Mechanisms of the Vertical Vortex Induced Vibration of the Storebælt Bridge

J.M. Terrés-Nícoli 1, Gregory A. Kopp 2

1 Wind Engineering Coordinator, CEAMA, University of Granada, Spain, jterres@ugr.es2 Professor, Faculty of Engineering, University of Western Ontario, London, Canada,

gakopp@.uwo.ca

ABSTRACT

Considerable research has focused on the vortex induced vibration of the circular andrectangular cylinders. There are however, fundamental questions that are still not clear. Modernbridges tend to shapes in between classical bluff bodies and airfoils. There is a need forfundamental physical investigations of the triggering mechanisms of the vortex induced responseof these shapes. The Storebælt Bridge in Denmark is a notable example of hexagonal shapecommon in modern suspension bridges for which the availability of extensive numerical andexperimental data at model and full scale makes it a remarkable case study. The fluid-structureinteractions which trigger the vertical vortex induced vibration of this bridge was investigated bymeans of section model tests. The study focuses on the simultaneously measured pressure fieldand response for different increasing and decreasing wind speeds.

INTRODUCTION

Even though scientific study of the flow induced vibrations of circular cylinders is vast, startingin the late second half of the 19th century with the work of pioneers such as Raleigh or Strouhal,there are still a number of fundamental questions for which there are no clear answers. Exceptfor creeping flow at low Re, the flow around a stationary cylinder involves boundary layerseparation and two corresponding free shear layers. The flow structure results from theinteraction of the shear layers in the wake region. At certain Re, the interaction of the two shearlayers results in organized, alternating vortex shedding from each side. The synchronizedshedding will lead to a fluctuating pressure field on the surface of a stationary body. The cylinderwill vibrate under this loading if it is flexible (or linked to sprung system of certain stiffness anddamping). It has been observed how the resulting body motion affects the above described flowstructure and forcing. Different vortex shedding modes which may involve single or pairs ofvortices shed on each side have been associated to different states (2S, 2P, S+P …, see [2, 28,29]). It is noted here how significant the differences of the flow structure around an oscillatingcylinder are compared to the stationary case and result in the development of forces of differentnature and magnitude.

Figure 1: Rectangular prism with superimposed triangular leading and trailing edges forming an hexagonalshape with reduced drag features common in modern flexible bridges (c/D=7).

The flow around a rectangular prism may involve more mechanisms depending on itschord to depth ratio, including trailing edge vortex shedding (TEVS), leading edge vortex

A

C D

B

shedding (LEVS), impinging leading edge vortex (ILEV) and interactions among them (vortexinteraction, VI) [15]. It has been observed that this vortex formation and structure is sensitive toperturbations across the mean flow, which, acting at a particular frequency, can trigger theinstability at the same frequency [18]. Such perturbations may originate by the buffetingbackground response of the body at its natural frequency. The vortices originating from thisexcitation can be referred to as motion induced vortices (MIV). The existence of MIV togetherwith the gradual development of a different flow structure due to the body motion, such as thoseobserved for the circular cylinder, enhance the different nature and magnitude of the pressurefield and resulting force compared to the stationary case.

Modern long and midspan bridges often incorporate aerodynamic leading and trailingedges in order to reduce the significant along wind loading and possibly the strength of thevortices shed at TE (see the recent Third Millenium Bridge, [24]). The most practical option is toplace a triangular fairing while maintaining the depth necessary to provide the needed verticalstiffness (see Figure 1 for c/D=7). This may justify why hexagonal shapes are becoming socommon in bridge decks. The addition of the leading and trailing edges will alter the flowstructure mechanisms that correspond to the rectangular prism. The effect maybe moresignificant at the leading edge where the BL flow over the leading edge surfaces may affect thestructure and instability of the impinging shear layer. Depending on the chord to depth ratio theseparated flow will or will not reattach along the afterbody. The shape of the trailing edge is notexpected to affect the flow structure to the same extent if the flow remains separated in thevicinity of “B” in Figure 1. If it does reattach permanently or intermittently, one can expect thetrailing edge shape may have some effect on the vortex formation and strength of vortices. Thecomplex aerodynamics of bridge decks with these shapes are normally evaluated through intenseexperimental programs. However, the fluid-structure mechanisms behind the different aeroelasticphenomena are not well known.

