Discrete vortex method simulations of the aerodynamic admittancein bridge aerodynamics

Johannes Tophøj Rasmussen a, Mads Mølholm Hejlesen a, Allan Larsen b, Jens Honore Walther a,c,!

a Department of Mechanical Engineering, Technical University of Denmark, Building 403, DK-2800 Kgs. Lyngby, Denmarkb COWI A/S, Parallelvej 2, DK-2800 Kgs. Lyngby, Denmarkc Computational Science and Engineering Laboratory, ETH Zurich, Universitatsstrasse 6, CH-8092 Zurich, Switzerland

a r t i c l e i n f o

Article history:Received 22 April 2010Received in revised form23 June 2010Accepted 29 June 2010

Keywords:Discrete vortex methodWind field simulationTurbulenceFluid structure interactionAerodynamic admittanceFlat plate

a b s t r a c t

We present a novel method for the simulation of the aerodynamic admittance in bluff bodyaerodynamics. The method introduces a model for describing oncoming turbulence in two-dimensionaldiscrete vortex method simulations by seeding the upstream flow with vortex particles. The turbulenceis generated prior to the simulations and is based on analytic spectral densities of the atmosphericturbulence and a coherence function defining the spatial correlation of the flow. The method isvalidated by simulating the turbulent flow past a flat plate and past the Great Belt East bridge. Theresults are generally found in good agreement with the potential flow solution due to Liepmann.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The discrete vortex method is widely used in academia and bythe industry to model two-dimensional (2D) bluff body flows(Smith and Stansby, 1988; Koumoutsakos and Leonard, 1995;Cottet, 1990; Larsen, 1998a; Larsen andWalther, 1998; Taylor andVezza, 2001; Zhou et al., 2002; Taylor and Vezza, 2009; Ge andXiang, 2008). The discrete vortex method is a Lagrangianformulation of the Navier–Stokes equations governing the flow.The equations are solved by convecting and diffusing vortexparticles carrying vorticity. A boundary element method may beused to enforce the no-slip condition at the solid boundaries andhence alleviates the time consuming mesh generation required inEulerian methods. Moreover, the simulation of flow past multiplebodies undergoing arbitrary motion is straightforward. The 2Ddiscrete vortex method implementation DVMFLOW (Walther, 1994)is an engineering tool, which has been used extensively by theconsulting company COWI1 to simulate the flow past bridgesections (Larsen and Walther, 1997; Larsen, 1998b; Vejrum et al.,2000; Larsen et al., 2008). The simulations provide detailedvisualisation of the flow field and time history of the aerodynamicforces. Moreover, the aerodynamic derivatives and associated

flutter limit are extracted from the simulations by imposing aprescribed heave and pitch motion of the bridge section (Larsenand Walther, 1998; Walther and Larsen, 1997). So far thesestudies have been limited to laminar flow simulations and only afew studies have considered the modeling of turbulence using 2Dparticle vortex methods cf. Alcantara Pereira et al. (2004) andPrendergast (2007). In the recent work of Prendergast andMcRobie (2006) and Prendergast (2007), oncoming turbulencewas modelled by seeding the free stream with vortex particlesand simulations were performed to study buffeting in bridgeaerodynamics. In the present work, we extend the model ofPrendergast and McRobie to enable, for the first time, simulationsof the aerodynamic admittance in bluff aerodynamics.The exten-sion is performed within the DVMFLOW implementation.

2. Aerodynamic admittance

The influence of turbulence on the aerodynamic forces can bequantified by the aerodynamic admittance which is the focus ofthe present study. In the analysis of the aerodynamic admittance(Larose, 1997), the lift coefficient

CL !L

12rU2C

, "1#

is assumed to be a linear function of the angle of attack (a), thus

CL"a# ! CL0 $CuLa, "2#

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

Journal of Wind Engineeringand Industrial Aerodynamics

0167-6105/$ - see front matter & 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.jweia.2010.06.011

! Corresponding author at: Department of Mechanical Engineering, TechnicalUniversity of Denmark, Building 403, DK-2800 Kgs. Lyngby, Denmark.Tel.: +4545254327; fax: +4545884325.

E-mail address: jhw@mek.dtu.dk (J.H. Walther).1 www.cowi.com

J. Wind Eng. Ind. Aerodyn. 98 (2010) 754–766

where L denotes the lift force per unit length, C the chord length ofthe bridge section, r the density of the fluid, U the horizontalmean free stream velocity, and CuL % @CL=@a. For a fixed rigidstructure, the instantaneous angle of attack (a) depends on U andthe horizontal and vertical velocity fluctuations u and w,respectively, thus

a&w

U$u: "3#

By inserting the instantaneous velocity

Ui !!!!!!!!!!!!!!!!!!!!!!!!!!!!!w2$"U$u#2

q, "4#

into Eq. (1) and combining it with Eqs. (2) and (3) the lift force canbe expressed as

L!rC2

"CL0U2$2CL0Uu$CuLwU#, "5#

where terms involving products of fluctuations have beenneglected. Thus, the lift force due to the wind fluctuationsbecomes

Lf !rUC2

'2CLu$CuLw(: "6#

Assuming that the force due to velocity fluctuations is a stationaryrandom process, the equation can be transformed into thefrequency domain (Larose, 1997):

SLL !rUC2

" #2

'4C2L Suuwu

L $Cu2L SwwwwL (, "7#

where wuL ! wu

L "o# and wwL ! ww

L "o# denote the admittances due tothe spectral density of the horizontal Suu ! Suu"o# and verticalSww ! Sww"o# velocity fluctuations, and o the angular frequencyof the fluctuation. It is difficult to distinguish between thecontribution from u and w, and instead the lumped aerodynamicadmittance is defined:

wL"o# !SLL"o#

"12rUC#2'4C2

L Suu"o#$Cu2L Sww"o#(: "8#

Often CL is small compared to CuL and the aerodynamicadmittance can be reduced to a relation between the spectraldensity of the vertical velocity fluctuations Sww and the lift forceSLL. That is, fluctuations in the lift force are mainly influenced bythe vertical velocity fluctuations. For the pitching moment M theaerodynamic admittance can be derived analogously (Larose,1997)

wM"o# !SMM"o#

12rUC

2$ %2

'4C2MSuu"o#$Cu2MSww"o#(

, "9#

where

CM !M

12rU2C2

, "10#

and CuM ! @CM=@a.In spite of the aerodynamic admittance generally being

frequency dependent the constant transfer functions

SLL"o# !12rCUCuL

" #2

Sww"o#, "11#

SMM"o# !12rC2UCuM

" #2

Sww"o#, "12#

are often being used, assuming proportionality between thespectrum of the vertical fluctuations Sww and the spectra SLL andSMM by CuL and CuM , respectively. This, in effect renders theaerodynamic admittance unity in the whole frequency range. Therelation stems from potential theory with the lift force being

the superposition of multiple lift force signals, each proportionalto a sinusoidal vertical velocity fluctuation.

