Numerical simulation of aeroelastic instability of a ...road.cnki.net/download/jtdh/1803201055070001.pdf · distance from left and right sides of the computational domain to the thin

Proceedings of the 2018 World Transport Convention Beijing, China, June 18-21, 2018

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Numerical simulation of aeroelastic instability of a streamlined-box

girder section with LES turbulence model

Z. W. Liu Hunan Provincial Key Laboratory for wind Engineering and Bridge Engineering, Hunan University

No.2, South Lushan Road, Changsha, China [email protected]

P. R. Xie Hunan Provincial Key Laboratory for wind Engineering and Bridge Engineering, Hunan University

No.2, South Lushan Road, Changsha, China [email protected]

Z. Q. Chen Hunan Provincial Key Laboratory for wind Engineering and Bridge Engineering, Hunan University

No.2, South Lushan Road, Changsha, China [email protected]

ABSTRACT

Aero-elastic instability is one of the important issues for wind-resistant design of large span suspension and cable-stayed bridges. In this paper, a novel numerical method with large-eddy simulation turbulence model based on user-defined functions (UDFs) and dynamic mesh techniques of ANSYS Fluent was presented. As a validation case, the numerical simulation of aero-elastic instability a thin plate with aspect ratio of 22.5 at 0o attack angle under smooth flow condition was conducted firstly. The numerical simulation results were compared with theory results and experimental results separately. Thereafter the aero-elastic instability of a box girder section of a large-span suspension bridge was investigated. The numerical results are compared with wind tunnel tests results. Finally, the aero-elastic instability of the box girder section with central stability plate was investigated. The mechanism of the aerodynamics countermeasures for flutter stability of the box girder section of suspension bridge was analysed. The research results shown that the numerical prediction flutter critical wind speed of the thin plate and bridge deck section with safety barriers and checking car rails agrees well with the experimental results separately. It can be concluded that the two-dimensional LES turbulence model is suitable to model the aerodynamic and aero-elastic characteristic of bridge deck with safety barriers and checking rails et al., which is of quasi-two-dimensional flow around the bridge deck.

KEY WORDS: Streamlined-box girder section; thin plate; aero-elastic instability; Large-eddy

simulation.

1. INTRODUCTION

Aerodynamic and aero-elastic performances, such wind loads, buffeting, vortex-induced vibrations, galloping and flutter stability, are the important issues for wind-resistant design of long span bridges. The wind actions on structures are used as the static loads before 1940s although about eleven bridges are destroyed by wind actions, including static and dynamic wind loads. In fact, as the original Tacoma Narrows suspension bridge neared completion, the vertical undulations of the bridge under wind action were observed. Thus a 1/20 scale wind tunnel model of the main deck was constructed to investigate the vertical vibrations of the bridge. The serious aerodynamic instability condition of the main deck was disclosed by Farquharson. The further tests of proposals for stabilization of the bridge were finished on November 2(Farquharson,1940). In November 7, the

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original Tacoma Narrows bridge was destroyed by 19m/s wind four months after opening to traffics. The research report on the original Tacoma Narrows bridge collapse was given by Othmar Amman, Von Karman and Glen B. Woodruff. According to their report, the aerodynamic forces and aero-elastic performances of suspension bridges should be tested in a wind tunnel. The free-oscillation experimental method was used to identification flutter derivatives of airfoil and bridge decks by Scanlan and Tomko. The negative damping of aerodynamic force of the main deck is the source of the original Tacoma Narrows bridge collapse (Scanlan and Tomko, 1971). A lot of experimental investigations on flutter instability, identification methods of flutter derivatives of bridge decks were conducted by many researchers in the last five decades (Sakar, Jones, et al., 1994; Jones, Scanlan et al., 1995; Singh, Jones et al.,1996). At present, wind tunnel test is still an important method for wind-resistant design of long-span bridges.

