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Journal of Wind Engineeringand Industrial Aerodynamics

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

Analysis of aeroelastic vibration of rectangular cylinder in a uniform flow bya Large Eddy Simulation formulated in a non-inertial moving coordinatesystem

Hiroshi Nodaa, Akihiko Nakayamab,⁎

a Department of Architecture, Kindai University, Higashiosaka, Osaka, Japanb Department of Environmental Engineering, Universiti Tunku Abdul Rahman, Kampar, 31900 Perak, Malaysia

A R T I C L E I N F O

Keywords:Aeroelastic vibrationLarge Eddy SimulationNon-inertial effectsRectangular cylinder

A B S T R A C T

A large-eddy simulation method of calculating the flow past an elastically-supported body, has been constructedbased on the finite difference method formulated in the coordinates fixed on the moving body. The boundaryconditions on the body can be applied in the same way as on a fixed body. Reconstruction of computationalmesh or interpolation of the computational points is not needed as the body moves. The coding of the method issimple and the computation is efficient, but the equations of motion must be written in the non-inertialcoordinates and the boundary conditions on the outer boundaries of the flow domain that also move must beadjusted as the orientation of the body changes. The method has been tested in the computation of the flow pasta rectangular cylinder supported by a spring and dashpot and the motion of the cylinder for which experimentaldata are available. Comparison with experimental data in the case of a rectangular cylinder of depth to breadthratio of D/B=2 indicates that the characteristics of the translational oscillation corresponding to the heaving oflong beams and the rotational oscillation corresponding to the torsional deformation are reproduced well.

1. Introduction

The recent advancement of Computational Fluid Dynamics (CFD)methods is making it possible to apply numerical simulation techni-ques to various engineering problems including those of wind engi-neering involving fluid-structure interaction (FSI). Calculation ofaerodynamic forces acting on fixed bodies in a given wind field cannow be done by different ways depending on the complexity of the bodygeometry and the boundary conditions. Among different methods,Large Eddy Simulation (LES) method has been shown to give goodresults in various conditions and have proved quite successful (e.g.Sagaut, 2006; Tamura, 2008; Grinstein et al., 2011; Hu et al., 2015).However, for analyzing the flow when the body moves due to theaerodynamic forces, effective and efficient FSI methods must bedeveloped that can be implemented in the LES methodology, and theapplicability and accuracies be verified. Efforts are being made in thatdirection so the problems involving FSI can be used in the design ofstructures like bridge decks of suspension or cable-stay bridges and tallbuildings (e.g. Tamura and Ito, 2002; Sun et al., 2008).

There are a number of numerical methods that are used to obtainthe vibration characteristics of bodies in a wind field. One common

method is the Arbitrary Lagrangian Eulerian method (Hirt et al., 1974;Hiejima et al., 1997; Facundo et al., 2007 Souli and Benson, 2010; Houet al., 2012) in which the computational points are moved as the bodymoves keeping the points on the body surface on it. The effects of themovement of the computational points are reflected in the equations ofmotion. Another method is to use a body-fitted moving grid imbeddedin a stationary grid extending to the free stream (e.g. Shimada andIshihara, 2012; Maruoka and Hirano, 2000; Nomura and Hughes,1992). This requires a special technique to smoothly connect the twonested grids. Either of these approaches moves the coordinates andreconstructs the computational mesh at every computational time stepand increases the computational loads.

In the Immersed Boundary (IB) method (Mittal and Iaccarino,2005; Sudhakar and Vengadesan, 2012; Frisani and Hassan; 2015), thecomputational points are kept fixed but special interpolation is appliedin the equations of motion so the boundary conditions are satisfied onthe surface of the moving body that may not necessarily coincide withthe computational points. Although re-meshing of the computationalpoints is not needed, interpolation points that move with the body stillneed to be kept track with. A finite element method that computes themotion of the fluid and the solid at the same time is also being

http://dx.doi.org/10.1016/j.jweia.2017.03.011Received 17 December 2015; Received in revised form 20 February 2017; Accepted 24 March 2017

⁎ Corresponding author.E-mail addresses: hnoda@arch.kindai.ac.jp (H. Noda), akihiko@utar.edu.my (A. Nakayama).

