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BBAA VI International Colloquium on:Bluff Bodies Aerodynamics & Applications

Milano, Italy, July, 20–24 2008

PROPERTIES OF UNSTEADY PRESSURE ACTING ON A CIRCULARCYLINDER OSCILLATING IN A WAKE OF A PRISM

Minoru Noda ∗, Fumiaki Nagao∗, Hiroyuki Wada † and Hiroko Naitou ‡

∗Institute of Technology and ScienceThe University of Tokushima, 2-1 Minami-Josanjima, Tokushima, 770-8506, Japan

e-mail: [email protected] , [email protected]

†Graduate School of Advanced Technology and ScienceThe University of Tokushima, 2-1 Minami-Josanjima, Tokushima, 770-8506, Japan

‡Graduate School of EngineeringKyoto University, Katsura Campus, Nishikyo-ku, Kyoto, 615-8540, Japan

Keywords: Wake Excitation, Forced Oscillation, Unsteady Aerodynamic Pressure Measure-ment, Flow Visualization

Abstract: The wake excitation of a circular cylinder in the wake of a prism, which was replacedinstead of a windward circular cylinder to reduce the effects of Reynolds number, was investi-gated by free vibration tests, unsteady pressure measurements and flow visualization tests.

The effects of the size of the windward prism, of the transverse offset between the circularcylinder and the windward prism and of DOF of the circular cylinder were examined. As theresults of the free vibration tests, it was clarified that the size of windward prism controls theresponse amplitude of the leeward circular cylinder, the transverse offset changes the prevailingresponse direction and the oscillation type.

On the other hands, the unsteady pressure measurement tests and the flow visualization testswere carried out with the 2 DOF forcedly oscillation method. The ensemble averaged pressurefor one cycle was calculated to investigate the mechanism of the wake excitation focusing tothe flow condition near the circular cylinder. The ensemble rms pressure, which was definedas an fluctuating pressure asynchronizing with the motion of the circular cylinder, reflects thecondition of the separation flow near the circular cylinder clearly. As the results of the unsteadypressure measurements and the flow visualization tests, it was clarified that the motion of theseparation flows from the windward prism plays an important role for the wake excitation of theleeward circular cylinder.

1

Minoru Noda, Fumiaki Nagao, Hiroyuki Wada and Hiroko Naitou

1 INTRODUCTION

The wake excitation of tandem or staggered cylinders, such as cables of cable-stayed bridge,chimneys and so on, is a serious problem. Many researchers have investigated this phenomenonto clarify the mechanism of this oscillation. [1, 2, 3, 4, 5] It has been pointed out that the char-acteristics of the wake excitation of a tandem or staggered cylinders depend on the arrangementof the circular cylinders and Reynolds number.

To clarify the mechanism of the wake excitation, it is necessary to separate the influence ofthe arrangement of the circular cylinders and that of Reynolds number from the experimentalresults. However, it is difficult to separate those influences while the circular cylinder was usedas the windward cylinder. Therefore, to reduce the effects of Reynolds number on the wakeexcitation, the windward circular cylinder was replaced by a square prism.

In this study, the wake excitation of a circular cylinder in the wake of a prism was investi-gated by following wind tunnel tests; 1) free vibration tests of 1DOF and 2DOF, 2) unsteadysurface pressure measurements in forced vibration, 3) flow visualization tests using a high speedcamera.

2 EXPERIMENTAL CONDITION

2.1 Arrangement properties of test models

Figure1 shows the arrangement of a windward prism and a leeward circular cylinder, andthe arrangement properties defined in this study. The first property was the distance betweenthe upstream edge of the windward prism and the center of the leeward circular cylinder,c. Thesecond one was the size of the windward prism,d. The third property was the distance betweenthe centers of the windward prism and of the circular cylinder normal to flow direction,e.

