ESTIMATION OF AERODYNAMIC ADMITTANCE BY NUMERICAL...

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BBAA VI International Colloquium on:

Bluff Bodies Aerodynamics & Applications

Milano, Italy, July, 20-24 2008

ESTIMATION OF AERODYNAMIC ADMITTANCE BY

NUMERICAL COMPUTATION

Hidesaku Uejima∗∗∗∗, Shinichi Kuroda

∗∗∗∗ and Hiroshi Kobayashi†

∗ Structural Strength Dept., Reserch Laboratory

IHI Corporation, 1,Shin-Nakahara-Cho, Isogo-Ku, Yokohama, Japan

e-mails: [email protected], [email protected]

† Department of Civil Engineering

Ritsumeikan University, 1-1-1 Noji Higashi, Kusatsu, Shiga, Japan

e-mails: [email protected]

Keywords: Gust force, CFD, RANS, Aerodynamic admittance

Abstract. The present study describes the evaluation using numerical computations of gust

forces acting on bridge decks. When predicting the gust response of long span bridges the

gust response analysis in frequency domain has been widely used. In order to improve the ac-

curacy of the computation, it is important to provide the aerodynamic admittance and span-

wise correlation of wind force with high accuracy. Using CFD based on a two-dimensional

RANS, the lift forces acting on several basic cross-sections in turbulent flow were evaluated.

First of all, the proposed computation method was validated by the calculation of a thin plate

of which gust force has been established by aerofoil theory. Then the aerodynamic admit-

tances of a rectangular section of B/D=5 and a shallow hexagonal section were calculated.

Computation results were in good agreement with experimental results of the previous study.

H.Uejima, S.Kuroda and H.Kobayashi

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1 INTRODUCTION

When evaluating the gust response of a bridge using gust response analysis in frequency

domain, aerodynamic admittance functions for the bridge deck and other members need to be

defined. The aerodynamic admittance function of a bridge deck depends on its cross sec-

tion[1][2], because the gust force acting on the bridge deck is generated not only from ap-

proaching gusty wind but also from the separated flow inherent in the geometry of the deck

cross section. A wind tunnel test is carried out for each cross section in the usual way to

evaluate the gust force, however, it is time-consuming and costly.

Lots of efforts have been made on points of applicability of CFD to wind resistant design

for structures. As for bridges, the estimation of flutter derivatives or steady aerodynamic force

coefficients of decks has been investigated as an alternative to preliminary wind tunnel

tests[3][4]. On the other hand, there are very few investigations on applications to gust force

in turbulent flow.

In this paper, the estimation procedure of the aerodynamic admittance function is discussed.

The aerodynamic admittance functions of several cross sections were calculated and were

compared with wind tunnel test results.

2 NUMERICAL METHOD[4]

The two-dimensional incompressible Navier-Stokes equations were used as the governing

equations. The numerical algorithm was based on the method of pseudocompressibility[5].

The convective terms were discretized by the fifth-order upwind differential scheme and the

implicit method of second-order was employed as a time integration algorithm. The k-ωSST

turbulence model[6] was used. An overset grid system was used in the present computation where an O-type grid around the body was overlapped on a background grid of H-type as

shown in Fig.(1).

This paper focuses on lift fluctuation in the vertical gust. At the inlet boundary of the com-

putational domain, the vertical gust was generated by controlling the inflow angle. At inlet

boundary, the specific dissipation rate ∞ω and the eddy viscosity coefficient ∞tν were set as

∞−

∞∞

∞ == ννω 310, tB

U

(1)

where, ∞U is the mean flow velocity, B is the width and ∞ν is the kinematic viscosity at

the inlet boundary. The Reynolds number ( ∞∞= ν/BURe ) was set as 140,000 for all compu-

tations. Thus the turbulent kinetic energy ∞k is given as follows:

282133 101010 ∞−

∞−−∞

∞−

∞∞∞ ≅=⋅== UURB

Uk et νων

(2)

Then, ∞k2 is sufficiently small compared to velocity fluctuations due to purposely-

generated gust. Values of k tended to be large in boundary layers or separated flow regions on

the body, however, in the regions ω which had dimensions (time)-1 was sufficiently high com-

pared to the relevant frequency range within which natural frequencies of structures fall. Ac-

tually, in the present computations, values of ω in the region were two orders of magnitude

higher than the significant frequency range relevant to the gust response analysis. Therefore,

the gust force properties of decks could be evaluated only from the unsteady solution of

mean-flow equation without taking into account the turbulent kinetic energy k obtained from

calculations.


