Experimental Comparisons to Assure the Similiarity Between VIM (Vortex-Induced Motion) and VIV (Vortex-Induced Vibration) Phenomena

1 Copyright 2011 by ASME

EXPERIMENTAL COMPARISONS TO ASSURE THE SIMILARITY BETWEEN VIM (VORTEX-INDUCED MOTION) AND VIV (VORTEX-INDUCED VIBRATION)

PHENOMENA

Rodolfo T. Gonalves1 ([email protected])

Csar M. Freire2 ([email protected])

Guilherme F. Rosetti1

([email protected]) Guilherme R. Franzini2

([email protected])

Andr L. C. Fujarra1 ([email protected])

Julio R. Meneghini2 ([email protected])

1TPN Numerical Offshore Tank Department of Naval Architecture and Ocean Engineering, Escola Politcnica University of So Paulo

So Paulo, SP, Brazil

2NDF Fluid & Dynamics Research Group Department of Mechanical Engineering, Escola Politcnica University of So Paulo

So Paulo, SP, Brazil

ABSTRACT Vortex-Induced Motion (VIM) is another way to

denominate the Vortex-Induced Vibration (VIV) in floating units. The main characteristics of VIM in such structures are the low aspect ratio (L/D < 4.0) and the unity mass ratio (m* = 1.0, i.e. structural mass equal water displacement). The VIM can occur in MPSO (Monocolumn Production, Storage and Offloading System) and spar platforms. These platforms can experience motion amplitudes of around their characteristic diameters. In such cases, the fatigue life of mooring and riser systems can be greatly reduced. Typically, the VIM model testing campaigns are carried out in the Reynolds range between 200,000 and 400,000.

VIV model tests with low aspect ratio cylinders (L/D = 1.0, 1.7 and 2.0) and unity mass ratio (m* = 1.0) have been carried out at the Circulating Water Channel facility available at NDF/EPUSP. The Reynolds number range covered in the experiments was between 10,000 and 50,000. The characteristic motions (in the transverse and in-line direction) were obtained using the Hilbert-Huang Transform method (HHT) and then compared with results obtained in experiments found in the literature.

The aim of this investigation is to definitely establish the similarity between the VIM and VIV phenomena, making possible to increase the understanding of both and, at same time, allowing some analytical models developed for VIV to be applied to the VIM scenario on spar and monocolumn platforms, logically under some adaption.

1. INTRODUCTION From the earliest evidence in spar platforms, the VIM

phenomenon has shown close similarity to the phenomenological aspects of VIV on remarkably slender structures, such as risers.

In fact, small-scale experiments on a truss spar platform performed by van Dijk et al. (2003) already showed that, despite the strong susceptibility to geometric aspects of the hull, in general terms, the VIM phenomenon clearly presents both the self-excitation and the self-controlled characteristics, similar to VIV on risers. Moreover, despite focused on mitigating the effects of VIM on such a type of platform by adding strakes, the authors also showed that without those suppressors the maximum response amplitude occurs in a range of reduced velocities between 5 and 9, making explicit mention

Proceedings of the ASME 2011 30th International Conference on Ocean, Offshore and Arctic Engineering OMAE2011

June 19-24, 2011, Rotterdam, The Netherlands

OMAE2011-490


to eight-shaped trajectories, typical characteristic of a coupling between in-line and transversal motions. Additionally, experiments by Finn et al. (2003) established a direct relationship between hull geometry and VIM response, highlighting the correlation between towing tests and those performed at water channels. Through these results, a formal procedure for testing VIM on spar-type platforms is presented in Irani & Finn (2004), including concerns about the restoring system in small-scale. By focusing on operating conditions and at the same time aware of the VIM susceptibility to other hydrodynamic mechanisms, Finnigan et al. (2005) started experiments considering VIM and free surface waves simultaneously. In general, those authors found that the waves have a mitigating effect on the VIM phenomenon by reducing its response amplitude. Recently, Roddier et al. (2009) conducted a study to verify the influence of the Reynolds number on VIM, according to which tests carried out at lower values of Reynolds are found to be conservative, i.e., they have maximum amplitudes of motion greater than those observed in full scale. Unlike all previous works, Wang et al. (2009) focused on testing models without appendices (smooth cylinder with low aspect ratio, L / D), being one of the earlier initiatives in establishing a direct relationship between VIM and VIV. Therefore, its results will be presented later in the comparisons of this work.

