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June 2010

Per-Åge Krogstad, EPT

Master of Science in Energy and Environment

Submission date:

Supervisor:

Norwegian University of Science and Technology

Three-dimensional wake measurements

Pål Egil Eriksen

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Problem Description

Objective

The student should perform measurements in the wake of a stationary body with cyclic vortex shedding (e.g. the wake behind a cylinder) to study such flow fields. The student is free to choose a suitable measurement technique for this purpose, the measurement technique chosen must however have both excellent spatial and temporal resolution.

The following questions should be considered in the project work:

1. The student shall investigate the performance of a suitable measurement technique for three-dimensional unsteady flows such as e.g. three component hot wire anemometry

2. If the method is deemed suitable, he should perform measurements in the wake of a blunt body to see if the periodic flow may be picked up by the measurement

technique.

Assignment given: 18. January 2010

Supervisor: Per-Åge Krogstad, EPT

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ThisMasterthesisiscompletedattheDepartmentofAppliedMechanics,Thermo-

andFluidDynamicsattheNorwegianUniversityofScienceandTechnology.

I ammostgratefulto my supervisor ProfessorPer-Åge Krogstadforhis guid-

ance andencouragementin theprocess ofcompleting this thesis. His insightand

constructiveadvisehasbeenofgoodhelp.

Trondheim,14juni2010

PålEgilEriksen

(8)

The performance ofahotwireprobewith threewires is investigated fortwodif-

ferent ow cases. The wires are made of a platinum/rhodium alloy, and has a

diameterof

5 µm

. Thethreewiresmakeaprobevolumewithacrosssectionofap-

proximately5mm. Acosinustusingtheeectiveanglemethodgivesadeviation

of

± 1

foravariationofyawangleequalto

± 20

. First theprobewastestedin a fullydevelopedturbulentpipeow,for

< D = 10 5

. Goodresultswereobtainedfor

|y/R| < 0 . 8

,bothformeanvelocitiesandturbulentstresses. Closertothewallthe mean owgradientwastoolargerelativetotheproberesolution, givinglargeer-

rors. Thesecondowcasewasacylinderwake. Atraverseoftheowat

x/D = 10

wasperformed at

< D = 3 · 10 3

. The mean velocities and turbulent stresseswas partlyfoundto bein qualitativeagreementwithresultsfound inlitterature. The

shear stresses

uw

and

vw

werehoweverfoundto beunphysicallylarge,this isbe- livedto bedue tothevelocitygradientin thewake. Conditional averagingofthe

wakeresultswithrespecttoshedding frequencywasalsoconducted.

(9)

Egenskapenetilenhotwireprobemedtretråderharblittundersøktfortoforskjel-

ligestrømningstilfeller. Trådeneerlagetavenplatinum/rhodiumlegeringoghar

endiameter på

5 µm

. Detretrådeneskapereitprobevolum medeittverrsnitt

ca 5mm. Eektivvinkelmetoden harblittbrukt ogen tilpassningtilen cosinus

funksjon gir et avik på

± 1

foren variasjon av yaw-vinkelenpå

± 20

. Først ble probentestetienfulltutvikletrørstrømning,med

< D = 10 5

. Resultateneerigodt samsvarmedteorioglitteraturfor

|y/R| < 0 . 8

,bådemhpmiddelhastigeterogtur- bulentespenninger. Nærveggenblegradiententilmiddelhastighetenstoriforhold

til probensrommelige oppløsning, noe somga storefeil. Den andrestrømningen

som bleundersøktvarvakenbakeinsylinder for

< D = 3 · 10 3

. Demåltemiddel- hastigheteneogturbulentespenningenevardelvisioverenstemmelsemedresultater

fra litteratur. Skjærspenningene

uw

og

vw

var ufysiskstore. Det antas at dette

er på grunn av den store hastighetsgradienten i vaken. Midling med hensyn på

virvelavløsningsfrekvensenerogforsøkt.

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1 Introduction 3

2 Theory 4

2.1 Theoryofmulticomponenthotwiremeasurements . . . 4

2.1.1 Eectivecoolingvelocity. . . 4

2.1.2 Theeectiveanglemethod . . . 4

2.1.3 Coordinatetransformation . . . 7

2.1.4 Obtainingtheangularresponse . . . 10

2.2 Probevolumeandfrequencyresponse . . . 10

2.3 Turbulentpipeow. . . 11

2.3.1 Thepressuregradient . . . 11

2.3.2 Meanvelocityprole . . . 11

2.3.3 Turbulentshearstresses . . . 13

2.3.4 Turbulentnormalstresses . . . 14

2.4 Cylinderwake. . . 14

3 Experimental setupand procedure 16 3.1 Thehot-wireprobe . . . 16

3.2 Measurementchains . . . 17

3.3 Signalsamplingrate . . . 17

3.4 Datareductionprogram . . . 18

3.5 Pipeowrig . . . 18

3.6 Windtunnel. . . 19

4 Resultsand disscusion 20 4.1 Calibrationandtesting. . . 20

4.1.1 Velocitycalibration. . . 20

4.1.2 Eectiveanglecalibration . . . 20

4.2 Turbulentpipeow. . . 23

4.2.1 Pressuregradient . . . 23

4.2.2 Meanvelocities . . . 24

4.2.3 Turbulentshearstresses . . . 27

4.2.4 Turbulentnormalstresses . . . 29

4.3 Cylinderwake. . . 30

4.3.1 Meanvelocityproles . . . 31

4.3.2 Turbulentshearstresses . . . 34

4.3.3 Turbulentnormalstresses . . . 36

4.3.4 Analysisofthetimevaryingwake . . . 36

4.4 Performanceoftheprobe . . . 40

5 Futurework and recommendations 42

6 Conclusion 43

Appendices 45

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Incomplex owsit can be dicultto obtaingood velocity measurements. Many

measurementtechniquesdependonknowledgeoftheowdirectionandisoflimited

value ifit is unknown orvarying with time. The wake behind awind turbine is

suchaowwhere theexactowdirectionisbothunknownandtimevarying. To

perform successful single-point measurementsin such a ow, a probe capable of

measuringthevelocityinallthreedimensionsisneeded.

Iftheturbulentcharacteristicsoftheowisofinterestthetemporalresolution

must be good. The number of three dimensional measurement techniques with

goodtemporal resolutionis limited. 3D laserdoppler anemometry isoneoption,

hotwireprobeswiththreeormorewiresisanother. Bothhavetheirstrengthsand

weaknesses.

In this project3-wire constant temperature hotwire anemometry is studied.

The goal is to learn if three component hot wire can be used in a complex and

time varying ow to measure themean velocities and turbulent stresses. Totest

theperformanceoftheprobetwodierentowcasesisexamined. Therstcaseis

turbulentpipeow. Forthisclassicowcase,themeasurementscanbecompared

withanalyticresults,anditisthereforesuitableasaninitialtest. Thesecondcase

isthenearwakeofacylinder. Thecylinderwakecanbeanalyzedbothwithrespect

to themeanowandasatimevaryingowwherevorticesareshedataconstant

frequencyfromthecylinder. Itisthereforesuitableforassessingthecapabilitesof

theprobeinadynamicow.