Journal of Wind Engineering & Industrial Aerodynamics 166 (2017) 29–36

0167-6105/ © 2017 Elsevier Ltd. All rights reserved.

MARK

developed and applied to fluid-structure interaction problems (e.g.Benk et al., 2012; Hou et al., 2012). Yet another recently studiedmethod is based on mesh free particle methods (Onate et al., 2004)which computes the motion of fluid and solid based on the purelyLangrangian equations of motion. These particle methods are veryflexible to treat the interaction between the fluid and solid body ordifferent phases of fluid but they further increase the computationaltime and often fast processors are needed. In any case, the calculationof the interaction increases the amount of computational loads.

For a rigid body that does not change its shape when it moves underthe influence of wind forces, methods that use the coordinate systemfixed on the body are efficient since the computational grids or theinterpolation points do not have to be adjusted for the moving body.However, the coordinates and the computational domain move relativeto the inertial system and the equations of motion and the boundaryconditions away from the body need to account for the effects of thebody motion.

Another point that needs consideration is when the flow iscalculated by the Large Eddy Simulation (LES) method withoutsufficiently resolving the viscous wall layer, in which case, additionalmodel equations are solved involving the distance and the directionfrom the in-flow computational points to the closest body surface (e.g.Pope, 2004; Sagaut, 2006; Grinstein et al., 2011). In the calculation offull-scale flow past real structures, such wall modeling is unavoidableand the numerical method must also be adapted for these additionalcomplications.

In this paper, we describe a numerical analysis method of fluid-structure interaction between the flow past a rigid body in two-dimensional motion with three degrees of freedom and the motion ofthe body induced by the aerodynamic forces. The method for calculat-ing the flow is a fully three-dimensional LES based on a finite-difference technique using the equations of motion in non-inertialcoordinate system fixed on the moving body. By using the non-inertialmoving coordinates fixed on the rigid body, the FSI effects can berepresented by adding a few extra inertia terms without modifying theboundary condition on the body surface, though the conditions on theouter boundaries must be modified. It is then used in the analysis ofvibration of a cylindrical body with rectangular cross section supportedby a spring and a dashpot. The results are examined by comparing withthe existing experimental data for a rectangular cross section of depthto breadth ratio of two in forced and free vibration modes in a uniformflow. It is found that the overall method predicts the steady andunsteady aerodynamic characteristics well and the obtained vibrationcharacteristics show good agreement with the experimental results.

2. LES equations in non-inertial frame

The equations of fluid motion around moving objects can beformulated either in the inertial frame of reference where the coordi-nates are fixed in space or in the non-inertial frame of reference wherethe coordinates are fixed on the moving object. The equations in non-inertial coordinate system contain terms related to the motion of thecoordinates. There are advantages and disadvantages of using either ofthe coordinate systems. If the coordinates are fixed on the body, theposition vectors of the points on the body surfaces and the equations ofthe boundary condition are unchanged by the motion of the body. It is agreat advantage especially when the near wall flow is modeled torepresent the unresolved turbulence effects in LES (e.g. Pope, 2004).However, the equations for the outer boundary conditions away fromthe body contain extra terms due to the motion of the coordinates. Ifthese terms are treated properly and effectively, the calculation loadcan be reduced considerably compared with the methods that requireeither identifying the instantaneous positions of the calculation pointson the moving body and/or re-meshing the calculation grids. Here wefirst summarize the equations in the non-inertial moving coordinatesand then the boundary conditions based on them are derived.

