tall chimneys circular section

Engineering Structures 23 (2001) 502–520www.elsevier.com/locate/engstruct

Across-wind aerodynamic parameters of tall chimneys with circularS. Arunachalam et al. /Engineering Structures 23 (2001) 502–520

S. Arunachalama, S.P. Govindarajub, N. Lakshmanana, T.V.S.R. Appa Raoa,*

a Structural Engineering Research Centre, Madras, 600 113, Indiab Indian Institute of Science, Bangalore, 560 012, India

Received 3 November 1999; received in revised form 16 May 2000; accepted 16 May 2000

Abstract

The prediction of across-wind response of circular cylinders remains a challenging task, despite extensive research efforts. Anattempt has been made to correlate the rms lift coefficient due only to vortex shedding,C9L,VS both in wind tunnel and full-scaleconditions by separating the local rms lift coefficient,C9L, into two components, one due to the lateral turbulence and the otherdue to the vortex shedding. Based on the literature, and also using test results measured by the authors, it is found that the finalvalue of C9L,V, as discussed in the paper, show a mean value of 0.089 with a coefficient of variation of 18%, independent ofReynolds number regime. The above value plus 1.66 times the standard deviation gives a value of 0.115 which is in excellentagreement with the value ofC9L,VS=0.12 recommended for design of chimneys under open terrain conditions, as per the IndianStandard Code of Practice: IS: 4998 Part-2, 1992. Further it is shown that the Griffin universal Strouhal number, G attains a meanvalue of 0.065 with a coefficient of variation (cov) of 8%, independent of sub-critical and trans-critical Reynolds number regime. 2001 Elsevier Science Ltd. All rights reserved.

Keywords:Circular cylinder; Across-wind response; Turbulence; Vortex shedding; Aerodynamic parameters; RMS lift coefficient

1. Introduction

Studies on wind induced vibrations of a circular cylin-der such as a chimney, have attracted extensive attentionby various researchers in the past but the subject stillremains one of the classical problems of bluff body aero-elasticity, particularly with respect to vortex inducedvibrations. Wind tunnel testing has been used towardspromoting the understanding, and assessment of theforces acting on the structure, and its response to vortexinduced oscillations. The response prediction depends onmany parameters, which describe details of the approachflow, the forces exerted on the structure due to windaction and dynamic sensitivities of the structure. Unfor-tunately these aspects are not adequately understood [1]and they are influenced by many factors including Reyn-olds number, Strouhal number, rms lift coefficient, free-stream turbulence, surface roughness, aspect ratio of the

* Corresponding author. Tel.:+91-44-235-2122; fax:+91-44-235-0508.

0141-0296/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved.PII: S0141-0296 (00)00060-2

structure, etc. Thus, there appears to be no single com-prehensive theoretical model developed from first prin-ciples to predict the vortex induced response of a circularbody such as chimney, as noted by Vickery, Simiu, Mel-bourne and Kareem among others [1–5,26]. The mostimportant pioneering research contributions towards pre-diction of crosswind response of isolated reinforced con-crete chimneys have been made by Scruton and Vickeryand his coworkers [6–10,14].

The Vickery and Basu model is currently regarded asthe most well developed model for predicting responseof RCC chimneys to vortex shedding [1] and hence hasbeen incorporated in several international codes of prac-tice [11–13]. While the above model is conceptuallyadvanced in addressing the problem of vortex inducedmotion of the structure using the random vibrationapproach, the predictions of responses of full-scale chim-neys using this method can vary as much as from 25 to30% [3,8]. Despite this fact, the above method is cur-rently being widely used. The alternative method of pre-diction of response of full-scale chimneys based on windtunnel studies on model chimneys has been reported to

503S. Arunachalam et al. / Engineering Structures 23 (2001) 502–520

Nomenclature

Cd mean drag coefficient (fD/1/2rU2d)C9d rms drag coefficientC9L rms lift coefficient (fL/1/2rU2d)Cpb base pressure coefficientd cylinder diameter, diameter of structured effective diameter of tapered chimney equal to average diameter of top 1/3rd heightf frequencyfs vortex shedding frequencyfL fluctuating lift force per unit lengthH height of cylinder/chimneyk wake parameterLc,i correlation length, in diameters, of fluctuating liftLux longitudinal length scale of turbulence foru-componentp pressureRe Reynolds number, (Ud/n)S Strouhal number, (fsd/U)Sz Strouhal number, (fsd/Uz)SRO Roshko universal Strouhal numberSB Bearman universal Strouhal numberG Griffin universal Strouhal numbersu rms longitudinal velocityU local mean wind speedUH mean wind speed at height of model/chimneyz height above basezref reference heightz/H relative heightIu turbulence intensity due tou-componentn kinematic viscosity of airr mass density of fluidC9L,turb component of rms lift coefficient only due to lateral turbulenceC9L,VS component of rms lift coefficient only due to vortex sheddingIu* modified turbulence intensityC9L,V modified value ofC9L,VS

IV turbulence intensity due tov-componentLc,ref reference correlation length=3.4d

be unsuccessful by Vickery [9,15] mainly because of theinability to match the Reynolds number in the wind tun-nel with full-scale values, and that the parametersdescribing the lift spectrum, viz., the rms lift coefficient,C9L,VS, the bandwidth parameterB, and the Strouhalnumber,S are primarily functions of Reynolds number.However, recently Kareem [4,24] reported that it is poss-ible to simulate the Reynolds number flow regime asin full-scale conditions, in a wind tunnel, by artificiallyaltering the flow characteristics using tripping wires andhence to predict the response of the full-scale RC chim-ney from the wind tunnel test results. A similar approachwas reported by Schnanbel and Plate [16] while compar-ing the wind induced response of a full-scale steel towerwith its model tower tested in a wind tunnel.

In other words, without resorting to some artificial

means of pseudo-simulation of the transcritical flowregime in the wind tunnel, currently it appears not poss-ible to predict the flow parameters such as rms lift coef-ficient and Strouhal number in full-scale conditionsbased only on wind tunnel measurements on a smoothcylinder.

In this paper, the authors present a new empiricalapproach for studying the across-wind forces acting ona circular cylinder/chimney. A modified version of localrms lift coefficient, only due to vortex shedding denotedas C9L,VS, evaluated from the local rms lift coefficientC9L inclusive of lateral turbulence in free stream, isshown to have a universal value of about 0.089 with acov of 18%, both in wind tunnel and in full-scale con-ditions, and is independent ofRenumber. Further, it isshown that the universal Strouhal number proposed by

504 S. Arunachalam et al. / Engineering Structures 23 (2001) 502–520

Griffin, ‘G’, which is directly related to the conventionalStrouhal number, ‘S’, attains a mean value of about0.065 with a cov of 8%, independent ofRenumber. Thisis based on analysing data from literature on differentcircular cylinders tested in boundary layer wind tunnelswith proper simulation of atmospheric boundary layer(ABL), and using data on two full-scale chimneys andtwo towers investigated in the field.

The present study makes use of the boundary layerwind tunnel data on circular cylinders published byKareem et al. [4], Garg and Niemann [21], Vickery andClark [6], Cheung and Melbourne [27] and also four full-scale sets of experimental data published by Waldeck[22], Sanada et al. [23], Ruscheweyh [36] and Davenport[38]. Further, details on the pressure measurement on acircular cylinder conducted by the authors are alsoincluded. It is demonstrated that for the purpose ofdetermining across-wind response, it is possible toreliably predict the aerodynamic force parameters in full-scale conditions based on wind tunnel tests on asmooth cylinder.