On the other hand, with the development of computational fluid dynamics (CFD), numerical simulations of aerodynamic and aero-elastic performances of bridge deck were conducted in the last three decades. The main methods of computational fluid dynamics are finite difference method (FDM), finite volume method (FVM), discrete vortex method (DVM) et al. The two-dimensional numerical simulation of flow around an elastically-supported bridge deck section with FDM was conducted by Fujiwara et al.(Fujiwara, 1993). In the mid-1990s Kuroda (1997) developed a laminar Navier-Stokes solver discretized based on FDM. The flow around a fixed girder section model of the Great Belt East Bridge was simulated. Pressure distributions and static coefficients of the deck section for Re of 3x105 agree well with the experimental results. Bruno et al. (2001) applied the finite volume method (FVM) to evaluate the aerodynamic behavior of equipped bridge deck sections. Bruno et al. (2002) conducted a study for evaluating the capability of 2D numerical simulation for predicting the vortical structure around a quasi-bluff bridge deck using laminar, Reynolds-Averaged Naviers-Stokes (RANS) turbulence models and LES model separately. The study confirms the importance of safety-barriers modeling in the analysis of bridge aerodynamics (Bruno, 2003). The fluid-structure interaction mechanisms due to wind action on either a stationary or a moving long-span bridge deck were investigated using the arbitrary Lagrangian-Eulerian (ALE) formulation by Frandsen. The bridge deck was modeled as a simplified model with lumped mass and three vertical springs and one horizontal spring to simulate the mass and stiffness of the main deck per unit length. It should be noted that the subsidiary facility of the main deck, such as crash barriers, are not considered in their research (Frandsen, 2004). Grid generation of FDM and FVM is a time-consuming process, especially for bridge decks. To avoid the grid generation, DVM was given by Sarpkaya and Chorin in 1960s. Larsen and Walther (1998) investigated aerodynamic characteristics of five generic bridge deck sections with two-dimensional DVM. The numerical simulation results agree well with the experimental results. Numerical simulation of flow past bridge deck sections with 2D-DVM also were conducted by Morgenthal and McRobie(2002), Taylor and Vezaa (2002), Zhou et al.(2002).

Finally, it should be mentioned that many researchers have investigated the aerodynamic and aero-elastic performances of the main deck sections without subsidiary facilities. As the results of the study by Bruno et al. (2003), it is important for analyzing aerodynamics of the bridge decks to model subsidiary facilities. This paper presented a novel numerical method using two-dimensional Large Eddy Simulation (LES) turbulence model to simulate flow past a fixed and oscillating bridge deck with subsidiary facilities to predict the self-excited flutter instability limit of it.

2. NUMERICAL METHOD

2.1 Governing equations for fluid

For bridge aerodynamics, it is assumed that the flow is incompressible for wind-induced bridge motions. In LES, large eddies are resolved directly, while small eddies are modeled. The filtered equations of incompressible fluid can be written as follows,

0)(=

∂∂

+∂∂

i

i

xu

tρρ

(1)

3

( ) ( )

∂

∂+

∂∂

∂∂

+∂∂

−=∂∂

+∂∂

i

j

j

i

jiji

ji x

uxu

xxpuu

xu

tυρ

(2)

where 2,1, =ji denote the Cartesian coordinate direction x, y, and the overbar denotes the spatial average over a grid volume (V) or the surface average over a cell face ( ji, ) of the volume. p is the filtered pressure, ρ is air density 3/225.1 mkg=ρ , υ is the uniform kinematic viscosity

which is assumed constant with a value of sm /105.1 25−× for air at Co20 , iu is the Cartesian components of the velocity field, and jiuu is the non-linear term. The non-linear term are given by Smagorinsky-Lilly model as

ijkkijTjiji RSuuuu δµρρ312 ++= ,

2/12 )2()( ijijST SSCu ∆= ρ ,

∂

∂+

∂∂

=i

j

j

iij x

uxuS

21

(3)

where sC is the sub-grid scale stress constant (taken here 0.1 as Ref.[14]) and 2/1)( yx∆∆=∆ defined by the grid spacing.