We consider the case when the body translates and rotates in aplane perpendicular to the axis of a cylindrical body. Let coordinates xibe the inertial frame of reference fixed in space. At time t the referencepoint (such as the center) of the body is at r0(t) and the direction of itsreference axis is θ (t) from x1 direction as shown in Fig. 1. Now let x∼i bethe coordinates of which the origin is located at r0(t) and the directionof x∼1 is inclined by θ(t) from the direction of x1, so the coordinatesmove with the body. As the body translates and rotates in x1-x2 planeso does x∼i. The position of field point P and the flow quantities at P suchas the velocity u and the pressure p can be described in either theinertial frame of reference xi or in the moving coordinates x∼i.

The equations of motion for the velocity components u∼i and thepressure p defined in x∼i are

⎛⎝⎜

⎞⎠⎟

DuDt ρ

px

τx

= − 1 ∂∂

+∂∂

∼∼ ∼

i

a i

ij

j (1)

where τij is the stress acting on a surface perpendicular to direction x∼iand (D/Dt)a denotes the material derivative in the absolute inertialframe. For the case where x∼i is translating with its origin at r0i (t) androtating with respect to xi, Du Dt( / )∼

i a is

⎛⎝⎜

⎞⎠⎟

DuDt

ut

u ux

d rdt

ε ω u ε ε ω ω x= ∂∂

+ ∂∂

+ + 2 +∼ ∼ ∼ ∼ ∼

∼∼i

a

ij

i

j

oiijk j k ijk jlm l m k

2

2 (2)

where ω1 =0, ω2 =0, ω3 = dθ/dt are the components of the angularvelocity of the rotation of the body and εijk is the permutation tensor(e.g. Piquet, 2001). The first and the second terms on the right-handside of this equation are the acceleration with respect to the movingcoordinates, the third term is the translational acceleration of themoving coordinates, the fourth term is the Coriolis acceleration and thelast term is the centrifugal acceleration, respectively. The stress τij

consists of the viscous stress and the sub-grid stress since we areformulating for the spatially filtered flow field. The expressions forthese stresses do not depend on the motion of the coordinates. If weuse the eddy viscosity model which is frame-independent,

⎛⎝⎜

⎞⎠⎟τ ν ν u

xux

= ( + ) ∂∂

+∂∂

∼ ∼∼ ∼ij sgs

i

j

j

i (3)

where νsgs is the sub-grid scale eddy viscosity coefficient and if theSmagorinsky model (Smagorinsky, 1963) is used, it is also independentof the motion of the coordinates as

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟ν C Δ u

xux

ux

ux

= ∂∂

+∂∂

∂∂

+∂∂

∼ ∼ ∼ ∼∼ ∼ ∼ ∼sgs S

i

j

j

i

i

j

j

i

21/2

(4)

where CS is the usual Smagorinsky constant and Δ is the grid spacing.The continuity equation for an incompressible flow in coordinates x∼i

is

ux

∂∂

= 0∼∼

i

i (5)

Next we derive the equations for the boundary conditions. To do it

Fig. 1. Relation between coordinates xi fixed in space and coordinates x∼i fixed on moving

body.

H. Noda, A. Nakayama Journal of Wind Engineering & Industrial Aerodynamics 166 (2017) 29–36

30

we write the relations between the variables defined in the non-inertialcoordinates and those defined in the inertial coordinates. First, thecoordinates xi and x∼i are related by

x r t Q t x x Q t x r t= ( ) + ( ) or = ( )( − ( ))∼ ∼i oi ij j i ji j oj (6)

where

⎡

⎣⎢⎢

⎤

⎦⎥⎥Q t

θ t θ tθ t θ t( ) =

cos ( ) − sin ( ) 0sin ( ) cos ( ) 0

0 0 1ij

(7)

is the transformation matrix for the coordinate rotation. The relationbetween the flow velocity components ui defined at xi and the velocitycomponents u∼i defined at x∼i is

u drdt

Q udQdt

x= + + .∼ ∼i

oiij j

ijj (8)

We analyze the flow around a body moving in a plane in a uniformflow as shown in Fig. 2, The computational domain is a rectangularregion, so the boundary conditions to be applied at the outer surfaces,x x=∼ ∼

in1 1 , x x=∼ ∼dn2 , x x=∼ ∼

up2 2 of the calculation region are that the flow isuniform with constant velocity Uin in x1 direction, i.e.

