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
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
ThisMasterthesisiscompletedattheDepartmentofAppliedMechanics,Thermo-
andFluidDynamicsattheNorwegianUniversityofScienceandTechnology.
I ammostgratefulto my supervisor ProfessorPer-Åge Krogstadforhis guid-
ance andencouragementin theprocess ofcompleting this thesis. His insightand
constructiveadvisehasbeenofgoodhelp.
Trondheim,14juni2010
PålEgilEriksen
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. Theshear stresses
uw
andvw
werehoweverfoundto beunphysicallylarge,this isbe- livedto bedue tothevelocitygradientin thewake. Conditional averagingofthewakeresultswithrespecttoshedding frequencywasalsoconducted.
Egenskapenetilenhotwireprobemedtretråderharblittundersøktfortoforskjel-
ligestrømningstilfeller. Trådeneerlagetavenplatinum/rhodiumlegeringoghar
endiameter på
5 µm
. Detretrådeneskapereitprobevolum medeittverrsnittpåca 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. Nærveggenblegradiententilmiddelhastighetenstoriforholdtil probensrommelige oppløsning, noe somga storefeil. Den andrestrømningen
som bleundersøktvarvakenbakeinsylinder for
< D = 3 · 10 3
. Demåltemiddel- hastigheteneogturbulentespenningenevardelvisioverenstemmelsemedresultaterfra litteratur. Skjærspenningene
uw
ogvw
var ufysiskstore. Det antas at detteer på grunn av den store hastighetsgradienten i vaken. Midling med hensyn på
virvelavløsningsfrekvensenerogforsøkt.
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
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.
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 thetangentialcomponentand
U b
isthebinormalcomponentwhichisnormaltothen-tplane. Thethreecomponentscontributeunequallytothecoolingofthewire. Thecoecients
kandhcorrectforthedierencesincoolingalongthethedierentaxes. Kandh
arenotconstantsbutfunctionsoftheowdirectiongivenby
α
andβ
,theyawandpitch angles,
k = k ( α )
andh = 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 binormalcooling, asshown in gure2(a). Theeective coolingvelocity can be related to
thevelocityvectorSin the n-tplane by thefunction
f ( α )
. Thisthe basisofthe eectiveanglemethod,thattheeectivecoolingvelocitycanberelatedtotheowvelocitytrough 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 angleα e
. Equation 4therefore givesus the component of thevelocityvectorS which is normalto the
wire,asafunctionof
α
andα e
.Whenavelocitycalibrationofthewireisconducted,theprobeaxis
x p
isalignedwith the ow direction, hence
α
is zero(see gure 2(a)). The measured voltageoutput,
E
,thencorrespondstoagivenvelocity,U
,whichisthesameastheknownvelocityvectorS. Equation4canforthiscasebewritten:
V e = S cos( α e ) = U cos( α e )
(5)Iftheprobeisyawedanangle
α
(seegure2(b)), forthesamevelocityS,thevoltageoutput
E yawed
willcorrespondtoavelocityU yawed
whichisobtainedfromthe velocity calibration.
U yawed
is dierent from S. Twodierentexpressions for the eective coolingvelocitymay now be written. Equation6gives theeectivecoolingvelocityasthecomponentofSwhichisnormaltothewire.
Figure2: Velocitydenitionsin thenormal-tangentialplane
V e = S cos( α e + α )
(6)Theeectivecoolingvelocity canalsobewritten asthecomponentof
U yawed
whichis normalto thewire. Thisrelation willnotdependon
α
sinceU yawed
hasaxedowdirectionrelativeto 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 willdenote
U yawed
inthemoregeneralformU(E),referringtothefactthatUisfoundfrom thehotwirevoltageEthroughthevelocitycalibration.
V e = S cos( α e + α ) = U ( E ) cos( α e )
(8)InthecasewhereSand
α
isunknown,equation8willhavetwounknownsandcan 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
) ratherthanrelativetotheindividualwire. Thecoordinatetransformationwillbediscussedin
section X. The reductionof unknowns resultsin aset of two equations and two
unknowns. Theequations areimplicitbutcaneasilybesolved. An alternativeto
theeectiveanglemethodcouldbetotabulateffordierentvaluesof
α
. Whentheequationsareto besolvedonecanrstguess avaluefor
α
,usethecorrespondingequal,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, thetransformationfromwirecoordinatesystemtoprobecoordinatesystemisdescribed
in thenextsection.
2.1.3 Coordinate transformation
Toreducethenumberofunknownsinequation10itisnecessarytoexpress
S i
andU bi
as functions of U,V and W which are dened in the probe xed coordinatesystem,(
x p , y p , z p
). Figure3denesthecoordinatesystemandtheanglesneededto relatetheprobewirestothecoordinatesystem.
