Galloping of square cylinders in cross-flow at low Reynolds numbers

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Journal of Fluids and Structures

Journal of Fluids and Structures 28 (2012) 232–243

0889-97

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/jfs

Galloping of square cylinders in cross-flow at low Reynolds numbers

A. Joly a,b, S. Etienne a,n, D. Pelletier a

a Ecole Polytechnique de Montreal, C.P. 6079, Succursale Centre-ville, Montreal, Canada H3C 3A7b Ecole des Ponts ParisTech, 6 et 8 avenue Blaise Pascal, 77455 Marne-la-Vallee cedex 2, France

a r t i c l e i n f o

Article history:

Received 14 December 2010

Accepted 1 December 2011Available online 19 December 2011

Keywords:

Square cylinder

Galloping

Added mass

Finite element method

46/$ - see front matter & 2011 Elsevier Ltd. A

016/j.jfluidstructs.2011.12.004

esponding author.

ail addresses: [email protected] (A. Joly

a b s t r a c t

Galloping of square cylinders is studied at low values of the Reynolds number using a

two-dimensional finite element method. A sinusoidal quasi-steady model allows

determination of the occurrence of galloping and its amplitude. Parameters of this

model are obtained via FEM unsteady simulations at different angles of incidence

between 01 and 101. The model efficiency is validated by comparing its predictions to

those of unsteady simulations of fluid–structure interaction of a spring mounted square

constrained to move in the direction transverse to the flow. Results show that the

model yields good predictions of both the onset of galloping and its amplitude as a

function of the Reynolds number at high values of the mass ratio. However, the quasi-

steady model fails to reproduce the sudden change of amplitudes observed in finite

element simulations at mass ratios below a critical value. Modifications to the model

are introduced to reproduce this low mass ratio effect.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Galloping is a flow-induced vibration that differs from the more common and well-known vortex-induced vibrationboth in its causes and its effects. It is a self-excited instability which may result in high-amplitude and low-frequencyoscillations. According to Kaneko et al. (2008), Den Hartog (1956) was the first to provide a theoretical explanation for it. Inhis study of galloping of iced electric transmission lines, he developed a stability criterion which we will be describe later.Paıdoussis et al. (2011) pointed out that the same criterion was developed earlier by Glauert (1919).

Galloping usually happens to long elastic structures of non-circular cross section. The square cylinder in particular isknown to be prone to the phenomenon. Galloping of square cylinders is relevant to offshore engineering, where bundles ofrisers can be sheathed in a square envelope (low values of the mass ratio), and to civil engineering (high values of the massratio). Parkinson and Brooks (1961) examined the validity of a polynomial development of forces as a function of the angleof attack for square cylinders; their theoretical explanation was largely bourne out by experimental evidence. Parkinsonand Wawzonek (1981) studied mutual effects of vortex-induced vibrations and galloping since they found that undercertain circumstances both phenomena could occur simultaneously. In the present paper, we show, by means of anexample for a large reduced velocity of 40, that vortex shedding could interfere with galloping while both phenomena arewell separated in terms of frequency.

Robertson et al. (2002) carried out a study of both rotational and transverse galloping of rectangular cylinders. Usingthe Den Hartog criterion, they determined which thickness ratios were susceptible to galloping and validated theirassumptions through simulations of fluid–structure interactions (FSI). Barrero-Gil et al. (2009) focused on the transverse

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), [email protected] (S. Etienne), [email protected] (D. Pelletier).

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A. Joly et al. / Journal of Fluids and Structures 28 (2012) 232–243 233

galloping of a square cylinder. Like most of their predecessors in the field, they used the quasi-steady (QS) theory; theircontribution was the use of the Krylov–Bogoliubov perturbation method to determine the amplitude of gallopingoscillations. They also highlighted a simple way of determining the critical Reynolds number for the onset of galloping.Closure of their QS model was achieved using data from Sohankar et al. (1998) giving the lift and drag coefficients of asquare cylinder at low Reynolds numbers (o200) and an incidence varying from 01 to 451. However, the data are notaccurate enough at low angles of incidence to study the onset of galloping. Furthermore, the characteristic length differsfrom the one used in the model by Barrero-Gil et al. (2009). The former used the projected width while the latter used thecylinder’s edge length.