The Storebaelt Bridge is a notable example of such hexagonal shape. Furthermore, theavailability of numerical and experimental data at full and model scale is remarkable. It is notedthat it may be among the few bridges where the numerical and physical modeling has beenperformed via such a wide variety of methodologies including: Section model tests at 1:60, 1:70and 1:80 scale, Full aeroelastic tests and Taut strip tests.

STOREBÆLT FULL SCALE OBSERVATIONS

Notwithstanding the above intensive and long term experimental program, the Storebælt Bridgeexperienced large amplitude, unacceptable vortex induced vibrations. The amplitudes were,however, higher than the predicted ones. Two reasons might be pointed out. First, the lowermeasured structural damping and, second, the lower than expected turbulence intensity levels.

Around the time of the inauguration of the 3.2 billion USD Storebælt Bridge, in May1998, the deck experienced vertical oscillations which excited the 3rd, 6th, but mainly the 5th

vertical modes of vibration. Accelerations were monitored for the period between 14 Jan. 1998and 16 May 1998 [3,13]. The vibrations were observed under the effect of moderate winds ofremarkably low turbulence intensity, Iu (as low as 2%, but up to 9%) and with wind directionsclose to the perpendicular of the bridge axis [14,13]. Wind speeds were in the range of 4 to 10m/s for all incidents, matching with predicted values from the wind tunnel tests. Simultaneouspressures and accelerations were recorded in a second campaign of full-scale monitoring [7, 8].The data collected correlated well with the full-scale data mentioned above. For oscillations inthe 5th mode structural damping values as low as =0.45% (as a fraction of the critical) werefound. A much higher correlation between pressures and accelerations at lock-in was measured

than for outside of lock-in. Correlations between pressures at different locations on the top flangeof the trailing edge were somehow insensitive to changes in wind direction. In contrast, pressure– acceleration correlations at this same location were highly affected by changes in thisparameter. Unfortunately, due to some constraints, pressures were measured only at threelocations. These were all on the top flange, and not near the edges. As will be shown in thecurrent investigation, and also according to the data of Larose [11], pressure in this area is notexpected to play a significant role in either the vertical or the torsional response [25].

The deck was ultimately retrofit by adding guide vanes along the main span (at the lowercorners E and F in Figure 2) with the subsequent additional cost.

METHODOLOGY

The self excited nature of VIV lies on the capability of the motion of the body in modifying theflow structure and hence inducing forces of different nature and magnitude. One example is the2P vortex shedding mode responsible for the maximum amplitude vibration of the cylinder or themotion induced instability of the impinging shear layer of leading edge separation bubble ofrectangular prism. The understanding of the mechanisms responsible for the vibrations of bluffbodies (such as hexagonal shapes) therefore requires the investigation of the forcing throughoutthe different phases of the response, which may differ from the stationary case. Consequentlysome analytical models have been derived by separately modeling a “stationary” force and a“motion induced force” [20, 27].

A 1:70 scaled section model was designed and built for this purpose. A set of 11 pressurescanners had to be installed inside the model and they had to be easily removable. A centralchannel under the top layer was made to house the scanners and all the tubes and wiresconnected to them. The scanners, together with these accessories, had to be tight so that theywould not move when the model was undergoing vibration. Scanners are insensitive toaccelerations parallel to their internal diaphragms; thus, they were located with the sensitive axisof the transducer in the restrained drag direction to avoid any induced error due to theaccelerations. The total mass of the scanners inside the model reached a value of over 10% thetarget mass. Consequently, the model had to be designed lighter than normal but still as stiff. Thedeck was made of a carbon fiber – cardboard sandwich laminated with the help of a foam mould.

12

3456789101112131415161718192122232420

2526

27

28

29

30

31 32 33 34 35 36 37 38 39 40 41 4243

4445

4647

48Center of rotationLeading edgeTrailing edge

Bottom flange

A

BC

D

E F

LINE #2LINE #1

LINE #4 LINE #5

LINE #3 Upper flange

Figure 2: Pressure measurement distribution of a typical ring. Three rings of taps distributed in the samefashion were used. One located at the midspan of the model and another two separated one chord and one

half of the chord respectively from the midspan. Another set of taps is distributed in five lines along the span(line #1,#2 and #3 at the upper flange, and line #4 and #5 at the bottom flange).

