Further results on the flow-induced streamwise vibration of cylinders

Journal of Fluids and Structures (1992) 6, 51-66

FURTHER RESULTS ON THE FLOW-INDUCED STREAMWISE VIBRATION OF CYLINDERS E. D. BMT Fluid Mechanics Limited,

OBASAJU

Teddington Middlesex

TW118LZ,

I/. K.

AND

R. Institut fiir Hydromechanik, (Received 20

ERMSHAUS AND E. NAUDASCHER

Universitiit Karlsruhe D 7500 Karlsruhe

1, Germany.

April 1990 and in revised form 27 April 1991)

Flow-induced streamwise vibrations of elastically mounted rectangular cylinders and cut-out square cylinders are investigated in the reduced velocity range 1 < U/ND’ < 13 (where D' is the dimension of the model perpendicular to the how) at angles of incidence, LX,in the range from 0 to 90”. The Scruton number, SC, of the models used for the investigation ranges from 0.68 to 714.4 and the structural damping is 6, = 0.0017. Here, SC = 2M6,/(pD2L), where M is the equivalent mass of the vibrating system, p is the fluid density, and D and L are cylinder diameter and length respectively. Sustained streamwise oscillation can occur at about one or two times the natural vortex-shedding frequency. When a line of symmetry of the cylinder cross-section is parallel to the approach flow and the separated shear layers do not reattach to the cylinder, sustained streamwise oscillation occurs at about twice the natural vortex-shedding frequency but only when there is an interaction between the shear layers and the afterbody. When the afterbody is small and there is little or no possibility of shear layer/trailing corner interaction, sustained streamwise oscillation can still occur, but over a very limited range of reduced velocity. It is suggested that these resonant vibrations are sustained either by vortex shedding on its own or by vortex shedding in combination with wake breathing. Beside the response to turbulence buffeting, if the flow around a rectangular cylinder is bi-stable, there can be an additional nonresonant vibration whose source is thought to be the highly unsteady wake and the change in the cylinder drag force accompanying the change in the flow pattern. 1. INTRODUCTION of the streamwise vibrations of elastically mounted bluff cylinders. It is a companion to Ermshaus et al. (1985), the extensive review given by Naudascher (1987), and our recent paper Obasaju et al. (1990). All are part of a research programme on flow-induced streamwise vibration in which structural shapes ranging from axisymmetric bodies (such as a sphere and a cylinder-stub) to sharp-edged bluff cylinders have been examined. The objectives are to investigate the possibility of self-sustained streamwise oscillations and to elucidate the mechanisms by which such oscillations are sustained. Some hydraulic structures are susceptible to streamwise vibration. For example, trashracks at hydraulic intakes frequently have component bars which are rectangular in cross-section, and severe vibration of the bars has been known to occur [see Knisely (1990) for references]. The cross-sections of the cylinders used in the experiments reported here are described in Figure 1. There are three rectangular cylinders and three cut-out squares. The cut-out square [Figure l(b)] was investigated by Naudascher et al. (1981), who examined the feasibility of suppressing the galloping vibrations of square-section THIS PAPER DESCRIBES WIND TUNNEL STUDIES

0889-9746/9’2/010051 + 16 $03.00

0

1992 Academic Press Limited

E. D. OBASAJU

ET AL

----

L:

(bl

(a )

Figure

1. Shape

of cylinders

cross-section:

(a) rectangular

cylinder;

(b) cut-out

square.

1981) for the same angle of incidence. Figure 2. Two possible flow patterns suggested (Naudascher et al. (y, when a is near 45”: (a) low CD; (b) high C,. (b) Lift and drag coefficients for the cut-out square for (a) e/D = 0, (b) e/D = 0.143, (c) e/D = 0.214; after Naudascher et al. (1981).