Therefore, we carry out finite element simulations to generate our own data for the lift and drag coefficients of a squarecylinder. We also opt to use them with an adapted version of the QS model of Barrero-Gil et al. (2009). Finally, FSIsimulations allow us to evaluate the reliability of the QS model. We also study the influence of the mass ratio on theamplitude of oscillation thanks to FSI simulations. We observed an unexpected decrease in the galloping amplitude. Wedevelop a simple analytical model to try to account for it.

Section 2 describes the adapted quasi-steady model. Section 3 presents the numerical characteristics and the validationtests for the code used in the study. Section 4 gives the results. The paper ends with the conclusion.

2. Quasi-steady model

We assume a two-dimensional incompressible and laminar flow and we make the quasi-steady assumption (de Langre,2002). The hypotheses are well detailed in Barrero-Gil et al. (2009) for the equation of motion of the square cylinder

mð €yþ2zoy _yþo2yyÞ ¼ Fy ¼

12rU2DCy, ð1Þ

where y is the displacement in the direction transverse to the flow, m the mass per unit length, oy the undamped naturalangular frequency, r the fluid density, U the incoming flow velocity, D the cylinder edge and Cy the fluid flow transverseforce coefficient (see Fig. 1). The overdot stands for differentiation with respect to time t.

This problem is characterized by four parameters: the mass ratio mn ¼m=ðrD2Þ ¼ 1=ð2mÞ, the reduced velocity

Ur ¼U=ðoyDÞ ¼UR=ð2pÞ, the structural damping ratio z¼ c=ð2moyÞ and the Reynolds number Re¼UD=n (where n is thekinematic fluid viscosity). A change of frame of reference is applied to account for the apparent velocity ~U ¼U

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þtan2a

p(see Fig. 2) with an angle of incidence a equal to tan�1ð _y=UÞC _y=U, a being relatively small, less than 101. Due tosymmetry, the transverse fluid force coefficient may be expressed as an odd series expansion of a. Considering angles ofincidence lower than 101, third order expansions were found to be adequate. Hence we use the following expression for Cy:

Cy ¼ a1

_y

U

� �þa3

_y

U

� �3

, ð2Þ

where a1 and a3 are Reynolds-dependent parameters. The dimensionless equation of motion is

Z00 þð2z�mUra1ÞZ0�ma3

UrðZ0Þ3þZ¼ 0, ð3Þ

where Z¼ y=D, t¼oyt and prime represents differentiation with respect to t.

Fig. 1. Transverse to the flow displacement of a square cylinder: notations and sign convention.

Fig. 2. Frame of reference modification: square cylinder with non-zero incidence angle.

A. Joly et al. / Journal of Fluids and Structures 28 (2012) 232–243234

We look for sinusoidal solutions, which is a strong hypothesis: ZðtÞ ¼ R cos ðtþfÞ where R¼A/D is the dimensionlessamplitude and A the actual amplitude. By multiplying Eq. (3) by Z0 and integrating over one period (from 0 to 2p), weobtain the dimensionless amplitude of galloping:

Rn¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4

3

a1Ur

�a3

� �Ur�

2zma1

� �s: ð4Þ

For the study of the onset of galloping, we can restrict ourselves to a first order expansion of Cy. Galloping will occurwhen the damping coefficient becomes negative, which implies that the critical reduced velocity is given by

Ucgr ¼

2zma1

: ð5Þ

Before we look for the critical Reynolds number, we need expressions for a1 and a3:

a1ðReÞ ¼ a10þa11Re,

a3ðReÞ ¼ a30þa31Re:

(ð6Þ

In the present context, first order expansions are sufficient. Introducing r¼ Re=Ur ¼oyD2=n, we obtain the followingexpression for the critical Reynolds number:

Recg ¼�a10þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2

10þ4ra11ð2z=mÞq

2a11, ð7Þ

which is minimal when there is no damping, assuming that a10o0:

Remincg ¼�

a10

a11: ð8Þ

Finding the values of the amplitude of oscillation Rn and the critical values Ucgr and Recg of parameters boils down to

determining a10, a11, a30 and a31. To do so, 30 numerical simulations with a fixed cylinder have been carried out, for nRe ¼ 6values of the Reynolds number and na ¼ 5 values of the angle of incidence. Let ðCy,ijÞ be the values of the force coefficientobtained through these simulations, and ~Cy the function that models this force coefficient ( ~Cy ða,ReÞ ¼ a10aþa11Reaþa30a3þa31Rea3). We use the least-squares method, which consists in solving the following problem:

minfa10 ,a11 ,a30 ,a31g

Xnai ¼ 1

XnRe

j ¼ 1

ðCy,ij�~Cy ðai,RejÞÞ

2: ð9Þ

3. Numerical details and validation

All computations are performed with a finite element method. The solver has been specifically designed and developedto treat fluid–structure interaction problems. Briefly, the solver possesses the following characteristics.

(i)
A mixed finite element method is used to solve the velocity–pressure form of the Navier–Stokes equations for anincompressible fluid. Note that pressure is the Lagrangian multiplier corresponding to the incompressibilityconstraint.
(ii)
Third order space accuracy is achieved thanks to the use of a P2–P1 Taylor–Hood element. (iii) An implicit Runge–Kutta time integrator of third order accuracy for the flow velocity components and structural
displacements on moving meshes.
(iv) The parallel sparse direct solver PARDISO (Schenk and Gartner, 2004, 2006) is used to solve the linear system. (v) An implicit method fully coupling all degrees of freedom.
The code has been thoroughly validated for unsteady flows on moving grids (Etienne et al., 2009).The computational domain is shown in Fig. 3. A convergence study has been performed for an angle of incidence of 01.

This study shows that a mesh of 15 000 P2–P1 elements and a non-dimensional time-step UDt=D of 0.2 are sufficient toensure an accuracy of 70.002 on Cy, as illustrated in Figs. 4 and 5. Fig. 6 shows a typical close-up view of a mesh near thecylinder (60 000 nodes), with refinement in the wake, near the edges and even more at the corners. Fig. 7 shows theinstantaneous vorticity field obtained at the final time of the simulation. The vortex shedding appears very clearly. Fig. 8shows the unsteady drag and lift force coefficients for this configuration. After a transient phase, both become periodic andwe evaluate the steady force coefficients by averaging the signal over an integer number of periods. The unsteady dragcoefficient is not sinusoidal in this case because of the angle of incidence.

Fig. 3. Domain size and fluid flow velocity orientation.

5 9 17 30 60

1.407

1.408

1.409

1.41

1.411

1.412

1.413

1.414

1.415

1.416

CD

5 9 17 30 60−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6C

L (x

10−2

)

number of nodes (x 103) number of nodes (x 103)

Fig. 4. Spatial convergence of drag (left) and lift (right) coefficients, for Re¼ 150, a¼ 01 and Dt ¼ 0;1.

0.05 0.1 0.2 0.4

1.407

1.408

1.409

1.41

1.411

1.412

1.413

1.414

1.415

1.416

Time step

CD

0.05 0.1 0.2 0.4−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

CL

(x 1

0−2)

Time step

Fig. 5. Temporal convergence of drag (left) and lift (right) coefficients, for Re¼ 150, a¼ 01, and 60 000 nodes.


Table 1 compares values obtained from two-dimensional computations at zero angle of attack and Re¼ 200 withsimilar results taken from the literature. Drag coefficients are very similar between all results. However, there areimportant differences with other sources concerning Strouhal and lift coefficient root mean square. Other references canbe found in Sohankar et al. (1999) showing Strouhal number experimental values ranging from 0.138 to 0.151.

Fig. 6. Close-up view of the mesh near the cylinder.

Fig. 7. Vorticity field in the wake of a square cylinder at a¼ 51 and Re¼ 200.

0 50 100 150 200 250 3001.35

1.4

1.45

1.5

1.55

1.6

1.65

1.7

1.75

0 50 100 150 200 250 300−1.5

−1

−0.5

0

0.5

1Drag coefficient CD Lift coefficient CL

Fig. 8. Unsteady drag (left) and lift (right) coefficients of a square cylinder at a¼ 51 and Re¼ 200.