The SIMPLEC algorithm is employed for pressure correlation. Second order and bounded central differencing are used for spatial discretization of pressure and momentum separately. Time discretization is second order implicit.

2.2 Governing equations for bridge deck

The bridge deck is simplified as a model with two-degree of freedom system which mounted on elastic springs with viscous dashpots appropriate to the structural mode of vibration under consideration. The structure equations of motion for the dynamic behavior of a damped linear mechanical system with 2DOF are

yyyy Fykycym =++ &&& (4)

αααα ααα MkcI =++ &&& (5)

where ym , yc and yk is the mass, damping and stiffness of the main deck section per unit length in vertical bending mode separately. αI , αc and αk is the mass moment of inertia，damping and stiffness of the bridge deck per unit length in torsional mode separately.

2.3 Modeling of fluid-structure interaction

A two-dimensional large eddy simulation of turbulent flow past the structures will be used to solve the equations of fluid. The dynamic mesh technique is employed to simulate the FSI of bridge deck under wind action. To analyze the structural responses of the bridge deck section with 2DOF under wind action, the equations of structural motion were integrated using Newmark-β method. The computational flow chart of numerical simulation of FSI of bridge deck used in present paper is given in Figure 1.

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Figure 1: Flow chart of numerical simulation of FSI of bridge deck

3. NUMERICAL SIMULATION OF FLUTTER INSTABILITY OF A THIN PLATE 3.1 Geometry and computational domain

The numerical simulations of aero-elastic responses of the thin plate with aspect ratio of 22.5 under different wind velocities were conducted to validate the proposed method in present paper. The width and height of the thin plate are 450mm and 20mm respectively. The main geometry parameters are given in Figure 2. The mass of the thin plate per unit length is mkgm /25.11= . The mass moment of inertia is mmkgIm /2828.0 2⋅= . The circular frequencies of the vertical and torsional vibration modes are 12.11rad/s and 19.0rad/s separately. The damping ratios of the vertical and torsional vibration modes are 0.5%. As shown in Figure 3, the computational domain is divided into three parts, namely rigid region, dynamic mesh region and stationary meth region separately. The distance from left and right sides of the computational domain to the thin plate are 5B and 20B separately. The distance from the upper and lower boundaries of the computational domain to the center of the thin plate is 5B. The block ratio of the computation domain is 0.0044, which is much less than 5.0%. The left and right boundaries of the computational domain are defined as velocity inlet and pressure outlet boundary conditions respectively. The upper and lower sides of the computational domain are defined as symmetry conditions. The surface of the structure is defined as no-slip wall condition.

Grid and boundary conditions

Numerical simulation of flow around the bridge deck

Is the lift and torque coefficients is stable?

Define the dynamic boundaries of the bridge deck using the UDFs

Solve the equations (1), (2) using LES model

Solve the equation (4), (5) separately by Newmark-βmethod and

write numerical results to the file.

Is the response of the bridge deck

stable?

N

Y

Renew the mesh by DEFINE-CG-MOTION

End

Y

5

In order to improve the quality of the grid, the chamfering of the radius of 1mm is set at the thin plate section edges. The mesh of the computational domain of the thin plate is given in Figure 4. The time steps used in present calculation are 0.0002s, 0.0005s and 0.0010s separately for the case of time step checking, which in non-dimensional form is 007.0/ == ∞ BdtUdτ , 0.018, 0.367 separately.

20

30

450

R1

Figure 2: Thin plate section (unit: mm)

4.5B B/2 B B/2 19.5B

4.5B

B4.

