CROSS-CORRELATION OF FLUCTUATING COMPONENTS OF WIND SPEED BASED ON STRONG WIND MEASUREMENT

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1 The Seventh Asia-Paifi Conferene on Wind Engineering, November 8-12, 29, Taipei, Taian COSS-COELATION OF FLUCTUATING COMPONENTS OF WIND SPEED BASED ON STONG WIND MEASUEMENT Maymi Fjimra 1 and Jnji Maeda 2 1 JSPS esearh Fello (Gradate), Gradate Shool of Hman-Environment Stdies, Kysh University, 6--1 Hakozaki, Higashi-k, Fkoka , Japan, fjimra@indmail.arh.kysh-.a.jp 2 Professor, Falty of Hman-Environment Stdies, Kysh University, 6--1 Hakozaki, Higashi-k, Fkoka , Japan, maeda@ind.arh.kysh-.a.jp ABSTACT We report on the strtre of atmospheri trblene, fosing espeially on the spatial orrelations of three omponents of fltating ind speed sing or reords taken from measrements at a transmission toer loated at the tip of a ape dring Typhoon KIK. Or reslts sho that the non-dimensional ross-spetrm of longitdinal omponent of fltating ind speed beteen to distantly-positioned points of the type has the best fit to the measrement reslts, and the approximate expression to the type ithot speial fntions an sffiiently explain several harateristis of the non-dimensional ross-spetrm of isotropi trblene. In addition, e sho that the ross-orrelation oeffiient beteen longitdinal and vertial lateral omponents of fltating ind speed is approximately expressed sing the spatial orrelation oeffiient beteen the omponents of fltating ind speed in the same diretion. KEYWODS: TUBULENCE STUCTUE, COSS-SPECTUM, COSS-COELATION Introdtion For the dynami response analysis of a strtre nder strong inds, the spetral response method in a freqeny domain or the step-by-step integration of motion eqation in a time domain is sed. The estimation of the ind response of a slender strtre reqires the longitdinal and/or lateral omponents (- and v-omponents) of fltating ind speed in a horizontal diretion, and the vertial lateral omponent (-omponent) is rarely onsidered. Hoever, e mst set p the ind field inlding -omponent in the ase of allating the response of a sspension-bridge or a long-span ondtor hih is strongly affeted by - omponent. The spetral response method postlates a linear deformation of a strtre, and so for the strtre for hih deformation is not linear, e generally have to perform the stepby-step integration of motion eqation. For example, for a opling system ith able, sh as a transmission toer, an inrease in the gradient angle of the ondtor-plane ith the ind speed enlarges not only the geometrial nonlinearity of the ondtor s rigidity bt also the effet of -omponent of ind speed on the response as revealed by [Fjimra and Maeda et al. (27)]. In this ase, it is neessary to generate the three-dimensional field of fltating ind speed inlding the three omponents in a time domain. And then the nonlinear response simlation of the opled system of toer and ondtor is performable sing the step-by-step integration method. In order to generate the three-dimensional ind field ith the three omponents of fltating ind speed nder strong inds, information abot the spatial strtre beteen the three omponents is needed. When the gst loading fator method as introded to the

2 The Seventh Asia Paifi Conferene on Wind Engineering November 8-12, 29, Taipei, Taian Photo 1: Vie of observed toer from sea. U1 (214.m) U2 (23.m) U3 (183.m) U4 (148.m) Photo 2: Arrangement of anemometer. U1~U8 : Ultrasoni anemometer Observed toer U (11m) U6 (83.m) U7 (3.m) 214.m : For three omponents of ind speed : For to omponents of ind speed U8 (3m) 2m Figre 1: elation beteen the observed toer and the azimth diretion. Figre 2: Position of anemometers. ind load design of strtres and bildings, many semi-empirial eqations onerning the spatial strtre of ind ere proposed in stdies by [ (1961) and Harris (1971)]. Crrently, s spetrm, hih as derived from the theory of isotropi trblene in the ork by [ (1948)], is often sed as the poer spetrm of fltating ind speed. This spetrm expression is in good agreement ith the measrement reslts as reported by [Maeda and Adahi (1983), Maeda and Makino (1988)]. [ESDU71 (197)] desribes s ross-spetrm as being the best expression for non-dimensional ross-spetrm of fltating ind speed beteen to distantly-positioned points, bt the exponential expression proposed in a stdy by [ (1961)] as an approximate expression is sed in pratie bease it is a simpler and more onvenient form of expression to se than s ross-spetrm. Hoever, the exponential expression is very different from s rossspetrm derived from the theory of isotropi trblene in that its orrelation vale is onsistently 1 at the point here freqeny is zero regardless of the distane beteen to points, and that it does not have any negative vales as pointed ot by [Maeda and Makino (198)]. As for the ross-orrelation beteen the longitdinal omponent (-omponent), horizontal lateral omponent (v-omponent) and vertial lateral omponent (-omponent) of fltating ind speed, [ESDU7431 (1974)] desribes the o-variane v and v as being small and hih an be ignored, bt this is not so for the o-variane near the grond, and the formlated vale of / σ σ as a fntion of height. Hoever, there are fe earlier reports hih desribe the o-varianes and ross-orrelation oeffiients beteen - and - omponents of fltating ind speed at to distantly-positioned points, and e annot find any later reports.