In the present work the flow past a flat plate and the flow pastthe Great Belt East bridge is simulated. By sampling time series ofthe velocity fluctuations, lift forces and pitching moments, thespectra Suu, Sww, SLL and SMM can be computed, and in turn theaerodynamic admittances wL and wM .

3. Vortex method

An incompressible flow with constant kinematic viscosityn is governed by the 2D Navier–Stokes equations in vorticityform

Dou

Dt!

@ou

@t$u )=ou ! nDou: "13#

The fluid velocity u is computed from the vorticity xu !ouez andthe stream function c:

xu !=* u, "14#

u!=* "cez#, "15#

Combining Eqs. (14) and (15) and assuming c to be divergencefree leads to the Poisson equation

Dc!+ou: "16#

Eq. (16) forms the basis of hybrid vortex particle-mesh methodssuch as the Vortex-In-Cell algorithm (Birdsall and Fuss, 1969;Sbalzarini et al., 2006; Morgenthal and Walther, 2007; Chatelainet al., 2008). In the present study, the Poisson equation is solvedusing Green’s function solution:

c"x# !C$Z

G"x+y#ou"y#dy, "17#

u"x# !U+Z

K"x+y#ou"y#dy, "18#

where C is the far-field stream function, and G and K thecorresponding 2D Green’s functions:

G!+12p logjxj, "19#

K !12p

x

jxj2* : "20#

The vorticity field is approximated by discrete vortex particlescarrying circulation:

G!Z

Aou dA!

Z

Su ) ds, "21#

where A is the area of the particle bounded by S. The vorticityfield is discretized by a superposition of N discrete vortexparticles:

oeu"x# !

XN

i

ze"xi+x#Gi, "22#

where ze"x# is a smooth approximate to the Dirac delta function:ze"x# ! "1=e2#z"jxj=e#, and e is the smoothing radius. The presentstudy uses the second order Gaussian kernel: z"r# ! "1=2p#e+r2=2

cf. e.g. Winckelmans and Leonard (1993).The discrete kinematic relation governing the flow is obtained

from Eqs. (18) and (22):

u"x# !U+XN

i

Ke"x+xi# *Giez, "23#

where Ke !+"qe"x#=jxj2#x is the smooth velocity kernel, andqe"x# ! q"jxj=e#, and q"r# ! "1=2p#"1+e+r2=2#.

J.T. Rasmussen et al. / J. Wind Eng. Ind. Aerodyn. 98 (2010) 754–766 755

The motion and strength of the discrete particles is solvedusing viscous splitting (Chorin, 1973). Hence, the particles are firstconvected:

dxpdt

! u"xp#, "24#

dxu

dt! 0, "25#

and subsequently diffused:

dxpdt

! 0, "26#

dxu

dt! nDxu: "27#

In the present DVMFLOW implementation, the convection step issolved using first or second order explicit time integrationschemes, and diffusion is modeled using the method of randomwalks (Chorin, 1973). Hence, after the convection step (Eq. (24)),the position of the vortex particles is perturbed with a randomdisplacement, drawn from a normal distribution with zero meanand variance 2nDt. Dt denotes the simulation time step.

The solid boundaries are discretized using a boundary elementtechnique (Wu, 1976; Wu and Gulcat, 1981). The no-penetrationcondition is enforced by determining the strength of the vortexsheets (g) on the panels, which forms a linear system of equations.A unique solution is obtained by imposing Kelvin’s circulationtheorem (Walther and Larsen, 1997):X

G! 0: "28#

The introduction of upstream vortex particles with a total non-zero circulation GT requires a modification of Eq. (28) such thatP

G!P

GT .The computational efficiency of the discrete vortex method is

closely related to the solution of the N-body problem implied byEqs. (23) and (24) which nominally scales as O(N2). The presentimplementation uses the fast multipole method (Greengard andRokhlin, 1987; Carrier et al., 1988) to achieve an optimal O(N)scaling.

The forces and moments are calculated from the pressuredistribution which in turn is calculated from the distribution ofthe vortex sheet along the boundary s:

1r@p@s

!+@g@t

: "29#

The derivatives CuL and CuM are found by finite differences of CLand CM measured at different angles of attack assuming either alaminar or turbulent oncoming flow. Laminar flow simulations areused similar to the practice of wind tunnel testing.

4. Synthesizing turbulence

To introduce turbulence into the oncoming flow of a simula-tion a time series of vortex particles is generated prior to thesimulation (Prendergast and McRobie, 2006). These particles areinserted into the flow during the simulation at a fixed insert rate.The vortex particles are convected downstream forming a band ofparticles which induce turbulent velocity fluctuations. Initiallyturbulent velocity series are generated on the nodes of a regularvertical grid using the Shinozuka–Deodatis method (Shinozukaand Jan, 1972; Deodatis, 1996), and in accordance with thespectral densities of the atmospheric velocity fluctuations. Thegrid consists of Np vertically aligned quadratic cells, see Fig. 1. Ineach point two velocity time series are generated: one in thestreamwise (horizontal) direction and another in the crosswise(vertical) direction, u and w, respectively. Presently these velocity

time series are based on the modified von Karman spectra of theEngineering Sciences Data Unit (1993) (ESDU), see Appendix A.Atmospheric turbulence is anisotropic, thus two spectra areneeded: one for the horizontal and another for the verticalfluctuations. The magnitude and distribution of the spectradepend on various physical parameters such as the surfaceroughness length scale (z0), the height above ground level (h),and the free stream velocity (U). However, the availableparameters can only be varied within a certain range and withlimited influence on the turbulence intensity

Iu !su

U,

Iw !sw

U: "30#

Hence, varying the physical parameters mostly influences themagnitude of the atmospheric velocity spectral density while thedistribution is largely unaffected cf. Clementson (1950) and Fung(1955). Therefore to obtain a specific turbulence intensity~Iw ! ~sw=U the analytic spectral density Sww"o# is scaled

~Sww"o# !sw

~sw

" #2

Sww"o#, "31#

to match the target intensity:Z

~Sww"o#do! ~s2w: "32#

The spectral densities are discretized in Nf discrete frequencies

ok ! kDo, k! 0,1, . . . :,Nf+1, "33#

with a uniform spacing

Do!omax

Nf: "34#

The upper cut-off frequency omax is chosen such that the energyof the discarded frequency range is negligible. Furthermore theNyquist criterion

omaxrp

Dtgen, "35#

is fulfilled to avoid aliasing. Dtgen is the time step at whichvelocities are generated on the grid, and 1=Dtgen is thecorresponding insert rate.