The pressure field was investigated in detail with the simultaneously measured responsethrough the different phases of the response from the onset up to the maximum amplitudeoscillations. The study was performed for both Vertical and Torsional Vortex Induced Vibration(V-VIV and T-VIV, respectively) and Flutter. This paper focuses on the first. The response wasstudied when the maximum amplitude wind speed was approached, increasing from lower anddecreasing from higher wind speeds. Hereinafter the different responses will be simply referredto as response to increasing and decreasing wind speeds.

The present test configuration, namely “BLWTL 3” is compared in Table 1 with thedifferent historical sectional model configurations of the V-VIV performed at differentlaboratories. This experimental setup was specifically designed for the study of the simultaneousmeasurement of the pressure field and response during vertical vortex induced vibration. Aremarkably stiff sprung system was designed to achieve a high velocity scale that, given the lowcritical wind speed (VCR

+ =1.32), would allow for the pressure field measurement. Consequentlythe rig support frame was stiffened to avoid any undesired contributions from rig flexibility.Similarly, a stiff design of the carbon fiber sandwich of the section model kept the model modesaway from the sprung system frequency so that these would not be excited. The fundamentalvertical bending mode was monitored and found at a frequency of 42 Hz, significantly above the9.78Hz of the simulated vertical mode. The first torsional mode was found at 48 Hz and, hence,no contribution is expected from the inherent model modes.The larger resulting velocity scale (1.44), allowed the investigation of the response domain witha higher resolution in the wind tunnel.

Table 1: Different section model test configurations available in the literatureLaboratory Date Model

ScaleAspectRatio

ModelNaturalBendingmode (Hz)

fV(Hz)

ModelV

ReD ReB Iu(%)

Damping(air)

Flutterlimit(f.s. m/s)

BLWTL 1 2001 1:70 4.81 42 2.28 0.34 5.6·103 3.9·104 0.5 0.18 73.3BLWTL 2 2002 1:70 4.81 42 3.38 0.50 8.3·103 5.8·104 0.5 0.15BLWTL 3 2003 1:70 4.81 42 9.78 1.44 2.4·104 1.7·105 0.5 0.33DMI 1 1993 1:80 6.5 n/a 3.54 0.46 7.4·103 5.1·104 3.5 1.7 73.9DMI 2 1998 1:60 4.94 20 6.94 1.19 2.1·104 1.5·105 0.3 0.1

RESULTS AND DISCUSSION

The pressure field could be measured at 14 different stages of the response from the onset windspeed (VOR

+) to the critical maximum amplitude oscillations (VCR+). Figure 4 presents the

observed response (rms values of yr) for different damping values and wind speed history. Thepresent results are in good agreement with previous experiments (DMI 1 and 2, as per Table 1).The vertical vortex induced response is found to have a hysteric behaviour: different response isobserved when the maximum amplitude oscillation wind speed (Vcr

+), point “C”, is approachedfrom lower wind speeds compared to when the wind speed is decreased from V->Vcr

+. Theformer correspond to the path marked as ABCDE, compared to EDFBA for the later, in thefigure. It is noted here that the modal mass was in every case matched to the scaled value of theprototype. Hence differences in the Scruton number between the different available data are onlydue to changes in damping. The strong sensitivity of V-VIV to the inherent damping can beclearly appreciated in the response domain. A reduction in the maximum amplitude oscillationsof around 30%, from 0.075 to 0.05 is observed when the damping was increased from air =0.15to 0.33%.Different response is observed for different rates of increasing wind speed. In general, highervalues of maximum amplitude oscillations are observed for slower rates in increasing ordecreasing wind speeds. This was also clearly observed in the T-VIV [25].

ONSET OF THE OSCILLATIONS

The onset windspeed (VOR+) was in every case around 0.8 based on the bridge chord (B) and

VORD+=5.64 based on the bridge depth (D). Similarly, maximum amplitude oscillations are

observed for VCR+ = 1.32 or VCRD

+=9.30. The model by Komatsu & Kobayashi [10] presented in

predicts a reduced windspeed of 8.96 which is derived for the rectangular prism based on thesign of the work done by the pressures along the chord. The onset of the oscillations is wellpredicted by the proposed relationship by Shirashi & Matsumoto [22] 0.83(B/D) which yields5.85 compared to the actual 5.64. These models are based on a flow structure scheme thatinvolves ILEV and TEV based on the chord to thickness ratio.