3 The Seventh Asia Paifi Conferene on Wind Engineering November 8-12, 29, Taipei, Taian Table 1: Mean harateristis of the measrement data. Sample No. Anemometer No. U (m/s) σ (m/s) σ v (m/s) σ (m/s) x L (m) x L v (m) x L (m) U No. 1 U U U No. 2 U U U No. 3 U U Japan Kysh Observed toer Figre 3: Path of Typhoon KIK. T9612 (KIK) In this paper, the athors estimate the non-dimensional ross-spetrm of fltating ind speed beteen to distantly-positioned points sing ind measrement data at a 2m high transmission toer loated at the tip of a ape dring the passing of Typhoon KIK (1996) And the estimated vales are ompared ith s exponential expression and an approximate expression to the type proposed by [Maeda and Makino (198)] ithot nontrollable speial fntions. Additionally, the athors disss the o-variane and ross-orrelation oeffiients beteen - and -omponents of fltating ind speed from their data and propose an approximate expression for the vales, hih is sefl for the three-dimensional simlation of a fltating ind field in a time domain. Otline of Strong Wind Observation Observed Toer and Terrain arond Observed Toer The observed toer as a 214.m tall kv transmission toer at the tip of a ape at a height of m above sea level, as shon in Photo 1. Figre 1 shos the relation beteen the observed toer and the azimth diretion. Hills stood to the north of the toer, and it as srronded by sea on its soth, east and est sides. Otline of Measrement Data Eight ltrasoni anemometers, for for three omponents of ind speed and for for to omponents, ere positioned on the toer for ind measrement, as shon in Figre 2. Every anemometer as loated several meters from a main post in a horizontal diretion, as shon in Photo 2. The sampling freqeny of measrement as 12 Hz. The strong ind observations ere ondted dring typhoons and non-typhoons from 1996 to 1997, and in this paper e seleted a total of three samples measred at U4, U6 and U8 hen the mean ind speed of U2 as more than 2m/s and the ind diretion as from the est here the observed toer as donind dring the passing of Typhoon KIK (No.12 in 1996), hose path is shon in Figre 3. Eah sample as 6 seonds long. Table 1 shos the mean ind

4 The Seventh Asia Paifi Conferene on Wind Engineering November 8-12, 29, Taipei, Taian speed, U, the standard deviation, σ, and the longitdinal sale of trblene, x L, at every anemometer allated sing every sample, and Figres 4(a), 4(b) and 4() sho the time histories reorded at anemometers U4, U6 and U8 for sample 1, respetively. Wind reords of U1 at the top of the toer ere not inlded in this stdy bease measrements ere affeted by a position light near anemometer U1. Non-dimensional Cross-spetrm of Fltating Wind Speed Proposed eqations for non-dimensional ross-spetrm The non-dimensional ross-spetrm of -omponent of fltating ind speed beteen vertially-distantly-positioned points A and B, as shon in Figre, is ritten as follos: S S =, (1a) S ( n) S ( n) ζ, n) ( ζ, τ )exp( j2πnτ ) A B S ( = dτ. (1b) Where ( ζ, τ ) = E[ A( t) B( t + τ )], S A (n) and S B (n) are the poer spetra at points A and B respetively, ζ is the vertial distane beteen points A and B and n is the freqeny. The non-dimensional ross-spetrm of and types and an approximate expression to the type proposed in the ork stdy by [Maeda and Makino (198)] are ritten as follos, respetively: nζ : S = exp kr, U (2a) /6 11/6 ~ θ θ : S = K/6( θ) K1/6( θ), 2 2 (2b) -omponent (m/s) -omponent (m/s) -omponent (m/s) v-omponent (m/s) v-omponent (m/s) v-omponent (m/s) (a) U (b) U () U8 Figre 4: Time histories of ind speed for sample 1. -omponent (m/s) -omponent (m/s) -omponent (m/s)