D

c

d

d

e

Wind direction

Windwardprism

Leewardcircularcylinder

x

y

Figure 1:Arrangement properties of test models

2.2 Free vibration tests

The free vibration test was carried out in a semi-closed circuit type wind tunnel, whose testsection was 1.0 m wide, 1.5 m high and 4 m long. Figure2 indicates the general view of the freevibration test. In this test, the diameter,D, and length,L, of the circular cylinder were 40 mmand 900 mm, respectively. The leeward circular cylinder was supported by 8 coil springs, andwas examined in longitudinal 1 DOF, lateral 1 DOF and 2 DOF, respectively. The windwardprism was fixed on the side walls of the wind tunnel at the point ofc = 3D or 9/4D. Itswidth, d, was changed from1/8D to 3/4D to investigate the effects of the width of the wakeon the aerodynamic response of the circular cylinder. Thee was also changed from 0 to6/5D

2


Wind

Axial load cell(Strain gauge)

Axial load cell(Strain gauge)

Leeward

Circular C

ylinder

Windward

Prism

Coil Springs

CoilSprings

CoilSprings

CoilSprings

Figure 2:General view of the free vibration test

transverse direction longitudinal directionNatural Frequency 1.91 Hz 1.80 HzLongitudinal damping ratio 0.0305 0.0170Scruton’s number 34.3 57.2

Table 1:Dynamic Properties of the Leeward Circular Cylinder (2 DOF)

Natural Frequency 1.91 HzLongitudinal damping ratio 0.0305Scruton’s number 34.3

Table 2:Dynamic Properties of the Leeward Circular Cylinder (1 DOF)

to examine the response of the circular cylinder under the staggered arrangement. In this study,the position of the leeward circular cylinder was set in the condition of no wind. Therefore,the neutral displacement was changed by the static aerodynamic forces. The displacementsof transverse and longitudinal direction were measured by axial resistant forces of supportingcoil springs. The dynamic properties were shown in Tables1 and2. The reduced wind speed,U/fD, were changed to 150 by changing the wind speed,U . Therefore the Reynolds number,UD/ν, was3 × 104 approximately.

2.3 Unsteady surface pressure measurement

In the unsteady surface pressure measurement tests, the leeward circular cylinder, whosediameter,D, and length,L, were 70 mm and 900 mm, respectively, was vibrated forcedly by 2DOF forced oscillator composed of two sets of two linear actuators, as shown in Figure3. Thisoscillator can reproduced the time record of the displacements measured in the free vibrationtests. Seventy two pressure holes, whose diameter was 1 mm, were installed on the surface of

3


Linear Actuator

Linear Actuator

Leeward Circular Cylinder

Clear supporting plate

Wind

Links to fix rotation

(a) Side view (b) Wind tunnel installed the oscillator

Figure 3:General view of the forced oscillation test

the center span of the circular cylinder. The unsteady pressure on the leeward circular cylinderwas measured by multi-channel pressure scanner in 2 DOF, 1 DOF for transverse or longitudinaldirection. The sampling frequency of pressure and the number of sampling were 200 Hz and60000, respectively. The measured pressure was ensemble averaged during one period.

The oscillation frequency,f , was about 0.98 Hz and the reduced wind speed,U/fD, wasabout 150. Therefore, the Reynolds number,UD/ν, was4.6 × 104 approximately.

2.4 Flow visualization tests

The visualization test of the flow between the windward prism and the leeward circularcylinder was carried out by the forced oscillation method, shown in Figure3. The flows wererecorded by a high speed camera to clarify the relation between the unsteady pressure and theflow. The flame rate of the high speed camera was 500 fps. Although the model setting wasthe same with the unsteady pressure measurement tests, the wind speed and the oscillation fre-quency were reduced keeping the reduced wind speed to take movies clearly. In this study, thewind speed and the oscillation frequency were 4 m/s and 0.4 Hz, respectively. Therefore, theReynolds number was1.8 × 104.