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Thissectionexplainsthetheorybehindmultiwirehotwiremeasurementsandthe

basicsoftheowsthat areinvestigated.

2.1 Theory of multi component hot wire measurements

Thetheoryofsinglehotwireanemometryalsoappliestotheindividualwiresofa2

or3componenthotwire. Inthissectiontheprinciplesofmulticomponenthotwire

anemometryareinvestigated,knowledgeofsinglehotwireanemometryisassumed

to beaprerequisite.

2.1.1 Eectivecoolingvelocity

Velocity measurements can be divided into two types. In many cases the ow

directionisknownoratleastassumedtobeknown,themagnitudeofthevelocity

isthenofinterest. Inmorecomplexowstheowdirectionisunknown,andmust

be determined by measurements. A single hotwire canonly determine the ow

velocityV whentheowdirectionisknown.

E 2 = A + BV n

(1)

Toalsodeterminetheowdirection,oneneedsaprobewithat leastone wire

per dimensionof interest. These wires are placedat angles to one another, and

experience dierent magnitudesof cooling. A useful denition in this context is

theeectivecoolingvelocitydenedbyJørgensen.

V e 2 = U n 2 + k 2 U t 2 + h 2 U b 2

(2)

Theeectivecoolingvelocityisdenedasthevelocitynormaltothewirewhich

has thesamecoolingeect astheactual velocity vector. Equation2decomposes

the eectivecoolingvelocityinto three components. Figure1 showshow thedif-

ferentcomponentsaredened.

U n

isthe normalcomponent,

U t

is thetangential

componentand

U b

isthebinormalcomponentwhichisnormaltothen-tplane. The

threecomponentscontributeunequallytothecoolingofthewire. Thecoecients

kandhcorrectforthedierencesincoolingalongthethedierentaxes. Kandh

arenotconstantsbutfunctionsoftheowdirectiongivenby

α

and

β

,theyawand

pitch angles,

k = k ( α )

and

h = h ( β )

. Thetangential component is signicantly lessecientatcoolingthewirecomparedtothenormalandbinormalcomponents.

Foran innitelylongwire, normaland binormal cooling should beequallyeec-

tive. Inthenitecasetheowwill beaected bythesupportingprongs,butthe

eectissmallandthenormalandbinormalvelocityisoftenassumedtobeequally

eective,whichimpliesthathcan besetto 1.

2.1.2 The eective angle method

Theeectivecoolingvelocitycannotbemeasureddirectly,butitcanbeestimated

as a function of

α

. Consider rst the two dimensional case, with no binormal

(14)

cooling, asshown in gure2(a). Theeective coolingvelocity can be related to

thevelocityvectorSin the n-tplane by thefunction

f ( α )

. Thisthe basisofthe eectiveanglemethod,thattheeectivecoolingvelocitycanberelatedtotheow

velocitytrough theyaw-angle

α

.

V e = ( U n 2 + k 2 U t 2 ) 1 2 = Sf ( α )

(3) Fcanbemanydierentfunctions,but acosineisanaturalchoice.

V e = S cos( α e + α )

(4)

In equation 4 a new constant,

α e

, is introduced. The wire in gure 2(b) is permanently yawed relative to the probe axis,

y p

by the angle

α e

. Equation 4

therefore givesus the component of thevelocityvectorS which is normalto the

wire,asafunctionof

α

and

α e

.

Whenavelocitycalibrationofthewireisconducted,theprobeaxis

x p

isaligned

with the ow direction, hence

α

is zero(see gure 2(a)). The measured voltage

output,

E

,thencorrespondstoagivenvelocity,

U

,whichisthesameastheknown

velocityvectorS. Equation4canforthiscasebewritten:

V e = S cos( α e ) = U cos( α e )

(5)

Iftheprobeisyawedanangle

α

(seegure2(b)), forthesamevelocityS,the

voltageoutput

E yawed

willcorrespondtoavelocity

U yawed

whichisobtainedfrom

the velocity calibration.

U yawed

is dierent from S. Twodierentexpressions for the eective coolingvelocitymay now be written. Equation6gives theeective

coolingvelocityasthecomponentofSwhichisnormaltothewire.

(15)

Figure2: Velocitydenitionsin thenormal-tangentialplane

V e = S cos( α e + α )

(6)

Theeectivecoolingvelocity canalsobewritten asthecomponentof

U yawed

whichis normalto thewire. Thisrelation willnotdependon

α

since

U yawed

has

axedowdirectionrelativeto theprobe.

V e = U yawed cos( α e )

(7)

Bycombiningequation 6and 7arelationbetween thetruevelocityvectorS,

the owdirection alphaand thevelocity

U yawed

is found. From now on we will

denote

U yawed

inthemoregeneralformU(E),referringtothefactthatUisfound

from thehotwirevoltageEthroughthevelocitycalibration.

V e = S cos( α e + α ) = U ( E ) cos( α e )

(8)

InthecasewhereSand

α

isunknown,equation8willhavetwounknownsand

can not be solved alone. By combining two wires at an angle twoone another,

aset of twoequationsis obtained. Toreduce thenumberofunknowns allangles

and velocities aredened in theprobecoordinate system,(

x p , y p , z p

) ratherthan

relativetotheindividualwire. Thecoordinatetransformationwillbediscussedin

section X. The reductionof unknowns resultsin aset of two equations and two

unknowns. Theequations areimplicitbutcaneasilybesolved. An alternativeto

theeectiveanglemethodcouldbetotabulateffordierentvaluesof

α

. Whenthe

equationsareto besolvedonecanrstguess avaluefor

α

,usethecorresponding

(16)

equal,thenalsolutionisfound,ifnotanotheriterationisneeded.

Inthecaseofthreedimensionalowthebinormalcoolingmustalsobeconsid-

ered. Equation8describestheeectofthenormalandtangentialcooling,andcan

beexpandedbyaddingthebinormalcoolingontherighthandsideoftheequation.

S isnowthevelocitycomponentinthenormal-tangentialplane.

U i 2 cos 2 ( α ei ) = S i 2 cos 2 ( α ei + α ) + U bi 2

(9)

Theindexiisusedtorefertothedierentwiresoftheprobe.

U ( E i )

isobtained from thevelocitycalibrationcurveasafunctionof thewirevoltage,

E i

.

Forathreewireprobewegetasystemofequations

U ( E 1 ) 2 cos 2 ( α e 1 ) = S 1 2 cos 2 ( α e 1 + α ) + U b 1 2

(10)

U ( E 1 ) 2 cos 2 ( α e 2 ) = S 2

2 cos 2 ( α e 2 + α ) + U b 2 2

(11)

U ( E 1 ) 2 cos 2 ( α e 3 ) = S 3 2 cos 2 ( α e 3 + α ) + U b 3 2

(12)

Tobeable to solvethis systemthe (

x p , y p , z p

) coordinate systemis used, the

transformationfromwirecoordinatesystemtoprobecoordinatesystemisdescribed

in thenextsection.

2.1.3 Coordinate transformation

Toreducethenumberofunknownsinequation10itisnecessarytoexpress

S i

and

U bi

as functions of U,V and W which are dened in the probe xed coordinate

system,(

x p , y p , z p

). Figure3denesthecoordinatesystemandtheanglesneeded

to relatetheprobewirestothecoordinatesystem.