5B

Thin plateRigid region

Dynamic meshStationary mesh

Symmetry

Symmetry

Pressure outletVelocity inlet

Figure 3: Computational domain and boundary conditions of the thin plate

Figure 4: Mesh of the thin plate

3.2 Numerical results

The time histories of the lift and pitch coefficients of the thin plate with different time steps at wind speed of 16.5m/s are given in Figure 5. The FFT magnitude of the lift forces and pitch moments of the thin plate with different time steps are given in Figure 6 to 8. From Figure 5, it can be seen that the lift forces and pitch moments of the thin plate with time step of 0.0002s are less than that of the thin plate with time step of 0.0005s and 0.0010s respectively. From Figure 6 to 8, It can be seen that there are two predominant frequencies for time step of 0.0002s, however there is only one predominant frequency for time step of 0.0005s and 0.0010s separately. The second predominant frequency of 205.97Hz for time step of 0.0002s may be derived from numerical calculation error. So, the time step of 0.0005s is used for the numerical simulation of the flutter stability of the thin plate and bridge deck in present paper.

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0 1 2 3 4 5 6 7 8-80

-60

-40

-20

0

20

40

60

80

Lift,

(N)

Time, (s)

t1=0.0002s t2=0.0005s t3=0.0010s

0 1 2 3 4 5 6 7 8-8

-6

-4

-2

0

2

4

6

8

Pitc

h m

omen

t, (N

.m)

Time, (s)

t1=0.0002s t2=0.0005s t3=0.0010s

(a) Lift force of the thin plate (b) Pitch moment of the thin plate

0 1 2 3 4 5 6 7 8-4

-3

-2

-1

0

1

2

3

4

Non

-dim

ensi

onal

dis

p., y

/D

Time, (s)

t1=0.0002s t2=0.0005s t3=0.0010s

0 1 2 3 4 5 6 7 8-10-8-6-4-202468

10

Tors

iona

l deg

ree,

(o )

Time, (s)

t1=0.0002s t2=0.0005s t3=0.0010s

(c) Non-dimensional displacement, y/D (d) Torsional angle Figure 5: Time histories of aerodynamic forces and displacements of the thin plate with

different time steps at 16.5m/s wind

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 3000

10000

20000

30000

40000

50000

Frequency,(Hz)

Mag

nitu

de

fs1=2.2770Hz

fs2=205.97Hz

0 60 120 180 240 3000

1100

2200

3300

4400

5500

Frequency,(Hz)

Mag

nitu

de

fs1=2.2771Hz

fs2=205.97Hz

(a) Magnitude of FFT of lift force (b) Magnitude of FFT of pitch moment

Figure 6: Magnitude of FFT of lift force and pitch moment of the tine plate with 0.0002s time step

0 60 120 180 240 3000

30000

60000

90000

120000

150000

Frequency,(Hz)

Mag

nitu

de

fs=2.3619Hz

0 60 120 180 240 3000

3600

7200

10800

14400

18000

Frequency,(Hz)

Mag

nitu

de

fs=2.3619Hz

(a) Lift force (b) Pitch moment

Figure 7: Magnitude of FFT of lift force and pitch moment of the tine plate with 0.0005s time step

7

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 3000

11000

22000

33000

44000

55000

Frequency,(Hz)

Mag

nitu

defs=2.3719Hz

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 3000

1300

2600

3900

5200

Frequency,(Hz)

Mag

nitu

de

fs=2.3719Hz

(b) Lift force (b) Pitch moment

Figure 8: Magnitude of FFT of lift force and pitch moment of the tine plate with 0.0010s time step

The time histories of non-dimensional vertical displacements and rotation angles of the thin

plate at various wind speeds are given in Figure 9. From Figure 9 it can be seen that the rotational angles of the thin plate are attenuated when wind speed is 14.0m/s. The rotational angle of the thin plate is increasing until a constant amplitude is reached when wind speeds are about 15.0 and 15.5m/s separately. The rotational angle of the thin plate is divergent when wind speed is about 16.0 m/s. Table 1shows the numerical prediction of the flutter limit of the thin plate, the experimental results and the theory results. The theory results of the flutter instability of the thin plate are calculated with aerodynamic derivatives of ideal thin plate. The relative error of the numerical result to experimental result and theory result for the flutter critical velocity of the thin plate are about 3.0~9.1% and 0.0~6.25% separately. The flutter critical wind speed and flutter critical circular frequency predicted by the present method agree well with the experimental and theory results separately.

