Study on Vortex Induced Characteristics of Four Square Columns at Different Spacing Ratio

LIU Wei-min,GU Jia-yang,QU Ji-shun

(School of Naval Architecture and Marine Engineering,Jiangsu University of Science and Technology,Zhenjiang 212003,China)

0 Introduction

Due to the viscosity of the fluid,when water flows past columns,alternating vortices will appear on the both sides of the column.Furthermore the vortex shedding will produce pulsating pressure in transverse and stream-wise direction,which causes serious fatigue damage to ocean structure.When the vortex shedding frequency approaches the natural frequency of the column,resonance will even happen,which gives rise to larger-amplitude vibration and greater harm to the structure.So it is necessary to reveal the formation,development,evolution of the VIM(vortex-induced-motion)of columns,and therefore also lays a theoretical foundation for VIM suppression.

There are two categories in VIM:self-motivated motion and forced motion.The former refers to the movement generated by the pulsing forces from vortex shedding;the latter refers to the movement generated by the external forces with a certain fluctuating amplitude and frequency.The research on the former is focused on the varying vibration amplitude at different reduced velocities.The research on the latter is focused on the‘lock-in’phenomena.In this paper,self-motivated motion is studied.In early 1996,a series of experimental studies on VIM of cylinder at low mass ratio was carried out by Williamson team[1](Khalak&Williamson[2],Govardhan&Williamson[3];Govardhan&Williamson 2002[4]).Their results have revealed that the mass ratio affects vibration amplitude definitely.When mass ratio is small,the transverse amplitude shows three branches:initial branch,lower branch and upper branch.Govardhan&Williamson[5]modified the Griffin fitting curve according to related experimental data,considering the effects of Reynolds number on vibration amplitude.Because not any assumption and amplification were applied in the modification function,so the variation rule of vibration amplitude with different Re and mass-damping parameter was considered to be more accurately.Sanchis et al conducted experiment on 2-dof(two-degree-of-freedom)VIM of rigid cylinder supported by spring.In this experiment,the mass ratio is relatively small(close to 1),and the mass-damping parameter α=（m*+1）ζ is big.In this experiment,the stream-wise motion did not affect transverse amplitude;and the‘super upper branch’which has been observed by Williamson did not appear either,only initial branch and lower branch were found[6].

The numerical simulation for VIM(vortex induced motion)is developed gradually from the simulation for wake flow of cylinders.1-dof and 2-dof VIM of cylinder was numerically simulated by discrete vortex method by Dong Jing[7].The response curve of lift coefficient and vortex shedding patterns at different spacing ratios and reduced velocities were obtained.The changing rule of amplitude with varying mass ratios was concluded,and the‘lock-in’region was found.The VIM of multi-cylinders with different arrangement was studied by Ou Jinping et al.Xu Feng used the finite volume for solving two-dimensional incompressible N-S equations and conducted the simulations for VIV of cylinders with equilateral triangle arrangement[8].The 2-dof VIV of cylinders in array was simulated by Zhao Ming and the finite volume method was adopted to solve two-dimensional incompressible N-S equation,the VIV at different approach angles and different velocities were compared and analyzed[9].The results showed that approach angle had great effects on VIM of four cylinders.The wake flow trajectory and the lift curve were studied as well.The VIM of two typical different cross-section columns[10],vortex induced characteristics of multi-columns with low mass ratio[11]and the effect of filleting radius on vortex induced characteristics of square column[12]are studied by Gu et al by using the finite volume method.

In this paper,VIM of four-square-columns with three typical spacing ratios and different reduced velocities is focused on.In order to find the effect of spacing ratio on VIM of four square columns,the stream-wise and transverse amplitude,lift coefficient,wake flow trajectory and vortex shedding patterns were investigated.

1 Numerical computation models

1.1 Governing equations for computational fluid dynamics

Mass and momentum conservation equations for incompressible viscous fluid are given by:

Equations(1)and(2)are written in tensor form,where x represents the position in the rectangular coordinate system,u represents the velocity component,i,j∈（1,2,3）,ρ represents the fluid density,t represents the time,and p represents the pressure.μ represents the dynamic viscosity,with ρ=1 025 kg/m3in this paper.Kinematic viscosity is υ=1.0×10-6m2/s,Sijis the tensor of strain rate,and

1.2 Dynamical governing equation in dimensionless forms

Fig.1(a) Structural model;(b)Close view of computational grid,L/D=4.0

The finite volume method combined with dynamic mesh tech is adopted to realize the coupling fluid-solid interaction.The first-layer grids are exactly arranged in viscous logarithmic layer to meet the complex boundary conditions.Flow field is obtained by DES model and unsteady first-order implicit solver.The time step is 0.05 s with total 600 s.The stable values of 100 s to 600 s are selected to perform statistical computations.The distance of two adjacent columns is labeled L,and the side length of square column is labeled D.The structural model and computation grid are shown in Fig.1.

