Articulo de Cfd

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SCIENCE CHINA

Physics, Mechanics & Astronomy

Science China Press and Springer-Verlag Berlin Heidelberg 2013 phys.scichina.com www.springerlink.com

*Corresponding author (email: [email protected])

Article February 2013 Vol.56 No.2: 277289

Special Topic: Fluid Mechanics doi: 10.1007/s11433-012-4982-4

Numerical aerodynamic analysis of bluff bodies at a high Reynolds

number with three-dimensional CFD modeling

BAI YuGuang1*, YANG Kai1, SUN DongKe1, ZHANG YuGuang1, KENNEDY David2,

WILLIAMS Fred2& GAO XiaoWei1

1 State Key Laboratory of Structural Analysis for Industrial Equipment, Faculty of Vehicle Engineering and Mechanics,

Dalian University of Technology, Dalian 116023, China;2

Cardiff School of Engineering, Cardiff University, Cardiff CF24 3AA, Wales, UK

Received July 2, 2012; accepted September 5, 2012; published online January 21, 2013

This paper focuses on numerical simulations of bluff body aerodynamics with three-dimensional CFD (computational fluid

dynamics) modeling, where a computational scheme for fluid-structure interactions is implemented. The choice of an appro-

priate turbulence model for the computational modeling of bluff body aerodynamics using both two-dimensional and

three-dimensional CFD numerical simulations is also considered. An efficient mesh control method which employs the mesh

deformation technique is proposed to achieve better simulation results. Several long-span deck sections are chosen as examples

which were stationary and pitching at a high Reynolds number. With the proposed CFD method and turbulence models, the

force coefficients and flutter derivatives thus obtained are compared with the experimental measurement results and computed

values completely from commercial software. Finally, a discussion on the effects of oscillation amplitude on the flutter insta-

bility of a bluff body is carried out with extended numerical simulations. These numerical analysis results demonstrate that the

proposed three-dimensional CFD method, with proper turbulence modeling, has good accuracy and significant benefits for

aerodynamic analysis and computational FSI studies of bluff bodies.

bluff body, aerodynamic analysis, fluid-structure interaction, three-dimensional CFD modeling, flutter

PACS number(s): 02.70.-c, 47.11.Df, 47.27.E-, 47.27.nb

Citation: Bai Y G, Yang K, Sun D K, et al. Numerical aerodynamic analysis of bluff bodies at a high Reynolds number with three-dimensional CFD modeling.

Sci China-Phys Mech Astron, 2013, 56: 277289, doi: 10.1007/s11433-012-4982-4

In wind engineering, investigations of flows around bluff

bodies have attracted wide attention. This challenging aero-

elasticity problem has been studied extensively through

wind tunnel tests or numerical simulations [15]. In the last

decade progress has been made to find a computational al-

ternative to partly replacing physical wind tunnel tests

which may be influenced by unpredictable factors [6] (e.g.,

incoming flow properties, model geometrical fidelity and

measurement complexity). CFD has been accepted as a po-

tentially powerful method for investigating various wind-

induced vibrations of bluff bodies.

In the case of wind-induced vibrations of bluff bodies,

such as bridge decks, the combination of wind turbulenceexcitation and aeroelastic effects can lead to new phenome-

na which are not always fully understood [4]. Especially for

high turbulent wind occurring close to the ground, the wind

gusts act more as sudden transient excitations than as sta-

tionary excitation. Computational fluid-structure interaction

(FSI) research in this field has been recognized as an effi-

cient tool. It has been the subject of many recent engineer-

ing applications.

Tamura [7], Tamura & Itoh [8] and Tamura & Ono [9]

presented the state-of-the-art for computational FSI research


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278 Bai Y G, et al. Sci China-Phys Mech Astron February (2013) Vol. 56 No. 2