φ i

istheangle betweentheprojectionof wireiin the(

y p − z p

)planeandthe

y p

axis. Figure4showstheprojectionofthewires inthe (

y p − z p

)planeandthe

corresponding

φ

angles.

If the wires are placed in a perfect triangle, the values of the angles will be

90 ◦ , 330 ◦

and

210 ◦

respectively. Twovelocitycomponentsaredenedinthe(

y p −z p

)

plane,

U bi

is the binormal cooling of wire i and

U T P i

is the projection of the

tangentialcoolingvelocityofwirei,tpreferstotangentialprojection.

U T P

and

U b

canbecalculatedfortheindividualwires. TheyarefunctionsofV,Wand

φ i

.

U T P i = V cos φ i + W sin φ i

(13)

U bi = V sin φ i − W cos φ i

(14)

Thevelocitycomponentinthenormal-tangentialplaneofwirei,

S i

,isafunction

ofU andtheprojectionofthetangentialcoolingvelocity.

S i 2 = U 2 + U T P i 2

(15)

Substitutingfor

U T P

inequation15yieldsS asafunction ofU,VandW.

(17)

Figure4: Velocitiesandanglesin the(

y p − z p

)projection

(18)

S i 2 = U 2 + ( V cos φ i + W sin φ i ) 2

(16)

Theow angle

α

in thenormal tangentialplane must also bedened. From

gureXanexpressionfor

α

iseasilyfound.

Figure 5: Denitionof

α

α = arctan U T P

U

= arctan

V cos φ i + W sin φ i U

(17)

Equations13,15and17maybesubstitutedintoequation10toyieldthenal

equationforthreedimensional owoverwirei.

U i 2 cos 2 ( α ei ) = ( U 2 + ( W cos( φ i ) − V sin( φ i )) 2 )

(18)

+cos 2

α ei + arctan

W cos φ i − V sin( φ i ) U

+( W sin( φ i ) + V cos( φ i )) 2

Forthe three wireprobea set of three equations with three unknownsis ob-

tained.

(19)

Forthe individual wires the eective angle

α e

must be found. This obtainedby

placingtheprobeinauniformowwithavelocityS, andmeasuringtheresponse

formultipleyawangles,

α

,whilethepitchangle

β

isheldconstantatzero.Equation

8describestherelationbetweentheowvelocityandthevelocitymeasuredbythe

wire,U(E).TheratiobetweenSandU(E)canthenbefoundfromequation8.

U ( E )

S = cos( α e + α )

U ( E ) cos( α e )

(19)

Theresultsfrom measurementsat dierentangles

α

canthenbecurvttedto

equation19byadjusting

α e

toobtainthebestttothedatapoints. Atypicalset of calibrationanglesis

α = − 20 : 5 : 20

. Insection 2.1.2

α e

waspresentedasthe

geometric angle betweenthe wirenormalin thenormal-tangentialplaneand the

x p

axis. This is notentirelytrue,

α e

will also beafunction of the properities of the individual wire and most importantly of the ow angle. The eective angle

approachassumesthat theeectiveangleisconstant,thisishowevernottruefor

large ow angles. A litterature review by Lekakis [5] found several estimates of

the limits for x-wireprobesranging from

± 12 ◦ − ± 20 ◦

. Russ and Simon found

that the range of valid angles were larger for three-wire probes, in the range of

± 30 ◦

(reportedinthelitteraturereviewofAanesland[1]).

2.2 Probe volume and frequency response

The spatialresolution isan importantproperty ofameasurementtechnique. All

measurementtechniqueshavealowerlimitforspatial resolution, the variationis

large. Thespatialresolutionofapitotequalsthediameteroftheprobeatleast,for

alaserdopplerthesizeofthecrossectionofthelaserbeamsisthelimit,inparticle

imagevelocimetryitwilldependonthewindowsizeandoverlappingamongother

factors.

ForasinglesubminiaturehotwireLigraniandBradshaw[6]foundtheidealra-

tiobetweenwirelengthanddiametertobeapproximately

L/D > 260

for

L < 1 mm

for measurentsin aturbulent boundarylayer. Forlonger wires 'eddy averaging'

wasreported. Thewires usedin thisprojectis notclosetothedimensionsofthe

wiresusedbyLigraniandBradshaw,andcanthereforenotbeexpectedtoresolve

thesmallestscalesintheowaccurately. Thephysicalsizeofthethreewireprobe

willhoweverbeagreaterlimitingfactorthanthedimesionsoftheindividualwires.

Avelocitygradientacrossthemeasurementvolumeoftheprobewillmeanthat

thewiresintheprobeexperiencedierentvelocities. Inaowwithalargevelocity

gradient, i.e. close to awall, this can resultin large dierences acrosstheprobe

volume and distort the result. The size of theprobevolume will therefore limit

howlargegradientswhichcanbemeasured.

Theresponseoftheindividualwires isalso importantto obtainagoodresult.

Ifthefrequencyresponseofthehotwireanemomtersaredierent,someturbulent

componentscanbeoverestimated. If forexamplethegoalof anexperimentis to

validate whether aowis isotropicornot,adierencein frequency response can

(20)

responses.Toreducetheeectofthis,thesignalsshouldallbelteredatthesame

cutofrequency.

2.3 Turbulent pipe ow

A conned ow such as apipe ow will develop until a steady state solution is

reached. Assumingthattheowenteringthepipeisuniform, theboundarylayer

willimmediatelystarttogrowatthewall. Thenalsteadystatesolutionisreached

whentheinviscidcoreisgone,theowisthensaidtobefullydeveloped. Theform

ofthevelocityprolewilldependonwhethertheowisturbulentorlaminar,the

wallroughnessand thepressuregradient.

Fully developedturbulentpipeowwill exhibitcertaincharacteristics. Inthis

sectionabriefreviewofsomeofthese characteristicsisgiven.

2.3.1 The pressuregradient

Auniformowenteringapipewillberetardedbytheshearstressfromthewalls.

The pressure gradient will be greatest in the beginning, and gradually decrease

untiltheowis fullydeveloped. Atsteadystatethedriving forceof thepressure

gradientwillbalancetheshearstressonthewall.

∂P

∂x = τ w 4

D

(20)

Thewallshear stress canberelated to the wall-friction velocity,

u

, which is

animportantparameterin pipeow.

τ w = ρu 2

(21)

Bycombiningequation20and 21thewall-frictionvelocitycanbe foundfrom

thepressuregradient.

u 2 = ∂P

∂X D

4 ρ

(22)

2.3.2 Mean velocityprole

Aturbulentpipeowwill consistofthreeregions.

Aninnerlayerclosetothewallwhere viscousshearisdominating

Anouterlayerwhere turbulentshearisdominating

Anoverlaplayermergingthetwolayerstogether,wherebothtypesofshear

isimportant.