2. Earlier studies

For evaluating the vortex-induced response of an iso-lated circular cylinder (corresponding to a tall chimney),several investigations have been reported in the literature[17–20]. Most of these experiments have been conductedunder uniform flow or turbulent boundary layer flowconditions, some of them with grid generated low turbu-lence. Since chimneys are exposed to the atmosphericboundary layer conditions, tests have been conducted inthe recent past two decades, by various researchers inthe boundary layer wind tunnels to study the aerody-namic forces and their effects on a chimney [4,6,21,27]by simulating height-dependent mean velocity, intensityprofiles and the spectrum of the longitudinal velocitycomponent. The vortex shedding force is caused by thechange of surface pressure distribution corresponding tothe alternate shedding of vortices. By integrating the cir-cumferential pressure distribution, Vickery and Clarkproposed the following empirical formula for predictingthe lift force spectrum based on wind tunnel tests [6]:

fSL(f)/s2L5

1

BÎpS ffsDexpF2S1−f/fs

B D2Gwhere

f frequencyfs shedding frequency given by the Strouhal num-

ber relationship,fs=S.U/dU mean wind speedd diameterB band width parameter

s2L variance of lift force {1/2C9LdrU2} 2

SL(f) spectrum of lift force

It is to be noted that the above across-wind forcemodel includes both contributions from vortex sheddingand free stream turbulence; i.e. the parameterC9L,includes the effects of both the parameters,C9L,VS thecontribution due to only vortex shedding andC9L,turb, thecontribution due to lateral turbulence only.

It is well established that the basic aerodynamic para-meters, viz., Strouhal number,S, and the local rms liftcoefficient,C9L, are influenced by the Reynolds number,aspect ratio and surface roughness. They also varydepending on the free stream turbulence, as reported byseveral investigators [1,8,9,15,18,25]. The latter altersthe transitional behaviour, namely the flow separationpoints, mean drag, base pressure coefficient and conse-quently the pressure distribution. Some of the effects offree stream turbulence on flow parameters as reportedby several investigators are as follows:

1. Both the scale and intensity of turbulence areimportant and they influence the values ofC9L. andS.

2. When the turbulence intensity is increased it tendsto increase the shear layer thickness, thus producingincreased vortex forces and greater organization ofvortex shedding resulting in an increased value ofC9L

[1]. Such a trend can be seen betweenC9L andIz fromfull-scale measurements reported by Sanada [23] andwhich are included in Table 6.

3. For Lux/d.10, whereLux represents the longitudinalturbulence length scale, the effects of large-scale tur-bulence are assumed to be as per quasi-steady theory.Since small-scale turbulence affects the boundarylayer and shear layer, the quasi-steady assumptionbecomes less valid particularly forLux/d#1.

4. An increase in the free stream turbulence causes adecrease in the narrow-band correlation length [4,29].

5. The presence of large-scale turbulence in the freestream broadens the lift force spectrum.

6. To account for the combined effect of intensity andscale of turbulence, Vickery [2] proposed an empiri-cal relation for the modified turbulence intensity,Iz*given by: I∗

z=Iz,(d/Lux)1/2 where Iz=sz/U(z) and otherfactors as defined earlier.

Based on the pioneering work by Vickery and Basu, thefollowing expression is used for computing across windresponse of chimneys with little or no taper, to vortexshedding [8] for {d(h)/d(o).0.5},


av5

gCLdj(h)8p2S2

rd2

me1 lÎp2(l+2)2

1/2

f(B,k1)

51hE

h

0

j2(z)dz6hbs−kard2/mej1/2

where

av normalized peak tip deflection due to vortexshedding

l aspect ratio,h/dd average diameter of top thirdd(o) mean diameter at the bottomh chimney heightC9L rms lift coefficient (inclusive of both due to lat-

eral turbulence and vortex shedding)S Strouhal numberj(h) mode shape ordinate at heighthme Em(z)j2(z)dz/Ej2(z)dz

m(z) mass/unit length at heightzr air densitybs structural damping as a fraction of criticall correlation length in multiples of diameterka aerodynamic damping coefficientf(B,k1) 1

√Bk1.5

1 expH20.5S1−k−11

B D2Jg peak factor (<4)B a spectral bandwidthk1 U/Uc

U mean wind speed at z=5/6 heightUc f0d/Sf0 fundamental frequency

The above equation is recommended in the IndianStandard Code of Practice, IS: 4998 Part 2, 1992 [13] onchimneys of circular cross-section for computing across-wind response due to vortex shedding. It can be seenthatC9L, Sandr are the flow parameters involved and allother parameters are related to geometric and dynamicproperties of the structure, such as aspect ratio, modeshape, damping etc. ThusC9L and S are the only flowparameters required to be determined. It is stated in theliterature that since these two parameters are signifi-cantly dependent on Reynolds number (besides aspectratio and surface roughness), prediction/extrapolation ofvaluesC9L and S from wind tunnel tests to prototypeconditions is very difficult and hence presently nomethod appears to be available for use. However, asshown in subsequent sections, the authors would like to

state that it is still possible to reliably make use of thewind tunnel results for predicting corresponding valuesof C9L and S in full-scale conditions.

3. Proposed approach

The local flow around an isolated circularchimney/cylinder (and its along wind response to windaction) at any given height mainly depends upon themean velocity and the turbulent intensity and scale atthat height, which are the characteristics of the approachflow. With regard to the cross wind response of a chim-ney, the vortex shedding mechanism and buffeting dueto lateral turbulence are the primary causes. The localrms lift coefficient,C9L and the Strouhal number,S areimportant parameters describing the vortex inducedresponse process. Direct measurements ofC9L andSaregenerally made both in wind tunnel and full-scale experi-ments through rings of pressure taps at various positionsalong the height. Time histories of pressure coefficientsat different pressure taps along the circumference areobtained and by resolving them into lift and drag direc-tions (the time histories of pressures and hence usingstatistics), values of local rms drag and local rms liftcoefficients are evaluated. Such a value of local rms liftcoefficient, C9L has contributions both from lateralcomponent of turbulence in oncoming flow and fromvortex shedding phenomenon.

The contribution due to lateral turbulence is spreadover a wide range of frequencies and is always present,whereas the contribution due to vortex shedding variesdepending on the vortex shedding frequency,fs, givenby the Strouhal number relationship, and is extendingonly for a finite region of frequencies with central fre-quency equal tofs. Thus the spectrum of lift force canbe treated as consisting of:

(a) the spectrum part due to the effect of the lateralcomponent of turbulence(b) the spectrum part due to only vortex shedding.This part can be viewed as a ‘running vehicle’ occu-pying some position on the spectrum part, due to lat-eral turbulence only, corresponding to a given valueof (fsd/UH). This is schematically shown in Fig. 1 fortwo different wind speed cases.

Such a trend is also seen to be supported by full-scalemeasurements of spectra of pressures on a 230 m tallreinforced concrete chimney, investigated by Kessler[33] and as shown in Fig. 2.

Since the lateral component of turbulence can betreated as wide banded and random in nature, while thevortex shedding is a narrow banded signal, it is sug-gested that the total variance of the lift force can be com-


Fig. 1. Schematic diagram showing contributions due to lateral turbu-lence and vortex shedding to the total force spectrum.

Fig. 2. Normalized power spectrum of pressure (from full-scale studyby Kessler on 230 m tall chimney).

puted as the sum of the squares of these two componentsand given by:

(C9L)25(C9L,turb)21(C9L,VS)2 (1)

where

(C9L,turb)2=area under that part of the spectrum con-tributed only by the lateral component of turbulence(C9L,VS)2=area under that part of the spectrum contrib-uted only by vortex shedding as a consequence ofshape of the cylinder

For any two random signals, which are uncorrelated Eq.(1) is exact [39]. The loads due the to lateral componentof turbulence and that due to vortex shedding are uncor-related [3]. Hence use of Eq. (1) for separating the localrms lift coefficient into two components due to lateralturbulence and vortex shedding is considered valid. Thisis further analogous to summing up the squares of rmsvalues of broad-banded background component and nar-row-banded resonant component to get the total varianceof along-wind response of a tower-like structure of anycross-section. The local rms lift coefficient,C9L as used

in this paper is the value ofC9L, based on referencedynamic pressure at the model height, as these valueshave been found to give uniform values for modifiedC9L,VS values, as discussed later.