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cylinders by altering the cylinder geometry. They reported that when the angle of incidence is close to 45”, two mean flow patterns, and hence two values of mean drag coefficients, CD, occurred at each angle of incidence-see Figure 2(b). They associated the CD values with a separated and a reattached shear layer configuration [see Figure 2(a)] and argued that, since a switch between the two flow states can be generated by body movements in the flow direction, a susceptibility to streamwise vibrations cannot be ruled out. It is for this reason that the cut-out square was included in the tests to be reported here. The results presented here will confirm that sustained streamwise oscillations can occur with a wide variety of structural shapes. As in our earlier work (e. g. Naudascher 1989), the vibration will be explained in terms of vortex shedding and wake breathing. It will be seen that when a line of symmetry of the cross-section of a cylinder is parallel to the direction of the approach flow and the separated shear layers do not reattach to the cylinder, sustained streamwise oscillation occurs only when the separated shear layers interfere with, but do not reattach (in a mean sense) to, the afterbody. If either a line of symmetry is not parallel to the flow direction or the final separation points of the shear layers are not fixed, sustained streamwise oscillation can still occur, although over only a very limited range of wind speed, even when the afterbody is so little that it cannot be expected to interfere with the separated shear layers.

Wind tunnel nozzle

.P.

Figure 3. Experimental

set-up.

Flow direction

c-

54

E7‘ Al

E. D. OBASAJU

1

TABLE

Table of parameters (a) Cut-out

D

elD

(mm)

Equivalent mass,

Eigenfrequency, N (Hz)

squares Structural damping. n>

M (kg)

x0 80 80

0.143 0.214 0.30

I.05 1.36

1.06

4.91 4.52 4.93

0w17 0~0017 0.0017

(b) Rectangular

D (mm)

;i 3

Side ratio, HID

4

Equivalent mass,

Eigenfrequency. N (Hz)

Structural damping,

4.40 4.34 4.00

2. EXPERIMENTAL

2027.4 1866.3 2035.6

Mass-damping parameter. K, = 2MA,/pD'L O.hX 0.X8 0.69

cylinders

M (kg)

1.46 1.55

Frequency parameter. D’Nlv

Frequency parameter.

Mass-damping parameter.

(8

D=Nlv

K,=2M6,/pDzL

0.0017 0.0017

255.5 252.0 2.323

6.73 6-73 714.4

ARRANGEMENT

Measurements were made in a low-speed wind tunnel with an open working section at the Institute of Hydromechanics, University of Karlsruhe, Germany. The turbulence level in the working section was about 0.5%. Models were suspended horizontally and were constrained to move only parallel to the undisturbed flow by an arrangement of wires and springs as shown in Figure 3. The purpose of the two fixed squares end-plates shown in the figure is to reduce any infhrence of end effects. The plates have sides of length 600mm. The leading edge of each plate was carefully streamlined in order to avoid flow separation. The axis of each model tested was located about 150 mm downstream of the plates leading edges, and there were gaps of about 1 mm between the plates and the ends of the model. The amplitude of streamwise vibration was monitored with the aid of a small rectangular disc attached to the base of one of the wires supporting the model. The disc was sandwiched between two inductive coils which formed part of a carrier-frequency bridge. Each model was made of hard Styrofoam and had an axial length between end plates of L = 683 mm. Cross-sections normal to the axis of a model are shown in Figure 1, and the dimensions and dynamic parameters are given in Table 1. The rectangular cylinders tested have side ratio H/D = 4,5 and 50. The basic section from which the cut-out squares were made had side length D = 80 mm and the sizes of the cut-out are e = 0.1430, O-2140 and O-30. 3. EXPERIMENTAL

RESULTS

For the rectangular cylinders, the root-mean-square

amplitude of streamwise vibration,

A r.m.s 7 is nondimensionalized using D', the width of the model normal to the flow direction (see Figure 1). D'/D is given by D'JD =(H/D) sin LY+COS(Y. For the cut-out square A,,,.is nondimensionalized square-section cylinder, and only (Y= 45” was tested.