The study of the different mechanisms involved in the VIV of rectangular prisms isnormally based on the chord over thickness parameter (B/D). The additional two edges of ahexagonal shape let argue whether such study shall be based on the 3 different ratios present. Forthe Storebaelt bridge this would lead to 3 B/D ratios based on AD, BC and DE: 7.05, 6.14, 4.32.The three of them would belong to the same group 3.2<B/D<7.6 where the separated leadingedge shear layer periodically separates and reattaches interacting with the vortex formation at thetrailing edge.

Figure 4: Observed Vertical Vortex Induced response observed for different damping ratios and wind speedhistory compared with previous wind tunnel and full scale data.

Naudascher [17] proposed a model domain which predicts possible transverse vibrationin mode 2 (n=2) due to ILEV for the present B/D ratios of the Storebaelt at VORD

+=1/StISLI,where StISLIis the Strouhal number corresponding to the vortex formation due to the instabilityand roll up of the impinging shear layer of 0.17 (based on B/D=7.05 and =0). Thecorresponding onset windspeed is VORD

+=5.87, VOR+= 0.83 which matches the experimental

observations in Figure 4. Naudascher’s Mode 2 corresponds to the coexistence of two ILEVsthrough one vibration cycle. Modes 1 and 3 for this B/D ratios appear less probable but possibledepending on the particular flow conditions (for example, Iu or surface roughness) affecting thevortex formation. This is controlled by the parameter in Naudascher formulation.

Cp=0.5 Cp=1 positive negative

flow

Mean Cp distribution. Vr=0.957+ .Ring B

Figure 5: Mean pressure field at onset increasing wind speed. Model is let free to vibrate.The instability of the impinging shear layer at large Re numbers requires a control mechanism.This control can be obtained by means of transverse sound [18] or by the actual body motion.Therefore, even leading edge movements at the natural frequency due to background buffetingresponse can act as the triggering mechanism of the impinging shear layer instability and thecorresponding ILEV formation. Such motion induced ILEV (MI-ILEV) may be responsible forthe V-VIV of the Storebaelt.Based on the mean pressure field presented in Figure 4.6, the separation bubble is anticipated tospan from a point located between A and B (tap #4) to a point downstream of the railings neartap #16. Remarkably, the span of the separation bubble (d) is found close to the bridge depth (D).The separation bubble located on the bottom flange, around point F, presents a comparable scale.To investigate the ISLI phenomena, the spectra of the pressures on the upper flange of theleading edge were examined. It is noted that at the onset wind speed no significant peak ispresent anywhere within the separation bubble (Figure 6) whereas at peak at frequency of 8.79Hz is present elsewhere downstream of it (Figure 7). This frequency corresponds to a St=0.16based on either (D) or the (d) in good agreement with the StISLI proposed by Naudascher.

Special attention is given to the area near the upper flange of the trailing edge (CD, taps24-27). The spectra at these locations present another peak at the exact natural frequency fV.Additionally, a broader peak is observed at a higher frequency corresponding to a higher St closeto St=0.20. A stronger peak corresponding to this higher St is observed in the spectra of hotwiremeasurements at different wind speeds in the near wake (Figure 9, top left). From left to right inthe spectra of the pressure on tap B27, the first peak would be related to the intermittentinteraction of the ISL, the second would motion induced and the third, TEV as it is postulatedbelow. Similar spectral density peaks at these three frequencies is observed too in the lift forcecoefficient CL in contrast with the only peak at fV present for the maximum amplitudeoscillations.

It is postulated here that this frequency, which is not present anywhere except for thetrailing edge and the wake is related to vortex shedding from the trailing edge (TEV). TEVwould be responsible for the initial vibration at the onset windspeed which will control and leadto the instability of the impinging shear layer, roll up and subsequent vortex formation.

Figure 6: Typical spectra of the pressure on locations within the separation bubble at the onset of theoscillations. Model is let free to vibrate amplitude is negligible. No peak is observed in contrast with the

spectra of the pressure for points right downstream of the separation bubble presented in Figure 7

A similar behavior is observed at the bottom flange which has its origin at a separation bubblelocated around point F and which, as will be shown, is out of phase with the upper flange ILEV.This is found consistent with “Group 2” mechanisms as proposed for the heaving motion byMatsumoto [22].Limited hotwire measurements B/2 downstream of the TE but further separated 3D/4 above andbelow the TE present a distinct peak which corresponds to a St=0.15 which matches the estimateStISL and that is here related to the same leading edge shear layer (Figure 8 top right). At higherwind speeds, as the magnitude of the oscillations builds up, the dominant frequency in the wakeis that of the vertical mode (fV) and its harmonics (Figure 8, bottom).