As interesting as the results from spar platforms was the similar occurrence of the VIM phenomenon on platforms with even smaller aspect ratios, L / D 0.5, called monocolumn platforms as presented and discussed in Gonalves et al. (2009b). Pioneer in developing such a type of platform, in 2005 the research group from USP University of So Paulo together with Petrobras started an intensive experimental research on VIM of monocolumn platforms. Cueva et al. (2006), Fujarra et al. (2007) and Gonalves et al. (2009a) established experimental procedures for VIM experiments on monocolumns, as well as proposed a more accurate technique for analyzing those experiments. Even for very small aspect ratios, those researches showed that VIM on monocolumns is very similar to the phenomenon action on spars with large amplitudes of transversal oscillation at reduced velocities above 4. Later, in Gonalves et al. (2010b) studied many mitigation aspects, such as the presence of appendages on the hull (stairs, fairleads, etc), the coexistence of waves and the damping promoted by risers, umbilicals and mooring lines. However, the most important aspect in the VIM mitigation was the aspect ratio, investigated by means of the draft variation of the small-scale monocolumns. In fact, in Gonalves et al. (2010a) the USP research group carried out intensive and preliminary comparison work between results from both VIM on monocolumn platforms and VIV on smooth cylinders, by using as a benchmark the aspect ratio / . As a result, the authors reaffirmed the hypothesis that the VIM and VIV phenomena are essentially the same, simply affected by the consequences of modifying the emission pattern of vortices in a given aspect ratio.

In order to definitively confirm the similarity between VIM and VIV phenomena, further experiments were carried out with cylinders of small aspect ratio in water channel, by using two distinct elastic supports and different mass ratios, . The comparison between results from different apparatus ensures the reliability for subsequent comparison with similar results reported in the literature.

For comparing the experiments described as follows, some results of VIV with two degrees of freedom were used, among them: Pesce & Fujarra (2000) by means a cantilevered flexible cylinder with 2.36 and / 94.50; Jauvtis & Williamson (2004) through a rigid cylinder elastically supported with 2.62 e / 10.00; Sanchis et al. (2008) by means a spring mounted cylinder with 1.04 and / 6.00; Stappenbelt & Lalji (2008) and Blevins and Coughran (2009) both based on elastically mounted rigid cylinders, the former with 2.36 and / 8.00 and the latter one with 1.00 or 2.36 and / 17.80; as well as Freire & Meneghini (2010) considering a rigid cylinder mounted in a pivoted beam with 2.80 e / 21.88.

Table 1 Results compared in the present work

Work Apparatus Pesce & Fujarra

(2000) 2.36 94.50 1.305 cantilevered

bar

Jauvtis & Williamson

(2004) 2.62 10.00 1.000 elastic base

Stappenbelt & Lalji (2008) 2.36 8.00 1.000 elastic base

Sanchis et al. (2008) 1.04 6.00 1.000 elastic base

Wang et al. (2009) 1.00 2.40 1.000

floating cylinder

Blevins & Coughran (2009)

1.00 2.56 17.8 1.000 elastic base

Gonalves et al. (2010b) 1.00 0.39 1.000

floating cylinder

Freire & Meneghini (2010) 2.80 21.88 1.291

pivoted pendulum

Present work 1.00 2.00 1.70 1.45

1.291 pivoted pendulum

Present work 1.00 1.00 2.62

2.00 1.50 2.00

1.305 cantilevered bar

Moreover, the results obtained in the present work were

also compared to those from VIM experiments by Wang et al. (2009) in floating rigid cylinders with / 2.40 and by


Gonalves et al. (2010b) in small-scale platforms with / =0.39. Obviously, since those are floating cylinders, in both cases the mass ratio is equal to one.

Table 1 summarizes all the important parameters for the

comparisons and conclusions presented later in this paper.

Importantly, the results with 2DOF are fairly recent and

thus, beyond those presented in Table 1, many other deserve to

be mentioned, since they have served as source of discussion

herein. Among them Fujarra et al. (2001), Jauvtis &

Williamson (2003); Flemming & Williamson (2005); Leong &

Wei (2008); Sanchis et al. (2008) and Sanchis (2009) should be

quoted.