3 Results and Discussions

3.1 Response of the leeward circular cylinder

In this section, the responses of the leeward circular cylinder in the wake of the prism are dis-cussed. Figures4 and5 show the responses of the leeward circular cylinder measured under the2 DOF condition. Figures (a) and (b) are the relation between the reduced wind speed,U/fD,and the double response amplitudes of the transverse direction,Y/D, and of the longitudinaldirection,X/D. The Lissajous curve of the response was indicated in figure (c), where twocurves starting from the leading edge of the prism indicate the wake area of the prism drawn bythe following formula obtained by the wind speed distribution[6].

b(x)

d=

√1.81

x

d+ 1 (1)

whereb(x) is the transverse width of the wake at the position ofx from the leading edge of aprism.

4


The results for 1 DOF of transverse direction were also indicated in Figure6. 1 DOF oflongitudinal direction had no oscillation.

3.1.1 Effects ofe/D on the response

Figures4 and5 show that the relation between the velocity and the response amplitude waschanged bye/D. The on-set velocity increased withe/D, and prevailing response directionchanged from transverse to longitudinal by increase ofe/D. Whene/D was less than 1/5 inthe case ofd/D = 1/8 or e/D was less than 3/5 in the case ofd/D = 1/2, an unstable limitcycle appeared near the onset velocity like a hard flutter. The type of response changed from ahard flutter type to a soft flutter type with increase ofe/D.

In the case of 1 DOF for transverse direction shown in Figure6, the relation betweenU/fDand the response amplitude also changed bye/D for eachd/D. The response amplitude in-creased, and the onset velocity decreased in comparison with those in 2 DOF. On the otherhand, the type of response was changed from a soft flutter type to a hard flutter type with in-crease ofe/D for eachd/D. This result indicates that the type of response depended on thelongitudinal motion ande/D, and the transverse response was decreased by the longitudinalmotion.

3.1.2 Effects ofd/D on the response

In the case ofe/D = 0 shown in Figures4 and5, the onset velocity ofd/D = 1/2 was lowerthan that ofd/D = 1/8, and the response amplitude of transverse direction ofd/D = 1/2 waslarger than that ofd/D = 1/8. On the other hand, the response amplitude of longitudinaldirection ofd/D = 1/8 was larger than that ofd/D = 1/2. Considering the relative size of thewake to the diameter of the leeward circular cylinder, it seems that these results were natural.

3.2 Unsteady surface pressure distributions

3.2.1 Definition of aerodynamic works

In this study, measured surface pressure,P (t) was investigated by ensemble averaging asfollows.

P (t∗) =1

Nt∗

Nt∗∑i=1

P ((i − 1 + t∗)T ) (2)

wheret∗ is relative time to the oscillation period,T , andNt∗ is a number of samples corre-sponding tot∗ in the time record.P (t∗) is a pressure component that synchronizes with themotion of the circular cylinder. On the other hand, a asynchronous pressure fluctuation can bedefined as following formula.

Prms(t∗) =

√√√√ 1

Nt∗

Nt∗∑i=1

(P ((i − 1 + t∗)T ) − P (t∗)

)2(3)

Prms(t∗) indicates the strength of the pressure fluctuation caused by asynchronous flow, suchas Karman vortex generated by the windward prism, turbulence in the wake, and so on. Thisparameter is called as ’ensemble rms pressure’ in this paper.

5


2Y/D

U/fD

e/D

0

1/5

2/5

0 50 100 150 200

0.5

1.0

1.5

2.0

2.5

e/D

0

1/5

2/5

U/fD

2X/D

0 50 100 150 200

0.5

1.0

1.5

2.0

2.5

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

-1.0 0.0 1.0 2.0 3.0 4.0 5.0

9/4

=0

=1/5

=2/5

wind

Y/D

c/D

e/De/De/D

(a) Transverse Response (b) Longitudinal Response (c) Lissajous curve at Max.U/fD

Figure 4:Response of the leeward circular cylinder in 2 DOF (d/D = 1/8, c/D = 9/4)