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3 CALCULATION CONDITIONS

3.1 Turbulent Characteristics of Generated Flow

The preliminary computations were performed to assure the accuracy of computed gusts.

Fig.(2) shows the time histories of velocity fluctuations computed on the condition that the

sinusoidal gusts are provided as inflow turbulence and no body is placed in the flow field. The

amplitude of the wind inclination angle at the inlet boundary is 2.0 degrees. Then it is equiva-

lent to turbulent intensity of 2.2%. As the amplitude of the wind inclination is sufficiently

small, the solution is expected to approach to the theoretical solution of the linear wave equa-

tion and the gust pattern is to move past without changing its wave shape. From an examina-

tion of Fig.(2), it is found that we should set two parameters suitably, the height of the

computational domain and ∆tU/B. In this investigation the height of the computational do-

main was set at 60B to weaken the effect of the slip walls at the upper and lower boundaries

particularly in low frequency regions. For low frequency waves, time steps were not crucial

and relatively large time steps could be taken. In high frequency regions, however, even a

relatively small time step could somewhat damp and diffract waves (Fig.2(4)). Employing a

smaller time step prolongs the computational time. Considering a trade-off between computa-

tional efficiency and accuracy, we set the time step ∆tU/B at 0.01 in the present computations.

Next, we tried to generate the random vertical gust which had the power spectrum of the

Karman type and had almost the same turbulence intensity as the sinusoidal gust described

above. As shown in Fig.(3), the power spectrum of the generated random gust agreed well

with the target except in the high frequency region.

w

U

B2B 5B

20B

8B

Body

Figure 1 : Computational domain

60B x

y


4

-0.04

-0.02

0.00

0.02

0.04

20 30 40 50

Theoretical

Present

tU/B

w/U

-0.04

-0.02

0.00

0.02

0.04

20 30 40 50

Theoretical

Present

tU/B

w/U

(1) k=0.16, Domain Height = 20B, dt=0.01 (2) k=0.16, Domain Height = 60B, dt=0.01

-0.04

-0.02

0.00

0.02

0.04

6 6.5 7 7.5

Theoretical

Present

tU/B

w/U

-0.04

-0.02

0.00

0.02

0.04

6 6.2 6.4 6.6

Theoertical

Present

tU/B

w/U

(3) k=3.0, Domain Height = 60B, dt=0.01 (4) k=6.0, Domain Height = 60B, dt=0.01

-0.04

-0.02

0.00

0.02

0.04

6 6.2 6.4 6.6

Theoretical

Present

tU/B

w/U

(5) k=6.0, Domain Height = 60B, dt=0.005

Figure 2 :Time histories of the vertical sinusoidal gust obtained at (x,y) =(2B,0), No body in the field

1.0E-07

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-02 1.0E-01 1.0E+00 1.0E+01

ωb/U

Sw(ω

)

Present

Karman

Figure 3 : Power spectrum of the random vertical gust at (x,y) =(2B,0) , b=B/2


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3.2 Computational Case

Three basic cross sections shown in Table 1 were addressed. Two kinds of turbulent flows,

random vertical gusts and sinusoidal gusts, were employed for evaluation of the aerodynamic

admittance function of the cross sections.

B/D=200 B/D=5 Hexagonal

Surrounding 285×83 293×45 265×43 Grid

Background 294×459

Time step (∆tU/B) 0.01

Reynolds Number

（Re=UB/ν） 1.4×10

5

Turbulent Intensity Random gust : Iu < 0.1%, Iw = 2.1%

Sinusoidal gust : Iu < 0.1%, Iw = 2.2%

Table 1: Computational case

4 AERODYNAMIC ADMITTANCE

4.1 Derivation Method of Aerodynamic Admittance

In the case of lift forces, the power spectrum can be expressed by

( ) ( )2

2

2

2

2

1)(

U

SXC

d

dCbUS w

LdL

L

ωω

αρω

+=

(3)

Where, )(ωwS is the spectrum of the lift force, ρ is air density, b is the half chord length,

( )ωLX is aerodynamic admittance, dL CddC +α is the derivative of lift force coefficient

with respect to the angle of wind incidence.

The aerodynamic admittance was assessed directly from power spectrums of the lift forces

in the generated gust and velocity at the assumed position of the body obtained from the com-

putation performed beforehand for free-field without the body. The dL CddC +α under the

smooth flow condition was also calculated beforehand.