φ i
istheangle betweentheprojectionof wireiin the(y p − z p
)planeandthey p
axis. Figure4showstheprojectionofthewires inthe (y p − z p
)planeandthecorresponding
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 andU T P i
is the projection of thetangentialcoolingvelocityofwirei,tpreferstotangentialprojection.
U T P
andU 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
,isafunctionofU andtheprojectionofthetangentialcoolingvelocity.
S i 2 = U 2 + U T P i 2
(15)Substitutingfor
U T P
inequation15yieldsS asafunction ofU,VandW.Figure4: Velocitiesandanglesin the(
y p − z p
)projectionS i 2 = U 2 + ( V cos φ i + W sin φ i ) 2
(16)Theow angle
α
in thenormal tangentialplane must also bedened. FromgureXanexpressionfor
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.
Forthe individual wires the eective angle
α e
must be found. This obtainedbyplacingtheprobeinauniformowwithavelocityS, andmeasuringtheresponse
formultipleyawangles,
α
,whilethepitchangleβ
isheldconstantatzero.Equation8describestherelationbetweentheowvelocityandthevelocitymeasuredbythe
wire,U(E).TheratiobetweenSandU(E)canthenbefoundfromequation8.
U ( E )
S = cos( α e + α )
U ( E ) cos( α e )
(19)Theresultsfrom measurementsat dierentangles
α
canthenbecurvttedtoequation19byadjusting
α e
toobtainthebestttothedatapoints. Atypicalset of calibrationanglesisα = − 20 : 5 : 20
. Insection 2.1.2α e
waspresentedasthegeometric 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 angleapproachassumesthat 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
forL < 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
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 isanimportantparameterin 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,wherebothtypesofshearisimportant.
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 +
andy +
respectively,givingu + = f ( y + )
. In the inner viscous shear dominated region turbulent shear can be ne- glected. Analysisof themomentum equation will thenyield thatu + = ( y + )
, see e.g. White[12]. Intheouterregionofthepipeowthevelocitynolongerdependson 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
. Theoverlaplawforagivenξ
canthenbefoundbymanipulating 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 theyareconsideredto benearlyuniversial.A willdepend on
ξ
.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 wouldthereforeexpect
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 thesimplied 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,seeingthatv 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 followingexpressionisfound.
∂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 variationofthetotalstresscanbefoundasafunctionofy.
−uv + µ ρ
∂U
∂y = u ∗ 2 1 − y
h
(37)
Equation37providesvaluableinformationabouthow
uv
vary asafunction ofy. 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
tov ¯ 2
andw ¯ 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 betweenf D
and U, where fisthevortexsheddingfrequencybehindthecylinder.
St = f D
U
(39)For
Re D > 1000
the Strouhalnumber startto decrease untill it stabilizes for10000 < Re D < 100000
atavaluecloseto0.21[11]. TheregionfromRe D = 1000
to
Re D < 200000
isnamedthesubrcriticalrange[13]. Inthesubcriticalrangethe boundarylayeronthecylinderremainslaminar. IftheReynoldsnumberincreasesfurthertheboundarylayerstartstodevelopfromlaminartoturbulent,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
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 wakeisexpanding. Inthisregion
dV
dy
willbepositive. Closertothecenterofthewakeweexpect
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 preservedalongthex-axis. Theshapeoftheprolecanbefoundbystartingwithequation41and
makingsomeadditionalassumptions.
3.1 The hot-wire probe
The probe consist of three wires on six supporting prongs, asshownin gure3.
Thechosengeometryisdenedbytwoproperties:
•
Theprojectionofthewires inthey p − z p
planeisatrianglewith60degreeangles
•
Thewires areinclinedanangleα e = 35 . 26 ◦
relativetothe
y p − z p
planeThesepropertiesgiveageometrywherethewires 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 aphoto can be seenin gure 6. Theangle
φ 1
is used to relate therotationof theprobein the
y p − z p
planetocoordinatesystem.Figure6: Pictureofthe
y p − z p
planetakentroughamicroscopeThelengthofthesupportingprongsischosensuchthattheowoverthewires
are notinuencedbytherest of theprobes. It isalso important that theprongs
are nottoolong,asthiscancausevibrationswhich inturn canbe interpretedas
aturbulentvelocitycomponent.
