.02m above the flume bottom. The end effects of the cylinder are believed to be small. There has been amore considered investigation of the end effects in the context of VIV[13]. The Reynolds numbers were all in the sub-critical range. It should be noted that, because of the different diameters used while the span length was identical for all the cylinders, the cylindermodels had different span to diameter ratios.

The VIV results presented in this paper are only for the two cylinders with the diameters of 0.16m and 0.08m, respectively. The two cylinders are in tandem with the larger cylinder upstream. Themass parameter, defined as the structuralmass divided by the displaced watermass is 3.0 for both the cylinders. The in-line and the cross-flow natural frequencies in water are identical at 0.31 Hz for the upstream cylinder (denoted as fn1), and 0.58 Hz for the downstream cylinder (denoted as fn2). The natural frequency ratio is therefore approximately 0.5 which was the targeted value. Decay tests were carried out in air in order to quantify the structural damping of the two elastically supported cylinders and the damping ratios were found to be at 0.6% for the 0.16m cylinder and 0.7% for the 0.08m cylinder. The parametersmeasured simultaneously in themodel test were the two horizontal force components on each cylinder as well as the acceleration and the displacement of each cylinder.

Table 1 Flow speeds and distances between the two tandem cylinders

2. Results and discussion

Tests were carried out for the conditions listed in Table 1. Some selected results of the two cylinders in tandem are presented in the paper. Unless otherw ise stated, the nom inal reduced velocity, Vr, is defined by using the free stream inflow velocity, the diameter and the natural frequency of the upstream cylinder. The reduced velocities are in the range from 1 to 10 covering the typical lock-in values. The actual reduced velocity for the downstream cylinder is not known as themean inflow velocity to the downstream cylinder was notmeasured. It is possible, however, to estimate approximately themean inflow velocity to the downstream cylinder based upon somemeasured wake velocities of a fixed upstream cylinder.

The hydrodynamic force components, i.e. the drag and the lift on the whole cylinder, are expressed in terms of the non-dimensional coefficients Cdand Cl. For the individual cylinder concerned, these two coefficients are based on the conventional definition, i.e.,

where D is the cylinder’s diameter, L its length, ρ the water density, U the free-stream velocity. In the follow ing figures, D1, Cd1and Cl1refer to the diameter, drag and lift coefficients of the upstream cylinder, sim ilarly, D2, Cd2and Cl2for the downstream cylinder. X is the non-dimensional in-line spacing between the two cylinders which is nondimensionalised by the use of the larger diameter of the two cylinders. X1and Y1are the in-line and cross-flow displacements of the upstream cylinder non-dimesnioanlised by its own diameter, similarly, X2and Y2are the in-line and cross-flow displacements of the downstream cylinder non-dimesnioanlised by its own diameter.

Fig.2 VIVmotion trajectories of both the upstream and downstream cylinders for the initial spacing X at 1.6, 2.0, 3.0, 5.0 and 10.0, respectively (from the top to the bottom). Vr=5

Fig.3 VIVmotion trajectories of both the upstream and downstream cylinders for the initial spacing X at 2.5, 3.0, 5.0 and 10.0, respectively (from the top to the bottom). Vr=9

Fig.4 Motion trajectories of upstream and downstream cylinders non-dimensionalised by each cylinder’s own diameter. Vr=5, D1/ D2=2, fn1/ fn 2=0.5, X= 5

Fig.5 Power spectral density of the cross-flow and the in-linemotions of the two cylinders. The nom inal reduced velocity Vr=5

2.1 Motion trajectories

The VIVmotions of both the upstream and the downstream cylinders, particularly in the transverse direction, can be large, as shown in Fig.2 and Fig.3. The figures give the trajectory plots of the two cylinders at the nom inal reduced velocities of 5 and 9, respectively, as an example. It should be noted that the cylinder’s cross-flow and in-line displacements are non-dimensionlised by using its own diameter while the spacing between the two cylinders is non-dimensionlised by the diameter of the upstream cylinder, hence these plots are not drawn exactly to scale. It is known that for an isolated cylinder permitting both the in-line and the cross-flow VIV responses, the response pe ak is

ty picall y del ayed to around Vr=9 or beyond. Itcanbeseenthatthemaximumtransversemotion amplitude to diameter ratio is as high as 1.5. The large amplitude responsemay be attributed to the lowmassdamping parameter value at about 0.02. It is also noted that both the upstream and downstream cylinders have appreciable VIVmotion in the in-line direction. Typically, the in-linemotion amplitude is about a quarter to a half of the transversemotion amplitude.It should be noted that the two cylinders have different natural frequencies in calm water with the downstream cylinder’s natural frequency twice as high. At the large spacing the effective reduced velocity for the downstream cylinder is approximately same as the nominal reduced velocity, because its diameter is half of that of the upstream cylinder and the wake shielding effects become weak.

Figure 4 presents the enlarged and separated trajectory plots for the nom inal reduced velocity at 5 and the non-dimensional spacing between the two cylinders X=5. Both the inline and the cross-flow displacements are non-diemsionalised by the use of the cylinder’s own diameter, and the plots are given individually without show ing the spacing between them. It can be seen that the trajectories are qualitatively different. Whilst the trajectory of the upstream cylinder has the classic “8”-shaped crescent, as observed bymany other researchers, the downstream cylinder appears not displaying any regular pattern.