(21)

commonapproachisto identifytheimportantparametersinthedierentregions

and apply dimensional analysis. In the inner region the velocity is assumed to

depend on the wall shear, uid properties and the distance from the wall. Free

stream conditionsareassumed notto be important. The wallshear will however

dependonfreestreampropertiessuchasthepressuregradient.

u ¯ = f ( τ w , ρ, µ, y )

(23) Dimensionalanalysisyieldstwodimensionlessparameters.

u ¯ u = f

yu ν

(24)

Thetwodimensionlessgroupsaredenoted

u +

and

y +

respectively,giving

u + = f ( y + )

. In the inner viscous shear dominated region turbulent shear can be ne- glected. Analysisof themomentum equation will thenyield that

u + = ( y + )

, see e.g. White[12]. Intheouterregionofthepipeowthevelocitynolongerdepends

on viscous shear, but on the freestream pressure gradient and the radius of the

pipe,R.

U cl − u ¯ = f ( τ w , ρ, R, ∂P

∂x ) , y

(25)

Dimensionalanalysisyieldsthreedimensionless groups.

U cl − u ¯ u = g

y R , R

τ w

∂P

∂x

(26)

Somewherebetweentheinnerandouterlayer,thetwolayersmustmerge,giving

thesamevelocity. Atagivenaxialpositioninthepipe,theshapeofgisassumed

tobeafunctionof

ξ = R

τ w

∂P

∂x

. Theoverlaplawforagiven

ξ

canthenbefoundby

manipulating equation24and 26.

u ¯ u = f

yu ν

= U cl

u − g y R

(27)

Thetwo regionscanonly bemerged ifthe fand g are logarithmic functions.

Theresultingrelationcan bewritten bothintermsofinnerandoutervariabels.

u ¯ u =

1 κ ln yu

ν + B

(28)

U cl − u ¯ u = − 1

κ ln y

R + A

(29)

Dierent valueshavebeen suggestedfor theconstants

κ

and B,but theyare

consideredto benearlyuniversial.A willdepend on

ξ

.

(22)

In a fully developed pipe ow the only mean velocity component is that in the

streamwisedirection,U.Insection2.3.2itwasshowedhowUvariesasafunction

ofy. Theshearstressesin aowarecloselylinkedtothemeanvelocitygradients.

ThegeneralizedBoussinesqeddyviscosityhypothesissuggestsarelation.

u i u j = ν T ∂U i

∂x j − 2

3 ρkδ ij

(30)

Basedonequation30,onecanmakesomassumptionsonthemagnitudeofthe

shear stressesin a pipe ow. Theonly mean velocity gradient is

∂U

∂y

, one would

thereforeexpect

uv

tobethedominantshearstressintheow.

The variation of

uv

as a function of y can be found from manipulation of the Reynolds averagedNavier-Stokesequations. Equations 31 and 32 show the

simplied RANSequationsforthepipeow.

∂p

∂x = ∂

∂y

µ ∂U

∂y − ρ uv ¯

(31)

∂p

∂y = ∂

∂y −ρ v ¯ 2

(32)

Byintegratingequation32withrespecttoy,from0toyanexpressionforthe

pressureatagivenycoordinateisfound.

P

ρ + ¯ v 2 = P 0

ρ

(33)

Ifonetakesthederivativeof thepressurewith respectto x onewill nd that

∂P

∂x

isconstantwithrespecty,seeingthat

v 2

isnotafunction ofx.

∂P

∂x = ∂P 0

∂x

(34)

The equation in the x-direction can be integrated in the same manner, with

respect to y from 0 to y. By using the fact that

dP

dx

is constant the following

expressionisfound.

∂P

∂x y = µ ∂U

∂y y − ∂U

∂y 0

− ρuv

(35)

Substitutingequation21for

dU dy 0

yieldsthefollowingequation.

0 = y ρ

∂P

∂x + µ ρ

∂U

∂y − uv − u 2

(36)

Atthecenterof thepipeat

y = h

,

µ ρ dU dy − uv ¯ = 0

dueto symmetry. Byusing theknownsituation at thecenter linean expressionforthe variationofthetotal

stresscanbefoundasafunctionofy.

(23)

−uv + µ ρ

∂U

∂y = u 2 1 − y

h

(37)

Equation37providesvaluableinformationabouthow

uv

vary asafunction of

y. In the inviscid region viscous shear stress is neglible and the turbulent shear

stress is expected to vary linearly with respect to y. And at the center line all

shearstressesareexpectedtobezero.

2.3.4 Turbulentnormal stresses

Boussinesq estimatesthe normalstressesto beonethird ofthe turbulentkinetic

energyk. Inturbulentpipeowthatisnotthecase. Equation30assumesisotropic

and homogeneous turbulence, but in a shear ow the production of the normal

stresses will vary. In the case of a turbulent ow the turbulent kinetic energy

equationin theaxial directionwillbetheonlyonewithaproductionterm.

P roduction = −uv ∂U

∂y

(38)

Theproductiondependsonthemeanowgradient,asmeanvelocityin they

andzdirectioniszeroforafullydevelopedpipeowtheproductionoftheturbulent

normalstressesis zero. Thisdoesnotmeanthattheothernormalstresseswillbe

zero. Energy is transfered from

¯ u 2

to

v ¯ 2

and

w ¯ 2

by nonlinear pressure-velocity interactions[8].

2.4 Cylinder wake

ThecylinderwakeisacomplexandReynoldsnumberdependentow. Forverylow

Reynoldsnumbers,Re<49,alaminar,symmetricalandsteadyrecirculationregion

is presentbehindthe cylinder. As the Reynoldsnumberincreaseslaminar vortex

shedding will begin. When the Reynoldsnumber reaches about 194 streamwise

vorticesbegintoform[4]. Uptoabout

Re D = 1000

theStrouhalnumberincreases [11]. TheStrouhal numberis dened astheratio between

f D

and U, where fis

thevortexsheddingfrequencybehindthecylinder.

St = f D

U

(39)

For

Re D > 1000

the Strouhalnumber startto decrease untill it stabilizes for

10000 < Re D < 100000

atavaluecloseto0.21[11]. Theregionfrom

Re D = 1000

to

Re D < 200000

isnamedthesubrcriticalrange[13]. Inthesubcriticalrangethe boundarylayeronthecylinderremainslaminar. IftheReynoldsnumberincreases

furthertheboundarylayerstartstodevelopfromlaminartoturbulent,movingthe

pointoftransitionupstreamandtheseparationpointdownstream. Thisresultsin

reduceddragandanarrowedwake.

Unlike apipe ow, the wake is continually evolving. The mean velocity eld

will continue to developuntil free stream conditions are reached. Momentum is

continuallytransportedtowardsthecenterofthewakewherethevelocitydecitis

(24)

form yieldsthefollowing.

∂V

∂y = − ∂U

∂x

(40)

The continuity equation tells us how the gradient of V with respect to y is

expected to vary. Far from the centerline

dU

dx

will benegative, sincethe wakeis

expanding. Inthisregion

dV

dy

willbepositive. Closertothecenterofthewakewe

expect

dU

dx

tobepositive,

dV

dy

mustthereforebenegative.

By performing an order of magnitude analysis, the x-direction Reynolds av-

eragedNavier-Stokesequation canbe simplied considerably. Thetwodominant

terms arethe U gradientwith respectto x and crossectionalgradientof thetur-

bulentshear stress

uv

.

U ∂U

∂x = − ∂

∂y ( uv )

(41)

Fardownstream from thecylinder

( x/D > 80)

theowcanbeassumedto be self-preserving[ref], which means that theshape ofthe proleis preservedalong

thex-axis. Theshapeoftheprolecanbefoundbystartingwithequation41and

makingsomeadditionalassumptions.