in the field of wind engineering. In previous work, similar

numerical simulations were already conducted with an

NACA airfoil [10,11]. Bai et al. [11] proposed a CFD

method with a computational scheme for FSI and provided

an effective assessment of aerodynamic and aeroelastic

performance of airfoil at high Reynolds numbers. However,

the dynamics and the aeroelasticity of a bluff body are quite

different from that of an airfoil. For example, most bridge

deck sections, except very streamlined ones, behave like

bluff bodies and the airflow is essentially separated down-

stream. Scanlan and Tomko [12] showed conclusively that,

though helpful, the Unsteady Airfoil Theory [13] has very

distinct limitations in cases of bluff bodies (e.g., aerody-

namic flutter derivatives calculated even for streamlined

bridge deck sections can show limited resemblance with

those of a symmetric airfoil). A very successful two-dimen-

sional CFD method for bluff bodies is the discrete vortex

method (DVM) as implemented by researchers such as

Walther & Larsen [14] and Taylor & Vezza [15]. These FSIstudies employed the classical grid-free method to investi-

gate the flows with moving boundaries and it is easy to pro-

gram. This has significant benefits in terms of efficiency but

has been criticized as a viscous flow model because it can-

not be readily extended to three-dimensional flows. Re-

cently, CFD software, e.g., Fluent and CFX, have become

widely used, but sometimes the precision is unsatisfactory

due to the difficulty of solving FSI problems in simulations

of flows around blunt bodies [16,17]. The block-iterative

coupling approach has been successfully applied to couple

simple fluid and solid mechanics behaviors [18]. Bai et al.

[11] have extended this method to study two-dimensionalFSI problems successfully with turbulence modeling,

boundary layer treatment and other features needed in air-

foil aeroelasticity. This approach combines the dynamic

structure analysis program system DDJ-W [19,20] devel-

oped at the Dalian University of Technology with a solver

of the commercially available CFD code CFX. Whether this

approach is efficient for bluff bodies will be investigated in

this paper.

Flows around long-span bridge decks are highly turbu-

lent, unsteady and three dimensional. The type of turbulence

model used is important for the computational modeling of

the bridge deck FSI. Bai et al. [11] used the k-turbulencemodel to compute the flutter derivatives of a two- dimen-

sional airfoil and investigated the effects of turbulence on

them. In addition to large eddy simulation (LES) or Reyn-

olds averaged Navier-Stokes (RANS) simulation, Detached-

eddy Simulation (DES) is a novel turbulence model [2123]

which is promising due to its ability to explicitly resolve

turbulent structures for massively separated high Reynolds

number flows around bluff bodies. Its behavior is similar to

LES but computationally cheaper, being closer to unsteady

RANS in terms of required CPU time. This paper is con-

cerned with the choice of an appropriate turbulence model

for the computational modeling of bluff body aerodynamics.

Three-dimensional wind flow past three bridge deck sec-

tions is investigated and simulations yield the aerodynamic

force coefficients and flutter derivatives obtained from sim-

ulating the motion-induced aerodynamic forces when the

deck cross sections oscillate within an incompressible flow

with a high Reynolds number. In addition, the effects of

oscillation amplitude on the flutter instability of them are

studied.

1 Turbulence modeling

Most engineering flows being studied are turbulent, thus we

have to consider how to represent or model the effects of

turbulence in simulating these flows. One issue for re-

searchers is to make appropriate choice of models for par-

ticular flows [24]. Though the Navier-Stokes equations can

describe a turbulent flow including all the turbulent eddy

details, the computational cost of direct numerical simula-

tion (DNS) is huge. There are mainly other three kinds of

turbulence models: RANS (Reynolds averaged Navier-

Stokes Equations), LES, and DES (= hybrid RANS/LES)

[21,25].

1.1 RANS models

In most engineering situations, it is the average velocity,

pressure, etc. that are of interest, and the details of all the

turbulent eddies are not required. RANS turbulence model

may strike the balance between computational efficiency

and accuracy in simulating the flow regime [25].

1.1.1 The standard k-model

The transport equations for the turbulent kinetic energy k

and its dissipation rate are given by the following equa-

tions in which the five constants needed are given the values

of 1.0,k

1.3, 0.09,C 1 1.44,C and 2C

1.92 [26].

( ) ( ) ,ti k k

i i k i

kk ku G Y

t x x x

(1)

( ) ( ) ti

i i i

u G Yt x x x

(2)

with the density of the fluid taken as a constant, the

previously undefined terms are now defined.

The production of turbulent kinetic energyk

G is com-

puted consistently with the Boussinesq hypothesis from

2

t,

kG S (3)

where S is the modulus of the mean rate-of-strain tensor,

defined as:


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Bai Y G, et al. Sci China-Phys Mech Astron February (2013) Vol. 56 No. 2 279

2ij ij

S S S

with

1.

