Mean velocities

Figure12showsthevelocityprolefortheaxialvelocity

U _x

^,forboth

φ 1 = 90

and

φ 1 = 180

.

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

datamatchesthechoiceof

κ = 0 . 41

. B=5.0assuggestedbyWhitegivesthebest tto themeasurementdatain thelog-lawregion.

10 ² 10 ³

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 variationoverthe}crossection 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

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15

Figure14: Radialmeanvelocity,

U _r

−0.5 0 0.5

Figure15: Circumferentialmeanvelocity,

U _θ

−5 0 5

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. Figure17showsthetheoreticalrelationandthe

experi-−1 −0.5 0 0.5 1

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

−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3

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 be}approximately0.85 for

< = 0 . 75 e 5

. TheresultsshouldbecomparableastheReynoldsnumbersisof thesame order. According to the resultsof Torbergsen,

u _x ⁺

^{is fairlyconstant onthe}

cen-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

−1 −0.5 0 0.5 1

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 gainmore}informationabout 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

−1 −0.5 0 0.5 1

Figure20: Turbulenceintensity

Reynoldsnumberandcomparable.

Emphasisis placedonassesmentof thephysical validityof theresult, aswell

ascomparisonwiththeresultsofOngandWallace.

In document Three-dimensional wake measurements (sider 33-40)