Aplatinum(90
%
),rhodium(10%
)alloyisusedinthewire. Thisalloygives agoodoxidationresistance,relativelyhightensilestrengthbuthasarelativlylowThe 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 theprobesisapproximatly4mm,ofthis approximatly1.75mmof thecoatingonthewirehas
beenetchedawayin thecentre. Thisgives
l/d ≈ 350
. Toreduce theinterference ontheowfromthesupportingprongsthedistancebetweenthesupportingprongsshould notbetosmall,theshapeoftheprongtipswillalsoaect theow[2].
Thecrossectionofthemeasurementvolumeis
≈ 5
mm,andthespatialresolu- tionoftheprobeisthereforassumedtobe5mm.3.2 Measurement chains
Figure7describesthemeasurementchainintheexperiment.
Figure7: Measurementchain
Thehot wireanemometersare optimized for
1 µm
notfor5 µm
which is usedin 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 acertaindegreebehelpedbychangingthebiassetting 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 thedampingintheloop.
3.3 Signal sampling rate
The sampling rate must be set accordingto the timescale of the smallesteddies
ofinterest. Whentherangeoftimescalesexpectedisunknown,thesamplingrate
mustbesetaccordingtothesmallesttimescaleoncanexpect. Kolmogorovsmicro
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
andU theta
and arethe axial, radialandcircumferentialvelocitiesrespectively.
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.
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 datawithoutthezerovelocitypointgavearesidualoftheorderof10 − 3
. Asecond order polynomialwasthereforefoundmostsuitableforvelocitycalibrationinthepiperig. 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.
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:
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. Totesttheeectiveanglesandtheφ
anglesthedatasetfromtheanglecalibrationofwire1canbesolved. Table2showsthetrueowangle,theU
velocitymeasuredbythepitot,thecalculatedvelocitycomponents,thecalculated
α
, thelength ofthe calculatedvelocityvector andthe relativeerrorbetweenthevelocitymeasuredbythepitotandthelengthofthecalculatedvelocityvector.
Table2showsthatthecalculated
α
fallswithin± 1 ◦
ofthetrueowangle. The relativeerrorbetweenU pitot
and|V |
is1 . 2%
atmost. TheY p
componentvelocity,V,should bezerobutshowsavariationwithrespectto
α
. ThemaximumvalueofV correspondstoaowangle of
2 . 9 ◦
ora
4 . 7%
relativeerror,whichisarelativly large error. The variation in V corresponds to an angle of2 . 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 mostlikelycausedbyawrongvaluechosenforφ 1
,since
φ 1
is determined by visual observation and is not likely to be exact. The variationinVcouldbeusedto ndthecorrectvalueofφ 1
,thecalculationscould thenbererunned,and theremainingconstanterrorin Vshould becausedbythepitch. Suchaprocedure hasnotbeenattempted in thisproject. Several authors,
i.e. Cantwell and Coles [3] uses yawingto perform similar corrections, Cantwell
andColesuseditforax-wireprobe.
Twoprolesof theturbulent pipeow hasbeentaken ata Reynoldsnumberof a
approximately
10 5
. Themain dierencebetweenthetwoprolesisthat theyare takenattwodierentvaluesofφ 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
toX/D = 70 . 5
. Based onthepressuregradient thefrictionvelocitymaybefoundfromequation22. Thefrictionvelocitycanalsobeestimatedfromthefrictionfactor,f.
u ∗ = U avg f
8 1
2
(42)WhentheReynoldsnumberandtheroughnessheightofthepipewallisknown
thefrictionfactorcanbefoundfromtheMoodydiagramorfromanequation. The
PVC pipe is hydraulically smooth, and thefriction factor cantherefor be found
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 1801 . 03 × 10 5
3.5487 0.3708 0.0179 0.3937Table3:
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. Inthefollowingthefrictionvelocityobtainedfromthepressuregradientisassumedtobecorrect.
4.2.2 Mean velocities
Figure12showsthevelocityprolefortheaxialvelocity
U x
,forbothφ 1 = 90
andφ 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
andφ 1 = 180
isplottedagainstthelogarithmic law. Torbergsen [9] didmeasurementsin the samepipe rig forRe = 75000
and obtained agood t with the logarithmic law usingκ = 0 . 41 andB = 5 . 5
. White [12] claimsthatκ = 0 . 41 andB = 5 . 0
giveabettert toexperimental data. Bothdatamatchesthechoiceof
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
andU θ
to be zero. Figures 14 and 15 show that neither of thevelocitycomponentsareexactlyzeroacrossthepipe. Theradialpipevelocity
U r
,isfairlyconstantoverthecrosssectionofthepipe,butshowsomevariation,especially
close tothepipe walls. Therangeof variationin velocityis about
± 0 . 125 m/s
orapproximatelyanangle of
± 0 . 8 ◦
relativeto the averageaxial velocity. Both the proles forφ 1 = 90
andφ 1 = 180 ◦
showthe samekindof variationwith respect
to
y/R
but theyare osetrelativeto one another. Theoset equalsabout1 . 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
withrespecttoy/R
followsthesamepatternforbothseriesof 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
−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
−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
andU θ
therelativeerrorbetweenthetwocenterlinepointsislarge. Whiletherepeatedmeasurementsgave
solutionsfortheaxialvelocity
U x
withinonepercentoftherstsolution,theradialand circumferential repeated solutions can vary up to
0 . 5 ◦
and
1 ◦
respectively.