5 The Seventh Asia Paifi Conferene on Wind Engineering November 8-12, 29, Taipei, Taian ~ 2 Maeda and Makino: S = exp( k1θ ) (1 k2θ ). (2). 4 x 2 2 1/ 2 Where kr = 13( ζ / zm), z m =.( za + zb ), θ = {(.747ζ L ) + (2πnζ U ) }, k 1 = 1., k2 =.2 and Kµ ( θ) is the modified Bessel fntion of the seond kind of order µ. Measrement eslts of Non-dimensional Cross-spetrm Figre 6 shos the non-dimensional ross-spetra of every non-dimensional distane, ζ x L, estimated from the measrement data in the ase of n=. The spetra and rossorrelation oeffiients hih are mentioned later are estimated sing the Ato-regressive Model and the A order is deided determined by the minimm FPE riterion as introded by [Maeda and Makino (1981), Maeda and Makino (1992)]. In the figre, the expressions ritten ith Eqs. (2a), (2b) and (2) have been added. As shon in the figre, the estimated vales of measrement reslts derease ith an inrease in the non-dimensional distane. s expression is in good agreement ith the measrement reslts, bt s expression is not in agreement ith them. And the expression proposed by Maeda and Makino is in very good agreement ith s expression. Figres 7(a) and 7(b) sho the non-dimensional ross-spetra of the lateral omponents (v- and -omponents) of fltating ind speed estimated from the measrement data in the ase of n=, respetively. The non-dimensional distane on the horizontal axis is defined as ζ. x L sing the harateristi of the theory of isotropi x x x trblene that is L : Lv: L = 2:1: 1. The estimated vales of measrement reslts of v- and - omponents derease ith an inrease in the non-dimensional distane and trae s expression as ell as those of -omponent. Figres 8(a), 8(b) and 8() sho the non-dimensional ross-spetra of -omponent x of every non-dimensional freqeny, n L U, in the ases that ζ x L.28,.44 and.82 U ζ B B A A z Figre : Positions of points A and B. S v ( ζ,) 1. v B v A B A y x Measred Maedaζ /. x L S ( ζ,) Measred Maeda ζ / x L Figre 6: Non dimensional ross-spetra of -omponent in the ase of n=. S ( ζ,) 1. (a) v-omponent. (b) -omponent. Figre 7: Non dimensional ross-spetra of v- and -omponents in the ase of n=. ~ S Measred Maedaζ /. x L (a) ζ / x L. 28 (b) ζ / x L. 44 () ζ / x L. 82 Measred Maeda n ζ / x L ~ S Measred Maeda n ζ / x L ~ S Measred Maeda Figre 8: Non-dimensional ross-spetra of -omponent for eah non-dimensional distane. n ζ / x L

6 The Seventh Asia Paifi Conferene on Wind Engineering November 8-12, 29, Taipei, Taian respetively. In the figres, the expressions ritten ith Eqs. (2a), (2b) and (2) have been added. egardless of non-dimensional distane, s expression is in good agreement ith the measrement reslts, and espeially in the ase of ζ x L.44 hen the measrement reslts inlde negative vales as shon in Figre 8(b), althogh it is lear that s ross-spetrm annot express this harateristi. The expression proposed by Maeda and Makino an sffiiently explain the harateristis of the measrement reslts. Cross-orrelation Coeffiient beteen - and -omponents of Fltating Wind Speed Definition of Cross-orrelation Coeffiient beteen - and -omponents The ross-orrelation oeffiient beteen - and -omponents of fltating ind speed at vertially-distantly-positioned points A and B, as shon in Figre, is ritten as follos: ( ζ, τ ) ( ζ, τ ) =. (3) σ Aσ B Where ( ζ, τ ) = E[ A( t) B ( t + τ )] and σ and A σ are the standard deviations of - and - B omponents at points A and B, respetively. Measrement eslts of Cross-orrelation Coeffiient beteen - and -omponents Figres 9(a), 9(b) and 9() sho the vales of / σ σ = (,), (, τ ) and ( ζ,) estimated from the measred data, respetively. As shon in the figres, the vales of / σ σ are beteen -.3 and -.4 at every height, z. The vales of (, τ ) are distribted