The velocity series in the points m and n on the grid arespatially correlated through the coherence function (Davenport,

H

C

U

x

z

Fig. 1. From the grid at the left vortex particles are released and convecteddownstream by the flow induced by the vortex particles of the flow and the freestream velocity U. Coordinates (x,z) are given relative to the center of the grid ofheight H, and (u,w) are the respective velocity components. Solid objects with achord length C inserted in the flow are placed on the x-axis.

J.T. Rasmussen et al. / J. Wind Eng. Ind. Aerodyn. 98 (2010) 754–766756

1968; Rossi et al., 2004)

Cohumun "o# ! e+f , "36#

with the decay function

f !o2p

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!C2ux"xm+xn#2$C2

uz"zm+zn#2q

0:5"U"zm#$U"zn##, "37#

where Cux ! 3 and Cuz ! 10 are decay coefficients (Simiu andScanlan, 1996). The spatial correlation for the vertical velocities isanalogous to Eqs. (36) and (37), and the decay coefficients areassumed Cwx ! Cux, Cwz ! Cuz (Prendergast, 2007). Hereby theinfluence of a velocity series in m to a series in n becomes

Sumun "o# !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Sumum "o#Sunun "o#

pCohumun "o#eiyumun "o#: "38#

The angular phase shift

yumun "o# !o xn+xmU

, "39#

is defined in correspondence with Taylor’s Hypothesis, i.e. thephase shift is equal to the duration for the flow to move betweenthe points m and n times the frequency. It is seen that Sumun isgiven directly from Suu for m=n. It is assumed that there is onlyspatial correlation between velocity components in the samedirection (Prendergast, 2007), i.e. that there is no correlationbetween the u and w velocities. The contribution of all velocitiesto each other is contained in the cross spectral matrix:

S"o# !'Sumun m,nrNp( 0

0 'Swmwn m,nrNp(

" #

, "40#

We let m be the index of a velocity process in a point and byvelocity process understand a velocity series either horizontal orvertical. It follows that there will be twice the number of velocityprocesses as the number of points in the grid Np. Then

fm"t# !!!!!!!!!!!!2Do

p X2Ng

n ! 1

XNf+1

k ! 1

jHmn"ok#jcosb"t#,

b"t# !okt$ymn"ok#$fnk, "41#

is the mth velocity process (Shinozuka and Jan, 1972) found bysumming a series of Nf cosine waves from each of the 2Np velocityprocesses, including m. In effect the summation over m is only toNp with no correlation between vertical and horizontal velocitiesin the cross spectral matrix S"ok# cf. Eq. (40). The amplitudes ofthe cosine waves are determined from the Cholesky decomposi-tion H"ok# of S"ok#. Furthermore

ymn"ok# ! arctanIm'Hmn"ok#(Re'Hmn"ok#(

" #, "42#

is the complex argument of Hmn"ok# and fnk is a random phase inthe interval '0;2p(.

The efficiency of the process is significantly improved by usingfast Fourier transforms (FFTs) to carry out the summation(Deodatis, 1996). The velocity series becomes

fm"tr# !!!!!!!!!!!!2Do

p XNp

n ! 1

Re'Cmnr(, "43#

where

Cmnr !X2Nf+1

k ! 0

cmnkei"2p=2Nf #kr , "44#

is the Fourier transform of

cmnk ! jHmn"ok#jei"ymn"ok#$fnk#: "45#

Using the maximum discrete time step that fulfills the Nyquistcriterion Eq. (35), the discrete time becomes

tr ! rDtgen, rA '0;2Nf+1(: "46#

For each of the quadratic cells of the grid the circulation isintegrated from the grid node velocities using the trapezoidal rule(Prendergast, 2007). The circulation is corrected in magnitude bya factor K such that

G! KGtrapezoidal: "47#

This accounts for the mismatch between the linear assumption ofthe integration and the actual circular velocity contours of a pointvortex. The actual distribution of circulation from a singular pointvortex of strength Gvortex on a cell with side length Dx is not linearbut given by

dGds

!Gvortex

pDx"s2$1#, sA '+1;1(, "48#

see Fig. 2. By integrating Eq. (48) around the cell boundary it isseen that G!Gvortex if K ! p=2. For each cell a particle isassociated with the corresponding circulation and insertedduring the flow simulation at every Dp !Dtgen=Dt time step. Theside length of the quadratic cells Dx!DpDtU corresponds to thedistance the particles are convected by the free stream velocitybetween subsequent particle releases. As the particles are beingconvected downstream they form a band of particles with a totalnon-zero circulation

PGT . The final condition for the panel

strength of the boundary elements Eq. (28) is modified such thatPG!

PGT . Imposing total zero circulation would otherwise

affect the panel strengths and render force calculation by surfacepressure distribution impossible. Alternatively the particlestrengths can be modified by subtracting the mean particleturbulence circulation from the particles being released such thatno net circulation is added. However, this disrupts the streamwisecorrelation between the particles and leads to strongly reducedenergies in the low frequency region of spectral density of theresulting simulation velocities.

5. Validation and results

We perform simulations of the turbulent flow past a flat plateand past the Great Belt East bridge to test and validate the currentimplementation. The admittance for the flow past the flat plateserves as a reference case and allow systematic variation of the keynumerical parameters. The reference parameters related to thediscretization of the spectral density of atmospheric turbulence is

!

Fig. 2. The circulation from a point vortex integrated with the corner pointapproximation on a cell with side length Dx is marked by the dark gray area. Theactual non-linear distribution dG=ds!Gvortex=pDx"s2$1#, sA '+1;1( is the unifica-tion of the dark and the light gray areas.

J.T. Rasmussen et al. / J. Wind Eng. Ind. Aerodyn. 98 (2010) 754–766 757

the upper angular frequency omax ! 36:7 rad=s and the number ofdiscrete frequencies Nf!4096. When integrating the grid velocitiesto particle strength a correction factor K ! p=2 is used. Thevariation of the spectra with respect to the point of sampling isexamined, and the dependency of the number of particles Np perrelease as well as the particle insert interval Dp is investigated.Reference values for these parameters are Np!120 and Dp ! 4.

The atmospheric turbulence is reconstructed from the ESDUspectra, defined in Appendix A using the reference parameters.The reference spectra are subsequently scaled to meet a specifiedvertical turbulence intensity of Iw!5%.

When introducing a solid structure into the flow we alsointroduce an additional length scale, i.e. the chord length (C), andhence the Reynolds number Re!UC=n. We study the influence ofRe and turbulence intensity Iw on the aerodynamic admittance. Asreference the flow is simulated at Re!10,000 to ensure arelatively thin boundary layer and to allow comparison with thepotential flow solution. For the boundary layer to remain stable(Walther and Larsen, 1997) and to ensure sufficient spatialresolution the simulations are not carried out at higher Reynoldsnumbers. The relevant Reynolds number for full scale bridges isO"108# and for the wind tunnel tests typically O"105#. However, toallow comparison with the results obtained for the flat plate wemaintain Re!10,000 for the bridge simulations.