2

ji

ij

j i

uuS

x x

(4)

The turbulent viscosityt

is computed from

2

t.

kC

(5)

The dissipation term in eq. (1) is ,k

Y while the

production and dissipation terms in eq. (2) are given by

1 kG C G

k

,

2

2.

kY C

k

(6)

1.1.2 The k-shear stress transport (SST) model

The constants needed below are given the values of

0.09, and 1 [26]. The transport equation for khasthe same form as in the standard k-model. For the specific

dissipation rate , this is

t

( ) ( )

.

i

i

i i

ut x

G Y Dx x

(7)

The production and dissipation terms of turbulent kineticenergy are

kG = min 2

t( , 10 )S k

, 2 .k

Y (8)

The specific dissipation is related to the dissipation

by ( / )k ; the production and dissipation equation

terms of are

t

.k

G G

(9)

The cross term in the dissipation terms of is

1 ,2

12(1 ) ,

i i

kD F

x x

(10)

where,2

is a constant and4

1 1tanh( )F is a blending

function with

1 2 2

,2

500 4min max , , ,

0.09

k k

y y D y

(11)

where,2

max[2 (1 / )(1 / )( / )( / ),j j

D k x x

1010]

and,2

1.168

.

Finally, the turbulent viscosityt

is computed as:

t

2 1

1,

max[(1 / ),( / )]

k

SF a

(12)

where1

0.31a and 22 2tanh( )F is a blending func-

tion with

2 2

2 500max , .

0.09

k

y y

(13)

The model factors , ,k

are interpolated as:

1 1 1 2

1 ,1 1 ,2

1 ,1 1 ,2

(1 ) ,

1,

/ (1 ) /

1

/ (1 ) /

k

k k

F F

F F

F F

(14)

with1

0.075, 2 0.0828, ,1 1.176,k ,2k 1.0,

,12.0,

and ,2 1.0.

1.2 LES models

The filtered Navier-Stokes equations for a constant-density

fluid are

0,i

i

u

x

(15)

,i i j ij ij

j j i j

u u u p

t x x x x

(16)

where filtered quantities are denoted by an overbar.ij

is

the filtered molecular viscosity stress tensor andij

is the

sub-grid scale stress tensor resulting from the filtering oper-

ation:ij i j i j

u u u u . Eqs. (15) and (16) are only cor-

rect for constant filter width. This means that for spatially

varying filter width, the spatial commutation errors that

generate supplementary terms in the equations are neglected

[27], as is commonly done in practically-oriented work. The

sub-grid scale stress tensor is modeled by

t

12 ,

3ij kk ij ij

S (17)

wheret is the sub-grid eddy viscosity and ijS is the

rate-of-strain tensor calculated with the filtered velocity

components. The sub-grid eddy viscosity has to be further

modeled. The isotropic part of the stresses is not modeled

but added to the pressure term.

More extensive details of LES models are not given in


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this paper because its calculations are too costly.

1.3 DES (hybrid RANS/LES) models

1.3.1 The DES k-model

The transport equations for k and

of the realizable k-

model are used to model the eddy viscosity in the RANS

zones and to model the sub-grid viscosity in the LES zones.

In the hybrid formulation, the dissipation term in eq. (1)

is computed from

3/ 2 / ,k des

Y k (18)

where min( , ),des rke les 3/ 2 / ,

rkek les desC and

max( , , ),x y z the maximum grid spacing. The

standard value ofdes

C is 0.65.

1.3.2 The DES k-SST model

This model is based on the k-SST model. The dissipation

term of the turbulent kinetic energy is modified into eq. (18),

where min( , )des k les

, 1/ 2/k

k

andles

,des

C as in the previous model.

The k-SST model, which belongs to RANS models, is

employed for two-dimensional CFD modeling in this paper,

because it allows direct integration through the boundary

layer. Also, it has been successfully applied to numerical

simulations of two-dimensional airfoil aerodynamics [11].

The DES k- SST model is employed in this paper for

three-dimensional CFD simulations because it has better

accuracy than RANS and much lower cost of computationthan LES [21].

2 Numerical algorithm for FSI

The computation of nonlinear FSI problems requires the

simultaneous solution of the strongly coupled fluid and

structural equations of motion. In partition methods [18],

the coupled problem is computed with a solution procedure

where the fluid and structure are separated and exchange

data in every time step or iteration of the coupling algorithm.