As far as the contribution to the lift force spectrumby the lateral component of turbulence is concerned, itis reasonable to assume quasi-steady aerodynamics [1–3]. Thus we can write:

sL51/2rU2d.CD.Iv (2)

or

sL551/2rU2d.(C9L,turb) (3)

where

sL standard deviation of fluctuating lift force perunit length

Iv v-component turbulence intensity

The ratio betweenIu and Iv can be approximatelyassumed as 0.68 [31], and we get

C9L,turb50.68Iu,.CD (4)

or

C9L,turb50.34C9D (5)

whereC9D is the fluctuating rms drag coefficientThe values ofC9L,VS using Eq. (1) have been evaluated

from different wind tunnel and full-scale test data oncircular cylinders/chimneys published in the literature.These include the wind tunnel experiments reported byKareem [4], Garg and Niemann [21], Melbourne [27],Vickery [6] and full-scale studies conducted by Waldeck[22], Sanada [23] and Ruscheweyh and Davenport [38],as stated earlier.

The experimental conditions in each case differ fromone another with respect to mean velocity and turbulenceprofiles, aspect ratio, correlation length, Reynolds num-ber etc. As a consequence of variations in the aboveparameters, it is seen that the value ofC9L,VS is not uni-versally constant but it depends on individual test con-ditions. However, the authors have found that by cor-recting the values ofC9L,VS in each case with itsrespective correlation length, through the following equ-ation, it is possible to obtain a modified parameter,C9L,V

with a mean value of 0.089. This value is found to beindependent of the Reynolds number regime, both inwind tunnel and in full-scale conditions.

Correction factor forC9L,VS5(Lc,i/Lc,ref)n (6)

where

Lc,i correlation length expressed in multiples ofdiameter, corresponding to any given test case

Lc,ref correlation length expressed in multiples ofdiameter corresponding to the reference case of


open terrain test case reported by Kareem [4].(Lc,ref=3.4d)

n exponent value taken as unity in this analysis

Within the range of test data analysed a linear vari-ation of (Lc,i/Lc,ref), (i.e. n=1) is found to yield values ofC9L,V which are close to 0.089. However, the exact valueof n requires further examination.

larger the region of vortex shedding and hence a highervalue will be obtained for the ratio of overall lift coef-ficient, C9L,VT to local lift coefficient C9L,VS. This willalso account for the effect of different aspect ratios.Hence if one were to deriveC9L,VS from C9L,VT, forexample from a force balance measurement, theseadditional factors are also to be considered.

In the literature it is emphasized that for a cylinder,the correlation length,Lc,i is very dependent on thelocation of the reference point and the direction in whichthe separation distance is taken [4,29]. In a turbulentflow, the presence of high turbulence reduces the corre-lation of the vortices and the direction of separation isof less importance. The reported values ofLc,i based ontests by Howell et al. [29], vary from approximately 1.8dat z/H=0.25 to 3.4d at z/H=0.5 and to 4.1d at z/H=0.85for a smooth terrain witha=0.186. The average valueof Lc,i for a rougher terrain witha=0.35 is reported as1.7d. A similar trend of variation ofLc,i with height hasbeen reported by Kareem et al. [4] and Garg et al. [21].At a given height on a cylinder, a larger value ofLc,i

implies that the eddies are highly correlated and hencethe spectrum of lift force based on pressure measure-ments at that level would indicate a narrow band vortexshedding with a higher peak. On the other hand, ifLc,i

is of relatively smaller value, then one might obtain abroad band lift force spectrum with reduced peak, shownas (A) and (B) respectively in Fig. 3.

From the general log–log plot of the lift force spec-trum with fSL(f) versusf, it may be inferred that withthe same mean velocity at the given height in both casesdiscussed above, the contribution toC9L,VS will be morewhenLc,i is smaller and vice versa as can be seen from

Fig. 3. Schematic diagram showing relative effect of high and lowcorrelation lengths on total variance of lift force spectrum.

Table 1. It may also be noted that between two terrains,for a given model at a given height, the value ofLc,i willbe smaller, when the model is located in a rougher ter-rain and the value ofLc,i will be relatively higher whenit is placed in a smooth terrain [4]. Hence, the computedvalues ofC9L,VS as above, are multiplied by a correctionfactor (Lc,i/Lc,ref) for respective correlation lengths. Acorrelation length of 3.4d is taken as the value for thereference correlation length. Many of the investigatorsreferred have given the correlation lengths and thesehave been used. A correlation length of one diameter hasbeen suggested by Vickery. However, this has beentaken as 1.6d based on the available literature [4,21,29].The reported low value of correlation length of onediameter was attributed to the effects of taper, finiteaspect ratio and also to the presence of shear flow [32].The resulting modified values ofC9L,VS denoted asC9L,V

from all the wind tunnel and full-scale data, is shown toattain a mean value of about 0.089 with a coefficient ofvariation (cov) of about 18%, independent of the Reyn-olds number regime. Thus it is hypothesized that whencircular cylinders are exposed to wind under atmosphericboundary layer flow conditions, either in a wind tunnelor at full-scale, at any height of the cylinder, (exceptvery close to tip and bottom of the cylinder) the valueof C9L,V as discussed above can be expected to approacha mean value of about 0.089 with a coefficient of 18%and this will be independent of Reynolds number. Thevalidation of this observation is discussed in the nextsection. The exactness of the above value 0.089 canhowever be improved when additional data becomeavailable. The above value of 0.089 will be valid whenthe reference correlation length is about 3.4d. In the caseof a given experiment with the value of correlationlength, Lc,i the value of 0.089 is to be multiplied by acorrection factor of (3.4d/Lc,i) to obtain correspondingC9L,V.

4. Validation of the hypothesis

Measurements of pressure and force fluctuations onisolated cylinders of finite height (l1=10 andl2=13.33)in two different simulated atmospheric boundary layers,have been studied by Kareem et al. [4]. Correlations ofpressure fluctuations, spectra of along-wind and across-wind forces and variations of rms lift coefficients andStrouhal number with model height have been reported.The tests were conducted in the sub-critical Reynoldsnumber range (Re=2.54 to 2.75×104). The salient fea-tures of the test results for the two models correspondingto the open terrain are suitably deduced and are givenin Table 1. The corresponding data for rougher terrainare given in Table 2. Since the values of local rms liftcoefficient, C9L, and Strouhal number are generallyreported as a variation of relative height,z/H, whereH


Table 1Values of different parameters deduced from wind tunnel results after Kareem et al. [4]. Open terrain:a=0.16

SI. No. Test case (z/H) z/zref U(z/H) I(z/H) C9L,turb C9L C9L,VS C9L,V S(m/s)

1 BL1-300 (l=10.0) 0.22 0.165 3.898 0.153 0.084 0.2 0.181 0.075 0.1340.54 0.405 4.500 0.086 0.047 0.088 0.074 0.074 0.1550.706 0.530 4.698 0.079 0.044 0.082 0.070 0.070 0.1450.84 0.630 4.829 0.079 0.044 0.085 0.073 0.073 0.133

2 BL1-400 (l=13.33) 0.22 0.22 4.08 0.126 0.069 0.258 0.249 0.102 0.1800.54 0.54 4.71 0.079 0.044 0.111 0.102 0.102 0.1840.706 0.706 4.92 0.079 0.044 0.106 0.097 0.097 0.1800.84 0.84 5.06 0.079 0.044 0.106 0.097 0.097 0.174