4. Numerical results

Table 2 presents the results of simulations with a fixed cylinder (Cy ¼�CL cos a�CD sin a).

Table 1Comparison of drag coefficient, lift coefficient root mean

square and Strouhal number with various results from the

literature at Re¼ 200.

Author CD CLrmsStrouhal

Sohankar et al. (1999) 1.46 0.32 0.170

Singh et al. (2009) 1.54 0.55 0.142

Present 1.44 0.42 0.151

Table 2Steady force coefficients.

Re a (1) CD CL Cy

130 0 1.41 0.001 �0.001

1 1.41 �0.020 �0.004

3 1.44 �0.058 �0.017

5 1.49 �0.084 �0.046

10 1.72 �0.077 �0.224

140 0 1.41 0.001 �0.001

1 1.41 �0.025 0.000

3 1.44 �0.071 �0.005

5 1.49 �0.101 �0.029

10 1.75 �0.089 �0.216

150 0 1.41 0.000 0.000

1 1.41 �0.030 0.006

3 1.44 �0.085 0.009

5 1.50 �0.118 �0.013

10 1.77 �0.099 �0.211

160 0 1.41 0.000 0.000

1 1.42 �0.036 0.011

3 1.44 �0.098 0.022

5 1.50 �0.136 0.004

10 1.80 �0.111 �0.203

170 0 1.42 0.000 0.000

1 1.42 �0.041 0.016

3 1.45 �0.111 0.035

5 1.51 �0.152 0.020

10 1.83 �0.123 �0.197

200 0 1.44 0.001 �0.001

1 1.44 �0.055 0.030

3 1.47 �0.150 0.073

5 1.54 �0.200 0.065

10 1.91 �0.167 �0.166


Solving the least-squares problem (9) with these values yields the following result:

a10 ¼�3:41,

a11 ¼ 2:43� 10�2,

a30 ¼ 50:56,

a31 ¼�0:65,

8>>><>>>:

ð10Þ

with the coefficient of determination R2¼ 0:993; see Fig. 9.

4.1. A simple and direct use of the results

The Den Hartog criterion (adapted to the present sign convention for Cy) provides a simple way to determine whengalloping occurs:

@Cy

@a

��a ¼ 0

40 ) unstableFgalloping will occur;

@Cy

@a

��a ¼ 0

o0 ) stableFno galloping:

8><>: ð11Þ

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

α (rad)

Cy

Fig. 9. CyðaÞ curves for Re¼ 200, 170, 160, 150, 140, 130 (from top to bottom).

140160

180200

5

10

15

200

0.51

1.52

2.53

3.5

Re ReU

r =UR /(2 π)

Ur =U

R /(2 π)

R*

140160

180200

5

10

15

200

0.51

1.52

2.53

3.5

R*

Fig. 10. QS model results; left: zero damping; right: 2z=m¼ 5.


Indeed, using this criterion on the data plotted in Fig. 9 indicates that the critical Reynolds number is approximately140. One can also find a simple way of estimating the amplitude of galloping: by assuming that (i) the displacement of thecylinder is of the form yðtÞ ¼ RnD cos ðoytþjÞ and (ii) the cylinder reaches its highest velocity when the incidence is a0

such that Cyða0Þ ¼ 0, one obtains Rn¼ a0Ur ¼Ur

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�a1=a3

p. Notice that, for zero damping, the only difference with Eq. (4) is

the factorffiffiffiffiffiffiffiffiffi4=3

pC1:15. However, as quick as this model might be, it does not take damping into account (neither does the

Den Hartog criterion).

4.2. The quasi-steady model

The QS model of galloping determines the critical Reynolds number for the onset of galloping using Eq. (8) and theparameter values given in Eq. (10). That is Remin

cg ¼ 140 as expected. Fig. 10 shows the galloping amplitude Rn as a functionof Re and Ur according to the sinusoidal QS model (Eq. (4)). The amplitude increases with both of these parameters. Forzero damping, we observe a linear relationship between Ur and Rn, as confirmed by Eq. (4). It is not the case any more fornon-zero damping, where the galloping amplitude is lower on the whole, and the critical values Ucg

r and Recg are higher.However, Rn becomes a linear function of Ur when Ur is high enough.