The governing equations for the FSI come from both thecorresponding flow and structural analyses [11]. All these

equations can be treated as time and space dependent and

need to be discretized before solution methods are applied.

The discretized incremental Navier-Stokes and structural

equations can be expressed as:

N(a,b) = 0, S(b) =f(a), (19)

where a and b are the field vectors consisting of the un-

knowns at the time step n+1 currently being solved for. Let

us denote the discretized vector of velocities in the fluid by

u, the corresponding pressures by p, the discretized dis-

placement vector in the structure by and the discretized

vector of structure velocities by ; a = u 1n , p 1n

contains the variables from the fluid domain; b =

1n , 1n contains the variables from the structural

domain. The field variables at the previous time step nare

assumed to be known, and are not reflected in eq. (19). The

two equation sets are fully coupled.Eq. (19) can be solved by the block-iterative method [18].

By using any available solvers for each of the two parts of

eq. (19) in conjunction with the block-Gauss-Seidel iterative

algorithm, the Navier-Stokes equations are solved first fora

and then forb. The iteration scheme can be written as:

( 1) ( ), 0,i i N a b (20)

( 1) ( 1) ,i i S b f a (21)

where iis the iteration counter, and it converges linearly.

For generality, eqs. (20) and (21) are both treated as non-linear. Therefore linearization methods like the New-

ton-Raphson method or the Picard (fixed point) method

must be used. For the latter, the linearization iteration is

( 1) ( 1) ( ) ( 1) ( 1) ( 1)( 1) ( ) ( 1) ( ) ( ), , , ,i i i i i ij j j j j a f a b b G a b (22)

wherejis the linearization iteration counter. The two layers

of iteration iandj in eq. (22) address both the field coupling

and the non-linearity. With global convergence checked at

every time step, the solution obtained should be identical to

that given by the direct coupled solution to eq. (19). Note

that the two iterations can be mixed, and when merged, the

equivalent iteration is

( 1) ( ) ( ) ( 1) ( 1) ( ), , , ,k k k k k k a a F a a b G b (23)

wherekis the merged iteration counter. Solvers in the form

of eq. (23) can be found in existing CFD and structural

analysis codes.

This paper uses a self-owned three-dimensional dynamic

structure analysis code DDJ-W, which is combined with a

solver of the commercially available finite volume code

CFX to conduct FSI problems. The accuracy of this way

will be compared to that of the popular commercial soft-

ware system ANSYS workbench in this paper.

3 Structure examples and numerical modeling

3.1 Structure examples

Three long-span bridge deck cross sections are taken as

examples in the present paper. They are named as sections

G1G3. Their geometrical features are shown in Figure 1,

in which B is the chord length. Every model has two de-

grees of freedom, namely vertical translation hand rotation

about its centre.

Section G1 can be treated as a streamlined structure, but


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Figure 1 The three long-span bridge deck sections G1G3 used. They all

have a vertical axis of symmetry.

G2 and G3 are typical bluff bodies with sharp edges. Partic-

ularly, the section G3, which has infamous aerodynamic

instability because it is the prototype of the Tacoma Nar-

rows Bridge in the USA, was destroyed in 1954 by a steady

wind with the small velocity of 20 m/s. So it is necessary toinvestigate these structures.

3.2 Three-dimensional CFD modeling

3.2.1 Mesh control method

Most failed simulations are caused by mesh failure, so an

efficient and robust mesh algorithm is crucial to FSI studies.

In addition to the question of coupling the fluid and struc-

tural analyses, mesh movement is an important issue in FSI

studies. Rewriting the fluid equations in the arbitrary La-

grangian Eulerian (ALE) formulation allows one to move

the mesh arbitrarily. In fact, this paper uses a mesh controlmethod which employs the mesh deformation technique

[11].