The velocity time series and particle strengths are generated inSI-units and non-dimensionalized before being used as input forthe DVMFLOW simulations. As characteristic length a typical valuefor the bridge chord length C!30m is used, for both bridgesection and the flat plate benchmark. The characteristic freestream velocity is U!35m/s which is a typical design wind speedin bridge engineering. The recorded velocities, forces and pitchingmoments are re-dimensionalized and analysed. As the aim of thistext is to investigate flow properties of solid objects all positionshave been given in units of C. For comparison purposes this is alsothe case when investigating the flow when no solid body ispresent.

Spectral densities are statistical properties of a sample or asignal and often contain a significant amount of noise. To reducenoise the signals have been subsampled with 50% overlap and theresulting spectra averaged. This significantly improves theconsistency of the spectra but at the cost of reduced low-frequency information. Further noise reduction can be achievedby applying window functions at the cost of uncertainty in themagnitude of the spectra (Harris, 1978). Window functionspreserve the shape of the spectrum and reduce noise, thusadmittances have been calculated from spectra based on wind-owed samples, as any change of the magnitude cancels out. Beforesampling any velocity and force the simulation is carried out untilboth the sampling points and the structures are well immersed inthe turbulent flow. Most signals in the present work consist of

38,000 samples which are divided into 2–10 subsamples. Thoughsome uncertainty remains, this reduces noise significantly at thecost of the frequency range. To be able to compare similar spectrain the same figure, approximations by N-point Bezier curves havebeen used to remove noise when necessary. N-point Bezier curvesuse all N samples for the approximation. Certain cases have beenrun extensively with a total of 380,000 samples. This allowsincreased subsampling and in turn accurate spectra in a widefrequency range. The typical range of interest for structuralanalysis of long span bridges is 0.05–1.0Hz. For pedestrianbridges the range of interest is typically in the range of0.5–5.0Hz. In angular frequency these intervals are 0.31–6.3and 3.14–31.4 rad/s, respectively. In the following the terms:low-, mid- and high-range frequencies refer approximately to thefrequency intervals separated by o! 0:8 and 10 rad=s.

5.1. Power spectral density of the velocity

When generating the upstream particles many parametersinfluence the simulated turbulent velocities. The optimal value ofthe numerical parameters is not necessarily obvious, nor can theybe chosen freely due to computational constraints. These para-meters are investigated in this section.

5.1.1. Sensitivity to upper cutoff and spectral resolutionThe upper cutoff of the frequency (omax) and the finite number

of discrete frequencies (Nf) at which the velocity spectrum isdefined will affect the resulting velocity signal. Both the velocitysignal on the grid and in turn the vortex induced velocities in thesimulation are affected. We study these parameters in thefollowing and compare the result with theoretical values.

The upper cutoff (omax) is chosen above the frequency range ofinterest such that the turbulent energies s2

u and s2w are conserved

to the extent possible. Fig. 3a shows the discarded energy

Z"omax# ! 1+1s2

Z omax

0Sdo

" #* 100%, "49#

as a function of omax. Thus, the reference cutoff omax ! 36:7 rad=sentails a loss of horizontal turbulent energy of Zu ! 1:7% andZw ! 6:7% vertical velocity energy. For the vertical velocity energyloss to be reduced to 1% the upper cutoff must be approximately20 times higher. Thus for a constant Do the complete energy ofthe spectrum cannot be expected to be preserved as the memoryrequirements scale as O"NfN

2p #.

To conserve the energy of the spectra for oromax a sufficientnumber of discrete frequencies (Nf) is required. Fig. 3b shows theturbulence intensities Iu and Iw of the grid velocity series as afunction of Nf. The turbulence intensities have been normalized byIu

!!!!!!!!!!!!1+Zu

p, the expected values corrected by the effect of the upper

0

2

4

6

8

10

12

10 30 50 70 90

! [%

]

"max [rad/s]

707580859095

100105

0 5000 10000 15000

I* [%

]

Nf

Fig. 3. (a) Deficiency of energy Z! "1+Romax

0 S do=s2# * 100% as a function of upper cutoff omax for the spectral density of the horizontal (——) and vertical (- - - - - -)velocities. (b) Normalized turbulence intensity I, ! IRMS=Itarget * 100% for the horizontal (——) and vertical (- - - - - -) velocity time series on the grid as a function of thenumber of discrete frequencies Nf. The energy loss due to omax has been accounted for in Itarget.

J.T. Rasmussen et al. / J. Wind Eng. Ind. Aerodyn. 98 (2010) 754–766758

cutoff. The vertical turbulence intensity Iw converges faster than Iuand for Nf!4096 they are 99.96% and 99.04% of the expectedvalues, respectively. The energy loss from the finite frequencyrange is more significant than that caused by the resolution of thespectrum. In addition we find that the grid velocities are in goodagreement with the prescribed spectra, both in terms of theturbulent energy s2

u and s2w and also the spectral distribution: Suu

and Sww (not shown).

5.1.2. Effects of frequency discretization and circulation integrationWhile the energy of the time series of the grid velocity is a first

indicator of the validity of the synthesis, the spectral density ofthe velocity time series sampled during the simulation showsmore detail of the method, cf. Fig. 4. In the following we study theeffect of the discrete frequencies (Nf) on the velocity spectrasampled during the simulation.

Both the horizontal (Suu) and vertical (Sww) spectra show asignificant dependency on the number of discrete frequencies Nf,see Figs. 4a and b, respectively. For both the horizontal andvertical spectra, the high frequency range has converged at just128 discrete frequencies. The higher the spectral resolution thefurther the spectra converge into the low frequency range and forNf Z2048 the spectra have converged for all but the very lowestfrequencies. The atmospheric turbulence spectra have beendiscretized with the lowest non-zero discrete frequency Do. Dois indicated for each spectral resolution by a vertical linepatterned similarly to the corresponding spectrum, and belowDo the sampled spectra contain little energy. For the coarsediscretization (Nf!128) the discrete frequencies are clearlyvisible, indicating that exactly the frequencies of interest arepreserved by the conversion from grid velocities to particleinduced velocities. We choose Nf!4096 as a compromise betweenaccuracy in the low frequency range and the required memoryand computational resources.

