Vortex-induced vibrations and lock-in phenomenon of ... vibrations and lock-in phenomenon of bellows structure subjected to fluid flow M. Watanabe & M. Oyama Department of …

Vortex-induced vibrations and lock-in phenomenon of bellows structure subjected to fluid flow

M. Watanabe & M. Oyama Department of Mechanical Engineering, Aoyama Gakuin University, Japan

Abstract

This paper deals with an experimental study on vortex-induced vibrations of bellows structures subjected to fluid flow. In the experiments, the bellows structure consists of flexible convolutions, and is subjected to fluid flow in a water channel. The vibration strains of the flexible convolutions are measured with increasing flow velocity. The vortex-induced responses are examined with changing the convolution pitch and number of the flexible convolutions. Moreover, the vortex shedding coupled with the vibrations of the convolutions is visualized. As a result, it is clarified that the vortex-induced vibration and lock-in phenomenon occur to the flexible bellows structures with large amplitude. The boundary of the lock-in region and Strouhal number are clarified, and detailed excitation mechanism of the vortex-induced vibration and lock-in phenomenon due to the vortex shedding is presented. Keywords: vortex-induced vibration, vortex shedding, bellows structure, lock-in phenomenon, Strouhal number.

1 Introduction

Flexible bellows are used widely in many industrial applications as expansion joints conveying fluid in piping and engine systems. Unfortunately, flexibility of the bellows makes them susceptible to vibration and, in particular, flow-induced vibration resulting from internal flow across the tips of the bellows convolutions. Up to this time, some research has been conducted on flow-induced vibrations of the bellows due to internal fluid flow. Gerlach [1,2] examined the detailed

Fluid Structure Interaction and Moving Boundary Problems 225

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flow-induced response in bellows, and concluded that the source of fluid excitation was vortex shedding from the convolution tips. He computed Strouhal number based on the width of a bellows convolution (diameter of the end), and reported Strouhal number varying from 0.1 to 0.25 with a mean value of 0.18. Moreover, from flow visualization studies, he recognized the fundamental fluid-elastic nature of the phenomenon. Bass and Holster [3] extended the work of Gerlach to bellows with internal cryogenic flows. Rockwell and Naudascher [4] and Weaver and Ainsworth [5,6,7] suggested the actual excitation mechanism is shear layer instability over the periodic cavities created by the bellows convolutions. Weaver and Ainsworth examined the flow-induced response in double bellows pipes and reported a Strouhal number based on convolution pitch of 0.45 which agree with that expected from research on deep rectangular cavities [4]. However, detailed excitation mechanism of the flow-induced vibration is not well understood. This paper presents the flow-induced vibration characteristics of the bellows structure and the visualization of fluid flow coupled with the bellows vibration. Moreover, the excitation mechanism of the flow-induced vibration due to the vortex shedding is presented.

2 Experimental apparatus and measurement

In the experiments, the bellows structure consists of flexible convolutions and is subjected to water flow in a channel. The area of the channel is 140 mm in height, and is 140 mm in width. The test section set in the water channel is shown in Figure 1. As flexible convolutions, round shaped tips are supported by flat springs, and are mounted on the rigid base. The diameter of the flexible convolution tip D is 10 mm, and its width is 136 mm. The cavity depth L is 50 mm. The thickness of the flat spring is 20 mm in length, 0.2 mm in thickness, and is 136 mm in width. Strain gauges are attached to the flat springs to measure the vibration strains. The vibration strains of the flexible bellows are measured with increasing flow velocity. The characteristics of the vortex-induced vibrations are examined with changing the convolution pitch p and number of the flexible convolutions n.

Figure 1: Test section of bellows structure.

226 Fluid Structure Interaction and Moving Boundary Problems


Moreover, flow visualization is conducted to examine the vortex shedding and dynamic behavior coupled with the convolution vibrations. To visualize the fluid flow around the vibrating convolutions, blue ink is injected into the fluid flow from the tip of an upstream rigid convolution. The dynamic behavior of vortexes is recorded by using a digital video camera.