using D, the width of the basic

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There are two types of cylinder response. There is the small irregular vibration generated by the unsteady buffeting loads induced by turbulence in the oncoming stream and in the wake of the cylinder. The root-mean-square (r.m.s.) amplitude of this background response increases continuously with the reduced velocity but is typically only about OXlO3D’ at U/ND’ = 12. The second type of cylinder response is sustained by flow processes such as vortex shedding and movement-induced periodic changes in the configuration of the separated shear layers (i.e. wake breathing). For the cases examined, sustained vibrations occur when U/ND’ is in the neighbourhood of the points l/S’ and 1/2S’, where S’ = nD’lU and n is the vortex-shedding frequency measured when the model is stationary. For the case H/D = 5, the values of S’ measured by Knisely (1985) are shown plotted against the incidence, LY,in Figure 4. The r.m.s. amplitudes of the rectangular cylinders are presented in Figures 5 to 9, for the cases H/D = 4 and 5. At LY= 0 and 90”, A,,,,,,lD’ is small because the response is mainly of the background turbulence buffeting type. In the range lo” < cx< 30”, higher-than-buffeting response occurs in a narrow range of reduced velocity near U/ND’ = l/S’-see Figures 5,6,8 and 9. For 30” < (Y< 70”, two types of response are possible: a low- and a high-amplitude response. When the low-amplitude or buffeting response occurs, A,,,.,_lD ’ increases with U/ND’ but is still only about 0*0004 when U/ND’ = 9. When the high-amplitude response occurs, there is a weak peak in A ,,,,,/D’ near U/ND’ = 1/2S’, a strong peak near U/ND’ = l/S’, and at UIND’ = 1987) that the 9-0, A,.,.,. /D’ =0$X)7. Flow visualization has revealed (Naudascher high-amplitude response is associated with vortex formations close to the cylinder, whereas the small-amplitude or buffeting response occurs when vortices are forming far downstream. The high-amplitude response was obtained either by increasing U/ND’

0 0

0

0

O

0

0

0

0 0 0

0 0

0

I

I

I

10

20

30

I

Angle

Figure 4. Variation

0

0

O

50

of incidence.

of S’ with CYin smooth

I

I

40

60

I

I

70

80

a (deg)

flow for H/D = 5; after Knisely (1981).

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0.016

ND' Figure

7. Root-mean-square

value

of

cylinder

amplitude

versus

reduced

velocity

for

H/D = 4.0:

A.cY=~O=';O,CY=~O"; x,cu=75"&85";0,(~=!90.

0.016

0

0

I

0

2

4

6

8

10

12

u ND' Figure

8. Root-mean-square

value

of

cylinder

amplitude

versus

reduced

velocity

O,n=O"; x.a=10":0,cu=20"; A,(u=30;0,cu=40°.

for

H/D = 5.0:

58

E. D. OBASAJU

ET AL.

0404 0

i

0

2

3

6

8

1 ND’ Figure

9. Root-mean-square A.a=50”;

value of cylinder amplitude versus X,(~=55~;0.cy=60”;0.cu=65”:-.cu=70.80and90.

reduced

velocity

for

H/D

= 5.0:

slowly or by inducing a large enough threshold amplitude (hard excitation). Switching from high to the low amplitudes can be induced by waving a ruler from side to side, parallel to the cylinder span, in the near wake. At (Y= 40”. it was difficult to induce switching whereas at (Y= 6.5”. the low amplitude response was the more stable and switching to low amplitudes tended to occur even when the how is not disturbed, particularly when U/ND’ 2 5. The flat plate (i.e. H/D = 50) was tested at (Y= 40 and 45”. The r.m.s. amplitudes of streamwise vibration are presented in Figures 10 and 11. Shown for comparison are results for H/D = 4 and 5. For the same angle of incidence, A,,,,,./D’ is smaller at HID = 50 than at HID = 4 and 5. When U/ND’ is between 5 and 6, there is a sharp peak in the amplitude of all the cylinders. The peak value of A,,,,,,/D’ measured with the plate is about half the values obtained when H/D = 4 or 5. Further comparison of the response of the rectangular cylinders is made in Figure 12. The magnitude of the peak that occurs in the vibration amplitude when U/ND’ is close to l/S’, and the value of U/ND’ at which the peak occurs have been plotted against the incidence. Also shown are square-section cylinder (H/D = 1) results taken from Obasaju et al. (1990). Furthermore, for H/D = 1 and 5, the variation of l/S’ with LYis number for the stationary model, is taken from presented where S’, the Strouhal Obasaju (1983) and Knisely (1985). For 10”~ CY< 80” the peak in the vibration amplitude occurs in the range 5.2 5 U/ND’ 5 6.5. Over the angles of incidence and Scruton numbers (SC) examined, the highest values of the peak amplitude are A ,.,,,,/D’ = OW6, 0.018, O-029 and O-066 for HID = 50,4,5 and 1, respectively. The r.m.s. amplitudes of the cut-out squares are presented in Figure 13. Also shown are the amplitudes of the basic square-section cylinder, that is the case e/D = 0. Except for a very sharp peak in the response curve, A,,,, lD is less than O-003. For e/D =O, 0.143, O-214 and 0.300, the peak amplitude is A,,,,/D = 0.006, 0.015, 0.018 and