The spectral densities of the time series of the grid velocityshow perfect match to the target spectra (not shown), whereasthe spectral density of the simulation deviate from the target.Hence the deviations are a result of the method of converting gridvelocity to particle strengths and the finite release rate of theupstream particles. The strongly deviating low frequency rangeenergy of Sww is observed only for the frequencies at which theatmospheric turbulence spectrum has been discretized, Fig. 4.This indicates that the deviation is primarily an effect ofconverting grid velocities to particle strengths and not an effectof the actual release of particles during the simulation. The latteris investigated in Section 5.1.6. It is worth noting how well thevalues of the discrete frequencies are preserved in the process ofconverting grid velocities to particle strengths. Sww is above target

for the whole frequency range, see Fig. 4b, as is also the case forSuu except for the low frequency region in which Suu is lower thanthe target, see Fig. 4a.

5.1.3. Influence of circulation correctionThe correction factor, K ! p=2 cf. Eq. (47), is based on the

simple comparison of the integral over the grid velocity inducedby a single vortex particle to the strength of the vortex particle.The spectra of the simulated velocities have been observed to begreater than their target, and we therefore investigate the effect ofvarying K. The variance of the sampled horizontal velocity signals2u is below target and above target for the vertical velocity signal

s2w. This corresponds to the deviations in the low frequency range

of Suu and Sww, respectively, see Fig. 4. Outside the low frequencyrange the spectra are generally above the target by a constantfactor & 1:25. However, the distribution is correct indicating thatthe conversion from grid velocities to particle strengths is valid inthis region and that the offset is caused by the circulationcorrection K. By varying K the spectra can be offset uniformly by afactor as seen in Fig. 5. Thus, K can be adjusted to achieve betteragreement between the spectral density of the sampled velocityand the target. As expected the variance scales proportionallywith K2, hence (su,sw) equals (4.87,4.17), (6.13, 5.36), and (11.0,8.23) for K ! 0:70p=2, 0:85p=2, and 1:00p=2, respectively.

5.1.4. Spatial dependencyThe velocity field induced from a vortex particle is purely

tangential relative to the position of particle cf. Eq. (23). Hence,the horizontal velocity component at a point is mainly induced byparticles vertically aligned with the point and vice versa. Ideally asample point should be surrounded with a larger number ofvortex particles to ensure convergence of the turbulent energiesof both the horizontal and vertical components. However, a finitedistance to the edge of the particle band is sufficient as othereffects, such as viscous diffusion, become more dominant than thecontribution from far-field particles. We study this effect byconsidering the variation of the spectral energy as a function ofthe position relative to the release grid. We do this by sampling atdifferent positions on the centerline of the particle band, down-stream of the release grid, x!1C,2C,4C,y,128C. As the horizontalvelocities mainly depend on particles vertically aligned with thesampling point, the energy s2

u increases and converges quicklywith respect to the downstream position x (not shown). Due tothe strong dependency of the vertical velocity to the particlesdownstream and particularly upstream, s2

w increases with x andconverges approximately 16 times farther downstream than s2

u, atx!32C (not shown).

S uu

[m2 /s]

" [rad/s]

10-3

10-2

10-1

100

101

102

10-3

10-2

10-1

100

101

102

10-3 10-2 10-1 100 10110-3 10-2 10-1 100 101

" [rad/s]

S ww

[m2 /

s]

Fig. 4. The spectral density (a) Suu and (b) Sww of the velocity from the simulations, using turbulence series based on analytic spectra with varying number of discretefrequencies Nf: 128 (—————), 512 (– – – – – –), 2048 (- - - - - -), and 16,384 (— -— -— -). The target spectrum (———) is shown as reference. The highest frequency inthe range is omax . The spectra are averages from 10 subsamples and have been smoothened by N-point Beziers.

J.T. Rasmussen et al. / J. Wind Eng. Ind. Aerodyn. 98 (2010) 754–766 759

The influence of x on s2w is predominant and the increased

energy is located mainly in the low frequency range of Sww

cf. Fig. 6b. Immediately downstream of the release grid, and untilx& 8C (not shown), the energy of the high frequency rangesgrows from approximately 70% below to 70% above target. At thispoint the spectrum has converged for all but oo0:25, in whichfrequency range the energy grows continuously as the flow isconvected. For sampling points in the range 16C–45C, Sww

remains unchanged and the sampling point can be consideredsufficiently immersed in the turbulent flow, and sufficiently farfrom the rotation-free flow upstream of the release grid. After 45Cthe spectral energy gradually decreases. Initially in the mid rangefrequencies but eventually extending to the entire spectrum.Fig. 7 indicates that this is due to mixing of rotation free flowtowards the centerline of the particle band.

The standard deviation of the horizontal fluctuations s2u is

below target which is primarily due to the energy of the lowfrequency range of Suu. It is conjectured that a higher particle bandwould cause a similar deviation of Suu to the target as that of Sww,

i.e. stronger low frequency energy. Due to memory constraints theparticle band height of the reference flow is limited to 12C, i.e. to15% of the distance required for s2

w to converge. Suu converges atx!2C, considerably faster than Sww. As the flow is convecteddownstream the horizontal turbulent energy diminishes as shownin Fig. 6a. From x!8C the energy of the mid-range frequenciesdecreases slightly and further downstream, at x!64C the entirefrequency range of Suu decays almost uniformly.

The aerodynamic admittance has a stronger dependence onSww than on Suu and any solid objects to be investigated in thepresent work is placed with the leading edge at x!20C. Herebythe object is immersed in a region of the flow with constantspectral properties and a representative velocity signal can besampled upstream of the object. When sampling 4C upstream ofthe leading edge neither the plate nor the Great Belt East bridgesection has any significant influence on the spectral densities ofthe sampled velocities.

Sampling will be performed on the centerline of the particleband. Sampling off of the centerline of the 12C high particle band

10-3

10-2

10-1

100

101

102

10-3

10-2

10-1

100

101

102

10-2 10-1 100 101 10-2 10-1 100 101

" [rad/s]

S uu

[m2 /s]

" [rad/s]

S ww

[m2 /s]

Fig. 5. Spectral density of the (a) horizontal velocity Suu and (b) vertical velocity Sww for circulation integral correction (Eq. (47)) of K ! 0:70p=2 (— — — — — —),K ! 0:85p=2 (- - - - - -), K ! 1:00p=2 (— -— -— -— -—) and target (———). The spectra have been plotted in the range of the discrete frequencies used for turbulencegeneration.

10-2 10-1 100 101

" [rad/s]

S uu

[m2 /s]

10-2 10-1 100 101

" [rad/s]

S ww

[m2 /s]

10-3

10-2

10-1

100

101

102

10-3

10-2

10-1

100

101

102

Fig. 6. Spectral densities (a) Suu and (b) Sww of the velocity field sampled on the particle band centerline: 2C (— — — — — —), 32C (- - - - - - - - -), 128C (— -— -— -)downstream of the release grid. The height of the grid is 12C. The signal has been subsampled twice and the spectra smoothened by N-point Beziers. The spectra have beenplotted in the range of the discrete frequencies used for turbulence generation.

x/H :0 1 2 3 4 5 6 7 8

Fig. 7. Flow visualization in the range from x!0C to 100C. The downstream distance is marked in units of particle band heights H (above). Below the particle band the x-coordinates of the sampling points of Section 5.1.4 have been marked. The height of the release grid is H!12C.