4.3. Fluid–structure interaction simulations and comparison with the model

To evaluate the reliability of the QS model, two-dimensional FSI simulations have been performed for a mass ratio mn

equal to 20. Fig. 11 shows comparisons of predictions by the QS model and by these simulations. Predictions obtained with

140 150 160 170 180 190 2000

0.5

1

1.5

2

2.5

3

3.5

Re=10Ur

R*

QS model with 2 ζ/μ=0FSI direct simulationQS model with 2 ζ/μ=5FSI direct simulation

Fig. 11. Comparison of the QS model results with FSI simulations.


the QS model are quite close to the maximum amplitudes obtained with FSI simulations. For the case where the dampingcoefficient is non-zero (2z=m¼ 5), the QS model yields good predictions of the amplitude, but with slight overestimation ofthe critical value of the Reynolds number. Thus, it can be inferred that the QS assumption is adequate to predict bothamplitude and critical values of the Reynolds number at large values of the mass ratio.

4.4. Influence of the mass ratio

We now focus on what happens at low values of the mass ratio (i.e. for values below 10). Numerical simulationsshowed that the galloping amplitude, at least for low values of the Reynolds number, is not independent of the mass ratio,even for zero damping ratio. Present results, at a Reynolds number of 200 and a reduced velocity UR of 40, show that theamplitude of oscillation decreases abruptly for values of mn close to 3 when performing simulation with a sequence ofdecreasing values of mn. We found no reference to this effect and we suspect that this observation was never reportedpreviously.

To confirm this unexpected behaviour we developed an analytical model based on physical grounds that reproducesthis phenomenon. Our starting point is the fact that galloping is a low frequency phenomenon compared to vortexshedding for UR¼40 (approximately five times as low). Thus, at such a value of the reduced velocity the vortex sheddingunsteady forces may be represented by a harmonic force of constant intensity. Hydrodynamic added mass ma is also takeninto account and deduced from simulations. Its mean value is around 3.5. This may appear quite high but it is corroboratedexperimentally (Molin, 2007). The resulting oscillatory force model is as follows:

m 1þma

m

� �€yþ2zoy _yþo2

yy� �

¼1

2rU2D a1

_y

Uþa3

_y

U

� �3 !

þF0 sin ðostÞ: ð12Þ

The characteristic angular frequency os ¼ 0:94 is obtained from fixed cylinder simulations that provided closurecoefficients of the QS model. To fit with FSI simulations, we take F0¼0.66 for the characteristic force. Note that wemeasured a lower value for its fixed simulation counterpart (F0¼0.3). This may reveal a more complex interaction withvortex shedding, which we would not deal with in this paper.

Figs. 12 and 13 compare results from the FSI simulations and numerical solutions of the oscillatory force model(Eq. (12)). The model reproduces quite well what is observed in the simulations. Therefore, vortex shedding appears to be agood candidate to explain why galloping amplitude decreases abruptly for decreasing values of the mass ratio close to 3.Changing the parameters in Eq. (12) shows that UR (provided it is high enough) and z have little effect on the mass ratiolimit, while F0 and os determine its value.

Fig. 14 compares amplitudes obtained from the sinusoidal QS model, FSI simulations, the oscillatory force model, andthe QS equation (1) solved numerically. This last corresponds to F0 ¼ 0. To see the effect of modifying the characteristicforce, oscillatory force model results with various values of F0 have also been plotted. Amplitudes predicted by theoscillatory force model agree well with those from the FSI simulations for F0¼0.66.

As the mass ratio increases, the oscillatory force model results converge to the QS model results. The harmonic forcingterm in Eq. (12) plays a decreasing role as the solid becomes heavier. Also, note that when the mass ratio is equal to one,amplitudes from the oscillatory force model and FSI simulations are higher than for mn ¼ 2 since vortex-induced vibrations

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

y/D

Numerical solution of oscillatory equation

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

y/D

Direct Simulation

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

y/D


−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

y/D

Direct Simulation

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Ut/D

y/D


−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Ut/D

y/D

Direct Simulation

0 100 200 300 400−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Ut/D

y/D


0 100 200 300 400

0 100 200 300 4000 100 200 300 400

Ut/DUt/D0 100 200 300 4000 100 200 300 400

Ut/DUt/D0 100 200 300 4000 100 200 300 400

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Ut/D

y/D

Direct Simulation

Fig. 12. Comparison of FSI simulations (left) with the oscillatory force model (right) for mn ¼ 20, 10, 5 and 4 (from top to bottom). Re¼ 200, UR¼40, z¼ 0,

F0¼0.66.