2. EXPERIMENTAL SETUP As commented above, the experiments herein presented

were performed making use of two different experimental

apparatus, namely a cantilevered bar and a pivoted pendulum

cylinder. The experimental setup of the former was presented in

Gonalves et al. (2010a) and Franzini et al. (2010) whereas the

latter was described in Freire & Meneghini (2010).

Both tests were performed at the Water Channel facility

available at NDF (Fluid & Dynamics Research Group) at the

University of So Paulo, Brazil. The channel has a 0.70 x 0.70

x 7.50m test section, presents a very low level of turbulence

(less than 1%) and can operate at well-controlled free-stream

velocity up to 0.60m/s. Further details concerning the Water Channel can be found in Assi et al. (2006).

Cantilevered bar

A vertical and rigid cylinder with 125mm in diameter was

attached to the end of a cantilevered bar, free to oscillate in two

degrees-of-freedom, cross-flow and in-line directions, see

example in Figure 1. The structural damping coefficient is very low with this apparatus, approximately = 0.1%. The stiffness of the system is defined by the length of the

cantilevered bar, selected in order to adjust the reduced velocity

range to the water channel capability.

Two values of aspect ratio, / = 1.5 and 2.0, were tested by changing the water level in the channel. Two values of

mass ratio parameter =4

were considered, = 1.00;

2.62 (where is the oscillating structural mass). By means of this arrangement, time series of displacement

were obtained by using two laser position sensors (LEUZE-

ODSL 8/V4 model).

Pivoted pendulum The same vertical rigid cylinder with 125mm in diameter

was also mounted in a pivoted pendulum base with two degrees

of freedom, see example in Figure 1. The base presents the

same natural frequency in both in-line and cross-flow

directions.

The base allows the model to rotate around the Cardan-

joint axis at the top of the rigid bar.

In all the experiments, the mass ratio () was kept constant equal to unity by adding small lumped masses to it.

Furthermore, three aspect ratios were considered in the

experiments, / = 1.45, 1.70 and 2.00.

For both arrangements, the acquisition time employed in

the experiments was 180 seconds and the time between

consecutive acquisitions was 120 seconds so that the water

channel had its current velocity stabilized.

Figure 1 Example of different apparatus for VIV study.

Source: Flemming & Williamson (2005).

3. NONDIMENSIONAL RESULTS All the model displacements were measured and then

transported to the tip of the model by employing a factor that is

a function of the base geometry.

In order to compare the results from different experimental

facilities, a factor was employed on the amplitudes values, as presented in detail, for example, in Blevins (1990). The value

of is given by the following equation.

= { 2()

0

4

0()

}

1/2

Where the () is the mode shape function and max{ (z)}=1. This equation is valid if the linear mass is constant

along the span.

The dimensionless mode factor is obtained from the solution of the wake oscillator problem. For a rigid cylinder

mounted on an elastic base, the mode shape function is constant

and equal to 1, then the mode factor is = 1. In the case in which the structural element is a rigid pivoted rod, (z) = z/L and = 1.291.

For a flexible element, the mode function and,

consequently, the mode factor depend on the oscillation

elastic

base

pivoted

pendulum

cantilevered

bar


frequency. For a cantilevered cylinder with bending stiffness and mass per unit length vibrating in its first eigenmode, the mode shape function is:

sin cos ,

where:

is the wavelength and 3.52 is the natural frequency. Integrating the mode shape function, the value of for the condition described above is 1.305. 4. RESULTS

The results presented in this section are twofold. Firstly, the experimental results performed with the cantilevered bar and pivoted pendulum cylinder are presented. Then these results are compared with the literature results of similar aspect and mass ratios.

Cantilevered bar and pivoted pendulum results Figure 2 presents nondimensional transverse amplitudes

for the experiments with the cantilever bar and pivoted pendulum apparatus. Firstly, one notices that the factor is, indeed, essential for establishing a common base for comparison of results from different experimental apparatus. Specifically, we refer to the comparison between pendulum and bar results of / 2.00 and 1.00; / 1.45/1.50 and 1.00. One notices that the amplitudes and trends are very similar as expected, since the mass and aspect ratios are common.

On the one hand, when comparing the results for different aspect ratios and mass ratio equal to unity, one can perceive two aspects. Firstly, that higher aspect ratios tend to present higher amplitudes, e.g., 1.10 for 2.00 and 1.00 for 1.45 1.50. It is likely that the tip of the cylinder disturbs the flow in a three-dimensional manner in such a way that the correlation is smaller for smaller aspect ratios. Also, there is a more subtle aspect related to a small shift in the curves to the right, which is related to the change in the Strouhal number for different aspect ratios.

