Analysis of the time varying wake

3.6 Wind tunnel

4.3.4 Analysis of the time varying wake

y/D cyl. [−]

u ² /U _ref ² v ² /U _ref ² w ² /U _ref ² u ² /U _ref ² , freestream

Figure27: Proleofnormalizedturbulentnormalstressesin cylinderwakeforRe

=30717,x/D=10

Ongand Wallacedoes notreport themagnitude of

w ²

^relative^to ^the ^others.

w ²

^exhibits^a^verycharacteristicplatouinthecentreregion.

4.3.4 Analysis ofthe timevarying wake

Thevortexsheddingbehindthecylindercausesthevelocityinthewaketovaryin

aorderlyfashionasthevorticespassby. Figure 28showsasmallpartofasolved

timeseriesonthecentrelineat

x/D = 10

All three velocitycomponentsexhibit variationsin magnitude, but it ismost

clear forV, which show aperiodic variationabouta meanvalueclose to 0. Ina

0 200 400 600 800 1000

Figure28: Thesolutionoftherst1000samplesonthecentrelineatx/D=10

perfectly two dimensional ow there should be no variation in W, which should

bezero. Fromgure 28one can howeversee that W varies overarange slightly

smallerthanthatofV.

The frequency of the periodic variation of V can be found by analyzing the

frequency content of the signal. This is doneby taking a fast Fouriertransform

(FFT) ofthe dataset,thebuiltin function in MATLABis used forthis purpose.

Figure29showstheFFTofthesolutionofatimeseriesmeasuredatthecenterline

ofV.

TheFFTshowsaclearpeakat

f = 42 . 8 Hz

^. ^F^rom^equation³⁹^the^shedding

fre-quencycanbeestimatedbyassumingaStrouhalnumberof0.21,andaspreviously

mentionedthefreestreamvelocityisassumedtobeabout

9 . 7 m/s

f =

0 . 21 U

d = 42 . 9 Hz

⁽⁴³⁾

Thisisaveryclosematch totheresultoftheFFT.It ishoweverworth

inves-tigatingtheshedding frequencyabit closer. Theanalyzeddataseriesismeasured

overa timeperiod of 20 seconds. If the vortex shedding process is perfectly

sta-ble, the peak frequency will be the shedding frequency. It is not likely that the

frequencyiscompletelystablebut ratherthatitwillvaryslightlyasafunctionof

time. Toinvestigatethis, subsetsofthemeasureddatawasinvestigated

indepen-dently. Analyzingasubset ofthemeasurementdatawill reducethesmoothingof

thedata andgivealargeramplitudeforthepeakfrequency. There ishoweveran

disadvantageofconsideringasmallsubset. WhenanFFTisfound,thenumberof

frequenciesanalyzedistakenasthelargestpoweroftwosmallerthanorequalto

thesamplesize,thisisdonetoincreasethecalculationspeed. Thismeansthatthe

10 ⁻² 10 ⁻¹ 10 ⁰ 10 ¹ 10 ² 10 ³ 10 ⁴

Figure29: FFTof solutionatcenterline,x/D=10,

f _s

^13000Hz,

T _s

⁼^20s

frequency resolution is reduced when the sample size is reduced. The timeseries

is sampledat 13kHz,theFFT canonlyndfrequenciesupto6.5 kHzaccording

to the Nyquist sampling criteria. To obtain aresolution of 1 Hzthe FFT must

betakenfor 6500frequencies. The smallestpowerof twolargerthan orequalto

6500 is

2 ¹³ = 8192

, which equalsa0.63secondssampling time, orapproximately 26 shedvortices. Analysisoftherst 0.63secondsyieldthe FFTshown in gure

30. Thesheddingfrequencywasfoundtobealmostthesame,buttheamplitudeof

thepeakfrequencyisalmostthree timesas large,indicatingthatfrequencyvaries

someoverthespanofonetimeseries.

The shedding frequency can be used to conditionally average the signal by

assuming that the shedding process is stable. This done by splitting the time

period into N number of bins. A sampled timeseries is then analyzed, and the

individual samples is placedin bins accordingto their temporal position relative

to theother samples. Ingure31theconditional averageofU, V andW forthe

rst second of a timeseries aquiredon thecenterlineis plotted. The meanvalue

of thedierentvelocities issubstracted. The dataisconditionally averagedusing

half the calculatedshedding frequency. I.e. the period averagedoverequalsthe

timeittakesfortwovortexestobeshed,onefromeachsideofthecylinder.