2Y/D

U/fD

0

1/5

2/5

3/5

4/5

e/D

0 50 100 150 200

0.5

1.0

1.5

2.0

2.5

e/D

0

1/5

2/5

3/5

4/5

U/fD

2X/D

0 50 100 150 200

0.5

1.0

1.5

2.0

2.5

3.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

-1.0 0.0 1.0 2.0 3.0 4.0 5.0

9/4

=0

wind

Y/D

c/D

e/D=1/5

=2/5

=3/5

=4/5

e/D

e/De/De/D

(a) Transverse Response (b) Longitudinal Response (c) Lissajous-dD12

Figure 5:Response of the leeward circular cylinder in 2 DOF (d/D = 1/2, c/D = 9/4)

2Y/D

U/fD

e/D

0

1/5

2/5

3/5

4/5

0 50 100 150 200

0.5

1.0

1.5

2.0

2.5

2Y/D

U/fD

e/D e/D0 1/5

2/5 3/54/5 16/5

0 50 100 150 200

0.5

1.0

1.5

2.0

2.5

(a)d/D = 1/8 (b) d/D = 1/2

Figure 6:Response of the leeward circular cylinder in 1 DOF (c/D = 9/4)

6


Wind

e

c

bx

y

Windward prism

Leeward circular cylinder

Neutral point in no wind

Figure 7:Definition ofβ

Ensemble averaged surface pressures on the leeward circular cylinder are expressed by fol-lowing formula.

P (β, t∗) =1

2ρU2Cp(β, t∗) =

1

2ρU2

{Cp(β) + C ′

p(β, t∗)}

(4)

Prms(t∗) =

1

2ρU2CPrms(β, t∗) (5)

where,Cp andC ′p are a mean pressure coefficient and a fluctuating pressure coefficient, respec-

tively. CPrms is a ensemble rms pressure coefficient.β is a central angle from the upstreamstagnation point to a point actingP (β, t∗) as shown in Figure7.

The non-dimensional transverse work of pressure,W ∗PY and that of lift,W ∗

L duringT , aredefined as follows, respectively.

W ∗PY (β) =

WP (β)12ρU2 · Y

(6)

=∫ 1

0−C ′

P (β, t∗) sin βy(t∗)T

Ydt∗

=∫ 1

0W ∗

PY (β, t∗)dt∗ (7)

W ∗L =

WL12ρU2 · Y · D

(8)

=∫ 1

0W ∗

L(t∗)dt∗ (9)

=∫ 1

0

∫ π

−πW ∗

PY (β, t∗)dβdt∗

=∫ 1

0

∫ π

−π−C ′

p(β, t∗) sin βy(t∗)T

Ydβdt∗

wherey(t∗) is an instantaneous velocity of the transverse direction.

3.2.2 Tandem arrangement condition (e/D = 0)

Figure8 indicates (a) time-record of displacement,x(t), y(t), (b) the ensemble averagedinstantaneous pressure distribution,Cp(β, t∗), (c) ensemble rms pressureCPrms(β, t∗), (d)the fluctuating pressure distribution,C ′

p(β, t∗), (e) the distribution of transverse work ofC ′p,

W ∗PY (β, t∗), (f) the transverse work ofC ′

L, W ∗L(t∗), and (g) transverse work ofC ′

p(β, t∗) duringone period in the cases ofe/D = 0.

Figure8 (b) shows that the positive high pressure area moved betweenβ=+10 degrees andβ=-10 degrees. This means that the leeward circular cylinder was out side of the wake of the

7


windward prism during appearing the positive pressure on the upstream surface of the circularcylinder. In this case, the circular cylinder went outside the wake of the windward prism twicea period. In Figure8 (c), the region including strong pressure fluctuation existed aroundβ= -60degrees duringt∗=0 – 0.2. This pressure fluctuation was generated by a shear flow starting fromthe upper leading edge of the windward prism. In the range oft∗ from 0.2 to 0.45, two regionsincluding strong pressure fluctuation appeared aroundβ=-60 – -10 degrees andβ=+10 – +60degrees. These two regions were caused by the lower and the upper separation flows of thewindward prism, respectively. The time disappearing these two regions was late for zero crossof y(t∗). This result shows that the wake of the prism transformed into avoiding the circularcylinder when the circular cylinder crossed the wake.