On the other hand, looking at different mass ratios, there is an unexpected increase in amplitudes with the larger mass ratio, e.g. 2.62 and 1.50; 1.00 and 1.10.

As a general trend observed in all results, one cannot identify an indication of a lower branch as a result of the low mass ratios.

Figure 3 presents the nondimensional in-line amplitudes for the same experiments.

Regarding the aspect ratio effect, no great difference is verified between the results from different cases, whereas for different mass ratios, there is a distinct behavior of the high-mass ratio case, with slightly lower amplitudes and a shift in the curve to the right, as result of the later synchronization, see Figure 5.

It can also be noticed that there is a smaller peak of amplitude in 2.50, which is due to in-line resonance as

previously mentioned by Blevins & Coughran (2009). The higher amplitudes, however, are noticed at the same range of as the transverse direction, as a result of the synchronization of the motions in the transverse and in-line directions.

Figure 4 and Figure 5 present, respectively, tranverse and in-line directions frequencies of response, made nondimensional by the system transverse natural frequency in still water and transverse response frequency, respectively.

In the region 3.00 both transverse and in-line frequencies are very similar to the system transverse natural frequency and one notices that in-line motions are predominant due to the in-line resonance. For 3.00, there is coupling between transverse and in-line motions and the double frequency pattern is observed between in-line and transverse frequencies. In Figure 4, the range of corresponding to increasing amplitudes, 5.00, is observed not to imply the classical definition of lock-in, 1, due to the low mass ratio. Also, there is no indication that the synchronization is near an end.

For the case of 2.62, both transverse and in-line frequencies synchronized later than for 1.00, consistent with the amplitudes results.

Comparison with the literature results Figure 6 presents our results compared to the literature

ones for 1.00. The literature results are from Wang et al. (2009), in which the authors report tests with a spar platform; Gonalves et al. (2010b), in which tests with the monocolumn platform are reported; Sanchis et al. (2008) and Blevins & Coughran (2009), in which bare cylinders are tested.

In general, it can be noticed that to higher aspect ratios, higher peak amplitudes correspond, as a result of higher correlation. Also, as commented before, the variability of Strouhal number highly influences the amplitude results by shifting the curves to the right for smaller aspect ratios (and smaller Strouhal numbers). This effect may easily be noticed when the smallest and largest aspect ratios are compared. For the largest, / 17.8 Blevins & Coughran (2009) there is an early synchronization as for infinite cylinder results. On the other hand, for the smallest / 0.39 by Gonalves et al. (2010b), the synchronization occurs later, due to the smaller Strouhal number.

Furthermore, owing to the stability of the flow, it is equally interesting and important to notice that, even though the results are from different experimental apparatus, namely platforms in model scales with appendages and bare cylinders on the other hand, the curves are very similar in trend and in values. Thus, the comparison is very good between these cases, showing that laboratory experiments may well be used in some practical situations.

Figure 7 presents the in-line amplitude results for the same 1.00 case. The literature result used for comparison is the one presented in Gonalves et al. (2010b) in which / 0.39 and other by Sanches et al. (2008) in which / 6.00. Consistently with the transverse results, the in-line ones show


that for the lower aspect ratio there is a later synchronization and lower amplitudes, 0.15 for 0.39 whereas 0.35 for the others. Again, the difference in Strouhal numbers causes the shift in the curve and lower correlation causes smaller forces and amplitudes of oscillation, see Gonalves et al. (2010a).

Figure 8 presents frequencies of transverse oscillations made nondimensional by the transverse natural frequency in still waters. Again, the literature result used for comparison is from Gonalves et al. (2010b), for which / 0.39 and Sanchis et al. (2008), for which / 6.00. The inclination of the linear curve out of the lock-in is related to the Strouhal number and, as can be noticed, for different aspect ratios, the inclinations are different and, therefore, the Strouhal numbers are also different, this result confirms the idea of Strouhal decreasing with the decreases. Discussion about this effect can also be found in Fox & Apelt (1993) and Stappenbelt & ONeill (2007).