The conditional averaging gives a clear variation of W with respect to the

periodoverwhichthesignalisaveraged.TherangeofWisabitsmallerthanthat

observedingure28. Thisindicatesthatvariationisdampenedbytheconditional

averagingprocessduetoavaryingsheddingfrequency. ThevariationinUandWis

lesssystematic,butshouldbestudiedfurther.Severalotherinterestingcorrelations

couldbeinvestigatedbyconditionalaveraging,but therewasnottimeforthat.

10 ⁰ 10 ¹ 10 ² 10 ³ 10 ⁴ 0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

X: 42.85 Y: 1.989

Frequency (Hz)

|Y(f)|

Figure 30: FFT ofsolutionatcenterline,x/D=10,

f _s

^13000Hz,

T _s

⁼^0.63s

0 0.005 0.01 0.015 0.02

−4

−3

−2

−1 0 1 2 3

U [m/s]

T [s]

U−U _mean V−V _mean W−W mean

Figure31: Conditionalaverageofmeanvelocities,y/d=0,30bins,1second

In this section an attempt to assess the properties of the hot wire probe based

on theresults presentedis made. For the turbulentpipe ow theresultswere in

reasonablygoodagreementwiththeory,majordiscrepancieswereonlyfoundclose

to the pipewall. Smallerrorsmostlikelydue to misalignmentof theprobe were

observedatseveraloccasions.Fromtheeectiveanglecalibrationitwasfoundthat

exactpositioningoftheprobevisuallycouldonlybedonewithalimitedaccuracy.

Thisaectedthecalculatedproperties,butmustbeexpected. Thenormalstresses

on the centerline of the pipe is such an example. Theoretically the crosstream

normalstressesshouldbeidenticalonthecenterline,butduetomisaligmentofthe

probetheresultsdeviated.

Thediscrepanciesclosetothewallarebelivedtobecausedbythelargevelocity

gradient. By makingan estimate using atwo-pointnumeric scheme the velocity

gradientisfoundandplottedin gure32.

−200 −150 −100 −50 0 50 100 150 200

Figure32: Gradientof

U _x

ⁱⁿ^the^turbulent^pipe^ow

The shape of the velocity gradient prole look suspicously like the prole of

U _r

^and

U _θ

^. ^This strenghtensthe hypothesis put forward in section X, that the variation of

U _r

^and

U _θ

^is ^linked ^to ^the ^velocity ^gradient. ^It ^is ^however ^worth

notingthatthetwodierentvaluesof

φ ¹

^gives^very^similar^results,^it^does^however

notseemobviousthat thelargevelocitygradientwillgivethesameerrorin both

cases. Based on the prole for

uw

^and

vw

^(gure ¹⁸⁾ ^a ^rough ^estimate ^for ^the

criticalmagnitudeofthevelocitygradientcanbefound. Assumingthattheresults

are good until

|x/D| ≥ 0 . 8

thecritical gradient canbeestimated to be

± 50 s ⁻ ¹

Overthecrossection of the hotwireprobethis will equal a

0 . 25 m/s

^dierence ⁱⁿ

velocity.

pect in the cylinderwake. As astart thegradient of themeanvelocityeld can

be estimated. The result is plotted in gure 33. A maximum mean gradient of

approximately

± 40 s ⁻ ¹

^is^foundⁱⁿ^the^wake,^which^is^just^below^what^is^estimated

as acritical gradient. The maximum value of the gradient could be largerthan

± 50 s ⁻ ¹

^as^the^real ^value^will ^vary^about^the^mean. ^Based^on^this^a^likely

conclu-sionisthatthespatialresolutionoftheprobecouldbiasthemeasurementsinthe

cylinderwake.

−50 0 50

−3

−2

−1 0 1 2 3

dU/dy [1/s]

y/D cyl. [−]

Figure33: GradientofU inthecylinderwake

Basingthe estimateof the gradientonthe streamwisevelocity obtainedfrom

the hotwireis ofcoursea sourceof errorin itself, asthegradientmight biasthe

streamwisevelocity. Howevertheresultsfromthepipeowgavearelativelygood

t to the logarithmic law even close to the wall, indicating that the streamwise

velocity is notseverly aected by thelarge gradient. From the pipeow results

one can conclude that all the turbulent properties are aected by the velocity

gradient. Inthecylinder wakethelargestdeviation from whatwasexpectedwas

foundfor

uw

^and

vw

^,^both^shear^stresses^had^a^magnitude^similar^to

uv

^. ^It ^seems

likely that the error is caused by the velocity gradient. Compared to litterature

thenormal stressesalso showedsomedeviation, butthe orderofthe resultswere

correct. Itdoesthereforeseemliketheturbulentshearstressesinthewakeismore

heavilyaectedbythelargegradientsthantheturbulentnormalstresses. Asimilar

conclusioncanto acertaindegreealso be madebystudying theresultsfrom the

pipe,butnodecissiveconclusioncanbemade,asthepipedataalsoshowsscatter

asaresultofvarying

φ ¹

Analysingtheresultsobtainedrevealedthatmoreworkshouldhavebeenputinto

aligningtheprobewiththeow. Designingof aholderwhich allowedfor yawing

andpitchingoftheprobeaswellastraversingcouldhelpthissituation. Byyawing

and pitching theprobeforaknown velocitythe resultscould be usedto ndthe

errorintheprobealignmentandcorrectforitinthedatareductionprogramwhen

analyzingtheresults.