(25)

3.1 The hot-wire probe

The probe consist of three wires on six supporting prongs, asshownin gure3.

Thechosengeometryisdenedbytwoproperties:

Theprojectionofthewires inthe

y p − z p

planeisatrianglewith60degree

angles

Thewires areinclinedanangle

α e = 35 . 26 ◦

relativetothe

y p − z p

plane

Thesepropertiesgiveageometrywherethewires areorientatedperpendicular

to one another. The geometry is the same as recommended by Aanesland [1].

It was chosen to reduce the probe volume and give a good cooling response in

all directions. The probesare manufacturedto t theabovedescription,but the

angleswillneverbeexactlycorrect. Theeectiveanglesmustbefoundtroughthe

procedure described in section 2.1.4. By taking a picture of the probe trough a

microscope,theorientationofthewiresin the

y p − z p

planecanbefound, such a

photo can be seenin gure 6. Theangle

φ 1

is used to relate therotationof the

probein the

y p − z p

planetocoordinatesystem.

Figure6: Pictureofthe

y p − z p

planetakentroughamicroscope

Thelengthofthesupportingprongsischosensuchthattheowoverthewires

are notinuencedbytherest of theprobes. It isalso important that theprongs

are nottoolong,asthiscancausevibrationswhich inturn canbe interpretedas

aturbulentvelocitycomponent.

Aplatinum(90

%

),rhodium(10

%

)alloyisusedinthewire. Thisalloygives agoodoxidationresistance,relativelyhightensilestrengthbuthasarelativlylow

(26)

The diameter also inuences the sensitivity of the probe. In this project a wire

with d =

5 µm

is used. This is arelativley thick wire, which reduces sensitivity but increases mechanicalstrength. Thelength ofthe wirebetween theprobesis

approximatly4mm,ofthis approximatly1.75mmof thecoatingonthewirehas

beenetchedawayin thecentre. Thisgives

l/d ≈ 350

. Toreduce theinterference ontheowfromthesupportingprongsthedistancebetweenthesupportingprongs

should notbetosmall,theshapeoftheprongtipswillalsoaect theow[2].

Thecrossectionofthemeasurementvolumeis

≈ 5

mm,andthespatialresolu- tionoftheprobeisthereforassumedtobe5mm.

3.2 Measurement chains

Figure7describesthemeasurementchainintheexperiment.

Figure7: Measurementchain

Thehot wireanemometersare optimized for

1 µm

notfor

5 µm

which is used

in this experiment. For the initial measurement setup a high frequency distur-

bance appearedonthesignalathighvelocities

( > 12 m/s )

. This isaresultofthe inability of the control circuitto regulate thewire voltage. It could to acertain

degreebehelpedbychangingthebiassetting ontheanemometer. Thisincreased

the dampingin the control loop at the cost of a lowerfrequency response. For

thepipemeasurementsthiswassucient,in thecylinder wakehoweverthelarge

uctuationsrequired that higher velocities couldbe measured. The solutionwas

to extendthecable, increasing

R cable

,and therebyincreasing thedampinginthe

loop.

3.3 Signal sampling rate

The sampling rate must be set accordingto the timescale of the smallesteddies

ofinterest. Whentherangeoftimescalesexpectedisunknown,thesamplingrate

mustbesetaccordingtothesmallesttimescaleoncanexpect. Kolmogorovsmicro

(27)

notbeendoneinthisproject.

Thelimitingfactorforthesamplingrateisinthiscaseisthefrequencyresponse

oftheanemometers. Thefrequencyresponsewasfoundtovarybetweenthewires

from approximatly6.3 kHzto 8.0 kHz. A lowpass ltercut o frequencyof 6.5

kHz waschosen. Thesamplingrate shouldbeset accordingto thesamplingrate

theoremorNyquistcriteria,whichstatesthatthesamplingrateshouldbegreater

thantwicethemaximumfrequencyexpectedtoavoidaliases[10].

A suitablesampling time should bechosensuch that repeated measurements

givethesameresult,averagingoverrelevanttimescalesintheow.

Inthecylinderwakemeasurementsasampling rateof 13kHzwasusedalong

with a sampling time of 20 seconds. For the pipe measurements the sampling

frequency wasset to 7 kHzand the sampling time to 10 seconds. This wasnot

intendedtobethenalmeasurements,but simplypreliminarymeasurements,the

reducednumberofdatapointsgavesignicantlyreduceddatasizeandwastherefore

chosenatthetime.

3.4 Data reduction program

The sampled signal from the velocity calibration, the eective angle calibration

and traverses, were stored in text les and imported into a Fortran script. The

script corrects the data for temperature change, ts polynomials to the velocity

calibration data, calculates the eective angles, and uses the calibrationdata to

calculatetimeseries ofvelocityvectorsfromthevoltagetimeseries.

The solution of the equations (Eqs. 18) was be found by using a zero point

nder. Initially afortranfunction called DNSQE from theSLATEC librarywas

used, this function had previouslybeenused byAanesland [1] with success. The

algorithmworkedneforaveragedvoltages,butconvergensproblemsarisedwhen

the turbulenttimeseries wereanalyzed. As analternativeMatlabs fzero function

wasused. TheMatlabfunctionisconsiderablyslowerthantheFortranroutinebut

it doesthe job. Simple constraintswere placed on the solutionto insure that a

physicallycorrectsolutionwasfound. TheFortranscriptusedfordataanalysisis

describedfurherinappendix6.

3.5 Pipe ow rig

ThepiperigconsistsofahydraulicallysmoothPVCpipe,with adiameterof 186

mm and a length of 83 diameters. Ten pressuretaps are mounted on the pipe,

making it easy to measure the pressuregradient. The pipe is tted such that a

traversecan be mounted ontop, making it possible to traverse the ow through

thecenterofthepipe. Velocitiesinthepiperigcouldbevariedfrom5to12.5m/s.

Thecoordinate systemusedin thepipehasits reference

( y = 0)

onthe centre lineofthepipe,yispositiveabovethecenterline,andnegativebelowthecentreline.

The velocities in thepipe aredenoted

U x

,

U r

and

U theta

and arethe axial, radial

andcircumferentialvelocitiesrespectively.

(28)

An openloop wind tunnel is used forthecylinder wakemeasurements. The test

sectionis45cmx45cmand110cmlong. Acylinderwithadiameterof47.5mm

is tted in the center of the test section, leaving 50 cm of distance downstream

for the ow to develop. Measurementsare takenat

x/D = 10

. Velocities in the windtunnelcouldbevariedfrom4to30m/s.

Inthewindtunnelthecentreofthewakeisthereference

( y = 0)

inthecoordi- nate system,yispositiveabovethecenterline,andnegativebelowthecenterline.

U,VandWaretheaxial,vertical,andtransversevelocitycomponentsrespectively.