Table 2Values of different parameters deduced from wind tunnel results after Kareem et al. [4]. Rough terrain:a=0.35

SI. no. Test case (z/H) z/zref U(z/H) I(z/H) C9L,turb C9L C9L,VS C9L,V S(m/s)

1 BL1-300 (l=10) 0.22 0.165 2.77 0.233 0.141 0.185 0.120 0.078 0.1280.54 0.405 3.79 0.198 0.121 0.185 0.140 0.090 0.1500.706 0.530 4.17 0.197 0.121 0.179 0.132 0.086 0.1420.84 0.630 4.42 0.185 0.113 0.176 0.135 0.087 0.137

2 BL1-400 (l=13.33) 0.22 0.22 3.06 0.221 0.135 0.195 0.141 0.091 0.1920.54 0.54 4.19 0.193 0.118 0.215 0.180 0.116 0.2110.706 0.706 4.60 0.184 0.113 0.200 0.165 0.107 0.2050.84 0.84 4.89 0.170 0.104 0.197 0.167 0.108 0.191

is the height of the cylinder/chimney, these values arelisted here against relative heights. Using Eq. (4), thelift coefficient part corresponding to lateral turbulencecomponent only, denoted asC9L,turb is worked out foreach relative height. The value ofC9L,VS is now com-puted using the suggested Eq. (1). The repotted valuesof correlation lengths are equal to 3.4d and 2.2d respect-ively, for smooth and rough terrain conditions and thesecorrespond to only average values. The final values ofmodified C9L,VS denoted asC9L,V are given in Tables 1and 2 respectively.

Based on a boundary layer wind tunnel investigationon fluctuating aerodynamic forces on a circular cylinderwith an aspect ratio of 8.5, Garg and Niemann [21]presented variation of mean and fluctuating componentsof pressure, drag and lift, and their correlation and vari-ation of C9L andS with height. The test data are shownin Table 3 with relevant deduced values and are used inthe present study.

Table 3Values of different parameters deduced from wind tunnel results after Garg and Niemann [21]. Open terrain:a=0.16

Test case SI. no. (z/H) z/zref U(z/H) I(z/H) C9L,turb C9L C9L,VS C9L,V S(m/s)

HJN (l=8.57) 1 0.208 0.125 10.88 0.182 0.177 0.269 0.203 0.082 0.11452 0.405 0.242 12.17 0.147 0.158 0.252 0.196 0.086 0.1343 0.612 0.367 13.02 0.124 0.130 0.254 0.218 0.103 0.1414 0.805 0.483 13.64 0.106 0.105 0.279 0.258 0.127 0.135

Further, the values ofC9L andSbased on wind tunnelresults published by Cheung and Melbourne [27] meas-ured on a horizontal cylinder (l=4.5) have also beenincluded in the present study and are shown in Table 4.Since it is hypothesized that at any given height Eq. (1)holds good, the above wind tunnel data have beenincluded even though the cylinder had been tested in ahorizontal position. Since values of fluctuating dragcoefficient,C9D have been directly reported, values ofC9L,turb have been computed based on quasi-steady aero-dynamics, using Eq. (5). The wind tunnel data corre-sponding to turbulence intensity levels between 4 and9% and corresponding to Reynolds number rangebetween 8×104 and 4×105 are considered in this analysis.

The vortex induced aerodynamic forces acting on avertical tapered cylinder and its response in simulatedatmospheric boundary layer conditions were reported byVickery and Clark [6]. The value of the power-lawcomponent,a, for the mean velocity profiles, based on


Table 4Values of different parameters deduced from wind tunnel results after Cheung and Melbourne [27].a=0.16 (assumed)

Test case SI. no. Re I C9D C9L,turb C9L C9L,VS C9L,V

WHM 1 8E04 0.044 0.204 0.0694 0.21 0.198 0.1052 1E05 0.044 0.159 0.0541 0.17 0.161 0.0853 2E05 0.044 0.063 0.021 0.086 0.083 0.0834 3E05 0.044 0.030 0.0102 0.07 0.068 0.0685 4E05 0.044 0.031 0.0105 0.083 0.082 0.0826 8E04 0.068 0.190 0.0646 0.18 0.168 0.0897 1E05 0.068 0.146 0.0496 0.14 0.131 0.0698 2E05 0.068 0.056 0.0190 0.086 0.084 0.0849 3E05 0.068 0.035 0.0119 0.095 0.094 0.094

10 4E05 0.068 0.037 0.0126 0.103 0.102 0.10211 8E04 0.091 0.184 0.063 0.16 0.147 0.07812 1E05 0.091 0.141 0.048 0.12 0.110 0.05813 2E05 0.091 0.052 0.0177 0.105 0.104 0.10414 3E05 0.091 0.044 0.015 0.11 0.110 0.11015 4E05 0.091 0.044 0.015 0.12 0.120 0.119

Figure 2 of their paper is deduced equal to 0.37, whichcorresponds to a rough terrain category. Based ondetailed pressure measurements on a rigid model, Vick-ery et al. have reported the variations of shedding fre-quency, fs and of local rms lift coefficient,C9L withheight. The range ofRenumber in their study is between2×104 and 7×104 (i.e. sub-critical flow). The aspect ratioof the model based on average diameter of the top 1/3rdheight of chimney is 25.7. These data are also includedin the present study as shown in Table 5. The resulting

Table 5Values of different parameters deduced from wind tunnel results after Vickery and Clark [6]. Rough terraina=0.37

Test case SI. no. z/H z(m) U(z/H)(m/s) I(z/H) C9L,turb C9L C9L,VS C9L,V

BJV (l=25.7) 1 0.278 0.254 4.74 0.186 0.110 0.204 0.171 0.0802 0.306 0.279 4.91 0.178 0.104 0.177 0.144 0.0683 0.333 0.305 5.07 0.170 0.099 0.189 0.160 0.0764 0.361 0.330 5.23 0.163 0.097 0.192 0.165 0.0785 0.389 0.356 5.37 0.157 0.096 0.199 0.174 0.0826 0.417 0.381 5.51 0.152 0.093 0.209 0.188 0.0887 0.444 0.406 5.64 0.147 0.093 0.223 0.203 0.0958 0.472 0.432 5.77 0.143 0.093 0.219 0.198 0.0939 0.500 0.457 5.90 0.140 0.094 0.227 0.206 0.097

10 0.528 0.483 6.02 0.136 0.0903 0.231 0.212 0.10011 0.556 0.508 6.13 0.133 0.087 0.233 0.216 0.10212 0.583 0.533 6.24 0.129 0.084 0.233 0.217 0.10213 0.611 0.559 6.35 0.125 0.080 0.211 0.195 0.09214 0.639 0.584 6.46 0.119 0.076 0.218 0.204 0.09615 0.667 0.610 6.56 0.112 0.071 0.235 0.224 0.10516 0.694 0.635 6.66 0.110 0.071 0.205 0.194 0.09117 0.722 0.660 6.76 0.107 0.069 0.201 0.190 0.08918 0.750 0.686 6.85 0.102 0.067 0.196 0.186 0.08819 0.778 0.711 6.94 0.099 0.063 0.194 0.185 0.08720 0.806 0.737 7.03 0.093 0.056 0.199 0.191 0.09021 0.833 0.762 7.12 0.088 0.052 0.166 0.158 0.07422 0.861 0.787 7.21 0.081 0.0496 0.186 0.179 0.08423 0.889 0.813 7.29 0.075 0.047 0.167 0.159 0.07524 0.917 0.838 7.38 0.064 0.041 0.188 0.183 0.086

values ofC9L,V can be seen to lie in the region of 0.07–0.10 with an average value of 0.088, as shown inTable 5.