4.2 Aerodynamic Admittances of The Basic Cross-Sections

Fig.(4) shows vertical velocity of generated gust and lift force coefficients in the same pe-

riod of time. Obtained unsteady lift force consists of two different wave patterns, that is to say,

the lift fluctuation caused by the variation of incident angle and the periodic shedding of vor-

tices. It is found that the gust force characteristics depend considerably on cross-section shape.

The aerodynamic admittance function of each cross section was evaluated based on these

data. First of all, the computational method described above was validated through the calcu-

lation of a thin plate whose aerodynamic property has been established by the aerofoil theory.

A rectangular section of B/D =200 was employed as a thin plate in the present computation.


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Fig.(5) shows the computed result of the aerodynamic admittance function together with the

Sears Function. These two are found to agree with each other well. In addition, it is clarified

that aerodynamic admittance functions in random gust and sinusoidal gust also correspond

with each other. This means that superimposition is applicable in estimating gust forces in

relatively small flow fluctuation.

As for the rectangular section of B/D=5 and hexagonal cross section, the feature of aero-

dynamic admittance simulated by the present computation is found to agree well with the

wind tunnel test results[7] obtained in two-dimensional turbulent flow. The reduced frequency

at the highest peak of each cross-section corresponds with the Strouhal number.

B/D=200

-0.60

-0.30

0.00

0.30

0.60

50.0 75.0 100.0 125.0 150.0

tU/B

CL

B/D=5

-0.60

-0.30

0.00

0.30

0.60

50.0 75.0 100.0 125.0 150.0

tU/B

CL

Hexagonal

-0.60

-0.30

0.00

0.30

0.60

50.0 75.0 100.0 125.0 150.0

tU/B

CL

Wind Velocity

-0.10

-0.05

0.00

0.05

0.10

50.0 75.0 100.0 125.0 150.0

tU/B

w/U

Figure 4 : Time histories of fluctuations of wind velocity and lift forces


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1.0E-2

1.0E-1

1.0E+0

1.0E+1

1.0E+2

1.0E-02 1.0E-01 1.0E+00 1.0E+01

Reduced Frequency　ωb/U

Aerodynamic Admittance |XL|2

Random Gust (Present)

Sinusoidal Gust (Present)

Sears Func.

Figure 5: Comparison of aerodynamic admittance functions of B/D=200

1.0E-2

1.0E-1

1.0E+0

1.0E+1

1.0E+2

1.0E-02 1.0E-01 1.0E+00 1.0E+01Reduced Frequency　ωb/U




Experiment [7]

Sears Func.

1.0E-2

1.0E-1

1.0E+0

1.0E+1

1.0E+2

1.0E-02 1.0E-01 1.0E+00 1.0E+01

Reduced Frequency　ωb/U




Experiment[7]

Sears Func.

(1) B/D=5 (2) Hexagonal

Figure 6: Comparison of aerodynamic admittance functions

5 FLOW PATTERNS AROUND THE CROSS SECTIONS

In order to discuss the difference of aerodynamic admittance in deck configurations, the

flow pattern around the each cross section was examined. Typical instantaneous flow patterns

at the moment that the upward and downward lift are at the maximum are shown in Fig.(7). In

these figures, the distributions of velocity vector and nondimensional pressure Cp =

p/(0.5*ρU2), where p is instantaneous pressure, are indicated by arrows and the shade of gray,

respectively.


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(1) At the maximum downward lift

1.0

-2.4

1.0

-2.4

Reattachment

wind

(2) At the maximum upward lift

Reattachment

1.0

-2.4

1.0

-2.4

wind

Figure 7: Instantaneous flow patterns around the bodies in sinusoidal gust (k=0.307)

As for the thin plate of B/D=200, a thin separated shear layer can be observed in a small

area around the leading edge on the upper surface when the flow is blowing up. On the other

hand, it is found that the massive separated shear layer is formed and that the flow reattached

near the trailing edge in the case of B/D=5. The reattach point tends to move back and forth

synchronously with vertical fluctuation of flow.

Fig.(8) shows the distributions for various reduce frequencies of unsteady pressures along

the surfaces of the cross-sections in the sinusoidal gusts, the magnitudes of pressure fluctua-

tions and the phase lags between the wind velocities at the leading edge and the pressures at

the points on the surfaces. In general, it is found that the hexagonal section has the similar

pressure distribution to the thin plate of B/D=200, that is to say, only the narrow region

around the leading edge is dominant in the formation of unsteady lift force, and the phase lag

tends to be flat except around the trailing edge in both cross-sections.