(29)

4.1 Calibration and testing

4.1.1 Velocity calibration

Inthepiperigthevelocitycalibrationwasperformedforvelocitiesbetween5and

12.5m/s. Athird order polynomialt to thecalibrationdata includingthezero

velocitypointgavearesidualoftheorderof

10 − 1

,whileasecond orderttothe datawithoutthezerovelocitypointgavearesidualoftheorderof

10 − 3

. Asecond order polynomialwasthereforefoundmostsuitableforvelocitycalibrationinthe

piperig. Theresidualisdened asthesumoftherelativedeviations betweenthe

polynomialt andthettingdata.

Inthewind tunnelthe velocity rangedfrom 4to 30 m/s. Figure 8showsthe

distributionofthemeasuredvelocityin apointinthecylinderwake.

0 2 4 6 8 10 12 14 16

0 1000 2000 3000 4000 5000 6000

U(E 1 ) hw [m/s]

# []

distribution lower calibration limit centre of distribution

Figure 8: Distribution of streamwise velocity,U, in a measurement point in the

wakebehindacylinder

The velocity scatter falls under the lowest freestream velocity obtainable in

the windtunnel. This isnotideal asthe polynomialt in that regionmost likely

will causeanerrorin theestimated velocitybut itcouldnotbeavoided. A third

order polynomialt tothe calibrationdata includingthe zerovelocity pointwas

chosenandgavearesidualof

10 1

. Thematchbetweenthepolynomialtandthe calibratondataforwire1isshowningure9.

4.1.2 Eectiveangle calibration

Figure10showsthecosinettingoftheeectiveanglecalibrationdata.

(30)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0

2 4 6 8 10 12 14 16 18

e − e 0 [V]

U [m/s]

polynomial calibration data

Figure 9:

−20 −15 −10 −5 0 5 10 15 20

0.7 0.8 0.9 1 1.1 1.2

U(E 1 ) hw /S [−]

Angle [ o ] Calibration data Cosine fit to data

Figure10:

(31)

Wire

α e

1 31.69

2 35.88

3 35.64

Table1: Eectiveanglesfoundfrom calibration

Trueangle

U pitot

U V W Calc.angle

|V |

Rel.error

-20.00 9.9692 9.3490 0.4565 -3.5968 -21.05 10.0275 0.0058

-15.00 9.8884 9.6096 0.4545 -2.7205 -15.81 9.9976 0.0110

-10.00 9.9434 9.7241 0.4218 -1.7365 -10.13 9.8869 0.0057

-5.00 9.9169 9.8504 0.2921 -0.8429 -4.89 9.8907 0.0026

0.00 9.9007 9.8726 0.1849 0.1024 0.59 9.8749 0.0026

5.00 9.9608 9.8039 0.1368 0.9367 5.46 9.8495 0.0112

10.00 9.8936 9.6796 0.0971 1.7844 10.45 9.8431 0.0051

15.00 9.9672 9.5099 0.1003 2.6105 15.36 9.8622 0.0105

20.00 9.9852 9.2656 0.1193 3.3928 20.12 9.8680 0.0117

Table2: Test ofsolutionondatasetforwire1

The residuals of the curvet for the wires was of order

10 − 2

. The eective anglesgivenbythecalibrationis givenin table1.

Theangles are in thevicinity ofthe idealvalue of

35 . 26

and vary within an aceptablerange. Totesttheeectiveanglesandthe

φ

anglesthedatasetfromthe

anglecalibrationofwire1canbesolved. Table2showsthetrueowangle,theU

velocitymeasuredbythepitot,thecalculatedvelocitycomponents,thecalculated

α

, thelength ofthe calculatedvelocityvector andthe relativeerrorbetweenthe

velocitymeasuredbythepitotandthelengthofthecalculatedvelocityvector.

Table2showsthatthecalculated

α

fallswithin

± 1

ofthetrueowangle. The relativeerrorbetween

U pitot

and

|V |

is

1 . 2%

atmost. The

Y p

componentvelocity,

V,should bezerobutshowsavariationwithrespectto

α

. Themaximumvalueof

V correspondstoaowangle of

2 . 9 ◦

ora

4 . 7%

relativeerror,whichisarelativly large error. The variation in V corresponds to an angle of

2 . 2 ◦

. Aanesland [1]

performedthe samemeasurements, and reportsa

3 − 4%

relativeerrorin V and W forsimilarconditions.

The largedeviation seemsto be a combination of misalignment of the probe

relativetotheowandadependencyofthesolutiononthetrueowangle. Aprobe

pitchangledierentfromzerowouldgivethemisalignment,producingapermanent

oset. Thevariationwith

α

is mostlikelycausedbyawrongvaluechosenfor

φ 1

,

since

φ 1

is determined by visual observation and is not likely to be exact. The variationinVcouldbeusedto ndthecorrectvalueof

φ 1

,thecalculationscould thenbererunned,and theremainingconstanterrorin Vshould becausedbythe

pitch. Suchaprocedure hasnotbeenattempted in thisproject. Several authors,

i.e. Cantwell and Coles [3] uses yawingto perform similar corrections, Cantwell

andColesuseditforax-wireprobe.

(32)

Twoprolesof theturbulent pipeow hasbeentaken ata Reynoldsnumberof a

approximately

10 5

. Themain dierencebetweenthetwoprolesisthat theyare takenattwodierentvaluesof

φ 1

. Thisisdonetoinvestigatetheeectofrotating theprobe.

4.2.1 Pressuregradient

The static pressure in the pipe is measured using the pressure taps distributed

along the pipe. Instead of measuring the absolute static pressure, the pressure

dierenceismeasuredbetweenthedierentpointsandachosenreferencepointat

X/D = 70 . 5

. Figure 11showsthedropin staticpressurealongthepipe. Theline drawnintheplotisastraightlinefrom therstmeasurementto thelast.

0 10 20 30 40 50 60

5 10 15 20 25 30 35 40 45 50

x/D []

dP [Pa]

Pressure drop, Re = 9.7413e+4 Pressure drop, Re = 1.0329e+5

Figure11:

AsexpectedthehighestReynoldsnumbergivesthebiggestpressuredrop. Both

measurementseriesshowanearlylineardropinpressure,asonewouldexpectfor

afully developed ow. Thepressuregradientisapproximatedby consideringthe

pressuredrop from

X/D = 11 . 3

to

X/D = 70 . 5

. Based onthepressuregradient thefrictionvelocitymaybefoundfromequation22. Thefrictionvelocitycanalso

beestimatedfromthefrictionfactor,f.

u = U avg f

8 1

2

(42)

WhentheReynoldsnumberandtheroughnessheightofthepipewallisknown

thefrictionfactorcanbefoundfromtheMoodydiagramorfromanequation. The

PVC pipe is hydraulically smooth, and thefriction factor cantherefor be found

(33)

φ 1 Re D dP dx [ P a/m ] u dP

dx

[ m/s ] f Re D [ − ] u f ReD [ m/s ]

90

9 . 74 × 10 4

3.2236 0.3534 0.0181 0.3736 180

1 . 03 × 10 5

3.5487 0.3708 0.0179 0.3937

Table3:

from Prandtls equation for smooth pipes [11]. Table 3 gives the results for the

dierentvaluesof

φ 1

The two means of calculating the friction velocity gives similar results, the

values given by the friction factor are about

6%

higher than that given by the pressuregradient. Inthefollowingthefrictionvelocityobtainedfromthepressure

gradientisassumedtobecorrect.

