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605

Elsevier

Vortex induced vibration of circular cylinder H.KAWAI Professor, Tokyo Denki University, Hatoyama, Saitama, Japan

Abstract

The flow field around a vortex induced vibrating cylinder is analyzed by the numerical simulation using the discrete vortex model. When the vortex shedding locks in the vibration, the separated shear layers sway in a body like a pendulum. The locked-in is not the necessary condition of the vortex induced vibration, but the vibration develops when the vortex does not shed synchronously with the vibration. At the unsynchronous state, the rolling up of the separated shear layers are prevented and promoted alternatively by the cylinder movement at every few cycles of the vibration. The energy which encourages the movement of the cylinder is supplied when the rolling up of the shear layer is promoted by the movement. The effect of the splitter plate in the wake was also investigated and the result of the simulation compared with the flow patterns experimentally obtained from the smoke wire method. The simulated flow patterns agree well with the one which were experimentally obtained.

1.INTRODUCTION The vortex induced vibration has been investigated mainly by wind tunnel experiments about the effect of various parameters such as damping, mass ratio and Scruton number using the velocity-amplitude diagram and the stability diagram etc. The various mathematical models have been proposed to clarify the self-exited characteristics of the vibration and have been checked in the light of the experimental results. However, the various points i.e. the vibration in the unsynchronized condition, is still unclear because we have not the clear image of the flow pattern around a body when the vortex induced vibration develops. The paper discusses how fluid interacts with vibration to induce the amplitude, what the locked-in state is, and how the energy of the fluid flow transfers to that of the mechanical vibration in the non-synchronously state by means of the numerical simulation of the flow around the vibrating circular cylinder using the discrete vortex model. In the paper, the effect of the splitter plate in the wake of the cylinder is also discussed. The validity of the numerical simulation is calibrated by the comparison with the flow patterns experimentally obtained by the smoke wire method. 0167-6105/93/$06.00 © 1993 - ElsevierSciencePublishers B.V. All rights reserved.

606 2. M E T H O D OF SIMULATION The discrete model employed here is the same as that of Kawai[1]. In the analysis, the separation point exists where the velocity is 96% of the maximum velocity according to Stansby[2]. The strength of the elementary vortices is reduced in the very similar way to be proposed by Sarpkaya[3]. The amplitude of the circular cylinder is estimated by the direct integration of the equation of motion by the linear acceleration method. In the simulation where the splitter plate is set at the various points in the wake of the cylinder, the plate is approximated by a series of equi-spaced sources. The calculation was carried out up to tU/D=125 at each time step A tU/D=0.0625, where U is the free stream velocity, D the diameter of the cylinder. 3. D I S C U S S I O N OF RESULTS 3.1 What is the locked-in state? Fig.1 shows the flow pattern around the cylinder vibrating at U/riD=5 which is almost resonance velocity. Fig.1 corresponds with the flow patterns from the beginning of the calculation to tU/D=19.75 where the amplitude, y/D, is about 0.025. The white and black circles show the elementary vortices shed from the separation points. The pressure around the cylinder are also shown together with the lift and the drag. When the amplitude is relatively small, the upper and the lower shear layers strongly interact and alternatively roll up to Karman vortex shedding into the wake. Fig.2 shows the flow patterns in the locked-in state when the amplitude of the vibration reaches y/D=0.1 and the vortex sheds into the wake fully synchronously with the vibration. In this state, the shear layers do not roll up like in Fig.1 but sway in a body synchronously with the vibration like a pendulum. When the sway of the shear layers reaches the peak, the vortex departs from the shear layer into the wake by the centrifugal force. From Fig.2, the vibration of the cylinder does not encourage the rolling up of the shear layers but prevents the rolling up so as to control the sway of the shear layer. Therefore, the locked-in state is defined as the one in which the movement of the shear layer is fully controlled by the vibration, while the interaction between the shear layers has no effect on the movement. 3.2 Why does the vortex induced vibration occur in the unsynchronized condition? Fig.3 shows the time variation of the amplitude of the cylinder together with the lift and the drag at the reduced velocity, U/nD=6.67. Here, the frequency of the vortex shedding does not coincide with that of the vibration of the cylinder, but the vortex induced vibration develops. Therefore, the locked-in is not the necessary condition for the vortex induced vibration. It is shown in Fig.3 that the amplitude of the displacement and the lift fluctuate slowly in every 2 or 3 cycles. Therefore, the three pairs of the vortex shed into the wake during two cycles of the vibration. Fig.4 shows the variation of the flow patterns in those cycles. At the beginning of these figures (T=83.8-88.15), the lower shear layer rolls up as the cylinder moves upward and the upper shear layer rolls up as the cylinder moves downward, thus the movement of the cylinder

607

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610 prevents the rolling up of the shear layer. Consequently, the vortex shedding is suppressed and the lift and the drag on the cylinder is small. On the other hand, the lower shear layer rolls up as the cylinder moves downward and the upper shear layer rolls up as the cylinder moves upward from T=91.3 to 96, thus the rolling up is promoted by the movement of the cylinder and the very strong vortex sheds into the wake. In this way the very large lift acts on the cylinder. In these cycles, the lift on the cylinder during T=83.8-88.15 has the positive damping effect and the lift during T=91.3-96 has the negative damping effect as is shown in Fig. 5. The total aerodynamic damping in these cycles is negative because of the imbalance of the positive and the negative damping, consequently the amplitude of the vibration is increased. 3.3 Effect of the splitter plate Fig.6 shows the flow patterns when the splitter plate of which length is 4 times of the diameter of the cylinder is set in the wake. The left hand figures show the streak lines and the patterns of the elementary vortices obtained by the numerical simulations. The right hand figures show the streak lines obtained by the experiment using by the smoke-wire method. It is seen that both flow patterns are very similar but the amplitude of the cylinder in the experiment is lager than the one in the numerical simulation. When the splitter plate is set just behind the cylinder, the flow-induced excitation at the resonant velocity is still observed but the amplitude of the vibration reduces considerably. In the case, the separated shear layers do not interact each other, but synchronizingly flap with the vibration. As the distance between the plate and the cylinder increases, the rolling up of the shear layers is promoted by the interaction of the layers. When the distance is about from one to two times of the diameter of the cylinder (L/D=1-2), the shear layer magnificently rolls up by the edge tone effect of the upstream end of the plate and the very strong vortex sheds onto the plate. By the comparison of the streak lines with the patterns of the elementary vortices, the vortices distribute inside of the streak line. The position of the vortex is within the region where no streak line exist, which is shown the black in the experimental photographs. 4. CONCLUSION It is investigated by the numerical simulation of the vortex discrete model, what the locked-in phenomenon is, why the vortex induced vibration occurs in the unsychronizing mode, What kind of effect the splitter plate has on the vibration. Through these investigation, the numerical simulation is the very effective tool to catch the physical image of the vortex induced vibration. REFERENCES 1. Kawai,H., A discrete vortex analysis of flow around a vibrating cylinder with a splitter plate, J. Wind Eng. Ind. Aerodyn., 35 (1990), pp.259-273. 2. Stansby, EK., A numerical study of vortex shedding from one and two cylinders, Aeron. Quat., 32 (1980), pp.48-71. 3. Sarpkaya,T and R.Schoaff, Inviscid model of two-dimensional vortex shedding by a circular cylinder, AIAA J., 11 (1979), pp.1193-1200.

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