100

101

10-6

10-4

tap B1, Vr=0.96+

S(f)

100

101

10-6

10-4

tap B2, Vr=0.96+

100

101

10-6

10-4

tap B3, Vr=0.96+

100

101

10-6

10-4

tap B4, Vr=0.96+

100

101

10-6

10-4

tap B5, Vr=0.96+

S(f)

100

101

10-6

10-4

tap B6, Vr=0.96+

100

101

10-6

10-4

tap B7, Vr=0.96+

100

101

10-6

10-4

tap B8, Vr=0.96+

100

101

10-6

10-4

tap B9, Vr=0.96+

S(f)

100

101

10-6

10-4

tap B10, Vr=0.96+

100

101

10-6

10-4

tap B11, Vr=0.96+

100

101

10-6

10-4

tap B12, Vr=0.96+

100

101

10-6

10-4

tap B13, Vr=0.96+

S(f)

freq.(Hz)10

010

1

10-6

10-4

tap B14, Vr=0.96+

freq.(Hz)10

010

1

10-6

10-4

tap B15, Vr=0.96+

freq.(Hz)10

010

1

10-6

10-4

tap B16, Vr=0.96+

freq.(Hz)

Figure 7: Typical spectra of the pressure on locations downstream of the separation bubble at reducedwindspeed of 0.96. A peak at a frequency of 8.79Hz was observed at every tap in contrast with the spectra ofpoints within the separation bubble in Fig. 6. The locations near the trailing edge presented a second distinct

peak at the vertical natural frequency, 9.78Hz.

COMPARISON WITH FULL-SCALE OBSERVATIONS

The full-scale response [7, 8] presented in Figure 4 compares well with the different modelresponse. The response points observed at reduced wind speeds around 1.8 correspond to broadband energy content and, therefore, are related to background buffeting response rather thanVIV. It is noted that the onset and maximum amplitude oscillations wind speeds and magnitudesare well predicted by the physical modeling. Especially significant are the hysteresis behaviourobserved in the full scale measurements. A full scale observation of VIV corresponding todecreasing wind speed is marked in Figure 4. Importantly, it is found that the presentexperiments for decreasing wind speeds are in good agreement with this unusual full scaleobservation.

Pressures were measured on the upper surface at 3 simultaneous locations between thecentral and side crash barriers of the downstream half. The St numbers derived from the spectra

100

101

10-6

10-4

tap B17, Vr=0.96+

S(f)

100

101

10-6

10-4

tap B18, Vr=0.96+

100

101

10-6

10-4

tap B19, Vr=0.96+

100

101

10-6

10-4

tap B20, Vr=0.96+

100

101

10-6

10-4

tap B21, Vr=0.96+

S(f)

100

101

10-6

10-4

tap B22, Vr=0.96+

100

101

10-6

10-4

tap B23, Vr=0.96+

100

101

10-6

10-4

tap B24, Vr=0.96+

101

10-6

10-4

tap B25, Vr=0.96+

S(f)

101

10-4

10-3

tap B26, Vr=0.96+

101

10-4

10-3

tap B27, Vr=0.96+

100

101

10-6

10-4

tap B28, Vr=0.96+

100

101

10-6

10-4

tap B29, Vr=0.96+

S(f)

freq.(Hz)10

010

1

10-6

10-4

tap B30, Vr=0.96+

freq.(Hz)10

010

1

10-6

10-4

tap B31, Vr=0.96+

freq.(Hz)10

010

1

10-6

10-4

tap B32, Vr=0.96+

freq.(Hz)

of the full scale pressures range from 0.08 to 0.15. The latter result is in good agreement withwhat has been previously related to ILEV. Nevertheless, as has been described, the region wherethe pressure measurements were conducted is unaffected by the pressure fluctuations related toTEVS which have been related to a higher St number around 0.20 and is held responsible for theonset of the oscillations.The limited pressure spectra in [8] present a motion induced sole peak at the natural modalfrequency which is in good agreement with the observed behavior in the present experiments forthe corresponding taps 21-22.