4.2.2 Mean velocities

Figure12showsthevelocityprolefortheaxialvelocity

U x

,forboth

φ 1 = 90

and

φ 1 = 180

.

0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

−1

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

U x /U x,max [−]

y/R [−]

U x,φ

1 = 90 U x,φ

1 = 180

Figure 12: Normalizedmean prole,U.

U x,max,φ 1 =90 = 8 . 75 m/s

,

U x,max,φ 1 =90 = 9 . 25 m/s

Thevelocityonthecenterlinewasmeasuredasecondtimeaftertheprolewas

taken, the resultwaswithin

1%

of therst measurement forbothproles. Both proleshavethesameshape.

Ingure13thedatafor

φ 1 = 90

and

φ 1 = 180

isplottedagainstthelogarithmic law. Torbergsen [9] didmeasurementsin the samepipe rig for

Re = 75000

and obtained agood t with the logarithmic law using

κ = 0 . 41 andB = 5 . 5

. White [12] claimsthat

κ = 0 . 41 andB = 5 . 0

giveabettert toexperimental data. Both

(34)

datamatchesthechoiceof

κ = 0 . 41

. B=5.0assuggestedbyWhitegivesthebest tto themeasurementdatain thelog-lawregion.

10 2 10 3

15 16 17 18 19 20 21 22 23 24 25

y + = y u * /ν [−]

u x + = U x /u* [−]

Velocityprofile, U x

φ 1 = 90 φ 1 = 180

u x + = (1/0.41)ln y + + 5.0 u x + = (1/0.41)ln y + + 5.5

Figure13: Logarithmicregion

Inafullydevelopedturbulentpipeowonewouldexpectthetransversevelocity

components

U r

and

U θ

to be zero. Figures 14 and 15 show that neither of the

velocitycomponentsareexactlyzeroacrossthepipe. Theradialpipevelocity

U r

,is

fairlyconstantoverthecrosssectionofthepipe,butshowsomevariation,especially

close tothepipe walls. Therangeof variationin velocityis about

± 0 . 125 m/s

or

approximatelyanangle of

± 0 . 8

relativeto the averageaxial velocity. Both the proles for

φ 1 = 90

and

φ 1 = 180 ◦

showthe samekindof variationwith respect

to

y/R

but theyare osetrelativeto one another. Theoset equalsabout

1 . 6 ◦

ofprobepitch,whichiswithintheerrorrangeonemustexpectwhentheprobeis

alignedwiththeowvisually.

Thecircumferentialvelocity

U θ

showapeculiar variationoverthecrossection of the pipe, varying overa rangeof

± 0 . 45 m/s

or

± 2 . 86 ◦

relativeto theaverage

velocity. If thedeviation of the circumferentialvelocity is compared to the local

axialvelocitytherangeofanglevariationexceeds

± 4

,this isshowningure16.

Thevariationof

U θ

withrespectto

y/R

followsthesamepatternforbothseries

of measurementsand closely resembles atypicalinverse tangent function. If the

variation were to be explainedphysically it would implythat the owinside the

pipe wasrotating about the centerline. The velocity doeshowevernot decrease

close to the wall, but increases rapidly, this implies an unphysically large shear

stressonthewall.

Sincetheowisassumedtobeunphysicaltheradialvariationmustbecaused

byoneormoreerrorsin thesetup,datareductionorcausedbylimitationsofthe

(35)

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15

−1

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

U r [m/s]

y/R [−]

U r,φ

1 = 90 U r,φ

1 = 180

Figure14: Radialmeanvelocity,

U r

−0.5 0 0.5

−1

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

U θ [m/s]

y/R [−]

U θ,φ

1 = 90 U θ,φ

1 = 180

Figure15: Circumferentialmeanvelocity,

U θ

(36)

−5 0 5

−1

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

arctan(U θ /U x ) [deg]

y/R [−]

U θ,φ

1 = 90 U θ,φ

1 = 180

Figure 16: Circumferentialmeanvelocity,

U θ

, expressedasangulardeviation rela-

tiveto

U x

probe. Thedistinctshapeoftheprolecanmakeonewonderifthisvariationcould

be linked to an error in the arctanexpression in equation 18, but no such error

hasbeenfound. Aninterestingobservationcanhoweverbemadebyobservingthe

dierenceinthetwocenterlinemeasurements,bothfor

U r

and

U θ

therelativeerror

betweenthetwocenterlinepointsislarge. Whiletherepeatedmeasurementsgave

solutionsfortheaxialvelocity

U x

withinonepercentoftherstsolution,theradial

and circumferential repeated solutions can vary up to

0 . 5 ◦

and

1 ◦

respectively.

Thisisalargedeviationcomparedtotherangeofthecalculatedvaluesfor

U r

and

U θ

. By re-examiningthe measurementdata there was found to be asmall drift in wirevoltage, whichcould notbecorrected forby consideringthe temperature

change. Ideally themeasurement seriesshould havebeenrepeated, but theerror

wasdiscoveredtolate. Basedonthisobservationpartsofthelargevariationfor

U r

and

U θ

mightbecausedbyvoltagedrift. Theshapeofthevariationof

U r

and

U θ

doeshoweverseemtobeafunctionofyorasomeotherpropertyrelatedtoy,not

onlyapossiblevoltagedrift. Butwhatpropertycouldthatbe?

U x

isafunctionof

y,butissymmtricaboutthecentreline.Thegradientof

U x

alsovariesasafunction

of y but is notsymmtric. Insection 2.2 the possible error ofmeasuring in large

velocitygradientswas dicussed, this couldpossiblybethecause. This discussion

iscontinuedin thenextsectiononshearstresses.

4.2.3 Turbulentshear stresses

As discussed in section 2.3.3, the

u x u r

shear stress is expected to be the domi-

nantshearstressandbehavelinearlyacrossalargeportionofthepipecrossection

accordingtoequation37. Figure17showsthetheoreticalrelationandtheexperi-

(37)

−1 −0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

−u x u r /u * 2 [−]

y/R [−]

φ 1 = 90 φ 1 = 180 Equation

Figure17: Turbulentshearstress

u x u r /u

Theexperimental resultsfollowthelinear relationwell. The gradientis abit

smallerthan1inthecentreregionwhereviscousshearisassumedtobeneglectable

but thedeviationis small,averysimilar resultwasfoundbyTorbergesen[9]. At

thecentrelinethemeasuredshearstressisclosetozero,asexpected.

Closeto the pipe wallthe turbulent shear stressesare expected to decay and

droptozerointheviscoussublayer. Thespatialresolutionoftheprobeisnotlarge

enoughtomeasure closerto thewallthanabout

y + = 70

whichisfar outsidethe viscoussublayer. Themeasurementdatashowlittletendencytodropofclosetothe

wall. For

φ 1 = 90

there isalittledropfor

y/R < − 0 . 9

,whilefor

y/R > 0 . 9

there is actually anincreasein shear stress measuredfor both datasets. Theincreased

shearstresscouldjust beoutliers,butithappensforbothdatasets.