Iftheprobeistobeusedindynamicows,suchase.g. aturbinwakeitwouldbe

necesarrytoinvestigatefurtherhowtheshedvorticesaectthemeasurements. This

could bedone by analysing acylinder wakefurter downstream, using both LDA

and hot-wire, perhaps combined with pressure measurements on the cylinder to

conditionallyaveragethedata. Goingdownstreamtheresultsarelikelytoconverge

at somepointwhen the spatial resolution of the probeis sucientcompared to

thegradient. Theresultsfromconditionalaveragingcouldthenbeusedtondthe

gradientsinthevorticesetc.

If possible it would of course be benecial to reduce the physical size of the

probe.

The eectiveangle of the individual wires has been found from calibration in a

turbulent pipe ow, the results show that the eective angle approach can be

applied within a range og

± 20 deg

with an uncertainty of

± 1 deg

in yawfor the individual wire. Thevelocities matched the referencevelocity obtained by pitot

withamaximumrelativeerrorof

1 . 1%

Measurementsintheturbulentpipeowgaveagoodmatchwiththe

logarith-mic law andthe theoreticaldistributionof

u _x u _r

^. ^The ^normal^shear ^stresses^were

in agreementwith theresultsfoundbyTorbergsen[9], but werefound tobe

sen-sitiveto probemisalignment. Outside

|y/R| > 0 . 8

theprobegavebadresultsfor shear stressesandnormal stresses,due tothe largegradientoftheaxial velocity.

The radialand circumferentialmeanvelocitywasalso foundto bebiased bythe

gradient,especiallyclosetothewallfor

|y/R| > 0 . 8

Aslongastheprobehasaphysicalsize anerrormustbeacceptedwhen

mea-suringin avelocitygradient. Aroughestimateofacriticalgradientfortheprobe

wassetto

501 /s

^based^on^assesment^of^the^variation^ofcrossstreamturbulentshear stresses. Theexactmagnitudeoftheerrorgivenbythegradientishardtondas

severalothersourcesoferroralsocontributetodeviationfromtheexpectedresult,

suchasprobeyawandpitchaswellasinaccuratlyestimatedvaluesfor

φ ¹

Measurementsintheturbulentwakeofthecylinderrevealedtheweaknessesof

themeasurementtechnique.Theresultsgivethatthecrosstreamturbulentstresses,

uw

^and

vw

^,^are ^of^the^same^order ^of^magnitude^as

uv

^. ^The^cause^is^belived^to ^be

spatialresolution oftheprobe.

Theresultsleadto theconclusionthat theprobeiscapableofmeasuringboth

mean velocities and turbulentstresses withgood accuracy in ows where the

ve-locitygradientissmallerthanthecritcalgradient. Aprerequisiteishoweveristhat

theprobeis carefullyalignedwiththe oworthat themisalignment iscorrected

forin thedatareductionprocess.

Furthertestingisrecommendedtoverifytowhatextenttheprobecanbeused

in owswherevorticesareshed,e.g. tipvorticesfrom windturbinemodels.

[1] AreAanesland. Utviklingavtre-komponentshetetrådsanemometri. 1998.

[2] T. Arts, H. Boerrigter, J.-M.Buclin, M.Carbonaro, G.Degrez, D. Fletcher

R. Dénos, D. Olivari, M.L. Riethmuller, and R.A. Van den Braembussche.

Measurement techniques in uid dynamics, An introduction,2nd revised

edi-tion. vonKarmanInstituteforFluidDynamics, 2004.

[3] BrianCantwellandDonaldColes. Anexperimentalstudyofentrainmentand

transportin theturbulentnear wakeof acircularcylinder. 1983.

[4] C.H.K.Williamson. Vortexdynamicsinthecylinderwake. 1996.

[5] I Lekakis. Calibration and signal interpretation forsingle and multiple

hot-wire/hot-lmprobes. 1996.

[6] P.M.Ligrani andP.Bradshaw. Subminiaturehot-wire sensors: development

anduse. 2008.

[7] L.Ongand J.Wallace. Thevelocityeldof theturbulentverynear wakeof

acircular cylinder. 2008.