100 101 102

10-6

10-5

10-4

10-3

10-2

freq(Hz)

S(f)

St=0.21Hw, B/2 downstream ofTEFlash with upper flange

101 102

10-5

10-4

10-3

10-2

10-1

freq(Hz)

S(f)

St=0.15windspeed=6.31HW B/2 downstream of TE3/4 above and below TE

0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

Endp

late

Acc

eler

omet

er (V

)Time (s)

Vr=1.3205+. Ring B

0 0.5 1 1.5-2

-1

0

1

2

Mid

span

Acc

eler

omet

er (V

)

Time (s)

Vr=1.3205+. Ring B

100

101

10-5

100

S(f)

f req.(Hz)

Endplate Acc. Spectra, Vr=1.3205+. Ring B

100

101

10-5

100

S(f)

f req.(Hz)

Midspan Acc. Spectra, Vr=1.3205+. Ring B

0 0.5 1 1.50.5

1

1.5

2

Hot

wire

pro

be 1

Time (s)

Vr=1.3205+. Ring B

0 0.5 1 1.50.8

1

1.2

1.4

1.6

1.8

2

Hot

wire

pro

be 2

Time (s)

Vr=1.3205+. Ring B

100

101

10-6

10-4

10-2

S(f)

f req.(Hz)

Hotw ire #1 spectra, Vr=1.3205+. Ring B

100

101

10-6

10-4

10-2

S(f)

f req.(Hz)

Hotw ire #2 spectra, Vr=1.3205+. Ring B

Figure 8: (top left) Spectra of windspeed measurements at a point located B/2 downstream of the trailingedge, flash with point C. The peak observed in the pressure on the trailing edge is consistently shown here as

well. (top right) Spectra of wind speed measured at points located B/2 downstream of TE but furtherseparated 3D/4 above and below the TE. The Strouhal number observed corresponds to the one that was

related to ISLI. (bottom) Same measurements at maximum amplitude oscillations.

COMPARISON OF THE PRESSURE FIELD AROUND THE OSCILLATING AND STATIONARY DECK

The pressure field at the onset wind speed compares well with the pressure around the fixed deckmeasured by Larose by means of a taut strip model [11]. The comparison of the mean pressurefield is presented in Figure 9, where the rms. values are compared as well. The predicted scalesof the separation bubbles at leading and trailing edge are in good agreement.

The minor observed differences could be partly due to the different spatial resolution, 48points of the present 1:70 scale section model compared to 32 of the smaller 1:300 taut strip. Adistinctive feature observed in both the mean and rms. results is the pressure distribution around

the central crash barrier. A characteristic slope, higher suctions and rms are observed in thesection model. This could be due to the slope in the roadways (2.5%) and the central barrier thatcould not be modeled in the taut strip model. The higher fluctuations around this area seemunimportant for the stationary state but may play a role in the ISLI mechanisms during deckmotion.

Figure 9: Comparison of the mean pressure distribution of steady sprung mounted deck (dotted line andbars) of the present study with the same around the fixed deck (solid blue line for suctions and black for

pressure), after Larose [11]

The predicted force coefficients at 0for the fixed deck from the different availableexperiments are in a range close to 0 of CL =0.08 to 0.1for smooth flow. Similar results were alsoobtained by the integration of the pressure field around the 1:300 taut strip model (Larose, 1992).These values are notably lower that the fluctuating coefficients measured at maximum amplitudeoscillation of the present study of CL 0.46,0.3 which emphasizes the nature and magnitudeof the motion-induced forces.

These values are compared to the static coefficients for the angle of attack due to theapparent wind speed considering the deck’s motion. An apparent angle of attack of 3.57 and2.27 are obtained for the maximum and mean deck speeds which correspond to staticcoefficients of approximately 0.27 and 0.18. Even though these are better estimates, they areclearly lower than the observed lift coefficients acting on the vibrating deck (Figure 10). Thisdifference stresses the significance of the motion induced mechanism responsible for doublingthe values derived from the above mentioned quasi-static approach

As has been described, the build-up of the oscillations involves motion inducedmechanisms. Therefore it is anticipated that the flow structure, pressure magnitudes and forcecoefficients of the oscillating deck differ from the stationary case. Not surprisingly different Stnumbers have been reported by different researchers. The full scale St in Table 2, for example, isbased on pressure measurements on the upper flange upstream the trailing edge (around tap #22)and, therefore, unable to capture the frequencies present at the onset that have been related toTEV and which are only observed further downstream or in velocity measurements in the wake.The value of 0.15 is in good agreement with the peaks observed in the present study that havebeen related to ISLI based on the previously mentioned observations from Nakamura,Naudascher, Shirashi and Matsumoto. Furthermore, most of the computational work simulates

the flow around the fixed deck and does not include many details that can significantly alter theflow structure (railings, barriers, surface roughness, etc.).