The two other shear stresses,

u x u θ

and

u r u θ

should theoretically be zero as there is no meanvelocity gradientresultingin productionof neitherof them. In

gure18thenormalizedstressesareplotted. Inthecentreregionthemagnitudeof

u x u θ

and

u r u θ

arerelativelysmallcomparedtothemaximumvalueof

u x u r

,about

3

%

, but not zero. Moving closer to the wall both shear stresses increase slowly until

|y/R| ≥ 0 . 8

,where theshearstressesincreasemorerapidly. Themagnitude and variation of the shear stresses can not be explained physically, and must be

relatedtothemeasurementprocess.

For

|y/R| ≥ 0 . 8

thevelocitygradientexperiencedbytheprobevolumeislarge.

The exact eect of an exessively large velocity gradient compared to the probe

volume is unknown. But it will result in calculated velocities dierent from the

truevelocity,as the wallis approached. Theresult canbe agradientofboth

U r

and

U θ

withrespecttoyasobservedinsection4.2.2. Subsequentlythisislikelyto

(38)

−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3

−1

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

[−]

y/R [−]

u x u θ /u * 2 u r u θ /u * 2

Figure18: Shearstresses

u x u θ

and

u r u θ

resultin unphysicalshear stresses. Insection4.2.2 thedeviation forboth

U r

and

U θ

wasfoundtoincreaserapidlyfor

|y/R| ≥ 0 . 8

,thismatchestheresultfoundfor

u x u θ

and

u r u θ

and supports the theory that a largevelocity gradientbiasesthe

result.

WhencomparingtheresultswiththatfoundbyAanesland[1],thedatascatter

fortheprolesobtainedin thisprojectislargeranddonotcollapseasneatlyinto

a line astheresults ofAanesland. What causes this is uncertain,but a to short

samplingtime couldpotentiallybethereason.

4.2.4 Turbulentnormal stresses

Figure 19 displaysthereduced normalstress forbothmeasurementseries, which

revealsthat

( u x ) +

isthelargestnormalstress,asonewould expect.

Torbergsen [9]found

u x +

onthe centrelineto beapproximately0.85 for

< = 0 . 75 e 5

. TheresultsshouldbecomparableastheReynoldsnumbersisof thesame order. According to the resultsof Torbergsen,

u x +

is fairlyconstant onthecen-

trelineforincreasingReynoldsnumbers,butincreasesclosertothewallduetothe

increasedvelocitygradient.

u x + cl. = 0 . 85

matchestheobtainedresultsfairlywell, there is however somescatter in the data as is already mentioned in the end of

section4.2.3. Thedataseriesforthetwodierentprobesalsogiveslightlydierent

results. Movingcloserto thewall

u x +

isunderestimatedcompared tothe results ofTorbergsen,butmatchestheresultsofAanesland[1]better.

u r +

and

u θ +

show

the same variation asreported by Torbergsen for

|y/R| ≥ 0 . 6

but the scatter is largeforvariationof

φ 1

.

Valuesfor

u x +

,

u r +

and

u θ +

cannot beestimated without givinga relativly

largepotentialerror.

u θ +

for

y/D = ± 0 . 6

,canforinstancebeestimatedas

1 . 1

but

(39)

−1 −0.5 0 0.5 1 0.8

1 1.2 1.4 1.6 1.8 2 2.2

[−]

y/R [−]

u x + φ 1 = 90 u r + φ

1 = 90 u θ + φ

1 = 90 u x + φ

1 = 180 u r + φ

1 = 180 u θ + φ

1 = 180

Figure19: Reducednormalstress

thescatter isof theorder0.2, andcould potentiallybelargerifmeasurementsat

morevaluesof

φ 1

weretaken. Thelargevaritioncanbecausedbypooradjustment

of

φ 1

,itisasmentionedearliersetvisually.

Ingure20theturbulenceintensityrelativetothelocalstreamwisevelocityis

plotted.

Thestreamwiseturbulence intensity onthe centreline,

u x + ≈ 3 . 3 − 3 . 6

. Tor- bergsenreporteda valueof approximately

3 . 5%

. On thecentreline

u r +

and

u θ +

should beequaldueto symmetry,there ishoweversomedierencewhich ismost

likelycausedbymisalignementoftheprobe.

4.3 Cylinder wake

Thecylinderwakecanbeanalyzedbothasameanowandasatimevaryingow.

Both approacheswillbetestedin thissection. Ideallyseveralmeasurementseries

for dierentvaluesof

φ 1

should havebeentakento gainmoreinformationabout theproberesponse. Onlyonemeasurement serieswastakenhowever. A velocity

prole wastakenin the test sectionbefore thecylinder wasinserted,to mapthe

referencefreestreamconditions.

Vortexsheddingfromacircularcylinderisamuchstudiedow,itdidhowever

provediculttondnearwakeresultsinthesamerangeofReynoldsnumbers

( 10 4 )

, and downstreamdistancex/D.A surveyusingafourwirehotwireprobebyOng

and Wallaceat

< = 3900

wastheclosestmatch found[7]. Theyusedafour wire hot wire probe with a crossection of 1 mm x 1 mm. For the ow investigated

in this project

< = 30717

, placing it in the subcritical range together with the resultsofOngandWallace. Theresultscanthereforebeassumedindependentof

(40)

−1 −0.5 0 0.5 1 0.02

0.04 0.06 0.08 0.1 0.12 0.14

u i /U [−]

y/R [−]

u x /U φ

1 = 90 u r /U φ

1 = 90 u Θ /U φ

1 = 90 u x /U φ

1 = 180 u r /U φ

1 = 180 u Θ /U φ

1 = 180

Figure20: Turbulenceintensity

Reynoldsnumberandcomparable.

Emphasisis placedonassesmentof thephysical validityof theresult, aswell

ascomparisonwiththeresultsofOngandWallace.

4.3.1 Mean velocityproles

The prole of the streamwise component U is shown in gure 21, the velocities

are plottedrelativeto theincomingvelocity. Fromgure21onecansee thatthe

velocityinthewakeexceedsthereferencefreestreamvelocity. Thisindicatesthere

is a speedup eect caused by the cylinder. The average free stream velocity is

9 . 7 m/s

whiletheaveragevelocityin the wakeproleis

9 . 3 m/s

. A loweraverage

velocityin thewakemeansthat thepresenceof thecylindercausesablockageof

theow. Theareaof thecrossection traverseddoeshowevernot covertheentire

crossection. Since thevelocityin thewakeproleishigher closeto thewall, this

means that the average velocity in the wake is higher than

9 . 3 m/s

. By simply

estimating the rest ofthe wake prolefrom the highestmeasuredvelocityin the

wake,the averagevelocityin the wake isfound to be

9 . 66 m/s

. This means that

blockage eects are neglectable, and that areference velocity of

9 . 7 m/s

may be

reasonable. BasedonthereferencevelocitytheReynoldsnumber,

Re D

,iscalculated

to be30717.

Asthecylinder wakeisanevolvingow,V isnotexpected tobezero,due to

the constraintsof continuity. Fora perfectly symmetrical wake V is expected to

bezeroin thecenter oftheowandnegativeoverthesymmetrylineandpositive

below. TheproleofV isplotted in gure22, alongwith thefreestreamprole.

Ideallythefreestreamconditionsshouldbezero. Duetomisalignmentoftheprobe

anosetfrom zerowouldnotbeunexpected, thevelocitydoeshowevervaryover

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