Figure 9 presents results for 2.62. Again, our results are compared with those from the literature. These are from tests with cantilevered bar - Pesce & Fujarra (2000); elastically mounted rigid cylinder - Jauvtis & Williamson (2004), Stappenbelt & Lalji (2008), Blevins & Coughran (2009); and pivoted pendulum cylinder - Freire & Meneghini (2010).

Except for our result, in which 2.00, the results resemble classical ones with upper and lower branches. For the same case, the peak amplitude, 1.50, is consistent with the other results; however, the small shift and less sharp inclinations of the curve suggest that 3D effects due to the cylinder tip play an important role, as commented before. Morse et al. (2008) discussed the presence of end plates for different and in one case without end plate, for 8.00, similar behavior was observed, namely comparable peak amplitudes but a shift in the curve and an apparent absence of a lower branch.

For the same case with 2.62, the in-line results show a peak amplitude of 0.35 not without some dispersion. One can again notice a lower branch in the results from Jauvtis & Williamson (2004) and Freire & Meneghini (2010) and not in our results due to the end effects. Also present in these results, there is a shift in the curve due to the smaller Strouhal number.

Finally, Figure 11 presents the transverse frequencies of oscillation made nondimensional by the transverse natural frequency in still waters for 3.00. In the result from Jauvtis & Williamson (2004) and Pesce & Fujarra (2000), the lower branch is characterized by a nearly constant 1.05, whereas in our results this is not observed. In this case, the ratio increases in a nearly linear manner. 5. GENERAL CONCLUSIONS

The main purpose was to investigate the VIV of short cylinders and the VIM of floating units drawing a connection between them. For that aim, we reported results from the experiments carried out with low aspect and mass ratios

cylinders and compared them to literature results, mainly in terms of amplitudes and frequencies of oscillations.

In terms of transverse amplitudes of response, when comparing the results for different aspect ratios, the higher aspect ratios were found to tend to present higher amplitudes, due to a larger vortex correlation and, thus larger force. Also, there is a small shift in the curves to the right, which is related to the change in the Strouhal number for different aspect ratios. Also, a general trend observed in all results is that one cannot identify an indication of a lower branch as a result of the low mass ratios and small aspect ratio (end effects).

Regarding the in-line amplitudes, consistently with the transverse results, the in-line ones show that for the lower aspect ratio there is a later synchronization and lower amplitudes.

In terms of frequencies, when there is coupling between transverse and in-line motions, large amplitudes occur. However, an important difference between lower and higher aspect ratio results is the shift in this synchronization. Also, for the former, there is no indication that the synchronization is near an end and no lower branch can be identified. The same pattern was identified by Morse et al. (2008) with the conclusion that the end effect was influencing the system behavior. In that case, comparable peak amplitudes between system with and without end plates with a shift in the curve and an apparent absence of a lower branch were also found.

Finally, as commented before, even though the results are from different experimental apparatus, namely platforms in model scales on the one hand and bare cylinders on the other hand, the curves are very similar in trend and values. The stability of the flow around the system is such that the presence of small appendages may not change the trends of the results.

Thus, the comparison is very good between these cases, showing that laboratory experiments may well be used in some practical situations.

NOMENCLATURE Wavelength Apparatus factor Mode shape function Natural frequency Water density Structural damping Characteristic nondimensional motion

amplitude in the in-line direction Characteristic nondimensional motion

amplitude in the transverse direction Characteristic diameter Bending stiffness

Natural frequency in the in-line direction in still water

Oscillation frequency in the in-line direction Oscillation frequency in the transverse

direction Immersed length


Mass per unit length Mass ratio Oscillating structural mass L / D Aspect ratio Flow velocity

Reduced velocity in still water

ACKNOWLEDGMENTS The authors would like to acknowledge FAPESP and

CAPES for the financial support.

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Figure 2 Nondimensional results for motions in the transverse direction of two degrees-of-freedom VIV in cylinders with low

aspect ratio ( . ) and low mass ratio ( . ).

Figure 3 - Nondimensional results for motions in the in-line direction of two degrees-of-freedom VIV in cylinders with low

aspect ratio ( . ) and low mass ratio ( . ).