3 Experimental results

3.1 Vibration strain and frequency

Figure 2 shows the variation of vibration strain with increasing flow velocity in the case of p/D = 2.5. These figures, from (a) to (e), show the flow-induced responses with changing the number of the flexible convolution from n = 1 to 5. In the experiments, the reduced flow velocity Vr is defined by Vr=V/fnD, where V is flow velocity. The first natural frequency of the bellows structure in water fN1 is used for the frequency fn. From this figure, it can be seen that flow-induced vibration does not occur in the case, (a) n=1, of one flexible convolution. In the cases, from (b) to (e), of more than two flexible convolutions, flow-induce vibrations occur to the flexible convolutions with the large amplitude. The flow-induced vibrations occur at about Vr=5.0, and the vibration amplitude increases with increasing the number of the flexible convolution. As a typical result, Figure 3 and Figure 4 show the frequency variations of the flow-induced vibration with increasing flow velocity V in the cases of p/D = 2.0 and 2.5, for three flexible convolutions n=3. Figure 5 shows the vibration modes of the bellows structure for three flexible convolutions in fluid. From these figures, it was found that the flow-induced vibration occurs in out-of-phase mode. In all cases from (b) n=2 to (e) n=5, it was observed that the flow-induced vibration occurs in out-of-phase mode, in which each flexible convolution moves, in the opposite direction, out-of-phase with the adjacent convolutions. The frequency of the flow-induced vibration was locked in the natural frequency of the out-of-phase mode, i.e., it was found that lock-in vibration occurred to the bellows. In particular, for p/D=2.5, it was observed that the vibration mode transfers from in-phase mode to out-of-phase mode with increasing flow velocity. The vibration amplitude of the in-phase mode was not large and was much smaller than that of out-of-phase mode.

3.2 Critical flow velocity and Strouhal number

Variation of the critical flow velocity, which is boundary of the lock-in region, of the flow-induced vibration with changing the convolution pitch p is shown in Figure 6. It can be seen that critical flow velocity decreases as convolution pitch decreases, and the critical flow velocity increases gradually with increasing the flexible convolution number n. Figure 7 shows the Strouhal number with changing the convolution pitch p. We based the Strouhal number computations on the width of the bellows convolution (diameter of the end, D). The Strouhal number is computed as



Std=fsD/V, where fs is the vibration frequency. From this figure, it can be seen that the Strouhal number Std is constant value of 0.18. This result agrees very well with that of Gerlach [1] for 2-D bellows model. On the other hand, the Strouhal number Stp (= fsp/V) based on the pitch convolution p ranges from 0.4 to 0.7. The value of 0.45 suggested by Weaver and Ainsworth [5] is in this range. Figure 2: Variation of vibration strain with increasing flow velocity in the case

of p / D = 2.5.



Figure 3: Variation of vibration Figure 4: Variation of vibration



frequency with increas-ing flow velocity in the

Case-3

(n = 3) , p / D =

frequency with increas-ing flow velocity in theCase-3 (n = 3) , p / D = 2.5. 2.0.

Figure 5: Vibration modes in the Case-3 (n = 3).

Figure 6: Variation of critical flow velocity rdV with p / D.

Figure 7: Variation of Strouhal number tdS with p / D.



3.3 Flow visualization

As a typical result, photographs of flow visualization around the flexible convolution are shown in Figure 8. Figure 8(a) and (b) show the flow pattern in the case (a) n=1 of one flexible convolution and case (e) n=5 of five flexible convolutions, respectively. It can be seen these is no periodic vortex shedding in the case of one flexible convolution. Here, the flexible convolution is in the absence of vibration. On the other hand, periodic vortex shedding and vortex street were observed around the vibrating flexible convolutions. In all cases from (b) n=2 to (e) n=5, it was observed that the flow-induced vibrations occur with periodic vortex shedding and vortex street. The dynamic behavior of vortex shedding and vortex street, in the case of three flexible convolutions, are shown in Figure 9. From these photographs, it can be seen that periodic vortex shedding is generated from the end of flexible convolution, and is coupled with the convolution vibration in out-of-phase mode. The synchronization of vortex shedding with convolution vibration in out-of- phase mode, and the vortex evolution in the shear layer on the periodic cavities were observed clearly. Moreover, it was observed that the vortexes move to the downstream across the cavities in combination with the vortexes generated in the downstream cavities and the vortex street is generated in the free shear layer on the periodic cavities. The similar dynamic behavior of vortex shedding was observed by Gerlach [2].