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OF CYLINDERS

10. Root-mean-square value of cylinder amplitude, measured at 40” incidence, velocity. 0. Flat plate (H/D = 50-O); X, H/D = 4.0;0, H/D = 5.0.

versus

reduced

versus

reduced

(@lilhllx x

u ND’ Figure

11. Root-mean-square value of cylinder amplitude, measured velocity. 0, Flat plate (H/D = 50.0); X, H/D

at 45” incidence, = 4.0.

60

E. D. OBASAJU

Incidence.

ET AL.

(I (dep)

Figure 12. Amplitude of peak that occurs near U/ND’ = l/S’ versus incidence; (Figure 12(a) shows the reduced velocity at which the peak occurs). Cl, Square-section cylinder (H/D = 1.0); SC = 1.60; x. H/D = 4, SC = 6.73; A, H/D = 5. SC = 6.73; 0, H/D = 50, SC = 714.4. Values of S’ are taken from Knisely (1985) and Obasaju (1983).

0.0154 respectively. As e/D increases, the peak occurs at lower values of U/ND, suggesting that the Strouhal number based on D increases. This result suggests, moreover, that the separated shear layers are drawn closer to the body. It is assumed that the shear layers reattach to the cylinder as sketched in Figure 2(a). For a cut-out square, Naudascher et al. (1981) found for angles of incidence approaching 45”, two values of steady lift and drag coefficients for each angle of incidence, and they suggested that two flow patterns (see Figure 2) are possible. This raises the question of whether there is more than one stable r.m.s. amplitude of streamwise vibration. For e/D = 0.124, this possibility was investigated at U/ND = 3.40 and 364 by deflecting the model to oscillate at high amplitude. The imposed oscillation was damped out. Furthermore for e/D = O-30 and U/ND = 2.98, the vibration was stopped and the cylinder was held stationary. The cylinder amplitude returned to its previous level when it was released. When a ruler was removed, after it had been waved from side to side just slightly above the vibrating model, the amplitude returned to its previous level.

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x

0.016 -

# A 0 0 0 x

A 0414

-

LQ

0.012 -

x0

A

0 0 0

O~(110 -

0

A

?C T

A

Woo8

-

0.006

-

X

0

Figure velocity.

O

0

13. Root-mean-square amplitude, measured with the cut-out square 0, e/D =O, SC= 6.73; 0, e/D = 0.143, SC= 0.68; x, e/D = 0.214, SC = 0.69.

cylinders, Sc=O%3;

versus reduced A, e/D = 0.30,

Figure 14 shows eight sample traces of the displacement of the vibrating cylinders of approximately 2 min each. For the cut-out square, the recording was made at U/ND = 3-55 (A,,,,,/D = O-0176) where the peak in the response of the cylinder occurred. Trace (a) shows that the vibration amplitude remained at one level for a long time before drifting to another level. The drift may have been caused by a minute change in the tunnel wind speed. Traces (b) to (f) c h aracterize the displacement of the rectangular cylinders when H/D = 4,5 and 50 and the angle of incidence is in the range 30” < IY< 70”. Traces (b) to (d) show that when U/ND = l/S’, the vibration amplitude was irregular and there were periods when there was little or no vibration. Near U/ND’ = 1/2S’, there was a burst-type response with rather long intervals between bursts-see trace (e). When U/ND’ was significantly higher than l/S’, the amplitude was irregular and had a low frequency modulation-see trace (f). Traces (g) and (h) characterize the displacement of the rectangular cylinder when H/D = 4 and 5 and the angle of incidence is in the range 10” < cx< 30”. When U/ND’ = l/S’, the vibration amplitude was fairly regular-see trace (g). Taking U/ND’ slightly away from l/S’ gave the “beating” type of vibration shown in trace (h). 4. DISCUSSION