nearly symmetrially against the vertial axis of τu x L = and lose to zero ith an inrease in absolte vales of non-dimensional time, x τ U L. The vales of ( ζ,) are lose to zero ith an inrease in non-dimensional distanes, x ζ L, and the absolte vales of ( ζ,) are smaller than.1 in the ase of ζ x L >.. In order to evalate the statistial natre of (, τ ) and ( ζ,), e define the ratio of the ross-orrelation oeffiient to / σ σ, * ( ζ, τ ), and Figres (a) and (b) sho the measrement reslts of * ~ (, τ ) and * ( ζ,) respetively. In Figre (a), the averages of * (, τ ) estimated from all the measrement data are disretely expressed. The longitdinal /σ σ 2 - Measred z(m) (, τ ) τu / x L Measred ( ζ,) (a) / σ σ (b) (, τ ) () ( ζ,) Figre 9: Co-variane and ross-orrelation oeffiients in the ases of τ = and ζ = beteen - and -omponents. 1. * (, τ ) Measred Eqation (4) τu / x L Measred Eqation () (a) * (, ) (b) * τ ( ζ,) Figre : atio of ross-orrelation oeffiient to / σ σ in the ases of ζ = and τ =. 1. * ( ζ,) ζ / x L ζ / Measred x L

7 The Seventh Asia Paifi Conferene on Wind Engineering November 8-12, 29, Taipei, Taian spatial orrelation oeffiient, f (r), and the lateral spatial orrelation oeffiient, g (r), beteen the omponents of fltating ind speed in the same diretion proposed by [ (1948)], hih are ritten as Eqs. (4) and () respetively, have been added in Figres (a) and (b) respetively: 3 f ( r) = f ( Uτ ) = α 1/ 1 au 1 τ K1/ 3( a1u τ ), (4) 1/ 3 g( r) = g( ζ ) = α1 a1ζ {K1/ 3( a1ζ ). a1ζ K 2/3( a1ζ )}, () here 2 2/ 3 x x α 1 = Γ(1/ 3). 926 and a1 = π Γ(/ 6) Γ(1/ 3) L L. As shon in Figre (a), Eq. (4) is in good agreement ith the measrement reslts. And as shon in Figre (b), the measrement reslts vary bt Eq. () traes the median measrement reslt. Derivation of Approximate Expression for Measrement eslt [Hinze (1987)] shoed that the ross-orrelation oeffiient beteen - and - omponents is defined sing the spatial orrelation oeffiient beteen the omponents of fltating ind speed in the same diretion, f (r) and g (r), as follos: ~ f ( r1 ) g( r1 ) ( ζ, τ ) = Uτ ζ. (6) 2 r Where r 1 = ( Uτ ) + ζ. Eq. (6) as derived by assming that the o-variane beteen the mtally-perpendilar omponents of fltating ind speed is zero in the field of isotropi trblene. Hoever, the measrement reslts of the o-variane beteen and omponents,, are not zero, as shon in Figre 9(a). Additionally, in onsidering that (, τ ) / σ σ f ( r) and ( ζ,) / σ σ g( r) as shon in Figre, it is thoght that the approximate expression for the measrement reslts of the ross-orrelation oeffiient beteen - and -omponents an be ritten as follos: ~ f ( r1 ) 2 g( r1 ) 2 ( ζ, τ ) = Uτ ζ + ( Uτ ) Uτ ζ ( ζ ) 2 2 (7) r1 σ σ r1 σ σ Figres 11(a) and 11(b) sho the omparisons of the measrement reslts of ( ζ, τ ) ith Eq. (7) in the ases of ζ x L =.2 and ζ x L = 1.4, respetively. In the ase of ζ x L =.2, the approximate expression is in good agreement ith the measrement reslts. On the other hand, in the ase of ζ x L = 1.4, the approximate expression inldes the positive vales and is different from the measrement reslts. Hoever, as shon previosly, there are fe orrelations beteen and omponents in the ase of ζ x L >., and so it is thoght that this distintion is not important. Conlsions Some of or findings onerning the spatial orrelation of three omponents of fltating ind speed based on the ind speed measrement at the transmission toer τu / x L τu / x L Measred Eqation (7) -.2 Measred Eqation (7) -.3 ( ζ, τ ) -.3 ( ζ, τ ) (a) ζ / x L =. 2 (b) ζ / x L = 1. 4 Figre 11: Comparison of measrement reslts of ross-orrelation oeffiients beteen - and -omponents ith Eq. (7) in the ases of ζ x L =. 2 and ζ x L =1. 4.