0 100 200 300 400−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Ut/D

y/D


0 100 200 300 400−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Ut/D

y/D

Direct Simulation

0 100 200 300 400

−0.5

0

0.5

1

1.5

2

Ut/D

y/D


0 100 200 300 400

−0.5

0

0.5

1

1.5

2

Ut/D

y/D

Direct Simulation

0 100 200 300 400

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Ut/D

y/D


0 100 200 300 400

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Ut/D

y/D

Direct Simulation

Fig. 13. Comparison of FSI simulations (left) with the oscillatory force model (right) for mn ¼ 3, 2 and 1 (from top to bottom). Re¼ 200, UR¼40, z¼ 0,

F0¼0.66.


are more effective. The numerical resolution of the QS equation (1) gives the result we had expected before discovering thedrop of amplitude phenomenon: as the mass ratio decreases, the galloping amplitude increases (F0 ¼ 0). The discrepancybetween the numerical resolution of the QS equation and the sinusoidal QS model is precisely that the oscillations havebeen assumed sinusoidal, while solutions of Eq. (1) become more triangular as mn decreases. Cases with structuraldamping are presented in Fig. 15. We have taken a particularly large damping ratio to see its effect. For the study of theinfluence of the mass ratio, no FSI simulation with damping has been carried out.

5. Conclusions

The study of transverse galloping at low values of the Reynolds number allows us to make the following observations:

(i)
Galloping for a square cylinder is observed only for Reynolds number larger than 140. (ii) A sinusoidal quasi-steady model accurately reproduces amplitudes obtained from fluid–structure interaction
simulations for moderately high values of the mass ratio.

0 5 10 15 200

0.5

1

1.5

2

2.5

m*

R*

ζ = 0 ma = 3.5

Sinusoidal QS modelFSI direct simulationOscillatory force model

F 0=0

.8

F0=0

F0=0.3F0=0.4

F 0=0.5

F 0=0.6

F 0=0.7

F0 : Sinusoidal perturbation strength

Fig. 14. Comparison of the amplitude with different methods of resolution and parameters, for Re¼ 200 and UR¼40. No structural damping.

0 5 10 15 200

0.5

1

1.5

2

2.5

m*

R*

ζ=10% ma=3.5

Sinusoidal QS modelOscillatory force model

F0 = 0

F 0=0

.4

F0 : Sinusoidal perturbation strength

F 0=0

.5F 0

=0.6

F 0=0

.7

Fig. 15. Comparison of the amplitude with different methods of resolution and parameters, for Re¼ 200 and UR¼40. Structural damping ratio of 10%.


(iii)
Galloping amplitude decreases abruptly for decreasing values of the mass ratio close to 3 at Re¼ 200. (iv) An oscillatory force term which models vortex shedding effects on the structure may be a good candidate to explain
this sudden drop in galloping amplitude at low values of the mass ratio.
(v) Without the effect of vortex shedding, the galloping amplitude would actually increase as the mass ratio decreases.
Finally, experimental results by Molin (2007) for a mass ratio mn � 3:75 did not exhibit the low amplitude gallopingphenomenon. The first explanation may be a Reynolds number effect. The experimental Reynolds number was in factmuch higher (Re¼ 300 000) than that of the simulations. The second explanation may be a too small value of the aspectratio of the square cylinder in the experiments. Indeed, the experimental aspect ratio L=D was approximately equal to 3,which may be inadequate for comparison with the two-dimensional simulation results.

Thus, an experimental study with either low values of the Reynolds number or high values of the aspect ratio, isrequired to evaluate the reliability of the low mass ratio effect on galloping amplitude.


References

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Galloping of square cylinders in cross-flow at low Reynolds numbers