OF RESULTS

The peaks that occur in the response curve of some of the cylinders when U/ND’ is near either 112s’ or l/S are associated with the formation and shedding of vortices.

62

E. D. OBASAJU

ET AL.

(hi

Figure 14. Time history of cylinder displacement. (a) Cut-out square, e/D = 0.214, U/ND = 3.55; (b) plate, H/D = 50, LY= 40”, U/ND’ = 5.67; (c) rectangular cylinder, H/D = 5, (Y = 40”, U/ND’ = 5.54; (d) rectangular cylinder, H/D = 5, LY= 60”. U/ND’ = 6.08; (e) rectangular cylinder, H/D = 5. a = 60”. U/ND’ = 3.29; (f) rectangular cylinder, H/D = 5, (Y = @Jo, U/ND’ = 8.48; (g) rectangular cylinder, H/D = 5, a = 30”, UJND’ = 5.55; (h) rectangular cylinder, H/D = 5, (Y = 30”. U/ND’ = 5.05.

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63

When a line of symmetry of the cross-section of a cylinder is parallel to the flow direction, the same streamwise force is associated with the formation of each vortex. Consequently, when vortices of opposite sense of rotation are shed alternately, the cylinder experiences a streamwise force oscillating at twice the vortex-shedding frequency. On the other hand, when a line of symmetry of the cylinder cross-section is not parallel to the flow direction, the streamwise forces induced by vortices of opposite sense of rotation are not equal, and there is a streamwise force oscillating at the vortexshedding frequency. Therefore, depending on the angle of incidence, vortex-induced streamwise oscillations can occur when U/ND’ is near either 1/2S’ or l/S’. The results obtained when a face of a rectangular cylinder is normal to the approach flow are presented in Figure 15 together with square-section cylinder results taken from Obasaju et al. (1990). In Figure 15, H is the depth of the cross-section in the stream direction and D is the width normal to the flow. For H/D = O-2, 0.25, 4, and 5, there are no peaks in the response curve when U/ND’ is close to either 1/2S’ or l/S’ because, conceivably, the excitation generated by vortex shedding is weak, and because the vortices may possibly even act to damp rather than excite streamwise vibration. Such a possibility is best explored by experiments where the cylinder vibration is forced rather than flow-induced as in the present case. An explanation for the high-amplitude oscillation for H/D = 1 is that there exist other mechanisms of sustained oscillation besides vortex shedding. Naudascher (1987) has suggested that such a mechanism is wake breathing, i.e., a periodic change in the configuration of the shear layers induced by the vibration of the cylinder. This wake breathing is expected to sustain streamwise oscillation in the range 0.6 I H/D I 2.5. As H/D increases beyond about 0.6 the shear layers begin to interact with the trailing edge corners. This interaction continues until the shear layers begin to reattach to the side faces near

u ND Figure

15. Root-mean-square value of cylinder amplitude, measured at 0” incidence, versus velocity. W, H/D = 0.20; A, HID = 0.25; 0, H/D = 1; x, H/D = 4; V, HID = 5.