0 2 4 6 8 10 12 14-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

Non

-dim

ensi

onal

dis

p, y

/D

Time, (s)

0 2 4 6 8 10 12 14-0.20-0.15-0.10-0.050.000.050.100.150.20

Rot

atio

n an

gle(

o )

Time, (s)

0 2 4 6 8 10 12 14 16 18 20

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

Non

-dim

ensi

onal

dis

p.,y

/D

Time, (s)

0 2 4 6 8 10 12 14 16 18 20-2.0-1.5-1.0-0.50.00.51.01.52.0

Rot

atio

n an

gle,

(o )

Time, (s)

(a) V=14.0m/s (b) V=15.0m/s

0 2 4 6 8 10 12 14 16 18 20 22 24

-1.0

-0.5

0.0

0.5

1.0

Non

-dim

ensi

onal

dis

p., y

/D

Time, (s)

0 2 4 6 8 10 12 14 16 18 20 22 24-4-3-2-101234

Rot

atio

n an

gle,

(o )

Time, (s)

0 1 2 3 4 5 6 7

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Non

-dim

ensi

onal

dis

p.,y

/D

Time, (s)

0 1 2 3 4 5 6 7-4-3-2-101234

Rot

atio

n an

gle,

(o )

Time, (s)

(c) V=15.5m/s (d) V=16.0m/s

Figure 9: Time histories of vertical displacement and rotation angle of the thin plate at different wind velocities

Table 1 Numerical, theory and experimental results of flutter critical wind velocities of thin plate

Parameters LES (present paper) Theory results Experimental results(Xiang

H.F., 2005) Flutter critical wind velocity, Vcr (m/s) 15.0~16.0 15.96 16.5

Flutter critical circular 14.9~15.4 15.6 15.6

8

frequency (rad/s)

4. NUMERICAL SIMULATION OF FLUTTER STABILITY OF A BRIDGE DECK SECTION

4.1 Lingding Bridge of Shenzhen-Zhongshan Links

The Lingding Bridge (LDB) is going to a part of the of Shenzhen-Zhongshan Links project motorway in Guangdong, China that will connect Shenzhen to Zhongshan. The structure is a three-span suspension bridge with a main span of 1660m and two side spans of 530m, it has two towers 266m high. The deck is a classical streamlined single box girder, 49.7m wide and 4.0m deep. Figure 10 shows the general arrangement and the deck cross-section of the bridge. The dynamic features of the LDB in service condition were calculated with finite element method (FEM). Table 2 shows the main dynamic parameters of the LDB.

(a) General arrangement of the bridge (unit: cm)

(b) The main deck cross-section (unit: mm)

Figure 10: General arrangement and main deck cross section of LDB

Table 2 Dynamic parameters of the LDB The parameters and units Values Mass per meter, m (kg/m) 42.411x103

Mass moment of inertia, mI (kg.m2/m) 9.838x106

The first symmetry vertical bending frequency, hf (Hz) 0.1000 The first symmetry torsional frequency, αf (Hz) 0.2228 Damping ratio of the first symmetry vertical bending mode, hξ (%) 0. 5 Damping ratio of the first symmetry torsional mode, αξ (%) 0.5

4.2 Computational domain, boundary conditions and mesh

In order to compare with the test results, it is determined that the geometric scale ratio of the main beam section is 1/70, and the wind speed scale is 1/5. The numerical simulation parameters of the bridge deck section model are given in Tab.3.

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The computational domain is also divided into three parts, namely rigid region, dynamic mesh region and stationary meth region separately. The distance from left and right sides of the computational domain to the thin plate are 4.5B and 10.5B separately. The distance from the upper and lower boundaries of the computational domain to center line of the main deck is 4B. The block ratio of the computation domain is 1%, which is much less than 5.0%. The left and right boundaries of the computational domain are defined as velocity inlet and pressure outlet boundary conditions respectively. The upper and lower sides of the computational domain are defined as symmetry conditions. The surface of the bridge deck is defined as no-slip wall condition. The computational domain and boundary condition are shown in Figure 11.