J.T. Rasmussen et al. / J. Wind Eng. Ind. Aerodyn. 98 (2010) 754–766760

at x! 16C shows only insignificant influence when e.g. sampling4C off the centerline, i.e. 2C from the edge of the particle band (notshown). The deviation is only for the very lowest frequency rangeand mainly in the horizontal velocity spectrum Suu. This suggeststhat the finite grid height H!12C leaves a broad vertical margin inwhich the flow is spectrally uniform. This does not mean that aparticle band height of 2*2C is sufficient as will be seen inSection 5.1.5.

5.1.5. Dependency of grid heightThe band of particles must be of a certain height for a structure

to be exposed to a flow with the properties of atmosphericturbulence. The required height of the particle band is investi-gated by keeping the insert interval Dp fixed thereby not changingthe particle density. The height of the release grid is varied from3C to 12C corresponding to 30 to 120 vortex particles per release.

The energy of the horizontal turbulent fluctuations increasesdue to the influence of the higher band of upstream particles. AsFig. 8a indicates increasing the grid height causes a consistentincrease of turbulent energy in the low frequency range of theenergy spectrum. This indicates that the height of the particleband limits large low frequency structures.

There is not a similar consistent relation between themagnitude of the vertical turbulent energy and the grid height,as shown in Fig. 8b. The vertical turbulent energy s2

w growsdownstream of the release grid until it converges. For all gridheights s2

w starts to decay at approximately 3.5 times the grid

height for the present settings. This corresponds well to themixing of rotation free flow towards the centerline of the particleband as shown in Fig. 7.

5.1.6. Dependency of inter-particle spacingThe band of particles must be sufficiently densely filled and the

distance between particles at the release grid is determined bythe width of the release grid cells. As described in Section 4 thewidth of the cells is proportional to the free-stream velocity andthe interval with which particles are inserted into the stream Dp.Thus Dp controls the density of particles and by decreasing theparticle insert interval Dp, particles are being inserted at a higherrate. Since the cells of the release grid are quadratic, the particleswill be more closely spaced in both the streamwise and verticaldirections. The number of particles per release strongly influencesthe computational requirements, thereby limiting the height ofthe particle band. Thus to be able to set Dp ! 1 a particle band ofheight 3C is used with the entailing energy deficiencies describedin Section 5.1.5.

By varying the rate with which particles are released into theflow it has been observed that an increased energy in the highfrequency range follows an increase of the particle insert intervalDp. When sampling velocities directly at the release grid thefrequency of the particle release

orelease !2pDtDp

, "50#

10-3

10-2

10-1

100

101

102

10-3

10-2

10-1

100

101

102

10-2 10-1 100 101 10-2 10-1 100 101

" [rad/s]

S uu

[m2 /s]

" [rad/s]

S ww

[m2 /s]

Fig. 8. Spectral density for the (a) horizontal Suu and (b) vertical Sww velocities, for varying release grid height 3C and 30 points (—————), 6C and 60 points (– – – – – –),9C and 90 points (- - - - - - - - -), 12C and 120 points (— -— -— -). The signal is sampled 10C downstream of the release grid and has been subsampled five times and the spectraaveraged. The spectra have been plotted in the range of the discrete frequencies used for turbulence generation.

" [rad/s]

S uu

[m2 /s]

10-4

10-3

10-2

10-1

100

101

102

10-4

10-3

10-2

10-1

100

101

102

10-2 10-1 100 101 10210-2 10-1 100 101 102

" [rad/s]

S ww

[m2 /s]

Fig. 9. Spectral density of the (a) horizontal velocities Suu and (b) vertical velocities Sww sampled 10C downstream of grid. The insert interval is varied between1 (— — — — — —), 4 (- - - - - - - - -), and 6 (— -— -— -). The vertical long-dashed line marks omax , the solid lines mark orelease corresponding to Dp ! 6, Dp ! 4, Dp ! 1 fromleft to right. Due to memory constraints a fixed release grid height of & 3C has been used to allow insertion every time step (Dp ! 1). The spectra are averages of fivesubsamples and have been smoothened by an N-point Bezier.

J.T. Rasmussen et al. / J. Wind Eng. Ind. Aerodyn. 98 (2010) 754–766 761

is visible as a well-defined spike. Further downstream the energyof the spike spreads to the surrounding frequency range obscuringthe spectral position of orelease. At x!10C the positions of theenergy spikes corresponding to Dp ! 4 and 6 can no longer bedistinguished, but the energy has spread to frequencies lowerthan the upper cutoff frequency omax as shown in Fig. 9. orelease

corresponding to Dp ! 1 is outside the visible range. Setting Dp ! 1removes the artificial high frequency energy for ooomax andthough this gives better agreement with the respective targets, itrestricts the height of the release grid due to memory constraints.Furthermore the added energy is far above the frequency range ofinterest when looking at cable bridges or pedestrian bridges. Inspite of the deviation of the high frequency range of the spectrathe aerodynamic admittance shows good agreement with targetcf. Section 5.2. In the present work large cable bridges are ofinterest and Dp ! 4 is chosen as this gives little deviation of thespectra in the high frequency range and ensures a sufficientheight of the particle band.

5.2. Aerodynamic admittance of a flat plate

5.2.1. Comparison with analytic solutionThe flow past an infinitely thin plate is well studied in

potential flow theory (Theodorsen, 1935; von Karman andSears, 1938), and by approximating the potential flow conditionsthe plate serves as a suitable benchmark. By assuming thevertical fluctuations to be small compared to the mean speedof the flow the admittance has been approximated by Liepmann(1952)

wL !1

1$"pC=U#o : "51#

In the present study, the potential flow past an infinitely thinplate subjected to an oncoming turbulent flow is approximated bythe viscous flow past a flat plate of finite length and thickness. Theviscous diffusion is modeled using random walks and hence theturbulent velocity fluctuation should be above a certain level forthe turbulent velocity fluctuations to dominate the fluctuations ofthe viscosity modeling at the solid surface. Due to the finitethickness of the plate, the viscous flow and the turbulentfluctuations causing instantaneous angles of attack of up to 121,separation occurs around the plate, see Fig. 10. In the presentwork a plate thickness D!1/200C has been used. Since therequirements for the potential flow solution are not fully met,some deviation is anticipated. The measured slopes of the lift (CuL)and pitching moment (CuM) are CuL ! 5:5 and CuM !+1:18,respectively. The experimental values (Larose and Livesey, 1997)obtained for a plate with a chord-to-thickness ratio of C=D! 16are CuL ! 5:8 and CuM !+1:43, and thus a deviation less than 5%and 17%, respectively.