Sanada et al. [23] published details of full-scalemeasurements on wind forces on a 200 m tall reinforcedconcrete chimney and their data were discussed by Vick-ery [8]. From these two references, the measured dataat a height of 140 m (z/H=0.706;l=13.5), pertaining toS, C9L, U and Re are taken and are given in Table 6.The deduced values ofC9L included in Table 6 are based


Table 6Values of different parameters deduced from full-scale experiments after Sanada [23]. Open terrain:a=0.16; z/H=0.706;l=13.5

SI. no. U(z/H) (m/s) I(z/H) C9L,turb C9L C9L,VS=C9L,V

1 13.51 0.106 0.045 0.134 0.1262 16.51 0.10 0.042 0.127 0.1193 16.89 0.067 0.028 0.064 0.0574 23.24 0.067 0.028 0.064 0.0575 24.79 0.059 0.025 0.057 0.0516 25.88 0.048 0.020 0.047 0.042

on dynamic pressure corresponding toUH. The powerlaw exponent is assumed to be equal to 0.16, since itsvalue is not available in Sanada’s paper [23].

A similar full-scale experiment on a 300 m tall chim-ney in an open terrain condition was reported by Wal-deck [22]. The relevant experimental data on mean velo-city, turbulence intensity, rms lift coefficient andStrouhal number, collected at a height of 252 m,(z/H=0.84, g=23) for different test runs are included inTable 7. It may be noted that the value ofC9L given inthe Table 7 corresponds to the total rms lift coefficient(i.e. rms lift coefficient due to lateral turbulence and dueto vortex shedding). In both these full-scale measure-ments, the value of mean drag coefficient is assumedequal to 0.62 based on values reported by Sanada et al.(as the corresponding value is not available in Waldeck’spaper). For the purpose of computingC9L,V in the presentanalysis, the correlation length,Lc,i has been assumed tobe equal to 3.4d for the above two full-scale studies. Thewind tunnel test data corresponding to a 300 high modelin smooth terrain tested by Kareem and included as ref-erence case in this study can be treated similar to theabove full-scale tests in terms of terrain features, power-law exponenta>0.16, and turbulence intensity level.The turbulence intensity level measured at the level ofz/H=0.706 by Sanada et al., and atz/H=0.84 by Waldeckis less than around 8% which is very similar thatobtained in the above wind tunnel case. Hence, it isreasonable to assumeLc,i=3.4d for the full-scale dataconsidered here. This implies that the correction factorto be applied for correlation factor becomes unity and

Table 7Values of different parameters deduced from full-scale experiments after Waldeck [22]. Open terrain ;a=0.16 (assumed);z/H=0.84;l=23.1

SI. no. U(z/H) (m/s) I(z/H) C9L,turb C9L C9L,VS=C9L,V

1 26.0 0.07 0.030 0.087 0.0822 7.9 0.07 0.030 0.097 0.0923 18.2 0.06 0.025 0.107 0.1034 16.1 0.04 0.017 0.089 0.0875 22.1 0.05 0.021 0.082 0.0796 17.7 0.03 0.013 0.077 0.0767 21.1 0.08 0.034 0.087 0.0808 20.7 0.05 0.021 0.077 0.074

values ofC9L,VS and C9L,V will be the same. The com-puted values ofC9L,V for both these cases are includedin Tables 6 and 7. It may be noted that for all the windtunnel and full-scale data considered here, the value ofC9L,V lies in the range of 0.07–0.11 with an averagevalue of 0.089. Only four points out of the total 93 datapoints are around 0.05 and three points are above 0.11.Hence, the authors propose that it is reasonable toassume that for a smooth circular cylinder, the values ofC9L,V as discussed above will exhibit a mean value ofabout 0.089.

Full-scale wind pressure measurements on the tele-vision tower, Hamburg, Germany was investigated byRuscheweyh [36]. The measurement level was at a rela-tive height ofz/H=0.29, (z=78.8 m) and the values ofRenumber, turbulent intensity, and rms level lift coefficient,(inclusive of both components due to lateral turbulenceand vortex shedding),C9L were given in their paper for20 records and are shown in Table 8. Since in the presentmethod, the value ofC9L,turb is deduced from the rms liftcoefficient, which is referenced to 1/2rU2

H, the reportedvalues given in their paper, which are referenced to1/2rU2

Z, are multiplied by (z/H)2α wherea is the power-law coefficient of mean velocity profile. As there is noexplicit mention of the value ofa in their paper, its valueis indirectly estimated. Based on the average value ofturbulence intensity from above 20 records, which isequal to 0.177, the mean value of roughness length isobtained asz0,mean=0.28 m using the expression,

Iz51/ln(z/z0) (7)


Table 8Values of different parameters deduced from full-scale experiments after Ruscheweyh [36].a=0.19 (deduced);z/H=0.29

SI. no. Reno. (×1024) I(z/H) C9L C9L,turb C9L,V

1 8.75 0.153 0.111 0.050 0.0992 8.54 0.185 0.107 0.061 0.0883 8.42 0.167 0.098 0.055 0.0814 6.67 0.151 0.115 0.050 0.1045 6.68 0.161 0.116 0.053 0.1036 10.90 0.167 0.096 0.055 0.0787 10.30 0.176 0.109 0.058 0.0928 6.51 0.208 0.130 0.068 0.1109 9.23 0.166 0.121 0.054 0.10810 9.57 0.194 0.116 0.064 0.09611 9.30 0.155 0.097 0.051 0.08212 8.70 0.163 0.101 0.053 0.08513 8.47 0.192 0.098 0.063 0.07514 10.0 0.207 0.099 0.068 0.07115 9.6 0.192 0.109 0.063 0.08816 12.0 0.211 0.117 0.069 0.09417 11.6 0.194 0.107 0.064 0.08618 10.9 0.170 0.104 0.056 0.08819 10.5 0.175 0.105 0.057 0.08820 9.91 0.162 0.097 0.053 0.081

which is generally an accepted procedure [37]. Using thefollowing relationship proposed by Counihan,

a50.096log10(z0)10.016{log10(z0)} 210.24 (8)

The value ofa is worked out to be equal to 0.19. Thusafter computing values of rms lift coefficient, referencedto dynamic pressure at the height of the tower, the valuesof C9L,turb and C9L,VS have been computed using pro-cedure described earlier. Since the value of correctionfactor for correlation length in full-scale studies can betaken as unity as discussed before, the values ofC9L,V

andC9L,VS are equal. It can be clearly seen from Table8 that these values are close to about 0.089. The averageof all the 93 values is equal to 0.089 with a standarddeviation of 0.01573. The accuracy of this value canhowever be improved, as more and more data becomeavailable.

It can be noted that the value of the mean plus 1.66times the standard deviation is equal to 0.115, which isin excellent agreement with the value of 0.12 forC9L,V

recommended for design in category-2, i.e. open terrainconditions as per the Indian Code of Practice [13]. Forcases, where chimneys are likely to be located in cate-gory-3 i.e. suburban terrains, the authors suggest that avalue of Lc,I=2.5d may be considered reasonable. Thusthe mean value ofC9L,V would be equal to 0.089(3.4d/2.5d)=0.121, and the design value ofC9L,V

(=mean+1.66*standard deviation) would be equal to0.147, which would be about 22% higher than itscounterpart value in open terrain conditions.