reduced

64

E. D. OBASAJU

ET AL

H/D = 2+-see Bearman & Trueman (1972). Ermshaus et al. (1986) tested a cylinder stub (that is a short circular cylinder with axis parallel to the flow), and found results analogous to those presented in Figure 15. They found that sustained streamwise oscillation occurred only in the range 0.9 < L/D -C 1.2 where L is the length of the cylinder and D is the diameter. They associated the upper boundary of the range with time-mean reattachment of the shear layers which is expected to occur near L/D = I .3 to 1.4. It seems clear from these results that when the separated shear layers do not reattach to the afterbody. shear layer/afterbody interaction is important for sustained streamwise oscillation near U/ND’ = 1/2S’. The afterbody may not be important for sustained streamwise oscillation near U/ND = l/S’. In Figure 10, there is still a strong peak in the response curve of the plate near U/ND’ = 6, even though the plate has H/D = 50. However, the oscillation occurs virtually at one wind speed, and it may be that when the afterbody is small, the body is not able to force vortices to be shed at the vibration frequency over a wide range of reduced velocity. The results for the cut-out squares give further insight into the role of the afterbody. We have suggested that the shear layers reattach to the flat faces of the cylinder as sketched in Figure 2(a). The afterbody (i.e. the part of the body downstream of the final separation points) is small and yet there is sustained vibration near U/ND’ = 1/2S’-see Figure 13. As in the case of a sphere, which was visualized by Ermshaus et al. (1986), we suggest that the separation points [S, and &--see Figure 2(a)] are tending to move upstream when the cylinder is moving downstream. and downstream when the cylinder is moving upstream. In other words, wake breathing could play a significant role in the case of a cut-out at 45” incidence as well. The conclusion we draw is that when the separation points of the shear layers are not fixed. sustained streamwise vibrations can occur near U/ND’ = l/X’. even when the afterbody is small. For H/D = 5, sketches indicating features of the near wake. which are thought to be important for streamwise vibration, are shown in Figure 16 for the regimes 10” < a < 30”, and 30” < (Y< 70”. The boundary of the regimes are based on the changes observed in the response of the cylinder. The sketches were inferred from Strouhal number measurements (Knisely 1985), from the visualization made during streamwise vibration at (Y=60”, and from stationary cylinder flow patterns (e.g., Geropp 8r Leder 1985; Perry & Steiner 1987). Rather than using the sketches to describe the weak breathing mechanism, we shall use them to describe how vortices help to sustain streamwise vibration. Figure 16(a) shows the shear layer behaviour expected in the range 10” < N < 30. where streamwise vibrations beyond the turbulence-buffeting response occurred only near U/ND’ = l/S’. For a stationary cylinder, Knisely (1985) reported that the Strouhal number, S’ peaked near CY= lo”, and he associated the peak with shear layer reattachment. In Figure 16(a), reattachment is shown to occur in the shear layer separating from the leading corner. We suggest that this case is similar to that of a square-section cylinder which is set at cx = 15”. Based on results for the square-section cylinder (Obasaju et al. 1990), we expect that a vortex forms on the side of the wake where reattachment occurs when the cylinder is moving downstream. The streamwise force generated by this vortex must be larger than that generated by a vortex formed on the opposite side, if the vortex-shedding process is to sustain streamwise oscillation. Figure 16(b) shows two flow patterns observed in the range 30” < N < 70”. The flow pattern indicated by the broken line is associated with vortex formation far downstream of the cylinder. Significant streamwise vibrations do not occur in this case. With

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65

OF CYLINDERS

Y

t

u

t

/ ,

Figure

16. Suggested

flow regimes

for

I

A’

H/D = 5:(a) 10 < (Y5 30; (b) 30”~ (Y5 70”.

the second flow pattern, associated with vortex formation close to the cylinder, the vortex formed by the shear layer separating from the leading edge dominates the near wake and should generate a larger streamwise force than the vortex formed by the opposing shear layer. Our visualization of the flow revealed that the leading edge vortex starts to develop just before the cylinder starts to move downstream and is shed before the end of the downstream stroke. We suggest that this is also the manner in which streamwise vibration is sustained when H/D = 50 and (Y= 40” and 45”. As reported in Section 3, when H/D = 4 or 5 and 30” < LY< 70, higher-than-background response can occur at other values of U/ND’. The source of this nonresonant vibration is thought to be the highly unsteady wake and the changes that occur in streamwise force when the flow switches between the two patterns sketched in Figure 16(b). We suggest that the vibration amplitude is irregular [see traces (b) to (f) of Figure 141 because the switching occurs more or less randomly. 5. CONCLUSION We have confirmed that a wide variety of structural shapes can be susceptible to sustained streamwise vibrations. For the cases examined, sustained streamwise oscillations occur when U/ND is near either 1/2S’ or l/S’. We have suggested that these resonant vibrations are sustained either wholly by vortex shedding or by vortex shedding in combination with wake breathing. Two examples of wake breathing are given. One is associated with shear layer-afterbody interaction and the other with oscillation of the flow separation points. Besides the normal response to turbulence buffeting, if the flow around a rectangular cylinder is bi-stable, additional nonresonant vibration can occur. The source of this vibration is thought to be the changes in the cylinder drag force associated with the change in flow pattern.