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.550

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Ver

tical

dis

p. (

z/D

), m

ax a

mpl

itude

Lift Coeficient Cl, max amplitude

Vertical Vortex Induced Response, Max response vs. max lift for diferent wind speeds

Vr+

Vr-

Figure 10: Maximum displacements observed for the maximum amplitude (within an oscillation cycle)through the build up phases of response.

BUILD UP

The build-up of higher amplitude oscillations is related to the synchronization of the flowstructure triggered by the deck motion, which results in an overall phase shift of the liftcoefficient. No significant difference other than the increase in the magnitudes of the pressureand force coefficients is observed throughout the build-up phases except for the region on theupper flange right at the leading edge (taps #1-3). It will postulated that (similarly for the T-VIV)the large amplitude fluctuations at this location play a significant role in the instability andstrength of the separated flow around the corner “B”.The phase averaged pressure field was investigated using the vertical displacement as areference. A clear sinusoidal response is observed for reduced wind speeds above 1.0 (Figure11). For higher wind speeds the frequency component that was associated to TEV was notpresent in the near wake velocity measurements nor the in the spectra of the pressures. Thedominant frequency in the flow is equal to fV.

Table 2: Different Strouhal numbers available in the literature compared to the present study.Scale Iu (%) ReD air St

Sectional model (present) 1:70 0.5 2.4·104 0.12-0.33 0.15 (ISLI)0.21 (TEV)

Larose, Taut strip model [11] 1:300 6 946 1.5 0.11Frandsen, CFD-FVE [7] 0 1 0.11Frandsen, Full scale [8] 1.1·106 0.08-0.15Morgenthal (flow vis.) 6 0.5 0.19Morgenthal and McRobie, CFD-DVM[16]

0.19

Selvam and Govindaswamy [21] 0.14-0.21Xiaoyang Wu and Kopp, G.A. (flow vis.) 1:200 8135 0.28Vezza and Taylor, CFD-DVM [26] 1: 0.16

0 45 90 135 180 225 270 315 360-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

Phase in increments of 18º

st01:ws=0.95692st09:ws=0.96048st02:ws=1.016st102:ws=1.0517st10:ws=1.052st03:ws=1.0794st11:ws=1.1854st05:ws=1.2134st06:ws=1.2822stmax:ws=1.317st07:ws=1.3179st30:ws=1.3205st12:ws=1.3427st13:ws=1.3793

Figure 11: Phase averaged sinusoidal vibration observed for the different phases of the response to increasingwind speed, as per Figure 4. The “ws” indicated in the legend correspond to the reduced wind speed, Vr

It is postulated that a typical Karman street of TEV acts as the control mechanism for theISLI. Alternate impinging leading edge vortices (ILEV) both on the upper and bottom flangesare responsible for the response. Similarly TEV will be related to the onset of torsional T-VIVoscillations. The lift responsible of maximum amplitude oscillations leads the response in aphase of 90, approximately (within the phase averaging resolution). The synchronization ofhigher amplitude oscillations is enhanced by a slowly varying phase shift of approximately 30-40 from 130 at the onset wind speed to 90 at maximum amplitude oscillations (Figure 12).This is consistent with a typical resonant response, where at resonance the force is out of phaseand therefore the total work per oscillation cycle is positive [4]. In this state, the work done bythe force is dissipated in damping. That explains why, for example, the lift coefficient atVr

+=1.38, with a considerably lower amplitude than at Vr+=1.18 causes a larger amplitude

response. The larger phase angle of the lower magnitude force results in a larger out of phasecomponent than that of the larger force and consequently the work, or energy transmitted to thebody is greater which if the damping remains constant eventually leads to a higher response.