Fig. 11 shows the spectral densities for the horizontal (Suu) andvertical (Sww) velocities, the lift force SLL and the pitching momentSMM, as well as the corresponding aerodynamic admittance of thelift force wL and pitching moment wM . The velocity spectra agree

with the results obtained in Section 5.1. The lift force spectrum SLLand pitching moment spectrum SMM have been plotted with thepredicted spectra from the frequency independent relation,Eqs. (11) and (12) as reference. The reference spectra are basedon the assumption of a frequency independent admittance andfrom Figs. 11c and d it is seen that the spectra cross the referencespectra. Except from the area of intersection of the spectra,neither their magnitudes nor their slopes match. In spite of theaerodynamic admittance generally being frequency dependentthe frequency independent assumption is widely used as an initialapproximation. In the low frequency range an increase of thespectral energy, similar to that of Sww, can be seen in both SLL andSMM. This indicates that the forces on the plate are reactions tothese added low frequency components of the flow. In spite ofthat the vertical spectrum deviates from the prescribedturbulence spectrum, the aerodynamic admittance of both thelift force wL and the pitching moment wM are in reasonableagreement with Liepmann’s approximate solution. The computedadmittances wL and wM are generally 75% higher than theapproximation. In the low frequency range the computedadmittances deviate below the profile of Eq. (51). This isconjectured to be due to large low frequency turbulent flowstructures resulting in instantaneous angles of attack outside thevalid range of Eq. (2). For frequencies above 10 rad/s the deviationof the computed admittances to the analytic solution increasesfurther. It is recalled that comparison is performed with thepotential flow solution entailing the above-mentionedrequirements.

In experiments the aerodynamic admittance has been found todepend on the spectral density of the turbulent fluctuations(Larose and Mann, 1998). When generating the wind tunnelturbulence by spires the measured admittance is generally aboveLiepmann’s approximation (Eq. (51)). However, at the lowestfrequencies the measured admittance is considerably belowEq. (51). We believe that the stronger admittance (and in turnthe lift signal) is due to body induced turbulence (Larose andMann, 1998). Both tendencies can be seen at Figs. 11e and f. In thepresent work the flow around the plate separates due to theturbulent gusts as shown in Fig. 10. Though the ratio of chord tothickness C/D!12.7 for the bridge section experiments (Laroseand Mann, 1998) is lower than C/D!200 for the flat plate, theagreement to of Liepmann’s approximation of the thin plate isbetter for the experimental results. However, the rectangularleading edge of the flat plate used in the present study is blunt,which may increase the aerodynamic admittance as demon-strated experimentally for a C/D!16 plate (Larose and Livesey,1997) cf. Fig. 11e.

5.2.2. Influence of Reynolds number and turbulence intensityThe deviation in the high frequency range of the computed

admittances to the analytic solution is suspected to be caused bythe viscosity modeling. That is, the standard deviation of thediffusion step length, Eq. (27), influences the magnitude of the high

Fig. 10. Flow visualization of turbulent flow past the flat plate at Re!10,000 and Iw!5%. The turbulent fluctuations result in flow separation.

J.T. Rasmussen et al. / J. Wind Eng. Ind. Aerodyn. 98 (2010) 754–766762

frequency deviation. The reference case is simulated at differentReynolds numbers, by varying viscosity and thereby the averagediffusion step lengths. As shown in Fig. 12 the high frequency rangeof both the lift force admittance wL and the pitching momentadmittance wM decrease with increasing Reynolds number.

As seen in Section 5.1.3 increasing the strength of the particlesinserted to generate turbulence increases turbulence intensity.Contrary to expectation the high frequency range deviation is notincreased as shown in Fig. 13. The deviation to the analyticsolution is decreased considerably and for specified vertical

10-3

10-2

10-1

100

101

102

103

10-2 10-1 100 101

S uu[

m2 /s]

" [rad/s]

10-3

10-2

10-1

100

101

102

103

10-2 10-1 100 101

" [rad/s]

S ww[

m2

/s]

103

104

105

106

107

108

10-2 10-1 100 101

S LL

[N2 s]

" [rad/s]

105

106

107

108

109

1010

10-2 10-1 100 101

" [rad/s]S

MM

[N2 m

2 s]

10-2

10-1

100

10-2 10-1 100 101

! L

" [rad/s]

10-2

10-1

100

10-2 10-1 100 101

" [rad/s]

! M

Fig. 11. Spectral densities (- - - - - -) (a) Suu and (b) Sww, and their target (———). (c) SLL and (d) SMM are spectral densities of the lift force and pitching moment (- - - - - -)measured on the flat plate and (———) is the predictions by Eqs. (11) and (12). Aerodynamic admittance of the lift force wL (- - - - - -) and experimental results (Larose andLivesey, 1997) for a C/D!16 plate (+), (e) and the pitching moment wM (f) for the flat plate (- - - - - -), compared to Liepmann’s (1952) approximation (———). The graphsare based on 380,000 samples subsampled 100 times. The spectra have been plotted in the range of the discrete frequencies used for turbulence generation.

10-2

10-1

100

10-2 10-1 100 101

! L

" [rad/s]

10-2

10-1

100

10-2 10-1 100 101

" [rad/s]

! M

Fig. 12. Aerodynamic admittance of the lift force wL (a) and the pitching moment wM (b) for the flat plate compared to Liepmann’s approximation (———). Results fromsimulation at Re!1000 (— — — — — —), Re ! 10,000 (- - - - - -), Re ! 100,000 (— -— -— -). The signals have been subsampled 10 times. The spectra have been plotted inthe range of the discrete frequencies used for turbulence generation.

J.T. Rasmussen et al. / J. Wind Eng. Ind. Aerodyn. 98 (2010) 754–766 763

turbulence intensities IwZ10% the admittance assumes theshape of Eq. (51) from the mid-range frequencies up to omax.This indicates that the influence of the random walk to the forcesignal is solely related to the particles generated and emitted from

the body surface to enforce the no penetration condition. For bluffbodies, vortex shedding appears in the admittance as a peak at theshedding frequency (not shown). By increasing the turbulenceintensity the forces from the turbulent vertical fluctuations

10-2

10-1

100

10-2 10-1 100 101

! L

" [rad/s]

10-2

10-1

100

10-2 10-1 100 101

" [rad/s]

! M

Fig. 13. Aerodynamic admittance of the lift force wL (a) and the pitching moment wM (b) for the flat plate compared to Liepmann’s approximation (———). By varying thespecified turbulence intensity from Iw!1.25% (—————), Iw!2.5% (– – – – – –), Iw!5.0% (- - - - - - - - -), Iw!10% (— -— -— -) it is seen that the influence from viscousdiffusion at the surface becomes less dominant. The signals have been subsampled 10 times. The spectra have been plotted in the range of the discrete frequencies used forturbulence generation.