66

E. D. OBASAJU ET AL

Our results indicate that when a line of symmetry of the cross-section of a cylinder is parallel to the approach flow and the separation points of the shear layers are fixed, sustained streamwise vibrations occur near U/ND’ = 1/2S’ only when there is an interaction between the shear layers and the afterbody. When the afterbody is small and there is little or no possibility of shear layer/afterbody interaction, sustained streamwise vibrations can still occur, but over a very limited range of wind speed. At an angle of incidence (Y= 45”, a cut-out square can be more susceptible to sustained streamwise vibration than the basic square section cylinder. However, with the Scruton number and structural damping used in our tests, the root-mean-square amplitude of the cut-out square was less than 0.020, where D is the width of the cross-section of the basic square-section cylinder. On the other hand, for a rectangular cylinder, root-mean-square amplitude of streamwise vibration up to about 0.070’ can occur, where D’ is the width of the model normal to the flow. ACKNOWLEDGEMENTS The research was carried out at the Institute of Hydromechanics, University of Karlsruhe, Germany. It was supported by the German Science Foundation (DFG) through the Sonderforschungsbereich 210. E. D. Obasaju would like to thank Professor E. Naudascher for allowing him the opportunity of participating in this project. REFERENCES BEARMAN, P. W. & TRUEMAN, D. M. 1972 An investigation of the flow around rectangular cylinders. Aeronautical Quarterly 23, 229-237. ERMSHAUS, R., KNISELY, C. & NAUDASCHER, E. 1985 Flow visualisation of the wake of three-dimensional bodies undergoing self-sustained oscillations. In Optical Methods in Dynamics of Fluids and Solids (ed. M. Pichal). Berlin: Springer-Verlag. ERMSHAUS, R., NAUDASCHER, E. & OBASAJU, E. D. 1986 Vortex-induced streamwise oscillation of prisms in a uniform stream of different incidence. Report No. SFB 210/E/25, Universitgt Karlsruhe, Karlsruhe, Germany. GEROPP, D. & LEDER, A. 1985 Turbulent separated flow structures behind body with various shapes. In Proceedings International Conference on Laser Anemometry-Advances and Applications, Paper 12, pp. 219-231. 16th-18th December 1985, Manchester UK. KNISELY, C. W. 1985 Strouhal number of rectangular cylinders at incidence. Report No. SFB 210/E/13, Universitat Karlsruhe, Karlsruhe, Germany. KNISELY, C. W. 1990 Strouhal number of rectangular cylinders at incidence: A review and new data. Journal of Fluiak and Structures, 4, 371-393. NAUDASCHER, E., WESKE, J. R. & FEY, B. 1981 Exploratory study on damping of galloping vibrations. Journal of Wind Engineering and Industrial Aerodynamics 8, 211-222. NAUDASCHER, E. 1987 Flow-induced streamwise vibrations of structures. Journal of Fluids and Structures, 1, 265-298. OBASAJU, E. D. 1983 An investigation of the effects of incidence on the flow around a square cylinder. Aeronautical Quarterly 34, 243-259. OBASAJU, E. D., ERMSHAUS, E. & NAUDASCHER, E. 1989 Vortex-induced streamwise oscillations of a square-section cylinder in a uniform stream. Journal of Fluid Mechanics 213, 171-189. PERRY, A. E. & STEINER, T. R. 1987 Large-scale vortex structures in turbulent wakes behind bluff bodies. Part 1. Vortex formation process. Journal of Fluid Mechanics 174, 233-270.