[8] H.TennekesandJ.L.Lumley. A rstcoursein turbulence. 1972.

[9] LarsEvenTorbergsen. Experimentsin turbulentpipeow. 1995.

[10] AnthonyJ. Wheeler and AhmadR. Ganji. Introduction to Engineering

Ex-perimentation,Secondedition. Pearson,PrenticeHall,2004.

[11] FrankM.White. FluidMeachanics, Fifth edition. McGraw-Hill,2001.

[12] FrankM.White. Viscousuidow,thirdedition. 2006.

[13] J.G. Wissinkand W. Rodi. Numerical study of the near wakeof a circular

cylinder. 1996.

A Data reduction program

A script (HW3Dv12.f95) hasbeen written in Fortran to perform all calculations

needed,from calibrationtocalculationof statistics. Theprogramiswrittento be

general, and easyto apply to dierent datasets. A textle (ledata.txt) species

where the input data is to be gathered and where to stor the output data. In

addtion everyset of angle calibrationor timeseries datasets isaccompanied by a

lespecifyingtransducerconstants,wiretemperatureetc(settings.txt).

TocurvetdataaFortranprogramkurve-mac.fwritten byPer-ÅgeKrogstad

isused,theprogramtsdatato anyequationspeciedandreportstheresultand

thematchbetweentheequationanddatapoints.

residual = X

ThedatareductionwasinitiallymeanttobeperformedusingaFortranroutine

called dnsqe.f from the Slatec library. Dnsqe.f uses the morecomplex dnsq.f to

ndthezeroofasystemofnonlinearfunctions,using"amodicationofthePowell

hybridmethod". Itdidhoweverprovediculttoobtainconvergenceusingdnsqe.f,

itworksformeanvaluesbutitdidnotndthesolutioninturbulentows. Asthe

routine wassuccessfully used by Aanesland [1] this was suprising, and probably

indicatesthat theroutinewasnotappliedcorrectly. Thesolutionwastouse

Mat-labsfzero function todothesamejob. Thiswashoweveramuchslowersolution,

but itworked.

ThescriptiscompilatedasaprojectinPlatotobeabletocombinefree-format

and xed-format Fortran les, since dnsqe.f is used to solvesome meanvalues in

thescriptdirectly.

Theoriginalideawasthattheprogramshouldbegeneralanduserfriendly,the

endresultworksasintendedbutistoocomplex. Furtherworkwouldbetoinclude

moreerrorchecking,butalsotodramaticallysimplythescript.

In document Three-dimensional wake measurements (sider 45-0)

Analysis of the time varying wake

3.6 Wind tunnel

4.3.4 Analysis of the time varying wake

y/D cyl. [−]

u 2 /U ref 2 v 2 /U ref 2 w 2 /U ref 2 u 2 /U ref 2 , freestream

w 2

w 2

x/D = 10

0 200 400 600 800 1000

f = 42 . 8 Hz

9 . 7 m/s

f =

0 . 21 U

d = 42 . 9 Hz

10 −2 10 −1 10 0 10 1 10 2 10 3 10 4

f s

T s

2 13 = 8192

10 0 10 1 10 2 10 3 10 4 0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

X: 42.85 Y: 1.989

Frequency (Hz)

|Y(f)|

f s

T s

0 0.005 0.01 0.015 0.02

−4

−3

−2

−1 0 1 2 3

U [m/s]

T [s]

U−U mean V−V mean W−W mean

−200 −150 −100 −50 0 50 100 150 200

U x

U r

U θ

U r

U θ

φ 1

uw

vw

|x/D| ≥ 0 . 8

± 50 s − 1

0 . 25 m/s

± 40 s − 1

± 50 s − 1

−50 0 50

−3

−2

−1 0 1 2 3

dU/dy [1/s]

y/D cyl. [−]

uw

vw

uv

φ 1

± 20 deg

± 1 deg

1 . 1%

u x u r

|y/R| > 0 . 8

|y/R| > 0 . 8

501 /s

φ 1

uw

vw

uv

residual = X

u ² /U _ref ² v ² /U _ref ² w ² /U _ref ² u ² /U _ref ² , freestream

w ²

w ²

10 ⁻² 10 ⁻¹ 10 ⁰ 10 ¹ 10 ² 10 ³ 10 ⁴

f _s

T _s

2 ¹³ = 8192

10 ⁰ 10 ¹ 10 ² 10 ³ 10 ⁴ 0

f _s

T _s

U−U _mean V−V _mean W−W mean

U _x

U _r

U _θ

U _r

U _θ

φ ¹

± 50 s ⁻ ¹

± 40 s ⁻ ¹

± 50 s ⁻ ¹

φ ¹

u _x u _r

φ ¹