The mesh algorithm can be conveniently described

through a circular cylinder, see Figure 2, which only has the

three degrees of freedom of heave, shove, and pitch, respec-

tively. The structural coordinate system differs from that of

the fluid domain.0

R is the radius of the cylinder and mesh

deformation is performed only in the cylindrical region with

2R R . This region is further divided into a rigid region

with1

R R and a buffer region with1 2

R R R . Fluid

grids falling in the rigid region are assumed to translate and

Figure 2 Rigid-plane-based mesh deformation for a cylinder.

rotate in the rigid plane which contains the shear centre re-

sponse of the cylinder. Grids falling in the buffer region are

updated with mesh movements that are interpolated from

those at1

R R and at2

R R . Bai et al. [11] improved

this algorithm through adding a wake zone in the buffer

region (1 2

R R R ) and got better computed results for

airfoils than those without wake zone, as shown in Figure 3.

The algorithm is summarized as:

0 0 1

2

0 0 1 2

2 1

0 2

, ,

, ,

, ,

S S S S

S S S S

S S

R R

R RR R R

R R

R R

x x T x

x x T x

x x

(24)

,S S

T x (25)

where

2 20 0sqrt ,S SR x y

cos sin 0

sin cos 0 .

0 0 1

z z

S z z

T

The left-hand column in Figure 4 shows the inner region

of the two-dimensional mesh for the three sections respec-

tively. It is cylindrical, is centered on the deck section, and

has radius2

16R B . The G1 meshes used for the rigid,

wake and remaining buffer region are, respectively, (594

88), (64160) and (10439), where the first and second

numbers are the number of hexahedral cells in, respectively,the tangential and radial directions. Hence there are 66568

Figure 3 Mesh geometry.


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cells, of which 44368 are in the rigid region and 10240 are

in the wake. Two-dimensional meshes for the other two

deck sections were similar to that for G1, and the cell num-

bers of the two-dimensional models for the other sections

are, respectively 65100 and 72728 (see the left-hand column

in Figure 4).

The main difference between two-dimensional and

three-dimensional CFD modeling is the model thickness

(i.e., perpendicular to the sections shown in Figure 1).

Meshes along the thickness direction of G1 are shown in

Figure 5. This influences the number of cells in the mesh

significantly. Many CFD methods use a very small thick-

ness or a two-dimensional model because of the limitations

of computation capacity and uncertainty about accuracy.

For example, increasing the thickness of 0.1 m of the

two-dimensional model of section G3 to 1 m for the

three-dimensional model leads to a huge number of cells

(4089792), which require parallel computing and much

CPU time. The right-hand column of Figure 4 shows three-dimensional meshes for the three sections when they all

have thickness 1 m. The numbers of cells for the three sec-

tions are respectively: 3695552, 3601600, and 4089792.

Figure 4 Enlargements of the inner regions for the two-dimensional

meshes (left-hand column) and the three-dimensional meshes (right-hand

column) used, with the densely populated circle being the rigid region.

Figure 5 Meshes along the thickness direction.

3.2.2 Boundary layer treatment

The viscous boundary layer over the structure surface is

well resolved by a fine mesh with the overall y less than

2, which eliminates the need for a boundary layer treatment

and the sub-viscous layer is resolved by the meshes [11].

The y

values of the three sections for the present CFD

simulations are given in the Appendix.

3.2.3 Parallel processing introduction

Parallel computing with 32 processors was used for the

three-dimensional numerical simulations and the fluid do-

main was divided into 32 blocks, with the computations for

each block delegated to one of the 32 processors and with

one processor used as the master and the rest as the slaves.

During the iteration, the master and the slave nodes per-

formed the mesh deformation for their own blocks using the

calculated structural responses. Solutions for both the

structural and the fluid analyses according to eq. (23) weresynchronized at each iteration loop.

4 Results and discussion

4.1 Flows around fixed sections

This analysis is to simulate the flow and aerodynamic forces

developed on fixed deck sections. Especially for G2, dif-

ferent angles of attack, 10, 8, 4, 0, 4, 8, 10, arechosen. The computed aerodynamic forces of drag D, liftL

and momentM

are expressed in the conventionalnon-dimensional forms [28].

d2

,1

2

DC

U Bl

l2

,1

2

LC

U Bl

m2 2

,1

2

MC

U B l

(26)

where is the fluid density, taken as 1.185 kg/m3 ; U is

the flow velocity; the stream flows from the front to the rear

of the sections for all of the simulations; for consistency

with the wind tunnel tests [28], the examples have a high

Reynolds number of 105, with 1.545U m/s and 1B m;

d,C lC and mC are the drag, lift, and moment coef-

ficients, respectively; and l is the thickness of the deck sec-

tions (the thickness is 0.1 m for the two-dimensional models

and 1 m for the three-dimensional models). The time incre-

ment used was T =0.002 s (i.e., approximately a fiftiethof the time unit 0.14B/U).