Thus it is shown that the value ofC9L,V is independentof Renumber regime and it attains a mean value of about

0.089 (cov=18%) both in tunnel and full-scale tests. Aplot has been drawn betweenC9L,V versus [Re.(z/H)] andthis removes bunching of points at certainRe values.The final results ofC9L,V plotted against [Re.(z/H)], areshown in Fig. 4. It is easily seen that the inclusion ofthe factor (z/H) in the abscissa is only to stretch the datapoints and to achieve a better presentation of the data.

5. Comparison of Strouhal number value betweenwind tunnel and full-scale test results

The Strouhal number,S=fd/U, (f=vortex shedding fre-quency,d=diameter of the cylinder andU=undisturbedupstream velocity) is one of the primary parametersdescribing the vortex frequency in the wake region of acylinder. It depends on Reynolds number, surface rough-ness value, and aspect ratio,l, of a cylinder [2–4]. Thewind tunnel data reported by Kareem [4], Garg and Nie-mann [21], Cheung and Melbourne [27] and Vickery [6]and the full-scale data published by Sanada [23] andWaldeck [22] which are considered for analysis in thispaper, show a value of Strouhal number varying between0.115 and 0.215 in the wind tunnel and between 0.17and 0.27 in full-scale studies. Even though there is con-siderable variation in the value ofS as stated above, theauthors propose that these values could be reduced to anear constant value, ifS is viewed in terms of the con-cept of a universal Strouhal number. Several investi-gations have been made of this universal non-dimen-sional number, which includes vortex sheddingfrequency. The dimension of the body,d, which isincluded in the conventional Strouhal number,S=fd/U,


Fig. 4. (a) Comparison between WT and FS results AK, HJN, BJV, RUSCH and FS. Data; 5899 mean=0.089; sig=0.01573; (b) Comparisonbetween WT and FS results AK, HJN, BJV, WHM, RUSCH and FS. Data mean=0.089; sig=0.01573.

is not related to the vortex street. The concept of a uni-versal Strouhal number is that same size vortex streetmay be expected to originate from different bluff bodies,when proper scaling for reference length and velocity isused, besides the vortex shedding frequency [30]. In theliterature, universal Strouhal numbers which include thedimensions of vortex streets of cylinders in uniformflow, were studied by Roshko et al. [28]. The effect ofsurface roughness on the universal Strouhal number hasbeen recently reported by Adachi [30]. He reported thatfor a cylinder with various roughness surfaces over awide range of Reynolds number 5×1024,Re,107, Bear-man’s number was the most uniform Strouhal number,and Griffin’s number,G, also is independent ofRerange,if data corresponding to about 5×105 to 2×106 are neg-

lected. A brief description of these three universal Strou-hal numbers is given below:

1. Roshko’s number,SRO:Roshko demonstrated that the wake Strouhal numbercan be defined in terms of wake width,d9, and velo-city at the point of separationUb. Thus, he definedthe following universal Strouhal number,SRO:

SRO5fs.dUb

The relation betweenUb andU`, is given by:

Ub5kU`5(12Cpb)1/2∗U`, (9)

where k is a wake parameter and can be obtained


using Bernoullis equation. Roshko also suggested thefollowing relation between mean drag coefficient,CD, and base pressure coefficient,Cpb,

d9

d5

CD

Cpb

whered9 is the wake-width.Thus, the universal Strouhal number proposed byRoshko takes the form:

SRO5(S/k)·(CD/(2Cpb))

2. Bearman’s Universal number:SB

The wake Strouhal number, according to Bearman isgiven by:SB=fsh/Ub whereh refers to lateral spacingof vortices. This can also be expressed as [30]:

SB51/k.(h/l).UN/U` (10)

where l is the longitudinal spacing of vortices ofopposite sign andUN is the velocity of the centres ofthe vortices relative to the body.

3. Griffin’s universal Strouhal number,G.Griffin demonstrated that the product of the wakeStrouhal number and wake drag coefficient,S*CD*,is a constant with a value of 0.073±0.005 for a widerange ofRe numbers,Re*=100–107, where

S∗5fsd9/U5(S/k).(d9/d)

Re∗5Ud9/v5Re k.(d9/d)

and

S∗C∗D5S.CD/k35G (11)

In the present study, the Griffin universal Strouhal num-ber is selected for comparing wind tunnel and full-scaledata. The Strouhal numberS, at any given heightz is

used for computation ofG (Sz=nDz

vz

). While the value of

Strouhal number,S, is available for a greater number ofz/H values in the test data reported in [4,6,21] data onbase pressure coefficient,Cpb, are reported only for lim-ited levels and are included in Table 9. It is clearly seen,that the computed values of Griffin number,G using Eq.(11) are equal to about 0.065, for the data reported byKareem (open terrain) and by Niemann, where the modelhas uniform diameter throughout its height. However,the value slightly increases to 0.07 for the case ofrougher terrain tested by Kareem. In the case of testresults reported by Vickery, the computed value ofGvaried from 0.044 to 0.074. Since the chimney modelhas a linear taper throughout the height, a taper correc-tion factor,d(z)/d(zref), was applied as a multiplying fac-

tor, whered(zref) is the diameter of the model at the 2/3rdreference height of the model. The resulting modifiedvalues ofG lie in the range between 0.055 and 0.072.In particular, the values ofG for heights above the midheight of the model have values closer to 0.065. Sincethe top 50 or 60% of the height of the chimney is gener-ally considered to contribute to the vortex sheddingphenomenon, it may be reasonably assumed that all thewind tunnel results discussed above, give a mean valueof G=0.065, with a cov of 7%, independent of the Reyn-olds number, 2×104,Re,7×104. This is in good agree-ment with test observations reported by Adachi [30].

The universality of Griffin Strouhal number,G, hasbeen established for smooth cylinders tested in wind tun-nels or for cylinders with various surface roughnessvalues as reported by Adachi only under uniform flowconditions. However, in the present study it is found thatwind tunnel data by Kareem, Niemann and Vickeryclearly indicate that even when cylinders are tested inwind tunnels with proper simulation of atmosphericboundary layer (ABL) (which is a more realistic case forfull-scale chimneys), the universality of Griffin Strouhalnumber,G, can be obtained. Thus, while the extent ofmean velocity, intensity and scale of turbulence at agiven height may vary between individual tests, theirinfluences on Strouhal number, can be indirectlyaccounted for through the Griffin universal Strouhalnumber,G.

Recently, Gu et al. [40], have reported time averagedpressures on two cylinders in various arrangements, aswell as fluctuating pressures in some cases, in uniformflow with a turbulence level of 10%. The relevant datapertaining to an isolated cylinder are taken from theirpaper and are included in Table 9. It is clear that thededuced value ofG is close to 0.065 even though thevalue of Strouhal number from this test is relatively highand equal to 0.263.

Fox et al. [41] have reported wind tunnel results onthe aerodynamic disturbance caused by the free ends ofa circular cylinder held horizontally and immersed in alow turbulence (|0.2%) steady, uniform flow. Mean andfluctuating surface pressures, local mean pressure drag,rms lift and rms drag and Strouhal number of vortexshedding have been measured corresponding to a Reyn-olds number ofRe=4.4×104. Using the above data, thevalues ofG are computed at locations ofy/d=15 and 20,(wherey is the distance from the free end,d is the diam-eter of the cylinder) and these are included in Table 9.It can be noted that the values ofG are found to be equalto 0.069 and it is in good agreement with theG valueobtained from other experiments discussed earlier.

Similarly full-scale data on a 265 m tall Mt. Isareported by Cheung and Melbourne [27,35] and on a 200m tall concrete chimney investigated by Sanada et al.[23] are included in Table 10. The values ofG computedusing these data also suggest a mean value ofG=0.065


Tab

le9

Val

ues

ofG

riffin

univ

ersa

lS

trou

hal

num

ber

dedu

ced

from

vario

usw

ind

tunn

elex

perim

ents

SI.

no.