Further the results of the spectral analysis of the inline and the cross-flowmotion responses are given in Fig.5 for both the cylinders. For the cross-flowmotion, the upstream cylinder has only one spectral peak, whilst the downstream cylinder can have two spectral peaks. These spectral peaks correspond to the vortex shedding frequency of the upstream cylinder and the lock-in vibration of its own vortex shedding. Further discussion is given in the next sub-section of the paper.

Fig.6(a) Cross-flowmotion time-history of the downstream cylinder. Vr=5 and X=2

Fig.6(b) High frequency part of (a)

The two frequency compo nents of the downstream cylinder shown in Fig.5(b) are split in the timedomain, and the results are given in Fig.6. The high frequency part is likely due to the downstream cylinder’s own vortex shedding. It has the classic feature of beating. In comparison, the low frequency part is rather steady. In separating the high and low frequency parts, the division is set at D2f/ U=0.12.

Fig.6(c) Low frequency part of (a)

Fig.7 Power spectral density of the lift and drag coefficients of the two cylinders. The nominal reduced velocity Vr=5

2.2 Hydrodynamic forces

Figure 7 gives the power spectral density functions of the drag and lift coefficients of the two cylinders. The follow ing observations aremade.

(1) At X=5, where the influence of the downstream cylinder upon the upstream cylinder is deemed to be negligible, the Strouhal number of the upstream cylinder is approximately at 0.15. The dominant dragfrequency is twice as large, i.e., for the drag peak frequency fD1/ U is approximately at 0.3. These results are typical of an isolated cylinder undergoing VIV in cross flow. There are some low frequency contents in the drag, and it is unclear as to the reason for this.

(2) At the small spacing, the lift frequency of the downstream cylinder is identical to that of the upstream cylinder. Itmay therefore be inferred that the lift variation is due to vortices shed from upstream cylinder and impinging on the downstream cylinder. It is also noted that there is a smaller high frequency component at around D2f/ U=0.18. With reference to Fig.5(b), it is clear that even though this high frequency force component is relatively small in itsmagnitude, its inducedmotion is greater because of resonance effects.

(3) As the spacing increases, the high frequency lift component at around D2f/ U=0.18 becomesmore significant for the downstream cylinder, probably reflecting the competing factors of the vortex shed upstream on the one hand and the vortex generated by itself on the other.

(4) The spectral features of the drag of the downstream cylinder appear farmore complex than its counterpart on the upstream cylinder.

Fig.8 Standard deviation of the in-line and the cross-flowmotion of the two cylinders. The horizontal axis is the spacing between the two cylinders non-dimensionalised by using the upstream cylinder’s diameter

2.3 Statistical parameters

From Fig.8, where the standard deviations of the VIV responses of both the cylinders are presented, it is clear that the VIVmotion of the upstream cylinder is not significantly affected by the existence of the downstream cylinder in this study. It should be emphasized again here that horizontal axis, X, is the non-dimensional spacing by the use of the upstream cylinder diameter which is twice as large as the downstream cylinder’s diameter. The VIV response of the downstream cylinder, on the other hand, appearsmore complex and sensitive to the spacing variation, particularly when the spacing is small.

Fig.9 Mean drag coefficients of the two cylinders versus the spacing for different Vr

Figure 9 gives themean drag coefficients of the two cylinders. For the purpose of comparison, themean drag coefficients of the two cylinders held stationary, i.e., without undergoing VIV, are also plotted in the figure. It is clear that for the upstream cylinder the VIV response gives rise to a drag amplification. The amplification is insensitive to the spacing, but dependent upon Vrwhich determines the VIV response. In other words, the drag amplification is VIV amplitude response dependent, as shown bymany other researchers in the past.

Fig.10 Mean drag coefficient versus standard deviation of the VIVmotion response

In comparison with the stationary cylinder results, it is clear that themean drag on the downstream cylinder is also increased. Itmay be tempting to attribute this increase to the samemechanism of drag amplification of the upstream cylinder. To this end, the relationship between the drag increase and the VIV ampli-tude response is plotted in Fig.10 for the two cylinders. For the upstream cylinder, it is clear that themean drag coefficient is almost linearly dependent upon the cross-flowmotion amplitude STD(Y1) where STD denotes the standard deviation. A least square fit yields the follow ing drag amplification expression.

Amplification factor=1+1.96× STD(Y)

where the amplification factor is the ratio between themean drag coefficient of the vibrating cylinder and the drag coefficient of the same cylinder held fixed in the flow. In comparison with the various expressions of the drag amplification given in Section 2.3[2], the above linear relationship yields drag coefficient values smaller than the curve-fit expression based upon the data of Sarpkaya, Tanida et al. and Torum et al., but greater than that of Vandiver’s expression. It appears there still is a large degree of uncertainty concerning VIV drag amplification, even though this is a rather important issue for the deepwater riser design[14]. It is also noted that for the upstream cylinder the drag amplification appears also linearly correlated with the inlinemotion amplitude. In contrast with the upstream cylinder, the drag of the downstream cylinder appears uncorrelated with its own VIVmotion response, as shown in Fig.10.