Figure 8: Photographs of flow visualization.

(a) Case-1 (n = 1)

(b) Case-5 (n = 5)



Figure 9: Vortex formation and shedding coupled with vibration of

convolutions in the Case-3 (n = 3), p / D = 3.0.

(1)

(2)

(3)

(4)

(5)



4 Discussion of excitation mechanism

Up to this time, Gerlach [1,2] suggested that the excitation mechanism is due to vortex shedding. He recognized the fundamental fluid-elastic nature of the phenomenon, but maintained his view that vortex shedding was the excitation mechanism. On the other hand, Weaver and Ainsworth [5] suggested the actual excitation mechanism is shear layer instability over the periodic cavities created by the bellows convolutions. From the experimental results and flow visualizations conducted in this study, we recognized the fluid-elastic nature of the lock-in phenomenon, and also observed vortex shedding and vortex street coupled with convolution vibrations. We think that the excitation mechanism is due to the feedback mechanism by the vortexes generated in the unstable shear layer on the periodic cavities. As shown in Figure 10, three feedback mechanisms by the vortex shedding can be considered. Figure 10(a) shows the self-excited feedback mechanism by the vortex shedding. However, we think this mechanism is not important for excitation mechanism of the bellows vibration, because there was no periodic vortex shedding in the case of one flexible convolution, as seen Figure 2(a). We think that two feedback mechanisms (loops) by the vortex moving from the

Figure 10: Feedback mechanism for lock-in phenomenon.

Figure 11: Block diagram of feedback mechanism for lock-in phenomenon.



upstream and to the downstream in the shear layer, as shown in Figure 10(b) and (c), play an important role for excitation mechanism. These feedback mechanisms consist of the inertia coupling between adjacent convolution motions, and the vortex movement in the shear layer. From these considerations, the excitation mechanism of the bellows vibration is presented as shown in Figure 11 with multiple feedback loops due to vortex street generated in the unstable shear layer.

5 Conclusions

The characteristics of the flow-induced vibration of the bellows were examined in 2-D model. The dynamic behavior of the vortex shedding and vortex street coupled with the convolution vibration was clarified by the flow visualization. The detailed excitation mechanism was considered and the excitation model of the flow-induced vibration with multiple feedback loops was presented.

References

[1] Gerlach, C.R., Flow-Induced Vibrations of Metal Bellows, ASME Journal of Engineering for Industry, 91, pp.1196-1202, 1969.

[2] Gerlach, C.R., Vortex Excitation of Metal Bellows, ASME Journal of Engineering for Industry, 94, pp.87-94, 1972.

[3] Bass, R.L. & Holster, J.L, Bellows Vibrations with Internal Cryogenic Flows, ASME Journal of Engineering for Industry, 94, pp.70-75, 1972.

[4] Rockwell, D. & Naudascher, E., Review Self-Sustaining Oscillations of Flow Past Cavities, ASME Journal of Fluid Engineering, 100, pp.152-165, 1978.

[5] Weaver, D.S. & Ainsworth, P., Flow-Induced Vibrations in Bellows, ASME Journal of Pressure Vessel Technology, 111, pp.402-406, 1989.

[6] Weaver, D.S., A Review of Flow-Induced Vibrations of Bellows Expansion Joints, Engineering Mechanics, 6, pp.323-336, 1999.

[7] Weaver, D.S., Flow-Induced Vibrations in Bellows Expansions Joints, Proceedings of the 3rd International Conference on Engineering Aero- Hydroelasticity, pp.77-84, 1999.



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Vortex-induced vibrations and lock-in phenomenon of ... vibrations and lock-in phenomenon of bellows structure subjected to fluid flow M. Watanabe & M. Oyama Department of …