u U= in1 (9)

u u= = 0.2 3 (10)

At the downstream boundary x x=∼ ∼out1 1 where the flow exits the

calculation region, we apply the free-outflow condition

ux

i∂∂

= 0, = 1, 2, 3.i

1 (11)

These conditions are given in terms of the coordinates xi. Theexpressions in terms of coordinates x∼i fixed on the body can beobtained by substituting Eq. (8) into Eqs. (9)–(11). Then Eqs. (9)and (10) are transformed to

u U θ dθdt

x drdt

θ drdt

θ= cos + − cos − sin ,∼ ∼in

o o1 2

1 2(12)

u U θ dθdt

x drdt

θ drdt

θ= − sin − + sin − cos∼ ∼in

o o2 1

1 2(13)

u drdt

= − = 0∼ o3

3(14)

at x x=∼ ∼in1 1 , x x=∼ ∼

dn2 2 , x x=∼ ∼up2 2 .

For the downstream boundary, x x=∼ ∼out1 1 , Eqs. (8)–(11) are trans-

formed to

ux

ux

θ dθdt

θ∂∂

= ∂∂

tan − tan ,∼ ∼∼ ∼

1

1

1

2 (15)

ux

ux

θ dθdt

∂∂

= ∂∂

tan − ,∼ ∼∼ ∼

2

1

2

2 (16)

ux

∂∂

= 0∼∼

3

1 (17)

This formulation applies only when the angle of rotation is small sothat the flow exits the calculation region through the downstreamplane.

The boundary conditions on the solid surface of the body are eithernonslip when the Reynolds number is small and the grid spacing iscomparable to the viscous length, but when the near-wall resolution isnot sufficient, the wall modeling based on the wall law is applied to thevelocity at the point closest to the surface. In this case, if u∼wi are thevelocity components at points closest to a solid wall perpendicular to x∼j

direction, then we assume that the wall shear stress given by

τ C ρV u=ji d w wi (18)

acts on this surface in the direction opposite to x∼i directions. Here

V u u u= + +∼ ∼ ∼w w w w1

22

23

2 , and Cd is a model resistance coefficient and isevaluated by the standard wall law

⎡⎣⎢

⎤⎦⎥C A z u

νB z u

ν= ln + , > 10,d w

τw

τ1−2

1

(19)

⎡⎣⎢

⎤⎦⎥C z u

νz u

ν= ≤ 10,d

τ τ1−2

1

(20)

where Aw(=2.5), Bw(=5.2) are constants of the standard logarith-mic law for smooth surfaces, uτ is the friction velocity defined by thetotal wall stress τw of which components are given by Eq. (18) and z1 isthe distance to the solid surface. The condition on the tangentialvelocity at the surface is not applied but shear stress on the wall is givenby the above equation. The normal velocity component is set equal tothe normal component of the velocity of the solid surface.

The boundary conditions on the surfaces perpendicular to thecylinder axis is the periodic boundary condition since we are consider-ing the case of two dimensional mean flow. The equations for thisboundary condition on these surfaces are not influenced by the motionof the body considered here, which is the motion in the planeperpendicular to these surfaces.

3. Equations of motion of rigid body supported by spring anddashpot

We assume that the body in the flow is a rigid body and undergoes arigid-body motion when subjected to aerodynamic forces. It is alsoassumed to be supported by a spring and a dashpot so that forappropriate values of the spring constant and the damping coefficient,the motion represents that of a section of a long elastic cylinder such asa bridge deck supported at both ends. The translational motion of thebody perpendicular to the free stream direction corresponds to heavingand the rotational motion corresponds to torsional deformation in asection near the center of a long span cylinder. If m is the mass of thebody per unit length, c1, c2 and k1, k2 are the damping coefficients andthe spring constant in the x1 and x2 directions, respectively, theequations for X1, and X2, the x1 and x2 positions of the body are

mX t c X t k X t f t ( ) + ( ) + ( ) = ( )1 1 1 1 1 1 (21)

mX t c X t k X t f t ( ) + ( ) + ( ) = ( )2 2 2 2 2 2 (22)

where f1 (t) and f2 (t) are the resultant fluid forces acting on the allsurfaces of unit length of the body in the x1 and x2 directions,respectively. It is noted that we assume that c1, c2, k1 and k2 areconstant independent of the positions or the orientation of the body.