Figure8 (d) indicates that the negative fluctuating pressure area moved synchronizing withthe movement of the circular cylinder. In Figure8 (e), the positive work area aroundβ=+60 -+90 degrees andβ=-30 – -90 degrees existed aroundt∗=0 – 0.3 andt∗=0.5 – 0.8. Figure8 (f)indicates that the total positive work was generated by the time lag of change from the positivework to the negative work aroundt∗=0.25 – 0.3 ort∗=0.75 – 0.8 in Figure8 (e).

Figure8 (g) shows that the distribution of the transverse work of surface pressure during oneperiod was almost symmetric and the excitation force was generated by almost the all surfaceof the circular cylinder.

As the result of this case, it is clarified that the excitation force was generated by the move-ment of the upper and lower separation flow from the leading edge of the prism.

3.2.3 Staggered arrangement condition (e/D = 3/5)

Figure9 shows the results in the case ofe/D = 3/5. In Figure9 (b), the positive highinstantaneous pressure area appeared once a period aroundβ=-30 – +20 degrees. Therefore, theleeward circular cylinder went outside the wake of the windward prism once a period.

In Figure9 (c) the ensemble rms pressure in the region ofβ = −90 degrees was keeping astrong value for one cycle. It seems that this pressure fluctuation was caused by a flow separationfrom the lower surface of the circular cylinder itself.

When the displacementY/D was maximum, that is, the position of the circular cylinder wasthe closest to the wake of the windward prism, the ensemble rms pressure aroundβ=-30 degreesbecame strong, because the separation flow from the lower leading edge of the prism attachedto the region ofβ= -30 degrees.

The instantaneous pressure aroundβ= -20 degrees in the range oft∗ from 0.25 to 0.75,became positive as shown in Figure9 (b) at the same time as the ensemble rms pressure aroundthis region was weak. It is also clarified that the circular cylinder went outside of the wake ofthe prism when the position of the circular cylinder was minimum. At this time, the ensemblerms pressure aroundβ = +50 degrees became stronger. This result indicates that the lowerseparation flow from the lower leading edge of the prism attached to this region.

Figure9 (e) indicates that the transverse work of the fluctuating pressure around the regionfrom β = +50 degrees toβ = +120 degrees became positive att∗ = 0 – 0.35 andt∗ = 0.5– 0.75. These works caused the positive work ofC ′

L in Figure9 (f). The positive work ofC ′p

shown in Figure9 (g) was also generated by the regionβ = +50 – +120 degrees. On the otherhands, the pressure fluctuation on the lower surface of the circular cylinder hardly contributedto the excitation force.

As the result of this investigation, it is clarified that the excitation force in the case of thestaggered arrangement was generated by the movement of the only one shearing flow, that is

8


-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

0.00 0.25 0.50 0.75 1.00 1.25 1.50

Y/D,X/D

t*

Y/D

X/D

(a) displacement

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

b(deg.)

t*

contourline pitch

0.5

CP(t

*)

0.00 0.25 0.50 0.75 1.00 1.25 1.50

-180

-150

-120

-90

-60

-30

0

30

60

90

120

150

180

0.0

0.1

0.2

0.3

0.4

0.5

b(deg.)

t*

contourline pitch

0.06

s

0.00 0.25 0.50 0.75 1.00 1.25 1.50

-180

-150

-120

-90

-60

-30

0

30

60

90

120

150

180

(b) ensemble averaged instantaneous pressure (c) ensemble rms pressure

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

b(deg.)

t*

contourline pitch

0.2

C'P(t

*)

0.00 0.25 0.50 0.75 1.00 1.25 1.50

-180

-150

-120

-90

-60

-30

0

30

60

90

120

150

180

-0.08

-0.04

0.00

0.04

0.08

b(deg.)

t*

contourline pitch

0.03

W'PY(t*,b)

0.00 0.25 0.50 0.75 1.00 1.25 1.50

-180

-150

-120

-90

-60

-30

0

30

60

90

120

150

180

(d) fluctuating pressure (e) transverse work of fluctuating pressure

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

0.00 0.25 0.50 0.75 1.00 1.25 1.50

WL

*(t

*)

t*

-180

-150

-120

-90

-60

-30

0

30

60

90

120

150

180

-10.0 -7.5 -5.0 -2.5 0.0 2.5 5.0 7.5 10.0 WP

*(b)

b(deg.)