Inve

stig

ator

(BLW

T(z/H

)I (

z/H

)C

pb

k=√1

−Cp

bC

DS H

S zC

orre

ctio

nfo

rT

aper

(dz/d

z re

f)G

=Sz.C

D/k

3R

e×10

4

expe

rimen

ts)

1K

aree

mB

LI-3

00[4

]0.

706

0.07

92

0.58

1.26

0.81

0.14

50.

161

–0.

066

2.39

2.K

aree

mB

L2-3

00[4

]0.

706

0.19

72

0.72

1.31

10.

900.

142

0.17

7–

0.07

12.

113.

Nie

man

n[2

1]0.

750

0.10

02

0.70

1.30

40.

950.

135

0.14

6–

0.06

36.

374.

Vic

kery

[6]

0.33

30.

170

20.

867

1.36

60.

857

–0.

131

1.3

0.05

71.

770.

361

0.16

32

0.87

61.

370.

876

–0.

138

1.27

50.

060

1.79

0.38

90.

157

20.

876

1.37

0.89

5–

0.13

61.

250.

059

1.80

0.41

70.

152

20.

886

1.37

30.

895

–0.

146

1.22

50.

062

1.81

0.50

00.

140

20.

895

1.37

70.

990

–0.

166

1.15

0.07

21.

820.

583

0.12

92

0.91

41.

383

0.95

2–

0.17

11.

075

0.06

61.

800.

667

0.11

22

0.82

91.

352

0.93

3–

0.18

61.

00.

070

1.75

0.75

00.

102

20.

781

1.33

50.

905

–0.

197

0.92

50.

069

1.69

0.83

30.

088

20.

714

1.30

90.

876

–0.

203

0.85

0.06

71.

610.

917

0.06

42

0.65

71.

287

0.95

2–

0.18

10.

775

0.06

31.

520.

500

0.10

02

0.55

1.24

50.

470

–0.

263

–0.

064

65.0

05.

Gu

etal

.,[4

0]0.

002

21.

133

1.46

01.

133

–0.

190

–0.

069

4.4

6.F

oxet

al.,

[41]

0.00

22

1.26

1.50

31.

24–

0.19

0–

0.06

94.

47.

Pre

sent

stud

y0.

580.

162

0.20

91.

100

0.49

4–

0.18

1–

0.06

719

.5


Table 10Values of Griffin universal Strouhal number deduced from various full-scale experiments. Mean=0.065; SD=0.005; cov=7.7%

SI no. Investigator (full-scale (z/H) I(z/H) Cpb k=√1−Cpb CD Sz G=Sz.CD/k3 Re×107

experiments)

1 Melbourne [35,27] 0.83 >0.07 20.50 1.225 0.60 0.20 0.065 0.482 Sanada et al. [23] 0.706 0.067 20.61 1.269 0.62 0.225 0.068 3 to 43 Sanada et al. [23] 0.706 0.059 20.61 1.269 0.62 0.210 0.064 3 to 44 Sanada et al. [23] 0.706 0.048 20.61 1.269 0.62 0.180 0.055 3 to 45 Erbacher and Plate [34] 0.37 - 20.86 1.364 0.93 0.200 0.0736 Erbacher and Plate [34] 0.652 - 20.86 1.364 0.77 0.216 0.0667 Davenport [38] - 20.917 1.385 0.70 0.250 0.0668 Ruscheweyh [36] 0.29 - 20.44 1.200 0.487 0.23 0.065 1.4

with a cov of 8%, even for the transcritical range, whereRe is of the order of 107. Thus the authors propose thatthe Griffin Strouhal number,G can be analysed to yielda mean value of 0.065, independent ofRenumber, bothin the subcritical regime and transcritical regime, inwhich wind tunnel tests and full-scale tests are normallybeing conducted respectively. The authors admit that thenumber of test data points discussed above are less innumber and they believe that the preciseness of the valueof G can be improved with addition of more test datawhen available.

Maier and Plate have investigated the velocity andpressure field on a prototype cylindrical tower locatedin an irregular terrain [34]. The power-law exponent var-ies from 0.21 to 0.36 depending upon the wind direction.The total height of the tower is 46 m and 16 differentialpressure transducers have been mounted at 17, 25.5 and30 m levels and based on the mean pressure distributionsmeasured at different directions and heights, shown inFigure 7 of their paper, the average value ofCpb is takenas equal to20.86. The reported values of mean dragcoefficient, CD decreases from 0.93 atz=17 m to 0.77at z=30 m. From the measured pressure spectra at theregion of flow separation atz=17 m and atz=30 m,shown in Figure 7 of their paper, the values ofSz havebeen evaluated asSz=0.20 at z=17 m andSz=0.216 atz=30 m. Using these data, the corresponding values ofG have been worked out to beG=0.073 andG=0.066 atz=17 m andz=30 m respectively.

Details of full-scale wind pressure measurements con-ducted by Jensen on a 240 m tall cylindrical tower, arereferred to by Davenport [38]. The reported values ofCpb, CD and Strouhal number are20.917, 0.7 and 0.25respectively. Thus the value of Griffin universal Strouhalnumber is computed as equal to 0.066 using Eq. (11).

Similarly based on the full-scale investigation of windpressures on the television tower, Hamburg, conductedby Ruscheweyh [36] and discussed earlier in Section 4,the reported values ofCpb, CD and S are 20.44, 0.487and 0.23 respectively. The computed value ofG usingEq. (11) works out to be 0.065. Thus these full-scale

data also yield a value ofG around 0.065 as exhibitedby other full-scale as well as wind tunnel test results.

The wind tunnel and full-scale test data included inTables 9 and 10 further show that the wake parameter,k, given by Eq. (9) is linearly related to the freestreamturbulence intensity,I(z/H). as shown in Fig. 5. The equ-ation of the best-fit line is given by :

k50.8393∗I(z/H)11.2251 (12)

6. Pressure measurement test on a circular cylinder

6.1. Boundary layer wind tunnel (BLWT) facility

Pressure measurements on a circular cylinder undersimulated open terrain conditions were carried out usingthe boundary layer wind tunnel available at the Struc-tural Engineering Research Centre, Madras. This state-of-the-art boundary layer wind tunnel is an open circuitand a blower type wind tunnel. It has a total length of52 m with a test section of size, 2.5 m(W)×1.8 m(H)×18m(L). The maximum speed that can be attained in thistunnel is 55 m/s. A schematic view of the wind tunnelis given in Fig. 6.

Fig. 5. Variation of wake parameter,k, with turbulence intensity.


Fig. 6. Three dimensional view of boundary layer wind tunnel structure.

6.2. Experimental set-up

The model cylinder had a diameter of 15 cm and atotal height of 60 cm (l=4). The rigid model was madeout of acrylic material. Twelve pressure taps were pro-vided at each of the two levels ofz=35 cm andz=24 cmat 30° uniform intervals along the circumference. Press-ure tubes made of PVC material with 1.2 mm ID, andwith a length of 50 cm each were used. In the presentstudy, the pressure measurement system with electronicpressure transducers supplied by M/s Pressure Systems,USA, was used for acquisition of pressure signals. Sincethe volume of these pressure transducers is very smallcompared to that of pressure transducers used in conven-tional system, no restrictors were used. The mean velo-city profile (a=0.16) and the turbulence intensity profilessimulated in the tunnel are shown in Fig. 7(a,b), whichare typical for an open terrain category. The simulatedspectrum of wind compared well with Karman’s spec-trum for longitudinal component of velocity. The press-ure signals were acquired at a scanning rate of 500 Hz,for a period of 10 s per channel. Data were acquired fortwo different values of free-stream velocity. The Reyn-olds number values, based on the free-stream velocitiesof 13.0 and 19.5 m/s, are 1.35×105 and 1.95×105 respect-ively. The spectra of pressure signals were suitably cor-rected for the frequency response of the tubing systemusing the analytically derived tubing frequency responseof the pneumatic circuit by means of available software.The software uses parameters such as diameter andlength of tubing, its flexibility, volume of electro-scanpressure transducers, and restrictors/manifolds includedin pneumatic circuit.