At each time step the section surface pressure distribu-

tion was computed and integrated along the contour to form

the time traces of drag, lift, and moment. For example, Fig-

ure 6 shows partial simulated time traces for the aerody-

namic force coefficients obtained from two-dimensional and

three-dimensional CFD simulations of section G1 with an-

gle of attack 0. Figure 7 shows the computed aerodynamic


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Figure 6 Time traces showing evolution with the time step of the aero-

dynamic force coefficients for the fixed deck section G1 at 0 angle of

attack: (a) two-dimensional CFD simulation; (b) three-dimensional CFD

simulation.

force coefficients d ,C lC and mC of section G2 for dif-

ferent angles of attack using two-dimensional () and

three-dimensional () CFD modeling. Clearly there aremany differences between the two-dimensional and

three-dimensional values of section G2. Table 1 shows the

comparison of drag coefficient values among the present

CFD method, wind tunnel test and DVM method, and it is

obviously that the present three-dimensional CFD method

has better accuracy than two-dimensional ones.

Though most wind tunnel test results could not be ob-

tained, it is reasonable to anticipate that the values using

three-dimensional CFD would be more accurate than those

using two-dimensional ones and that three-dimensionalCFD simulations have important practical significance for

blunt bodies.

The features of the flow field at different parts can be

shown visually, which is another advantage of three-dimen-

sional CFD simulations. For example, the three-dimensional

wake flows for the three deck cross sections are shown pic-

torially in Figure 8. Hence it can be seen that section G1 has

the best aerodynamic stability, while section G3 has the

worst. Such visualization of wake through three-dimen-

sional CFD simulations is of direct benefit for aerodynamic

analysis of structures.

Figure 7 Computed aerodynamic force coefficients versus angle of at-

tack for the five generic deck sections: = three-dimensional CFD; =

two-dimensional CFD.

Table 1 Comparisons drag coefficient using different method for Tacoma

section

The present computed results

(two/three dimensional)

Wind

tunnel test

DVM

method

0.279/0.296 0.3 0.27

Figures 911 show the pressure contours of the three

sections (i.e., the upper part from two-dimensional simula-

tions and the lower part from three-dimensional simula-

tions). For the streamlined section G1, as shown in Figure 9,

the head contours are nearly the same but the downstream

ones have significant differences. For sections G2 and G3,

as shown in Figures 10 and 11, pressure distributions from

three-dimensional simulations are quite different from those

from two-dimensional simulations. From the contour of the

lower half of Figure 10, it can be found that around section

G2, the flow reattachment occurs at the middle of the deck,

and a large vortex region can be found at the rear part,

where the section has a concave angle. For the two kinds of

contours for section G3, though the flow separations in the

front have both been observed, the flow reattachment region


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Figure 8 Pictures of the three-dimensional wake flow for the three sec-

tions G1G3.

Figure 9 Pressure contour of section G1.

and the variation of vortex are obviously different. There-

fore, only three dimensional numerical simulations can be

used to implement qualitative analysis of bluff body aero-

dynamics.

4.2 Computation of flutter derivatives

In order to evaluate the safety of a long-span bridge against

flutter instability, it is very important to accurately obtain

the flutter derivatives of the bridge deck. Usually, the flutter

analysis is carried out by using flutter derivatives obtained



from wind tunnel experiments on a scaled model of the

bridge deck [1,12]. There are two conventional methods for

flutter derivative identification. One is the free vibration

method [1,12,28], which is based on analysis of the varia-

tions of the apparent damping ratios and natural mechanical

frequencies of the structure when placed in free flow. Thealternative forced vibration method uses identification of

the motion-related fluid forces exerted on the structures, and

a formulation of the motion induced aerodynamic forces L

and moment M has been proposed which is suitable for

cross-sections in cross wind bending and twisting motion

[12]. Hence there has been much discussion on how to iden-

tify flutter derivatives accurately through wind tunnel tests,

e.g., that of Jones et al. [29].

The free vibration approach is easy to deploy in wind

tunnel laboratories, but suffers from poor quality, since

buffeting and vortex shedding factors may be mixed in ref.