The evolution of the in-phase and out-of-phase components are better studied in thephase plane by means of Lissajou diagrams. The corresponding diagram, for all phases of thebuild-up, is presented in Figure 13. The hysteric nature of the oscillation cycle is clear in thisfigure. As can be anticipated, the somewhat elliptical curves are read clockwise, the upperbranch corresponding to the loading path. It can be observed how, as the amplitude grows, thephase (the ellipse major axis) rotates from a nearly vertical orientation (force in phase withdisplacement) to an orientation close to the horizontal (force in phase with the velocity) leadingto larger amplitude response. As previously stated that explains how large increases in thedisplacement follow to lower relative increments in the force magnitude. This could be describedas parametric transformation of the ellipse (rotation, scaling and translation). Interestingly, thediagrams do not present any kink as observed in the corresponding cylinder diagram [1].

0 45 90 135 180 225 270 315 360-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Phase in increments of 18º

Pha

se a

vera

ged

norm

aliz

ed li

ft co

efic

ient

Phase averaged lift coeficient through the build-up. Max amplitude @ Vr=1.32

st01:ws=0.95692st09:ws=0.96048st02:ws=1.016st102:ws=1.0517st10:ws=1.052st03:ws=1.0794st11:ws=1.1854st05:ws=1.2134st06:ws=1.2822stmax:ws=1.317st07:ws=1.3179st30:ws=1.3205st12:ws=1.3427st13:ws=1.3793

Figure 12: Evolution of the force coefficients through the build up phases of the response.The remaining question is how the flow structure around the deck leads to the net forcespresented in Figures 12 and 13. The pressure fluctuation on the bottom flange of the leading edgeup to the end of the separation bubble are all in phase with the response and therefore do notcontribute to the energy transfer or the increase in the response. This can be observed in Figure14 (left) and will be consistent with the body motion controlling the instability of ISL.Downstream of it, a phase shift is observed in the pressure along the bottom flange Figure 4.14(right). The resulting phase with respect to the rotation when the ILEV approaches the trailingedge is 90, which results in optimum contribution to the increase of the oscillations magnitude.This would explain the effectiveness of the guide vanes located around point E. The vanes on theleading edge corner F would distort the structure of the separation bubble and on the trailingedge corner would act complicating the interaction with the body surface as they leave thebottom flange and the consequent out of phase force.

- 0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Pha

se a

vera

ged

lift c

oefic

ient

P hase averaged norm alized vertical d isplacem ent

Phase averaged li ft coeficient vs vertical displacem ent through the build-up. Max amplitude @ Vr=1.32

st01:ws=0.95692st09:ws=0.96048st02:ws=1.016st102:ws= 1.0517st10:ws=1.052st03:ws=1.0794st11:ws=1.1854st05:ws=1.2134st06:ws=1.2822stmax:ws=1.317st07:ws=1.3179st30:ws=1.3205st12:ws=1.3427st13:ws=1.3793

Figure 13: Lissajou phase diagrams for the different phase of vertical vortex induced response

Figure 14: Left, pressure coefficients at maximum amplitude oscillation on the lower flange of the leadingedge. Constant phase 0. Right, pressure coefficients at maximum amplitude oscillations along the bottom

flange, progressive phase shift up to 90º near the trailing edge corner.

CONCLUSIONS

The vertical vortex induced vibration mechanisms of the Storebælt were investigated. Thesimultaneously measured pressure field around a section model at the full range of increasingand decreasing wind speeds within the lock-in range was analyzed in detail. Vortex shedding atthe trailing edge (TEV) seems to be the only source of significant pressure fluctuation of themotionless deck. The motion induced by these trailing edges vortices may be responsible for thecontrol of the impinging shear layer instability at the leading edge. The onset of the oscillation istherefore attributed to TEV. Significant pressure fluctuations are observed at the leading edgewhen the motion takes place. The build-up of larger amplitude vibrations appears to be causedby leading edge vortices generated by motion induced instability at the natural frequency. Amechanism that has been referred to as vortex interaction of these vortices and those forming atthe trailing edge could be responsible for the enhancement of maximum amplitude oscillations,however this requires further investigation. An increasing phase is observed in the force, withrespect to the response, reaching values of /2 consistent with a typical resonant response.Finally it is noted that significant hysteresis behavior is observed in the response.

ACKNOWLEDGEMENTS

The authors gratefully acknowledge the financial support from the University of WesternOntario’s Academic Development Fund for the wind tunnel models used in this study. J.M.Terrés-Nícoli acknowledges the support by the GDFA Research Group at the University ofGranada. G.A. Kopp gratefully acknowledges the support provided by the Canada ResearchChairs Program.

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