6.3. Results and discussion

The mean and fluctuating pressure distributions meas-ured on the cylinder atz=35 cm are given in Figs. 8and 9. The value of dynamic pressure atz=35 cm wasexperimentally found to be 157 Pa. By integration ofpressure coefficients, through the following equations,the mean drag coefficient,CD and the rms lift coefficient,C9L have been computed and these values are equal to0.494 and 0.22 respectively.

CD5p/12∗SCpIcos 30(i21) (13)

C9L5p/12∗SC9pIsin 30(i21) (14)

The value of turbulence intensity atz=35 cm wasequal to 0.16. Hence using Eq. (4), the value ofC9L,turb

is worked out to be 0.048. Further, the value ofC9L,VS

from Eq. (6) is calculated as equal to 0.215.Based on measured fluctuating pressure data from 12

corresponding taps at the two levels ofz=35 cm andz=24 cm, (Dz/d=r/d=11 cm), the values of correlationcoefficient,R(r) have been obtained and they are plottedin Fig. 10 as a function of azimuthal angle. The abovecurve can be seen to be reasonably symmetric. The aver-age value of correlation coefficient, forq=60° to q=180°is found to be 0.61. Using this value in Eq. (17) dis-cussed in Appendix A, the value ofc is found to be 0.67.The corresponding value of correlation length for thetested cylinder withl=4 atz=35 cm is found to be 1.39d,using Eq. (18). As described earlier, the correction factorfor C9L,VS is given by the ratio (1.39d/3.4d) and themodified value,C9L,V for tested cylinder is obtained as,


Fig. 7. (a) Mean velocity profile; (b) turbulence intensity profile.

Fig. 8. Mean pressure distribution on the circular cylinder,z=35 cm.

Fig. 9. Rms pressure distribution on the circular cylinder,z=35 cm.

Fig. 10. Variation of correlation coefficient with azimuthal angle(PRTW300p1.dat).

C9L,V5(1.39d/3.4d)∗(0.215)50.088 (15)

This is in good agreement with the value of 0.089 sug-gested by the authors.

Further, the value ofCpb is found to be20.209. Thisgives a value of the wake parameter,k=1.10. From thepower spectrum of pressure shown in Fig. 11, for a tapin the wake (Tap no. 6), it is found that the peak occursat the shedding frequency of 23.5 Hz. The correspondingStrouhal number is computed as 0.181. Hence using Eq.

Fig. 11. Spectrum of pressure for a tap in the wake (z=35 cm, tapno. 6).


(11), the Griffin universal Strouhal number,G, is com-puted as equal to 0.067. Thus the experimental data onthe circular cylinder measured by the authors undersimulated atmospheric flow conditions, exhibit values ofC9L,V and G as equal to 0.088 and 0.067. This furthersupports the validity of the hypothesis discussed earlier.

From the foregoing discussions, it may be stated thatsince the value ofG is close to 0.065 (with a cov of 8%),independent of Reynolds number, both in subcritical andtranscritical regimes, it becomes possible to compute thevalue of conventional Strouhal number,S, for any indi-vidual test case, be it in a wind tunnel or in a full-scalestudy, provided corresponding values ofCD and Cpb ork are known. Further, as discussed earlier, the rms valueof lift coefficient due to vortex shedding,C9L,V shows amean value equal to 0.089 (with a cov of 18%), inde-pendent ofRe number based on both wind tunnel andfull-scale test data published.

In other words, the observations that the values ofC9L,V andG being almost invariant with change in Reyn-olds number and thatG is directly related toSz in everyindividual case are adequate reasons to support the claimthat the wind tunnel tests can be considered as a reliablemethod (even though conducted with relaxation ofRenumber similarity) to predict the values of Strouhal num-ber Sz and C9L,V corresponding to full-scale chimneyconditions.

6.4. A procedure for prediction of Strouhal numberand C9L,V

The following procedure is suggested to determine thevalues of C9L,V and G from a pressure measurementstudy on a circular cylinder carried out either in windtunnel or in full-scale conditions.

Step 1: From the pressure measurement test data thefollowing input parameters are initially evaluated.(a) mean drag coefficient(b) turbulence intensity corresponding to the heightof the measurement(c) local rms lift coefficient,C9L

(d) correlation length,Lc,I corresponding to theheight of the measurement (discussed in AppendixA)(e) base pressure coefficient,Cpb

(f) Strouhal number based on spectrum of pressurefrom a tap in the wake

Step 2: Using Eq. (4),C9L,turb is computedStep 3: Using Eq. (6),C9L,VS is computedStep 4: C9L,V=(Lc,i/3.4d)*C9L,VS and this value isexpected to be close to 0.089 (with a cov of 18%),independent of the test Reynolds numberStep 5: Evaluate wake parameter,k using Eq. (12)Step 6: Compute Griffin universal Strouhal number,

G using Eq. (11) and this is expected to be close to0.065 with a cov of 8%

7. Conclusions

A new empirical method is presented for correlatingthe values of rms lift coefficient,C9L, and Strouhal num-ber,S relevant to full-scale chimney conditions based oncorresponding values on circular cylinders in properlysimulated boundary layer wind tunnel results. It is hypo-thesized that at any given height, the modified value ofrms lift coefficient due only to vortex shedding,C9L,V

attains a mean value 0.089 (with a cov of 18%), inde-pendent ofRe number regime. This hypothesis is vali-dated using the test data measured by the authors andalso the published test results in the literature byKareem, Garg and Niemann, Cheung and Melbourne,Vickery and Clark, Waldeck, Sanada et al., Ruscheweyhand Davenport which include both wind tunnel and full-scale experiments. Similarly, all these data yield a Grif-fin universal Strouhal number equal to about 0.065 (witha cov of 8%), independent ofRenumber. In view of theabove, the authors conclude that wind tunnel experi-ments can be viewed as reliable tools forextrapolating/predicting values ofC9L,V and S corre-sponding to full-scale conditions. Also, the wake para-meter,k is found to be linearly related to the turbulenceintensity. A procedure for estimating Griffin universalStrouhal number,G andC9L,V is suggested.

Acknowledgements

This paper is published with the kind permission ofthe Director, Structural Engineering Research Centre,Madras.

Appendix A. Correlation length

The fluctuating lift force at any location is generallyimperfectly correlated with fluctuating lift with someother location. This correlation is usually represented bya correlation length, which is expressed in multiples ofdiameter of the structure. The correlation length betweentwo points,z1 and z2 is evaluated by the integration ofthe correlation coefficient with respect to the separationdistance, r=|z22z1|. The correlation length can beobtained using the correlation coefficient,R(r), as fol-lows:

Lc,i5E`

0

R(r)dr (16)


In the present study, the spatial coherence of the press-ures in the flow has been determined by the two-pointlateral correlations at zero time lag. The correlation coef-ficient, R(r), between two signals X and Y is evaluatedas the ratio of the cross covariance,Cxy, to the productof individual rms values, (sxsy). In practice, for narrowband correlations such as vortex shedding, the correal-tion coefficient can be expressed by a negativeexponential function as [29]:

R(r)5exp(2c(r/d)) (17)

in which

c dimensionless coefficientr/d normalized separation distance

in which

c From Eqs. (14) and (15), for a cylinder withfinite height,h, we get

Lc,i/d5[(12exp(2c(h/d)))/c] (18)

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