(f) transverse work ofC ′L (g) transverse work ofCp for one period

Figure 8:Ensemble averaged unsteady surface pressure and its work (2 DOF,d/D = 1/2, c/D = 9/4, e/D = 0,U/fD = 145)

9


the lower separation flow from the leading edge of the prism.

3.3 Flow between the windward prism and the leeward circular cylinder

Figure10 and11 show the flows fort∗=0.200, 0.250, and 0.450 in the case ofd/D=1/2 ande/D=0, and fort∗=0.200 and 0.500 in the case ofd/D=1/2 ande/D=3/5. These figures indicatethe flows visualized by smoke and the high speed camera. In these figures, the instantaneousflow shows the both of the outer flow and the inner flow, the instantaneous wake flow indicatesthe instantaneous flow of the inner flow, and the mean wake is the ensemble averaged image ofthe wake flow, furthermore the ensemble instantaneous and rms pressures show the distributionsof the ensemble averaged instantaneous pressure (red line) and of the ensemble rms pressure(blue line) for eacht∗, respectively.

Figure 10 (a) indicates that the ensemble rms pressures on the regions ofβ=-120 – -30degrees and ofβ=0 – +30 degrees att∗=0.200 were caused by the upper and the lower separationflows from the leading edges of the prism, respectively.

Figure10 (b) shows the flows att∗=0.250. In this time, it is clear that the ensemble rmspressures on the lower and the upper surfaces of the circular cylinder were generated by thelower and the upper separation flows from the prism, respectively.

At t∗ = 0.450 shown in Figure10(c), the lower separation flow from the lower leading edgeof the prism streamed the upper side of the circular cylinder. Therefore, it is clarified that theensemble rms pressure aroundβ = +50 – +120 degrees were generated by the lower separationflow from the lower leading edge of the prism.

On the other hands, Figure11 (a) indicates that the ensemble rms pressure on the region ofβ = +50 – +120 degrees, and that aroundβ = −30 degrees were caused by the upper and thelower separation flows of the prism, respectively. Whent∗ = 0.500, it is clear that the ensemblerms pressure on the upper surface of the circular cylinder was generated by the lower separationflow from the lower leading edge of the windward prism.

As the results of the flow visualization tests, it is clarified that the distributions of the ensem-ble rms pressure were reflected the separation flows from the prisms.

4 Conclusions

From the investigations of the wake excitation of the circular cylinder in the wake of theprism by the free vibration test, the unsteady pressure measurements of the circular cylinderand the visualization of the flow around the circular cylinder, following results were obtained.

• d/D controls the response amplitude of the circular cylinder by changing the wake size.

• e/D changes the oscillation type between the hard flutter type and the soft flutter type.The prevailing response direction in 2 DOF was also controlled bye/D.

• The degree of freedom for the longitudinal direction decreases the transverse response.

• The excitation forces were generated by the pressure fluctuation controlled by the sepa-ration flows from the windward prism.

• The ensemble rms pressure reflects clearly the condition of the separation flow near thecircular cylinder.

• The motion of the separation flow from the windward prism plays an important role forthe wake excitation of the leeward circular cylinder.