Fig. 2. Boundary conditions for flow past a two-dimensional body moving in a plane.

H. Noda, A. Nakayama Journal of Wind Engineering & Industrial Aerodynamics 166 (2017) 29–36

31

The equation for the angle θ (t) of the orientation of the body is then

Iθ t c θ t k θ t f t ( ) + ( ) + ( ) = ( )θ θ θ (23)

where I is the moment of inertia of the section of the body and cθ andkθ are the torsional damping and spring coefficients, and fθ is themoment of the fluid forces on the body surfaces around the center ofthe body.

4. Numerical methods

The numerical solution method that we use here is essentially thesame as that we used for flow past fixed body (Nakayama and Noda,2000). It is constructed on a rectangular grid with variable spacing andthe variables are defined at the staggered positions. The pressurecoupling method is based on the HSMAC (Highly-Simplified Markerand Cell) iteration (Hirt and Cook, 1972) where the pressure and thevelocity components are iteratively corrected until the continuityequation is satisfied. The spatial derivatives are evaluated by thesecond-order conservative central difference, and the time advancingis done by the third-order Adams-Bashforth method. The extra inertiaterms are similarly evaluated explicitly. The additional terms in theboundary conditions at the downstream section are also evaluatedexplicitly. The values next to the body surfaces are determined by theequations for the boundary conditions as explained above. For the caseof translational motion of the body, its accuracy has been evaluatedearlier by Nakayama et al. (1998). The treatment of the effects ofrotation is examined in the following validation calculations.

The equations of the rigid body motion Eqs. (21)–(23) are second-order ordinary differential equations and integrated by the Newmark-βmethod at every time step of the overall calculation.

5. Calculation results

5.1. Verification in case of fixed rectangular cylinder

As a verification of the present formulation and the coding,calculation was first conducted for the case of flow past a fixedrectangular cylinder without motion. In this case the coordinates donot move and it is meant to verify the basic performance of the LESmethod. One of the experimental conditions investigated by Okajimaand Mizota, 1988, who obtained detailed surface pressure distribu-tions, is considered. The depth to breadth ratio of the rectangularsection is D/B=2.0 and the Reynolds number based on the free streamvelocity and B of 22000. The size of the computational grid is 187 in thestreamwise direction, 123 in the transverse direction and 21 in thespanwise direction. The variable grid spacing is used to make thespacing small near the cylinder. The grid spacing normal to the cylindersurface is 0.05B but since the flow is separated at the upstream cornerand flow on the side surfaces are slow and the viscous sublayer is thick,in the present case of D/B=2. The time averages are taken over theduration of 500B/Uin after the flow development period of 100 B/Uin.

Fig. 3 shows the distributions of the calculated mean pressure andthe RMS of the fluctuating pressure in terms of the pressure coefficientsCP and CP’

Cp p

ρU=

−1/2

,pin

in2 (24)

Cp p

ρU′ =

( − )1/2p

in

2

2 (25)

on all surfaces at the center span. Here the overbar indicates thetime average and pin is the pressure in the on-coming uniform flow. Itis seen that the present LES results are in reasonable agreement withthe experiment except that the pressure on the rear surface is lower andthe RSM pressure fluctuation is a little larger than the experiment. A

similar LES calculation with a dynamic Smagorinsky model with alarger spanwise calculation domain done by Kuroda et al. (2007) showsessentially the same mean pressure results as the present one for thesimilar Reynolds number of 20,000. Therefore the present LES modelis sufficient to be extended to calculation with moving body.