[6]. In contrast, the forced vibration approach has strongly


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and clearly defined input and output signals and so gives

flutter derivatives of high quality. However it depends on

actuation systems, which are expensive and hard to build.

For the CFD technique, the forced vibration method is

generally more convenient than the free vibration one, and

Larsen and Walther [28] proposed the expansions:

2 2 * * 2 * 2 *

1 2 3 4 ,h B h

M U B KA KA K A K AU U B

(27)

2 * * 2 * 2 *

1 2 3 4 ,h B h

L U B KH KH K H K HU U B

(28)

where /K B U is the dimensionless reduced frequencyof the motion; 2 f ; f is the forced vibration fre-

quency; hand h are the vertical cross wind motion and its

time derivative; and are the section rotation (twist)

in degrees and its time derivative; and*

jA and*

jH

( 1, 2, 3, 4)j are the flutter derivatives, which in general

are functions of K. The time increment T used was stillequal to 0.002 s. Another parameter needed is the dimen-

sionless reduced velocity*

/U U fB , for which this paper

uses the six values of 2, 4, 6, 8, 10 and 12 [28] when com-

puting the dimensionless reduced frequency K.

Assume forced vibration of the form:

0 0( ) exp(i ), ( ) exp(i ).h t h t t t (29)

The motion induced forces are also assumed to be har-

monic, with identical but a phase shift relative to themotion. Replacing exp( i ) by ( cos i sin ) to deter-

mine the flutter derivatives from eqs. (27) and (28) gives

* *

1 22 2 2 2 2

0 0

* *

3 42 2 2 2 2

0 0

* *

1 22 2 2 2

0 0

* *

3 42 2 2 2

0 0

( )sin ( )sin, ,

( )cos ( )cos, ,

( )sin ( )sin, ,

( ) cos ( ) cos, ,

h

h

h

h

M t M tA A

h K U B K U B

M t M tA A

K U B h K U B

L t L tH Hh K U K U B

L t L tH H

K U B h K U

(30)

where L

and M

are, respectively, the section lift and

section moment caused by the forced twisting vibration; and

hL and

hM are, respectively, the section lift and section

moment caused by the forced vertical vibration. Thus,* * *

1 4 1, ,A A H and *

4H can be computed from the forced

vertical motion, and * * *

2 3 2

, ,A A H and*

3

H can be com-

puted from the forced twisting motion.To identify the flutter derivatives of the three sections for

zero angle of attack, forced motion simulations were con-

ducted using the driving signal amplitudes of eq. (29) with

( ) 0.05 sin and ( ) 3sinh t B t t t [28]. However the

flutter derivatives *4

A and*

4H are not presented below

because wind tunnel results [28] are not available for them.

Each individual simulation was run for enough time in-

crements for the simulated time traces of L and M to be

stable. The analysis of the simulations involved the least-

squares fitting of a sinusoid to the simulated L and M

time traces. An example of this procedure is shown in Fig-ure 12, obtained for section G1 with 0=3 and U

*=6.

Figures 1315 show the values computed by the proposed

Figure 12 Three-dimensional simulated motion-induced aerodynamic force time traces (----) and corresponding sinusoidal least-squares fit (solid curve),

for section G3 with 0=3 and U*=6.


10/13


Figure 13 Comparison of the three-dimensional () and two-dimensional () CFD computed values for the flutter derivatives of section G1 with windtunnel test results () , computed values from ANSYS workbench () and the curve obtained via DVM.

Figure 14 Comparison of the three-dimensional () and two-dimensional () CFD computed values for the flutter derivatives of section G2 with wind

tunnel test results (

) , computed values from ANSYS workbench () and the curve obtained via DVM.


11/13


Figure 15 Comparison of the three-dimensional () and two-dimensional () CFD computed values for the flutter derivatives of section G3 with windtunnel test results () and the curve obtained via DVM.

two-dimensional () and three-dimensional () CFD simu-lations. The results given by DVM (solid lines), ANSYS

workbench () and from the wind tunnel test () are also

given for comparison, noting that the wind tunnel test re-

sults are incomplete, and for section G3 none are available

for *1

A and *3

A . The meshes and parameters used by

ANSYS workbench are the same as those of the present

CFD simulations. However, ANSYS has its own mesh de-

formation algorithm which is different from that of sect.