8 The Seventh Asia Paifi Conferene on Wind Engineering November 8-12, 29, Taipei, Taian loated at the tip of a ape dring Typhoon KIK (1996) are smmarized as follos. (1) The measrement reslts of the non-dimensional ross-spetra are in good agreement ith s expression derived from the theory of isotropi trblene. (2) The approximate expression to the type ithot speial fntions an sffiiently explain several harateristis of the measrement reslts of the non-dimensional rossspetra. (3) The ross-orrelation oeffiients beteen the longitdinal and vertial lateral omponents of fltating ind speed are lose to zero ith an inrease in absolte vales of non-dimensional time and non-dimensional distanes. (4) The ross-orrelation oeffiients beteen the longitdinal and vertial lateral omponents of fltating ind speed are approximately expressed sing the spatial orrelation beteen the omponents of fltating ind speed in the same diretion. The spatial orrelation of the three omponents of fltating ind speed, as desribed in this paper, an be sefl for the three-dimensional simlation of a ind field in a time domain. Aknoledgement This ork as spported by Grant-in-Aid for JSPS Fellos, , Japan Soiety for the Promotion of Siene. eferenes, A. G. (1961), The Spetrm of Horizontal Gstiness near the Grond in High Winds, Q. J. oy. Met. So., 87, ESDU7431 (1974), Charateristis of Atmospheri Trblene near the Grond Part II: Single Point Data for Strong Winds (netral atmosphere), Engineering Sienes Data Unit, Item No ESDU71 (197), Charateristis of Atmospheri Trblene near the Grond Part III: Variations in Spae and Time for Strong Winds (netral atmosphere), Item No. 71. Fjimra, M., Maeda, J., Morimoto, Y. and Ishida, N. (27), Aerodynami Damping Properties of a Transmission Toer Estimated Using a Ne Identifiation Method, 12th International Conferene on Wind Engineering Preprints-Vol.1, Harris,. J. (1976), The Natre of the Wind, Proeedings of CIIA Seminar on the Modern Design of Windsensitive Strtres, Constrtion Indstry esearh and Information Assoiation, London. Hinze, J. O. (1987), Trblene, MGra-Hill, , T. v. (1948), Progress in the Statistial Theory of Trblene, Proeedings of Natl. Aad. Si., Vol.34, Maeda, J. and Makino, M. (198), Classifiation of Cstomary Proposed Eqations elated to the Component of the Mean Wind Diretion in the Strtre of Atmospheri Trblene and these Fndamental Properties, Trans. Arhitet. Inst. Japan, No.287, (in Japanese). Maeda, J. and Makino, M. (1981), On the Methods of Spetral Analyses of Trblent Winds Adaptation of Ato-regressive Model, Trans. Arhitet. Inst. Japan, 3, (in Japanese). Maeda, J. and Makino, M. (1983), On the Spatial Strtres of Longitdinal Wind Veloities near the Grond in Strong Winds, Jornal of Wind Engineering and Indstrial Aerodynamis,, Maeda, J. and Makino, M. (1988), Poer Spetra of Longitdinal and Lateral Wind Speed near the Grond in Strong Winds, Jornal of Wind Engineering and Indstrial Aerodynamis, 28, Maeda, J. and Makino, M. (1992), Charateristis of Gsty Winds Simlated by an A..M.A. Model, Jornal of Wind Engineering and Indstrial Aerodynamis, 41-44,