5.2. Forced vibration of rectangular cylinder

Next, in order to verify that the present treatment of the motion ofthe body is correctly implemented, a calculation of flow past arectangular cylinder that oscillates in a known manner has beenperformed. In this case the motion of the rectangular cylinder is notcalculated but set to the known position and the velocity. It is one stepcloser to the case of full flow-structure interaction. The cases of forcedvibration experiments conducted by Washizu et al. (1978) and Washizuet al. (1980) are taken as the verification case. In these experiments arectangular cylinder again of D/B =2 is forced to oscillate in a uniformflow so that the lateral position of the cylinder X2(t) and the angle of itsorientation θ(t), the angle between the direction of the long surface ofthe cylinder and the flow direction, to be

X t A ω t( ) = sin ,h m2 0 (26)

θ t A ω t( ) = sinθ m0 (27)

where Ah0 and Aθ0 are the amplitudes of the lateral displacement andthe rotation angle, respectively and ωm is the angular frequency 2πnm.

Sample results of calculated instantaneous velocity fields are shownin Figs. 4 and 5. These are the results of calculation done for rotationangles larger than the experiment with the amplitude of the oscillationof the rotation angle Aθ0 of 10° and no translational displacement Aη0=0 to examine the effects of the rotation of the coordinates. The valueof the non-dimensional velocity Vr=Uin/nmB is 5, so that the max-imum angular velocity is 12.6°/s.

Fig. 4(a) shows the distribution of the computed velocity vectorsu u( , )∼ ∼

1 2 in the center plane x x= = 0∼3 3 of the non-inertial moving

coordinates x x x( , , )∼ ∼ ∼1 2 3 at the instant when the rotation angle is near

maximum of 10° and the angular velocity is nearly zero. Fig. 4(b) showsthe distribution of velocity (u1,u2) at (x1,x2,0) at the same instant,converted from u u( , )∼ ∼

1 2 using Eqs. (6) and (8). Dr0i/dt and dQij/dt inEq. (8) are zero and the transformation is that of rotation by thetransformation Qij. The sub figures on the right hand side show theenlarged velocity distributions in the vicinity of the cylinder. It is seenin the plot of u u( , )∼ ∼

1 2 (Fig. 4(a)) that the velocity vectors near thecylinder surface are consistent with the wall boundary condition. Thevectors (u1,u2) plotted in the inertial coordinates (Fig. 4(b)) showcorrect approach to the uniform flow near the outer boundaries. Thevelocity near the downstream surface is seen to satisfy the free outflowcondition.

Fig. 5 is the similar plots for the time when the rotational angle isnear zero and the rotational velocity is near maximum of 12.6°/s. At

Fig. 3. Calculated pressure distribution on cylinder surfaces compared with experiment.○, Cp, □, C′p Experiment, Okajima and Mizota (1988); •, Cp, ■, C′p Present LES.

H. Noda, A. Nakayama Journal of Wind Engineering & Industrial Aerodynamics 166 (2017) 29–36

32

this instant, Qij is nearly zero and dQij/dt is maximum. Again it is seenthat u u( , )∼ ∼

1 2 satisfy the body surface conditions and (u1,u2) satisfy theouter boundary condition. These results indicate that the computationin the non-inertial coordinates is performed properly.

Next, calculations that are done with the same condition as theexperiment of Washizu et al. (1978) are shown, and the results arecompared with the same experiment. Fig. 6 shows the calculatedaerodynamic force coefficients for various values of the non-dimen-sional flow velocity Vr in terms of the displacement-proportionalcomponent CLR and the velocity-proportional component CLI of theaerodynamic force f2(t) defined by

∫CρU BL T

f t ω tdt= 11/2

2 ( )sin ,LRin

Tm2 0

2 (28)

and

∫CρU BL T

f t ω tdt= 11/2

2 ( )cosLIin

Tm2 0

2(29)

and the phase difference βL

β C C= tan ( / )L IL LR−1 (30)