3.2.1.

It can be seen that the present three-dimensional CFD

simulations mostly give better results than the DVM method

and ANSYS workbench, though sometimes all results are in

good agreement. For *2

A , which is well-known to be a crit-

ical parameter for flutter [29], the present three-dimensional

CFD method has obtained exact results when compared to

wind tunnel results. It can be concluded that no matter if the

structure belongs to streamline body or bluff body, the pre-

sent three-dimensional CFD method gives better predictions

for flutter derivatives of bluff bodies, which have relatively

poor aerodynamic stabilities than the DVM method and

ANSYS workbench.

4.3 Forced vibration amplitude influences on the flut-

ter derivatives

Noda et al. [30] focused on the effects of forced vibration

amplitudes on the flutter derivatives of a thin rectangular

cylinder, and found that flutter derivatives identified using

small forced vibration amplitudes are quite different from

those using large forced vibration amplitudes, especially for*

2A and *

2H . Bai et al. [11] investigated this effect on an

airfoil, but found *2

A and *2

H were not affected strongly

until the maximum amplitudes that Noda et al. [30] had

used had been reached. Here two increasing amplitudes are

chosen to study this effect which is related to dynamic stall

[11].

The computed values of the flutter derivatives for section

G2 with the twist amplitudes,0

8 and 0 12 , are

plotted in Figure 16. The results of0

3 are also shown

for comparison. All these results are from three-dimensional

CFD simulations.There are not obvious differences between them, though

the maximum amplitudes that Noda et al. [30] used were

reached. It can be concluded that although there are inherent

flow separation around some bluff bodies like the deck sec-

tion G2 used here, amplitude effects on the flutter deriva-

tives are not significant if the amplitudes are not large

enough.

5 Conclusions

The main aim of the current study is to investigate the bluff

body aerodynamics with three-dimensional CFD modeling

at a high Reynolds number and proper turbulence models.

The results presented are encouraging and demonstrate the

accuracy of the proposed three-dimensional CFD method. It

has been shown that the present three-dimensional CFD

method is an effective numerical tool for evaluating the

aerodynamic stability instability of bluff bodies.

Three long-span bridge deck cross sections were investi-

gated, using both two-dimensional and three-dimensional

CFD modeling. Comparisons of flutter derivatives were

given by the CFD method, along with those given by the

DVM method, the commercial software system ANSYS


12/13


Figure 16 Influence of increasing forced vibration amplitudes. Computed results are shown for*

A and*

, ( 2, 3)jH j for 0 = 3 (); 8 (); and 12

(). Wind tunnel test results () and the curve obtained via DVM are also shown for comparison.

workbench, and by the wind tunnel tests, and showed that

the present three-dimensional CFD method is better overall

than the DVM method and ANSYS workbench, especially

for bluff bodies with relatively poor aerodynamic stability.

There are some differences between the two-dimensional

and three-dimensional CFD simulation results in the com-

puted aerodynamic force coefficients of section G2 for dif-

ferent angles of attack. The features of the flow field at dif-

ferent parts of the long-span deck cross section can be dis-

played pictorially via the three-dimensional CFD simula-

tions and the results show that section G1 has the best aer-

odynamic stability, while section G3 has the worst. Finally,

an investigation with different forced vibration amplitudes

shows that amplitude effects on the flutter derivatives of

bluff bodies are not significant if the amplitudes are not

large enough.

Being treated as a hybrid RANS/LES model, DES was

successfully applied in the three-dimensional CFD simula-

tions of bluff body with different aerodynamic stability, and

has shown significant benefits in efficiency and accuracy.

So it can be widely used to investigate engineering structure

aeroelasticity when turbulence effect is also included.

Appendix

The values of y+ obtained from both two-dimensional and

three-dimensional CFD simulations can be seen in Table A1.

Table A1 The values of y+ obtained from both two-dimensional and

three-dimensional CFD simulations

Section G1 G2 G3

Two-dimensional CFD

Three-dimensional CFD

1.7655

1.8236

1.6397

1.7214

1.6027

1.9574

This work was supported by the National Natural Science Foundation of

China (Grant No. 11172055) and the Foundation for the Author of Nation-

al Excellent Doctoral (Grant No. 2002030).

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