As for the rectangular section of B/D=5, the magnitudes of the pressure fluctuations is rela-

tively large in a wide region which corresponds to the separated flow region shown in Fig.(7).

The separated flow region which expands and contracts in sync with vertical gust tends to

amplify the fluctuating lift force. As a result, its aerodynamic admittance differs from the

Sears Function.

B B/4

Cp Cp

Cp Cp


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0.00

0.10

0.20

0.30

0.40

0.50

0.0 0.2 0.4 0.6 0.8 1.0X/B

Cpd

k=0.077

k=0.153

k=0.307

k=0.920

k=2.45

-270.

-180.

-90.

0.

90.

180.

0.0 0.2 0.4 0.6 0.8 1.0X/B

Phase (deg)

k=0.077

k=0.153

k=0.307

k=0.920

k=2.45

X/B X/B

(1) B/D= 200

0.00

0.10

0.20

0.30

0.40

0.0 0.2 0.4 0.6 0.8 1.0X/B

Cpd k=0.077

k=0.153

k=0.307

k=0.920

k=2.45

-270.

-180.

-90.

0.

90.

180.

0.0 0.2 0.4 0.6 0.8 1.0X/B

Phase (deg)

k=0.077

k=0.153

k=0.307

k=0.920

k=2.45

X/B X/B

(2) B/D= 5

0.00

0.10

0.20

0.30

0.40

0.50

0.0 0.2 0.4 0.6 0.8 1.0X/B

Cpd k=0.077

k=0.153

k=0.307

k=0.920

k=2.45

-270.

-180.

-90.

0.

90.

180.

0.0 0.2 0.4 0.6 0.8 1.0X/B

Phase (deg)

k=0.077

k=0.153

k=0.307

k=0.920

k=2.45

X/B X/B

(3) Hexagonal

Figure 8: Unsteady pressure distribution along the surface of the cross sections in the sinusoidal gust,

magnitudes of unsteady pressure (Cpd) and phase lags

between the wind gust at the leading edge and the pressures

6 CONCLUSIONS

• Aerodynamic admittances due to vertical wind gust for basic cross-sections were evalu-

ated by computation based on the two-dimensional RANS. The aerodynamic admittance

for a thin plate of B/D=200 predicted by the present computations agreed well with the

Sears Function. As for the rectangular and hexagonal sections, the calculation results


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agreed well with an experimental result, too. The present method seems like a promising

tool for predicting gust force characteristics of bridge decks.

• The aerodynamic admittance function of a bridge deck depends on its cross section. The

feature of aerodynamic admittance of the hexagonal section was in agreement with that

of a flat plate. On the other hand, it was found that the separated flow from the leading

edge had a great influence on the lift force property, so that there was a big difference in

aerodynamic admittance between the B/D=5 and the flat plate.

REFERENCES

[1] J.D.Holmes: Prediction of the Response of a Cable Stayed Bridge to Turbulence, Pro-

ceeding of 4th International Conference on Wind Effects on Building and Structures,

187-197, 1975.

[2] G.L.Larose, H.Tanaka, N.J.Gimsing and C.Dyrbye : Direct Measurements of Buffeting

Wind Forces on Bridge Deck, Journal of Wind Engineering and Industrial Aerodynam-

ics, 809-818, 1998.

[3] A.Larsen, J.H.Walther: Discrete Vortex Simulation of the Flow around Five Generic

Bridge Deck Sections, Proceedings of 8th US National Conference on Wind Engineering,

591-602, 1997.

[4] S.Kuroda: Numerical Computation of Unsteady Flows for Airfoils and Non-airfoil

Structure, AIAA 2001-2714,31st AIAA Fluid Dynamics Conference & Exhibit, 2001

[5] S.E.Rogers and D.Kwak: Upwind Differencing Scheme for the Time-Accurate

Incompressible Navier-Stokes Equations, AIAA Journal, Vol.28, No.2, 253-262, 1990.

[6] F.R.Menter : Two-Equation Eddy-Viscosity Turbulence Models for Engineering Appli-

cations, AIAA Journal, Vol.32, No.8, 1598-1605, 1994.

[7] T.Ihara, A.Hatanaka and H.Kobayashi : Behavior of negative pressure affecting on the

characteristics of lift force of rectangular cylinders in two-dimensional vertical gust ,

Proceedings of annual meeting of JSCE Kansai branch (2006) , I-52, 2006(in Japanese)

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