The values of Ah0 and Aθ0 used in the calculation are 0.1B and1.91°, respectively in accordance with the experiment. The calculationresults are in good agreement with the experiment except for the lowCLR at Vr=2.0. The positive values of the velocity-proportionalcomponent CLI, which is an indication of the instability, is seen nearVr=2.75 and above Vr=8.0 in agreement with the experiment. Theycorrespond to the vortex-induced vibration and the galloping, respec-

tively. As to the discrepancy between the experimental CLR and thecalculation results near Vr=2.0, experimental data by Katagiri et al.(2001) show large positive value, and the present calculation may notaltogether be incorrect.

Fig. 7 shows the similar results for the case of forced rotationaloscillation. The coefficients shown in these figures are defined by

∫CρU B L T

f t ω tdt= 11/2

2 ( )sin ,MRin

T

θ m2 2 0 (31)

∫CρU B L T

f t ω tdt= 11/2

2 ( )cos ,MIin

T

θ m2 2 0 (32)

and

β C C= tan ( / ).M ML MR−1 (33)

Similar trend seen in the forced translation case is seen here also.Other than the value of CMR near Vr=2, all parameters CMR, CMI andβM are in good agreement with the experiments. Particularly, the valueof the velocity-proportional component CMI is seen to take positivevalues near Vr=3.0 where vortex induced instability occurs, and nearVr=7.0 to 8.0 as the experimental results.

The comparisons of the present calculation with experiment shownabove indicate that the present method does calculate the aerodynamicforces on a rectangular body oscillating sinusoidally in the lateraldirection and rotating around the axis in the spanwise direction verywell. In the present method, the boundary conditions on the surfacesfar away from the body can involve large velocities due to the rotationof the coordinates. The conditions given by Eqs. (12)–(17) are shownto represent these boundary conditions properly.

Fig. 4. Calculated velocity vectors in non-inertial coordinates and converted velocity vectors in the inertial coordinates at time when the inclination angle θ(t) is near maximum of 10°.

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Fig. 5. Calculated velocity vectors in non-inertial coordinates and converted velocity vectors in the inertial coordinates at time when the rotation velocity θ t ( ) is near maximum of 12.6°/

s.

Fig. 6. Unsteady aerodynamic force coefficients of rectangular cylinder in forced lateral vibration. ○, Experiment, Washizu et al. (1978); •. Present LES.

Fig. 7. Unsteady aerodynamic force coefficients of rectangular cylinder in forced rotational vibration. ○, Experiment, Washizu et al. (1980); •, Present LES.

H. Noda, A. Nakayama Journal of Wind Engineering & Industrial Aerodynamics 166 (2017) 29–36

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5.3. Free vibration of rectangular cylinder

Next, we show the results of the free vibration calculation where thecylinder moves due to the aerodynamic forces with a support of aspring and a dashpot. It is the simulation of the motion of a section oflong cylindrical structure undergoing heaving and torsion. The calcula-tion is done in the three dimension with a finite span as in the fixed andthe forced vibration cases. Disturbances to the initiate motion is givenby forcing motion of small amplitude of 0.1–0.2B or 0.01–0.09 rad fora short duration of t=0 to 10B/Uin. The calculation conditions for thetwo cases are summarized in Table 1.

Table 1calculation conditions.

translation rotation

Sc 3.0 5.5Damping coefficient h 0.000637 0.00325Ratio of density ρb of cylinder and that of air ρ 375.0I – 539 kg m2/mInitial forcing 0.1D − 0.2D 0.01 – 0.09 rad

Fig. 8. Amplitude of translational oscillation. ○, Experiment, Miyazaki (1982); △,Takeda and Kato (1992); •, present LES.

(a) Vr=2.0

(b) Vr=2.75

(c) Vr=10.0

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0 200 400 600 800 1000

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0 200 400 600 800 1000

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0 200 400 600 800 1000

tUin/B

tUin/B

tUin/B

X2/B

X2/B

X2/B

Fig. 9. Time series records of lateral position X2 in heaving mode free oscillation.

