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.

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)

inthe coordi-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 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.

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. 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.

Twoprolesof theturbulent pipeow hasbeentaken ata Reynoldsnumberof a

approximately

10 ⁵

. Themain dierencebetweenthetwoprolesisthat theyare takenattwodierentvaluesof

φ 1

^{. Thisisdoneto}investigatetheeectofrotating 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

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

φ ¹ Re _D ^dP _dx [ P a/m ] u ^{∗ dP}

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

.

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.

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

0.75 0.8 0.85 0.9 0.95 1 1.05 1.1

−3

−2

−1 0 1 2 3

U/U _r ef [−]

y/D cyl. [−]

U _wake /U _ref U _freestream /U _ref

Figure21: NormalizedproleofUin cylinderwakeforRe=30717,x/D=10

thecrossection. Whetherthis isatruepropertyof theowornotisunclear,the

rangeofvariationinVequals

2 . 4 deg

probepitchrelativetothereferencevelocity.

Ifthevelocityproleiscorrectedbysubtractingthelocalfreestreamvelocity,the

prolewillbeasin gure23. Thiscorrectionassumesthattheowvelocities can

besuperpositioned. The resultlooksmorelikewhatonewouldexpect,but when

comparedtotheresultsofothersi.e. OngandWallace[7]theshapeofthevelocity

proleisnotaperfectmatch.

The velocity component in the z-direction is plotted in gure 24 along with

the freestream measurements. The samevelocity prole correctedfor freestream

conditionsisplottedin gure23.

Thefreestreamvelocityvariationcannotbeexplainedbythenitesizeofthe

probeandit is notaconstantoset asayawangle would give. The variationof

Winfreestreamconditionsishoweverrathersmall,itequalsapproximately

2 ◦

. In

thewakeofthecylinderthevariationofWis sligtlylarger.

Itisnotstraightforwardtounderstandhowthetimevaryingvelocitygradients

Itisnotstraightforwardtounderstandhowthetimevaryingvelocitygradients

In document Three-dimensional wake measurements (sider 27-0)