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Reduced Velocity (Vr)

Ay /

( D

)

Pendulum L/D = 2.00m* = 1.00Pendulum L/D = 1.70m* = 1.00Pendulum L/D = 1.45m* = 1.00Bar L/D = 2.00m* = 1.00Bar L/D = 2.00m* = 2.62Bar L/D = 1.50m* = 1.00

0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5


Ax /

( D

)



Figure 4 - Nondimensional results of transverse frequency oscillation and natural transverse frequency in still water for two

degrees-of-freedom VIV in cylinders with low aspect ratio ( . ) and low mass ratio ( . ).

Figure 5 - Nondimensional results of in-line frequency oscillation and transverse one for two degrees-of-freedom VIV in

cylinders with low aspect ratio ( . ) and low mass ratio ( . ).

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


f y / f

0


St = 0.20

St = 0.10

St = 0.05

0 5 10 15 200

0.5

1

1.5

2

2.5

3


f x / f

y


fx = 2 fy


Figure 6 Comparison between nondimensional results for motions in the transverse direction of two degrees-of-freedom VIV

in cylinders with mass ratio approximately equal to the unity (m*=1.00).

Figure 7 - Comparison between nondimensional results for motions in the in-line direction of two degrees-of-freedom VIV in

cylinders with mass ratio approximately equal to the unity (m*=1.00).

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


Ay /

( D

)

Pendulum L/D = 2.00m* = 1.00Pendulum L/D = 1.70m* = 1.00Pendulum L/D = 1.45m* = 1.00Bar L/D = 2.00m* = 1.00Bar L/D = 1.50m* = 1.00Sanchis et al. (2008)L/D = 6.00 m* = 1.04Wang et al. (2008)L/D = 2.40 m* = 1.00Blevins & Coughran (2009)L/D = 17.80 m* = 1.00Gonalves et al. (2010)L/D = 0.39 m* = 1.00

0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5


Ax /

( D

)

Pendulum L/D = 2.00m* = 1.00Pendulum L/D = 1.70m* = 1.00Pendulum L/D = 1.45m* = 1.00Bar L/D = 2.00m* = 1.00Bar L/D = 1.50m* = 1.00Sanchis et al. (2008)L/D = 6.00 m* = 1.04Gonalves et al. (2010)L/D = 0.39 m* = 1.00


Figure 8 - Comparison between nondimensional results of transverse frequency oscillation and natural transverse frequency in

still water for two degrees-of-freedom VIV in cylinders with mass ratio approximately equal to the unity (m*=1.00).

Figure 9 Comparison between nondimensional results for motions in the transverse direction of two degrees-of-freedom VIV

in cylinders with low mass ratio ( . ).

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


f y / f

0

Pendulum L/D = 2.00m* = 1.00Pendulum L/D = 1.70m* = 1.00Pendulum L/D = 1.45m* = 1.00Bar L/D = 2.00m* = 1.00Bar L/D = 1.50m* = 1.00Sanchis et al. (2008)L/D = 6.00 m* = 1.04Gonalves et al. (2010)L/D = 0.39 m* = 1.00

St = 0.20St = 0.10

St = 0.05

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


Ay /

( D

)

Bar L/D = 2.00m* = 2.62Pesce & Fujarra (2000)L/D = 94.50 m* = 2.36Jauvtis & Williamson (2004)L/D = 10.00 m* = 2.62Stappenbelt & Lalji (2008)L/D = 8.00 m* = 2.36Blevins & Coughran (2009)L/D = 17.80 m* = 2.56Freire & Meneghini (2010)L/D = 21.88 m* = 2.80


Figure 10 - Comparison between nondimensional results for motions in the in-line direction of two degrees-of-freedom VIV in

cylinders with low mass ratio ( . ).

Figure 11 - Comparison between nondimensional results of transverse frequency oscillation and natural transverse frequency

in still water for two degrees-of-freedom VIV in cylinders with low mass ratio ( . ).

0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5


Ax /

( D

)

Bar L/D = 2.00m* = 2.62Jauvits & Williamson (2004)L/D = 10.00 m* = 2.62Freire & Meneghini (2010)L/D = 21.88 m* = 2.80

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


f y / f

0

Bar L/D = 2.00m* = 2.62Pesce & Fujarra (2000)L/D = 94.50 m* = 2.36Jauvtis & Williamson (2004)L/D = 10.00 m* = 2.62

Documents

Experimental Comparisons to Assure the Similiarity Between VIM (Vortex-Induced Motion) and VIV (Vortex-Induced Vibration) Phenomena