10


-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

0.00 0.25 0.50 0.75 1.00 1.25 1.50

Y/D,X/D

t*

Y/D

X/D

(a) displacement

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

b(deg.)

t*

contourline pitch

0.5

CP(t

*)

0.00 0.25 0.50 0.75 1.00 1.25 1.50

-180

-150

-120

-90

-60

-30

0

30

60

90

120

150

180

0.0

0.1

0.2

0.3

0.4

b(deg.)

t*

contourline pitch

0.06

s

0.00 0.25 0.50 0.75 1.00 1.25 1.50

-180

-150

-120

-90

-60

-30

0

30

60

90

120

150

180

(b) ensemble averaged instantaneous pressure (c) ensemble rms pressure

-0.4

-0.2

0.0

0.2

0.4

b(deg.)

t*

contourline pitch

0.15

C'P(t

*)

0.00 0.25 0.50 0.75 1.00 1.25 1.50

-180

-150

-120

-90

-60

-30

0

30

60

90

120

150

180

-0.050

-0.025

0.000

0.025

0.050

b(deg.)

t*

contourline pitch

0.025

W'PY(t*,b)

0.00 0.25 0.50 0.75 1.00 1.25 1.50

-180

-150

-120

-90

-60

-30

0

30

60

90

120

150

180

(d) fluctuating pressure (e) transverse work of fluctuating pressure

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

0.00 0.25 0.50 0.75 1.00 1.25 1.50

WL

*(t

*)

t*

-180

-150

-120

-90

-60

-30

0

30

60

90

120

150

180

-10.0 -7.5 -5.0 -2.5 0.0 2.5 5.0 7.5 10.0 WP

*(b)

b(deg.)

(f) transeverse work ofC ′L (g) transverse work ofCp for one period

Figure 9:Ensemble averaged unsteady surface pressure and its work (2 DOF,d/D = 1/2, c/D = 9/4, e/D =3/5, U/fD = 145)

11


instantaneous flow instantaneous wake flow mean wake ensemble instantaneous and rms pressures

wind

+1.0

0.0

-1.0

-2.0

0.00

0.20

0.40

0.60

(a) t∗ = 0.200

wind

+1.0

0.0

-1.0

-2.0

0.00

0.20

0.40

0.60

(b) t∗ = 0.250

wind

+1.0

0.0

-1.0

-2.0

0.00

0.20

0.40

0.60

(c) t∗ = 0.450

Figure 10: Flow between the prism and the circular cylinder (2 DOF,d/D = 1/2, c/D = 9/4, e/D = 0,U/fD = 145)

instantaneous flow instantaneous wake flow mean wake ensemble instantaneous and rms pressures

wind

+1.0

0.0

-1.0

-2.0

0.00

0.20

0.40

0.60

(a) t∗ = 0.200

wind

+1.0

0.0

-1.0

-2.0

0.00

0.20

0.40

0.60

(b) t∗ = 0.500

Figure 11: Flow between the prism and the circular cylinder (2 DOF,d/D = 1/2, c/D = 9/4, e/D = 3/5,U/fD = 145)

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REFERENCES

[1] A. Simpson ”On the Flutter of a Smooth Circular Cylinder in a Wake”Aeronautical Quar-terly. 25 – 41, 1971.

[2] K. R. Cooper and R. L. Wardlaw. ”Aeroelastic instabilities in wakes”Proceedings of Thirdconference on Wind Effects on Buildings and Structures, 647–655, 1971.

[3] H. Ruscheweyh.Proceedings of IAHR/IUTAM Symposium, Practical Experiences withFlow-Induced Vibrations, 115–125, 1979.

[4] H. Utsunomiya and Y. Kamakura.Proceedings of Japan Society of Civil Engineering, 336,1–8, 1983. (in Japanese)

[5] F. Nagao, H. Utsunomiya, M. Noda, M. Imoto and R. Sato. ”Aerodynamic propertiesof closely spaced triple circular cylinders”Journal of Wind Engineering and IndustrialAerodynamics, 91, 75–82, 2003.

[6] H. Utsunomiya, F. Nagao, M. Noda, M. Katayama and T. Koori. ”Study on the WakeExcitation of a Circular Cylinder in the Wakes of Rectangular Cylinders”Proceedings ofthe 18th National Symposium on Wind Engineering, 395 – 400, 2004. (in Japanese)

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