Thisisalargedeviationcomparedtotherangeofthecalculatedvaluesfor
U r
andU θ
. By re-examiningthe measurementdata there was found to be asmall drift in wirevoltage, whichcould notbecorrected forby consideringthe temperaturechange. Ideally themeasurement seriesshould havebeenrepeated, but theerror
wasdiscoveredtolate. Basedonthisobservationpartsofthelargevariationfor
U r
and
U θ
mightbecausedbyvoltagedrift. TheshapeofthevariationofU r
andU θ
doeshoweverseemtobeafunctionofyorasomeotherpropertyrelatedtoy,not
onlyapossiblevoltagedrift. Butwhatpropertycouldthatbe?
U x
isafunctionofy,butissymmtricaboutthecentreline.Thegradientof
U x
alsovariesasafunctionof 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-
−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. Themeasurementdatashowlittletendencytodropofclosetothewall. For
φ 1 = 90
there isalittledropfory/R < − 0 . 9
,whilefory/R > 0 . 9
there is actually anincreasein shear stress measuredfor both datasets. Theincreasedshearstresscouldjust beoutliers,butithappensforbothdatasets.
The two other shear stresses,
u x u θ
andu r u θ
should theoretically be zero as there is no meanvelocity gradientresultingin productionof neitherof them. Ingure18thenormalizedstressesareplotted. Inthecentreregionthemagnitudeof
u x u θ
andu r u θ
arerelativelysmallcomparedtothemaximumvalueofu x u r
,about3
%
, 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 berelatedtothemeasurementprocess.
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−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
andu r u θ
resultin unphysicalshear stresses. Insection4.2.2 thedeviation forboth
U r
andU θ
wasfoundtoincreaserapidlyfor|y/R| ≥ 0 . 8
,thismatchestheresultfoundforu x u θ
andu r u θ
and supports the theory that a largevelocity gradientbiasestheresult.
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 ofsection4.2.3. Thedataseriesforthetwodierentprobesalsogiveslightlydierent
results. Movingcloserto thewall
u x +
isunderestimatedcompared tothe results ofTorbergsen,butmatchestheresultsofAanesland[1]better.u r +
andu θ +
showthe same variation asreported by Torbergsen for
|y/R| ≥ 0 . 6
but the scatter is largeforvariationofφ 1
.Valuesfor
u x +
,u r +
andu θ +
cannot beestimated without givinga relativlylargepotentialerror.
u θ +
fory/D = ± 0 . 6
,canforinstancebeestimatedas1 . 1
but−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. Thelargevaritioncanbecausedbypooradjustmentof
φ 1
,itisasmentionedearliersetvisually.Ingure20theturbulenceintensityrelativetothelocalstreamwisevelocityis
plotted.
Thestreamwiseturbulence intensity onthe centreline,
u x + ≈ 3 . 3 − 3 . 6
. Tor- bergsenreporteda valueof approximately3 . 5%
. On thecentrelineu r +
andu 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 velocityprole wastakenin the test sectionbefore thecylinder wasinserted,to mapthe
referencefreestreamconditions.
Vortexsheddingfromacircularcylinderisamuchstudiedow,itdidhowever
provediculttondnearwakeresultsinthesamerangeofReynoldsnumbers
( 10 4 )
, and downstreamdistancex/D.A surveyusingafourwirehotwireprobebyOngand Wallaceat
< = 3900
wastheclosestmatch found[7]. Theyusedafour wire hot wire probe with a crossection of 1 mm x 1 mm. For the ow investigatedin this project
< = 30717
, placing it in the subcritical range together with the resultsofOngandWallace. Theresultscanthereforebeassumedindependentof−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 wakeproleis9 . 3 m/s
. A loweraveragevelocityin 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 simplyestimating the rest ofthe wake prolefrom the highestmeasuredvelocityin the
wake,the averagevelocityin the wake isfound to be
9 . 66 m/s
. This means thatblockage eects are neglectable, and that areference velocity of
9 . 7 m/s
may bereasonable. BasedonthereferencevelocitytheReynoldsnumber,
Re D
,iscalculatedto 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