Fig.11 Lift coefficient versus standard deviation of the VIVmotion response

In Fig.11, the lift coefficient is plotted against the cross-flowmotion amplitude. For the upstream cylinder, as the cross-flowmotion increases from zero, so does the lift. However, the lift decreases as the crossflowmotion further increases. The trend of the lift on the downstream cylinder appears less clear, even though the lift also appears to decrease once themotion increases beyond a certain limit. It should be noted that the lift presented here is the to tal hydrodynamic loading on the cylinder in the cross-flow direction and its includes both the components in phase with the acceleration and with the velocity, respectively.

2.4 Motion amplitude versus reduced velocity

For the cases investigated here, the effects of the downstream cylinder on the upstream cylinder VIV are generally very small. As such, it can be expected that the upstream cylinder VIVmotion is correlated with the nom inal reduced velocity as in the case of an isolated VIV cylinder in cross flow. As discussed in the foregoing, on the other hand, the dominant spectral peak of the lift on the downstream cylinder is largely due to its own vortex shedding when the spacing is large, and a combination of its own vortex shedding and the vortices shed from the upstream cylinder for small and intermediate spacing. The force component with its frequency closer to the natural frequency tends to give greatermotion response due to the resonance effects. Itmay therefore be reasonable to expect the VIV response of the downstream cylinder ismore dependent upon the actual reduced velocity of the downstream cylinder. Some evidence to support the conjecture can be found in Fig.12. In the left plot, the standard deviations of the transverse VIVmotions of the downstream cylinder are plotted against the nom inal reduced velocity defined by the use of the free stream velocity and the diameter and the natural frequency of the upstream cylinder. The trend is hard to discern here. The nom inal reduced velocity is not representative of the downstream cylinder as the inflow velocity to the downstream cylinder is smaller than the nominal free stream velocity to the upstream cylinder due to the wake shielding effects. To estimate the “true” reduced velocity of the downstream cylinder, a correction to the free stream in flow velocity is required. In the present study, this was done based upon experimental data obtained by Cantwell and Coles[15], which is reproduced in Fig.13. It should be noted that the result of the velocity reduction in the wake presented in Fig.13 is for a fixed cylinder (not for a vibrating cylinder) and at a different Reynolds number. The result is only used here for an approximate estimate of the “true” reduced velocity of the downstream cylinder which is denoted as Vrw. In Fig.12, it is clear that STD(Y2) is not correlated well with the nom inal reduced velocity Vr, but better correlated with the “true” reduced velocity. This strongly suggests that VIV of the downstream cylinder is primarily influenced by its own reduced velocity.

Fig.12 Standard deviation of the cross-flowmotion of the downstream cylinder versus the nom inal reduced velocity Vr(a) and the “true” reduced velocity Vrw(b)

Fig.13 Mean wake velocity uclalong the wake centreline behind a fixed cylinder of diameter d, non-dimensionalised by using the free stream velocity ui. Re=1.4× 105

Fig.14 Upstream cylinder VIV response STD(Y1) vs nom inalreduced velocity Vrand downstream cylinder VIV response STD(Y2) vs it own “true” reduced velocityVrw

Figure 14 compares the cross-flow VIV amplitude responses of both the cylinders with their own reduced velocities. Whilst the overall trends are similar, it is also observed that at large reduced velocities the upstream cylinder has greater VIVmotion response in comparison. The reason for this is unclear and requires further investigation.

3. Conclusions

Based upon themodel test results presented in the paper, the following conclusions can be drawn.

(1) The VIVmotion trajectories are qualitatively different between the upstream cylinder in steady uniform flow and the downstream cylinder situated in the wake of the upstream cylinder.

(2) The lift force on the downstream cylinder has the dominate spectral peak at either the shedding frequency of the upstream cylinder or its own shedding frequency depending upon the spacing between the two cylinders. The influence of the upstream cylinder ismore pronounced when the in-line spacing is small.

(3) Like the upstream cylinder, the downstream cylinder also experiences a certain degree of drag amplification. But unlike the upstream cylinder, this drag increase appears not correlated with the crossflow VIVmotion and it remains a challenging task as to how to quantify this drag amplification.

(4) Large VIVmotions are observed in the test results presented in the paper with themaximum transversemotion amplitude around 1.5 diameters. For the upstream cylinder, the VIV is similar to the classical lock-in response. The test data indicate that the VIV response of the downstream cylinder is significantly dependent upon its own reduced velocity based upon the reducedmean wake velocity.

Acknowledgements

The authors would like to acknow ledge the support of the Norwegian Deepwater Programme and BP Exploration Operating Company Ltd as well as their permissions for publishing the paper. Themodel tests were carried out by DHI.

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July 30, 2011, Revised August 28, 2011)

10.1016/S1001-6058(10)60147-3

* Biography: HUANG Shan (1963-), Male, Ph. D., Professor