10-3

10-2

10-1

100

101

102

103

10-2 10-1 100 101

S uu[

m2 /s]

" [rad/s]

10-3

10-2

10-1

100

101

102

103

10-2 10-1 100 101

" [rad/s]

S ww[

m2

/s]

103

104

105

106

107

108

10-2 10-1 100 101

S LL[

N2 s]

" [rad/s]

105

106

107

108

109

1010

10-2 10-1 100 101

" [rad/s]

S M

M[N

2 m2 s]

10-2

10-1

100

10-2 10-1 100 101

! L

" [rad/s]

10-2

10-1

100

10-2 10-1 100 101

" [rad/s]

! M

104

Fig. 14. Spectral density (- - - - - -) (a) Suu and (b) Sww, and their target (———). (c) SLL and (d) SMM are the spectral densities of the lift force and pitching moment(- - - - - -) measured on the Great Belt bridge section and (———) is the predictions by the frequency independent assumption Eqs. (11) and (12). Aerodynamic admittanceof the lift force (- - - - - -) wL (e) and the pitching moment wM (f) for the Great Belt bridge section (- - - - - -), compared to Liepmann’s approximation (———). The graphsare based on 380,000 samples subsampled 100 times. The spectra have been plotted in the range of the discrete frequencies used for turbulence generation.

J.T. Rasmussen et al. / J. Wind Eng. Ind. Aerodyn. 98 (2010) 754–766764

become dominant and the admittance tends to the analyticsolution (Liepmann, 1952).

5.3. Aerodynamic admittance of Great Belt East bridge section

5.3.1. Comparison with LiepmannBridge sections are typically bluff bodies with increased vortex

shedding compared to the flat plate. The thickness of the GreatBelt East bridge section is D!0.14C. Fig. 14 shows theaerodynamic admittance of the lift force wL and pitchingmoment wM as well as the corresponding spectral densities. Thevelocity spectra are equal to those of the flat plate, as shown inFig. 11. SLL and SMM for the bridge section is of the same order ofmagnitude as the frequency independent assumption Eqs. (11)and (12) in the frequency interval from 0.8–10 rad/s. This is due tovortex shedding. Therefore using the frequency independentapproximation for an initial estimate of the force spectrum maygive reasonable results. Outside the interval the agreement issimilar to that of the flat plate. Both wL and wM are stronger thanLiepmann’s approximation (51) and similar to the admittances ofthe flat plate. However, the vortex shedding is stronger for thebridge section which manifests itself as a peak in both wL and wM

thereby deviating from the analytic solution that does not takeinto account vortex shedding. As seen experimentally (Larose andMann, 1998) the admittance is lower than Eq. (51) at lowerfrequencies.

6. Conclusions

We have presented a novel technique for the calculation of theaerodynamic admittance in bluff body aerodynamics. The methodis based on the two-dimensional discrete vortex method andintroduces a turbulent oncoming flow through the insertion ofupstream vortex particles modeling the anisotropic turbulentvelocity spectra. The admittances of the lift and pitching momentare obtained from the measured spectra of the turbulent flow fieldand the corresponding spectra of the aerodynamic loads. Themethod has been validated through detailed simulations of theturbulent flow past a flat plate and past the Great Belt East bridge.The results were found in good agreement with the semi-analytical model of Liepmann and wind tunnel experiments. Themethod is expected to be a useful engineering tool in bridgeaerodynamics.

Acknowledgements

The research has been supported in part by the COWIfoundation, and the Danish Research Council (Grant. No. 274-08-0258). Computational resources at the Department of Physicsat DTU have been made available through the Danish Center forScientific Computing (DCSC). The authors wish to acknowledgethe collaboration with Sanne Poulin.

Appendix A. ESDU atmospheric turbulence spectra

The Engineering Sciences Data Unit (1993) (ESDU) providesspectral densities of the horizontal and vertical atmosphericturbulent fluctuations, Suu and Sww, respectively. Parameter valuesused in the present work are stated as the parameters areintroduced:

nSuus2u

! b12:987nu=a

'1$"2pnu=a#2(5=6$b2

1:294nu=a'1$"pnu=a#2(5=6

F1, "52#

nSww

s2w

! b12:987'1$"8=3#"4pnw=a#2("nw=a#

'1$"4pnw=a#2(11=6

$b21:294nw=a

'1$"2pnw=a#2(5=6F2, "53#

with the reduced frequencies

nu !n"xLu#U

, "54#

nw !n"xLw#

U: "55#

We use tabulated values of the coefficients a, b1 and b2 cf. Harris(1990). In the present work these are a! 0:662, b1 ! 0:80 andb2 ! 0:20. The longitudinal standard deviations

su !7:5Zu, 0:538$0:09ln z

z0

$ %h i2

1$0:156ln u,fz0

$ % , "56#

sw ! su 1+0:45cos4p2zh

$ %h i, "57#

where

Z! 1+6fzu,

, "58#

p! Z16, "59#

depend of the friction velocity u* from the logarithmic law profile

U"z# !u,

kln

zz0

" #: "60#

k is the von Karman constant assumed to be 0.4, z0 is the surfaceroughness length. u* is determined from the velocity U!35m/s atz!70m above ground level when z0!0.003m for open water isused (Engineering Sciences Data Unit, 2001). f!10+4Hz is themid-latitude Coriolis frequency and h!660m the atmosphericboundary layer thickness. The longitudinal length scales of thehorizontal and vertical fluctuations are given by

xLu !A3=2k

suu,

$ %3z

2:5K3=2z 1+z

h

& '21$5:75z

h

& ' , "61#

xLw ! xLu 0:5sw

su

" #3" #

, "62#

and with the present configuration xLu ! 229:6m and xLw ! 21:8m.Also

Ak ! 0:115 1$0:315 1+zh

$ %6( )

, "63#

Kz ! 0:19+"0:19+K0#exp +Bkzh

$ %Nk( )

, "64#

K0 !0:39

Ro0:11, "65#

Bk ! 24Ro0:155, "66#

Nk ! 1:24Ro0:008, "67#

Ro!u,

fz0, "68#

F1 ! 1$0:455exp +0:76nu

a$ %+0:8

( ), "69#

F2 ! 1$2:88exp +0:218nw

a$ %+0:9

( ): "70#

J.T. Rasmussen et al. / J. Wind Eng. Ind. Aerodyn. 98 (2010) 754–766 765

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