2 Calculation results and analyses

2.1 Amplitude response

To sum up the vibration amplitude,the maximum statistics and normal value statistic were used.

Fig.2 Stream-wise amplitude with reduced velocity(with maximum amplitude statistics)

Fig.3 Stream-wise amplitude with reduced velocities(with nominal-value statistics)

Fig.2 shows the stream-wise amplitude with different reduced velocities at spacing ratios 3.0,4.0 and 5.0 using maximum amplitude statistics.When the reduced velocity is less than 5.0,the stream-wise amplitude increases with increasing reduced velocity,furthermore the stream-wise amplitude at spacing ratios 3.0 increases fastest.With continuously increasing velocity,the stream-wise amplitude fluctuates within a certain range.At spacing ratios 3.0 and 4.0,the stream-wise amplitudes both reach their peaks at reduced velocity 7.0;at spacing ratio 5.0,the stream-wise amplitude reaches its peak 0.39D at reduced velocity 10.0.When reduced velocity is greater than 7.0,at same velocities,the stream-wise amplitude of spacing ratio 5.0 is greater than that of spacing ratio 3.0,and the stream-wise amplitude of spacing ratio 4.0 is the smallest.

The results of stream-wise amplitude using normal value statistic differ from that of using maximum statistic.The changing trends of stream-wise amplitude with three typical spacing ratios at different reduced velocities are distinct from each other,as shown as Fig.3.At spacing ratio 3.0,the stream-wise amplitude reaches its peak 0.17D as reduced velocity 5.0,and subsequently fluctuates with increasing reduced velocity.At spacing ratio 4.0,the stream-wise amplitude fluctuates within 0.12-0.14D at reduced velocity greater than 6.0.At spacing ratio 5.0,the stream-wise amplitude always increases with increasing reduced velocity generally,and arrives at its peak 0.22D at reduced velocity 11.0.

The stream-wise equilibrium has significant effects on stream-wise amplitude:the farmer the stream-wise equilibrium deviates from the initial position,the larger force the stream-wise motion needs.From Fig.4,the offset distance from the equilibrium to the initial position proximately increases linearly with increasing reduced velocity.At the same reduced velocity,the offset distance of spacing ratio 4.0 is greater than that of spacing ratio 3.0,and the offset distance of spacing 5.0 is the smallest,exactly matching the stream-wise amplitudes at the three spacing ratios.

Fig.5 gives the transverse amplitude by maximum statistic with changing reduced velocity in three typical spacing ratios.At spacing ratio 3.0,the transverse amplitude reaches the maximum value 0.61D at reduced velocity 12.0.At spacing ratios 4.0 and 5.0,the transverse amplitudes reach their peaks at reduced velocity 10.0 respectively.The general regularity of transverse amplitude of three typical spacing ratios is:at first,it increases,and then fluctuates.However,the amplitude fluctuation characteristics of the three working conditions differ from each other:zigzag fluctuation of spacing ratio 5.0 is more obvious than that of spacing ratios 3.0 and 4.0.

Fig.5 Transverse amplitude with reduced velocities(with maximum amplitude statistics)

Fig.4 The equilibrium position of stream-wise motion

From the normal amplitude results in Fig.6,it obviously reveals that the transverse amplitude gradually trends to be stable with increasing reduced velocity.At spacing ratio 5.0,the transverse amplitude declines slightly.As spacing ratios 3.0 and 5.0,the transverse amplitudes both reach their peaks 0.44D and 0.25D respectively at the same reduced velocity 4.0.As spacing ratio 4.0,the transverse amplitude reaches its peak 0.35D at reduced velocity 9.0.

Fig.7 Time history of stream-wise displacement and transverse displacement at L/D=4.0 and L/D=5.0

Fig.7 shows the time histories of stream-wise and transverse displacements of spacing ratio 4.0 under different reduced velocity when the system remains steady state.At reduced velocity 3.0,the stream-wise and transverse displacements of four-square-columns both are relatively small.With the increase of velocity,the stream-wise and transverse displacements increase.The transverse displacement is much larger than stream-wise displacement,so the transverse motion has received more attention in ocean engineering field.