Fig. 10. Amplitude of torsional rotational oscillation. ○, Experiment, Shimada andIshihara (2012); •, present LES.

H. Noda, A. Nakayama Journal of Wind Engineering & Industrial Aerodynamics 166 (2017) 29–36

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In the first case the cylinder translates in the lateral direction, andthe Scruton number defined by Sc=2mδ/ρBD, (where δ=2πh and h isthe damping coefficient) is set at 3.0 same as the experiments.

The results of the calculation of the motion of the cylinder are givenin terms of the amplitudes of the oscillation for various values of thenon-dimensional velocity Vr in Fig. 8. The amplitude Ah of oscillationX2 is obtained by the RMS fluctuation of the position multiplied by 2 .The experimental results obtained by Miyazaki (1982) and Takeda andKato (1992) are also shown in the same figure. The data show that theamplitude of oscillation takes a local peak of about 0.16 near Vr=2.7corresponding to the vortex induced vibration and drops to nearly zeroamplitude at about Vr=3.2 and monotonically increases for larger Vrleading to galloping. These characteristics are well reproduced by thepresent calculation, except the amplitude in the range of Vr=6 to 10 issomewhat larger than the experiment.

Fig. 9 shows samples of time variations of the lateral position X2 ofthe cylinder for the non-dimensional velocity of Vr=2.0, 2.75 and 6.0.It is seen that the oscillation amplitude settles at about 0.05B at Vr=2.0but is as large as 0.15 at Vr=2.75. It corresponds to the vortex-inducedvibration. At Vr=10.0, the oscillation amplitude is large and unstablewhich occurs in the range of Vr larger than about 6 and it correspondsto the galloping.

Calculation corresponding to the case of free torsional oscillationwas performed for the case of the Scruton number of Sc=2Iδ/ρB2D2=5.5 to compare with the experiment of Shimada and Ishihara(2012). The results are shown in Fig. 10. The amplitude At of theangular oscillation is obtained by multiplying 2 to the RMS of theangular fluctuations. The vortex induced instability near Vr=3.0 andthe increasing amplitude for larger Vr are seen and agree well with theexperiment.

The computational time for these free vibration cases is about oneand a half times the fixed cylinder case done in Section 5.1, whichindicates that the fluid-structure interaction computation is only about50% of the flow computation without the FSI interaction. Thecomputational time of the extra terms is almost negligible and it ismainly the increased pressure iteration time due to these added terms.So, unless the body motion becomes very large and the significance ofthe extra inertia terms increases, the computational time will notincrease significantly. This is the great advantage compared withmethods that compute the motion and move the computational gridpoints, though it is limited to the case of rigid body motion.

6. Conclusion

A numerical method of calculating the flow past a cylindrical bodyoscillating due to the unsteady aerodynamic forces has been con-structed based on the finite difference method formulated in thecoordinates fixed on the body moving with the three degrees offreedom. In this method, extra inertia terms due to the motion of thecoordinates need to be included in the equations of motion and theboundary conditions on the flow outer boundary must be adjusted asthe body moves. However, the method does not require reconstructingthe computational mesh or interpolating points near the surface of themoving body and is very efficient. It is also advantageous to formulateimplicit boundary conditions needed in the wall turbulence modeling ofthe LES or in Reynolds-averaged Navier Stokes equation methods.Comparison with available experimental data for the case of thebreadth to the depth ratio of D/B=2 indicates the characteristics ofthe translational oscillation corresponding to the deflection of longbeams and the rotational oscillation corresponding to the torsion arereproduced well. LES methods for calculating aeroelastic stability ofreal structures need to be efficient and accurate to be used as a practicaldesign tool and the present method is an alternative method when thestructure moves as a rigid body.

Acknowledgement

The present work was conducted with a partial support of JapanSociety of Promotion of Science Grant in Aid (c)25420599.

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