The grid near the deck section is divided into two parts. The triangular mesh is adopted in the deformation area near the deck section, and the structured grid is used in the outer area of the computation domain, so as to improve the quality of mesh generation. The first height of the grid close to the main deck section is about B5100.4 −× . The total number of the grid is 358871, and the grid of the main deck of LDB is given in Figure 12.

The Smagorinsky-Lilly turbulence model of two-dimensional large eddy simulation is chosen to model the turbulence wind around the bridge deck section. The Smagorinsky constant is Cs=0.10. The time step is 0.0002s.

Table 3 Numerical simulation parameters of the bridge deck model

Parameters and units Values Mass per meter, m(kg/m) 8.6553 Vertical frequency, hf (Hz) 1.4000 Torsional frequency, αf (Hz) 3.1180 Damping ratio of vertical mode, hξ (%) 0.5 Damping ratio of vertical mode, αξ (%) 0.5 Wind speed scale, Vλ 1/5

4B B 10B

Symmetry

Pressure outletVelocity inlet

Regig region

Deformation region

4B4B

Symmetry

Figure 11: Computational domain and boundary conditions of the LDB deck section

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(a) The grid near the main deck section

(b) The grid near the windward (c) The grid near the central safety barriers

Figure 12: Grid of the LDB deck section

4.3 Numerical Results

The time histories of the vertical displacements and rotational angles of the bridge deck section under 15.0m/s and 15.5m/s wind at 0o attack angle are given in Figure 13 separately. From figure 13(a) it can be seen that there is no divergence in the response of the vertical and torsional displacement of the bridge deck section under 15.0m/s wind action. While the torsional divergence of the bridge deck section under 15.5m/s wind action is found as shown in Figure 13(b). The time histories of the vertical displacements and rotational angles of the bridge deck section under 15.0m/s and 15.5m.s wind at +3o attack angle are given in Figure 14. The rotational divergence of the bridge deck section under 15.5m/s wind action at +3o attack angle is found as shown in Figure 14(b). The numerical results and the wind tunnel experimental results of flutter critical wind speeds of the bridge deck section are given in Table 4. The numerical results agree well with the experimental results.

Table 4 Numerical and experimental results of flutter critical wind speeds of the bridge deck

Wind attack angle(o) Flutter critical wind speed, Vcr (m/s) Relative error (%) Numerical results Experimental results 0 77.5 74.9 3.47

11

0 2 4 6 8 10 12 14 16 18-0.08-0.06-0.04-0.020.000.02

Verti

cal d

ispl

acem

ent,

y(m

)

Time,（s）

0 2 4 6 8 10 12 14 16 18

-3.0

-1.5

0.0

1.5

3.0

Rot

atio

nal a

ngle

,α( °

)

(a) V=15.0m

0 2 4 6 8 10-0.10

-0.05

0.00

0.05

Verti

cal d

ispl

acem

ent,

y(m

)

Time, (s)

0 2 4 6 8 10-12-8-4048

12

Rot

atio

nal a

ngle

,α( °

)

(b) V=15.5m/s Figure 13: Wind-induced vibration responses of the bridge deck section under 15.5m/s and 15.0m/s

wind at 0o attack angle

5. CONCLUSIONS

The numerical simulation method of FSI of the bridge deck with two-dimensional LES turbulence model using dynamic mesh techniques is presented in this paper. The numerical prediction flutter critical wind speed of the thin plate and bridge deck section with safety barriers and checking car rails agrees well with the experimental results separately. It can be concluded that the two-dimensional LES turbulence model is suitable to model the aerodynamic and aero-elastic characteristic of bridge deck with safety barriers and checking rails et al., which is of quasi-two-dimensional flow around the bridge deck.

6. ACKNOWLEDGEMENTS

This research works has been supported by National Natural Science Foundation of China (No.51478180, 51778225). The authors fully acknowledge the support received.

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