From Fig.7,in smaller spacing ratio,because of strong interference from the shear layers of adjacent columns,the lift and drag forces acting on the columns are irregular,which furthermore leads to irregular stream-wise and transverse movements of four square columns.With increasing spacing ratio,the interference from adjacent column shear layers weakens,and thereby the motion becomes more regular.

2.2 Frequency spectrum characteristics

Fast Fourier transform(FFT)can convert the lift coefficient function with respect to time into the density distribution function of system energy with respect to frequency,which can put the results in time domain into the frequency domain.Fig.8 shows the varying main lift coefficient frequencies with spacing ratios.fL*is the ratio of the main lift coefficient frequency to natural frequency.As shown as Fig.8,the changing trends of main lift coefficient frequencies at three typical spacing ratios are similar:the lift coefficient frequencies all increase with increasing reduced velocity.When the reduced velocity ranges from 0 to 6.0,the main lift coefficient frequencies of three spacing ratios do not exhibit obvious regulation.When the reduced velocity is greater than 6,at the same velocity,the main lift coefficient frequency of spacing ratio 5.0 is greater than that of others.

Fig.8 Main lift coefficient frequency with reduced velocity,L/D=4.0,5.0,6.0

In all three working conditions,when the reduced velocity is near 4.0-5.0,the main lift coefficient frequency approaches the natural frequency,and at this time,the transverse amplitude curve starts oscillating.

Fig.9 Main lift coefficient frequency with different reduced velocity,L/D=4.0

Fig.9 shows the lift coefficient frequency spectrum analysis of VIM of four-squarecolumns with spacing ratio 4.0.At reduced velocities 4.0,7.0 and 9.0,the energy distribution of lift coefficients is clearly divided into two parts:at main frequency and at secondary frequency.When the reduced velocity is 12.0,there is only one peak in system energy,and it occurs at frequency of 0.14 Hz.

Fig.10 shows the lift coefficient frequency spectrum analysis of VIM of four-squarecolumns with spacing ratio 5.0.At the reduced velocities 4.0,7.0 and 9.0,the system energy is very concentrated,and there is only one peak,and the peak becomes greater with increasing reduced velocity.At reduced velocity 12.0,the system energy is distributed in two frequencies:at main frequency of 0.17 Hz;at secondary frequency of 0.12 Hz.

When the velocity is greater and the main frequency of the system energy distribution is far away from the natural frequency,the extreme value is relatively larger.

2.3 The vortex trajectory and wake flow characteristics

It is well-known that the trajectory of VIM for single-cylinder is the classic‘8’shape,but the trajectory of VIM for four-square-columns is obviously distinct from that:regularity is very vague,and only when the velocity is very small,vortex trajectory appears‘0’shape.

From Figs.11-12,obvious vortex shedding phenomenon has been observed in the wake region of upstream column.External shear layer rolls up and joins into the wake region of downstream column.Therefore different-scale messy vortices appear in the wake region of downstream column.At small spacing ratio,under the strong flow interference,the complex vortex shedding patterns give rise to irregular hydrodynamic forces.

With the increase of spacing ratio,as shown in Figs.11-12,each column has its own single vortex street gradually,but the interference phenomenon still exists at this time.

Fig.12 Vortex shedding mode of four square columns,L/D=5.0

3 Conclusions

The VIM of four-square-columns with three typical spacing ratios and different reduced velocities is numerically simulated to investigate the effects of spacing ratio.

(1)The general regularity of transverse and stream-wise amplitudes at three typical spacing ratios is:at first,it increases,and then fluctuates.The transverse amplitude is much greater than the stream-wise amplitude.

(2)The time histories of stream-wise and transverse amplitudes exhibit:in smaller spac-ing ratio,because of strong interferences from the shear layers of adjacent columns,the lift and drag forces acting on the columns are irregular,which leads to irregular stream-wise and transverse movements of four square columns.With increasing spacing ratio,the interference weakens.

(3)The main lift coefficient frequency increases with increasing reduced velocity at three typical spacing ratios.At reduced velocities near 4.0-5.0,the main lift coefficient frequency approaches the natural frequency,and the transverse amplitude curve starts oscillating.As spacing ratio 4.0,at reduced velocities 4.0,7.0 and 9.0,the energy of lift coefficients is distributed at two frequencies,at reduced velocity 12,there is only one peak.This is contrast to the condition at spacing 5.0.

(4)The trajectory of VIM for four-square-columns is obviously distinct from that of cylinders.At small spacing ratio,regularity is very vague,complex vortex shedding patterns give rise to irregular hydrodynamic forces.With the increase of spacing ratio,each column has its own single vortex street gradually.

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